Kunio KATAOKAi

PREFACEThis volume is supposed to serve as a progressive book advanced for the second-phase study of“Transport Process Chemical Engineering” in the graduate course of Chemical Engineering.
The previous course in the undergraduate course began with derivation of the fundamental laws in PARTI to be followed by understanding the rate-based concept of transport processes. In the successive stage,
we learned the models of interphase transport of momentum, heat, and mass applicable to a variety ofchemical equipment. In the latter half stage, we considered how to combine these models with thepractical design of popular unit operations. As the fundamental consideration of mass transfer problems,
we studied gas absorption in Chapter 10 of PART I and distillation in Chapters 11 and 12 of PART Ⅱ.
We treated such complicated processes macroscopically in the form of interphase mass transport, but
those simple models did not take into consideration fully the effect of gas or vapor bubbles dispersed in
the continuous liquid or vapor phase. In addition, some transport processes accompanied with phasetransformation or chemical reaction were treated in the successive chapters. In the remaining portion,some topics of fluid mechanics (e.g. turbulent flow) in transport phenomena were treated to grasp more
precisely the role of fluid motion in transport processes.
Several advanced, comprehensive themes are incorporated for enlarging the boundaries of chemicalengineering knowledge in this volume.
We will therefore begin the Chapter 21 of PART III with reconsidering the interphase mass transport
in the dispersed system. In the following chapter, we will take an interest especially in liquid-liquid
extraction as an appropriate example of dispersed system process.
In Chapter 23, we will be concerned a little with the effect of dramatically-changing thermophysicalproperties of supercritical fluid on the transport process as a new technology.
The primary objective of this volume is to inspire students with courage studying positively sometransport processes in relation to the practical issues of energy and resource conservation. The chemical
process industries consume a lot of energy for executing various practical chemical processes. Chapter
24 of PART IV will offer some effective energy-saving technologies for global issues of energy
conservation from a thermodynamic viewpoint. In particular, the heat pump technology is worthy to bestudied. As its example, we will try to understand the heat pump technology for a distillation plant. Therean example of innovative development due to HIDiC technology for practical energy saving problems
will be reviewed. This volume will pay a little bit bigger attention to the study of concrete concepts ofpractical chemical equipment design. In Chapter 25, as an example, it will be instructive to reconsider
the concept of the practical design of a real packed distillation column from a viewpoint of simultaneousinterphase transfer of enthalpy and mass. This design concept is based upon the control volume
consideration which bridges the gap between the ideal and real chemical equipment design models.
In the next chapter (Chapter 26), we had better consider some topics in thermal engineering from athermodynamic viewpoint. When the transport process engineering deals with very complicatedprocesses, we are very often saved by the concept of thermodynamics based on a quasi-static change
from an ideal state to another ideal state. The practical design of steam ejectors will be treated based on
the thermodynamical concept because the transport phenomena occurring in an ejector are toocomplicate to analyze precisely.

ii
In Chapter 27, we will study several examples of visualization techniques as to how to observe various
dynamic transport phenomena.
It can be considered to be very interesting to observe time-dependent and/or local flow structures
accompanied with heat and mass transfer. The observation experience of this kind will be a help to ouranalytical understanding of the chemical engineering fundamentals.
In the final chapter, we would like to end this course considering some nonlinear dynamics in relation
with the instability and complex dynamic processes accompanied with liquid-phase chemical reactions.
This course makes it an educational rule to encourage readers and/or students to make efforts for how
to make creative problems rather than for how to solve questions.
We would like to emphasize that it is much more important to make creative problems than to solve
given problems.
We would not deny that AI (Artificial Intelligence) programs can answer a variety of questions
intelligently, present an appropriate advice, and make deductions about real-world facts. When the AI
machine receives a new observation, that observation is classified based on his previous experiences for
integrating a wide range of problem-solving techniques. The AI does not work unless any problems are
given. There is an undesirable possibility that the AI may give us an inappropriate answer from theinsufficiently accumulated databases.
We emphasize that it is of great importance how to accumulate various useful actual data and facts
observed into the data bases of the AI not only for machine learning but also for data mining. Machinelearning has intimate relations with optimization, e.g. minimization of some loss function or
maximization of efficiency in transport processes. Data mining may give knowledge discovery in databases, e.g. discovery of unknown properties in the transport process data.
Human intelligence stands on mind, consciousness and aspiration. It is human beings that create many
challenging problems throughout industry and academia. From that consciousness, we are also studying
how to make creative problems in this kind of chemical engineering courses.
It is hoped that this volume will provide a valuable aid to those who are interested in more challenging
creative problems.
If a reader wants to understand the fundamental ideas and/or concept of the preceding volume for therespective topics of this volume, he can access the following URL web site for the previous volume:
Kansai Chemical Engineering: https://www.kce.co.jp/en/library
Kobe University: https://doi.org/10.24546/90008260
KUNIO KATAOKA*Emeritus Professor
Dept. of Chemical Science andEngineering,
Kobe University*Honorary Advisor
Kansai Chemical Engineering Co., Ltd.
Kobe
November 2023

iii

CONTENTSPART ⅢCHAPTER 21 TRANSPORT PROCESSES IN DISPERSED SYSTEMS
21.1
Transport Phenomena in Particles/Droplets Suspended or Dispersed
in a Flowing Fluid,
21.1-1 Solid or liquid particles suspended in a flowing fluid,
21.1-2 Interphase mass transfer from a single liquid droplet
21.1-3 Surface tension acting on liquid droplets
21.2
Structure of Dispersed System
21.2-1 Effect of surfactant adsorbed on liquid surface
21.2-2 Alignment structure of adsorbed surfactant molecules
21.3
Viscosity of Dispersed Systems
21.3-1 Definition of viscosity
21.3-2 Slurry and solid-in-liquid mixtures
21.3-3 Bingham plastic fluids
21.3-4 Emulsion and liquid-in-liquid mixtures
21.4
Pipe Flow of Dispersed Systems
21.4-1 Pipe flow of solid particles dispersed system
21.4-2 Pipe flow of liquid droplets dispersed system
21.5
Emulsion
21.5-1 Emulsification and emulsion viscosity
21.5-2 Rheological behavior of emulsion
21.5-3 Velocity distribution of non-Newtonian fluid pipe flow
21.5-4 Shear-thinning emulsion

1
1
1
4
9
9
9
10
11
11
12
13
14
16
16
18
23
23
25
28
29

CHAPTER 22 EXTRACTION ENGINEERING
33
22.1
Liquid Extraction
33
22.1-1 Liquid-liquid equilibrium
33
22.2
Extraction Technology
3722.2-1 Liquid-liquid equilibria of ternary system
37(a) Triangular coordinates(b) Rectangular coordinates
22.2-2 Unsteady mass transfer problems in liquid-liquid extraction
41
22.2-3 Continuous extractors
4222.2-4 Stage-by-stage design calculation of a continuous extractor
43
22.3
Solid Extraction – Leaching49
22.3-1 Equilibrium stage
49
22.3-2 Design calculation of a counter-current leaching system52
22.3-3 McCabe-Thiele graphical method for design of a continuous leaching system54
22.3-4 Continuous leaching process equipment
57
CHAPTER 23 SUPERCRITICAL FLUID TECHNOLOGY
23.1
Characteristics of Supercritical Fluid

59
59

iv

23.2

23.3

23.1-1 Properties of supercritical fluids
23.1-2 Supercritical fluid flow and heat transfer in a horizontal pipe
23.1-3 Nusselt correlations of supercritical fluid tube flow
23.1-4 Heat transfer equipment for supercritical fluids
Applicability of Supercritical Fluids
23.2-1 Supercritical fluid extraction
23.2-2 Principle of supercritical fluid extraction
23.2-3 Modeling diffusion of dissolved solute inside a solid particle
Leaching by Supercritical Fluids
23.3-1 Leaching equipment
23.3-2 Rapid expansion of supercritical fluid solutions

60
62
63
65
67
67
68
70
71
71
71

PART ⅣCHAPTER 24 ENERGY SAVING TECHNOLOGY
24.1
Energy Requirement in Continuous Distillation
24.2
Heat Pump Technology
24.2-1 Principle of heat pump
24.2-2 Vapor recompression technology for distillation
24.2-3 New heat-pumped distillation system
24.3
Vapor Recompression Technology for a Jacketed Tank Evaporator
24.3-1 A new batch tank evaporator
24.3-2 Saving energy by a vapor recompression technology
24.4
Energy Saving by Heat Integration in a Distillation System
24.4-1 Principle of internally heat integrated distillation column (HIDiC)
24.4-2 Operation modes of HIDiC
24.4-3 Heat transfer characteristics of a trayed-column HIDiC system
24.5
Application of HIDiC technology
24.5-1 HIDiC system applied for a bioethanol distillation process
24.5-2 Test operation of a HIDiC bench plant

75
75
77
77
81
83
85
85
86
95
95
97
98
100
100
101

CHAPTER 25 TRANSFER MODEL FOR DESIGN OF PACKED DESTILLATION COLUMNS
105
25.1
Modeling Concept by Control Volume Method
105
25.1-1 Definition of control volume105
25.1-2 Allocation of theoretical profiles of temperature and composition107
25.1-3 Definition of local film coefficients of mass and enthalpy transfer
10725.1-4 Process simulation based on equilibrium-stage model
108
25.2
Experiment of a Packed Column Distillation
109
25.2-1 Test column and experimental method
10925.2-2 Determination of local HETPs
111
25.3
Correlation of Mass and Enthalpy Transfer
11625.3-1 Local variation of mass and enthalpy transfer
11625.3-2 Generalization of mass and enthalpy transfer correlations
117
CHAPTER 26 THERMODYNAMIC APPROACH FOR THERMAL ENGINEERINGPROCESSES
123
26.1
Fundamental Laws of Thermodynamics
123
26.2
Thermodynamic Diagram
124
26.3
Joule-Thomson Expansion
125

v
26.4
26.5

Actual Issues of Joule-Thomson Expansion Process
Ejector Engineering
26.5-1 Working principle of ejectors
26.5-2 Usage of steam ejectors
26.5-3 Thermodynamic consideration of steam ejectors
26.6
Steam Ejectors for Effective Energy Conservation
26.6-1 Effective application of a steam ejector for evaporation processes
26.6-2 Thermodynamic calculation model for steam ejectors
26.7
Diffuser Efficiency of a Steam Ejector
26.8
Motive Steam Required for a Steam Ejector
26.9
Multi-stage Ejector System
26.10 Special Matters of Heed to Ejector Engineering
26.11
Macroscopic Momentum Balance for Steam Ejectors

131
133
133
134
135
136
136
137
139
140
146
150
151

CHAPTER 27 VISUALIZATION OF DYNAMIC TRANSPORT PHENOMWENA
27.1
Observation of an Organized Structure of Stirred Tank Reactor
27.2
Transitional Route of Laminar to Chaotic Flow
27.2-1 Instability of circular Taylor-Couette flow
27.2-2 Visualization of time-independent and time-dependent vortex flows
27.2-3 Transitional scenario to generation of chaos in Taylor-Couette flow
27.3
Visualization of Mass Transfer Distribution
27.3-1 Mass transfer distribution in Taylor-Couette flow
27.4
Visualization of Dynamically Organized Jet Flows
27.4-1 Hydrogen bubble method for a free jet development
27.4-2 Hydrogen bubble method for a two-dimensional impinging jet
27.4-3 Local time-dependent wall shear stresses in jet impingement region
27.5
Local convective heat transfer on the wall of jet impingement plate

153
153
157
157
158
162
163
163
166
166
167
169
172

CHAPTER 28 NONLINEAR BEHAVIORS IN COMPLEX CHEMICALPROCESSES
28.1
Why Do We Study Nonlinear Dynamics in Complex Chemical Processes?
28.1-1 A simplified chemical model
28.1-2 Phase space analysis
28.2
Nonlinearity of Brusselator Model
28.2-1 Bifurcation analysis of Brusselator model for a CSTR
28.2-2 Bifurcation scenario of Brusselator model
28.3
Instability of a Complex Reaction in a CSTR
28.3-1 Belousov-Zhabotinskii reaction
28.3-2 Chemical oscillations sustained in a continuous B-Z reaction
28.4
Dynamical Behavior of an Emulsion Polymerization Process
28.4-1 Experimental method of continuous emulsion polymerization
28.4-2 Dynamical behavior of continuous emulsion polymerization processes
28.4-3 Temporal variation in particle size distribution of polymer particles

177
177
177
178
179
179
180
181
181
183
184
185
186
187

vi

Chapter 21

Transport Processes in Dispersed Systems

1

PART ⅢCHAPTER 21
TRANSPORT PROCESSESIN DISPERSED SYSTEMS
21.1 Transport Phenomena in Particles/Droplets Suspended or
Dispersed in a Flowing Fluid
This course is basically concerned with the macroscopic or bulk behavior of fluids rather than withthe microscopic or molecular behavior. This is the concept of a continuum. Even in this viewpoint, there
are quite a few dispersed systems used for various industrial purposes: small liquid or solid particles
suspended in gas (e.g. aerosol), fine liquid droplets in liquid (e.g. emulsion), solid-in-liquid slurry, andliquid-liquid and/or solid-liquid colloids.
For example, we can consider a liquid mixture agitated in a tank, which consists of many liquid
droplets/gas bubbles/solid particles suspended in an agitated bulk liquid. This kind of flow systems arevery often utilized for various chemical reaction, extraction, absorption, and crystallization processes.
Especially in case that the number of suspended solid particles and the solid percentage are very large,
the dispersed system should be regarded as a slurry. We should consider the effective physical properties
such as apparent viscosity resulting from the interaction of suspended particles.
Furthermore, if the size of particles or droplets is very small, the dispersed system is called “colloid,”
which can be defined as a mixture in which one substance consisting of microscopically dispersed
insoluble particles/droplets is suspended in another substance. The difference between the solution andcolloid is that a colloid has a dispersed phase and a continuous phase, while in a solution, the solute andsolvent constitute only one phase. Therefore, the interface between the dispersed particle/droplet and thecontinuous phase is a very important factor.
Firstly, we will begin by studying the basic properties of dispersed system.
21.1-1 Solid or liquid particles suspended in a flowing fluid
Heat or mass transfer between a fluid and suspended particles is very often encountered in various
industrial situations. Firstly only for simplicity, heat and mass transfer from solid particles suspended ina fluid flow will be studied here. Beside the physical properties of solid and fluid, another importantfactor, i.e. the shape of particles can be treated as sphere of equivalent diameter.
As we already learned in Chapter 9.7-3, the heat transfer around a submerged sphere in a uniform
flow field of velocity 𝑣 is given by the following Ranz and Marshall correlation1):⁄= 2.0 + 0.60⁄

1 < 𝑅𝑒 < 7 × 10 , 0.6 < Pr < 400 (9.7-8)

2Transport Phenomena in Particles/Droplets Suspended

The similar form of the mass transfer correlation is also given by⁄⁄(21.1-1)= 2.0 + 0.60

These equations indicating an analogy between heat and mass transfer have been obtained for a single
droplet or particle placed in a uniform flow field. The Eq.(21.1-1) can be used for analyzing theinterphase mass transfer such as evaporation from a liquid droplet, dissolution from a liquid droplet or asolid particle, and deposition to a droplet or particle.
If you want to use the above correlations in an agitated vessel, the fluid velocity 𝑣 should beinterpreted as a representative relative velocity between the moving particle and the flowing fluid
velocity.
Since the structure of the agitated flow is very complicated, it may be a problem what definition should
be adopted as the representative relative velocity taking into account the turbulence.
1.

Ranz, W. E. and W.R. Marshall, Jr.: Chem. Eng. Prog., vol.48, 141, 173 (1962)

Terminal Velocity
If the size of particles is very small (e.g. of the order 0.1 mm), the velocity of the fluid relative to theparticle, sometimes called slip velocity, can be taken as the terminal velocity in the gravitational field byusing Stokes’ law:
𝑢 =

(

)

𝑅𝑒 =

<2

(21.1-2)

This indicates the balance between the gravity force accompanied by the buoyancy force and the dragforce exerted by the flowing fluid.
For example, a 0.1 mm dia. solid particle of density 2.5 × 10 kg/m3 falling in water at 20 ˚C shouldhave the following terminal velocity:𝑢 =( . )( .

)×

( .

( .

)

)

= 0.00817 m⁄s

where
𝑅𝑒 =

(1,000)(0.00817)(0.0001)= 0.817 < 2
0.001

Behavior of a single liquid droplet flying in a hot gas stream
Let us consider a liquid droplet flying in a hot gas stream. In the case of a spray dryer, liquid droplets
ejected from a nozzle placed at the ceiling center of the dryer are falling downwards by gravity in contact
with an upward-flowing hot gas, in which the droplet size is reducing accompanied with evaporation
due to the heat transfer from the hot gas. For the calculation of the droplet trajectory within the dryer
vessel and the evaporation time required, we should consider the force balance including the dragresistance force 𝑅 𝑅 , 𝑅 and the heat transfer between the falling droplet (temperature 𝑇) and the
hot gas stream (temperature 𝑇 ) taking into account the reduction of droplet size.
We are supposed here to consider the evaporation of the liquid droplet in the first half time period.
It should be kept in mind for spray dryers that the droplet of aqueous solution will undergo the drying
process changing into a porous solid particle of solute in the latter half time period.

Transport Processes in Dispersed Systems

3

Fig.21.1-1 Trajectory of a liquid droplet falling downward in a spray dryer

Let us consider a single droplet issuing at the initial velocity 𝑣 (𝑣 , 𝑣 ).
The force balance equation for a single droplet (diameter 𝐷 , density ρ) is given by𝜌𝐷= −𝜌 𝑅(21.1-3)𝜌 𝐷= −𝜌 𝑅 + 𝜌− 𝜌𝐷 𝑔
(21.1-4)
The y coordinate is parallel to the gravity acceleration and the x coordinate is horizontal.
The drag resistance force can be expressed as the following function of relative velocity 𝑣:
𝑅= 𝐷 𝐶𝑣(21.1-5)(21.1-6)𝑣= 𝑣 + 𝑣
where 𝐶 is the drag coefficient for spheres moving relative to a fluid with velocity 𝑣 , 𝑣 ,which has a meaning similar to friction factor 𝑓 for pipe flows.

Fig.21.1-2 Force and heat balances of a liquid droplet falling in a hot gas stream

4Transport Phenomena in Particles/Droplets Suspended

The drag coefficient is known to be a function of Reynolds number2) (see Chapter 7.5):
𝐶 = 𝑓 𝑅𝑒The Stokes law followed by the two empirical correlations is available:
𝑅𝑒 < 0.1

𝐶 =
𝐶 =

.
⁄

(21.1-7)

2 < 𝑅𝑒 < 500 (approximate correlation for intermediate region) (21.1-8)

𝐶 ≅ 0.44 (constant)
2.

(Stokes’ law)

500 < 𝑅𝑒 < 2 × 10

(Newton’s law of drag force) (21.1-9)

Bird, R. B., Stewart, W. E., and Lightfoot, E. N., “Transport Phenomena,” Wiley, New York, p.193 (1960)

The heat balance equation for a single solid particle can be expressed as𝐷= ℎ 𝜋𝐷𝑇 − 𝑇(21.1-10)
𝜌𝐶𝑝
One example of the heat balance equations for a single droplet can also be written by−𝜌 𝐷= ℎ 𝜋𝐷𝑇 − 𝑇 ⁄𝜆(21.1-11)
Here λ is the latent heat of evaporation. The left side indicates the mass reduction of the droplet due
to evaporation.
This equation can be obtained assuming constant density 𝜌 of the liquid from the condition that the
droplet temperature 𝑇 is kept constant at the saturation temperature such as the boiling point or the wet
bulb temperature.(see Chapter 13.2)
The heat transfer coefficient ℎ averaged over the droplet surface can be evaluated by the above
Ranz and Marshall correlation.
When the droplet size is reduced to a certain critical size, the droplet terminal velocity becomes equal
to the hot gas velocity, and then the droplet becomes stationary. After that, the droplet will turn to flying
upward with the hot gas because the terminal velocity of the droplet becomes smaller than the velocityof the hot gas.
For the case of spray dryers in the latter half time period, we should deal with a drying process of aliquid droplet containing solute substances. The droplet is dried to reach a certain particle size of driedporous solid.
[PROBLEM 21.1-1]
A benzene droplet of 𝐷 = 3 mm is falling vertically downward at the initial velocity 𝑣 = 0 m/s
and temperature 𝑇 = 80℃ (boiling point at normal pressure) in the stationary superheated vapor ofbenzene (normal pressure, 𝑇 = 105 ℃) in a big vessel. The size of the falling droplet 𝐷 decreases
due to the evaporation by the heat transfer with the superheated vapor. In this case, the liquid droplet is
kept at the normal boiling point. Obtain the temporal variation of the droplet velocity and size.

21.1-2 Interphase mass transfer from a single liquid droplet
Fig.21.1-3 shows a single droplet suspended in an extracting solvent S. The droplet contains an
extractable component (solute) B and the original solvent (diluent) A. The solute concentration is given
by 𝐶 . The bulk concentration of the solute in the solvent S is given by 𝐶 .

Transport Processes in Dispersed Systems

5

Fig.21.1-3 Interphase mass transfer from a single liquid droplet of solvent A suspended in a solvent
S with uniform velocity

Regarding the correlation of mass transfer around a sphere, the analogy between heat and masstransfer holds as shown by the Ranz-Marshall equation3):⁄= 2.0 + 0.60⁄

(21.1-12)[EXAMPLE 21.1-1]
Let us consider how to experimentally evaluate the mass transfer coefficient in the solvent S-phasefilm around liquid droplets containing a solute B in the original solvent (diluent) A.
(1) Firstly, the extracting solvent S (volume 𝑉 ) is put in a turbine-agitator tank.
(2) The original mixture B in A to be extracted has the initial concentration 𝐶
in mass %. By
using a pipette-type droplet feeder, its liquid droplets of approximately constant size 𝑑 are
carefully poured into the solvent S.
(3) The total volume of the original mixture supplied amounts to 𝑉 .
Therefore, the total number 𝑁 of droplets fed is given by
𝑁 = 𝑉 ⁄(𝜋⁄6) 𝑑(21.1-E1)
It can be assumed for simplification that the number of uniform-size droplets are suspended in thecontinuous phase solvent S with uniform velocity.
At time 𝑡 = 0, the liquid extraction is started by rotating the turbine agitator at a rotation number forthe droplets so as not to be collided and broken up.
The suspended droplet size can be assumed constant during the whole extraction period. We also
assume an isothermal condition.
Explain how to estimate the film coefficient of mass transfer by assuming uniform concentration of
B inside those droplets.
Solution:
Let us pay attention to a single droplet being suspended in the solvent S.
Only for simplicity, we assume uniform velocity stream with the relative velocity (slip velocity) 𝑈
between the droplet and the ambient fluid. For example, the representative velocity of the agitated liquidshould be defined as the time-averaged turbulence intensity.
We can also regard 𝑈 as the terminal velocity.
In this condition, time-dependent mass balance can be expressed as− (𝜋⁄6) 𝑑= 𝑘𝜋𝑑(𝐶 − 𝐶 )(21.1-E2)
𝑉 𝐶

+ 𝑉 𝐶

= 𝑉 𝐶

(21.1-E3)

6

Transport Phenomena in Particles/Droplets Suspended

where 𝑘 is a S-phase film coefficient of mass transfer around the droplet. For simplification, thesevolumes 𝑉 , 𝑉 of the two phases can be assumed to be constant during the whole extraction period.
The concentration B in the S-phase will increase with time while the concentration B in the A-phase
droplet will decrease as the extraction proceeds.
As distinct from Fig.21.1-3, we assume the concentration of solute B kept uniform inside the droplet.
Since the uniform concentration of component B in the droplet is in equilibrium with the interfacialconcentration 𝐶 , we assume a linear equilibrium relationship given by a distribution coefficient 𝐾
in the limited range of solute concentration:
𝐶 =𝐾𝐶(21.1-E4)
Here the distribution coefficient 𝐾 is defined as the solute concentration ratio of the extract phase to
the raffinate phase in an equilibrium condition.
In a limited concentration range of solute B, K can be assumed to be constant.
Substituting Eq.(21.1-E3) with Eq.(21.1-E4) into Eq. (22.1-E2),(𝑉 ⁄𝑉 )
6𝑘
𝑉
𝑑𝐶
𝐶 −
=
𝐾+
𝐶
−
𝐾 + (𝑉 ⁄𝑉 )𝑑𝑉𝑑𝑡
That is= 𝑎(𝐶 − 𝑏)
(21.1-E5)
−
where
𝑎=

𝐾+

and 𝑏 =

⁄

(
(

)
⁄

)

(21.1-E6)

𝐶

I.C. at 𝑡 = 0
𝐶 = 𝐶
The following solution is obtained𝐶− 𝑏 = (𝐶− 𝑏) exp(−𝑎𝑡)Or= exp(−𝑎𝑡)
⁄ )
⁄ )
(
⁄ )
(
⁄ )
(

(

= exp −

𝐾+𝑡

(21.1-E7)

If we plot the experimental data of the time-dependent solute concentration 𝐶
as in semilogarithmic coordinates of Fig.21.1-E1, the film coefficient of mass transfer 𝑘 can be evaluated fromthe slope of experimental line.

Transport Processes in Dispersed Systems

7

Fig.21.1-E1 Calculation of mass transfer coefficient from the slope of the straight line plotted in thesemi-logarithmic coordinates

It should be kept in mind that the above example should be regarded as a simplified case of batch
operation with an assumption of smooth mass transfer from a single droplet suspended in a mild streamof the extracting solvent.
The mass transfer coefficient 𝑘 can be expressed as the Sherwood number:
𝑆ℎ = 𝑘 𝑑 ⁄𝐷 .
(21.1-13)
For a simple case, the Sherwood number can be considered to depend on the droplet Reynolds number
as follows:
𝑆ℎ = 𝑎 𝑅𝑒 𝑆𝑐(21.1-14)
When 𝑅𝑒 is sufficiently low, the following Ranz and Marshall equation3) can be applied with anassumption of the Stokes’ terminal velocity 𝑢 .

where 𝜌 , 𝜇 , 𝐷
3.

⁄

⁄

= 2.0 + 0.60

(21.1-15)

are liquid density, viscosity, and diffusivity of the extract phase.

Bird, R. B., Stewart, W. E., and Lightfoot, E. N., “Transport Phenomena,” Wiley, New York, p.647 (1960)

However in the flow condition of agitated vessel, the Reynolds number of turbine agitator may be
adopted as follows:
𝑅𝑒(21.1-16)=

where 𝑑, 𝑁 are impeller diameter and rotation number of the turbine agitator.
When the Reynolds number is very low, there is a possibility of obtaining the following simple
correlation:
⁄

=𝑎

⁄

(21.1-178Transport Phenomena in Particles/Droplets Suspended

According to the literature, many empirical correlations are available for the film coefficient of masstransfer 𝑘 around a droplet.
For the film coefficient of mass transfer 𝑘 around droplets suspended in an agitated stream, thefollowing empirical equation is also available1):.

= 0.052.

(21.1-18)

where 𝐷 is the agitated tank diameter, 𝑑 the diameter of impeller, 𝑁 the rotation number of agitator,
𝜌 , 𝜇 , and 𝐷 the density, and viscosity, and diffusivity of the continuous phase.
1)

Barker, J.J. and Treybal, R.E., AIChE Journal, 6, 289 (1960)

For a single rigid droplet suspended in the uniform stream with slip velocity 𝑢 , the Ranz-Marshallcorrelation can be applied for a single rigid droplet:𝑆ℎ = 2 + 𝐶 𝑅𝑒 ⁄ 𝑃𝑟 ⁄(21.1-19)
Here the coefficient 𝐶 varies slightly from 0.6, depending on investigators.

Fig.21.1-4 Circulating stream inside a liquid droplet rising in a continuous-phase liquid

In the case when such a circulating flow exists inside a droplet, another film coefficient of masstransfer 𝑘 in the droplet diluent A phase is added in the consideration based on the two-film theory:=+(21.1-20)
This implies that the overall resistance 1⁄𝐾 to mass transfer consists of the S-phase film resistanceand the A-phase film resistance lying in series.
In this condition, the mass transfer coefficient 𝑘 is given by taking into account the effect of insidecirculating flow as:⁄𝑆ℎ = 2 + 4⁄√6𝜋 𝑓
𝑃𝑒 ⁄(21.1-21)
Here the Peclet number for mass transfer is defined as 𝑃𝑒 = 𝑅𝑒 𝑆𝑐 = 𝑑 𝑢 ⁄𝐷 , where therelative velocity 𝑢 between the droplet and the continuous phase S is used.
You can understand that Eq.(22.1-15) is similar in form to Ranz-Marshall equation for a rigid particle.
The coefficient 𝑓 is a coefficient modifying the Peclet number talking into account the effect of
inside circulating flow.
Regarding the film coefficient 𝑘 of mass transfer inside a droplet, various equations are reported.
One simple equation is a function of the droplet Peclet number defined asTransport Processes in Dispersed Systems

9𝑆ℎ = 4𝑑 ⁄𝑙 𝜑 𝑃𝑒
and 𝑃𝑒 = 𝑅𝑒 𝑆𝑐 = 𝑑 𝑢 ⁄𝐷
(21.1-22)
The coefficient 𝜑 is also a coefficient modifying the droplet Peclet number and 𝑙 is thecharacteristic moving distance of droplets. The functional form of those coefficients 𝑓 and 4𝑑 ⁄𝑙 𝜑
are too complicated to discuss in detail here.
21.1-3 Surface tension acting on liquid droplets
Surface tension is acting on the surface of droplets in contact with gas or immiscible liquid.
As shown in Fig.21.1-5, within the bulk liquid, interior molecules have many neighboring molecules,
so that intermolecular forces work every direction and keep balance. However the molecules at thesurface are pulled inward and sideways by other molecules, but not outward away from the surface. The
tangential force is referred to as the surface tension. The surface tension works to reduce the surface area
to a minimum. Droplets are pulled into a spherical shape by the imbalanced cohesive forces. This force
balance creates some internal pressure. That is, if the pressure on the inside surface differs from theoutside pressure, the pressure difference times surface area results in additional pressure rise. Therefore
the boiling point of the droplet liquid becomes somewhat higher than the environment.

Fig.21.1-5 Generation of surface tension from intermolecular force balance near the gas-liquidinterface of a droplet

Water has a surface tension (72.8 mN/m at 20˚C) higher than most other liquids.
The pressure difference between inside and outside of the spherical droplet has the following relation
with the curvature radius 𝑅:∆𝑝 = 2 𝜎 𝑅(21.1-23)
This equation can be derived by the following force balance:∆𝑝 = 4 𝜎 ⁄𝐷 = 2 𝜎 𝑅(𝜋⁄4) 𝐷 ∆𝑝 = 𝜋 𝐷 𝜎
For the case of water at 20˚C and 1 atm, a water droplet of 1 𝑚𝑚 diameter has 145.6 Pa in the internalpressure rise.
Surfactants adsorbed on the outside surfaces of droplets can keep them stable without agglomeration
in a liquid bulk. Surfactants reduce the surface tension of water. Emulsions are a kind of colloid in whichsurface tension plays a role.

21.2

Structure of Dispersed System

21.2-1 Effect of surfactant adsorbed on liquid surface
Due to the adsorption of surfactant molecules on the liquid surface, the droplet surface is very muchhigher in surfactant concentration than the continuous liquid volume phase.
A simplest emulsion structure is accompanied with ions such as OH- adsorbed on the emulsion
droplet producing a charge. A dilute O/W emulsion is stabilized due to the electrostatic repulsion.
As shown in Fig.21.2-1, a surfactant molecule has a long hydrophobic tail and a hydrophilic head of
surfactant. For the case of cationic surfactants and anionic surfactants, the surfactant head has surface

10Structure of Dispersed System

active ions.
Surfactants adsorbed to the surface improve the dispersion in the oil and water phases.
Molecular alignment at gas-liquid or oil-water interface forms monolayer structure.

Fig.21.2-1 Monolayer structure due to molecular alignment at oil-water interface

Regarding the emulsion formation by using turbulent flows, viscous forces of the turbulent flow causeshear stresses to act on the interface between the droplets and the continuous phase. If the turbulent
eddies are much larger than the droplets, they exert shear stresses on the interface of droplets, and thencause the droplet breakup into smaller droplets.
21.2-2 Alignment structure of adsorbed surfactant molecules
When the droplet surface is fully covered with adsorbed surfactant molecules, micelles are formed.
A micelle is an aggregate of surfactant molecules dispersed in a liquid colloid. That is, micelles form
only when the concentration of surfactant is greater than the critical micelle concentration (CMC), andthe temperature of the system goes beyond the critical micelle temperature. Regarding a normal phasemicelle (oil-in-water micelle), the micelle in aqueous solution forms an aggregate with the hydrophilic
head region in contact with surrounding solvent (water), confining the hydrophobic tail regions inside.
Fig.21.2-2 shows a structure of normal phase micelle of an oil droplet in water, where a fatty acidsoap serving as the surfactant forms such a micelle structure.
For example, when the concentration of stearate ions goes beyond the critical value (CMC), stearate ionscome inside the bulk phase with their heads (-COO-) pointing outward and the tail pointing toward thecenter of droplet sphere. This aggregation forms the ionic micelles in the continuous phase solution.

Transport Processes in Dispersed Systems

11

Fig.21.2-2 Structure of normal phase micelle of an oil droplet and ionic micelle of fatty acid.

Inverse micelles have the hydrophilic headgroups directed to the center with the hydrophobic tails
extending out (water-in-oil).
The CMC (Critical Micelle Concentration) is minimum concentration at which surfactant molecules
begin to form micelles in the continuous phase solution.

21.3 Viscosity of Dispersed Systems
21.3-1 Definition of viscosity
In Chapter 2.5, we dealt with non-Newtonian Fluids. Fig.21.3-1 (or Fig.2.5-1) shows their rheologicalmodels indicating the relation between the shear stress and the velocity gradient (i.e. rate of strain). We
know that several kinds of fluids which do not obey Newton’s law of viscosity have an important relation
with the dispersed system to be studied in this chapter.

Fig.21.3-1. Schematic picture of shear stress against velocity gradient for non-Newtonian fluids

For simplicity, in this chapter, the steady-state rheological behavior of most fluids is expressed by thefollowing generalized form:(2.5-1)
𝜏 = −𝜂

12

Viscosity of Dispersed Systems

Here the proportional constant 𝜂 is called “apparent viscosity.”
shear, the behavior reduces to Newtonian, with 𝜂 = 𝜇.

If 𝜂 is independent of the rate of

21.3-2 Slurry and solid-in-liquid mixtures1)
A slurry consists of solid particles suspended in a liquid. For example, a slurry transporting pipelineis used for a coal mine (mixture of coal and water) from its pond to its destination such as a coal power
plant. When the particle concentration of solid goes beyond 10 wt%, the slurry generally begins tobehave as non-Newtonian.
The density of slurry can be calculated by the following equation:
𝜌 = ( ⁄ ) [()⁄ ](21.3-1)

where𝜌 : density of slurry mixture (kg/m3)
𝐶 : solid concentration (wt%)
𝜌 : density of solid in mixture (kg/m3)
𝜌 : density of liquid in mixture(kg/m3)

If the concentration is expressed in volume per cent,𝐶 : concentration of solids by vol%Φ: volume fraction, Φ = 𝐶 ⁄100
The viscosity of a dilute suspension can be approximated by the Einstein equation:𝜇 = 𝜇 (1 + 2.5 Φ)for Φ < 0.01
(21.3-2)
For higher-concentration suspension, the viscosity of the slurry is given by the modified equation1):
𝜇 = 𝜇 [1 + 2.5 Φ + 10.05 Φ + 0.00273 exp(16.6 Φ)](21.3-3)

1.

Thomas, D.G., Transport characteristics of suspension: VIII
particles,

A note of the viscosity of Newtonian suspensions of uniform spherical

J. Colloid Science, 267-277 (1965)[EXAMPLE 21.3-1]
Calculate the viscosity of a slurry mixture of solid particles (50 wt%, density= 2,100 kg/m3) in water.
The density and viscosity of water are given by 1,000 kg/m3 and 0.001 Pa s
Since 𝐶 = 50 𝑤𝑡%,
= 1,355 (kg/m3)
𝜌 = ( ⁄
) [()⁄]
The volume per cent is
𝐶 = 50 (1355⁄2100) = 32.3 vol% Then Φ = 𝐶 ⁄100 = 0.323
Using the water viscosity 𝜇 = 0.001 Pa ∙ s,
the viscosity of the slurry calculated by Eq.(21.3-3) becomes
𝜇 = 0.001 [1 + 2.5 ∗ 0.323 + 10.05 ∗ 0.323 + 0.00273 exp(16.6 ∗ 0.323)]= 3.44 × 10 Pa ∙ s.
This implies that the slurry mixture has 3.4 times the water viscosity.

(21.3-E1)(21.3-E2)(21.3-E3)[PROBLEM 21.3-1]
Consider the production of the iron ore slurry to be supplied to the pelletizing process of a steel maker.

Transport Processes in Dispersed Systems

13

The particles of the iron ore crushed have average size of 40 μm and density of 3,600 kg⁄m . If 40
ton of the iron ore is used for the slurry, how much water (1,000 kg⁄m , 0.001 Pa ∙ s) is required for
producing the solid concentration of 30 wt%.? Calculate the viscosity of the iron ore-in-water slurry.
Only for simplification, the particle shape can be assumed to be spherical. Usually a small quantity ofadditive is necessary but its effect can also be neglected here.
[PROBLEM 21.3-2]
We would like to prepare a slurry consisting of Magnetite powder in water at 25℃. The solid weightper cent is given by 𝐶 = 50 wt%. The Magnetite solid density 𝜌 = 5,200 kg⁄m .
The average particle size is about 0.3 μm.
Calculate the volume ratio of the slurry Φ, the density of the slurry 𝜌 , and the slurry viscosity 𝜇 .
21.3-3 Bingham plastic fluids
Slurries and emulsions often exhibit the non-Newtonian behavior.
The Bingham model is expressed as+ 𝜏
if 𝜏 > 𝜏𝜏 = −𝜇=0
if 𝜏 < 𝜏(21.3-4)
where 𝜏 is the yield stress and 𝜇 is a Bingham model dynamic viscosity similar to the Newtonianviscosity. (This model is firstly introduced in Chapter 2.5 of PART I)
Similarly to the Newtonian flow model, the pressure drop over the length L of a horizontal pipe (Inside
diameter D) can be expressed in the form of friction factor definition as𝜌𝑣(21.3-5)∆𝑝 = 4 𝑓
1)
Swamee & Aggarwal propose the following friction factor for laminar Bingham plastic flow:
𝑓 =

1+

.

(21.3-6).

The original equation has 64 instead of 16 owing to the different definition of friction factor.
The constant 16 of this equation was modified for the above definition of friction factor.
Here the Hedstrom number is defined for Bingham plastic model by:(21.3-7)

𝐻𝑒 =[EXAMPLE 21.3-2]
A Bingham plastic fluid ( 𝜌 = 1,100 𝑘𝑔⁄𝑚 at 25 ℃ ) is flowing in laminar flow regime withvolumetric flow rate 0.0025 𝑚 ⁄𝑠 in a horizontal circular pipe of D = 0.10 m. The pressure drop overL = 20 m is measured as ∆𝑝 = 0.07 MPa. The parameters of Bingham plastic fluid are given as𝜏 = 10 Pa and 𝜇 = 0.035 Pa ∙ s
Calculate the friction factor in this flow condition.
Solution:
The Hedstrom number becomes
𝐻𝑒 =

=

( ,

)( .
.

)(

)

= 9.0 × 10

(21.3-E4)

The average velocity is calculated as.
𝑣 = ( ⁄ )= ( ⁄ )( . ) = 0.318 m/s
The Reynolds number is given by( . )( .
)( ,
)
𝑅𝑒 =
=
= 1,000.
The friction factor by Swamee & Aggarwal1) is calculated as(21.3-E5)

14

𝑓 =

Viscosity of Dispersed Systems

1+

.
.

=

,

1+

. ×
.

× ,

.

= 0.223(21.3-E6)[PROBLEM 21.3-3]
A Bingham plastic fluid with 𝜏 = 8.0 Pa, 𝜇 = 30 cP, and 𝜌 = 1,200 kg⁄m is flowing
isothermally in laminar flow regime in a horizontal circular pipeline of inside diameter 𝐷 = 0.20 m.
Calculate the power requirement of the slurry pump for transporting the volumetric flow rate of 0.0040
m ⁄s in the pipeline of length 25 m.
1.

Swamee, P.K. and Aggarwal, N.: doi:10.1016/j. petrol.2011.01.015.

21.3-4 Emulsion and liquid-in-liquid mixtures
Usually emulsion viscosity is greater than the viscosity of either the oil or the water. When theconcentration of dispersed phase becomes large, the emulsion exhibits non-Newtonian behavior. The
effective viscosity of oil-in-water emulsions depends mainly on the volume fraction of dispersed phaseand temperature, along with several minor effects, such as shear rate, average droplet size, droplet size
distribution, viscosity and density of oil.
Similarly to the slurry viscosity, a simple viscosity correlation of water-in-oil emulsion has thefollowing form:𝜇= 𝜇 (1 + 2.5 Φ + 10 Φ )(21.3-8)
is emulsion viscosity, 𝜇 the viscosity of pure oil (continuous phase), and Φ the volume
where 𝜇fraction of dispersed water phase (vol. of water/mixture )This equation can also be regarded as the modified Einstein equation.
Generally the emulsion viscosity tends to increase with the droplet size decreasing. An empiricalcorrelation of emulsion viscosity with droplet size is reported in the low concentration region for the
O/W emulsions.
Fig.21.3-2 shows an experimental result1) of the relative viscosity of corn oil-in-water emulsion in
relatively low volume fraction of oil phase. Here the relative viscosity 𝜂⁄𝜂 can be defined as the ratioof emulsion viscosity to the viscosity of continuous phase fluid (water in this case).
The diagram indicates that the emulsion viscosity tends to certainly decrease with emulsion droplet size
and increases with total surface area of dispersed phase (Corn oil in water emulsion, Surfactant: fatty
acid ester of poly-glycerol, 25℃)1)
The total surface area is calculated as the total surface area of all the oil droplets contained in unit
volume of emulsion:()×()(21.3-9)𝑆=

Transport Processes in Dispersed Systems

15

Fig.21.3-2 Effects of droplet size and droplet surface area on viscosity of corn oil-in-water emulsion

1)

Asano, Y. and Sotoyama, K., The Viscosity Characteristics of The O/W Emulsions, Nippon Shokuhin Kagaku Kogaku Kaishi, Vol. 45,
435-439 (1998)

Inverse emulsion (water-in-oil) is mostly encountered as one of the important problems in oiltransporting pipelines. Undesirable effects caused by the emulsions are related to higher viscosity, larger
pressure drop, and more difficulty in oil-water separation.
At low concentration of emulsion droplets, the emulsion system can be assumed to behave as
Newtonian fluid. When the oil droplet concentration goes beyond a critical volume fraction, the emulsion
system will start to exhibit non-Newtonian pseudoplastic behaviors.
The oilfield emulsions behave as shear-thinning or pseudoplastic fluids (i.e., as shear rate increases,
viscosity decreases). It is known that there is significant delay in the transition from laminar to turbulent
regime in water-in-oil emulsion flows but such transitional delay is not observed for the case of oil-inwater emulsions. It is known that the emulsion exhibits non-Newtonian behavior when the dispersed
phase (water) volume fraction becomes greater than approximately 40%. However when the volume
fraction of water (dispersed phase) increases above a certain critical value, e.g. 90%, the emulsion"inverts" to an oil-in-water emulsion, and the water, which was the dispersed phase, now becomes thecontinuous phase.
[PROBLEM 21.3-3]
It is necessary to emulsify 0.02 𝑚 of soy bean oil with 0.1 𝑚 of water by adding a small quantityof surfactant. The pure soy bean oil has viscosity of 0.054 Pa ∙ s at 25℃.
Calculate the viscosity of soy bean oil-in-water emulsion at 25℃.
The effect of oil droplet size on the emulsion viscosity, which is not so large, may not be taken into
account.
[PROBLEM 21.3-4]
For the preparation of emulsion polymerization of styrene monomer, we are planning to make thestyrene monomer emulsion by using an agitated tank equipped with turbine-blade agitator. Styrene
monomer 0.5 ton (density 902 kg⁄m , viscosity 0.006 Pa ∙ s at 25℃) and 1 kg of sodium dodecyl
sulfate as emulsifier agent are used to produce 1 ton of styrene monomer emulsion. The emulsion droplet
size is controlled to be 0.1 μm on the average. For simplicity, it can be assumed that double emulsion(water-in-oil-in-water) droplets are not formed.
Calculate the number of emulsion droplets produced, the quantity of water required, the volume
fraction of dispersed phase, and the emulsion viscosity at 25℃.

16

Pipe Flow of Dispersed Systems

21.4 Pipe Flow of Dispersed Systems
21.4-1 Pipe flow of solid particles dispersed system
Slurries can be classed as either ”homogeneous” or “heterogeneous”: a homogeneous slurry such as
high pulp density slurry does not tend to separate immediately into solid and liquid when it becomes still
and a heterogeneous one such as ion-powder slurry tends to separate immediately when its flow is halted.
The subject of slurry rheology is quite complex, so we will deal with an example of simple heterogeneousslurry being transported in a horizontal pipeline.
When the velocity of slurry flow reduces to a certain critical value, the solid particles form a bed at
the bottom of the pipe. Therefore it is important to avoid settling of solids by keeping the flow velocities
above a certain level. This is called “critical slurry velocity” or “limiting settling velocity”. In addition,
when the solid concentration becomes above several percent, the friction losses or pressure drops
increase above 1.5 ~ 4 times the friction losses of water flow.
The calculation of slurry pipe flows will be studied with the next example.
[EXAMPLE 21.4-1]
We want to transport 𝑀 = 6 kg/s of solid powder (density 𝜌 = 2,100 𝑘𝑔⁄𝑚 , particle size
𝐷 = 100 μm) with water at 25℃ by using a horizontal pipeline of length 𝐿 = 100 m. A steel pipewith relative roughness 𝜖 ⁄𝐷 = 0.001 will be used. Its inside diameter 𝐷 should be determined by atrial-and-error method. Fig.21.4-E1 shows a horizontal pipeline for transporting the slurry of solid
powder-in-water mixture.
Calculate the inside diameter of the pipeline and the power requirement for pumping the 𝐶 = 35
wt% slurry.

Fig.21.4-E1 Horizontal pipeline transporting slurry mixture

Solution:
We should begin by calculating the water required to add to the solids to achieve the required solid
concentration.
The mass flowrate of water required for making the slurry is calculated as𝑀 =− 𝑀 = .
− 6 = 11.14 kg⁄𝑠
(21.4-E1)⁄
The volumetric flow rate of the slurry is
.
𝑄
=
+
=
+
= 14.0 × 10
The mean density of the slurry ism ⁄s(21.4-E2)

Transport Processes in Dispersed Systems

𝜌

.

=

=

. ×

= 1,220 kg⁄m

The volumetric concentration of solid in the slurry is
( ⁄ )
( ⁄
)
𝐶 =
= ( . ⁄
= 0.204)17(21.4-E3)(21.4-E4)

In this condition of the solid concentration, the slurry can be assumed to exhibit Newtonian viscosity.
It is necessary to consider the flow velocity in slurry transport systems to avoid settling of solid
particles.
The limiting settling velocity 𝑉
is expressed by the following equation with the Durand factor1) 𝐹(dimensionless):
𝑉 = 𝐹 2 𝑔 𝐷 (𝜌 − 𝜌 )⁄𝜌
(21.4-E5)
This equation can be empirically derived for turbulent flows by considering the following force balanceof the downward gravitational force with the upward force acting on particles:𝐹= (𝜌 − 𝜌 )𝑔 (𝜋⁄6)𝐷 × (1 − 𝐶 )(21.4-E6)
𝐹 = 𝜌 (𝜋⁄4)𝐷 𝑉
where 𝑉is the turbulent fluctuation velocity.
From these two equations, we obtain
𝑉

=

(

)

𝐷 × (1 − 𝐶 )𝑔(21.4-E7)

The turbulent fluctuation velocity can be related with the limiting settling velocity 𝑉
consideration the power 𝑃 dissipated per unit mass of fluid:
𝑃 = 𝜆where 𝜆 is a kind of friction factor (called “Darcy Weisbach friction factor”)
Many various models are found in most texts on slurry pumping, if required.
1.

taking into

(21.4-E8)

Durand, R., The Hydraulic Transportation of Coal and Other Materials in Pipes, Colloq. of National Coal Board, London, pp.39-55(1952)

Only for convenience and simplicity, the following empirical equation for the Durand factor isutilized:⁄

𝐹 =4𝐶 ⁄
(21.4-E9)
where 𝐶 is the volumetric concentration of the slurry, 𝐷 the particle diameter , and 𝐷 the pipe
inside diameter.
From here, let us start the trial-and-error calculation by assuming the pipe diameter 𝐷 = 0.1 m.
The Durand factor is calculated as.

⁄= 0.921𝐹 = 4 (0.204) ⁄.
Therefore substituting the Durand factor, the limiting settling velocity becomes𝑉 = (0.921) 2 (9.8) (0.10) (2100 − 1000)⁄1000 = 1.61 m⁄s
The average velocity of the slurry is given by𝑉= ( ⁄ ). ×= ( ⁄ ) .

= 1.78 m⁄s

Therefore the result 𝑉> 𝑉
for the stable slurry pumping.

(21.4-E10)(21.4-E11)(21.4-E12)

implies that the assumed pipe diameter 𝐷 = 0.1 m is appropriate

18

Pipe Flow of Dispersed Systems

The slurry viscosity with relatively high solid concentration is calculated by Eq.(21.3-3)
𝜇 = 𝜇 [1 + 2.5 Φ + 10.05 Φ + 0.00273 exp(16.6 Φ)],(21.3-3)Here Φ = 𝐶 = 0.204 and 𝜇 = 0.001 Pa ∙ s in this case.
Then𝜇= (0.001)[1 + 2.5 × 0.204 + 10.05 × 0.204 + 0.00273 exp(16.6 × 0.204)] = 0.00201 Pa ∙ s(21.4-E13)
The Reynolds number is calculated as( . )( . )()== 108,000
𝑅𝑒=.

(21.4-E14)

As shown in Fig.21.4-E2, the friction factor is obtained from the dotted line of Fig.21.4-E2 (see Fig.7.31 of PART I),
f = 0.0055
for 𝜖 ⁄𝐷 = 0.001
at 𝑅𝑒= 108,000

Fig.21.4-E2 Friction factor vs. Reynolds number for slurry flows

The pressure drop over the horizontal pipeline of 100 m is calculated by the following equation:(1220)(1.78 ) = 42,500 Pa
𝜌 𝑉= 4(0.0055) .
(21.4-E15)∆𝑃 = 4 𝑓= 42,500 kg⁄(m s )Finally the theoretical power requirement for the slurry pump is given by
𝑊 = −𝑊 = 𝑄 (𝑚 ⁄𝑠) × ∆𝑃 (𝑘𝑔⁄𝑚 𝑠 ) = (14.0 × 10 ) (42,500) = 595 W ≅ 0.6 kW.
(21.4-E16)
It should be kept in mind that if possible, we should consider the pump efficiency as well as the friction
losses of pipe fittings such as valves, bends and so on.
[PROBLEM 21.4-2] (see [PROBLEM 21.3-1])
We are planning to transport the iron ore slurry through a long steel pipeline to the pelletizing plant.
As in Problem 21.3-1, the particles of the iron ore crushed have average size of 40 μm and density of
3,600 kg⁄m . If 𝑀 = 18 ton/h of the iron ore is transported with the solid concentration of 30 wt%.,
how much 𝑀 ton⁄h of water (1,000 kg⁄m , 0.001 Pa ∙ s) is required ? Determine the appropriate
inside diameter of the pipeline.
21.4-2 Pipe flow of liquid droplets dispersed systems
Regarding the emulsion flow at low concentration of emulsion droplets, the emulsion system exhibits
Newtonian behaviors. However there is a possibility of undesirable adverse effects related to higher
viscosity and larger friction loss. For the case of water-in-oil emulsion flow, there may be a significant

Transport Processes in Dispersed Systems

19

delay in the transition from laminar to turbulent regime. However such a delay does not seem to beobserved for the case of oil-in-water emulsion. It can be said that the emulsion viscosity strongly depends
on the viscosity of the continuous phase. When the water concentration increases beyond a certain
critical value, the water-in-oil emulsion will undergo the phase inversion into the oil-in-water one.
According to the surface tension theory, emulsification takes place by reduction of interfacial tension
between the dispersed phase and the continuous phase. An appropriate surfactant can increase the kinetic
stability of an emulsion so that the size of the emulsion droplets is kept unchanged over time. Emulsion
stability can be regarded as the system ability to resist changes in physicochemical properties over time.
This indicates the effect of surfactant adsorbed layer on the emulsion stabilization.
The effect of emulsions on the flow friction factor of pipe flows1) exhibits different behaviors forturbulent flow regimes depending upon unstable emulsions or surfactant-stabilized emulsions. For thecase of unstable emulsions in turbulent flow, the experimental friction factor may become much lower
than the usual values of single-phase flow calculated by the Blasius equation (see Eq.(7.3-12)). This iscalled “drag-reduction behavior”. However for the case of surfactant-stabilized emulsions in turbulent
flow, it is known that no drag-reduction behavior appears. This may be due to the fact that the dispersed
droplets smaller than turbulent eddies flow along with them.
For simplicity, we do not deal here with multiple emulsions such as “water-in-oil-in-water” and “oilin-water-in oil” emulsions.
1)

Dol, S-S and Sen, L-J, The effect of flow-induced oil-water emulsions on pressure drop, J. Theoretical and Applied Mechanics, Vol.2,
73-78 (2019)[EXAMPLE 21.4-2]
Consider a crude oil containing some volume of water being transported by a turbine pump in apipeline shown in Fig.21.4-E3. The inside diameter and the length of the pipeline are 100 mm and 100
m, respectively. The density of the pure oil with no water content is 𝜌 = 880 kg/m3. The flowrate of
the oil mixture is 𝑉 = 30 m3/h. Its water content is 20% in volume. The density of the oil mixture
becomes 𝜌= 900 kg/m3. Small amount of surfactant is added into the crude oil mixture forstabilization. It has been confirmed at the flow condition that the oil-water mixture can become a waterin-oil emulsion in the downstream region. The viscosity of pure crude oil (0% water content) is measured
in advance as 𝜂 = 0.008 Pa∙ s at 25℃. It can be considered that the pure crude oil exhibits Newtonian
property.

Fig.21.4-E3 Crude oil transporting pipeline

The pressure drop measured over the straight pipe of length 10 m in the downstream region is measured
as 1,760 Pa. Calculate the effective viscosity of the oil mixture.

20

Pipe Flow of Dispersed Systems

Solution:
We begin by calculation of the pressure drop for the pure crude oil in the same condition as for thereal oil mixture of interest.
The pressure drop is expressed with the definition of friction factor as∆𝑃 = 4𝑓
𝜌〈𝑣〉(7.2-10)
The average velocity in the straight pipe region is calculated as
〈𝑣〉 =

⁄
( ⁄ )

= 1.06 m⁄s

The Reynolds number is given by( . )( . )()
𝑅𝑒 === 11,660.

(21.4-E17)(21.4-E18)

According to Fig.21.4-E4 (see Fig.7.3-1), the friction factor for Newtonian fluids is given at the
Reynolds number to be 𝑓 = 0.0075.

Fig.21.4-E4 Friction factor vs. Reynolds number for pipe flows

Therefore the pressure drop for the pure oil in the fully-developed region can be calculated as follows:(880)(1.06) = 1,480 Pa∆𝑃 = 4(0.0075) .
(21.4-E19)
The real pressure drop for the oil mixture measured in the downstream region is given by ∆𝑃=
1,760 Pa
It can be assumed in the fully-developed region with the same flowrate that×==
= 1.19∆×∆(21.4-E20)

The pressure drop of the crude oil mixture becomes 1.19 times as large as the pressure drop for pure
crude oil (0% water content).
Therefore the friction factor of the oil mixture becomes
×
∆
.
×
𝑓
=
=
× 1.19 = 0.0087(21.4-E21)∆
This value is obtained at 𝑅𝑒= 7,000 in Fig.7.3-1.
The viscosity of the oil mixture is calculated as〈 〉( . )( . )()
𝜂==
= 0.0136(21.4-E21)

𝜂 ⁄𝜂 = 0.0136⁄0.008 = 1.7(21.4-E22)
The viscosity of the oil mixture with water content of 20% becomes 1.7 times the viscosity of the pureoil with no water content.

Transport Processes in Dispersed Systems

21

This kind of emulsion (water-in-oil) is mostly encountered as one of the important problems in oiltransporting pipelines. Undesirable effects caused by the emulsions are related to higher viscosity, larger
pressure drop, and more oil-water separation difficulty.
An emulsion polymerization will be studied in Chapter 28.3 of this volume.
[EXAMPLE 21.4-3]
Consider an agitated tank baffled for producing an emulsion of fatty acid methyl ester in water.
0.1 m3 of oleic acid methyl ester and 1.0 m3 of water are put in the tank (1.5 m3).
A 6 bladed disk turbine as the agitator is rotated at 80 rpm to make an emulsion. The temperature is kept
at 40℃ with the jacket outside the tank.
The dimensions of the agitator and vessel are
Diameter of impeller 𝑑 = 0.8 m, tank diameter 𝐷 = 1.0 m, impeller width 𝑏 = 0.1 m, depth of liquid
in vessel 𝐻 = 1.4 m, height of impeller above tank floor 𝐻 = 0.7 m, width of baffles 𝑗 = 0.05 m,
By the preliminary examination, the effective viscosity and density of the stable emulsion of the samecontent are confirmed to be 𝜂 = 0.00105 Pa ∙ s and 𝜌 = 970 kg⁄m at 40℃.
Calculate the power requirement at steady emulsion state.
It may be assumed that the sample emulsion for the preliminary test should have the same
properties as the real emulsion of interest.

Fig.21.4-E5 Agitated vessel with a 6-blade disk turbine impeller for emulsification

Solution:
The Reynolds number defined by Eq.(20.2-1) of PART II is calculated as
𝑅𝑒 =

=

( .

⁄

)(
.

)(

)

= 788,300(21.4-E23)

According to Fig.20.2-3 of PART II, the power number is obtained as shown in Fig.21.4-E6:
𝑁𝑝 = 6.0 at 𝑅𝑒 = 788,300 for 6-bladed disk turbine with baffled tank
From Eq.(20.2-2) of PART II, the power requirement is given by
𝑃 = 𝜌 𝑁 𝑑 𝑁𝑝 = (970 𝑘𝑔⁄𝑚 ) (80⁄60 𝑠) (0.8 𝑚) (6.0) = 4,520 W(21.4-E24)
Theoretical power requirement is given by 4.52 kW.
This emulsion can be considered to exhibit Newtonian behavior because of the low concentration (ca.
9%) of emulsion droplets.
It can be understood that this emulsion should have 1.5 times the viscosity of pure water (0.0007 Pa ∙ s
at 40℃).

22

Pipe Flow of Dispersed Systems

Fig.21.4-E6 Power number vs. Reynolds numberThis is an example of oil-in-water emulsion. Roughly speaking, the emulsion exhibits non-Newtonian
behavior when the volume fraction of the dispersed phase goes beyond 40%. When the fraction is greaterthan 90%, the phase inversion to the water-in-oil will occur.
[PROBLEM 21.4-3]
One of the pasteurization processes of cows’ milk is conducted by using a batch jacketed agitator tank.
Cows’ milk is an emulsion whose constituents are about 87% water, 4% fat, 3.4% protein 4.8% lactose,and 0.8%minerals. It can be considered that the raw milk exhibits Newtonian behavior.
The agitator is a 6-bladed disk turbine impeller. The first treatment is heated keeping at 62℃ for 30 min
by the 70℃ hot water flowing in the jacket. The next treatment is cooled by the cooling water replacedin the same jacket.
Let us consider that an emulsion of cow’s raw milk 2 ton is heated for pasteurization at 62℃ in thejacketed agitator tank. The dimensions of the agitator tank are given by
Turbine impeller diameter 𝑑 = 0.5 m, width 𝑏 = 100 mm,
Tank diameter 𝐷 = 1 m, and baffle plate width 𝑗 = 50 mm
The rotation number of the impeller is set at 𝑁 = 50 RPM.
The physical properties are given at 62℃ are: density 𝜌 = 1,020 kg⁄m , viscosity 𝜇 = 1.2 mPa ∙ s,heat capacity 𝐶𝑝 = 3.93 kJ⁄kg K, thermal conductivity 𝜅 = 0.6 W⁄m K,
The correlation of the heat transfer coefficient on the inside wall of the agitator tank is given by
Eq.(20.3-1) of PART II:(20.3-1)
𝑁𝑢 = 0.44 𝑅𝑒 ⁄ 𝑃𝑟 ⁄ (𝜇 ⁄𝜇 ) .
Since the temperature difference between the bulk and the jacket wall is small, the viscosity correction
term can be assumed to be unity.
For simplification, the heat transfer coefficient of the jacket-side wall can be assumed as ℎ = 1,160
W/m2K. The hot water temperature in the jacket is also assumed as 70℃ averaged between the inlet andthe outlet of the jacket.
(1) Calculate the power requirement of the agitator.
(2) Calculate the heat transfer area required for the pasteurization process by heating.

Transport Processes in Dispersed Systems

21.5

23

Emulsion

Let us consider a stable suspension of very small droplets of one liquid in another with which it isimmiscible. This is called “emulsion.” That dispersed mixture of two or more immiscible liquids areusually characterized by the terms water-in-oil (dispersed water droplets in oil) and oil-in-water(dispersed oil droplets in water). The bulk liquid surrounding these dispersed droplets is called“continuous phase.” For example, an oil phased emulsion features water small droplets, where thedispersed phase uniformly distributes throughout the continuous phase of oil.
Almost all emulsions are inherently unstable due to the thermodynamic incompatibility of the oil andwater phases at the interface. Therefore a surfactant is usually used for emulsion stability preventing thedispersed droplets from collision and coalescence.
Many food mixtures such as mayonnaise, yogurt, homogenized milk, and vinaigrette can be regarded
as stable oil-in-water food emulsions, and butter is one of the water-in-oil emulsion.
The size of dispersed liquid droplets usually lies in 100 ~ 400 nm.
Usually an emulsion mixture is made using some kinds of surfactants as an emulsifier to stabilize
dispersed droplets. The mechanical method of emulsion formation usually requires high shear mixing or
homogenization. Emulsions appear white when all light is scattered equally, but high-frequency (shortwavelength) light will be scattered more for very dilute emulsions, so the emulsion will appear bluer(Tyndall effect).
21.5-1 Emulsification and emulsion viscosity
Generally speaking, higher mixing speed and longer mixing duration can produce more stableemulsions with smaller dispersed droplets. Usually an emulsifier is a substance called surfactant or“surface-active agent” that stabilizes an emulsion by reducing the surface tension at the oil-waterinterface.
In the emulsification process, one immiscible liquid is dispersed in another immiscible liquid.
Emulsifiers having more or less solubility either in oil or water as emulsifiers are those compounds thathave a polar or hydrophilic means water-soluble part and a non-polar or hydrophobic means waterinsoluble part.
The following three mechanisms of emulsification are available: (1) Reduction of interfacial tension
between the dispersed phase and the dispersion medium, (2) Repulsion force acting between thedispersed droplets where a repelling film is formed over the one phase by the emulsifying agent, and (3)Viscosity increased by some emulsifying agents forms globules of the dispersed phase stable in thedispersion medium.
Emulsification is accomplished by lowering the interfacial tension between the two phases.
Successively a repelling film created over a single phase separates into globules by the emulsifying agent.
If some viscous substances such as glycerol or CMC (carboxymethyl cellulose) are added, the emulsion
is preserved due to the viscosity increase.
Fundamentally the viscosity of an emulsion depends on the initial viscosity of the bulk phase 𝜇 andthe volume fraction of the droplets 𝜑. The equation of the emulsion is expressed as
𝜇 = 𝜇 (1 + 𝑎 𝜑)(21.5-1)1)Here the Einstein equation gives 𝑎 = 2.5 for solid rigid particles dispersed.
However we should be careful about the unfavorable results obtained because the viscosity tends toextremely rise in the volume fraction above a certain upper limit value.
If the concentration of dispersed-phase liquid is low, the effective viscosity of the emulsion can be
expressed as the Taylor equation1) for the emulsion of solid particles suspended in the continuous phaseof water.

24

EmulsionIf the dispersed liquid droplets having a circulating flow inside has very large viscosity 𝜂′, the internal
circulating flow is suppressed. The constant 𝑎 of Eq. (21.5-1) approaches 2.5.
This indicates that such liquid droplets behave like solid rigid particles.
For the case of liquid droplets having circulating flows inside, the emulsion viscosity 𝜇 is lowered
and the constant 𝑎 is given by the equation:.
(21.5-2)𝑎=Here 𝜂′ is the viscosity acting within the dispersed oil droplets.
According to Taylor (1934)1), for dilute simple emulsion with no interaction between the droplets, therelative viscosity of emulsion is given by.
= 1 + 2.5
𝜑
(21.5-3)𝜂 =where 𝜂 is the effective viscosity of emulsion and 𝑘 = 𝜂 ⁄𝜂 is the ratio of the viscosity of thedispersed phase (𝜂 ) to that pf the continuous phase (𝜂 ).
1.

Taylor, G. I., Proc. Roy. Soc., (London), A146, 501 (1934).

[PROBLEM 21.5-1]
An O/W emulsion at 25℃ consists of 10 vol% corn oil (viscosity 52.3 mPa∙s) and 90 vol% water(viscosity 1 mPa∙s) with small quantity of surfactant (PGE). Calculate the relative viscosity 𝜂 of theemulsion at 25℃.
Except for very viscous mixtures, the power requirement of emulsification can be calculated by thesame as the usual power number correlation 𝑁𝑝 = 𝑓(𝑅𝑒) for single-liquid agitation (see Chapter 20.2).
Usually the emulsification by agitation utilizes a rotating impeller immersed in the liquid mixture. The
basic requirements regarding the shape and arrangement of the vessel, the arrangement of the impeller,
and the baffle plates are essentially the same as those for dispersing finely divided solids in liquids,
which are discussed in Chapter 20 of Vol.1.
For the case of emulsion, the size of droplets dispersed should have a wide range varying locally withlocation in the vessel. There are different shear stress zones distributed within the agitated vessel.
There are many small droplets near the impeller and many larger droplets in the low shear stress region
far from the impeller. If possible, it is much better to practically measure the emulsion viscosity by aviscometer. We can check whether the emulsion tested should have non-Newtonian properties or not.
The mixing condition and agitator power requirement should be calculated by using the measured
viscosity of the emulsion in the equation of the power requirement correlation database.
[EXAMPLE 21.5-1]
By using an agitated vessel with an immersed six-bladed disk turbine impeller, a fatty acid ester andwater are agitated to obtain an emulsion with 𝜂′⁄𝜇 = 4.8. The conditions of agitation by the disk
turbine are: 𝑁 = 180 rpm, blade diameter 𝑑 = 400 mm , tank diameter 𝐷 = 800 mm, depth ofliquid 𝐻 = 800 mm, impeller height from tank floor 𝐻 = 250 mm. The viscosity of the ester dropletsof fatty acid ester is assumed to be 0.0038 Pa s. The water viscosity 𝜇 = 0.001 Pa s (assumed 25℃).
The volume fraction of the ester in the mixture is approximately 15%, i.e. 𝜑 = 0.15.
Calculate the power requirement.
Solution:
The densities of the dispersed-phase and continuous-phase are
998 kg⁄m .

𝜌 = 870 kg⁄m , 𝜌 =Transport Processes in Dispersed Systems

25

The constant of Eq.(21.5-2) relating with the circulating flow inside a droplet) becomes
.
. ( .
)
.
=
= 2.24(21.5-E1)𝑎=.
.
The average density and viscosity of the emulsion are calculated by Eqs.(21.5-3) and (21.5-4):𝜌 = 𝜌 𝜑 + 𝜌 𝜑 = (998)(0.85) + (870)(0.15) = 978.8 kg⁄m
(21.5-E2)
𝜇 = 𝜇 (1 + 𝑎 𝜑 ) = 0.001(1 + 2.24 × 0.85) = 0.0029 Pa s
(21.5-E3)
The Reynolds number is calculated as
𝑅𝑒 =

=

(

. )( .
.

) (

⁄

)

= 162,000(21.5-E4)

According to Fig.20.2-3 in Chapter 20 of PART II, the unbaffled 6 blades disk turbine impeller gives thepower number 𝑁𝑝 = 1.1 at 𝑅𝑒 = 162,000.
Therefore the power requirement becomes
𝑃 = 𝑁𝑝 × (𝜌 𝑁 𝑑 ) = 1.1 × [(978.8)(3.0) (0.40) ] = 2980 kg m ⁄s = 2.98 kW(21.5-E5)[PROBLEM 21.5-2]
We are requested to produce a corn oil emulsion by using an agitated vessel equipped with a sixbladed disk turbine impeller. A corn oil and water are mixed at 25℃ to obtain an emulsion with therelative viscosity of 𝜂 = 1.3. The conditions of agitation by the disk turbine are: 𝑁 = 200 rpm, blade
diameter 𝑑 = 300 mm, tank diameter 𝐷 = 700 mm, depth of liquid 𝐻 = 700 mm, impeller heightfrom tank floor 𝐻 = 250 mm. The viscosity of the corn oil droplets is assumed to be 0.054 Pa s. The
water viscosity 𝜇 = 0.001 Pa s (assumed 25℃). The effect of the added surfactant on the viscosity
behavior can be assumed to be negligible. The volume fraction of the corn oil in the mixture isapproximately 15%, i.e. 𝜑 = 0.15.
We can consider the rheological behavior of the emulsion to be Newtonian because of the small
volume fraction of the dispersed phase.
Calculate the power requirement.
At any rate, it is outside of this text territory to describe further the rheological and/or colloidal
behavior of various viscous emulsions, which becomes very difficult especially in the developing termof non-steady state variation.
21.5-2 Rheological behavior of emulsion
The rheological effect of microstructural mechanism of emulsions is so complicated that thetheoretical approaches are beyond the territory of this course. Therefore some of the simple models will
be studied here. The flow field of a highly concentrated O/W emulsion exhibits laminar regime but does
not form a parabolic velocity profile in a pipe flows. For example, a dense emulsion like mayonnaise or
yogurt exhibits non-Newtonian behaviors. Emulsion viscosities 𝜂 are also a function of shear rate 𝛾̇ .
Oil-in-water emulsion
Let us consider an oil-in-water emulsion such as mayonnaise.
If the emulsion viscosity 𝜂 decreases with increasing the shear rate 𝛾̇ , the emulsion is called “shearthinning”. The emulsions made up from small droplets are usually more viscous than the emulsions
made up from bigger droplets. When the oil fraction in volume increases, the viscosity is increased. Thisis due to the fact that the continuous-phase fluid film surrounding the dispersed droplets becomes thin.
Figure 21.5-1 indicates a schematic picture of the apparent viscosity of the shear-thinning emulsion
varying with the shear rate.

26

EmulsionFig.21.5-1 Typical variation of apparent viscosity with shear rate for shear-thinning emulsions

The region of shear rate where the emulsion viscosity decreases with increasing shear rate is called“Power-model region”:
𝜂 = 𝐾 𝛾̇(21.5-4)
The HB consistency index K changes with oil concentration but the flow behavior index n hardly changes.
As shown in Fig.21.5-2, however there are two shear rate regions turning away from the Power law.

Fig.21.5-2 Zero-shear-rate viscosity and infinite-shear-rate viscosity

As shown in Fig.21.5-2, when the shear rate approaches zero, shear stress does not approaches zero,that is, some yield stress 𝜂 exists. On the other hand, at very high shear rates, the apparent viscosity
becomes constant. This is called “limiting viscosity”, i.e. infinite shear rate viscosity.
It is known that as the shear rate increases with time, the apparent viscosity of emulsions decreases.

Transport Processes in Dispersed Systems

27

This is also called “shear thinning behavior”, which is termed “pseudoplastic”. The shear thinningbehavior occurs due to a gradual orientation of fluid elements such as liquid droplets or molecules in theflow direction to reduce friction and the deformation of the microstructures.
The shear thinning behavior is described by the following Herschel-Bulkley model1):𝜏 = 𝜏 + 𝐾 𝛾̇(21.5-5)This can be regarded as the power-law model accompanied with a yield stress.
The apparent viscosity is given by the Power-law model based on the same concept:
𝜂 = 𝑘 𝛾̇(21.5-6)
where 𝜏, 𝜏 , 𝛾̇ , and 𝜂 are shear stress (Pa), yield stress (Pa), shear rate (1/s), and apparent viscosity(Pa ∙ s), respectively..
The 𝐾 is a constant called the HB consistency index.
The shear rate 𝛾̇ can also be expressed as the velocity gradient: − (𝑑𝑣 ⁄𝑑𝑦).
The yield stress is an important parameter for stability of foods such as mayonnaise.
Usually the exponent n of pseudoplastic fluids lies in the range 0.4 < 𝑛 < 0.7.
Mayonnaise consists of very many oil droplets forming the dispersed phase and small quantity ofcontinuous-phase water which forms a very thin film around those oil droplets.
1)

Herschel, W.H. and Bulkley, R., Kolloid Zeitschrift, vol.39(4), 291-300 (1926)

Regarding time-dependent shear thinning behavior, when the shear rate increases with time, the shear
stress becomes higher than when the shear rate decreases. The shear stress results in the reordering of
fluid elements or molecules to reduce the overall stress. This hysteresis shown in Fig.21.5-3 can beconsidered as one of the thixotropic behaviors.2)

Fig.21.5-3 Hysteretic property of shear-thinning behavior of high-concentrated emulsion

It should also be kept in mind that there exist shear-thickening fluids. A dilatant (termed shear
thickening) has the viscosity increasing with shear rate. Shear thickening is observed only in somesuspension mixtures. Shear thickening viscosity can also be expressed by the Power-law model:
𝜂 = 𝑘 𝛾̇(21.5-7)
A mixture of cornstarch and water is an example of shear-thickening.
The dilatant behavior occurs when n is greater than unity.

28

2.

Emulsion

Yuceer, M., Ilyasoglu, H., Oezcelik, B., J. Appl. Poult. Res., 25, 518-527 (2016)21.5-3 Velocity distribution of non-Newtonian fluid pipe flow
It is comprehensible to reconsider the rheological characteristics of the dispersed systems before
closing this chapter.
Consider laminar flow in a circular pipe with the pressure drop 𝑝 − 𝑝 from pipe length 𝑧 = 0 to 𝐿.
Here 𝑣 is axial velocity component and 𝑅 = 𝐷⁄2 is the inner radius of the pipe.
[Power law model]
As in Chapter 6.4, the power low model for a circular pipe flow is given by(21.5-8)=𝑚 −According to Eq. (6.4-E8) in Chapter 6.4, the laminar flow of the power law model in a circular pipe
has the following velocity distribution:

𝜏

⁄

𝑣 =

𝑅

(21.5-9)1−

By using Eq. (21.8-9), the average velocity can be calculated as
𝑣

=

∫

⁄

=

∫

(21.5-10)

𝑅

Therefore=(21.5-11)

1−[Bingham model]
As in Chapter 2.5, the Bingham model is given by+ 𝜏
if 𝜏 > 𝜏𝜏 = −𝜇=0
if 𝜏 < 𝜏
where 𝜏 is the yield stress and 𝜇 is a Bingham model parameter similar to the Newtonian viscosity.
The velocity distribution is given by
𝑣 =

𝑅− 𝑟

1−

𝑟 <𝑟 <𝑅

𝑣 = 𝑣

=

𝑅− 𝑟

0 <𝑟 < 𝑟where 𝑟 is the radial position called the yield radius where the shear stress becomes 𝜏
The average velocity is given by(𝑝 − 𝑝 ) 1𝑣 =(3 𝑅 − 4 𝑟 𝑅 + 𝑟 )24 𝜇 𝐿 𝑅
Then𝑟− 𝑟
1− 𝑅− 𝑟
𝑅−𝑟
𝑣
=6
𝑅
𝑣
3𝑅 −4𝑟 𝑅 + 𝑟= 𝜏 .

By using the above equations, we can compare various non-Newtonian flids in velocity distribution
with Newtonian fluid. These results are given in Fig.21.5-4, where two cases of the pseudoplastic fluids
depending on the power model with n = 0.6 and 0.25 and the Bingham plastic fluid case with the yieldradius 𝑟 ⁄𝑅 = 0.25 are shown. Mayonnaise (n = 0.6) and ketchup (n = 0.25) are given as two examplesof the pseudoplastic.

Transport Processes in Dispersed Systems

29

Fig.21.5-4 Comparison of non-Newtonian flows with Newtonian flow with respect to velocity profile
of laminar flow in a circular pipe.

[PROBLEM 21.5-3]
We know as the Bingham plastic behavior that a toothpaste is pushed out in the form of cylindricalsolid from its tube. Calculate the velocity profile of 𝑣 ⁄𝑣
vs. 𝑟⁄𝑅 when the yield radius by𝑟 ⁄𝑅 = 0.45 and plot its result as in Fig.21.8-4.
21.5-4 Shear-thinning emulsion
Since shear rate 𝛾̇ in the agitated vessel varies with distance from the impeller, it depends on threedimensional velocity gradients. As shown in Fig.21.5-5, local shear rate becomes lowest near the vessel
walls and at the free surface of the agitated fluid while the apparent viscosity becomes lowest near the
impeller and highest at the walls. Local shear rates become so high near the edges of the impeller thatlarger dispersed droplets undergo large stresses enough to be broken into smaller droplets.

Fig.21.5-5 Local distributions of shear rate and apparent viscosity of shear-thinning emulsion in an
agitated unbaffled vessel

Figure 21.5-6 gives several rheological models of dispersed systems summarized. Many shear-thinning
emulsions may have small yield stresses at the point of zero shear rate.

30

Nomenclature

Fig.21.5-6 Non-Newtonian properties of various dispersed systems

Nomenclature
A
𝑎
𝐶
𝐶
𝐶𝑝
𝐶 ,𝐶
𝑑
𝐷
𝐷
𝐷𝑝
𝐹
𝑓
𝑓, 𝜑
𝑔
𝐻𝑒
ℎ
𝑘
𝐾
𝐿
𝑙
N
𝑁𝑢
𝑁
Pe
𝑃𝑟
𝑟
𝑅 ,𝑅
𝑅𝑒
𝑆𝑐
Sh
𝑇
t
𝑢
𝑣

mass transfer area, [m2]constant of Eq.(21.5-1) for viscosity of dispersed sytems,, [ - ]concentration, [kg/m3]
drag coefficient, [ - ]heat capacity, [J/kmol K]
solid concentration, [vol%], [wt%]impeller diameter, [m]tube diameter or tank diameter, [m]diffusivity of B in S, [m2/s]
diameter of a particle or droplet, [m]Durand factor, [ - ]friction factor, [ - ]coefficient modifying Peclet number, [ - ]gravitational acceleration, [m/s2]
Hedstrom number defined by Eq.(21.3-7)
heat transfer coefficient averaged, [W/m2K]
mass transfer coefficient averaged, [kmol/m2s]
overall mass transfer coefficient, [m/s] or HB model consistency index, [ - ] (Herschel-Bulkleymodel)
tube length, [m]characteristic moving distance, [m]number of droplets, [ - ] or rotation number, [1/s]Nusselt number, [ - ]Power number
Peclet number, [ - ]Prandtl number, [ - ]yield radius, [m]resistance force, [N]Reynolds number, [ - ]Schmidt number, [ - ]Sherwood number, [ - ]temperature, [K]time, [s]terminal velocity, [m/s]
fluid velocity, [m/s]Transport Processes in Dispersed Systems
𝑉
𝑥, 𝑦
𝛾̇
𝜂
𝜅
𝜆
𝝀𝑳
𝜇
𝜌
𝜎
𝜏 ,𝜏
𝜏
Φ
𝜑

limiting settling velocity, [ - ]horizontal or vertical coordinate, [m]shear rate, [1/s]apparent viscosity, [Pa s]
thermal conductivity, [W/m K]
latent heat of evaporation, [J/kg]
Darcy-Weisbach friction factor, [ - ]viscosity, [Pa s]
density, [kg/m3]
surface tension, [N/m]shear stress, [Pa]yield stress, [Pa]volume fraction, [ - ]volume fraction of droplets, [ - ]Subscripts
Bsoluteem
emulsionexextract
𝑓film or fluidhg
hot gasiinterfaceLliquid phasemix
mixturepparticle or droplet
s
solid
S
solvent∞
uniform flow

31

32

Chapter 22 Extraction Engineering

33

CHAPTER 22
EXTRACTION ENGINEERING
22.1 Liquid Extraction
Separation of close-boiling mixtures often becomes a technical problem as to which method may be
used, distillation or extraction. When either distillation or extraction may be used, the choice is usually
distillation in spite of the large heating and cooling duties. Extraction utilizes differences in thesolubility of the components rather than the differences in their volatility. The liquid extraction is used
to separate two mutually soluble liquids by adding a third liquid (solvent) having a preferential solubility
for one of the components of the mixture. The liquid-liquid extraction can be used if at least one of theliquid components (e.g. solvent) must be sufficiently insoluble so that two liquid phases are formed. In
usual cases the liquid-liquid extraction is conducted in isothermal operation with negligibly small
thermal energy consumption.
22.1-1 Liquid-liquid Equilibrium
The phase equilibria in liquid extraction give the distribution of a solute B between two immiscible
liquids A and S. Firstly only for simplicity, let us consider the situation where the mutual solubility ofliquids A and S is negligible even in the presence of component B.

Fig.22.1-1 Batch extractor of baffled agitation tank

For a batch tank-type extractor shown in Fig.22.1-1, the original solution (feed) after adding thesolvent is mixed by a turbine or propeller stirrer installed in the tank. During that period, the solutetransfers in some degree depending on their solubility from the continuous phase (the original solution)
to the dispersed liquid droplets (the added solvent). When the stirrer is shut off, two immiscible layers
are formed by gravity. The solvent layer is called extract and the other layer composed of the remainderof the original solution plus a small amount of the solvent is termed the raffinate.

34

Liquid Extraction

In this Chapter, we will study the ternary case where the solvent S is immiscible with the initialsolution diluent A. This system consists of three components where one phase has some solute B
dissolved in diluent A and the other phase has some other solute B extracted in solvent S.
One example is the recovery of acetic acid from its dilute aqueous solution by isopropyl ether used as
the extracting solvent. Isopropyl ether and water are practically immiscible in mixture. When isopropyl
ether is added to the aqueous solution, some of the acetic acid dissolves in this extracting solvent and
two liquid layers are formed. The extract (isopropyl ether) layer contains only acetic acid and ether,while the raffinate (water) layer contains acetic acid and water. At equilibrium conditions, there must bea definite relationship in the acetic acid concentration between the extract and the raffinate.
Let 𝑋 represent the kg of acetic acid (i.e., the extractable component) per kg of water in the initialsolution, 𝑆 the kg of isopropyl ether (i.e., the extracting solvent) added, 𝐻 the kg of water in theinitial solution. After mixing, 𝑌 and 𝑋 the equilibrium concentrations of acetic acid in the extract andthe raffinate as kg/kg of isopropyl ether and kg/kg of water.
The overall mass balance is given by
𝑋 𝐻 = 𝑋 𝐻 + 𝑌𝑆
(22.1-1)
The phase equilibrium relationship is usually expressed on an equilateral triangular graph drawn with
an isothermal condition and at an appropriate pressure keeping the whole system liquid.
A typical simplest example is given by Fig.22.1-2, where A, B, S are the feed solvent (diluent), solute,and solvent, respectively. The liquid pairs B-A and B-S are completely miscible but A and S can givethe mutually saturated solutions at G and H owing to their limited dissolubility.

Fig.22.1-2 Triangular diagram for ternary system with partial miscibility of solvent and diluent.

As seen in Fig.22.1-2, if some of B is added to such mixtures G and H, the two mixtures A and S
become more soluble. In this case, the saturated composition gives a solubility curve GMPNH.
The concentrations represented by the triangular diagrams are based on the entire mixture. Therefore
the concentration of the three components must add to unity:𝑥 + 𝑥 + 𝑥 =1
The two phases become one at the point P called “plait point.” There the solute B provides equilibrium
liquids distributing between the saturated phases at points M and N joined by a tie line.
When the solvent S is added into the feed with composition given by F, the mixture composition
moves along the line from F to S’. If the mixture composition lying at D within the curve GMPNH is
settled, the mixture solution will is separated into the two equilibrium phases. Their equilibriumcompositions are given by points R and E, respectively. At this condition, the mass balance is given by
𝑅 + 𝐸 = 𝐷 and 𝑅𝐷 ⁄𝐷𝐸 = 𝐸 ⁄𝑅
(22.1-2)
The S-phase composition moves along the curve HEP whereas the A- phase composition moves along
the curve GRP.

Extraction Engineering

35

Using the concentration of B at points M and N, the distribution coefficient 𝑚 is defined in units ofmole fraction as:(22.1-3)𝑚=
The distribution coefficient 𝐾 is also expressed as the mass fraction ratio:
𝐾=(22.1-4)

Finally the point M coincides with the point N at the plait point P, where the A phase (raffinate) andS phase (extract) are combined into a single-phase mixture.
If the solvent S is removed from one phase of extracted mixture of composition E, an A-phase of
mixture with concentration 𝑃 is obtained at the point PE. Similarly if the solvent S is removed fromthe other phase of extracted mixture of composition R, another A-phase of mixture with concentration
𝑃 is obtained at the point PR.
[EXAMPLE 22.1-1]
An aqueous solution of acetic acid contains 40 kg of acetic acid in 1000 kg of water. The initialsolution is to be extracted by 800 kg of isopropyl ether. The batch extraction is to be performed at 20℃,
where the equilibrium relationship between the acetic acid concentration in the extract and raffinate may
be expressed in a low concentration range as
𝑌 = 0.225 𝑋 + 0.95 𝑋(22.1-E1)
If an equilibrium condition is attained between the two phases in a single batch extraction process,how many kg of acetic acid can be recovered by the isopropyl ether?
SolutionFirstly we use Eq.(21.1-1),
𝑋 𝐻 = 𝑋 𝐻 + 𝑌𝑆
0.04 × 1000 = 1000 𝑋 + 800 𝑌
Substituting the equilibrium relationship 𝑌 = 0.225 𝑋 + 0.95 𝑋 ,the following equation is obtained:
𝑋 + 1.5525 𝑋 − 0.05263 = 0
From the equation, we can obtain the following solution:
𝑋 = 0.0332 and 𝑌 = 0.00851
𝑌 𝑆 = (0.00851)(800) = 6.81 kg.

(22.1-1)(22.1-E2)(22.1-E3)

The result implies that 6.81 kg of acetic acid is removed from the initial aqueous solution by a single(one-stage) extraction operation.
The recovery factor for one stage is given by𝑟 = 𝑌 𝑆 ⁄𝑋 𝐻 = (0.00851)(800)⁄(0.04)(1000) = 0.17
(22.1-E4)
It should be kept in mind that this equilibrium composition can be obtained in acid concentration
between the extract and raffinate if the two liquid layers formed by a batch extraction are settled down
into two phases of extract and raffinate after a sufficiently long time. It is possible to combine a seriesof these stages so that the recovery factor for the extracting component can be improved.
One example of countercurrent cascade systems of mixer and settler combinations is shown in
Fig.22.1-3.

36

Liquid Extraction

Fig.22.1-3 A mixer-settler extractor

In extraction processes, it is essentially important to promote mass transfer between two liquids.
Extractors can be made compact if one liquid is dispersed and suspended in the second as small droplets.
The advantage lies in the establishment of a very large contacting-surface between the two liquid phases.
While mixing the solution, the interphase mass transfer of solute takes place between the original
solution and the solvent.
Continuous flow extractors comprise a train of the mixer and settler. Fig.22.1-3 shows an example
of a three-stage continuous flow extractor called “Mixer-settler.”
The mixer is usually a small agitated tank provided with supply lines and a draw-off line. The settler
is a simple continuous decanter.
Consider the mass transfer of component B from droplets of the original solution solvent (diluent) A
to the continuous phase of solvent S. For a baffled vessel extractor, the shape of droplets often becomes
non-spherical and the droplet size varies locally.
The average droplet size can be expressed as𝑑 ,∑(22.1-5)= ∑For simplicity, let us consider a single spherical droplet suspended in the stream of continuous-phaseextracting solvent. The relative velocity of the droplet is given as the slip velocity 𝑣 .
If the droplet size 𝑑 is of the order 1 mm or less, the slip velocity in the gravitational field becomes
Stokes’ terminal velocity, as in solid particles:
𝑣 =

(

)

(22.1-6)

The droplet is so small that it does not break up into two or more smaller droplets.
The mass-transfer rate is increased with the slip velocity and further enhanced with the intra-mixing
within the droplet.
Therefore the so-called two-film theory should be applied. We should consider the mass transfer inthe liquid film of continuous-phase as well as inside the droplet
Assume the clean condition without surface-active agents and impurities on the interface of droplet
surface,
Except for the time of droplet formation, unless the droplet size is relatively larger than a few mm,
the mass transfer correlation will become similar to that for solid spheres, e.g., Ranz-Marshall equation.

Extraction Engineering37

22.2 Extraction Technology
Liquid-liquid extraction takes place with two liquids immiscible in each other. One liquid is of organic
nature (oil) while the other is aqueous (water).
The liquid extraction is often used when distillation and rectification are difficult or ineffective in
relation to the energy saving. Regarding a simple batch operation, a quantity of feed mixture containing
solute B in diluent A may be mixed with a quantity of the extracting solvent S in an agitated vessel, after
which the two layers are settled and separated into the extract (B in S) and raffinate (B in A).
For the case of extracting solute B in a diluent A into organic phase S, the distribution coefficient isdefined as⁄()(22.2-1)
𝐾=⁄()
The distribution coefficient can be obtained from the slope by plotting 𝑦 against 𝑥 . The large
values are desirable since less solvent is required for a given degree of extraction.
If several contacts are necessary, continuous flow operation becomes economical. A mixer-settler
extractor (see Fig.22.1-3) is adequate in simple extractions.
The batch extraction is usually operated in a stirred tank.
22.2-1 Liquid-liquid equilibria of ternary system(a) Triangular coordinates
When the mutual solubility of the original solvent A and the extracting solvent S cannot be neglected,
the solubility and equilibrium are expressed on triangular coordinates shown in Fig.22.2-1.

Fig.22.2-1 Triangular diagram for a ternary system

For a ternary mixture, a solute B is placed at the apex. Usually mass fractions or per cent may be usedas the concentration units for extraction.
According to the geometric principle, Fig.22.2-1 indicates that the sum of the perpendicular distancesfrom any point to the three sides of an equilateral triangle equals the altitude of the triangle.
𝑀𝑁1 + 𝑀𝑁2 + 𝑀𝑁3 = 1(22.2-2)

38

Extraction Technology

For an example, the point M indicates a mixture of 𝑋 = 0.2, 𝑋 = 0.34, and 𝑋 = 0.46 in massfraction.
The sum of those concentrations must be unity,
𝑋 + 𝑋 + 𝑋 =1
(22.2-3)
Concentrations given by triangular diagrams are based on the entire mixture. For example,
the point P of Fig.22.2-2 indicates a mixture of 𝑋 =0.41, 𝑋 = 0.03 and 𝑋 =0.56 in mass fraction.

Fig.22.2-2 Triangular phase diagram for a ternary system of ethanol-cyclohexene-water in mass %Fig.22.2-2 is a triangular diagram for the ternary system of ethanol B, cyclohexene A, and water S in
the isothermal condition at 25˚C. The water S is the solvent for extracting the solute ethanol B from thediluent cyclohexene A. The solubility curve can be obtained by the following procedure.
In this ternary system, the points E and F existing almost at the bottom ends of the triangle imply theimmiscibility between A and S.
Firstly if ethanol is added to a two-layer mixture of water and cyclohexene, the ethanol distributes
between the two layers, and the composition of the layers give two solubility curves, one for the
cyclohexene layer (raffinate phase) and the other for the water layer (extract phase). The curve EGTPHF
is called “solubility curve.” The line E→G→T→P gives the compositions of the saturated water phase,and line F→H→P compositions of the saturated cyclohexene phase. The point P where the two phases
become identical is called a plait point.
Straight lines connecting points on the two solubility curves is called a tie line, the ends of whichindicate the points for two phases in equilibrium. The tie lines shown in the figure are given by someexperimental data. As can be seen from the figure, the line length decreases as the plait point isapproached. Each point within the space below the equilibrium curve EGTPHF called a dome lies on atie line, and therefore an infinite number of such tie lines exist.
If the ethanol concentration in one phase (water) 𝑦 is plotted against the ethanol concentration in
the other phase (cyclohexene) 𝑥 , the distribution curve with slope m can be obtained as shown below
in the figure.

Extraction Engineering

39

Fig.22.2-3 Distribution curve for a ternary system of water-ethanol-benzene(Solute: ethanol, Diluent: water, Solvent: benzene)

Local slope of the curve indicates the distribution coefficient: 𝑚 = 𝑦 ⁄𝑥 .
This equilibrium relationship expressed in mole fraction can be understood as follows.
Since the fugacity of solute B, 𝑓 in the A-phase is equal to 𝑓 in the S-phase in an equilibrium
condition,
𝑓 = 𝑓(22.2-4)
If the pressure is low enough, the fugacity becomes equal to the pressure of an ideal gas.
The ratio of mole fraction gives the reciprocal of activity coefficient ratio, i.e. the reciprocal of
distribution coefficient:== 1⁄𝑚(22.2-5)(b) Rectangular coordinates
A disadvantage of the triangular diagrams is that their scales can be neither changed nor enlarged
when it is necessary to follow changes in a narrow concentration range. Therefore the rectangulardiagram shown in Fig.22.2-4 is very convenient and recommendable. The composition of point P isgiven by solute = 𝑥 , solvent= 𝑥 , and diluent 𝑥 = 1 − 𝑥 − 𝑥 .

Fig.22.2-4 Rectangular coordinate diagram for a ternary system

40

Extraction Technology

Center-of- Gravity Principle (Principle of Leverage)
If 𝑊 kg of liquid mixture (𝑥 , 𝑥 , 𝑥 ) is mixed with 𝑊 kg of liquid mixture(𝑥 , 𝑥 , 𝑥 ), the obtained mixture 𝑊 = 𝑊 + 𝑊 kg should have the following material balance:
𝑊 = 𝑊 + 𝑊
(22.2-6)
𝑊 𝑥 = 𝑊𝑥 + 𝑊𝑥
(22.2-7)
𝑊 𝑥 = 𝑊𝑥 + 𝑊𝑥
(22.2-8)
From these equations,==(22.2-9)
As can e seen from Fig.22.2-4, the above equation indicates that the position of the obtained mixture
is located at point M on the line PQ where the line length ratio 𝑀𝑄 ⁄𝑀𝑃 is given by the mass ratio
𝑊 ⁄𝑊 . This rule is called “Center-of-Gravity Principle.”
The composition of the obtained mixture is given by
𝑥

=

𝑥

=

⁄
⁄
⁄
⁄

(22.2-10)(22.2-11)[Example 22.2-1]
A batch agitated tank extractor is used to separate ethanol from its mixture withcyclohexene by using clean water. The liquid-liquid equilibrium diagram for the ternary
system at 25℃ is shown below.
The feed of cyclohexene mixture 𝑊 = 50 kg contains 30 wt% of ethanol only. The 𝑊 kg
feed mixture and the 𝑊 kg of the solvent (water) are put into the agitated tank extractor
and then after the sufficient agitation, the upper cyclohexene layer settled down is takenout as the raffinate product.
Its composition is required to have 99 wt% cyclohexene.
How much water 𝑊 is necessary for one batch operation?

Fig.22.2-E1 Rectangular coordinates of liquid-liquid equilibrium diagram for a ternary system of
ethanol, cyclohexene, and water.

Extraction Engineering

41

Solution:On the rectangular diagram, the feed is located at Point F and the solvent (cleanwater).is located at the right corner of the triangle S. The composition of the raffinate
product should be 𝑥 = 0.01 , 𝑥 = 0.99, and 𝑥 = 0. Therefore the point R is located at
𝑥 = 0.01 just on the vertical axis.

Fig.22.2-E2 Calculation of solvent quantity required using experimental tie lines on a rectangulardiagramFig.22.2-E2 is the equilibrium diagram region below 40 wt% ethanol enlarged. The tie line RE drawn
along the experimental tie line from the point R can get the point E on the equilibrium curve, whichindicates 𝑥 = 0.397.
If we assume that the solvent (water) is immiscible with the diluent (cyclohexene), all the cyclohexeneof the feed is left into the raffinate phase.
𝑊 𝑥 = 𝑅𝑥
That is,
𝑅= 𝑊 𝑥 + 𝑅𝑥
(22.2-E1)
Therefore( )( . )= ( . ) = 35.4 kg
𝑅= ()
The following two equations are obtained by the principle of leverage:
𝑊 (𝑥 − 𝑥 ) = 𝑊 (𝑥 − 𝑥 )(22.2-E2)
𝑅 (𝑥 − 𝑥 ) = 𝐸 (𝑥 − 𝑥 )(22.2-E3)
𝑅+𝐸 = 𝑊 + 𝑊(22.2-E4)
Since
𝑥 = 0, 𝑥 = 1.0, 𝑥 = 0, 𝑥 = 1 − 𝑥 = 0.60, 𝑊 = 50, 𝑅 = 35.4
From these equations
𝑊 = 21.1 kg, 𝐸 = 35.7 kg, and 𝑥 = 0.297.
The 35.4 kg of raffinate contains 99 wt% of cyclohexene and 1 wt% of ethanol is obtained by using 21.1
kg of clean water.
22.2-2 Unsteady mass transfer problems in liquid-liquid extraction
Actually speaking, as distinct from the above example, there is a possibility of collision and breakingup of droplets depending upon the Reynolds number.

42

Extraction Technology

In particular, mass transfer is greatly strengthened owing to unsteady strong shearing stresses in thevicinity of the agitator blade tips or at the instant when droplets are formed. The mechanism for the masstransfer in this unsteady condition is very difficult to investigate but very attractive to utilize fordesigning practical commercial-scale extractors.
Mass transfer in the period of droplet formation due to shear stresses
The radical variation in the fluid velocity and droplet surface makes the analysis and modeling difficult.
The following mass transfer coefficients can be assumed according to penetration theory.
𝑘 = 𝐶 𝐷 ⁄𝜃(continuous phase) and𝑘 = 𝐶 𝐷 ⁄𝜃 (dispersed phase)where 𝐶 and 𝐶 are empirical constants to be determined by experiment.
Here 𝜃 and 𝜃 are time-constants characteristic for periodical surface renewal.
This suggests that when these time constants become small, the mass transfer is enhanced.
You can call to mind that the mass transfer coefficient is proportional to the square root of diffusivity
divided by contact time. (see Eqs.(6.7-14)and (16.1-13))
22.2-3 Continuous extractors
Various extraction columns similar to distillation columns are available depending on therequirements of individual extraction processes.
The tie lines for design computations of extractors are obtained by interpolation with the aid of the
distribution curve from experimentally observed tie lines
Countercurrent multistage liquid-liquid extraction systems
Let us consider the design method for a stripping packed column shown in Fig.22.2-5. In a manner
similar to distillation column design (PART II, p.128~133), firstly the number of theoretical stages
required for the specification of a desired composition change is stage-by-stage calculated. Secondly theactual height of the packed column is determined by use of the height equivalent to a theoretical plate,i.e. HETP (see PART II, p.147~148).
However a difficult problem may arise that the assumption of constant molar flow rates of the streams
is often impermissible in extraction calculations with mass fraction units. In addition, since the diluent
and solvent usually have small mutual miscibility, the extract phase contains solute B in solvent S withconcentrations 𝑦 , 𝑦 , and small 𝑦 , while the raffinate phase contains solute B in diluent A withconcentrations 𝑥 , 𝑥 , and small 𝑥 . Consider the case when the raffinate is heavier in fluid density
than the extract. The original mixture (feed rate 𝑅 = 𝐹) of solute B dissolved in diluent A is fed fromthe top of the column. A pure solvent S (feed rate 𝑄) is supplied from the bottom of the column.

Fig.22.2-5 Countercurrent multi-staged extractor with no reflux

Extraction Engineering

43

22.2-4 Stage-by-stage design calculation of a continuous extractor
The overall mass balance gives 𝐹 + 𝑆 = 𝐸 + 𝑅
This equation indicates that the straight line 𝐹𝑆 intersects the straight line 𝐸 𝑅
triangular diagram.
The mass balance set up over each shell from the top is given by
1st stage (shell 1): 𝐹 + 𝐸 = 𝑅 + 𝐸
1st ~2nd stage (shell 2): 𝐹 + 𝐸 = 𝑅 + 𝐸------------------------------------------------1st ~Mth stage (shell M), 𝐹 + 𝐸= 𝑅 + 𝐸 , and 𝐸=𝑆
Therefore
𝐹− 𝐸 = 𝑅 − 𝐸 = 𝑅 − 𝐸 = ⋯= 𝑅 −𝐸
Similarly for the component B
𝐹𝑥 −𝐸 𝑦 = 𝑅 𝑥 − 𝐸 𝑦 = 𝑅 𝑥 − 𝐸 𝑦at the point M on the≡ 𝐿 (constant)(22.2-12)(22.2-13)(22.2-14)(22.2-15)= ⋯= 𝑅 𝑥 −𝐸𝑦
≡ 𝐿 𝑥 (constant)(22.2-16)
These two equations imply that as shown in Fig.22.2-6 , the extensions of operating lines 𝐹𝐸 , 𝑅 𝐸 ,𝑅 𝐸 ,--- 𝑅 𝐸
cross at a common point O.
The mass balance at steady state is set up over the shell (enclosed by the broken line 2) between thebottom and stage m as 𝑅 + 𝐸 = 𝑅+𝑆𝑅−𝐸 = 𝑅 −𝑆 =𝐿(22.2-17)
Here 𝐸= 𝑆 is the solvent being supplied from the bottom, and 𝐿 implies the net downward flow.
The mass balance for the solute B: 𝑅 𝑥 + 𝐸 𝑦 = 𝑅
𝑥
or
⁄𝐸 )𝑥
− (𝑅 ⁄𝐸 )𝑥(22.2-18)
𝑅𝑥− 𝐸 𝑦 = 𝑅 𝑥or 𝑦 = (𝑅⁄𝐸 .
This is an equation of operating line with slope 𝑅
In the case when the raffinate does not contain any solvent S, i.e. 𝑥 = 𝑥 = 0 and 𝑦= 1, themass balance at each stage 𝑚 for the extracting solvent S reduces to:𝐸 𝑦 =𝑆(22.2-19)

Fig. 22.2-6 Stage-by-stage calculation of a countercurrent multi-staged extractor

44

Extraction Technology

If we assume an equilibrium curve shown in Fig.22.2-6, we can calculate stage-by-stage the numberof equilibrium stages required for attainment of the raffinate concentration 𝑥 as the bottom product.
Eqs.(22.2-15) and (22.2-16) give the stage-by-stage changes in composition of the countercurrent
streams.
The following operating line equation can also be obtained:,⁄𝐸 )𝑥𝑦 = (𝑅− (𝑅 ⁄𝐸 )𝑥(22.2-20)[EXAMPLE 22.2-2]
We need to extract propanoic acid from its mixture with heptane by means of clean water as theextracting solvent. The extraction method is adopted for energy saving instead of distillation. The
heptane solution as the feed includes 30 wt% of propanoic acid but does not contain any water.
A continuous countercurrent column shown in Fig.22.2-E3 is used. The propanoic acid concentration of
the final effluent raffinate is required to be 3 wt% or less. Here we assume the distribution of a solute C
between two immiscible liquid B (water) and A (heptane).
As distinct from the last example, the raffinate phase is lighter in density than the extract phase.
Therefore the heptane mixture to be extracted is supplied from the bottom at a feed rate of 100 kg/h.
Water as the extracting solvent is fed from the top at a rate of 30 kg/h. The entire system can be assumed
to be kept isothermal at 50˚C.

Fig. 22.2-E3 Countercurrent extractor

The data of equilibrium curve and tie line are listed in Table 22.2-E1.
Table 22.2-E1 Liquid-liquid equilibrium data for ternary systemof propanoic acid, heptane, and water (50℃)(Tie line data)
Composition in mass fraction
propanoic acid
0.06018
0.119200.206590.278290.35494
0.41980LEFT PHASE
Water
0.00174
0.002290.005430.00874
0.01432
0.01887

heptane0.93808
0.878510.78799
0.71297
0.63074
0.56133

propanoic acid0.26968
0.41475
0.565830.649510.70801
0.73535

RIGHT PHASE
Water
0.712080.55794
0.40638
0.320880.24795
0.20205

Heptane0.018240.027310.027780.029620.04404
0.06259Extraction Engineering(Equilibrium curve data)
propanoic acid
0.037764
0.076646
0.116696
0.160141
0.203963
0.250630
0.298177
0.349324
0.403104
0.471757
0.554265
0.693489

LEFT PHASE
Water
0.001837
0.002144
0.002460
0.005843
0.006946
0.009145
0.010362
0.012746
0.015253
0.0241000.036771
0.073963

45

Composition in mass fraction
heptane0.960400.921210.880840.834020.789090.740220.69146
0.637930.581640.504140.408960.23255

propanoic acid
0.69349
0.70746
0.726800.735150.751600.75075
0.763500.74260
0.718070.68882
0.63792
0.50684
0.31355

RIGHT PHASE
Water
0.073960.081670.089900.111170.13543
0.16801
0.19346
0.21275
0.257640.31118
0.36208
0.493160.68645

Heptane0.232550.210870.18330
0.15367
0.112970.081240.04303
0.044650.024280000The liquid-liquid equilibrium data for the ternary system is given in the triangular coordinate diagram
shown in Fig.22.2-E4.

Fig.22.2-E4 Triangular diagram of a ternary system of propanoic acid, heptane, and water at 50˚C

Calculate the number of ideal stages with no reflux required for this specification of extraction byusing the ideal stage concept.
Solution:
Let us designate propanoic acid, water, and heptane to be component C, component B, and component
A, respectively.
The compositions of the extract phase and the raffinate phase are expressed by x and y in mass fraction,
respectively.

46

Extraction Technology

According to the condition of specification,
Heptane mixture (feed): 𝐹 = 𝑅= 100 kg/h, (solute) 𝑦 = 𝑦= 0.30
Final effluent raffinate: 𝑅 and (solute) 𝑦 = 0.03
Extracting solvent (water): 𝑆 = 𝐸 = 30 kg/h, (solute) 𝑥 = 𝑥 = 0, (solvent) 𝑥 = 1.0,
Effluent extract: 𝐸 , 𝑥
First of all, let us draw the line FS from point F (𝑦= 0.3, 𝑦= 0.7) to point S (𝑥 = 𝑥1.0) on the triangular or rectangular coordinates.

=

The feed is located at Point F (𝑦 = 𝑦= 0.70, 𝑦 = 𝑦= 0.30, 𝑦 = 𝑦= 0) . The
clean water as the solvent is located at Point S (𝑥 = 0, 𝑥 = 0, 𝑥 = 1. ) .
From the material balance of solute between the top and bottom of the entire system
𝐹+𝑆 = 𝐸 + 𝑅 =𝑀
(22.2-E5)
𝐹𝑦 +𝑆𝑥 = 𝐸 𝑥 + 𝑅 𝑦 =𝑀𝑥(22.2-E6)
From the first equation, 𝐹 + 𝑆 = 𝑀 = 100 + 30 = 130 kg⁄h
The solute concentration of mixture M is given by Point M:()( . ) ( )( )== 0.231(22.2-E7)
𝑥 =The final raffinate condition is specified at 𝑅 as
𝑦 = 0.03
Since the point 𝑅 indicates 𝑦 = 0.03 and 𝑦 = 0.97, the mass balance of diluent A gives the
final raffinate flow rate 𝑅 :𝐹𝑦 = 𝑅 𝑦or 𝑅 = 𝐹 𝑦 ⁄𝑦 = (100)(0.70)⁄0.97 = 72.2 kg⁄h(22.2-E8)
From Eq.(22.2-E5) and Eq.(22.2-E6),
𝑥 =(22.2-E9)𝑥(22.2-E10)=

Since 𝑀 = 𝐹 + 𝑆 = 100 + 30 = 130 kg/h,
𝐸 = 𝑀 − 𝑅 = 130 − 72.2 = 57.8 kg/h
Therefore, as shown in Fig.21.6-E6, the solute concentration of the extract is given by the Point Em
𝑥

=

=

=

(

)( .

) (

)( ) (.

. )( .

)= 0.482

The mass balance set up over each shell from the top is given by
1st stage (shell 1): 𝑆 + 𝑅 = 𝑅 + 𝐸
1st ~2nd stage (shell 2): 𝑆 + 𝑅 = 𝑅 + 𝐸------------------------------------------------1st ~mth stage (shell m), 𝑆 + 𝐹 = 𝑅 + 𝐸 , and 𝑅=𝐹
Therefore
𝑆− 𝑅 = 𝐸 − 𝑅 = 𝐸 − 𝑅 = ⋯= 𝐸 −𝑅
≡ 𝐿 (constant)
Similarly for the component C
𝑆𝑥 −𝑅 𝑦 = 𝐸 𝑥 − 𝑅 𝑦 = 𝐸 𝑥 − 𝑅 𝑦 = ⋯= 𝐸 𝑥 −𝑅
≡ 𝐿 𝑥 (constant)(22.2-E11)(22.2-E12)(22.2-E13)(22.2-E14)(22.2-E15)

𝑦≡(22.2-E16)

The straight line given by the final extract and raffinate 𝑅 𝐸 must pass through point M. The
extension of the line passing through the points 𝑅 and M crosses the equilibrium curve at point 𝐸 .
This point M indicates the solute concentration:
𝑥

=

=

(

. )( .

) (. )( .

)= 0.231(22.2-E17)

Extraction Engineering

47

Fig.22.2-E5 Countercurrent multi-stage extractor for stage-by-state calculation

The material balance set over each stage of the countercurrent multi-stage extractor is calculated stageby-stage as follows. These results are also shown in Fig.22.2-E6.
he final extract flow solute concentration𝑥 = (100 × 0.30 − 72.2 × 0.03)⁄57.8 = 0.482(22.2-E18)
The necessary number of stages m will be determined by the following step-by-step calculation.

Fig.22.2-E6 Step-by-step method for calculating the number of stages

In the equilibrium stage model, the extract stream 𝐸
stage are in equilibrium.

and the raffinate stream 𝑅

leaving the mth

48

Extraction Technology

Since 𝐸 = 57.8 kg/h and 𝑅 = 72.2 kg/h, the concentration of C (propanoic acid) in the final extract
Em is calculated as
𝑥 = (100 × 0.30 − 72.2 × 0.03)⁄57.8 = 0.482(22.2-E19)
Therefore by reference to the experimental tie lines, as shown in Fig.21.6-E6, the tie line expressed by
the broken line 𝐸 𝑅 gives 𝐸 : 𝑥 = 0.25 and 𝑅 : 𝑦 = 0.03. The next operating line obtained from
𝐸 𝑅 using the operating point O indicates 𝑦 = 0.14 at point 𝑅 .
According to the assumption of immiscible liquids A and S, the mass balance equations regarding thediluent A (heptane) give the following quantities:
𝑅 𝑦 = 𝑅 𝑦or 𝑅 = 𝑅 𝑦 ⁄𝑦 = (72.2)(1 − 0.03)⁄(1 − 0.14) = 81.4 kg⁄h
𝐸 =𝑆+𝑅 −𝑅
or 𝐸 = 30 + 81.4 − 72.2 = 39.2 kg/h
(22.2-E20)
The tie line 𝑅 𝐸 has 𝐸 : 𝑥 = 0.47 at 𝐸 . The operating line 𝐸 𝑅 gives 𝑦 = 0.29 at point 𝑅 .
That is, 𝑦 = 0.71.
𝑅 = 𝑅 𝑦 ⁄𝑦 = (72.2)(1 − 0.03)⁄0.71 = 98.6 kg⁄h(22.2-E21)
𝐸 = 𝑆 + 𝑅 − 𝑅 = 30 + 98.6 − 72.2 = 56.4 kg/h
(22.2-E22)
As can be seen from Fig.22.2-E6, the next tie line 𝑅 𝐸 gives 𝐸 : 𝑥 = 0.68 but this value has gonevery much beyond the point 𝐸 (𝑥 = 0.482).
It can be considered from the result that the number of ideal stages should be more than 2 but lie near 2.
[PROBLEM 22.2-1]
Let us reconsider the same problem as EXAMPLE 22.2-2. An organic solution 100 kg/h of heptane
containing propanoic acid is fed from bottom into the counter-current extractor shown in Fig.22.2-P1.
The solute propanoic acid needs to be recovered by clean water which is supplied at a rate of 50 kg/h
into the top of the counter-current extractor. The heptane solution as the feed includes 40 wt% of
propanoic acid but does not contain any water.

Fig.22.2-P1 Continuous counter-current extractor

The propanoic acid concentration of the final effluent raffinate is required to be 1 wt% or less. Here weassume the distribution of a solute C between two immiscible liquid B (water) and A (heptane).

Extraction Engineering

49

Fig.22.2-P2 Rectangular diagram of a ternary system of propanoic acid, heptane, and water at 50˚C

As shown in Fig.22.2-P2, the rectangular diagram of equilibrium and some tie lines arealso obtained from Table 22.2-E1 of EXAMPLE 22.2-2..
The entire system is also kept isothermal at 50˚C.
Calculate the number of ideal stages with no reflux required for this specification of extraction by using
the ideal stage concept.
As in the above example and problem, we can assume that the raffinate stream leaving each stage
does not include any solvent owing to the immiscible liquid pair of A and S. Since the solvent rate 30
kg/h or 50 kg/h may be considerably small as compared to the feed rate 100 kg/h, we should be carefulof preventing the solvent stream from flooding in the practical operation.
As distinct from distillation, the difference in density between the extract liquid and raffinate liquid is
not so large. It should be kept in mind as a practical problem that it is also considerably difficult toseparate the extracting mixture into the extract and raffinate in each stage.

22.3 Solid Extraction – Leaching
Leaching or solid extraction is a unit operation obtaining valuable soluble matters from the inside of
an insoluble solid body. The properties of the solids may change considerably in the leaching operation.
Granular or porous-matrix solids may be effective when soluble material is removed.
As an example of leaching processes, we can give a tea bag containing dried tea leaves to be soaked
in hot water. The tea flavor substance is extracted from the tea leaf solid by the hot water used as thesolvent. Leaching process is controlled by the following three subprocesses: (1) dissolution of solutefrom solid in solvent, (2) diffusion of solute through pores of the solid particles and (3) transfer of soluteinto bulk solvent.
22.3-1 Equilibrium stage
In leaching, if sufficient solvent is present to dissolve all the solute in the entering solid and there is

50

Solid Extraction - Leaching

no adsorption of solute by the solid, equilibrium is attained when the solute completely dissolves andthe concentration of the solution becomes uniform. Ideal single stage can also be expressed as in
Fig.22.3-1.
If the concentration of the liquid retained by the solid (underflow) leaving the stage of interest is equal
to that of the concentrated liquid (overflow) leaving the same stage, that is, 𝑥 = 𝑦 , this stage iscalled “equilibrium (ideal) stage.”

Fig.22.3-1 Definition of equilibrium stage in a continuous leaching system

This definition is similar to the definition of ideal stage in multi-staged distillation column. (seeFig.11.2-10 in Chapter 11)
Fig.22.3-2 shows a continuous countercurrent cascade. The V phase is the liquid solution (solute +solvent) overflowing from stage to stage in the right direction (from stage N toward stage 1) counter tothat of the underflowing slurry (solute + solvent + solid) of the solute-retained solid. The L phase is
the liquid solution included in the solid flowing from stage 1 toward N. Exhausted solids
are discharged from stage N while concentrated solution issues from stage 1 as theoverflow product. The solute-free solid is insoluble and inert in the solvent. Therefore theflow rate of the solid is constant throughout the cascade on the dry solute-free basis.
Two definitions of solute concentration in the overflow and underflow, respectively.
𝑥: solute concentration in the underflowing solution
𝑦: solute concentration in the solution included in the underflowing slurry
𝑉: flow rate of overflow solution
𝐿: flow rate of underflow of the liquid retained by the solid based on a definite flow of dry
solute-free solid
𝑥 : solute concentration of the solution included in the entering solid slurry𝑥 : solute concentration of the solution included in the leaving solid slurry𝑦 : solute concentration in the concentrated solution leaving stage 1
𝑦 : solute concentration in the fresh solvent entering the stage N

Extraction Engineering

51

Fig.22.3-2 Continuous countercurrent leaching cascade

22.3-2 Design calculation of a counter-current leaching system
In a similar manner to the step-by-step calculation for trayed distillation column, a design calculation
method can be derived for staged counter-current leaching systems.
Operating Line
Mass balance set up over the system boundary indicated by the dotted line in Fig.22.3-2 gives𝑉+ 𝐿 = 𝑉 + 𝐿(22.3-1)
𝑉 𝑦+ 𝐿 𝑥 = 𝑉𝑦 + 𝐿 𝑥(22.3-2)
From these two equations, we obtain the following operating line equation:𝑥 +(22.3-3)
𝑦
=
(
)
(
)
The line of this equation passes through points (𝑥 , 𝑦 ) and (𝑥 , 𝑦 ).
Usually the density and viscosity of the solution vary remarkably with soluteconcentration. The liquid retained by the solid from the lower-numbered stages is morethan that from the higher-numbered stages: 𝐿 > 𝐿. Therefore the slope of theoperating line may vary from unit to unit. However if we can assume that the mass of theretained solution is independent of solute concentration, 𝐿 = constant, the operating line
is straight.
This is called “constant underflow condition.”𝐿= 𝐿 = −−−= 𝐿 =𝐿
(22.3-4)
𝑉= 𝑉 = −−−= 𝑉 =𝑉
(22.3-5)
We will consider the constant underflow case only for understanding the fundamentals.
Firstly the operating line equation is simplified as𝑦= 𝑥 + 𝑦 − 𝑥(22.3-6)
The following material balance is given for the first stage:
𝐿 + 𝑉 = 𝐿 + 𝑉 =𝑀
(22.3-7)
𝐿 𝑥 + 𝑉 𝑦 = 𝐿 𝑥 + 𝑉𝑦 =𝑀𝑥(22.3-8)
For the dry inert solid movement,
𝐵 = 𝑁 𝐿 +0= 𝑁 𝐿 +0= 𝑁 𝑀(22.3-9)
Similarly the material balance equation for an arbitrarily chosen stage n becomes𝐿+ 𝑉= 𝐿 + 𝑉 =𝑀
(22.3-10)
𝐿 𝑥+ 𝑉 𝑦= 𝐿 𝑥 + 𝑉𝑦 =𝑀𝑥(22.3-11)
The dry inert solid movement is given by
𝐵= 𝑁 𝐿= 𝑁 𝐿 = 𝑁 𝑀(22.3-12)

52

Solid Extraction - Leaching

A very important parameter is the underflow function 𝑈(𝑥) kg solution/kg solid.
The definition means how much the solution is carried by the solid in the underflow.
The function depending on the properties of the solid, of the solution, and of the equipment used toseparate the overflow (filtration, decantation, thickening, etc.) should be experimentally determined.
[EXAMPLE 22.3-1]
A soybean crude oil is produced by the leaching process of a solid feed 𝐹 ＝1,000 kg/h (e.g. flaked
soybeans) containing 75wt% insoluble solid (including proteins) and approximately 25wt% oil to be
extracted. A continuous countercurrent leaching battery is used. Pure solvent (e.g. n-hexane) 900 kg/h is
supplied to extract 95% of the solute (e.g., soybean oil). The solid discharging as the slurry from thefinal stage N entrains a limited rate of liquid solution. By a preliminary test, we get such an empirical
underflow function that each kg of solid carries (0.9 + 0.3 x) kg of the solution of solute mass fraction x.
We assume that the leaching process can be treated as an ideal stage-to-stage cascade neglecting theeffect of the substances other than the oil contained in the soybeans.
Calculate how many ideal stages are necessary for the desired extraction.
Solution:Fig.22.3-E1 indicates the entire cascade system with the operating condition.

Fig.22.3-E1 Entire system of countercurrent leaching cascade with operating conditions(Subscript a: solid feeding entrance and exit of overflow solution, subscript b: solvent entrance andexit of underflow slurry)

The countercurrent cascade consists of a series of (thickener type) liquid-solid separator units, in eachof which the solid from the previous unit is mixed with the liquid from the succeeding unit and the
mixture is allowed to settle. The solid is then transferred with the entrained solution in the form of slurry
to the next succeeding unit by slurry pumps. The separated liquid is transferred in the form of overflow
to the previous unit. An appropriate mixer and settler should be chosen for the individual units.
Usually the cascade system has a decanter to separate the entrained solution from the ex-service solid
after stage N.
Let us begin by material balance calculation:
The feed F= 1,000 kg/h for the underflow contains 250 kg/h of solute in solid with no solvent.
Solid: 750 kg/h, Solute: 250 kg/h, Solvent: 0 kg/h, therefore 𝐿 = 250 kg/h and 𝑥 = 1.0
That is, the composition of the solution in the feed is 𝑥 = 250⁄250 = 1.0 on the solid-free basis.
Pure solvent to be supplied has the following quantities:
Solid: 0 kg/h, Solute: 0 kg/h, Solvent: 900 kg/h, therefore 𝑉 = 900 kg/h and 𝑦 = 0

Extraction Engineering

53

The underflow function given by the problem can be applied as the underflow solution leaving eachstage:
𝐿 = (𝐹 − 𝐿 ) 𝑈(𝑥) = (750)(0.9 + 0.3 𝑥 ) kg/h
(22.3-E1)
The recovery condition indicates the flow of unrecovered solute leaving stage N with the solid as
𝐿 𝑥 = (1 − 0.95)𝐿 𝑥 = (0.05)(250)(1.0) = 12.5 kg⁄h(22.3-E2)
Because the underflow leaving Stage N should also follow the underflow function,
𝐿 𝑥 = (750)(0.9 + 0.3 𝑥 ) 𝑥 = 12.5 kg⁄h
From this equation with respect to 𝑥 , we obtain𝑥 = 0.0184(22.3-E3)
Then 𝐿 = (750)(0.9 + 0.3 × 0.0184) = 679 kg/h
(22.3-E4)
This rate of the solution is discharged with the underflow from stage N.
The rate of the overflow issuing from stage 1 is given by the total mass balance:
𝑉 = 𝐿 + 𝑉 − 𝐿 = 250 + 900 − 679 = 471 kg⁄h(22.3-E5)
Therefore the overflow concentration is calculated from the solute balance()( . ) ()( ) ()( .
)== 0.5043
𝑦 =(22.3-E6)[First stage]
Since the feed to the battery (at Stage 1) does not obey the underflow relationship, the feed point F is
not on the operating line. Therefore the mass balance around the first stage should be consideredseparately from the other stages (2 to N):

Fig.22.3-E2 Material balances around stage 1

The first stage is also an equilibrium stage expressed by
𝑥 = 𝑦(22.3-E7)
Therefore 𝑥 = 0.5043
𝐿 = (750)(0.9 + 0.3 × 0.5043) = 788.5 kg/h
(22.3-E8)
The total balance around Stage 1 gives
𝑉 = 𝑉 + 𝐿 − 𝐿 = 471 + 788.5 − 250 = 1,010 kg/h
(22.3-E9)
The solute balance around Stage 1 gives the solute concentration of the overflow entering Stage 1 as
(
)( .
) (
) (
. )( .
)( . )== 0.3814(22.3-E10)𝑦 =,[Stages 2 to N]
From here, we can begin the calculation for the stages 2 to N after checking if the operating line canbe assumed to be straight.
Let us consider the control volume indicated by the dotted line in Fig.22.3-E3. Here the right-side endof the control volume is indicated by A.

54

Solid Extraction - Leaching

At the end A: 𝐿 ⁄𝑉 = 788.5⁄1,010 = 0.781
At the end b: 𝐿 ⁄𝑉 = 679⁄900 = 0.754
Although the slope changes by 3.46% from the ends A to b, we assume that the operating line isapproximately straight.

Fig.22.3-E3 Control volume for calculation of stages 2 to N

In this condition, we can estimate the required number of ideal stages by the Kremser equation1) forstages 2 to N:
∗
∗

𝑁′ =
∗

∗

.

=

=

.
.

.

.

= 7.84(22.3-E11).

Here since we are considering the region of Stages 2 to N,
𝑥 ∗ = 𝑦 = 0.3814
𝑥∗ = 𝑦 = 0
𝑥 = 0.5043
𝑥 = 0.0184(22.3-E12)
The calculated result N = 7.84 means the required number of stages except for stage 1. Therefore weshould add 1 for the plus 1 meaning the first stage.
As a result, the required number of equilibrium stages is given by
𝑁 = 𝑁 + 1 = 7.84 + 1 = 8.84
(22.3-E13)
The required number of equilibrium stages is determined to be approximately 9.
We can also determine the required number of ideal stages by the McCabe-Thiele graphical method2).
You can remember the McCabe-Thiele method as a very convenient design concept of distillation
columns. (see Chapter 11.2-2-4 of PART II)
22.3-3 McCabe-Thiele graphical method for design of a continuous leaching system
We can also gain a better understanding of the above leaching process by the McCabe-Thiele step-bystep calculation procedures shown below. 2)[Stage 1]
The stage is an equilibrium stage with 𝑦 = 𝑥 = 0.5043s(471)(0.5043) + (788.5)(0.5043) − (250)(1.0)
𝑉 𝑦 + 𝐿 𝑥 − 𝐿 𝑥𝑦 === 0.3814
𝑉
1,010
Therefore 𝐿 = 788.5 kg/h and 𝑉 = 𝑉 + 𝐿 − 𝐿 = 471 + 788.5 − 250 = 1,010 kg/hExtraction Engineering

55[Stage 2]
Stage 2 is also an equilibrium stage: 𝑦 = 𝑥 = 0.3814
From the underflow function, we obtain the underflow rate from Stage 2:
𝐿 = (𝐹 − 𝐿 )(0.9 + 0.3 𝑥 ) = (750)(0.9 + 0.3 × 0.3814) = 760.8 kg/h
Then the overflow is 𝑉 = 𝑉 + 𝐿 − 𝐿 = 1,010 + 760.8 − 788.5 = 982.3 kg/h
The operating line gives
(
)( .
) (
)( . ).
𝑥 +=× 0.3814 += 0.2827𝑦 =.
.
[Stage 3]
Stage 3 is also an equilibrium stage: 𝑦 = 𝑥 = 0.2827
From the underflow function, we obtain the underflow rate from Stage 3:
𝐿 = (𝐹 − 𝐿 )(0.9 + 0.3 𝑥 ) = (750)(0.9 + 0.3 × 0.2827) = 738.6 kg/h
Then 𝑉 = 𝑉 + 𝐿 − 𝐿 = 982.3 + 738.6 − 760.8 = 960.1 kg/h
The operating line gives
(
)( .
) (
)( . ).
𝑦 =𝑥 +=× 0.2827 += 0.2045.
.
[Stage 4]
Stage 4 is also an equilibrium stage: 𝑦 = 𝑥 = 0.2045
From the underflow function, we obtain the underflow rate from Stage 4:
𝐿 = (𝐹 − 𝐿 )(0.9 + 0.3 𝑥 ) = (750)(0.9 + 0.3 × 0.2045) = 721.0 kg/h
Then 𝑉 = 𝑉 + 𝐿 − 𝐿 = 960.1 + 721.0 − 738.6 = 942.5 kg/h
The operating line gives(471)(0.5043) − (250)(1.0)
𝑉 𝑦 − 𝐿 𝑥
721.0
𝐿
𝑥 +
=
× 0.2045 += 0.1432
𝑦 =
𝑉
942.5
942.5
𝑉
[Stage 5]
Stage 5 is also an equilibrium stage: 𝑦 = 𝑥 = 0.1432
From the underflow function, we obtain the underflow rate from Stage 5:
𝐿 = (𝐹 − 𝐿 )(0.9 + 0.3 𝑥 ) = (750)(0.9 + 0.3 × 0.1432) = 707.2 kg/h
Then 𝑉 = 𝑉 + 𝐿 − 𝐿 = 942.5 + 707.2 − 721.0 = 928.7 kg/h
The operating line gives(471)(0.5043) − (250)(1.0)
𝐿
𝑉 𝑦 − 𝐿 𝑥
707.2
𝑦 =
𝑥 +
=
× 0.1432 += 0.0956
𝑉
𝑉
928.7
928.7
[Stage 6]
Stage 6 is also an equilibrium stage: 𝑦 = 𝑥 = 0.0956

From the underflow function, we obtain the underflow rate from Stage 6:
𝐿 = (𝐹 − 𝐿 )(0.9 + 0.3 𝑥 ) = (750)(0.9 + 0.3 × 0.0956) = 696.5 kg/h
Then 𝑉 = 𝑉 + 𝐿 − 𝐿 = 928.7 + 696.5 − 707.2 = 918.0 kg/h
The operating line gives(471)(0.5043) − (250)(1.0)
𝑉 𝑦 − 𝐿 𝑥
696.5
𝐿
𝑦 =
𝑥 +
=
× 0.0956 += 0.0589
𝑉
918.0
918.0
𝑉
[Stage 7]
Stage 7 is also an equilibrium stage: 𝑦 = 𝑥 = 0.0589
From the underflow function, we obtain the underflow rate from Stage 7:
𝐿 = (𝐹 − 𝐿 )(0.9 + 0.3 𝑥 ) = (750)(0.9 + 0.3 × 0.0589) = 688.3 kg/h
Then 𝑉 = 𝑉 + 𝐿 − 𝐿 = 918.0 + 688.3 − 696.5 = 909.8 kg/h
The operating line gives(471)(0.5043) − (250)(1.0)
𝐿
𝑉 𝑦 − 𝐿 𝑥
688.3
𝑦 =
𝑥 +
=
× 0.0589 += 0.0308
𝑉
𝑉
909.8
909.8
[Stage 8]
Stage 8 is also an equilibrium stage: 𝑦 = 𝑥 = 0.0308

56

Solid Extraction - Leaching

From the underflow function, we obtain the underflow rate from Stage 8:
𝐿 = (𝐹 − 𝐿 )(0.9 + 0.3 𝑥 ) = (750)(0.9 + 0.3 × 0.0308) = 681.9 kg/h
Then 𝑉 = 𝑉 + 𝐿 − 𝐿 = 909.8 + 681.9 − 688.3 = 903.4 kg/h
The operating line gives(471)(0.5043) − (250)(1.0)
𝐿
𝑉 𝑦 − 𝐿 𝑥
681.9
𝑦 =
𝑥 +
=
× 0.0308 += 0.0094
𝑉
903.4
903.4
𝑉
The goal concentration of the underflow from the final stage N is given by 𝑥 = 0.0184 as thecondition required for the recovery rate 0.95.
The above result 𝑦 = 0.0094 indicates that the goal value is stepped over when the underflow passes
through Stage 9. This suggests the final stage.
The McCabe-Thiele stage-by-stage calculation is expressed by Fig.22.3-E4.

Fig.22.3-E4 McCabe-Thiele graphical calculation for leaching

It has been found that the same result is also attained by the McCabe-Thiele method.2) This suggeststhat the required number of equilibrium stages is approximately 9.
Although the calculation method for variable underflow is a little bit difficult and different from thatfor constant underflow, we will close the section of leaching with this simplest example.
1)
2)

Kremser, A., Natl. Petr. News, vol.22(21), 42 (1930)
McCabe, W. L. and Thiele, E. W., Ind. Eng. Chem., vol.17, 605 (1925)

Extraction Engineering

57

22.3-4 Continuous leaching process equipment
Finally an example of real continuous leaching process equipment is shown in Fig.22.3-3.
This equipment is a kind of counter-current multi-staged column which is equipped with rotating conicaldisks perforated. A pair of upright cone and upside-down cone constitutes one stage. The feed consistingof insoluble solid and its contained solute is supplied from the top of the column while the solvent is fed
into the bottom. The underflow containing the solid, solute, and solvent flows downward stage-to-stage
along the slopes of conical disks inward and outward. The overflow containing the dissolved solute withthe solvent flows upward passing not only in the space between conical disks but also through theperforation holes. The final effluent of the overflow is obtained as the product from the nozzle set at the
top of the column. The final underflow discharging in the form of slurry from the bottom of the column
is moved to a decantation or filtration system for recovering the dissolved-solute solution entrained withthe solid.

Fig.22.3-3 An example of continuous leaching process equipment[PROBLEM 22.3-1]
A sunflower crude oil is produced by the leaching process of a solid feed 𝐹 ＝1,000 kg/h (e.g.
flaked seeds after pressing) containing 80wt% insoluble solid and approximately 20wt% oil to be
extracted. A continuous countercurrent leaching battery is used. Pure solvent (e.g. n-hexane) 800 kg/h is
supplied to extract 95% of the solute (i.e., sunflower seed oil). The flaked solid seeds discharging as theslurry from the final stage N entrains a limited rate of liquid hexane solution. By a preliminary test, we
get such an empirical underflow function that each kg of solid carries (1.6 + 0.4 x) kg of the solution ofsolute mass fraction x. We assume that the leaching process can be treated as an ideal stage-to-stage
cascade neglecting the effect of the substances other than the oil contained in the sunflower seeds.

58

Nomenclature

Calculate how many ideal stages are necessary for the desired extraction.
(The treatment of producing the sunflower seed oil comprises (1) grinding and flaking of the seeds into
meal, (2) pressing and squeezing, (3) leaching, (4) solvent recovering, and (5) oil refining. Roughly
speaking, the sunflower seed oil contains about 40% oleic acid, mono-unsaturated fat and 60% linoleic
acid, polyunsaturated fat. The value 20 wt% given above means the average composition of the total
fatty components.)

Nomenclature
𝑑
E
𝐹
𝑓
H
𝐾
𝑘 ,𝑘
L
𝑚
N
N’
R
𝑆
V
X
Y
𝑥, 𝑦
𝑥, 𝑦
𝑣
𝜃 ,𝜃

droplet diameter, [m]extract rate, [kg/s]
feed rate, [kg/s]
fugacity of B in A-phase, [Pa]mass of feed mixture, [kg]distribution function defined by Eq.(22.2-1), [ - ]mass transfer coefficients of continuous phase and dispersed phase [kmol/m2s]
underflow rate in continuous leaching, [kg/s]
distribution function defined by units of mole fraction, [ - ]total number of equilibrium stages, [ - ]number of ideal stages calculated by Kremser equation, [ - ]raffinate rate, [kg/s]
mass of solvent, [kg]overflow rate in continuous leaching, [kg/s]
mass of solute in raffinate or feed, [kg]mass of solute in extract or solvent, [kg]concentrations of solute in raffinate and extract of continuous liquid extraction, [kg/kg]
concentrations of solute in underflow and overflow of continuous leaching, [kg/kg]
slip velocity or terminal velocity, [m/s]time constant of continuous-phase and dispersed-phase characteristic for periodical surface renewal, [s]SubscriptsAdiluent
B
solute𝑖interfaceSsolvent

Chapter 23

Supercritical Fluid Technology

59

CHAPTER 23
SUPERCRITICAL FLUID TECHNOLOGY
23.1 Characteristics of Supercritical Fluid
If a gas or a liquid is heated and compressed above its critical temperature and pressure, it will reacha dynamic equilibrium and become a supercritical fluid. For example, when the temperature of water
goes beyond 374 °C and the pressure exceeds 22 MPa, it goes into a supercritical state. There is nodifference between the liquid and gas of water in the supercritical state, where they are completely
blended at constant temperature and pressure. Therefore there is no phase boundary. In the region ofsupercritical fluid, no matter how much pressure or temperature is raised, the material cannot transformfrom gas to liquid or from liquid to gas phase. It also has no surface tension.
Thermophysical properties vary sharply with temperature in a narrow temperature range called the“pseudo-critical range”.
Carbon dioxide also behaves as a supercritical fluid above its critical temperature (304.1 K) andcritical pressure (7.38 MPa). It is very useful fluid as a solvent because its good volatility is very
convenient for separation of solute from the extract (mixture of solute and CO2 obtained by extraction).
The properties of supercritical fluids are in between pure liquid and gas as follows.
(1) Liquid like densities (100 ~ 1,000 times larger than gases)(2) Diffusivities higher than liquids (of the order of 10 ~ 10 m ⁄s )(3) Good solvating power(4) Low viscosity (10~100 times less than liquid)(5) Reduction in surface tension(6) Gas like compressibility
For example, a supercritical fluid is a good solvent for solid extraction (leaching) because of great
possibilities of solubilization of compounds owing to its high density and good abilities of penetration
into solids owing to its low viscosity.

Fig.23.1-1 Schematic explanation of an idealized phase diagram

60

Characteristics of Supercritical Fluid

As will be mentioned later, the supercritical fluids exhibit continuous dramatic variations of physicalproperties when going beyond the critical point. There are many possibilities of trying various fruitfulutilizations for various chemical processes.
23.1-1 Properties of supercritical fluids
Supercritical fluids have low viscosity like gas and high density like liquid. The thermos-physicalproperties such as density, specific heat, thermal conductivity, and diffusivity vary drastically when
exceeding the critical point.
As distinct from the subcritical fluids, at supercritical pressure, the fluid phase changes from liquid togas continuously. As shown in Fig .23.1-2, when the temperature rises across the pseudo-critical point
𝑇 , the density 𝜌, the thermal conductivity 𝜅, and viscosity 𝜇 decrease drastically, and the specificheat capacity 𝐶𝑝 shows a sharp peak in a very narrow temperature range about the pseudo-critical point.
Table 23.1 indicates the critical temperature and pressure with density of several fluids.
Substance
𝐶𝑂
𝑁𝑂
𝑁𝐻
𝐻𝑂
𝐶𝐻

Table 23.1 Critical Parameter of Common Fluids
𝑇 (°𝐾)𝑃 (MPa)
304.1
7.38
309.6
7.34
405.5
11.40
647.3
22.12
190.4
4.6

𝜌 (kg⁄m )
470
450
240
340
162

Supercritical CO2
Supercritical carbon dioxide is very useful in various industries as a green, non-flammable, non-toxic,
recyclable, and powerful solvating medium.
Fig.23.1-3 indicates continuous variation of fluid density around pseudo-critical point.

Fig.23.1-2
Schematic picture of drastic variations of fluid density, specific heat, thermal
conductivity, and kinetic viscosity in the region near the critical point (e.g. CO2)

Supercritical Fluid Technology

61

Fig.23.1-3 Variation of fluid density with temperature in regions of subcritical and supercritical statesFig.23.1-4 Schematic illustration of mixtures of solute molecules with solvent molecules

Fig.23.1-4 expresses the microstructure of the mixture of solute molecules and solvent molecules
imagined in various phases.
We can expect that it is very interesting to utilize the transitional region where the critical point exists
for chemical processes.
Supercritical H2O
As can be seen from Fig.23.1-5, the thermos-physical properties of sH2O undergo significant change
in the temperature range of the pseudo-critical region (for 25 MPa between 372 and 392℃).

62

Characteristics of Supercritical Fluid

Fig.23.1-5 Thermo-physical properties of water in the vicinity of critical point

The supercritical water sH2O is used for nuclear power plants. The thermal efficiency of supercriticalwater-cooled reactor with sH2O as a coolant is better than that of light-water reactors.
There is also a possibility of the transformational technology using high-pressure sCO2 instead ofsteam in a closed power cycle because of the unique properties of sCO2 superior to steam in closed
cycle to absorb thermal energy.
23.1-2 Supercritical fluid flow and heat transfer in a horizontal pipe
A supercritical fluid such as CO2, water, and kerosene serving as the heat transfer medium plays animportant role for heat transfer enhancement or deterioration in a horizontal circular pipe.

Fig.23.1-6 Thermophysical properties in the neighborhood of critical point for carbon oxide

Development of a horizontal tube flow with heat transfer section
According to Tian et al. (2021)1), as distinct from the constant property flow, a three-stage streamwise
development of flow and heat transfer occurs in a horizontal pipe:

Supercritical Fluid Technology

63Stage I: Rapid reduction of heat transfer coefficient influenced by the development of thermalboundary layer.
Stage II: Flow stratification and secondary flow owing to the strong buoyancy effect caused by apeculiar entrance effect. The stratified flow is formed by the two different effects of the high temperatureand low density fluid in the upper part of the pipe and the low temperature and high density fluid in thelower part. Owing to the strong variation of thermos-physical properties, the wall temperature of thesupercritical fluid also varies circumferentially in a horizontal heated-wall tube.
Stage III: Pseudo-developed state free from the influence of the entrance condition.
At any rate, the thermos-physical properties vary remarkably not only in the stream direction but also
in the radial direction. Strictly speaking, the flow characteristics of supercritical fluid will never reach afully-developed condition. For supercritical fluid, it is technically too difficult to realize a fullydeveloped region under an isothermal condition. Many attempts have been made to correlate convective
heat transfer data in terms of conventional dimensionless groups with some modifications to allow for
the effects of thermos-physical property variations. It is still very difficult to analyze the effects ofinteraction of highly non-linear thermos-physical property variations (viscosity, thermal conductivity,
heat capacity and density) on the convective heat transfer. That is due to the fact that the flow and heattransfer processes are linked. Very different results are also obtained depending upon whether the pipe
is horizontal or vertical, and whether the flow is upward or downward in the vertical pipe.
1.

Tian, R., He, S-S, Wei, M-S., and Shi, L., The staged development of a horizontal pipe flow at supercritical pressure, Int. J. Heat
Mass Transfer, vol.168, Article 120841 (2021)[EXAMPLE 23.1-1]
Let us consider a supercritical fluid flow in a horizontal heated pipe shown in Fig.23.1-7.

Fig.23.1-7 Three-stage development of flow and heat transfer in a horizontal heat transfer tube

According to a rough estimation due to Fig.23.1-6, density 𝜌 = 330 kg⁄m , viscosity 𝜇 =2.3 × 10 Pa ∙ s , heat capacity 𝐶𝑝 = 8 kJ⁄kg ∙ K , thermal conductivity 𝜅 = 4.6 × 10 W⁄m ∙ K
at 307 K.
The Prandtl number is calculated as
𝑃𝑟 = 𝐶𝑝 𝜇 ⁄𝜅 = (8 × 10 )(2.3 × 10 )⁄(4.6 × 10 ) = 4(23.1-E1)
We can understand that the supercritical CO2 at 307 K a little bit above the pseudo-critical temperature
has the Prandtl number Pr = 4 , which lies between liquid and gas.
For a capillary tube of 3 mm ID, when the flow velocity of supercritical CO2 goes beyond 0.05 m/s,
the flow will become unstable leading to the transition to turbulent flow. The critical Reynolds numberis64

Characteristics of Supercritical Fluid

( .

)( .

)(

)

≅ 2,100(23.1-E2)
𝑅𝑒 =. ×
This is due to the fact that a supercritical fluid such as sCO2 has very low viscosity for large density.
It is known in supercritical chromatography that a neat supercritical CO2 owing to its significantly
low viscosity can experience a transition from laminar to turbulent flow at a flow rate 1.3 ml/min in a
microchannel of 127 μm ID . The high density with the low viscosity allows for highly compact,
microchannel-based heat exchanger technology.
Vertical pipe flow systems for supercritical fluids are often encountered in nuclear or various power
plants. For example, SCWR (Supercritical Water-cooled Reactor) is expected for the next generation of
nuclear reactor. Supercritical water is used there as a coolant. Their flow condition in a vertical pipe isdifferent from that of horizontal flow system in that the buoyancy force is in parallel to the main flow.
As a result, peculiar phenomena of flow laminarization and heat enhancement or deterioration may
appear.
We will study here the supercritical fluid flow focusing on a horizontal pipe flow system.
As distinct from the subcritical fluid flow, the flow and heat transfer does never reach an axiallyasymptotic condition because of the significant variation of thermos-physical properties.
This phenomenon still remains to be examined in various conditions.
It is said that the flow and heat transfer exhibit a three-stage development, which has two entrance effectstages and in the downstream region an asymptotic development stage.
Stage 1: Rapid reduction of heat transfer coefficient is caused by the developing thermal boundary layer.
When the inlet temperature is close to the pseudo-critical value, flow laminarization will take place.
Stage 2: Strong buoyance effect will play a role leading to stratification with secondary flow. Thisphenomenon has not been observed for the case of subcritical fluid flows.
Stage 3: The flow and heat transfer seem to be free from the influence of the entrance condition.
Due to the large density change with temperature, the fluid flow is strongly influenced by thebuoyancy force perpendicular to the main flow. As a result, the mixed convection controls the dominantheat transfer mode. This could be explained in the downstream region of a horizontal circular pipe withconstant wall heat-flux as follows. The buoyancy force inside the pipe drives the high-temperature andlow-density fluid upward along the circumferential wall while the high-density cooler fluid downward.
As can be seen from Fig.23.1-7, a secondary flow is formed. This fluid motion causes stratification,
which makes the wall temperature different between the top and the bottom.
The low-velocity flow near the circumferential pipe wall is heated firstly and transported toward the top
region by the secondary flow while the high-velocity bulk fluid is concentrated at the bottom owing to
the high-density of the fluid. This implies non-axisymmetric flow and heat transfer field.
In this non-uniformity in the circumferential distribution of the wall temperature, the heat transfer isenhanced at the bottom but deteriorated at the top by buoyancy.

Supercritical Fluid Technology

65

Fig.23-1-7 Stratification due to buoyancy effect in downstream region in a horizontal heated pipe
(Schematic picture of pipe cross-sectional view)

For constant property flow, the entrance region (i.e. developing region) is controlled by the axial
development of hydraulic and thermal boundary layers. However the flow and heat transfer at
supercritical pressure does never reach a fully-developed condition.
Even in the far downstream region, the behaviors of flow and heat transfer still depend on the history offlow conditions from the entrance. Therefore it is very difficult to obtain a generalized correlation ofheat transfer.
23.1-3 Nusselt correlations of supercritical fluid tube flow
The heat transfer performance of supercritical fluid is different from conventional fluids, especiallyin the determination of the convective heat transfer correlations. Many attempts have been made tocorrelate convective heat transfer data in terms of conventional dimensionless groups. However various
correlations are available for different geometries and dimensions but there is a lack of a unique universal
correlation. Only a few correlations are introduced below for reference.
Very early, Bringer and Smith presented the following correlation for a vertical tube flow1):
𝑁𝑢 = 0.0375 𝑅𝑒 . 𝑃𝑟 .
(23.1-1)
Yoon et al.2) proposed the Nusselt correlation for horizontal small-channel flow in the case when thebulk temperature is higher than the pseudocritical temperature:
𝑁𝑢 = 0.14 𝑅𝑒 . 𝑃𝑟 .
(23.1-2)
Here the subscript b means bulk temperature.
According to Jackson (2011)3) and Jackson et al.(1975)4), some modifications for the effect of physicalproperty variation are made to get the Dittus-Boelter correlation form for constant properties forced
convection:(23.1-3)
𝑁𝑢 = 0.0183 𝑅𝑒 . 𝑃𝑟 . (𝜌 ⁄𝜌 ) . (𝐶𝑝⁄𝐶𝑝 )𝜂 = 0.4
for 𝑇 < 𝑇 < 𝑇𝜂 = 1.2
for 𝑇 < 𝑇 < 𝑇
Here b, w, and pc mean bulk, wall, and pseudocritical, respectively and 𝐶𝑝 is mean specific heataveraged between 𝑇 and 𝑇 .
1.
2.
3.
4.

Bringer, R.P. and Smith, J.H., AIChE J., vol.3, pp.49-55 (1957)
Yoon, S.H., et al., Int. J. Refrig., vol.26, pp.857-864 (2003)
Jackson, J.D., DOI:10.1615/Ato Z.s. super_critical_heat_transfer (2011)
Jackson, J.D. et al., Heat transfer to supercritical pressure fluids. H.T.F.S. Design Report No.34 (1975)

66

Characteristics of Supercritical Fluid

23.1-4 Heat transfer equipment for supercritical fluids5)
Except in the region of heat transfer deterioration, the heat transfer coefficient of supercritical fluid
tube flow can usually be made very high owing to its turbulent flow at a high Reynolds number since its
viscosity is small for large density in the vicinity of critical point. Therefore in relation with the relief
requirement, the heat exchanger operated under high pressure for supercritical fluids is usually compact
as the heat transfer rate is relatively high. . We can refer to the Kwon et al. review paper of sCO2 PCHE
(Printed Circuit Heat Exchanger). The PCHE is manufactured by the following procedures: (1) Etching
metal plates to obtain flow channels, (2) stacking the etched plates to form exchanger blocks, (3)diffusion welding the blocks under high pressure and temperature, and (4) bonding making the stacked
blocks as strong as the metal itself.

5.

Kwon, J.G., Son, S., Heo, J.Y., and Lee, J-LK., Compact heat exchanger for supercritical CO2 power cycle application, Energy
Convers. Manage., vol.209, Article 112666 (2020)

As shown in Fig.23.1-8, a multi-stacked sheets of tube-bundle structure with staggered tube array canbe considered as an example of compact sCO2 heat exchangers. The sCO2 is placed on the tube side
because of its high pressure.

Fig.23.1-8 Configuration of a heat exchanger block unit to be stacked to form a compact sCO2 heatexchanger.

Supercritical Fluid Technology

67

Fitg.23.1-9 shows an example of semi-circular channel layout of PCHE.
The cross-section of the channels involves semicircular, rectangular, triangular and airfoil-shaped.

Fig.23.1-9 Configuration of etched channel layout of PCHE (Printed Circuit Heat Exchanger).
Dimensions are given as an example.

Liao and Zhao (2002)6) measured the convective heat transfer in horizontal mini- /micro-channels andreported a Nusselt correlation as a function of the Reynolds number and the Prandtl number:
𝑁𝑢 = 0.12 𝑅𝑒 . 𝑃𝑟 . 𝐵(23.1-4).
.
.
(𝜌 ⁄𝜌 )
(𝐶𝑝 ⁄𝐶𝑝 )𝐵 = (𝐺𝑟⁄𝑅𝑒)(23.1-5)
The subscripts b and w indicate the bulk fluid and tube wall temperatures, respectively.
The above correlation equation is applicable in the pseudo-developed state region reported by Tian et al.
because the entrance condition does not appear in it.
The part of Eq.(23.1-4) in front of B resembles the following constant-property flow correlation bySieder and Tate (1936) . (see Eq.9.2-4 in Chapter 9.2).
= 0.023.

⁄(23.1-5)

The term B can be considered as the effect of buoyancy plus the correction term of thermophysicalproperty variations characterized by the inner wall temperature with respect to bulk fluid temperature.
6.

Liao, S-M and Zhao, T-S, Measurement of Heat Transfer Coefficients from Supercritical Caron Dioxide Flowing in Horizontal
Mini/Micro Channels, J. Heat Transfer, 124 (2002)

23.2

Applicability of Supercritical Fluids

Various applications of supercritical fluid are exhibited in many industries as well as thermal andpower engineering plants. Supercritical carbon dioxide is a green, economic, non-flammable, non-toxic,and recyclable medium, especially as an environmentally benign solvent.
23.2-1 Supercritical fluid extraction
It was found in 18797) that supercritical fluids could dissolve selected compounds and that theirdissolving power was pressure-dependent, the higher the pressure, the higher the dissolving power.
7.

Hannay, J. B. and Hogarth, J., On the Solubility of Solids in Gases, Proc. Roy. Soc., London, Vol.29, 324-326 (1879)

It is an advantageous property that a supercritical fluid shares some common features with both gases
and liquids. The special combination of gas-like viscosity and diffusivity, and liquid-like density andsolvating properties of a supercritical fluid indicates an excellent solvent for various applications,especially in food and pharmaceutical industries. These properties can also be used as anenvironmentally friendly solvent for dry cleaning.

68

Applicability of Supercritical Fluids

The unique physical properties enable supercritical fluid extraction to be utilized for the extraction
processes that cannot be executed by liquids due to their high density and low diffusivity and by gases
due to their inadequate density.
The higher diffusivity and lower viscosity of supercritical fluids, as compared to traditional extraction
liquids, help the components to be extracted faster than other techniques. In addition, when the criticalpressure is removed, a supercritical fluid easily transforms to gas phase.
Therefore the most commonly used solvent is supercritical CO2.
In particular, supercritical carbon dioxide is a very useful solvent for industrial extraction processes
because of chemically stable, low-cost, non-flammable, non-poisonous working fluid. In addition, it iseasy to separate the solute from the extract solution by evaporation of the supercritical CO2 solvent.
The solid extraction “leaching” is used to dissolve soluble matter from its mixture with an insoluble
solid. The leaching technology under the usual fluid condition has already been studied in Chapter 22.3.
Supercritical fluid extraction is very advantageous for such leaching processes. In particular,
supercritical CO2 is becoming an important commercial and industrial solvent due to its role in chemical
extraction. Fig.23.2-1 is an example of the supercritical fluid extraction systems.

Fig.23.2-1 Supercritical fluid extraction system having a stationary solid-bed extraction vessel

Usually the extraction condition by supercritical CO2 is above the critical temperature 31℃ andcritical pressure 7.4 MPa. The supercritical CO2 is pumped to a heater, where it is heated tosupercritical conditions.
Solids such as coffee beans, soybeans, and corn-meal, etc. loaded into the pressurized extraction
vessel are stationary on the perforated false bottom as a packed column. This packed bed is usuallycalled “solid matrix.” The supercritical fluid solvent passes through the solid matrix in the upward
direction, where the supercritical CO2 spreads into the solid matrix and dissolves the material to be
extracted. The dissolved material swept out of the extractor undergoes the vaporization of the solvent
sCO2 through the pressure release valve, and flows into a separator at low pressure. The extracted
material settles out there. Then the CO2 can be cooled, recompressed , and recycled.
The solvent is cooled before pumping to maintain liquid conditions, and then heated afterpressurization. For larger industrial extractions, the energy required during each stage of the process canbe calculated using the thermodynamic properties of the supercritical fluid.

Supercritical Fluid Technology

69

Supercritical fluid extraction by sCO2 is popularly used in various separation processes, especially for
relatively non-volatile solids and liquids because the sCO2 is non-flammable and non-hazardous: Various
application to agricultural and food products such as coffee decaffeination and hops extraction.
23.2-2 Principle of supercritical fluid extraction1)
One example of supercritical fluid extraction systems consists of (1) Fluid reservoir (gas cylinder incase of CO2), (2) Pump (usually reciprocating pump), (3) Extraction vessel (stationary phase for solids,
high pressure 300~600 atm) or packed columns (for liquids), (4) Restrictor for maintaining pressurechange inside the extraction vessel (variable nozzle, back pressure regulator), and (5) Collector (trapping
system).
The supercritical fluids can dissolve some solid-like involatile substances from non-volatile solid.
In order to understand the principle of supercritical fluid extraction, we very often consider the solubilityof naphthalene (𝐶 𝐻 , Melting point 80.2℃) in supercritical CO fluid shown in Fig.23.2-2.
Dependency of pressure on solubility of supercritical fluids7)
As can be seen in Fig.23.2-2, the solubility of naphthalene in sCO2 changes considerably withpressure. This suggests why the rapid expansion is convenient for precipitation of the solute dissolved
in sCO2.

Fig.23.2-2 Solubility of naphthalene in supercritical CO2

Let us consider a solid extraction system recovering naphthalene from a solid mixture shown in
Fig.23.2.-3. The solid extraction, i.e. leaching technology and the rapid expansion method will be studied
in Chapters 23.2 and 23.3. The supercritical CO2 at selected pressure –temperature conditions of 300 bar
and 55 ℃ is passed through the extraction column in which naphthalene admixed in a solid state withsome other non-volatile solid. As the sCO2 percolates through the column contents, naphthalene isdissolved, up to its solubility limit 15 wt%, and it is extracted from the solid mixture. The fluid solutionof CO2 and naphthalene issuing from the extractor is passed through the depressurizing valve anddropped in pressure to 90 bar and in temperature to 35 ℃. At 90 bar and 35 ℃, the solubility ofnaphthalene is only about 2 wt%. More than 90% drops out like a snow in the separator (Precipitationchamber), that is, naphthalene is separated as solid from the CO2 gas stream. Then the gas stream isrecompressed for further extraction. The solid naphthalene product is precipitated due to the rapid
expansion in the precipitation chamber.

70

Applicability of Supercritical Fluids

Fig.23.2-3 Extraction system of naphthalene by rapid expansion method

At high pressures, supercritical CO2 exhibits liquid-like behavior since the solubility increases withtemperature while at low pressures, the solubility rapidly decreases with temperature owing to a steep
decrease in density of the gas. It seems that the intermediate pressure exists around 120 bar.
The principle of supercritical fluid extraction can be considered that the solubility of a compound (solute)
above the critical temperature is sufficient to consider the supercritical fluid as a extracting medium(solvent).
One great advantage of supercritical fluid extractions is complete separation of solvent from extract
and raffinate. It should be kept in mind that adequate contact time is necessary for penetration of solvent
into solid particles of packed matrix.
There are two essential steps to supercritical fluid extraction: (1) transport of solid particles to be
extracted toward the outside surface of particles, and dissolution of the material in the supercriticalsolvent and (2) diffusion of the supercritical solvent to the inside of the particle and counter-diffusion of
the dissolved material from inside the particles.
This reversible release is very difficult, so not dealt with in detail here in this course.
The solid extraction by supercritical fluids is a diffusion-based process, in which the solvent fluid isrequired to diffuse deeply into the solid matrix and the extracted material to diffuse out of the matrix
into the bulk solvent. Fortunately diffusivities are much faster in supercritical fluids than in liquids anddue to the lack of surface tension and the negligibly small of viscosities, the solvent can penetrate more
into the matrix inaccessible to liquids

23.2-3 Modeling diffusion of dissolved solute inside a solid particle1)
Fig.23. 2-4 shows the stages of the concentration profile during extraction inside a solid particle.

Supercritical Fluid Technology

71

Fig.23.2-4 Schematic profile of concentration of extracted material in solvent during extraction
process inside a single solid particle for two extreme cases

Profile (a) is the case of diffusion limited extraction that the material is carried away from the outside
surface by the bulk fluid of solvent faster than it can diffuse from the center to the outside surface.
Therefore the concentration at the outside surface drops to zero.
Profile (b) is the case of solubility limited extraction where solubility is low relative to diffusion. The
dissolved material is able to diffuse to the outside surface faster than it can be carried away by the solvent,and the concentration profile becomes flat.
One great advantage of supercritical fluid extractions is complete separation of solvent from extract
and raffinate. It should be kept in mind that adequate contact time is necessary for penetration of solvent
into solid particles of packed matrix.
1)

Sapkale, G.N., Patil, S.M., Surwase, U.S. and Bhatbhage, P.K., A review of Supercritical Fluid Extraction, India Int. J. Chem. Sci.,
Vol.8(2), 729-743 (2010)23.3 Leaching by Supercritical Fluids
A supercritical solvent can be used effectively for the leaching processes. The supercritical carbon
dioxide is the most useful for supercritical extraction because it is very easy to separate solvent from theextract mixture obtained by leaching operation. Supercritical fluids dramatically improve not only thesolubility of matters but also the permeability of cellular channels by swelling.
23.3-1 Leaching equipment
Pure supercritical fluid as the solvent is fed into the solid bed equipped in a vessel or a tank. The
supercritical fluid may be percolated through stationary solid beds due to the pressure drop. There thedissolution and diffusion/mass transfer of soluble matters are also influenced very much by the surficial
and internal structure of solid bodies.
The materials dissolved in the bulk of supercritical fluid are carried away downstream.
Fig.23.3-1 shows an example of decaffeination systems by using a supercritical carbon dioxide.
Decaffeination process by supercritical CO2 takes place prior to roasting. After steaming, the green
coffee beans are stacked up as a packed bed in a high-pressure vessel. The sCO2 containing a little wateris circulated through the leaching system at about 30 MPa and 65℃. Caffeine can be dissolved into thesCO2 but the insoluble flavor compounds remain in the beans. The caffeine extracted is scrubbed fromthe sCO2 with additional water, and then recovered in a distillation system.

72

Leaching by Supercritical Fluids

The supercritical fluid percolates and dissolves the solid substance in the extractor, and then resulted
solution with sCO2 is rapidly depressurized through a heated nozzle at supersonic velocity into a low
pressure chamber. This is called “Rapid expansion process.”

Fig.23.3-1 Leaching equipment for decaffeination by supercritical CO2 used as a solvent

23.3-2 Rapid expansion of supercritical fluid solutions2)
The formation of nano- and micro-structured energetic particles results from the rapid expansion ofa supercritical solution. In order to produce very fine particles, a supercritical solution of organic
substances dissolved in sCO2 is rapidly expanded through an expansion nozzle. Fig.23.3-2 shows aschematic configuration of leaching equipment accompanied with the rapid expansion precipitation
system. Fig.23.3-3 shows typical variation of fluid temperature occurring in the rapid expansion ofsupercritical CO2 solution.
During the rapid expansion, the density and solvent power significantly decrease, and then result insupersaturation of the solution and consequently precipitation of the solute free of a residual solvent.
The precipitation of the solute from the sCO2 solution occurs due to the rapid reduction of solute’s
solubility. The precipitated solute particles have narrow particle size distribution. The higher thepressure of supercritical solution, the smaller the size of precipitated particles.

Fig.23.3-2 Schematic configuration of leaching plant accompanied with rapid expansion ofsupercritical carbon dioxide solution

Supercritical Fluid Technology

73

Fig.23.3-3 Variation of fluid temperature caused by rapid expansion

After leaching process, the rapid expansion of supercritical solution is also used to separate thesolute as solid particles from the carbon dioxide.
Supercritical antisolvent technique using supercritical CO2 is also very useful for microencapsulationin pharmaceutical industry.
2.

Debenedetti, P.G., Tom, J.W., Kwank, X. and Yeo, S-D, Rapid expansion of supercritical solutions: fundamentals and applications,
Fluid Phase Equilibria, vol. 82, 311-321 (1993)

Nomenclature
𝐶𝑝
𝐷
𝐺
Gr
ℎ
𝑁𝑢
P
Pr
R
𝑅𝑒
𝑇
𝜂
𝜅
𝜇
𝜈
𝜌

heat capacity, [J/kmol K]
tube diameter, [m]mass velocity, [kg/m2s]
Grashof number, [ - ]heat transfer coefficient on the tube inside wall, [W/m2K]
Nusselt number, [ - ]pressure, [Pa]Prandtl number, [ - ]tube radius, [m]Reynolds number, [ - ]temperature, [K]exponent of Eq.(23.1-3), [ - ]thermal conductivity. [w/m K]
viscosity, [kg/m s]
kinematic viscosity, [m2/s]
density, [kg/m3]

Subscripts
b
bulk
C, cr
criticalpc
pseudo-critical
w
wall

74

Chapter 24 Energy Saving Technology

75

PART ⅣCHAPTER 24
ENERGY SAVING TECHNOLOGY
Any chemical processes cannot be executed without consumption of energy.
It is an economically important issue to reduce the energy requirement as much as possible. We knowthat distillation is a dominant separation technology in the chemical process industries, and at the sametime, one of the largest energy-consuming unit operations. The distillation process proceeds by directcontact of the flowing-downward liquid with the flowing-upward vapor. In the case of column-type
distillation systems, at the bottom reboiler, therefore, we need the heating duty for producing the vapor
to be fed back into the bottom of the distillation column while the cooling duty is required at the overheadcondenser to produce the reflux liquid by condensing the vapor coming out from the column top.
It is a very important issue to consider how to effectively reduce the consumption of energy.
In this chapter, we will study the energy conservation problem of a distillation column as an example.

24.1 Energy Requirement in Continuous Distillation
We already studied the energy balance in a single continuous distillation column (shown below) in
Chapter 11.6（PART I）.
Let us begin this chapter by going over that lesson with Fig.11.6-1.
A binary mixture consisting of 𝑥 (mole fraction) of the more volatile component X and (1 − 𝑥 )of the less volatile component Y is fed at the feed rate 𝐹 (kmol/h) into the central portion (i-stage) of
the distillation column. The steady-state operation with reflux ratio 𝑟 = 𝑅 ⁄𝐷 obtains the overhead
product 𝐷 (kmol/h) containing 𝑥 of component X and the bottom product B (kmol/h) containing(1 − 𝑥 ) of component Y. There are several heat exchangers around the distillation column. Eachexchanger should choose an appropriate pair of heating medium and cooling medium for the sake ofeffective energy exchange.

76

Energy Requirement in Continuous Distillation

Fig.11.6-1. Flow sheet of distillation column system

First of all, the overall mass balances can be expressed as
𝐹 =𝐷+𝐵𝐹𝑥 =𝐷𝑥 +𝐵𝑥(11.6-1)(11.6-2)

The feed is to be heated from 𝑡 to 𝑡 (K) by exchanger I, from 𝑡 to 𝑡 (K) by exchanger
II, and from 𝑡 to 𝑇 (K) by exchanger III, so that it can be introduced into the column as the saturated
liquid at its bubbling point.
The molar latent heat and heat capacity of the mixture can be approximated as
Δ𝐻 = 𝑥 Δ𝐻 + (1 − 𝑥)Δ𝐻
(J/kmol)
𝐶𝑝 = 𝑥 𝐶𝑝 + (1 − 𝑥)𝐶𝑝
(J/kmol K)
The vapor leaving the top of the column is to be completely condensed at its dew point 𝑇 (K) by
the exchanger II followed by the overhead condenser.
Since the vapor flow rate in the rectifying section is given by
𝑉 = (𝑟 + 1)𝐷,the total rate of heat released in condensation of the vapor becomes(𝑟 + 1)𝐷 [𝑥 ∆𝐻 + (1 − 𝑥 )∆𝐻 ]Assuming that the average heat capacity 𝐶𝑝 does not depend on temperature, the rate of heattransfer at exchanger II is given by
𝐹 [𝑥 𝐶𝑝 + (1 − 𝑥 )𝐶𝑝 ](𝑡 − 𝑡 )If a cooling water enters the overhead condenser at 𝑡 (K) and leaves at 𝑡 (K), the heat balance
set up at the condenser gives the water flow rate 𝑊 (kg/h) as(𝑟 + 1)𝐷 [𝑥 ∆𝐻 + (1 − 𝑥 )∆𝐻 ] − 𝐹 [𝑥 𝐶𝑝 + (1 − 𝑥 )𝐶𝑝 ](𝑡 − 𝑡 )= 𝑊 𝐶𝑝 (𝑡 − 𝑡 )(11.6-7)
where 𝐶𝑝 is the heat capacity of water in (J/kg K).
The bottom product leaves the reboiler at its bubbling point 𝑇 (K). If the heating duty is provided
by condensation of steam at pressure 𝑃 (Pa) in the jacket of the reboiler, the heat required forvaporizing 𝑉′ (kmol/h) of the bottom liquid is given byEnergy Saving Technology

77(11.6-8)

𝑉 [𝑥 ∆𝐻 + (1 − 𝑥 )∆𝐻 ] = 𝑆 ∆𝐻′
From this equation, the required steam, 𝑆 (kg/h) can be calculated.
By heat balance set up at exchanger III, the outlet temperature 𝑡(K) can be calculated.

𝐹 [𝑥 𝐶𝑝 + (1 − 𝑥 )𝐶𝑝 ](𝑇 − 𝑇 ) = 𝐵[𝑥 𝐶𝑝 + (1 − 𝑥 )𝐶𝑝 ](𝑇 − 𝑡 )(11.6-9)
We already learned the heat balance up to this position in Chapter 11.6.
Take a look at Figure 11.6-1. We employed three exchangers to preheat the feed mixture to be
distilled.
[PROBLEM 24.1-1]
Regarding the flow sheet of heat exchangers around the distillation system in Fig.11.6-1, we canconsider the following two cases: the exchanger I is placed (1) between the exchangers II and III and (2)between the exchanger III and the feeding inlet. Which position is more profitable?
From a viewpoint of energy saving, the latent heat of the vapor leaving the top of the column isutilized in exchanger II to heat the feed mixture up to 𝑡 and the sensible heat of the bottom productdischarged from the bottom reboiler is utilized in exchanger III to heat the feed mixture up to 𝑇 . Thesemethods can be regarded as the prerequisite utilization of energy. However it is more important todrastically reduce the heating duty of the reboiler and the cooling duty of the overhead condenser. We
would like to consider whether or not the heat released by vapor condensation in the overhead condenser
can be utilized for the heating duty for vaporization in the bottom reboiler. A problem is how to pumpup the thermal energy from the low-temperature overhead condenser to the high-temperature bottom
reboiler.
We shall study the principle of heat pump in the next section as another technology of effectiveenergy saving.
In the latter half of this chapter, we will study a heat pump system only for the energy-saving
technology of distillation and evaporation plants.

24.2

Heat Pump Technology

24.2-1 Principle of heat pump
There are various heat pump systems available depending on their purposes.
First of all, let us begin by considering the vapor-compression cycle as an example used by many
refrigeration systems. We know well from the second law of thermodynamics that thermal energy
cannot be freely transferred from a lower-temperature environment to a higher-temperatureenvironment. It will become possible only if an external high-quality energy such as mechanical shaft
work obtained by a compressor is applied from the outside to the system. We know that thetemperature of gases can be raised by adiabatic compression due to a compressor. Therefore a standard
room air conditioning system works by the vapor-compression cycle called “refrigeration cycle.”
Fig. 24.2-1 is a basic heat pump flow configuration using a refrigerant as the working fluid.

78

Heat Pump Technology

Fig. 24.2-1 Basic flow configuration of a heat pump air-conditioner

The amount of work 𝑊 required to pump an amount of heat 𝑄 from a lower-temperature reservoir
such as ambient outdoor air to a higher-temperature reservoir such as room air is given by𝑊=(24.2-1)

where 𝐶 is called as instantaneous coefficient of performance for the heat pump; a ratio of usefulheating or cooling provided to work required.
Regarding the refrigeration cycle in cooling mode of Fig.24.2-1, the compressor sends the highpressure and high-temperature refrigerant vapor to the condenser coil where it is converted to a liquid.
In this condenser section, the latent heat is released for heating the outside air. Then by isenthalpic
expansion traveling through the expansion valve, the refrigerant becomes low pressure and low
temperature liquid with some flashed vapor. At the next evaporator section, the low-pressure and lowtemperature refrigerant liquid is evaporated absorbing heat from the indoor air. A fan is necessary
there to improve the efficiency of heat exchange with the condensing or evaporating refrigerant in thelooped system. The fan blows the indoor air across the cold evaporator coil where the heat inside theroom is absorbed into the refrigerant.
In this manner, the cooled air is then circulated throughout the room while the evaporated lowtemperature refrigerant vapor is sent back to the compressor. There the low-pressure and lowtemperature refrigerant vapor is adiabatically compressed to become high-pressure and hightemperature vapor.
This refrigeration cycle can be reversed; if the indoor evaporator coil switches roles, it becomes thecondenser coil producing heat while the outdoor condenser unit also switches roles to serve as theevaporator discharging cold air.
That is, a usual sink or source for heat is the outdoor or room air, as used by an air-source heat pump.

Energy Saving Technology

79

A typical diagram of heat pumping by vapor-compression refrigeration is expressed as in Fig.24.2-2.

Fig.24.2-2 Principle of Reversed Heat Pumping

In the case of an ideal heat pump, the first law of thermodynamics gives(24.2-2)

𝑄 + 𝑊 = 𝑄
Its coefficient of performance is defined as
𝐶(refrigerator) = 𝑄 ⁄𝑊 = 𝑄 ⁄(𝑄(24.2-3)− 𝑄 )If the heat transfers are reversible and isothermal at temperatures 𝑇
expressed as
𝐶(refrigerator) = 𝑇⁄(𝑇𝐶(heat pump) = 𝑄⁄𝑊 = 𝑇− 𝑇

and 𝑇, the COP is(24.2-4)

)
⁄(𝑇

− 𝑇

)

(24.2-5)

Let us understand the principle of heat pump by using an ideal Carnot heat engine cycle
schematically shown in the T-S diagram (Fig.24.2-3).
Consider the following ideal heat engine cycle using steam as an example vapor.

Fig. 24.2-3 Ideal heat engine cycle for heat pumping

80

Heat Pump Technology

Fig.24.2-3 indicates an ideal heat engine cycle for pumping thermal energy from the temperature 𝑇
of cool environment to the temperature 𝑇 of hot environment.
The basic loop consists of the following four sub-processes:
A→B

Isentropic expansion (Throttle valve)

B→C

Constant pressure heating (Evaporation)

C→D Isentropic compression
D→A Constant pressure cooling(Condensation)

Practically speaking, since it is difficult to compress wet vapor, the refrigerant is usually dried
before compression. The vapor compressed by isentropic compression is usually somewhat
superheated. The throttle does not produce a useful work. The enthalpies of the refrigerant vapor (orsteam) before and after a throttle are the same. This is called “isenthalpic process” but the entropy
increases over the throttle.
Fig.24.2-4 shows an actual Entropy-Temperature diagram of the heat pump system.

Fig. 24.2-4 S-T diagram for heat engine cycle for actual heat pumping

The purpose of this chapter is to provide a basic understanding of heat pump theory and a morepractical approach to the solution of problems which may arise in conjunction with energy
consumption of distillation processes.
By using Fig.24.2-4, an analysis would start at the evaporator (A→B) where cold liquid at low
pressure is vaporized as it picks up heat from the room air passing over the coil of the exchanger. This
sub-process is constant-pressure evaporation at its saturation temperature. Cool vapor from theevaporator then flows to the electric compressor (B→C), where its pressure and temperature are
elevated due to the isentropic compression. The hot refrigerant vapor containing heat of compression
and latent heat of vaporization (superheated vapor) is pumped to the condenser (C→D) where heat isremoved by the outdoor air passing over it.

Energy Saving Technology

81

Thus heat has been pumped up from the evaporator (room air) to the condenser (outdoor air). The
refrigerant liquid (condensate) leaving the condenser passes through the throttle (D→A) accompaniedwith isenthalpic expansion and then goes back to the evaporator.
In this manner, the heat pump working principle is that a device transfers heat energy from a sourceof heat to the thermal reservoir. This implies a possibility that in a distillation column, the overheadcondenser can be the source of heat and the bottom reboiler can be the thermal reservoir. In a usual
situation, however, the overhead condenser is lower in temperature than the bottom reboiler. Therefore
we need an electric compressor for the overhead condenser to become the heat source whereas thebottom reboiler serves as the heat sink. This method is also called “Vapor Recompression” or VRC
method. However it should be kept in mind that the electric energy is much more valuable in quality
than the thermal energy. From this point of view, therefore, we should compare how much the electricenergy applied with how much the thermal energy saved.

24.2-2 Vapor recompression technology for distillation
An example of the VRC (vapor recompression) system for a distillation columns is given by
Fig.24.2-5.

Fig. 24.2-5 VRC system of distillation column constructed by using heat pump technology

In this system, a heat exchanger plays a role of the condenser for the high-pressure overhead vapor
as well as of the reboiler for the low-pressure bottom liquid. If we set up the energy balance over thearea enclosed by the broken line in Fig.24.2-6, the following equation is obtained:𝐹 𝐻 + (−𝑊 ) = 𝐷 𝐻 + 𝑊 𝐻(24.2-6)This implies that we should consider the adiabatic work done by the electric compressor as the surplus
energy from the outside. As a result, the heating duty for the bottom reboiler can be covered by thecooling duty for the overhead condenser by virtue of the additional energy input supplied by theelectric compressor.

82

Heat Pump Technology

Fig. 24.2-6 Energy balance of VRC system of distillation column

For an ideal gas, the adiabatic compression work (−𝑊 ) is given from the outside by(−𝑊 ) = ∫ 𝑉 𝑑𝑝 = 𝑝

⁄

𝑉 ∫

⁄

=

1−

(24.2-8)This equation can be employed in case when steam used as the refrigerant with the volumetric flowrate 𝑉 is compressed from 𝑝 to 𝑝 by the compressor.
Here 𝜅 = 𝐶𝑝⁄𝐶𝑣 is the specific heat ratio of the gas. For the case of steam, it can be assumed as𝜅 ≅ 1.32.

In this system, the more-volatile vapor (usually organic) must be directly compressed by an electriccompressor. Practically speaking, we do not use a turbo-compressor which cannot give a highcompression ratio. In order to get a high compression ratio, we very often employ a positive
displacement compressor such as a screw-type compressor rather than a turbo-compressor. For
safety, therefore we often utilize a steam compressor for the VRC system to be described in thefollowing section.

[PROBLEM 24.2-1]
By using the VRC system shown in Fig.24.2-6, the overhead vapor of a distillation column having
the following operation data is adiabatically compressed by a screw-type compressor.
Overhead vapor data:
Composition: ethanol 55.8 mol% and water 44.2 mol%, Temperature 82.3℃, Pressure 0.1 MPa,
Vapor rate 𝑉= 4.5 kmol/hEnergy Saving Technology

83(1) Calculate the temperature of the vapor compressed with the compression ratio of 2.0.
Is the compressed vapor superheated?
(2) Calculate the theoretical work of the compressor.
(3) The liquid has the composition of ethanol 2.0 mol% and water 98 mol% in the bottom reboiler
operated at 0.1 MPa and 95.5℃. Obtain the condition of the heat exchanger as the reboiler ifusing the compressed vapor as the VRC heating medium.

24.2-3 New heat-pumped distillation system
Let us consider how to pump up heat from low-temperature condensate of the overhead condenser tohigher-temperature liquid of the bottom reboiler.
Fig.24.2-7 shows a standardized heat pump unit.

Fig. 24.2-7 Inside structure of a standard heat pump unit.

Consider the heat pump system using a refrigerant as the working fluid
This system has a heat engine cycle which consists of condenser section, evaporator section connectedwith a compressor and a throttle valve.
Regarding the red line loop shown in Fig.24.2-7, the high-pressure and high-temperature refrigerant
vapor from the compressor (side B) is condensed in the condenser section operated at high pressure𝑝 , where the latent heat released is given to the hot water line for the reboiler. Usually thecondensation takes place under the constant high temperature: 𝑇 = 𝑇 . Then isenthalpic expansion
without energy input is conducted by a throttle valve, where the refrigerant temperature and pressure
go down to 𝑇 and 𝑝 accompanied with some partial vaporization.
After the throttle valve, the two-phase fluid of the refrigerant is evaporated in the evaporator section(side A) operated at constant low temperature 𝑇 = 𝑇 and low pressure 𝑝 , where the latent heatfor evaporation is obtained from the cooling water line for the overhead condenser. Downstream of theevaporator, it is usually necessary to get dry vapor for good operation of the compressor. The saturated
vapor is adiabatically compressed from pressure 𝑝 to 𝑝 by the electric compressor.

84

Heat Pump Technology

This isentropic compression work (−𝑊 ) is put into the system by electricity from the outsideenvironments.
Except for the heat input 𝑄 at the evaporator and the heat output 𝑄 at the condenser in the heatpump unit, the compression work should be treated as the additional energy input into the heat pumpunit. In this manner, thermal energy can be pumped from the condenser cooling water to the reboiler
heating water by adding the mechanical shaft work into the system.
Setting an energy balance over this heat pump unit
𝑄 + (−𝑊 ) = 𝑄(24.2-7)

Fig.24.2-8 shows a distillation column system inside of which a heat pump unit is installed.
It should be kept in mind that the surplus energy, i.e. the work −𝑊 given to the heat pump unit
by the electric compressor needs to be cancelled by a cooler on the cooling water line shown in
Fig.24.2-8.
Usually the heat duty of the bottom reboiler becomes slightly greater than the cooling duty of theoverhead condenser. Therefore there is a possibility working without the surplus energy remover.

Fig. 24.2-8 Distillation column installed by a heat pump unit.

For the case of ideal solutions such as benzene-toluene binary system, the heating duty 𝑄 of thereboiler becomes approximately equal to the cooling duty 𝑄 of the condenser.
Therefore additional work is required in VRC distillation to pump up heat from the low temperaturecondenser to the higher temperature reboiler by using a heat pump unit.
The surplus energy is given as the compression work 𝑊 by means of the compressor of the heatpump unit.
In the region enclosed by the red broken line as shown in Fig.24.2-9, the following energy balancecan be obtained taking into account the work (−𝑊 ) by the compressor and the cooling duty 𝑄′ forthe surplus energy.

Energy Saving Technology

𝑄 + 𝑄′ = 𝑄 + (−𝑊 )85(24.2-9)

Roughly speaking, for an ideal system, the reboiler heating duty is approximately equal to thecondenser cooling duty: 𝑄 ≅ 𝑄 . The above equation indicates that the cooling device for removing
the surplus energy 𝑄′ is necessary to keep the distillation system steady.

Fig. 24.2-9 Compensation for the surplus energy resulting from the heat pump unit.

24.3

Vapor Recompression Technology for a Jacketed Tank
Evaporator

24.3-1A new batch tank evaporator

By the way, we would like to introduce an effective batch evaporator shown below.

Fig.24.3-1 A batch system of Wall-Wetter evaporator

86

Vapor Recompression Technology

This is an our original “Wall-Wetter” batch evaporator consisting of two or four half-cut pipes (orgutters) which are fixed obliquely to the rotating shaft within a jacketed tank. (Kansai Chemical
Engineering Co., Ltd)

Fig.24.3-2

A configuration of two-wing Wall-Wetter.

The Wall-Wetter system has been devised noticing that the process liquid pooled in the tank bottom canbe pumped up within the vessel due to the centrifugal effect of the rotating Wall-Wetter wings. As
shown in Fig.24.3-2, the Wall Wetter wing has a form of inclined gutters (half-cut pipes), inside of which
the process liquid crawls up toward their upper edges, and then the liquid jet issuing radially from theupper edges. It forms a liquid film falling on the inside wall of the evaporator tank.1~4)
Therefore the evaporating rate can be kept at a large constant value during the whole evaporation process
time by maintaining the whole heat transfer surface constantly wet with the falling liquid film of theprocess liquid.
The process time required for attaining the request of evaporation is usually shortened very much.
The evaporative characteristics concerning the Wall Wetter can be obtained referring to the relevant
literature.1~4)

1.
2.
3.
4.

Yamaji, H., Noda, H., and Kataoka, K., Proc. 6th World Congress of Chemical Engineering, (CD-ROM), Melbourne, Australia(2001)
Yamaji, H., Doctoral Dissertation, Kobe Univ. (2002)
Mukaida, T., Kataoka, K., Yamaji, H., Noda, H., and Ohmura, N., Kagaku Kogaku Ronbunshu, 44(2), pp.107-112 (2018)
Mukaida, T., Kataoka, K., Noda, H., and Yamaji, H., Chem. Eng. Transactions, 86, (2021)

24.3-2 Saving energy by a vapor recompression technology
From a viewpoint of heat pump, we can consider the so-called vapor-recompression (VRC) method
applied for this batch evaporator system shown in Fig. 24.3-3.

Energy Saving Technology

87

Fig.24.3-3 A batch system of Wall-Wetter evaporator arranged for VRC method.

Let us consider the case when water is evaporated for concentration of an aqueous solution of nonvolatile substance such as salt and sugar.
The water vapor evaporated from the liquid film falling over the jacket wall as well as the liquid
solution pooled in the bottom leaves the top of the evaporator and is compressed from 𝑃 to 𝑃 by
an electric screw-type vacuum pump or compressor, so that the vapor temperature is raised from 𝑇
to 𝑇.
The basic idea of VRC in this evaporator is expressed in Fig.24.3-4.

Fig.24.3-4 Basic idea of VRC for a Wall Wetter batch evaporator

Roughly speaking, the above figure states that one mole of evaporated water vapor can evaporate
one mole of water of the same aqueous solution after the adiabatic compression. The energy
requirement is the work of isentropic compression by the compressor only.

88

Vapor Recompression Technology

However it is usually necessary to utilize a cooling water system equipped inside a compressor or avacuum pump. Otherwise the outlet vapor temperature raised due to the isentropic compression will
become considerably higher than the saturation temperature at the compressor outlet pressure.
Because the latent heat of evaporation decreases in some degree with increasing pressure, the thermalenergy for the evaporation cannot be covered with the thermal energy released by the condensation,that is, some extra steam or an auxiliary heater may be required.
Practically this pressurized water vapor a little higher than the saturation temperature is supplied into
the jacket of the same evaporator for reuse as the heating medium.
Strictly speaking, owing to the effect of pressure on heat of evaporation, the thermal energy released
by condensation of the pressurized vapor is a little less than the thermal energy required forevaporation of the solution. Therefore a very small auxiliary heater or an extra steam (make-up steam)should be utilized for compensating the energy deficit 𝑄 .

Fig.24.3-5 shows the temperature profile on both sides of the tank wall for heat exchange.

Fig.24.3-5 Schematic profile of the temperature distribution in the neighborhood of the jacket wallof the WW evaporator

For simplicity, the overall driving force for evaporating the process liquid is given by∆𝑇 = 𝑇(24.3-1)− 𝑇

Assuming no fouling effect, the overall heat transfer resistance 1⁄𝑈 can be expressed as the sum of
the resistances lying in series:
∆𝑅 =

+

+

+

=

+

+

=

(24.3-2)

The evaporative heat transfer coefficient ℎ is obtained from the resistance of the falling liquid film
and the fictitious vapor film contacting the surface of the falling liquid film.

Energy Saving Technology

89

The thermal energy transferred from the jacket side to the tank side for evaporation of the process
liquid is𝑄(24.3-3)= 𝑈 𝐴 Δ𝑇

In the case of non-volatile solute, the basic equation of evaporation by a batch WW evaporator can be
expressed as(24.3-4)=−

where 𝐿 is the molar mass of the liquid solution contained in the evaporator tank, and λ the molar
latent heat for vaporizing the solvent (i.e. water) in the tank.
The boiling point of the evaporating solution can be calculated at the operating pressure by Antoine
equation𝑇=(24.3-5)−𝐶

where the units of temperature and pressure are 𝑡 (℃) and 𝑃 (mmHg).
The change in specific volume of the vapor by the adiabatic compression obeys the following perfectgas law:𝑣 = 𝑣 (𝑃 ⁄𝑃 ) ⁄(24.3-6)

The theoretical power requirement is calculated by the following adiabatic compression:(−𝑤 ) =

𝑃𝑣

1−

(24.3-7)

where the specific heat ratio 𝜅 = 𝐶𝑝⁄𝐶𝑣 becomes 1.4 for perfect gas or 1.32 for steam.
The vapor temperature raised by compression is given by=(24.3-8)

As aforementioned for ideal adiabatic compression by positive displacement compressors, this outlet
temperature 𝑇 will usually become extremely higher than the saturation temperature.
In this example, a small amount of liquid water injected into the compressed water vapor line justbefore the entrance to the jacket is evaporated by the sensible heat above the saturation temperature of
the compressed water vapor. That is, a proper amount of make-up water serves as the make-up water
vapor.
As a result, the proper amount of water vapor in total is supplied at the saturation temperature into thejacket.
The condensation heat transfer occurs in the major part of the jacket. It is also possible to utilize thesensible heat of the condensate drain in the lower part of the jacket.
However the effect of condensate drainage are not so large. The heat capacity effect 𝑄
evaporator construction system should also be considered.

of the90

Vapor Recompression Technology

In the practical system design, the heat loss 𝑄
from the evaporator system to the surrounding
atmosphere should also be taken into consideration. The evaluation of the energy saving should bedone comparing with the heat duty of the original batch evaporator in the same operating condition.

In this case, the energy-saving rate can be considered as
𝐸𝑠 (%) = 1 −× 100 =

((

)⁄

)

× 100

(24.3-9)

where 𝜂 is the overall efficiency of the compressor.
The consumption of electric energy of the compressor should take into consideration the generation
efficiency of electric power besides its own efficiency.
[EXAMPLE 24.3-E1]
Let us consider the energy-saving problem of the Wall Wetter evaporator arranged for the VRC
method shown in Fig.24.3-3. The cylindrical evaporator tank which can contain 60 litre of aqueous
solution of non-volatile substance (ex. sugar) has the total effective heat transfer area 𝐴 = 0.34m ,the inside diameter 𝐷 = 400 mm of the cylindrical tank and the thickness 𝛿 = 4 mm of the tankwall, respectively. The construction material of the system is assumed as stainless steel (SUS304).
A screw-type dry vacuum pump belonging to a positive displacement vacuum pump is employed tocontrol the operating pressure for evaporating water from aqueous solution of non-volatile solute.
Owing to the given discharge rate of the vacuum pump, the evaporator tank pressure is raised a little
bit gradually from 148 mmHg. The suction-side and discharge-side nozzles of the vacuum pump have
60 mm and 40 mm in inside diameter, respectively.
Actually the latent heat of evaporation decreases a little bit as the pressure is raised. The energy
deficit due to the difference in latent heat of evaporation is compensated by the make-up water injected
into the compressed water vapor line. Therefore the proper but small amount of make-up water added
serves as the make-up saturated water vapor obtained by the sensible heat above the saturation
temperature. The operation pressure is kept constantly about 148 mmHg so as to prevent the solutefrom becoming solidified.
In this way, the proper rate of the saturated water vapor is supplied into the jacket.
In this example, we shall deal with the batch evaporation required for the following specification.
The aqueous solution of non-volatile solute is to be concentrated from 0.10 to 0.90 in mass fraction.
The operation is begun by supplying the 60 kg (60℃) of the feed solution with the soluteconcentration of 10 wt% into the evaporator tank at the initial condition (temperature 60℃ andpressure 148 mmHg).
Because of a positive displacement type vacuum pump, the volumetric flow rate condition is
automatically set to be 𝑉 = 0.020 m ⁄s on the suction-side of the vacuum pump. This is equal to
the rate of evaporation. The outlet pressure of the vacuum pump is also maintained as one atmosphericpressure. In the jacket maintained at 1 atm, the water vapor from the vacuum pump accompanied by
the make-up water vapor is condensed to give the released energy to the tank for evaporation of
aqueous solution.

Energy Saving Technology

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In the case of a usual batch evaporator which does not have any VRC equipment, the total thermalenergy requirement can be expressed as the sum of the latent heat 𝑄 for evaporating the solution,
the sensible heat 𝑄
for heating the evaporator construction system, and the heat loss 𝑄
throughthe outside surface of the equipment to the surrounding atmosphere:
𝑄

= 𝑄

+ 𝑄

(24.3-E1)+ 𝑄

The isentropic compression work (−𝑤) by the vacuum pump can be obtained by the following
equation:(−𝑤) =

𝑃𝑣(24.3-E2)

1−

The total efficiency of the vacuum pump is assumed as 𝜂 = 0.30 , which comes from the electric
power generation efficiency as well as the vacuum pump efficiency.
The overall heat transfer coefficient is assumed to be kept at 𝑈 =524 W⁄m K which is assumed
taking into account our experimental data.
Only for simplicity, the boiling-point rise due to the non-volatile solute is assumed to be negligibly
small in this example.
Obtain the temporal variations of energy saving rate, salt concentration, vacuum pump electricity
requirement, and make-up water.
[Solution] The rate of evaporation in the tank is expressed in the form of decrease in mass of theliquid solution and the heat input from the jacket side:(24.3-E3)=−

Here the heat input 𝑄 should include not only the condensation of the water vapor compressed by
the vacuum pump but also an additional heat input from the make-up water vapor.
That is, the sensible heat 𝑄
up water.
𝑄= 𝑉

𝐶𝑝 (𝑇 − 𝑇of the superheated vapor can be consumed for evaporation of the make)(24.3-E4)

This effect is considered to be small (of order 10% or less).
The initial conditions are expressed as
𝐿 = 𝐿 (= 60 𝑘𝑔) , 𝑇 = 𝑇 (= 60℃), and 𝑥 = 𝑥 (= 0.10) at 𝑡 = 0(24.3-E5)

The initial tank pressure is assumed to be the saturation pressure (148 mmHg) for the initial solution.
The saturation temperature of water is calculated by the Antoine equation.
𝑇=−𝐶(24.3-E6)

As the result of rough calculation, the overall heat capacity of the evaporator construction material is
assumed to be 650 kJ⁄kg ℃.
The overall heat transfer coefficient 𝑈 is assumed to be constant at 𝑈 = 524 W⁄m K.

92

Vapor Recompression Technology

As the tank-side pressure or the boiling point rises gradually by the vacuum pump, the molar latent
heat of water depending on the saturation temperature 𝑇
is evaluated by the approximate equation:
𝜆= 𝜆 + 𝜆 (𝑇where 𝑇

(24.3-E7)−𝑇 )is the saturation temperature of the solution in the tank and 𝑇

the normal boiling point.

The heat transfer area 𝐴 is allocated to the heat transfer due to the condensation and the remaining
area 𝐴 to the heat transfer from the sensible heat of the drain: 𝐴 + 𝐴 = 𝐴.
For simplicity, the vapor density in the tank is evaluated by the perfect gas law.
𝜌 = 𝑀

(24.3-E8)( )The evaporation rate 𝑉 = 𝜌 𝑉 kg⁄s is automatically controlled by the volumetric suction rate
𝑉 = 0.020 m ⁄s of the vacuum pump.
The specific heat ratio 𝜅 = 𝐶𝑝⁄𝐶𝑣 is assumed to be 1.3 for steam in this condition.
Let us consider from here the calculation results of the process simulation analysis summarized in
Figs.24.3-E1 to E5.
(1) Solute concentration, liquid solution temperature, tank pressure, and mass of liquid solution
Fig.24.3-E1 shows temporal variation of the pressure of the evaporator tank and the temperature of
the liquid solution in the tank.
It has been found that owing to the inevitable small deviation between the vapor discharging rate of
the vacuum pump employed and the evaporating rate of the WW evaporator, the liquid solution
temperature is rising gradually with the tank pressure.

60 ℃

68.3 ℃

Fig.24.3-E1 Temporal variation of pressure and solution temperature in evaporator tank.

As shown in the figure, it has been found that the process time required for concentrating thesolution from 10 to 90wt% is 4.75 h.

Energy Saving Technology

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90 wt%

10 wt%

4.75
Fig.24.3-E2 Temporal variation of mass concentration and liquid solution volume.

The rate of energy saving is plotted taking into account the vacuum pump efficiency and the electric
power generation efficiency in Fig.24.3-E3.

97.8 %

2.68 kW2.86 kW

Fig.24.3-E3 Temporal variation of the energy-saving rate, the electric power requirement, and theenergy fraction consumed for evaporation.

Under an assumption of 30% as the total efficiency of the vacuum pump, it has been found that thevery large rate of energy saving by the VRC method is attained in between 58.9 and 67.7% for 4.75 h.

94

Vapor Recompression Technology

Fig.24.3-E4 Temporal variations of liquid solution in the tank, operation pressure, and thetemperature difference as the driving force.

The condensation temperature in the jacket is kept at 100℃ at 1 atm.
The electric power required for the vacuum pump varies gradually from 2.68 to 2.86 kW during theprocess time (4.75 h).
The fraction of thermal energy consumed for evaporation is constantly maintained at 98% during thewhole operation time. This implies that the remaining fraction comes from heat loss and heatrequirement for the evaporator construction system.
As the operating pressure of the tank side is gradually raised by the vacuum pump, the boilingtemperature of the liquid solution is very slowly raised.
It should be kept in mind that a small rate of makeup steam 0.846 kg/h is required for the energy
deficit of evaporation latent heat.

Fig.24.3-E5 Temporal variations of power requirement with available thermal energy

Energy Saving Technology

95

It is practically important to consider the fraction of the thermal energy used for evaporation of theliquid solution in the total thermal energy transferred from the jacket to the tank side.
Although no elevation of boiling point due to the concentrated solute in this condition is assumed, therate of evaporation and the evaporation temperature are a little bit rising gradually with the operationpressure rise.
The theoretical work for isentropic compression of vapor by a screw-type vacuum pump varies slowly
from 7.0 kW through 6.2 kW during the evaporation time.
The detailed explanation of calculation results is omitted here but the heat loss and the injected
makeup water amount to 0.08% and 0.41%, respectively for the overall energy requirement.
The fraction of the energy saving rate amounts to 54.7 % if the efficiency of the vacuum pump isassumed to be 30%.
The contribution to the evaporation rate of the concentrated solution by the condensate drain in thejacket becomes 3.1% only.

You can understand that this is a good example for energy-saving of the batch evaporator by using aheat pump idea.
As the second law of thermodynamic law states, heat energy naturally transfers from warmer places
to colder spaces. However, a heat pump can reverse this process, by absorbing heat from a cold spaceand releasing it to a warmer place. Therefore heat is not conserved in this process and requires someamount of energy such as electricity from the outside. In heating, ventilation and air conditioning(HVAC) systems, the term heat pump usually refers to vapor-compression refrigeration devices
optimized for high efficiency in both directions of thermal energy transfer. These heat pumps can bereversible, and work in either direction to provide heating or cooling to the internal space. Most of theenergy for heating comes from the external environment, only a fraction of which comes from electricity(or some other high-grade energy source required to run a compressor). In electrically-powered heatpumps, the heat transferred can be three or four times larger than the electrical power consumed, giving
the system a coefficient of performance (COP) of 3 or 4, as opposed to a COP of 1 for a conventional
electrical resistance heater, in which all heat is produced from input electrical energy.
Heat pumps use a refrigerant as an intermediate fluid to absorb heat where it vaporizes, in theevaporator, and then to release heat where the refrigerant condenses, in the condenser. The refrigerant
flows through insulated pipes between the evaporator and the condenser, allowing for efficient thermalenergy transfer at relatively long distances.

24.4 Energy Saving by Heat Integration in a Distillation System
24.4-1 Principle of internally heat integrated distillation column (HIDiC)
A configuration of internal heat integration method originally proposed by Mah et al.1) andFitzmorris and Mar2) for the continuous distillation system of a single column is given by Fig.24.4-1.

96

Energy Saving by Heat Integration

The concentric double-tubed HIDiC system consists of an inner column serving as the rectifying
section and a concentric outer column serving as the stripping section. The vapor from the top of the
stripping section is compressed into the bottom of the rectifying section by a compressor.
The internal heat integration takes place as the transfer of heat from the high-pressure and hightemperature rectifying section to the low-pressure and low-temperature stripping section. Therefore theliquid flow rate in the stripping section is decreased in the downward direction by partial vaporizationof the internal reflux liquid flowing down due to the heat input from the rectifying section. This effectresults in the big decrease in the heat duty of the bottom reboiler. At the same time, the vapor flow ratein the rectifying section is decreased in the upward direction by partial condensation of the internal
vapor flowing upward due to the heat transfer to the stripping section. This effect results in the big
decrease in the cooling duty of the overhead condenser. This suggests the great reduction of theexternal reflux.
One advantage of HIDiC is that HIDiC can be operated at significantly low compression ratio(pressure ratio of the rectifying to stripping sections) as compared to a conventional heat pump systemsuch as the vapor recompression. In addition, if the thermal design is improved, the HIDiC column
could be made much lower in column height than normal columns.3)

1)2)

Mah, R. S. H., Nicholas, Jr., J. J., Wodnik, R. B.: Distillation with Secondary Reflux and Vaporization. A Comparative Evaluation,
AIChE Journal, Vol.23, 6521-658 (1977).
Fitzmorris, R. E. and Mar, R. S. H.: Improving Distillation Column Design Using Thermodynamic Availability Analysis, AIChE
Journal, Vol.26, 265-273 (1980).

Fig. 24.4-1 Basic configuration of a single column HIDiC system

The partial condensation of the vapor flowing upward occurs at temperature 𝑇 at the jth stage ofthe rectifying section, whereas the partial vaporization of the liquid flowing downward occurs attemperature 𝑇 at the ith stage of the stripping section.
The overall heat transfer coefficient can be defined as shown in Fig.24.4-2. For convenience, thefictitious heat transfer area 𝐴 is defined as the surface area of the partition wall corresponding to one
tray-to-tray spacing. However it is very difficult to estimate the area of the partition wall that causes
vapor condensation in the rectifying section and liquid vaporization in the stripping section. In
addition, usually for effective effect of heat transfer, the stages of the rectifying section and thestripping section are piled up in stagger arrangement: the condensing area on the jth stage of theEnergy Saving Technology

97

rectifying section is located at the same height with the evaporating bubbling layer on the ith stage ofthe stripping section.
3.

De Rijke, A., Development of a concentric internally heat integrated distillation column (HIDiC), Ph.D. Dissertation, (2007)

Fig.24.4-2 Internal heat exchange between the rectifying and stripping sections in a HIDiC trayeddistillation column

24.4-2 Operation modes of HIDiC
The internal heat transfer is controlled by the pressure ratio 𝑃 ⁄𝑃 between the rectifying andstripping sections. In relation to the boiling points of volatile components for distillation, as shown by
Fig.24.4-3, there are two operation modes possible: (1) Pressurized mode and (2) Depressurized mode.
(1) Pressurized mode (𝑃 > 1 atm & 𝑃 = 1 atm) is operated by a compressor while (2)Depressurized mode (𝑃 = 1 atm & 𝑃 < 1 atm) is operated by a dry vacuum pump.
In particular, a fouling problem may arise in the stripping section as well as in the bottom reboiler.
Therefore if it is necessary to lower the boiling points, the depressurized mode is recommendable.

Fig.24.4-3 Two operation modes for single column HIDiC systems

98

Energy Saving by Heat Integration

24.4-3 Heat transfer characteristics of a trayed-column HIDiC system
For reference, some experimental data of internal heat transfer characteristics observed by a single
concentric HIDiC column are introduced here.
Fig.24.4-4 shows the experimental distillation column consisting of concentric double-tube trayedcolumn.
The column structure with dimensions was given as follows:
Inner column diameter = 508 mm OD, outer column diameter = 800 mm ID
Tray: Lift Tray
The tray-to-tray spacing was 400 mm
Heat transfer area 3.97 𝑚

Fig.24.4-4 Experimental apparatus obtaining HIDiC heat transfer data

As can be seen from Fig.24.4-2, because of very complicated phenomena occurring on both sides, it
is not easy to determine the internal heat transfer from the rectifying section to the stripping section.
Only for simplicity, the overall heat transfer coefficient is defined as(24.4-1)
𝑈 =where the heat transfer area 𝐴
one plate-to-plate spacing 𝐿 :𝐴 = 𝜋𝐷 𝐿

is defined as the cylindrical surface area of the inner tube given by(24.4-2)

The distillation experiment for the pressurized and depressurized modes was conducted by using two
kinds of binary systems: (1) benzene-toluene system and (2) ethanol-water system.
Both experiments were conducted under the following two conditions: (1) 𝑃 > 1 atm & 𝑃 = 1 atm
for the pressurized mode and (2) 𝑃 = 1 atm & 𝑃 < 1 atm for the depressurized mode.

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99

The experimental data of internal heat transfer are shown for benzene-toluene binary system in Figs.
24.4-5 & 24.4-6 and for ethanol-water system in Figs.24.4-7& 24.4-8, respectively.
The temperature differences between the rectifying to the stripping section averaged over thecolumn height increases linearly with the pressure ratio. This must be due to the boiling point rise withpressure.
The overall heat transfer coefficients averaged over the column height slightly decrease with thepressure ratio. This can be considered to be due to the change in the physical properties such as
viscosity and thermal conductivity of the two countercurrent flows in the rectifying and stripping
sections.
The heat flux between the rectifying and stripping sections also increases almost linearly with thepressure ratio 𝑃 ⁄𝑃 mainly owing to the boiling point rise with the increasing pressure.

Fig.24.4-5 Variation of experimental temperature difference with pressure ratio for benzenetoluene system

Fig.24.4-6 Variation of overall heat transfer coefficient with pressure ratio for benzene-toluene
system100

Energy Saving by Heat Integration

Fig.24.4-7 Variation of experimental temperature difference with pressure ratio for ethanol-water
system

Fig.24.4-8 Variation of overall heat transfer coefficient with pressure ratio for ethanol-water
system

It can be considered that the overall heat transfer coefficients are larger for ethanol-water systemthan for benzene-toluene system.

24.5

Application of HIDiC Technology

24.5-1 HIDiC system applied for a bioethanol distillation process1)
A concentric HIDiC system of a single trayed column was developed for dehydration process in the
production process of ethanol from non-food biomass. It is very serious problem that the fermented
mash liquid (6 wt% ethanol) to be fed still contains various fouling substances such as proteins andcarbohydrates even after filtration. Therefore in order to prevent the stripping section and bottom
reboiler from becoming fouled, this bench plant places the stripping section inside as the inner column
and the rectifying section outside as the outer column. In addition to that, the operating pressure of the
stripping section is lowered to 225 mmHg by a dry vacuum pump while the rectifying section isEnergy Saving Technology

101

operated at normal pressure. If the stripping section temperature is raised above 100 ℃, the depositionof solidified substances by Maillard reaction occurs on the inside heat transfer surface of the stripping
section. For easy cleaning and maintenance, an actual double-tube HIDiC plant should have thestripping section placed inside as the inner column.
From the same reason, the steam distillation supplying live steam should be used instead of a bottom
reboiler. Taking into account that non-volatile substances do not usually invade the rectifying section,
which is equipped with structured packings.
On the basis of the process simulation results obtained by a commercial simulator package
based on an algorithm of equilibrium-stage model, a HIDiC bench plant was designed for producing an
overhead product of > 90 wt% ethanol from 50 kg/h of fermented mash (5 wt % ethanol).
The flow sheet is shown in Fig.24.5-1.

Fig.24.5-1 Flow sheet of the HIDiC bench plant

The rectifying section is kept at normal pressure. A dry vacuum pump is necessary to keep thepressure of the stripping section lower than 250 mmHg. The stripping section is equipped with dualflow Lift Trays inside with no downcomer while the rectifying section is filled with structured packing.
To avoid fouling, the bottom of the stripping section has a nozzle for live steam supply instead of abottom reboiler.

24.5-2 Test operation of a HIDiC bench plantThis bench plant successfully recovered 95.4 % of the ethanol in the feed, gave an ethanol
concentration in the overhead product of 94.4 wt%, and left only 0.024 wt% ethanol in the bottom of
the stripping section. Owing to the azeotropic solution, the ethanol concentration of the overhead
product amounted to 94.4 wt% a little bit lower than the azeotropic concentration 95.5 wt%.

102

Application of HIDiC Technology

Fig.24.5-2 HIDiC distillation bench plant for bio-ethanol dehydration1)

Kataoka, K/., Noda, H., Mukaida, T., Nishimura, G., and Yamaji, H.: Boost to Bioethanol Distillation by Internal Heat-Integrated
Distillation Column (HIDiC), Advanced Chemical Engineering Research, Vol.3, 48-57 (2014).

The depressurizing mode of operation should be used to prevent the stripping section from becoming
fouled due to the Maillard reaction. In addition, the bottom reboiler also has the same problem.
The fouling test shown in Fig. 24.5-3 suggests that when the stripping section is operated at normalpressure (0.1 MPa), the boiling point goes beyond 100℃ and the fermented mash is scorched due to
Maillard reaction. As can be seen from the right photo, the Maillard reaction can be suppressed if
operated at 235 mmHg.

Energy Saving Technology

103

Fig.24.5-3 Fouling test examining the fermented mash scorching due to Maillard reaction

A live steam is supplied into the bottom of the stripping section instead of a bottom reboiler for a
prevention against fouling. Fortunately, there was no need for supplying live steam under the conditionof optimal operation for energy saving. This means that the total energy consumed by the HIDiC
distillation plant amounts to the work requirement of the dry vacuum pump only.
Fig.24.5-4 is a test operation result of the HIDiC bench plant designed for dehydration process of thefermented bio-ethanol liquid mash which is filtrated in advance.
The energy rate 2.86 MJ/L- ethanol consumed in the dehydration process for producing 1 L ethanol
can be considered to be sufficiently small in comparison with the thermal energy for combustion of 1 Lethanol 23.4 MJ/L-ethanol.

Fig.24.5-4 Test operation result of HIDiC bench plant for dehydration of fermented bio-ethanol mash

104

Nomenclature

It is necessary to take into account the practical electric generation efficiency in power plant in orderto evaluate the energy requirement of the compressor.
If we compare this HIDiC system in energy requirement with a conventional dehydration system inthe same condition, we obtain by calculation as
𝐸 − 𝐸
× 100 = 68 %𝐸 =𝐸
[PROBLEM 24.5-1]
Explain qualitatively in the above HIDiC bench plant how the energy consumption rate changes if theethanol concentration 𝑧 of the fermented bio-ethanol mash becomes higher than 5 wt%.

Nomenclature𝐴
A, B,C
𝐶
𝐶𝑝
𝐷
𝐸
𝐹
𝐻 , 𝐻
ℎ
L
Lpp
𝑚
𝑃
Q
Q’
r
R
𝑅𝑒
𝑆𝑐
𝑇
t
U
v
−𝑊
𝑥 ,𝑦
𝛿
𝜆
𝜅
𝜇

heat transfer area, [m2]Antoine constantcoefficient of performance, [ - ]heat capacity, [J/kmol K]
overhead product, [kmol/s]
energy saving rate, [%]feed rate, [kmol/s]
specific enthalpy, [J/kmol]
heat transfer coefficient, [W/m2K]
mass of solution, [kg]tray-to-tray spacing for trayed column, [m]slope of equilibrium curve, 𝑑𝑦 ⁄𝑑𝑥 ,pressure ratio, [ - ]heat input, [J] or [W]surplus energy by heat pump, [W]reflux ratio, [ - ]resistance to heat flow, [m2K/W]
Reynolds number, [ - ]Schmidt number, [ - ]temperature, [K]time, [s]overall heat transfer coefficient, [W/m2K]
specific volume, [m3/kg] or [m3/kmol]
theoretical work of compressor, [W]mole fraction of more-volatile component A, liquid- and vapor-phase, [ - ]film thicknesses for heat transfer, [m]latent heat of evaporation, [J/kg]
thermal conductivity. [w/m K]
viscosity, [kg/m s]

Subscripts
Bbottomccondenser or condensationDoverheadev
evaporation
F
feed
fl
vapor filmooutside surfacerrectifying sec. or reboilerssolstripping sec.
solutionChapter 25 Transfer Model for Design of Packed Distillation Columns

105

CHAPTER 25
TRANSFER MODEL FOR DESIGNOF PACKED DISTILLATION COLUMNS
We know that distillation is by far the most applied but the most energy-consuming separation
technology. It is of great significance to reconsider the practical design of a real packed distillation
column from a viewpoint of the characteristics of simultaneous energy and mass transfer.
There are two modeling concepts available for analyzing the packed column distillation1): the idealstage (or equilibrium-stage) model and the rate-based transfer model. As described in Chapter 11 ofVolume 1, the ideal-stage model, e.g. the McCabe-Thiele model is very simple and convenient to apply
to various column design calculations. However the ideal-stage model still remains to be extended to thereal interphase mass transfer in packed column distillation processes. A generalized model for distillation
efficiency has not yet been constructed.
On the other hand, the rate-based model2, 3, 4, 5) based on the very strict nonequilibrium thermodynamic
concept has to make the theoretical approach solving the differential transport equations, which rely
heavily on empirical correlation data of heat and mass transfer coefficients as well as the counter-current
two-phase flow conditions for various boundary conditions of the differential governing equations.
Owing to the complicatedness of packed column distillation phenomena, the rate-based model is not yet
so fruitful for analyzing the practical packed column distillation phenomena. In this chapter, we study
the modeling of a real packed distillation column by applying the control volume method from aviewpoint of interphase mass and enthalpy transfer. In other words, we would like to construct the designconcept for a real packed distillation column based not only on the equilibrium model but also on theinterphase mass and enthalpy transfer model.

25.1 Modeling Concept by Control Volume Method1,2)
25.1-1 Definition of control volume
The control volume method1) is a very convenient tool for bridging the gap between the ideal column
and the real column. We already studied it a little bit in Chapter 12.2 ~ 3 of PART II.
The control volume can be defined taking into consideration the interphase mass and enthalpy transfer
occurring within it. As in Fig.25.1-1, the equilibrium stage of an ideal column is converted into thecorresponding control volume of a real column where local HETP (i.e. the reciprocal of distillation
efficiency) can be utilized as the control volume height.
You can review the definition of HETP for packed distillation columns in Chapter 12.2-2.

106Modeling Concept by Control Volume Method

Fig.25.1-1 jth control volume corresponding to jth equilibrium stage

Figure 25.1-2 gives how to define a control volume taking into account the composition profiles within
the control volume. The bulk and interface compositions should vary in the vertical direction within thecontrol volume. From the control volume definition, the bulk compositions of the vapor and liquid
𝑦 and 𝑥 , respectively leaving the control volume should have the equilibrium relationship: 𝑦 =𝐾 𝑥 because the jth control volume also corresponds to jth equilibrium stage having local HETP as
the height.
The more-volatile component A gives rise to the mass transfer from liquid-phase to vapor-phase withthe concentration profile of 𝑦 ∗ > 𝑦 = 𝑚 𝑥 > 𝑦 . The simulation analysis based on the equilibrium
stage model can give the interfacial composition 𝑦 , 𝑥
and temperature 𝑇 of jth control volume.
In addition, as in Fig.25.1-2, we can also assume that the vapor bulk concentrations at the top and bottomof the jth control volume become respectively:𝑦= 𝑦 = 𝑦
and
𝑦
= 𝑦
= 𝑦
Therefore at the central height 𝑧 = (ℎ + ℎ )⁄2 , the vapor bulk concentration 𝑦=⁄2 and the vapor-phase driving force of component A 𝑦 − 𝑦
𝑦 + 𝑦are given for masstransfer analysis. Similarly the liquid-phase driving force becomes 𝑥− 𝑥 .
Then the film coefficient of mass transfer 𝑘 𝑎 can be calculated within the jth control volume.

Fig.25.1-2 Definition of control volume and HETP for a packed distillation column

Transfer Model for Design of Packed Distillation Columns

10725.1-2 Allocation of theoretical profiles of temperature and composition
The theoretical profiles pf temperature and composition are obtained by the process simulation
calculation (see Chapter 25.1-4) based on the equilibrium-stage model.
Considering similarly to the McCabe-Thiele method, as shown in Fig.25.1-3, the temperatures of thevapor phase, liquid phase, and interface 𝑇 , 𝑇 , 𝑇 calculated on jth equilibrium stage are assigned
step-by-step by the control volume method.

Fig.25.1-3 Allocation of vertical variation in temperature with vapor and liquid compositions

For example, at the top and bottom of the jth control volume corresponding to the jth equilibrium stage,
the vapor and liquid bulk concentrations can be considered as follows:𝑦= 𝑦 = 𝑦
𝑦
= 𝑦
= 𝑦
𝑥
= 𝑥
= 𝑥
𝑥
= 𝑥 = 𝑥1)
2)

Kataoka, K., Transport Process Chemical Engineering (2021), https://doi.org/10.24546/90008260
Nishimura, G., Kataoka, K., Noda, H., and Ohmura, N., Control Volume Approach for Modeling the Interphase Mass Transfer in a
Packed Column Distillation, J. Adv. Chem. Eng., Vol.11, Issue 2, 1-14 (2021)

25.1-3 Definition of local film coefficients of mass and enthalpy transfer
As can be seen from the definition of jth control volume in Fig.25.1-4, the vapor-phase film coefficientsof enthalpy and more-volatile component A accompanied with the effective vapor-liquid contacting
interfacial area can be expressed as follows:
𝐸 = 𝑉 𝐻
− 𝑉 𝐻 = (ℎ 𝑎)ℎ(25.1-1)
𝑁

= 𝑉𝑦

− 𝑉

𝑦

= 𝑘 𝑎

ℎ

(25.1-2)

These equations suggest that as long as the information of a generalized correlation of HETP is available,

108

Modeling Concept by Control Volume Method

both the film coefficients can be determined with the calculation results of temperature and vapor
composition based on the equilibrium-stage model.

Fig.25.1-4 Definition of vapor-phase film coefficients of mass and enthalpy transfer

This indicates that the evaluation of local film coefficients of mass and enthalpy transfer needs local
variation of HETPs for defining a control volume.
25.1-4 Process simulation based on equilibrium-stage model
The practical evaluation of HETP as the reciprocal of distillation efficiency is usually conducted by
the distillation experiment under the total-reflux condition.
For determining local HETPs, the real column distillation experiment is collaborated with thefollowing process simulation analysis based on the ideal-stage model. Its algorithm is consisted with thefollowing set of equations.
Mass and enthalpy balance around jth equilibrium stage under the total-reflux condition
𝑀 = 𝑉𝑦 , + 𝐿 𝑥 , − 𝑉 𝑦 , − 𝐿 𝑥 , = 0(25.1-3)
𝐸 = 𝑉𝐻 , + 𝐿 𝐻 , − 𝑉 𝐻 ,− 𝐿 𝐻,=0(25.1-4)
Overhead condenser (stage 1)
𝑀 = 𝐿 𝑥 , − 𝑉𝑦 , =0(25.1-5)
Cooling duty: 𝑄 = 𝐿 𝐻 − 𝑉 𝐻(25.1-6)
Bottom reboiler (stage N)
𝑀 = 𝑉 𝑦 , − 𝐿
𝑥 , =0(25.1-7)
Heating duty:
𝑄 = 𝑉 𝐻 − 𝐿
𝐻(25.1-8)
Equilibrium relation
𝐾 , = 𝑝 , (𝑇 )⁄𝑃(25.1-9)
Vapor-liquid equilibrium condition
𝑆, = 𝐾, 𝑥 , − 𝑦 , = 0(25.1-10)
Mole fraction condition of liquid composition𝑊 = ∑𝑥, −1=0(25.1-11)
Mole fraction of vapor composition𝑊 = ∑𝑦, −1=0(25.1-12)

Transfer Model for Design of Packed Distillation Columns

109

In this calculation, the NRTL model is used to evaluate activity coefficients and physical propertiesof vapor and liquid of a ternary mixture (𝑚 = 3) such as methanol-ethanol-iso-propanol system. The
specification and the operating conditions used for the process simulation are given from theexperimental condition and the distillation performance data obtained.
Usually a commercial process simulator package is employed. This process simulation analysis will
give local values of temperature, enthalpy, flowrate and concentration of vapor and liquid at eachequilibrium stage.
1)

Nishimura, G., Kataoka, K., Noda, H., and Ohmura, N., Proc. 30th European Symposium on Computer Aided Process Engineering(ESCAPE30), 1 – 6, May 24-27, Italy (2020)

25.2 Experiment of a Packed Column Distillation
25.2-1 Test column and experimental method1)
A generalized data base of HETPs or distillation efficiency of packed distillation columns has not yet
been constructed. The distillation experiment under the total-reflux condition should be usually enforced
with a test column for the practical design of a packed distillation column.
The experimental method and result of ternary system1) are introduced here to understand the new designconcept. The packed column distillation data and calculated result introduced in this chapter come fromthe experimental and theoretical investigation of total-reflux distillation for a ternary system of
methanol-ethanol-iso-propanol1).
Flow channel of structured packing (see Chapter 7.4-3)
In a packed column distillation, the liquid flows down the channel in counterflow to the vapor andspreads into thin liquid film. Therefore the vapor velocity is much larger than the liquid velocity.
It is necessary to define an equivalent diameter of a flow channel. For example, the hydraulicequivalent diameter of a structured packing of triangular cross section is defined as(triangular cross section)𝑑 = 4 flow cross-sectional area/wetted perimeter = 2𝐵 ℎ⁄(2𝑆 + 𝐵)
(25.2-1)(diamond cross section)𝑑 = 𝐵 ℎ⁄𝑆
(25.2-2)
For the structured packing having the corrugated mixed structure of triangular and diamond crosssections, the following arithmetic mean can be adopted2):
𝑑 = 𝐵 ℎ⁄[1⁄(𝐵 + 2𝑆) + 1⁄2𝑆](25.2-3)
where S = channel side length, h = crimp height, and B = channel base length1)
2)

Nishimura, G., Kataoka, K., Noda, H., and Ohmura, N., Control Volume Approach for Modeling the Interphase Mass Transfer in a
Packed Column Distillation, J. Adv. Chem. Eng., Vol.11, Issue 2, 1-14 (2021)
Bravo, J. L., Rocha, J. A., and Fair, J. R., Mass Transfer in Gauze Packings, Hydrocarbon Processing, Jan. 91-95 (1985)

110

Experiment of a Packed Column Distillation

Fig.25.2-1 Experimental packed distillation column for determining local variation of HETP

Figure 25.2-1 shows an experimental packed distillation column equipped with three beds of wiremesh corrugated structured packing stacked up in series. Those packing sections are 1,900 mm, 1,900mm, and 1,520 mm in vertical height. The mesh-type corrugated structured packing has the hydraulicequivalent diameter 𝑑 = 0.00625 mm and the specific surface area 𝑎 = 500 m ⁄m .
The two empty spaces between those three beds are installed with a funnel-type liquid collector and achannel-type liquid distributor in order to avoid maldistribution of liquid stream. In the empty space
above the upper edge of the top packing section, the channel-type distributor is also equipped for the
reflux liquid.
The ternary system of methanol, ethanol and iso-propanol was selected as an ideal mixture. The key
component methanol (the most volatile component A) was adjusted high in concentration compared withthe remaining two components (component B: ethanol and C: iso-propanol). The performance dataobtained by the total-reflux distillation experiment were used for the specification of the processsimulation analysis. The reason why the total-reflux condition was adopted is that in a usual way for thepractical design of packed columns, the HETP measurement makes it a rule to conduct the total-reflux
distillation.
The primary purpose of the distillation experiment is to observe local variation of HETPs under the
total-reflux condition. For that purpose, nine sets of sheathed thermocouples (wire dia.=1 mm OD) wereembedded so as to locate their hot junctions just on the column centerline at an equal interval (every 475
mm in the top and middle beds and every 380 mm in the bottom bed). It was assumed that thosethermocouples could measure local temperatures of internal reflux liquid because of its heat capacity
much larger than the flowing up vapor. The distillation experiment was performed at normal pressure
under the total-reflux conditions accompanied with 4 variables of F-factor (𝐹 = 0.5, 1.0, 1.5, and 2.0),
i.e. by changing the heating duty of the bottom reboiler. The representative F-factors based on the
superficial vapor velocity were defined at the top stage of the experimental column.

Transfer Model for Design of Packed Distillation Columns

111

Fig.25.2-2 indicates the vertical variations in the equilibrium coefficient for each component andliquid temperature calculated by the process simulator when 𝐹 = 2.0. All the data tend to decrease
hyperbolically toward the column top.

Fig.25.2-2 Vertical variations of equilibrium coefficients of ternary system and liquid temperature

25.2-2 Determination of local HETPs
It is a little bit difficult to measure local values of HETP.1)
Fig.25.2-3 indicates the liquid temperature measured by the embedded thermocouples compared withthe liquid temperature calculated by the process simulation based on the equilibrium-stage model.
Since the locations of the embedded thermocouples in the test column are known, local values of
HETPs can be evaluated comparing step-by-step the theoretical temperature profiles 𝑇
at theoretical stage 𝑛 calculated for an ideal column with the experimental temperature
profiles 𝑇 at a vertical position 𝑍 observed by the embedded thermocouples in a realcolumn. The local HETP of each control volume is determined by the following equation:ℎ=× (𝑍 − 𝑍)(25.2-1)

Fig.25.2-3 Comparison of the temperatures observed by thermocouples with the temperatures
calculated by the process simulation based on the equilibrium stage model for determination of local
HETPs

112

Experiment of a Packed Column Distillation

It is very important to obtain a generalized correlation model of HETPs for the practical design ofpacked distillation columns.

Local variation of HETPs

Fig.25.2-4 shows the experimental result of HETP obtained for the ternary system, where theobserved HETPs are plotted against the dimensionless vertical height divided by equivalent diameterof test structured packing. In this figure, each run of the experiment is specified by the F-factor definedas the column top vapor flow rate.

Local values of HETP as the distillation efficiency vary largely in the vertical direction. Actually the
local HETPs in a real column tend to give very large values in the upper region of the packing section,
especially near the column top in the rectifying section. This suggests the necessity of observing local
variation in distillation efficiency, i.e. HETP in a real packed column.
Therefore the practical design of real packed distillation columns should be constructed taking into
account the local variation of distillation efficiency.

Fig.25.2-4 Vertical variation of local HETPs for ternary system

It has been found that the distillation efficiency (i.e., the reciprocal of HETP) remarkably
goes down vertically upward in the packing section. This result suggests that it is possible
to analyze local variation of mass and enthalpy transfer in a real packed column byutilizing those HETP data as the height of each control volume. The figure indicates that
we should be careful of very low distillation efficiency near the column top in the rectifying
section where the HETP becomes very large.
Variation of HETPs with vapor velocity
Judging from the correlation of Reynolds number dependency on interphase mass transfer resistance,
we can conjecture that the HETPs usually decrease with increasing F-factor. Therefore for the correlation
analysis of HETPs, we should consider that the HETPs are related not only with the distribution function
in vertical variation but also with the F-factor defined at the leading edge (i.e. bottom) of the packing
section. Here we regard the bottom edge of the packing section as the leading edge of boundary layer
flow along a flat plate.
Our distillation experiment of the ternary system of methanol, ethanol, and iso-propanol gives theexperimental HETPs plotted against local F-factor in Fig.25.2-5, which indicates the local variation of
HETPs from bottom to top of the packing section. If these data are divided by a distribution function,

Transfer Model for Design of Packed Distillation Columns

113

they will converge on the correlation curve obtained at the leading edge. For an example, thoseconverged HETPs denoted by hmin (EXP) can be correlated with the F-factor at the leading edge by thefollowing equation:.
(25.2-2)ℎ=.
.
where 𝐹 is the F-factor defined at the leading edge (bottom ) of the packing section.
If the ℎis calculated by the above equation, local values of HETP can be obtained by using thefollowing distribution function:𝑓(𝜁) = 1 + 0.7 (𝜁 + 4) /(25.2-3)
where 𝜁 = 𝑧⁄𝑑 is vertical distance from the bottom of the packing section. The number 4 in theparenthesis indicates the deviation of a virtual origin of the vertical development from the leading edge.

Fig.25.2-5 Experimental data of local HETPs and the HETP data converged on the correlation curve
defined at bottom of packing section

As a result, local HETPs can be estimated by the following correlation equation:ℎ= ℎ× 𝑓(𝜁)(25.2-4)
If the F-factor is given at the leading edge of the packing section, local values of HETP
can be calculated by this equation. Then by means of the control volume method, we canestimate local values of vapor rate and composition, interphase mass and enthalpy
transfer coefficients, and vapor and liquid temperatures and enthalpies necessary for thepractical design of packed distillation columns.
If the local HETPs estimated by Eq.25.2-4 are plotted against the dimensionless vertical distance fromthe bottom of the packing section in Fig.25.2-6, it has been found that the vertical variation of local
HETPs is similar to the streamwise variation of laminar boundary layer along a flat plate.
For this kind of conceptual consideration, we had better review the boundary layer theory in Chapter
18 (see Chapter 18.2 from page 212).
If the HETP correlation curve is expressed similarly to the boundary layer flow type functional form,
the following equation can be obtained:

114

Experiment of a Packed Column Distillationℎ= 0.05 + 0 .02 𝜁 ⁄(25.2-5)This equation can be derived from streamwise development of the boundary layer thickness expressedby Eq.18.2-21 of Chapter 18.2 in PART II.
The mass transfer in the boundary layer flow indicates the similar functional form:
𝑆ℎ = 0.343 𝑅𝑒 ⁄ 𝑆𝑐 ⁄(18.2-38)⁄
and 𝑅𝑒 =, 1⁄𝑘 becomes proportional to 𝑥 .
Here since 𝑆ℎ =The Reynolds number is called “length Reynolds number” which means the streamwise distance fromthe leading edge of the flat plate. (see Chapter 18.2)
The exponent 1/2 indicates the laminar boundary layer.
This kind of conceptual consideration is also done in Chapter 12.3 of PART II. (see on Fig.12.3-3 on
p.150).

Fig.25.2-6 Boundary layer plots of local HETPs in comparison with interphase mass transfer
resistance of the boundary layer flow along a flat plate.

The comparison of the packed column distillation with the boundary layer flow along aflat plate is shown in Fig.25.2-7.
The bulk vapor-phase velocity (superficial velocity) is kept almost constant similarly tothe free stream uniform velocity of boundary layer flow along a heated flat plate. However
as distinct from the free stream temperature of the boundary layer flow, the bulk vaporphase temperature and composition vary in the vertical direction.
For simplified modeling the packed distillation column, we consider by separating thecounter-current two-phase flow in the packing section into the vapor-phase section and theliquid-phase section. It is marvelously interesting that the inter-phase mass transfer ofthe simplified counter-current flow in the packing section is similar in the streamwise
development to the laminar boundary layer flow along a flat plate.

1)

Lockett, M. J., Easily predict structured-packing HETP, Chem. Eng. Prog., Jan., 60-66 (1998)

Transfer Model for Design of Packed Distillation Columns

115

Fig.25.2-7 Comparison of packed column distillation with boundary layer flow along a flat plate.

Total height of packing section for practical design of a real packed distillation column
As indicated in Fig.25.2-8, if a generalized form of the HETP correlations is given
separately in the rectifying and stripping sections, we can calculate the total height of thepacked distillation column as
𝑍 = ∑

ℎ

+ ∑

ℎ

(25.2-2)

Fig.25.2-8 Calculation of total height of packing section in a real packed distillation column

116

Correlation of Mass and Enthalpy Transfer

For a continuous distillation process feeding the liquid mixture, the rate of the internal
reflux liquid is different between the rectifying and stripping sections. Therefore, for thepractical design of a real packed column operated by continuous distillation, we will haveto evaluate the HETPs separately in the rectifying and stripping sections, respectively.

25.3 Correlation of Mass and Enthalpy Transfer25.3-1 Local variation of mass and enthalpy transfer
Fig.25.3-1 shows the vertical variation in molar fluxes of the three components. For convenience, thedistillation experiment of this study was conducted using the following mass balance based on thecondition of equimolar counter-diffusion:
𝑁 = 𝑁 + 𝑁
(25.3-1)
In this condition, the two components B and C transfer in a direction opposite to component A.
In the experimental range of three components (𝑥 ≫ 𝑥 > 𝑥 ) , the ethanol and iso-propanolcomponents transfer together from the vapor bulk toward the interface for condensation whereas themethanol component vaporized at the position near the interface of the liquid phase transfers in the
opposite direction in the vapor stream.

Fig.25.3-1 Vertical variation of molar fluxes of methanol, ethanol, and iso-propanol. (F = 2.0)

Fig.25.3-2 shows the overall, vapor-phase film, and liquid-phase film coefficient evaluated by thecontrol volume analysis. This suggests that these vertical distributions can be expressed in the hyperbolic
function form.

Transfer Model for Design of Packed Distillation Columns

117

Fig.25.3-2 Vertical distribution of the overall, vapor-film, and liquid-film volumetric mass transfer
coefficients (𝑭 = 𝟐. 𝟎)

Similarly to the correlation analysis of local HETPs in Chapter 25.2-2, this suggests that it isnecessary to construct the correlative model dividing the mass and enthalpy transfer data into the two
following groups: Reynolds number dependency and vertical variation.
There are not so many experimental investigations on mass transfer in packed distillation columns.1,2)
1)
2)

Bravo, J. L., Rocha, J. A., and Fair, J. R., Mass transfer in gauze packings, Hydrocarb. Process. Jan. 91-95 (1985)
Kosuge, H., Matsudaira, J., and Asano, K., Ternary mass transfer in packed distillation column, J. Chem. Eng. Jaspan, Vol.24, 455460 (1991)

25.3-2 Generalization of mass and enthalpy transfer correlations
We can conjecture that distillation process should make progress under the analogy between
simultaneous mass and enthalpy transfers.
The following dimensionless groups are defined as
𝑁𝑢 =
𝑆ℎ =
𝑅𝑒 =

(

)

(

and
)

(

)

𝑁𝑢 =

(

)

and

𝑆ℎ =

and

𝑅𝑒 =

(25.3-2)()(

Here the superficial velocity difference 𝑢
approximated as 𝑢 because 𝑢 ≫ 𝑢 .

(25.3-3))− 𝑢(25.3-4)
between vapor and liquid streams can be

In the interphase transfer of this process, the vapor-phase film exerts the rate-controlling contribution to
the interphase enthalpy and mass transfer.
Therefore the generalized correlation of transfer coefficients will be described focusing the mass andenthalpy transfer through the vapor-phase film.
If you desire, the liquid-phase mass and enthalpy transfer could be treated in a similar manner.
In order to construct a simplified transfer model, all the experimental and theoretical data should beordered separating the correlative equations into the two parts: local variation effect and the Reynolds
number dependency.

118

Correlation of Mass and Enthalpy Transfer

𝑁𝑢 = 𝑎 𝑅𝑒 𝑃𝑟 𝑓 (𝜁 )(25.3-5)
𝑆ℎ = 𝑎 𝑅𝑒𝑆𝑐
𝑓 (𝜁 )(25.3-6)
where the two distribution functions 𝑓 (𝜁 ) and 𝑓 (𝜁 ) are expressed as a dimensionless vertical
position from the bottom edge of the packing section in the upward direction: ζ = 𝑍⁄𝑑 .
The Reynolds number dependency was analyzed as the dimensionless correlation functions giving the
maximal coefficients when 𝑓 (𝜁) = 1 and 𝑓 (𝜁) = 1
This position should have corresponded to the bottom edge 𝜁 = 𝜁 = 0 of the packing section.
However the leading edge position deviates to some degree 𝜁 from the bottom edge of the packing
section.
Vertical distribution function of vapor-phase mass and enthalpy transfer
Fig.25.3-3and 4 indicate the two distribution functions expressed as 𝑆ℎ ⁄𝑎 𝑅𝑒
𝑆𝑐
vs. 𝜁
and 𝑁𝑢 ⁄𝑎 𝑅𝑒 𝑃𝑟vs. 𝜁
For simplification of modeling concept, we may consider one
common distribution function curve indicated by dotted line.

Fig.25.3-3 Distribution function for mass transfer through vapor-phase film

Fig.25.3-4 Distribution function for enthalpy transfer through vapor-phase film

Transfer Model for Design of Packed Distillation Columns

119

After making trial-and-error calculations using a parameter fitting method, the following distributionfunctions are obtained:.
𝜁 = 31.8(25.3-7)
𝑓 (𝜁) = () ⁄𝑓 (𝜁) = ( .

.
) ⁄𝜁 = 35(25.3-8)

Here 𝜁 means that the fictitious origin of vertical distribution deviates downward from the lowest edgeof the packing section.
It should be noticed that the local variation in HETPs (see Eq.25-2-59) is also similar to these
distribution functions.
In the case of enthalpy transfer, the effect of thermal conduction going back toward vapor-upstreamside below the lowest edge of the packing section seems considerably larger than expected.

Fig.25.3-5 Reynolds number dependency of mass transfer

Fig.25.3-6 Reynolds number dependency of enthalpy transfer

120

Correlation of Mass and Enthalpy Transfer

Reynolds number dependency
Finally the following correlation equations are obtained from Figs.24.3~6 analyzing the Reynolds
number dependency:
𝑁𝑢 = 0.0022 𝑅𝑒 . 𝑃𝑟 ⁄ 𝑓 (𝜁)(25.3-9)⁄.
𝑆ℎ = 0.028 𝑅𝑒𝑆𝑐
𝑓 (𝜁)(25.3-10)
Both the exponents of the Reynolds number given by 𝑏 and 𝑏 = 1.0 are the same as for random
packings.
Regarding the vertical distribution functions, the term 𝜁 implies that the vertical origin deviates on
the downside below the lowest edge of the packing section. For energy transfer, in particular, it is very
difficult to prevent the leading edge of the vertical development from going up a vapor stream below thelowest edge of the packing section.
As in Chapter 12.3, the j-factor for enthalpy and mass transfer in the vapor-phaser are defined as(25.3-11)
𝑗 =
⁄
𝑗

=

⁄

(25.3-12)

As can be seen from the functional form of the above equations, the j-factor analogy comes toexistence between the simultaneous mass and enthalpy transfers. It can be interpreted as the analogy thatexcept for conductive energy transfer, this packed column distillation makes progress with the vaporphase mass transfer of the less--volatile component accompanied with the enthalpy transfer.
If the distillation conditions necessary for the design are given, we can start with (1) the processsimulation based on the equilibrium stage model. At the next step, we determine (2) HETP variationfrom the generalized HETP correlation and evaluate (3) the Reynolds number dependency accompaniedwith the vertical distribution functions. (4) The packed column height required can be calculated by
Eq.(25.2-2). (5) Local distribution of temperature, compositions, and mass and enthalpy transfer
coefficients can be calculated for a real packed column.
At any rate, as a result, the practical design concept based on the control volume model expressed asEq.(25.3-7) through Eq.(25.3-10) has been constructed. Still for that purpose, we will have to constructa generalized data bank accumulating very many experimental investigations.
For generalizing the control volume method for practical design of various packed distillation columns,
systematic accumulation of many experimental data on HETP for various packings is hoped. As in
Lockett article3), there is no doubt about the dependency of the HETPs on the vapor velocity, i.e. F-factor.
3.

Lockett, M. J., Easily predict structured-packing HETP, Chem. Eng. Prog., Jan. 60-66 (1998)[PROBLEM 25.3-1]
Discuss the similarity between the vertical development of simultaneous mass and enthalpy transfer
in a packed column distillation and the streamwise development of the heat and mass boundary layer
along a flat plate which plays a role of the mass and energy source. Assume that in the free stream outside
the boundary layer, the profiles of velocity, temperature, and concentration are kept uniform.
[PROBLEM 25.3-2]
We would like to construct a test packed distillation column and treat an experiment of distillation
under a total-reflux condition operated at normal pressure by using a binary system of benzene andtoluene. The distillation performance data expected are: (a) operating pressure 𝑝 = 0.1 MPa, (b) rateof the column top vapor rate is 𝑉
= 2.6 kmol/h, (c) desirable F-factor based on column top vapor isTransfer Model for Design of Packed Distillation Columns

121𝐹= 1.2 , and (d) benzene concentrations of the overhead product and the bottom product 𝑥 =0.99 and 𝑥 = 0.97. The specific surface area and equivalent diameter of the structured packing are
given by 𝑎 = 400 m ⁄m , 𝑑 = 0.02 m. For simplicity, the internal reflux liquid is assumed to beequal to the vapor flow rate, i.e. 𝐿 = 2.6 kmol/h.
As the preliminary test, the vertical variation of local HETPs is given by the following distributionfunction:(25.3-P1)
𝐻𝐸𝑇𝑃 =⁄. × ⁄where ℎ.

is the maximum HETP ℎ= 1.1 m evaluated at the column top 𝑧 = 0.

(1) Determine the inside diameter of the test column.
(2) Calculate the Reynolds number and the Schmidt number assuming that the vapor rate is kept
constant over the total height of the column.
(3) Calculate by using the McCabe-Thiele step-by-step method how many ideal stages are necessary.
(4) Obtain the vapor composition profile within each control volume from the above graphicalcalculation.
(5) Calculate vapor-phase film coefficient of more-volatile component at each control volume(6) Give a result of the required total height of the test packed column.

Nomenclature
𝑎, 𝑎
𝐶𝑝
𝑑
𝐷 ,𝐷
𝐸
𝐹
𝐹
f
𝐺
𝐻 , 𝐻
ℎ
ℎ
j
𝐾
𝐾 ,𝐾
𝑘 ,𝑘
L
𝑚
𝑁
𝑛
Pr
𝑅𝑒
𝑆𝑐
Sh
𝑇
u
V
𝑥 ,𝑦
𝑧
𝜁
𝜇

effective interfacial area per unit packed volume, [m2/m3]
heat capacity, [J/kmol K]
hydraulic equivalent diameter of packing, [m]diffusivity in gas- and liquid-phase, [m2/s]
Enthalpy transfer rate, [W]F-factor, [(m/s)(kg/m3)0.5]
F-factor specified at leading edge of packing section, [(m/s)(kg/m3)0.5]
distribution function, [ - ]superficial molar gas-mass velocity, [kmol/m2s]
specific enthalpy of liquid and vapor, [J/kmol]
HETP (Height Equivalent to a Theoretical Plate), [m]HETP evaluated at leading edge of packing section, [m]stage number, or j-factor, [ - ]vapor-liquid equilibrium constant or coefficient, [ - ]overall mass transfer coefficients defined by vapor-phase and liquid-phase concentrations [kmol/m2s]
mass transfer coefficients of vapor-phase and liquid-phase film [kmol/m2s]
liquid flow rate, [kmol/s]
slope of equilibrium curve, 𝑑𝑦 ⁄𝑑𝑥 ,mass transfer rate, [kmol/s]theoretical plate/stage number, [ - ]Prandtl number, [ - ]Reynolds number, [ - ]Schmidt number, [ - ]Sherwood number, [ - ]temperature, [K]velocity, [m/s]vapor flow rate, [kmol/s]
mole fraction of component A, liquid- and vapor-phase, [ - ]vertical height of packing section, [m]dimensionless vertical distance, [ - ]viscosity, [kg/m s]122Nomenclature

Subscripts
A
component Aiinterfacesv
superficial
L
liquidV
vapor∞
free stream
Superscripts*

equilibrium

Chapter 26 Thermodynamic Approach for Thermal Engineering Processes

123

Chapter 26
Thermodynamic Approachfor Thermal Engineering Processes

Historically speaking, “Thermodynamics,” the basic structure of which wasconstructed about 100 years earlier than Transport Science, still continues to exercise its
authority over extensive scientific and technological fields. In particular, the first law ofthermodynamics is serving as the basic concept of various sciences.
In this course, welearned the first law of thermodynamics in Chapter 4 of PART I as the fundamental ideaof the conservation of energy. You can go back to Chapter 4 whenever you want to know
the fundamentals of thermodynamics.

26.1 Fundamental Laws of Thermodynamics
We begin by reviewing the first law of thermodynamics for total energy given by𝒅𝒅𝒕∫𝑽 𝝆𝑬𝑻 𝒅𝑽 = − ∆ 𝑯𝒆 + 𝑬𝒌 + 𝑬𝒑 + 𝑸 − 𝑾(4.4-5)You can see the derivation procedure in the macroscopic energy balance in Chapter 4.4
of PARTⅠ.
A simple steady-state form of the mechanical energy balance can be obtained forincompressible fluid flows with no heat input (𝑄 = 0) into the flow system:+ ∆ 〈𝑣〉 + 𝑔 ∆ℎ = − 𝐹𝑟 − 𝑊(4.5-5)A steady-state form of thermal energy balance for non-isothermal system is given by− ∆ 𝐻 + 〈𝑣〉 + 𝑔 ℎ + 𝑄 − 𝑊 = 0(4.6-E13)
In moderate non-isothermal conditions, the kinetic and potential energy terms in thefirst term of Eq. (4.6-E13) 〈𝑣〉 + 𝑔 ℎ and the work term 𝑊 are much smaller than theheat input term 𝑄 for the case of thermal energy systems such as heat exchangers,
evaporator, condenser and so on,
Finally Eq.(4.6-E13) reduces to

124

Thermodynamic Diagram

𝑄 = ∆𝐻(26.1-1)
For a heat exchanger without phase transformation, this equation gives the following
heat balance expressed with heat flux:
𝑄 = 𝑤 𝑄 = 𝑤 𝐶𝑝 (𝑇 − 𝑇 )(26.1-2)
The heat exchange rate is proportional to the change in temperature of the flowing fluid
between the inlet and outlet of the heat transfer pipe.
The variables of these equations can be referred to the Notation of Chapter 4.
In Chapter 4, the loss of energy by irreversible viscous dissipation such as − 𝐹𝑟 was
taken into consideration in the energy balance. The friction loss is an important factor for
determining the power requirement of the pump transporting the fluid through a pipeline.
However the friction loss is too difficult to theoretically analyze for complicated flow
systems. We have already studied that the friction loss and heat loss should be evaluatedalong the route of fluid stream over the boundary wall between the system and thesurroundings. This chapter will study the role of thermodynamics in complicated flow
phenomena from a viewpoint of quasi-static change in the ideal condition.

26.2

Thermodynamic Diagram

In thermodynamics, there are various convenient diagrams available which represent many
thermodynamic states of materials at the condition of thermodynamic variables such astemperature, pressure, specific volume, enthalpy, entropy, etc. A thermodynamic diagram
illustrates the change or relation between two or more thermodynamic variables during athermodynamic process. One of the thermodynamic diagram is shown Fig.26.2-1. This is also called“Mollier diagram”, where the vertical axis gives specific enthalpy while the horizontal axis gives
specific entropy.

Fig.26.2-1 Mollier diagram indicating an adiabatic change and an isothermal change.

Thermodynamic Approach for Thermal Engineering Processes

125

An isentropic change from 𝑝 to 𝑝 is indicated as the adiabatic (constant-entropy) compression by
the vertical arrow. During this compression, the enthalpy change is given by 𝐻 − 𝐻 . An isothermalchange from 𝑝 to 𝑝 is indicated as the isothermal (constant-temperature) compression by the arrowalong the isotherm of 𝑇 . During the compression, the enthalpy change is given by 𝐻 − 𝐻 . These
two evaluations of enthalpy change are done under an assumption of quasi-static change with neither
heat loss nor friction loss.

26.3

Joule-Thomson Expansion

Let us consider adiabatic abrupt expansion through a horizontal flow system shown in Fig.26.3-1,
where a real gas flowing at high pressure 𝑝 in the region A is released into the region B of low
pressure 𝑝 without significant change in kinetic energy. This is called “Joule-Thomson expansion”.
Throttling is done through a throttling valve or a porous plug where no kinetic energy loss is assumed..
This is one of the quasi-static processes treated as the thermodynamic approach. In the quasi-static
process, no loss is assumed in the route from state 1 to state 2.

Fig.26.3-1 Throttling flow system for Joule-Thomson expansion

The expansion is inherently irreversible. During this expansion, enthalpy remains unchanged(see below). Unlike a free expansion, work is done, causing a change in internal energy. Whether theinternal energy increases or decreases is determined by whether work is done on or by the fluid; that isdetermined by the initial and final states of the expansion and the properties of the fluid.
Applying the thermodynamic law to this expansion,
𝑈 + 𝑝 𝑣 + 〈𝑣 〉 ⁄2 + 𝑔 𝑍 + 𝑞 = 𝑈 + 𝑝 𝑣 + 〈𝑣 〉 ⁄2 + 𝑔 𝑍 + 𝑊(26.3-1)

You may notice that the frictional loss term 𝐹𝑟 vanishes from the above equation. This is thethermodynamic consideration.
In this flow system,
𝑍 = 𝑍 , i.e. negligible potential energy effect.

126

Joule-Thomson Expansion

𝑞 = 0 , i.e. no heat input and output from the insulated wall. In thermodynamic consideration, no heatloss is assumed.
𝑤 = 0 , i.e. no shaft work such as pumps, turbines and impellers.
The above equation reduces to
𝐻 + 〈𝑣 〉 ⁄2 = 𝐻 + 〈𝑣 〉 ⁄2
(26.3-2)
〉.
〈𝑣
〉
〈𝑣
≪
Since 𝑝 ≫ 𝑝 ,In this flow system dealing with the enthalpy change, 𝐻 , 𝐻 ≫ 〈𝑣 〉 ⁄2.
As a result,𝐻 ≅ 𝐻(26.3-3)
It has been found that the Joule-Thomson expansion is done by the equi-enthalpy (Isenthalpic) change.
Thermodynamic approach can consider a change in the physical state between the starting point andthe reaching point. What route traced from the start to the goal is not taken into account. It is necessary
to consider the friction loss and heat loss along the route.
For this Joule-Thomson expansion,
𝐻 ≅ 𝐻
Therefore(26.3-4)∫ 𝑑𝐻 = 𝐻 − 𝐻 = (𝐻 − 𝐻 ) + (𝐻 − 𝐻 ) = 0
Generally the enthalpy change can be expressed by the following formula based on the change intemperature and pressure:(26.3-5)𝑑𝑝

𝑑𝐻 = 𝐶𝑝 𝑑𝑇 + 𝑉 − 𝑇

As shown in Fig.26.3-2, from a thermodynamic viewpoint, the change from 1 to2 directly is equivalent
to the change from 1 through 2 via 3.

Fig.26.3-2 Two routes for enthalpy change calculation in Joule-Thomson expansion

We can calculate the enthalpy change separately:
𝐻 − 𝐻 = ∫𝑉− 𝑇

𝐻 − 𝐻 = ∫ (𝐶𝑝)
Therefore
𝐻 − 𝐻 = ∫𝑑𝑝

(constant temperature)(𝑇 − 𝑇 ) (constant pressure)𝑑𝑇 = (𝐶𝑝 )(𝑇 − 𝑇 ) = 0

𝑑𝑝 + (𝐶𝑝 )𝑉− 𝑇(26.3-6)

From this equation, we obtain the following relation:∫

𝑇 − 𝑇 =

(

)

=

(

)

,
(

(
)

,

)

(26.3-7)

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127

where (𝐶𝑝 ) is the specific heat averaged over the range of 𝑇 and 𝑇 .
This result indicates that the gas temperature is lowered or raised by the Joule-Thomson expansion
depending upon the 𝑝𝑉 work.
The enthalpy is defined as 𝐻 = 𝑈 + 𝑝 𝑉.
Under the condition of a Joule-Thomson expansion, the change in 𝑝𝑉 represents the work done by
the fluid. This is called “flow work”. (see Chapter 4.4) If 𝑝 𝑉 increases doing the work on its
surroundings, 𝑈 should decrease to keep 𝐻 constant. This means the cooling effect. However for an
ideal gas, 𝑝𝑉 does not change during a Joule-Thomson expansion. That is, there is no change ininternal energy. The numerator of the above equation reduces to zero if the perfect gas law issubstituted.
The adiabatic cooling which occurs by the Joule-Thomson expansion when a compressed gas at 𝑝
is suddenly allowed to expand into a low pressure space in the condition of no external work done.
This indicates that the Joule-Thomson expansion can produce larger heating or cooling than observedin a free expansion.
The Joule-Thomson coefficient is defined as(26.3-8)𝜇 =This coefficient may be either positive or negative.
Fig.26.3-3 shows a schematic picture of T-p mapping diagram of Joule-Thomson coefficients for N2
gas.

Fig.26.3-3 T-p mapping diagram of Joule-Thomson coefficients （N2 gas）

The coefficient 𝜇 is negative in the region of very high and very low temperatures. At temperatures
below the gas-liquid coexistence curve, N2 condenses to form a liquid and 𝜇 becomes negative.
Within the region bounded by the red line, 𝜇 > 0. This means cooling effect.

128

Joule-Thomson Expansion

Outside the bounded region, all regions indicate 𝜇 < 0. This implies that the adiabatic expansion
causes heating effect. The dashed line indicates the demarcation line for becoming a supercriticalfluid (see Fig,23.1-1). All gases have an inversion point at which the 𝜇 value changes sign. For
example, N2 gas has inversion temperature of 621 K as shown in Fig.26.1-3.
[EXAMPLE 26.3-1]
In the flow system shown in Fig.26.3-E1 , a steam stream A (𝑝 = 5 MPa, 𝑇 = 280 ℃) undergoes
a Joule-Thomson expansion with pressure change from 𝑝 to 𝑝 = 0.1 MPa. Calculate thetemperature change.
Solution:Fig.26.3-E1 is a thermodynamic diagram (p-H diagram), where a Joule-Thomson expansion isindicated by an arrow from the point of 𝑝 = 5 MPa and 𝑇 = 280℃ to a pressure line of 𝑝 =0.1 MPa. Since this is an adiabatic expansion with H constant, the arrow reaches an isotherm of 230℃.
Therefore the temperature drop reduces to 50℃. (It may be somewhat difficult to precisely find thecorrect isotherm from this entangled diagram.)

Fig.26.3-E1 Calculation of Joule-Thomson expansion by means of a thermodynamic diagram

Let us reconsider a similar problem by another diagram (Mollier diagram).
A steam stream (𝑝 = 1 MPa, 𝑇 = 250 ℃) undergoes a Joule-Thomson expansion with pressurechange from 𝑝 to 𝑝 = 0.1 MPa. We will calculate the temperature change.
The constant-pressure heat capacity averaged at 𝑝 is given by
𝐶𝑝 = 1.96 kJ/kg K.

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129

Fig.26.3-E2 Calculation of Joule-Thomson expansion using a Mollier diagram.

The initial state (𝑝 = 1 MPa, 𝑇 = 250 ℃) is located at the star indicated in Fig.26.3-E2. The red
horizontal line implying the Joule-Thomson expansion with constant enthalpy crosses the isobaric lineof 𝑝 = 0.1 MPa. We can consider that an isotherm of 220℃ should pass through that crossing point.
That is, the temperature of the steam stream at the final state is 220℃. The temperature drop is 𝑇 −𝑇 = − 30℃.
Another solution: From the above Mollier diagram, the specific enthalpy of the steam stream(𝑝 = 1 MPa, 𝑇 = 250 ℃) is 𝐻 = 704 kcal/kg.
From a thermodynamic viewpoint, the route 1 → 3 → 2 gives the same result of isenthalpic
change as the direct route 1→ 2.

Fig.26.3-E3 Two routes of isenthalpic change from 1 to 2 for a Joule-Thomson expansion.

130

Joule-Thomson Expansion

As shown in the Mollier diagram, the specific enthalpy at the fictitious condition (𝑝 = 0.1 MPa,
𝑇 = 250 ℃) is given by
𝐻 = 718 kcal/kg.
The constant-pressure heat capacity averaged is given by 𝐶𝑝 = 1.96 kJ/kgK = 0.47 kcal/kg℃ .
Therefore∫

𝑇 − 𝑇 =

(

)

=

(

)

,
(

(

,

)

)

=

.

= −29.8℃(26.3-E1)

This solution with the aid of the (𝐶𝑝 ) value seems to be somewhat easier than the first one.
Derivation of Joule-Thomson coefficient
Let us study using the laws of thermodynamics about what the Joule-Thomson coefficient is.
The isenthalpic condition of a Joule-Thomson expansion is applied in the defining equation ofenthalpy
𝑑𝐻 = 𝑑𝑈 + 𝑑(𝑝𝑉) = 𝑇 𝑑𝑆 + 𝑉 𝑑𝑝 = 0 for 𝑑𝑈 = 𝑄 − 𝑝 𝑑𝑉 = 𝑇 𝑑𝑆 − 𝑝 𝑑𝑉
(26.3-E2)
From the above equation we obtain𝑑𝑆⁄𝑑𝑝 = − 𝑉 ⁄𝑇
(26.3-E3)
The following relation could be obtained by somewhat difficult derivation.
= −(26.3-E4)
Only for simplification, we will omit the derivation process.
Since 𝑑𝐻 = 𝑇 𝑑𝑆 in the constant pressure condition, the definition of constant-pressure heat
capacity is given by=(26.3-E5)
𝐶𝑝 =Therefore=(26.3-E6)

Differentiating the relation 𝑑𝐻 = 𝑇 𝑑𝑆 + 𝑉 𝑑𝑝 with respect to 𝑝 in the condition of constant 𝑇
gives=𝑇+ 𝑉 = −𝑇+𝑉
(26.3-E7)
Using this relation, the enthalpy change can be expressed as
𝑑𝐻 =

𝑑𝑇 +

𝑑𝑝 = 𝐶𝑝 𝑑𝑇 + 𝑉 − 𝑇𝑑𝑝(26.3-E8)

Differentiating with respect to 𝑝 in the condition of 𝑑𝐻 = 0, the Joule-Thomson coefficient isobtained:
𝜇==(26.3-E9)

This coefficient indicates the rate of temperature change with the pressure for an isenthalpic condition.
For perfect gases (𝑝 𝑉 = 𝑛 𝑅 𝑇),==(26.3-E10)
Therefore the numerator of the defining formula becomes zero for perfect gases
𝜇 =0(perfect gas)
It has been demonstrated that even if the Joule-Thomson expansion process is performed with aperfect gas, the fluid temperature would not change.

Thermodynamic Approach for Thermal Engineering Processes

26.4

131

Actual Issues of Joule-Thomson Expansion Process

According to the macroscopic energy balance obtained from the first law of thermodynamics in
Chapter 4.6:
(4.6-E13)− ∆ 𝐻 + 〈𝑣〉 + 𝑔 ℎ + 𝑄 − 𝑊 = 0
For the case of Joule-Thomson expansion from 𝑝 to 𝑝 , the flow system exhibits the following
isenthalpic change
𝐻 = 𝐻
In the thermodynamic diagram shown in Fig. 26.4-1, the initial state and the final state are indicated.
The route between the initial and final states for the Joule-Thomson expansion is not taken into
account except the condition of isenthalpic change.

Fig.26.4-1 Mollier diagram indicating an isenthalpic Joule-Thomson expansion process.

This means an ideal isenthalpic process due to the quasi-static change which can be analyzed by asimple calculation method assuming neither friction loss nor heat loss. This is one of thethermodynamic approaches.

132

Actual Issues of Joule-Thomson Expansion Process

Fig.26.4-2 Friction losses in an actual Joule-Thomson expansion process

However these irreversible losses should be evaluated along the flow direction. The heat loss
from the outside surface of the system to the surroundings should be reduced as much as possible by athermal insulation. On the other hand, the friction losses occur not only due to the shear stress acting
on the inside surface of the flow system but also due to the viscous dissipation of turbulent flow in thebulk flow region. The latter effect is very difficult to evaluate except by experiment.
In particular, the fluid stream undergoes very large loss of kinetic energy due to the orifice effect at thethrottling valve.
Strictly speaking, an actual Joule-Thomson process should proceed accompanied with some degreeof enthalpy loss, i.e. 𝐻 < 𝐻 . The friction loss is converted to the generation of internal energy.
For a flow system without heat exchange (𝑄 = 0), The enthalpy change reduces to∆𝐻 = ∆𝐸 + ∆(𝑝 𝑉 ) = 𝑇 ∆𝑆 + 𝑉 ∆𝑝
(4.5-1)
The friction loss renders the process irreversible owing to the frictional heating. Therefore
𝑇 ∆𝑆 = 𝑄 + 𝐹𝑟
That is, ∆𝐻 = 𝑄 + 𝐹𝑟 + 𝑉 ∆𝑝
(4.5-2)
As a result, the mechanical energy balance equation reduces to(4.5-3)∫ 𝑉 𝑑𝑝 + ∆ 〈𝑣〉 + 𝑔 ∆ℎ = − 𝐹𝑟 − 𝑊
This generalized Bernoulli equation should be used for a strict analysis of practical Joule-Thomson
expansion processes. If the flow system does not have any shaft work, 𝑊 = 0.
The friction loss can be evaluated by experiment with the aid of Eq. (4.5-3).

[PROBLEM 26.4-P1]
Fig.26.4-P1 is a p-H diagram of carbon dioxide.
A compressed carbon dioxide gas (𝑝 = 12 MPa, 𝑇 = 80℃) undergoes the first step of JouleThomson expansion from 𝑝 = 12 MPa to 𝑝 = 3 MPa. Obtain the temperature drop.

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133

Fig.26.4-P1 P-H diagram of carbon dioxide

26.5

Ejector Engineering

26.5-1 Working principle of ejectors
There are various kinds of ejectors available for many chemical engineering purposes: (1) waterejector or aspirator, (2) air ejector, (3) vacuum ejector, (4) steam ejector, etc.
An ejector is constituted of a suction chamber, a jet nozzle section, and a diffuser section.
As shown in Fig.26.5-1, a working fluid A (motive fluid) flows through a jet nozzle into a suction
chamber with high velocity where another fluid B is drawn in due to a low pressure (obtained by
Bernoulli principle). Then the mixed fluid A+B entering the diffuser section is very much accelerated
due to the convergent section. Then the velocity of the mixed fluid becomes maximum at the diffuserthroat. They undergo an adiabatic expansion after passing through the throat. In the next divergent
section, they are decelerated with the recovery of pressure up to the outlet pressure.
The strength of the vacuum depends on the working fluid velocity and pressure, the shape of the jet
nozzle, and the shape of the construction and mixing section.
Water ejector (aspirator) is commonly used when not strong vacuum is needed in chemistry andbiology laboratories. If a liquid is used as the working fluid like an aspirator, the strength of thevacuum is limited by the vapor pressure of the liquid. Therefore steam ejectors are the most effectively
utilized for industrial processes. In this chapter, we will study mainly the steam ejector behavior anddesign method from a thermodynamic viewpoint.

134

Ejector Engineering

Fig.26.5-1 Standard structure of ejectors

Steam ejectors using high-pressure steam as the working fluid A have a simple structure with nomoving part such as rotating machinery. In addition, they have sufficient durability with advantages ofeasy maintenance and cheap price.
The steam ejector operates by utilizing the pressure energy of a motive steam A, converting it to thekinetic energy, and then ejecting the high-velocity jet from a jet nozzle. As a result, another fluid B(usually low-pressure steam) is drawn in into a suction chamber by Bernoulli principle. Then themixed fluid A+B flowing into the downstream tube called “diffuser” is accelerated in the convergent
section, and reaches the maximum velocity at the diffuser throat. After the throat, the mixed fluid A+B
is decelerated in the divergent section with increasing pressure. Steam ejectors can be regarded not
only as a vacuum pump but also as a compressor recovering low-pressure steam to medium-pressure
steam.
26.5-2 Usage of steam ejectors
As shown in Fig.26.5-2, usually a steam ejector has two purposes for usage except for
depressurization as a vacuum pump. Generally speaking, low-pressure or low-temperature steam
implies thermally low effectiveness for thermal engineering. One purpose is to compress a lowpressure steam up to a medium-pressure steam. The pressure of the motive steam is lowered down to
the medium pressure instead. It should be considered that we can obtain thermally valuable steam
𝑤 + 𝑤 . The second purpose is to use the medium-pressure steam produced by an ejector as aneffective heating medium.
Steam ejectors are very often used in a variety of applications. Typical duties involve distillation,
filtration, mixing, vacuum packaging, freeze drying, etc. Steam ejectors are very simple and reliablewith low cost in that there are no moving parts such as rotating machinery and in that ejectors can bemade highly resistant to corrosion and erosion.

Thermodynamic Approach for Thermal Engineering Processes

135

Fig.26.5-2 Usage of a steam ejector

26.5-3 Thermodynamic consideration of steam ejectors
It is very difficult to theoretically analyze the flow behavior in a steam ejector. In particular, the lossof mechanical energy generated within the flow passage of the ejector and the loss of thermal energy
from the surface of the ejector system to the surroundings are complicated engineering issues. As
treated in Joule-Thomson expansion processes, the thermodynamic approach has a very simple andconvenient concept (Fig.26.5-3): A change 1 to 2 in thermodynamic variables such as enthalpy,
entropy, specific volume, pressure, and temperature is given by a difference only between the initialstate and the final state, regardless of the tracing routes from 1 to 2.
Regarding an ejector shown in Fig.26.5-1, the initial state corresponds to A and B whereas the final
state corresponds to C. For example, a change in enthalpy becomes 𝐻 − (𝐻 + 𝐻 ).
A Mollier H-S diagram shown in Fig.26.5-4 is very convenient to use for consideration ofthermodynamic behavior of ejectors.

Fig.26.5-3 Thermodynamic concept of change 1 to 2 of thermodynamic variables

136

Steam Ejectors for Effective Energy Conservation

Fig.26.5-4 Schematic picture of Mollier diagram for calculating a thermodynamic change

26.6

Steam Ejectors for Effective Energy Conservation

26.6-1 Effective application of a steam ejector for evaporation processes
Let us consider about the utilization of a steam ejector for effective energy conservation of
steam lines in a factory. Fig.26.6-1 shows a continuous evaporator supported by a steam ejector. The
evaporator is one of our original Wall Wetter evaporators (see Chapter 24.3-1&2). The water vapor
produced by evaporation of aqueous solution of non-volatile substance is recompressed by a steam
ejector which works with the aid of a motive steam taken from the steam line of another plant.
In a usual condition of the evaporator operated without any ejector, the evaporator needs the heating
duty 𝑄 = 5,390 kcal/h which is covered with 𝑤 = 10.1 kg⁄h of saturated water vapor (1 atm) inthe jacket. The motive steam requirement 𝑤 = 28.0 kg⁄h is calculated by the trial-and-error
method to be described in the later section. At this stage, 𝑤 = 28.0 kg⁄h can be regarded as an
assumed value.
𝑤 = 28.0 kg⁄h of motive steam (𝑝 = 0.7 MPa, 𝑇 = 166℃) and 𝑤 = 10 kg⁄h of water
vapor (𝑝 = 0.04 MPa, 𝑇 = 75℃) are mixed inside the ejector and comes out as the medium-pressure
steam 𝑤 + 𝑤 = 38.0 kg⁄h in the condition of 𝑝 = 0.1 MPa, 𝑇 = 100℃, and quality 𝑥 =100% (saturated steam). Strictly speaking, the outlet steam is obtained in the somewhat superheatedcondition.
Only for simplicity, we ignore the boiling point elevation due to the solute of the aqueous solution in
the evaporator and the heat loss from the outside surface of the steam jacket.
The 𝑤 = 10.1 kg⁄h of the outlet steam can be reused for the heating duty of the evaporator.
This can be considered as a kind of vapor-recompression method (VRC). As a result, the remaining
steam 𝑤 = 27.9 kg/h at 1 atm can be used for another purpose. Finally only the thermal energy due tothe pressure change of the motive steam from 0.7 to 0.1 MPa is utilized for evaporation without
consumption in mass quantity. In addition, the pressure 𝑝 = 0.04 MPa of the evaporator iscontrolled by the sucking strength of the ejector.
For the low-pressure steam to be sucked into the ejector, the compression ratio is
𝛽 = 𝑝 ⁄𝑝 = 0.1⁄0.04 = 2.5.

Thermodynamic Approach for Thermal Engineering Processes

137

Fig.26.6-1 Energy saving method of an evaporator depressurized by a steam ejector

Practically the necessary supplying rate of the motive steam should be calculated in advance by thetrial-and-error method to attain the appropriate condition of the ejector outlet. The practical design
calculation method will be described later.
From now, let us study first the thermodynamic calculation based on unit mass.
According to the H-S diagram, the enthalpies of the motive steam and the water vapor from theevaporator are obtained by 𝐻 = 2,765 kJ⁄kg and 𝐻 = 2,620 kJ⁄kg, respectively.
As aforementioned, the following enthalpy balance can be used with neither friction loss nor heat loss:
𝑤 𝐻 + 𝑤 𝐻 = (𝑤 + 𝑤 )𝐻
(26.6-1)
This indicates the enthalpy balance of the ejector between the initial state A+B and the final state C.
From the above equation, the enthalpy at the ejector outlet is calculated as(28.0)(2,765) + (10)(2,620) = (28.0 + 10)𝐻
(26.6-2)
The enthalpy of the outlet steam is obtained as 𝐻 = 2,726.8 kJ⁄kg.
If 𝑤 becomes a different quantity as a result of the practical design calculation, the above
calculation should be revised with a new value.
The 10.1 kg⁄h of the mixed steam obtained from the ejector outlet 𝑤 + 𝑤 = 38.0 kg⁄h can bereused as the heating medium for the evaporator and the remaining steam 27.9 kg⁄h can be used as
the effective steam for another usage. This suggests that the evaporator can be operated by using thewater vapor obtained by itself as the necessary steam for the jacket.
26.6-2 Thermodynamic calculation model for steam ejectors
Let us consider how to calculate the enthalpy change in a steam ejector.
As aforementioned, it is one of the great advantages of the thermodynamics to calculate a change ofthermodynamic variables from the initial and final state without considering any routes.
We introduce “Two-flow-passage model” for the calculation of enthalpy change for a steam ejector.

138

Steam Ejectors for Effective Energy Conservation

The two-flow-passage model shown in Fig.26.6-2 assumes that two streams A and B are separately
flowing without interaction between them. Therefore we can consider that the stream A related withthe motive steam undergoes adiabatic (isentropic) expansion with the pressure change from 𝑝 to 𝑝
whereas the stream B undergoes adiabatic (isentropic) compression with the pressure change from 𝑝
to 𝑝 .

Fig.26.6-2 Two-flow-passage model for thermodynamic calculation of a steam ejector

If we adopt the two-flow-passage model for this problem, we can calculate the enthalpy changeon the following Mollier diagram.
The stream B experiences isentropic change from 𝑝 = 0.04 MPa to 𝑝 = 0.10 MPa. Therefore
the stream B requires the enthalpy change 𝐻 − 𝐻 = 2,800 − 2,620 = 180 kJ⁄kg.
However in Fig.26.6-3, the stream A has available energy 𝐻 − 𝐻 = 2,765 − 2,285 =480 kJ⁄kg. Be careful about 𝐻 − 𝐻 as the available energy based on the 0.04 MPa water vapor
although the stream A also experiences isentropic change from 𝑝 to 𝑝 .

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139

Fig.26.6-3 Thermodynamic enthalpy calculation on Mollier diagram

From here, let us consider the practical method determining the rate of the motive steam 𝑤 kg/h.
The flow and energy behaviors inside the ejector are so complicated that we cannot but rely on theempirically constructed data base.

26.7

Diffuser Efficiency of a Steam Ejector

The capacity of a steam ejector must be investigated by experiment.
In a practical design procedure, the diffuser efficiency empirically correlated with the suction
pressure shown below is usually used.

140

Diffuser Efficiency of a Steam Ejector

Fig.26.7-1 Empirical correlation of diffuser efficiency with suction pressure

The diffuser efficiency of a steam ejector depends on the structural design of an individual ejector.
We should have our own correlation data accumulated by experiment.
In the problem of interest, the suction pressure is given as 𝑝 = 0.04 MPa. Therefore the diffuser
efficiency lies between 0.725 and 0.83.
As shown in Fig.26.7-1, the diffuser efficiency can be interpreted in a relation with the friction loss
factor 𝜍 as follows.
The diffuser efficiency is defined as(1 − (𝐴 ⁄𝐴 ) )(26.7-1)
𝜂 = ( ⁄ )This implies how to convert the kinetic energy (1⁄2) 𝜌𝑣 into the pressure energy 𝑝 − 𝑝 .
For the case of suddenly enlarged flow passage (see Chapter 7.4-2), the pressure change based on
the friction loss can be expressed as𝑝 − 𝑝 = 𝜌 𝐹𝑟 = 𝜍 𝜌〈𝑣 〉(26.7-2)
where the friction loss factor is given by 𝜍 = 1 −Therefore we can consider that the friction loss factor of the diffuser can be expressed as𝜍=(26.7-3)( ⁄ )Thermodynamic Approach for Thermal Engineering Processes

141

Fig.26.7-2 Physical meaning of diffuser efficiency

26.8 Motive Steam Required for a Steam Ejector
In order to obtain the necessary motive steam rate 𝑤 , it is necessary to consider a relation with theejector efficiency based on the enthalpy change ratio defined as∆(26.8-1)
𝜂 = ∆ =
Here the denominator should be 𝐻 − 𝐻 from a viewpoint of available energy (enthalpy difference)higher than the steam with pressure 𝑝 to be sucked.
Fig.26.8-1 indicates an empirical correlation of the necessary motive steam rate 𝑤 ⁄𝑤 with theejector efficiency (enthalpy efficiency) 𝜂 , where each correlative curve designates the diffuser
efficiency 𝜂 as the parameter.

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Motive Steam Required for a Steam Ejector

Fig.26.8-1 Empirical correlation of motive steam rate with ejector efficiency

These two correlation data should be empirically provided as the data bank for practical design
work. (Figs. 26.7-1 & 26.8-1 )Let us try to determine the necessary motive steam rate from these correlation data.
[EXAMPLE 26.8-E1]
Regarding the problem treated above, we will determine the necessary rate of the motive steam.
The suction pressure is given by 𝑝 = 0.04 MPa from the operating condition of the evaporator.
According to Fig.26.7-1, the diffuser efficiency lies between 0.725 and 0.83.
We will adopt 𝜂 = 0.725 for our safe-side design principle.
The ejector efficiency is given by,,= 0.375(26.8-E1)
𝜂 = ,,Fig.26.8-E1 Determination of necessary motive steam rate

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143

Using Fig.26.8-1, the necessary motive steam rate can be obtained as 𝛽 = 2.8, as shown in theabove diagram. Therefore
𝑤 = 𝛽 𝑤 = 2.8 × 10.0 = 28.0 kg⁄h(26.8-E2)
The motive steam rate 𝑤 = 28.0 kg⁄h assumed at the beginning has been confirmed to be valid.
Usually the obtained motive steam rate is very large owing to the energy efficiency of steam
ejectors.

Fig.26.8-E2 Momentum balance around the ejector

The following momentum balance check can also be tested:𝑤 𝑣 = 𝛾(𝑤 + 𝑤 )𝑣
(26.8-E3)
When the constant (a kind of correction factor) 𝛾 = 1.0, the momentum balance is completely kept.
However usually 𝛾 = 0.8 ~ 0.9 owing to the kinetic energy loss.
[EXAMPLE 26.8-E2]
Fig.26.8-E3 shows a continuous evaporating system equipped with a steam ejector for producing
distilled water from the city water (20℃). A motive steam (𝑝 = 1.2 MPa) with quality 98% can beutilized for the steam ejector. The evaporator is operated at normal pressure. A portion of the mediumpressure steam (𝑝 = 0.15 MPa) issuing from the ejector is supplied into the coil for evaporation of thecity water. Roughly speaking, from a viewpoint of VRC, the water vapor evaporated can be used after
compression by the ejector as the heating medium for the same evaporator. Actually the latent heat ofvaporization decreases with increasing pressure. For example, the heat of evaporation is 540 kcal/kg at
0.1 MPa and 532.7 kcal//kg at 0.15 MPa.
This is a problem how to determine the rate of the motive steam appropriate for compression by theejector. As shown in the figure, the water (saturation temperature 110℃) condensed in the coil isused for preheating the city water up to 100℃ by the preheater.

Fig.26.8-E3 Coil evaporator equipped with a steam ejector producing distilled water

144

Motive Steam Required for a Steam Ejector

Heat balanceThis project is planning how efficiently to produce distilled water from 𝑤 = 200 kg/h of the citywater with the aid of the ejector.
The heat duty for the evaporator producing 200 kg/h of water vapor (0.1 MPa) is given by
𝑄 = 𝜆 𝑤 = (540)(200) = 108,000 kcal/h.
(26.8-E4)
We can firstly calculate how much the steam (0.15 MPa) compressed by the ejector is necessary for theevaporator:
𝑤 = 𝑄 ⁄𝜆 = 108,000⁄532.7 = 202.7 kg/h
(26.8-E5)
The 202.7 kg/h of hot water (100℃) condensed in the coil is used to preheat the 200 kg/h of city waterfrom 20℃ to 100 ℃ in the preheater. The heat exchange rate is given by(26.8-E6)
𝑄 = 𝑤 𝐶𝑝 (𝑇 − 𝑇 ) = 𝑤 𝐶𝑝 (𝑇 − 𝑇 )The outlet temperature of the distilled water is calculated as()(100 − 20) = 31.1 ℃
𝑇 = 𝑇 −𝑇 − 𝑇
= 110 −
(26.8-E7).
Enthalpy calculation for determining the rate of motive steam 𝒘𝟏
The water vapor discharged from the evaporator is compressed by the ejector using 𝑤 kg/h of the
motive steam (𝑝 = 1.2 MPa). The compression ratio of the ejector is 𝛼 = 𝑝 ⁄𝑝 = 1.5. Determine
how much motive steam is required for this purpose.
The Mollier diagram shown below expresses the isentropic enthalpy change in the two-flow-passage
model.

Fig.26.8-E4 Mollier diagram of an enthalpy change of a steam ejector for a process producing
distilled water from the city water

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145

The available energy of the motive steam (1.2 MPa) based on the water vapor (0.1 MPa) from theevaporator is given by
𝐻 − 𝐻 = 2,760 − 2,350 = 410 kJ⁄kg(26.8-E8)
On the other hand, the isentropic enthalpy change of the stream B from 𝑝 to 𝑝 is given in the
Mollier diagram:
𝐻 − 𝐻 = 2,740 − 2,650 = 90 kJ⁄kg(26.8-E9)
For convenience, the enthalpy calculation can be schematically redrawn as Fig.26.8-E3.
According to TheFig.26.7-1, the diffuser efficiency is selected as 𝜂 = 0.725 for the safe-side design.
The ejector efficiency is given by== 0.22(26.8-E10)
𝜂 =Fig.25.7-1 gives the following steam flow ratio 𝛽 for 𝜂 = 0.725:
𝛽 = 𝑤 ⁄𝑤 = 1.1(26.8-E11)
Therefore the rate of the motive steam is determined as
𝑤 = 1.1 𝑤 = 1.1 × 200 = 220 kg⁄h(26.8-E12)
As a result, 𝑤 + 𝑤 = 420 kg/h of mixed steam (0.15 MPa) issues from the ejector.
From the heat balance calculation, 202.7 kg/h of the mixed steam should be supplied into the coil.
Finally the 202.7 kg/h of distilled water (31.1℃) is produced.
The 2.7 kg/h of the motive steam is consumed for producing the 202.7 kg/h of the distilled water.
For this condition, it should be kept in mind that the motive steam should also be pure enough.
Since the remaining mixed steam 217.3 kg/h still has 0.15 MPa, it can be utilized for another purpose.
The specific enthalpy 𝐻′ of the mixed steam issuing from the ejector exit can be calculated as
𝑤 𝐻 + 𝑤 𝐻 = (𝑤 + 𝑤 )𝐻′(26.8-E13)(220)(2,760) + (200)(2,650) = (220 + 200)𝐻′As a result, 𝐻′ = 2,708 kJ/kg.
This value lies a little bit above the vapor-liquid saturation curve. This suggests that the mixed steam isslightly superheated.

Fig.26.8-E5 Schematical picture of enthalpy change calculation on Mollier diagram

146

Motive Steam Required for a Steam Ejector[PROBLEM 26.8-P1]
Fig.26.8-P1 shows a continuous evaporating system of coil evaporator accompanied with a steam
ejector. An aqueous dilute solution of non-volatile organic substance (𝑤 = 20 kg/h, 𝐶 = 10 wt%) is
fed into the coil evaporator after preheating it from 25℃ to 81.6℃ (= boiling point at 0.05 MPa) by
the steam 𝑤 obtained from the ejector. In order to avoid thermal decomposition of the organic
substance (solute), the evaporator is operated at 𝑇 = 81.6℃ and 𝑝 = 0.05 MPa by condensing
the steam 𝑤 supplied into the coil from the steam ejector. Only for simplification, the elevation of
the boiling point due to the solute can be neglected. A motive steam 𝑤 (𝑝 = 1.0 MPa, quality 𝑥 =1.0) is utilized for the ejector. The concentrated product 𝑤 is continuously withdrawn with theconcentration of 𝐶 = 50 wt% from the bottom nozzle. Therefore 𝑤 = 16 kg/h of water should be
evaporated, and then a part of 𝑤 , 𝑤 kg/h is condensed in an overhead condenser in order to savethe rate of motive steam. The remaining water vapor 𝑤 kg/h is sucked into the ejector.
The latent heat of evaporation is assumed to be 550 kcal/kg at 0.05 MPa. The outlet pressure of theejector should be set at the same as the pressure inside the coil, i.e. 0.10 MPa. The operating pressure
inside the coil can be assumed to be 0.1 MPa constant. The latent heat of evaporation is 539 kcal/kg at
0.1 MPa. The compression ration 𝑝 ⁄𝑝 is given by 2.0. The mixed steam 𝑤 issuing from theejector is consumed in the coil of the evaporator, and then its condensate is used in the preheater forheating the feed solution up to 81.6℃. The residual steam shown in Fig.26.8-P1) should be controlled
to zero.
(1) What is the heat duty 𝑄 of the evaporator?
(2) Calculate the necessary steam 𝑤 issuing from the ejector for evaporation.
(3) Calculate the rate 𝑤 of water vapor condensed in the overhead condenser.
(4) Determine the diffuser efficiency 𝜂 .
(5) Complete the Mollier H-S diagram (shown below) for determining the steam enthalpies
𝐻 ,𝐻 , 𝐻 ,𝐻 .
(6) Calculate of the enthalpy efficiency 𝜂 = (𝐻 − 𝐻 )⁄(𝐻 − 𝐻 ).
(7) Determine the rate of the motive steam 𝑤(8) Set up a heat balance of the preheater between the hot water issuing from the evaporator and the
supplying stream of the aqueous solution feed. Then obtain the outlet temperature of the hotwater issuing from the preheater.

Fig.26.8-P1 Continuous evaporation system of coil evaporator accompanied with a steam ejector.

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147

Fig.26.8-P2 Mollier diagram to be completed for PROBLEM 25.8-P1

26.9 Multi-stage Ejector System
Staging of steam ejectors becomes necessary for economical operation as the absolute suction
pressure decreases since there is an upper limit for compression ratio of a single ejector system.
Fig.26.9-1 shows a standard system of two-stage steam ejector.
The two-stage ejectors comprise a primary high-vacuum ejector and a secondary low-vacuum
ejector. The general operating range of two-stage steam ejector usually lies between 3 mmHg and 130
mmHg. The calculation method of two-stage steam ejector is the same as for a single steam ejector.
The mixed steam from the outlet of the primary ejector should be treated as the steam to be sucked
into the secondary ejector. The same motive steam is usually supplied into both of the ejectors.

Fig.26.9-1 Standard type of two-stage ejector

148

Multi-stage Ejector System[EXAMPLE 26.9-E1]
Fig.26.9-E1 shows a two-stage steam ejector for keeping the operating pressure of an evaporator
at 𝑝 = 3.0 mmHg. The compression ratio is divided into two ejectors, i.e. 𝛼 and 𝛼 are given to
be 3.0 and 5.0, respectively. The primary ejector is made smaller in the compression ratio 𝛼 than thesecondary ejector.
A high-pressure steam of 0.7 MPa can be utilized as the motive steam 𝑤 and 𝑤 for the twoejectors. The water vapor 𝑤 = 15 kg/h which is generated at 3 mmHg in the evaporator should besucked by the #1 ejector using the motive steam 𝑤 . The mixed steam 𝑤 + 𝑤 discharging fromthe outlet of the #1 ejector is supplied into the inlet of the #2 ejector. Calculate the necessary rate of the
motive steam 𝑤 + 𝑤 assuming no heat loss from the system to the surroundings.

Fig.26.9-E1 Two-stage ejector for depressurizing an evaporator

Solution:[Primary ejector]
A thermodynamic enthalpy-entropy diagram (Mollier H-S diagram) for water vapor is given by
Fig.26.9-E2.

Fig.26.9-E2 Thermodynamic H-S diagram for enthalpy calculation of primary ejector

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149

Following the two-flow-passage model introduced in Chapter 26.6-2, the isentropic change in the
motive steam indicates the downward arrow from state 1 to 2. Regarding this stream A, the motive
steam enthalpy becomes 𝐻 = 2,763 kJ/kg at 𝑝 = 0.7 MPa and 𝐻 = 1,790 kJ/kg at 𝑝 =0.0004 MPa . Regarding stream B, the isentropic change is expressed as the upward arrow from state 0
to 4. The vapor enthalpy of stream B becomes 𝐻 = 2,490 kJ/kg at 𝑝 = 0.0004 MPa and 𝐻 =2,650 kJ/kg at 𝑝 = 0.0012 MPa .
The isentropic enthalpy difference of the stream A meaning the available energy for stream B becomes
𝐻 − 𝐻 = 2,763 − 1,790 = 973 kJ⁄kg.
The isentropic enthalpy change of the stream B from 𝑝 to 𝑝 is given in the Mollier diagram:
𝐻 − 𝐻 = 2,650 − 2,490 = 160 kJ⁄kg.
According to Fig.26.7-1, since the suction pressure is 0.0004 MPa, the diffuser efficiency can beassumed as 𝜂 = 0.55 for the safe-side design.
The ejector efficiency defined by Eq.(26.8-1) is calculated as= 0.164𝜂 =From the curve of 𝜂 = 0.55 in Fig.26.8-1, the ratio of motive steam to water vapor is obtained as
𝛽 = 𝑤 ⁄𝑤 = 1.1. Then the motive steam for the primary ejector amounts to
𝑤 = 𝛽 𝑤 = 1.1 × 15 = 16.5 kg/h
The enthalpy of the mixed steam 𝑤 + 𝑤 is obtained by the following enthalpy balance:𝑤 𝐻 + 𝑤 𝐻 = (𝑤 + 𝑤 )𝐻′That is𝐻′ == 2,633 kJ/kg K
In this condition, 𝑇′ = 65 ℃ (superheated steam)[Secondary ejector]
The next enthalpy-entropy diagram (Mollier H-S diagram) for water vapor is given by Fig.26.9-E3.

Fig.26.9-E3 Thermodynamic H-S diagram for enthalpy calculation for secondary ejector

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Multi-stage Ejector System

The vapor to be sucked into the secondary ejector is𝑤′ = 𝑤 + 𝑤 = 31.5 kg/h
The compression ratio is 𝛼 = 5.
Since 𝑝 = 0.0012 MPa, the diffuser efficiency can be assumed as 𝜂 = 0.57 for the safe-side
design.
The isentropic enthalpy difference of the stream A of the secondary ejector is given by
𝐻 − 𝐻′ = 2,763 − 1,890 = 873 kJ⁄kg.
The isentropic enthalpy change of the stream B from 𝑝 to 𝑝 is given in the Mollier diagram:𝐻′ − 𝐻′ = 2,785 − 2,525 = 260 kJ⁄kg.
𝜂′ == 0.30From Fig.26.8-1, 𝛽′ = 𝑤′ ⁄𝑤′ = 3.0. Therefore 𝑤 = 𝛽 𝑤′ = 94.5 kg/h.
The total motive steam necessary for the two ejectors amounts to
𝑤 = 16.5 + 94.5 = 111 kg/h,
The mixed steam issuing from the secondary ejector has
𝑝′ = 0.006 MPa = 45 mmHg, 𝐻′′ = 2,730.5 kJ/kg,
Its outlet temperature 𝑇 = 120 ℃ implies a superheated steam but its saturation temperature
reduces to 37℃ at 45 mmHg.
It can be considered that this steam temperature is appropriate for the condenser using the city water(25℃) as the cooling water. As a result, the condenser can be compact as the work required of thevacuum pump is reduced very much.
The vacuum pump to be used for the 45 mmHg condenser can be made very much profitable in thework requirement as compared with the vacuum pump originally used for the evaporator.
Fig.26.9-E4 gives the summary of calculation results.

Fig.26.9-E4 Summary of calculation results

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151

26.10 Special Matters of Heed to Ejector Engineering
We have been studying the engineering aspect of steam ejectors by the thermodynamic consideration. Let us consider why a steam ejector cannot be considered by the fundamental laws of
momentum and enthalpy conservations.
The behavior of the flow and enthalpy transfer in a steam ejector is actually so difficult to analyze
by theory that we have to rely on the empirical investigations.
Fig.26.7-1 of the diffuser efficiency and Fig.26.8-1 of the calculation of the rate of motive steam
are an example of empirical databases.
In particular, Fig.26.8-1 is worthy to reconsider.
According to the two-flow-passage model, as shown in Fig.26.10-1, a steam ejector can be
regarded as a double-tube exchanger shown below:

Fig.26.10-1 Exchange of enthalpy between stream A and stream B in an ideal state

The two-flow-passage model gives the enthalpy balance between the stream A (motive steam) andstream B (low-pressure steam):𝑤 (𝐻 − 𝐻 ) = 𝑤 (𝐻 − 𝐻 )(26.10-1)
The following relation is obtained:=(26.10-2)

This relation implies that the ratio 𝛽 = 𝑤 ⁄𝑤 of motive steam to low-pressure steam to be suckedis proportional to the enthalpy efficiency 𝜂 .
As shown in Fig.26.10-2, for comparison with the empirical data, the above relation can be plottedin the schematic picture of Fig.26.8-1 which is usually employed for the practical design procedure inindustry.
From a fundamental consideration, the above enthalpy balance should have been done with 𝐻
instead of 𝐻 .
In addition, the two-flow-passage model does not take into account the interactions such as mixing
and friction loss/viscous dissipation between the two streams.

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Macroscopic Momentum Balance

Fig.26.10-2 Comparison of empirical data on ejector efficiency with ideal state variation

At any rate, it has been confirmed from the figure that a steam ejector is usually operated with anefficiency much lower than the ideal operating condition without any losses.
It should be kept in mind that ejectors still have various advantages of low cost, easy maintenance,
safe operation, simple structure, and sufficient durability.
Still much remains to be done for improving the ejector efficiency.

26.11 Macroscopic Momentum Balance for Steam Ejectors
It is worthy to reconsider the above steam ejector problems by means of macroscopic momentum
balance. (see Chapter 4.3 of PART I)
According to Eq.(4.3-7) of Chapter 4.3 of PATRT I, the macroscopic momentum balance set up overa steam ejector can be expressed as− 𝐹⃗ = 𝑤 〈𝑣 ⃗〉 + 𝑤 〈𝑣⃗ 〉 − 𝑤 〈𝑣 ⃗〉 − 𝑝 𝑆 𝑛 ⃗ − 𝑝 𝑆 𝑛 ⃗ − 𝑝 𝑆 𝑛 ⃗ + mg⃗(26.11-1)
Here 𝐹⃗ means the total holding force against steam stream including the frictional force acting on
the fluid stream.
As can be seen from the figure, the remaining variables 𝑤 , 〈𝑣⃗〉, 𝑝 , 𝑛⃗ , 𝑆 , are mass flow rate ofsteam, velocity vector, pressure, unit vector, and cross-sectional area of inlet and outlet, respectively.
The outlet mass flow rate is
𝑤 = 𝑤 + 𝑤
(26.11-2)
The gravitational effect mg⃗ can be neglected because the steam ejector is placed horizontally.
With respect to the main stream direction, the equation is simplified as−𝐹 = 𝑤 𝑣 − 𝑤 𝑣 + 𝑝 𝑆 − 𝑝 𝑆(26.11-3)
𝑣 = 𝑤 ⁄𝜌 𝑆 and 𝑣 = (𝑤 + 𝑤 )⁄𝜌 𝑆(26.11-4)
If the total holding force is known, the mass flow rate of the motive steam required can be
determined.

Thermodynamic Approach for Thermal Engineering Processes

Fig.26.11-1 Macroscopic momentum balance set up around a steam ejector

Nomenclature
𝑎
𝐶𝑝
𝐷
𝐷 ,𝐷
𝐹
𝐺
𝐻 , 𝐻
ℎ
𝐾
𝐾 ,𝐾
𝑘 ,𝑘
𝑚
𝑛
𝑅𝑒
𝑆𝑐
𝑇, 𝑡
𝑥 ,𝑦
𝑧
𝛼
𝛿 ,𝛿
𝜆
𝜇
𝜇

effective interfacial area per unit packed volume, [m2/m3]
heat capacity, [J/kmol K]
overhead product, [kmol/s]
diffusivity in gas- and liquid-phase, [m2/s]
feed rate, [kmol/s] or F-factor [(m/s)(kg/m3)0.5]
superficial molar gas-mass velocity, [kmol/m2s]
latent heat (enthalpy) of x- and y- component, [J/kmol]
HETP (Height Equivalent to a Theoretical Plate), [m]vapor-liquid equilibrium constant, [ - ]overall mass transfer coefficients defined by vapor-phase and liquid-phase concentrations [kmol/m2s]
mass transfer coefficients of vapor-phase and liquid-phase film [kmol/m2s]
slope of equilibrium curve, 𝑑𝑦 ⁄𝑑𝑥 ,theoretical plate/stage number, [ - ]Reynolds number, [ - ]Schmidt number, [ - ]temperature, [K]mole fraction of component A, liquid- and vapor-phase, [ - ]height of packing section, [m]relative volatility, [ - ]thicknesses of gas-phase and liquid-phase film, [m]stripping factor, [ - ]viscosity, [kg/m s]
Joule-Thomson coefficient, [ - ]Subscripts
A, B
component A, B
G, L
gas-phase, liquid-phase𝑖interface

153

154

Chapter 27 Visualization of Dynamic Transport Phenomena

155

CHAPTER 27
VISUALIZATION OF DYNAMIC
TRANSPORT PHENOMENA

27.1 Observation of an Organized Structure of Stirred Tank
Reactor1,2)
In any kinds of liquid-phase chemical reactions, any reactants participating in a reaction must betransferred to collide with each other. This implies that any chemical reaction makes progress withmicro-mixing due to the coupling of two or more nonlinear dynamics. In particular, complex reactions
are apt to become hysteretically unstable depending upon the degree of micro-mixing.
We have an experience in observing nonlinear dynamical structure of chaotic mixing motion at very
low Reynolds numbers in a disk-turbine stirred tank reactor with the aid of flow-visualization
technique.1)
The Reynolds number is defined using the rotation number of the disk turbine impeller as
𝑅𝑒 =(27.1-1)

Fig.27.1-1 Experimental apparatus for visualization of a stirred tank flow
(1) LDV measuring system, (2) Ar-laser system, (3) Digital video camera, (4) Motor control system,(5) Disk turbine impeller (a: six-bladed, b: four-bladed), (6) Cylindrical test vessel

156

Observation of an Organized Structure

The experimental apparatus shown in Fig.27.1-1 consists of a transparent cylindrical vessel (200 mmID) with no baffle plates which is equipped with four-bladed or six-bladed disk turbine impeller. The
upper end of liquid solution is fixed with a flat plate, so that the ratio of the liquid height to the innerdiameter of the vessel is fixed at unity and the impeller is placed at height ℎ = 100 mm, i.e. ℎ⁄𝐷 =0.5 . An aqueous glycerin solution (𝜇 = 0.85~1.5 Pa ∙ s ) which contains a small amount of basic
solution (0.5 N-NaOH + glycerin) is colored with fluorescent green dye. (Laser fluorescent method)
After the basic solution filled in the vessel reaches a certain steady flow state, a small amount of acidic
solution (0.5 N-HCl + glycerin) is poured from the top of the vessel, so that the fluid region locallyneutralized due to the mixing of acidic and basic solutions can be decolored. The cross-section of theflow field is illuminated by a plane sheet of Ar-laser light.

Fig.27.1-2 Ring-doughnut-shaped mixing regions appearing above and below the rotating impeller

Fig.27.1-2 shows two ring-doughnut-shaped toroidal regions accompanied by three satellites (named“isolated mixing region”) appearing respectively above and below the rotating impeller when 𝑅𝑒 < 100.
From the video images, the toroidal motion can be observed.

Visualization of Dynamic Transport Phenomena

157

Fig.27.1-3 Cross-sectional views of the lower isolated mixing regions observed with four-bladed andsix-bladed turbine impellers.

Fig.27.1-3 shows two cross-sectional views of the lower isolated mixing region observed with the sixbladed and four-bladed turbine impellers. The torus structure depends on the number of turbine blades.
Four satellite-like filaments go around the central core for the case of four-bladed turbine (𝑅𝑒 = 29).
Three filaments can be seen to go around the central core for the case of six-bladed turbine (𝑅𝑒 = 31).
A set of stable satellite islands circulate around the torus with a constant period depending on the
period of the impeller rotation. It can be conjectured that this filament-torus structure depends on the
periodical perturbation produced by the rotating turbine blades.

Fig.27.1-4 Multi-structure inside an isolated mixing region (six-bladed turbine impeller)

158

Observation of an Organized Structure

Another set of islands have been found inside the core region in the case of six-bladed turbine (𝑅𝑒 =29). This flow structure is formed by unsteady zigzag startup operation (𝑅𝑒: 29 → 0 → 29) . Thissuggests that the isolated mixing region has a complex multi-structure consisting of various KAM tori.
Fig.27.1-5 shows the streak lines of the circulating flow surrounding the isolated mixing region
visualized by suspending small MMA particles. This picture is taken at the same time by suspending
small MMA particles in the stream of the basic solution.
The trajectories of the tracer particles show the streak lines of the outer circulating flow.
It can be considered that the primary effect of mixing is controlled in the main flow region outside theisolated mixing region.
The isolated mixing region surrounded by the circulating flow shows a turnstile-lobe like structure
generated by each stroke of the passing turbine blades, which causes the chaotic mixing (alternate
stretching-folding) mixing at the boundary between the circulating flow region and the isolated mixing
region. This implies that the dynamically-chaotic micro-mixing occurs even in a very complex fractal/multi-structure. The streak lines near the outer edge of turbine blades are oscillating with the fluid
ejected by the turbine blades.

Fig.27.1-5 Streak lines of circulating flow surrounding an isolated mixing region (six-bladed turbine
impeller)

Much remains to be done for observing at higher Reynolds numbers.
1)
2)

Makino, T., Kaise, T., Ohmura, N., and Kataoka, K., Lase-optical observation of chaotic mixing structure in a stirred vessel, Proc. of
10th Int. Symp. on Application of Laser Techniques to Fluid Mechanics, July 10-13, Lisbon, CD-ROM (Session 21-2) (2000)
Makino, T., Ohmura, N., and Kataoka, K., Observation of Isolated Mixing Regions in a Stirred Vessel, J. Chem. Eng. Japan, Vol.34(5),574 (2001)

Visualization of Dynamic Transport Phenomena

159

27.2 Transitional Route of Laminar to Chaotic Flow3)
27.2-1 Instability of circular Taylor-Couette flow
If the degree of flow intensity goes beyond a critical point, many kinds of shear flow generally
become unstable due to non-linear dynamic phenomena. We know that a laminar flow in a straight
circular tube becomes unstable when the Reynolds number reaches 𝑅𝑒 = 2,100. We study in
Chapter 3.2 of PART I that when the flow rate is increased enough, the velocity gradient near the tube
wall becomes so steep that an unstable rotating motion due to the moment effect of shear stress occurs
to generate turbulent eddy motions.
In the case of the straight tube flow, there is a transitional region in between 2,100 < 𝑅𝑒 <10,000. The width of the transitional region depends on various conditions such as pipe roughness,tube length from its entrance, and perturbation superimposed in the approaching stream. The microstructures in the transitional region are so complicated that the route to turbulence generation has not
yet realized enough.
At any rate, in this kind of parallel flow, the Reynolds number can be considered as a universal
dimensionless parameter indicating the flow states.
Shear flow between two concentric cylinders with the inner one rotating is a very interesting shear
flow for investigation of the transitional scenario to chaos in the pure physical science. This flow is
firstly introduced at page 51 of Chapter 6.2. In this flow system, the Reynolds number is defined forthis flow system as
𝑅𝑒 =(27.2-1)
Here the annular gap width d is a very important controlling parameter when the flow condition is
specified because there is a distinction in flow structure between the wide-gap and small-gap systems.
Therefore the flow problem is considered by separating into a wide-gap problem and a small-gap
problem.
Fig.27.2-1 gives a schematic picture of time-independent laminar Taylor vortex flow and timedependent wavy Taylor vortex flow.
When the rotation speed of the inner cylinder, i.e. the Reynolds number goes beyond a certain criticalpoint (𝑅𝑒 ∗ = 𝑅𝑒⁄𝑅𝑒 = 1), the basic laminar shear flow in the annulus between the concentric
cylinders becomes unstable to lead to the transition to a laminar cellular vortex flow, referred to as
Taylor vortex flow.4)
This time-independent flow regime is drawn on the left side of Fig.27.2-1.
As the rotation speed is further raised, this flow system exhibits a sequence of distinct timeindependent followed by time-dependent flow regimes until chaotic turbulence occurs.
The first time-dependent wavy vortex flow is drawn on the right side of Fig.27.2-1.
This transitional scenario is of great importance for the fundamental concept of non-linear mechanics
in physics.

160

Transitional Roue of Laminar to Chaotic Flow

Fig.27.2-1 Schematic pictures of time-independent and time-dependent Taylor vortex flows

27.2-2 Visualization of time-independent and time-dependent vortex flows3)
For observing such time-dependent vortical structures, the flow visualization technique suspending
fine aluminum platelets in the viscous test fluid is very often utilized. The flow behavior can beobserved clearly as variations in the transmitted or reflected intensity from the scattered light of thesuspending aluminum platelets aligned with the stream. We can observe dynamic behaviors of theoutside surfaces of flow structure. (Aluminum platelet flow visualization method)Fig.27.2-2 Flow visualization by suspending fine aluminum platelets in a stream

All the following photographs3) of Taylor vortex flows visualized by fine aluminum platelets aretaken in a wide annular gap between two concentric rotating cylinders. The radius ratio of the innercylinder to the outer cylinder is 𝜂 = 𝑅 ⁄𝑅 = 0.68.

Visualization of Dynamic Transport Phenomena

161(1) Laminar Taylor vortex flow
The Taylor-Couette flow system has the first instability from purely laminar Couette flow to astable laminar cellular vortex flow called “Taylor vortex flow.”

Fig.27.2-3 Time-independent laminar Taylor vortex flow visualized by suspending fine aluminum
platelets

The dark horizontal lines indicate the boundary between the neighboring two vortices. Ringdoughnut-shaped counter-rotating vortices are arrayed along the cylinder axis. The first transition to atime-independent laminar vortex flow called “Taylor instability”4) depends upon the radii 𝑅 ⁄𝑅 of
the two cylinders, the critical rotation speed of the inner cylinder and the kinematic viscosity of fluid.
As distinct from the Reynolds number for straight pipe flows, the critical Reynolds number cannot
easily be determined for the Taylor-Couette flow system.
Thus it is very difficult to uniquely define the flow system since the Reynolds number still depends on
𝑑 ⁄𝑅 .
Usually we consider this flow problem by separating into small-gap and large-gap problems.
The critical Reynolds number for small annular gap is given by𝑅𝑒(27.2-2)𝑑 ⁄𝑅 = 41.2 if 𝑑 ⁄𝑅 ≪ 1
This critical Reynolds number group 𝑅𝑒𝑑 ⁄𝑅 for the small gap problem can be predicted by thelinear theory of hydrodynamic stability.
We observe for the present equipment with 𝜂 = 𝑅 ⁄𝑅 = 0.68 that the critical Reynolds number
becomes 𝑅𝑒 = 86.2, i.e. 𝑅𝑒 ∗ = 1.
Since the critical Reynolds number still varies with the annular gap, the Reynolds number ratio 𝑅𝑒 ∗ =𝑅𝑒⁄𝑅𝑒 is used as the dimensionless flow parameter.
The axisymmetric laminar vortex flow continues to keep the stabile structure over some range of the
Reynolds number above the critical point. All the flow properties such as local pressure and velocity
components are kept time-independent everywhere in this flow regime.

162

3.
4.

Transitional Route of Laminar to Chaotic Flow

Kataoka, K., “Taylor vortices and instabilities in circular Couette flows,” Encyclopedia of Fluid Mechanics, Gulf Pub., Houston, ed.
By N.P. Cheremisinoff, Chap.9, 236-274 (1986)
Taylor, G. I., Phil. Trans., A223, 289 (1923), Proc. Roy. Soc. (London), A151. 494 (1935)(b) Singly-periodic wavy vortex flow
As the rotation speed is increased further, the axisymmetric Taylor vortex flow becomes unstable andthe instability leads to a new time-dependent vortex flow called “wavy vortex flow.”

Fig.27.2-4 Time-dependent singly periodic wavy vortex flow (m= 3)

Fig.27.2-4 shows time-dependent wavy vortex flow observed when 𝑅𝑒 ∗ > 𝑅𝑒
The number of azimuthal waves is 3.

∗

As can be seen in the photograph of Fig.27.2-4, the vortex boundaries show time-dependent S-shapedsinusoidal waves. This instability is known as wave instability. The second critical Reynolds number𝑅𝑒
depends on the ratio of the gap to the inner cylinder diameters. The wavy vortex flow with mwaves in the azimuthal direction shows many stable states at a given Reynolds number. The timedependent wavy vortex flow having a single fundamental frequency and its harmonics is called “singly
periodic wavy vortex flow.”
The fundamental frequency corresponds to the frequency of traveling azimuthal waves passing a pointof observation.
( c) Doubly-periodic wavy vortex flow
As the Reynolds number is raised further, the wavy vortex flow becomes more complex at 𝑅𝑒
where the amplitude of the azimuthal waves varies periodically with time.

,Visualization of Dynamic Transport Phenomena

163

Fig.27.2-5 Doubly periodic wavy vortex flow

This amplitude-modulated wavy vortex flow called “doubly periodic wavy vortex flow” has two
fundamental frequencies; the first fundamental frequency comes from the basic wave motion and thesecond fundamental frequency from the periodicity of amplitude-modulation. This transition appears at
about 𝑅𝑒 ∗ = 16.
These quasi-periodic wavy vortex flow regimes exhibit various bifurcation properties depending not
only upon the equipment dimensions such as the annular gap width and height but also upon the startup operation as to how to reach a steady state.
(d) Weakly turbulent wavy vortex flow
Then when 𝑅𝑒 > 𝑅𝑒 , the doubly periodic vortex flow undergoes the next transition to weakly
turbulent wavy vortex flow where chaotic turbulence is superimposed on the wavy vortex structure.
It can be considered that this transition implies the generation of chaotic turbulence.
It is still a question as to whether or not the generation of chaos results from nonlinear interaction of
fundamental frequencies and their harmonics of azimuthal wave motion.

Fig.27.2-6 Weakly turbulent wavy vortex flow (m = 6)164Transitional Route of Laminar to Chaotic Flow(e) Fully turbulent Taylor vortex flow
As the Reynolds number goes beyond a critical value 𝑅𝑒, the azimuthal waves disappear but the cellularvortex structure is still preserved.
The author obtained a triply-periodic like spectrum from an electrochemical method but its existence
remains denied.
For the present, it is generally considered that no triply periodic wavy vortex flow has yet confirmed.

Fig.27.2-7 Fully turbulent Taylor vortex flow

27.2-3 Transitional scenario to generation of chaos in Taylor-Couette flow1,3)
For the present, as indicated in Fig.27.2-8, the following scenario of transition to chaos is approved
from many investigations observed by various observation techniques:
The axisymmetric circular shear flow (Couette shear flow) becomes unstable to form a timeindependent cellular vortex flow (Taylor vortex flow) when the Reynolds number reaches its criticalvalue 𝑅𝑒 .2) This first transition to the time-independent Taylor vortex flow can be analyzed by thelinear instability model. For very small gap width system (𝑑 ⁄𝑅 ≪ 1), the critical Reynolds number
𝑅𝑒 is given by Eq.(27.2-2). The Reynolds number is further raised beyond 𝑅 ∗ ≥ 𝑅𝑒 ⁄𝑅𝑒 , theTaylor vortex flow changes into a time-dependent vortex flow called “Singly-periodic wavy vortex flow.”
A regularly periodical wavy motion appears with a characteristic frequency in the cellular structure.
The second transition to the wavy vortex flow is analyzed by the supercritical non-linear analysis. The
characteristic wave frequency and the number of azimuthal waves are investigated by very many
researchers from the bifurcation problem.3) The next transition occurs when 𝑅 ∗ ≥ 𝑅𝑒 ⁄𝑅𝑒 . Thistransition from the singly- to the doubly-periodic wavy vortex flow can be interpreted by the non-linear
interaction of two independent frequencies of wave motion. In this quasi-periodic flow regime, thecellular vortical motion exhibits more complicated wave motion amplitude-modulated with two
independent characteristic frequencies 𝑓 and 𝑓 : the first one comes from the basic azimuthal wave
motion and the other one from the periodicity of amplitude-modulation.
However regarding the next transition, there is still a question as to whether the triply-periodic wavy
vortex flow exists or not. This fourth transition which this author observed by the electrochemical
method is not approved. The physics researchers are wondering that it seems a very long transitional
state. When 𝑅 ∗ ≥ 𝑅𝑒 ⁄𝑅𝑒 , the quasi-periodic wavy vortex flow exhibits not so large irregular
disturbances inside the stable wavy vortex structure.

Visualization of Dynamic Transport Phenomena

165

That can be considered as the generation of chaotic turbulence. For the present, they think that the chaotic
turbulence appears after the doubly-periodic wavy vortex flow has become unstable. When 𝑅 ∗ ≥𝑅𝑒 ⁄𝑅𝑒 , the weakly-turbulent wavy vortex flow changes into the fully-turbulent vortex flow where
the cellular structure is still alive but the azimuthal wave motion cannot clearly be detected.

Fig.27.2-8 Scenario of transition to chaos for Taylor-Couette flow
1.
2.
3.

Kataoka, K., Chapter 9 Taylor Vortices and Instabilities in Circular Couette Flows, Encyclopedia of Fluid Mechanics, Vol.1, 236-274,
Gulf Pub., Houston (1986)
Taylor, G.I., Phil. Trans. Roy. Soc. London, A223, 289 (1923)
Coles, D., J. Fluid Mech., Vol.14, 385 (1965).

27.3 Visualization of Mass Transfer Distribution
27.3-1 Mass transfer distribution in Taylor-Couette flow1)
Chapter 14.4 introduces that the electrochemical reaction can be made use of for measuring local
and time-dependent mass transfer under the limiting current condition. Fig.27.3-1 indicates theelectrochemical method measuring the axial distribution of mass transfer in a Taylor-Couette flow
system. The inner rotating copper cylinder served as the anode and the outer stationary copper cylinder
serves as the main cathode. In the main cathode wall, many isolated cathodes are embedded forobserving local mass transfer of copper ions.
The following electrolytic reaction is utilized for observing axial distribution of mass transfer on the
inner wall of the outer stationary cylinder in a Taylor vortex flow.
(anode)Cu – 2 𝑒 → Cu
(cathode)
Cu + 2 𝑒 → Cu
The principle of this ionic mass transfer method is explained in Chapter 14.2.

166

Visualization of Mass Transfer Distribution

As shown in Fig.27.3-1, a circular copper wire is inserted into the hole drilled in the main cathode,and fixed with resin adhesive used as the electric insulator, and then the electrode surface is smoothly
flattened.
By making isolated point cathode at the same potential (voltage) as the main cathode, we canobserve local film coefficients of mass transfer on the inner wall of the outer cylinder.
Fig.27.3-2 is the experimental apparatus for measuring local mass transfer of Cu++ ions to the innerwall surface of the outer cylinder at rest.
The working fluid is an aqueous solution of CuSO4 (1.15~1.75 mol/m3) which contains
2 × 10 mol/m3 H2SO4 as the supporting electrolyte.
An appropriate quantity of glycerol is added to adjust the fluid viscosity.
The mass flux on the wall of the outer cylinder wall is measured at the limiting current condition byusing the following equation:
𝑁 = 𝐼 ⁄(𝐹 𝑆) = 𝑘 𝐶
where 𝐶 at tis the bulk fluid concentration he limiting current condition, the concentration of copper
ion becomes zero
The viscosity of the working solution is adjusted by adding glycerine in thw range of Sc = 3,000 to 7.7
× 10

Fig.27.3-1 Electrochemical measuring system for two concentric copper cylinders with the inner
one rotating1)

Visualization of Dynamic Transport Phenomena

167

Fig.27.3-2 Experimental apparatus of electrochemical method for two concentric cylinders with theinner one rotating.1)

1)

Kataoka, K., Doi, H., and Komai, T., Heat/mass transfer in Taylor vortex flow with constant axial flow rates, Int. J. Heat Mass
Transfer, Vol.20, 57-63 (1977)

Fig.27.3-3 Axial variation of mass transfer corresponding to cellular vortex array

As shown in Fig.27.3-3, the axial distribution of the mass transfer coefficient on the outer cylinder
corresponds to the axially periodic array of Taylor vortices. The maximal position of mass transfer
corresponds to the vortex outflow cell boundary while the minimal position corresponds to the inflow
cell boundary. The peak-to-peak distance is equal to the axial spacing of a couple of vortices. The
wavelength ∆ is defined as the axial length of a pair of Taylor vortices.

168

Visualization of Mass Transfer Distribution

Fig.27.3-4 shows time-dependency and local variation of Cu++ ionic mass transfer observed byelectrochemical methos.1,2)

Fig.27.3-4 Time-dependent limiting current and time-averaged local mass transfer coefficient
observed by electrochemical method2)

In this experiment, we could not notice the transition from singly-periodic to doubly-periodic wavy
vortex flow regimes.
2.

Kataoka, K., Chapter 9 Taylor Vortices and Instabilities in Circular Couette Flows, Encyclopedia of Fluid Mechanics, Vol.1, 236-274,
Gulf Pub., Houston (1986)

27.4 Visualization of Dynamically Organized Jet Flows
27.4-1 Hydrogen bubble method for a free jet development1)
The hydrogen bubble method is a convenient visualization technique making use of an electrolysis
of water. A small amount of NaOH is added for increasing the electrical conductivity of water. Owing
to the buoyancy effect of hydrogen bubbles, this visualization technique is limited to the slow velocity
flow condition.
A round free jet issuing from a convergent nozzle into quiescent surroundings of the same fluid
exhibits a beautiful structural development accompanied with dynamically organized structuresgenerated due to the shear forces between the jet flow and the surrounding fluid at rest.
Fig.27.4-1 is a photograph of an isothermal submerged water jet visualized by a hydrogen-bubble
technique. The white streaklines of hydrogen bubbles are produced by applying a 40-60 V d.c.
potential to a nickel zigzag-wire electrode (100 μm diameter, zigzag pitch = 1.73 mm). The
hydrogen-bubble timelines are simultaneously produced by applying 140 V pulse potential at a chosenfrequency (30-50 Hz) to a platinum straight-wire electrode (50 μm diameter). The anode plate
generating oxygen bubbles is placed far downstream of the main flow region.

Visualization of Dynamic Transport Phenomena

169

From the successive photographs of the visualized flow pattern, we can draw a schematic picture ofdeveloping jet with organized structures shown on the right of the figure.
In the well-designed convergent nozzle, close to the jet exit, there is a potential core region in thecenter where the centerline velocity remains unchanged from the initial value. Then the core
diminishes in width in the downstream direction. The instability of the initial shear layer near thenozzle lip results in a migration of vorticity to form periodic axisymmetric coherent concentrations andleads to the physical appearance of the rolling-up of the laminar shear layer. That results in the periodic
street of ring-shaped vortices. In this manner, large-scale eddies are produced in the late-transitional
region as a result of disintegration of line- or ring-vortices.
The free jet development in the axial direction is a very important problem for the convective heattransfer where a free jet impinges normally on a flat plate. The flow structure in the jet impingement
region depends on the flow structure of the approaching free jet.

Fig.27.4-1 Cross-sectional view of a free jet flow pattern visualized by the hydrogen-bubble
methodIt is considered that jet impingement heat transfer is largely controlled by unsteady large-scalecoherent structures of the approaching free jet.
Many convective heat transfer researchers paid attention to the role of large-scale eddies in theenhancement of the jet impingement heat transfer. Therefore it is important to observe dynamic change
in the coherent structures in the developing region of a free jet.
As a similar technique, a smoke wire method is available for visualization of slow gas flows.
1.

Kataoka, K., Mihata, I., Maruo, M., Suguro, M., and Chigusa, T., Quasi-periodic large-scale structure responsible for the selective
enhancement of impinging jet heat transfer, Proc. 8th Int. Heat Transfer Conf., vol.3, 1193-1198 (1986)

27.4-2 Hydrogen bubble method for a two-dimensional impinging jet2)
If a convergent slot nozzle is employed for convective heat transfer processes, jet development
accompanied by instability of the free shear layer downstream of the nozzle mouth is a very important170

Visualization of Dynamically Organized Jet Flows

factor to control the stagnation point flow behavior. By using the same hydrogen-bubble method, we
can observe the downstream instability of an intermittently generated large-scale vortex structure in thenear-field of the free jet.
It is known that the stagnation-point heat transfer usually becomes maximal when a heat transfer
surface (impingement plate) is placed at 6-8 nozzle widths, i.e. H/D = 6-8 downstream of a nozzle
mouth. As shown in Fig.27.4-2, we would like to pay attention to the unsteady coherent structuresintermittently appearing in the stagnant region of the impingement plate.
It is necessary to observe how unsteady coherent structures appear in the jet impingement region
from a viewpoint of the heat transfer enhancement in such an impinging flow.

Fig.27.4-2 Impinging jet arrangement and generation of large-scale eddies in stagnation region

A two-dimensional jet issuing from a well-designed convergent slow nozzle (exit width D) impinges
normally on the impingement plate. The nozzle-to-plate distance H is an important jet development
parameter. A zigzag Pt wire is placed 50 mm upstream in parallel to the impingement plate.

Fig.27.4-3 Intermittently formed longitudinal vortex pairs visualized by the streaklines of hydrogen
bubbles when H/D = 6

Visualization of Dynamic Transport Phenomena

171

Fig.27.4-3 is a photograph of organized vortical motions intermittently appearing across the stagnation
line.
Note: Impinging jet heat transfer is dealt with a little in Chapter 18.5 of the previous volume.

Fig.27.4-4 Schematic picture of cross-sectional view of longitudinal vortex pairs

As shown in Fig.27.4-2, when the impingement plate is in the late-transitional region (4 ≤ 𝑧⁄𝐷 ≤ 8))of a two-dimensional free jet, longitudinal vortex pairs intermittently appear across of the stagnation
line. The schematic picture of vortex pairs is indicated by Fig.27.4-4. The size of the vortex pairs tends
to become large with the development length z from the jet exit. It can be conjectured that theaugmentation of stagnation point heat transfer is controlled by these intermittent large-scale vorticalmotion.
1.
2.

Kataoka, K., Kawasaki, H., Tsujimoto, M., and Ohmura, N., Effect of longitudinal vortices on heat transfer surfaces a twodimensional jet strikes against, Proc. 10th Int. Heat Transfer Conf., Vol.3, 31-36 (1994)
Kataoka, K., Ochi, T., Uyama, K., and Ohmura, N., Time-dependent structures of a two-dimensional turbulent impinging jet
accompanied by longitudinal vortex pairs, J. Chem. Eng. Japan, Vol.37(2), 299-303 (2004)

27.4-3 Local time-dependent wall shear stresses in jet impingement region1,2,3)
It was considered to be very difficult to observe local distribution of shear stresses acting on a flat
plate. The electrochemical method is also utilized to measure local and time-dependent behaviors
occurring on a flat plate on which a free round jet impinges perpendicularly. The measuring principle
is introduced in Chapter 14.4.
Fig.27.4-5 shows that a free jet issuing from a convergent nozzle impinges normally on a flat plate.
Fig.27.4-6 is layout of two kinds of test cathodes where dimensions given are in mm. The U-electrodeis used or velocity measurement similarly to the hot-wire anemometry.
Local measurement of the mass transfer coefficient k (m/s) is accomplished using circular point
cathodes (K-electrode) which are fabricated by inserting 0.3 mm diameter nickel wires into 0.5 mm
diameter holes drilled in the main cathode wall. The K-electrodes electrically isolated from the main
cathode are held at the same potential as the main cathode for the following electrode reaction under
the diffusion-controlling condition:

172

Visualization of Dynamically Organized Jet Flows

Cathode:Fe(CN)63- + e- = Fe(CN)64Anode:Fe(CN)64= Fe(CN)63- + eLocal measurement of local, time-dependent velocity gradient on the impingement plate is also
accomplished using small rectangular cathodes (S-electrode) which are fabricated by inserting 0.2 mm
thick nickel sheets into 0.7 mm long slits normally to the radial main flow direction in the wall of themain cathode.
𝑘 = 𝑖 ⁄𝐹𝑎 𝐶(14.3-2)
The velocity gradient measurement is carried out by measuring the limiting current between each Selectrode and the anode under the condition of the main cathode inactive.
Local velocity gradient on the impingement flat plate can be obtained as the limiting current density
𝑖 on each S-electrode:𝑠 = 1.90 (𝑖 ⁄𝐹𝑎 𝐶 ) (𝐿⁄𝐷 )(14.4-1)

Fig.27.4-5 Schematic picture of a free round jet impinging normally on a flat plateFig.27.4-6 Layout of test cathodes for measuring local velocity-gradients and mass transfer
coefficients on jet impingement plate1)

Visualization of Dynamic Transport Phenomena

173

The heat or mass transfer at the stagnation point becomes maximal when the jet exit-to-impingement
plate distance H/d is H/d = 6 ~ 8. Some experimental results when H/d = 6 are shown here.

Fig.27.4-7 Lateral variation of mass transfer coefficients when H/d = 61)

Fig.27.4-7 is lateral distribution of mass transfer coefficient in the form of Sherwood number.
This indicates a typical bell-shaped distribution but the mass transfer coefficient seems to have a small
dent at the stagnation point. This behavior is still a question for discussion. As distinct from the twodimensional impinging jet, it has been not yet elucidated well for the circular impinging jet what
intermittent organized motion is formed in the stagnant region. It is observed that when H/d > 4,
instantaneous stagnation point hastily moves around the geometrical stagnation point.

Fig.27.4-8 Lateral variation of time-averaged wall shear stress and the relative turbulence intensity1)

174

Local Convective Heat Transfer on the Wall of Jet Impingement Plate

In the impingement region, the velocity gradient increases almost linearly due to the accelerating
lateral flow along the impingement plate. In the outside region of the impingement region (r/d > 0.5)
with no static pressure change, the lateral velocity is decelerated owing to the widely developing
boundary layer.
3.

Kataoka, K., Kamiyama, Y., Hashimoto, S., and Komai, T., Mass transfer between a plane surface and an impinging turbulent jet:the influence of surface-pressure fluctuations, J. Fluid Mech., vol.119, 91-105 (1982)

Note: The principle of the electrochemical method is also treated in Chapter 14 of the previous
volume.
27.5 Local convective heat transfer on the wall of jet impingement plate4,5)
An isothermal air jet, issuing vertically upward from a well-designed convergent nozzle, impinges
normally on a horizontal, circular flat plate. The jet exit has a uniform velocity distribution with low
initial turbulence (<0.5%). The circular flat plate made of copper (420 mm dia. 10 mm thick) has a hotwater jacket on the backside so as to keep it at a constant uniform temperature 𝑇 (about 52℃).
The heat transfer distribution in the jet impingement region can be measured by a heat-flux sensor
when the circular flat plate is cooled by the impinging air jet.

Fig.27.5-1 Circular copper flat plate for convective heat transfer measurement and the embeddedplug-type heat-flux sensor

Visualization of Dynamic Transport Phenomena

175

Gardon Heat-flux Sensor
A plug-type heat-flux sensor constituted by thermocouples shown in Fig.27.5-2 is available to measure local valuesof convective heat transfer coefficient.

Fig.27.5-2 Measurement of local heat transfer coefficient by wall heat-flux sensor

The hot junction (i.e. the periphery of the constantan foil) is kept at the same temperature 𝑇 as theflat copper plate. The cold junction (i.e. the spot welded with copper lead at the center of theconstantan foil) is cooled by air jet (temperature 𝑇 ). The temperature difference between these twojunctions is proportional to local coefficient of heat transfer:ℎ = 𝐾 ∆𝑇
(27.5-1)
Local wall heat-fluxes at the periphery and center of the constantan foil are respectively expressed as
𝑞 = ℎ (𝑇 − 𝑇 )(27.5-2)
𝑞 = ℎ (𝑇 − 𝑇 )(27.5-3)
The rate of heat conduction occurring within the foil from the periphery to the center is given by∆(27.5-4)
𝑄 =𝜅 ∆ 𝑆
Since the heat loss 𝑞 is rapidly compensated by the heat conduction,∆𝜅 ∆ 𝑆 = ℎ 𝑆 (𝑇 − 𝑇 )(27.5-5)
In the operating condition, ∆𝑇 = (𝑇 − 𝑇 ) ≪ 𝑇 − 𝑇 .
Therefore time-dependent wall heat-flux and heat transfer coefficient have the following linear
relationship with ∆𝑇:ℎ = 𝐾 ∆𝑇
(27.5-6)
𝑞 = 𝐾 ∆𝑇
(27.5-7)
where
𝐾 = (
(27.5-8))∆
𝐾 = ∆(27.5-9)

These two constants are determined by experiment.
Local heat-flux on the surface of the flat plate can be measured by means of a Gardon-type heat-flux
sensor5). The sensor consists of a copper hollow male-screw as a heat sink and a circular constantan
foil (14𝜇m thick) which has a spot-welded thermocouple junction of 50𝜇m dia. copper lead at its
center. The other thermocouple junction is formed at the periphery of the constantan foil. This sensor isembedded in the copper flat plate, 50 mm from the rotation axis.

176

Local Convective Heat Transfer on the Wall of Jet Impingement Plate

Another copper plug heat-flux sensor is also embedded in the same flat plate.
The axis of the convergent nozzle is fixed 50 mm away from the rotation axis of the flat plate so thatthe heat-flux sensor can be placed at an arbitrary radial position from the geometrical stagnation point.
The turbulent velocities of the approaching air jet are measured by a constant-temperature X-probe
hot-wire anemometry.
The radial distribution of wall heat-flux measured by experiment is shown in Fig.27.5-3.

Fig.27.5-3 Lateral variation of wall heat-flux with lateral distance from geometrical stagnation point
normalized by stagnation-point heat-flux and half-radius of wall heat-flux.5)

Fig.27.5-3 suggests that the radial distribution of the normalized heat transfer coefficient normalizedwith the stagnation point heat-flux becomes similar when 𝐻 ⁄𝑑 ≥ 6 if the radial coordinate is madedimensionless with respect to the radius 𝑟 ⁄where the wall heat -flux becomes 1/2 𝑞 .
The summarized correlations of stagnation point heat transfer coefficients are given by Fig.27.5-4.
It can be confirmed that the stagnation point heat transfer coefficient becomes maximal when 𝐻 ⁄𝑑 =6 ~ 8.

Fig.27.5-4 Correlations of stagnation-point heat transfer coefficients with nozzle-to- jet
impingement plate distances

Visualization of Dynamic Transport Phenomena

4.
5.

177

Kataoka, K., Sahara, R., Ase, H., and Harada, T., Role of large-scale coherent structures in impinging jet heat transfer, J. Chem. Eng.
Japan, Vol.20(1), 71-76 (1987)
Kataoka, K., Harada, T., and Sahara, R., Mechanism of heat transfer in turbulent impinging jets, Current Research in Heat and Mass
Transfer, A compendium and a Festschrift for Professor A. Ramachandran, ed. M.V. Krishna Murthy et al., Hemisphere, 81-95(1986)

Nomenclature𝑎effective interfacial area per unit packed volume, [m2/m3]
𝐶𝑝
heat capacity, [J/kmol K]𝐷overhead product, [kmol/s]
𝐷 ,𝐷diffusivity in gas- and liquid-phase, [m2/s]
𝐸
Murphree plate efficiency, [ - ]𝐸 ,𝐸point efficiency for vapor- and liquid-phase, [ - ]𝐹
feed rate, [kmol/s] or F-factor [(m/s)(kg/m3)0.5]
𝐺
superficial molar gas-mass velocity, [kmol/m2s]
𝐻 , 𝐻
latent heat (enthalpy) of x- and y- component, [J/kmol]ℎ
HETP (Height Equivalent to a Theoretical Plate), [m]ℎoutlet weir height, [m]𝐾vapor-liquid equilibrium constant, [ - ]𝐾 ,𝐾overall mass transfer coefficients defined by vapor-phase and liquid-phase concentrations [kmol/m2s]
𝑘 ,𝑘mass transfer coefficients of vapor-phase and liquid-phase film [kmol/m2s]𝑚slope of equilibrium curve, 𝑑𝑦 ⁄𝑑𝑥 ,𝑁 , 𝑁 , 𝑁 number of transfer unit, OG:overall, G:gas phase, L:liquid phase, [ - ]𝑛theoretical plate/stage number, [ - ]𝑅𝑒Reynolds number, [ - ]𝑆𝑐Schmidt number, [ - ]𝑇, 𝑡temperature, [K]𝑥 ,𝑦mole fraction of component A, liquid- and vapor-phase, [ - ]𝑧height of packing section, [m]𝛼relative volatility, [ - ]𝛿 ,𝛿thicknesses of gas-phase and liquid-phase film, [m]𝜆stripping factor, [ - ]𝜇viscosity, [kg/m s]
Subscripts
A, B
component A, B
G, L
gas-phase, liquid-phase𝑖interface

178

Chapter 28

Nonlinear Behaviors in Complex Chemical Processes

179

CHAPTER 28NONLINEAR BEHAVIORSIN COMPLEX CHEMICAL PROCESSES
28.1 Why Do We Study Nonlinear Dynamics in Complex Chemical
Processes?
In general, chemical processes proceed under an effect of coupling between chemical dynamics andmass transport controlled by fluid dynamics. In any kinds of liquid-phase chemical reactions, any
reactant solution should be mixed for increasing the frequency of collision of reactant molecules. In
industrial chemical processes. if elementary reaction steps are taken into account for reaction mechanism,
most of chemical reactions can often become difficult to control. Nonlinear dynamics of complex
chemical processes in an open system (i.e. flow system) is so complicated that we will study a little
about its simple examples in this Chapter for the design methodology of system and operation.
Any chemical reaction cannot proceed without the coupling of two or more nonlinear dynamics suchas the complex interaction of elementary reaction steps with micro-mixing of reactant solution. This
Chapter brings up the necessity of operation design as well as system design from a viewpoint of
nonlinearity in chemical dynamics coupled with fluid mixing.
28.1-1 A simplified chemical model
The Brusselator model1,2) is very interesting as a model of nonlinear chemical reactions. This model
is given by an irreversible reaction of the following form:
𝐴+𝐵 →𝐶+𝐷(28.1-1)
The elementary steps including two intermediates X and Y are expressed as
𝐴 →𝑋
𝐵+𝑋 → 𝐷+𝑌
𝑌+2𝑋 → 3𝑋
𝑋 → 𝐸(28.1-2)
The third elementary step can be considered to be autocatalytic. The intermediate component X is
produced from 2 to 3 moles by itself.
Assuming that the concentrations of A and B are almost constant since the reactant components A and
B are supplied sufficiently, the reaction rate equations for a batch system can be given by= 𝑘 𝐶 − 𝑘 𝐶 𝑥+ 𝑘 𝑥 𝑦− 𝑘 𝑥(28.1-3)= 𝑘 𝐶 𝑥− 𝑘 𝑥 𝑦
Only for simplification, the following quantities are given to those rate constants.
𝑘 𝐶 = 1, 𝑘 𝐶 = 3, 𝑘 = 𝑘 = 1(28.1-4)

180

Why Do We Study Nonlinear Dynamics?

The dynamic equations of this reaction model for a continuous stirred tank reactor (CSTR) as an open
system can be expressed as= 1− 4𝑥+ 𝑥 𝑦− 𝑓𝑥(28.1-5)= 3𝑥− 𝑥 𝑦−𝑓𝑦
(28.1-6)
where 𝑓 = 𝐹 ⁄𝑉 denotes a space velocity of efflux from the CSTR shown in Fig.28.1-1.

Fig.28.1-1 Open system of perfect mixing tank reactor

The space velocity can be adopted as a bifurcation parameter. Such a kind of open system can beassumed to be in a perfect mixing condition.
28.1-2 Phase space analysis
The nonlinear behavior of the ordinary differential equations can be analyzed in phase space if timedependent concentrations of the intermediate components X and Y are drawn as a trajectory in phasespace.
Firstly solving the governing equations for a batch system (𝑓 = 0) under the initial conditions 𝑥(0) =𝑦(0) = 0, a limit cycle attractor can be obtained in Fig.28.1-2. This indicates that both intermediates
have a singly-periodic oscillation in concentration. It should be noted that the trajectory of the dynamic
solution never converges on the stationary-state solution: 𝑥 = 1 and 𝑦 = 3.
It has been confirmed that this chemical dynamics gives the same attractor regardless of the initialcondition.
What is an attractor?
Attractors are subsets of the phase space of a dynamical system. As its simple examples, attractors arethought as being simple geometric subsets of the phase space, like points, lines, and surfaces (e.g. tori)
as simple regions of three-dimensional space. For example, a limit cycle is a periodic orbit of acontinuous dynamical system. As shown below, a trajectory can be obtained in the dynamical phasespace by a temporal variation plotting the X concentration against the Y concentration.

Nonlinear Behaviors in Complex Chemical Processes

181

Fir.28.1-2 Attractor of Brusselator model in a batch system.

28.2 Nonlinearity of Brusselator Model
28.2-1 Bifurcation analysis of Brusselator model for a CSTR
The governing equations for an open system Eqs (28.1-5 and 6) can be solved for various spacevelocities under the common initial condition: 𝑥(0) = 𝑦(0) = 0. The space velocity 𝑓 = 0.1 implies
that 10% of the reacting fluid in the reactor is exchanged for the fresh feed fluid every unit time. Fig.28.21 shows the temporal variation in concentration of those two intermediates X and Y.
Both intermediates seem to converge on the stationary-state point attenuating the oscillating behavior.

Fig.28.2-1 Temporal variation of concentration of two intermediate components X and Y.

Figure 28.2-2a indicates that when f =0.100, the trajectory converges attenuating its own original
oscillation towards the stationary-state solution point. It has, however, been confirmed that if the spacevelocity is increased to 0.11, the trajectory converges very fast and directly towards the stationary-state
point without oscillation. Figure 28.2-2b shows the trajectory converges on the stationarystate point after very small oscillations. This suggests that there is a bifurcation point between
f = 0.105 and 0.11. This implies that only 5% deviation from the feed rate in this region gives rise toquite a big difference in transient phenomenon.

Fig.28.2-2 Attractor of an open system when (a) f = 0.1 and (b) 0.105.

182

Nonlinearity of Brusselator ModelOn the other hand, if f is a little bit less than 0.1, the convergence of the oscillating trajectory towards
the stationary-state point is decelerated, as shown in Fig.28.2-3a.
This diagram does not show the further trajectory towards the stationary-state point owing to very slow
convergence. Figure 28.2-3b indicates that the trajectory shows a limit cycle when f = 0.095. Thissuggests that the other bifurcation point lies between f = 0.095 and 0.098. For the case of practicalcommercial-size reaction systems, there very often appear disturbances superimposed on the feed rate.
Stable operation mode should be considered taking into account the effect of the periodical
disturbances superimposed on the space velocity from a viewpoint of such a delicate bifurcation behavior.

Fig.28.2-3 Attractor of an open system when (a) f = 0.098 and (b) 0.095.

The dynamic equation can also be solved for the case when a certain amplitude (fraction) fa of
periodical disturbances are superimposed on the space velocity f = 0.1:
(28.2-1)
f  0 .1(1  f a sin 2t )The results calculated for fa = 0.05 and 0.02 are shown in Fig.28.2-4.

Fig.28.2-4 Doubly-periodic attractor with periodical disturbances in space velocity.

Both diagrams show that the trajectory of doubly-periodic oscillation is gradually attenuated but theoscillation coming from the forced oscillation of the feed rate would not disappear.
28.2-2 Bifurcation scenario of Brusselator model
Table 28.2-1 gives a bifurcation scenario depending on space velocity as a control parameter. It has
been found that the limit cycle oscillation occurs only in the limited range of space velocity. In orderto guarantee a very stable stationary-state, the space velocity should be set at a considerably high value
without disturbance for this process.

Nonlinear Behaviors in Complex Chemical Processes

183

Table 28.2-1 Bifurcation Scenario of an Open System of Brusselator
Space Velocity :f = F/V
0
0.08
0.095
0.098
0.1
0.105
0.11
0.2

Type of Trajectory
A
A
A
B
B
B
C
C

A: Singly-periodic oscillation (Limit cycle) without attenuation
B: Singly-periodic attenuating oscillation converging
on the stationary-state fixed point
C: Transient trajectory without oscillation converging directlyon the stationary-state fixed point

1.
2.

Prigogine, I., Thermodynamic Theory of Structure, Stability, Fluctuations, Wiley-Interscience, London (1971)
Scott, S. K., Chemical Chaos, Clarendon, Oxford Chapter 3, p.60 (1991)

28.3 Instability of a Complex Reaction in a CSTR
28.3-1 Belousov-Zhabotinskii reaction
As an example of complex reactions, the following Belousov-Zhabotinskii (B-Z) reaction areinvestigated by many researchers.
Elementary Steps of Belousov-Zhabotinskii reaction (FKN scheme)
Br- + HOBr + H+ ↔ Br2 + H2O
HBrO2 + Br- + H+ ↔ 2HOBr
BrO3- + Br- + 2 H+ ↔ HBrO2 + HOBr
2 HBrO2 ↔ HOBr + BrO3- + H+
BrO3- + HBrO2 + H+ ↔ 2 BrO2∙ + H2O
Ce3+ + BrO2∙ + H+ ↔ Ce4+ + HBrO2
Ce4+ + BrO2∙ + H2O ↔ Ce3+ + BrO3- + 2 H+
Br2 + CH2(COOH)2 ↔ BrCH(COOH)2 + Br- +H+
6 Ce4+ + CH2(COOH)2 + 2 H2O ↔ 6 Ce3+ + HCOOH + 2 CO2 + 6 H+
4 Ce4+ + BrCH(COOH)2 + 2 H2O ↔ 4 Ce3+ + HCOOH + Br- + 2 CO2 + 5 H+

(R1)
(R2)
(R3)
(R4)
(R5)
(R6)
(R7)
(R8)
(R9)
(R10)

This reaction involves the following autocatalytic step:(R5+2R6)
2 Ce3+ + BrO3- + HBrO2 + 3 H+ ↔ 2 Ce4+ + 2 HBrO2 + H2OThis autocatalytic step switches on or accelerates the overall rate of reaction.
We are interested in an experimental work from a viewpoint of reactor design and operation.
Fig.28.3-1 shows an experimental setup. Three kinds of CSTRs different in volume but similar inshape are used for examining the effect of reactor volume. Their effective volumes are 49 ml, 99 ml,
and 159 ml, respectively.

184

Instability of a Complex Reaction

As described in the Brusselator model (28.2), this kind of complex reaction is so sensitive to theinitial condition that the reaction starts from the condition of empty reactor filled with nitrogen gas
only. Four kinds of feed solutions shown in Table 28.3-1 are supplied separately at a specified feed rate
into the reactor by a common peristaltic pump. The reaction temperature is kept at 25℃. The mean
residence time based on the total rate of the four feeding streams is changed as a controlling parameter.
Each CSTR used is well mixed with a three-bladed propeller rotating at 2,300 rpm.

(1) Feed containers (2) Peristaltic pump (3) Magnetic pump (4) Cylindrical glass reactor ,(5) Stirrer (6) Pt-electrode (7) Saturated KNO3 agar bridge (8) Ag-AgCl electrode (9) DC
amplifier (10) Disc recorder (11) mV recorder (12) Constant temperature water bath (13) Cuwire (14) Glass tube
Fig.28.3-1 Experimental setup for observing dynamic chemical oscillations of BZ reaction

The redox potential based on the activity ratio of Ce4+/Ce3+ is measured as the dynamical signal of
the reaction with a platinum electrode. At very small mean residence times (large space velocity) , thereaction state becomes steadily dynamically-balanced, so that any chemical oscillations do not occur.
At very large mean residence times, the reaction field becomes in chemical equilibrium. In a relatively
wide range of mean residence time between these two limits of reaction state, there appear two typesof oscillations depending upon the mean residence time: (1) single-peak periodic and (2) chaotic
oscillations. Some of the sustained chemical oscillations experimentally observed are shown in
Figs.28.3-2~4. Any CSTR shows chaotic oscillations when the mean residence time becomes large.
Table 28.3-1 Four feed solutions supplied separately into CSTRs
Feed
Feed 1
Feed 2Feed 3
Feed 4

Reagent
Malonic acid CH2(COOH)2
Sodium bromate NaBrO3
Sulphuric acid H2SO4
Potassium bromide KBr
Cerium sulphate
Ce2(SO4)3

Concentration (kmol/m3)
1.20
0.56
0.80
1.4 × 100.40 × 10

Nonlinear Behaviors in Complex Chemical Processes

18528.3-2 Chemical oscillations sustained in a continuous B-Z reaction
It can be seen from Figures 28.3-2 to 4 that the temporal variation of the redox potential measured
by the platinum electrode exhibits three-different chemical oscillations at the respective mean
residence times.

Fig.28.3-2 Chemical oscillations sustained in a small CSTR (V = 39 ml)

Fig.28.3-3 Chemical oscillations sustained in a middle-sized CSTR (V = 99 ml)

Fig.28.3-4 Chemical oscillations sustained in a large CSTR ( V = 159 ml)

186

Instability of a Complex Reaction

When 𝜏 = 35 min in the two smaller reactors, there appears an intermittent oscillation between the
single-peak and chaotic oscillations.
In the case of the largest reactor (V = 159 ml), the stable single-peak chemical oscillations survive tosome larger region of the mean residence time.
Table 28.3-2 Bifurcation scenario of chemical oscillations depending upon reactor volume
Mean Residence Time
Reacting Liquid Volume V (ml) in CSTR
49 ml
99 ml
159 ml𝜏 (min)20
Single PeakSingle PeakSingle Peak
30
Single PeakSingle PeakSingle Peak
35
Intermittent
Intermittent
Single Peak
40
Chaotic
Chaotic
Single Peak
45
Chaotic
Chaotic
Single Peak
50
Chaotic
Chaotic
Chaotic60
Chaotic
Chaotic
Chaotic
Table 28.3-2 shows the bifurcation scenario of chemical oscillations obtained with the threedifferent-sized CSTRs. The largest reactor indicates a transition point different from the other two
smaller ones.
It should be considered that there is a narrow transition window having peculiar bursting-likeintermittent oscillation between the single peak and chaotic chemical oscillations. Since the reciprocalof mean residence time means space velocity, this bifurcation scenario may be contrasted with that of
the Brusselator model (Table 28.2-1). We can conjecture that any complex chemical reactions tend tohave chaotic chemical oscillations when the mean residence time becomes large.
Apart from temperature control, what reaction state should be attained by steady operation depends
upon the following various mixing conditions: (1) initial condition and start-up operation, (2) degree of
mixing/micromixing, and (3) way and position for feeding and effluxing.
1.
2.

Kataoka, K., Ohmura, N., Watanabe, S., Sumida, K., and DEki, S., Electrochemical observation of instabilities of the BelousovZhabotinskii reaction in CSTR, J. Appl. Electrochem., vol.24, 647-651 (1994)
Kataoka, K., Plenary Lecture , 2000 Annual Meeting AIChE, Los Angeles, Nov.12-17 (2000)

28.4 Dynamical Behavior of an Emulsion Polymerization Process
Nonlinear dynamics such as self-sustained chemical oscillations is also encountered in continuous
emulsion polymerization processes. Usually in order to stabilize an emulsion polymerization process, asingle CSTR is very often employed in industrial large-scale productions.
Even at steady operation, there can appear self-sustained oscillations in theelementary steps.
It is known in the process of emulsion polymerization that self-sustained oscillations observed intime-dependent monomer conversion result from intermittent nucleation of primary polymer particles.
As distinct from water-insoluble monomer such as styrene, an emulsion polymerization of vinyl
acetate exhibits an interesting mechanism owing to solubility in aqueous phase. The self-sustained
oscillations observed in time-dependent monomer conversion can be considered to result fromintermittent nucleation of the primary polymer particles. The main interest in this section is focused on
the instability of continuous polymerization of water-soluble monomer due to the water-soluble
emulsifier below CMC (Critical Micelle Concentration, see Chapter 21.2).

Nonlinear Behaviors in Complex Chemical Processes

187

Reaction Scheme of Emulsion Polymerization
Decomposition of water-soluble initiator: I →Initial step of propagation: R ∙ + M → P ∙Subsequent propagation:
P ∙ + M → P
Termination (1):P ∙ + P ∙ →Termination (2):P ∙ + P ∙ →Entry:
P + latex particle

(S1)
(S2)
(S3)
(S4)
(S4)’
(S5)

2R∙
∙
P
P
→

+ P
entry

28.4-1 Experimental method of continuous emulsion polymerization
The recipe of the continuous emulsion polymerization in a CSTR is indicated in Table 28.3-1.
Table 28.4-1 Recipe of continuous emulsion polymerization(A) Initial contents of reactor(B) Initiator solution(C) Initiator & emulsifier
In solution tank(D) Monomer tank

Reagent
Distilled water
Vinyl acetate monomer
Sodium dodecyl sulfate
Ammonium peroxodisulfate
Distilled water
Ammonium peroxodisulfate
Sodium dodecyl sulfate
Vinyl acetate monomer

Quantity
164 ml
22 ml
0.131 g0.218 g/10 ml water
4,000 ml
5g
3g
600 mlFig.28.4-1 shows an experimental setup for the continuous emulsion polymerization. The CSTR
(effective volume = 196 ml) (1) consists of a cylindrical glass vessel with water jacket and a stirrer (2)made of Teflon. Firstly the monomer is fed directly into the CSTR and emulsified by adding theemulsifier. After the initial operation adding the initiator, the monomer and the emulsifier and initiatorsolution are continuously supplied from the monomer tank (6) and the solution tank (7), respectively.

Fig.28.4-1 Experimental setup for continuous emulsion polymerization of water-soluble monomer.
(1) CSTR, (2) Stirrer, (3) Pump, (4) Thermocouple, (5) Nitrogen gas, (6) Monomer, (7) Aqueous solution ofemulsifier and initiator, (8) mV recorder, (9) Sampling collector, (10) Temperature-controlled water, (11)Computer

188

Dynamical Behavior of an Emulsion Polymerization

Since this reaction is very sensitive to the initial condition, the following operation procedure isfulfilled. Initially the reactor is filled with an emulsified mixture (A) and then the reaction is started
with adding the initiator solution (B) into the reactor. Ammonium peroxodisulfate (NH4)2S2O8 (watersoluble initiator) and sodium dodecyl sulfate CH3(CH2)11OSO3Na (emulsifier) are premixed at 50℃
in another stirred tank (7). The vinyl acetate monomer CH3COOCH:CH2 (D) and the aqueous
solution of initiator and emulsifier (C) are fed separately and continuously into the reactor at a constantratio of feed rates, i.e. C/D = 8. The mean residence time is defined from the total feed rate of these
two streams. The reaction temperature is kept at 55℃.
Figure 28.4-2 is a PSD of monomer droplets emulsified in an initial solution of the reactor at 800
rpm. The PSD is obtained by use of the laser diffraction particle size analyzer. This gives the mean
droplet diameter to be 2.712 m.

Fig.28.4-2 Particle size distribution of monomer droplets emulsified in the initial solution at 800 rpm.

(Mean size = 2.m)
28.4-2 Dynamical behavior of continuous emulsion polymerization processes
Figure 28.3-3 shows temporal variation of monomer conversion obtained at the same mean
residence time (   27.2 min) with various revolution numbers of the stirrer and the corresponding mean
diameters of monomer droplets (given in the brackets) of the initial emulsified solution measured by
the laser-diffraction particle-size analyzer.

Fig.28.4-3 Temporal variation of monomer conversion observed at various stirrer revolutions. The
number given in bracket means initial mean diameter of emulsified monomer droplets.

Nonlinear Behaviors in Complex Chemical Processes

189

The monomer conversion attained at steady operation depends neither on the stirrer revolution number
nor on the mean diameter of emulsified monomer droplets. This implies that the total surface area ofmonomer droplets does not influence very much the overall reaction rate. It can be considered that theelementary process for dissolving monomer molecules into the aqueous phase is not a rate-controlling
step and that the aqueous phase is saturated with monomer. The homogeneous nucleation begins withthe radical reaction of dissolved monomer with water-soluble initiator radicals. When the oligomeric
radicals become hydrophobic because of their increased molecular weight, primary solid polymer
particles appear due to precipitation. The primary particles grow by the further propagation reaction. If
the mass transfer process of the dissolved monomer and oligomer toward the primary growing particles
were a rate-controlling step, the monomer conversion should have become high owing to the large
mixing effect when the revolution number is high.
As can be seen from Fig28.4-4, the emulsion polymerization reaction would not occur at thereaction temperatures below 53℃.
28.4-3 Temporal variation in particle size distribution of polymer particles

Fig.28.4-4 Variation of time-averaged monomer conversion with reaction temperature.
(mean residence time = 24.2 min, stirrer revolution = 600 rpm)

Fig.28.4-5 Latex particle size distributions sampled in efflux solutions observed by laser diffraction
particle size analyzer. (Revolution = 800 rpm, mean residence time 𝝉 = 27.2 min.)190

Dynamical Behavior of an Emulsion Polymerization

Fig.28.4-5 shows temporal variation in the PSDs of latex particles sampled at various times.
It can be considered that the PSDs indicate a kind of cyclic behavior in the continuous emulsion
polymerization.
The PSDs from b to f indicate the continuous growth of secondary particles due to selective
precipitation of hydrophobic nuclei to the surfaces of the secondary particles. The number of thegrowing secondary particles decreases owing to the continuous discharge of the reacting solution fromthe CSTR. The process seems to end at the critical particle size of about 1 μm. For 90 min in the firstterm, a steady PSD is maintained with the primary particles only growing due to polymerization
reaction. For the next 3 hours, unsteady PSDs appear. This indicates a process of selective
agglomerative precipitation combined with continuous nucleation due to polymerization reaction. In
such a manner, agglomerative precipitation process is repeated.
It can be considered that the following cyclic process begins with the first initiation of homogeneous
nucleation under the condition below the CMC of free emulsifier:1) Initiator radicals are produced by decomposition in the aqueous phase2) Polymerization starts with making oligomeric radicals due to the initial propagation reaction of thewater-soluble initiator radicals with the monomer dissolved in the aqueous phase. The
homogeneous nucleation step is controlled by free emulsifier in the aqueous phase when its
concentration is relatively high.
3) When the growing oligomeric radicals become hydrophobic, a large number of primary particles
are formed via nucleation due to the precipitation of hydrophobic oligomeric radicals.
4) As the primary particles grow, i.e. as the total surface area of the particles increases rapidly, theconsumption rate of free emulsifier for covering over the particle surfaces becomes larger than the
net feed rate of emulsifier. At this stage, the reaction rate reaches a local maximum and the PSD
becomes very narrow and has a very small average diameter.
5) When the concentration of free emulsifier becomes extremely low, the homogeneous nucleation issuppressed while the agglomeration of the primary grown-up particles occurs very actively to
become very large secondary particles owing to the shortage of emulsifier covering over the
primary particle surfaces. At this stage, the reaction rate goes down to a local minimum and the
PSD is expanded with its peak shifting toward a large diameter. At the end of this stage, theaverage particle size becomes maximum.
6) Most of the secondary particles are discharged with the continuous efflux of the CSTR reactor.
When the net feed rate of emulsifier becomes larger than the consumption rate of free emulsifier,the homogeneous nucleation becomes active again. At the beginning of this stage, the PSD
becomes bimodal because of the mixture of the newly-produced primary particles and theremaining old secondary particles owing to the effect of residence time distribution in the CSTR.
This result suggests that the PSD of polymer particles can be controlled by controlling theintermittent nucleation and agglomeration, i.e. the concentration of the free emulsifier in the aqueous
phase.
TEM (transmission electron microscope) photographs of polymer particles are shown in Fig.28.4-6.

Nonlinear Behaviors in Complex Chemical Processes

191

Fig.28.4-6 TEM photographs of latex particles sampled at 120, 180, 240, and 280 min.
(Revolution = 800 rpm, mean residence time 𝝉 = 27.2 min.)

The TEM photographs indicate a cyclic change controlled by the agglomerative precipitation
process. After 260 min, the large secondary particles are discharged from the CSTR and then the PSD
returns to the state of 120 min containing secondary particle cores only. This time period (from 120 to
260 min) implies the second stage of secondary particle growth.
From 𝜏 = 280 min, the next agglomerative precipitation proceeds, but some grown-up secondary
particles (largest particles) are still left behind. After the first stage of primary particles production, thesecond stage of complex process combined between homogeneous nucleation and agglomerative
precipitation is repeated with a large time scale of 3 hr. The selective agglomerative precipitation
regarded as a kind of mass transfer influenced by an electrostatic effect in a diffusion electric double
layer. The electrostatic attraction and repulsion occur unbalanced in an electrostatic charge due to theadsorption of ionic emulsifier.

192

Dynamical Behavior of an Emulsion Polymerization

Fig.28.4-7 TEM photographs of latex particles obtained when mean residence time t = 31.1 min.
The diameter of the largest particles when t = 320 min is approximately 1 𝝁m.

Fig.28.4-7 clearly indicates the cyclic second stage process of the primary particles growing to thelarge secondary particles. This implies that a primary grown-up particle become a core of thesecondary particle. It can be seen that the relatively small primary particles growing when t = 240 min
in the early stage are undergoing the successive growing process due to selective agglomerative
precipitation for about 1.5 hr. When t = 320 min, the next stage process starts by the very small
primary particles but some of the grown-up secondary largest particles are still left behind in the CSTR
reactor.

Fig.28.4-8 Second stage of emulsion polymerization controlled by selective agglomerative
precipitation

Nonlinear Behaviors in Complex Chemical Processes,

Nomenclature

193

Fig.28.4-8 indicates the second stage process of the growth of the secondary particles due to theselective agglomerative precipitation.
We would like to add a very fair possibilities of periodical or stepwise switching operation forcontrolling the PSDs of latex particles.2)

1.
2.
3.

Ohmura, N., Kataoka, K., Watanabe, S., Okubo, M., Controlling Particle Size by Self-Sustained Oscillations in Continuous
Emulsion Polymerization of Vinyl Acetate, Chem. Eng. Sci., Vol.53, 2129 (1998)
Ohmura, N.,.Kitamoto, K., Yano, T., and Kataoka, K., Novel Operating Method for Controlling Latex Particle Size Distribution in
Emulsion Polymerization of Vinyl Acetate, Ind. Eng. Chem. Research, Vol.40, 5177-5183 (2001)
Kataoka, K., Consideration of Instabilities of Nonlinear Transport and Reaction in the Synthesis of Complex Chemical Process
Systems, Plenary Lecture Presentation Record, 2000 Annual Meeting of AIChE, Los Angeles, Nov. 12-17 (2000)

Nomenclature
𝐶 , 𝐶
𝐸
𝐹
𝑓
𝑘 ,𝑘 ,𝑘
𝑇
𝑡
𝑉
𝑥, 𝑦
𝜏

concentration of reactant A and B, [kmol/m3]
redox potential, [V]feed rate, [kmol/s] or [m3/s]
space velocity, [1/s]reaction rate constant,temperature, [K]time, [s]reactor volume, [m3]concentration of intermediate component,
specified time or mean residence time, [s]Subscripts
A, B
component A, B
𝑠𝑠stationary state

194Index

195

INDEX
Agglomerative precipitation, 189~194
Aluminum platelet flow visualization method, 160
Taylor-Couette flow system, visualization, 159
transitional scenario to chaos, 164
Antoine equation, 89
Apparent viscosity, 12 (see non-Newtonian fluid)
Attractor, 179
Autocatalytic reaction step, 179, 183
Belousov-Zhabotinskii reaction, 183
autocatalytic step, 183
chemical oscillation, 181, 185
Bifurcation, 180, 184 (see Chemical oscillation)bifurcation scenario, 182, 184
Bingham model, 3, 13, 28, (see non-Newtonian)Bingham plastic, 3, 52,
Swamee & Aggarwal friction factor, 13
yield stress, 13
Brusselator model, 179, 181
attractor, 179
bifurcation, 181
limit cycle, 181
phase space analysis, 182
Chemical oscillation, 181, 185(Brusslator model, B-Z reaction)
chaotic, 181
doubly-periodic, 182)
intermittent, 181
singly-periodic, 181
Compressor, 82 (see Heat Pump)
adiabatic compression, 82, 89, 91
Control volume/control volume approach, 105
enthalpy and mass transfer coefficient, 107
HETP (Height Equivalent to a Theoretical Plate), 105interphase transfer, 108, 117packed column model, 147,
Dispersed system, 1drag coefficient, 4drag resistance force, 3
droplet, 2
emulsion, 23
heat transfer, 1interphase mass transfer, 4mass transfer, 2
Ranz-Marshall equation, 1, 2
Reynolds number, 1
slurry,Stokes law, 2, 4
terminal velocity, 2
trajectory, 3Distribution coefficient, 6, 35
Drag coefficient, 4drag resistance force, 3Einstein equation, 12, 23
Ejector, 133diffuser, 138diffuser efficiency, 139
Mollier diagram, 136 (see Thermodynamic diagram)
momentum balance, 143, 152
motive steam, 135, 141
multi-stage ejector, 147
steam ejector, 135
thermodynamic calculation, 137
two-flow-passage model, 138
Electrochemical method, 165 (see Chapter 14.3,4)
jet impingement mass transfer, 170
limiting current, 165,
mass transfer coefficient, 166
Taylor vortex flow, 165
mass transfer measurement, 165,
velocity-gradient measurement, 167,
wall shear stress, 171 (see Chapter 14.4)
Elementary step, 179, 183 (see BZ reaction)
Emulsion, 23
emulsification, 23
friction factor, 20
HB consistency index, 26
Hertschel-Bulkley model, 27
pipe flow, 19
power law model, 26 (see non-Newtonian)
power requirement, 18, 21, 25
pseudoplastic, 27
Reynolds number, 20
rheology, 25
shear-thinning emulsion, 27, 29
viscosity, 23
Emulsion polymerization, 186
agglomerative precipitation, 189~193
initiator, 187, 188
latex particle, 189
monomer droplet, 188
nucleation, 189
primary particle, 190
secondary particle, 190
propagation step, 187
termination step, 187
Energy saving in distillation process, 76, 81(see VRC ad HIDiC)
Equivalent diameter, 110 (structured packing)
Extraction, 33
center-of-gravity principle, 40
continuous extractor, 42
counter-current extractor, 42, 44
distribution coefficient, 35
equilibrium diagram, 34
leaching, solid extraction, 49
mixer-settler extractor, 36
operating line, 44
raffinate, 37, 42
stage-by-stage design calculation, 43
tie line, 38
triangular diagram, 36

196
rectangular coordinates, 39, 40
triangular coordinates, 36
Enthalpy balance, 107
control volume model, 105
packed distillation column, 105
First law of thermodynamics, 123 (see Chapter 4.4)
Flow visualization, 155
aluminum platelet method, 160
hydrogen bubble method, 169
laser fluorescent method, 156
organized flow structure, 155
stirred tank reactor flow, 156
isolated mixing region, 156
torus, 156
Taylor-Couette flow, 159
doubly-periodic wavy vortex flow, 162
fully-turbulent vortex flow, 164
laminar Taylor vortex flow, 161
singly-periodic wavy vortex flow, 162
transitional scenario to chaos, 164
weakly-turbulent wavy vortex flow, 163
Fouling, 103,
Maillard reaction, 102
Gardon sensor, 175
Heat integration, 95
HIDiC, 95
Heat pump, 77
adiabatic compression, 82, 89
coefficient of performance,
entropy-temperature diagram, 80
heat engine cycle, 79
ideal heat pump, 89
isentropic compression, 82, 89
refrigeration cycle, 78
reversed heat pump, 79
surplus energy, 84
work requirement, 82
Hedstrom number, 13
HETP (Height Equivalent to a Theoretical Plate), 106
HETP measurement, 111
packed distillation column, 111
HIDiC (Heat Integrated Distillation Column), 95
bio-ethanol dehydration, 100
compression ratio, 97
energy saving rate, 104
fouling, 103
Maillard reaction, 102
operation mode, 97
overall heat transfer coefficient, 98
pressure ratio, 97
rectifying sec., 96 (see Chapter 11.2-2)
stripping sec., 96
Hydrogen bubble method, 169(see Flow visualization)
free jet development, 169
intermittent organized structure, 170
longitudinal vortex pair, 170
streakline, 169
time line, 169
two-dimensional impinging jet, 169
Instability, 159

Indexagitated tank, 156,
rotating coaxial cylinders, 159
Taylor-Couette flow, 159
Interphase mass transfer, 4, 107
single droplet or particle, 2mass transfer coefficient, 2, 5Peclet number, 8
Ranz-Marshall equation, 1, 2
Reynolds number, 1
Sherwood number, 7terminal velocity, 2
two-film theory, 8Interphase transfer (packed column), 116
heat or enthalpy transfer, 85, 143, 145,
mass transfer, 106, 146, 155,
Ionic mass transfer, 163,
electrochemical method,
limiting current, 165,
molar-flux of ions, 163,
supporting electrolyte, 165,
Isolated mixing region (stirred tank), 156
Jet flow, 168hydrogen bubble method, 169
impinging jet heat transfer, 174
impinging jet mass transfer, 170
stagnation point heat transfer, 176
j-factor, 120, 168 (see Packed column)
j-factor for enthalpy transfer, 120
j-factor for heat transfer, 120
j-factor for mass transfer, 120
Joule-Thomson expansion, 125isenthalpic expansion, 126
Joule-Thomson coefficient, 127, 130
throttling expansion, throttling valve, 132
Laser fluorescent method, 156
isolated mixing region, 156
organized structure, 155
stirred tank observation, 155
streakline, 158
toroidal motion, 156
Torus, 156
Leaching, 49,
counter-current leaching cascade, 51
Kremser equation, 54
operating line, 51
McCabe-Thiele graphical method, 54
overflow, 50
stage-by-stage calculation, 54
supercritical fluid extraction, 70
underflow, 50
Limit cycle, 181
Limiting current, 165, (see Ionic mass transfer)
Mass & heat transfer visualization, 165
cellular vortex flow, 166
convective heat transfer, 174
electrochemical method, 165
Gardon sensor, 175
wall shear stress measurement, 171
Mass transfer (packed column), 116
transfer coefficient (definition),107, 108, 114,
correlations, 116,
interphase transfer, 116,

Indexmass & enthalpy transfer5 model,
Mass transfer visualization, 165
McCabe-Thiele method, 43, 54
for continuous extraction cascade, 43
for continuous leaching cascade, 54
Mechanical energy balance, 123
Micelle, 11
CMC (critical micelle concentration), 11hydrophilic head, 10hydrophobic tail, 10monolayer, 10
surface tension, 9Mixer-Settler extractor, 36
Mollier diagram, 124
Enthalpy-entropy diagram, 124
Nonlinear dynamic process, 179
Non-Newtonian fluid, 14, 28
apparent viscosity, 12Bingham plastic model, 3, 13, 28
dilatant, 11,
Herschel-Bulkley model, 27
HB consistency index, 26
power law model, 26 (Ostwald-de Waele Model)
pseudoplastic, 27
Yield stress, 13
Operating line, 44
in extraction, 44
in leaching, 51
Overall heat-transfer coefficient, 118,
Overall mass-transfer coefficient, 118(packed column distillation)
Packed distillation column, 105
control volume model, 105
correlation of enthalpy and mass transfer, 116, 120
distribution function, 113 (HETP),
distribution function, 118 (mass & enthalpy transfer),enthalpy transfer coefficient, 108
equilibrium-stage model, 108
F-factor, 112
HETP (Height Equivalent to a Theoretical Plate), 106, 111
j-factor, 120 (see Chapter 8.2)
mass transfer coefficient, 107, 114
Nusselt number, 117
Prandtl number, 117
Reynolds number, 117
Schmidt number, 117
Sherwood number, 114, 117
structured packing, 110
superficial velocity, 112
total-reflux distillation condition, 109
Particle Size Distribution, 189
latex particle PSD, 189
monomer droplet PSD, 188
Peclet number, 8Power law model, 14, 26 (see non-Newtonian)Ostwald-de Waele model, 26Power number, 21 (see Chapter 20.2)
Prandtl number, 1, 63, 118
Pseudoplastic, 27
Ranz-Marshall equation, 1
Rapid expansion method, 72 (supercritical fluid)

197
Reynolds number, 1
Droplet or particle, 1in agitation, 21
in circular pipe flow, 16
in coaxial rotating cylinder, 159
in packed column, 116
Roughness, 16
Relative roughness in pipes, 16,
Schmidt number, 117
Selective agglomerative precipitation, 189
Shear stress, 12
Sherwood number, 114, (definition)
Slurry, 12
Einstein equation, 12
critical slurry velocity, 17
Darcy Weisbach friction factor, 17
Durand factor, 17
friction factor, 14
Hedstrom number, 13
heterogeneous slurry, 16
homogeneous slurry, 16
limiting settling velocity, 17
non-Newtonian viscosity, 11pipe flow, 16
pressure drop, 18
Reynolds number, 18
Sphere,2drag force, 3, heat transfer, 1
Steam ejector, 134, 136
compression ratio, 136energy saving, 137
isentropic compression or expansion, 136, 138
momentum balance, 143, 152
motive steam, 141
multi-stage ejector, 147
two-flow passage model, 138
Stirred tank mixing, 155
isolated mixing region, 156
laser fluorescent method, 156
Reynolds number, definition, 155toroidal motion, 156
torus, 156
Stagnation point, 176
Heat transfer, 176, Impinging jet, 176,
Stokes law, 2, 4
Supercritical fluid, 59
critical parameter, 60
extraction, 67
heat transfer, 62
heat exchanger, 66
leaching, 71
Nusselt number, 65, 67
pipe flow, 63
Prandtl number, 63, 67
pseudocritical point or range, 60
Reynolds number, 64, 67
solubility, 69
supercritical fluid flow, 63
flow stratification, 63, 65
thermodynamical property, 60
Supercritical extraction, 67
rapid expansion method, 72
Surface tension, 9hydrophilic head, 10198hydrophobic tail, 10
Sustained chemical oscillation, 181
Taylor-Couette flow, 160
aluminum platelet visualization method, 160
amplitude modulated wavy vortex flow, 163
cellular vortex array, 167
critical Reynolds number, 161
doubly-periodic wavy vortex flow, 162
electrochemical method, 165
inward-flow cell boundary, 161
laminar Taylor vortex flow, 161
mass transfer distribution, 167, 168
outward-flow cell boundary, 161
singly-periodic wavy vortex flow, 162
Taylor instability, 161
toroidal motion, 156, 161transitional scenario to chaos, 165
weakly-turbulent wavy vortex flow, 163
turbulent Taylor vortex flow, 164

IndexTerminal velocity, 2
Thermodynamic diagram, 80, 124
Mollier diagram, 124
Thermophysical property, 60
Transitional scenario to chaos, 165
doubly-periodic wavy vortex flow, 162
laminar Taylor vortex flow, 161
singly-periodic wavy vortex flow, 162
Taylor-Couette flow, 159
Weakly-turbulent wavy vortex flow, 163
Viscosity, 11apparent viscosity, 12Bingham plastic, 13
yield stress, 13
VRC (Vapar Recompression), 81
VRC distillation column, 82
VRC evaporator, 85

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POST FACE
The author has felt very happy to find his way to this place. He has been confining himself to an
educational concept as to how to make creative problems rather than how to solve the given problems.
He intends to have given many examples for application of fundamental theories and models toengineering. He hopes that the readers and students will bring themselves to feel like making morechallenging problems.
Several special topics given as the starting point for development in PART III will encourage thereaders to take further interest for their progressive development.
In particular, in the last half of the text, several miscellaneous topics are dealt to widen the all-roundknowledge for chemical engineering for future challenge.
The PART IV gives various complicated topics that the graduates students should have as thefundamental background serviceable for their advanced challenges.
It is hoped that active and practical application of those concepts will be developed in the future formore challenging problems.
This book is placed as the PDF file in the following web site:
Kobe university: https://doi.org/10.24546/0100485268
Kansai Chemical Engineering Co., Ltd: https://www.kce.co.jp/en/library/phase2/
Copyright:@2023 Kunio Kataoka. This is an open-access book. Any teachers and/or students, andengineers can freely access the URL web site to get any useful knowledge/information and materials forstudy and investigation.

KUNIO KATAOKA