Investigations on driving flow expansion characteristics inside ejectors
Zuozhou CHEN, Chaobin DANG*, Eiji HIHARA

Institute of Environmental Studies, Graduate School of Frontier Sciences, The University of Tokyo, 51-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8563, Japan
Email: dangcb@k.u-tokyo.ac.jp
Abstract: This research investigates the Mach wave structure of the driving flow under off-design working
conditions by both numerical and experimental methods. By adopting the method of characteristics as the simulation
model, prediction of the driving flow regime inside an ejector is obtained. The simulation results are further
validated by an experimental visualization method conducted using a Schlieren system. Through this investigation,
the influence of Mach wave on the driving flow boundary development is discussed. The expansion wave from thenozzle exit increases the driving flow regime in the under-expanded condition, which has a negative impact onejector performance. The results show that the Mach wave should be considered when the ejector is operated under
off-design working conditions. The results also demonstrate that an appropriate nozzle structure design was able torestrain the effect of the expansion wave, which improves ejector performance. The results are significant forachieving a comprehensive understanding of the mechanism of an ejector, as well as for the applications, such asejection refrigeration cycles.
Keywords: ejector, Schlieren photography, method of characteristics, numerical simulation, visualization
Nomenclature
Notations
A
A
AR
C
Cp
D
ER
h
I
Iter
M
N
P
R
T
U
V
V
Q
O
S

Units
Cross-sectional area
Local sonic velocity
Area ratio
Characteristic curve
Specific heat capacity
Diameter
Entrainment Ratio
Specific enthalpy
Information of a grid point (x, y, u, v)
Iteration number
Mach number
Mass flow rate
Grid number
Pressure
Gas constantTemperature
Velocity component in x direction
Velocity component in y direction
Velocity
Terminologies applied in the finite
differential equations
Specific heat ratio
Mach angle

subscripts
b
d
e
m
n
max
s
t
x
y
+
0
1, 2, 3…

Back pressure
Driving flow
Nozzle exit
Mixing section
Surface number inside the ejector
Maximum value
Suction flow
Nozzle throat
x Direction
y Direction
Left-running Curve
Right-running Curve
Static condition
Numbers of iteration

Velocity angle
Mach line angle
Flow deflection angle
Constant for axisymmetric flowΔ

1. Introduction
Large-scale applications for air conditioning and refrigeration systems consume huge amounts of energy andcause environmental problems. Efforts to reduce the level of energy consumption in these applications have led torenewed interest in heat recovery systems. Heat recovery refrigeration systems are an alternative to vaporcompression refrigeration systems. In these systems, low-grade heat such as solar energy or exhausted heat can beutilized as the driving energy. The ejection refrigeration cycle is one example of such heat recovery systems, and it
has the following advantages: simple-structure, reliability, and low-cost. In recent years, the number of journal
papers focusing on ejectors or ejection refrigeration cycles has grown rapidly [1]. Ejectors have been investigatedin the field of waste-heat utilization [2] and in ejector–vapor-compression hybrid cycles [3]. In CO2 heat pumpsystems, ejectors are employed as expansion devices to reduce throttling losses [4]. In addition, a number of studies
on solar-driven ejection refrigeration cycles [5]-[10] have been conducted.
Suction Chamber
Boundary

Mixing Section

Diffuser

Shear-Layer

Input Energy
ThroatDriving flow

Compression shock waves
Suction flow
Non-mixing Section

Fig. 1 Schematic of the ejector and the Mach wave positions
Fig.1 shows the structure of an ejector with the flow regime and Mach wave inside. The ejector comprises of amotive nozzle and a suction chamber. High pressure refrigerant, known as the driving flow, is accelerated throughthe motive nozzle and converted into high velocity flow with low pressure. The suction flow is entrained into theejector from the suction flow inlet. The flow from the motive nozzle exit is then divided into two regimes: thedriving flow regime and the suction flow regime. On the shear layer of the driving flow boundary, part of the kinetic
energy from the driving flow is transferred to the suction flow. The two flows will finally mix in the mixing section
and jet outward from the diffuser. The performance of the ejector is described by the parameter ER (the Entrainment
Ratio) and PR (the Pressure Ratio) as shown by Eq. (1), (2).
ER  m suctionflow / m drivingflow(1)

PR  Pb / Ps ,0(2)

and , represent the pressure of the driving flow and the suction flow at the nozzle exit,
In Fig. 1,
respectively. There are two locations inside of the ejector where Mach waves may occur. In those locations, theMach wave could manifest as an expansion wave or shockwave based on the expansion or compression effect on
the supersonic flow. The driving flow Mach wave may occur at the nozzle exit, and the mixed flow shockwave may
occur in the diffuser, where the mixed flow changes to subsonic from supersonic. There are three conditions for theis larger than , , the driving flow is in an under-expanded condition and expansion waves will
driving flow: If
occur. On the other hand, if
is smaller than , , the driving flow is in over-expanded condition, and shockwaves

will occur. The ideally-expanded condition is reached if
is equal to , . The occurrence of a Mach wave isusually avoided by optimizing the design of the motive nozzle to reach the ideally-expanded condition. In the onedimensional theoretical model, the driving flow condition is usually assumed near the ideally-expanded condition.
The constant-pressure mixing theory proposed by J.H. Keenan et al. [11] and the non-mixing process between thedriving and suction flows proposed by J.T. Munday and D.F. Bugster [12] were adopted to describe the working
process of the ejector. In the models, ER is obtained from the cross-sectional flow areas of the driving and suction
flow in the non-mixing section. Since the ejector structure is fixed, the relationship of the flow areas could beobtained by Eq. (3), and the calculation process of ER was introduced in the one-dimensional model developed by
Huang et al [13].
(3)As,n + Ad,n = AmIn applications such as waste heat utilization, a relatively stable heat source temperature could maintain the
optimum-expand condition for the ejector. Yet, in other cases, especially for solar energy utilization, the driving
flow will be in either the over-expanded or under-expanded condition. This is due to the fact that the ideallyexpanded condition cannot be maintained because the solar energy input is fluctuating. Under off-design conditions,
Mach waves may develop and influence the ejector performance. Shockwaves in the over-expanded condition cause
irreversible energy loss in the driving flow. On the other hand, expansion waves in the under-expanded condition
creates radial velocity components in the driving flow, which will reduce the flow area of the suction flow regime.
To employ ejection cycles in solar energy utilization, the influence of Mach wave should be considered. However,there have not been many studies aimed toward the occurrence of Mach waves and its influence on ejector
performance.
In this research, the Mach wave in the gas-ejector at the off-design working condition is discussed. The influenceof Mach wave on the driving flow expansion, as well as the ejector performance, is investigated numerically andexperimentally. A numerical approach using the method of characteristics model is adopted to predict the driving
flow expansion inside an ejector. The simulation results are further validated by visualization experiments conductedusing the Schlieren photography method. The research reveals the influence of Mach wave on the ejector
performance, which is significant for the application of solar-driven ejection–refrigeration cycles.

2. Prior Work on Model Development of an Ejector
Following the models proposed by J.H. Keenan et al. [11], J.T. Munday, and D.F. Bugster [12], Huang et al.
established and validated a one-dimensional model in which an isentropic process was considered for the driving
flow expansion inside of an ejector [13]. Eames also proposed an ejection–refrigeration cycle evaluation method
using the isentropic process, and validated it experimentally with a steam ejection cycle [14]. B.J. Huang et al.
conducted a series of experiments and validated the choking assumption for the one-dimensional model [15]. Fig.
2 (a) shows the working process assumed in the one-dimensional model. Since the driving flow is in the ideallyexpanded condition, the isentropic expansion process described by Eq. (4), (5) is adopted to predict the flow area
for the driving flow regime. As the driving flow expands from the nozzle exit, a hypothetical converging tunnel forthe suction flow is formed by the driving flow boundary and the ejector wall. The suction flow accelerates inside
the tunnel until it reaches sonic velocity.
2 / (  1)
Pn (1  M d ,n (  1) / 2)
 / (  1)
2(1  M e (  1) / 2)
Pe

Αd,n
Ae

=

(4)

2
(  1) / 2) (1 / M d,n )  2 / (  1)(1  M d,n(1 / M e )  2 / (  1)(1  M e2 (  1) / 2) (  1)/(2(  1))

( 1)/(2( 1))

(5)

Expansion wave

Nozzle exit

Nozzle exit

Driving flow
Hypothetic boundary
Suction flow

Shear layer,
Expansion Fan Reflected shockwave

Section n

Suction flow(a) Driving flow boundary by the non-Mach wave(b) Driving flow boundary affected by the Mach
isentropic expansion assumption
waves in under-expanded condition
Fig. 2 The driving flow expansion inside the ejector
Fig. 2 (b) shows other factors that may occur inside the ejector, such as a Mach wave and a shear layer. Those
factors are discussed in the following studies using both experimental and numerical methods. S.K. Chou et al.
introduced efficiency factors for the nozzle structure and the shear-layer thickness into a one-dimensional model[16]. K. Mohammed et al. considered the friction loss inside the ejector, and employed polytropic efficiencies todetermine the dimensions of the ejector [17]. Y.H. Zhu et al. presented a two-dimensional model for the suction
flow velocity distribution [18], and further developed the model for both dry and wet vapor ejectors [19]. H. El
Dessouky et al. proposed a model for both choked and un-choked conditions under various backpressures [20]. S.J.
Chen et al. discussed a model for the ejector with a converging mixing cone [21]. J.G. del Valle et al. proposed a
prediction model for driving flow boundary development based on a turbulence model, and built the prediction
model into the one-dimensional model [22]. J.Y. Chen et al. proposed a model that considers the suction flow
velocity at the ejector entrance [23]. K. Matsuo et al. studied nozzle structures and claimed that the nozzle’s exitto-throat area ratio was an important factor affecting ejector performance [24]. D.A. Pounds et al. predicted theejection refrigeration system performance from the model they proposed with a quasi-1-D assumption for the mixing
section [25]. D. Butrymowicz et al. developed a calculation methodology for an ejection refrigeration system withan internal heat exchanger [26]. W.N. Fu et al. conducted an investigation on nozzle structures using numericalsimulation, and claimed that nozzle structure design was essential for optimizing an ejector [27]. N.B. Sag and H.K.
Ersoy also conducted experiments on the nozzle structure, and claimed that an optimal nozzle enhanced system
performance by 8 % - 13 % [28].The driving flow Mach wave is considered to be one of the factors that influence
ejector performance. Currently, investigations on driving flow Mach waves are conducted using visualization
experiments and numerical simulations. Y.H. Zhu and P.X. Jiang studied the shockwave characteristics in a Schlieren
system. The driving flow Mach wave from both supersonic and convergent nozzles were observed, and they claimed
that by reducing the first Mach disk length, the ejector performance would be enhanced [29]. The empiricalequations for Mach disk length are further obtained using visualization experiments in the following studies [30]. J.
Gagan et al. combined PIV and numerical simulation methodologies to obtain the appropriate turbulence simulation
model for a gas ejector [31]. A. Bouhanguel et al. conducted visualization experiments using laser tomography
techniques that showed the supersonic flow regime [32]. A.B. Little and S. Garimella adopted the shadow methodand discussed the condensation effect caused by shockwaves inside of the ejector [33].
The under-expanded condition for the driving flow is discussed in this research because in the over-expanded
condition; the driving flow velocity is reduced after passing through the shockwave. At the same time, driving flow
may detach from the nozzle wall and the negative factors make the over-expanded condition inappropriate for theoperation of gas ejector. A.L. Addy proposed the empirical prediction of driving flow boundary from a convergent
nozzle into a free space [34]. A.J. Ruggles and I.W. Ekoto adopted Schlieren photography to investigate the Machwave structure of supersonic flow in free space [35]. The method of characteristics (MOC) model is commonly
employed in simulations involving supersonic flow. A.L. Addy and W.L. Chow attempted to employ the MOC
model to predict the driving flow expansion inside of the ejector. They combined the numerical simulation method

with a one-dimensional model, and gave a reasonable prediction of the ejector performance [36]. In this research,
the MOC methodology is also adopted to predict the driving flow expansion inside the ejector. A Schlieren opticalexperimental setup is constructed to validate this simulation method. The Mach wave is discussed in terms of thesimulation and experiment results, and the importance of employing an optimized nozzle inside the ejector is also
discussed by simulation.

3. Simulation Methodology (The Method of Characteristics)
3.1 Expansion process of the driving flow
As shown in Fig.3, the driving flow field is divided into three parts, marked as Ⅰ, Ⅱ, and Ⅲ. The three parts
represent the flow field before, inside, and after the expansion fan. In the under-expanded condition, an expansion
fan is formed by a group of expansion waves propagating from the nozzle exit rim. The expansion fan depressurizes
the driving flow pressure to suction flow pressure at the nozzle exit, and gives an out-turning flow angle to thedriving flow. The driving flow pressure will further decrease in flow regime Ⅲ because of this flowing angle. In
regime Ⅲ, when the driving flow pressure is smaller than the environmental pressure, a shockwave will originate
from the boundary to correct the driving flow pressure and flow direction. In summary, the Mach wave is the
propagation of physical disturbances caused by a pressure difference, and the driving flow regime is influenced byboth expansion waves and shockwaves.
The Mach wave could be considered a simple wave if the pressure difference between the driving flow and theenvironment on the boundary was infinitesimal. The driving flow undergoes an isentropic process through a simple
wave. In the method of characteristics, because the finite differential method is adopted, the expansion wave andshockwave are all considered simple waves. The simple wave could be described by the characteristic curve, because
along the characteristic curve direction for each point in the supersonic flow field, the derivatives of the physicalproperties of the flow are discontinuous, while the properties are continuous. Therefore, the flow field could be
described by creating a grid of units linked by the characteristics curves as illustrated in Fig. 3. The method
introduced by M.J. Zucrow and J.D. Hoffman is adopted in this study [37] and several assumptions are clarified:
1. The outflow from the nozzle is parallel with unified velocity.
2. The flow field is axisymmetric to the x-axis.
3. The flow is supersonic throughout the driving flow regime.
4. Friction between driving flow and suction flow is neglected.
5. The calculation proceeds under an ideal gas assumption with constant
and .
6.
7.

The suction flow is considered as one-dimensional flow.
The driving flow pressure is equal to the suction flow pressure on the boundary.

Mixing sectionyBoundaryNozzle/2…
/2

Throat Exit

x
Ⅰ

Ⅱ

Ⅲ

Fig.3 Flow fields presented by the method of characteristics model in the under-expanded condition
3.2 Finite difference method
The governing equations for two-dimensional, irrotational, and inviscid supersonic flow of a compressible gas
are presented by Eq. (6) - (8).
is a constant equal to 1 in axisymmetric flow. a is the local sonic velocity; u, v arevelocities in the x, y direction; R is the gas constant, and T is the temperature. The units for velocity and temperature
are m/s and K, respectively.
(u 2 - a 2 )ux  (v 2 - a 2 )vy  2uvuy - 

a 2v
0
y

(6)

u x  vy  0

(7)

 RT

(8)

a

As shown in Fig. 3, two characteristic curves from two points upstream are necessary to obtain the objective
parameters of (x , y ). The two characteristic curves are defined as the left-running curve (C ) and the right running
curve (C ), and are considered as lines when the finite difference method is adopted.
x

x

Right-runningcharacteristic curve (Tangent line to)Left-runningcharacteristic curve ()

Left-runningcharacteristic curve (

y

)

y

(a) Inner grid(b) Boundary gridFig. 4 Unit process by the Method of Characteristics
The Mach angle, β , could be defined by Eq. (9) since the characteristic curve is equal to a simple wave.

 1

M 

   arctan (9)As the velocity angle, θ, is obtained by Eq. (10), the tangent of the absolute angle of the characteristic curve, λ , canbe obtained by Eq. (11).

v
 

  arctan  u(10) dy      tan(   ) dx  (11)

By the finite difference method, the position of the objective point is obtained by Eq. (12-a).
y c   xc  y1    x1
y c    xc  y 2    x 2(12-a)

To obtain the parameters of velocity, the governing equations of the flow field and the total differential equationsof u and v are integrated into a matrix, as shown by Eq. (13-a).
could be obtained by Cramer’s Rule, as shownin Eq. (13-b).

 u 2  a2

 0
 dx

 0

2uv v 2  a 2
1
1
dy
0
0

dx

0   ux    a 2 v / y   0   uy   0 
0   vx   du 





dy   vy   dv (13-a)

 a 2v / y 2uv v 2  a 2
ux 

0
du

1
dy

1
0

0 u 2  a2
0
0
/
0
dx

dv

0

dx

dy

0

2uv v 2  a 2
1
1
dy
0

0
0
0

0

dy

dx

(13-b)

Since
is discontinuous but not infinite along the characteristic curve, both the denominator and the numerator
in Eq. (13-b) are 0. The equation can be then simplified as Eq. (13-c), and the finite difference forms are shown in
Eq. (14).
(u 2  a 2 )d u   2uv  (u 2  a 2 )  dv  (a 2v
) d x  0yQ1uc  O1vc  S1 ( xc  x1 )  Q1u1  O1v1Q 2 u c  O 2 v c  S 2 ( x c  x 2 )  Q2 u 2  O 2 v 2(13-c)(14)Where
Q  u2  a2
O  2uv  (u 2  a 2 ) 

a 2v
y

S 

The position and velocity for each unit could be obtained by Eq. (6)-(14). The Euler corrector algorithm is adopted
to improve the accuracy, and is shown in Fig. 5 (a). The input parameters are renewed until the results are withintolerance. Based on the unit process, the entire flow field could be predicted with additional boundary conditions.
3.3 Boundary conditions
3.3.1. The initial driving flow velocity
Since the driving flow is assumed to be parallel from the nozzle exit with uniform velocity, the velocity and thepressure are obtained by Eq. (15) - (17).
2 Ae 
1  2   1 2  Me     2
1 
2
 At  M e    1   1 2  1 
Me 
Pe
2



Pd,0

Me 

( 1)/( 1)(15)

 /(  1)

Ve
ae

(16)

(17)

The inlet boundary condition of the driving flow is,
0.
3.3.2. The flow field inside the expansion fan
As shown in Fig. 3, a group of simple waves perform as expansion waves originating from the nozzle rim, andform an expansion fan. The driving flow undergoes Prandtl–Meyer expansion through the expansion fan, and theturning angle is obtained by Eq. (18).

   b arctan

M 2 1
 arctan M 2  1   0
b

(18)

 1
 1

b

is a constant that represents the initial flowing angle. Eq. (18) is the turning angle as the gas accelerates from astatic condition to a supersonic condition with Mach number, M. When the driving flow pressure changes fromto , at the nozzle exit, the turning angle can be obtained by Eq. (19).
(19)
   s1   e
 /( 1)  1 2  1
Md,s1 Ps,1 
2


Pd,0

(20)

Assuming that the expansion fan is formed by a number ( ) of expansion waves, and an average turning angle isobtained through every expansion wave, the turning angle and velocity could be obtained through each expansion
wave by Eq. (18) - (20). The boundary condition is x0, yd /2, and for each expansion wave: n   d,s1  n ( d,s1   e ) / N e (n  1, 2,..., N e )(21)

3.3.3. Driving flow boundary after the expansion fan
At the boundary point, the driving flow pressure is equal to the suction flow pressure,process on the boundary is isentropic, the boundary velocity can be obtained by Eq. (22).
2
p,n

v

2 RTp,0   Ps,n  1    1   Pd,0  ( 1)/ 22
 ud,n vd,nAs shown in Fig. 4 (b), since C
previous boundary point (The equation for C
dy
v
 
dx
u

′

,

, . Since the expansion

′

(22)

does not exist for the driving flow boundary point, ( ,), and C) is substituted by theis substituted by a line satisfying the condition illustrated by Eq. (23).

is then presented as Eq. (12-b)
(23)

v'
v'
yc  1' xc  y1'  1' x1'
u1
u1

3.3.4. Axis points
Since C does not exist for the axis point, (0.
,(12-b),) is substituted by (,) at axis points:,

0,

3.4 General iteration for boundary simulation
Based on the unit process shown by Fig. 5 (a), the flow field process could be conducted as shown in Fig. 5 (b).
As the suction flow is considered one-dimensional, by assuming the ejector inlet suction flow velocity and using
the suction flow cross-sectional area obtained from Eq. (3), pressure distribution of the suction flow could beobtained by Eq. (23), (24). The iteration continues until both the driving flow boundary and the suction flow pressure
distributions converge.
m suctionflow   sVs As,nhs,0 

Vs,n22

 f h ( Ps,n , Ts,n )

(23)

(24)

,

Left/Right runningcharacteristics curves
Eq. (8) – (11)

,

,

,

Parameters of the point downstream
Eq. (12-a), (14),,Correct the initial conditions:,,Eq. (8) - (11)
,

,

,

,

,

,

,

Eq. (12-a), (14)No
Yes
Complete(a) Iteration algorithm for unit process
Nozzle exit conditions:Eq. (15) – (17),Grid number

Flow field Ⅰ

Expansion Fan: Eq. (18) - (21),Flow field Ⅱ(Initial boundary condition for Flow field Ⅲ)
Substitute the suction
flow pressure
Flow field Ⅲ,,…,

Assuming suction flow accelerate in aconverging nozzle

AND

NO

YES
Complete(b) Iteration algorithm for driving flow expansion inside the ejector
Fig. 5 Iteration algorithms for the MOC model

4. Visualization Experimental Setup
As shown in Fig. 6, the Schlieren experimental setup is constructed to obtain images of flow inside of the ejector.
Light originates from an LED light source, and a point light source is formed after light passes through a spatial
filter. A convex lens is placed to produce a parallel light beam that passes through the ejector. The density

discontinuities inside of the ejector caused by the Mach waves or the jet boundary cause the refraction of part of thelight. When another convex lens is placed after the ejector to focus the light, the refracted light will be focused at
different locations. A knife-edge is employed to block the refracted light. A high-speed camera is used to capture
the Schlieren photos. In Schlieren photos, places where density discontinuity occurs are illustrated by different
brightness compared to the rest of the parts. On the ejector side, a nitrogen gas tank is used as the gas supply. The
working condition is controlled by three regulators placed on the ejector inlets and outlet. In addition, pressure
sensors are placed at the inlets and outlet of the ejector. A volume flow rate meter is placed at the suction flow inlet.
Fig.7 shows the schematic of the ejector utilized in the visualization experiment. The mixing section of the ejector
is rectangular in shape and has a width of 4.60 mm. The height of the part with constant area is 3.50 mm. The front
and back of the ejector are hollow. The main body is manufactured by a 3D printer, and two pieces of glass 0.30
mm thick are attached to the front and back walls of the ejector to form the mixing section. A convergent nozzle anda convergent-divergent supersonic (CD) nozzle are adopted in the experiment. The convergent nozzle has aconverging duct with an outlet diameter of 1.30 mm. The CD nozzle has a throat diameter of 1.30 mm and an outlet
diameter of 1.36 mm. It has an exit-to-throat area ratio of 1.094. The area from the nozzle exit is observed during
the experiment.
High Speed
CameraKnife Edge

Convex 2

Volume Flow
Rate Meter P

Convex 1’

P

P

Regulator

NozzleConvex 1
Sight Glass

Nitrogen Gas Tank

Spatial filter

Light source

Fig. 6 Schematic of the Schlieren System

Fig.7 Schematic of visualization ejector

5. Results and Discussion
5.1 Influence from working conditions
In the experiment discussed in this section, the CD nozzle is employed in the ejector while the driving flowpressure is adjusted from 400 kPa to 600 kPa, and the suction flow pressure is adjusted from 80 kPa to 90 kPa. The
Schlieren photos are shown in Fig. 8. Information including the driving flow boundary and the Mach wave arehighlighted in the photos. In the first Mach disk, the maximum diameter of the driving flow () and its distancefrom the nozzle exit () are considered as the criteria for the driving flow expansion.
As seen in Fig. 8, the driving flow expansion is magnified as the driving flow pressure increases from 400 kPa to
600 kPa. On the other hand, the expansion is slightly restrained as the suction flow pressure increases from 80 kPa
to 90 kPa. This behavior implies that large pressure differences between the driving and suction flow at the nozzle
exit will amplify the driving flow expansion. The triangles or trapezoids observed in the driving flow are formed by
the shockwave, and the shockwave intercepts the propagation of the expansion wave. In the experiment, a normal
shockwave is formed when the driving flow pressure is higher than 500 kPa. The normal shockwave turns supersonicflow directly into subsonic flow, producing large and irreversible energy losses, which has a negative effect on theperformance of the ejector.

In addition,
are obtained from the MOC model and are compared with the measured values from the
Schlieren photos in Fig. 9. During the simulation, the grid number is increased until thevalue is independentof the grid number. From Fig. 9 (a), it can be seen that the simulation results from the MOC model agree well withthe measured values. In Fig. 9 (b),increases along with, since u in the flow field is much larger than v
in the analysis. The grid number needed for a gradually changing
is much larger than
, and the resultsof
are not completely gradual. In this study,
is considered as the main criteria for the evaluation,
therefore the grid number is not further increased for.
, (kPa)
,
(kPa)
400
500
600

80

85

90

Fig. 8 Schlieren pictures under various driving and suction pressures
2.1

400 kPa
500 kPa
Simulation Results

2.0

600 kPa

1.9

600 kPa

1.8

Lmax (mm)dmax (mm)400 kPa
500 kPa
Simulation Results

2.0

1.8

1.6

1.7

1.4

1.6
1.5
75

80

85

Suction Pressure (kPa)

90

95

1.2
75

80

85
90
Suction Pressure (MPa)95(a) Maximum driving flow diameter(b) Distance of thesurface from the nozzle exit
Fig. 9 Comparison between the MOC simulation results and the measured dimensions from the Schlieren Photos
5.2 Influence of nozzle structures
The Schlieren photos of the CD and convergent nozzles are compared in Fig. 10. The driving flow pressure is
500 kPa and the suction flow pressure is adjusted from 80 kPa to 90 kPa. Because the outflow pressure from a
convergent nozzle is higher than the outflow from a CD nozzle, a stronger expansion occurs in the convergent nozzle.
The first Mach disk from the convergent nozzle is shorter and wider than the Mach disk from the CD nozzle. Fig.
11 shows aand entrainment ratio comparison between the different nozzles. From Fig. 11 (b), it can be seenthat the CD nozzle achieves higher performance. The results imply that appropriate expansion of the driving flow
benefits the ejector performance. However, vertical shockwaves still occur in the driving flow from both nozzles,
which means that the nozzle structure is yet not optimal in this experiment.
Types,
Convergent-divergent nozzle
Convergent nozzle
(kPa)

80

85

90

Fig. 10 Comparison between the convergent-divergent and convergent nozzle when the driving pressure is 500
kPa
2.0

Convergent-Divergent nozzle
Convergent nozzle
Simulation Results (CD)
Simulation Results (C)2.0

dmax (mm)Entranment Ratio

1.9

Convergent nozzle
Convergent-divergent nozzle

1.5

1.8

1.0

1.7

1.6
75

0.5
80

85

Suction Pressure (kPa)

90

95

75

80

85
90
Suction Pressure (kPa)95(a) Maximum diameter of the driving flow(b) Entrainment Ration comparison
Fig. 11 Comparison between the convergent-divergent nozzle and the convergent nozzle
5.3 Discussion of appropriate nozzle design
In this section, the influence of nozzle structure on the driving flow expansion is discussed based on the MOC
model. When the driving and suction flow pressures are set to 500 kPa and 80 kPa by adjusting the nozzle exit
diameter ( ), the dimensionless/ results are obtained by the MOC model. These results are presented inFig. 12 (a). The simulation results are also compared with the isentropic expansion model given by Eq. (4), (5).
From the figure,decreases as the nozzle exit diameter increases. When the driving flow pressure at the nozzle
exit is equal to the ambient pressure, the driving flow reaches optimal condition. This behavior implies thatappropriate nozzle structure can restrain the effects of a Mach wave. If the nozzle exit diameter increases further,an over-expanded condition occurs with an oblique shockwave. As shown in Fig. 12 (b),/ increases when
the nozzle exit diameter increases. The isentropic model given by Eq. (4), (5) cannot predict thevalue.

1.5

Convergent Nozzle

MOC simulation results
fitting
1.3

1.3

Lmax/dt

shockwaves occur

dmax/dt

1.4

1.4

MOC simulation results
Isentropic expansion
fitting

1.2

1.2

1.1

1.1
1.0

1.0
1.00

shockwaves occur
Convergent Nozzle

1.1

1.2

de/dt

1.05

1.10

1.15

de/dt

(a) Dimensionless maximum driving flow diameter(b) Dimensionless distance of theFig. 12. Simulation Results obtained by the MOC model on nozzle structures

1.20

surfaceFig. 13 shows the simulation results from the MOC model of the driving flow boundary with two different CD
nozzle structures. The different CD nozzle structures are marked as A and B. The simulation is conducted when thedriving and suction flow pressures are 500 kPa and 80 kPa. Nozzle A is the nozzle adopted in the experiment, withan exit-to-throat ratio of 1.094. Nozzle B is an optimized CD nozzle with an exit-to-throat ratio of 1.475. In the
figure, the boundary and expansion fan are illustrated, while the ejector wall and the grid in the driving flow regime
are omitted. By adjusting the throat-to-exit ratio of the ejector, the driving flow pressure at the nozzle exit could be
adjusted to equal the ambient pressure.
Unit: mm
80 kPa1.01670 kPa
A

0.8

Driving flow boundary

0.6

1.77 mm

0.4

mm

0.2

Unit: mm

0.0
0

1

B

2

mm
80 kPa3mmNozzle exit
80 kPaFig. 13 Simulation results from the MOC for different nozzle exit diameters ()

5.4 The driving flow expansion in the off-design condition
Fig. 14 (a) and (b) show theandof nozzle A and B under off-design working conditions. As thedriving flow pressure increases, the deviation between the isentropic expansion model and the MOC model increases.
This behavior implies that it is necessary to consider the Mach wave when the ejector is operated under off-design
conditions. From the figures, the larger exit diameter in nozzle B always creates more appropriate driving flowboundary development compared to nozzle A. This means that appropriate nozzle structure design can minimize theinfluence of the expansion waves.

1.7

1.6

Nozzle aNozzle b
isentropic expansion

1.6

Nozzle aNozzle b

1.5

dmax/dt

Lmax/dt

1.5
1.4

1.4

1.3

1.3
1.2

1.2
1.1

0.50

0.55
0.60
0.65
Driving Flow Pressure (MPa)

0.70

1.1

0.50

0.55
0.60
0.65
Driving Flow Pressure (MPa)0.70

Fig. 14 Driving flow boundary prediction by the MOC when pressure changes

6. Conclusions
In this research, experimental and numerical investigations on driving flow inside the ejector were conducted,
and the main conclusions are listed as follows:
1. The Method of Characteristics is applied as the numerical approach to predict the driving flow boundary
development inside the ejector in the under-expanded condition. The influence of expansion waves at the nozzle
exit was considered.
2. The numerical simulation results were validated by visualization experiments conducted using the Schlieren
optical system. The results show good agreement between the MOC simulation results and the measured values
from the experiment.
3. The values forandare affected by the nozzle structure. The expansion waves from the nozzle
exit strengthen the driving flow expansion, which creates a negative effect on ejector performance. However, with
an appropriate nozzle design, the performance of the ejector can be enhanced.
4. Under the off-design condition,
deviates from the predicted value according to the isentropic expansion
equations. It is necessary to consider the influence of Mach wave on the ejector performance when an ejector isapplied in a refrigeration system with an unstable heat source.

Acknowledgement
This work is partly financially supported by the Japan Science and Technology Agency (JST) under the project“Research and Development to Find Solutions to Environmental and Energy Issues in Urban Areas”

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