CRREL REPORT 95-20

Effect of Condensation on

Performance and Design of

Extended Surfaces

Virgil J. Lunardini and Abdul Aziz November 1995

Condensate

Film

D

Saturated Vapor, Tsat

L

Tfb

h (Circumferential Average)

X

q kf g
Abstract

Heat transfer surfaces operating in cold regions often involve condensation. The analytical and experimental progress made in understanding the

process of condensation on extended surfaces (fins) is reviewed in detail.

The review covers condensation of pure vapor as well as dehumidification

of air. The analytical models discussed range from simple Nusselt-type

analysis to the three-dimensional conjugate approach, in which the conservation equations for the condensate film are tightly coupled to conduction in the fin. A separate section discusses the topic of dehumidification of

air on finned cooling coils. Other topics reviewed include condensation on

horizontal integral-fin tubes, convective condensation in internally finned

tubes, and condensation in micro-fin tubes. Although condensation on

horizontal integral-fin tubes appears to be well understood, our understanding

of convective condensation in internally finned tubes, particularly the micro-fin tubes, is very limited. Furthermore, there exists no established methodology for designing extended surfaces for condensation applications.

This report contains several examples illustrating the theoretical results that

provide some insight into the design process.

For conversion of SI units to non-SI units of measurement consult ASTM

Standard E380-93, Standard Practice for Use of the International System

of Units, published by the American Society for Testing and Materials,

1916 Race St., Philadelphia, Pa. 19103.
CRREL Report 95-20

Effect of Condensation on

Performance and Design of

Extended Surfaces

Virgil J. Lunardini and Abdul Aziz November 1995

Prepared for

ARMY RESEARCH OFFICE

Approved for public release; distribution is unlimited.

US Army Corps


of Engineers

Cold Regions Research &


Engineering Laboratory
PREFACE

This report was prepared by Dr. Virgil J. Lunardini, Mechanical Engineer, Applied

Research Division, Research and Engineering Directorate, U.S. Army Cold Regions Research and Engineering Laboratory (CRREL), Hanover, New Hampshire, and Dr. Abdul

Aziz, Professor, Department of Mechanical Engineering, Gonzaga University, Spokane,

Washington. This study was primarily funded by the Army Research Office Battelle

Summer Faculty Program, Contract DAAL03-91-C-0034, TCN 94-077, no. 1140.

The authors thank Paul Richmond and F. Donald Haynes of CRREL for their technical

review of this report.

The contents of this report are not to be used for advertising or promotional purposes.

Citation of brand names does not constitute an official endorsement or approval of the use

of such commercial products.

ii
NOMENCLATURE

a slope of saturation curve or a constant

A heat transfer area or a constant

b slope of enthalpy–temperature curve or base distance

B a constant

Bi Biot number

c a constant

cp specific heat

Ca Ackerman correction factor

Cf interface enhancement factor

d tube diameter

D pin fin diameter

Dh hydraulic diameter

e fin height

f stream function

F,F1,...F4 parameters

g acceleration due to gravity or dimensionless temperature

G mass flux (velocity)

h local heat transfer coefficient

 

h average heat transfer coefficient

hfg latent heat of vaporization

hm mass transfer coefficient

 

hN average heat transfer coefficient over N tube rows

H fin depth

k thermal conductivity

L fin length

Lf average fin height over the diameter do

m exponent or fin parameter

m1 a constant defined by eq 87

 

m ̇ condensate mass flow rate

n exponent or number of fins

N number of tubes in a column or fin parameter

N* wet fin parameter

p pitch

P total pressure

Prl Prandtl number for liquid phase

Ps,Tr saturation pressure at temperature Tr

Pva partial pressure of water vapor at temperature Ta

q heat transfer to the base of the fin

q” heat flux

R dimensionless radial coordinate or the ratio of sensible to total heat flux

iii
Rel condensate Reynolds number

Rv gas constant for water vapor

s gap between fins

sm maximum length of the condensate interface

T temperature

Ua airstream velocity

w fin thickness

wt weight of the tube

wtp weight of the plain tube

x axial distance or vapor quality

X dimensionless axial distance

z transverse coordinate

Z dimensionless transverse coordinate

Greek Symbols

α helix angle

β condensate flooding angle or fin included angle

∆ dimensionless film thickness

∆* dimensionless parameter ∆4/Z

δ condensate film thickness

η total surface efficiency

ηf fin efficiency

θ dimensionless temperature

θm rotation angle from fin tip to base

μ absolute viscosity

ν kinematic viscosity

ξ similarity variable or interface shape parameter

ρ density

σ surface tension

φ relative humidity

ψ a parameter equal to ∆4

ωs specific humidity of saturated air

Subscripts

a ambient or air

b base

bs saturated at base temperature

c classical

cr critical

d dry

f fin

fb fin base

ft fin tip

h horizontal or hydraulic

i condensate-air interface

l liquid phase

o outside

iv
p plain

r reference or fin root

s sensible or saturated air

sat saturated

t total

v vapor phase

w wet

ws sensible for wet fin

wt total for wet fin

∞ freestream or ambient

v
CONTENTS

Page

Preface ................................................................................................................................ ii

Introduction........................................................................................................................ 1

Condensation on single fins ............................................................................................ 2

Nusselt-type models .................................................................................................... 2

Conjugate models ......................................................................................................... 7

Dehumidification of air on fins ....................................................................................... 17

Simple models ............................................................................................................... 17

Conjugate models ......................................................................................................... 20

Experimental studies ................................................................................................... 26

Optimum fin design ..................................................................................................... 27

Horizontal integral-fin tubes ........................................................................................... 30

Condensate flooding .................................................................................................... 30

Theoretical models for heat transfer coefficient ...................................................... 33

Experimental heat transfer coefficients ..................................................................... 37

Effect of interfacial shear ............................................................................................. 38

Effect of tube bundle geometry .................................................................................. 39

Effect of tube thermal conductivity ........................................................................... 40

Internally finned tubes ..................................................................................................... 41

ILLUSTRATIONS

Figure

1. Condensation on a horizontal pin fin .................................................................. 2

2. Temperature distributions in a horizontal pin fin with condensation ........... 3

3. Efficiencies of horizontal and vertical fins with condensation........................ 4

4. Pin fin in upward and downward vertical orientations ................................... 5

5. Temperature distributions in vertical pin fins ................................................... 6

6. Condensation on a vertical fin of rectangular profile ....................................... 11

 7. Two-dimensional vertical fin of rectangular profile ......................................... 12

8. Nonsimilar temperature profiles in a vertical rectangular fin ........................ 15

9. Similar and nonsimilar distributions of condensate film thickness ............... 15

10. Comparison of similarity and nonsimilarity results for the fin efficiency .... 16

11. Temperature distributions in a radial fin with moisture condensation

from surrounding air ...................................................................................... 19

12. Efficiencies of dry and wet radial fins ................................................................. 20

13. Dehumidification of air on a vertical rectangular fin ....................................... 21

14. Effects of dry bulb temperature, relative humidity, and fin base temperature on the temperature distributions in a vertical rectangular fin ............ 23

15. Sensible, latent, and total heat transfers for wet vertical rectangular fins .... 23

16.Vertical rectangular fins, and vertical triangular fins ........................................ 28

17. Optimum fin parameter, Nopt, for a vertical rectangular fin with moisture

condensation ........................................................................................................ 29

18. Comparison of Nopt for vertical rectangular and vertical triangular fins

under dry and wet conditions .......................................................................... 29

vi
Figure Page

19. Horizontal integral-fin tube .................................................................................. 30

20. Condensate flooding on a horizontal integral-fin tube .................................... 30

21. Effect of porous drainage strip on condensate flooding .................................. 31

22. Typical film profile for condensation on a fin with small tip radius, with

increasing radius along the arc length............................................................. 34

23. Effect of fin spacing on the enhancement ratio for steam condensing on

horizontal integral-fin tubes.............................................................................. 37

24. Effect of fin spacing on the enhancement ratio for R-113 condensing on

horizontal integral-fin tubes.............................................................................. 38

25. Condensation heat transfer coefficient for condensation of steam in

internally finned horizontal tubes.................................................................... 42

26. Pressure drop during the condensation of steam in internally finned

horizontal tubes ................................................................................................... 42

27. Cross sections of Hitachi Thermofin tubes ......................................................... 43

28. Heat transfer coefficient for R-22 condensing in micro-fin tubes ................... 44

TABLES

Table

1. Tip temperatures for a vertical rectangular fin .................................................. 8

2. Condensation efficiency of a rectangular vertical fin ....................................... 9

3. Effect of tube material on enhancement ratios ................................................... 40

4. Geometric data for internally finned tubes ........................................................ 42

5. Characteristics of micro-fin tube designs............................................................ 44

vii
Effect of Condensation on Performance and

Design of Extended Surfaces

VIRGIL J. LUNARDINI AND ABDUL AZIZ

INTRODUCTION

Extended surfaces or fins have been traditionally employed to reduce the convective

resistance associated with low values of the heat transfer coefficient h such as those

encountered in convection to and from gases. In condensation, the typical values of h are

high and the need for and the effectiveness of fins for augmentation may not be immediately apparent. However, in the past 50 years, several engineering situations have been

identified where augmenting condensation can be beneficial. Consider, for example, the

condensation of organic vapors where the poor thermophysical properties result in comparatively low values of h and consequently offer room for enhancement. Even with

fluids having favorable thermophysical properties, the condensing side resistance may be

significant and warrant reduction, especially if the cold side is augmented. The widespread use of integral fin tubes in surface condensers for the refrigeration and so-called

process industries clearly demonstrates the usefulness of fins for enhancing condensation. The use of finned tubes in the cold regions of the world will almost always occur

with condensation and the effects of the condensed liquid must be carefully examined.

Vapor on a surface condenses if the temperature of the surface is kept below the vapor

saturation temperature. Although four basic mechanisms (homogeneous, direct contact,

drop, and film) occur, most condensers are designed to operate under the film condensation mode. The process is characterized by the formation of a thin film of liquid that

drains under the action of gravity or surface tension or both. The presence of a film creates

a barrier between the vapor and the cooled surface and thus retards the condensation

process. If condensation is to be enhanced, the film thickness must be reduced. This

reduction can be achieved by using, among other methods, finned surfaces instead of

plain surfaces.

The purpose of this report is to serve as a comprehensive review of the published

literature on condensation on extended surfaces. The authors hope that the review will be

useful to researchers and practicing engineers alike. The report contains several examples

that serve to demonstrate the applicability of the material to engineering analysis and

design.

To facilitate a systematic presentation, the report has been organized as follows. First,

the theory of film condensation on extended surfaces is introduced. The theory utilizes

the well-known Nusselt model for the heat transfer coefficient to analyze condensation on

three fin configurations: horizontal cylindrical (pin) fin, vertical cylindrical fin, and vertical rectangular fins. This is followed by a discussion of conjugate conduction–condensa-
tion theory, and includes condensation of pure vapor as well as condensation of humid

air. Next, a section is devoted to the design of optimum fins for condensation applications.

The concluding part of the report refers briefly to vapor space condensation on horizontal

integral fin tubes, convective condensation in internally fined tubes, and condensation in

micro-fin tubes.

CONDENSATION ON SINGLE FINS

Nusselt-type models

The main difficulty in the analysis of film condensation on fins is the variability of the

heat transfer coefficient, h. Unlike the classical fin analysis that assumes h to be constant, h

for laminar condensation is a function of the difference between the local fin temperature

and the saturation of the condensing vapor. Another difficulty is that the surface (fin) is

nonisothermal, whereas the simple Nusselt theory applies to an isothermal surface. Despite these difficulties, it will be shown in the following subsections that a localized

application of Nusselt theory can be used at least for preliminary analysis and design.

Horizontal cylindrical (pin) fin

Consider a horizontal cylindrical fin of diameter D, length L, and thermal conductivity

kf as shown in Figure 1. The fin is in contact with a pure saturated, quiescent vapor at

temperature Tsat. The fin is attached to a cooled surface at fin base temperature Tfb (< Tsat).

Thus, the fin provides a cooled surface for the adjoining vapor to condense upon. Under

steady-state conditions, the latent heat extracted from the vapor is conducted into the

colder base. The condensate film formed on the surface of the fin drips down under the

action of gravity.

Let 

h be the circumferentially averaged condensation heat transfer coefficient at any

axial location on the fin. Since the values of 

h are usually large, high values of the Biot

number Bi = hD/2k occur which, in turn, induce two-dimensional thermal effects in the

fin. We ignore this fact to avoid further complication, and assume that axial conduction is

the dominant mode of heat transfer through the fin. The equations governing the temperature distribution in the fin can be written as

 

d

dX

hL

k D

2

2

2 4 0 θ − = θ

f

(1)

 

X X d

dX == = = 01 1 0 ,; , θ θ (2a,b)

where 

θ= − ( ) T T T T X xL sat f / ,/ ( ) sat − fb = , and the boundary condition (eq 2b) implies an

Figure 1. Condensation on a horizontal pin fin.

Condensate

Film

Saturated Vapor, Tsat

L

k f D g q

X

Tfb

h (circumferential average)

2
Figure 2. Temperature distributions in a horizontal pin fin with condensation.

insulated fin tip. To obtain 

h , we apply the Nusselt theory locally and use the well-known

expression for laminar condensation on a single, horizontal tube with zero interfacial

shear on the condensate film (Webb 1994). Thus, in terms of θ, we write

 

h g kh T T D = − 0 728[ ] ( ) ( ) − 3 1 4. / /ρρ ρ μ θ l l v l fg l sat fb (3)

where g = acceleration due to gravity

ρl = density of condensate

ρv = density of vapor

kl = thermal conductivity of the condensate

μl = absolute viscosity of the condensate

hfg = latent heat of condensation.

Substituting for 

h from eq 3 in eq 1 gives

 

d

dX

N

2

2

3 4 0 θ − = θ / (4)

where the number of tubes on a fin is

 

N g kh L k T T D v = − ( ) ( ) − [ ] 2 912 38 5 1 4. / /ρρ ρ μ l l l hg f l

4

sat fb . (5)

Since an analytical solution of eq 4 subject to boundary conditions (eq 2a,b) is not feasible,

a numerical procedure involving the combination of quasi-linearization and superposition (Na 1979) was used to obtain the solution. Figure 2 displays the numerical results for

N = 5, 10, 50 and 100. These results match those of Lienhard and Dhir (1974) who used a

shooting method to generate the numerical solutions.

Figure 2 provides some insight into fin design. For N = 50 and 100, the last half of the fin

does not sustain any condensation because the temperature difference, Tsat – Tf, is virtually zero. Such a design is a waste of fin material. At low values of N, say N = 5, the

entire fin surface is effective in supporting condensation, but then the fin is too short for

substantial condensation augmentation. It is therefore reasonable to conclude that for

good design, N should be of the order of 10.

1.0

0.5

0 0.5 1.0

X

θ

N = 5

10

50

100

3
1.0

0.8

0.6

0.4

0.2

0 20 40 60 80 100

N

η

Horizontal

Fin

Upward Pointing

Vertical Fin

Downward Pointing

Vertical Fin

The efficiency, ηf, of the fin can be found by computing the heat conducted into the base

of the fin and dividing it by 

q h x DL T T ideal == − ( ) sat fb ( ) 0 π . The resulting expression for ηf is

 

η

θ

f = −

=

1

N 0

d

dX X

 . (6)

The efficiency values calculated from the numerical solution and using eq 6 are plotted in

Figure 3. The figure also contains the results for the vertical fins that are discussed in the

next section. If N = 10 is selected to design the fin, then the corresponding ηf read from

Figure 3 is 0.34 or 34%, which is rather low. This low value for η is inevitable if a

significant increase in condensation rate is to be realized.

Example 1

Saturated steam at 0.15 bar condenses on a surface at 25°C. It is desired to enhance the

rate of condensation by attaching a cylindrical fin made of brass to the primary surface.

Suggest some suitable design options.

Solution

For saturated steam at 0.15 bar, the following data apply:

 

T h sat v =° = = 54 0 098 2373 C, kg/m fg kJ/kg 3 ρ . , .

Evaluating the properties of condensate at a mean temperature of (54 + 25)/2 = 39.5°C, the

following values for properties result:

 

ρ μ l = =× = l l − 992 kg/m 663 10 s/m , 0.631 W/m K 3 62 , N k .

The thermal conductivity of brass is taken as kf = 61 W/m K.

For good design, take N = 10. Using eq 5, a relationship between length L and diameter

D can be established as follows:

 

L

D

N g kh L

kTT

v

5 8

1 2 3 8

4

1 8

3 8

2 912

0 2244 /

/ /

/

. = . 



 



 ( ) −

( ) −















 = ρρ ρ

μ

l l l

l

fg

sat fb

m

f

.

Figure 3. Efficiencies of

horizontal and vertical fins

with condensation.

4
Figure 4. Pin fin in upward

and downward vertical orientations.

D

L

X

Tfb

Kf

q

Saturated

Vapor, Tsat

g

δ

(a) Upward Pointing

Saturated

Vapor, Tsat

D

Kf

Tfb

X

(b) Downward Pointing

δ(L)

g

q

L

Selecting a number of values for D, the corresponding values of L can be found. The results are summarized below:

D (mm) L (mm)

5 8.18

7.5 10.54

10 12.60

12.5 14.51

15 16.26

Vertical cylindrical (pin) fin

Figure 4 shows a cylindrical fin in two vertical orientations—upward and downward.

For the upward pointing fin, the distance x is measured from the tip, while for the

downward pointing fin, the same is measured from the base. The essential difference

between the horizontal and vertical fins is that for the former, the surface was isothermal

along the direction (circumferential) of condensate flow, while in the latter, the temperature decreases for upward configuration or increases for downward configuration along

the condensate flow direction. Thus the original Nusselt’s theory, which assumes isothermal conditions along the condensate flow direction, is not directly applicable for the

vertical fins of Figure 4. However, Lienhard and Dhir (1974) have shown that the Nusselt’s theory, if appropriately modified to account for a nonisothermal surface, gives

results that are close to the predictions of the full boundary layer equations. For a nonisothermal flat vertical surface, the modified Nusselt’s theory gives the following expression

for the value of (circumferential average) at any location x from the leading edge:

 

h g kh

T T dx x = ( ) −

( ) −















 ∫

0 7071

3

0

1 4

.

/

ρρ ρ

μ

l l l

l

v fg

sat f

. (7)

Equation 7 applies to a cylindrical surface if the curvature effects are small, that is,

 

δ ( ) L D << 1 2 . Using the definitions of θ and X from Horizontal Cylindrical Fin, eq 7 can be

(a) Upward Pointing (b) Downward Pointing

5
recast as follows

 

h g kh

L T T dX X = ( ) −

( ) −















 ∫

0 7071

3

0

1 4

.

/

ρρ ρ

μ θ

l l l

l

v fg

sat fb

. (8)

Substituting h from eq 8 in eq 1, the following integro-differential equation for θ is

obtained:

 

d

dX

N dX X 2

2 0

1 4

0 θ − θ θ [ ] ∫ = − /

(9)

where

 

N g kh L

k D =  ( ) −















2 8284

3 7

4

1 4

.

/

ρρ ρ

μ

l l l

l

v fg

f

4 . (10)

Equation 9 applies to both configurations of Figure 4, but the boundary conditions are

different. For the upward pointing fin, these are

 

X d

dX = = == 0 0 11 , ;, X θ θ . (11a,b)

For the downward pointing fin, the boundary conditions are

 

X X d

dX === = 01 1 0 ,; , θ θ . (12a,b)

Numerical solutions of eq 9 subject to boundary conditions (eq 11a,b or 12a,b) have

been reported by Lienhard and Dhir (1974). Figure 5 is an adaptation of their results. A

close examination of the left and the right portions of Figure 5 reveals that the temperaFigure 5. Temperature distributions in vertical pin fins. Adapted from Lienhard

and Dhir (1974).

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1.0

X

0

100

50

10

N = 5

Downward

Pointing

1.0

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1.0

Upward

Pointing

N = 5

10

50

100

θ

X

6
ture profiles for the two orientations are slightly different. As in the case of the horizontal

fin, a good design value of N is on the order of 10.

Equation 6 for the fin efficiency also applies to the vertical fins. The results for the

efficiency of vertical fins are shown in Figure 3 to allow a comparison with the horizontal

fin. Figure 3 shows that vertical fins are more efficient than horizontal fins and, of the two

vertical arrangements in Figure 4, the downward pointing fin has a higher efficiency than

the upward pointing fin.

Vertical rectangular fin

The results of the foregoing section are also applicable to a vertical rectangular fin if the

definition of N is modified appropriately to represent the rectangular geometry.

Conjugate models

In the conjugate models, the heat conduction equation for the fin and the condensate

boundary layer equations are solved simultaneously. In the simple model, which has been

used by Nader (1978), Burmeister (1982) and Acharya et al. (1986), both the Tf and δ are

allowed to vary along the condensate flow direction only; that is, a one-dimensional fin

model is used with a two-dimensional condensate film. The improved model proposed by

Patankar and Sparrow (1979) considers a two-dimensional fin with a three-dimensional

condensate layer.

Simple conjugate model

Consider a vertical fin of rectangular profile as shown in Figure 6. The fin has length L,

thickness w, and thermal conductivity kf. Both faces of the fin are exposed to a saturated

vapor at temperature, Tsat > Tfb. The boundary conditions for the fin are those of constant

base temperature Tfb < Tsat and no heat flow through the tip. To establish the conservation

equations, consider a slice (fin and two films) of thickness dx. Equating the heat conducted through the two condensate films to the net heat conducted through the fin slice gives

 

d T

dx

kT T

kw

2

2

2 f sat f = − − l ( )

δ (13)

where w is fin thickness and δ the condensate film thickness at x. In deriving eq 13, the

temperature distribution through the condensate film has been assumed to be linear. The

film thickness δ can be related to the local temperature difference, Tf – Tsat, through the

application of the momentum equation, giving

 

d

dx

kT T

g h

( ) δ μ

ρρ ρ

4 4 = ( ) −

( ) −

l l

l l

sat f

v fg

 . (14)

It is convenient to introduce the following dimensionless quantities:

 

θ δ = − ( ) T T T T X xL L sat f / , /, / ( ) sat − fb = = ∆ (15)

 

F g hL

kT T

F k w

k L

v

1

3

2 2 = ( ) −

( ) − = ρρ ρ

μ

l l

ll l

fg

sat fb

f ,

into eq 13 and 14 to give

7
 

d

dX F

2

2

2

θ θ = ∆ (16)

 

d

dX F

( ) ∆4

1

4 = θ . (17)

The boundary conditions for eq 16 and 17 are

 

X d

dX = = = == 0 0 0 11 , , ;, ∆ X θ θ . (18a,b)

Nader (1974) obtained a numerical solution of eq 16–18, while Burmeister (1982) developed an approximate analytical solution of the same equations. Acharya et al. (1986)

solved the dimensional equations (13 and 14) using an iterative numerical scheme. They

also extended their computations to six other fin shapes (triangular, trapezoidal, convex

parabolic, concave parabolic, cylindrical, and conical) and developed simple correlations

for the fin efficiency.

Nader’s solution

Nader (1974) introduced a new variable ψ = ∆4 which enabled him to transform eq 16

and 17 into two, coupled first-order differential equations. These are

 

d

dX F

ψ θ = 4

1

(19)

 

d

dX

F

F

θ = ψ 1

2

3 4

3

/ (20)

subject to

X = 0, ψ = 0; X = 1, θ = 1. (21a.b)

Solving the foregoing equations numerically, Nader obtained the values of tip temperature, θ(0) for a range of values of F1 and F2. These values are recorded in Table 1.

The rate of heat conduction into the base of the fin, q can be obtained as

Table 1. Tip temperatures for a vertical rectangular

fin.

 F1/F2 104 103 102 10

107 0.9970 0.9703 0.7460 0.0875

108 0.9947 0.9479 0.6011 0.0169

109 0.9905 0.9096 0.4182 0.0007

1010 0.9832 0.8461 0.2307 —

1011 0.9703 0.7460 0.0875 —

1012 0.9479 0.6011 0.0169 —

1013 0.9096 0.4183 0.0070 —

1014 0.8461 0.2307 — —

1015 0.7460 0.0875 — —

1016 0.6011 0.0169 — —

1017 0.4183 — — —

8
 

q kT T F d

dX X

= − ( ) =2 2

1

l sat fb

θ . (22)

The ideal heat transfer, qideal, occurs when the entire fin is at temperature Tfb and is given

by

 

q hL T T ideal = − ( ) sat fb 2 (23)

where 

h is the average heat transfer coefficient for condensation on an isothermal vertical

surface. The expression for 

h based on Nusselt’s theory is

 

h g kh

T TL

v

b

= ( ) −

( ) −

















0 943

3 1 4

.

/

ρρ ρ

μ

l l l

l

fg

sat f

(24)

which in terms of F1 becomes

 

h k

L = 0 943 F1

1 4 . l / . (25)

Combining eq 23 and 25 and noting that η = q/qideal, we have

 

η

θ

f =

=

1 0604 2

1

1 4

1

. /

F

F

d

dX X

 . (26)

The efficiency values obtained from eq 26 appear in Table 2.

The film thickness δ at x = L, the condensate flow rate 

m ̇ at x = L, and the heat conducted

into the base of the fin q, can all be expressed in terms of F1 and ηf giving

 

δ η () . / / L LF = − 1 412 1 3

1

1 4

f (27)

 

 ̇ () . / / mL k T T F h = − 1 8856 ( ) 1

1 4

l sat fb f fg η (28)

 

q kT T F = − 1 8856 ( ) 1

1 4 . /

l sat fb f η . (29)

Table 2. Condensation efficiency of a rectangular vertical fin.

F1/F2 104 103 102 10

107 0.9988 0.9857 0.8745 0.4603

108 0.9979 0.9747 0.7981 0.3612

109 0.9955 0.9563 0.6969 0.2823

1010 0.9919 0.9254 0.5776 —

1011 0.9857 0.8745 0.4603 —

1012 0.9747 0.7989 0.3612 —

1013 0.9562 0.6969 0.2823 —

1014 0.9254 0.5776 — —

1015 0.8745 0.4603 — —

1016 0.7989 0.3612 — —

1017 0.6969 — — —

9
Example 2

Pure saturated steam at 60°C condenses on the outside surface of a 50-mm diameter

horizontal tube maintained at a uniform temperature of 34°C. Calculate the rate at which

heat is transferred to the tube and the condensate flow rate. To enhance the rate of

condensation, the tube is fitted with a vertical fin of rectangular profile. The fin is 2 mm

thick and 7.5 mm long, and has a thermal conductivity of 48 W/m K. Calculate the rate at

which steam condenses on the two faces of the fin and compare this with the rate of

condensation on the fin’s base area if the fin is absent.

Solution

For saturated steam at 60°C, we have

ρv = 0.129 kg/m3, hfg = 2358 kJ/kg .

Evaluating the properties of condensate at a mean temperature of 47°C, we have

ρl = 989.1 kg/m3, μl = 577 × 10–6 N s/m2, kl = 0.640 W/m K.

The average heat transfer coefficient for laminar film condensation on an isothermal

horizontal tube is given by eq 3 with θ = 1 (isothermal):

 

h g kh

T TD = ( ) −

( ) −

















0 728 6864 =

3 1 4

.

/

ρρ ρ

μ

l l l

l

v

b

W/m K . 2 fg

sat f

The rate at which heat is transferred to the tube is given by

 

q h DT T = −= π ( ), sat f b 28 033 W/m.

The condensate flow rate m ̇ is given by

 

m ̇ . q

h

==× −

fg

1 19 10 2kg/s m .

To evaluate q and m ̇ (L), we first calculate F1 and F2 as follows:

 

F g kL

kT T 1

3

8 9 = 9 94 10 10 ( ) −

( ) − = ×≅

ρρ ρ

μ

l l

l l

v fg

sat fb

.

 

F k w

k L 2 2 = = 10 f

l

.

For F1 = 109 and F2 = 10, Table 2 gives η = 0.2823. Using eq 28, the condensate flow rate,

 

m ̇ (L) is

 

 ̇ () . ( ) / . . / mL k T T F h = − =× 1 8854 6 67 10 1

1 4 4

l sat fb fg η kg/s m

The condensation rate on the base of the fin in the absence of fin is (1.19 × 10–2) (0.002)/

(π)(0.05) = 1.51 × 10–4 kg/s m. Comparing this with the figure of 6.67 × 10–4 kg/s m, the fin

is seen to enhance condensation by a factor of 4.4.

10
Burmeister’s solution

Burmeister (1982) combined eq 16 and 17 into a single equation and obtained an

approximate analytical solution for it. His solutions for the tip temperature θ(0), heat

transfer rate q, and fin efficiency are

 

θ( ) /cosh 0 1 = F (30)

 

q k T T FF F = − 1 8263 ( ) ( ) 1 2

3 1 7 6 7 . tanh / / l sat fb (31)

 

η = 



 



 tanh / F

F

6 7

(32)

where

 

F FF = 1 038( ) 1 2

4 1 8 . / . (33)

The efficiency values predicted by eq 32 are in close agreement with the values given in

Table 2.

Archarya et al. solution

Archarya et al. (1986) considered vertical fins (Fig. 6) of seven profile shapes, namely

rectangular, triangular, trapezoidal, concave parabolic, convex parabolic, cylindrical and

conical. For each geometry, they solved eq 13 and 14 numerically and obtained the results

for η. For all seven shapes, the efficiency could be represented by a simple relationship of

the form

 

η η = c

0 855 . (34)

where ηc is the efficiency of the fin calculated from classical one-dimensional fin theory

and assuming the heat transfer coefficient to be constant for all shapes, its value being given by eq 24.

Expressions for ηc for different shapes can be found

in Kern and Kraus (1972).

Improved conjugate model

Kazeminejad (1993) improved the simple conjugate model described by eq 16–18 by including the

effect of vapor velocity. As expected, the effect of

vapor shear was to reduce the condensate film thickness and hence increase the fin surface temperature.

The heat transfer to the fin and its capability to support condensation is considerably enhanced.

Patankar and Sparrow (1979) considered film condensation on a vertical rectangular fin, which is attached to a cooled vertical base at temperature Tfb

(Fig. 7). The fin has length L, thickness w and depth

H, and is made of a material with thermal conductivity kf. The fin is immersed in a pure saturated

vapor at temperature Tsat (> Tfb).

Figure 6. Condensation on a vertical

fin of rectangular profile.

w

y

L

X

kf

Tfb

dx Saturated

Vapor, Tsat

Film

11
The analysis assumes the fin temperature Tf and the film thickness δ to be functions of

x and z; that is, Tf = Tf (x,z) and δ = δ (x,z). These assumptions can be justified as follows.

As the condensate film flows downward along the fin, more condensate is added to it, and

its thickness increases along z to accommodate the increased flow rate. Along the x

direction, the temperature differential, Tsat – Tf, decreases from the base (x = 0) of the fin to

the tip (x = L) of the fin. Consequently, δ also decreases along the x direction. Thus the

physics of the process dictates that δ is a function of x and z. The effect of the growth of δ

with z is to increase the thermal resistance of the film, thereby decreasing the heat conduction into the fin. The decreased heat flow implies that the fin temperature at a given x

must decrease along the z direction. Thus the temperature distribution in the fin is also

two-dimensional; that is, Tf = Tf (x,z).

Considering a fin element of dimensions dx, dy and w and making an energy balance

gives

 

∂

∂ δ

2

2

T 2

x

kT T

k w

f f sat

f

= l ( ) − . (35)

Equation 35 assumes that in the fin, conduction in the x direction is dominant, while in the

film, conduction in the y direction is dominant. The momentum equation for the z direction takes the form

 

∂

∂

δ μ

z ρρ ρ

kT T

h ( ) 4 4 = ( ) −

( ) −

l l

l l

sat f

g v fg

 . (36)

Equations 35 and 36 constitute two coupled partial differential equations for Tf (x,z)

and δ (x,z).

For convenience, the following dimensionless quantities are introduced into eq 35

and 36:

 

θ= − ( ) T T T T X xL f sat / ,/ ( ) fb − sat =

Tsat

Tfb

y

x

z

H

k f

w

L

Figure 7. Two-dimensional vertical fin of

rectangular profile.

12
 

Z

kT T

h

k w

k L

z

k w

k L = ( ) −

( ) −





 



 = 











μ

ρρ ρ

δ l l

l l l l

sat fb

v fg

f f

4g 2 2

4

2 , ∆ (37)

which then become

 

∂ θ

∂

θ 2

X2 = ∆ (38)

 

∂

∂

θ ( ) ∆4

Z = . (39)

The boundary conditions on θ and ∆ are

 

X X

X == = = 01 1 0 ,; , θ ∂θ∂ (40a,b)

Z = 0 , ∆ = 0 . (41)

Patankar and Sparrow (1979) sought a similarity solution of eq 38–41 by arguing as

follows. The Nusselt’s theory on a vertical isothermal shows that the local heat transfer

coefficient hZ is proportional to Z–1/4, giving high values for hZ at small values of Z. The

high values of hZ cause the fin temperature to increase rapidly from Tb at x = 0 to Tsat,

significantly before x = L. Thus the behavior of the fin closely approximates that of an

infinitely long fin, permitting the condition eq 40b to be replaced by

X = ∞ , θ = 0 . (42)

In the limit Z = 0, hZ becomes infinite and the temperature distribution in the fin takes the

form of a step increase from Tb to Tsat. Mathematically, this means

Z = 0 , X > 0 , θ = 0 . (43)

Examining the behavior of δ, one notes that at small values of Z, δ must diminish quite

rapidly with X to reflect the rapid decrease of Tsat – T with X. This permits us to write

X = ∞ , ∆ = 0 . (44)

The behavior of θ and ∆ at Z = 0 and X = ∞ indicates the possibility of a similarity

solution.

Similarity solutions

The introduction of a similarity variable ξ as

 

ξ = X Z/ 1 8 / (45)

with dependent variables

 

θξ θ ξ * * () , () / = = ∆ ∆4 Z (46)

reduces the partial differential eq 38 and 39 to the following ordinary differential equations

13
 

d

d

2

2

1 4 0 θ

ξ

θ * ** / − = ( )− ∆ (47)

 

d

d

∆ ∆ * * * ( ) ξ −− = θ ξ− 8 0 1 (48)

with the boundary conditions

 

ξθ ξ θ = = =∞ = = 01 0 , ;, * ** ∆ . (49a,b)

Patankar and Sparrow (1979) observed that an analytical solution of eq 47–49 was not

possible, but Wilkins (1980) showed that an analytical solution does exist and can be

written as

 

θ θ

ξ ξ

ξ

= = − 



 



 ≤ ≤

>











* ,

,

1

42

0 42

0 42

7

(50)

 

∆* ,

,

= − 



 



 ≤ ≤

>











1

42

0 42

0 42

8 ξ ξ

ξ

(51)

Using eq 50, the temperature gradient

 

∂θ

∂X X=0

can be found. Integrating

 

∂θ

∂X X=0

from Z = 0 to Z = Z, the heat conducted into the base of the fin over a distance Z can be

evaluated. The final result is

 

q Z

kh L

k w () . Z / =  ( ) −













 4 9371

3 7

3

7 8 g v fg

f

3

ρρ ρ

μ

l l l

l

 . (52)

The ideal heat transfer qideal (Z) can be found by assuming the entire fin to be isothermal at temperature Tb and using 

h from eq 34. This gives

 

q Z

kh L

k w ideal Z v fg

f

g () . / =  ( ) −















5 333

3 7

3 3

3 4 ρρ ρ

μ

l l l

l

 . (53)

The fin efficiency η expressed as the ratio of q(Z)/qideal(Z) is found to be

 

η = 0 9257 1 8 . / Z . (54)

14
The similarity solutions for θ, ∆*, and η are shown in Figures 8, 9, and 10, respectively.

These figures also show the nonsimilarity solutions that are discussed next.

Nonsimilarity solutions

The applicability of the similarity solutions is limited to those z locations for which the

boundary condition eq 42 is justified, that is, locations where the tip temperature is nearly

equal to the vapor saturation temperature (θ = 0). For z locations where this condition is

not met, eq 38–41 were solved numerically by Patankar and Sparrow (1979). These results

appear in Figures 8–10.

Figure 8 shows the temperature profiles at various Z locations. For low values of Z, the

temperature distribution in the fin is quite steep as envisioned earlier. As Z increases, the

profiles become less and less steep, since larger values of θ= − ( ) TT T T f sat /( ) fb − sat mean

lower values for T (note that Tb – Tsat is negative); one concludes from Figure 8 that the

general level of fin temperature decreases as Z increases. This confirms the earlier hypothesis about the dependence of T on Z. Since the nonsimilarity temperature profiles must

terminate at X = 1, the terminal point on 

ξ = X Z/ 1 8 / scale occurs at lower and lower

values of ξ as Z increases. The similarity solution, on the other hand, extends up to

 

ξ= = 42 6 48 . .

θ

1.0

0.8

0.6

0.4

0.2

0 1 234

ξ

Z = 10

1.0

0.1

0.01

0.001

0.0001

Similarity

Solution Figure 8. Nonsimilar temperature profiles

in a vertical rectangular fin. Adapted from

Patankar and Sparrow (1970).

1.0

0.8

0.6

0.4

0.2

0 1234

ξ

Z = 10

1.0

0.1

0.01

0.001

0.0001

Similarity

Solution

∆*

1

/

4

5 6

Figure 9. Similar and nonsimilar

distributions of condensate film

thickness. Adapted from Patankar

and Sparrow (1970).

15
The condensate film thickness results are shown in Figure 9. For a fixed Z location, the

film thickness decreases as x or ξ increases, and this is consistent with the temperature

differential (Tsat – T) decreasing along X. For small Z, the film is highly nonuniform along

the X direction but becomes more and more uniform as Z increases. This clearly shows

that the assumption δ = δ(Z) employed in previous sections is not strictly valid.

Figure 10 shows the fin efficiency as a function of Z. The lower two curves cover the

range of Z from 0.0001 to 0.1, while the upper two curves cover the Z values from 0.01 to

10. Interestingly the similarity and nonsimilarity solutions for η are virtually identical up

to Z = 0.01. Thus for Z ≤ 0.01, eq 54 for η and hence eq 52 for q(Z) give accurate predictions.

However, this is not true of the similarity results for θ and ∆*. For example, Figure 8 and 9

shows that for Z = 0.01, there is a significant difference between the similarity and the

nonsimilarity solutions.

Example 3

A vertical rectangular fin (k = 400 W/m K) is attached to a cooled vertical surface (Fig.

7). The fin dimensions are L = 1.5 cm, w = 1.5 mm, and H = 25 cm. The environment

surrounding the fin is saturated steam at 50°C. Calculate (i) the rate at which heat is

removed by the cooled surface, (ii) the condensation rate supported by the fin, (iii) the fin

temperature and the film thickness at H = 1.5 cm, z = 25 cm.

Solution

The density and heat of vaporization for steam at 50°C are

 

ρv = =× kg m J/kg fg 0 082 2 383 10 3 6 . /, . . h

Evaluating the properties of water (condensate) at a mean temperature of (24 + 50)/2 =

37°C, we have

 

ρ μ ll l = =× = − 993 694 10 0 628 3 6 kg m N s/m W/m K 2 /, , . . k

Using the above properties in eq 37, Z can be evaluated:

 

Z

kT T

h

k w

k L = z ( ) −

( ) −





 



 = μ

ρρ ρ

l l

l l l

sat fb

v fg

f

4g

0 01 2

4

. .

1.0

0.1

η

Non-similarity

Solution Similarity 

Solution

0.01 0.1 1.0 10

(a)

1.0

0.1

0.0001 0.001 0.01 0.1

Z

η

(b)

Non-similarity

Solution Similarity 

Solution

Figure 10. Comparison of similarity and nonsimilarity results for the fin

efficiency. Adapted from Patankar and Sparrow (1970).

16
(i) Since Z is within the limit of applicability of the similarity solution, eq 52 can be

used to compute the rate at which heat is removed by the cooled surface:

 

q

kh L

k W = Z

 ( ) −















4 9371 571 =

3 7

3 3

7 8 . / gρρ ρv fg

μ

l l l

l

W.

(ii) The condensate rate supported by the fin is given by

 

m ̇ . q

h

== × −

fg

2 396 10 kg/s = 0.86 kg/h 4 .

(iii) Reading the terminal value of θ for the curve marked Z = 0.01 in Figure 8, the

dimensionless tip temperature is qt = 0.25. Thus

 

θt = −

− = T TT T

ft sat

fb sat

0 25 .

or 

T TT T ft fb = − 0 25 43 5 . . ( ) sat sat + = oC.

Similarly, the terminal value of ∆*1/4 for the curve marked Z = 0.01 in Figure 9 gives ∆*1/4 =

0.65. Using eq 46,

∆ = ∆*1/4 (Z)

1/4 = (0.65) (0.01)1/4 0.2055 .

Invoking the definition of ∆, that is, eq 37, the film thickness δ is given by

 

δ= = 2 0 0968 2 k L kw l ∆/ . mm

DEHUMIDIFICATION OF AIR ON FINS

In air conditioning applications, finned cooling coils are often used to cool and dehumidify air. The thermal performance of these coils is not only affected by geometry,

materials and psychrometric conditions, but also by the efficiency of the fin. If the fin

temperature is lower than the dew point of air passing over the coil, then the moisture is

condensed on the fin surface and affects the fin efficiency.

This section considers the performance of fins operating in moist air streams, with

moisture condensation occurring on their surface. Simple Models discusses simple models

in which the classical fin theory for dry fins is modified to take into account the effect of

mass transfer. Conjugate Models describes two conjugate models for simultaneous heat

and mass transfer to a cooling and dehumidifying vertical rectangular fin. Experimental

studies of dehumidification in finned coil heat exchangers are covered in Experimental

Studies. The design of optimum-dimensional rectangular and triangular fins with condensation is covered in Optimum Fin Design.

Simple models

Longitudinal fins

McQuiston (1975) considered moisture condensation on a longitudinal fin of rectangular profile having a length L, thickness w, and thermal conductivity kf. Let hd be the

17
average heat transfer coefficient for dry operating conditions. For the moisture condensation situation, McQuiston, neglecting the thermal resistance of the condensate, postulated

that the local driving potential for simultaneous heat and mass transfer was the difference

between the enthalpy of air adjacent to the fin and that of saturated air at the local fin

temperature. By approximating the saturation curve on the psychrometric chart by a

straight line over a small range of temperatures, he expressed the slope a as

 

a TT =− − ( )/( ) ω ω s s , , 2 1 21 (54)

where ωs is the specific humidity of saturated air. The heat transfer coefficient hw for wet

conditions was expressed in terms of hd and a as follows:

 

h h

ah

c w d

fg

p

= +











1  (55)

where cp is the specific heat of moist air at constant pressure.

In an earlier paper, Ware and Hacha (1960) recommended the following expression

for hw:

hw = hd b/cp (56)

where b is the slope of the enthalpy–temperature curve for saturated air.

With hw specified by either eq 55 or 56, the conventional fin theory can be employed to

obtain the efficiency of a wet fin. For boundary conditions of constant base temperature

and insulated tip, the fin efficiency can be expressed as

 

ηw = tanh *

*

N

N (57)

where 

N h kw L * / = ( ) 2 / 1 2

w f .

Radial fins

Elmahdy and Biggs (1983) considered a radial fin of base radius rb, tip radius rt,

thickness w, and thermal conductivity k, exposed to a stream of moist air at temperature Ta

and with specific humidity ωa. If the average heat and mass transfer coefficients are h and

hm, respectively, then the differential equation governing the temperature distribution in

the fin can be written as

 

d T

dr r

dT

dr

h

kw

T T h

kw

h

2 1 2 2 0 f

2

f a f m +− − ( ) −−= a T,s fg ( ) ω ω (58)

where ωT,s is the saturated specific humidity of air corresponding to the local fin temperature Tf. Assuming a constant fin base temperature Tfb and insulated fin tip, the boundary

conditions for eq 58 can be written as

 

rr T T rr dT == = = b f fb t

f , ;, . dr 0 (59)

Next ωT,s is assumed to be a linear function of temperature Tf, that is,

ωT,s = c + a Tf (60)

 .

18
1.0

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1.0

R

θ

Dry

Surface

80% RH

90

99

Figure 11. Temperature distributions in a radial fin with moisture

condensation from surrounding air. Adapted from Elmahdy and

Biggs (1983).

where the constant a (given by eq 54) and c are to be determined from the psychrometric

data for the range of temperatures considered. Substituting for ωT,s from eq 60 into eq 58,

the differential equation for Tf becomes

 

d T

dr r

dT

dr

h

kw

T T h

kw

c aT h

2

2

1 2 2 0 f f a f m +− − ( ) − −− ( ) ωa f fg = . (61)

A sample of numerical solution of eq 61 subject to boundary conditions eq 59 is shown

in Figure 11. In this figure, the dimensionless temperature 

θ= − ( ) TT TT a f /( ) a − fb is plotted

against dimensionless radius R = (r – rb)/(rt – rb) for dry as well as wet operating

conditions. The results are based on the following data: Ta = 16°C, Tfb = 7°C, h = 57 W/m2

°C, and 

N r r h kw = − ( )( / ) /

t b 2 1 2 = 0.82. It can be seen that the temperature profiles for a wet

fin lie below those of a dry fin. As the relative humidity of air increases, the driving

potential for mass transfer increases, which leads to a higher latent heat transfer and

higher fin temperature. Note that lower values of θ mean higher fin temperatures

Considering a typical fin surface element 2πrdr, the heat transfer dq to the element can

be expressed as

 

dq h T T h h r dr = −+ − [ ] ( ) ( ) ( ). a f m a T,s ωω π fg 2 (62)

Allowing for heat transfer to both faces of the fin and integrating eq 62, the total heat

transfer to the fin is found to be

 

q h T T h h r dr r

r

b

= −+ − ∫ [ ] t

a f m a T,s fg 4π ωω ( )( ) . (63)

The maximum or ideal heat transfer to the fin occurs if the entire fin surface is maintained at temperature Tfb, and is therefore given by

 

q r r hT T h ideal = − 2 ( ) t b [ ] ( ) a − fb + − m a ( ) 2 2 π ωωb,s . (64)

19
1.0

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1.0

N

η

Dry

Surface 

80% RH

90

99

Figure 12. Efficiencies of dry and wet

radial fins. Adapted from Elmahdy

and Biggs (1983).

The ratio of q/qideal gives the efficiency of the fin. Figure 12 shows the efficiency as a

function of N for both dry and wet fins. This figure is based on the same data as used in

Figure 11. The efficiency of a wet fin can be seen as lower than that of a dry fin, and

decreases as the relative humidity increases. This can be explained as follows. As the

relative humidity increases, the driving potential for mass transfer increases that, in turn,

causes qideal to increase. The corresponding actual q, however, does not increase by the

same amount. The net result is a decrease in η.

Conjugate models

This section describes two conjugate models for a cooling and dehumidifying vertical

fin of rectangular profile. The first model from Coney et al. (1989a) allows for the coupling

between the fin temperature and the condensate film, but assumes the convective heat

transfer coefficient to be constant. The approach is essentially the same as in Simple

Models, except that the model of Coney et al. also includes the effect of mass transfer in

writing the energy balance for the fin. The second model described by Kazeminejad et al.

(1993) neglects the thermal resistance of the condensate film but allows for the heat

transfer coefficient h to vary along the fin. The model finds the variation of h through the

solution of boundary layer equations. Both models are discussed in sections that follow.

Coney et al. model

The model considers a vertical rectangular fin as depicted in Figure 13. Taking a slice of

fin of volume bwdz and equating the net energy conducted through the slice to the energy

convected to the surface 2(b + w)dz by simultaneous heat and mass transfer, gives

 

d T

dz

b wq

kwb

2 2 f

2

t = − ( ) + ′′ (65)

where qt′′ is the total heat flux through the condensate film. Assuming a linear temperature profile for the condensate film, 

q′′ can be expressed as

 

′′ = − q

kT T

t

l ( ) i f

δ (66)

where Ti is the condensate/air interface temperature.

The presence of condensate film can enhance the heat and mass transfer at the condensate-air interface due to increased turbulence and effectiveness roughness. This can be

20
x

Air

φ

T , U, a

y

z

w

L

b

Condensate

Film

Figure 13. Dehumidification of air on a vertical

rectangular fin.

taken into account by multiplying the single-phase heat transfer coefficient h by an interface enhancement factor Cf. Cf depends on geometry and flow conditions, and has to be

determined experimentally. Since the minimum value of Cf is unity (for smooth interface

at low vapor velocity), the use of Cf = 1 would be conservative. The effect of mass transfer

on the temperature profile is taken into account by introducing the Ackermann correction

factor Ca. Thus, the sensible heat flux, qs′′ , between air and condensate film can be expressed as

 

qs′′ = CfCah(Ta–Ti) . (67)

Eliminating Ti between eq 66 and 67 and denoting the ratio 

q q ′′ ′′ s / t by R, the expression

for 

q′′t becomes

 

′′ = −

+

q

T T

k

R

CC h

t

( ) a f

f a

δ

l

 . (68)

Substituting for 

q′′t from eq 68 into eq 65, the differential equation governing the temperature distribution in the fin becomes

 

d T b w

kwb k

R

CC h

T T

2 1

2 0 f

2 f a a f dz

+

+

+ 



 



 ( ) − =

− ( ) δ

l

 . (69)

The momentum eq 14 for the condensate film can be adapted for the present analysis as

follows:

 

δ δ μ

ρρ ρ

2 d 1

dz

R q

g h = − ′′

−

l

l l

( )

( )

t

v fg

 . (70)

21
Eliminating 

q′′t between eq 68 and 70, the equation governing δ can be written as

 

δ δ μ

ρρ ρ

δ 2

1

1 0 d

dz

R

g hk

R

CC h − T T −

− + 



 



 ( ) − =

−

l

l l l

( )

( ) v fg f a a f . (71)

The simultaneous solution of eq 69 and 71 gives the fin temperature Tf(z) and condensate

film thickness δ(z).

Introducing the following dimensionless variables

 

θ δξ

ρρ ρ

μ

= − ( ) ( ) − = =

= −

( ) − = = +

= == +















T T T T L zL

F g hL

kT T

F Bi C C hL b w

k bw

F k

C C hL

F

F F

k bw

kLb w

a f a fb

v fg

a b

f a

f

f a

f

/ , /, /

( ) , ( )

, ( )

∆

1

3

2

2

3 4

2 3

2

1

2

l l

l l

l

l













(72)

into eq 69 and 71 gives

 

d

d F FR

2

2

4 3

θ

ξ

θ = ( ) ∆ +

(73)

 

∆ ∆

∆

2

4 3

d 1

d

R

ξ F FR

θ = −

+

( )

( ) (74)

The case of R = 0 represents the condensation of pure vapor on a fin, and eq 73 and 74

reduce to eq 16 and 17 of Simple Models. Note for a thin fin, w/b << 1 and F4 = 

kw k2 l ,

which equals F2 in eq 15. It is also interesting to note that the case of purely convecting fins

with no condensation is represented by R = 1 and ∆ = 0. The range 0 < R < 1 represents the

case of simultaneous heat and mass transfer.

Figure 14 shows typical results for θ obtained from a numerical solution of eq 73 and 74

subject to the boundary condition of constant base temperature (ξ = 0, θ = 1) and insulated

tip 

ξ θξ = = 1 0 , d d . The parameters F1, F2, F3 and F4 were calculated assuming moisture

condensation on a copper fin (kf = 380 W/m K) having dimensions of L = 240 mm, b = 220

mm, w = 20 mm. The value of h was calculated using the correlation of Motwani et al.

(1985) and assuming the free-stream velocity of U∞ = 4 m/s, which is typical of airconditioning systems. Figure 14a illustrates the effect of dry bulb temperature Ta with Tfb

= 0°C and φ = 50%. As Ta increases, θ decreases, indicating an increase in fin temperature

Tf. The increase in Tf reflects higher sensible and latent heat transfer to the fin surface. The

effect of relative humidity φ shown in Figure 14b is similar to that for a radial fin (Fig. 11).

Finally, Figure 14c shows that as the fin base temperature Tfb is reduced, the driving

potential for both heat and mass transfer is increased causing the fin temperature to

increase or θ to decrease.

A comparison of dry and wet fin heat transfer is presented in Figure 15. Figure 15a

shows the ratio of total heat transfer for wet conditions, qwt, to that for dry conditions, qd.

This ratio decreases with an increase in Biot number. The ratio of sensible heat transfer for

wet conditions, qws, to qd is plotted in Figure 15b. It can be seen that the sensible heat

transfer during condensation is appreciably reduced as Bi increases. Figures 15c,d show

22
1.0

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1.0

ξ

θ

Dew Points

Dry

T = 25 a °C

30

35

(a)

φ = 50 %

T = 0 °C

U = 4 m/s

fb

0 0.2 0.4 0.6 0.8 1.0

ξ

(b)

Dry

60 70

T = 25 °C

T = 0 °C

U = 4 m/s

a

fb

φ = 50 %

0 0.2 0.4 0.6 0.8 1.0

ξ

(c)

Dry

T = 25 °C

φ = 50 %

U = 4 m/s

a

T = 0 fb °C

10

5

Figure 14. Effects of dry bulb temperature, relative humidity, and fin base temperature on the

temperature distributions in a vertical rectangular fin. Adapted from Coney et al. (1989).

wt q

qd

2.4

2.0

1.6

1.2

30

25

T =35 a °C

T = 0 fb °C

φ = 50%

(a)

qws

qd

1.0

0.9

0.8

(b)

25

30

35

qws

qwt

0.8

0.6

0.4

0.2

0 0.4 0.8 1.2

Bi

(c)

25

30

35

qlw

qwt

0.8

0.6

0.4

0.2

0 0.4 0.8 1.2

Bi

(d)

35

30

25

Figure 15. Sensible, latent, and

total heat transfers for wet vertical rectangular fins. Adapted

from Coney et al. (1989).

how the sensible and latent heat transfers, as a fraction of the total heat transfer for wet

conditions, qwt, are affected by Bi and Ta.

The paper by Coney et al. (1989a) also gives the results for the condensate film thickness ∆. These results indicate that ∆ increases as Ta and φ increase, or as Tfb decreases.

However, the study concluded that the effect of film thermal resistance can be neglected

without introducing significant error for normal conditions encountered in practice. On

23
the other hand, if dropwise condensation occurs, the increased surface roughness and the

resultant high turbulence intensity of the fin surface can slightly enhance the heat and

mass transfer to the fin.

Kazeminejad et al. model

Kazeminejad et al. (1993) considered a downward pointing vertical fin of rectangular

profile (Fig. 4b) with moist air (temperature Ta, relative humidity φ) flowing upward with

a uniform velocity Ua. They neglected the condensate film thickness ∆ in eq 73 but

allowed h, appearing in the definition of F3, to be a function of x, that is h = h(x) where x is

measured from the fin tip. To obtain h(x) the nonsimilar boundary layer equations for

upward flow over a nonisothermal vertical surface were written as

 

′′′ + ′′ = − ′′ 



 

 f ff x f  f

x

f f

x

1

2

∂

∂

∂

∂ (75)

 

1 1

2

1

Pr ′′ + ′ + − ( ) ′ = ′ − ′ 



 

 g fg n g f x f  g

x

g f

x

∂

∂

∂

∂ (76)

 

fx f x f x gx gx (,) (,) , (, ) ; (,) , (, ) 0 00 1 00 1 = ′ = ′ ∞= = ∞= (77)

where

 

f Ux yUx ( ) /( ) , /( ) / / ηψ ν η ν = = a a 12 12

 

g T x Tx T x T = − [ ] f f [ ] − a ( ) ( , )/ ( ) η (78)

 

h x

T T

dT T

dx = −





 



 −

f a

f a ( )

and primes denote differentiation with respect to η. The local heat transfer coefficient h(x)

relates to g’(0) as follows

 

hx k U

x

() () g

/

= 



 



 ′ ∞ a ν

1 2

0 . (79)

The coupled problem consisting eq 73 with ∆ = 0, and the boundary layer equations

(75–77) were solved numerically by Kazeminejad et al. (1993) to obtain the fin temperature distribution, total heat transfer to the fin, and the fin efficiency. These results show

that the conjugate conduction–boundary layer analysis gives higher fin temperatures and

higher fin efficiencies than those predicted by the Coney et al. (1989a) model. This conclusion applies to both dry and wet fins.

Example 4

Air at 25°C and 4 m/s flows over a vertical rectangular fin as shown in Figure 13. The

fin that is 240 mm long, 220 mm wide, and 20 mm thick is made of copper (kf = 380 W/m

K). The base of the fin is cooled and maintained at a temperature of 0°C. Assuming that

the results of Figures 14 and 15 apply to this fin, calculate the tip temperature and total

24
heat transferred to the fin for (i) perfectly dry air and (ii) moist air with a relative

humidity of 50 %. Also for moist air, calculate the sensible heat and latent heat contributions to the total heat transfer. Use the following correlations for calculating the average

convective heat transfer coefficient h:

 

hD

k

h U D

a

h

a

= 



 



 ∞ 0 590

0 60

.

.

ν for dry conditions

 

hD

k

h U D

a

h

a

= 



 



 ∞ 0 231

0 69

.

.

ν

for wet conditions

where Dh is the hydraulic diameter and equals 2bL/(b+L). These correlations are discussed in the next section.

Solution

The hydraulic diameter Dh is given by

 

D bL

b L h = + = 2 0 23 . m .

The thermal conductivity and kinematic viscosity of air at 25°C are ka = 0.0255 W/m K, νa

= 15.5 × 10–6 m2/s.

(i) For dry conditions, the average heat transfer coefficient h is given by

h = 47.83 W/m K.

The Biot number Bi can now be calculated as follows:

 

Bi = + = 2 0 79

2 hL b w

k bw

( ) .

f

Using Figure 14c to read θ at ξ = 1 (fin tip) on the dry fin curve, we get

 

θt a ft

a fb

= −

− = T TT T

0 8.

or 

T T TT ft = − − =° a a fb ( . )( ) 08 5 C.

The efficiency of a dry fin is given by the equation

 

ηd = = tanh( ) .

/

/

Bi

Bi

1 2

1 2 0 8

The ideal heat transfer, qideal, is given by

 

q hb wLT T ideal = + −= a fb 2 137 75 ( ) ( ) . W.

Thus the actual heat transfer qd is

qd = ηd qideal = 110.2 W.

25
(ii) For the wet conditions, the average heat transfer coefficient h is given by

 

h k

D

U D = 



 



 = ∞ 0 231 50 36

0 69

. .

.

a

h

h

νa

W/m K2 .

The Biot number for the wet conditions is 0.83.

Reading Figure 14c for θ at ξ = 1 (fin tip) on the curve for Tfb = 0°C, we have

 

θt

T T

T T = −− = a ft

a fb

0 72 .

or 

T T TT ft = C a a − − 0 72 7 . ( ) fb = o .

For Bi = (0.83)1/2 = 0.91, so that Figure 15a for Ta = 25°C gives

 

q

q

wt

d

= 1 45 .

or qwt = 1.45 qd = 159.8 W.

Reading the curve for Ta = 25°C in Figure 15c, the ratio qws/qwt for Bi = 0 91 . is

 

q

q

ws

wt

= 0 63 .

or qws = 100.7 W.

The latent heat transfer is 59.1 W.

Experimental studies

Experimental studies of finned coiled heat exchangers have been carried out by several

workers including Bryan (1962), Bettanini (1970), Yoshi et al. (1971) and Guillory and

McQuiston (1973). These studies have confirmed that the performance of finned coils is

significantly reduced when dehumidification occurs. This reduction is the consequence of

lower fin efficiency for wet conditions.

Kazeminejad (1987) and Coney et al. (1989b) conducted an experimental study to investigate the performance of a vertical rectangular fin (Fig. 13) when moist air in turbulent

flow dehumidifies on the surface of the fin. The study revealed that the wet fin surface

temperature increases with increase in free stream velocity, relative humidity and dry

bulb temperature. The increase in wet fin surface temperature also occurs when the fin

base temperature is decreased. These observations confirm the theoretical predictions

shown in Figure 14. The study also noted that smooth and clean surface copper fins

promote dropwise condensation rather than filmwise condensation. Although the effect

of mass transfer on the heat transfer coefficient was small, the fin efficiency was markedly

reduced under wet conditions. Another interesting conclusion was that the shape of the

leading edge of the fin affected the heat and fluid flow. The heat transfer to the fin was

higher when the leading edge of the fin was blunt (with attendant flow separation and

reattachment) than when the leading edge was elliptical.

26
Based on their experimental work, Kazeminejad (1987) and Coney et al. (1989b) presented the following correlations for the convective heat transfer coefficient for vertical

rectangular fins.

Blunt-edged dry fin:

 

hD

k

h U D

a

h

a

= 



 



 ∞ 0 590

0 60

.

.

ν

(80)

Blunt-edged wet fin:

 

hD

k

h U D

a

h

a

= 



 



 ∞ 0 231

0 69

.

.

ν

. (81)

Elliptical-edged dry fin:

 

hD

k

h U D

a

h

a

= 



 



 ∞ 0 420

0 60

.

.

ν

. (82)

Elliptical-edged wet fin:

 

hD

k

h U D

a

h

a

= 



 



 ∞ 0 146

0 69

.

.

ν

. (83)

Optimum fin design

This section considers the design of optimum dimensioned fins for use in a moist air

stream. The discussion will be based on the works of Kilic and Onat (1981) and Toner et al.

(1983), and will cover longitudinal fins of rectangular and triangular profiles.

Rectangular fins

Kilic and Onat (1981) considered a vertical rectangular fin as shown in Figure 16a, and

modified the classical convecting fin equation to allow for simultaneous heat and mass

transfer. The modified equation can be expressed as

 

d T mTT

h h

T

PP A B

T

2

2

f 2

f a

m fg

v a va dx hR f

= −− − − 



 







































 ( ) exp (84)

where m = 2h kw /

hm = mass transfer coefficient

Rv = gas constant for water vapor

Pva = partial pressure of water vapor at temperature T,

P = total pressure

A and B = constants.

A model similar to eq 84 has also been used by Karniven et al. (1990) to study moisture

on fins. The Karniven et al. model allows for radiative heat transfer in addition to convective heat and mass transfer. Furthermore, the model considers a partially wet fin rather

than a fully wet fin, and also determines the line separating the wet and dry regions.

27
Assuming a constant base temperature and an insulated tip, an approximate analytical

solution of eq 84 was derived using the quasi-linearization technique. For a fixed profile

area (wL), the expression for heat transfer rate was maximized to yield the following

implicit relations for w and L:

 

w

L

m

hL

k = 1 (85)

 

L wL k

m h = 



 





f

1

1 3 /

(86)

where

 

m

Bh h P

hR T T

s Tr

1 = +1 m fg

var2

, (87)

and 

TT TT mL

mL ra a =− − fb ( ) tanh . (88)

Here Ps, Tr is the saturation pressure at reference temperature Tr.

A sample solution of eq 85–88 is shown in Figure 17 for atmospheric air with a relative

humidity of 50%. The details of evaluation of hm/h and B can be found in Kilic and Onat

(1981). Figure 17 shows that the optimum value of N = mL for a wet fin is lower than that

for a dry fin (Nopt = 1.4192). Also, Nopt for a wet fin decreases as the fin base temperature

Tb and the free-stream air temperature Ta increase.

Tfb

kf

dx

L

x

x = 0

Air

Flow

b

T , φ a

(b)

Tfb

kf

dx

L

x

x = 0

Air

Flow

b

T , φ a w

(a)

Figure 16. a) Vertical rectangular fins; (b) Vertical triangular

fins.

28
No Condensation

1.6

1.4

1.2

1.0

0.9

0.8

0.7

25 30 35 40

T ( a °C)

Nopt

5

10

T = 0 fb °C

Figure 17. Optimum fin parameter, Nopt,

for a vertical rectangular fin with moisture condensation. Adapted from Kilic and

Onat (1981).

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

25 30 35 40

Ta

Nopt

Dry Triangular

Dry Rectangular

Wet Triangular

Wet Rectangular

10

0

10

T = 0 fb °C

Figure 18. Comparison of Nopt for vertical

rectangular and vertical triangular fins under dry and wet conditions. Adapted from

Toner et al. (1983).

Triangular fins

Toner et al. (1983) extended the analysis of the foregoing section to a vertical fin of

triangular profile (Fig. 16b). Figure 18 gives their final results showing how the optimum

value of N is affected by the variations of the ambient temperature Ta and fin base

temperature Tb. Also shown for comparison are the results for the optimum rectangular

fins. It can be seen that for a given Ta and Tb, the optimum value of N for a wet triangular

fin is lower than that of a dry triangular fin, and that Nopt decreases as Ta and Tb increase.

As noted earlier, the wet rectangular fin exhibits the same behavior. It is also interesting to

note that the values of Nopt for triangular fins (dry and wet) are higher than those of the

rectangular fins (dry and wet).

29
w s e

dr

Retained

Condensate

do

(a)

β

(b)

HORIZONTAL INTEGRAL-FIN TUBES

The use of horizontal, low profile integral-fin tubes (Fig. 19) to enhance condensation is

quite common in the design of surface condensers in the refrigeration and “process”

industries. These tubes have, therefore, been studied extensively for the past fifty years.

An excellent review of the pertinent literature has been provided by Marto (1988). The

recently published book by Webb (1994) also contains a substantial discussion on the

analysis and design of integral-fin tubes.

Commercial integral-fin tubes are made of different materials (aluminum, stainless

steel, titanium, copper and its alloys, etc.) and are available in different densities ranging

from 433–1675 fins/m or 11–40 fins/in. Although the standard integral-fin design provides a significant enhancement in condensation over plain tubes, more advanced designs

such as Hitachi Thermoexcel-C, Wieland GEWA-SC, Wolverine Turbo-C, and Sumitomo

Tred-26D improve the performance still further. These designs use a saw-toothed fin

shape.

Since the topic of condensation on integral-fin tubes has been comprehensively covered by Marto (1988) and Webb (1994), the following sections will highlight only the

important results.

Condensate flooding

When a high-surface-tension fluid such as steam condenses on a horizontal finned

tube, the surface tension (capillary) force causes the condensate to be retained between the

fins on the lower side of the tube as shown in Figure 20a. This phenomenon, which is

known as condensate flooding, increases the thermal resistance in the flooded region,

thereby adversely affecting the performance of the tube.

Katz et al. (1946) were the first to investigate the phenomenon of liquid retention in

horizontal finned tubes. Their measurements, which were performed under static (no

condensation) conditions, revealed that in extreme cases, flooding could cover the entire

surface of the tube. Rudy and Webb (1981)

measured retention of refrigerant R-11, n-pentane, and water in tubes of three different densities (748, 1024 and 1378 fins/m). For the tube

with 1024 fins/m, they found that R-11, npentane and water flooded 26, 42 and 100 %

of the tube circumference, respectively. They

also found that flooding under dynamic (condensation) conditions was only slightly different from that under static (no condensation) conditions. The same conclusion was

Figure 19. Horizontal integral-fin tube.

di r d do

Figure 20. Condensate flooding on a horizontal integral-fin tube.

30
reached by Honda et al. (1983) when they found their data for methanol and R-113 under

static and dynamic conditions to be essentially the same.

More recent experiments on flooding by Masuda and Rose (1987) have shown that the

liquid is not only retained on the lower side of the tube, but also on the upper portion in

the form of a small liquid wedge between the flanks of the fins and the tube surface

between adjacent fins.

Rudy and Webb (1983a, 1985) and Honda et al. (1983) developed a theoretical expression for the condensate flooding angle b (Fig. 20b) by equating the surface tension and

gravity forces on the liquid retained between the fins. The expression is

 

β σ

ρ = − 









 − cos 1 1 4

d gs o l

(89)

where σ = surface tension

do = tube outside diameter

ρl = condensate density

g = acceleration due to gravity

s = gap between fins.

Equation 89 shows that the flooding angle β increases with increasing surface tension σ

and with decreasing fin spacing s.

The amount of condensate retained between the fins can be reduced by attaching a

rectangular porous plate to the bottom of the tube. This attachment, known as a drainage

strip, increases the downward capillary force, drawing the condensate into its pores.

Figure 21 illustrates schematically the reduction in condensate flooding achieved with a

porous drainage strip. A solid plate can also serve as a drainage strip, but it is not as

effective as the porous plate. For example, Honda and Nozu (1987a) measured the effect

of different drainage strips for R-113 condensing on an 18.9-mm-diam. (do) tube having a

fin density of 2000 fins/m and a fin height of 1.13 mm. With no drainage strip, they found

that 62 % of the tube circumference was flooded. When a polyvinyl chloride strip (solid)

of height 12.6 mm was used, the percentage of circumference flooded was reduced to 57, a

reduction of about 8 %. However, when a porous nickel strip of the same height (12.6 mm)

was attached, only 32 % of the tube circumference was flooded, giving a reduction of

about 48 %. Clearly the porous strip is much more effective than a solid strip. The

Condensate Flooding

(no drainage strip)

b = b c

Porous

Drainage

Strip

Condensate

Flooding

(porous drainage strip)

Figure 21. Effect of porous drainage strip on condensate flooding. Adapted

from Marto (1988).

31
measured heat transfer coefficients for no strip, a PVC strip, and a porous nickel strip

were 6200 W/m2 K, 6900 W/m2 K, and 10,200 W/m2 K, respectively. The effectiveness of

drainage strips for controlling condensate flooding was also demonstrated by others

including Yau et al. (1986) and Marto et al. (1988). Although drainage strips have proved

effective under laboratory conditions, their use in an actual condenser tube bundle may

not be feasible.

Example 5

Horizontal integral-fin tubes are to be designed to condense steam, ethylene glycol and

R-113 at atmospheric conditions.

(i) If the maximum circumference flooded is to be 50 %, calculate the number of fins

per meter that should be provided if the outer tube diameter is 19 mm and rectangular fins of 0.25 mm thickness are used.

(ii) If the outer diameter and fin thickness are changed to 21.05 mm and 0.5 mm,

respectively, find the new fin density for each of the three fluids.

(iii) Calculate the maximum fin density for steam, ethylene glycol and R-113, condensing on 19-mm outer-diameter tubes, fitted with 0.25-mm-thick rectangular fins.

Solution

(i) For 50 % flooding, β = 90° in eq 89. The interfin spacing s is then given by

 

s

d g = 4σ

o ρl

 .

The values of the ratio σ/ρl at atmospheric pressure for the three fluids have been quoted

by Marto (1988) as follows:

Steam 61 × 10–6 m3/s2

Ethylene glycol 34 × 10–6 m3/s2

R-113 11 × 10–6 m3/s2.

Thus for steam, the spacing is s = 1.3 mm.

The fin density is 645 fins/m.

The above calculations repeated for ethylene glycol and R-113 give fin densities of 1021

and 2057, respectively.

(ii) Repeating the calculations with 21.05-mm-outer diameter and 0.5-mm-thick fins

gives densities of 595 fins/m for steam, 863 fins/m for ethylene glycol, and 1402

fins/m for R-113.

(iii) For total flooding, the angle b in eq 89 would be 180°; then the fin spacing s is given

by

 

s

d g = 2σ

o ρl

 ,

which shows the fin spacing is one-half the values obtained in part (a). Thus for steam

condensing on 19-mm outer-diameter tube, s = 0.65 mm. For a fin thickness of 0.25 mm,

the fin density is 1111 fins/m. The corresponding figures for ethylene glycol and R-113

work out to be 1626 and 2717. The value of 1111 fins/m for steam confirms Webb’s (1994)

statement that condensation of steam on a 19-mm-diam. tube having a 0.25-mm-fin thick32
ness would result in total flooding if the fin density were greater than 1000 fins/m.

Indeed, the experimental work of Jaber and Webb (1993) on enhanced tubes for steam

condensers shows that integral-fin tubes for steam condensation should use no more than

630 fins/m (16 fins/in.).

Theoretical models for

heat transfer coefficient

Perhaps the first theoretical model for predicting the condensation heat transfer coefficient for integral-fin tubes was proposed by Beatty and Katz (1948). They assumed the

condensate flow to be purely gravity driven and applied a Nusselt type analysis for both

the fin surface and the tube surface between the fins. The average heat transfer coefficient,

 

h , for the configuration of Figure 20 was expressed as

 

h h A h Aη η = + h r f f

f

A A (90)

where A = heat transfer area

= Ar + Af, Ar

= surface area of tube at base of fins

Af = fin surface area

ηf = fin efficiency

η = total surface efficiency

hh = heat transfer coefficient for horizontal tube surface

hf = heat transfer coefficient for the fin surfaces.

The expressions for hh and hf, and the relationship between η and ηf are given by

 

h

k h

T Td h

v fg

sat b r

g = ( ) −

( ) −

















0 728

3 1 4

.

/

ρρ ρ

μ

l l l

l

(91)

 

h

k h

T TL f

v fg

sat b f

g = ( ) −

( ) −

















0 943

3 1 4

.

/

ρρ ρ

μ

l l l

l

(92)

 

η η =− − 1 1( )f

f

A

A (93)

where Lf is the average fin height over the diameter do equals 

π dd d o

2 r2 ( ) − /4 o .

Beatty and Katz (1948) used eq 90 to predict their test data for six low surface tension

fluids condensing on finned tubes ranging in fin density from 433 to 630 fins/m. They

found that if the coefficient in eq 91 was changed from 0.728 to 0.689, eq 90 predicted their

data within 10%. Despite the fact that the Beatty and Katz’s model does not account for

condensate retention and surface tension effects, the model has enjoyed success in the

refrigeration industry for many years.

The first model to include the effect of tension in draining condensate from horizontal

finned tubes was presented by Karkhu and Borovkov (1971). They postulated that condensate drainage from the fin surface was purely surface tension driven, while that from

the tube surface between the fins was purely gravity driven. From experiments in which

R-12 was condensed on the face of a cylinder with machined in rectangular fins, Webb et

33
al. (1982) concluded that surface tension was indeed the force controlling the condensate

drainage from the fin surface. Using a linear surface tension model, Rudy and Webb

(1983b) developed the following expression for the heat transfer coefficient on the fin

surface:

 

h

k h

T T d d w s

f

fg

sat b o r

= ( ) −















 ( ) −

+ 



 



















 0 943 8 11 3 1 4

2

1 4

.

/ /

ρ

μ

l l σ

l

. (94)

Pursuing the surface tension drainage approach, Adamek (1981) considered a family of

convex liquid/vapor interfaces that promote surface tension drainage. The radius of

curvature of the family of profiles is described by the equation

 

1 1 1

r

s

s = 



 





 +



 



 − 



 





















θ ξ

ξ

ξ

m

sm m

(95)

where r = local radius of interface,

s = distance along the curved interface profile,

sm = maximum length of the curved interface,

θm = rotation angle from tip to fin base

ξ = interface shape parameter, – 1 < ξ < ∞.

These quantities are illustrated for a typical convex liquid-vapor interface in Figure 22.

The relationship between sm, θm, ξ, and tip radius ro is given by

 

s

r

m

o mθ

ξ

ξ = + 1 . (96)

For the condensate profiles described by eq 96, Adamek (1981) obtained the following

expression for the condensation heat transfer coefficient:

 

h k

s

h s

kT T f m

fg m m

b sat

= ( ) − ⋅ +

+

















2 149 1

2 3

1 4

. ( )

/

l l

l l

ρσ θ

μ

ξ

ξ . (97)

Sm

r

o

S

Base

Surface

δ

θ m

θ r

Condensate surface

for a typical value of ξ

Figure 22. Typical film profile for condensation

on a fin with small tip radius, with increasing

radius along the arc length.

34
A new family of practical fin profiles for surface tension drained condensation has been

described by Kedzierski and Webb (1990). More precise models for surface tension controlled condensation have been developed by Honda and Nozu (1987b), Honda et al.

(1987) and Adamek and Webb (1990).

To account for the condensate flooding, Owen et al. (1983) and Webb et al. (1985)

suggested that the heat transfer coefficients for the unflooded and flooded parts be

computed separately. They recommended that eq 90 should be modified as follows:

 

h B h A

A

h A

A

B

η h

π

η

π = − 



 



 + 



 

 1 h  + r f f

f

b (98)

where 1 – (B/π) and B/π represent the fractions of the circumference unflooded and flooded, respectively, and hb is the heat transfer coefficient for the flooded region. Webb et al.

(1985) found that for steam condensing on a 19-mm-diam. tube having 203 fins/m, heat

transfer through the flooded region was only 1.6 % of the total. Thus the second term in eq

98 can be neglected for most practical purposes.

In the most recent work, Rose (1994) used some simplifying assumptions, together

with dimensional analysis, to develop an equation for calculating the ratio hfined tube/

hplain tube for condensation on horizontal trapezoidal integral-fin tubes. In a contemporaneous paper, Briggs and Rose (1994) modified the equation given by Rose (1994) to

include the effect of fin efficiency.

Example 6

Saturated refrigerant R-12 at 32°C condenses on a horizontal integral-fin tube of outer

diameter do = 19.1 mm and a root diameter dr = 15.88 mm. The tube has 748 fins/m. The

temperature at the root of the fin is maintained at 22°C. The fins are trapezoidal in shape,

0.38 mm thick at the base and 0.23 mm thick at the tip. Assuming the fin efficiency ηf = 1,

calculate the condensation heat transfer coefficient, and the enhancement ratio using 1)

the Beatty and Katz (1948) gravity drained model, and 2) the Webb et al. (1985) surface

tension drained model. For the surface tension drained model, assume that the condensate film shape is described by the Adamek profile parameters, ξ = –0.857, sm = 1.5936

mm, and θm = 85 degrees (1.4835 radians). The properties of R-12 at the mean film

temperature of 27°C (300 K) are

ρl = 1305.8 kg/m3

ρv = 40 kg/m3

νl = 19.5 × 10–8 m2/s

kl = 0.072 W/m K

σ = 0.0158 N/m

hfg = 133.79 kJ/kg.

Solution

The fin surface per meter of tube length is

 

A d d n d wn f = − 2 ( ) o r ot + =

4

0 1458 π 2 2 π . m /m2 .

The unfinned surface area of the tube per meter is

 

A d n dw rr r =− = π π b 0 0357 2 . m /m.

The total surface of the finned tube per meter is

35
A = Af + Ar = 0.1815 m2/m.

The heat transfer coefficient for the horizontal (unfinned) surface of the tube is given by

eq 91

 

h g kh

T Td h

v fg

sat b r

= ( ) −

( ) −

















0 728 1540 =

3 1 4

.

/

ρ ρ

ν

l l

l

W/m K2 .

Beatty-Katz model:

The average fin height over the diameter do is

 

L ddd f o

2 r2 = − π( ) 4 0 0046 o = . m.

The heat transfer coefficient for the fin surface is

 

h g kh

T TL f

v fg

sat b f

= ( ) −

( ) −

















0 943 2719 =

3 1 4

.

/

ρ ρ

ν

l l

l

W/m K2 .

Because ηf = 1, η = 1 from eq 93. Thus the Beatty-Katz model, eq 90 becomes

 

h h A

A

h A

A

= += h r f

f 2487 W/m K2 .

The enhancement ratio is

 

h

h

h

h p h

= = 1 63 . .

Based on the envelope area over the fins, πdoL, the average heat transfer coefficient is

 

h = 7568 W m K2 .

Webb et al. model:

For the Adamek condensate film profile, the Webb et al. model uses eq 97 for hf. Thus

 

h k

S

h S

kT T f m

fg m m

sat b

= −

+

+





 



2 419 ⋅  = 1

2

4696 3

1 4

. ( ) ( )

/

l

l l

σ θ

ν

ξ

ξ W/m K2 .

The average interfin spacing s is calculated by assuming the fin to have an average

thickness of (0.38 + 0.23)/2 = 0.305 mm. Thus

s = 1.032 mm.

The flooding angle β can now be calculated using eq 89:

 

β σ

ρ = − 









 = − cos 1 1 4

d gs o l

41.44 degrees or 0.72 radians .

Finally using eq 98 and neglecting the last term, 

h is given by

36
 

h h A

A

h A

A = − 



 



 + 



 



 1 = β

π

h r f

f 3141W/m K2 .

The enhancement ratio is

 

h

h

h

h p h

= = 2 04 . .

Based on the envelope area over the fins, the average heat transfer coefficient is

 

h h = = 9501 W/m2 K.

The Beatty-Katz model predicts a lower value of 

h than the Webb et al. model. The

value of 9501 W/m2 K is about the same as the value as measured by Webb et al. for

condensation of R-11 on 748 fins/m tube (see Fig. 6 in Webb et al. 1985).

Experimental heat transfer coefficients

The review article by Marto (1988) provides a comprehensive listing and discussion of

references that report experimental heat transfer data for horizontal finned tubes. Consequently, the brief review here will discuss the difficulties and uncertainties in experimental investigations, present some sample results, and assess the predictive capabilities of

the theoretical models in the light of experimental data.

Experimental investigations of film condensation on horizontal integral-fin tubes entail many difficulties. For example, the presence of noncondensable gases, partial dropwise condensation or a substantial vapor velocity near the tube can affect the

data significantly, but is rarely brought

out in published work. In many cases,

the uncertainties in measurements remain obscure. The technique used to determine the average condensation heat

transfer coefficient can itself introduce a

10 to 15 % discrepancy between different data. The lack of consistency in the

choice of the surface area on which is

based often confuses the end user. The

interpretation and use of experimental

data must therefore be made with great

care. The difficulties and uncertainties

associated with the measurement of condensation on horizontal integral-fin

tubes are discussed in detail by Marto

(1992).

Attention is now turned to the discussion of the experimental data. The

last fifty years of activity has generated

a vast amount of data for condensation of

various refrigerants and steam on tubes

having different fin geometries and spacing. Only a representative sample can be

discussed here. Figure 23 shows the enFigure 23. Effect of fin spacing on the enhancement

ratio for steam condensing on horizontal integralfin tubes. Adapted from Honda et al. (1987).

Honda et al.

(1987)

Honda-Nozu

(1987)

Beatty-Katz

(1948)

Wanniarachchi et al. (1985)

T = 100 °C; T = 24 °C

u = 3.3 m/s

sat c

c

Steam

10

8

6

4

2

0

hfinned

hplain

T = 48 °C; T = 22 °C

u = 3.3 m/s

sat c

c

6

4

2

0 246

δ (mm)

hfinned

hplain

37
hancement ratio ε = hfined tube/hplain tube

data of Wanniarachchi et al. (1985) for

steam condensing on a family of cooled

copper tubes (coolant temperature Tc,

coolant velocity Uc) with a root diameter

of 19.05 mm and having rectangular integral fins, 1 mm thick and 1 mm high.

The figure also shows the predictions of

theoretical models of Beatty and Katz

(1948), Honda and Nozu (1987b), and

Honda et al. (1987). The Beatty-Katz model overpredicts the experimental data at

small fin spacings, but underpredicts it

at large fin spacings. The Honda-Nozu

model underpredicts the data, and the

discrepancy gets worse as the fin spacing increases. The predictions of Honda

et al. (1987) appear to fit the data best,

except at small fin spacings where the

flooding is very nearly complete. Although not shown here, the model proposed by Adamek and Webb (1990) predicts the steam data within ±10 to ±15 %

over the complete range of fin spacings.

The same family of copper tubes as

used by Wanniarachchi et al. (1985) for

steam were tested by Marto et al. (1988)

for R-113. These results together with the theoretical predictions of Beatty and Katz (1948),

Honda et al. (1987) and Webb et al. (1985) are shown in Figure 24. The Beatty-Katz model

vastly underpredicts the data. The Webb et al. (1985) theory provides a much better

prediction at ζ = –0.85 than at ζ = –0.95. However, the closest agreement with the data is

achieved by using the Honda et al. (1987) model.

From the brief discussion here and more extensive coverage elsewhere, e.g., Marto

(1988), it appears that virtually all available experimental data pertain to steam and

currently popular refrigerants such as R-11, R-12, R-22 and R-113. The data for R-152a,

which is a promising alternative to CFCs, are just beginning to appear. Cheng and Tao

(1994) are perhaps the first to report experimental work on condensation of R-152a on

plain and finned tubes. They conclude that i) the simple Nusselt theory predicts within 15

% the data for condensation on a single smooth tube, ii) the performance of a single

smooth tube with R-152a condensing on its outside is 20–25 % better than that obtained

with R-12, and iii) the integral fin tube provides enhancement of 4 to 10 times that of a

plain tube.

Effect of interfacial shear

It is well known that the interfacial shear stress of the liquid/vapor interface increases

the heat transfer coefficient if the vapor and liquid (condensate) flow in the same direction, and decreases it if the two flow in opposite directions. For condensation on plain

(smooth) tubes, the effect of vapor velocity has been studied extensively, see for example,

Memory and Rose (1986), Honda et al. (1986) and Fujii (1991). However, the correspondhfinned

hplain

0 123

δ (mm)

10

8

6

4

2

Area

Ratio

Honda et al. (theory)

(1987)

Webb et al. (theory)

(1985)

Beatty and

Katz (theory)

(1948)

ζ = – 0.85

ζ = – 0.95

e = 1.0 mm

w = 1.0 mm

R-113 Marto et al. (1988)

Figure 24. Effect of fin spacing on the enhancement ratio for R-113 condensing on horizontal integral-fin tubes. Adapted from Marto et al. (1988).

38
ing information for the integral-fin tubes is very limited. Webb (1984), Gogonin and

Dorokhov (1981) report data for R-12 condensing on two different 800-fins/m tubes,

under a range of operating pressure and heat fluxes, and a maximum vapor velocity of 8

m/s. Their data indicate that the effect of vapor velocity for finned tubes is very small

compared with the effect for smooth tubes. This observation is in contradiction with the

results obtained by Yau et al. (1986), who measured the condensation coefficient for steam

condensing on a family of finned tubes and found that the heat transfer enhancement due

to assisting vapor shear stress was essentially the same for plain and finned tubes. This

latter result has been confirmed in a recent study by Bella et al. (1993) who condensed R-11

and R-113 on a finned horizontal tube, with vapor velocity ranging from 2 to 30 m/s. The

work of Bella et al. (1993) concludes that the enhancement due to the vapor velocity

begins to appear when the vapor Reynolds number exceeds 10. They also found that the

heat transfer coefficient at a vapor velocity of 30 m/s was 50% more than the value for

stagnant vapor. Studies by Lee and Rose (1984) and Michael et al. (1989) also underscore

the beneficial effect of vapor shear in finned tubes.

Effect of tube bundle geometry

When condensation occurs over a vertical column of tubes, the condensate drains from

tube to tube, causing the film thickness to increase on the lower tubes. The net effect is to

lower the overall heat transfer coefficient compared with a single tube. The classical

Nusselt analysis takes this condensate inundation effect into account by multiplying the

single tube heat transfer coefficient by a factor N–m giving

 

h hN m N 1 = − (99)

when 

hN = the average heat transfer coefficient for a N-tube column

 

h1 = the average heat transfer for a single tube, given by eq 91

m = an exponent which equals 1/4 for the Nusselt analysis.

The experimental values of 

hN are usually higher than those predicted by eq 99. The

enhancement in 

hN has been attributed by Fujii (1991) to the splashing of the condensate

from one tube as it impinges on the next tube. Kern (1958) recommended m = 1/6 to

account for the enhancement effect.

Several studies have attempted to apply eq 99 to a column of finned tubes. Katz and

Geist (1948) obtained data for R-12, n-butane, and steam condensing on a 6-tube column

of finned tubes, each having a fin density of 590 fins/m and 1.6-mm-high fins. They found

that eq 99 best fit their data for m = 0.04. Marto (1986) also recommends m = 0.04 based on

his data for a column of integral-fin tubes. Pearson and Withers (1969) used two identical

60-tube condensers to condense R-22. Both condensers were equipped with integral-fin

tubes, one having tubes with 748 fins/m and other have tubes with 1024 fins/m. These

authors suggested that the average heat transfer coefficient for tube bundles can be

estimated by multiplying the Beatty-Katz 

hN , given by eq 90–93, with a correction factor

CN/N1/4, where CN = 134 for the 748-fins/m-density tubes and CN = 1.31 for the 1024-

fins/m density tubes.

More recently, Webb and Murawski (1990) have conducted experiments on four enhanced tube geometries (1024 standard integral fin, Tred-26D, Turbo C, and GEWA-SC).

For each geometry, they arranged five tubes in a column and measured the heat transfer

coefficient with R-11 condensing over the column. The data were correlated by an equation of the form

39
 

h a n N = − Rel (100)

where the condensate Reynolds number Rel equals 4m L  ̇ l l /μ . The constants a (W/m2 K)

and n for the four geometries tested together with values of 

hN for Rel = 100 are given

below:

1024 standard integral fin: a = 12.90 × 103, n = 0, 

hN = 12,900 W/m2 K

Tred-26D: a = 269.90 × 103, n = 0.576, 

hN = 18,956 W/m2 K

Turbo C: a = 257.80 × 103, n = 0.507, 

hN = 24,885 W/m2 K

GEWA-SC = a = 54.14 × 103, n = 0.22, 

hN = 19,657 W/m2 K.

The 1024 standard integral fin exhibits zero row effect (n = 0) for the same tube using R113. The results for 

hN show that the Turbo C tubes gives the best heat transfer performance. GEWA- SC and Tred-26D tubes perform comparably. The lowest performance is

displayed by the 1024 standard integral-fin tubes.

Theoretical models for the effect of tube bundle geometry have been expounded by

Ishihara and Palen (1983), El-Meghazy (1986) and Honda et al. (1987).

The combined effects of interfacial shear and tube bundle geometry with integral-fin

tubes have been studied for vapor downflow by Smirnov and Lukanov (1972). Their

results show that the effect of tube bundle geometry is more pronounced in finned tubes

than in plain tubes. The paper by Webb (1984) reviews the work done on the effects of

interfacial shear and tube bundle geometry for both plain and integral-fin tubes. The

paper recommends that designers of shell-side refrigeration condensers should maintain

a certain minimum vapor velocity in all regions of the tube bundle.

Effect of tube thermal conductivity

The bulk of the experiments described in Experimental Heat Transfer Coefficients pertain

to condensation on copper tubes. Because copper has a high thermal conductivity, the “fin

efficiency” effects due to the temperature drop in the fin are small. However, this effect

can become more pronounced if tubes of lower thermal conductivity are employed. To

determine the effect of thermal conductivity on the performance of horizontal integral-fin

tubes, Huang et al. (1994) condensed steam and R-113 on tubes made of copper, brass, and

bronze. Each tube had a root diameter of 12.7 mm, with rectangular fins 1 mm thick and

spaced 0.5 mm apart. Four tubes of each material were tested with fin heights of 0.5, 0.9,

1.3 and 1.6 mm. For comparison, plain tubes (outside diam. = 12.7 mm) of each material

were also tested.

The measured data expressed as the enhancement ratio are given in Table 3. As one

might have expected, the effect of lowering the thermal conductivity is to reduce the

enhancement ratio. This effect is stronger for steam than for R-113. Consider, for example,

the condensation of steam on copper and bronze tubes with 1.6-mm-high fins. The enTable 3. Effect of tube material on enhancement ratios, k = 315

W/m K, 112 W/m K, and 78 W/m K for copper, brass and

bronze, respectively.

Fin height Steam R-113

(mm) Copper Brass Bronze Copper Brass Bronze

0.5 1.74 1.50 1.50 3.16 3.15 2.96

0.9 1.90 1.63 1.43 4.24 4.35 3.90

1.3 2.05 1.68 1.37 4.60 4.72 4.28

1.6 2.40 1.77 1.39 5.16 5.09 4.91

40
hancement ratio of 1.39 for a bronze tube is 42 % less than the enhancement ratio of 2.4 for

a copper tube. For R-113, the corresponding figure is less than 5 %. The enhancement ratio

increases as the fin height increases except for steam condensing on bronze tubes when

the trend is opposite. It is also worth noting that the enhancement ratios for R-113 are

much higher than those for steam.

INTERNALLY FINNED TUBES

When saturated vapor flows into a cooled tube, vapor condenses on the tube wall

forming a condensate film on the tube wall. A variety of flow patterns can arise depending on the mass flux (velocity). For example, when the condensate layer is symmetric and

the liquid/vapor interface is sharply defined, the flow is classified as annular flow. Other

flow possibilities include bubbly flow, dispersed flow, wavy flow, stratified flow, plug

flow, and slug flow. Even for a smooth (plain) tube, the flow is quite complex and difficult

to analyze. The presence of internal fins complicates the situation further, and makes it

extremely difficult, if not impossible, to develop an accurate physics based model describing the heat and fluid flow processes. Consequently, the discussion in this section, including the pressure drop and heat transfer correlations, will be largely based on experimental

studies.

In 1974, two papers, one by Reisbig (1974) and the other by Vrable et al. (1974) reported

on the condensation of R-12 in internally finned tubes. Reisbig found the condensation

heat transfer coefficient to be 20–40 % greater than the smooth-tube value. However, he

did not propose any correlation. Vrable et al. (1974), on the other hand, conducted 26

experiments with two different internally finned tubes, varying the inlet reduced pressure

(p/pcr) from 0.18 to 0.46 and the mass flux from 86.7 to 853 kg/m2 s. The more effective

tube was found to give a maximum enhancement of 300 %. The following correlation

represented their data within ±30 %:

 

h k

d

d G p

p = 



 



 











−

0 01 2 0 8

0 33

0 65

. Pr

.

.

.

l

l

l

h

h e

μ cr

(101)

where

 

G x xG l

l = 



 



 + −

















ρ

ρv

0 5

1

.

( ) (102)

G = mass flux (velocity)

dh = tube hydraulic diameter

x = vapor quality

p = pressure

pcr = critical pressure.

Royal and Bergles (1978) condensed low pressure steam in four internally finned

copper tubes, three of which had spiral fins. The geometric data for these fins are given in

Table 4. Figures 25 and 26 represent the heat transfer and pressure drop data, respectively.

The heat transfer data for finned tubes indicate a significant improvement over the smooth

(plain tube). The highest enhancement ratio of 2.3 is achieved with a 15.9-mm outsidediameter tube containing 16 spiraled fins of height 1.45 mm and a helix angle of 3.22

degrees (tube 5). It is also interesting to note that the area ratio (A/Ap) for tubes 3 and 4 is

41
the same, but the heat transfer performance of tube 3 is superior to that of tube 4. Clearly,

the area ratio is not the controlling factor. The pressure drop characteristics (Fig. 26) show

that the highest pressure drop occurs in tube 3.

Based on their experimental data, they proposed a correlation that represented 95 % of

the data within ±30 %. The Royal-Bergles correlation is

 

h k

d

d G e

sd = 



 



 









 +















 0 0265 160 1

0 8

0 33

2 1 91

. Pr

.

.

.

l l

l

l

h

h

μ h

(103)

where s is the average fin spacing. It is important to note that this correlation was derived

for steam at 1.0 bar and may not be accurate for other fluids or pressures. Furthermore, the

correlation does not account for the surface tension effects or the temperature gradients in

the fin.

Luu and Bergles (1979) tested the tubes of Table 4 with R-113 as the condensing fluid,

and found that tube 2, with shortest fins and highest number of fins, exhibited the best

heat transfer performance at low flow rates. At high mass flow rates, the measured heat

transfer coefficient ranged between 70 and 120 % of the smooth tube value. In a subsequent paper, Luu and Bergles (1980) tested eq 103 against their R-113 data and found it to

be unsatisfactory. A new correlation was

developed to predict condensation of R-113

in internally finned tubes. However, this

correlation overpredicted the data of Said

and Azer (1983), who therefore developed

their own correlation. In doing so, they excluded the data of Luu and Bergles which

was rather unfortunate. Kaushik and Azer

(1988) applied regression analysis to the

steam data of Royal and Bergles (1978), the

Figure 26. Pressure drop during the condensation of steam in internally finned horizontal

tubes. Adapted from Royal and Bergles (1978).

Table 4. Geometric data for internally

finned tubes.

Tube do e n α

 no. (mm) (mm) (°) A/Ap

1 15.9 — — — 1.00

2 15.9 0.60 32 2.95 1.70

3 12.8 1.74 6 5.25 1.44

4 12.8 1.63 6 0 1.44

5 15.9 1.45 16 3.22 1.73

Tube 1

 2

 3

 4

 5

G (kg/m /s) 2

200 300 400 500 600 700

100

80

60

40

20

h(W/m /K x 10 ) – 3 2

4.0

0.1

0.4

0.6

0.8

200 400 600 800 1000

G (kg/m /s) 2

∆P (BAR)

Tube 1

 2

 3

 4

 5

2.0

Figure 25. Condensation heat transfer coefficient for condensation of steam in internally

finned horizontal tubes. Adapted from Royal

and Bergles (1978).

42
R-113 data of Luu and Bergles (1979) and of Said and Azer (1983), and the R-11 data of

Venkatesh (1984) to develop a general heat transfer correlation. This correlation was

claimed to predict 71% of the data points within ±30%. An analytical model to predict

condensation heat transfer in internally finned tubes was proposed by Kaushik and Azer

(1989), but the model remains unvalidated against experimental data, and as such can

only be recommended with caution.

The information on pressure drop correlation for condensation in internally finned

tubes is limited. Kaushik and Azer (1990) used the data of several workers and developed

a correlation using the least squares regression technique. This correlation predicted 68 %

of the data for steam and R-113 within ±40 %. Sur and Azer (1991) proposed an analytical

model, which is claimed to predict the effects of changing the number, height, and thickness of fins on the pressure drop.

So far, the discussion has been focused on condensation of pure vapor. When internally

finned tubes are used in the condensers of vapor compression refrigeration systems, the

refrigerant vapor is often mixed with small quantities of compressor lubricating oil. This

situation has been studied by Schlager et al. (1990a,b) in a two-part paper. In the first part

(1990a), the authors reviewed the heat transfer and pressure drop correlations for singlephase and two-phase flows in smooth and internally finned tubes. They found that there

was no published correlation for flow of refrigerant-oil mixtures in finned tubes. This

conclusion prompted them to embark on an experimental program to study the evaporation and condensation of refrigerant-oil mixtures in smooth and internally finned tubes.

The results of this study revealed that the presence of oil lowers the condensation heat

transfer coefficient for both smooth and finned tubes. The degradation of heat transfer

performance increases as the oil concentration increases. The pressure drop in finned

tubes was also found to increase with the increase in oil concentration.

The discussion in this section clearly indicates a need for better understanding of the

condensation process in internally finned tubes, so that accurate theoretical models for

heat transfer and pressure drop can be developed. There is also a need for correlations,

which are more generally applicable and not confined to specific fluids and operating

conditions. Similarly, more work is needed to understand and predict condensation of

refrigerant-oil mixtures in internally finned tubes.

MICRO-FIN TUBES

The micro-fin tube is currently the most popular enhancement device for residential

air-cooled air conditioners (window and central) and automotive refrigerant condensers.

This tube geometry was first developed by Hitachi Cable, Ltd., and described in a patent

by Fujie et al. (1977). The tube is commercially available in diameters between 4 and 16

mm. The original design, trade named Thermofin, has now been superseded by Thermofin EX, Thermofin HEX, and Thermofin HEX-C. The last design, Thermofin HEX-C, was

specially developed for condensation enhancement applications. Figure 27, adapted from

(a) (b)

(c) (d)

Figure 27. Cross sections of Hitachi Thermofin tubes: (a) Thermofin,

(b) Thermofin EX; (c) Thermofin HEX; and (d) Thermofin HEX-C.

Adapted from Yasuda et al. (1990).

43
10

1

100 200 300 400 500 600

G, Mass Velocity (kg/m /s) 2

Plain

EX

HEX

HEX-C

9.52 mm o.d.

7.54

2 Heat Transfer Coefficient (kW/m /K)

Figure 28. Heat transfer coefficient for R-22

condensing in micro-fin tubes.

Yasuda et al. (1990), shows the cross-sections of different micro-fin tubes. Table 5 gives

information on geometry, weight, enhancement ratio (for condensation of R-22) for the

four micro-fin tube designs (tube outside diameter = 9.52 mm).

The last column of Table 5 shows that the original micro-fin tube improves the condensation coefficient for R-22 by 80% over a plain tube. The subsequent designs, EX and HEX,

raise the improvement figure to 140 and 150%, respectively. The HEX-C gives the highest

performance with an improvement of 210% over a plain tube. The improved designs have

greater fin heights (e) and smaller fin included angles (β). Condensation experiments with

micro-fin tubes conducted by Shinohara and Tobe (1985), Shinohara et al. (1987), and

Schlager et al. (1990c) have shown that the heat transfer coefficient for R-22 gradually

increases with helix angle (α), from 7°, up to 30°. Although these results do not clearly

identify the optimum helix angle (α), it does appear that α = 30° results in the best condensation performance.

Figure 28 shows the R-22 condensation heat transfer coefficient as a function of mass

flux (velocity) for the tube geometries in Table 5, and demonstrates graphically the

superiority of micro-fin tubes over plain tubes. The HEX-C design stands out as distinctly

superior to HEX and EX designs for R-22 condensation. The tube outside diameter appears to have little effect on the heat transfer performance.

Table 5. Characteristics of micro-fin tube designs.

e* α β

Design (mm) p/e (° ) (° ) n A/Ap wt/wtp h/hp

Original 0.15 2.14 25 90 65 1.28 1.22 1.8

EX 0.20 2.32 18 53 60 1.51 1.19 2.4

HEX 0.20 2.32 18 40 60 1.60 1.19 2.5

HEX-C 0.25 2.32 30 40 60 1.73 1.28 3.1

*e = fin height in mm, p = fin pitch, α = helix angle, β = included angle of fin cross-section,

n = number of fins in the tube, A = surface area of micro-fin tube, Wt = weight of the tube,

h = condensation heat transfer coefficient, and the subscript p denotes the corresponding

values for the plain tube.

G, Mass Velocity (kg/m2 s)

44
Pressure drop measurements for R-22 condensing in 12.7-mm-diameter micro-fin tubes

have been reported by Schlager et al. (1990c). Their results show that the pressure drop in

micro-fin tubes is somewhat higher than that of the plain tube. Because of the uncertainties in the pressure drop measurements, the relative differences between the different

microfin tubes could not clearly be established.

The effect of small concentrations of oil in R-22 condensation in micro-fin tubes has

been studied by Schlager et al. (1989). The presence of oil generally adversely affects the

heat transfer and pressure drop in micro-fin tubes.

Despite the proven effectiveness of the micro-fin tube in practice, detailed work on

understanding the enhancement mechanism involved is lacking. Webb (1994) has proposed that both vapor shear and surface tension have important roles to play in augmenting the heat transfer process in micro-fin tubes. There is a great need for experimental

work on fluids other than refrigerants so that general purpose correlations for heat transfer and pressure drop in micro-fin tubes can be developed.

CONCLUDING REMARKS

This report described the progress that has been achieved with the analysis and design

of extended surfaces for applications involving condensation. The report covered condensation of pure vapor, as well as the condensation of moisture from humid air. The simple

models based on Nusselt-type correlations furnish the basic information about the performance of single fins. However, a more detailed picture about the condensate film characteristics and the thermal response of a fin is obtained with conjugate models. Considerable progress has been made in understanding and predicting condensation on horizontal

integral-fin tubes. On the other hand, the existing correlations for predicting the performance of internally finned tubes are not only limited in number but also unsatisfactory in

many cases. The micro-fin tube, which is currently very popular with condenser designers, remains virtually unexplored both theoretically and experimentally.

The bulk of the information appearing in the literature pertains to analysis. The various

analyses have established general guidelines for designing fins for condensation application, but the designer is still left to exercise judgment for designing condensing extended

surfaces. This is true of both single fins and an ensemble of fins. Despite 60 years of

studies on horizontal integral-fin tubes, these enhanced surfaces are still designed based

on past experience. Basic design questions as to the optimum fin shape, size, and number

for a specified condensation rate and vapor-tube-coolant combination are still a challenge

for researchers. The development of analytical and design tools for the micro-fin tube is

still in its infancy.

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45
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November 1995

Effect of Condensation on Performance and Design of

Extended Surfaces

ARO Contract

DAAL03-91-C-0034

TCN 94-077 No. 1140

Virgil J. Lunardini and Abdul Aziz

U.S. Army Cold Regions Research and Engineering Laboratory

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Army Research Office

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Available from NTIS, Springfield, Virginia 22161

59

Condensation Heat transfer

Finned surfaces Refrigeration

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

Heat transfer surfaces operating in cold regions often involve condensation. The analytical and experimental

progress made in understanding the process of condensation on extended surfaces (fins) is reviewed in detail. The review covers condensation of pure vapor as well as dehumidification of air. The analytical models

discussed range from simple Nusselt-type analysis to the three-dimensional conjugate approach, in which

the conservation equations for the condensate film are tightly coupled to conduction in the fin. A separate

section discusses the topic of dehumidification of air on finned cooling coils. Other topics reviewed include

condensation on horizontal integral-fin tubes, convective condensation in internally finned tubes, and condensation in micro-fin tubes. Although condensation on horizontal integral-fin tubes appears to be well understood, our understanding of convective condensation in internally finned tubes, particularly the micro-fin

tubes, is very limited. Furthermore, there exists no established methodology for designing extended surfaces

for condensation applications. This report contains several examples illustrating the theoretical results that

provide some insight into the design process.