bertsch & garimella 2008 review & comparative analysis of studies on sat fb in small channels

8/11/2019 Bertsch & Garimella 2008 Review & Comparative Analysis of Studies on Sat FB in Small Channels

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Review and Comparative Analysis of Studies on

Saturated Flow Boiling in Small Channels

Stefan S Bertsch1,2, Eckhard A. Groll1,2, Suresh V. Garimella2,*

1Ray W. Herrick Laboratories and 2Cooling Technologies Research Center

School of Mechanical Engineering, Purdue University

585 Purdue Mall, West Lafayette, IN 47907-2088 USA

Supplement/Appendix: Heat transfer Correlations

IntroductionThis document is intended as supplemental information to the paper entitled Review and

Comparative Analysis of Studies on Saturated Flow Boiling in Small Channels. It catalogs

several flow-boiling heat transfer correlations found in the literature. In addition, the complete

nomenclature and specific citations are included. Several correlations were updated and

corrected according to errata and based on personal communications with the correspondingauthors. To minimize errors in the listed correlations, the predictions from each correlation were

compared to the datasets that were published in the original paper, whenever possible. The

degree to which each correlation was validated with the original data is discussed in the

introduction to each correlation. The presentation of correlations is ordered by date of

publication, with the exception of two frequently referenced pool-boiling correlations which are

included at the end. For each correlation, the same nomenclature as in the originating paper was

used. This measure should simplify the comparison between this summary and the original

papers that use different symbol conventions and units.

Reference should be made to the original sources for further details of the correlations and

their applicability.

Table of Contents Page

1. Chen (1966) and Edelstein et al. (1984) ................................................................................. 22. Bennett and Chen (1980)........................................................................................................ 33. Lazarek and Black (1982)....................................................................................................... 44. Shah (1982)............................................................................................................................. 55. Gungor and Winterton (1986)................................................................................................. 66. Kandlikar (1990, 1991)........................................................................................................... 77. Liu and Winterton (1991) ....................................................................................................... 98. Steiner and Taborek (1992) .................................................................................................. 109. Tran et al. (1996) .................................................................................................................. 1210. Yan and Lin (1998)............................................................................................................... 13

11. Lee and Lee (2001)............................................................................................................... 1412. Warrier et al. (2002) ............................................................................................................. 1513. Yu et al. (2002)..................................................................................................................... 1614. Haynes and Fletcher (2003).................................................................................................. 1615. Sumith et al. (2003) .............................................................................................................. 18

*To whom correspondence should be addressed: [email protected], 765-494-5621

This supplement/appendix is being provided to the Editor simultaneously with the parent paper, for

inclusion in the journal issue, or on the journal website at the discretion of the Editor.


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16. Balasubramanian and Kandlikar (2004) ............................................................................... 1917. Thome et al. (2004)............................................................................................................... 2018. Lee and Mudawar (2005)...................................................................................................... 2219. Zhang et al. (2005)................................................................................................................ 2320. Yun et al. (2006)................................................................................................................... 2521. Liu and Garimella (2007) ..................................................................................................... 26

22. Saitoh et al. (2007)................................................................................................................ 2723. Lee and Garimella (2008)..................................................................................................... 2924. Cooper (pool boiling) (1984)................................................................................................ 3025. Gorenflo (pool boiling) (1993) ............................................................................................. 31

1. Chen (1966) and Edelstein et al. (1984)

References:J. C. Chen, 1966, Correlation for boiling heat transfer to saturated fluids in convective flow,

I&EC Process Design and Development, 5(3), pp. 322-329.

S. Edelstein, A. J. Perez and J. C. Chen, 1984, Analytic representation of convective boiling

functions, American Institute of Chemical Engineers Journal, 30(5), pp. 840-841.

Comments: The correlation is based on a superposition approach that includes contributions from

saturated nucleate boiling and two-phase forced convection for vapor qualities below 0.7.

Original correlation (as cited below) is in IP units an alternative form in SI units is shown in

Chen and Bennett (1980) in section 2 of this paper.

Comparison of the predictions from the correlation to data published in the paper could not be

achieved due to the absence of a plot or table with heat transfer data. However, the

correlation is well established in the literature and the predictions have been compared

successfully to published data in several textbooks.

Fluids: Water, Methanol, Pentane, Heptane, Benzene

Application range:

P = 55 kPa 3.5 MPa; x = 0.01 0.71; v = 0.06 4.5 m s

-1

; q = 0.6 240 W cm

-2

Equations:

NcB conh h S h F= +

0.8 0.4 Lconv L L

kh 0.023 Re Pr

D=

0.79 0.45 0.49 0.25

L pL L 0.24 0.75

NcB 0.5 0.29 0.24 0.24

L fg V

k c gh 0.00122 ( T) ( P)

i

=

L

L

D G (1 x)Re

=

( )

1.780.5

tt

F 1 X = +

0.10.50.9

V Ltt

L V

1 xX

x

=

1

4

ReS 0.9622 0.5822 tan

6.18 10

=

1.25

LRe Re F=


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Nomenclature:cPL heat capacity of liquid, Btu lbm

-1F-1

D diameter, ft

F enhancement factor

g gravitational constant, ft h-2

G mass flux, lbm ft-2

h-1

h heat transfer coefficient, Btu h-2ft-2ifg latent heat of vaporization, Btu lb

-1

k thermal conductivity, Btu h-1 ft-1F-1

PrL Prandtl number of the liquid

Re Reynolds numberS suppression factor

x vapor quality

Xtt Martinelli parameter

P difference in vapor pressure corresponding to T, psfT superheat, Tw-Ts, F viscosity, lb ft-1 h-1

density, lbm ft-3

surface tension, lb ft-1

2. Bennett and Chen (1980)

References:D. L. Bennett and J. C. Chen, 1980, Forced convective boiling in vertical tubes for saturated

pure components and binary mixtures, American Institute of Chemical Engineers Journal, 26

(3), pp. 454-461.

D.L. Bennett, M.W. Davies, and B.L. Hertzler, 1980, The suppression of saturated nucleate

boiling by forced convective flow, American Institute of Chemical Engineers Symposium

Series, 76(199), pp. 91-103.

Comments:

The correlation is based on a superposition approach that includes contributions fromsaturated nucleate boiling and two-phase forced convection in vertical channels for vapor

qualities below 0.7.

Comparison of the predictions from the correlation to the data published in the paper showed

good agreement.

A summary of the correlation can also be found in the textbook by V. P. Carey,Liquid-Vapor

Phase-Change Phenomena, 2008, Taylor & Francis.

Fluids: Water, glycol, mixtures, etc.

Application range:G = 0.16 1600 kg m

-2s

-1; x = 0.0 0.3; q = 0 300 W cm

-2

Equations:

TP NcB conh h S h F= +

0.8 0.4 Lconv L L

kh 0.023 Re Pr

D=

0.79 0.45 0.49

L pL L 0.24 0.75

NcB 0.5 0.29 0.24 0.24

L fg V

k ch 0.00122 ( T) ( P)

i

=


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0.10.50.9

V LL tt

L L V

D G (1 x) 1 xRe ; X

x

= =

( )0.444

1.78-0.5ltt

Pr +1F= 1+X

2

0.5

0

l v

X =0.041g ( - )

conv 0 l

conv 0 l

1-exp(-F h X /k )S=

F h X /k

Nomenclature:cPL heat capacity (liquid), J kg

-1K

-1

D diameter, m

g gravitational constant, m s-2

G mass flux, kg m-2s-1

h heat transfer coefficient, W m-2

K-1

ifg latent heat of vaporization, J kg-1

kL thermal conductivity (liquid), W m

-1K

-1

PrL Prandtl number (liquid)

Re Reynolds number

x vapor quality


P difference in vapor pressure corresponding to T, PaT superheat, Tw-Ts, K viscosity, kg m-1s-1

density, kg m-3 surface tension, N m

-1

3.

Lazarek and Black (1982)Reference:G. M. Lazarek and S. H. Black, 1982, Evaporative heat transfer, pressure drop and critical heat

flux in a small vertical tube with R-113, Int. J. Heat Mass Transfer, 25(7), pp. 945-960.

Comments: Flow boiling correlation for saturated boiling in vertical tubes for vertical upward and

downward flow.


good agreement.

Fluid: R113

Application range:

x = 0.0 0.6; Re = 860 5500; G = 125 750 kg m-2s-1; q = 1.4 38 W cm-2 ; D = 3.1 mm

Equations:0.857 0.714Nu 30 Re Bo=

l

G DRe

=

fg

BoG h

=


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lkh NuD

=

Nomenclature:Bo Boiling number

D diameter, m

G mass flux, kg m-2

s-1

hfg phase change enthalpy, J kg

-1

kl conductivity (liquid), W m-1

K-1

Nu Nusselt number

Re Reynolds number

l dynamic viscosity (liquid), kg m-1s-1

heat flux, W m-2

4. Shah (1982)

Reference:M. M. Shah, 1982, Chart correlation for saturated boiling heat transfer: equations and further

study, ASHRAE Transactions, 88, pp.185-196.

Comments: Flow boiling equation for saturated boiling in horizontal and vertical tubes and annuli.


achieved due to the absence of a plot or table with heat transfer data. However, the

correlation is well established in the literature and the calculated heat transfer results were

compared successfully to some plots in textbooks attributed to the Shah (1982) correlation.

Fluids: several refrigerants including R11, R12, R22, R502 (data from 19 independent

studies)

Application range:x = 0.0 0.7; ReL= 3000 345000; q = 9 122 W cm

-2; Dh= 5 16 mm

Equations:TP

L

h

h =

0.50.8

V

L fg

1 qCo 1 ; Bo

x G i

= =

2

L L2

L L

G D G (1 x)Fr ; Re

g D

= =

0.8 0.4 LL L L

kh 0.023 Re Pr Dittus Boelter

D

=

0.3

L

horizontal tubes : N 0.38 Fr Co=

cb 0.8

1.8

N =

for N 1.0 :> 0.5 4

nb230 Bo for Bo 0.3 10

= > 0.5 4

nb1 46 Bo for Bo 0.3 10 = + <

nb cbmax( , ) =


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for 0.1 N 1.0 :< 0.5 0.1

bsF Bo exp(2.74 N ) =

bs cbmax( , ) =

for N 0.1: 0.5 0.15

bsF Bo exp(2.47 N )

=

bs cbmax( , ) = 4F 14.7 for Bo 11 10=

4F 15.43 for Bo 11 10= <

Nomenclature:Bo Boiling numberCo Convective number

D diameter, m

F constant

FrL Froude number (liquid)

g acceleration due to gravity, m s-2

G mass flux, kg m-2

s-1

h heat transfer coefficient, W m

-2K

-1


K-1

k conductivity, W m-1

K-1

N constantPrL Prandtl number of the liquid

q heat flux, W m-2

ReL Reynolds number (liquid)

x vapor quality

density, kg m-3

ratio of two-phase to single-phase heat transfer coefficient

Subscripts

bs bubble suppressioncb convective boiling

l liquidnb nucleate boiling

tp two-phase

v vapor

5. Gungor and Winterton (1986)

Reference:K. E. Gungor and R. H. S. Winterton, 1986, A general correlation for flow boiling in tubes and

annuli, Int. J. Heat Mass Transfer,29(3), pp. 315-358.

Comments: Saturated flow boiling heat transfer correlation for horizontal and vertical tubes.

Correlation fitted based on measurements from 28 studies.

Comparison of the predictions from the correlation to data published in the paper could not beachieved due to the absence of a plot or table with heat transfer data.

Fluids: water, six different refrigerants (data from 28 authors).

Application range:G = 67 61518 kg m

-2s

-1; x = 0.0 1.0; q = 0.11 228 W cm

-2; Dh= 2.95 32.0 mm


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Equations:

tp l poolh E h S h= +

0.8 0.4 ll l l

h

kh 0.023 Re Pr

D=

( ) ( )0.55 0.670.12 0.5

pool r 10 rh 55 P log P M q =

0.86

1.16

tt

1E 1 24000 Bo 1.37

X

= + +

6 2 1.17

l

1S

1 1.15 10 E Re

=+

For horizontal tubes and the Fr < 0.05 then E and S should be multiplied by:

( )0.1 2 F r2 2

E Fr and S Fr

= =

2

hL 2

L fg L h

D G (1 x) q" GRe ; Bo ; Fr

G i g D

= = =

0.10.50.9

v ltt

l v

1 xX

x

=

All properties are calculated at saturation temperature.


Dh hydraulic diameter, m

E enhancement factor

Fr Froude numberG mass flux, kg m-2s-1


h heat transfer coefficient, W m-

K-1

ifg latent heat of vaporization, J kg

-1

kl conductivity (liquid), W m-1

K-1

M molecular weight, kg kmol-1

Pr reduced pressure

Prl Prandtl number (liquid)

q" heat flux, W m-2

Rel Reynolds number (liquid)

S suppression factor

x vapor quality


viscosity, kg m-1

s-1

density, kg m-3

6. Kandlikar (1990, 1991)

References:S. G. Kandlikar, 1990, A general correlation for saturated two-phase flow boiling heat transfer

inside horizontal and vertical tubes, J. Heat Transfer, 112, pp. 219-228.

S. G. Kandlikar, 1991, A model for correlating flow boiling heat transfer in augmented tubes

and compact evaporators, J. Heat Transfer, 113, pp. 966-972.


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Comments: Saturated two-phase flow boiling heat transfer correlation inside horizontal and vertical tubes

for water and refrigerants.

Comparison of the predictions from the correlation to the data published in the paper from

1990 showed reasonable agreement.

Fluids: Cryogens, several refrigerants, water

Application range:G = 13 8179 kg m-2s-1; x = 0.00 0.98; q = 0.03 228 W cm-2; Dh= 4.6 32 mm

Equations:0.50.8 2

g

Lo Lo 2

fg L L L

q" G D 1 x GBo ; Re ; Co ; Fr

G i x g D

= = = =

( )( )2

Lof 1.58 ln Re 3.28

=

For 2'300 < Re < 10'000: (Gnielinski)

( ) ( )

( ) ( )

Lo L

Lo 0.52/ 3

L

Re 1000 f / 2 PrNu

1 12.7 Pr 1 f / 2

=

+

For 10'000 < Re < 500'000: (Petukhov, Popov)

( )

( ) ( )Lo L

Lo 0.52/ 3

L

Re Pr f / 2Nu

1.07 12.7 Pr 1 f / 2

=

+

LLo Lo

kh Nu

D=

( ) ( ){ }0.8 0.80.2 0.7nbd 2 Fl Loh 0.6683 Co 1 x f 1058.0 Bo 1 x F h= +

( ) ( ){ 0.8 0.80.9 0.7cbd 2 Fl Loh 1.136 Co 1 x f 667.2 Bo 1 x F h= +

LoFor vertical tubes and horizontal tubes with Fr > 0.4:

2f 1=

LoFor horizontal tubes with Fr < 0.4:

( )0.3

2 Lof 25 Fr=

nbd cbdh = max(h , h )

The constant FFLcan be found in the following table.

Fluid FFL

Water 1.00

R-11 1.30

R-12 1.50

R13B1 1.31

R-22 2.20R-113 1.30

R-114 1.24

V. Gnielinski, 1976, New equations for heat and mass transfer in turbulent pipe and channel flow,

International Chemical Engineer, Vol. 16, pp. 359-368.B.S. Petukhov and V.N. Popov, 1963, Theoretical calculation of heat exchange and frictional resistance

in turbulent flow in tubes of an incompressible fluid with variable physical properties, Teplofiz. Vyosk.

Temperature (High Temperature Heat Physics), Vol. 1, No. 1.


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R-134a 1.63

R-152a 1.10

R-32/R-132 3.30

R-141b 1.80

R-124 1.00

R-123 0.616

Nitrogen 4.7

Neon 3.5

Kerosene 0.488

all fluid in conjunction with

a stainless steel surface1.00


Co Convection number

D tube diameter, m

f friction factorFfl fluid-dependent constant

FrLo Froude number (all flow liquid)G mass flux, kg m

-2s

-1



K-1

ifg latent heat of vaporization, J kg

-1K-1

k conductivity, W m-1K-1

Nu Nusselt number

q heat flux, W m-2

PrL Prandtl number (liquid)ReL Reynolds number (liquid)

x vapor fraction

dynamic viscosity, kg m-1s-1

density, kg m-3

Subscripts:

cbd convective boiling dominant

g gaseous

l liquid

Lo liquid only

nbd nucleate boiling dominant

7. Liu and Winterton (1991)

Reference:Z. Liu and R. H. S. Winterton, 1991, A general correlation for saturated and subcooled flow

boiling in tubes and annuli, based on a nucleate pool boiling equation, Int. J. Heat MassTransfer, 34(11), pp. 2759-2766.

Comments: Correlation for saturated flow boiling in vertical and horizontal tubes.


achieved due to the absence of a plot or table with heat transfer data.

Fluids: Water, refrigerants (compiled from 30 papers)

Application range:


10/33

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G = 12.4 8157 kg m-2

s-1

; x = 0.0 0.95; q = 0.35 262 W cm-2

; Re = 570 87500;Dh= 2.95 32.0 mm

Equations:

( ) ( )222

TP L poolh F h S h= +

0.8 0.4LL L Lkh 0.023 Re Pr

d

=

( )0.550.12 2 /3 0.5

pool r 10 rh 55 p q log (p ) M =

0.35

LL

v

F 1 x Pr 1

= +

( )1

0.1 0.16

LS 1 0.055 F Re

= +

For a horizontal tube and Fr < 0.05 multiply F and S with

( 0.1 2 F r )

f se Fr and e Fr

= =

where

2

L2

L L

G G dFr ; Re

g d

= =

Nomenclature:d diameter, m


Fr Froude number


G mass flux, kg m-2

s-1

h heat transfer coefficient, W m-2K-1

k thermal conductivity, W m-1

K-1


pr reduced pressurePr Prandtl number (liquid)

q heat flux, W m-2

ReL Reynolds number (liquid)


x vapor quality dynamic viscosity, kg m

-1s

-1

density, kg m-3

Subsripts

L liquid

pool pool boiling

TP two-phasev vapor

8. Steiner and Taborek (1992)

Reference:D. Steiner and J. Taborek, 1992, Flow boiling heat transfer in vertical tubes correlated by an

asymptotic model, Heat Transfer Engineering, 13(2), pp. 43-68.

Comments:


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11/33

Heat transfer correlation for nucleate and convective flow boiling in vertical tubes.


good agreement.

Fluids: water, refrigerants, cryogens

Application range:

G = 4 4850 kg m

-2

s

-1

; x = 0.0 1.0; q = 0.08 460 W cm

-2

; P = 10 10800 kPa;Dh= 1 32 mm

Equations:

( ) ( )1/ n

1/ n nnn n

fb nbf cb nb,o nbf LO tpF F (n = 3 is recommended) = + = +

LO GOUse any appropriate single-phase heat transfer correlation to determine and .

for x 0.6:

( )

1.10.35

1.5 0.6 Ltp

G

F 1 x 1.9 x = +

for x > 0.6 or low heat flux:

( ) ( )

( )

0.52.20.35

1.5 0.010.6 L

G

tp 20.67

0.70.01GO L

LO G

1 x 1.9 x 1 x

F

x 1 8 1 x

+ + =

+ +

r0,133nf (p ) 0.4

anbf pf

0f 0 a,0

Rq dF F F(M)

q d R

=

a ,0 0R 1 m; d 0.01m= =

r

0.45 3.7

PF r r7

r p 0.95

1.7F 2.816 p 3.4 p

1 p

= + +

( )rnf 0.8 0.1 exp 1.75 p (all fluids except cryogenics)=

( )rnf 0.7 0.13 exp 1.105 p (cryogenics)= 0.27F(M)=0.36 M (for M < 10)

( ) 22.5

F(M)= 0.377+0.199 ln M +2.8427E-5 M (for M > 10)

The value of 0for selected fluids is included in the table presented in section 25. Additional

details are available in the original source.

Nomenclature:

d diameter, mF multiplication factor


n arbitrary exponent

nf(Pr) nucleate flow boiling exponent

Pr reduced pressure

q heat flux, W m-2

Ra wall roughness, m


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12/33

x vapor quality heat transfer coefficient, W m-2K-1

density, kg m-3

Subscripts:

0 reference

cb convective boilingfb effective flow boiling

g gas

GO flow assumed to be totally gas

l liquid

LO flow assumed to be totally liquid

nbf nucleate flow boiling

tp two-phase

9. Tran et al. (1996)

Reference:T. N. Tran, M. W. Wambsganss and D. M. France, 1996, Small circular and rectangular-channel

boiling with two refrigerants, Int. J. Multiphase Flow, 22, pp. 485-498.

Comments: Flow boiling heat transfer correlation for small tubes and rectangular channels.


good agreement. A typographical error in the original paper in the constant in the calculation

of h was corrected from the listed 8.4 x 10-5

to 8.4 x 10+5

Fluids: R12 and R113

Application range:G = 44 8832 kg m-2s-1; x = 0.00 0.94; q = 0.36 12.9 W cm-2; P = 510 820 kPa;

Dh= 2.4 and 2.92 mm

Equations:

( ) ( )0.4

0.35 2 l

TP l

v

h 8.4 10 Bo We

=

2

l

l

G DWe

=

fg

q"Bo

G i=


D diameter, m

h heat transfer coefficient, W m-2K-1G mass flux, kg m-2s-1

ifg enthalpy of vaporization, J kg-1

q heat flux, W m-2

Wel Weber number (liquid) density, kg m-3

surface tension, N m-1

Subscripts


13/33

13/33

l liquidTP two-phase

v vapor

10.Yan and Lin (1998)

Reference:Y. Yan and T. Lin, 1998, Evaporation heat transfer and pressure drop of refrigerant R-134a in a

small pipe, Int. J. Heat Mass Transfer, 41, pp. 4183-4194.

Comments: Heat transfer correlation for saturated flow boiling in horizontal tubes.

An error in the definition of hlin the original paper has been corrected below.


reasonable agreement for some plots.

Fluid: R134a

Application range:G = 50 200 kg m-2s-1; x = 0.0 0. 9; q = 0.4 1.9 W cm-2; Dh= 2.0 mm

Equations:

( ) ( )2 40.8C C

tp 1 3 lo m lh C Co C Bo Fr 1 X h= +

Ll

h

kh 4.364

D=

hL

L

D GRe

=

0.8 0.52

g wmlo 2

m l fg l h

q1 X GCo ; Bo ; Fr

X i G g D

= = =

m ,2 m,3C C

m m,1 l RC C Re T using the table below=

m Cm,1 Cm,2 Cm,3

1 933.6 0.0757 26.19

2 -0.2 0 0

3 41700 0.5731 34.98Co>

0.5

4 14.84 -0.0224 13.22

1 47.3 0.3784 14.67

2 2612.8 0 37.27

3 100150 0 24.3710.1

5

200 use Kandlikar (1990) correlation, otherwise equations below:

TP Lh F h= 0.398 0.598

LF 10.3=

2

L 2

C 11

X X = + +

0.726

LoC 0.06185 Re=

LL

h

Nu kh

D

=

where Nu = 4.36 for constant heat flux and Nu = 3.66 for constant surface temperature

0.50.50.5

v L

L v

1 xX

x

=

hLo

L

G DRe

=

Nomenclature:F enhancement factor

C constant



K-1


15/33

15/33


K-1

Nu Nusselt number

Re Reynolds number

x vapor quality

X Martinelli parameter aspect ratio

viscosity, kg m-1s-1 density, kg m-3L two phase multiplier

Subscripts

L liquid

TP two-phase

V vapor

12.Warrier et al. (2002)

Reference:G. R. Warrier, V. K. Dhir and L. A. Momoda, 2002, Heat transfer and pressure drop in narrow

rectangular channels, Experimental Thermal Fluid Science, 26, pp. 53-64.

Comments: Enhancement factor correlation for saturated flow boiling valid for 0.00027 < Bo < 0.00089.


good agreement.

Correlation was fitted to data obtained in horizontal channels with top and bottom wallsheated.

Fluid: FC-84

Application range:G = 557 1600 kg m-2s-1; x = 0.03 0.55; q = 0 6 W cm-2; ReL= 418 2015; Dh= 0.75 mm

Equations:

1/16 0.65TP2

sp_FD

h1 6.0 Bo f (Bo) (x)

h= + +

( )2f (Bo) 5.3 1 855 Bo=

fg

q"Bo

G i=

Lsp_FD

h

kh Nu

D=

where Nu = 4.36 for constant heat flux and Nu = 3.66 for constant surface temperature

Nomenclature:

Bo Boiling numberDh hydraulic diameter, m

G mass flux, kg m-2

s-1

hsp_FD single-phase flow heat transfer coefficient, W m

-2K-1

hTP two-phase heat transfer coefficient, W m-2

K-1


kL liquid conductivity, W m-1

K-1

Nu Nussel number (single phase, laminar)


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16/33

q" heat flux, W m-2

x vapor quality

13.Yu et al. (2002)

Reference:

W. Yu, D. M. France, M. W. Wambsganss and J. R. Hull, 2002, Two-phase pressure drop,boiling heat transfer, and critical heat flux to water in a small-diameter horizontal tube, Int.J. Multiphase Flow, 28, pp. 927-941.

Comments: Update to the heat transfer correlation by Tran et al. in section 9.

Comparison of the predictions from the correlation to the data published in the paper could

not be achieved due to the absence of a plot or table with heat transfer data. The accuracy of

the correlation listed below was confirmed by personal communication with the author.

Nucleate boiling dominates over a large mass flux and quality range.

Fluid: Water

Application range:

G = 50 200 kg m

-2

s

-1

; x = 0.0 0.9; P = 200 kPa; Dh= 2.98 mmEquations:

( )0.2

0.272 l

TP l

v

h 6400000 Bo We

=

fg

q"Bo

G i=

2

l

l

G DWe

=

Nomenclature:

Bo Boiling numberD diameter, m

G mass flux, kg m-2

s-1

h two-phase heat transfer coefficient, W m-2K-1


q heat flux, W m-2

Wel Weber number (liquid) density, kg m

-3


Subscripts

l liquid

TP two-phaseV vapor

14.Haynes and Fletcher (2003)

Reference:B. S. Haynes and D. F. Fletcher, 2003, Subcooled flow boiling heat transfer in narrow

passages, Int. J. Heat Mass Transfer, 46, pp. 3673-3682.

Comments:


17/33

17/33

Heat transfer correlation for subcooled and saturated flow boiling.


good agreement.

Fluids: R11 and R123

Application range:

G = 110 1840 g m

-2

s

-1

; x = 0.0 1.0; q = 1.1 17 W cm

-2

; Dh= 0.92 and 1.95 mmEquations:

satTP conv pb

mean

Th h h

T

= +

convh should be calculated with an equation for single phase liquid only:

i.e. for laminar flow:

lz 1/ 3

st st h

k0.0273h 4.364

z 0.0236 z D

= +

+

hst

l

z / Dz

Re Pr=

i.e. for transitional flow:

( )

( )

ll

2/ 3h

l

fRe 1000 Pr

k 2hD f

1.07 12.7 Pr 12

=

+

0.25f 0.0396 Re2

=

sat w sat mean w meanT T T ; T T T = =

l

G dRe

=

For the nucleate boiling heat transfer coefficient use the Gorenflo (1993) equation with:pb conv meanq q h T=

Nomenclature:Dh hydraulic diameter, m

f friction factor


K-1


K-1

Nu Nusselt number

Pr Prandtl number of liquid

q heat flux, W m-2

Re Reynolds number (liquid)

T temperature, Kz distance from inlet, m

viscosity, kg m-1

s-1

Subscripts

conv convectionl liquid

pb pool boiling

sat saturation


18/33

18/33

tp two-phasew wall

15.Sumith et al. (2003)

Reference:

B. Sumith, F. Kaminaga and K. Matsumura, 2003, Saturated flow boiling of water in a verticalsmall diameter tube, Experimental Thermal and Fluid Science, 27, pp 789-801.

Comments: Flow boiling heat transfer correlation for vertical pipe flow.


reasonable agreement.

Fluid: Water

Application range:G = 23.4 152.7 kg m

-2s

-1; x = 0.0 0.7; q = 1 17.5 W cm

-2; Dh= 1.45 mm

Equations:

GFor Re < 2000:

z LFh 2.36 h=

LLF

m

kh =

( )0.21 0.25 0.12m h L GD 0.082 exp 0.0594 Re Fr x =

GFor Re > 2000:

LLO DB

h

kh Nu

D=

4 /5 0.4

DB L LNu 0.023Re Pr= 0.736

z LO

tt

1h 2.83 0.213 hX

= +

( )Gh h

L G G G0.5

L g Gh

JG (1 x) D G x D G xRe ; Re ; Fr ; J

g D

= = = =

0.9 0.1 0.5

L L Ltt

G G G

GX =

G

Nomenclature:Dh hydraulic diameter, m

Fr Froude number

g gravitational acceleration, m s-2

G mass flux, kg m-2

s-1



Nu Nusselt number

Pr Prandtl number

Re Reynolds numberx vapor quality

Xtt Martinelli parameterm liquid film thickness, m


19/33

19/33

dynamic viscosity, kg m-1

s-1

density, kg m-3

Subscripts

DB Dittus-Boelter

G gaseous

L liquidG gas

z local

16.Balasubramanian and Kandlikar (2004)

Reference:P. Balasubramanian and S. G. Kandlikar, 2004, An extension of the flow boiling correlation to

transition, laminar, and deep laminar flows in minichannel and microchannels, Heat

Transfer Engineering, 25(3), pp. 86-93.

Comments:

Extension of the Kandlikar (1990) equation to low Reynolds number (small channel) flows.

Comparison of the predictions from the correlation to the data published in the paper showedreasonable agreement.

Fluids: Water, several refrigerants

Application range:G = 50 570 kg m-2s-1; x =0.00 0.98; q = 0.5 9.1 W cm-2; Dh= 0.19 2.92 mm

Equations:

LO TP TP,NBDfor Re 1000 : h h< =

( )LO TP TP,NBD TP,CBDfor Re 1000 : h max h , h> =

( ) ( )0.8 0.80.2 0.7

TP,NBD LO Fl LOh 0.6683 Co 1 x h 1058.0 Bo 1 x F h= +

( ) ( )0.8 0.80.9 0.7

TP,CBD LO Fl LOh 1.136 Co 1 x h 667.2 Bo 1 x F h= +

0.5 0.8

G hLO

L LG

G D1 x qCo ; Bo ; Re

x G i

= = =

LOfor turbulent region (Re 3000) : Gnielinski eqn.

( )

( ) ( )LO L L h

LO 0.52/ 3

L

Re 1000 Pr (f /2) (k / D )h

1 12.7 Pr 1 f / 2

=

+

( )2

LOf 1.58 ln Re 3.28

=

LOfor laminar region (Re 1600) : Nu C =

LLO

h

Nu kh (with Nu 4.36 for const. heat flux and Nu = 3.66 for const. surface temperature

D

= =

LOfor transient region (1600 < Re 3000):<

Linear interpolation between laminar and turbulent regime

The constant FFLcan be found in the table presented in section 6.


Co Convection number


20/33

20/33

Dh hydraulic diameter, mf fanning friction factor

FFl fluid surface parameter



K-1

iLG latent heat of vaporization, J kg-1

k conductivity, W m-1K-1Nu Nusselt number

Pr Prandlt number

q" heat flux, W m-2

Re Reynolds number (liquid)x vapor quality

viscosity, kg m-1s-1

density, kg m-3

Subscripts

CBD convective boiling dominantLO liquid only

NBD nucleate boiling dominantTP two-phase

17.Thome et al. (2004)

References:J. R. Thome, V. Dupont and A. M. Jacobi, 2004, Heat transfer model for evaporation in

microchannels. Part I: presentation of the model, Int. J. Heat Mass Transfer, 47, pp. 3375-

3385.

J. R. Thome, V. Dupont and A. M. Jacobi, 2004, Heat transfer model for evaporation in

microchannels. Part II: comparison with the database, Int. J. Heat Mass Transfer, 47, pp.

3387-3401.

Comments:

Three-zone boiling model to predict heat transfer coefficient of elongated slugs in saturated

flow boiling.


good agreement.

Fluids: R11, R12, R113, R123, R134a, R141b, CO2

Application range:G = 50 564 kg m-2s-1; x = 0.01 0.99 ; q = 0.5 17.8 W cm-2; Dh= 0.7 3.1 mm

Equations:

dryl filml film v

tt th(z)= h (z) h (z) h (z) + +

3lam,z

d ReNu 2 0.455 Pr (for vapor and liquid)L(z)

=

( ) ( )

( )

2/ 3

trans,z 2 /3

/ 8 Re 1000 Pr dNu 1 (for vapor and liquid)

L(z)1 12.7 /8 Pr 1

= +

+

( )2

101.82 log (Re) 1.64

=


21/33

21/33

( )1/ 4

4 4

lam transh= Nu Nu

d

+

lfilm

0 min

2h (z)

(z) (z)

=

+

( )( )00.84

1/ 88

0.41 80 l

p

C 3 0.07 Bo 0.1d U d

= +

2lp p total

v l

d x 1 xBo U ; U G

= = +

0

l lv

q(z, t) (z) t

h =

vll v

l vp p

v l

LLt ; t

x 1 xU U1 1

1 x x

= = = =

+ +

( )l lv

dry film 0 min

h

t (z) (z)q

=

For tdry, film> tvwe use

tfilm= tvand end(z) = (z,tv) and tdry= 0

For tdry, film< tvwe use

tfilm= tdry, filmand end(z) = minand tdry= tv-tfilm

opt

1

f =

fn 0.5

satopt ref

ref crit

Pqf ; q 3328

q P

= =

6

f 0 minn 1.74; C 0.29; 0.3 10 m

= = =

l v

l l v v

G d (1 x) G d xRe ; Re

= =

Nomenclature:Bo Bond number

C0 correction factor for initial film thickness

d diameter, m

f frequency, s-1



L length, m

Nu Nusselt numberP pressure, Paq heat flux, W m-2

Pr Prandtl number

Re Reynolds number

t time, s

U velocity m s-1

x vapor quality

z longitudinal abscissa, m


22/33

22/33

liquid film thickness, mhlv latent heat of vaporization, J kg

-1

thermal conductivity, W m-1K-1

kinematic viscosity, m2s-1

density, kg m-3


pair period, s drag coefficient

Subscripts

0 initial

crit critical

dry dryout zone

end end of liquid film

film film

l liquid

lam laminar flow

min minimum

ref referencesat saturation

v vapor

18.Lee and Mudawar (2005)

Reference:J. Lee and I. Mudawar, 2005, Two-phase flow in high-heat flux micro-channel heat sink for

refrigeration cooling applications: Part II heat transfer characteristics, Int. J. Heat Mass

Transfer, 48, pp. 941-955.

Comments: Three different heat transfer correlations for nucleate boiling (x < 0.05), bubbly and slug flow

(0.05 < x < 0.55) and annular flow (x > 0.55).


achieved due to missing parameters for the calculation of the heat transfer data provided.

Fluids: R134a, water


-2s

-1; x = 0.26 0.87; q = 15.9 93.8 W cm

-2; Dh= 0.35 mm

Equations:

efor x 0 0.05 :=

0.267

tp sp.f h 3.856 X h=

( )

( )

2 3 ffsp,f

hg

dp/dz Nu kX , h

dp / dz d

= =

0.5 0.50.5

ef fvv

g e g

1 xX ;

x

=

0.50.5 0.50.25

f g e fvt

e g

f Re 1 xX

0.079 x

=

efor x 0.05 0.55 :=


23/33

23/33

0.522 0.351 0.665

tp fo sp,f h 436.48 Bo We X h=

2

f hfo

fg

G dqBo ; We

G h

= =

efor x 0.55 1.0 :=

( )( )1.665

tp sp,g sp,gh max 108.6 X h ,h=

3 g

sp,g

h

Nu kh (for laminar gas flow)

d

=

g0.8 0.4

sp,g g g

h

k h 0.023 Re Pr (for turbulent gas flow)

d=

( )2 3 4 53Nu 8.235 1 1.883 3.767 5.814 5.361 2.0= + +

e h e hg f

g f

G x d G (1 x ) dRe ; Re

= =

ff 16 / Re=

Nomenclature:Bo boiling number

dh hydraulic diameter, m

ff fanning friction factor

G mass velocity, kg m-2s-1

hfg latent heat of vaporization, J kg-1


K-1


K-1

Nu3 Nusselt number (3 sided wall heating)

q" heat flux, W m-2

Re Reynolds number

Wefo Weber number (liquid)

xe equilibrium quality

X Martinelli parameter

Xvt Martinelli parameter (liquid laminar, gas turbulent)

Xvv Martinelli parameter (liquid laminar, gas laminar)

channel aspect ratio (< 1) dynamic viscosity, kg s

-1m

-1

specific volume, m3kg

-1


Subscripts

f liquid

g vapor

sp,f single-phase liquidtp two-phase

19.Zhang et al. (2005)

Reference:W. Zhang, T. Hibiki and K. Mishima, 2005, Correlation for flow boiling heat transfer at low

liquid Reynolds number in small diameter channels, J. Heat Transfer, 127, pp. 1214 -1221.

Comments:


24/33

24/33

Heat transfer correlation for saturated flow boiling in small diameter channels at lowReynolds numbers for thermodynamic qualities below 0.7.


achieved due to the absence of a plot or table with heat transfer data.

Fluids: Water, R11, R12 and R113

Application range:G = 23.4 560 kg m-2

s-1

; q = 0.3 80.3 W cm-2

; Dh= 0.78 6.0 mm

Equations:

tp pb f sp,vh h h= +

0.79 0.45 0.49

f pf f 0.24 0.75

pb sat sat0.5 0.29 0.24 0.24

f fg g

k ch 0.00122 T p

h

=

2

eq g2 ff 2

g eq f

1 xfC 11 ; X

X X f x

= + + =

h hf g

f g

D G (1 x) D G xRe ; Re

= =

n

k k kf C Re where k = f or g=

k kwith C 16 and n 1 ( laminar flow, Re < 1000)= =

k kand C 0.046 and n 0.2 (turbulent flow, Re > 2000)= =

0.64 (adjustable constant determined from the database) =

For rectangular channels:

( )2 3 4 5 fsp,vh

kh 8.235 1 2.042 3.085 2.4765 1.058 0.186

D= + +

The values for the constant C in the two-phase multiplier can be found in the following table:

Rel Rev C

> 2000 > 2000 20< 1000 > 2000 12

> 2000 < 1000 10

< 1000 < 1000 5

Nomenclature:cp heat capacity, J kg

-1K-1



K-1

hfg latent heat of vaporization, J kg-1


K-1

Re Reynolds number

x vapor quality

X Martinelli parameterTsat superheat, Tw-Ts, Kpsat vapor pressure difference corresponding to Tsat, Pa

two-phase multiplier channel aspect ratio (< 1)

viscosity, kg m-1

s-1

density, kg m

-3



25/33

25/33

Subscripts

f fluid

g gas

pb pool boiling

sat saturation

sp single-phase

tp two-phase

20.Yun et al. (2006)

Reference:R. Yun, J. H. Heo and Y. Kim, 2006, Evaporative heat transfer and pressure drop of R410A in

microchannels, Int. J. Refrigeration, 29, pp. 92-100.R.Yun, J. H. Heo and Y. Kim, 2007, Erratum to Evaporative heat transfer and pressure drop of

R410A in microchannels, [Int. J. Refrigeration 29 (2006) 92-100], Int. J. Refrigeration, 30,

pp. 1468.

Comments:

Flow boiling heat transfer correlation for R410A.

Comparison of the predictions from the correlation to the data published in the paper showedgood agreement after correcting the correlation as suggested in the Erratum, and correcting a

typographical error in the definition of Reynolds number not covered in the erratum.

Fluid: R410A


-2s

-1; x = 0.1 0.85; q = 1 2 W cm

-2; Dh= 1.36 and 1.44 mm

Equations:

( )0.1993 0.1626

tp l l136876 Bo We Re =

hl

L

G D (1 x)Re

=

fg

qBo

G h=

2

hl

l

G DWe

=

Nomenclature:Bo boiling number


G mass flux, kg m-2

s-1

hfg latent heat of vaporization, kJ kg-1

q heat flux, W m-2

Rel Reynolds number based on liquidWel Weber number (liquid)

x vapor quality

tp two-phase heat transfer coefficient, W m-2

K-1

surface tension, N m

-1

dynamic viscosity, kg s-1m-1

density, kg m-3


26/33

26/33

21.Liu and Garimella (2007)

Reference:D. Liu and S. V. Garimella, 2007, Flow boiling heat transfer in microchannels, J. Heat

Transfer, 129(10), pp 1321-1332.

Comments:

Flow boiling heat transfer correlation for saturated flow boiling. The authors recommend use of an interpolation to calculate the constant C in the transitional

flow regime for 1000 < Re < 2000.

Fluid: Water

Application range:G = 221 1283 kg m-2s-1; x = 0.0 0.2; q = 0 129 W cm-2; Dh= 0.38 and 0.59 mm

Equations:

tp conv boiling sp nbh h h F h S h= + = +

0.141/ 3

h f fSP f f

w h

D kh 1.86 Re Pr

L D

=

( ) h hf v

f v

G 1 x D G x DRe ; Re

= =

( )1/ 40.105 3/ 4

1/ 4 tp p,tp tp2 0.167

f f

f p,f f

c kF Pr

c k

=

2 2 v ff 2

f v

C 1 1 x2; 1 ; X

X X x

= = + + =

Determination of the two-phase value for any variable:

tp v f x (1 x) (replace with any variable) = +

0.133n

p

nb 0 PF

0 p0

Rqh h F (Gorenflo - 1993)

q R =

0.27 2

PF r r

r

0.68F 1.73 p 6.1 p (for water)

1 p

= + +

0.27

PF r r

r

1F 1.2 p 2.5 p (all other fluids)

1 p

= + +

0 p0 0 0Please see Gorenflo equation for the determination of n, Pr , R , q and h .

S in empirical form:

( )3

f3f

55746

S exp 36.57 3.4 ln Re FRe F

=

S in analytical form:

1/ 22

conv 0 h0 w

f w f

h y D GS 1 ; y C ; 2 f

2 k

= = =

2 3 4 5

f

96 1.3553 1.9467 1.7012 0.9564 0.2537f 1

Re

= + +


27/33

27/33

The according values for the constant C can be found in the following table:

Rel Rev C

> 2000 > 2000 20

< 1000 > 2000 12

> 2000 < 1000 10

< 1000 < 1000 5

Nomenclature:C constant depending on flow regime

cp specific heat, J kg-1K-1



G mass flux, kg m-2

s-1



L channel length, m

Pr Prandtl number

Pr reduced pressure

q" heat flux, W m-2

Rp surface roughness, m

Re Reynolds number



x vapor quality microchannel aspect ratio (


28/33

28/33

Comparison of the predictions from the correlation to the data published in to the papershowed reasonable agreement.

Fluid: R134a

Application range:x = 0.2 1.0; Dh= 0.5 10.9 mm

Equations:

TP,pre l poolh F h S h= +

l

m

g

1

XF 1

1 We

= ++

with l = 1.05 and m = -0.4

2

g

g g l g l

g g l

G D G x D G (1 x) DWe ; Re ; Re ; G G x; G G (1 x)

= = = = =

0.10.50.9

g l

l gl g

1 x

X (for Re 1000 and Re 1000)x

= > >

0.5 0.5 0.50.5

g0.4l l lg l g

g g l g

C GX Re (for Re 1000 and Re 1000)

C G

= < >

g lC 0.046; C 16= =

( )n

4

TP

1S

1 a Re 10=

+

with a = 0.4 and n = 1.41.25

TP lRe Re F= 0.745 0.581

g 0.533blpool l

b l l l

q dh 207 Prd T

=

( )

0.5

b

l g

2d 0.51

g

=

4 /5 0.4 ll l

kh 0.023 Re Pr

D=

Nomenclature:a fitting parameter

C friction factor

D (hydraulic) diameter, mdb bubble departure diameter, mF enhancement factor


G mass flux, kg m-2

s-1


K-1


K-1

l fitting parameter

m fitting parameter


29/33

29/33

n fitting parameterPr Prandtl number

q heat flux, W m-2

Re Reynolds number


T temperature, K

We Weber numberx vapor quality


thermal conductivity, W m-1

K-1


s-1

density, kg m-3


Subscripts

g gas

l liquid

pre pre-dryout

pool pool boilingTP two-phase

23.Lee and Garimella (2008)

Reference:P. S. Lee and S. V. Garimella, 2008, Saturated flow boiling heat transfer and pressure drop in

silicon microchannel arrays, Int. J. Heat Mass Transfer, 51(3-4), pp 789-806.

Comments: Flow boiling heat transfer correlation for saturated flow boiling (modified from Steiner and

Taborek, 1992).

Fluid: Water

Application range:x = 0.0 0.2; q = 0 80 W cm-2; Dh= 0.16 0.57 mm

Equations:

( ) ( )3 33

tp conv sp nb nbh F h F h= +

Forced convection boiling:

0.378

h fsp f f

h

D kh 1.766 Re Pr

L D

0.1224

=

( )0.2743 0.7257

0.2743 p,tp tp2

conv f

p,f f

c kF

c k

=

2

f 2

vv vv

C 11

X X = + +

hf

f

G DRe

=

( )h0.5 0.50.5

319 D0.5466 0.8819 e f fh vv

e g g

1 xC 2566 G D 1 e ; X

x

= =


30/33

30/33

Two-phase properties are calculated as:

tp g f x (1 x) = +

0 W,0Nucleate boiling: (For h and q see Steiner & Taborek, 1992):

n

0.15Wnb 0 PF r

W,0

qh h F ; n 0.9 0.3 pq

= =

0.27 2

PF r r

r

0.68F 1.73 p 6.1 p (for water)

1 p

= + +

0.27

PF r r

r

1F 1.2 p 2.5 p (all other fluids)

1 p

= + +

w hnb 10 6

q DF 4.6809 0.6705 log 3.908

1 10 0.001

= +

Nomenclature:

C constant depending on flow regimecp specific heat, J kg-1K-1


F boiling correction factor

G mass flux, kg m-2

s-1


K-1


L channel length, m

Pr Prandtl number

pr reduced pressure

q heat flux, W m-2

Re Reynolds number

xe thermodynamic equilibrium qualityXvv Martinelli parameter channel aspect ratio (< 1)


s-1

specific volume, m

3kg

-1

f two-phase multiplier

arbitrary variable

Subscripts:

0 reference

conv convective

f fluid

g gas

nb nucleate boilingr relative

sp single-phase

tp two-phase

w wall

24.Cooper (pool boiling) (1984)

Reference:


31/33


32/33

32/33

* * 0.27 * 2

*

0.68F(p ) 1.73 p 6.1 p (for water)

1 p

= + +

The value of 0can be found for some selected fluids in the following table.

Fluid q'of, W cm-2

nb,of,calc , W m-2

K-1

nb,of,calc , W m-2

K-1

Methane 2 8060 7000

Propane 2 4000 4000Acetone 2 3270 3200 - 4700

R-11 2 2690 2800

R-12 2 3290 4000

R-13 2 3910 3900

R-13 B1 3 3380 3500

R-22 2 3930 3900

R-23 2 4870 4400

R-113 2 2180 2650

R-123 2 2600 -

R-134a 2 3500 4500

Nitrogen 2 7360 5000 - 10000Ammonia 2 8090 7000

CO2 2 4170 5100

Water 2 6400 5600

If 0is not given in the following table or the original paper the following equations can be used.0.674 0.3500.371 0.160.156 2 2

p0 0 v 0

2

s 0

cq d h d aNu 0.1

T a d

=

( )

0.5

0

2d 0.0149

g

=

The numerical values of the contact angle are 45 for water, 1 for cryogenics and 35 for otherfluids.

Nomenclature:a diffusivity, m

2s

-1

cp specific heat capacity, J kg-1

K-1

Cw relation of surface roughness

d0 bubble diameter at departure, m

F(p*) constant


n exponent

Nu Nusselt number

p* reduced pressureq heat flux, W m-2

Ra Roughness measure (assume 0.4m in case no value is available), m

Rp Roughness measureas defined in DIN 4762/1 (German standard)T Temperature

heat transfer coefficient, W m-2

K-1

contact anglehv latent heat of vaporization, J kg

-1


s-1


33/33

thermal conductivity, W m-1

K-1

density, kg m-3


Subscripts

property of the boiling liquid

property of the saturated vapor0 reference

s surface

bertsch & garimella 2008 review & comparative analysis of studies on sat fb in small channels

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