J. of Thermal Science Vol.2, No.4

Journal of Thermal Science Science Press 1993

A n I m p r o v e m e n t on t h e M e t h o d for C a l c u l a t i n g t h e Capillary Limit of A x i a l - G r o o v e d H e a t P i p e s Chen Huanzhuo

M a Tongze

Institute of Engineering Thermophysics, Chinese Academy of Sciences, P.O. Box 2706, Beijing 100080, China

A new model has been developed to predict the capillary limit of axial-grooved heat pipe. In the model the concepts of liquid saturation or liquid fraction of the cross-sectional area of groove,the modified relative permeability, absolute permeability of groove and Leverrt's function are used. The Leverrt's function is well represented by the function f ( s ) = ( 1 / x / 5 ) ( 1 / s - 1) °'1v5. In the model the effects of gravitational force, capillary force and viscous force are considered. The calculated results are in good agreement with existing experimental data reported in the literature. K e y w o r d s : l i q u i d s a t u r a t i o n , r e l a t i v e permeability, capillary limit .

INTRODUCTION The axial-grooved heat pipe has the potential of relatively high heat transfer rates and its inherent simplicity. Since a long time ago, the research of axialgrooved heat pipe has attracted much attention and has made substantial progress. The capillary limit of axial-grooved heat pipe is an important parameter for heat pipe design. When a heat pipe is in a normal operating condition, the amount of working liquid in the groove is varied along the liquid flow direction due to evaporation and condensation. As dryout occurs the amount of working liquid approaches zero at the beginning of evaporator, and, in the mean time, the groove is full of working liquid at the end of condenser. In practice the permeability and curvature radius will be variant along the flow direction. In some of previous work the permeability and curvature radius between liquid and vapor were considered constant. In this case the calculated capillary limit often deviated from the experimental results. In Ref.[1], the permeability for flow in groove was considered as variant. But the permeability has to be predetermined as average value prior to calculation. Recently, in the study of heat pipe with sintered wick, some concepts and methods for two phase flow in Received November, 1993

porous media were utilized to determine the capillary limit[2]. In Ref.[3], the similar method has been modified and used to predict the maximum performance of micro heat pipe. In present work the m e t h o d has been developed to predict the capillary limit of common axial-grooved heat pipe.

ANALYTICAL

MODELLING

For theoretical modelling the following assumptions are made: (a) local thermal equilibrium is present between phases. (b) the thermophysical fluid properties are constant. (c) the evaporator and condenser are uniformly heated and cooled. (d) operation is steady (e) the flow is dominated by vicious force and can be described by Darcy's law. (f) cross -sectional area of vapor core is much large compared with cross-sectional area of grooves. (g) when dryout occurs the working liquid is full of the groove at the condenser end. The mass flux of liquid can be related to the pressure gradient through modified Darcy's equation dP~ dx -

Vlml AKKI

gpl

sin 0

(1)

From Ref. [4] ,the vapor pressure drop in the heat pipe is described by

Chen Huanzhuo et al

A dh g h.f9 hv K L, L¢, L¢ Leff P q

An Improvement on the Method for Calculating the Capillary Limit

Nomenclature cross sectional area of groove hydraulic radius gravitational acceleration latent heat of vaporization tilt permeability length of heat pipe, evaporator and condenser, respectively effective length of heat pipe pressure heat flow, heat input minimum radius of curvature

dPv

S # v p 0 a l v c r

8Vvmv

dx- -

r%r

in which the gravitational effect is negligible. energy equation can be written as

q = mvhlg

(2)

The

249

liquid saturation dynamic viscosity kinematic viscosity density wetting angle inclination angle surface tension Subscripts liquid vapor capillary relative

sectional area of working liquid will be decreased. This means the flow permeability will be decreased. T h e concept of the relative permeability will be utilized to describe this situation. The definition of liquid saturation is

(3)

At any location the mass conservation requires t h a t the net mass flux be zero for steady-state conditions. m, ----my

(4)

The curvature of the liquid-vapor interface induces a pressure difference

st=

cross-sectional area occupied by liquid total cross-sectional area of groove

The relative permeability K~ is related only to the liquid saturation and m a y be taken to have various values in the calculation of different cases, as shown in Table 1 T a b l e 1 Relative permeability

Pc=P.-Pz Differentiating with respect to x

dPc dx

dPv dx

dPt dx

(5)

Subtraction of Eq.(2) from Eq.(1) and incorporation of Eq.(5) and Eq.(4) result in the capillary pressure gradient

dP~ • 8v~ vt -~x - rn(~-~ + K l ~ r A ) +gptsinO

(6)

On the right hand of Eq.(6), the first t e r m represents the vapor pressure drop, the second t e r m is liquid pressure drop, the third t e r m is gravitational effect. In which Kt, K~ are the absolute and relative permeability respectively. For two phase flow in porous media, liquid and vapor relative permeabilities account for the decrease in the mobility of one phase due to the presence of the other. In grooved heat pipes, the amount of liquid in the groove will be decreased due to evaporation, so t h a t the flow cross-

Unction

Kr

Linear Verma etal [5] Cubic [8]

S~ S~3 Si.3

Corey [7]

S~4

It should be noted t h a t in two phase flow in porous media, St* is the reduced liquid saturation, defined in terms of irreducible liquid and gas saturation in various ways for the different relative permeability function. For the flow in the groove, all irreducible saturation will be zero, t h a t is, no liquid is ever immobile under two phase conditions. So S~ ----St. The relative permeability is correlated as a linear function of liquid saturation K r -- S. T h e liquid saturation S can vary from one to zero. W h e n S is one, or say, K r is also one, it means t h a t the cross section is fully occupied by liquid and permeability is for the single phase flow. Conversely, when S and K r are approaching zero, most of the cross section of groove is occupied by vapor, and, in the mean time, the liquid permeability is very low.

250

Journal of Thermal Science, Vol. 2, No.4, 1993 As shown in Fig.1.

T h e capillary pressure Pc depends on the g e o m e t r y of t h e groove, local liquid s a t u r a t i o n and t h e surface properties. It is a s s u m e d t h a t the capillary pressure follows the e q u a t i o n Pc = C 2 [ sin

~f(S)

1.0

(r)

l'~mi n

0.5

W h e r e a is the fluid surface tension, ~ t h e wetting angle, R,~in the m i n i m u m curvature radius of the groove. D e p e n d i n g on the groove shape, Rmin can have different values [4]. For example, Rmin = w for rectangular a n d trapezoidal grooves.(w is the o p e n w i d t h of the groove.) T h e coefficient C is taken as ( L / R m i n ) "245. f ( S ) is Leverrt's function. T h e Leverrt's function f ( S ) depends only on S a n d it has various formulations, as shown in Table 2. T a b l e 2. Various formulations of f ( S ) Writer

f(S) 1

Lip inski~<

VT~(

1 _ 1).1r5

Udell [6] 1.417(1 - S) - 2.12(1 - S) 2 + 1.263(1 - S) 3 1(4 - 1) 1 Ot Sm

Plumb 191

0.0 0.0

1

1 _ 1).175

f ( S ) - - v'5--~-(S

0.4

0.6

0.8

S

T h e flows in p o r o u s m e d i a and in heat pipe grooves are b o t h d o m i n a t e d by viscous force. T h e y can be described by D a r c y ' s law. For c o m p a r i s o n b e t w e e n modelling for flow in p o r o u s m e d i a and modelling for flow in heat pipe, the governing equations and some of its terms are listed in Table 3. Considering the assumption(c), the mass flux of working fluid is

f m(x) = I

(8)

qx/(hygLc)

o
q/(hyg)

Lc < x < L - Le

q(L - x ) / ( h y g L ~ )

L-Le
study in porous media [6]

present work

governing equation IAPc > AP, + h p .

dP~,

8mu,~

dx

717"4

dPl = mL~lful dx KA

Pc K

K~ f(S)

23 sin q~ Rrnin

dP~ dx

(9)

In c o r p o r a t i n g equation(8) and (9) into (6), we have

T a b l e 3. Comparison between the governing equations heat pipe study [4]

1.0

Fig.1 The Leverrt's function vs liquid saturation

Note: a = 3.25, n = 24, m = 1 - 1/n In this work application of Lipinski's formulation t o the grooved heat pipe s t u d y seems to be justified by the agreement between experiment and theory. So

I

0.2

K

(

+

) + (pl + P,,)g

K-.5 °-f(S)

Eq.(6)

C~f(S)

4

d2ea

dh 2

32

150(1 -- e) ~

32

Sa 1.417(1 - S) -- 2.12(1 -- S) 2 + 1.263(1 - S) a [ 1

1 _l).lr s

Chen Huanzhuo et al

q__(Sv.

An Improvement on the Method for Calculating the Capillary Limit

+ ~ )~'i

A non-dry.out (100~ & 110~) A dryout (100~. & 110~) . .. O non-dryout (,charge 74~) A A'~ryout (charge 74~)

•

+Plg sine

dSd___= x h/g "rr 42a f ' ( S" 2) (4 5 zL~ ° ~) Rmin Rmin

(10)

5' = .01

10.0E E

in where f'(S) is the derivative of Leverrt's function with respect to S. As shown by the visualization in experiment, in axial-grooved heat pipes, the dryout will begin at the beginning of evaporator. This means that at this point the local liquid saturation approaches zero at the beginning of dryout. For computer calculation, the liquid saturation S is taken to be 0.01 to approximate zero. Thus the following boundary condition holds: x = O,

251

"\

O•

&A

~ 5.0 --0.0 0,0

pre t Kosowski's

,

i

J

i

"\

i

,

,

"~

~ ,

I

100,0

- . ~

~

200.0

,

I

i

~.,

300,0

Q _ (w) Fig.2 Comparison between present calculation and experimental results as well as Kosowski's calculation

(11) 3.0

Equation(10) and boundary condition(ll) can be calculated by the fourth order Runge-Kutta scheme. In calculation, zero wetting angle is assumed. The integration of Eq.(10) is started by guessing q, then the calculation proceeds step by step to the condenser end. If the filling charge in heat pipe is enough, we can assume that the working liquid is full of the grooves at the condenser end, which can be described as S = 1. When the calculated saturation S at the condenser end approaches one, the maximum heat flux is determined.

\

\\ \\\\\

,-, 2.0 E E v 1,0 0 A ---0.0 02

i

i

non-dryout xx X dryout x \ Kosowski't theory x \ present theory ~" X ,

I

5.0

i

I

I

i

I

i

I 0.0

i

i

I

I

i

15.0

J

i

J

I

~XJ

20,0

I

I

I (~x~

25.0

.~.

'

I

50,0

Q_ (w) MODEL SION

VERIFICATION

AND

DISCUS-

Though there have been many papers which are related to experimental research for maximum performance, results available for comparing with the theoretical model are very few, because the experimental parameters provided are not adequate. Now, some experimental results are collected for comparison. 1. C a s e 1 [1°1 The total length of heat pipe L = 889 mm. The rectangular groove is of width w = 0.889 mm and depth y = .762 mm. In the heat pipe the number of grooves is 30. The vapor channel diameter is 10 mm. The operating temperature is 21°C. Fig.2 shows the results for working liquid ammonia. Fig.3 is for Freon-ll3, and Fig.4 for Freon-21. In these figures the experimental and calculated maximum performances of heat pipe against different tilts are presented.

Fig.3 Comparison between present calculation and experimental results as well as Kosowski's calculation

In Fig.2 we can find that Kosowski's calculations significantly underpredict the maximum performance for experiment with filling charge 100% or 110%. The calculated results by the present theory agree well with experimental data for filling charge 100% or 110%. It should be noted that the calculation overpredicts the maximum performance for experiment with filling charge 74%. When filling charge in the test heat pipe is not enough, the local liquid saturation at the condenser end may not reach one. This will be a gross violation of assumption(g) used in the calculation In Fig.4 a similar situation occurs. The results calculated by the present method are in agreement with experimental results for filling charge 108%. But there is considerable discrepancy between present theoretical and experimental results for filling charge 89%.

252

Journal of Thermal Science, Vol. 2, No.4, 1993 non-dry.out (charge 108~) 0 dryout (chorge 108~)

4.0

\\

0 °C. T h e calculated results by the present theory is in good agreement with experimental data for filling charge 104% and 119%, as shown in Fig.6. Similar to case 1, the present theoretical results deviate from experimental data at fillingcharge 90%.

O~

\N\\\

E E "-1 2.0

- -

15.0

\\\

Kosowski's

\

,oo

~.

present, 13 (charge)

"~

"\

L

I

I

t

0.0

I

l

I

I

20.0

I

I

'C

J,.~'J

40.0

,

~)'

LI

60.0

'

-.:%

E E

" x 0.0

D

k'

80.0

Q ~ (w) 5.0

Fig.4 Comparison between present calculation and experimental results as well as Kosowski's calculation

--0 D

0.0

I

*

0.0

2. C a s e 2 []1] The total length of heat pipe L -- 1000 mm. The trapezoidal groove has open width 0.6 mm, b o t t o m width 0.9 m m and depth 1.2 m m . The number of grooves is 19. T h e heat pipe has vapor channel diameter 6.2 mm. The working liquid is a m m o n i a and operating t e m p e r a t u r e is 30 °C. The comparison between the present m e t h o d and experiment as well a Brost's calculations are shown in Fig.5. T h e calculation results by present m e t h o d are in reasonable agreement with experimental results. 10.0

-.

0 \

o\

\

5.0 Brost's present's experiment

-0 0.0

,

0.0

i

,

,

I

50.0

J

i

I

100.0

,

i

,

,

Xl~"

i

i

I

50.0

i

1

I

J

I

i

l

100.0

~

t

I

i

150,0

I

J

I k I Xl

200.0

Q~, (w) F i g . 6 Comparison between the present calculation

results and experimental results as well as Schlitt's calculation

SUMMARY

AND CONCLUSIONS

(1) T h e concept of two phase flow in porous media with modified relative permeabilities and the Leverrt's function were used to calculate the m a x i m u m performance of axial-grooved heat pipe. T h e predicted resuits agree well with available experimental d a t a in literature. (2) The calculation by the present theory is susceptible to filling charge. W h e n filling charge is 100% or a little higher, the calculated results are in good agreement with experimental results.

Acknowledgment This work is supported by the National Science Foundation of China.

\ \ " ~, 0 ~,, ~

present's \ -. Schut/'s \ \ experiment (charge 90~), "~ experiment (charge 119~) % x '-

~)

150.0

Q ~ (w) Fig.5 Comparison between present calculation results and experimental results as well as Brost's calculation

3. C a s e 3 [121 The heat pipe total length is 914 m m . T h e rectangular groove has open width 0.61 ram, depth 1.02 ram. There are 27 grooves in the heat pipe. The working liquid is a m m o n i a and operating t e m p e r a t u r e is

REFERENCES

[1] Hou Zengqi et al, "Performance Investigation and Application of Grooved Heat Pipes," AIAA 14th Thermophysics Conference, June 4-6, (1979). [2] F.Ruel, "Heat Transfer Limitations of Porous Sintered Wicks with Arteries," ASME Pro. National Heat Transfer Conference, 1, (1988). [3] Huanzhuo Chert, M.Groll and S.Rosler, "Micro Heat Pipes: Experimental investigation and Theoretical modelling," Pro. the 8th Heat Pipes Conference, Beijing, (1992).

Chen Huanzhuo et al

An Improvement on the Method for Calculating the Capillary Limit

253

[9] O.A. Plumb, "Heat Transfer during Unsaturated Flow [4] Tongze Ma et al, <>, Science Press, (1983), in Porous Media," NATO Advanced Study Institute on (in Chinese) Convective Heat & Mass Transfer in Porous Media [5] A.K.Verma et al, "A Study of Two Phase Concurrent Flow of Steam and Water in an Unconsolidated Porous [10] N. Kosowski, "Experimental Performance of Groove Heat Pipes at Moderate Temperatures," AIAA 6th Medium," Pro. 23rd. ASME/AIchE National Heat Thermophysics Conference, AIAA paper, No.71-409, Transfer Conf. HTD., 46~ (1985). (1971). [6] K.S.Udell, "Heat Transfer in Porous Media Heated from above with Evaporation, Condensation and Capillary [11] O.Brost, M.Groli and W.D. Munzel, "Technical Applications of Heat Pipes," Proc. of 3rd Int. Heat Pipe effect," J. Heat Transfer, 105, (1983). Conference, (1978). [7] A.T.Corey, "The Interrelation between Gas and Oil Relative Permeabilities," Producers Monthly, Nov.38-41, [12] K.R.Schlitt and J.P.Kirkpatrick, "Parametric Performance of Extruded Axial Grooved Heat Pipes from (1954). 100 to 300°K, " AIAA/ASME Thermophysics and Heat [8] R.J.Linpinski, "A Coolability Model for Post accident Transfer Conference, Boston, (1974). Nuclear Reactor Debris", Nuclear Technology, 65~ April (1984).