Vapor pressure distribution of a flat plate heat pipe

Pergamon

Int. Comr~ HeatMass Transfer, Vol. 23, No. 6, pp. 789-797, 1996 Copyright © 1996 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/96 $12.00 + .00

PII S0735-1933(96)00062-0

V A P O R P R E S S U R E D I S T R I B U T I O N OF A F L A T P L A T E H E A T PIPE

K.C. Leong and C.Y. Liu School of Mechanical and Production Engineering, Nanyang Technological University Nanyang Avenue, Singapore 639798, Republic of Singapore K.H. Sun SGS-Thomson Microelectronics (Pte) Ltd 28 Ang Mo Kio Industrial Park 2, Singapore 569508, Republic of Singapore

(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT The vapor pressure distributions of an asymmetric vapor flow of a flat plate heat pipe were calculated using the concept of suction and injection principle. Both symmetrical and asymmetrical flow conditions were considered. Comparisons were made with existing results. The agreement was reasonably good. Introduction An accurate pressure distribution in both the vapor core and wick regions of a heat pipe is essential to the calculation of its capillary limit. The analysis of the vapor flow in heat pipe is a very complicated problem since the flow can be either compressible or incompressible, and heat and mass transfer are being involved as well. Many researchers have proposed methods to calculate the pressure distribution of the vapor flow. Cotter [1] used the results of Yuan and Finkelstein [2] and Knight and Mclnteer [3] for radial Reynolds numbers less than and greater than unity, and applied to a circular heat pipe. In addition, Busse [4] solved the pressure field for laminar flow using a modified Poiseuille velocity profile. Moreover, a Poiseuille approximation was used by van Ooijen and Hoogendoorn [5] to calculate the pressure distribution of an asymmetric fiat plate heat pipe with an adiabatic top wall. They also performed an experimental study on it. Bycomparing the analytical results with their experimental results, the agreement was found to be reasonable. However, since a Poiseuille profile is symmetric, it is not able to account 789

790

K.C. Leong, C.Y. Liu and K.H. Sun

Vol. 23, No. 6

fully for the pressure drop inside the asymmetric flow field. In fact, the actual velocity profiles in the evaporator and condenser sections are not symmetric at all. Many numerical techniques have also been reported to obtain the pressure distribution inside a heat pipe, namely, Ooijen and Hoogendoorn [6], Tien and Rohani [7] and Narayana [8]. A brief review of the various techniques was compiled and presented by Chen and Faghri [9]. They have also solved the two-dimensional elliptic governing equations in conjunction with the thermodynamic equilibrium relation. The solutions found were in good agreement with the experimental results. Most of the results reported are on symmetrical vapor flow condition. There are not many works available for the case of a fiat plate heat pipe with one surface completely adiabatic. This type of heat pipe was used by van Ooijen and Hoogendoorn [5,6]. In addition, Marn et al. [10] measured the temperature and concentration of non-condensable gas in a rectangular heat pipe with the wick structure mounted on one surface only. It is the objective of this note to present a method of determining the vapor pressure distribution along an asymmetric fiat plate heat pipe using the model proposed by Berman [11]. Both symmetrical and asymmetrical conditions were considered. Comparisons were also made with the experimental, theoretical and numerical results of van Ooijen and Hoogendoorn [5,6]. Generally, a good agreement was found. Mathematical Formulation The continuous evaporation and condensation of the vapor flow in a flat plate heat pipe can be imagined as constant injection and suction of channel flow. It is assumed that the vapor flow in the channel is steady, two-dimensional and incompressible, with constant fluid properties. The momentum equations are then written as

cTu 8x

u--

6v

+

v 6u hS,~

-

vOv

U~x + h 6 2

-

10P ( d2u l c72u~ -p,Ox +v~Yxcx~+h2 822) 1 6P [c92v l_l_d2v~ ph62 +v~-x2 +h2622)

(1)

(2)

where 2 is the dimensionless y coordinate and is equal to ~ , u(x,2) and v(x,2) are velocity components along the x and y directions of the vapor channel, respectively, h is the half-width of the channel as shown in Figure 1, P is the static pressure, while v and p are the kinematic viscosity and density of the vapor, respectively.

Vol. 23, No. 6

VAPOR PRESSURE OF A HEAT PIPE

791

-V•

,0,.h

'--I------T V,

(-

Vc

..... 0 ,Jmy

I

rn

-1 .

m

.

.

.

.

.

.

.

.

.

.

.

-V,

I

L•

.

etxi¢) La

I'

.

.

(=

.

.

0 ,amy

.

.

m

m

.

.

• trie) I

L,

FIG. 1 Comparison of asymmetrical and symmetrical velocity distributions of a flat plate heat pipe If the heat flux is uniform and one surface (the lower one) is completely adiabatic, the boundary conditions are u(x,_i)

=

0

(3a)

v(x,-1)

=

0

(3b)

- Ve

v (x,1)

f o r the evaporator

t

=

0

f o r the adiabatic section

Vc

for the condenser

(3c)

where Ve a n d Vc are the injection and suction velocities in the evaporator and condenser sections, respectively. These velocities can be calculated using the heat power input Q of the heat pipe, i.e., Q

=

(4)

pAVhf~

where P is the density of the vapor, A is the surface area of the evaporator or condenser section exposed to the vapor space, V is either the injection or suction velocity depending on the position, and hfg is the latent heat of vaporization. Using the same approach as postulated by Berman [11], the stream function is written as a product of two functions, i.e., ~F(x,2)

=

g(x).f(2)

(5)

Thus, the velocities u and v are found to be equal to Oud u

-

Oy

v

=

-0~

dud

1 -

-hgf -

g'f

(6a) (6b)

792

K.C. Leong, C.Y. Liu and K.H. Sun

Vol. 23, No. 6

As there are mass addition and removal in the evaporator and condenser sections, the function g(x) would be different in these sections. Based on the given boundary conditions, it was found that f Vox

g(x)

=

0_
VoL,

L,<_x<_L, +Lo

2h o-

L,

(7)

where g, is the mean axial velocity in the adiabatic section, and L,, L~ and L~ are the lengths of the evaporator, adiabatic and condenser sections, respectively. The boundary conditions for the functions f ( 2 ) and f ' ( 2 ) are then written as f'(-1)

=

0

(8a)

f'(1)

= 0

(8b)

f(-1)

= 0

(8c)

f0)

:

(8d)

1

By substituting equation (5) into equations (1) and (2), and differentiating the resulting equation with respect to ~, we obtain

R [ f '2- ff'']+ f'''

= k(R)

(9)

Here k(R) is the constant of integration and it is a function of R only. R is known as the "wall Reynolds number" and found to be equal to

hv(x,1) v

I f R is known, the solution of

equation (9) can give the velocity profiles of u and v in the whole vapor space of the heat pipe. This was solved by first differentiating it with respect to L, followed by using Newton's method [12] together with a fourth order Runge-Kutta method. Once u and v are known, the vapor

[

pressure distribution in the three sections of an asymmetric flat plate heat pipe can be found. They are

i

P(x,2)=

5 k(n) 2

P(L~,2)+~-Z:2~(x-L.)k(R) +I t

L, <_x<_L~+L~ i 2

l 2

where P(O,)~) is a reference pressure at the beginning (x=O) of the evaporator section.

(lO)

Vol. 23, No. 6

VAPOR PRESSURE OF A HEAT PIPE

793

If heat is being applied and removed from both the upper and lower surfaces, the vapor flow is now symmetrical with respect to its centerline. By considering only half of the vapor space, the boundary conditions for the functions f ( 2 ) and f " ( 2 ) at the centre of the channel are f(O)

=

0

(lla)

f"(0)

:

0

(llb)

As only half of the heat pipe is being considered, the injection and suction velocities would reduce to half of their values of the asymmetrical case for the same power input. The forms of the boundary condition of equations 3(a) and 3(c) are still valid in this case. In addition, the function g(x) remains the same for both the evaporator and adiabatic sections, except in the condenser, where g(x)

=

h~o-Vc[x-(L-Lc)

(12)

]

Following the same procedure as before, the vapor pressure distribution in the three sections of a symmetric flat plate heat pipe is found to be equal to u

P(x, 2 ) =

x:

P(0,,~)+ ~vo 5-k(n)

O<_x<_L~

P(L,, ~. ) + ,u ~ (x - L , ) k ( R ) h 2 ,,

L e <-x<-Le+L a

(13)

L e + L <_x<_L

Given the parameters of a flat plate heat pipe, the velocity and pressure distributions of the vapor can be calculated from the above equations. Results and Discussions van Ooijen and Hoogendoom [5,6] used a symmetric Poiseuille velocity profile and derived an equation to calculate the vapor pressure distribution of an asymmetric flat plate heat pipe with an adiabatic top surface. However, the real velocity profile is not symmetrical with respect to the centre of the heat pipe at all. Figure 1 gives a comparison of the velocity profiles for both symmetrical and asymmetrical cases calculated by the present method. The calculation was based on a power input of 295 W which was the same as that presented by van Ooijen and Hoogendoorn [5,6]. This figure also shows the comparison of the velocity profiles as the vapor flows from the evaporator section to the condenser section of the symmetric and asymmetric heat pipe.

794

K.C. Leong, C.Y. Liu and K.H. Sun

Vol. 23, No. 6

Figure 2 compares the results obtained using the present method and the experimental data measured by van Ooijen and Hoogendoorn [5,6]. Generally, the agreement was found to be reasonable. It can be seen quite clearly from this figure that the results of using the asymmetrical case fits better to the experimental data compared to that of the symmetrical case. Figure 3 shows the comparison between the present results (asymmetrical case) and the numerical results of van Ooijen and Hoogendoorn [5,6]. It is fairly obvious that the asymmetrical case can give a better prediction for the vapor pressure drop over that made by the twodimensional PoiseuiUe approximation, especially in the condenser section. This is because, for the same value of wall Reynolds number, suction produces a more profound effect on the form of velocity distribution compared to its injection counterpart.

0

--

Analytical results (Asymmetrical ease)

--

Analytical results (Symmetrical case)

2 :

%,

4

165 W /

i°

'~..~

- - L e - -

6

"

o

8

L

"x

2,5 w

10

0.1

0.2

i

•

*

•

*

5o . . . . " ;; . i I

I

I

0

"\

/.'%,!

/ 295 W

12

iO O • Experimentalresults i of van Ooijen and ~, Hoogcndoom [51

/

0.3

Heat Pipe Length

0.4

•

•

I

"

"

0.5

(m)

FIG. 2 Comparison of analytical results with the experimental values

Vol. 23, No. 6

VAPOR PRESSURE OF A HEAT PIPE

0

795

Analytical results (Asymmetrical case)

0.2

I i

~

165 W

" 2-D Poiseuill¢ Appro~drnaUon • o • Numerical data of

0.4

0.8 1 1.2

0

0.1

0.2

0.3

0.4

0.5

Heat Pipe Length (m)

FIG. 3 Comparison of analytical results with the numerical values Concluding Remarks The pressure distribution calculated by using an asymmetrical velocity profile can give a good pressure variation of a flat plate heat pipe with a completely adiabatic surface. The results can also be used to calculate the velocity and pressure distribution of a flat plate heat pipe with a asymmetrical heat source. Nomenclature A

surface area [m2]

B

width of heat pipe [m]

f

function of X

g

function ofx

h

half-width of the vapor region [m]

hfg

latent heat of vaporization [J/kg]

k(R)

constant of integration

796

K.C. Leong, C.Y. Liu and K.H. Sun

Za

adiabatic length [m]

Lc

condenser length [m]

L,

evaporator length [m]

P

pressure [q'q/m2]

Q

heat power input [W]

R

wall Reynolds number

11

velocity in the x direction [m/s]

~a

mean axial velocity in the adiabatic section [m/s]

V

velocity in the y direction [m/s]

vo vc

injection velocity [m/s]

I

coordinate in the axial direction of heat pipe

Y

coordinate in the transverse direction of heat pipe

Vol. 23, No. 6

suction velocity [m/s]

Greek symbols A

dimensionless y coordinate

/.t

dynamic viscosity [Ns/m 2]

V

kinematic viscosity [m2/s]

9

density [ kg / m 3]

~F

stream function References

1.

T.P. Cotter, Theory_of heat pipes, Los Alamos Scientific Lab. Rept., LA-3246-MS (1965).

2. S.W. Yuan and A.B. Finkelstein, Trans. ASME 78, 719 (1956). 3. B.W. Knight and B.B. Mclnteer, Laminar incompressible flow in channels with porous walls, LADE-5309 (1953). 4. C.A. Busse, Pressure drop in the vapor phase of long heat pipes, Thermionic Conversion Specialist Conference, 391 (1967). 5. H. van Ooijen and C.J. Hoogendoorn, Experimental pressure profiles along the vapour channel of a flat-plate heat pipe, Advances in Heat Pipe Technology, D.A. Reay (ed.), 415 (1982). 6. H. van Ooijen and C.J. Hoogendoorn, AIAA Journal 17, 1251 (1979).

Vol. 23, No. 6

VAPOR PRESSURE OF A HEAT PIPE

797

7. C.L. Tien and A.R. Rohani, Int. J. Heat Mass Transfer 17, 61 (1974) 8. K.B. Narayana, Numerical Heat Transfer 10, 79 (1986). 9. M.M. Chert and A Faghri, Int. J. Heat Mass Transfer 33, 1945 (1990). 10. J. Marn, F. Issacci and I. Catton, Measurements of temperature and concentration fields in rectangular heat pipe, Heat Transfer in Space Systems HTD Vol. 135, 33 (1990). 11. A.S. Berman, J. of Applied Physics 24 (9), 1232 (1953). 12. T.Y. Na, Computational Methods in Engineering Boundary_ Value Problems, p. 70 Academic Press, London (1979).

Received January 30, 1996