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Liquid flow in the anisotropic wick structure of a flat plate heat pipe under block-heating condition
-
Aoolied Thermal ..
Pergamon PII: S13594311(96)00044-0
Vol. 17. No. 4. DD. 339-349. 1997 Co&right 0.1997 Eidvier Science Ltd Printed in Great Britain. All rights reserved 1359-4311/97 $17.00 + 0.00
Ennineerinn
LIQUID FLOW IN THE ANISOTROPIC WICK STRUCTURE OF A FLAT PLATE HEAT PIPE UNDER BLOCK-HEATING CONDITION W. Qin and C. Y. Liu* Thermal
& Fluids Engineering Division, School of Mechanical and Production Engineering, Technological University, Nanyang Avenue, Singapore 639798, Singapore (Received
13 August
Nanyang
1996)
Abstract-An analytical solution of liquid flow in the anisotropic permeability wick of a flat plate heat pipe is presented. Using the method of Fourier expansion, the pressure distribution and velocity field of liquid phase are calculated. It was found that the pressure and the velocity distributions depended strongly on the anisotropic property of the wick. The effects of anisotropic property and the heater location on the maximum pressure difference are also discussed. Copyright 0 1997 Elsevier Science Ltd Keywords-Anisotropic
wick, flat plate,
heat pipe, block heater,
Fourier
series.
NOMENCLATURE
zl b z&l OXEYE K P P rcl P 4 W U,V X,Y x,y
length of the flat plate heat pipe [m] Fourier coefficients of p width of the flat plate heat pipe [m] length of the heater [m] y-coordinates of the heater [m] width of the heater [m] distribution function of the condensation rate Fourier coefficients of f(x,y) wick permeability [m’] pressure [Pa] reference pressure [Pa] non-dimensional pressure x-coordinates of the heater [m] velocity components in the x- and y-directions [m s- ‘1 non-dimensional velocity components in the x- and y-directions axial and transverse coordinates non-dimensional axial and transverse coordinates
Greek lerters tl aa+ B E 9 p P
general condensation rate in the wick [kg s- ’m-‘1 evaporation rate [kg s- ’m-‘1 condensation rate [kg s- ’m-‘] = pa+/pK., [kg s-* m-‘1 ratio of the permeability in the y-direction to the permeability ratio of the condensation area to the heating area dynamic viscosity of liquid [Ns m-‘1 surface density of the liquid [kg m- ‘1
in the x-direction
INTRODUCTION An accurate determination of the pressure distribution for both vapour core and wick region of a heat pipe is essential for the calculation of the capillary limit [l]. The calculation of the liquid flow field of a flat plate heat pipe under normal operation conditions, i.e. when the heater and the condenser cover the whole evaporator section, is quite straightforward. However, when the heater partially covers the evaporator section or when multiple heat sources are applied to the surface of the heat pipe, the flow of liquid in the wick structure is no longer one-dimensional. It may be *Author
to whom
correspondence
should
be addressed. 339
340
W. Qin and C. Y. Liu
two-dimensional or even three-dimensional. This makes it more difficult to obtain an analytical solution of the liquid flow. In practical applications, heaters such as electronic components applied on the surface of the heat pipe may be discrete. It is of practical use to have some methods to calculate the pressure distribution of the working fluid in heat pipes under block-heating condition. Gernert [2] analyzed the flow fields of heat pipes with multiple heat sources with the use of superposition and the extension of the existing theories for a single evaporator heat pipe. The experimental study on a cylindrical pipe with a narrow block-heated evaporator was reported by Rosenfeld [3]. He presented an approximate method to calculate the steady-state wall temperature profile in the azimuthal direction. Faghri and Buchko [4] made experimental and numerical analysis of low-temperature heat pipes with multiple heat sources. The tests show that the maximum total heat load on the heat pipe varied greatly with the location of the local heat fluxes. Schmalhofer and Faghri [5] studied both the circumferentially heated and block-heated circular heat pipes analytically and experimentally. An approximate method to calculate the effective length was also proposed. Following Schmalhofer and Faghri’s method, Sun et al. [6] presented an approximate method to estimate the effective length of a flat plate heat pipe with localized distribution of the heat source. More recently, Huang and Liu [7] reported an analytical method to calculate the velocity and pressure distributions of the liquid phase in a flat plate heat pipe under localized heating conditions, in which the heater was fixed at the end of the flat plate heat pipe. They assumed that the working fluid evaporated uniformly over the heat input zone and condensed uniformly over the condensation zone and developed a two-dimensional analytical method to calculate the pressure distribution and the velocity field of liquid phase with a localized heater. All the studies cited above are based on isotropic wick structure. However, several types of wick structures commonly used for heat pipes are of anisotropic permeability, such as a groove wick covered with meshes. If the liquid flow in the wick is one-dimensional or if the heating zone completely covers the evaporator section of the heat pipe, there is no effect of the anisotropic property on the liquid flow, since there is no transverse velocity component or the transverse velocity component serves as a secondary role to distribute the working fluid uniformly in all locations. However, when the heater partially covers the evaporator section, the flow of liquid becomes two-dimensional. There exists a transverse velocity component. The velocity and pressure fields will be quite different from that of an isotropic wick. The objective of this paper is to present some results of the velocity and pressure fields in an anisotropic wick of a flat plate heat pipe under partially block-heating condition. The heater in this study is arbitrarily placed on the surface of the flat plate heat pipe. Both the velocity and pressure fields were solved using Fourier series method for various permeability ratios and different heater locations. The maximum pressure differences in the whole liquid phase region were also calculated. The results indicate that the anisotropic property of the wick structure can yield a significant effect on the capillary limit. THE
GOVERNING
EQUATIONS
AND
SOLUTIONS
The basic model of the present study is shown in Fig. 1. The flat plate heat pipe has a dimension of a x b and the heat input zone occupies an area of R(q I x I q + c,d 5 y I h), as shown by the shaded patch in Fig. 1. Outside the heating zone is the condenser of the heat pipe. The working fluid in the flat plate heat pipe is assumed to be evaporated uniformly over the heat input zone at an evaporation rate GI- (kg s - ’m -*), while over the rest of the surface the fluid is uniformly condensed into the wick at rate o!+ (kg s - ’m -‘). Referring to the configuration shown in Fig. 1, a non-dimensional area ratio of condensation-evaporation q is introduced:
’
ab S =-_=---R ce
”
(1)
where S and R are the areas of the condensation zone and the evaporation zone, respectively. Subject to the mass conservation of the fluid, the area ratio r~ is actually the ratio of evaporation rate 01- to condensation rate LX +:
u- =
a+r].
(2)
341
Liquid flow in the wick structure of a heat pipe
1 heater
h
0
a
9
x
Fig. 1. An illustration of the flat plate heat pipe with a block heater.
A distribution
function of the heater is defined as (3)
It is assumed that the fluid evaporated from the heat input zone may be considered to be condensed into this zone at a negative condensation rate - a + . Thus, the condensation rate over the whole heat pipe may be written as a = a+f(x,y)
.
(4)
The flow of liquid in the wick structure is assumed to follow Darcy’s law. The governing equations for the liquid flow in the wick structure can be written as
(6)
where U, u are the liquid flow velocity components in the x and y directions respectively, p is the pressure, K, and K, are the permeability of the wick structure in the x- and y-directions, respectively, E is the permeability ratio, p is the viscosity and p is the density of the liquid and a is the condensation rate defined in equation (4). The boundary conditions are
x=ou=~d) ax 3
x=a,u=
ap=0
z
342
W. Qin and C. Y. Liu
2
y=b,v=
co.
ay
Substituting equation (5) into the continuity equation (6), yields an equation for the pressure p: (8)
where j3 is defined as
5. ‘i
p=
(9)
Following the technique presented by Huang and Liu [7], both the pressure distribution and the distribution functionf(x,y) can be expanded in the form of a Fourier cosine series as shown below:
P = pref +
flab f A dcos(~x)+
f
fA,cos($y)+
m=I
fA,cos(~x)cos(~y)
m=,n=,
“=I
(10) f(x,y)=
+’+ ; f
Fmocos(~x)+
;
~F,.cos($y)+ n=,
m=l
F,,,,cos ( 7
2 f m=,
n=l
x)cos ( 5 y) .
(11)
Substituting equations (10) and (11) into equation (8), comparing the terms on both sides of equation (8) and transforming all variables into non-dimensional forms by p = P - Prcr
Bab
the non-dimensional P =
pressure distribution and the coefficient A,, can then be obtained as follows:
f A,cos(mnX)
+ i A,cos(nnY) n=,
m=l
A,a,
=
_
2a(h- d, (1 b*(mn)’
Ao,,=
-
_26c_(l &a (nn)
2 2 A,nncos(m~X)cos(~~Y)
+
n==ln=l
+q){
+q)
sin[mn(qy’c)]-
sin(F)-
sin(y)}
sin(F)
1
(12)
Liquid flow in the wick structure
Amn =x
of a heat pipe
343
+rj){ (mn)(nn)’ b’(nx)(mn)‘?&a’ (1
[
The non-dimensional to X and Y, U= - g
sin ( F
)-
sin ( F
)
sin[ mn(:+c)]-
sin(y)}
1.
velocity components can be obtained by taking derivatives of P with respect
= 2 A,(mn) m=l
sin (m7rX) + f 2 A,,(mrr) sin (m7rX) cos (n7rY) m=ln=l
V = - (Ez ) Fy = (E% ) F A&r) i n=,
(13)
sin (n7cY) + T f A,,(nrc) cos (m7rX) sin MY) . (14) Ill=,“=, i
Equations (12)-(14) can be used to calculate both the pressure and velocity vectors in the wick structure. The results are presented in the next section. RESULTS
AND DISCUSSION
Results of the velocity vector and pressure distribution for the liquid flow in the anisotropic wick are presented in this section. The result of maximum pressure drop for various permeability ratios is also presented for the understanding of the effect of the anisotropic property on the capillary limits. For all the calculations, the infinite series of equations (12)--(14) are truncated at m = n = 40. A test was conducted and verified that for any further increase in the number of terms, there was a negligible increase in accuracy. The area of the heater, c x e, and the condensation rate c1were kept constant. The pressure fields are illustrated by contours of the non-dimensional pressure difference AP(X, Y) = (P - P,,,)I(P,,, - Pmi,), where Pminand P,,, are the non-dimensional minimum and maximum pressures, respectively, in the whole wick structure. The velocity fields are represented in the velocity vector ‘v = Ui + Vj. Figure 2 shows the constant pressure contours and velocity vectors when the heater is mounted on the top corner of the heat pipe. The results for the permeability ratios, E = 0.5 and E = 1.0, are presented in Fig. 2(a) and (b), respectively, in which the dashed lines represent the heater. As can be seen, when the permeability ratio E is less than 1.0, i.e. the permeability in the y-direction is smaller than that in the x-direction, the non-dimensional pressure gradient in the y-direction becomes larger. This is the reason that the liquid needs to overcome a large pressure loss to flow back from the condenser region to the evaporator. The transverse velocity component becomes smaller when the permeability ratio is less than unity. Figure (3) shows the results for different heater locations. Case 1 is for the heater mounted on the centre left end of the heat pipe, while case 2 is for the heater located at a distance q = 3/l&2 from the left end. Three permeability ratios were considered. Figure 3(a) shows the isotropic case. At a large distance from the left-hand side, the constant pressure curve approaches a straight line. The results for E = 0.5 are shown in Fig. 3(b). The flow rate near the centre is larger than that of the isotropic case. It is more obvious for the case when E = 0.05. The wick structure near the two sides will not give much help for the return of the working fluid from the right-hand side condensation region to the evaporator of the heat pipe. Comparing the constant pressure lines, it can be seen that the non-dimensional pressure gradient in the x-direction decreased when the heater moved from the centre left end of the heat pipe to the central part of the heat pipe. When the heater is placed at the centre left end of the heat pipe, the minimum pressure or the stagnation point is located at the left end of the heat pipe. This yields a maximum length of the path for the return of the working fluid. If the heater moves a distance q = 3/16a from the left end of the heat pipe, which is designated as case 2, there occurs a reverse flow and the minimum pressure or the stagnation point moves to inside of the evaporation zone. As a result, the fluid may need to
W. Qin and C. Y. Liu
344
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Fig. 2. Velocity fields P = Ui + vj and pressure contours of AP(X, Y) = (P - Pm,.)/@‘,,,,,- Pm,,) at different permeability ratios E. (a) E = 0.5, (b) E = 1.0.
overcome smaller pressure loss to flow back to the evaporator from the condenser as compared with that for case 1. The constant pressure contours shown in Figures (2) and (3) give an indication of the relative pressure distribution in the whole heat pipe. In order to see the effect of the anisotropic permeability on capillary pressure required for the returning of the working fluid, the calculated maximum
345
Liquid flow in the wick structure of a heat pipe
* 1.o
0.8
0.6
AP= 0.4
0.2
0.0 0.2
0.0
0.4
0.6
0.8
1.0 X
(i)Case 1
0.6
0.0
A
0.2
0.0
0.4
0.6
0.0
1.0
X
(ii)Case 2 _.
-
.(a) d.0
_
_ __.
Fig. 3. Velocity fields P = Ui + Vj and pressure contours of AP(X,Y) = (P - P,,,,.)/(Pm., - P&. (a) E = 1, (b) E = 0.5, (c) E = 0.05. (i) Case 1: the heater is mounted on the centre left end of the heat pipe. (ii) Case 2: the heater is mounted at a distance 4 = 3/16a from the centre left end.
346
W. Qin and C. Y. Liu
0.4
(i)Case “1”
0.4
. . . . . . . . . . . . . . . . . . .
02
_ . . . . . . . . . . . . 0.0 0.0
02
0.4
0.6
(ii)Case 2 (b) E=os Fig. 3. continued.
0.8
1.0 X
341
Liquid flow in the wick structure of a heat pipe
*1.0
-0
0.0
0.6
-0
.a
-0
!7
-0
.6
LO
8.6
-0
I.7
-0
.I
-0
19
AR
0.4
0.2
. 0.0
.
.
. A
0.2
0.0
0.6
0.4
1.0 X
0.8
(i)Case 1
........................ . . . ... . . . . , . . , ,
, . . . ......
a,,
,
,
,
,
,
.,
. . . . ..,
,
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a8 0.7 0.6 0.5 0.4 0.4 0.5 0.6 a7
a.
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(@Case 2 (c) so.05 Fig. 3. continued
ATE
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..,
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ar
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X
348
W. Qin and C. Y. Liu
2.5
AL2
/! case
4
/
!
1.5 1 0.5 0 0
1 &
0.5
1.5
2
Fig. 4. Maximum pressure differences AP,,. = P,,,, - P,,. vs the permeability ratio E. Case 1: the heater is mounted on the centre left end of the heat pipe. Case 2: the heater is mounted at a distance q = 3/16a from the centre left end. Case 3: the heater is mounted at the centre of the heat pipe [q = l/2(0 - c)].
pressure difference (P,,, - P,,,$“)for different permeability ratios when the heater is located at three distinct positions are depicted in Fig. 4. It is evident that when the permeability ratio is very small, the maximum pressure difference in the wick structure is much larger than that for the isotropic wick. This is because the wick structure near the two sides cannot give much help for the return of the working fluid. The majority of the working fluid is pumped back by the capillary pressure produced by the centre portion of the wick. For the same heat input, the mass flow rate near the centre is large and thus yields a large pressure difference. As for E > 0.5, the maximum pressure difference drops gradually. For any further increase in E beyond unity, there is only a slight decrease in the maximum pressure difference. This shows that the decrease in resistance in the transverse direction still helps the return of the working fluid, but only slightly. When the heater moves away from the centre left end of the heat pipe to some distance inside, the maximum pressure difference decreases since the effective length is smaller than that for case 1. When the heater is located at the centre of the flat plate heat pipe, the maximum pressure difference is minimum. This shows that, at this location, the capillary limit will be maximum. CONCLUSION
The analytical study presented in this paper indicates that the anisotropic permeability of the wick structure can give a significant effect on the velocity and pressure fields of the working fluid for the case of localized heating condition. Decreasing the permeability in the transverse direction will increase the maximum pressure difference in the wick structure. This indicates that there is a decrease in capillary operation limit. However, when the permeability in the transverse direction is larger than that in the longitudinal direction, the effect is negligible for any further increase in permeability in the transverse direction. REFERENCES 1. P. D. Dunn and D. A. Reay, Heat Pipes, 4th Edition. Pergamon Press, Oxford (1994). 2. N. J. Gernert, Analysis and performance evaluation of heat pipes with multiple heat sources.
AIAA/ASME
4th Joint
Thermophysics and Heat Transfer Conf. (1986). 3. J. H. Rosenfeld, Modeling of heat transfer Heat Transfer 83, 71-76 (1987).
in a heat pipe for a localized
heat input zone. Proc. AIChE Symp. Series,
Liquid flow in the wick structure of a heat pipe
349
4. A. Faghri and M. Buchko, Experimental and numerical analysis of low temperature heat pipes with multiple heat sources. ASME J. Heat Transfer 113, 728-734 (1991). 5. J. Schmalhofer and A. Faghri, A study of circumferentially heated and block-heated heat pipes--I. Experimental analysis and generalized analytical prediction of capillary limits. Inc. .I. Heat Mass Transfer 36, 201-212 (1993). 6. K. H. Sun, C. Y. Liu and K. C. Leong, The effective length of a flat plate heat pipe covered partially by a strip heater on the evaporator section. Hear Recovery Systems & CHP 15, 383-388 (1995). 7. X. Y. Huang and C. Y. Liu, The pressure and velocity fields in the wick structure of a localized heated flat plate heat pipe. Inr. J. Heat Muss Transfer 39(6), 1325-1330 (1996).
Aoolied Thermal ..
Pergamon PII: S13594311(96)00044-0
Vol. 17. No. 4. DD. 339-349. 1997 Co&right 0.1997 Eidvier Science Ltd Printed in Great Britain. All rights reserved 1359-4311/97 $17.00 + 0.00
Ennineerinn
LIQUID FLOW IN THE ANISOTROPIC WICK STRUCTURE OF A FLAT PLATE HEAT PIPE UNDER BLOCK-HEATING CONDITION W. Qin and C. Y. Liu* Thermal
& Fluids Engineering Division, School of Mechanical and Production Engineering, Technological University, Nanyang Avenue, Singapore 639798, Singapore (Received
13 August
Nanyang
1996)
Abstract-An analytical solution of liquid flow in the anisotropic permeability wick of a flat plate heat pipe is presented. Using the method of Fourier expansion, the pressure distribution and velocity field of liquid phase are calculated. It was found that the pressure and the velocity distributions depended strongly on the anisotropic property of the wick. The effects of anisotropic property and the heater location on the maximum pressure difference are also discussed. Copyright 0 1997 Elsevier Science Ltd Keywords-Anisotropic
wick, flat plate,
heat pipe, block heater,
Fourier
series.
NOMENCLATURE
zl b z&l OXEYE K P P rcl P 4 W U,V X,Y x,y
length of the flat plate heat pipe [m] Fourier coefficients of p width of the flat plate heat pipe [m] length of the heater [m] y-coordinates of the heater [m] width of the heater [m] distribution function of the condensation rate Fourier coefficients of f(x,y) wick permeability [m’] pressure [Pa] reference pressure [Pa] non-dimensional pressure x-coordinates of the heater [m] velocity components in the x- and y-directions [m s- ‘1 non-dimensional velocity components in the x- and y-directions axial and transverse coordinates non-dimensional axial and transverse coordinates
Greek lerters tl aa+ B E 9 p P
general condensation rate in the wick [kg s- ’m-‘1 evaporation rate [kg s- ’m-‘1 condensation rate [kg s- ’m-‘] = pa+/pK., [kg s-* m-‘1 ratio of the permeability in the y-direction to the permeability ratio of the condensation area to the heating area dynamic viscosity of liquid [Ns m-‘1 surface density of the liquid [kg m- ‘1
in the x-direction
INTRODUCTION An accurate determination of the pressure distribution for both vapour core and wick region of a heat pipe is essential for the calculation of the capillary limit [l]. The calculation of the liquid flow field of a flat plate heat pipe under normal operation conditions, i.e. when the heater and the condenser cover the whole evaporator section, is quite straightforward. However, when the heater partially covers the evaporator section or when multiple heat sources are applied to the surface of the heat pipe, the flow of liquid in the wick structure is no longer one-dimensional. It may be *Author
to whom
correspondence
should
be addressed. 339
340
W. Qin and C. Y. Liu
two-dimensional or even three-dimensional. This makes it more difficult to obtain an analytical solution of the liquid flow. In practical applications, heaters such as electronic components applied on the surface of the heat pipe may be discrete. It is of practical use to have some methods to calculate the pressure distribution of the working fluid in heat pipes under block-heating condition. Gernert [2] analyzed the flow fields of heat pipes with multiple heat sources with the use of superposition and the extension of the existing theories for a single evaporator heat pipe. The experimental study on a cylindrical pipe with a narrow block-heated evaporator was reported by Rosenfeld [3]. He presented an approximate method to calculate the steady-state wall temperature profile in the azimuthal direction. Faghri and Buchko [4] made experimental and numerical analysis of low-temperature heat pipes with multiple heat sources. The tests show that the maximum total heat load on the heat pipe varied greatly with the location of the local heat fluxes. Schmalhofer and Faghri [5] studied both the circumferentially heated and block-heated circular heat pipes analytically and experimentally. An approximate method to calculate the effective length was also proposed. Following Schmalhofer and Faghri’s method, Sun et al. [6] presented an approximate method to estimate the effective length of a flat plate heat pipe with localized distribution of the heat source. More recently, Huang and Liu [7] reported an analytical method to calculate the velocity and pressure distributions of the liquid phase in a flat plate heat pipe under localized heating conditions, in which the heater was fixed at the end of the flat plate heat pipe. They assumed that the working fluid evaporated uniformly over the heat input zone and condensed uniformly over the condensation zone and developed a two-dimensional analytical method to calculate the pressure distribution and the velocity field of liquid phase with a localized heater. All the studies cited above are based on isotropic wick structure. However, several types of wick structures commonly used for heat pipes are of anisotropic permeability, such as a groove wick covered with meshes. If the liquid flow in the wick is one-dimensional or if the heating zone completely covers the evaporator section of the heat pipe, there is no effect of the anisotropic property on the liquid flow, since there is no transverse velocity component or the transverse velocity component serves as a secondary role to distribute the working fluid uniformly in all locations. However, when the heater partially covers the evaporator section, the flow of liquid becomes two-dimensional. There exists a transverse velocity component. The velocity and pressure fields will be quite different from that of an isotropic wick. The objective of this paper is to present some results of the velocity and pressure fields in an anisotropic wick of a flat plate heat pipe under partially block-heating condition. The heater in this study is arbitrarily placed on the surface of the flat plate heat pipe. Both the velocity and pressure fields were solved using Fourier series method for various permeability ratios and different heater locations. The maximum pressure differences in the whole liquid phase region were also calculated. The results indicate that the anisotropic property of the wick structure can yield a significant effect on the capillary limit. THE
GOVERNING
EQUATIONS
AND
SOLUTIONS
The basic model of the present study is shown in Fig. 1. The flat plate heat pipe has a dimension of a x b and the heat input zone occupies an area of R(q I x I q + c,d 5 y I h), as shown by the shaded patch in Fig. 1. Outside the heating zone is the condenser of the heat pipe. The working fluid in the flat plate heat pipe is assumed to be evaporated uniformly over the heat input zone at an evaporation rate GI- (kg s - ’m -*), while over the rest of the surface the fluid is uniformly condensed into the wick at rate o!+ (kg s - ’m -‘). Referring to the configuration shown in Fig. 1, a non-dimensional area ratio of condensation-evaporation q is introduced:
’
ab S =-_=---R ce
”
(1)
where S and R are the areas of the condensation zone and the evaporation zone, respectively. Subject to the mass conservation of the fluid, the area ratio r~ is actually the ratio of evaporation rate 01- to condensation rate LX +:
u- =
a+r].
(2)
341
Liquid flow in the wick structure of a heat pipe
1 heater
h
0
a
9
x
Fig. 1. An illustration of the flat plate heat pipe with a block heater.
A distribution
function of the heater is defined as (3)
It is assumed that the fluid evaporated from the heat input zone may be considered to be condensed into this zone at a negative condensation rate - a + . Thus, the condensation rate over the whole heat pipe may be written as a = a+f(x,y)
.
(4)
The flow of liquid in the wick structure is assumed to follow Darcy’s law. The governing equations for the liquid flow in the wick structure can be written as
(6)
where U, u are the liquid flow velocity components in the x and y directions respectively, p is the pressure, K, and K, are the permeability of the wick structure in the x- and y-directions, respectively, E is the permeability ratio, p is the viscosity and p is the density of the liquid and a is the condensation rate defined in equation (4). The boundary conditions are
x=ou=~d) ax 3
x=a,u=
ap=0
z
342
W. Qin and C. Y. Liu
2
y=b,v=
co.
ay
Substituting equation (5) into the continuity equation (6), yields an equation for the pressure p: (8)
where j3 is defined as
5. ‘i
p=
(9)
Following the technique presented by Huang and Liu [7], both the pressure distribution and the distribution functionf(x,y) can be expanded in the form of a Fourier cosine series as shown below:
P = pref +
flab f A dcos(~x)+
f
fA,cos($y)+
m=I
fA,cos(~x)cos(~y)
m=,n=,
“=I
(10) f(x,y)=
+’+ ; f
Fmocos(~x)+
;
~F,.cos($y)+ n=,
m=l
F,,,,cos ( 7
2 f m=,
n=l
x)cos ( 5 y) .
(11)
Substituting equations (10) and (11) into equation (8), comparing the terms on both sides of equation (8) and transforming all variables into non-dimensional forms by p = P - Prcr
Bab
the non-dimensional P =
pressure distribution and the coefficient A,, can then be obtained as follows:
f A,cos(mnX)
+ i A,cos(nnY) n=,
m=l
A,a,
=
_
2a(h- d, (1 b*(mn)’
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-
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2 2 A,nncos(m~X)cos(~~Y)
+
n==ln=l
+q){
+q)
sin[mn(qy’c)]-
sin(F)-
sin(y)}
sin(F)
1
(12)
Liquid flow in the wick structure
Amn =x
of a heat pipe
343
+rj){ (mn)(nn)’ b’(nx)(mn)‘?&a’ (1
[
The non-dimensional to X and Y, U= - g
sin ( F
)-
sin ( F
)
sin[ mn(:+c)]-
sin(y)}
1.
velocity components can be obtained by taking derivatives of P with respect
= 2 A,(mn) m=l
sin (m7rX) + f 2 A,,(mrr) sin (m7rX) cos (n7rY) m=ln=l
V = - (Ez ) Fy = (E% ) F A&r) i n=,
(13)
sin (n7cY) + T f A,,(nrc) cos (m7rX) sin MY) . (14) Ill=,“=, i
Equations (12)-(14) can be used to calculate both the pressure and velocity vectors in the wick structure. The results are presented in the next section. RESULTS
AND DISCUSSION
Results of the velocity vector and pressure distribution for the liquid flow in the anisotropic wick are presented in this section. The result of maximum pressure drop for various permeability ratios is also presented for the understanding of the effect of the anisotropic property on the capillary limits. For all the calculations, the infinite series of equations (12)--(14) are truncated at m = n = 40. A test was conducted and verified that for any further increase in the number of terms, there was a negligible increase in accuracy. The area of the heater, c x e, and the condensation rate c1were kept constant. The pressure fields are illustrated by contours of the non-dimensional pressure difference AP(X, Y) = (P - P,,,)I(P,,, - Pmi,), where Pminand P,,, are the non-dimensional minimum and maximum pressures, respectively, in the whole wick structure. The velocity fields are represented in the velocity vector ‘v = Ui + Vj. Figure 2 shows the constant pressure contours and velocity vectors when the heater is mounted on the top corner of the heat pipe. The results for the permeability ratios, E = 0.5 and E = 1.0, are presented in Fig. 2(a) and (b), respectively, in which the dashed lines represent the heater. As can be seen, when the permeability ratio E is less than 1.0, i.e. the permeability in the y-direction is smaller than that in the x-direction, the non-dimensional pressure gradient in the y-direction becomes larger. This is the reason that the liquid needs to overcome a large pressure loss to flow back from the condenser region to the evaporator. The transverse velocity component becomes smaller when the permeability ratio is less than unity. Figure (3) shows the results for different heater locations. Case 1 is for the heater mounted on the centre left end of the heat pipe, while case 2 is for the heater located at a distance q = 3/l&2 from the left end. Three permeability ratios were considered. Figure 3(a) shows the isotropic case. At a large distance from the left-hand side, the constant pressure curve approaches a straight line. The results for E = 0.5 are shown in Fig. 3(b). The flow rate near the centre is larger than that of the isotropic case. It is more obvious for the case when E = 0.05. The wick structure near the two sides will not give much help for the return of the working fluid from the right-hand side condensation region to the evaporator of the heat pipe. Comparing the constant pressure lines, it can be seen that the non-dimensional pressure gradient in the x-direction decreased when the heater moved from the centre left end of the heat pipe to the central part of the heat pipe. When the heater is placed at the centre left end of the heat pipe, the minimum pressure or the stagnation point is located at the left end of the heat pipe. This yields a maximum length of the path for the return of the working fluid. If the heater moves a distance q = 3/16a from the left end of the heat pipe, which is designated as case 2, there occurs a reverse flow and the minimum pressure or the stagnation point moves to inside of the evaporation zone. As a result, the fluid may need to
W. Qin and C. Y. Liu
344
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overcome smaller pressure loss to flow back to the evaporator from the condenser as compared with that for case 1. The constant pressure contours shown in Figures (2) and (3) give an indication of the relative pressure distribution in the whole heat pipe. In order to see the effect of the anisotropic permeability on capillary pressure required for the returning of the working fluid, the calculated maximum
345
Liquid flow in the wick structure of a heat pipe
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346
W. Qin and C. Y. Liu
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Liquid flow in the wick structure of a heat pipe
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348
W. Qin and C. Y. Liu
2.5
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Fig. 4. Maximum pressure differences AP,,. = P,,,, - P,,. vs the permeability ratio E. Case 1: the heater is mounted on the centre left end of the heat pipe. Case 2: the heater is mounted at a distance q = 3/16a from the centre left end. Case 3: the heater is mounted at the centre of the heat pipe [q = l/2(0 - c)].
pressure difference (P,,, - P,,,$“)for different permeability ratios when the heater is located at three distinct positions are depicted in Fig. 4. It is evident that when the permeability ratio is very small, the maximum pressure difference in the wick structure is much larger than that for the isotropic wick. This is because the wick structure near the two sides cannot give much help for the return of the working fluid. The majority of the working fluid is pumped back by the capillary pressure produced by the centre portion of the wick. For the same heat input, the mass flow rate near the centre is large and thus yields a large pressure difference. As for E > 0.5, the maximum pressure difference drops gradually. For any further increase in E beyond unity, there is only a slight decrease in the maximum pressure difference. This shows that the decrease in resistance in the transverse direction still helps the return of the working fluid, but only slightly. When the heater moves away from the centre left end of the heat pipe to some distance inside, the maximum pressure difference decreases since the effective length is smaller than that for case 1. When the heater is located at the centre of the flat plate heat pipe, the maximum pressure difference is minimum. This shows that, at this location, the capillary limit will be maximum. CONCLUSION
The analytical study presented in this paper indicates that the anisotropic permeability of the wick structure can give a significant effect on the velocity and pressure fields of the working fluid for the case of localized heating condition. Decreasing the permeability in the transverse direction will increase the maximum pressure difference in the wick structure. This indicates that there is a decrease in capillary operation limit. However, when the permeability in the transverse direction is larger than that in the longitudinal direction, the effect is negligible for any further increase in permeability in the transverse direction. REFERENCES 1. P. D. Dunn and D. A. Reay, Heat Pipes, 4th Edition. Pergamon Press, Oxford (1994). 2. N. J. Gernert, Analysis and performance evaluation of heat pipes with multiple heat sources.
AIAA/ASME
4th Joint
Thermophysics and Heat Transfer Conf. (1986). 3. J. H. Rosenfeld, Modeling of heat transfer Heat Transfer 83, 71-76 (1987).
in a heat pipe for a localized
heat input zone. Proc. AIChE Symp. Series,
Liquid flow in the wick structure of a heat pipe
349
4. A. Faghri and M. Buchko, Experimental and numerical analysis of low temperature heat pipes with multiple heat sources. ASME J. Heat Transfer 113, 728-734 (1991). 5. J. Schmalhofer and A. Faghri, A study of circumferentially heated and block-heated heat pipes--I. Experimental analysis and generalized analytical prediction of capillary limits. Inc. .I. Heat Mass Transfer 36, 201-212 (1993). 6. K. H. Sun, C. Y. Liu and K. C. Leong, The effective length of a flat plate heat pipe covered partially by a strip heater on the evaporator section. Hear Recovery Systems & CHP 15, 383-388 (1995). 7. X. Y. Huang and C. Y. Liu, The pressure and velocity fields in the wick structure of a localized heated flat plate heat pipe. Inr. J. Heat Muss Transfer 39(6), 1325-1330 (1996).