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Analysis of flat heat pipes with various heating and cooling configurations
Applied Thermal Engineering 31 (2011) 2645e2655
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Analysis of flat heat pipes with various heating and cooling configurations Maziar Aghvami, Amir Faghri* Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 January 2011 Accepted 19 April 2011 Available online 6 May 2011
A simplified analytical thermal-fluid model including the wall, and both liquid and vapor flows is developed for flat heat pipes or vapor chambers with different heating and cooling configurations. The results for the two-dimensional temperature distribution inside the wall, the vapor velocity profiles, the axial pressure distributions in the vapor and liquid phases, and the vapor temperature are presented. The capillary pressures are calculated analytically and compared to the maximum capillary pressures to determine the dry-out limitations of the heat pipes. The reduction of the maximum effective capillary pressures due to the mass flux across the liquid/vapor interface is considered. The proposed analytical solution agrees well with a two-dimensional numerical simulation, which consists of the wall, wick and vapor regions. The parametric investigations show that the assumptions of previous studies that evaporation and condensation occur uniformly in the axial direction are valid only if the solid thermal conductivity is small and the axial wall conduction can be neglected. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Flat heat pipe Analytical solution
1. Introduction Flat miniature heat pipes are an effective passive heat transfer technology in the cooling of high heat flux electronic devices such as computer chips and thyristors due to their high thermal conductivity, reliability and low weight penalty. A flat heat pipe is an enclosed chamber whose inner surface is lined with a capillary wick structure, with the remaining volume containing the vapor. Heat sources and heat sinks are located on the chamber, with the other parts being thermally insulated. The working fluid is vaporized by the heat source at the evaporator, and the generated pressure difference drives vapor from the evaporator to the condenser, where it condenses and enters the wick structures. The capillary pressure in the wick pumps the condensed liquid back to the evaporator where it is vaporized. This internal phase-change circulation will continue as long as flow passage is not blocked and sufficient capillary pressure is maintained [1]. For the cooling of electronic components, the more commonly used working fluids are water, acetone, ethanol or methanol. Investigations on flat heat pipes have been conducted to characterize the thermal and hydrodynamic performances of miniature heat pipes used for the cooling of electronics. For predicting thermal characteristics of heat pipes, many researchers have suggested simplified one-dimensional theoretical models [2e7].
* Corresponding author. E-mail address: [email protected] (A. Faghri). 1359-4311/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2011.04.034
Vafai and Wang [8] developed a pseudo-three-dimensional analytical model for incompressible vapor and liquid flows within the asymmetrical flat heat pipe without including heat conduction in the wall. The parabolic velocity profiles were used for vapor flow within the heat pipe. The results for the vapor velocity profile and axial pressure distributions in vapor and liquid phases, and the axial vapor temperature distribution were presented. Lefèvre and Lallemand [9] presented an analytical solution for the conduction inside the heat pipe wall linked to a hydrodynamic model inside the flat heat pipe. The results for wall temperature, liquid and vapor isobars and velocities and maximum heat transfer capability were presented. Fully developed expressions for vapor flow between two parallel plates were used. Prasher [10] introduced a simplified conduction based modeling scheme to assess the effective thermal conductivity of the vapor. The results were compared with experimental data for vapor resistance and wall temperature distribution. Leong et al. [11] developed an analytical model to capture the variation of the vapor pressure distribution of a flat heat pipe with a symmetric and an asymmetric heat source and heat sink. The model used a stream function that is calculated as a product of two functions, each in an orthogonal coordinate direction, and only considered the vapor region. Do et al. [12] developed a one-dimensional mathematical model for predicting the thermal performance of a flat heat pipe with a rectangular grooved wick structure. The numerical results were obtained for the wall temperatures and the maximum heat transport rate. It was found that the axial variations of the wall
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heat sources as well as constant, convective, and radiative heat sinks. The maximum capillary pressure under a given heating load and the capillary dry-out limitations were predicted in their simulations. The existing analytical models employed major assumptions in order to simplify the analysis. First, that evaporation and condensation occur uniformly in the axial direction; second, that neither evaporation nor condensation occurs in the adiabatic section inside the heat pipe; and third, that the wall temperature is either constant or that its variations can be excluded from the analysis. In addition, the vapor flow inside the heat pipe was not taken into account or was modeled using inappropriate expressions. Therefore, there is a need for a simplified analytical model which includes the wall, wick, and vapor regions. In the present work, a thermal-fluid model is presented in order to analytically predict the temperature distribution in the wall linked to the velocity and pressure fields in the wick and vapor regions of flat heat pipes with various heating and cooling configurations. The capillary pressures are obtained and the dry-out limitations of the heat pipes are determined analytically. The two-dimensional variations of the wall temperature, the inertial and viscous effects for the vapor flow, and the Darcian effects for the liquid flow through the porous wick are taken into account. The effects of the evaporative heat input, the wall thermal conductivity, and the size of heat source on the thermal and hydrodynamic performance of the flat heat pipes are also investigated. The analytical results were compared with experimental data and numerical simulations. 2. Thermal-fluid model
Fig. 1. Flat heat pipes with different heating/cooling configurations: a) configuration I, single heat source and sink at top; b) configuration II, multiple heat sources and sink at top; c) configuration III, heat source at the bottom and heat sink on the top; and d) configuration IV, multiple heat sources and sink at both top and bottom.
temperature and the evaporation and condensation rates should be taken into account to accurately predict the thermal performance of a miniature heat pipe. Zhu and Vafai [13] performed a steady-state, incompressible, three-dimensional analytical and numerical investigation for an asymmetrical flat heat pipe to reveal the vapor flow in the vapor channel, as well as the liquid flow in the wick. The top, bottom and vertical wicks were assumed to be isotropic and saturated with liquid. The vapor injection and suction rates were assumed to be uniform for the top and bottom wicks, and negligible for the vertical wicks. It was found that the transverse pressure variations had an insignificant effect on the vapor flow. Vadakkan et al. [14] numerically analyzed the transient and steady-state performance of a flat heat pipe with multiple discrete heat sources. Three-dimensional flow and energy equations were solved for the vapor and wick regions, along with conduction in the wall. The evaporation and condensation rates were locally calculated using kinetic theory, which required an empirical accommodation coefficient with an overall energy balance at the liquid/ vapor interface. Rice and Faghri [15] developed a numerical model to investigate cylindrical/flat heat pipes with screen wick for single and multiple
The proposed configurations of the flat heat pipes, which consist of the wall, the porous wicks saturated with working fluid, and the vapor space are shown in Fig. 1. Fig. 1aed presents the four configurations for flat heat pipes with a single heat source, multiple heat sources, a heat source at the bottom and a heat sink on the top of the heat pipe, and multiple heat sources with symmetric boundary conditions, respectively. The underlying assumptions are two-dimensional wall temperature and vapor flow, steady-state, incompressible and laminar flow, a saturated wick, constant properties and saturation temperature, and linear temperature profile across the thin wick structure. 2.1. Conduction in wall The two-dimensional dimensionless steady-state heat conduction equation in the wall with constant thermal conductivity can be written as:
v2 q 1 v2 q þ ¼ 0 2 vY 2 vX 2 Hw
(1)
The boundary conditions at the end caps (X ¼ 0 and X ¼ 1) are:
vq ¼ 0 vX
(2)
At the interface between the wall and the wick (Y ¼ 0), the thermal boundary condition determined by assuming linear temperature profile across the thin wick structure and constant saturation temperature at the liquidevapor interface can be written as:
vq k Hw q ¼ eff vY kw H[
(3)
The boundary conditions at the outer wall (Y ¼ 0) are constant heat fluxes in the evaporators and the condenser, which can be located at
M. Aghvami, A. Faghri / Applied Thermal Engineering 31 (2011) 2645e2655
different locations on the heat pipe, and a heat flux equal to zero in the adiabatic sections at the rest of the surface. This is shown as:
8 < 1 vq 0 ¼ : vY g
evaporator adiabatic condenser
(4)
where the non-dimensional temperature, heat flux, and coordinates are defined as:
q ¼
kw ðT Tsat Þ qe hw
product in the x- and y-directions:
" 2 # h[ 2 h[ h[ y y þ þ þ 1þ2 hv hv hv hv hv
uv ðx;yÞ ¼ 6Uv ðxÞ
Zh[ ðh[ þhv Þ
N X
Am ðYÞ cos ðmpXÞ
(7)
The non-dimensional input heat flux is dependent on the location of heat sources and sinks, and can be written as follows:
(8)
where
Bm ¼
2 fsin ðmpL2e Þ sin ðmpL1e Þ þ sin ðmpL4e Þ mp sin ðmpL3e Þ þ sin ðmpL6e Þ sin ðmpL5e Þ
Zh[ ðh[ þhv Þ
vuv dy vx
þ vv;I ðh[ Þ vv ð ðh[ þ hv ÞÞ ¼ 0 where vv,I denotes the vapor interfacial velocity. The heat flux normal to the liquidevapor interface, qI, can be calculated from the conduction model with the assumption of linear temperature profile across the wick: N mpx qI k hw X ¼ qe eff Am ð0Þcos rv hfg rv hfg kw h[ m ¼ 1 l vv ð ðh[ þ hv ÞÞ ¼ 0
vv;I ðh[ Þ ¼
(15)
Integrating Eq. (14) with respect to x and using the boundary condition at the beginning of the evaporator, uv ¼ 0, will result in an expression for Uv(x):
Uv ðxÞ ¼ qe
N mpx X keff hw l Am ð0Þsin rv hv kw h[ hfg m ¼ 1 mp l
(16)
Therefore, the quasi-two-dimensional velocity profile is:
þ sin ðmpL8e Þ sin ðmpL7e Þ a½sin ðmpL2c Þ sin ðmpL1c Þg
vuv vvv þ dy ¼ vx vy
(14)
m¼1
N X vq Bm cos ðmpXÞ ¼ vY m¼1
(6)
The non-dimensional temperature for configurations I and II can be expanded in the form of an infinite Fourier series as:
qðX; YÞ ¼
(13)
where Uv(x) is the local mean velocity along the x-axis. The continuity equation for the two-dimensional incompressible vapor flow can be integrated with respect to y to determine Uv(x):
(5)
x y hw hl le lc Le ;Hw ¼ ;Hl ¼ ;Le ¼ ;Lc ¼ ; g ¼ X ¼ ;Y ¼ l l l l Lc l hw
2647
(9)
in which L1e, L2e, L3e, L4e, L5e, L6e, L7e, L8e, L1c and L2c express the locations of the heat sources and heat sink. Substituting Eqs. (7) and (8) into Eqs. (1), (3) and (4), yields the following expression for coefficient Am:
N mpx X keff hw l Am ð0Þ sin rv hv kw h[ hfg m ¼ 1 mp l " 2 # 2 h[ h[ h[ y y þ þ 1þ2 þ hv hv hv hv hv
uv ðx; yÞ ¼ 6qe
k k mp þ eff exp ðmpHw YÞ þ mp eff exp ðmpHw YÞ kw H[ kw H[ Am ðYÞ ¼ Bm k k exp ðmpHw Þ mp eff exp ðmpHw Þ mpHw mp þ eff kw H[ kw H[
ð17Þ
For configuration III, the non-dimensional temperature can be expanded in the form of an infinite Fourier series as:
qðX; YÞ ¼ A0 ðYÞ þ
N X
Am ðYÞ cos ðmpXÞ
The boundary layer form of the x-momentum equation for steady-state incompressible laminar vapor flow can be integrated to obtain the vapor pressure distribution:
(11)
m¼1
Zx
where
"
kw H[ A0 ðYÞ ¼ ðL2e L1e Þ Y þ keff Hw
(10)
Zh[
rv
#
0 ðh[ þhv Þ
(12)
and Am can be obtained from Eq. (10).
vðuv Þ2 dxdy þ rv vx Zx
Zh[
¼ 0 ðh[ þhv Þ
Zx
Zh[
0 ðh[ þhv Þ
vpv dxdy þ mv vx
vðuv vv Þ dxdy vy Zx
Zh[
0 ðh[ þhv Þ
v2 uv dxdy vy2 (18)
2.2. Vapor flow A parabolic velocity profile is used for vapor flow within the heat pipe. The velocity distribution is represented by a functional
Assuming the constant vapor pressure at each cross-section, and utilizing the boundary conditions at the beginning of the evaporator, uv ¼ 0 and pv ¼ pref, yields:
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N h mpx i 12mv keff hw X l 2 Am ð0Þ cos 1 3 p m l rv hv kw h[ hfg m ¼ 1 # " 3 h[ 4 h[ 38 h[ 2 17 h[ 6 þ þ16 þ þ 36rv Uv2 8 hv hv 3 hv 3 hv 5
utilizing the pressure boundary condition at the end of the condenser (x ¼l), p[ ¼ pv:
pv ¼ pref þ qe
N h mpx i m[ keff hw X l 2 Am ð0Þ cos cosðmpÞ 2 p m l r[ Kkw h[ hfg m ¼ 1 N 12mv keff hw X l 2 Am ð0Þ½cosðmpÞ 1 þpref þ qe 3 rv hv kw h[ hfg m ¼ 1 mp
p[ ¼ qe
(19) where Uv is obtained from Eq. (16). For configuration III, the interfacial velocity (normal to the liquidevapor interface), as shown in Eq. (15), becomes:
vv;I ðh[ Þ ¼
qI
rv hfg
¼
qe
rv hfg
" ðL2e L1e Þþ
qc qe ðL2e L1e Þ vv ððh[ þhv ÞÞ ¼ ¼ rv hfg rv hfg
N X
keff hw Am ð0Þcos kw h[ m¼1
mpx
#
For configuration III, the interfacial velocity (normal to the liquidevapor interface) for the liquid can be expressed as:
v[ ð0Þ ¼ 0 qI qc hfg r[
l
v[;I ðh[ Þ ¼ (20)
The vapor velocity profile and pressure distribution for configurations III and IV can be obtained in a similar manner. The temperature drop in the vapor can be related to the pressure drop in the vapor region by applying the Clapeyron equation and using the ideal gas law [1]:
RT 2 DTv ¼ Dpv hfg pv
(27)
(21)
where the vapor pressure drop is obtained from Eq. (19).
The velocity profile and pressure distribution in the wick for configurations III and IV can be calculated in a similar manner.
2.4. Capillary pressure Rice and Faghri [15] performed a limiting analysis of the reduction of the capillary pressure due to the increased resistance of the liquid film for a single idealized cylindrical pore, as presented in Fig. 2. The pressure gradient needed to drive the flow to the microfilm region is created by the change in surface curvature. The effective pore radius in a screen mesh wick is given by Faghri [1] as:
2.3. Liquid flow The continuity equation for the incompressible liquid flow can be integrated with respect to y to obtain the liquid axial velocity, u[(x):
Z0 h[
vu[ vv[ du[ dy ¼ v[;I h[ þ ¼ 0 vx vy dx
(22)
where v[,I denotes the interfacial velocity (normal to the liquidevapor interface) for the liquid and is related to the vapor interfacial velocity by:
r[ v[;I ¼ rv vv;I
(23)
The axial liquid velocity can be calculated by integrating Eq. (22) with respect to x using the velocity boundary condition at the beginning of the evaporator (x ¼ 0), u[ ¼ 0:
u[ ðxÞ ¼ qe
N mpx X l Am ð0Þsin 2 l r[ kw h[ hfg m ¼ 1 mp keff hw
(24)
(28)
r0 ¼
1 2N
(29)
where N is the screen mesh number. The effective pore radius can be used to calculate the static capillary pressure:
pcap;static ¼
2s[v r0
(30)
If there is no flow within a capillary tube, the effective capillary pressure is equivalent to the static capillary pressure. However, as shown by Ma et al. [17], the mass flow rate in a capillary tube can also affect the effective capillary pressure. From previous investigations, the mass flow rates are significantly reduced in the region where the change in surface curvature and the disjoining pressure effects both have comparable magnitudes in the pressure gradients. The disjoining pressure can be modeled by an equation presented in [16] for water:
" # d b pd ¼ r[ RTd ln a 3:3
(31)
The one-dimensional steady-state conservation of momentum for incompressible liquid flow in the wick can be expressed by Darcy’s law, assuming negligible inertial effects in comparison to viscous losses [16], as:
m u[ dp[ ¼ [ dx K
(25)
in which the permeability K is calculated for a mesh screen [1] as:
K ¼
d2 43
122ð1 4Þ2 1:05pNd 4z1 4
(26)
The liquid pressure is obtained by integrating Eq. (25) and
Fig. 2. An idealized single pore at the wick/liquid/vapor interfacial region.
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2649
Table 1 Specifications of the heat pipe and the working fluid for four configurations. Configuration I
Configurations II & IV
Configuration III
Wall (copper)
Thickness (m) Thermal conductivity (W/(mK))
0.000265 400
0.0017 400
0.00089 400
Wick/Water
Thickness (m) Effective thermal conductivity (W/(mK)) Specific volume (m3/(kg)) Viscosity (Ns/m2) Latent heat (J/kg)
0.00014 1.3 0.0010078 0.00065 2406700
0.00075 1.2 0.001026 0.00037 2321400
0.00076 26 0.0010078 0.00065 2406700
Vapor (water)
Thickness (m) Specific volume (m3/(kg)) Viscosity (N/m2)
0.0023 19.523 0.0000096
0.01025 4.131 0.000011
0.00635 19.523 0.0000096
The disjoining pressure has a negligible effect when pd is equal to zero. At this value the film thickness is:
1=b 1 dd ¼ 3:3 a
(32)
The capillary pressure across the liquid/vapor interface is:
0 B pcap ¼ pv p[ ¼ s@
1 00
d 02
d þ1
3=2
1 C pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 0 ðr0 dÞ d 2 þ 1
(33)
Eqs. (34) and (35) can be solved with the initial conditions:
dð0Þ ¼ 0:99999r0
(36)
d0 ð0Þ ¼ 1 109
(37)
pcap ¼ pcap;0
(38)
For each heat input, with an associated effective capillary pressure pcap,0, the solution will produce a minimum film
where the vapor and liquid pressures can be obtained from the analytical solutions. The pressure gradient is given by Ref. [15] considering constant vapor pressure, knowing the shear stress at the wall, and neglecting the inertial effects:
dpcap ¼ ds
Qinput 6reff m[ 2 3 r[ hfg N 2 Ah 2reff d d
(34)
where Qinput is the total heater power, and Ah is the area of the heater. Rearranging Eq. (33), the second derivative of d with respect to x is:
d00 ¼
2 3=2 p 6 cap d0 2 þ 1 4
s
3 ðr0 dÞ
1 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 d0 2 þ 1
(35)
Fig. 3. The variation of heat flux at the interface along the heat pipe with respect to the wall thermal conductivity, for configuration I (Qe ¼ 30 W, Tsat ¼ 40 C).
Fig. 4. Variation of wall temperature along the heat pipe with respect to the a) input power and b) heat source size, for configuration I (Tsat ¼ 40 C).
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Fig. 5. Effect of the input power on a) capillary pressure, b) vapor velocity, and c) vapor temperature, for configuration I.
thickness dmin . If dmin is greater than dd , then the overall capillary pressure is too large and needs to be reduced. If dmin is less than dd , then the disjoining pressure effects become significant, and it may be possible for the film to reach an equilibrium film thickness. Therefore, the maximum effective capillary pressure for a given heat input is reached when dmin is equal to dd .
Fig. 6. Variation of wall temperature along the heat pipe with respect to the a) input power, b) wall thermal conductivity, and c) heat source size, for configuration II (Tsat ¼ 71 C).
3. Results The specifications of the heat pipe wall, wick, and vapor core for the four configurations are given in Table 1.
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2651
48 Present Analytical Experimental (Prasher, 2003)
46
T (C)
44
42
40
38 10
20
30
40
50
60
70
80
x (mm) Fig. 8. Wall temperature profile along the heat pipe, for configuration III (Qe ¼ 30 W, Tsat ¼ 40 C).
Fig. 7. Effect of the input power on a) capillary pressure, b) vapor velocity, and c) vapor temperature, for configuration II.
3.1. Configuration I: flat heat pipe with single heat source and sink at top The heat pipe, the evaporator and the condenser lengths are 0.04 m, 0.01 m and 0.015 m, respectively. The operating temperature is 40 C. Profiles of the heat flux along the axial direction and the influence of the solid thermal conductivity on the heat flux profiles are shown in Fig. 3. As expected, evaporation and condensation rates are not uniform, but instead vary along the axial direction. Even in the adiabatic section, heat transfer by evaporation or condensation takes place inside the heat pipe, as also
pointed out in [12,14]. The figure shows that as the solid thermal conductivity decreases, the evaporation (condensation) rate becomes more uniform along the axial direction and heat transfer by evaporation or condensation in the adiabatic section becomes negligible. Therefore, the assumption that neither evaporation nor condensation occurs in the adiabatic section is valid only if the effect of the axial wall conduction is negligible. The outer wall temperature distributions of the flat heat pipe for different input powers applied to the heat source are shown in Fig. 4a. The temperature increases faster from the condenser to the evaporator and the maximum temperature at the surface increases about 18% when the heat input is changed from 20 W to 50 W. Fig. 4b shows the jumps in the maximum wall temperature for 30 W power input due to shortening of the evaporator length. The ratio of the heat source length to the overall heat pipe length is an important parameter controlling the resistance to the spreading of heat from the evaporator to the condenser section. The smaller the heat source, the higher the maximum wall temperature. As the heater size is reduced, the heat flux in the evaporator region increases drastically. This causes a jump in the heat pipe maximum wall temperature. Fig. 5 demonstrates the variations of the capillary pressure, vapor mean velocity and temperature along the flat heat pipe for different input powers. It can be seen that the vapor velocity increases in the region corresponding to the evaporator section and decreases in the region corresponding to the condenser section, due to the vapor injection (increasing mass) and suction (decreasing mass) over the corresponding regions. As the mass flow rate is increased at the vapor/liquid interface by the increasing condensation of vapor, pressure drop and velocity increase considerably as the heat input increases. It should be noted that the pressure differences in the y-direction are negligible [13]. The vapor pressure decreases in the evaporator zone due to friction and acceleration of the vapor flow caused by mass injection from the wick, while the vapor pressure increases in the condenser zone owing to the deceleration of the vapor flow by mass suction. It can also be seen that the difference in vapor temperature is quite small.
3.2. Configuration II: flat heat pipe with multiple heat sources and sink at top There are four heat sources with a length of 0.0635 m beginning at x locations of 0.020, 0.1588, 0.2976 and 0.4364 m, respectively.
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Fig. 9. Effect of the input power on a) capillary pressure, b) vapor velocity, and c) vapor temperature, for configuration III.
The beginning of the heat sink is at an x location of 0.680 m and it has a length of 0.300 m. The rest of the outer wall is adiabatic, and the total length of the heat pipe is 1.000 m. The outer wall temperature distributions of the flat heat pipe for different input powers applied to the heat source are shown in Fig. 6a. The temperature increases faster from the condenser to the evaporator, and the maximum temperature at the surface increases
Fig. 10. Variation of wall temperature along the heat pipe with respect to the a) input power; b) wall thermal conductivity and c) heat source size, for configuration III (Tsat ¼ 40 C).
about 3% when the heat input is changed from 50 W to 70 W. Fig. 6b and c shows the effect of the wall thermal conductivity and heat source size on the wall temperature. For the same heat input level, the heat pipe with higher wall thermal conductivity is more
M. Aghvami, A. Faghri / Applied Thermal Engineering 31 (2011) 2645e2655
a
95 90
95 Present Analytical
90
Numerical (Rice and Faghri, 2007)
85
85
80
80
T (C)
T (C)
b
Present Analytical
75
Numerical (Rice and Faghri, 2007)
75
70
70
65
65 60
60
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
X
c
0.6
0.8
1
X
d
95 Present Analytical
90
95 Present Analytical
90
Numerical (Rice and Faghri, 2007)
85
85
80
80
T (C)
T (C)
2653
75
75
70
70
65
65
60
Numerical (Rice and Faghri, 2007)
60 0
0.2
0.4
0.6
0.8
1
X
0
0.2
0.4
0.6
0.8
1
X
Fig. 11. Wall temperature profiles for configuration IV with a) single heat source, b) two heat sources, c) three heat sources, and d) four heat sources.
effective in dissipating heat. The smaller the heat source, the higher the maximum wall temperature. Fig. 7 shows the variations of the capillary pressure, vapor mean velocity and temperature along the flat heat pipe, for different input powers. As the mass flow rate is increased at the vapor/liquid interface by increasing condensation of vapor, pressure drop and velocity increase considerably as the heat input increases. The vapor temperature difference is negligible. 3.3. Configuration III: flat heat pipe with heat source at the bottom and heat sink on the top Fig. 8 presents the comparison of the thermal model with the experimental data of Prasher [10]. The length of the heat pipe was
0
-Pcap (Pa)
-50
0.089 m, and the testing was done for a heat source 0.01 m in length at the center of the heat pipe and a saturation temperature of 40 C. The results for the wall temperature profile agree with the experimental data. Fig. 9 demonstrates the variations of the capillary pressure, vapor mean velocity and temperature along the flat heat pipe, for different input powers. As the mass flow rate is increased at the vapor/liquid interface by increasing the condensation of vapor, the pressure drop and velocity rise considerably with heat input. The outer wall temperature distributions of the flat heat pipe for different input powers applied to the heat source are shown in Fig. 10a. The temperature increases faster from the condenser to the evaporator and the maximum temperature at the surface increases about 11% when the heat input is changed from 30 W to 50 W. Fig. 10b and c shows the effect of the wall thermal conductivity and heat source size on the wall temperature. For the same heat input level, the heat pipe with higher wall thermal conductivity is more effective in dissipating heat. The smaller the heat source, the higher the maximum wall temperature.
-100 3.4. Configuration IV: flat heat pipe with multiple heat sources and sink at both top and bottom
-150 -200
Present Analytical Numerical (Rice and Faghri, 2007)
-250 0
0.2
0.4
0.6
0.8
X Fig. 12. Capillary pressure for configuration IV (Qe1 ¼ 97 W).
1
A complete two-dimensional numerical analysis of the flat heat pipe, which consists of the vapor flow in the core region, the liquid flow in the wick, and the heat conduction in the wall, is presented in Ref. [18] to validate the analytical model. There are four heat sources with a length of 0.0635 m beginning at x locations of 0.020, 0.1588, 0.2976 and 0.4364 m, respectively. The beginning of the heat sink is at an x location of 0.680 m, and it has a length of 0.300 m. The rest of the outer wall is adiabatic. The total length of the heat pipe is 1.000 m.
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a
Analytical
0.25
0.5 0.45 0.4
Numerical (Aghvami, 2010)
0.35
0.2
Uv (m/s)
Uv (m/s)
b
0.3
0.15 0.1
0.3 0.25 0.2 0.15 Analytical
0.1
0.05
Numerical (Aghvami, 2010)
0.05 0
0 0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
0.6
c
d
0.7 Analytical
1
0.8
1
0.5 0.45
0.6
0.4
Numerical (Aghvami, 2010)
0.5
0.35
Uv (m/s)
Uv (m/s)
0.8
X
X
0.4 0.3
0.3 0.25 0.2 0.15
0.2
Analytical
0.1 0.1
Numerical (Aghvami, 2010)
0.05
0
0 0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
0.6
X
X
Fig. 13. Vapor axial velocities for configuration IV with a) single heat source, b) two heat sources, c) three heat sources, and d) four heat sources.
A comparison of the outer wall temperature profiles for different heating configurations of the present analytical model and the numerical results of Rice and Faghri [15] is shown in Fig. 11. Fig. 11a presents the outer wall temperatures resulting from the application of 97 W to evaporator 1 at an interfacial saturation temperature of 76 C. The saturation temperature at the liquidevapor interface is assumed to be constant and nearly equal to the centerline or axis temperature for the heat pipe presented by Rice and Faghri [15]. Fig. 11b shows the outer wall temperatures resulting from the application of 98 W to evaporators 1 and 2 at a saturation temperature of 71 C. Fig. 11c presents the outer wall temperatures resulting from the application of 75 W to evaporators 1, 2, and 3 at a saturation temperature of 82 C. Fig. 11d presents the outer wall
temperatures resulting from the application of 50 W to evaporators 1, 2, 3, and 4 at a saturation temperature of 71 C. Fig. 12 shows the capillary pressure resulting from the application of 97 W to evaporator 1 and a comparison with the numerical results of Rice and Faghri [15]. A comparison of the vapor axial mean velocities for different heating configurations of the numerical simulations [18] and the analytical results is presented in Fig. 13. It can be seen that these results match up very closely. 3.5. Capillary pressure and limitations The maximum capillary pressure versus heating load for a lowtemperature water heat pipe is presented in Fig. 14, which is the intersection of the root of disjoining pressure and the minimum film thickness. The pressure drop in the wick must be known in order to calculate the dry-out limitations of a heat pipe. For the cases listed in Table 2, the total evaporator heat input is increased until the total capillary pressure drop bounds the maximum capillary pressure in Fig. 14. Table 2 Capillary dry-out limitations of the flat heat pipe with multiple heat sources (configuration IV). Qinput, dry out, W
Evaporator heat input
Fig. 14. The maximum effective capillary pressure as a function of heat input.
Case
1
2
3
4
Numerical [16]
Analytical
1 2 3 4 5 6 7
Qinput 0 0 0 Qinput/2 Qinput/3 Qinput/4
0 Qinput 0 0 Qinput/2 Qinput/3 Qinput/4
0 0 Qinput 0 0 Qinput/3 Qinput/4
0 0 0 Qinput 0 0 Qinput/4
223 266 333 446 245 272 305
259 314 400 552 284 314 352
M. Aghvami, A. Faghri / Applied Thermal Engineering 31 (2011) 2645e2655
4. Conclusions A simplified linked thermal-fluid model is presented in order to analytically predict the thermal and hydrodynamic performance of flat heat pipes with different heating and cooling configurations. The results for the flat heat pipes show that, the assumptions that evaporation and condensation occur uniformly in the axial direction, that evaporation occurs only in the evaporator section, and that condensation occurs only in the condenser section, are valid only if the solid thermal conductivity is small and the axial wall conduction can be neglected. Higher evaporative heat input increases maximum surface temperature, pressure drop and fluid velocities due to the increasing mass flow rate at the vapor/wick interface. Shortening the size of the heat source can degrade the thermal performance of the heat pipe, due to the jump in the peak wall temperature. The maximum effective capillary pressure can be decreased while the static capillary pressure remains constant by increasing the heat input. The capillary pressures needed to support a steady-state operation can be calculated and compared to the computed maximum capillary pressures for given heat inputs to analytically determine the dry-out limitations of the heat pipe. Nomenclature A0, Am d H h hfg K k L l N p Q q r0 T
Fourier coefficients diameter of the mesh fiber, m dimensionless thicknesses thickness, m latent heat of vaporization, J kg1 permeability, m2 thermal conductivity, W m1 K1 dimensionless lengths lengths, m mesh number pressure, Pa heat transfer rate, W heat flux, W m2 effective pore radius, m temperature, K q dimensionless temperature u x velocity, m s1 v y velocity, m s1 x, y, z, s coordinates, m X, Y, Z dimensionless coordinates Greek letters ratio between heat source and heat sink lengths dynamic viscosity, Pa s density, kg m3 surface tension, N m1 porosity film thickness, m
g m r s 4 d
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Subscripts and superscripts a adiabatic c condenser cap capillary eff effective e evaporator I interface [ liquid sat saturation v vapor w wall
References [1] A. Faghri, Heat Pipe Science and Technology. Taylor and Francis, Washington, DC, 1995. [2] R. Hopkins, A. Faghri, D. Khrustalev, Flat miniature heat pipes with micro capillary grooves, ASME J. Heat Transfer 121 (1999) 102e109. [3] S.J. Kim, J.K. Seo, K.H. Do, Analytical and experimental investigation on the operational characteristics and the thermal optimization of a miniature heat pipe with a grooved wick structure, Int. J. Heat Mass Transfer 42 (2003) 3405e3418. [4] L. Lin, R. Ponnappan, J. Leland, High performance miniature heat pipe, Int. J. Heat Mass Transfer 45 (2002) 3131e3142. [5] B.R. Babin, G.P. Peterson, D. Wu, Steady-state modeling and testing of a micro heat pipe, ASME J. Heat Transfer 112 (1990) 595e601. [6] D. Khrustalev, A. Faghri, Thermal analysis of a micro heat pipe, ASME J. Heat Transfer 116 (1994) 189e198. [7] J.P. Longtin, B. Badran, F.M. Gerner, A one-dimensional model of a micro heat pipe during steady-state operation, ASME J. Heat Transfer 116 (1994) 709e715. [8] K. Vafai, W. Wang, Analysis of flow and heat transfer characteristics of an asymmetrical flat plate heat pipe, Int. J. Heat Mass Transfer 35 (1992) 2087e2099. [9] F. Lefèvre, M. Lallemand, Coupled thermal and hydrodynamic models of flat micro heat pipes for the cooling of multiple electronic components, Int. J. Heat Mass Transfer 49 (2006) 1375e1383. [10] R.S. Prasher, A simplified conduction based modeling scheme for Design Sensitivity Study of thermal solution utilizing heat pipe and vapor chamber technology, ASME J. Electron. Packag. 125 (2003) 378e385. [11] K.C. Leong, C.Y. Liu, K.H. Sun, Vapor pressure distribution of a flat plate heat pipe, Int. J. Heat Mass Transfer 23 (6) (1996) 789e797. [12] K.H. Do, S.J. Kim, S.V. Garimella, A mathematical model for analyzing the thermal characteristics of a flat micro heat pipe with a grooved wick, Int. J. Heat Mass Transfer 51 (2008) 4637e4650. [13] N. Zhu, K. Vafai, Vapor and liquid flow in an asymmetrical flat plate heat pipe: a three-dimensional analytical and numerical investigation, Int. J. Heat Mass Transfer 41 (1) (1998) 159e174. [14] U. Vadakkan, S.V. Garimella, J.Y. Murthy, Transport in flat heat pipes at high heat fluxes from multiple discrete sources, ASME J. Heat Transfer 126 (2004) 347e354. [15] J. Rice, A. Faghri, Analysis of screen wick heat pipes, including capillary dryout limitations, J. Thermophys. Heat Transfer 21 (3) (2007) 475e486. [16] A. Faghri, Y. Zhang, Transport Phenomena in Multiphase Systems. Elsevier Inc., 2006. [17] H.B. Ma, G.P. Peterson, D.M. Pratt, Disjoining pressure effect on the wetting characteristics in a capillary tube, Nanoscale Microscale Thermophys. Eng. 2 (1998) 283e297. [18] M. Aghvami. Thermal-fluid analysis of heat pipes, M.S. thesis, University of Connecticut, 2010.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Analysis of flat heat pipes with various heating and cooling configurations Maziar Aghvami, Amir Faghri* Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 January 2011 Accepted 19 April 2011 Available online 6 May 2011
A simplified analytical thermal-fluid model including the wall, and both liquid and vapor flows is developed for flat heat pipes or vapor chambers with different heating and cooling configurations. The results for the two-dimensional temperature distribution inside the wall, the vapor velocity profiles, the axial pressure distributions in the vapor and liquid phases, and the vapor temperature are presented. The capillary pressures are calculated analytically and compared to the maximum capillary pressures to determine the dry-out limitations of the heat pipes. The reduction of the maximum effective capillary pressures due to the mass flux across the liquid/vapor interface is considered. The proposed analytical solution agrees well with a two-dimensional numerical simulation, which consists of the wall, wick and vapor regions. The parametric investigations show that the assumptions of previous studies that evaporation and condensation occur uniformly in the axial direction are valid only if the solid thermal conductivity is small and the axial wall conduction can be neglected. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Flat heat pipe Analytical solution
1. Introduction Flat miniature heat pipes are an effective passive heat transfer technology in the cooling of high heat flux electronic devices such as computer chips and thyristors due to their high thermal conductivity, reliability and low weight penalty. A flat heat pipe is an enclosed chamber whose inner surface is lined with a capillary wick structure, with the remaining volume containing the vapor. Heat sources and heat sinks are located on the chamber, with the other parts being thermally insulated. The working fluid is vaporized by the heat source at the evaporator, and the generated pressure difference drives vapor from the evaporator to the condenser, where it condenses and enters the wick structures. The capillary pressure in the wick pumps the condensed liquid back to the evaporator where it is vaporized. This internal phase-change circulation will continue as long as flow passage is not blocked and sufficient capillary pressure is maintained [1]. For the cooling of electronic components, the more commonly used working fluids are water, acetone, ethanol or methanol. Investigations on flat heat pipes have been conducted to characterize the thermal and hydrodynamic performances of miniature heat pipes used for the cooling of electronics. For predicting thermal characteristics of heat pipes, many researchers have suggested simplified one-dimensional theoretical models [2e7].
* Corresponding author. E-mail address: [email protected] (A. Faghri). 1359-4311/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2011.04.034
Vafai and Wang [8] developed a pseudo-three-dimensional analytical model for incompressible vapor and liquid flows within the asymmetrical flat heat pipe without including heat conduction in the wall. The parabolic velocity profiles were used for vapor flow within the heat pipe. The results for the vapor velocity profile and axial pressure distributions in vapor and liquid phases, and the axial vapor temperature distribution were presented. Lefèvre and Lallemand [9] presented an analytical solution for the conduction inside the heat pipe wall linked to a hydrodynamic model inside the flat heat pipe. The results for wall temperature, liquid and vapor isobars and velocities and maximum heat transfer capability were presented. Fully developed expressions for vapor flow between two parallel plates were used. Prasher [10] introduced a simplified conduction based modeling scheme to assess the effective thermal conductivity of the vapor. The results were compared with experimental data for vapor resistance and wall temperature distribution. Leong et al. [11] developed an analytical model to capture the variation of the vapor pressure distribution of a flat heat pipe with a symmetric and an asymmetric heat source and heat sink. The model used a stream function that is calculated as a product of two functions, each in an orthogonal coordinate direction, and only considered the vapor region. Do et al. [12] developed a one-dimensional mathematical model for predicting the thermal performance of a flat heat pipe with a rectangular grooved wick structure. The numerical results were obtained for the wall temperatures and the maximum heat transport rate. It was found that the axial variations of the wall
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heat sources as well as constant, convective, and radiative heat sinks. The maximum capillary pressure under a given heating load and the capillary dry-out limitations were predicted in their simulations. The existing analytical models employed major assumptions in order to simplify the analysis. First, that evaporation and condensation occur uniformly in the axial direction; second, that neither evaporation nor condensation occurs in the adiabatic section inside the heat pipe; and third, that the wall temperature is either constant or that its variations can be excluded from the analysis. In addition, the vapor flow inside the heat pipe was not taken into account or was modeled using inappropriate expressions. Therefore, there is a need for a simplified analytical model which includes the wall, wick, and vapor regions. In the present work, a thermal-fluid model is presented in order to analytically predict the temperature distribution in the wall linked to the velocity and pressure fields in the wick and vapor regions of flat heat pipes with various heating and cooling configurations. The capillary pressures are obtained and the dry-out limitations of the heat pipes are determined analytically. The two-dimensional variations of the wall temperature, the inertial and viscous effects for the vapor flow, and the Darcian effects for the liquid flow through the porous wick are taken into account. The effects of the evaporative heat input, the wall thermal conductivity, and the size of heat source on the thermal and hydrodynamic performance of the flat heat pipes are also investigated. The analytical results were compared with experimental data and numerical simulations. 2. Thermal-fluid model
Fig. 1. Flat heat pipes with different heating/cooling configurations: a) configuration I, single heat source and sink at top; b) configuration II, multiple heat sources and sink at top; c) configuration III, heat source at the bottom and heat sink on the top; and d) configuration IV, multiple heat sources and sink at both top and bottom.
temperature and the evaporation and condensation rates should be taken into account to accurately predict the thermal performance of a miniature heat pipe. Zhu and Vafai [13] performed a steady-state, incompressible, three-dimensional analytical and numerical investigation for an asymmetrical flat heat pipe to reveal the vapor flow in the vapor channel, as well as the liquid flow in the wick. The top, bottom and vertical wicks were assumed to be isotropic and saturated with liquid. The vapor injection and suction rates were assumed to be uniform for the top and bottom wicks, and negligible for the vertical wicks. It was found that the transverse pressure variations had an insignificant effect on the vapor flow. Vadakkan et al. [14] numerically analyzed the transient and steady-state performance of a flat heat pipe with multiple discrete heat sources. Three-dimensional flow and energy equations were solved for the vapor and wick regions, along with conduction in the wall. The evaporation and condensation rates were locally calculated using kinetic theory, which required an empirical accommodation coefficient with an overall energy balance at the liquid/ vapor interface. Rice and Faghri [15] developed a numerical model to investigate cylindrical/flat heat pipes with screen wick for single and multiple
The proposed configurations of the flat heat pipes, which consist of the wall, the porous wicks saturated with working fluid, and the vapor space are shown in Fig. 1. Fig. 1aed presents the four configurations for flat heat pipes with a single heat source, multiple heat sources, a heat source at the bottom and a heat sink on the top of the heat pipe, and multiple heat sources with symmetric boundary conditions, respectively. The underlying assumptions are two-dimensional wall temperature and vapor flow, steady-state, incompressible and laminar flow, a saturated wick, constant properties and saturation temperature, and linear temperature profile across the thin wick structure. 2.1. Conduction in wall The two-dimensional dimensionless steady-state heat conduction equation in the wall with constant thermal conductivity can be written as:
v2 q 1 v2 q þ ¼ 0 2 vY 2 vX 2 Hw
(1)
The boundary conditions at the end caps (X ¼ 0 and X ¼ 1) are:
vq ¼ 0 vX
(2)
At the interface between the wall and the wick (Y ¼ 0), the thermal boundary condition determined by assuming linear temperature profile across the thin wick structure and constant saturation temperature at the liquidevapor interface can be written as:
vq k Hw q ¼ eff vY kw H[
(3)
The boundary conditions at the outer wall (Y ¼ 0) are constant heat fluxes in the evaporators and the condenser, which can be located at
M. Aghvami, A. Faghri / Applied Thermal Engineering 31 (2011) 2645e2655
different locations on the heat pipe, and a heat flux equal to zero in the adiabatic sections at the rest of the surface. This is shown as:
8 < 1 vq 0 ¼ : vY g
evaporator adiabatic condenser
(4)
where the non-dimensional temperature, heat flux, and coordinates are defined as:
q ¼
kw ðT Tsat Þ qe hw
product in the x- and y-directions:
" 2 # h[ 2 h[ h[ y y þ þ þ 1þ2 hv hv hv hv hv
uv ðx;yÞ ¼ 6Uv ðxÞ
Zh[ ðh[ þhv Þ
N X
Am ðYÞ cos ðmpXÞ
(7)
The non-dimensional input heat flux is dependent on the location of heat sources and sinks, and can be written as follows:
(8)
where
Bm ¼
2 fsin ðmpL2e Þ sin ðmpL1e Þ þ sin ðmpL4e Þ mp sin ðmpL3e Þ þ sin ðmpL6e Þ sin ðmpL5e Þ
Zh[ ðh[ þhv Þ
vuv dy vx
þ vv;I ðh[ Þ vv ð ðh[ þ hv ÞÞ ¼ 0 where vv,I denotes the vapor interfacial velocity. The heat flux normal to the liquidevapor interface, qI, can be calculated from the conduction model with the assumption of linear temperature profile across the wick: N mpx qI k hw X ¼ qe eff Am ð0Þcos rv hfg rv hfg kw h[ m ¼ 1 l vv ð ðh[ þ hv ÞÞ ¼ 0
vv;I ðh[ Þ ¼
(15)
Integrating Eq. (14) with respect to x and using the boundary condition at the beginning of the evaporator, uv ¼ 0, will result in an expression for Uv(x):
Uv ðxÞ ¼ qe
N mpx X keff hw l Am ð0Þsin rv hv kw h[ hfg m ¼ 1 mp l
(16)
Therefore, the quasi-two-dimensional velocity profile is:
þ sin ðmpL8e Þ sin ðmpL7e Þ a½sin ðmpL2c Þ sin ðmpL1c Þg
vuv vvv þ dy ¼ vx vy
(14)
m¼1
N X vq Bm cos ðmpXÞ ¼ vY m¼1
(6)
The non-dimensional temperature for configurations I and II can be expanded in the form of an infinite Fourier series as:
qðX; YÞ ¼
(13)
where Uv(x) is the local mean velocity along the x-axis. The continuity equation for the two-dimensional incompressible vapor flow can be integrated with respect to y to determine Uv(x):
(5)
x y hw hl le lc Le ;Hw ¼ ;Hl ¼ ;Le ¼ ;Lc ¼ ; g ¼ X ¼ ;Y ¼ l l l l Lc l hw
2647
(9)
in which L1e, L2e, L3e, L4e, L5e, L6e, L7e, L8e, L1c and L2c express the locations of the heat sources and heat sink. Substituting Eqs. (7) and (8) into Eqs. (1), (3) and (4), yields the following expression for coefficient Am:
N mpx X keff hw l Am ð0Þ sin rv hv kw h[ hfg m ¼ 1 mp l " 2 # 2 h[ h[ h[ y y þ þ 1þ2 þ hv hv hv hv hv
uv ðx; yÞ ¼ 6qe
k k mp þ eff exp ðmpHw YÞ þ mp eff exp ðmpHw YÞ kw H[ kw H[ Am ðYÞ ¼ Bm k k exp ðmpHw Þ mp eff exp ðmpHw Þ mpHw mp þ eff kw H[ kw H[
ð17Þ
For configuration III, the non-dimensional temperature can be expanded in the form of an infinite Fourier series as:
qðX; YÞ ¼ A0 ðYÞ þ
N X
Am ðYÞ cos ðmpXÞ
The boundary layer form of the x-momentum equation for steady-state incompressible laminar vapor flow can be integrated to obtain the vapor pressure distribution:
(11)
m¼1
Zx
where
"
kw H[ A0 ðYÞ ¼ ðL2e L1e Þ Y þ keff Hw
(10)
Zh[
rv
#
0 ðh[ þhv Þ
(12)
and Am can be obtained from Eq. (10).
vðuv Þ2 dxdy þ rv vx Zx
Zh[
¼ 0 ðh[ þhv Þ
Zx
Zh[
0 ðh[ þhv Þ
vpv dxdy þ mv vx
vðuv vv Þ dxdy vy Zx
Zh[
0 ðh[ þhv Þ
v2 uv dxdy vy2 (18)
2.2. Vapor flow A parabolic velocity profile is used for vapor flow within the heat pipe. The velocity distribution is represented by a functional
Assuming the constant vapor pressure at each cross-section, and utilizing the boundary conditions at the beginning of the evaporator, uv ¼ 0 and pv ¼ pref, yields:
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N h mpx i 12mv keff hw X l 2 Am ð0Þ cos 1 3 p m l rv hv kw h[ hfg m ¼ 1 # " 3 h[ 4 h[ 38 h[ 2 17 h[ 6 þ þ16 þ þ 36rv Uv2 8 hv hv 3 hv 3 hv 5
utilizing the pressure boundary condition at the end of the condenser (x ¼l), p[ ¼ pv:
pv ¼ pref þ qe
N h mpx i m[ keff hw X l 2 Am ð0Þ cos cosðmpÞ 2 p m l r[ Kkw h[ hfg m ¼ 1 N 12mv keff hw X l 2 Am ð0Þ½cosðmpÞ 1 þpref þ qe 3 rv hv kw h[ hfg m ¼ 1 mp
p[ ¼ qe
(19) where Uv is obtained from Eq. (16). For configuration III, the interfacial velocity (normal to the liquidevapor interface), as shown in Eq. (15), becomes:
vv;I ðh[ Þ ¼
qI
rv hfg
¼
qe
rv hfg
" ðL2e L1e Þþ
qc qe ðL2e L1e Þ vv ððh[ þhv ÞÞ ¼ ¼ rv hfg rv hfg
N X
keff hw Am ð0Þcos kw h[ m¼1
mpx
#
For configuration III, the interfacial velocity (normal to the liquidevapor interface) for the liquid can be expressed as:
v[ ð0Þ ¼ 0 qI qc hfg r[
l
v[;I ðh[ Þ ¼ (20)
The vapor velocity profile and pressure distribution for configurations III and IV can be obtained in a similar manner. The temperature drop in the vapor can be related to the pressure drop in the vapor region by applying the Clapeyron equation and using the ideal gas law [1]:
RT 2 DTv ¼ Dpv hfg pv
(27)
(21)
where the vapor pressure drop is obtained from Eq. (19).
The velocity profile and pressure distribution in the wick for configurations III and IV can be calculated in a similar manner.
2.4. Capillary pressure Rice and Faghri [15] performed a limiting analysis of the reduction of the capillary pressure due to the increased resistance of the liquid film for a single idealized cylindrical pore, as presented in Fig. 2. The pressure gradient needed to drive the flow to the microfilm region is created by the change in surface curvature. The effective pore radius in a screen mesh wick is given by Faghri [1] as:
2.3. Liquid flow The continuity equation for the incompressible liquid flow can be integrated with respect to y to obtain the liquid axial velocity, u[(x):
Z0 h[
vu[ vv[ du[ dy ¼ v[;I h[ þ ¼ 0 vx vy dx
(22)
where v[,I denotes the interfacial velocity (normal to the liquidevapor interface) for the liquid and is related to the vapor interfacial velocity by:
r[ v[;I ¼ rv vv;I
(23)
The axial liquid velocity can be calculated by integrating Eq. (22) with respect to x using the velocity boundary condition at the beginning of the evaporator (x ¼ 0), u[ ¼ 0:
u[ ðxÞ ¼ qe
N mpx X l Am ð0Þsin 2 l r[ kw h[ hfg m ¼ 1 mp keff hw
(24)
(28)
r0 ¼
1 2N
(29)
where N is the screen mesh number. The effective pore radius can be used to calculate the static capillary pressure:
pcap;static ¼
2s[v r0
(30)
If there is no flow within a capillary tube, the effective capillary pressure is equivalent to the static capillary pressure. However, as shown by Ma et al. [17], the mass flow rate in a capillary tube can also affect the effective capillary pressure. From previous investigations, the mass flow rates are significantly reduced in the region where the change in surface curvature and the disjoining pressure effects both have comparable magnitudes in the pressure gradients. The disjoining pressure can be modeled by an equation presented in [16] for water:
" # d b pd ¼ r[ RTd ln a 3:3
(31)
The one-dimensional steady-state conservation of momentum for incompressible liquid flow in the wick can be expressed by Darcy’s law, assuming negligible inertial effects in comparison to viscous losses [16], as:
m u[ dp[ ¼ [ dx K
(25)
in which the permeability K is calculated for a mesh screen [1] as:
K ¼
d2 43
122ð1 4Þ2 1:05pNd 4z1 4
(26)
The liquid pressure is obtained by integrating Eq. (25) and
Fig. 2. An idealized single pore at the wick/liquid/vapor interfacial region.
M. Aghvami, A. Faghri / Applied Thermal Engineering 31 (2011) 2645e2655
2649
Table 1 Specifications of the heat pipe and the working fluid for four configurations. Configuration I
Configurations II & IV
Configuration III
Wall (copper)
Thickness (m) Thermal conductivity (W/(mK))
0.000265 400
0.0017 400
0.00089 400
Wick/Water
Thickness (m) Effective thermal conductivity (W/(mK)) Specific volume (m3/(kg)) Viscosity (Ns/m2) Latent heat (J/kg)
0.00014 1.3 0.0010078 0.00065 2406700
0.00075 1.2 0.001026 0.00037 2321400
0.00076 26 0.0010078 0.00065 2406700
Vapor (water)
Thickness (m) Specific volume (m3/(kg)) Viscosity (N/m2)
0.0023 19.523 0.0000096
0.01025 4.131 0.000011
0.00635 19.523 0.0000096
The disjoining pressure has a negligible effect when pd is equal to zero. At this value the film thickness is:
1=b 1 dd ¼ 3:3 a
(32)
The capillary pressure across the liquid/vapor interface is:
0 B pcap ¼ pv p[ ¼ s@
1 00
d 02
d þ1
3=2
1 C pffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 0 ðr0 dÞ d 2 þ 1
(33)
Eqs. (34) and (35) can be solved with the initial conditions:
dð0Þ ¼ 0:99999r0
(36)
d0 ð0Þ ¼ 1 109
(37)
pcap ¼ pcap;0
(38)
For each heat input, with an associated effective capillary pressure pcap,0, the solution will produce a minimum film
where the vapor and liquid pressures can be obtained from the analytical solutions. The pressure gradient is given by Ref. [15] considering constant vapor pressure, knowing the shear stress at the wall, and neglecting the inertial effects:
dpcap ¼ ds
Qinput 6reff m[ 2 3 r[ hfg N 2 Ah 2reff d d
(34)
where Qinput is the total heater power, and Ah is the area of the heater. Rearranging Eq. (33), the second derivative of d with respect to x is:
d00 ¼
2 3=2 p 6 cap d0 2 þ 1 4
s
3 ðr0 dÞ
1 7 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 d0 2 þ 1
(35)
Fig. 3. The variation of heat flux at the interface along the heat pipe with respect to the wall thermal conductivity, for configuration I (Qe ¼ 30 W, Tsat ¼ 40 C).
Fig. 4. Variation of wall temperature along the heat pipe with respect to the a) input power and b) heat source size, for configuration I (Tsat ¼ 40 C).
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Fig. 5. Effect of the input power on a) capillary pressure, b) vapor velocity, and c) vapor temperature, for configuration I.
thickness dmin . If dmin is greater than dd , then the overall capillary pressure is too large and needs to be reduced. If dmin is less than dd , then the disjoining pressure effects become significant, and it may be possible for the film to reach an equilibrium film thickness. Therefore, the maximum effective capillary pressure for a given heat input is reached when dmin is equal to dd .
Fig. 6. Variation of wall temperature along the heat pipe with respect to the a) input power, b) wall thermal conductivity, and c) heat source size, for configuration II (Tsat ¼ 71 C).
3. Results The specifications of the heat pipe wall, wick, and vapor core for the four configurations are given in Table 1.
M. Aghvami, A. Faghri / Applied Thermal Engineering 31 (2011) 2645e2655
2651
48 Present Analytical Experimental (Prasher, 2003)
46
T (C)
44
42
40
38 10
20
30
40
50
60
70
80
x (mm) Fig. 8. Wall temperature profile along the heat pipe, for configuration III (Qe ¼ 30 W, Tsat ¼ 40 C).
Fig. 7. Effect of the input power on a) capillary pressure, b) vapor velocity, and c) vapor temperature, for configuration II.
3.1. Configuration I: flat heat pipe with single heat source and sink at top The heat pipe, the evaporator and the condenser lengths are 0.04 m, 0.01 m and 0.015 m, respectively. The operating temperature is 40 C. Profiles of the heat flux along the axial direction and the influence of the solid thermal conductivity on the heat flux profiles are shown in Fig. 3. As expected, evaporation and condensation rates are not uniform, but instead vary along the axial direction. Even in the adiabatic section, heat transfer by evaporation or condensation takes place inside the heat pipe, as also
pointed out in [12,14]. The figure shows that as the solid thermal conductivity decreases, the evaporation (condensation) rate becomes more uniform along the axial direction and heat transfer by evaporation or condensation in the adiabatic section becomes negligible. Therefore, the assumption that neither evaporation nor condensation occurs in the adiabatic section is valid only if the effect of the axial wall conduction is negligible. The outer wall temperature distributions of the flat heat pipe for different input powers applied to the heat source are shown in Fig. 4a. The temperature increases faster from the condenser to the evaporator and the maximum temperature at the surface increases about 18% when the heat input is changed from 20 W to 50 W. Fig. 4b shows the jumps in the maximum wall temperature for 30 W power input due to shortening of the evaporator length. The ratio of the heat source length to the overall heat pipe length is an important parameter controlling the resistance to the spreading of heat from the evaporator to the condenser section. The smaller the heat source, the higher the maximum wall temperature. As the heater size is reduced, the heat flux in the evaporator region increases drastically. This causes a jump in the heat pipe maximum wall temperature. Fig. 5 demonstrates the variations of the capillary pressure, vapor mean velocity and temperature along the flat heat pipe for different input powers. It can be seen that the vapor velocity increases in the region corresponding to the evaporator section and decreases in the region corresponding to the condenser section, due to the vapor injection (increasing mass) and suction (decreasing mass) over the corresponding regions. As the mass flow rate is increased at the vapor/liquid interface by the increasing condensation of vapor, pressure drop and velocity increase considerably as the heat input increases. It should be noted that the pressure differences in the y-direction are negligible [13]. The vapor pressure decreases in the evaporator zone due to friction and acceleration of the vapor flow caused by mass injection from the wick, while the vapor pressure increases in the condenser zone owing to the deceleration of the vapor flow by mass suction. It can also be seen that the difference in vapor temperature is quite small.
3.2. Configuration II: flat heat pipe with multiple heat sources and sink at top There are four heat sources with a length of 0.0635 m beginning at x locations of 0.020, 0.1588, 0.2976 and 0.4364 m, respectively.
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Fig. 9. Effect of the input power on a) capillary pressure, b) vapor velocity, and c) vapor temperature, for configuration III.
The beginning of the heat sink is at an x location of 0.680 m and it has a length of 0.300 m. The rest of the outer wall is adiabatic, and the total length of the heat pipe is 1.000 m. The outer wall temperature distributions of the flat heat pipe for different input powers applied to the heat source are shown in Fig. 6a. The temperature increases faster from the condenser to the evaporator, and the maximum temperature at the surface increases
Fig. 10. Variation of wall temperature along the heat pipe with respect to the a) input power; b) wall thermal conductivity and c) heat source size, for configuration III (Tsat ¼ 40 C).
about 3% when the heat input is changed from 50 W to 70 W. Fig. 6b and c shows the effect of the wall thermal conductivity and heat source size on the wall temperature. For the same heat input level, the heat pipe with higher wall thermal conductivity is more
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Fig. 11. Wall temperature profiles for configuration IV with a) single heat source, b) two heat sources, c) three heat sources, and d) four heat sources.
effective in dissipating heat. The smaller the heat source, the higher the maximum wall temperature. Fig. 7 shows the variations of the capillary pressure, vapor mean velocity and temperature along the flat heat pipe, for different input powers. As the mass flow rate is increased at the vapor/liquid interface by increasing condensation of vapor, pressure drop and velocity increase considerably as the heat input increases. The vapor temperature difference is negligible. 3.3. Configuration III: flat heat pipe with heat source at the bottom and heat sink on the top Fig. 8 presents the comparison of the thermal model with the experimental data of Prasher [10]. The length of the heat pipe was
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0.089 m, and the testing was done for a heat source 0.01 m in length at the center of the heat pipe and a saturation temperature of 40 C. The results for the wall temperature profile agree with the experimental data. Fig. 9 demonstrates the variations of the capillary pressure, vapor mean velocity and temperature along the flat heat pipe, for different input powers. As the mass flow rate is increased at the vapor/liquid interface by increasing the condensation of vapor, the pressure drop and velocity rise considerably with heat input. The outer wall temperature distributions of the flat heat pipe for different input powers applied to the heat source are shown in Fig. 10a. The temperature increases faster from the condenser to the evaporator and the maximum temperature at the surface increases about 11% when the heat input is changed from 30 W to 50 W. Fig. 10b and c shows the effect of the wall thermal conductivity and heat source size on the wall temperature. For the same heat input level, the heat pipe with higher wall thermal conductivity is more effective in dissipating heat. The smaller the heat source, the higher the maximum wall temperature.
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A complete two-dimensional numerical analysis of the flat heat pipe, which consists of the vapor flow in the core region, the liquid flow in the wick, and the heat conduction in the wall, is presented in Ref. [18] to validate the analytical model. There are four heat sources with a length of 0.0635 m beginning at x locations of 0.020, 0.1588, 0.2976 and 0.4364 m, respectively. The beginning of the heat sink is at an x location of 0.680 m, and it has a length of 0.300 m. The rest of the outer wall is adiabatic. The total length of the heat pipe is 1.000 m.
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Fig. 13. Vapor axial velocities for configuration IV with a) single heat source, b) two heat sources, c) three heat sources, and d) four heat sources.
A comparison of the outer wall temperature profiles for different heating configurations of the present analytical model and the numerical results of Rice and Faghri [15] is shown in Fig. 11. Fig. 11a presents the outer wall temperatures resulting from the application of 97 W to evaporator 1 at an interfacial saturation temperature of 76 C. The saturation temperature at the liquidevapor interface is assumed to be constant and nearly equal to the centerline or axis temperature for the heat pipe presented by Rice and Faghri [15]. Fig. 11b shows the outer wall temperatures resulting from the application of 98 W to evaporators 1 and 2 at a saturation temperature of 71 C. Fig. 11c presents the outer wall temperatures resulting from the application of 75 W to evaporators 1, 2, and 3 at a saturation temperature of 82 C. Fig. 11d presents the outer wall
temperatures resulting from the application of 50 W to evaporators 1, 2, 3, and 4 at a saturation temperature of 71 C. Fig. 12 shows the capillary pressure resulting from the application of 97 W to evaporator 1 and a comparison with the numerical results of Rice and Faghri [15]. A comparison of the vapor axial mean velocities for different heating configurations of the numerical simulations [18] and the analytical results is presented in Fig. 13. It can be seen that these results match up very closely. 3.5. Capillary pressure and limitations The maximum capillary pressure versus heating load for a lowtemperature water heat pipe is presented in Fig. 14, which is the intersection of the root of disjoining pressure and the minimum film thickness. The pressure drop in the wick must be known in order to calculate the dry-out limitations of a heat pipe. For the cases listed in Table 2, the total evaporator heat input is increased until the total capillary pressure drop bounds the maximum capillary pressure in Fig. 14. Table 2 Capillary dry-out limitations of the flat heat pipe with multiple heat sources (configuration IV). Qinput, dry out, W
Evaporator heat input
Fig. 14. The maximum effective capillary pressure as a function of heat input.
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223 266 333 446 245 272 305
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4. Conclusions A simplified linked thermal-fluid model is presented in order to analytically predict the thermal and hydrodynamic performance of flat heat pipes with different heating and cooling configurations. The results for the flat heat pipes show that, the assumptions that evaporation and condensation occur uniformly in the axial direction, that evaporation occurs only in the evaporator section, and that condensation occurs only in the condenser section, are valid only if the solid thermal conductivity is small and the axial wall conduction can be neglected. Higher evaporative heat input increases maximum surface temperature, pressure drop and fluid velocities due to the increasing mass flow rate at the vapor/wick interface. Shortening the size of the heat source can degrade the thermal performance of the heat pipe, due to the jump in the peak wall temperature. The maximum effective capillary pressure can be decreased while the static capillary pressure remains constant by increasing the heat input. The capillary pressures needed to support a steady-state operation can be calculated and compared to the computed maximum capillary pressures for given heat inputs to analytically determine the dry-out limitations of the heat pipe. Nomenclature A0, Am d H h hfg K k L l N p Q q r0 T
Fourier coefficients diameter of the mesh fiber, m dimensionless thicknesses thickness, m latent heat of vaporization, J kg1 permeability, m2 thermal conductivity, W m1 K1 dimensionless lengths lengths, m mesh number pressure, Pa heat transfer rate, W heat flux, W m2 effective pore radius, m temperature, K q dimensionless temperature u x velocity, m s1 v y velocity, m s1 x, y, z, s coordinates, m X, Y, Z dimensionless coordinates Greek letters ratio between heat source and heat sink lengths dynamic viscosity, Pa s density, kg m3 surface tension, N m1 porosity film thickness, m
g m r s 4 d
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Subscripts and superscripts a adiabatic c condenser cap capillary eff effective e evaporator I interface [ liquid sat saturation v vapor w wall
References [1] A. Faghri, Heat Pipe Science and Technology. Taylor and Francis, Washington, DC, 1995. [2] R. Hopkins, A. Faghri, D. Khrustalev, Flat miniature heat pipes with micro capillary grooves, ASME J. Heat Transfer 121 (1999) 102e109. [3] S.J. Kim, J.K. Seo, K.H. Do, Analytical and experimental investigation on the operational characteristics and the thermal optimization of a miniature heat pipe with a grooved wick structure, Int. J. Heat Mass Transfer 42 (2003) 3405e3418. [4] L. Lin, R. Ponnappan, J. Leland, High performance miniature heat pipe, Int. J. Heat Mass Transfer 45 (2002) 3131e3142. [5] B.R. Babin, G.P. Peterson, D. Wu, Steady-state modeling and testing of a micro heat pipe, ASME J. Heat Transfer 112 (1990) 595e601. [6] D. Khrustalev, A. Faghri, Thermal analysis of a micro heat pipe, ASME J. Heat Transfer 116 (1994) 189e198. [7] J.P. Longtin, B. Badran, F.M. Gerner, A one-dimensional model of a micro heat pipe during steady-state operation, ASME J. Heat Transfer 116 (1994) 709e715. [8] K. Vafai, W. Wang, Analysis of flow and heat transfer characteristics of an asymmetrical flat plate heat pipe, Int. J. Heat Mass Transfer 35 (1992) 2087e2099. [9] F. Lefèvre, M. Lallemand, Coupled thermal and hydrodynamic models of flat micro heat pipes for the cooling of multiple electronic components, Int. J. Heat Mass Transfer 49 (2006) 1375e1383. [10] R.S. Prasher, A simplified conduction based modeling scheme for Design Sensitivity Study of thermal solution utilizing heat pipe and vapor chamber technology, ASME J. Electron. Packag. 125 (2003) 378e385. [11] K.C. Leong, C.Y. Liu, K.H. Sun, Vapor pressure distribution of a flat plate heat pipe, Int. J. Heat Mass Transfer 23 (6) (1996) 789e797. [12] K.H. Do, S.J. Kim, S.V. Garimella, A mathematical model for analyzing the thermal characteristics of a flat micro heat pipe with a grooved wick, Int. J. Heat Mass Transfer 51 (2008) 4637e4650. [13] N. Zhu, K. Vafai, Vapor and liquid flow in an asymmetrical flat plate heat pipe: a three-dimensional analytical and numerical investigation, Int. J. Heat Mass Transfer 41 (1) (1998) 159e174. [14] U. Vadakkan, S.V. Garimella, J.Y. Murthy, Transport in flat heat pipes at high heat fluxes from multiple discrete sources, ASME J. Heat Transfer 126 (2004) 347e354. [15] J. Rice, A. Faghri, Analysis of screen wick heat pipes, including capillary dryout limitations, J. Thermophys. Heat Transfer 21 (3) (2007) 475e486. [16] A. Faghri, Y. Zhang, Transport Phenomena in Multiphase Systems. Elsevier Inc., 2006. [17] H.B. Ma, G.P. Peterson, D.M. Pratt, Disjoining pressure effect on the wetting characteristics in a capillary tube, Nanoscale Microscale Thermophys. Eng. 2 (1998) 283e297. [18] M. Aghvami. Thermal-fluid analysis of heat pipes, M.S. thesis, University of Connecticut, 2010.