Three-dimensional simulation on heat transfer in the flat evaporator of miniature loop heat pipe

International Journal of Thermal Sciences 54 (2012) 188e198

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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Three-dimensional simulation on heat transfer in the flat evaporator of miniature loop heat pipe Xianfeng Zhang a, b, Xuanyou Li c, Shuangfeng Wang a, c, * a

Key Laboratory of Enhanced Heat Transfer and Energy Conservation of the Ministry of Education, South China University of Technology, Guangzhou 510640, PR China CETC No. 38 Research Institute, Hefei 230088, PR China c Industrial Energy Conservation Center of Shandong Academy of Sciences, Jinan 250013, PR China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 March 2011 Received in revised form 29 November 2011 Accepted 1 December 2011 Available online 31 December 2011

A complete three-dimensional model is developed to investigate the flow and thermal transport in the flat evaporator of a miniature loop heat pipe. The model is involved in flow and heat transfer in the wick and vapor groove, which is conjugated with heat conduction in the wall. The pressure and temperature of boundary conditions are coupled with the operating status of the LHP. Effect of structural parameters of vapor groove in the flat evaporator is investigated in details under the specific heat flux. The results show that heat transfer coefficient of evaporator with vapor groove inside the sintered wick is greater than that inside the solid wall when the wick is in fully saturated operating status. The results predict that the optimum of form factor(a) of vapor groove is 1, and when ratio(b) between interval of vapor grooves and width of vapor groove is the less, wall temperature of evaporator (i) is the lower. The suitable range of b is from 0.5 to 1. Ó 2011 Elsevier Masson SAS. All rights reserved.

Keywords: Miniature loop heat pipe Flat evaporator Three-dimensional simulation Heat transfer Vapor groove

1. Introduction Due to cooling requirements for the space and commercial electronic devices with high heat flux, loop heat pipe (LHP), introduced in 1980s’ by Russian scientist Y. F. Maydanik, has been developed as an attractive system in recent decades [1e4]. As a passive cooling device, LHP has some unique features, such as high heat capability, efficient operation in the gravity field, long distance heat transport capability and flexibility in design. Compared with conventional heat pipes, LHP offers more applications for thermalemanagement in both space and ground environments. The detailed reviews on operating characteristics and working mechanisms in the LHP can be found in Maydanik [1], Ku [2] and Launay et al. [3]. To cool the compact electronic devices, the miniaturization of LHP has been devoted many efforts to and can offer wide flexibility in the structure of evaporator. Two types of LHP with traditional cylindrical evaporator and flat evaporator are developed. And flat evaporator can be considered as an optimum design for some

* Corresponding author. Key Laboratory of Enhanced Heat Transfer and Energy Conservation of the Ministry of Education, South China University of Technology, Wushan RD.,Tianhe District, Guangzhou, Guangdong 510640, PR China. Tel.: þ86 2022236929. E-mail address: [email protected] (S. Wang). 1290-0729/$ e see front matter Ó 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.12.002

advantages, such as more convenience to be integrated into compact enclosures of electronics system, more opportunities for miniaturization and better heat transfer efficiency [5]. Z. C. Liu [5], Ji Li [6] and Singh et al. [7] presented some studies on miniature LHP with flat evaporator, which were layer structure between compensation chamber and wick. Evaporator with the layer structure has good heat transfer efficiency due to less heat leak but its thickness is above 10 mm. This size is too big to be applied to some narrow space in notebook computer and GPU. A compact miniature LHP with series structure evaporator has been presented by Wang [8,9]. The sketch of LHP and evaporator is shown in Fig. 1. The flat evaporator is made of sintered copper powder and compensation chamber is located in the extension of the evaporator. The LHP with 8 mm thick evaporator has been applied to cooling of GPU [10]. Plenty of investigations on operating characteristics of LHP have been carried out with theoretical analyses and experimental studies. Tests and simulations of the whole loop have been extensively focused on [6e12]. However, there has been relatively little research conducted on the operating principle in the evaporator. As the core component in LHP, the evaporator with the capillary porous wick can offer the driving force of LHP through phase change of working fluid. It is worthwhile to investigate the thermodynamic behavior of working fluid in the wick. Cao and Faghri [13,14] presented analytical solution and numerical analysis on the

X. Zhang et al. / International Journal of Thermal Sciences 54 (2012) 188e198

Nomenclature

dynamic viscosity, Ns/m2 porosity thickness of wall, m local resistance coefficient

m 3

Cp d H h hfg K k L _ m n P Q q R S T u V W X Y Z

specific heat, J/(kg K) diameter, m height, m heat transfer coefficient, W/(m2 K) latent heat of vaporization, J/kg permeability, m2 thermal conductivity, W/(m K) length, m mass flux, kg/s unit normal vector pressure, Pa heat load, W heat flux, W/m2 thermal resistance, K/W surface area, m2 temperature, K velocity component, m/s velocity vector, m/s width, m Cartesian coordinate, m Cartesian coordinate, m Cartesian coordinate, m

189

d x

Subscripts cc compensation chamber cd condenser e outlet of vapor groove eff effective h heating wall i inlet in inner surface l liquid max maximum value min minimum value out outer surface s solid sat saturation sc subcooled region t total sink heat sink tube tube wall v vapor w wick

Greek symbols r density, kg/m3

wick saturated with liquid in the CPL evaporator and studied the flow and heat transfer of working fluid in the wick. Demidov and Yatsenko [15], Figus [16], Kaya and Goldak [17] and Chuan Ren et al. [18] developed the two-dimensional numerical model with the inverted meniscus principle and analyzed the phenomena of vapor zone formed in the wick. Ji Li et al. [19] presented the quasi threedimensional model and investigated the thermal characteristics of LHP under low heat loads, but in this model, the study on the flow and heat transfer in the vapor groove is insufficient. There are some experimental researches on the structure and material of wick and heat transfer in porous media [20e22]. Based on the works mentioned above, it is seen that the operating mechanics and thermal transport characteristics of working fluid in the evaporator has been widely concerned. However, investigation of heat transfer in the wick coupled with the loop’s operation is insufficient, and structural optimization of evaporator hasn’t been made profound study on. Both of them are very important. Because the evaporator is the core part of LHP as the driving component, its performance has significant influence on the operating characteristics of the LHP. In this work, a complete three-dimensional model is developed to analyze the thermal/flow characteristics in the evaporator of LHP. Temperature and pressure in the compensation chamber and evaporator outlet are determined by coupling the heat transfer and flow in the whole loop. The governing equations for the wall, wick and vapor groove are solved as a conjugated problem. Heat transfer and flow in the flat evaporator presented by Wang [8,9] are investigated in details. Effect of structural parameters of vapor groove is focused on under the low and medium heat flux. 2. Mathematical models The schematic of computational domain of the evaporator in LHP is shown in Fig. 2, which is based on a vapor groove and consists of the wall, wick and vapor groove. Because of the asymmetry of structure, the model is calculated in a half domain as illustrated in

Fig. 2(a). In this work, the wick is perfectly saturated with the liquid working fluid. Heat load is applied on the top wall of the evaporator. The liquidevapor interface is thought to be fixed at the interface between the wick and vapor groove with zero thickness. As shown in Fig. 2(b), there are three liquidevapor interfaces located in YZ-plane, XY-plane and XZ-plane. The mathematical model is similar to the previous studies [13e19] and based on the following assumptions: (1) (2) (3) (4)

the steady process is only considered. the capillary structure is homogeneous and isotropic. the gravitational effect and radiation is negligible. the working fluid is Newtonian and has constant properties at each phases, and (5) there is local thermal equilibrium between the porous structure and working fluid. According to the upper assumption, the governing equations of heat and mass transfer in the wick, vapor groove and wall can be written as follow. (i) The vapor groove region For the steady flow in the vapor groove, the continuity, momentum and energy equations are:

V,ðrv VÞ ¼ 0

(1)

V,ðrv VVÞ ¼ VP þ Vðmv VVÞ

(2)


V, rv VCp;v T ¼ V,ðkv VTÞ

(3)

The density of the vapor is calculated by the ideal gas law,

rv ¼

p RT

(4)

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X. Zhang et al. / International Journal of Thermal Sciences 54 (2012) 188e198

Fig. 1. Schematic of LHP and evaporator.

R is the ideal gas constant, which is the universal gas constant divided by the molecular weight. (ii) The wick region In the wick region, the volume-averaging method is applied in the continuity, momentum and energy equations. The wick is assumed to be saturated with incompressible, liquid working fluid. For the momentum equation, the viscous interaction between fluid and the solid matrix is modeled by Darcy’s Law. The governing equations in the wick become:

V,ðrl VÞ ¼ 0 V,ðrl VVÞ ¼ VP þ Vðml VVÞ      V, rl VCp;l ¼ V, keff VT

ml K

V

here, V is the superficial velocity in the wick. keff is the effective thermal conductivity of wick. There are some mathematical models to estimate keff, such as the parallel model, series model, Maxwell’s model and Chaudary and Bhandari’s model [12,19]. Compared with experimental data of [21,22], the Chaudary and Bhandari’s model shows better agreement. So that model is adopted in this work and written as follow, g

keff ¼ ðkmax Þ ,ðkmin Þ1

g

where,

(5)

kmax ¼ 3 ,kl þ ð1  3 Þ,ks

(6)

kmin ¼ ks ,kl =½3 ,ks þ ð1  3 Þ,kl 

(7)

l ¼ 0:42

(8)

X. Zhang et al. / International Journal of Thermal Sciences 54 (2012) 188e198

191

vP vT vV ¼ 0; ¼ 0; ¼ 0 vX vX vX

(12)

At X ¼ Lw, for the vapor groove

Pe ¼ Pi þ DPloop ;

vT ¼ 0 vX

(13)

here, DPloop is pressure drop from vapor line to compensation chamber. At Y ¼ Hw þ 2d, for heated region ((Lw  Lv) < X < Lw)

ks

 vT  ¼ q vn

(14)

At Y ¼ Hw þ 2d, for other region (X < (Lw  Lv))

vT ¼ 0 vX

(15)

The walls at Y ¼ 0 is adiabatic. At Z ¼ 0 and Z ¼ Ww/2, where are the symmetry plane,

vP vT vV ¼ 0; ¼ 0; ¼ 0 vZ vZ vZ

(16)

Three liquidevapor interfaces, such as XY-plane, YZ-plane and XZ-plane in Fig. 2(b), are located at the interfaces between wick and vapor groove. At these interfaces, the conservation of mass and energy are applied with the following equations:

Fig. 2. Numerical domain and coordinate system.

(iii) The solid wall region For the solid wall, the steady-state energy balance is determined by heat conduction.

V,ðks VTÞ ¼ 0

(9)

un;l rl ¼ un;v rv

(17)

    vT  vT  ¼ un;l rl hfg  keff   kv  vn G vn G

(18)

And the interface temperature is equal to the saturation temperature which is calculated by the ClausiuseClapeyron equation.

Tl ¼ Tv ¼ Tsat

(vi) Boundary conditions Heat transfer among the wall, wick and vapor groove is coupled as a coupled problem. The boundary conditions of momentum and energy are shown as following. At X ¼ 0, for the wick region (d < Y < Hw þ d),

Ti ¼ Tcc ; Pi ¼ Psat ðTi Þ

(19)

To determine Pi, Ti and DPloop, flow and heat transfer in the loop need be considered. As the LHP operates stably, the total pressure drop (DPt) in the loop is equal to capillary force formed at the liquidevapor interface in the wick.

DPt ¼ DPvg þ DPvl þ DPcd þ DPll þ DPcc þ DPw þ DPg

(20)

(10)

At X ¼ 0, for the wall region, (Hw þ d < Y < Hw þ 2d and Y < d)

vT ¼ 0 vX

360

(11)

At X ¼ Lw, for the wick and the wall where is adiabatic

Experimental Result Present Numerical Result

Parameters

Value

Unit

Length  Wide  Height of the wick (Lw  Ww  Hw) Length  Wide  Height of vapor groove (Lv  Wv  Hv) Thickness of Wall(d) Wick permeability Wick porosity Thermal Conductivity of Cu Inside/outside diameter of Vapor and liquid pipe Length of condenser Length of vapor line Length of liquid line

35  2  4

mm

30  1  1

mm

1 5.2  1011 50％ 387.6 3/4 130 140 180

mm m2 e W/(m K) mm mm mm mm

Th (K)

350

Table 1 Structure parameters of LHP and evaporator in the numerical domain.

340

330

320

40

60

80

100

Heat Load (W) Fig. 3. Temperature of heated wall comparison between the numerical and experimental results under different heat load.

192

X. Zhang et al. / International Journal of Thermal Sciences 54 (2012) 188e198

where, DPvg is the pressure drop through the vapor groove, DPvl is the pressure drop through the vapor line, DPcd denotes the pressure drop through the condenser, DPll is the pressure drop through the liquid line, DPcc is the local pressure drop in the compensation chamber, DPw is the pressure drop through the wick, DPg is the pressure drop by the gravity. In this work, DPvg and DPw can been calculated in the model. Others need be calculated by the other theory. And

DPloop ¼ DPt  DPvg  DPw

(21)

If Q ¼ 100 W and d ¼ 3 mm, the maximum Reynolds Number in the vapor line and liquid line is about 18. And the vapor and liquid flow is laminar. The pressure drop in the vapor line and liquid line is calculated as:

Dp ¼

_ 128mmL 4 prd

(22)

_ denotes the mass flow rate in the loop and can be where, m calculated as follows,

_ ¼ m

Q hfg

(23)

In the condenser,

DPcd ¼ DP24;cd þ DPsc;cd

(24)

DP24,cd and DPsc,cd represent the pressure drop in the two-phase region and subcooled region of condenser, respectively. To simplify the computational process, DP24,cd is computed with properties of vapor [11,19].

Dp24;cd ¼

_ 24 128mv mL prd4

(25)

where, L24 is the length of two-phase region in the condenser and computed according to Nima Atabaki [12].

In the compensation chamber,

Dpcc ¼ x

ru2

(26)

2

According to Chernysheva et al. [11], temperature of working fluid in compensation chamber is obtained through the following functions:

 Tcc ¼ Tsink þ

 X dT  1 1  DP $Q þ þ Rtube þ hout Sout hin Sin dP T i

(27)

i

3. Numerical process In this model, the finite volume method is introduced to solve the governing equations with the boundary conditions in different regions. The SIMPLE algorithm is used for the velocityepressure coupled relations among the governing equations. The upwind scheme is applied for the convective term. The coupled problems are considered, such as flow and heat transfer between liquid and vapor, heat transfer among the wick, vapor groove and wall. The overall numerical procedures can be summarized as follows: 1. Determine the temperature (Ti) and pressure (Pi) in the compensation chamber according to Eqs. (19)e(27). _ 1 Þ of the loop from Eq. (23) and 2. Calculate the mass flow rate ðm compute the pressure drop from the exit of evaporator to compensation chamber using Eqs. (20) and (21). Determine the pressure of vapor groove outlet from Eq. (13). 3. Specify the thermophysical properties and boundary conditions. Initialize the field of temperature, pressure, density and velocity. 4. Solve the mass and momentum equations (Eqs. (1),(2),(5) and (6)) over the wick and vapor groove. 5. Solve the energy equation (Eqs. (3),(7) and (9)) over the whole regions.

Fig. 4. Temperature contours at Z ¼ 0 and Z ¼ Ww/2.

X. Zhang et al. / International Journal of Thermal Sciences 54 (2012) 188e198

a 36000

Y=0.0015 Y=0.0025 Y=0.0035 Y=0.0045

Pressure (Pa)

6. Repeat steps 4e5 until residuals of mass, momentum and energy equations are within 103. _ e Þ in the outlet of vapor groove 7. Calculated the mass flow ðm _ e =m _ e j  5%. If it is _ 1m according results of steps 4e5. Check jm _1 ¼ m _ e . And compute the not satisfied, calculate DPloop with m pressure of vapor groove outlet (Pe) and update this boundary condition. 8. Return to step 4 until all criteria are satisfied.

193

4. Results and discussions 4.1. Model validation

35000

According to results in Section 4.1, numerical results and experimental data are coincident well under heat load of 60 W to 80 W. The following numerical cases are conducted with heat load of 80 W (q ¼ 10 W/cm2). Fig. 4 shows the temperature contours at Z ¼ 0 and Z ¼ Ww/2 where are located in the symmetry plane of computational domain. It is seen that temperature distributions of the wall at Z ¼ 0 and Z ¼ Ww/2 are similar. Temperature in the top wall increases smoothly in X direction and the highest temperature of the evaporator is 350.5 K, where is on the top surface. Due to the effect of evaporation, temperature on the interface of wick and vapor groove is lower than that of wick below vapor groove. From Fig. 4(a) and Fig. 4(b), it is found that temperature distributions in the wick from bottom wall to the region below vapor groove are similar at Z ¼ 0 and Z ¼ Ww/2 and there is slight temperature

0.01

0.02

0.03

X (m)

Pressure

b Y=0.0015 Y=0.0025 Y=0.0035 Y=0.0045

Temperature (K)

348

347

346

345

0

0.01

0.02

0.03

X (m)

Temperature

c

Y=0.0015, Liquid Y=0.0025, Liquid Y=0.0035, Liquid Y=0.0045, Liquid Y=0.0045, Vapor

0.0003

20

15

0.0002 10

0.0001

0

5

0

0.01

0.02

0.03

0

X (m)

Velocity Fig. 5. Flow and temperature profiles for different height at Z ¼ Ww/2.

Vapor Velocity (m/s)

4.2. Field of flow and temperature in the evaporator

0

Liquid Velocity (m/s)

To validate the model, the numerical solutions are compared with the experimental results in [9]. In the experiment, the wick is made of sintered copper powder and structural parameters of LHP and evaporator in the computational region are given in Table 1. Condensing line in the LHP is cooled by isothermal water jacket and temperature of cooling water is 298 K. Water is the working fluid, and thermal properties taken in the computations are: kl ¼ 0.6 W/ (m K), kv ¼ 0.0261 W/(m K), ml ¼ 1.003  103/(m s), mv ¼ 1.34  105 kg/(m s) and hfg ¼ 2.424  106 J/kg. According to Eq. (8), effective thermal conductivity of the wick is 10.2 W/(m K). The details of experimental setup are presented in [9]. Fig. 3 shows the comparison of temperature in the heated wall of evaporator between the numerical results and the experimental ones under different heat load. In Fig. 3, heat flux density is from 3.75 W/cm2 to 12.5 W/cm2 with heat load from 30 W to 100 W. From Fig. 3, it can be seen that the numerical result is coincident with the experimental data well with heat load of 60 W to 80 W. Under low heat load (Q < 60 W), the numerical results are lower than the experimental results. For high heat load (Q > 80 W), the result is adverse. For the present model, heat leak to compensation chamber and environment through evaporator wall is neglected. Under the low heat load, mass flux in the loop is smaller. The fraction of heat leak to compensation chamber and environment is larger in the experiments with lower operational efficiency in the LHP. This results that temperature of heated wall in the experiments is higher. Because the assumption of the liquidevapor interface fixed in the surface of wick is reasonable under low heat load, the present model is considered to be valid. Mass flux in the loop rises with heat load increasing. The efficiency of heat transfer in the evaporator is improved with less heat sink. Under high head load (Q > 80 W), evaporation interface area enlarges and liquidevapor interface can be formed inside the wick. Efficiency of heat transfer in the evaporator is enhanced. This results that results of the present model is lower than the experimental ones. The present model is invalid.

194

X. Zhang et al. / International Journal of Thermal Sciences 54 (2012) 188e198

a

b

0.006

5X10 m/s

Y (m)

0.004

0.002

Y (m)

351 350.7 350.4 350.1 349.8 349.5 349.2 348.9 348.6 348.3 348 347.7 347.4 347.1 346.8 346.5 346.2 345.9 345.6

0.00

5

0.00

4

0.00

3

0.00

2

0. 00

0.00

0 0

Z(

1 0

m)

1

X

0.001

Z (m)

temperature contours

c

3-D velocity vector in the wick

d

0.005

10 -4 m/s

12 m/s

Y (m)

0.005

0.003

Y (m)

0.0045

0.001

0.0008

0.004

0.0006

Z(

m)

X

0.001 0

3-D velocity vector in the vapor groove

Z (m)

0.001

Two-dimensional velocity vector with the Y and Z component

Fig. 6. Temperature contours and velocity vector at X ¼ 0.02 m.

difference along X direction in this region. And it is clear that the evaporator has good temperature uniformity in the Z direction. Fig. 5 represents flow and temperature distributions for different height in the symmetry plane of Z ¼ Ww/2. Y ¼ 0.0015 m, 0.0025 m and 0.0035 m, which are the distance from the bottom wall, are located in the wick. Y ¼ 0.0045 m passes through the center line of vapor groove. As shown in Fig. 5(a), in the region of Y  0.0035 m pressure distribution in X direction is the same. There is small pressure drop in wick from X ¼ 0 m to X ¼ 0.035 m and total

pressure drop is about 129 Pa when Y  0.0035 m. From Fig. 5(a), it is also seen that pressure for Y ¼ 0.0045 m jump abruptly at x ¼ 0.005 m, where is the location of the evaporation interface in the YZ-plane. This results in abrupt pressure rise of 1060 Pa which is formed by the capillary force in the evaporation interface. From Fig. 5(b), it is found that temperature differences among another three lines are little and less than 1 K where X > 0.005 m and Y  0.0035 m. When X > 0.005 m, temperature along Y ¼ 0.0045 m keeps on increasing due to the heat from the top wall. Velocity

X. Zhang et al. / International Journal of Thermal Sciences 54 (2012) 188e198

a

195

b wall

wall

Vapor Groove

wick

wick

evaporator (i)

evaporator (ii)

Fig. 7. Evaporator with different vapor groove.

distribution is shown in Fig. 5(c). The signs ‘liquid’ and ‘vapor’ represent the state of working fluid and the corresponding velocity distributions are pointed to the ‘liquid velocity’ axis and the ‘vapor velocity’ axis, respectively. From Fig. 5(c), it is illustrated that where x ¼ 0.005 m velocity along Y ¼ 0.0045 m suddenly increases from 1.18  104 m/s to 0.162 m/s with phase change from liquid to vapor. This is the result of big ratio of density between liquid and vapor. From Fig. 5(c), it can be found that velocity is close to 20 m/s at the outlet of vapor groove, while the maximum of velocity in the wick is about 3.5  104 m/s. Fig. 6 illustrates the temperature contours and velocity vector in the plane of X ¼ 0.02 m. In Fig. 6(a), temperature distributions are 0.006

356.4 355.8 355.2 354.6 354 353.4 352.8 352.2 351.6 351 350.4 349.8 349.2 348.6 348 347.4 346.8 346.2 345.6

Y (m)

0.004

0.002

Fig. 9. Average temperature of the heated surface with different a.

similar with the one in the Fig. 4. Temperature at the evaporation interface is the smallest and there is little temperature difference in the region below the vapor groove. Three-dimensional velocity vector in the wick is given in Fig. 6(b). It illustrates that large amounts of liquid flows in the X direction and the X component of velocity is the largest. Fig. 6 (c) shows three-dimensional velocity vector in the vapor groove. It is seen that velocity of working fluid in the vapor groove is far greater than the one in the wick. To analyze the flow on the evaporation interface, two-dimensional velocity vector with the Y and Z component is shown in Fig. 6(d). It’s seen that the Y-component of velocity is greater than the Z component of one with attribution of the evaporation of working fluid in the liquidevapor interface. The major portion of working fluid flows to the XY-plane. For this case fraction of mass flux in the XY-plane to total evaporation flux in all planes is 96.37%, while the fractions in the XZ-plane and in the YZ-plane are 2.21% and 1.42%, respectively. Due to low evaporation flux in the XZ-plane, temperature gradient near the XZ-plane in the wick is small. This results in the little

a

b

0 0

0.001

Z (m) Fig. 8. Temperature contours at x ¼ 0.02 m for the evaporator (ii).

Fig. 10. Temperature contours in the plane of x ¼ 0.02 m for a ¼ 0.5 and a ¼ 2.

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X. Zhang et al. / International Journal of Thermal Sciences 54 (2012) 188e198

between wick and vapor groove. Therefore, the structural parameters of vapor groove are paid particular attention in this work. The following cases are also carried out under q ¼ 10 W/cm2. There are two types of evaporators as shown in Fig. 7, such as vapor groove machined inside the wick (Fig. 7(a)) and inside the wall (Fig. 7(b)). This results in different heat transfer processes in the evaporator. In this work, the case of the evaporator (ii) is that the top wall is thickened with 1 mm and the apparent size of the evaporator is fixed. Fig. 8 illustrates temperature contours in the plane of X ¼ 0.02 m for the evaporator (ii). Compared with the case of the evaporator (i) in Fig. 6(a), temperature of top wall in the evaporator (ii) is higher and the maximum is 356.2 K in Fig. 8. Because the evaporating interface for the evaporator (ii) is only located at the bottom of vapor groove, there is bigger temperature gradient in the wick where is below vapor groove. Heat is transported to the evaporating interface by this gradient in the wick. This results in higher temperature of working fluid in the wick as shown in Fig. 8. Accordingly, temperature of heating wall for evaporator (ii) is higher. According to definition in [22], heat transfer coefficient (h) of evaporator is defined as

Table 2 Fraction of mass flux at different evaporating surfaces.

a

XY-plane (%)

YZ-plane (%)

XZ-plane (%)

0.5 0.8 1 1.2 1.5 2

89.01 94.96 96.37 96.81 97.41 97.73

1.25 1.32 1.42 1.57 1.72 1.84

9.74 3.72 2.21 1.62 0.87 0.43

h ¼ q=ðTh  Tv Þ

Fig. 11. Average temperature of the heated surface for different b.

temperature difference in the region below vapor groove as shown in Figs. 4 and 6(a). 4.3. Effect of the vapor groove structure From the results in Section 4.2, it is found that there are notable change for temperature, velocity and pressure on the interface

where, Th is temperature of heated wall and Tv is vapor temperature in the outlet of vapor groove. Evaporator (i) and (ii) are 2.802 W/(cm2K) and 2.51 W/(cm2K), respectively. The trend of h is opposite to results of experiments in [22], which are conducted with sintered nickel wick. The possible reason is that the liquidevapor interface was formed inside the wick when the experiment was conducted with higher heat load. The wick between vapor grooves is partly filled with vapor for the evaporator (i). This resulted in different heat transfer processes in the evaporator with result of the present model. This problem needs to be solved by a new mathematical model. In this work, effect of sectional shape of vapor groove is investigated by changing height of vapor groove while other structural parameters of evaporator are fixed. Form factor (a) is defined by ratio of height and width of vapor groove, and a ¼ Hv/Wv. Fig. 9 shows average temperature of heated wall in the evaporator for different a. It is found that average temperature decreases firstly,

b 0.006

c 0.006

0.006

Y (m)

0.004

0.002

0

351 350.7 350.4 350.1 349.8 349.5 349.2 348.9 348.6 348.3 348 347.7 347.4 347.1 346.8 346.5 346.2 345.9 345.6

0.004

Y (m)

351 350.7 350.4 350.1 349.8 349.5 349.2 348.9 348.6 348.3 348 347.7 347.4 347.1 346.8 346.5 346.2 345.9 345.6

0.002

0 0

0.0005

Z (m)

=0.3

351 350.7 350.4 350.1 349.8 349.5 349.2 348.9 348.6 348.3 348 347.7 347.4 347.1 346.8 346.5 346.2 345.9 345.6

0.004

Y (m)

a

(28)

0.002

0 0

0.0005

Z (m)

=0.5

0

0.001

Z (m)

=2

Fig. 12. Temperature contours in the plane of x ¼ 0.02 m for b ¼ 0.3, 0.5 and 2.

X. Zhang et al. / International Journal of Thermal Sciences 54 (2012) 188e198

a

b

c 0.005

0.005

0.005 -4

10 m/s

Y (m)

Y (m)

Y (m)

10 m/s

0.003

0.001

-4

10 m/s

-4

0.003

0.001

0

0.0005

Z (m)

β =0.3

197

0.003

0.001 0

0.0005

0

0.001

Z (m)

Z (m)

β =0.5

β =2

Fig. 13. Two-dimensional velocity vector in the wick at x ¼ 0.02 m for b ¼ 0.3, 0.5 and 2.

and increases afterwards with a from 0.5 to 2. It reaches its minimum when a ¼ 1. Fig. 10 illustrates the temperature contours in the plane of X ¼ 0.02 m for a ¼ 0.5 and a ¼ 2. Compared with Fig. 6(a), it can be seen that temperature of the wick below vapor groove slightly drops with a increasing. Table 2 shows the fraction of mass flux in different evaporating interface for different a. With the increase of a from 0.5 to 2, it is noted that fraction of mass flux in the XY-plane and YZ-plane increases while one in the XZ-plane decreases. As a < 1, due to smaller channel height more heat is transferred to the wick below the vapor groove. This results that fraction of mass flux through the XZ-plane increases and temperature gradient in the wick near the XZ-plane rises. Because pressure of vapor groove for different cases differs little with each other, there are slight different for evaporation temperature in the liquidevapor interface. As a result, temperature of wick increases and average temperature in the heated surface is larger. While a > 1, mass flux in the XY-plane rises up with vapor groove height increasing. And more heat in the XYplane is absorbed through the evaporation of working fluid. Larger temperature gradient is formed in the wick between the vapor grooves. This results in higher temperature of the top wall. For the evaporator (i) in Fig. 7, heat is transported to the wick through the region of between vapor grooves. The effect of wick width (Ww  Wv) between vapor grooves is one of important parameters. b is the ratio of wick width between vapor grooves and width of vapor groove, and b ¼ (Ww  Wv)/Wv. Fig. 11 shows the average temperature of the heated surface for different b. As can be seen from Fig. 11, average temperature increases with b increasing for b > 0.5 and there is little difference for average temperature as b < 0.5. Fig. 12 illustrated temperature contours in the plane of X ¼ 0.02 m for b ¼ 0.3, 0.5 and 2. As shown in Fig. 12, temperature distributions in the wick below vapor groove only shows little

difference when b ¼ 0.3 and 0.5, while the temperature in this region is higher for b ¼ 2. Fig. 13 shows two-dimensional velocity vector with the Y and Z component in the wick at X ¼ 0.02 m for b ¼ 0.3, 0.5 and 2. It can be seen that velocity of working fluid between the vapor grooves increases with (Ww  Wv) decreasing. As a result, less heat is transported to working fluid below vapor groove and temperature in this region is lower with b decreasing, which is shown in Fig. 12. So temperature of the top wall is reduced with b decreasing. When b < 0.5, change of velocity has little influence on the heat transfer and temperature of the top wall has slight difference with b. From the point of view of manufacture, interval between vapor grooves cannot be too small for the evaporator (i). Otherwise, the wick has insufficient strength for machining and assembling because the wick is porous media structure. For the evaporator (i), the range of b is suggested from 0.5 to 1. 5. Conclusions Thermal transport characteristics in the flat evaporator of miniature LHP is investigated with a complete three-dimensional model, which is involved in flow and heat transfer in the wick and vapor groove, heat conduction in the wall as a conjugated problem. The boundary conditions of pressure and temperature are determined by coupling the operating status of the system. The results of the present model are in good agreement with the experimental results of miniature LHP with given evaporator under specific heat flux. The flat evaporators in the LHP are intensively investigated. The flow and temperature field in the wick and the structural optimization of the evaporator are discussed in details under specific heat load. The conclusions are drawn as following:

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(1) Heat transfer coefficient of evaporator with vapor groove inside the wick is higher than that with vapor groove inside the wall when the wick is saturated with liquid working fluid. (2) The results predict the optimum of form factor(a) of vapor groove in the evaporator (i). When ratio(a) of height and width of vapor groove is 1, thermal transport performance of the evaporator is the best. (3) For the evaporator (i), when ratio(b) of wick width between vapor grooves and width of vapor groove is the less, temperature in the wick is the lower. This results in better heat transfer characteristics of the evaporator with lower wall temperature. Considering the design reliability, the range of b is recommended from 0.5 to 1. Acknowledgments This work was supported by the International Cooperation and Exchange Program from the Ministry of Science and Technology of China (Grant No. 2011DFA60290), the Program from the Science and Information Technology of Guangzhou (Grant No. 2010U1D00201) and Funding from Breakthroughs in Key Areas of Guangdong and Hong Kong Project (Grant No.2010Z21). References [1] Y.F. Maydanik, Loop heat pipes, Appl. Therm. Eng. 25 (5e6) (2005) 635e657. [2] J. Ku, in: Operating Characteristics of Loop Heat Pipes, 29th International Conference on Environmental System, Paper No. 1999-01-2007, Denver, USA, 1999. [3] S. Launay, V. Sartre, J. Bonjour, Parametric analysis of loop heat pipes operation: a literature review, Int. J. Therm. Sci. 46 (2007) 621e636. [4] Y.F. Maydanik, S.V. Vershinin, M.A. Korukov, J.M. Ochterbeck, Miniature loop heat pipesea promising means for cooling electronics, IEEE Trans. Compon. Pack. Technol. 28 (2) (2005) 290e296. [5] Z.C. Liu, W. Liu, A. Nakayama, Flow and heat transfer analysis in porous wick of CPL evaporator based on field synergy principle, Heat Mass Transfer 43 (2007) 1273e1281. [6] J. Li, D. Wang, G.P. Peterson, Experimental studies on a high performance compact loop heat pipe with a square flat evaporator, Appl. Therm. Eng. 30 (6e7) (2010) 741e752.

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