Numerical analysis of heat and mass transfer in the capillary
亭的组词structure of a loop heat pipe
Tarik Kaya *,John Goldak
Carleton University,Department of Mechanical and Aerospace Engineering,1125Colonel By Drive,Ottawa,Ont.,Canada K1S 5B6
Received 24February 2005;received in revid form 20December 2005
Available online 31March 2006
Abstract
The heat and mass transfer in the capillary porous structure of a loop heat pipe (LHP)is numerically studied and the LHP boiling limit is investigated.The mass,momentum and energy equations are solved numerically using the finite element method for an evapo-rator cross ction.When a parate vapor region is formed inside the capillary structure,the shape of the free boundary is calculated by satisfying the mass and energy balance conditions at the interface.The superheat limits in the capillary
structure are estimated by using the cluster nucleation theory.An explanation is provided for the robustness of LHPs to the boiling limit.Ó2006Elvier Ltd.All rights rerved.
Keywords:Two-pha heat transfer;Boiling in porous media;Boiling limit;Loop heat pipes;Capillary pumped loops
1.Introduction
Two-pha capillary pumped heat transfer devices are becoming standard tools to meet the increasingly demand-ing thermal control problems of high-end electronics.Among the devices,loop heat pipes (LHPs)are particu-larly interesting becau of veral advantages in terms of robust operation,high heat transport capability,operabil-ity against gravity,flexible transport lines and fast diode action.As shown in Fig.1,a typical LHP consists of an evaporator,a rervoir (usually called a compensation chamber),vapor and liquid transport lines and a con-denr.The cross ction of a typical evaporator is also shown in Fig.1.The evaporator consists of a liquid-pas-sage core,a capillary porous wick,vapor-evacuation grooves and an outer casing.In many LHPs,a condary
wick between the rervoir and the evaporator is also ud to ensure that liquid remains available to the main wick at all times.Heat is applied to the outer casing of the evapo-rator,leading to the evapor
ation of the liquid inside the wick.The resulting vapor is collected in the vapor grooves and pushed through the vapor transport line towards the condenr.The meniscus formed at the surface or inside the capillary structure naturally adjusts itlf to establish a capillary head that matches the total pressure drop in the LHP.The subcooled liquid from the condenr returns to the evaporator core through the rervoir,completing the cycle.Detailed descriptions of the main characteristics and working principles of the LHPs can be found in Maidanik et al.[1]and Ku [2].
In this prent work,the heat and mass transfer inside the evaporator of an LHP is considered.The formulation of the problem is similar to a previous work performed by Demidov and Yatnko [3],where the capillary struc-ture contains a vapor region under the fin parated from the liquid region by a free boundary as shown in Fig.2.Demidov and Yatnko [3]have developed a numerical procedure and studied the growth of the vapor region under increasing heat loads.They also prent a qualitative
0017-9310/$-e front matter Ó2006Elvier Ltd.All rights rerved.doi:10.1016/j.ijheatmasstransfer.2006.01.028
*
Corresponding author.
E-mail address:tkaya@mae.carleton.ca (T.Kaya),jgoldak@mrco2.carleton.ca (J.Goldak).
/locate/ijhmt
analysis of the additional evaporation from the meniscus formed in thefin–wick corner when the vapor region is small without exceeding thefin surface.They report that the evaporation from this meniscus could be much higher than that from the surface of the wick and designs facilitating the formation of the meniscus would be desir-able.Figus et al.[4]have also prented a numerical solu-tion for the problem pod by Demidov and Yatnko[3] using to a certain extent similar boundary conditions and a different method of solution.First,the solutions are obtained for a single pore-size distribution by using the Darcy model.Then,the solution method is extended to a wick with a varying pore-size distribution by using a two-dimensional pore network model.An important conclusion of this work is that the pore network model results are nearly identical to tho of the Darcy model for an ordered single pore-size distribution.On the basis of this study,we consider a capillary structure with an ordered pore distri-bution posssing a characteristics single pore size.A sim-ilar problem has also been studied analytically by Cao and Faghri[5].Unlike[3,4],a completely liquid-saturated wick is considered.Therefore,the interface is located at the sur-face of the wick.They indicate that the boiling limit inside the wick largely depends on the highest temperature under thefin.This statement needs further investigation espe-cially when a vapor region under thefin is prent.In a later study,Cao and Faghri[6]have extended their work to a three-dimensional geometry,where a two-dimensional liquid in the wick and three-dimensional vaporflow in the grooves parated by aflat interface at the wick s
urface is considered.A qualitative discussion of the boiling limit in a capillary structure is provided.They also compare the results of the two-dimensional model without the vapor flow in the grooves and three-dimensional model and con-clude that reasonably accurate results can be obtained by a two-dimensional model especially when the vapor velo-cities are small for certain workingfluids such as Freon-11 and ammonia.Bad on the results,in our work,we con-sider a two-dimensional geometry to simplify the formula-tion of the problem.All the referenced works assume a steady-state process.Dynamic phenomena and specifically start-up is also extensively studied[7,8].The superheat at the start-up and temperature overshoots is still not well understood.In this work,the transient regimes and start-up are not investigated.
One of the goals of the prent study is a detailed inves-tigation of the boiling limit in a capillary structure.There-fore,the completely liquid-saturated and vapor–liquid wick cas are both studied.The boiling limit in a porous struc-ture is calculated by using the method developed by Mish-kinis and Ochterbeck[9]bad on the cluster nucleation theory of Kwak and Panton[10].Our primary interest in this study is LHPs.In comparison,the previously refer-enced works focus primarily on capillary pumped loops (CPLs),a cloly related two-pha heat transfer device to an LHP.Unlike in a CPL,the proximity of the rervoir to the evaporator in an LHP ensures that the wick is con-tinuousl
y supplied with liquid.However,there is no signif-icant difference in the mathematical modeling of both devices especially becau only a cross ction of the evap-orator is studied.The main difference here is that LHPs easily tolerate the u of metallic wicks with very small pore sizes,with a typical effective pore radius of1l m,resulting in larger available capillary pressure heads.
Nomenclature
c p specific heat at constant pressure[J kgÀ1KÀ1] h c convection heat transfer coefficient[W mÀ2KÀ1] h i interfacial heat transfer coefficient[W mÀ2KÀ1] h fg latent heat of evaporation[J kgÀ1]
J nc critical nucleation rate[nuclei mÀ3sÀ1]
k thermal conductivity[W mÀ1KÀ1]
K permeability[m2]
L length[m]
p pressure[Pa]
D p pressure drop across wick[Pa]
Pe Peclet number
Q b heat load for boiling limit[W]
q in applied heatflux[W mÀ2]
Q in applied heat load[W]
r radius[m]
r p pore radius[m]
安神补脑液的功效Re Reynolds number
t thickness[m]
T temperature[K]
u velocity vector[m sÀ1]Greek symbols
h angle[degrees]
l viscosity[Pa s]动静脉瘘
q density[kg mÀ3]
u porosity
r liquid–vapor surface tension[N mÀ1] Subscripts
c casing
effeffective
g groove
in inlet
int interface
l liquid
max maximum
n normal component
sat saturation
v vapor
w wick
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2.Mathematical formulation
A schematic of the computational model for the wick gment studied is shown in Fig.3.Becau of the symme-try,a gment of the evaporator cross ction is considered,
which is between the centerlines of the fin and adjacent vapor groove.The numerical solutions for this
晒的组词
geometry
Fig.2.Schematic of evaporation inside the
evaporator.
Fig.1.Schematic of a typical LHP and cross ction of the evaporator.
T.Kaya,J.Goldak /International Journal of Heat and Mass Transfer 49(2006)3211–32203213
are obtained for two parate configurations.At low heat loads,the wick is entirely saturated by the liquid.At higher heat loads,the wick contains two regions divided by an interface as shown in Fig.3:an all-vapor region in the vicinity of thefin and a liquid region in the remaining part of the wick.Heat is applied on the exterior walls of the cas-ing and it is transferred through thefin and wick to the vapor–liquid interface.This leads to the evaporation of the liquid at the interface and thus theflow of the vapor into the grooves.For the vapor–liquid wick,the vapor formed inside the wick is pushed towards the grooves through a small region at the wick–groove border.In both of the cas,as a result of the pressure difference across the wick,the liquid from the core replaces the outflowing vapor.Under a given heat load,the system reaches the steady state and the operation is maintained as long as the heat load is applied.
The mathematical model adopted in this work is bad on the following assumptions:the process is steady state; the capillary structure is homogenous and isotropic;radia-tive and gravitational effects a
re negligible;thefluid is Newtonian and has constant properties at each pha; and there is local thermal equilibrium between the porous structure and the workingfluid.Many of the assump-tions are similar to tho made in Demidov and Yatnko [3]and Figus et al.[4].In addition,we also take into account convective terms in the energy(advection–diffu-sion)equation.The validity of the Darcy equation for the problem studied is also discusd.Under the assump-tions,the governing equations for vapor and liquid phas (continuity,Darcy and energy)are as follows:
rÁu¼0ð1Þ
u¼ÀK
l
银边天竺葵r pð2Þ北京大学深圳校区
q c p rðu TÞ¼k eff r2Tð3ÞIt should be noted that the Darcy solverfirst calculates the pressure from the Laplace equation for pressure($2p=0), which is obtained by combining Eqs.(1)and(2).The vapor flow in the groove region is not solved to simplify the prob-lem.The boundary conditions for the liquid-saturated wick are described as follows:
At r=r i
p¼p
core
;T¼T satð4ÞAt r=r o and h A6h6h C
u n¼À
k eff
q l h fg
o T
o n
;k eff
o T
o n
¼h iðTÀT vÞð5Þ
At r=r o and h C6h6h D
o p o n ¼0;k c
o T
o n
¼k eff
o T
o n
ð6Þ
At r=r g and h A6h6h C
Àk c o T
o n
¼h cðTÀT vÞð7Þ
At r=r c
k c
o T
o n
¼q inð8Þ
At h=h A and r i6r6r o and r g6r6r c
o p
o h
¼0;
o T
o h
¼0ð9Þ
At h=h C and r o6r6r g
Àk c
o T
o n
¼h cðTÀT vÞð10Þ
At h=h D and r i6r6r c
o p
o h
¼0;
o T
o h
¼0ð11Þ
In the equations above,(o/o n)reprents the differential
operator along the normal vector to a boundary.The
boundary conditions for the wick with the parate vapor
and liquid regions are identical to the above equations ex-
cept along the wick–groove boundary and for the vapor–
liquid interface inside the wick.The following equations
summarize the additional boundary conditions for the
vapor–liquid wick:
At r=r o and h A6h6h B
u n¼À
k eff
q l h fg
o T
o n
;k eff
o T
o n
¼h iðTÀT vÞð12Þ
At r=r o and h B6h6h C
p¼p
v
;
o T
o n
¼0ð13Þ
The interface is assumed to have zero thickness.Sharp
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discontinuities of the material properties are maintained
across the interface.The interfacial conditions are written
as follows:
The mass continuity condition
ðu nÞ
v
q v¼ðu nÞ
l
q lð14Þ
The energy conrvation condition
ðk effÞ
v
o T v
o n
Àðk effÞ
l
o T l
o n
¼ðu nÞ
v
爸爸的胡子q
v
h fgð15Þ
For the interface temperature condition,we assume
local thermal equilibrium at the interface inside the wick:
T int¼T v¼T lð16Þ
Here,we assume that the interface temperature T int is given
by the vapor temperature.This condition is ud to locate
the vapor–liquid interface as explained in the following
ction.
For the interface at the wick–groove border,a convective
boundary condition is ud,Eqs.(5)and(12).A temper-
ature boundary condition ignoring the interfacial resistance
is also possible.The interfacial heat transfer coefficient is
calculated by using the relation given in Carey[11]bad
on the equation suggested by Silver and Simpson[12].
The heat transfer coefficient h c between the cover plate
and the vaporflow is calculated by using a correlation sug-
3214T.Kaya,J.Goldak/International Journal of Heat and Mass Transfer49(2006)3211–3220
gested by Sleicher and Rou[13]for fully developedflows in round ducts.It is extremely difficult to experimentally determine the heat transfer coefficient h c and a three-dimen-sional model is necessary to solve the vaporflow in the grooves.A convective boundary condition is more realistic since the u of temperature boundary condition implies h c!1.The convective boundary condition here with a reasonable heat transfer coefficient also allows some heat flux through the groove rather than assuming the entire heat load is transferred to the wick through thefin.
3.Numerical procedure
The governing equations and associated boundary con-ditions described previously are solved by using the Galer-kinfinite element method.The computational domain under consideration is discretized with isoparametric and quadratic triangular elements.
The numerical solution quence for the all-liquid wick is straightforward.As the entire process is driven by the liquid evaporation at the vapor–liquid front,the energy equation isfirst solved.The numerical solution quence is as follows:
1.Initialize the problem by solving the energy equation
assuming zero velocity inside the wick.
2.Calculate the normal component of the outflow velocity
at the interface between the wick and groove from the results of the energy equation,which is then ud as an outflow boundary condition for the Darcy solver.
3.Solve the Darcy equation to obtain the liquid velocity
field inside the wick.
4.Solve the energy equation on the entire domain with the
Darcy velocities.
5.Return to step2until all equations and boundary con-
ditions are satisfied to a desired level of accuracy.
At high heat loads,when a parate vapor region devel-ops in the wick,the numerical procedure is more compli-cated since the location of the interface is also an unknown of the problem.Therefore,a more involved iter-ative scheme is necessary.The numerical solution proce-dure is summarized as follows:
1.Initialize the problem by solving the Laplace equation
for temperature($2T=0)on the entire domain for a liquid-saturated wick.
2.Choo an arbitrary temperature isoline clo to thefin
as the initial guess for the location of the vapor–liquid interface.
3.Solve the energy equation for two parate domains:
casing-vapor region and liquid region.Calculate the normal conductive heatflux at the vapor–liquid interface.
4.Solve the Darcy equation parately in the vapor and
liquid regions to calculate the vapor and liquid velocities inside the wick.5.Solve the energy equation with the Darcy velocities on
the entire domain by imposing the energy conrvation boundary condition at the interface.
6.Check if the temperature condition at the interface is
satisfied.If it is not satisfied,the interface shape needs to be modified.
7.Return to step3until all equations and boundary con-
ditions are satisfied to obtain a pret level of accuracy.
After each interface update at step6,the solution domain needs to be remeshed.As the transient terms are not maintained in the governing equations,the numerical procedure prented is not a moving boundary technique and only the converged solutions have a physical meaning. For each solution,the static pressure drop across the inter-face is calculated to make sure that the difference in pres-sures is less than the maximum available capillary pressure in the wick(P vÀP l62r/r p),where
the normal viscous stress discontinuity and inertial forces are neglected.Thus,the momentum jump condition across the interface is satisfied as long as the maximum capillary pressure is not exceeded.
The accommodation coefficient for all the calculations is assumed to be0.1,leading to a typical value of h i=3.32·106W mÀ2KÀ1.To test the influence of this parameter,the results are also obtained with the accommodation coeffi-cients of0.01and1.Since the resulting interfacial heat transfer coefficients are sufficiently large,the change in the maximum temperature is negligibly small,on the order of less than0.01%.A typical value for the convection heat transfer coefficient h c is100W mÀ2KÀ1.The change of h c from100to50results in an increa of less than3%in the cover plate maximum temperature.However,the over-all change in the wick temperatures is negligibly small. 4.Results and discussion
Numerical calculations are performed for the evaporator ction with an outer diameter of25.4·10À3m as shown in Fig.3.The porous wick inside the evaporator has an outer diameter of21.9·10À3and a thickness of7.24·10À3m. The wick permeability and porosity are K=4·10À14m2 and u=60%,respectively.The workingfluid is ammonia. The LHP saturation temperature and pressure difference on both sides of the wick are calculated by using a one-dimensional mathematical model.
The model is bad on the steady-state energy conrvation equations and the pressure drop calculations along thefluid path inside the LHP.The details of this mathematical model are prented in[14].Fig.4reprents the calculated saturation temper-ature and pressure drop values across the wick as a function of the applied power.The pressure drops and heat transfer coefficients in the two-pha regions of the LHP are calculated by using the interfacial shear model of Chen [15].Incompressible fully developedfluidflow relations are ud to calculate the pressure drop for the single pha regions.
T.Kaya,J.Goldak/International Journal of Heat and Mass Transfer49(2006)3211–32203215
