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Analysis of local thermal non-equilibrium condition for unsteady heat transfer in porous media with closed cells: Sparrow number
Accepted Manuscript
Analysis of local thermal non-equilibrium condition for unsteady heat transfer in porous media with closed cells: Sparrow number C.Y. Wang , M. Mobedi , F. Kuwahara PII: DOI: Reference:
S0020-7403(18)34047-5 https://doi.org/10.1016/j.ijmecsci.2019.04.022 MS 4869
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
6 December 2018 15 March 2019 13 April 2019
Please cite this article as: C.Y. Wang , M. Mobedi , F. Kuwahara , Analysis of local thermal nonequilibrium condition for unsteady heat transfer in porous media with closed cells: Sparrow number, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.04.022
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ACCEPTED MANUSCRIPT
Highlights
Instead of Nusselt number, Sparrow number appears in the dimensionless governing equations by using equilibrium thermal diffusivity.
The Sparrow number is interpreted as the equilibrium conduction thermal resistance to the convection thermal resistance in the pores.
A good agreement between the pore scale and volume averaged results for the working fluid of water and air is observed.
A chart is presented in terms of Sparrow number, solid thermal diffusivity and thermal
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capacitance to estimate the local thermal equilibrium condition.
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ACCEPTED MANUSCRIPT Analysis of local thermal non-equilibrium condition for unsteady heat transfer in porous media with closed cells: Sparrow number C. Y. Wang, M. Mobedi*, F. Kuwahara
Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu-shi, Japan ＊
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Corresponding author：[email protected]
Abstract
The local thermal non-equilibrium condition for a porous medium with closed cells under unsteady state heat transfer is analyzed. Although the fluid circulates in the closed pores due to the buoyancy effect, the volume averaged velocity for a closed pore is zero causing excluding of the continuity and
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momentum equations from the volume averaged governing equations. The volume averaged governing equations are non-dimensionalized by using the equilibrium thermal diffusivity and Sparrow number appears in the dimensionless governing equations automatically. The Sparrow number is interpreted as the equilibrium conduction thermal resistance to the convection thermal resistance for the entire domain. For the high values of Sparrow number (such as 1000), the
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convection heat transfer resistance is considerably smaller than the heat conduction resistance resulting in high possibility of the local thermal equilibrium. Dimensionless governing equations show
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that four dimensionless parameters as porosity, Sparrow number, solid (or fluid) dimensionless thermal capacitance and thermal conductivity play important roles on the prediction of the local thermal non-equilibrium state in a closed cell porous medium. A chart for the prediction of local thermal
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non-equilibrium is presented for Sparrow number of 1, 50, 100 and 500. Furthermore, a pore scale study for a closed cell porous medium with working fluids as water and air is performed. A good
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agreement between the pore scale and volume averaged results is observed. The obtained pore scale results support the results of the volume averaged parametric study and also the suggested chart for the
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prediction of local thermal non-equilibrium state.
Keywords: Sparrow number, local thermal non-equilibrium, closed cell porous media.
1. Introduction The voids of a porous medium can be interconnected (open pore) or unconnected (closed cell or isolated pore). Although most of studies in the literature were performed on the porous media with open pores (Varol et al. [1], Celik et al. [2], Ozgumus and Mobedi [3]), many applications for the closed cell porous media particularly in industry exist that requires further researches. For instance, a 2
ACCEPTED MANUSCRIPT closed cell metal foam is used for insulation against fire, blast energy adsorption, sound absorber, electromagnetic wave shield. Transverse heat transfer in a honeycomb structure or heat transfer through the walls constructed by hollow bricks can be other examples for the closed cell porous media. Furthermore, the closed cell structure can also be faced in materials such as plastic based insulation material (for instance polyethylene or polyurethane insulation plates) or even foods (for instance breads or cakes). Our literature survey showed that most of the studies on porous media with closed pore were
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performed on the mechanical properties and mechanical behavior of the porous media rather than heat transfer response. Some examples of the limited number of studies for heat transfer analysis in the closed cell porous media are the studies of Ma and Tzeng [4] who investigated heat transfer in multi-channels located in the closed cell aluminum foams, Vesenjak et al. [5] who performed a parametric computational study on the heat conduction in a closed-cell cellular metal foam under steady state condition, or Wang et al. [6] who discussed the thermal conductivity of closed-cell
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aluminum foams for various cell configurations and different porosities. In all of these studies, it is assumed that a local thermal equilibrium exists between the fluid and solid phases. Researchers showed that the results between the local thermal equilibrium and non-equilibrium can be considerably different, and that is why the validity of the local thermal equilibrium assumption was investigated and reported in literature. For instance, Vafai and Sozen [7] found that there might be
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significant discrepancies between the fluid and solid phase temperature distributions in a heat storage packed bed. Amiri and Vafai [8] also analyzed the thermal dispersion and local thermal
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non-equilibrium effects through porous media and presented error maps to assess the commonly used assumptions. Minkowycz et al. [9] analyzed the departure from local thermal equilibrium in porous media due to a rapidly changing heat source. They used Sparrow number and declared that a local
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thermal equilibrium can exist for rapidly changing heat sources under sufficiently large Sparrow number. Alazmi and Vafai [10] studied the effect of using different boundary conditions for the case of
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constant wall heat flux under LTNE (local thermal non-equilibrium) conditions. Haji-Sheikh and Minkowycz [11] performed an analytical study to describe the role of dimensionless quantities that identify the early departure from local thermal equilibrium in the presence of a rapidly changing heat
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source. Further theoretical studies on the heat transfer under local thermal non-equilibrium state such as the studies of Neild et al. [12], Neild and Kuznetsov [13], Vadasz [14], Dukhan and Hooman [15], Hoghoughi et al. [16], Izadi et al. [17], Bayrak et al. [18], Ejlali et al. [19] and Sheremet and Pop [20] exist in literature. Among the available studies in literature, the study done by Minkowycz et al. [9] has taken the attention of authors of the present study due to the use of Sparrow number for assessing of the thermal equilibrium assumption, and it motivated the authors to perform the present study. The volume averaged heat transfer equations are non-dimensionalized by using equilibrium thermal diffusivity and 3
ACCEPTED MANUSCRIPT Sparrow number appears automatically. Although the fluid motion exists in the closed pores due to the buoyancy effect, the volume averaged velocity in the closed pore is zero removing the volume averaged continuity and momentum equations. The dimensionless forms of the volume averaged heat conduction equations for the solid and fluid phases are solved and the effects of Sparrow number as well as the solid dimensionless thermal capacitance and thermal conductivity on the heat transfer mechanism are investigated. Furthermore, a porous medium with 25 closed pores under one dimensional heat transfer is considered and both the pore scale governing equations (continuity,
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momentum and energy equations) and the volume averaged governing equations (heat conduction equations for the solid and fluid phases) are numerically solved to support the parametric study results and the suggested charts. To the best of our knowledge, this is the first comprehensive study on the validation of local thermal equilibrium in a closed cell porous media based on the definition of Sparrow number.
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2. Considered problem
The schematic views of the considered pore scale domain and volume averaged domain are shown in Figure 1. Both the pore scale and volume averaged domains are described below. Pore scale domain: The porous media is a square domain with length of L . The details of closed cells can be seen in the Figure 1(a). It consists of 25 square closed cells with the length of l . The cells are
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not connected to each other while the solid phase (i.e. aluminum frame) interconnected in the entire domain. The top and bottom surfaces of the porous media are insulated and the left and right vertical
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surfaces are maintained at constant temperatures of Th and Tc , respectively. The gravity affects in –y direction and the radiation heat transfer is neglected. The working fluids are water and air and their thermal properties are given in the Table 1. The density, specific heat capacity and thermal
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conductivity of the solid phase, which is aluminum alloy, are 2700(kg/m3), 900(J/kg.K) and 203(W/m.K), respectively. Based on the thermophysical properties and the geometry of the studied
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domain, the value of dimensionless parameters such as Prf , Ra f and are calculated for the studied cases with air and water and given in Table 1.
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Volume averaged domain: The volume averaged domain is also shown in Figure 1(b). A continuous domain exists for the volume averaged equations since the governing equations are volumetrically integrated. No velocity field exists in the volume averaged domain and the conduction heat transfer is the only mechanism of heat transfer. The volume averaged governing equations are non-dimensionalized and a parametric study is performed by changing the dimensionless governing parameters in the range of 0.01 s 1 , 0.01 s 1 and 1 Sp 1000 . The definitions of these dimensionless parameters ( s , s , Sp ) are given in next section.
3. Governing equations and boundary conditions 4
ACCEPTED MANUSCRIPT The governing equations for the pore scale and volume averaged approaches are presented in this section, separately.
3.1 Governing equations for the pore scale analysis For the pore scale study, the continuity and momentum equations are solved to find the velocity and pressure distributions for fluid while the energy equations for the solid and fluid are solved to find the temperature distribution for the entire domain. The dimensional governing equations for the pore scale analysis of heat and fluid flow are:
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V 0
V 1 (V. )V p f 2 V g(T Tref ) j t f
(C p ) s
Tf V. Tf ） k f 2 Tf t
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(C p ) f (
Ts k s 2 Ts t
(1) (2)
(3)
(4)
where and Tref are the thermal expansion coefficient of working fluid and reference temperature (initial temperature which is equal to the cold wall temperature in this study), respectively. The
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equilibrium thermal diffusivity can be employed and the above governing equations can be written in dimensionless form as follows, * V * 0
(5) (6)
f * * V . f f *2 f
(7)
s s *2 f
(8)
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V* (V* . * )V* *P* Pre *2 V* Ra e 3 Pre f j
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where V and P* represent the dimensionless velocity vector and pressure. The dimensionless temperatures of the solid and fluid phases are shown by f and s , respectively. , Ra e and Pre are the dimensionless time, Rayleigh number and Prandtl number based on the equilibrium thermal diffusivity, respectively. is the thermal diffusivity ratio between the fluid or solid phase with equilibrium one. The employed dimensionless parameters can be mathematically expressed as,
* L
,
* VL V e
,
L l
,
t e 2
L
5
,
*
P
PL2 f e2
,
Pre f e
,
(9)
ACCEPTED MANUSCRIPT T Tc f g(T Tc )l 3 , , f f , s s Ra e Th Tc e e ef where e is the equilibrium thermal diffusivity. It should be mentioned that the following relations are valid for the Prantl and Rayleigh numbers.
Prf Pre
f , Ra f Ra e f e e
(10)
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where Prf and Ra f are Prantl and Rayleigh numbers based on the fluid thermal diffusivity.
3.2 Governing equations for volume average analysis
For the volume average method, a scalar or vector quantity can be integrated in a representative discontinues space and consequently the continuity of that space can be provided. Since two phases as
1 dV VV
x
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solid and fluid exist in the porous media, two volume averaged quantities can be defined:
1 dV Vx V x
x
(12)
are the volume average and intrinsic volume average of quantity, respectively.
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and
where
(11)
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V and Vx are the total volume and the corresponding volume of solid or fluid phase. Taking volume average of the dimensional pore scale governing equations (Eq. (1-4)) yields the dimensional volume averaged governing equations as:
(13)
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V 0
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f f V 1 C ( V . ) V P f 2 V V f V V t K K 2 g(T Tref ) j
Cf (
Cs
f
T
t
T
t
s
f V . T ） k eff ,f 2 T
f
s
f
hv( T T )
s s f k eff ,s 2 T h v ( T T )
(14)
(15)
(16)
where Cs and C f are the thermal capacitance of solid and fluid phases (i.e. Cs (Cp )s and Cf (Cp ) f ), respectively. h v is the volumetric interfacial heat transfer coefficient, k eff ,s and
k eff ,f are the solid and fluid effective thermal conductivity. k eff ,s and k eff ,f can be calculated from
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ACCEPTED MANUSCRIPT the following equations ((Kuwahara et al. [21]) :
k eff , f *k f k disp
(17)
k eff , s (1 * )k s
(18)
k s and k f are the thermal conductivity of solid and fluid, and * is the effective porosity that can
be found from the following equation.
k kf * eff ks kf
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(19)
The value of k eff which is the effective thermal conductivity (for entire domain) can be numerically calculated by considering a representative volume, imposing temperature difference and applying Fourier law. In this study, the value of * is found as 0.95 for the porous media with 0.9 . For a closed cell porous media, the volume averaged velocity can vanish.
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1 V V dV 0 VV
(20)
The continuity and momentum equations (Eqs. (13-14)) disappear since the volume averaged velocity for the cell porous media is zero. Hence, the volume averaged equations (13-16) take the following
f
t
(1 ) C s
k eff ,f 2 T
T
s
t
f
s
f
hv( T T )
s s f k eff ,s 2 T h v ( T T )
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Cf
T
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form for a closed cell porous medium,
(21)
(22)
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The heat conduction equations for the solid and fluid phases are the only equations for the volume average approach in the closed cell porous media. h v which is the volumetric interfacial heat transfer
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coefficient can be defined as h v h s A v , where h s (W / m 2 K) is the surface interfacial heat transfer coefficient and A v (m 2 / m3) is the volumetric heat transfer area. It is possible to write the volumetric
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heat transfer area as A v / rh , where rh is hydraulic diameter of the fluid phase. It can be defined as rh Vf / Aint where A int is the interface heat transfer area in the fluid volume of Vf . Then, the Eqs. (21) and (22) become as:
Cf
T t
(1 ) C s
f
k eff ,f 2 T
T t
s
f
hs s f (T T ) rh
(23)
h s s f k eff ,s 2 T s ( T T ) rh
(24)
The above equations can be non-dimensionalized by using dimensionless variables of Eq. (9) as: 7
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f s
f
s
f *2
f
s
f
Sp ( )
(25)
s s f s *2 Sp ( )
(26)
, are the dimensionless thermal conductivity and thermal capacitance, respectively, and Sp is
the Sparrow number. The employed dimensionless parameters can be written mathematically as (27)
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k eff , s k eff , f C s (1 ) Cf hL2 s , f , s , f , Sp k e rh C e Ce k e k e
C e and e are the thermal capacitance and thermal conductivity of local thermal equilibrium state.
They can be expressed as:
Ce 1 Cs Cf
(28)
k e k f (1 )k s
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(29)
Eqs. (25) and (26) can be solved for this problem, however two additional equations can also be written by using definitions in Eq. (27)
s f
1
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s f
1
(30)
(31)
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If the Eqs. (30) and (31) are considered with Eqs. (25) and (26), it is seen that the number of governing parameters decreases to four as solid (or fluid) dimensionless thermal conductivity and thermal capacitance, Sparrow number and porosity. In this study, the values of f and f are written in
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terms of s and s . Hence the dimensionless governing parameters are s , s and Sp while porosity is constant as 0.9.
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Furthermore, for the thermal equilibrium case, s and
f
are close to each other ( s f ),
therefore Eqs.(25) and (26) take the following form:
*2
(32)
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3.3 On the physical definition of sparrow number Sparrow number plays an important role on the results of the governing equations (Eqs. 30-31). In order to understand the physical meaning of Sparrow number, a cube porous medium with length of L can be considered, the Sparrow number can be written as,
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Sp
hA v L2 hL2 k e rh k e
hA int L 2 L3 k e
L 1 L 2 k e L k L k A R e e cr conduction 1 1 1 R convection hA int hA int hA int
(34)
where R conduction is the conduction thermal resistance through the length of the cube and R convection is convection thermal resistance. Both R conduction and R convection concerns with entire domain. Aint is the total area of the interfacial heat transfer of porous medium while Acr is the cross section area
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of the porous media with cube shape. Eq. (34) shows that Sparrow number can be defined as the ratio of conduction to convection heat transfer resistance through the porous media. For the high values of Sparrow number (such as Sp 1000 ), the convection heat transfer resistance is considerably smaller than the heat conduction resistance and comparatively large amount of heat can be transferred from the fluid to solid resulting in high possibility of local thermal equilibrium. The local thermal
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non-equilibrium can exit for the low values of Sparrow number (such as Sp 50 ) implying that the convection heat transfer resistance is comparable or less than heat conduction resistance. In other words, the propagation of heat by conduction through the porous media is faster than heat transfer by convection from fluid to the solid phase.
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3.4 Initial and Boundary Conditions
Initially, the temperature of the entire domain is Ti both for the pore scale and volume averaged
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domains. Suddenly the temperatures of right and left walls of the porous media jump to Th and Tc , respectively and they are maintained at the same temperature for the entire process. In this study, the initial and cold wall temperature are taken equal. The top and bottom surfaces of the cavity are
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thermally insulated. For the pore scale domain, a continuous heat flux and unique temperature conditions are applied onto the interface between solid and fluid phases. For the volume averaged
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domain, the heat exchange between the solid and fluid phases is provided by Sparrow number. The dimensionless form of the initial and boundary conditions is given in Eq. (35). Pore scale
Volume averaged
s f 0
s
f
0
x* 0
s 1, f no need
s
f
1
x* 1
s 0, f no need
s
f
0
s 0, f no need n
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0
*
y 0 and 1
n
s
n
(35)
f
0
At the interface between the solid and fluid for the pore scale analysis, the following relation is valid: 9
ACCEPTED MANUSCRIPT s k f f , s f n k s n
(36)
where n is the normal direction to the considered surface.
4. Numerical solution for the pore scale and volume averaged governing equations For the pore scale study, a commercial software, COMSOL Multiphysics (V5.3a COMSOL Multiphysics), based on the finite element method is used to solve the coupled heat and fluid flow
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equations. All steps to obtain the results are described in the flow chart of Figure 2. A mesh refinement analysis is done and it is found that 57600 number of domain elements (triangular), 3840 edge elements, and 136 vertex element are sufficient to obtain the accurate results for this problem. The Multiphysics module used to solve motion equation is “Laminar flow” while for solving heat transfer equations in solid and fluid the module of “Heat transfer in fluids” is used. The software was run in a workstation with Intel(R) Xeon(R) CPU E5-2630 v3, 32G RAM and it took around 8 hours to obtain
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the accurate results. The relative tolerance was 10 3 . The algorithm of the employed software is written in Ref [22] in details.
The volume averaged governing equations are solved numerically by using an in-house code in Fortran platform based on the finite difference method. It consists of inner and outer loops. The inner loop is used to solve conduction equations for the solid and fluid for each time step. The relative
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convergence criterion for inner loop is 10 6 . The outer loop is used to increase time since the problem is unsteady.
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The solution of the volume averaged heat conduction equations for the solid and fluid (Eqs. (30) and (31)) requires to know the value of Sp (consequently, h v should be known). Unfortunately, no
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study relating to the transient value of the interfacial heat transfer coefficient for the closed pores of porous media could be found. In this study, the value of interfacial heat transfer coefficient is obtained
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from the pore scale study. The employed definition for determination of the average interfacial heat transfer coefficient is,
A int t
q" ( t )dA
t A int s 0( T
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hv
1
f
T )
(37)
dt
q" is the heat flux between solid and fluid phases at any point of the interface and at any instant such
as t , and A int is the total interfacial heat transfer area. t is the required time for the system to become steady state ( ( T
s
f
T ) 0.05 is the condition for the steady state). As it can be seen
from Eq. (37), the average of interfacial heat transfer coefficient is obtained by integration of the local interfacial heat transfer coefficient both in time and space.
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ACCEPTED MANUSCRIPT The average of the solid and fluid temperatures of the entire domain for the solid and fluid phases are found at any time step to compare the solid and fluid phase temperatures.
f
m () where
vf
f () m
f
()dv s
, m ()
dv vf
and
s () m
vs
s
()dv (38)
dv vs
are the solid and fluid phase mean (i.e., surface average for the entire
parameters of is defined as following:
s () m
f () m
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domain) temperatures. Finally, in order to understand the order of local thermal non-equilibrium, the
(39)
The above equation shows that the temperature difference between the mean solid and fluid phases for
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the whole domain during the entire heat transfer process.
5. Results and discussions
The grid independency study for the pore scale analysis was done and it was found that 61440 elements is sufficient to obtain the accurate results. For the validation of numerical study, the study of K.D. Antoniadis et al. [19] done for a hollow brick (two dimensional porous media with closed cell)
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for different values of thermal conductivity ratio is used. An excellent agreement (below 1% difference) was observed between our study and their reported study. As it can be seen from Eqs. (5-8), the pore scale dimensionless temperatures of the solid and fluid phases depend on four parameters as
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Pre , Ra e , , . The dimensionless volume averaged solid and fluid temperatures depend on three
parameters as s , s and Sp . The values of all dimensionless parameters are given in Table 1 for air
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separately.
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and water. The results of volume averaged parametric study and pore scale study are explained
5.2 Results for the volume averaged parametric analysis
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A parametric study is performed to investigate the effects of dimensionless numbers ( s , s and Sp ) on the temperatures of the solid and liquid phases. Since the transient behavior of temperature is important, the average of solid and liquid temperatures for the entire domain are calculated (i.e., Eq. (38)) and plotted. Figure 3 shows the change of mean temperature of solid and fluid for the case of s 0.01 , s 1 , Sp 50, 250 and 1000 . As it can be seen from Figure 3(a), there is an obvious temperature difference
between the solid and fluid mean temperatures before they become steady state. Since the value of Sparrow number is low (referring to high convection thermal resistance), the heat exchange between the solid and fluid is weak and a non-equilibrium heat transfer in the porous media is expected. By 11
ACCEPTED MANUSCRIPT increasing the Sparrow number from 50 to 250, the heat exchange between the solid and fluid phases increases and the temperatures of the solid and fluid phases approach to each other. Further increase in Sparrow number (from 250 to 1000) causes the fluid and solid phase mean temperatures become considerably close to each other and almost a local thermal equilibrium can be observed. The difference between the dimensionless temperature of solid and fluid (i.e., ) can be a good parameter to show the magnitude of local thermal non-equilibrium. This difference for four Sparrow number is shown in Figure 3(d). As can be seen for Sp 1000 , the maximum temperature difference between the
Sp 50 is 0.13 indicating a local thermal non-equilibrium state.
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solid and liquid is 0.016 showing almost an equilibrium state, while the maximum value of for
The parameter values of Figure 4 are the same with Figure 3 except the solid dimensionless thermal conductivity s 0.5 . The values of *s (i.e., *s s / s ) and *f (i.e., *f f / f ) can be calculated by using Eqs. (22, 30 and 31). By decreasing the solid dimensionless thermal conductivity
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from 1 to 0.5, the value of *s / *f decreases from 991 to 91 causing heat propagation in the solid and fluid phases becomes closer to each other and the temperature difference between the solid and fluid phases becomes smaller. This fact can be seen by the comparison of Figures 3(a) and 4(a). Similar to the Figure 3, by increasing Sp from 50 to 100, and from 100 to 1000, the solid and fluid dimensionless mean temperatures become closer due to the increase of heat exchange between the
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solid and fluid phases. The temperature differences between the solid and fluid are shown in Figure 4(d) to observe the magnitude of local thermal non-equilibrium. The comparison of 3(d) and 4(d)
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indicates the effect of s on the local thermal equilibrium condition. By decreasing the value of s from 1 to 0.5, the difference between the solid and fluid temperatures decreases.
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The change of solid and fluid phases temperature for further decrease of s from 0.5 to 0.01 is shown in Figure 5. Further decrease in s from 0.5 to 0.01causes the value of *s / *f decreases from 91 to 1. An equal heat propagation in the solid and fluid phases exists for Figure 5. Therefore, for the all
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studied Sparrow number ( Sp 50, 250, 1000 ), a local thermal equilibrium can be observed. The results of Figures 3, 4 and 5 clearly show that the heat exchange between the solid and fluid phases increases
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by increasing Sparrow number however, the critical Sparrow number (can be defined to distinguish the local thermal equilibrium from non-equilibrium) strongly depends on s . A parametric study on the various values of s , s and Sp was done to separate the region of local thermal equilibrium from non-equilibrium. The results of the parametric study is shown in Figure 6 for Sp 50, 100 and 500 . The x axis of Figure 6(a) shows s while the y axis refers to s . It is assumed
that a local thermal equilibrium for the heat transfer in the closed cell porous media exits if 0.05 . This is the assumption of present study and it might be changed based on the application. Figure 6(b) is presented in order to interpret the results of Figure 6(a). It shows the value of *s / *f for the 12
ACCEPTED MANUSCRIPT studied case of Figure 6(a). For the studied cases of Figure 6(a), the value of *s / *f changes from the large value of 991 to small value as 0.001. Figure 5 provides important information for the prediction of a local thermal non-equilibrium heat transfer process in a cavity with closed cell as follows; -
For the value of *s / *f 1 , the propagation of heat in the solid and fluid are close to each other causes local thermal equilibrium state becomes independent of Sparrow number (which is considered in this study). Further increase or decrease in the value of *s / *f (such as 991 or 0.001) causes the possibility
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-
of local thermal non-equilibrium increases due to different speed of heat propagation in the solid and fluid phases. For instance, when Sp =50, a local thermal non-equilibrium exists for the points of s 0.01 and s 1 due the large difference in heat propagation of solid and fluid (faster heat prorogation exists in the solid , *s / *f 991 ).
In addition to s and s , the Sparrow number plays an important role on the character of heat
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-
transfer through the closed cell porous media. By increasing the value of Sp , the source terms in Eqs. (30) and (31) becomes more active, and consequently the heat exchange between the solid and fluid increases. This increase of heat exchange increases the possibility of occurrence of local thermal equilibrium state. That is why, number of unfilled markers in the Figure 6(a) for Sp 100
For Sp 500 , all markers are unfilled showing considerable heat transfer between the solid and
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-
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is more than Sp 50 , or number of unfilled markers for Sp 500 is greater than Sp 100 .
fluid phases despite of extremely high or low values of *s / *f (such as 991 or 0.001). It seems that *s / *f and Sp are two important parameters for the prediction of local thermal
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-
non-equilibrium for the unsteady of a heat transfer in closed cell of porous media. .
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The above comments are only valid for the range of dimensionless parameters considered in this study.
5.3 The pore scale and volume average results for real cases
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A concept for specifying non-equilibrium heat transfer in a closed cell porous media is proposed and explained in the previous section. A volume average study for two different working fluids as water and air is done to validate the suggested graphics (Figure 6) and validate the explained comments. The volume average study is also validated by the results of pore scale analysis, otherwise the results of the volume average for two different fluids might be questionable 5.3.1. Results for the porous media with water in cells Figure 7 shows the temperature and velocity distributions for both the pore scale and volume averaged domains when s 0.068 , s 1.06 , Sp 46.7 and Ra f 105 , as given in Table 1. The value of 13
ACCEPTED MANUSCRIPT Sparrow number is calculated from the pore scale results when Ra f 105 . The values of fluid thermal capacitance and thermal conductivity are found as s 0.068 and s 1.06 , and the thermal diffusivity ratio is calculated as 300 (i.e. *s / *f 300 ). This value of *s / *f is considerably larger than 1 and furthermore the Sparrow number is low ( Sp 46.7 ). Based on the previous section comments, the possibility of the existence of local thermal non-equilibrium under this condition is high and this fact is investigated below. From Figure 7(a), it is seen that the solid phase temperature is faster than the fluid phase due to the
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faster propagation of heat in the solid at the starting period ( 0.032 ). By increasing the time to 0.16 and 0.32 , the pore scale temperatures of both solid and fluid phases become closer to
each other. The volume averaged temperatures (middle and bottom cavities of Figure 7) also show the same behavior of the pore scale one. It should be mentioned that heat conduction equations for the solid and fluid phases are solved and that is why no velocity is seen in the results of the volume
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averaged domains.
Figure 8 shows the comparison of solid and fluid dimensionless mean temperatures (
s m
,
f m
)
obtained from the pore scale and volume averaged analysis for Ra f 10 2 and 105 , respectively. A good agreement between the volume averaged and pore scale analysis can also be observed proving the correctness of our volume average study. The small differences between the pore scale and the
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volume averaged results may be due to the definition of interfacial heat transfer coefficient (Eq. (32)) in which the average of local heat transfer coefficient both in time and space was considered. A local
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thermal non-equilibrium for a large period at the beginning of the heat transfer process can be seen, supporting the comments of the parametric volume averaged study. Large difference in heat
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propagations due to the high value of *s / *f (i.e. 300) and weak heat exchange between the solid and
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fluid phases causes a local thermal non-equilibrium exists.
5.3.2 Results for porous media filled with air in cells Figure 9 shows the temperature and velocity distributions of a closed cell porous media when the
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voids are filled with air for s 1.11 , s 1.08 , Sp 1.91 and Ra f 105 , as given in Table 1. The value of thermal conductivity ratio ( *s / *f ) is calculated as 2.09. The value of thermal diffusivity ratio takes the attention since it is around 1. Although the Sparrow number is extremely low (conduction and convection resistances are almost equal), a local thermal equilibrium can be expected due to almost equal propagation of heat in the solid and fluid. Figure 9 indicates that the propagation of heat in the solid and fluid are the same for the solid and fluid phases. This fact can be clearly seen from both pore scale and volume average temperature distributions. Figure 10 shows the solid and fluid dimensionless mean temperatures ( 14
s m
,
f m
) for
ACCEPTED MANUSCRIPT Raf 102 and 105 at the center of the domain for different time steps. Almost, a local thermal
equilibrium state can be seen from this figure both from volume average and pore scale results. The values of s and s are known and the location of the above two cases in the local thermal equilibrium map (Figure 6) can be found. The location of two cases are marked in Figure 6(a) for the case of air ( s 1.11 , s 1.08 , Sp 1.91 ) and for the case of water ( s 0.068 , s 1.06 , Sp 48.6 ). The location of marker for air case is in right place validating Figure 6. A local thermal equilibrium exists for the case of air due to the small values of *s / *f 2.09 referring to almost identical heat
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propagation in solid and fluid in spite of low value of Sparrow number ( Sp 1.91 ). Similar to the case of air, the location of marker for water case is correct, and a local thermal non-equilibrium must exist for the water case due to faster propagation of heat in the solid comparing to fluid phase ( *s / *f 300 ).
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6. Conclusions
A numerical study on the local thermal non-equilibrium state for the transient heat transfer through the closed cell porous media is performed. Based on the obtained results and performed discussions, the following remarks can be concluded. -
If the governing equations are non-dimensionalized according to the equilibrium thermal
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diffusivity, six dimensionless parameters as porosity, the solid and fluid thermal diffusivity and thermal capacitance ratio and Sparrow number are obtained. Sparrow number appears -
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automatically.
Two relations between the dimensionless solid and fluid thermal diffusivity and thermal
-
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capacitance are found resulting in decreasing of dimensionless governing parameters to four. The Sparrow number is interpreted as the ratio of conduction to convection thermal resistance through the porous media. High value of Sparrow number (such as Sp 1000 ) referring to the
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higher convection in the pores compared to heat conduction through the porous media while low values ( Sp 50 ) refers to a large convection resistance in the pores and consequently a weak heat
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exchange between the solid and fluid.
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When the value of Sparrow number is low (such as 50), a local thermal non-equilibrium heat transfer occurs for the cases in which the thermal diffusivity ratio is considerably greater or smaller than 1 (such as 100 or 0.01). A local thermal equilibrium independent of the value of Sparrow number exists if the thermal diffusivity ratio is around 1.
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For high values of Sparrow number (such as 500), a local thermal equilibrium is observed for the all studied parameters due to the considerable heat exchange between the solid and fluid.
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The results of parametric study are supported by the pore scale of two real cases of closed cell porous media. For closed cell porous media when the solid is aluminum and working fluid is water, 15
ACCEPTED MANUSCRIPT a local thermal non-equilibrium is seen due to large value of thermal diffusivity ratio ( *s / *f 300 ) and small Sparrow number ( Sp 38 ). For the closed cell porous media with air a local thermal equilibrium is observed because almost the same propagation of heat in the solid and fluid ( *s / *f 2 ) Although many studies on the predication of local thermal non-equilibrium can be found in literature, number of studies based on Sparrow number is limited. Authors believe that the non-dimensionlization of the solid and fluid energy equations according to the equilibrium thermal diffusivity is an
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appropriate method for finding appropriate governing parameters and consequently Sparrow number appears automatically. However, the determination of critical Sparrow number may be used to separate local thermal equilibrium state from non-equilibrium state requires further studies.
Acknowledgement
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The first author gratefully acknowledges the financial support from China Scholarship Council (No. 201808050059), which has sponsored his PhD study at the Shizuoka University in Japan.
NOMENCLATURE Heat transfer area, m2
C
Thermal capacitance , J/ m3 K
Cp
Specific heat capacity, J/ kg K
h
Interfacial heat transfer coefficient, W/m2K
K
Permeability, m2
k
Thermal conductivity, W/m K
L
Length of cavity, m
LT NE
Local thermal non-equilibrium
l
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Local thermal equilibrium Length of cell, m Normal direction to the surface
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n
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LT E
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A
P
Pressure, N/m2
Pr
Prantl number
q"
Heat flux, W⋅m−2
Ra
Rayleigh number
r
Diameter, m
Sp
Sparrow number
T
Temperature, K 16
t
Time, s
V
Velocity vector, m/s
x, y
Cartesian coordinates, m Thermal diffusivity, m2/s
Thermal expansion coefficient, 1/K
Function for the level of non-equilibrium condition
Porosity
Any dependent parameter
Thermal conductivity in equilibrium state
Density, kg/m3
Ratio between the characteristic lengths
Dimensionless time
Thermal diffusivity ratio between fluid and solid
Dimensionless thermal conductivity
Dimensionless thermal capacitance
cr
Cross section
e
Equilibrium
eff
Effective
f
Fluid
h
Hot, Hydraulic
i
Initial
int
Interface
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Cold
s
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c
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Subscripts and superscripts
v
Volumetric
x
Solid or fluid phase
Reference Solid
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ref
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Dynamic viscosity, Pa.s Kinematic viscosity, m2/s Dimensionless temperature
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References [1] Y. Varol, H.F. Oztop, I. Pop, Conjugate heat transfer in porous triangular enclosures with thick bottom wall, International Journal of Numerical Methods for Heat and Fluid Flow 19 (5) (2009) 650-664. 17
ACCEPTED MANUSCRIPT [2] H. Celik, M. Mobedi, A. Nakayama and U. Ozkol, A numerical study on determination of volume averaged thermal transport properties of metal foam structures using x-ray microtomography technique, Numerical Heat Transfer (in press). [3] T. Ozgumus and M. Mobedi, Effect of pore to throat size ratio on thermal dispersion in porous media, International Journal of Thermal Sciences 104 (2016) 135-145. [4] W.P. Ma and S.C. Tzeng, Heat transfer in multi-channels of closed cell aluminum foams, Energy Conversion and Management 48 (3) (2007) 1021–1028.
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[5] M. Vesenjak, Z. Zunic, A. O. chsner, M. Hribersek and Z. Ren, Heat conduction in closed-cell cellular metal, Mat. -wiss. u. Werkstofftech 36 (10) (2005) 608–612. [6] H. Wang, X.Y. Zhou, B. Long, J. Yang and H.Z. Liu, Thermal properties of closed-cell aluminum foams prepared by melt foaming technology, Transactions of Nonferrous Metals Society of China 26 (12) (2016) 3147−3153.
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[7] M. Sözen and K.Vafai, Analysis of oscillating compressible flow through a packed bed, International Journal of Heat and Fluid Flow 12 (2) (1991) 130-136. [8] A. Amiri and K. Vafai, Analysis of dispersion effects and non-thermal equilibrium, non- Darcian, variable porosity incompressible flow through porous media, International Journal of Heat and Fluid Flow 37 (6) (1994) 939-954.
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[9] W.J. Minkowycz, A. Haji-Sheikh and K. Vafai, On departure from local thermal equilibrium in porous media due to a rapidly changing heat source: the Sparrow number, International Journal of Heat and Mass Transfer 42 (1999) 3373-3385.
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[10] B. Alazmi and K. Vafai, Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions, International Journal of Heat and Mass Transfer 45 (2002) 3071-3087. [11] A. Haji-Sheikh and W.J. Minkowycz, Heat transfer analysis under local thermal non-equilibrium conditions, Emerging Topics in Heat and Mass Transfer in Porous Media (2008) 39-62.
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[12] D.A. Nield, A.V. Kuznetsov and M. Xiong, Effect of local thermal non-equilibrium on thermally developing forced convection in a porous medium, International Journal of Heat and Mass Transfer 45 (2002) 4949-4955.
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[13] D.A. Nield and A.V. Kuznetsov, Local thermal non-equilibrium effects in forced convection in a porous medium channel: a conjugate problem, International Journal of Heat and Mass Transfer 42 (1999) 3245-3252.
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[14] P. Vadasz, Explicit Conditions for Local thermal equilibrium in porous media heat conduction, Transport in Porous Media 59 (2005) 341–355. [15] N. Dukhan and K. Hooman, Comments on two analyses of thermal non-equilibrium Darcy– Brinkman convection in cylindrical porous media, International Journal of Heat and Mass Transfer 66 (2013) 440–443. [16] G. Hoghoughi, M. Izadi, H. F. Oztop, N. Abu-Hamdeh, Effect of geometrical parameters on natural convection in a porous undulant-wall enclosure saturated by a nanofluid using Buongiorno's model, Journal of Molecular Liquids 255 (2018) 148–159. [17] M. Izadi, S.A.M. Mehryan, M.A. Sheremet, Energy and exergy analyses of porous baffles 18
ACCEPTED MANUSCRIPT inserted solar air heaters for building applications, Energy and Buildings 57 (2013) 338–345. [18] F. Bayrak, H. F. Oztop, A. Hepbasli, Energy and exergy analyses of porous baffles inserted solar air heaters for building applications, Energy and Buildings 57 (2013) 338–345. [19] A. Ejlali, A. Ejlali, K. Hooman, H. Gurgenci, Application of high porosity metal foams as air-cooled heat exchangers to high heat load removal systems, International Communications in Heat and Mass Transfer 36 (2009) 674–679.
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[20] M. A. Sheremet and Ioan Pop, Effect of local heater size and position on natural convection in a tilted nanofluid porous cavity using LTNE and Buongiorno's models, Journal of Molecular Liquids 266 (2018) 19–28. [21] F. Kuwahara, C. Yang, K. Ando, A. Nakayama, Exact Solutions for a thermal non-equilibrium model of fluid saturated porous media based on an effective porosity, Journal of Heat Transfer 133 (2011) 112602.
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[22] Comsol Multiphysics official website: https://www.comsol.jp/multiphysics [accessed 15 March 2019]..
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ACCEPTED MANUSCRIPT Figure captions Figure 1. The schematic view of the considered study (a) pore scale domain (b) volume averaged domain Figure 2. The computational flow chart of COMSOL Multiphysics Figure 3. The change of dimensionless mean temperatures of solid and fluid and the difference between them with dimensionless time when s 1 , s 0.01 (a) Sp 50 (b) Sp 250 (c) Sp 1000 (d) value for Sp 50，100，250，and 1000
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Figure 4. The variation of dimensionless mean temperatures of solid and fluid phases and the difference between them with dimensionless time when s 0.5 , s 0.01 (a) Sp 50 (b) Sp 250 (c) Sp 1000 (d) value for Sp 50，100，250，and 1000
Figure 5. The variation of dimensionless mean temperatures of solid and fluid phases with
dimensionless time when s 0.01 , s 0.01 for Sp 50，250 and 1000 (two curves overlaps)
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Figure 6. Occurrence of local thermal equilibrium for the studied cases (a) local thermal equilibrium map according to s and s (assumption for the local thermal equilibrium is 0.05 ) (b) the variation of thermal diffusivity ratio ( *s / *f ) with respect to s and s
Figure 7. The temperature and velocity distributions of the water in the cells of domain with Raf 105 and the same domain for volume averaged when s 0.068 , s 1.06 and Sp 46.7 for three
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different time steps (a) 0.0320 (b) 0.1602 (c) 0.3205
Figure 8. The comparison of pore scale and volume averaged results for the centerline of the porous
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media in which voids filled with water for s 0.068 , s 1.06 (a) Sp 39.2 (for Ra f 10 2 ) (b) Sp 46.7 (for Raf 105 )
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Figure 9. The temperature and velocity distributions of air in the cells of domain with Ra f 105 and for the same domain for volume averaged when s 1.11 , s 1.08 and Sp 1.91 for three different
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time steps (a) 0.0510 (b) 0.1275 (c) 0.2550 Figure 10. The comparison of pore scale and volume averaged results for the centerline of the porous
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media in which voids filled with air for s 1.11 , s 1.08 (a) Sp 1.74 (for Ra f 102 ) (b) Sp 1.91 (for Ra f 105 )
Table Caption Table 1. Thermophysical properties and dimensionless governing parameters for water and air in the pore scale and volume averaged studies.
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(a)
(b)
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Figure 1. The schematic view of the considered study (a) pore scale domain (b) volume averaged domain
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Figure 2. The computational flow chart of COMSOL Multiphysics
22
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(b)
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(a)
(c)
(d)
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Figure 3. The change of dimensionless mean temperatures of solid and fluid and the difference between them with dimensionless time when s 1 , s 0.01 (a) Sp 50 (b) Sp 250 (c) Sp 1000 (d) value for Sp 50，100，250，and 1000
23
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(b)
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(a)
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(c)
(d)
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Figure 4. The variation of dimensionless mean temperatures of solid and fluid phases and the difference between them with dimensionless time when s 0.5 , s 0.01 (a) Sp 50 (b) Sp 250 (c) Sp 1000 (d) value for Sp 50，100，250，and 1000
24
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Figure 5. The variation of dimensionless mean temperatures of solid and fluid phases with dimensionless time when s 0.01 , s 0.01 for Sp 50，250 and 1000 (two curves overlap)
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(a)
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(b)
Figure 6. Occurrence of local thermal equilibrium for the studied cases (a) local thermal equilibrium map according to s and s (assumption for the local thermal equilibrium is 0.05 ) (b) the
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variation of thermal diffusivity ratio ( *s / *f ) with respect to s and s
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Volume averaged Analysis
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Pore Scale Analysis
f
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Volume averaged temperature for fluid phase
s
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Volume averaged temperature for solid phase
(b)
(c)
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(a)
Figure 7. The temperature and velocity distributions of the water in the cells of domain with Raf 105 and the same domain for volume averaged when s 0.068 , s 1.06 and Sp 46.7 for three different time steps (a) 0.0320 (b) 0.1602 (c) 0.3205
27
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(a) (b) Figure 8. The comparison of pore scale and volume averaged results for the centerline of the porous media in which voids filled with water for s 0.068 , s 1.06 (a) Sp 39.2 (for Ra f 10 2 )
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(b) Sp 46.7 (for Raf 105 )
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Volume averaged Analysis
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Pore Scale Analysis
f
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Volume averaged temperature for fluid phase
s
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Volume averaged temperature for solid phase
(a)
(b)
(c)
Figure 9. The temperature and velocity distributions of air in the cells of domain with Ra f 105 and for the same domain for volume averaged when s 1.11 , s 1.08 and Sp 1.91 for three different time steps (a) 0.0510 (b) 0.1275 (c) 0.2550
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(a)
(b)
Figure 10. The comparison of pore scale and volume averaged results for the centerline of the
(b)
Sp 1.91 (for Ra f 105 )
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Ra f 102 )
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porous media in which voids filled with air for s 1.11 , s 1.08 (a) Sp 1.74 (for
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ACCEPTED MANUSCRIPT Table 1. Thermophysical properties and dimensionless governing parameters for water and air in the
Water
Air
k (W/m K)
0.61
0.026
( kg/m )
996
1.177
C p ( J/ kg K)
4180
1005
( Pa.s)
8.544e-4
1.86e-5
k eff ,f (W/m K)
0.58
0.025
Prf
5.85
0.71
Ra f
10 2 , 10 5
10 2 , 10 5
Pre
0.30
0.34
Ra e
5.066, 5065.9
28.88
*s
15.60
s
0.068
*f
0.053
0.50
f
1.04
0.0048
48, 47996.1 1.82 0.98 1.11
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Parameters
46.75 for Ra f 105
1.91 for Ra f 105
39.28 for Ra f 102
1.74 for Ra f 102
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Sp
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pore scale and volume averaged studies.
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Analysis of local thermal non-equilibrium condition for unsteady heat transfer in porous media with closed cells: Sparrow number C.Y. Wang , M. Mobedi , F. Kuwahara PII: DOI: Reference:
S0020-7403(18)34047-5 https://doi.org/10.1016/j.ijmecsci.2019.04.022 MS 4869
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
6 December 2018 15 March 2019 13 April 2019
Please cite this article as: C.Y. Wang , M. Mobedi , F. Kuwahara , Analysis of local thermal nonequilibrium condition for unsteady heat transfer in porous media with closed cells: Sparrow number, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.04.022
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights
Instead of Nusselt number, Sparrow number appears in the dimensionless governing equations by using equilibrium thermal diffusivity.
The Sparrow number is interpreted as the equilibrium conduction thermal resistance to the convection thermal resistance in the pores.
A good agreement between the pore scale and volume averaged results for the working fluid of water and air is observed.
A chart is presented in terms of Sparrow number, solid thermal diffusivity and thermal
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capacitance to estimate the local thermal equilibrium condition.
1
ACCEPTED MANUSCRIPT Analysis of local thermal non-equilibrium condition for unsteady heat transfer in porous media with closed cells: Sparrow number C. Y. Wang, M. Mobedi*, F. Kuwahara
Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu-shi, Japan ＊
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Corresponding author：[email protected]
Abstract
The local thermal non-equilibrium condition for a porous medium with closed cells under unsteady state heat transfer is analyzed. Although the fluid circulates in the closed pores due to the buoyancy effect, the volume averaged velocity for a closed pore is zero causing excluding of the continuity and
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momentum equations from the volume averaged governing equations. The volume averaged governing equations are non-dimensionalized by using the equilibrium thermal diffusivity and Sparrow number appears in the dimensionless governing equations automatically. The Sparrow number is interpreted as the equilibrium conduction thermal resistance to the convection thermal resistance for the entire domain. For the high values of Sparrow number (such as 1000), the
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convection heat transfer resistance is considerably smaller than the heat conduction resistance resulting in high possibility of the local thermal equilibrium. Dimensionless governing equations show
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that four dimensionless parameters as porosity, Sparrow number, solid (or fluid) dimensionless thermal capacitance and thermal conductivity play important roles on the prediction of the local thermal non-equilibrium state in a closed cell porous medium. A chart for the prediction of local thermal
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non-equilibrium is presented for Sparrow number of 1, 50, 100 and 500. Furthermore, a pore scale study for a closed cell porous medium with working fluids as water and air is performed. A good
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agreement between the pore scale and volume averaged results is observed. The obtained pore scale results support the results of the volume averaged parametric study and also the suggested chart for the
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prediction of local thermal non-equilibrium state.
Keywords: Sparrow number, local thermal non-equilibrium, closed cell porous media.
1. Introduction The voids of a porous medium can be interconnected (open pore) or unconnected (closed cell or isolated pore). Although most of studies in the literature were performed on the porous media with open pores (Varol et al. [1], Celik et al. [2], Ozgumus and Mobedi [3]), many applications for the closed cell porous media particularly in industry exist that requires further researches. For instance, a 2
ACCEPTED MANUSCRIPT closed cell metal foam is used for insulation against fire, blast energy adsorption, sound absorber, electromagnetic wave shield. Transverse heat transfer in a honeycomb structure or heat transfer through the walls constructed by hollow bricks can be other examples for the closed cell porous media. Furthermore, the closed cell structure can also be faced in materials such as plastic based insulation material (for instance polyethylene or polyurethane insulation plates) or even foods (for instance breads or cakes). Our literature survey showed that most of the studies on porous media with closed pore were
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performed on the mechanical properties and mechanical behavior of the porous media rather than heat transfer response. Some examples of the limited number of studies for heat transfer analysis in the closed cell porous media are the studies of Ma and Tzeng [4] who investigated heat transfer in multi-channels located in the closed cell aluminum foams, Vesenjak et al. [5] who performed a parametric computational study on the heat conduction in a closed-cell cellular metal foam under steady state condition, or Wang et al. [6] who discussed the thermal conductivity of closed-cell
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aluminum foams for various cell configurations and different porosities. In all of these studies, it is assumed that a local thermal equilibrium exists between the fluid and solid phases. Researchers showed that the results between the local thermal equilibrium and non-equilibrium can be considerably different, and that is why the validity of the local thermal equilibrium assumption was investigated and reported in literature. For instance, Vafai and Sozen [7] found that there might be
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significant discrepancies between the fluid and solid phase temperature distributions in a heat storage packed bed. Amiri and Vafai [8] also analyzed the thermal dispersion and local thermal
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non-equilibrium effects through porous media and presented error maps to assess the commonly used assumptions. Minkowycz et al. [9] analyzed the departure from local thermal equilibrium in porous media due to a rapidly changing heat source. They used Sparrow number and declared that a local
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thermal equilibrium can exist for rapidly changing heat sources under sufficiently large Sparrow number. Alazmi and Vafai [10] studied the effect of using different boundary conditions for the case of
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constant wall heat flux under LTNE (local thermal non-equilibrium) conditions. Haji-Sheikh and Minkowycz [11] performed an analytical study to describe the role of dimensionless quantities that identify the early departure from local thermal equilibrium in the presence of a rapidly changing heat
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source. Further theoretical studies on the heat transfer under local thermal non-equilibrium state such as the studies of Neild et al. [12], Neild and Kuznetsov [13], Vadasz [14], Dukhan and Hooman [15], Hoghoughi et al. [16], Izadi et al. [17], Bayrak et al. [18], Ejlali et al. [19] and Sheremet and Pop [20] exist in literature. Among the available studies in literature, the study done by Minkowycz et al. [9] has taken the attention of authors of the present study due to the use of Sparrow number for assessing of the thermal equilibrium assumption, and it motivated the authors to perform the present study. The volume averaged heat transfer equations are non-dimensionalized by using equilibrium thermal diffusivity and 3
ACCEPTED MANUSCRIPT Sparrow number appears automatically. Although the fluid motion exists in the closed pores due to the buoyancy effect, the volume averaged velocity in the closed pore is zero removing the volume averaged continuity and momentum equations. The dimensionless forms of the volume averaged heat conduction equations for the solid and fluid phases are solved and the effects of Sparrow number as well as the solid dimensionless thermal capacitance and thermal conductivity on the heat transfer mechanism are investigated. Furthermore, a porous medium with 25 closed pores under one dimensional heat transfer is considered and both the pore scale governing equations (continuity,
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momentum and energy equations) and the volume averaged governing equations (heat conduction equations for the solid and fluid phases) are numerically solved to support the parametric study results and the suggested charts. To the best of our knowledge, this is the first comprehensive study on the validation of local thermal equilibrium in a closed cell porous media based on the definition of Sparrow number.
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2. Considered problem
The schematic views of the considered pore scale domain and volume averaged domain are shown in Figure 1. Both the pore scale and volume averaged domains are described below. Pore scale domain: The porous media is a square domain with length of L . The details of closed cells can be seen in the Figure 1(a). It consists of 25 square closed cells with the length of l . The cells are
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not connected to each other while the solid phase (i.e. aluminum frame) interconnected in the entire domain. The top and bottom surfaces of the porous media are insulated and the left and right vertical
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surfaces are maintained at constant temperatures of Th and Tc , respectively. The gravity affects in –y direction and the radiation heat transfer is neglected. The working fluids are water and air and their thermal properties are given in the Table 1. The density, specific heat capacity and thermal
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conductivity of the solid phase, which is aluminum alloy, are 2700(kg/m3), 900(J/kg.K) and 203(W/m.K), respectively. Based on the thermophysical properties and the geometry of the studied
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domain, the value of dimensionless parameters such as Prf , Ra f and are calculated for the studied cases with air and water and given in Table 1.
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Volume averaged domain: The volume averaged domain is also shown in Figure 1(b). A continuous domain exists for the volume averaged equations since the governing equations are volumetrically integrated. No velocity field exists in the volume averaged domain and the conduction heat transfer is the only mechanism of heat transfer. The volume averaged governing equations are non-dimensionalized and a parametric study is performed by changing the dimensionless governing parameters in the range of 0.01 s 1 , 0.01 s 1 and 1 Sp 1000 . The definitions of these dimensionless parameters ( s , s , Sp ) are given in next section.
3. Governing equations and boundary conditions 4
ACCEPTED MANUSCRIPT The governing equations for the pore scale and volume averaged approaches are presented in this section, separately.
3.1 Governing equations for the pore scale analysis For the pore scale study, the continuity and momentum equations are solved to find the velocity and pressure distributions for fluid while the energy equations for the solid and fluid are solved to find the temperature distribution for the entire domain. The dimensional governing equations for the pore scale analysis of heat and fluid flow are:
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V 0
V 1 (V. )V p f 2 V g(T Tref ) j t f
(C p ) s
Tf V. Tf ） k f 2 Tf t
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(C p ) f (
Ts k s 2 Ts t
(1) (2)
(3)
(4)
where and Tref are the thermal expansion coefficient of working fluid and reference temperature (initial temperature which is equal to the cold wall temperature in this study), respectively. The
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equilibrium thermal diffusivity can be employed and the above governing equations can be written in dimensionless form as follows, * V * 0
(5) (6)
f * * V . f f *2 f
(7)
s s *2 f
(8)
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V* (V* . * )V* *P* Pre *2 V* Ra e 3 Pre f j
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where V and P* represent the dimensionless velocity vector and pressure. The dimensionless temperatures of the solid and fluid phases are shown by f and s , respectively. , Ra e and Pre are the dimensionless time, Rayleigh number and Prandtl number based on the equilibrium thermal diffusivity, respectively. is the thermal diffusivity ratio between the fluid or solid phase with equilibrium one. The employed dimensionless parameters can be mathematically expressed as,
* L
,
* VL V e
,
L l
,
t e 2
L
5
,
*
P
PL2 f e2
,
Pre f e
,
(9)
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Prf Pre
f , Ra f Ra e f e e
(10)
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where Prf and Ra f are Prantl and Rayleigh numbers based on the fluid thermal diffusivity.
3.2 Governing equations for volume average analysis
For the volume average method, a scalar or vector quantity can be integrated in a representative discontinues space and consequently the continuity of that space can be provided. Since two phases as
1 dV VV
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solid and fluid exist in the porous media, two volume averaged quantities can be defined:
1 dV Vx V x
x
(12)
are the volume average and intrinsic volume average of quantity, respectively.
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and
where
(11)
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V and Vx are the total volume and the corresponding volume of solid or fluid phase. Taking volume average of the dimensional pore scale governing equations (Eq. (1-4)) yields the dimensional volume averaged governing equations as:
(13)
PT
V 0
AC
CE
f f V 1 C ( V . ) V P f 2 V V f V V t K K 2 g(T Tref ) j
Cf (
Cs
f
T
t
T
t
s
f V . T ） k eff ,f 2 T
f
s
f
hv( T T )
s s f k eff ,s 2 T h v ( T T )
(14)
(15)
(16)
where Cs and C f are the thermal capacitance of solid and fluid phases (i.e. Cs (Cp )s and Cf (Cp ) f ), respectively. h v is the volumetric interfacial heat transfer coefficient, k eff ,s and
k eff ,f are the solid and fluid effective thermal conductivity. k eff ,s and k eff ,f can be calculated from
6
ACCEPTED MANUSCRIPT the following equations ((Kuwahara et al. [21]) :
k eff , f *k f k disp
(17)
k eff , s (1 * )k s
(18)
k s and k f are the thermal conductivity of solid and fluid, and * is the effective porosity that can
be found from the following equation.
k kf * eff ks kf
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(19)
The value of k eff which is the effective thermal conductivity (for entire domain) can be numerically calculated by considering a representative volume, imposing temperature difference and applying Fourier law. In this study, the value of * is found as 0.95 for the porous media with 0.9 . For a closed cell porous media, the volume averaged velocity can vanish.
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1 V V dV 0 VV
(20)
The continuity and momentum equations (Eqs. (13-14)) disappear since the volume averaged velocity for the cell porous media is zero. Hence, the volume averaged equations (13-16) take the following
f
t
(1 ) C s
k eff ,f 2 T
T
s
t
f
s
f
hv( T T )
s s f k eff ,s 2 T h v ( T T )
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Cf
T
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form for a closed cell porous medium,
(21)
(22)
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The heat conduction equations for the solid and fluid phases are the only equations for the volume average approach in the closed cell porous media. h v which is the volumetric interfacial heat transfer
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coefficient can be defined as h v h s A v , where h s (W / m 2 K) is the surface interfacial heat transfer coefficient and A v (m 2 / m3) is the volumetric heat transfer area. It is possible to write the volumetric
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heat transfer area as A v / rh , where rh is hydraulic diameter of the fluid phase. It can be defined as rh Vf / Aint where A int is the interface heat transfer area in the fluid volume of Vf . Then, the Eqs. (21) and (22) become as:
Cf
T t
(1 ) C s
f
k eff ,f 2 T
T t
s
f
hs s f (T T ) rh
(23)
h s s f k eff ,s 2 T s ( T T ) rh
(24)
The above equations can be non-dimensionalized by using dimensionless variables of Eq. (9) as: 7
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f s
f
s
f *2
f
s
f
Sp ( )
(25)
s s f s *2 Sp ( )
(26)
, are the dimensionless thermal conductivity and thermal capacitance, respectively, and Sp is
the Sparrow number. The employed dimensionless parameters can be written mathematically as (27)
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k eff , s k eff , f C s (1 ) Cf hL2 s , f , s , f , Sp k e rh C e Ce k e k e
C e and e are the thermal capacitance and thermal conductivity of local thermal equilibrium state.
They can be expressed as:
Ce 1 Cs Cf
(28)
k e k f (1 )k s
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(29)
Eqs. (25) and (26) can be solved for this problem, however two additional equations can also be written by using definitions in Eq. (27)
s f
1
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s f
1
(30)
(31)
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If the Eqs. (30) and (31) are considered with Eqs. (25) and (26), it is seen that the number of governing parameters decreases to four as solid (or fluid) dimensionless thermal conductivity and thermal capacitance, Sparrow number and porosity. In this study, the values of f and f are written in
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terms of s and s . Hence the dimensionless governing parameters are s , s and Sp while porosity is constant as 0.9.
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Furthermore, for the thermal equilibrium case, s and
f
are close to each other ( s f ),
therefore Eqs.(25) and (26) take the following form:
*2
(32)
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3.3 On the physical definition of sparrow number Sparrow number plays an important role on the results of the governing equations (Eqs. 30-31). In order to understand the physical meaning of Sparrow number, a cube porous medium with length of L can be considered, the Sparrow number can be written as,
8
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Sp
hA v L2 hL2 k e rh k e
hA int L 2 L3 k e
L 1 L 2 k e L k L k A R e e cr conduction 1 1 1 R convection hA int hA int hA int
(34)
where R conduction is the conduction thermal resistance through the length of the cube and R convection is convection thermal resistance. Both R conduction and R convection concerns with entire domain. Aint is the total area of the interfacial heat transfer of porous medium while Acr is the cross section area
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of the porous media with cube shape. Eq. (34) shows that Sparrow number can be defined as the ratio of conduction to convection heat transfer resistance through the porous media. For the high values of Sparrow number (such as Sp 1000 ), the convection heat transfer resistance is considerably smaller than the heat conduction resistance and comparatively large amount of heat can be transferred from the fluid to solid resulting in high possibility of local thermal equilibrium. The local thermal
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non-equilibrium can exit for the low values of Sparrow number (such as Sp 50 ) implying that the convection heat transfer resistance is comparable or less than heat conduction resistance. In other words, the propagation of heat by conduction through the porous media is faster than heat transfer by convection from fluid to the solid phase.
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3.4 Initial and Boundary Conditions
Initially, the temperature of the entire domain is Ti both for the pore scale and volume averaged
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domains. Suddenly the temperatures of right and left walls of the porous media jump to Th and Tc , respectively and they are maintained at the same temperature for the entire process. In this study, the initial and cold wall temperature are taken equal. The top and bottom surfaces of the cavity are
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thermally insulated. For the pore scale domain, a continuous heat flux and unique temperature conditions are applied onto the interface between solid and fluid phases. For the volume averaged
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domain, the heat exchange between the solid and fluid phases is provided by Sparrow number. The dimensionless form of the initial and boundary conditions is given in Eq. (35). Pore scale
Volume averaged
s f 0
s
f
0
x* 0
s 1, f no need
s
f
1
x* 1
s 0, f no need
s
f
0
s 0, f no need n
AC
0
*
y 0 and 1
n
s
n
(35)
f
0
At the interface between the solid and fluid for the pore scale analysis, the following relation is valid: 9
ACCEPTED MANUSCRIPT s k f f , s f n k s n
(36)
where n is the normal direction to the considered surface.
4. Numerical solution for the pore scale and volume averaged governing equations For the pore scale study, a commercial software, COMSOL Multiphysics (V5.3a COMSOL Multiphysics), based on the finite element method is used to solve the coupled heat and fluid flow
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equations. All steps to obtain the results are described in the flow chart of Figure 2. A mesh refinement analysis is done and it is found that 57600 number of domain elements (triangular), 3840 edge elements, and 136 vertex element are sufficient to obtain the accurate results for this problem. The Multiphysics module used to solve motion equation is “Laminar flow” while for solving heat transfer equations in solid and fluid the module of “Heat transfer in fluids” is used. The software was run in a workstation with Intel(R) Xeon(R) CPU E5-2630 v3, 32G RAM and it took around 8 hours to obtain
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the accurate results. The relative tolerance was 10 3 . The algorithm of the employed software is written in Ref [22] in details.
The volume averaged governing equations are solved numerically by using an in-house code in Fortran platform based on the finite difference method. It consists of inner and outer loops. The inner loop is used to solve conduction equations for the solid and fluid for each time step. The relative
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convergence criterion for inner loop is 10 6 . The outer loop is used to increase time since the problem is unsteady.
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The solution of the volume averaged heat conduction equations for the solid and fluid (Eqs. (30) and (31)) requires to know the value of Sp (consequently, h v should be known). Unfortunately, no
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study relating to the transient value of the interfacial heat transfer coefficient for the closed pores of porous media could be found. In this study, the value of interfacial heat transfer coefficient is obtained
CE
from the pore scale study. The employed definition for determination of the average interfacial heat transfer coefficient is,
A int t
q" ( t )dA
t A int s 0( T
AC
hv
1
f
T )
(37)
dt
q" is the heat flux between solid and fluid phases at any point of the interface and at any instant such
as t , and A int is the total interfacial heat transfer area. t is the required time for the system to become steady state ( ( T
s
f
T ) 0.05 is the condition for the steady state). As it can be seen
from Eq. (37), the average of interfacial heat transfer coefficient is obtained by integration of the local interfacial heat transfer coefficient both in time and space.
10
ACCEPTED MANUSCRIPT The average of the solid and fluid temperatures of the entire domain for the solid and fluid phases are found at any time step to compare the solid and fluid phase temperatures.
f
m () where
vf
f () m
f
()dv s
, m ()
dv vf
and
s () m
vs
s
()dv (38)
dv vs
are the solid and fluid phase mean (i.e., surface average for the entire
parameters of is defined as following:
s () m
f () m
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domain) temperatures. Finally, in order to understand the order of local thermal non-equilibrium, the
(39)
The above equation shows that the temperature difference between the mean solid and fluid phases for
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the whole domain during the entire heat transfer process.
5. Results and discussions
The grid independency study for the pore scale analysis was done and it was found that 61440 elements is sufficient to obtain the accurate results. For the validation of numerical study, the study of K.D. Antoniadis et al. [19] done for a hollow brick (two dimensional porous media with closed cell)
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for different values of thermal conductivity ratio is used. An excellent agreement (below 1% difference) was observed between our study and their reported study. As it can be seen from Eqs. (5-8), the pore scale dimensionless temperatures of the solid and fluid phases depend on four parameters as
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Pre , Ra e , , . The dimensionless volume averaged solid and fluid temperatures depend on three
parameters as s , s and Sp . The values of all dimensionless parameters are given in Table 1 for air
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separately.
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and water. The results of volume averaged parametric study and pore scale study are explained
5.2 Results for the volume averaged parametric analysis
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A parametric study is performed to investigate the effects of dimensionless numbers ( s , s and Sp ) on the temperatures of the solid and liquid phases. Since the transient behavior of temperature is important, the average of solid and liquid temperatures for the entire domain are calculated (i.e., Eq. (38)) and plotted. Figure 3 shows the change of mean temperature of solid and fluid for the case of s 0.01 , s 1 , Sp 50, 250 and 1000 . As it can be seen from Figure 3(a), there is an obvious temperature difference
between the solid and fluid mean temperatures before they become steady state. Since the value of Sparrow number is low (referring to high convection thermal resistance), the heat exchange between the solid and fluid is weak and a non-equilibrium heat transfer in the porous media is expected. By 11
ACCEPTED MANUSCRIPT increasing the Sparrow number from 50 to 250, the heat exchange between the solid and fluid phases increases and the temperatures of the solid and fluid phases approach to each other. Further increase in Sparrow number (from 250 to 1000) causes the fluid and solid phase mean temperatures become considerably close to each other and almost a local thermal equilibrium can be observed. The difference between the dimensionless temperature of solid and fluid (i.e., ) can be a good parameter to show the magnitude of local thermal non-equilibrium. This difference for four Sparrow number is shown in Figure 3(d). As can be seen for Sp 1000 , the maximum temperature difference between the
Sp 50 is 0.13 indicating a local thermal non-equilibrium state.
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solid and liquid is 0.016 showing almost an equilibrium state, while the maximum value of for
The parameter values of Figure 4 are the same with Figure 3 except the solid dimensionless thermal conductivity s 0.5 . The values of *s (i.e., *s s / s ) and *f (i.e., *f f / f ) can be calculated by using Eqs. (22, 30 and 31). By decreasing the solid dimensionless thermal conductivity
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from 1 to 0.5, the value of *s / *f decreases from 991 to 91 causing heat propagation in the solid and fluid phases becomes closer to each other and the temperature difference between the solid and fluid phases becomes smaller. This fact can be seen by the comparison of Figures 3(a) and 4(a). Similar to the Figure 3, by increasing Sp from 50 to 100, and from 100 to 1000, the solid and fluid dimensionless mean temperatures become closer due to the increase of heat exchange between the
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solid and fluid phases. The temperature differences between the solid and fluid are shown in Figure 4(d) to observe the magnitude of local thermal non-equilibrium. The comparison of 3(d) and 4(d)
ED
indicates the effect of s on the local thermal equilibrium condition. By decreasing the value of s from 1 to 0.5, the difference between the solid and fluid temperatures decreases.
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The change of solid and fluid phases temperature for further decrease of s from 0.5 to 0.01 is shown in Figure 5. Further decrease in s from 0.5 to 0.01causes the value of *s / *f decreases from 91 to 1. An equal heat propagation in the solid and fluid phases exists for Figure 5. Therefore, for the all
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studied Sparrow number ( Sp 50, 250, 1000 ), a local thermal equilibrium can be observed. The results of Figures 3, 4 and 5 clearly show that the heat exchange between the solid and fluid phases increases
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by increasing Sparrow number however, the critical Sparrow number (can be defined to distinguish the local thermal equilibrium from non-equilibrium) strongly depends on s . A parametric study on the various values of s , s and Sp was done to separate the region of local thermal equilibrium from non-equilibrium. The results of the parametric study is shown in Figure 6 for Sp 50, 100 and 500 . The x axis of Figure 6(a) shows s while the y axis refers to s . It is assumed
that a local thermal equilibrium for the heat transfer in the closed cell porous media exits if 0.05 . This is the assumption of present study and it might be changed based on the application. Figure 6(b) is presented in order to interpret the results of Figure 6(a). It shows the value of *s / *f for the 12
ACCEPTED MANUSCRIPT studied case of Figure 6(a). For the studied cases of Figure 6(a), the value of *s / *f changes from the large value of 991 to small value as 0.001. Figure 5 provides important information for the prediction of a local thermal non-equilibrium heat transfer process in a cavity with closed cell as follows; -
For the value of *s / *f 1 , the propagation of heat in the solid and fluid are close to each other causes local thermal equilibrium state becomes independent of Sparrow number (which is considered in this study). Further increase or decrease in the value of *s / *f (such as 991 or 0.001) causes the possibility
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-
of local thermal non-equilibrium increases due to different speed of heat propagation in the solid and fluid phases. For instance, when Sp =50, a local thermal non-equilibrium exists for the points of s 0.01 and s 1 due the large difference in heat propagation of solid and fluid (faster heat prorogation exists in the solid , *s / *f 991 ).
In addition to s and s , the Sparrow number plays an important role on the character of heat
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-
transfer through the closed cell porous media. By increasing the value of Sp , the source terms in Eqs. (30) and (31) becomes more active, and consequently the heat exchange between the solid and fluid increases. This increase of heat exchange increases the possibility of occurrence of local thermal equilibrium state. That is why, number of unfilled markers in the Figure 6(a) for Sp 100
For Sp 500 , all markers are unfilled showing considerable heat transfer between the solid and
ED
-
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is more than Sp 50 , or number of unfilled markers for Sp 500 is greater than Sp 100 .
fluid phases despite of extremely high or low values of *s / *f (such as 991 or 0.001). It seems that *s / *f and Sp are two important parameters for the prediction of local thermal
PT
-
non-equilibrium for the unsteady of a heat transfer in closed cell of porous media. .
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The above comments are only valid for the range of dimensionless parameters considered in this study.
5.3 The pore scale and volume average results for real cases
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A concept for specifying non-equilibrium heat transfer in a closed cell porous media is proposed and explained in the previous section. A volume average study for two different working fluids as water and air is done to validate the suggested graphics (Figure 6) and validate the explained comments. The volume average study is also validated by the results of pore scale analysis, otherwise the results of the volume average for two different fluids might be questionable 5.3.1. Results for the porous media with water in cells Figure 7 shows the temperature and velocity distributions for both the pore scale and volume averaged domains when s 0.068 , s 1.06 , Sp 46.7 and Ra f 105 , as given in Table 1. The value of 13
ACCEPTED MANUSCRIPT Sparrow number is calculated from the pore scale results when Ra f 105 . The values of fluid thermal capacitance and thermal conductivity are found as s 0.068 and s 1.06 , and the thermal diffusivity ratio is calculated as 300 (i.e. *s / *f 300 ). This value of *s / *f is considerably larger than 1 and furthermore the Sparrow number is low ( Sp 46.7 ). Based on the previous section comments, the possibility of the existence of local thermal non-equilibrium under this condition is high and this fact is investigated below. From Figure 7(a), it is seen that the solid phase temperature is faster than the fluid phase due to the
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faster propagation of heat in the solid at the starting period ( 0.032 ). By increasing the time to 0.16 and 0.32 , the pore scale temperatures of both solid and fluid phases become closer to
each other. The volume averaged temperatures (middle and bottom cavities of Figure 7) also show the same behavior of the pore scale one. It should be mentioned that heat conduction equations for the solid and fluid phases are solved and that is why no velocity is seen in the results of the volume
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averaged domains.
Figure 8 shows the comparison of solid and fluid dimensionless mean temperatures (
s m
,
f m
)
obtained from the pore scale and volume averaged analysis for Ra f 10 2 and 105 , respectively. A good agreement between the volume averaged and pore scale analysis can also be observed proving the correctness of our volume average study. The small differences between the pore scale and the
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volume averaged results may be due to the definition of interfacial heat transfer coefficient (Eq. (32)) in which the average of local heat transfer coefficient both in time and space was considered. A local
ED
thermal non-equilibrium for a large period at the beginning of the heat transfer process can be seen, supporting the comments of the parametric volume averaged study. Large difference in heat
PT
propagations due to the high value of *s / *f (i.e. 300) and weak heat exchange between the solid and
CE
fluid phases causes a local thermal non-equilibrium exists.
5.3.2 Results for porous media filled with air in cells Figure 9 shows the temperature and velocity distributions of a closed cell porous media when the
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voids are filled with air for s 1.11 , s 1.08 , Sp 1.91 and Ra f 105 , as given in Table 1. The value of thermal conductivity ratio ( *s / *f ) is calculated as 2.09. The value of thermal diffusivity ratio takes the attention since it is around 1. Although the Sparrow number is extremely low (conduction and convection resistances are almost equal), a local thermal equilibrium can be expected due to almost equal propagation of heat in the solid and fluid. Figure 9 indicates that the propagation of heat in the solid and fluid are the same for the solid and fluid phases. This fact can be clearly seen from both pore scale and volume average temperature distributions. Figure 10 shows the solid and fluid dimensionless mean temperatures ( 14
s m
,
f m
) for
ACCEPTED MANUSCRIPT Raf 102 and 105 at the center of the domain for different time steps. Almost, a local thermal
equilibrium state can be seen from this figure both from volume average and pore scale results. The values of s and s are known and the location of the above two cases in the local thermal equilibrium map (Figure 6) can be found. The location of two cases are marked in Figure 6(a) for the case of air ( s 1.11 , s 1.08 , Sp 1.91 ) and for the case of water ( s 0.068 , s 1.06 , Sp 48.6 ). The location of marker for air case is in right place validating Figure 6. A local thermal equilibrium exists for the case of air due to the small values of *s / *f 2.09 referring to almost identical heat
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propagation in solid and fluid in spite of low value of Sparrow number ( Sp 1.91 ). Similar to the case of air, the location of marker for water case is correct, and a local thermal non-equilibrium must exist for the water case due to faster propagation of heat in the solid comparing to fluid phase ( *s / *f 300 ).
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6. Conclusions
A numerical study on the local thermal non-equilibrium state for the transient heat transfer through the closed cell porous media is performed. Based on the obtained results and performed discussions, the following remarks can be concluded. -
If the governing equations are non-dimensionalized according to the equilibrium thermal
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diffusivity, six dimensionless parameters as porosity, the solid and fluid thermal diffusivity and thermal capacitance ratio and Sparrow number are obtained. Sparrow number appears -
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automatically.
Two relations between the dimensionless solid and fluid thermal diffusivity and thermal
-
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capacitance are found resulting in decreasing of dimensionless governing parameters to four. The Sparrow number is interpreted as the ratio of conduction to convection thermal resistance through the porous media. High value of Sparrow number (such as Sp 1000 ) referring to the
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higher convection in the pores compared to heat conduction through the porous media while low values ( Sp 50 ) refers to a large convection resistance in the pores and consequently a weak heat
AC
exchange between the solid and fluid.
-
When the value of Sparrow number is low (such as 50), a local thermal non-equilibrium heat transfer occurs for the cases in which the thermal diffusivity ratio is considerably greater or smaller than 1 (such as 100 or 0.01). A local thermal equilibrium independent of the value of Sparrow number exists if the thermal diffusivity ratio is around 1.
-
For high values of Sparrow number (such as 500), a local thermal equilibrium is observed for the all studied parameters due to the considerable heat exchange between the solid and fluid.
-
The results of parametric study are supported by the pore scale of two real cases of closed cell porous media. For closed cell porous media when the solid is aluminum and working fluid is water, 15
ACCEPTED MANUSCRIPT a local thermal non-equilibrium is seen due to large value of thermal diffusivity ratio ( *s / *f 300 ) and small Sparrow number ( Sp 38 ). For the closed cell porous media with air a local thermal equilibrium is observed because almost the same propagation of heat in the solid and fluid ( *s / *f 2 ) Although many studies on the predication of local thermal non-equilibrium can be found in literature, number of studies based on Sparrow number is limited. Authors believe that the non-dimensionlization of the solid and fluid energy equations according to the equilibrium thermal diffusivity is an
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appropriate method for finding appropriate governing parameters and consequently Sparrow number appears automatically. However, the determination of critical Sparrow number may be used to separate local thermal equilibrium state from non-equilibrium state requires further studies.
Acknowledgement
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The first author gratefully acknowledges the financial support from China Scholarship Council (No. 201808050059), which has sponsored his PhD study at the Shizuoka University in Japan.
NOMENCLATURE Heat transfer area, m2
C
Thermal capacitance , J/ m3 K
Cp
Specific heat capacity, J/ kg K
h
Interfacial heat transfer coefficient, W/m2K
K
Permeability, m2
k
Thermal conductivity, W/m K
L
Length of cavity, m
LT NE
Local thermal non-equilibrium
l
ED
PT
Local thermal equilibrium Length of cell, m Normal direction to the surface
AC
n
CE
LT E
M
A
P
Pressure, N/m2
Pr
Prantl number
q"
Heat flux, W⋅m−2
Ra
Rayleigh number
r
Diameter, m
Sp
Sparrow number
T
Temperature, K 16
t
Time, s
V
Velocity vector, m/s
x, y
Cartesian coordinates, m Thermal diffusivity, m2/s
Thermal expansion coefficient, 1/K
Function for the level of non-equilibrium condition
Porosity
Any dependent parameter
Thermal conductivity in equilibrium state
Density, kg/m3
Ratio between the characteristic lengths
Dimensionless time
Thermal diffusivity ratio between fluid and solid
Dimensionless thermal conductivity
Dimensionless thermal capacitance
cr
Cross section
e
Equilibrium
eff
Effective
f
Fluid
h
Hot, Hydraulic
i
Initial
int
Interface
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Cold
s
CE
PT
c
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Subscripts and superscripts
v
Volumetric
x
Solid or fluid phase
Reference Solid
AC
ref
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Dynamic viscosity, Pa.s Kinematic viscosity, m2/s Dimensionless temperature
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ACCEPTED MANUSCRIPT [2] H. Celik, M. Mobedi, A. Nakayama and U. Ozkol, A numerical study on determination of volume averaged thermal transport properties of metal foam structures using x-ray microtomography technique, Numerical Heat Transfer (in press). [3] T. Ozgumus and M. Mobedi, Effect of pore to throat size ratio on thermal dispersion in porous media, International Journal of Thermal Sciences 104 (2016) 135-145. [4] W.P. Ma and S.C. Tzeng, Heat transfer in multi-channels of closed cell aluminum foams, Energy Conversion and Management 48 (3) (2007) 1021–1028.
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[5] M. Vesenjak, Z. Zunic, A. O. chsner, M. Hribersek and Z. Ren, Heat conduction in closed-cell cellular metal, Mat. -wiss. u. Werkstofftech 36 (10) (2005) 608–612. [6] H. Wang, X.Y. Zhou, B. Long, J. Yang and H.Z. Liu, Thermal properties of closed-cell aluminum foams prepared by melt foaming technology, Transactions of Nonferrous Metals Society of China 26 (12) (2016) 3147−3153.
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[7] M. Sözen and K.Vafai, Analysis of oscillating compressible flow through a packed bed, International Journal of Heat and Fluid Flow 12 (2) (1991) 130-136. [8] A. Amiri and K. Vafai, Analysis of dispersion effects and non-thermal equilibrium, non- Darcian, variable porosity incompressible flow through porous media, International Journal of Heat and Fluid Flow 37 (6) (1994) 939-954.
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[9] W.J. Minkowycz, A. Haji-Sheikh and K. Vafai, On departure from local thermal equilibrium in porous media due to a rapidly changing heat source: the Sparrow number, International Journal of Heat and Mass Transfer 42 (1999) 3373-3385.
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[10] B. Alazmi and K. Vafai, Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions, International Journal of Heat and Mass Transfer 45 (2002) 3071-3087. [11] A. Haji-Sheikh and W.J. Minkowycz, Heat transfer analysis under local thermal non-equilibrium conditions, Emerging Topics in Heat and Mass Transfer in Porous Media (2008) 39-62.
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[12] D.A. Nield, A.V. Kuznetsov and M. Xiong, Effect of local thermal non-equilibrium on thermally developing forced convection in a porous medium, International Journal of Heat and Mass Transfer 45 (2002) 4949-4955.
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[13] D.A. Nield and A.V. Kuznetsov, Local thermal non-equilibrium effects in forced convection in a porous medium channel: a conjugate problem, International Journal of Heat and Mass Transfer 42 (1999) 3245-3252.
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[14] P. Vadasz, Explicit Conditions for Local thermal equilibrium in porous media heat conduction, Transport in Porous Media 59 (2005) 341–355. [15] N. Dukhan and K. Hooman, Comments on two analyses of thermal non-equilibrium Darcy– Brinkman convection in cylindrical porous media, International Journal of Heat and Mass Transfer 66 (2013) 440–443. [16] G. Hoghoughi, M. Izadi, H. F. Oztop, N. Abu-Hamdeh, Effect of geometrical parameters on natural convection in a porous undulant-wall enclosure saturated by a nanofluid using Buongiorno's model, Journal of Molecular Liquids 255 (2018) 148–159. [17] M. Izadi, S.A.M. Mehryan, M.A. Sheremet, Energy and exergy analyses of porous baffles 18
ACCEPTED MANUSCRIPT inserted solar air heaters for building applications, Energy and Buildings 57 (2013) 338–345. [18] F. Bayrak, H. F. Oztop, A. Hepbasli, Energy and exergy analyses of porous baffles inserted solar air heaters for building applications, Energy and Buildings 57 (2013) 338–345. [19] A. Ejlali, A. Ejlali, K. Hooman, H. Gurgenci, Application of high porosity metal foams as air-cooled heat exchangers to high heat load removal systems, International Communications in Heat and Mass Transfer 36 (2009) 674–679.
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[20] M. A. Sheremet and Ioan Pop, Effect of local heater size and position on natural convection in a tilted nanofluid porous cavity using LTNE and Buongiorno's models, Journal of Molecular Liquids 266 (2018) 19–28. [21] F. Kuwahara, C. Yang, K. Ando, A. Nakayama, Exact Solutions for a thermal non-equilibrium model of fluid saturated porous media based on an effective porosity, Journal of Heat Transfer 133 (2011) 112602.
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[22] Comsol Multiphysics official website: https://www.comsol.jp/multiphysics [accessed 15 March 2019]..
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ACCEPTED MANUSCRIPT Figure captions Figure 1. The schematic view of the considered study (a) pore scale domain (b) volume averaged domain Figure 2. The computational flow chart of COMSOL Multiphysics Figure 3. The change of dimensionless mean temperatures of solid and fluid and the difference between them with dimensionless time when s 1 , s 0.01 (a) Sp 50 (b) Sp 250 (c) Sp 1000 (d) value for Sp 50，100，250，and 1000
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Figure 4. The variation of dimensionless mean temperatures of solid and fluid phases and the difference between them with dimensionless time when s 0.5 , s 0.01 (a) Sp 50 (b) Sp 250 (c) Sp 1000 (d) value for Sp 50，100，250，and 1000
Figure 5. The variation of dimensionless mean temperatures of solid and fluid phases with
dimensionless time when s 0.01 , s 0.01 for Sp 50，250 and 1000 (two curves overlaps)
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Figure 6. Occurrence of local thermal equilibrium for the studied cases (a) local thermal equilibrium map according to s and s (assumption for the local thermal equilibrium is 0.05 ) (b) the variation of thermal diffusivity ratio ( *s / *f ) with respect to s and s
Figure 7. The temperature and velocity distributions of the water in the cells of domain with Raf 105 and the same domain for volume averaged when s 0.068 , s 1.06 and Sp 46.7 for three
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different time steps (a) 0.0320 (b) 0.1602 (c) 0.3205
Figure 8. The comparison of pore scale and volume averaged results for the centerline of the porous
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media in which voids filled with water for s 0.068 , s 1.06 (a) Sp 39.2 (for Ra f 10 2 ) (b) Sp 46.7 (for Raf 105 )
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Figure 9. The temperature and velocity distributions of air in the cells of domain with Ra f 105 and for the same domain for volume averaged when s 1.11 , s 1.08 and Sp 1.91 for three different
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time steps (a) 0.0510 (b) 0.1275 (c) 0.2550 Figure 10. The comparison of pore scale and volume averaged results for the centerline of the porous
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media in which voids filled with air for s 1.11 , s 1.08 (a) Sp 1.74 (for Ra f 102 ) (b) Sp 1.91 (for Ra f 105 )
Table Caption Table 1. Thermophysical properties and dimensionless governing parameters for water and air in the pore scale and volume averaged studies.
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(a)
(b)
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Figure 1. The schematic view of the considered study (a) pore scale domain (b) volume averaged domain
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Figure 2. The computational flow chart of COMSOL Multiphysics
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(b)
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(c)
(d)
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Figure 3. The change of dimensionless mean temperatures of solid and fluid and the difference between them with dimensionless time when s 1 , s 0.01 (a) Sp 50 (b) Sp 250 (c) Sp 1000 (d) value for Sp 50，100，250，and 1000
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(b)
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(a)
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(c)
(d)
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Figure 4. The variation of dimensionless mean temperatures of solid and fluid phases and the difference between them with dimensionless time when s 0.5 , s 0.01 (a) Sp 50 (b) Sp 250 (c) Sp 1000 (d) value for Sp 50，100，250，and 1000
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Figure 5. The variation of dimensionless mean temperatures of solid and fluid phases with dimensionless time when s 0.01 , s 0.01 for Sp 50，250 and 1000 (two curves overlap)
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(a)
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(b)
Figure 6. Occurrence of local thermal equilibrium for the studied cases (a) local thermal equilibrium map according to s and s (assumption for the local thermal equilibrium is 0.05 ) (b) the
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variation of thermal diffusivity ratio ( *s / *f ) with respect to s and s
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Volume averaged Analysis
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Pore Scale Analysis
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Volume averaged temperature for fluid phase
s
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Volume averaged temperature for solid phase
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Figure 7. The temperature and velocity distributions of the water in the cells of domain with Raf 105 and the same domain for volume averaged when s 0.068 , s 1.06 and Sp 46.7 for three different time steps (a) 0.0320 (b) 0.1602 (c) 0.3205
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(a) (b) Figure 8. The comparison of pore scale and volume averaged results for the centerline of the porous media in which voids filled with water for s 0.068 , s 1.06 (a) Sp 39.2 (for Ra f 10 2 )
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(b) Sp 46.7 (for Raf 105 )
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Volume averaged Analysis
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Pore Scale Analysis
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Volume averaged temperature for fluid phase
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Volume averaged temperature for solid phase
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Figure 9. The temperature and velocity distributions of air in the cells of domain with Ra f 105 and for the same domain for volume averaged when s 1.11 , s 1.08 and Sp 1.91 for three different time steps (a) 0.0510 (b) 0.1275 (c) 0.2550
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(a)
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Figure 10. The comparison of pore scale and volume averaged results for the centerline of the
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Ra f 102 )
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porous media in which voids filled with air for s 1.11 , s 1.08 (a) Sp 1.74 (for
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ACCEPTED MANUSCRIPT Table 1. Thermophysical properties and dimensionless governing parameters for water and air in the
Water
Air
k (W/m K)
0.61
0.026
( kg/m )
996
1.177
C p ( J/ kg K)
4180
1005
( Pa.s)
8.544e-4
1.86e-5
k eff ,f (W/m K)
0.58
0.025
Prf
5.85
0.71
Ra f
10 2 , 10 5
10 2 , 10 5
Pre
0.30
0.34
Ra e
5.066, 5065.9
28.88
*s
15.60
s
0.068
*f
0.053
0.50
f
1.04
0.0048
48, 47996.1 1.82 0.98 1.11
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Parameters
46.75 for Ra f 105
1.91 for Ra f 105
39.28 for Ra f 102
1.74 for Ra f 102
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Sp
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pore scale and volume averaged studies.
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