A new formulation for nondimensionalization heat transfer of phase change in porous media: An example application to closed cell porous media

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A new formulation for nondimensionalization heat transfer of phase change in porous media: An example application to closed cell porous media Chunyang Wang a, Moghtada Mobedi a,b,∗ a b

Graduate School of Science and Technology, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu-shi 432-8561, Japan Mechanical Engineering Department, Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu-shi, 432-8561, Japan

a r t i c l e

i n f o

Article history: Received 13 August 2019 Revised 21 October 2019 Accepted 16 November 2019 Available online xxx Keywords: Solid/liquid phase change Local thermal non-equilibrium Sparrow number Closed cell porous media Pore scale study Volume average study

a b s t r a c t The problem of solid/liquid phase change in a cavity assisted by a closed cell porous medium is analyzed by pore scale and volume average approaches. The volume average governing equations are nondimensionalized by using the porous media stagnant thermal diffusivity and Sparrow number appears automatically. The study consists of four parts as the derivation of new formulation based on Sparrow number, proving the validation of volume average method based on the given formulation for the closed cell porous media, discussion on the change of interfacial heat transfer coefficient with time and finally presenting a thermal equilibrium chart in terms of Sparrow number and thermal diffusivity ratio for prediction of local thermal equilibrium state. A good agreement between the pore scale and volume average results is observed. Based on the established thermal equilibrium chart, it is found that the possibility of local thermal equilibrium is high for the large values of Sparrow number (such as 500) and for the thermal diffusivity ratio around 1. The established equilibrium chart is verified by prediction of local thermal equilibrium state for two different closed cell porous media and then the validation of predication is proved by using the results of pore scale study. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Thermal energy storage is one of the hot topics in heat transfer area in recent years due to the limitation of conventional energy sources and global warming problems. Among many thermal storage methods, the solid/liquid phase change has become too popular recently. Nowadays, the solid/liquid phase change thermal storage is used in many areas such as the cooling storage in air conditioning systems during the night and using the stored energy in daytime when the peak consumption of electricity occurs. The solid/liquid cooling storage is also used to reduce the inlet air temperature of gas turbines and increases its performance in daytime when the ambient temperature is high. Further applications on the use of solid/liquid phase change such as cooling of electronic equipment, thermal storage in solar energy systems, cooling of batteries in car industry or the using of PCM (phase change material) in building for insulation can be found in literature.

∗ Corresponding author at: Shizuoka University, 3-5-1 Johoku, Hamamatsu-shi, 432-8561, Japan. E-mail address: [email protected] (M. Mobedi).

Naka-ku,

Solid/liquid phase change thermal storage has many advantages compared to other storage methods such as sensible or chemical thermal storage. The use of latent heat, simple principle working and design, environmental friendly working materials are some advantages of the solid/liquid phase change thermal storage. However, different applications have different working temperature and phase change period forcing researchers to investigate new phase change materials. One of the main problems of solid/liquid PCM is the values of thermal diffusivity and conductivity which are generally low. This causes the increase of melting period and decrease of storage power. To overcome this difficulty, many methods for the heat transfer enhancement have been studied and reported in literature such as the use of fins, porous media and metal nanoparticles. Among them, the use of high thermal conductive metal foam (with high porosity) has attracted the attentions of many researchers and many studies on this method have been done in literature [e.g., 1–3]. In the most of the reported studies in literature, the open cell metal foam is used to accelerate heat transfer in PCM. In the open cell metal foam whose voids are filled with a phase change material, the fluid can freely circulate inside the metal foam during the phase change process. Some of the reported studies on

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Nomenclature A C CF Cp Fo h K k L LTNE l n P PCM Pr PSM Ra r Sp T t V x, y

α β σ ɛ

φ μ ν θ ρ τ ψ 

Heat transfer area, m2 Thermal capacitance, J/m3 K Inertial resistance coefficient Specific heat capacity, J/kg K Fourier number Interfacial heat transfer coefficient, W/m2 K Permeability, m2 Thermal conductivity, W/m K Length of cavity, m Local thermal non-equilibrium Length of cell, m Normal direction to the surface Pressure, N/m2 Phase change material Prandtl number Pore scale method Rayleigh number Diameter, m Sparrow number Temperature, K Time, s Velocity vector, m/s, Volume, m3 Cartesian coordinates, m Thermal diffusivity, m2 /s Thermal expansion coefficient, 1/K Function of local thermal non-equilibrium condition Porosity Any dependent parameter Dynamic viscosity, Pa.s Kinematic viscosity, m2 /s Dimensionless temperature Density, kg/m3 Dimensionless time Thermal diffusivity ratio Dimensionless thermal conductivity Dimensionless thermal capacitance

Subscripts and superscripts e Stagnant eff Effective f Fluid h Hot, Hydraulic int Interface r Ratio ref Reference s Solid V Volumetric x Solid or fluid phase

the use of open cell metal foam for the heat transfer enhancement are the study of Alipanah and Li [4] who investigated the thermal management systems of lithium-ion battery by pure octadecane, pure gallium and octadecane aluminum foam composite materials, Jourabian et al. [5] who studied instantaneous melting of ice inside a horizontal rectangular cavity by using metallic porous matrix made of Ni-Steel alloys or the study of Feng et al. [6] who investigated Docosane (paraffin wax with C22 H46 composition) and aluminum foam. Mancin et al. [7] performed an experimental analysis on the phase change phenomenon of paraffin waxes embedded in copper foams. Wang et al. [8] performed a numerical study on the melting of ice assisted by open cell aluminum foam under forced convection in a channel.

There are limited studies on the closed cell metal foam which can be used for the purpose of heat transfer enhancement in solid/liquid phase change. Kovacik et al. [9] studied closed cell aluminum foam whose voids were filled with phase change material. Ohsenbrugge et al. [10] investiga0ted the thermal behavior of closed cell metal foam infiltrated with paraffin wax as latent heat storage system. Some techniques can be used for filling PCM in the closed cells such as the infiltration methods explained by Kovacik et al. [9] and Ohsenbrugge et al. [10] or even two tiny holes can be provided in sides of a closed cell and melted PCM can be injected into the cells through the tiny holes. In general, the closed cell metal foam may have significant advantages making them attractive for heat transfer enhancement in thermal energy storage application compared to the open cell metal foam such as (a) the heat transfer area of the closed cell metal foam is the larger than that of open cell metal foam, (b) the structure of a closed cell porous medium is more consolidated compared to an open cell porous medium since heat can transfer in different ways freely and consequently the solid phase thermal resistance is lower, (c) temperature change in the gravity direction can omit and melting front moves uniformly. The main disadvantage of phase change in the closed cell porous media is the conduction heat transfer is dominant and fluid cannot flow freely, which may reduce the interfacial heat transfer coefficient between the solid and fluid phases. There are two approaches for analyzing heat and fluid flow in porous media as the local thermal equilibrium and nonequilibrium approaches. Studies in literature for distinguishing the local thermal equilibrium from non-equilibrium can be found such as the study of Alazmi and Vafai [11] or Amiri and Vafai [12]. However most of the reported studies relate to the single phase heat transfer in porous media for which the mechanism of heat transfer is very different from the phase change in porous media. It seems that more studies should be done on the separation of local thermal equilibrium from non-equilibrium region by using dimensionless parameters. Furthermore, there are reported studies in literature in which the dimensionless forms of the volume average heat transfer equations for the solid and fluid phases are solved. In some of reported studies (such as the study of Buonomo et al. [13]), the dimensionless governing equations are based on the fluid thermal diffusivity and the interfacial Nusselt number appears in the coupled solid and fluid heat transfer equations. However, Dukhan and Hooman [14] made the governing equations dimensionless based on the solid thermal conductivity and Biot number was found in their energy equations. In the study of Wang et al. [15] the dimensionless governing equations were obtained by using the porous media stagnant thermal diffusivity and Sparrow number appears in solid and fluid energy equations automatically. The Sparrow number was discussed in literature by some researchers such as Minkowycz et al. [16] who declared that for a porous medium subjected to a rapid transient heating, the existence of the local thermal equilibrium depends on the magnitude of the Sparrow number and the rate of change of the heat input. Nnanna et al. [17] performed an experimental study on the local thermal non-equilibrium phenomena during the phase change in porous media. They stated that early during the phase change process, the Sparrow number is relatively small and that is why the solid matrix is not at the local thermal equilibrium with the pore materials. It seems that the use of Sparrow number and making the governing equations dimensionless by using the porous media stagnant thermal diffusivity may be meaningful than that of using solid or liquid thermal diffusivity since Sparrow number is a favorable parameter for distinguishing local thermal equilibrium state from the non-equilibrium. This motivated authors to do the present study and suggest a new formulation for phase change in the closed cell porous media. In this study, the melting of PCM

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Fig. 1. The schematic views of the studied cavity, (a) pore scale domain, (b) volume average domain.

existing in the voids of a two dimensional closed cell porous medium is analyzed. The governing equations are nondimensionalized by using the porous media stagnant thermal diffusivity and the Sparrow number appears automatically. The aim of the present study is to do a comprehensive study on the solid/liquid phase change in a closed cell porous medium to (a) derive a new formulation based on Sparrow number and apply it onto the phase change in the closed cell porous media, (b) discuss the advantages of using Sparrow number in the phase change problems, (c) establish a thermal equilibrium chart for the prediction of local thermal equilibrium. Based on the best knowledge of authors, the present study is the first study on deriving the new formulation in terms of Sparrow number and applying to the solid/liquid phase change in the closed cell porous media.

∂ Tf    2 T − ρ ∂ω H + V.∇ Tf ) = kf (T )∇ f f ∂t ∂t ∂ Ts  2 Ts ( ρ C p )s = ks ∇ ∂t ( ρ Cp )f (

(4)

where ω is the liquid fraction and its value changes from 0 (no melting) to 1 (fully melted). H, β and Tref are the latent heat, volumetric thermal expansion coefficient of working fluid and reference temperature, respectively. The reference temperature is equal to the initial temperature in this study. The porous media stagnant thermal diffusivity can be defined as

αe =

ke

(ρ Cp )e

=

ε kf + ( 1 − ε )ks ε (ρ Cp )f + (1 − ε )(ρ Cp )s

(5)

The above governing equation can be written in dimensionless form as follows,

2. The considered problem The schematic views of studied pore scale and volume average domains are shown in Fig. 1(a). The pore scale domain consists of 25 square closed cells in which PCM exists. All surfaces of the cavity are insulated, except the left vertical surface maintaining at the constant temperature. The gravity affects in –y direction and its effect is included in the pore scale study. The effect of radiation is neglected. The working fluid is water and the solid frame is aluminum with thermal conductivity of 237 W/mK. The occurrence of maximum density of water at 4 °C is not included in the formulation. Fig. 1(b) shows the volume average domain for the solid and fluid phases, the domain is continuous since the independent parameters are integrated over the entire domain. The study is limited to Raf ≤ 104 . 3. Governing equations In this section, the pore scale and volume average equations are presented separately.

 ∗·V ∗ = 0 ∇ ∗ ∂V  ∗ )V  ∗2 V  ∗ P∗ + M(θ )Pr∇  ∗ .∇  ∗ = −∇  ∗ + Rae Prθfj + (V e e ∂ Fo ∂θ  ∗  ∗  ∗2 θ − ∂ω 1 (θ ) f + V .∇ θf = (θ )ψf ∇ f ∂ Fo ∂ Fo Ste ∂ θs ∗2  θs = ψs ∇ ∂ Fo

Pore scale of heat and fluid flow including phase change in the closed cell porous media is analyzed. The effect of gravity is considered and Boussinesq approximation is employed to include the buoyancy effect. The continuity, momentum and energy equations are solved for the voids in which PCM exits while the conduction heat transfer equation is solved for the solid frame. The dimensional form of pore scale governing equations can be written as follows:

(6) (7) (8) (9)

 ∗ and P∗ are the dimensionless velocity vector and preswhere V sure. The dimensionless temperatures of fluid and solid phases are shown by θ f and θ s , respectively. Fo , Rae and Pre are the Fourier number, Rayleigh number and Prandtl number based on the porous media stagnant thermal diffusivity, respectively. ψ is the thermal diffusivity ratio between the fluid or solid phase and porous media stagnant thermal diffusivity. The dimensionless governing parameters can be expressed mathematically as, 2   ∇  ∗ = V L , Fo = tαe , P∗ = PL , Pr = νf , ,V 2 L αe L ρf αe2 e αe ρf g β ( T − T i ) L 3 T − Ti α αs Rae = ,θ = , ψf = f , ψs = αe νf Th − Ti αe αe

∗= ∇

3.1. Pore scale governing equations

 ·V  =0 ∇  ∂V  )V  p + ν ( T )∇  2 .V  .∇  =−1∇  + gβ (T − Tref )j + (V f ∂t ρf

(3)

(10)

The functions of M(θ ), (θ ) and (θ ) change by temperature, they are the necessary coefficients to describe PCM properties from frozen to melted state and defined as:

M (θ ) =

ν (T ) Cp(T ) k (T ) , (θ ) = , (θ ) = νf Cpf kf

(11)

(1)

It should be mentioned that the following relations for the Prandtl and Rayleigh numbers exist.

(2)

Pr = Pr f

e

αe αe , Raf = Rae αf αf

(12)

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3.2. Volume average governing equations

( 1 − ε ) Cs

In the volume average method, a scalar or vector quantity can be integrated in a representative discontinues space and consequently the continuation of the space can be provided for the considered quantity. There are two phases as solid and fluid in the porous media, therefore two volume average quantities can be defined:

φ =

1 V



φ dV

V

φ

x

1 = Vx



φ dV

(13)

(14)

Vx

φx are

where φ and the volume average and intrinsic volume average of φ quantities, respectively. V and Vx are the total volume and the corresponding volume of solid or fluid phase. The dimensional volume average governing equations can be obtained by taking volume average of the dimensional pore scale governing equations Eq. (1)–((4)):

 

 · V  =0 ∇

(15)

     μf ( T )   ρf ∂ V 1       Pf + μf (T ) ∇ 2V  −  + 2( V .∇ ) V = −∇ V ε ∂t ε K ε CF    −ρf √  V V + ρf gβ (T − Tref )j (16)

ε Cf ( T )

∂ Tf  2 Tf + hs ε (Ts − Tf ) = keff,f (T )∇ ∂t rh ∂ω −ρf ε H ∂t

( 1 − ε ) Cs

( 1 − ε ) Cs

∂ Ts  2 Ts − hv (Ts − Tf ) = keff,s ∇ ∂t

(18)

where Cs and Cf are the thermal capacitances of the solid and fluid phases (i.e.Cs = (ρ Cp )s and Cf = (ρ Cp )f ), respectively. hv is the volumetric interfacial heat transfer coefficient, keff, s and keff, f are the solid and fluid effective thermal conductivities, respectively. keff, s and keff, f can be calculated from the following equations [18]:

∂ Ts  2 Ts − hs ε (Ts − Tf ) = keff,s ∇ ∂t rh

(25)

(26)

The above equations can be non-dimensionalized based on the dimensionless parameters of Eq. (10):

(θ ) f

s (17)

(24)

The above equations show that for the volume average approach for the closed cell porous media only the heat conduction equations for the solid and fluid phases should be solved. hv is the volumetric interfacial heat transfer coefficient defined as hv = hs Av , where hs (W/m2 K) is the surface interfacial heat transfer coefficient and Av (m2 /m3 ) is the volumetric heat transfer area. It is possible to write the volumetric heat transfer area as Av = ε /rh , where rh is hydraulic diameter of a void containing the fluid phase (i.e., rh = Vf /Aint , where Aint is the interfacial heat transfer area in the volume of Vf ). Then, the Eqs. (23) and (24) become as:

K

∂ T  f     f  2 Tf ε Cf (T )( + V .∇ T ) = keff,f (T )∇ ∂t ∂ω +hv (Ts − Tf ) − ρf ε H ∂t

∂ Ts  2 Ts − hv (Ts − Tf ) = keff,s ∇ ∂t

∂ θ f  ∗2 θ f + Sp(θ s − θ f ) = N(θ )f ∇ ∂ Fo 1 ∂ω − f Ste ∂ Fo

∂ θ s  ∗2 θ s − Sp(θ s − θ f ) = s ∇ ∂ Fo

(27)

(28)

 and are the dimensionless thermal conductivity and thermal capacitance, respectively, and Sp is the Sparrow number. The employed dimensionless parameters can be defined mathematically as

s =

keff,f (T ) keff,s Cs ( 1 − ε ) C , f = f , N ( θ ) = , s = , ε Ce Ce keff,f ε ke

f =

keff,f hs L2 , Sp = ε ke ke rh

(29)

∗

(19)

Ce and  e are the porous media stagnant thermal capacitance and thermal conductivity, and they can be written as:

keff,s = (1 − ε ∗ )ks

(20)

Ce = (1 − ε )Cs + ε Cf

(30)

e = s + f

(31)

keff,f = ε kf + ε kdisp

ks and kf are the thermal conductivities of the solid and fluid phases, and ɛ∗ is the effective porosity which can be found as:

ε∗

k − ks = eff kf − ks

(21)

The value of keff which is the effective thermal conductivity can be numerically calculated by considering a representative volume, imposing temperature difference and applying the Fourier law. The calculation method will be explained in the next section. For a closed cell porous medium, the volume average velocity can vanish.

 

 = 1 V V



 dV = 0 V

(22)

V

Therefore, the continuity and momentum equations Eqs. (15)– ((16)) disappear since the volume average velocity for the closed cell porous media is zero. Hence, the volume average Eqs. (15)– (18) take the following form for a closed cell porous medium:

ε Cf ( T )

∂ Tf  2 Tf + hv (Ts − Tf ) − ρ ε ∂ω H(23) = keff,f (T )∇ f ∂t ∂t

The values of (θ ) and N(θ ) adjust thermal properties from the frozen to melted state and their values do not play important roles compared to other dimensionless parameters. For many PCMs, kf |frozen ~ kf |melted and Cf |frozen ~ Cf |melted and hence N(θ ) ≈ 1 and (θ ) ≈ 1 leading Eqs. (27) and (28) into:

∂ θ f  ∗2 θ f + Sp(θ s − θ f ) − 1 ∂ω = f ∇ f ∂ Fo Ste ∂ Fo s ∂ θ  s s f  ∗2 θ  − Sp(θ  − θ  ) s = s ∇ ∂ Fo f

(32) (33)

Therefore, the dominant dimensionless parameters for the present study are s , f ,  s ,  f , Ste and Sp. Our numerical experience shows that it is possible to combine four coefficients as s , f ,  s ,  f by defining thermal diffusivity ratio as

αr =

s / s (1 − ε ∗ )ε αs αs = ∗ ≈ f / f ε (1 − ε ) αf αf

(34)

Hence, the results of Eqs. (32) and (33) are controlled by three parameters which are α r , Ste and Sp.

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Fig. 2. Temperature distribution and boundary conditions for a cell of pore scale domain for determination of the effective thermal conductivity.

It should be mentioned that although the convection term is removed in the volume average equations Eqs. (25), ((26)), it does not mean that the natural convection in the cells does not play any role on the volume average heat transfer through the cavity. Strong natural convection in the cells causes the increase of interfacial heat transfer coefficient between the solid and fluid phases, consequently the heat transfer through the domain increases. Obviously, the vice versa situation is also valid. 3.3. Initial and boundary conditions The initial temperature of the whole domain is Ti both for the pore scale and volume average domains. Suddenly the vertical left wall temperature jumps to Th , and it is maintained at the same temperature for the entire process. The right, top and bottom surfaces of the cavity are thermally insulated. For the pore scale domain, the continuous heat flux boundary condition is applied to the interface between the solid and fluid phases. For the volume average domain, the Sparrow number provides the heat exchange between the solid and fluid phases. The dimensionless form of the initial and boundary conditions is given in Eq. (35).

Fo = 0 X∗ = 0 X∗ = 1, Y∗ = 0 and 1

Pore scale

Volume average

θs = θf = 0 θs = 1, θf = no need ∂ θs ∂ n = 0, θf = no need

θ s = θ f = 0 θ s = θ f = 1

∂ θ s ∂ θ f ∂n = ∂n = 0

(35) For the interface boundary conditions between the solid and fluid phases and for the pore scale analysis, the following relation is valid:

∂ θs k f ∂ θf = , θ = θf ∂ n ks ∂ n s

(36)

where n is the normal direction to the considered surface. 4. Solution and validation 4.1. Computational details In this study, the enthalpy method based on fixed grid is used to find out the solution for both pore scale and volume average equations. The dimensional forms of the governing equations are solved and then the obtained results have been transferred to the dimensionless form by using Eq. (10). As it was mentioned before, in order to handle the slurry region, the values of viscosity, thermal conductivity and thermal capacitance changed from solid to

fluid by using the following formula,

φ (T ) = φsolid + (

T − Tm )(φfluid − φsolid ) T

(37)

Tm is the melting point and T is the phase change temperature range. In this study, these values are Tm = 273.15K and T = 1K. The parameter of φ (T) can be thermal conductivity, thermal capacitance or dynamics viscosity whose values changes in the phase change temperature range. One of the important parameters influences the volume average results is the effective thermal conductivities of the solid and fluid phases (i.e., keff, f , keff, s ). In order to obtain the effective thermal conductivities, a representative cell of pore scale domain is considered. The representative cell is shown in Fig. 2. A temperature gradient is imposed horizontally while other surfaces are insulated. Pure heat conduction equations are solved for the cases of ice and water (without phase change) and effective thermal conductivity of the cell consisting of the solid and fluid phases are obtained. Then, by using Eq. (21) the value of ɛ∗ is found and finally the values keff, f , keff, s are calculated by using Eq. (19) and (20). Many attempts are done in this study to find out a criterion for distinguishing local thermal equilibrium from local thermal non-equilibrium. Our numerical results showed that actually, always there is a local thermal non-equilibrium at the beginning of the phase change process particularly in the region close to the hot surface (surface in which rapid temperature change occurs). However, for some cases the initial period of local thermal nonequilibrium is very short and almost the entire process is in local thermal equilibrium state, while for other cases the period of local thermal non-equilibrium is very long and those cases can be accepted as local thermal non-equilibrium. In this study the pore scale domain is divided into five equal columns and the first column touching the hot wall is excluded, and the condition for the local thermal equilibrium can be assumed as

 s  f  θ  − θ  ∗ < 0.1 x > 0. 2,Fo > 0

(38)

For the pore scale and volume average studies, a commercial software, COMSOL Multiphysics (V5.4 COMSOL Multiphysics), based on the finite element method is used to solve the coupled heat and fluid flow equations. The Multiphysics module used to solve motion equation is “Laminar flow” while for solving the heat transfer equations in solid and fluid, the module of “Heat transfer in fluid and solid” is applied. The software was run in a workstation with Intel(R) Xeon(R) CPU E5-2630 v3, 32 G RAM. A grid refinement analysis is done (Fig. 3) in which the temperature profile at the horizontal center line (Y∗ = 0.5 ) for the cavity with ε = 0.9, Raf = 103 and Ste = 1 is shown. The program did not converge for small number of element and that is why the mesh independency

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excellent agreement between our study and their reported studies (below 7% difference for the third validation and below 1% for the fourth validation) was observed. 5. Result and discussion

Fig. 3. A sample of grid independency for a closed cell cavity with ε = 0.9, Raf = 103 and Ste = 1.

started from 17,058 total elements. Hence, the results for different number of grids are close to each other. It is found that 41,684 total elements (37,708 number of domain elements (triangular), 3840 edge elements, and 136 vertex elements) are sufficient to obtain the accurate results for this problem. 4.2. Validation Four studies have been performed to validate the method and program of the present study. The first validation was done for one dimensional melting problem under pure condition heat transfer. The temperature for fluid phase can be found as [19]:



erf(x/2 α t ) T ( x, t ) − Tw = Tm − Tw erf(β )

(39)

where α f is the thermal diffusivity of fluid phase and the value of β can be found from following relation,

Ste

β eβ erf(β ) = √ π e

(40)

The comparison of the results of present study and analytical solution is shown in Fig. 4(a) for water melting at t = 20 0 0 sec. The second comparison was done between the results of our program and the results of Beckrmann and Viskanta [20] for the melting in a square cavity filled with porous media under natural convection. The Fig. 4(b) shows the comparison of melting interface positions for different dimensionless time step. As it can be seen from Fig. 4 our results are very close to the reported results in literature indicating that the employed method and program for solving of the phase change problem is correct. Furthermore, a comparison between our results and the results of Huber et al. [21] for the melting of clear ice without metal foam and also a comparison between our results with the study for pure convection in the cells of a hollow brick (two dimensional porous media with closed cell) of Antoniadis et al. [22] are done and an

The mechanism of heat transfer in closed cell cavities with different shapes such as right-angle porous trapezoidal enclosure, triangular enclosure and even partially opened cavity were discussed in literature [23–25]. In all of those studies, there is one Rayleigh number defined based on the boundary or initial temperatures in the domain. Similar to other studies on heat transfer in a cavity filled with porous media (e.g. Hooman and Merrikh [26]) two kinds of Rayleigh number as pore scale Rayleigh number and the whole cavity Rayleigh number can be defined. In this study, Rayleigh number (i.e., Raf ) refers to the entire cavity Rayleigh number. Fig. 5(a) shows the temperature and velocity distributions of the studied domain for ε = 0.9 and Raf = 103 at Fo = 0.67. Fig. 5(b) shows the temperature profile in the center line of the second row of the studied cavity for different Rayleigh number as Raf = 0, 10, 102 , 103 , 104 at the same time step. All temperature profiles overlap each other showing that Rayleigh number does not have significant effect on the heat and fluid flow when it is less than 104 . That is why, the continuity and momentum equations are discarded in the pore scale study and only the conduction heat transfer equations for the solid and fluid phases are taken into account. 5.1. Comparison of pore scale and volume average results In this study, many runs were done to validate the results of volume average method based on the new formulation for phase change process assisted with closed cell porous media. In general, a good agreement between the volume average method (VAM) and pore scale method (PSM) was observed. In this section, two examples of the preformed studies are presented. Fig. 6 shows the temperature distributions of PSM and VAM for the closed cell porous media when ε = 0.9 and Ste = 1 for the pore scale analysis and Sp = 35, αr = 668, and Ste = 1 for the volume average equations. These figures represent the temperature distributions for the solid and liquid phases at the different dimensionless time of Fo = 0.13, Fo = 0.34 and Fo = 0.67. High speed heat propagation in the solid frame can be seen clearly in dimensionless time steps of Fo = 0.13, Fo = 0.34. Fig. 6(b) shows the results of VAM for the solid and liquid phases. The temperature increases by passing time in both solid and fluid phases, however it seems that the heat transfer in the solid frame is the faster compared to the fluid phase. In order to show the consistence of the PCM and VAM results, Fig. 6(c)

Fig. 4. Validation studies, comparison of the present study results with the reported studies for (a) pure analytical solution [19], (b) Beckrmann and Viskanta [20].

Please cite this article as: C. Wang and M. Mobedi, A new formulation for nondimensionalization heat transfer of phase change in porous media: An example application to closed cell porous media, International Journal of Heat and Mass Transfer, https://doi.org/10. 1016/j.ijheatmasstransfer.2019.119069

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Fig. 5. Domination of heat conduction for the phase change in a closed cell porous medium (a) Temperature and velocity distributions for Raf = 103 in cavity with ε = 0.9. (b) The temperature profile in the center line of the second row of cavity for four different values of Ra number at Fo = 0.67.

is presented in which the temperature profiles in the middle line of the cavity (Y∗ = 0.5 ) for the solid and fluid phases at the same time step are plotted. The cell average of temperatures calculated from PSM results is also shown in the same figure. At the initial stages, the heat transfer in the solid phase is the faster, however the temperatures become equilibrium at the end of melting process. The same comparisons between the pore scale and volume average results under the same condition of Fig. 6 but for porosity of 0.7 is shown in Fig. 7. Good agreement between the PSM and VAM can be observed when ε = 0.7. The values of the Sparrow number and thermal diffusivity ratio for the volume average equations are 17 and 668, respectively. The melting period is shorter due to the larger amount of solid frame distributed in the entire domain. The faster heat transfer in the solid phase in the initial stages can be observed more clearly due to the increase of cross section of the solid frame and reduction of thermal resistance in the solid phase. Both Figs. 6 and 7 show the good agreement between PSM and VAM results proving the validation of VAM for analyzing of heat and fluid flow for melting in the closed cell porous media. The comparisons of melting fraction of PSM and VAM approaches for the three different porosities shown in Fig. 8. The melting fraction is shown in two diagrams for the dimensional and dimensionless time. In both figures, a good agreement between the VAM and PSM can be seen. It should be mentioned that the melting fraction curves in Fig. 8(a) varies with porosity since the closed cell porous media with ε = 0.7 has more metal amount compared to the porous media with ε = 0.9. However for the Fig. 8(b), the curves overlap each other since the values of αr = 668 and Ste = 1 are the same for three cases and the values of Sp are close to each other for three cases (Sp = 17,Sp = 22,Sp = 35) .

5.2. Effective thermal conductivity and local Sparrow number Effective thermal conductivity and interfacial heat transfer coefficient are two parameters which should be known to obtain the VAM results. The changes of effective thermal conductivity and the effective porosity with the porosity are calculated and shown in Fig. 9 for the different porosity value when kf /ks = 0.0093 for the cavity with frozen PCM and kf /ks = 0.0026 for the cavity with melted PCM. There is small differences between keff and ɛ∗ of melted PCM (water) and no melted PCM (ice). The value of keff decreases with the increase of porosity due to the decreasing of metal amount while the value of ɛ∗ increases since the domain approaches to pure PCM cavity. It should be mentioned that

keff, f and keff, s are calculated in terms of keff and ɛ∗ by using Eqs. (19) and (20). The interfacial heat transfer coefficient is an important parameter in the volume average method, unfortunately the number of studies on the interfacial heat transfer coefficient for the closed cell porous media is limited [27]. In this study, the value of the interfacial heat transfer coefficient is determined from the pore scale results and used for calculation of the Sparrow number. Fig. 10 shows one sample of the obtained diagrams in which the change of the local heat transfer coefficient with dimensionless time for five different columns of the pore scale cavity and for ε = 0.9, Ste = 1 and αr = 668 is drawn. As can be seen from the Fig. 10, the interfacial heat transfer coefficient is not constant during the melting process and it changes by time. If the change of interfacial heat transfer coefficient with time for the second column is considered, three different regions can be observed as no melted region (pure conduction heat transfer) in which the value of hv is almost constant, slurry region in which the value of hv suddenly increases due to the phase change and then it decreases by the melting of PCM and finally fully melted region in which the value of hv refers to the single phase pure natural convection heat transfer and consequently the value of hv is constant. Furthermore, the change rate of hv not only depends on time but also varies with the location of cells. For the closest cell to the hot wall, the value of hv rapidly increases and decreases while for the last cell the increase or decrease rate of hv is slower. The change trend of hv with time is almost the same for all cells however a time lag exists between different columns. The increase of hv in the first column is observed just after starting of the melting period while after a long time lag the same trend is seen for the last column. The final value of hv is very close to each other after the melting process in the entire PCM referring to the value of hv for the pure convective heat transfer. As it can be seen, the change of hv with dimensionless time is complicated to be formulized, however in this study the time and space average value of hv is calculated and used in the volume average energy equations for the solid and fluid phases. The good agreement between the volume average and pore scale results shown in the previous section proving that the time and space average of hv provides sufficiently accurate volume average results. 5.3. Analysis of local thermal equilibrium and non-equilibrium assumptions In this section, a parametric study is performed to investigate the effects of three dimensionless parameters as α r , Ste and Sp on

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Fig. 6. The comparisons of the pore scale and volume average temperatures of solid and fluid phases at different time steps with ε = 0.9 and Ste = 1 for the pore scale equations and Sp = 35, αr = 668 and Ste = 1 for the volume average equations.

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Fig. 7. The comparisons of the pore scale and volume average temperatures of solid and fluid phases at different time steps for ε = 0.7 and Ste = 1 for pore scale equations, and Sp = 17, αr = 668 and Ste = 1 for the volume average equations.

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Fig. 8. Comparisons of the melting fraction between the pore scale and volume average results for three different porosities as ε = 0.7, 0.8, 0.9 when Ste = 1.

Fig. 9. The change of effective thermal conductivity and effective porosity with the different porosity values for ice and water.

the local thermal equilibrium state. The volume average equations Eqs. (27) and ((28)) are solved for different values of three dimensionless parameters in the range of 1 < α r < 500, 1 < Sp < 500, Ste = 1 and ε =0.9. Fig. 11 shows the change of the volume average temperatures for the solid and fluid phases with αr = 500 and Sp = 1, 10, 100, 500 when Ste = 1. Fig. 11(a) shows the volume average solid and fluid temperature profiles at Y∗ = 0.5 when αr = 500 and Sp = 1 for different time steps as Fo = 0.67, 1.68 and 2.08. In this study a parameter as σ (σ = θ s − θ f ) is defined to show the level of local thermal equilibrium. As it can be seen, there is big difference between θ s and θ f showing an obvious local thermal non-equilibrium condition. In the same Figure (Fig. 11(a)), the change of σ for three different time is shown. The maximum difference between θ s and θ f belongs to the initial stage (Fo = 0.67 ). By increasing of time, the degree of local thermal nonequilibrium also decreases. The main reason of the high degree of local thermal non-equilibrium in Fig. 11(a) is the considerable high speed of heat propagation in the solid phase compared to the fluid

phase due to high value of thermal diffusivity ratio (αr = 500 ) and also the high convection thermal resistance between the solid and fluid phases due to low value of Sparrow number (Sp = 1 ). The same diagram of Fig. 11(a) is plotted for the same value of α r (i.e., αr = 500) but for Sp = 10, and presented in Fig. 11(b). As it can be seen, still a local thermal non-equilibrium state between the solid and fluid phases exists and the same trend of variation of θ s and θ f can be observed, however the values of σ become smaller compared to Fig. 11(a). The increase of Sparrow number increases the heat interaction between the solid and fluid phases and that is why the degree of local thermal non-equilibrium becomes smaller. The Fig. 11(c) and 11(d) show the change between θ s and θ f and also σ in three different time steps of the same thermal diffusivity ratio when Sp = 100 and 500, respectively. By increasing the value of Sparrow number, the maximum value of σ becomes smaller and its value is 0.3 for Sp = 300 when Fo = 0.27 and 0.1 for Sp = 500 when Fo = 0.07. As it can be seen from Fig. 11, a local thermal equilibrium can be observed for Sp = 500 in the entire melting period. Fig. 11 clearly shows that Sparrow number plays an important role on the melting process and by increase of Sparrow number value, the possibility of a local thermal equilibrium increases. The same runs for different values of α r as 1.7, 10, 100 and 300 are done and based on the definition of local thermal equilibrium condition given by Eq. (38), a local thermal equilibrium chart is obtained and shown in Fig. 12 when Ste = 1. This chart clearly shows the effects of thermal diffusivity ratio and Sparrow number. By decreasing of the value of α r , the possibility of the local thermal equilibrium increases since the rate of propagation of heat in the solid and fluid phases becomes closer to each other. Similarly, the local thermal equilibrium can exist for the high

Fig. 10. The change of local heat transfer coefficient and Sparrow number for five different columns of the pore scale cavity when Ste = 1 and αr = 668.

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Fig. 11. The change of the solid and fluid volume average temperatures and σ for different values of Sparrow number when αr = 500 and Ste = 1.

value of Sparrow number such as Sp = 500 due to considerable heat exchange between PCM and solid frame. 5.4. Validation of the thermal equilibrium chart As it can be seen from the chart of Fig. 12, the regions for the local thermal equilibrium and non-equilibrium can be separated. A pore scale study is performed to support and to validate the results of chart presented in Fig. 12. The pore scale results for Case 1

(Ti-6Al-4V-water) and Case 2 (Al 2024-T6-water) are obtained. The thermophysical properties of the two metals are given in Table 1. Before doing the pore scale study, the interfacial heat transfer coefficient can be calculated roughly from the study of Wang et al. [14]. The values of hv are calculated as 53917 W/m3 K and 180226 W/m3 K for the Case 1 and Case 2. Based on these values of interfacial heat transfer coefficients, the values of Sparrow number for the Case 1 and 2 are calculated as Sp = 367 and Sp = 18, respectively. Furthermore, the values of thermal diffusivity ratio

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Fig. 12. Local thermal equilibrium condition in terms of different values of Sparrow number and thermal diffusivity ratio when Ste = 1.

Table 1 The thermophysical parameters of the porous media for Case 1 and 2. Parameters

Case 1

Case 2

k(W/mK) ρ (kg/m3 ) Cp (J/kg K) keff (W/mK) C(J/m3 K)

7.6 4420 537 0.779 2,373,540 0.659 0.068 9.61 0.9

177 2780 875 87.69 2,432,500 1.98 0.72 2.74 0.5

 α (m2 /s)

ɛ

for the Case 1 and Case 2 are αr = 22 and αr = 499.The positions of two studied cases in the local thermal equilibrium chart are shown in Fig. 12. It can be clearly seen that Case 1 is in the local thermal equilibrium region while Case 2 is in the local thermal non-equilibrium region. Fig. 13 shows that the pore scale results for the studied cases. Fig. 13(a) shows the temperature distribution and temperature profile of the solid and fluid phases at Y∗ = 0.5 for Case 1 at Fo = 0.15 and Fig. 13(b) shows the same temperature distribution and temperature profiles at Y∗ = 0.5 for the Case 2 at Fo = 0.36. A clear local thermal non-equilibrium can be seen for Case 2 while a perfect thermal equilibrium exists for Case 1. These two examples validate

Fig. 13. Temperature distribution and temperature profile at Y∗ = 0.5, (a) Fo = 0.15 (b) Fo = 0.36.

Please cite this article as: C. Wang and M. Mobedi, A new formulation for nondimensionalization heat transfer of phase change in porous media: An example application to closed cell porous media, International Journal of Heat and Mass Transfer, https://doi.org/10. 1016/j.ijheatmasstransfer.2019.119069

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the presented local thermal equilibrium chart of Fig. 12 and show that just knowing of Sparrow number and thermal diffusivity ratio may be sufficient to determine the state of local thermal equilibrium for phase change in a closed cell porous medium in which melting occurs. 6. Conclusions The heat transfer in a closed cell porous medium whose voids are filled with PCM is studied. A new formulation is suggested in terms of Sparrow number. Both the pore scale and volume average equations are solved and the obtained results are compared to validate the suggested volume average formulations. Based on the obtained results following remarks can be concluded. - The good agreement between the pore scale and volume average results shows the suggested volume average formulation can represent heat transfer during the melting of PCM in a porous medium. - The suggested volume average formulation is based on three dimensionless parameters as Stefan number, thermal diffusivity ratio and Sparrow number. Sparrow number plays an important role on heat transfer between the solid and fluid phases and appears automatically if the dimensional governing equations are made dimensionless by using stagnant thermal diffusivity. - Based on the pore scale results, the interfacial heat transfer coefficient is not constant during the phase change process. During phase change process, the interfacial heat transfer coefficient takes a maximum value and it becomes almost constant after finishing the phase change in the cell. However, the time and space average of the interfacial heat transfer coefficient can be used in the volume average equations. It provides sufficiently accurate volume average results. - The Sparrow number and thermal diffusivity ratio are employed and a local thermal equilibrium chart is established to predict the local thermal equilibrium state of the phase change process, if the values of Sparrow number and thermal diffusivity ratio are known. - The established local thermal equilibrium chart is validated by giving two examples of different closed cell porous media having different Sparrow number and thermal diffusivity ratio. The obtained pore scale results validated the prediction done by the suggested thermal equilibrium chart. - It is found that the possibility of local thermal non-equilibrium is high for the low values of Sparrow number (such as Sp = 1) and high values of thermal diffusivity ratio (such as αr = 500). Declaration of Competing Interest The authors declare no conflict of interest between the contents of this work and any reported studies. Acknowledgement The first author (Chunyang Wang) gratefully acknowledges the financial support from China Scholarship Council (No. 201808050059), which has sponsored his PhD study at the Shizuoka University in Japan.

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