Solar energy latent thermal storage by phase change materials (PCMs) in a honeycomb system

Accepted Manuscript Solar Energy Latent Thermal Storage by Phase Change Materials (PCMs) in a Honeycomb System Assunta Andreozzi, Bernardo Buonomo, Davide Ercole, Oronzio Manca PII: DOI: Reference:

S2451-9049(17)30371-2 https://doi.org/10.1016/j.tsep.2018.02.003 TSEP 131

To appear in:

Thermal Science and Engineering Progress

Received Date: Revised Date: Accepted Date:

15 October 2017 4 February 2018 4 February 2018

Please cite this article as: A. Andreozzi, B. Buonomo, D. Ercole, O. Manca, Solar Energy Latent Thermal Storage by Phase Change Materials (PCMs) in a Honeycomb System, Thermal Science and Engineering Progress (2018), doi: https://doi.org/10.1016/j.tsep.2018.02.003

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Solar Energy Latent Thermal Storage by Phase Change Materials (PCMs) in a Honeycomb System Assunta Andreozzi1*, Bernardo Buonomo2, Davide Ercole2*, Oronzio Manca2 1

Dipartimento di Ingegneria Industriale, Università degli Studi di Napoli Federico II, Piazzale Tecchio, 80, 81025 (Napoli, Italy) *2 Università degli Studi della Campania, “Luigi Vanvitelli”, Dipartimento di Ingegneria Industriale e dell’Informazione, Via Roma, 29, 81031 (Aversa, Italy)

Abstract:

A computational investigation of a honeycomb system with Phase Change Materials

(PCM) for solar energy applications is accomplished. The system is a solid honeycomb structure made in checkerboard matrix using parallel squared channels, half of them are filled with PCM and in the other the Heat Transfer Fluid (HTF) passes through. Transient regime numerical simulations are created for different channels per unit of length (CPL). The Solid-liquid PCM is paraffin wax. A comparison between the direct honeycomb model (Model A) and a porous medium model (Model B) is carried out. The model B is modelled with the extended Darcy-Brinkman law using the Local Thermal Equilibrium (LTE) assumption for heat exchange between solid and liquid zones. By the results of the direct honeycomb model, the characteristics such as permeability, effective thermal conductivity and interfacial heat transfer are evaluated and then compared with the porous medium model. Numerical simulations were carried out using the Ansys-Fluent code. Results in terms of melting time and temperature fields as function of time are presented. * Corresponding author: Davide Ercole, [email protected] Keywords: Thermal storage; PCM; Phase Change material; Porous media; Honeycomb; DarcyBrinkman law.

Nomenclature A

Cross section Area, m2

c,cp

Specific heat, J kg-1 K-1

H

Height of a single elementary channel, m

HL

Latent heat of fusion, J kg-1

k

Thermal conductivity W m-1 K-1

K

Porous Permeability, m2

L

Length of a parallel channel, m

n

channel Per Unit of length (CPL)

p

Static pressure, Pa

t

Time, s

T

Temperature, K

u

Air x-Velocity, m s-1

U

Unit of length, m

v

Air y-Velocity, m s-1

V

Volume, m3

w

Air z-velocity, m s-1

x

Cartesian axis direction, m

y

Cartesian axis direction, m

z

Cartesian axis direction, m

Greek Symbols αsf

Area Surface density, m

β

Liquid fraction

μ

viscosity

δ

Thickness of a single elementary channel, m

ΔT

Melting range temperature, K

ρ

Density, kg m-3

Δp

Air Pressure drop, Pa

Subscripts air

air

avg

average

cord

cordierite

D

Darcy

eff

effective

f

Porous fluid phase, air

m

PCM melting

PCM

Phase change material

total

packaging

n

Log2 of Channel Per Unit of length (CPL)

s

Solid phase of the porous zone

1. Introduction The always-growing demand of thermal energy, the energy storage problems and the environmental worries are important problems that employ the researcher in energetic fields. About the renewable energy, especially the solar energy, there is the problem of the mismatch between the thermal energy demand and its supply, because the energy supply is not constant due to the weather conditions and the day cycles. In order to overcome these problems or at least to reduce their impact, a Thermal Energy System (TES) could solve the difference between the solar energy supply and the energy demand [1]. A TES works as a thermal buffer that store the exceed energy and release it when required. A classification of the Thermal Energy Systems divides them in Sensible Heat Thermal Energy Storage System (SHTES) and Latent Heat Thermal Energy System (LHTESS). In this study a Latent Heat Thermal Energy System (LHTESS) is employed to store the thermal energy. It based on the use of phase change materials (PCMs) to store thermal energy by changing phase keeping the temperature nearly constant [2]. The PCM utilized in this study is a solid-liquid PCM because it represents a good compromise between the solid-solid phase change PCM where the capacity heat storage is limited (due to the lower value of their latent heat) and the liquid-gas phase change PCM where the capacity heat storage is higher but the volume change is not negligible. Therefore, the Potassium carbonate (K2CO3) is used in this study as solid-liquid PCM. The worse disadvantage about the solid-liquid PCMs is the lower value of their thermal conductivity. In literature [[3]-[4]] are discussed some enhancement systems for thermal storage applications, as the addition of metal foam [5], fins [6] or using a honeycomb structure. A honeycomb structure has the advantage to increase the heat exchange area between the PCM and a heat source and in the same time reduces the mechanical stress due to the PCM expansion during the phase change process [7]. In fact, there are several applications of the honeycomb as thermal building [8] and thermal management [[9]-[10]]. Pal and Joshi [10] experimentally and numerically had studied a hexagonal-cross-section honeycomb structure

with phase change materials heated down below. The numerical method is modelled with the enthalpyporosity method. By the results they had affirmed that the natural convection is negligible inside the honeycomb. An inorganic PCM mixture of KNO3/NaNO3 inside a silica ceramic honeycomb was prepared by Li et al. [11] for an experimental study. By the results they found a delay during the PCM melting when the honeycomb is present but, in the same time, the thermal energy storage rate was increased. A numerical and experimental investigation was carried out by Luo et al. [12]. A comparison between the one-dimensional model and the experimental data was made and by the results they had concluded that larger channels and thinner walls lead to a faster increase of exit temperature for the charging phase and higher decrease for the discharging phase. About solar applications, a Thermal Energy storage with a honeycomb structure was investigated by Andreozzi et al. [13], where the honeycomb was modelled as a porous media. An experimental investigation on ceramic honeycomb for high thermal energy storage was accomplished by Srikanth et al. [14]. The performance of the ceramic honeycomb was investigated in a temperature range between 773 K and 1273 K for charging and discharging phase and an equivalent numerical model was developed in order to compare it with the experimental results. There was a good agreement between the experimental apparatus and the numerical model. Mekaddem et al. [15] had used a honeycomb structure with phase change materials for an experimental latent energy storage study in wall building. A comparison between a simple panel and a honeycomb panel filled with paraffin RT27 was made; moreover a 3D simulation was created. By the results, the effect of the PCM on the temperature evolution has showed that the cycle of charging and discharging phase becomes more stationary without abrupt spikes of temperature and the presence of the honeycomb has slightly increased the heat flux density into the panel. An aluminum honeycomb with phase change material for thermal management was numerically and experimentally investigated in [16]. The reliability of the numerical model was confirmed by the experimental results and the presence of the honeycomb structure has decreased the temperature variation of the heating

source. The complexity of the honeycomb geometry could be a problem for numerical simulation due to the increase of the computational cost for complex and regular geometry with numerous channels. Therefore in literature there are some works where the honeycomb is modelled as a porous media such as Andreozzi et al. [17], where a transient numerical analysis was accomplished for an honeycomb with parallel squared channels. It is related to a sensible TES where the PCM is not present. A first comparison between a honeycomb structure with a porous model where the PCM is also considered in [18]. The honeycomb was treated at first as a conjugate heat transfer problem and then as a porous media using the Local Thermal Equilibrium model. Nevertheless, the porous model in this comparison depends on the final results given by the direct model, because the effective thermal conductivity is calculated using the results of the real honeycomb model. Therefore, it is not possible to create a porous model without the results of a corresponding direct model. This paper aims to build a porous medium model. The honeycomb is made with parallel squared channels, half of them are filled of PCM and in the others the heat transfer fluid (air) passes through in checkerboard way. Different channels per unit of length (CPL) are investigated in transient regime. The numerical model of the honeycomb structure is compared with an equivalent porous model structure. The porous model is modelled with the extended Darcy-Brinkman law and the heat exchange between the fluid phase and the solid phase is evaluated using the Local Thermal Equilibrium (LTE) assumption. By the results of the direct honeycomb model, the characteristics such as permeability, effective thermal conductivity and interfacial heat transfer are evaluated and then compared with the porous medium model.

2. Honeycomb structure The characteristics of the honeycomb are depicted in figure 1. It is designed with parallel squared channels, half of them are filled with PCM and in the others the air passes through. The height

of a single elementary channel is H, the thickness is δ and the Length is L. The cross section area is H × H. The solid walls of the honeycomb are made of the ceramic cordierite. The PCM is enclosed inside the channels of the honeycomb in checkerboard way and it is considered fixed without any movement as it showed in figure 2a. The porosity in honeycomb configuration is:

f 

Vair 1 H2  Vtotal 2  H  2 2

(1)

Where Vair is the air volume and Vtotal is the packaging volume. The factor ½ is presented because half of channels are closed. Various honeycomb system for different channel per unit of length (CPL) are studied at the same porosity and volume. The honeycomb system with different CPL is reported in figure 2b in a unit length U. The relation between the channel height H and the thickness δ for different CPL is:

Hn 

H0  ;  n  0n n 2 2

(2)

Where H0 and δ0 are the values for 1 CPL (only one channel, not possible because the model is created in checkerboard way) and n = log2CPL. The unit length U is 0.2 m, and the porosity of the system is 0.405. Therefore, given that U=H0+2δ0 and using the relation 3, H0 is 0.18m and δ0 is 0.01m. The PCM used in this simulation is Potassium carbonate (K2CO3). with a mass flux of 0.6 Kgm-2s-1. The thermal properties of the various material are listed in table 1. The variation of thermal properties for the air is described by the following equations [20]:

c p  1.06 103  0.449T  1.14 103T 2  8 107 T 3  1.93 1010 T 4

(3)

k  3.93 103  1.02 104 T  4.86 108 T 2  1.52 1011T 3

(4)

3. Physical System In order to numerically study the physical model presented in figure 1, two domain modeling approaches are presented. The direct honeycomb model – called Model A – is a conjugate heat transfer problem where each channel is geometrically and numerically shaped in base of the number of CPL. The porous honeycomb model – called Model B – is modelled with the extended Darcy-Brinkman law using the Local Thermal Equilibrium (LTE) assumption for heat exchange between solid and liquid zones. Because of the thermal and dynamic symmetry, a repetitive unit is used as computational domain of the model A as showed in figure 3. Therefore the computational 3D domain of the model A has a height and width of the repetitive unit (100 ×100 mm for CPL=2; 50 x 50 mm for CPL=4 and 25 x 25 for CPL=8) and a length of 1 m. The surfaces of the domain are the following: inlet surfaces where the air flows; outflow surfaces where the air exit from the domain and wall frontal surfaces; wall back surfaces where the PCM is encapsulated and the other lateral surfaces where the symmetry is applied (figure 3c.). The model B is introduced because very often it is computationally not convenient to simulate the direct model A for higher CPL, due to the complexity geometry or the computational costs. Fortunately, the honeycomb matrix could be considered as a porous medium with a certain value of permeability and effective thermal conductivity. The model B, therefore, is a porous media model with the equivalent properties of the model A, such as length, cross-section, initial conditions and boundary conditions.

About the domain modeling of the model B, a uniform parallelepiped

computational domain is presented with the same size of the model A. In figure 4a is presented the computational domain of the model B. The model A is then compared with the model B in order to understand if the honeycomb structure with PCM could be simulate with a porous media model. The size is 200 x 200 x 1000 mm, with an inlet condition at left surface, outlet condition at right surface and wall adiabatic lateral surfaces. Varying the CPL in the model B there is a variation of the permeability in the Darcy-Brinkman law.

4. Mathematical Description And Governing Equations The mathematical description about the both models is based on the enthalpy porosity method [22]. This method describes the melting of the phase change material and there is not an explicit separation surface between the solid and liquid zone, but this method introduces a mixed solid-liquid phase change zone, called mushy region. It is useful for that materials melting in a temperature range. The mushy region is described using a parameter – liquid fraction β – varying from 0 to 1. In the fully solid zone the parameter is 1 and in the fully liquid zone its value is null:   0  T  Tm  T / 2    T     1

for T  Tm  T / 2 for Tm  T / 2  T  Tm  T / 2 for

(5)

T  Tm  T / 2

T is the local temperature, Tm is the central value of the PCM temperature melting range, ΔT is the size of the temperature melting range, in this study is 4 kelvin. The PCM is completely encapsulated inside the channels while in the other channels the air is free to move. Pal and Joshi [10] have found that in a honeycomb structure the natural convection of the PCM could be neglected. The governing equations for

the

direct

honeycomb

model

(model

A)

are

the

following:

Equation for the Air:

u v w   0 x y z  u

u

u

u 

(6)

p

  2u

 2u

 2u 

f  u v  w     f  2  2  2  x y z  x z   t  x y

(7)

 2w 2w 2w   w w w w  p f  u v  w     f  2  2  2  x y z  z y z   t  x

 f c p, f

T f

T T     T f    T f    T f    T   f c p, f  u f  v f  w f     k f  kf  kf  t  x  y  z  x  x  y  y z          z 

(8)

(9)

Energy equation for the PCM:

  c PCM

TPCM   2TPCM  2TPCM  2TPCM     kPCM  k PCM  k PCM    f  f HL 2 2 2 t x y z  t 

(10)

Energy equation for the Cordierite:

  c cord

Tcord   2Tcord  2Tcord  2Tcord    kcord  k  k  cord cord t x 2 y 2 z 2  

(11)

The interaction between them is a conjugate heat transfer problem. The main problem of the direct model is related to the increment of the nodes number when the CPL is higher. The boundary conditions of the model A are the following: -

The inlet air sections: mass flux of 0.6 Kgm-2s-1 at 1473.15 K

-

The outside air sections: outflow condition.

-

Frontal surfaces: isothermal condition at 1473.15 K

-

The other surfaces are adiabatic.

Initial condition: -

The system is assumed to be at 1073.15K

About the mathematical description of the model B, the porous zone has a fluid zone (air) and a solid zone. The solid zone has cordierite and PCM, because the last one is encapsulated inside the channels and the natural convection is neglected. The porous model is anisotropic with an effective thermal conductivity along z direction different respect to the x and y directions. The permeability is null along the x and y directions while it has a positive value along z direction. The permeability K is independent by the PCM because it depends only by the dynamic effects inside the channels where the air flows. To calculate the value of the permeability along z, Kzz, the work of Bahrami et al. [23] is taken as a reference. In this work, the average velocity uavg in a single channel with a square cross section is: p  H   1 64         5 tanh     f L  2  3   2  2

uavg

(12)

Δp is the pressure drop along the channel, L is the channel length and μf is the dynamic viscosity of the fluid. For low values of Reynolds number, the Darcy law is used to descried the dynamic behavior of a porous media:

f p  f  uD   uavg L K K

(13)

Therefore, for a single channel the permeability Kzz along z direction is the following:

 H   1 64    K zz       5 tanh     2  3   2  2

(14)

While the total permeability for different CPL is

 H   1 64    K zz    n    5 tanh     2   2  3  2

(15)

where uD is Darcy velocity. The relation between Darcy velocity and average velocity is uD=εuavg. The Local Thermal Equilibrium (LTE) assumption is used to simulate the heat exchange between the air and the solid zone in the porous model. Obviously, the PCM is treated as a solid zone in the porous model because the PCM is encapsulated in the channels and the natural convection could be neglected [10]. The governing equations for 3D LTE porous model are the following:

u v w   0 x y z

(16)

 1 u u u v u w u  p 1  2  2  2     x  f   f t  f x  f y  f z  (17)

   u    u    u    f u  f   f   f   x  y  y  z  z   K xx  x 

 1 v u u v v w v  p 1  2  2  2     y  f   f t  f x  f y  f z  (18)

   v    v    v    f v  f   f   f    x  x  y  y  z  z   K yy

f  

f  

 1 w u u v w w w  p 1  2  2  2     z  f   f t  f x  f y  f z  (19)

f  

   w    w    w    f w  f   f   f   x  y  y  z  z   K zz  x 

Energy equation for the porous media in the case local thermal equilibrium (LTE), Ts=Tf=TPCM=Tcord=T is [24]:

  c eff

T   f c p, f t

 T T T v w u y z  x

   T    T    T      kx    kz    ky    PCM H L t  x  x  y  y  z  z  (20)

u, v and w are the air velocity and x, y and z are the Cartesian coordinates. ρf is the air density, p is the relative pressure, μf is the air dynamic viscosity, K is the permeability of the porous zone. Cp,f and kf

are respectively the air specific heat and air thermal conductibility. The effective heat transfer capacity is:

  c eff     c p  f     c p PCM  1  2    c p cord

(21)

The component of the equivalent thermal conductivity of the solid phase are: 

Parallel for heat conduction in the longitudinal direction (x-direction):

k z  k pcm 

Apcm A

 kf

Af A

 ks

As A  k pcm  k f   ks s   k pcm  k f    ks 1  2  A A

(22)

Serial for heat conduction in the transversal direction (x and y directions):

kx  k y 

kcord k pcm k f

 k pcm kcord   k f kcord  (1  2 ) k pcm k f

(23)

The effective thermal conductivities are used in LTE model, where there the thermal equilibrium is assumed. HL is the latent heat of fusion. Model A and model B will be compared for different CPL using the same porosity and boundary conditions.

5. Numerical Procedure Ansys-Fluent 15.0 is selected as the computational software to solve the governing equations both for model A and model B. A grid dependence test is accomplished in order to evaluate the best grid. The grid for the model A is made of uniform rectangular prism cells. The grid of the model B is a set of 30000 rectangular prisms cells and it is showed in figure 4b. In figure 5 there is a comparison between various grids about model A. Four different grids are employed in this operation, respectively with

7776, 33984, 246328 and 543900 nodes. To compare the grids, for each discretization it has been calculated the Average Temperature of the entire model at a fixed time and percentage deviation is calculated between two consecutive meshes. The analysis shows that the mesh with 246528 nodes is chosen for this study because it present a percentage deviation equals to 0.02% respect to the mesh with 543900, ensuring a good accuracy. Increasing the number of CPL, a scaling operation is applied on the model A in order to store the same number of cells for different CPLs. For the both models, a transient analysis is made with a time step size of 0.1 s. The SIMPLE algorithm is used for the pressure-velocity coupling, while PRESTO algorithm is used for the pressure calculation. Second order upwind scheme is used for energy and momentum equation and the residuals are set at 10-4 and 10-7 for continuity, momentum equation and energy equation. The inlet air temperature is 1473.15 K and the initial temperature of the system is 1073.15 K.

6. Results and discussion In figure 6 there are the average temperature profiles of air, PCM and air+PCM as function of time for different CPLs for the model A. In general, for CPL=2 and CPL=4 the evolution of temperature present a different behavior: the average air temperature has higher values at the beginning times whereas for the PCM the average temperature values are lower for lower CPLs. For CPL=8 and CPL=16 the average temperature profiles evolve in the same way for both air and PCM material, this behavior could be justify by the fact that the exchange contact area between air-solid and solid-PCM is higher. In figure 7 there is the average liquid fraction evolution for the model A at different CPLs. It can be possible to see that for CPL=2 the evolution is not linear.. In figure 8 there is a comparison between the temperatures of air, pcm, cordierite for different CPL of the model A. It is possible to see that for

CPL=8 and CPL=16 the temperature difference is nearly null, suggesting that for higher value of CPL there is a possibility to simulate the direct model (model A) using the porous media formulation. In fact, this can be seen in figure 9, where there is a comparison between the model A and the model B in term of average temperature evolution as function of time. For CPL=2 the two models has different output in term of temperature and liquid fraction, while for CPL = 4 the differences are becoming lower. Finally for CPL =8 and CPL=16 the two models yield the same results. The total stored energy profiles for different CPLs as function of time is reported in figure 10. It is observed for lower CPLs the stored energy profiles are higher while for CPL=8 and CPL=16 the two profiles are overlapped. The liquid fraction fields for model A and model B at different times are reported in figure 11 for CPL=2, where it is observed that for the model A (figure 11a) the melting evolution along z direction is not parallel, because the mushy region is sloped due to natural convection. Instead, for the model B (figure 11b), the melting progress is linear because the melting evolution progresses along z direction in a parallel way. This consideration is very important because it shows the differences between model A and model B for lower CPLs. The PCM temperature fields for model A and model B at different times are reported in figure 12 for CPL=2 and it shows that for the porous medium model the temperature evolution is linear and parallel along z direction while the direct model is affected by the lateral walls which are heated by the air.Therefore it is possible to use the model B (LTE porous model) to simulate the direct honeycomb structure (model A) for higher CPLs., in fact just for CPL=8 the differences are negligible.

7. Conclusions A computational investigation of a honeycomb system with Phase Change Materials (PCM) for solar energy applications is carried out. A comparison between the direct honeycomb model and a LTE

porous medium model is accomplished for different channel per unit of length (CPL). The differences are evident only for CPL ≤ 4, while for higher values, there is an overlapping about the results of the two models. This aspect is related that for higher CPLs there are higher values of the total contact area between air and PCM. This allows to have a greater surface heat transfer which is close to a bulk heat transfer and it determines a more convenient computational model employing the porous medium one. Some interesting developments could be implemented to simulate the honeycomb as thermal storage for charging and discharging cycles. Moreover, it could be removed the adiabatic condition on the lateral surfaces implementing the heat losses toward the external ambient.

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Figure Captions Figure 1. Honeycomb sketch with the single channel size. Figure 2. Figure 2. Cross section of the model A: (a) arrangement of the PCM inside the honeycomb structure and (b) dimensions of the honeycomb system for 2 and 4 channels per unit of length (CPL).. Figure 3. Frontal section of the Domain A and the repetitive unit: (a) single unit cell with repetitive unit, (b) repetitive unit for different CPLs, (c) computational domain of the model A. Figure 4. LTE Porous Model (Model B): (a) Computational domain and (b) computational grid. Figure 5. Independence test grid of model A. Figure 6. Profile Temperature as function of time for different CPLs related to: (a) Air, (b) PCM and (c) average profile for the model A. Figure 7. Profile PCM liquid fraction evolution as function of time for the model A at different CPL. Figure 8. Comparison between the temperature profiles of air, pcm and cordierite at different CPLs of the model A. Figure 9. (a) Average temperature profiles and (b) PCM liquid fraction profiles during the time for both models. Figure 10. Total stored energy of the honeycomb model for the model A as function of time. Figure 11. Figure 11. Liquid Fraction fields at different times, 5s, 5000s, 10000s, 15000s, 25000s for: (a) porous model yz plane; (b) direct model Figure 12. Figure 12. Temperature fields at different times, 5s, 500s, 5000s, 20000s, 65000s for: (a) porous model yz plane; (b) direct model

Table 1. Thermal properties of the materials Thermal properties Density[kg/m3] Specific Heat [J/kg K] Thermal Conductivity [W/m K] Dynamic Viscosity [kg/m s] Thermal expansion Coefficient [1/K] Melting Heat [J/kg] Melting Temperature [K]

Cordierite 2300 900 2.5 -

Air Ideal gas law Eq. 1 Eq. 2 Sutherland law -

K2CO3 [20] 2290 1513.69 [21] 2 0.00011 235800 1170.15

Figure 1. Honeycomb sketch with the single channel size

PCM

Air

(a)

(b)

Figure 2. Cross section of the model A: (a) arrangement of the PCM inside the honeycomb structure and (b) dimensions of the honeycomb system for 2 and 4 channels per unit of length (CPL).

(a)

(b)

(c)

Figure 3. Frontal section of the Domain A and the repetitive unit: (a) single unit cell with repetitive unit, (b) repetitive unit for different CPLs, (c) computational domain of the model A

y wall Inlet

Porous zone

Outlet

z x

(a) A

A

Sez. AA Plane xy

(b)

Figure 4. LTE Porous Model (Model B): (a) Computational domain and (b) computational grid

Figure 5. Independence test grid of model A

(a)

(b)

(c) Figure 6. Profile Temperature as function of time for different CPLs related to: (a) Air, (b) PCM and (c) average profile for the model A.

Figure 7. Profile PCM liquid fraction evolution as function of time for the model A at different CPL

Figure 8. Comparison between the temperature profiles of air, pcm and cordierite at different CPLs of the model A

(a) (b) Figure 9. (a) Average temperature profiles and (b) PCM liquid fraction profiles during the time for both models.

Figure 10. Total stored energy of the honeycomb model for the model A as function of time

(a)

(b) Figure 11. Liquid Fraction fields at different times, 5s, 5000s, 10000s, 15000s, 25000s for: (a) porous model yz plane; (b) direct model

(a)

(b) Figure 12. Temperature fields at different times, 5s, 500s, 5000s, 20000s, 65000s for: (a) porous model yz plane; (b) direct model

Highlights Develop thermal porous medium model for analysis of a complex geometry Comparison between a porous medium model and a direct model at varying of the number of channels Reporting the average temperature profiles and liquid fraction at varying the number of channels