Analysis of Natural Convection in Porous Media for Thermal Storage Using Darcy-Brinkman-Forcheimer Formulation

Analysis of Natural Convection in Porous Media for Thermal Storage Using Darcy-Brinkman-Forcheimer Formulation

Mohamed Belhadj* Aissa Atia Ahmed Benchatti

Laboratory of Mechanics, University Amar Telidji, Laghouat 03000, Algeria

Corresponding Author Email: 
m.belhadj@lagh-univ.dz
Page: 
73-78
|
DOI: 
https://doi.org/10.18280/mmep.070109
Received: 
18 May 2019
|
Accepted: 
3 January 2020
|
Published: 
31 March 2020
| Citation

OPEN ACCESS

Abstract: 

This numerical study deals with heat convection in a porous media. The heat transfer is modeled using Darcy-Brinkman-Forcheimer formulation. A CFD code based on the finite volume method is used to solve the mathematical set of equations governing air flow and convective heat transfer in the porous media. The numerical predictions are validated by comparison with results available in the literature. Parameters governing heat transfer and fluid flow are investigated (Rayleigh number, Darcy number and porosity). Increasing Rayleigh and Darcy numbers resulted in enhancing the heat transfer rate but the convective heat transfer rate decreased with increasing porosity.

Keywords: 

Darcy-Brinkman-Forcheimer, heat convection, porosity, porous media

1. Introduction

Heat transfer by convection in porous media has attracted researchers for the last decades. It has various applications such as geothermal, crystal growth, oil industry, drying and building thermal applications. Nield and Bejan has reviewed a detailed literature related to convective heat transfer in porous media [1]. Numerical studies are focused then on a differentially heated porous cavity. Loganathan and Sivapoornapriya [2] studied heat transfer and fluid flow in porous medium. They considered an impulsively started vertical plate. Zheshu et al. [3] investigated a natural convection problem in a square cavity filled with porous media. They used the Lattice Boltzmann method to solve numerically the mathematical model. Horton and Rogers [4] and Lapwood [5] are the pioneers to investigate heat transfer in porous media. Revnic et al. [6] analyzed heat convection in infinite rectangular cavity of bi-dispersed porous medium (BDPM), they used a mathematical model proposed by Nield and Kuznetsov [7] and Rees et al. [8]. Rayleigh numbers investigated were less than 103. This study showed that heat transfer is done mainly by conduction for lower Ra values. Both thermal conductivity and Darcy number have great influence on heat convection central cells. Bahloul [9] studied fluid flow in a vertical cavity with porous media. The vertical walls were differentially heated whereas the horizontal walls were kept adiabatic. Rayleigh number and the aspect ratio had significant effect on flow structure. Kalla et al. [10] investigated numerically heat convection in a very long horizontal porous layer. The porous layer walls were exposed to uniform heat flux. The model of Darcy was used. Finite difference was used to solve the numerical model. The effect of Rayleigh number on heat transfer in the porous layer was investigated. Nawaf et al. [11] conducted a numerical study on transient heat transfer in porous media. The model of Darcy was used with Boussinesq approximation of a standard fluid. The hot wall was exposed to fluctuated heat flux whereas the cold wall was kept at constant temperature. The other two walls were kept adiabatic. Streamlines, isothermals and Nusselt number were analyzed. 

However, there are only few works that deal with the effect of porosity in porous media. Thus, the objective of this study is to investigate both heat transfer and fluid flow in a cavity filled with porous media using Darcy Brinkman-Forcheimer formulation.

2. Mathematical Model

The air flow is assumed to be two-dimensional in a cavity filled with porous media (Figure 1). Fluid properties are supposed to be constant. The fluid is considered to be incompressible and Newtonian. Boussinesq approximation is applied. Heat generation and radiation effects are neglected in the present study.

Figure 1. Schematic diagram of the cavity and boundary conditions

The mathematical model is composed of [12]:

Continuity equation

$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$                   (1)

X- Momentum equation

$\begin{align}  & \frac{1}{{{\varepsilon }^{2}}}U\frac{\partial U}{\partial X}+\frac{1}{{{\varepsilon }^{2}}}V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{\Pr }{\varepsilon }\left( \frac{{{\partial }^{2}}U}{\partial {{X}^{2}}}+\frac{{{\partial }^{2}}U}{\partial {{Y}^{2}}} \right) \\ & \text{                                     }-\frac{\Pr }{Da}jU-\frac{1.75}{\sqrt{150}}\frac{\sqrt{{{U}^{2}}+{{V}^{2}}}}{\sqrt{Da}}\frac{U}{{{\varepsilon }^{{3}/{2}\;}}} \\\end{align}$              (2)

Y- Momentum equation

$\begin{align}  & \frac{1}{{{\varepsilon }^{2}}}U\frac{\partial V}{\partial X}+\frac{1}{{{\varepsilon }^{2}}}V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial Y}+\frac{\Pr }{\varepsilon }\left( \frac{{{\partial }^{2}}U}{\partial {{X}^{2}}}+\frac{{{\partial }^{2}}U}{\partial {{Y}^{2}}} \right) \\ & \text{                                   }-\frac{\Pr }{Da}jV-\frac{1.75}{\sqrt{150}}\frac{\sqrt{{{U}^{2}}+{{V}^{2}}}}{\sqrt{Da}}\frac{V}{{{\varepsilon }^{{3}/{2}\;}}}+Ra\Pr \theta  \\\end{align}$            (3)

Energy equation

$U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{{{\partial }^{2}}\theta }{\partial {{X}^{2}}}+\frac{{{\partial }^{2}}\theta }{\partial {{Y}^{2}}}$            (4)

Boundary conditions

At X = 0: U = V = 0, θ = 1                   (5a)

At X = 1: U = V = 0, θ = 0                   (5b)

At Y = 0: U = V = 0, $\frac{\partial \theta }{\partial Y}$= 0              (5c)

At Y = 1: U = V = 0, $\frac{\partial \theta }{\partial Y}$= 0               (5d)

3. Numerical Solution

Eqns. (1-4) are subjected to the boundary conditions (Eq. 5). They are integrated numerically using a CFD based on the finite volume method.

3.1 Effect of mesh size

The effect of mesh size is shown in Table 1. For Ra=105, Da= 10-2 and ε = 0.4, the maximum horizontal velocity, vertical velocity and average Nusselt number are presented with different mesh sizes. The grids have been optimized where a uniform mesh of (182x182) is used in X and Y directions. 

Table 1. Effect of mesh size on maximum horizontal velocity, vertical velocity and average Nusselt number for Ra=105, Da= 10-2 and ε = 0.4

Mesh size

Umax

Vmax

Nu

82×82

14.069

20.3699

5.033571

102×102

13.85677

19.64757

4.941667

122×122

13.747

19.19972

4.837619

142×142

13.69577

18.96857

4.761693

162×162

13.68824

18.91078

4.705161

182×182

13.68114

18.83855

4.666403

202×202

13.68114

18.83855

4.666303

222×222

13.68114

18.83855

4.666302

 
3.2 Validation

The present numerical model has been validated by comparison with results due to Nithiarasu et al. [13]. The mid-height temperature distribution for Ra=105, Da= 10-2 and ε = 0.4 have been plotted and compared with results of Nithiarasu et al. (Figure 2). Good agreement has been obtained. In addition, Figure 3 shows for the two results the streamlines and isothermal lines and this for Da = 10-6, Ra = 108 and ε = 0.8. The two results compare well as can be seen. Finally, Figure 4 presents another comparison of the two works based on streamlines and isothermal lines and this for Da = 10-2, Ra = 104 and ε = 0.6. Good accordance has also been obtained.

Figure 2. Comparison of the mid-height temperature distribution for Ra=105, Da= 10-2 and ε = 0.4 with results due to Nithiarasu et al. [13]

Figure 3. Comparison of streamlines and isothermal lines for Da = 10-6, Ra = 108, ε = 0.8. (a) present work, (b) Nithiarasu et al. (1997)

Figure 4. Comparison of streamline and isothermals (Da = 10-2, Ra = 104, ε = 0.6) a) present work, b) Nithiarasu et al. [13]

4. Results

4.1 Effect of Rayleigh number

Figure 5 shows the effect of Rayleigh number on the velocity profiles at the middle of the cavity for Da=10-3 and ε = 0.7). Greater values of Ra lead to increasing velocity components. Stronger natural convection is observed near the hot wall. Figure 6 shows the average Nusselt number as function of Rayleigh number for a fixed value of Darcy number Da = 10-3 and a given porosity ε = 0.7. One can notice that for Ra>105, increasing Ra increases strongly the average Nusselt number. Figure 7 shows streamlines and isothermals for different Ra values and this for fixed Darcy number and porosity i.e. Da = 10-3 and ε = 0.7. For low Ra numbers the isotherms are vertical and start getting deformed. The presence of significant convection currents is observed with clockwise cell. The heated wall works to move the fluid upward while the fluid moves downward near the cold wall. Greater values of Rayleigh number result in stronger circulation and the temperature contours are notably deformed which means that natural convection dominates the heat transfer.

Figure 5. Profiles of velocity components horizontal velocity (a) and vertical velocity (b) in the middle of the cavity for various values of Ra (Da = 10-3, ε = 0.7)

Figure 6. Average Nusselt number for various values of Ra (Da = 10-3, ε = 0.7)

Figure 7. Isothermal lines (a) and streamlines (b) for various values of Ra (Da = 10-3, ε = 0.7)

4.2 Effect of Darcy number

In order to investigate the effect of Darcy number on the fluid flow and heat transfer in the porous cavity, simulations with different values of Darcy number are carried out (Da = 10-5, 10-4, 10-3, 10-2) where Rayleigh number and porosity are kept constant (Ra = 105, ε = 0.7).

Figure 8. Profiles of velocity components horizontal velocity (a) and vertical velocity (b) in the middle of the cavity for various values of Da (Ra = 105, ε = 0.7)

Figure 8 shows the effect of Da on the velocity profiles at the middle of the cavity. The velocity profiles are symmetric about the middle of the cavity. Greater values of Ra result in increasing velocity components. The maximum values are shown in the vicinity of the hot wall which is explained by stronger natural convection at this region. Increasing Darcy number resulted in lower velocities.

Figure 9. Average Nusselt number for various values of Da (Ra = 105, ε = 0.7)

Figure 10. Isothermal lines (a) and streamlines (b) for various values of Da (Ra = 105, ε = 0.7)

Figure 9 shows the average Nusselt number as function of Darcy number for Ra=105 and ε = 0.7. For 10-4<Da<10-2, the average Nusselt number increases exponentially. Then there is practically no effect at lower values (10-4≥Da≥10-6). For these values of Darcy the medium is no longer porous.

Figure 11. Effect of porosity on the heat transfer rate (Nu-Ra) for different Darcy number values 

Figure 10 shows streamlines and isotherms patterns in the cavity. One can see that Darcy number has an effect on fluid flow and heat transfer. This effect is related to the cavity porosity. However, for low Darcy values Da = 10-6-10-4, heat convection is not significant due to the friction effect associated with low porosity (Brinkman term). Isotherms are vertical which means that conduction transfer mode is dominant in this case. But, for greater values of Darcy number (i.e. 10-3 à 10-1), convection heat transfer is significant which is accompanied with significant viscous effect that can be seen with distorted isotherms. The porosity of the cavity increases with increasing Da which results in decreasing viscous effect and hence fluid circulation. Streamlines show significant circulation near the cold wall.

4.3 Effect of porosity

The influence of porosity on fluid flow and heat transfer is shown in Figure 11. Different values of porosity and Darcy numbers are investigated. At higher values of Da, significant effect of porosity on heat transfer is observed. But, for lower Da this effect is not significant (Da=10-5) which could be neglected for Da=10-6.

5. Conclusion

Fluid flow and heat transfer have been investigated in a cavity filled with a storage porous media. The results of this numerical investigation have showed the effect of parameters governing heat transfer and fluid flow (i.e. Rayleigh number, Darcy number and porosity). Greater Rayleigh numbers resulted in stronger heat transfer by natural convection. Increasing Darcy numbers led to enhancing heat transfer rate whereas heat transfer by convection decreased with increasing porosity.

Acknowledgment

This work is supported by the Algerian Ministry of Higher Education and Scientific Research (DGRSDT-MESRS).

Nomenclature

CP

Specific heat, J. kg-1. K-1

Da

Darcy number

g

h

k

Gravitational acceleration, m.s-2

Heat transfer coefficient

Thermal conductivity, W.m-1. K-1

Nu

Nusselt number

Pr

Prandtl number

Ra

Rayleigh number

T

Temperature, °C

u, v

Velocity components, m/s

x, y

Coordinate axes, m

Greek symbols

 

a

Thermal diffusivity

b

Thermal expansion

e

Porosity

μ

Dynamic viscosity

υ

Kinematic viscosity

ψ

Streamline function

ρ

Density

  References

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