Transp Porous Med DOI 10.1007/s11242-016-0747-2

Natural Convection Coupled with Thermal Radiation in a Square Porous Cavity Having a Heated Plate Inside C. Sivaraj1 · M. A. Sheremet2,3

Received: 20 March 2016 / Accepted: 13 July 2016 © Springer Science+Business Media Dordrecht 2016

Abstract This study reports a numerical simulation of the natural convection combined with thermal radiation in a square porous cavity with a thin isothermally heated plate placed horizontally or vertically at its center. The vertical walls of the cavity are cooled while the horizontal ones are adiabatic. The governing equations were solved using a finite volume method on a uniformly staggered grid system. The computational results are presented in the form of isotherm and streamline plots and Nusselt numbers. The effects of the Darcy number (10−5 ≤ Da ≤ 10−2 ), plate length (0.25 ≤ D ≤ 0.75) and the radiation parameter (0 ≤ Rd ≤ 2) are investigated for the Rayleigh number Ra = 107 . It is found that the Darcy number, the plate length and the radiation parameter enhance the overall heat transfer rate across the cavity. Keywords Natural convection · Thermal radiation · Porous cavity · Brinkman-extended Darcy model · Heated plate · Numerical simulation

List of symbols Da g K L Nu

B

Darcy number Gravitational acceleration Permeability of the porous medium Size of the cavity Local Nusselt number

M. A. Sheremet [email protected]

1

Department of Mathematics, PSG College of Arts and Science, Coimbatore, Tamil Nadu 641014, India

2

Department of Theoretical Mechanics, Tomsk State University, Tomsk, Russia 634050

3

Department of Nuclear and Thermal Power Plants, Tomsk Polytechnic University, Tomsk, Russia 634050

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Nu P p Pr qr x , qr y Ra Rd T t Th Tc u, v U, V x y X, Y

Average Nusselt number Dimensionless pressure Dimensional pressure Prandtl number Radiation heat fluxes along horizontal and vertical axis Rayleigh number Thermal radiation parameter Dimensional fluid temperature Dimensional time Dimensional temperature at heated plate Dimensional temperature at cooled vertical walls Dimensional velocity components along horizontal and vertical directions Dimensionless velocity components along horizontal and vertical directions Dimensional Cartesian coordinate measured along the bottom wall of the cavity Dimensional Cartesian coordinate measured along the vertical wall of the cavity Dimensionless Cartesian coordinates

Greek symbols αm β βr ε μ θ λm ρ   ρC  p m ρCp f σ τ

Overall thermal diffusivity of the porous medium Coefficient of thermal expansion Extinction coefficient Porosity of the porous medium Dynamic viscosity Dimensionless temperature Overall thermal conductivity of the porous medium Fluid density Overall heat capacity of the porous medium Heat capacity of the fluid Stephan–Boltzmann constant Dimensionless time Dimensionless stream function

Subscripts c h m

Cold Hot Porous medium

1 Introduction Convective heat transfer in porous enclosures has important applications in industry and nature. These applications include building heating, packed bed reactors, grain storage, porous heat exchangers, geothermal systems, underground contaminant transport, cooling of electronic devices and many others. Comprehensive analysis of heat and mass transfer in porous media can be found in the books by Ingham et al. (2004), Ingham and Pop (2005), Vafai

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Natural Convection Coupled with Thermal Radiation in a Square...

(2005), Nield and Bejan (2013) and Shenoy et al. (2016). Only several studies concerning natural convection in porous enclosures include analysis of heat and mass transfer at the presence of internal discrete heaters or solid blocks. At the same time, such analysis plays an important role in different engineering applications. Oztop et al. (2015) have prepared a comprehensive review on natural convective heat transfer and fluid flow in different types of cavities filled with clear, porous and nanofluids with local heat sources. It should be noted that all heat sources in the considered papers are mounted on the solid walls. Datta et al. (2016) have analyzed natural convection in a differentially heated non-Darcy porous square cavity with an adiabatic block. They have found that the presence of solid obstacle can enhance heat transfer and the maximum heat transfer rate is characterized by optimal block size. Sankar et al. (2011) have studied the convective flow and heat transfer in a square porous cavity with partially active thermal walls. It has been revealed that the position of local heaters and coolers has an essential influence of the fluid flow and heat transfer inside the cavity. Aleshkova and Sheremet (2010) have analyzed numerically unsteady conjugate natural convection in a square porous enclosure having heat-conducting solid walls and an internal heat source of constant temperature. They have ascertained that an increase in the Rayleigh number leads to a formation of an unstable thermal plume above the local heater and heat conduction starts to dominate over convective heat transfer at reduction of the medium permeability. Badruddin et al. (2012a, b) have studied free convection in a porous square duct using local thermal equilibrium model (see Badruddin et al. 2012a) and thermal nonequilibrium model with thermal dissipation (see Badruddin et al. 2012b). It has been revealed that the upper part of the duct is dominated by heat conduction due to weak fluid motion and the viscous dissipation leads to reduce the heat transfer rate from the hot walls to the porous medium. Lam and Prakash (2014) have analyzed natural convective fluid flow and heat transfer in a porous square cavity with two local heaters mounted on top and bottom adiabatic walls. They have found that the average Nusselt number is significantly higher for bottom heat source than that of top heater for both in-line and staggered arrangement of heaters. The role of thermal radiation is of major importance in the design of many advanced energy systems. Thermal radiation within the systems is usually the result of emission by hot walls and the working fluid. The considered configuration with radiation in porous media involves a wide range of industrial applications such as heat exchangers, chemical reactors, nuclear waste disposal, grain storage, etc. Thus, Ahmed et al. (2014a) have investigated numerically free convection coupled with thermal radiation in an inclined porous enclosure with a corner heater. They utilized non-Darcy model with Boussinesq and Rosseland approximations for analysis of heat transfer and fluid flow within the cavity. It has been found the heat transfer enhancement with Darcy number and radiation parameter. Badruddin et al. (2006) have studied numerically natural convection coupled with thermal radiation and viscous dissipation in a vertical square cavity. Mathematical model has been formulated on the basis of Darcy, Boussinesq and Rosseland approaches. They have shown an increase in the average Nusselt number with radiation parameter. Ahmed et al. (2014b) have analyzed MHD natural convective heat transfer and fluid flow inside a vertical square cavity under the effects of thermal radiation and viscous dissipation. Governing equations have been written taking into account Darcy model, local thermal equilibrium between the porous medium and the fluid, Boussinesq approximation for natural convection and Rosseland approach for thermal radiation. It has been revealed that the radiation parameter has a significant effect on streamlines and isotherms and also increases the role of conduction regime. Mahapatra et al. (2012) have examined an effect of thermal radiation on the basis of Rosseland approximation on natural convection of heat-generating fluid within a porous square cavity. Governing equations have been formulated using the Darcy–Brinkman–Forchheimer model with Boussinesq approxi-

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C. Sivaraj, M. A. Sheremet Fig. 1 A scheme of the system: 1—fluid-saturated porous medium; 2—heated plate

mation. They have found that average Nusselt number increases with increase in the thermal radiation parameter. Hossain et al. (2013) have investigated natural convection coupled with thermal radiation of optically thick fluid in a porous cavity. Mathematical model has been written using Darcy–Brinkman–Forchheimer approach, Boussinesq and Rosseland approximations. It has been ascertained that flow reduces and heat transfer rate growths with increase in the thermal radiation parameter. However, these above-mentioned papers did not take into account the effect of heated plates located in the center of the porous cavity on convective fluid flow and heat transfer. It is known that such heated plates can essentially modify flow patterns and temperature fields in the case of clear fluid (see Saravanan and Sivaraj 2013, Saravanan and Sivaraj 2015). The main purpose of the present paper is a numerical simulation of natural convection combined with thermal radiation in a non-Darcy porous square cavity with a heated plate.

2 Mathematical Formulation The physical model of natural convection combined with thermal radiation in a porous enclosure and the coordinate system is schematically shown in Fig. 1. The domain of interest includes the fluid-saturated porous medium (1 in Fig. 1) and a heated plate (2 in Fig. 1). During all time of process, the temperature of the heated plate Th is constant. The horizontal walls (y = 0, y = L) are assumed to be adiabatic while the vertical walls (x = 0, x = L) are kept at constant temperature Tc where Tc < Th . It is assumed in the analysis that the thermophysical properties of the fluid are independent of temperature, and the flow is laminar. The fluid is viscous, heat-conducting, Newtonian, and the Boussinesq approximation is valid. Further, it is assumed that the temperature in the fluid phase is equal to the temperature of the solid structure everywhere in the porous region (1 in Fig. 1), and local thermal equilibrium model is applicable in the present investigation. The porous medium is considered to be homogeneous and isotropic. All walls are assumed to be impermeable. Radiation heat flux inside the fluid-saturated porous cavity is modeled on the basis of the Rosseland approximation (see Martyushev and Sheremet 2012).

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Natural Convection Coupled with Thermal Radiation in a Square...

In the present investigation, the Brinkman-extended Darcy model has been adopted in the governing equations of the problem in the following way (see Aleshkova and Sheremet 2010; Astanina et al. 2015): ∂u ∂v + =0 ∂x ∂y     1 ∂u ∂ 2u u ∂u v ∂u μ ∂ 2u ∂p μ + ρ + 2 + 2 + =− − u ε ∂t ε ∂x ε ∂y ∂x ε ∂x2 ∂ y2 K     μ 1 ∂v ∂ 2v u ∂v v ∂v μ ∂ 2v ∂p + + 2 + 2 + =− + ρgβ (T − Tc ) − v ρ ε ∂t ε ∂x ε ∂y ∂y ε ∂x2 ∂ y2 K   2      ∂T    ∂qr y ∂ T ∂qr x ∂2T ∂T ∂T ρCp m + − + ρCp f u +v + = λm ∂t ∂x ∂y ∂x2 ∂ y2 ∂x ∂y

(1) (2) (3) (4)

with the initial and boundary conditions t =0:

u = v = 0,

T = Tc ,

at 0 ≤ x ≤ L and 0 ≤ y ≤ L

t >0:

u = v = 0,

T = Tc , ∂T = 0, ∂y T = Th ,

at x = 0, L and 0 ≤ y ≤ L

u = v = 0, u = v = 0,

at y = 0, L and 0 < x < L at the plate

(5)

Invoking Rosseland approximation (see Martyushev and Sheremet 2012) for radiation 4σ ∂ T 4 4σ ∂ T 4 , qr y = − 3β . Expanding T 4 in Taylor series about Tc and neglecting qr x = − 3β r ∂x r ∂y higher order terms, we have T 4 ≈ 4T Tc3 − 3Tc4 . Introducing the following dimensionless variables, y u v x , Y = , U= , V = L L εαm /L εαm /L t T − Tc p , τ= 2 , θ= P= 2 2 ραm /L L /εαm Th − Tc

X =

(6)

the governing Eqs. (1)–(4) can be written in dimensionless form as ∂U ∂V + =0 ∂X ∂Y   2 ∂ U ∂ 2U ∂U Pr ∂U ∂U ∂P + − +U +V =− + Pr U ∂τ ∂X ∂Y ∂X ∂ X2 ∂Y 2 Da   2 Pr ∂V ∂2V ∂V ∂V ∂P ∂ V + RaPrθ − + +U +V =− + Pr V ∂τ ∂X ∂Y ∂Y ∂ X2 ∂Y 2 Da  2   ∂ θ ∂θ ∂θ 4 ∂ 2θ ∂θ +U +V = 1 + Rd + ∂τ ∂X ∂Y 3 ∂ X2 ∂Y 2

(7) (8) (9) (10)

with the corresponding initial and boundary conditions τ =0:

U = V = 0,

θ = 0,

at 0 ≤ X ≤ 1 and 0 ≤ Y ≤ 1

τ >0:

U = V = 0,

θ = 0, ∂θ = 0, ∂Y θ = 1,

at X = 0, 1 and 0 ≤ Y ≤ 1

U = V = 0, U = V = 0,

at Y = 0, 1 and 0 < X < 1 at the plate

(11)

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C. Sivaraj, M. A. Sheremet

The dimensionless parameters appearing in the Eqs. (7)–(11) are defined as Pr =

ν K gβ (Th − Tc ) L 3 h 4σ Tc3 , Ra = , D = , Da = , R = d αm ναm L εL 2 λβr

(12)

The stream function is calculated from its definition as ∂

∂

and V = − (13) ∂Y ∂X In order to determine the total heat transfer rate, we need to define the local heat transfer rate along the right wall by the local Nusselt number as follows:   ∂θ 4 Nu = 1 + Rd (14) 3 ∂X U=

Thus, the average Nusselt number along the right wall is defined by 1 Nu =

Nu dY

(15)

0

3 Numerical Procedure The governing Eqs. (7)–(10) were discretized by the finite volume method on a uniform staggered grid system using the SIMPLE algorithm of Patankar (1980). The third-order QUICK scheme of Hayase et al. (1992) and the second order central difference scheme were used for the convection and diffusion terms, respectively. In order to keep consistent accuracy over the entire computational domain, a third-order-accurate boundary condition treatment was adopted as suggested by Hayase et al. (1992). The set of discretized equations were then solved by a line-by-line procedure of the tri-diagonal matrix algorithm (TDMA). The steady-state results alone were considered when the convergence criteria    m m−1  i, j ϕi, j − ϕi, j  ≤ 10−7 (16)   m  i, j ϕi, j  are achieved. Here ϕ represents the variables U, V or θ , the superscript m refers the iteration number and (i, j) refers the space coordinates. The numerical code was firstly validated for the case of pure natural convection in a differentially heated square cavity containing a heat-conducting solid block (see House et al. 1990). Table 1 gives the average Nusselt number for different Rayleigh numbers, solid block size (ζ ) and thermal conductivity ratios (λ∗ ). It can be clearly seen that the agreement is highly satisfactory. When porous medium is taken into account, a comparison was made with the previous numerical results given by Nithiarasu et al. (1997), Guo and Zhao (2005) and Liu et al. (2014). A fairly good agreement is observed from Table 2. These results validate the present computational output indirectly. Moreover, a grid independency study of the solution was made, and the average Nusselt numbers for the grid of 143 × 143 points are found to deviate from that of 163 × 163 points by <1 % (see Table 3). Hence, the numerical results presented in this study were restricted to the grid system of 143 × 143 points.

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Natural Convection Coupled with Thermal Radiation in a Square... Table 1 Comparison of average Nusselt number with House et al. (1990)

λ∗

ζ

Ra

Average Nusselt number House et al. (1990)

Present study

Error %

105

0.5

0.2

4.624

4.634

0.22

105

0.5

1.0

4.506

4.512

0.13

105

0.5

5.0

4.324

4.326

0.05

106

0.9

0.2

2.402

2.437

1.46

106

0.9

5.0

3.868

3.816

1.34

Table 2 Comparison of average Nusselt number for Pr = 1.0 and various Ra, ε and Da Da

Ra

Average Nusselt number Nithiarasu et al. (1997)

10−2

10−4

10−6

Guo and Zhao (2005)

Liu et al. (2014)

Present study

103

1.010

1.008

1.007

1.008

104

1.408

1.367

1.362

1.362

105

2.983

2.998

3.009

3.000

105

1.067

1.066

1.067

1.065

106

2.550

2.603

2.630

2.602

107

7.810

7.788

7.808

7.790

107

1.079

1.077

1.085

1.078

108

2.970

2.955

2.949

3.043

109

11.460

11.395

11.610

11.787

Table 3 Grid independence results of Nu for Ra = 107 , Da = 10−2 , D = 0.50 and Rd = 1 Grid

83 × 83

103 × 103

123 × 123

143 × 143

163 × 163

Horizontal plate

15.423

15.147

14.930

14.758

14.619

Vertical plate

23.840

23.619

23.478

23.381

23.311

4 Results and Discussion The present study is made to analyze the effect of thermal radiation on buoyancy-induced flow in a porous square cavity containing a heated thin plate. The thickness of the plate δ was taken as one grid spacing in the computational domain. The computations were carried out for Pr = 0.72. The effects of the Darcy number Da, varied from 10−5 to 10−2 , the dimensionless plate length D, varied from 0.25 to 0.75 and the radiation parameter Rd , varied from 0 to 2, on the fluid flow and heat transfer are investigated for Ra = 107 . The computational results are discussed in terms of isotherm and streamline plots and Nusselt numbers. The isotherms and streamlines at the steady state are plotted side by side exploiting the symmetry about the vertical center line. The effect of Darcy number on the thermal and flow fields is displayed in Figs. 2 and 3 for the horizontal and vertical plates, respectively. The presented figures clearly exhibit two symmetric counter rotating cells both rising up at the centre of the cavity and falling down

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C. Sivaraj, M. A. Sheremet

Fig. 2 Isotherms and streamlines for horizontal plate with D = 0.5 and different Rd . a Rd = 0, b Rd = 1, c Rd = 2

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Natural Convection Coupled with Thermal Radiation in a Square...

Fig. 3 Isotherms and streamlines for vertical plate with D = 0.5 and different Rd . a Rd = 0, b Rd = 1, c Rd = 2

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C. Sivaraj, M. A. Sheremet

along the vertical cold walls due to symmetric boundary conditions maintained there. The corresponding isotherms also maintain the symmetry. Figure 2 shows the situation of the horizontal plate for different Rd . For Da = 10−5 a moderate convective flow occupies the entire cavity due to low permeability of the porous medium. When Da increases to 10−2 convection gets strengthened in the top half of the cavity and conduction becomes prominent below the heated plate due to underlying stable setup. We can also observe strong temperature gradients around the heated plate and along the top portion of the cold walls resulting in the development of the thermal plume above the heated plate. This is due to the fact that when Da increases, the permeability of the porous medium increases which results in the reduction of resistance to the fluid flow inside the porous cavity. This can be clearly seen by observing the | |max values. The effect of thermal radiation is now examined. It should be noted from Fig. 2 that the temperature gradients in the top of the vertical walls and above the heated plate are weakened with increasing the radiation parameter Rd . The reason is that the ascending fluid particles from the center of the cavity dissipate their radiative heat energy to the solid matrix of the cavity which produces an increase in the density. This leads to the resistance of the upward fluid flow. We can also notice crowded isotherms below the heated plate due to the penetration of radiative flux from the plate which make a thermally active region in the bottom of the cavity. Thus, the overall convective pattern in the cavity is further augmented with a weak secondary vortices just above the plate for the higher values of Rd = 2 and Da = 10−2 . The isotherms and streamlines for the vertical plate are illustrated in Fig. 3. These plots show that the convection becomes more active almost the entire cavity compared to the situation of the horizontal plate. In the absence of radiation, an increase in the value of Da from 10−5 to 10−2 produces a vertical thermal stratification on either side of the vertical plate which leads to the enhancement of convection in the cavity. In the presence of thermal radiation, this thermal stratification becomes less pronounced owing to the penetration of the radiative flux from the heated plate to the cavity. It may be noted that a thin thermal boundary layer near the vertical walls is formed resulting in a strong convective pattern with a distortion in the bottom of the cavity. Figures 4 and 5 show the effect of plate length on the fluid flow and heat transfer characteristics for Da = 10−3 and different Rd . In general, an increase in D adds more amount of heat energy to the fluid in the cavity leading to an improved convective flow moving with higher momentum near the cold walls. From Fig. 4, one can find a strong and weak convective flow occupying regions above and below the horizontal plate, respectively, for increasing D. This is because the horizontal plate offers a mechanical resistance i.e., an obstruction effect to upward fluid flow for increasing D. It is also to be noted that the temperature in the top region of the cavity is maintained within θ = 0.42–0.47 for D = 0.25 and θ = 0.63–0.68 for D = 0.75 in the absence of thermal radiation. When radiation is taken into account, the radiative heat added to the cavity increases the average temperature of the fluid in the cavity. The size of the primary eddy in the core region decreases with an increase in Rd . For the case of vertical plate, a region of thermal stratification on both sides of the plate increases with D. The corresponding streamlines exhibit strong convective cells with the core region moving from the top to the center of the cavity for increasing D from 0.25 to 0.75. We further notice that the thermal stratification becomes less pronounced and the thickness of the thermal boundary layer along the vertical walls decreases in the presence of radiation. The variation of the average Nusselt number Nu against Da is depicted in Fig. 6a, b for various values of Rd and D. It is clearly seen that for a fixed Rd and D, Nu increases with Da. This is because an increase in Da makes the reduction of the solid structure volume which results in an enhanced convection in a cavity. Let us now examine the effect of Rd and D on

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Natural Convection Coupled with Thermal Radiation in a Square...

Fig. 4 Isotherms and streamlines for horizontal plate with Da = 10−3 and different Rd . a Rd = 0, b Rd = 1, c Rd = 2

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C. Sivaraj, M. A. Sheremet

Fig. 5 Isotherms and streamlines for vertical plate with Da = 10−3 and different Rd . a Rd = 0, b Rd = 1, c Rd = 2

the overall heat transfer rate. When Rd increases, the thermal energy added to the cavity, viz., radiative mechanism increases. An increase in D extends the region of the hot layer which invokes more amount of thermal energy to the cavity, viz., conductive as well as radiative

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Natural Convection Coupled with Thermal Radiation in a Square...

Fig. 6 Variation of average Nusselt number with Da for horizontal and vertical plates

mechanisms. These two mechanisms increase the average temperature of the fluid and lead to a rise in the intensity of the convective flows. It means that the extra heat energy added to the cavity has to be dissipated through the cold walls. Thus, an increase in Rd and D produces an overall increase in the heat transfer across the cavity. It is further interesting to analyze the heat transfer rates within the cavity by comparing the horizontal and vertical plates. In general, we observed that for fixed values of D and Da, the heat transfer rate for the vertical plate is higher than that for the horizontal plate in the absence of radiation. This is natural behavior because when the plate is placed vertically, a strong fluid flow is observed almost occupying the entire cavity in contrast to the horizontal one. A different behavior is observed when radiation is taken into account. It is to be noted that for low values of Da = 10−5 , an increase in Nu for the vertical plate compared to the horizontal plate decreases with Rd , whereas it is marginal for increasing Da.

5 Conclusions In the present study, the combined effect of natural convection and thermal radiation in a porous square cavity with an isothermally heated plate placed horizontally or vertically at its center has been numerically investigated. The following conclusions may be drawn from the above-mentioned study: • The Darcy number, the radiation parameter and the plate length enhance the overall heat transfer rate across the cavity.

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• For low values of the Darcy number, a moderate convective flow is observed, whereas an increase in the Darcy number enhances the strength of the convective motion due to the reduction of the solid structure volume. • In the absence of thermal radiation, the overall heat transfer rate for the vertical plate is higher than that for the horizontal plate. When thermal radiation is considered, an increase in the overall heat transfer rate for the vertical plate compared to that of the horizontal plate gets reduced for low Da and is marginal for high Da.

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