Analysis of natural convection heat transfer and entropy generation inside porous right-angled triangular enclosure

International Journal of Heat and Mass Transfer 65 (2013) 500–513

Contents lists available at SciVerse ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Analysis of natural convection heat transfer and entropy generation inside porous right-angled triangular enclosure Saurabh Bhardwaj, Amaresh Dalal ⇑ Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India

a r t i c l e

i n f o

Article history: Received 23 February 2013 Received in revised form 30 May 2013 Accepted 13 June 2013 Available online 12 July 2013 Keywords: Natural convection Porous media Nusselt number Triangular enclosure Undulation

a b s t r a c t Natural convection in a two-dimensional porous right-angled triangular enclosure having undulation on the left wall is analyzed numerically. The bottom wall is having a sinusoidal temperature and other two walls are maintained at isothermal cold temperature. The stream function-vorticity equations are solved using finite-difference technique. The computations are done on a structured non-orthogonal body fitted mesh. Standard QUICK scheme is used for convective term discretization and diffusive term is discretized using central difference scheme. In this study, the effect of Rayleigh number (Ra = 103  106), Darcy number (Da = 104  102) and undulations on the left wall is investigated on the heat transfer and fluid flow. It is found that for small Ra, the heat transfer is dominated by conduction across the fluid layers but with the increase of Ra, the process began to be dominated by convection and the undulation on the left wall play an important role in increasing heat transfer. It is also found that heat transfer is also a strong function of Darcy number. As Darcy number increases, heat transfer increases in an appreciable manner. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The study of natural convection heat transfer in a porous enclosure has aroused a great interest in many researchers as this phenomenon is very common in several engineering and environmental problems. The main areas of its application are electronic equipment cooling, nuclear reactor systems, geothermal exploitation, solar heating, atmospheric study etc. Varol et al. [1] has investigated the steady state free convection heat transfer in a porous media filled right-angled triangular enclosure. Heat transfer increases with the decreasing aspect ratio. Natural convection in porous triangular enclosure with centered conducting body is analyzed by Varol [2]. It is found that height and width of the body and thermal conductivity ratio play a huge role in increasing heat transfer. Anandalakshmi et al. [3] have analyzed the heat distribution and thermal mixing during steady laminar convection flow inside right-angled triangular enclosure filled with porous media. The heat transfer distribution and thermal mixing enhances due to isothermal heating of walls. Numerical analysis of the two-dimensional laminar natural convection in a pitched roof of triangular cross-section under summer day boundary conditions has been done by Asan and Namli [4]. Considerable amount of heat transfer across the base wall take place near the intersection of the cold horizontal wall and hot ⇑ Corresponding author. Tel.: +91 361 2582677; fax: +91 3612582699. E-mail addresses: [email protected] (S. Bhardwaj), [email protected] (A. Dalal). 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.06.020

inclined wall. Varol et al. [5] have analyzed the natural convection in porous non isothermally heated triangular cavity. Three boundary conditions were used. It was observed that heat transfer enhances when vertical and inclined walls were isothermal while bottom wall was at non-uniform temperature. Basak et al. [6] have analyzed the natural convection flow in an isosceles triangular enclosure with various thermal boundary conditions. It is analyzed that at low Darcy number the heat transfer is primarily due to conduction irrespective of Rayleigh number and Prandtl number. Natural convection heat transfer has been analyzed numerically in a triangular enclosure with flush mounted heater on vertical wall by Varol et al. [7]. It is found that the most important parameter on heat transfer and flow field is the position of heater. The effect of undulations on the heat transfer in square cavity with sinusoidal temperature boundary condition has been analyzed by Dalal and Das [8,9]. Entropy generation in a tilted saturated porous cavity for laminar natural convection has been analyzed by Baytas [10]. Darcy’s law and Boussinesq approximation was used to write the governing equations and solved numerically by using ADI and finite difference method. the effect of inclination angle and Darcy–Rayleigh number has been investigated. It is observed that as Rayleigh number decreases, heat transfer irreversibility begins to dominate the fluid friction irreversibility. Magherbi et al. [11] have investigated the entropy generation due to heat transfer and fluid friction in transient state for laminar natural convection numerically. The effect of Rayleigh number and irreversibility distribution ratio on entropy generation has been studied. The total entropy generation has a

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

501

Nomenclature Be Da g H J k K L Nu Pr Ra T

Bejan number Darcy number gravitational acceleration (ms2 ) height of the enclosure (m) Jacobian thermal conductivity (Wm1 K 1 ) permeability (m2 ) length of the enclosure (m) Nusselt number Prandtl number Rayleigh number dimensionless temperature

To

bulk temperature

u; v U; V

dimensionless velocity components dimensionless contravariant velocity components in n and g direction dimensionless Cartesian coordinates

x; y

ðT h þT c Þ 2

(K)

Greek symbols k wave amplitude a thermal diffusivity (m2 s1 ) b volumetric expansion coefficients (K 1 ) m kinematic viscosity (m2 s1 )

maximum value on the onset of the transient state which increases with the Rayleigh number and irreversibility distribution ratio. Basak et al. [12] have analyzed the entropy generation due to natural convection in right-angled triangular enclosure saturated with porous media. Galerkin finite element method has been used to solve the governing equations. Analysis has been done for Pr = 0.025 and Pr = 1000 in the range of Darcy number 105 to 103 with two different top angles 15° and 45° for various thermal boundary conditions. In this analysis, it is analyzed that for triangular cavities, / = 15° is the optimal top angle for thermal processing due to its high heat rate of heat transfer and total entropy generation. The effect of aspect ratio on entropy generation in a rectangular cavity is numerically analyzed by Ilis et al. [13]. The total entropy generation due to fluid friction and total entropy generation number increase for a cavity with high Rayleigh number (Ra = 105), as aspect ratio increases, reached a maximum value and then decrease. Bejan [14,15] has given a vast information on entropy generation through heat transfer and fluid flow and also entropy generation minimization. In the present study, natural convection inside a two-dimensional right-angled triangular enclosure having three undulations on the left wall filled with porous media has been studied numerically. In most of the engineering application we can see the undulated geometry e.g., regenerative heat exchanger. The other applications of natural convection heat transfer in porous media are also found in solar power collectors, air saturated fibrous insulation material surrounding a heated body, pollutant dispersal in aquifers, storage of nuclear waste in deep geological repositories and heat loss from underground energy storage systems. So, based on past study, the number of undulations for this study has been chosen as 3 because it has been seen that for three undulations, more heat transfer can be achieved. The left and inclined walls are maintained at isothermally cold temperature and bottom wall is non-isothermally heated. Air has been taken as working fluid (Pr = 0.71). Effect of Rayleigh number, Darcy number and undulations on the heat transfer and fluid flow has been analyzed using finite difference method and stream function-vorticity formulation. The results are presented in the form of streamlines, isotherms, local and average Nusselt number for different Ra and

/

U

s w

x n; g

irreversibility distribution ratio general variable dimensionless time streamfunction vorticity dimensionless curvilinear coordinates

Subscripts avg average c cold h hot Abbreviations FFI Fluid Friction Irreversibility HTI Heat Transfer Irreversibility QUICK Quadratic Upstream Interpolation for Convective Kinetics Superscript ⁄ dimensional quantity

Da. The analysis of entropy generation has also been done for all the cases. The aim of this study is to increase the heat transfer in the localized area of interest in the cavity by the use of undulated wall geometry where more heat generation takes place. It is necessary to analyze the natural convection heat transfer in the irregular geometry also because many researcher till now have focussed on regular geometries and very few of them have focussed on undulated triangular geometry. 1.1. Problem specification The problem considered in the present study is a two-dimensional right-angled triangular cavity saturated with porous media having wavy left wall. The left and right inclined walls are considered to be of isothermally cold wall temperature, T c and the bottom horizontal wall is considered to be of spatially varying sinusoidal temperature, T w ðxÞ. The dimensional form of the sinusoidal temperature distribution on the bottom wall is given as [16]

T w ðx Þ ¼ T c þ

   DT  2px 1  cos 2 H

ð1Þ

where DT is the temperature difference between the maximum and minimum temperatures of the bottom walls. The non-dimensional temperature distribution on the bottom wall can be written by using scale parameters [17]. The non-dimensional form of Eq. (1) is as follows:

T w ðxÞ ¼

1 ð1  cosð2pxÞÞ 2

ð2Þ

The expression for sinusoidal wavy left wall is given by

f ðyÞ ¼ ½1  k þ kðcos 2pnyÞ

ð3Þ

where, n is the number of undulations and k is the amplitude [18]. The amplitude for all cases in this study has been taken as 0.05. The Rayleigh number is varied from 103 to 106 and Darcy number is varied from 104 to 102.

502

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

dimensional governing equations in the transformed non-orthogonal boundary-fitted coordinate system are

2. Governing equations and boundary conditions The equations governing fluid flow and heat transfer considering laminar, incompressible and two-dimensional flow are written below in stream function-vorticity formulation. Thermo– physical properties of the fluid in the flow field are assumed to be constant except the density variations causing a body force term in the vorticity equation. Boussinesq approximation is invoked in the fluid properties involving the density variation. Darcy–Forchheimer model is used to simulate the momentum transfer in the porous medium neglecting inertia term. The dimensional form of streamfunction-vorticity equations are given below. Vorticity equation: 2

2

@x @ðu x Þ @ðv x Þ @ x @ x þ þ ¼m þ 2 @x @y @t  @x2 @y 









þ gb





! 

m K

x

1 J

xs þ ððU xÞn þ ðV xÞg Þ ¼ Prr2 x 

1 T s þ ððUTÞn þ ðVTÞg Þ ¼ r2 x J

ð14Þ

r2 w ¼ x

ð15Þ

u¼

1 ðxg wn þ xn wg Þ; J

ð4Þ

r2 U ¼

Energy equation:

ð5Þ

¼ ¼

x ; L

v  ¼ wx



y¼

y ; L

u¼

u L

a

v¼

;

vL a

;

T¼

T   T c ; T h  T c

m2

;

w¼

p¼

p L2

qa2

x

ð8Þ

Energy equation:

u ¼ wy ;

v ¼ wx

i

ð9Þ ð10Þ

For solving the above given governing equations (Eqs. (8)–(11)), the physical plane ðx; yÞ is transformed into the computational plane ðn; gÞ by using boundary-fitted coordinate system. The set of non-

V ¼ xn v  yn u;

J ¼ xn yg  yn xe ta

The heat transfer rate in an enclosure can be obtained from the Nusselt number calculation. Once the temperature distribution is calculated, the local Nusselt number along any surface can be determined from its definition

hL @T ¼ k @n

ð18Þ

@ where @n is the dimensionless derivative along the direction of the outward drawn normal to that surface. Since all the calculations have been performed on the computational domain, this derivative @ should be expressed in terms of the n; g independent variables. @n The local Nusselt numbers in different walls are expressed as Bottom wall Nul ¼  Jp1ffiffic ðcT g  bT n Þ.

Nul ¼  Jp1ffiffia ðcT n  bT g Þ.

The average Nusselt number is the average of local Nusselt number along a wall and is defined as

Nuav g ¼

1 L

ð11Þ

ð12Þ

ð17Þ

2.1. Nusselt number calculation

Left wall

In addition to these governing equations, the temperature boundary conditions take the below given form. Velocities in two directions are zero at all solid boundaries (u = v = 0) and the value of stream function has been taken as unity ðw ¼ 1Þ at all solid boundaries. On the bottom wall, 0 6 x 6 L; T ¼ 12 ð1  cosð2pxÞÞ. On the Left wall, 0 6 y 6 H; T ¼ 0. On the inclined wall, T ¼ 0. The vorticity boundary condition can be calculated by its basic definition,

@ v @u x¼  @x @y

ðyg A þ xg BÞUn þ ðyn A  xn BÞUg

where,

Nu ¼

The non-dimensional governing equations are: Vorticity equation:

wxx þ wyy ¼ x

J

3

U ¼ yg u  xg v ;

a

! @ x @ðuxÞ @ðv xÞ @2x @2x Pr @T þ þ ¼ Pr þ x þ RaPr  @x @y Da @x @t @x2 @y2

1h

ð7Þ

w

@T @ðuTÞ @ðv TÞ @ 2 T @ 2 T þ þ ¼ 2þ 2 @t @x @y @x @y

ð16Þ

ðaUnn þ bUng þ cUgg Þ

a ¼ x2g þ y2g ; b ¼ xn xg þ yn yg ; c ¼ x2n þ y2n A ¼ axnn þ bxng þ cxgg ; B ¼ aynn þ byng þ cygg

x L2 at  K ; t ¼ 2 ; Da ¼ 2 ; Ra a L L gbðT h  T c ÞL3 Pr

1 ðyg wn þ yn wg Þ J

ð6Þ

For non-dimensionalizing the governing equations, the below given scale parameters are considered [17]: x¼

1 J2 þ

!

wx x þ wy y ¼ x u ¼ wy ;

v¼

The Laplacian of the generic scalar U in transformed plane is given by

@T   T c @x

@T  @ðu T  Þ @ðv  T  Þ @2T  @2T  þ þ ¼a þ @x @y @t  @x2 @y2

Pr 1 x þ RaPrðyg T n  yn T g Þ Da J ð13Þ

Z

L

Nul dl

ð19Þ

0

2.2. Entropy generation The irreversible nature of heat transfer and viscosity effects, within the fluid flow and at the boundaries, cause the generation of entropy. The non-dimensional form of local entropy generation rate due to heat transfer ðSHTI Þ and fluid friction ðSFFI Þ for a two dimensional flow is given as follow [12]

SHTI

"   2 # 2 @T @T ¼ þ @x @y "

SFFI ¼ / ðU 2 þ V 2 Þ þ Da 2

ð20Þ  2  2 !  2 !# @U @V @U @V þ þ þ @x @y @y @x ð21Þ

503

Bejan number (Be) is an alternative to irreversibility distribution and is defined as the ratio of heat transfer irreversibility to entropy generation number (Eq. 24). This factor shows the importance of heat transfer irreversibility in the domain. Be  12 shows that irreversibility due to heat transfer dominates in the flow. Fluid friction irreversibility dominates when Be  12 and when Be ¼ 12, the heat transfer and fluid friction entropy generation are equal.

ld Co

Cold Wall (T = 0) H=1

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

l( al W T = 0)

g

Be ¼

L=1 Hot Wall, Tw(x)

SHTI Ns

ð24Þ

3. Numerical procedure

(a)

Numerical methods used in the present study is discussed in this section. 3.1. Grid generation In the present study curvilinear body-fitted grids have been generated by solving Poisson equations for the curvilinear coordinates given as

r2 n ¼ Pðn; gÞ

ð25Þ

r2 g ¼ Q ðn; gÞ

ð26Þ

where P and Q are the grid control functions. The above two equations can be written in the below given form with the physical coordinates as the dependent variables

(b) Fig. 1. (a) Schematic of the flow domain with bounday conditions on the three walls, (b) Mesh representation.

where / is the irreversibility distribution ratio and is defined as:

/¼

 2 a pffiffiffiffi k K DT

lT o

ð22Þ

The value of / in the present study is taken as 102 for all the cases. The local entropy generation number ðNs Þ can be defined as the addition of local entropy generation rate due to heat transfer ðSHTI Þ and fluid friction ðSFFI Þ.

Ns ¼

"   2 # 2 @T @T þ @x @y "

 2  2 !  2 !# @U @V @U @V þ þ / ðU þ V Þ þ Da 2 þ þ @x @y @y @x 2

2

ð23Þ

Table 1 Comparison of present solution with the results of Lauriant and Prasad [19] for average Nusselt Number. Case

Da

Ra

(a)Lauriant and Prasad [19]

(b)Present solution

ab a

axnn þ bxng þ cxgg þ J2 ðPxn þ Qxg Þ ¼ 0

ð27Þ

aynn þ byng þ cygg þ J2 ðPyn þ Qyg Þ ¼ 0

ð28Þ

Here, grid control functions are used to maintain a desired grid density near the left undulated wall. SOR method is used to solve Eqs. (27 and 28). All the derivatives are discritized using second order central difference scheme. In computational plane, inclined wall of the triangular geometry in physical plane is split into two walls (i.e., top wall and right wall) and then numerical schemes are applied considering four different walls. A typical representation of generated grid in physical plane is shown in Fig. 1(b). 3.2. Discretization technique The governing equations (Eqs. (13)–(16)) are discritized on a non-orthogonal body fitted grid using finite difference method. Second-order central difference scheme is used for all the spatial discritization and convective terms are discritized using QUICK scheme [17]. A semi-implicit time integration scheme is adopted in the present study to advance the vorticity and temperature values in time. Linearized form of the vorticity and energy equations can be written as:

J

2

102

104

ð29Þ

 100

J 1

xnþ1  xni;j i;j þ ðU n xnþ1 Þi;j þ ðV n xnþ1 Þi;j Ds ¼ JPrðr2 xnþ1 Þi;j þ RaPrðyg T n  yn T g Þnþ1 i;j

103

1.02

1.02

0

104

1.70

1.73

1.76

105

4.26

4.30

1.07

5  105

7.10

7.31

2.95

105

1.06

1.08

1.88

106

2.84

2.95

3.87

T nþ1  T ni;j i;j þ ðU n T nþ1 Þi;j þ ðV n T nþ1 Þi;j ¼ Jðr2 T nþ1 Þi;j Ds

ð30Þ

For solving the discretized form of Eqs. (29 and 30), SOR technique is used for taking the advantage of its faster convergence due to diagonal dominance. Stabilized Bi-Conjugate Gradient Method (BiCGSTAB) is used to solve the system of linear equations arising from the discretization of the Poisson equation for the streamfunction.

504

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513 Table 3 Comparison of average Nusselt Number on heated bottom wall for different grid size considering present problem.

Table 2 Comparison of present solution with the results of Kaluri et al. [20] for average  Nusselt Number on the left wall, Nul . Case

Pr

1

0.015 7.2

(a)Kaluri et al. [20]

(b)Present solution

ab a

 100

103

5.421

5.243

3.26

104

5.931

5.534

6.69

103

5.712

5.251

8.05

104

6.031

5.635

6.53

105

7.882

7.602

3.55

Case

Grid size

Average Nusselt number,Nuav g

1 2 3 4

41  41 81  81 121  121 161  161

1.909720 1.910252 1.918503 1.917341

enclosure with non-porous media. In this test problem, the left wall of the triangle is taken as isothermally hot and right inclined wall is taken as isothermally cold keeping bottom wall adiabatic. In this problem, computation are done on a 57  57 non-orthogonal body-fitted mesh for all cases. The result of this test is compared with the solution of Kaluri et al. [20]. The aspect ratio (length to height ratio) is taken as unity (A.R. = 1) and angle h = 45°. In this model, u = v = 0 for all solid boundaries. For the comparison of the result, average Nusselt number on the left wall has been taken and compared with the solution of Kaluri et al. [20]. In Table 2, average Nusselt number on the left wall for two different Prandtl number, Pr = 0.015 and Pr = 7.2 and for different Rayleigh numbers has shown.

3.3. Code validation The present code is validated for the natural convection laminar flow by comparing the results with the differentially heated square cavity with heated left wall and isothermally cold right wall keeping top and bottom walls adiabatic. Solutions are calculated on an orthogonal mesh with 41  41 mesh for all cases. Average Nusselt number for different Rayleigh numbers and for different Darcy numbers was compared and given in Table 1 with the solution of Lauriat and Prasad [19]. The present code is also tested for triangular cavity problem which models the right-angled triangular

SHTI

SFFI 0.

02

0.1

2

1.3

9 0.

0.1

1.6

0. 6

0.05

1.9

0. 0

0. 0 2

0.2

(a)

1.3

0.9

0. 0

2

1.9

0.6

0.05

SFFI 40

SHTI

20

5

50

10

200

10

(b)

50

10

5

20

10

2

Ra

Fig. 2. Local entropy generation due to heat transfer (on the left) and fluid friction (on the right) for a square cavity with / ¼ 104 (a) Ra ¼ 103 , (b) Ra ¼ 105 .

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

1. 0

505

(a)

05

15 1. 0

0.0

1.05

0. 1

0. 2

0.9 95 0.985

3 1.0 1.04

5

0. 3 0. 4

0.6 0.85

1

1. 0

5

(b)

1.15

0.0

5

1.3 1.4

0.1

1.5

0. 2

0.9

1.6

0.3 0. 8

0.5

0.7 0.9

0.95

1.5

(c)

2.5

0.05 0.1 0.2

4.5 3 0.

6

5 7. -0.5

0. 5

0.5

0.6 0.7

0.05

0 -2

(d) 4 10 16 20

0.1

0.2

0.3 0.4 5 0.

-2 -6

0.6 0

Fig. 3. Streamlines (on the left) and Isotherms (on the right) for different Rayleigh numbers at Da ¼ 102 (a) Ra ¼ 103 , (b) Ra ¼ 104 , (c) Ra ¼ 105 , (d) Ra ¼ 106 .

For the entropy generation point of view, results for entropy generation due to heat transfer and entropy generation due to fluid flow is compared with the results of Ilis et al. [13] in which a differentially heated cavity with hot left wall and

cold right wall keeping adiabatic top and bottom wall is considered and the results are shown in Fig. 2. The results for all cases are in very good agreement with the compared solutions.

506

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

1. 2 1.4

0.05 0.1

1. 8

(a)

2. 4

0.2

0.8

2.8

0. 3

4 0.

3. 2

0

0. 5 0. 7 0.8 0.9

-0.4

2

0.1

0. 2

-1 4

(b)

0

0. 3

7

1

0. 4

0. 5

-5

10

0.6

-3

0.7

0

0.05

0 -2

0.1

(c) 4 10 16 20

0. 2

0.3 0. 4 5 0.

-2 -6

0.6 0

Fig. 4. Streamlines (on the left) and Isotherms (on the right) for different Darcy numbers at Ra ¼ 106 (a) Da ¼ 104 , (b) Da ¼ 103 , (c) Da ¼ 102 .

3.4. Grid independence study of the problem considered Grid independent test has been carried out for the natural convection flow in a porous right-angled triangular enclosure with sinusoidally heated bottom wall and isothermally cold left and inclined walls with 41  41, 81  81, 121  121 and 161  161 grid size. The parameters taken for this study are Ra = 104, Da = 102 and Pr = 0.71. Average Nusselt number on the heated wall is calculated and compared results of this study are tabulated in Table 3. For computations in the present study 121  121 grid size has been chosen for all the cases. 4. Results and discussion A parametric study has been carried out to determine the effect of Rayleigh number and Darcy number on the flow field and the effect of undulation on heat transfer. The results are presented for Rayleigh number 103 to 106, Darcy number 104 to 102 and the

undulation amplitude is taken as k = 0.05. Entropy generation due to heat transfer and fluid friction is also analyzed in this study. 4.1. Streamtraces and isotherms Fig. 3 shows the streamline (on the left) and isotherms (on the right) in a porous right-angled triangular enclosure with bottom wall sinusoidally heated. The fluid near the bottom portion of the enclosure is hotter than the fluid near to the inclined wall and left wall and hence the fluid near to the bottom wall have lower density than those near to the left wall and inclined wall. Thus, double circulation cells were formed in different rotating directions, left circulation rotates in anticlockwise direction and right circulation rotates in clockwise direction. Streamlines and isotherms for Darcy number 102 and Rayleigh number 103 are shown in Fig. 3(a). The fluid circulation in this case is seen to be very weak as observed from streamline contours and the strength of the vortex in this case is found to be 1.05. Here the cell on the left side is dominating

507

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

(a)

14 12

8

(a)

Ra = 103 4 Ra = 10 Ra = 105 6 Ra = 10

Da = 10-4 -3 Da = 10 Da = 10-2 6

10 8

Nul

Nul

4

6

2

4 2

0

0 -2

(b)

0

0.2

0.4

X

0.6

0.8

-2

1

0

0.2

0.4

0.6

0.8

1

X

0 0

(b) -1

-0.5

-1

-3

Nul

Nul

-2

-1.5

-4 3

Ra = 10 4 Ra = 10 Ra = 105 6 Ra = 10

-5

-6

0

0.2

0.4

Y

0.6

0.8

-2

1

Fig. 5. Variation of local Nusselt number at Da ¼ 102 and Ra ¼ 103  106 (a) on heated bottom wall, (b) on left wavy wall.

on the right side cell and also because of the presence of undulations on the left wall, the left cell is somewhat squeezed. It can be seen from isotherms that they are distributed almost smooth and parallel to each other and this illustrates that the heat transfer is purely due to conduction and viscous forces are dominating. As Ra increases to 104, the intensity of the recirculation patterns increases and the strength of the vortex increases to 1.6 which can be shown in Fig. 3(b). In this case, viscous forces are comparable to buoyancy forces and convection mode of heat transfer is coming into picture. But beyond Rayleigh number Ra = 104, transition from conduction to convection mode of heat transfer takes place. Figs. 3(c)–(d) show the streamlines and isotherms for Ra = 105 and Ra = 106. As Ra increases to 105, the buoyancy forces start dominating on viscous forces and the stream contours on right side is squeezed by the left side contours and the strength of the vortex is reached up to 7.5. Now the isotherms are no more smooth and a feathered structure is observed and because of the inclined wall, isotherms are observed to have skewness towards left wavy wall. Convection mode of heat transfer is completely dominated on conduction mode of heat transfer when Ra reaches to 106. In this case

Da = 10-4 -3 Da = 10 Da = 10-2 -2.5

0

0.2

0.4

0.6

0.8

1

Y Fig. 6. Variation of local Nusselt number at Ra ¼ 105 and Da ¼ 104  102 (a) on heated bottom wall, (b) on left wavy wall.

viscous forces are no more in the flow and only buoyancy forces are leading the flow. As shown in Fig. 3(d), the streamline contours are distributed almost throughout the cavity and also strength of the vortex increases up to 20 and temperature gradient near bottom wall in this case are more due to proper mixing of fluid. The effect of Darcy number on heat transfer and fluid flow for a particular Rayleigh number is shown in Fig. 4. At a fixed Rayleigh number, Ra = 106, as Darcy number increases from 104 to 102, a huge variation in streamlines and isotherms can be seen. This is because of the fact that at low Darcy number, the porosity is low and it provides more resistance to the flow. But as Darcy number increases, this resistance decreases and fluid easily flow through the pores and the high buoyancy forces are applied on the fluid flow. This can be seen from the increased value of strength of the vortex and high temperature gradients as shown in Figs. 4(a)–(c). In first case when Ra = 106, Da = 104, the strength of the vortex is 3.2 (Fig. 4(a)) but as Darcy number increases to 102 the strength of the vortex is also increases and reached up to 20 (Fig. 4(c)).

508

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

(a)

7

(a) 7

-2

Da = 10 -3 Da = 10 -4 Da = 10

6

6 5

Nuavg

Nuavg

5

4

4

3

3

2

2

1

undulation no-undulation

10

3

4

5

10

1

6

10

10

103

104

Ra (b)

(b)

-0.5

0 -0.5

-1

-1

-1.5

Nuavg

Nuavg

106

Ra

0

-2 -2.5

-1.5 -2 -2.5

Da = 10-2 Da = 10-3 Da = 10-4

-3 -3.5

105

3

10

-3

4

10

10

5

6

10

Ra

-3.5

undulation no-undulation

103

104

105

106

Ra

Fig. 7. Average Nusselt number (a) on heated bottom wall, (b) on left wavy wall. Fig. 8. Comparison of average Nusselt number for present results with noundulation case for Da ¼ 102 (a) on heated bottom wall, (b) on left wavy wall.

4.2. Nusselt number 4.2.1. Local Nusselt number The variation of the local Nusselt number along the heated bottom wall for different Rayleigh numbers and different Darcy numbers are shown in Figs. 5(a) and 6(a). As it is known that higher Rayleigh number means larger heat addition to the system i.e. intensification of fluid convection increases with increasing of Rayleigh number. The value of local Nusselt number increases with increasing Rayleigh number due to adding of large amount of heat to the system at a particular Darcy number as shown in Fig. 5(a). At small Rayleigh numbers, heat transfer is possible due to conduction because viscous forces are dominating and single peak is observed up to Ra = 104. Beyond Ra = 105, the viscous forces become weak in the flow and buoyancy forces coming into picture. In this case, value of Nusselt number increases in an appreciable manner and two peaks appear which is because of the sinusoidal temperature heating of bottom wall and variation of local Nusselt number is not symmetrical due to presence of undulation on left wall. At Ra = 106, heat transfer is abruptly increased because of the complete dominance of convection heat transfer (i.e. buoyancy

forces). Maximum Nusselt number in this case is equal to 13.244 and occured in between 0.2 to 0.4 from the left corner. Fig. 6(a) shows the variation of local Nusselt number with distance for Ra ¼ 105 and different Darcy numbers. Because of the non-isothermal heating at bottom wall, sinusoidal type of local heat transfer rate produces. Nusselt number increases from a negative value at the left corner because of heat loss towards the center with its maximum value at around X ¼ 0:4 and then decreases. The same pattern is repeated in the next half of the wall with low peak value as compared to previous peak. This behaviour of heat transfer rate is due to the higher value of stream function (i.e., high flow rate). Fig. 5(b) shows the variation of local Nusselt number with distance along the left wall for Darcy number, Da ¼ 102 and different Rayleigh numbers. Here ()ve sign with Nusselt number values indicates the heat loss from the fluid to the wall. The higher the Rayleigh number, the larger the amount of heat rejected from the fluid to the left wall near the bottom corner of the wall. This pattern of Nusselt number is due to the undulations present on the left wall. The examination of isotherms and local Nusselt

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

0 -1

Numax

-2 -3 -4 -5 -6

undulation no-undulation 103

104

105

106

Ra Fig. 9. Comparison of maximum Nusselt number on left wavy wall with noundulation case for Da ¼ 102 .

number distribution on the left wall reveals that the thermal boundary layer thickness on the side of the undulated wall increases and decreases just before a crest or just after a trough. These undulations introduce boundary layer interuptions and therefore improve the heat transfer. The highest peak of the local Nusselt number occurs around Y ¼ 0:3 i.e. for the middle undulation. Here heat rejection increases with increase of Ra as shown in Fig. 5(b). For low Rayeigh numbers, the values of Nusselt number are almost same because of the conduction heat transfer but for high Ra values, Nusselt number increases in an appreciable manner. The effect of Darcy number at a particular Rayleigh number, Ra ¼ 105 are shown in Fig. 6(b). In that case also with the increase of Darcy number, the flowability increases and because of that more heat is lost through left wall. For low Darcy number, it is clear from isotherms that gradients are less near the left wall but as Darcy number increases to 102 , the temperature gradients are more because of that Nusselt number values are more in negative direction i.e., heat loss is more. 4.2.2. Average Nusselt number The overall effects upon the heat transfer rates are presented in Fig. 7 where the distribution of the average Nusselt number vs the logarithmic Rayleigh number for bottom and left walls are shown. The average Nusselt number is calculated using Eq. 17 where the integral is evaluated using Simpson’s 1=3 rule. In Fig. 7(a), average Nusselt number on the bottom wall for Da ¼ 104 remains almost constant during entire Rayleigh number regime and also for Ra ¼ 103 Ra ¼ 104 , average Nusselt number is almost same. This is due to dominance of viscous forces in conduction heat transfer mode. Beyond Ra ¼ 104 , transition from conduction to convection takes place. In this region, as Rayleigh number and Darcy number increases, the conduction dominant heat transfer regime is narrowed down and convection starts dominating. It is observed that larger Rayleigh number region along with lower Darcy number produces lower average Nusselt numbers. Fig. 7(b) shows the variation of average Nusselt number for left wall. In this case also average Nusselt is almost same for low Rayleigh number, it can be explained by examining the isotherms for Ra ¼ 103 and Ra ¼ 104 . The gradient of temperature near left wall is not high because of which lesser heat is lost fromthe fluid to the left wall and

509

Nusselt number values are almost constant. Beyond Ra ¼ 104 , because of the transition, values changes rapidly and a huge difference in the values of Nusselt number is observed because of the convection heat transfer which can be easily observed in Figs. 7(a)–(b). But for Ra ¼ 105 and Ra ¼ 106 case, isotherms are more distorted near left wall and more heat is lost, the effect of which can be seen from the average Nusselt number values in Fig. 7(b). Also as Darcy number increases, pores resistance is decreases and fluid flow occurs easily and throughtout the domain. The maximum value of average Nusselt is found to be 6:6 on the heated bottom wall for Ra ¼ 106 and Da ¼ 102 . Fig. 8 shows the comparison of average Nusselt number for the present problem with the problem having no-undulation on the left wall for Da ¼ 102 and for different Rayleigh numbers. This comparison shows the effect of undulation on the heat transfer. Average Nusselt number on the heated bottom wall has been compared with no-undulation case showing in Fig. 8(a). Here it can be seen that higher values of average Nusselt number are found in all cases. Undulation on the left wall creates proper vortex in the flow and because of that proper mixing of fluid takes place which increases the heat transfer rate. The same pattern is seen for the left wall as shown in Fig. 8(b). For low values of Ra, the difference in the average Nusselt number values between two cases (i.e., undulation and no-undulation) is not large but for higher Rayleigh number values, Ra ¼ 106 , the difference in average Nusselt number values is too large which is clearly seen in Fig. 8(b). Th effect of undulation on the heat transfer can also be shown by comparing the maximum Nusselt number on left undulated wall with that of no-undulation case which is shown in Fig. 9. Here, if we compare the magnitude of the maximum Nusselt number, a huge difference can be seen in the maximum Nusselt number value for Da ¼ 102 and Ra ¼ 106 . At low Ra, the difference is not much but as Ra increases, this difference also increases which shows that because of undulations on the left wall, heat transfer enhances. 4.3. Entropy generation The entropy generation results are produced here for Darcy number 104 and 102 and Rayleigh number 103 to 106 considering irreversibility distribution ratio as 102 . Figs. 10 and 11 show the plots for local entropy generation due to heat transfer and local entropy generation due to fluid friction. For Da ¼ 104 and Ra ¼ 103 , the local entropy generation due to heat transfer is dominating in the cavity because of temperature gradients as compared to entropy generation due to fluid flow as shown in Fig. 10(a). The maximum value of entropy generation in this case due to heat transfer is, SHTImax ¼ 21:97 and due to fluid flow is, SFFImax ¼ 3:22  106 which is very low as compared to SHTI . The average Bejan number, Be  12 which also shows that the total entropy generation due to heat transfer is dominating throughout the cavity. As Rayleigh number increases, for Da ¼ 104 , the heat transfer irreversibilities increases in the domain and fluid flow irreversibilities also increases as stream function value increases (Figs. 10(b)–(c)). When Ra increases to 106 , the irreversibilities cover the more part of the domain. These irreversibilities are more near to heated boundary and at the corners where bottom wall is in contact with the left wall and inclined wall as shown in Fig. 10(d). It is because of more temperature gradients and high fluid flow. This we can judge by the increased values of maximum entropy generation due to heat transfer, SHTImax ¼ 36:86 and due to fluid flow is, SFFImax ¼ 5:34. But the value of average Bejan number is still greater than 12. These irreversibilities also increase as Darcy number increases because in that case pore resistance is decreased and flow strength increases which can be seen in Fig. 11. At Da ¼ 102 and Ra ¼ 103 , the maximum value of entropy generation

510

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

SFFI

SHTI

36 4548 0.00

3

66 0.8

826

2E -0 7

2E-07

0

(a)

98 5 . 11

4E-07

6E-07

4

6 10

4

8

16

SHTI

4E-0

2E-07

3E-06

7

SFFI

(b) 79

2

6

6

05

0.001 0.0015

5 0.004

4

12 18

0.0 01 5

0.00

02 0.0

0.39 310 2

4659 0.98

0.0005

03 16 0.0

SHTI

SFFI

738

215 0.00

0. 0

1 306

(c)

55

0. 4

15 8

0.00173403

49

0.00456625

2

0.

6

10 14

6 0.03

2

SHTI

01 2

0.00456625

SFFI

0. 0

351

225

22

(d)

3

0.5

1

34

0.5

0. 6

2

10

1.30659

1

2

2.5

4

0.5

8

6

18

4

0.634223

8

4

2

1

1.5

1

Fig. 10. Local entropy generation due to heat transferðSHTI Þ and fluid flowðSFFI Þ for / ¼ 102 at Da ¼ 104 (a) Ra ¼ 103 , (b) Ra ¼ 104 , (c) Ra ¼ 105 , (d) Ra ¼ 106 .

511

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

SFFI

SHTI

0.01

164

58

01

4 003 0. 2 0. 0

0.001

0.395026

2 6

(a)

12

6

16

0.004

0.002

0.001

SFFI

SHTI

58 61

3

4 0.0

87 87

0. 2

0. 4

0 0.8

2 4

6

8

10

14

0.2

(b)

6

0. 2 0.4

2

SHTI

SFFI

0. 1

4 066

20

1.32441 4 24 1.3

40

15 40

25

5

1

5

10

15

20

10 5

(c)

SHTI

20 81.1176

40

20

SFFI

0.3 7

23

95

2.64873 500

. 25

10

50

(d)

07

0.788232

16

2.6 487 3

2.64 873

0 50

57 27 0.0

805.029

Fig. 11. Local entropy generation due to heat transferðSHTI Þ and fluid flowðSFFI Þ for / ¼ 102 at Da ¼ 102 (a) Ra ¼ 103 , (b) Ra ¼ 104 , (c) Ra ¼ 105 , (d) Ra ¼ 106 .

due to heat transfer is found to be SHTImax ¼ 20:64 and due to fluid flow is, SFFImax ¼ 0:0172. If we compare this case with the previous case at same Ra and Da ¼ 104 , it is clearly observed that the heat transfer irreversibilities reduces and fluid flow irreversibilities increase. A huge difference in the fluid flow irreversibility is ob-

served. At same Darcy number and Ra ¼ 106 , the maximum value of fluid flow irreversibilty is greatly increased (SFFImax ¼ 7696:78). At small values of Darcy number (i.e. Da ¼ 104 ), as Rayleigh number increases, Beav g decreases but it will remain greater than 1 for all Ra values. As Darcy number increases to Da ¼ 102 , for 2

512

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513

Table 4 Comparison of average Nusselt number and average Bejan number for Ra ¼ 106 and different Daracy number. Case

Da

Nuav g

Beav g

1

104

2.8796

0.79182

2

103

5.7350

0.18493

3

102

6.4395

0.02878

Table 5 Comparison of irreversibilities, SHTI and SFFI for undulation case with no-undulation case for Ra ¼ 106 and different Daracy number. Case

Da

undulation

no-undulation

SHTI

SFFI

SHTI

SFFI

1

104

36.86

5.34

36.73

5.28

2

103

133.57

561.06

132.53

257.23

3

102

171.87

7696.78

170.77

4318.1

smaller Rayleigh numbers (i.e., Ra ¼ 103 ; Ra ¼ 104 ), the value of average Bejan number, Be  12, but for higher Ra (i.e., Ra ¼ 105 ; Ra ¼ 106 ), the value of average Bejan number decreases and for these cases, Be  12. It reveals that at high Darcy number and at high Rayleigh numbers the total entropy generation due to fluid friction is dominating in the cavity. This is because of the fact that at high Darcy and at high Rayleigh number, the fluid friction near the boundary increases and the value of maximum stream function also increases which intern increases the total entropy generation due to fluid friction in the cavity. Table 4 shows the comparison of average Nusselt number with the average Bejan number for Ra ¼ 106 and for different Darcy number.This comparison shows how much available energy is destructed with average heat transfer. At a particular Ra, when Da increases, average Bejan number decreases which shows that fluid friction irreversibility is dominating the flow and hence average Nusselt number also increases. This is because of the fact that as Da increases, hydraulic resistance decreases and hence fluidity improves. For the better understanding of undulation effect on heat transfer and fluid flow, a comparison of SHTI and SFFI for undulation case with no-undulation case has been done and presented in Table 5. We can see from this comparison that as Darcy number increases at a particular Rayleigh number, Ra ¼ 106 , fluid friction irreversibility, SFFI in undulation case increases with a much heigher value (SFFI ¼ 7696:78) than no-undulation case (SFFI ¼ 4318:1). Undulations present on the left wall help in distorting the boundary layers and a better mixing of fluid occurs in the cavity. Heat transfer irreversibility does not affect much with the undulations present on the left wall. It is found to be almost equal in undulation case as well as no-undulation case.

5. Conclusions A numerical study had been done to analyze the effect of Rayleigh number and Darcy number on heat transfer and fluid flow in a right-angled triangular cavity having three undulations on the left wall. The bottom wall was considered to be at sinusoidal temperature boundary condition and other two walls are kept at isothermally cold. Rayleigh number was varying from 103 to 106 and Darcy number was taken in the range of 104 to 102 . The streamlines and isotherm contours were shown in the results and the local and average Nusselt numbers were presented here for left and bottom walls. The below given conclusions can be made on the basis of this study:

 For small Ra, the heat transfer was dominated by conduction across the fluid layers but with the increase of Ra, the process began to be dominated by convection. The transition from conduction mode of heat transfer to convection mode of heat transfer occurs beyond Ra ¼ 104 .  Heat transfer is a strong function of Darcy number. As Darcy number increases, heat transfer also increases in terms of increased values of Nusselt number. At Da ¼ 102 and Ra ¼ 106 , the strength of the flow and heat transfer is maximum in this parametric study.  The maximum Nusselt number on heated bottom wall is found to be equal to 13.244 for Da ¼ 102 and Ra ¼ 106 case.  For the low values of Rayleigh numbers the average Nusselt number for left and bottom walls are found to be almost same and as Rayleigh number increases average Nusselt increases abruptly and also as Darcy number increases, average Nusselt number increases to a higher value. The maximum value of average Nusselt number is found to be equal to 6.5 for Da ¼ 102 and Ra ¼ 106 .  Undulations on the left wall plays an important role in increasing heat transfer rate in the cavity. An increase of 51% in the maximum Nusselt number for left wall was observed for Da ¼ 102 and Ra ¼ 106 case when compared with no-undulation case. It is also observed that fluid friction irreversibility for higher Rayleigh number and for higher Darcy number is greatly increased in undulated wall case compared to no-undulation case. Undulations on left wall do not affect heat transfer irreversibility much.  At low Darcy and low Rayleigh numbers, the total entropy generation due to heat transfer is dominating but at high Rayleigh numbers, the irreversibility due to heat transfer and fluid friction are comparable. As Darcy number increases for a fixed Rayleigh number, fluid friction irreversibility greatly increases.  As Darcy number increases to Da ¼ 102 , the total entropy generation due to fluid friction is dominating in the cavity for high Rayleigh number. This can be seen by the low values of Bejan number. The average Bejan numbers for Ra ¼ 105 and Ra ¼ 106 are found to be equal to 0.2451 and 0.2487 respectively. It shows that Be  0:5 and hence fluid friction irreversibility dominates.

Acknowledgement The helpful comments by the reviewers are gratefully acknowledged by the authors. References [1] Y. Varol, H.F. Oztop, A. Varol, Free convection in porous media filled right-angle triangular enclosures, Int. J. Heat Mass Transfer 33 (2006) 1190–1197. [2] Y. Varol, Natural convection in porous triangular enclosure with a centered conducting body, Int. Commun. Heat Mass Transfer 38 (2011) 368–376. [3] R. Anandalakshmi, R. Kaluri, T. Basak, Heatline based thermal management for natural convection within right-angled porous triangular enclosures with various thermal conditions of walls, Energy 36 (2011) 4879–4896. [4] H. Asan, L. Namli, Laminar natural convection in a pitched roof of triangular cross-section: summer day boundary conditions, Energy Buildings 33 (2000) 69–73. [5] Y. Varol, H. Oztop, M. Mobedi, I. Pop, Visualization of natural convection heat transport using heatline method in porous non-isothermally heated triangular cavity, Int. J. Heat Mass Transfer 51 (2008) 5040–5051. [6] T. Basak, S. Roy, C. Thirumalesha, Finite element analysis of natural convection in a triangular enclosure: effects of various thermal boundary conditions, Chem. Eng. Sci. 62 (2007) 2623–2640. [7] Y. Varol, A. Koca, H.F. Oztop, Natural convection in a triangle enclosure with flush mounted heater on the wall, Int. J. Heat Mass Transfer 33 (2006) 951– 958. [8] A. Dalal, M.K. Das, Laminar natural convection in an inclined complicated cavity with spatially variable wall temperature, Int. J. Heat Mass Transfer 48 (2005) 3833–3854.

S. Bhardwaj, A. Dalal / International Journal of Heat and Mass Transfer 65 (2013) 500–513 [9] A. Dalal, M.K. Das, Numerical study of laminar natural convection in a complicated cavity heated from top with sinusoidal temperature and cooled from other sides, Comput. Fluids 36 (2007) 680–700. [10] A.C. Baytas, Entropy generation for natural convection in an inclined porous cavity, Int. J. Heat Mass Transfer 43 (2000) 2089–2099. [11] M. Magherbi, H. Abbassi, A.B. Brahim, Entropy generation at the onset of natural convection, Int. J. Heat Mass Transfer 46 (2003) 3441–3450. [12] T. Basak, P. Gunda, R. Anandalakshmi, Analysis of entropy generation during natural convection in porous right-angled triangular cavities with various thermal boundary conditions, Int. J. Heat Mass Transfer 55 (2012) 4521–4535. [13] G.G. Ilis, M. Mobedi, B. Sunden, Effect of aspect ratio on entropy generation in a rectangular cavity with differentially heated vertical walls, Int. Commun. Heat Mass Transfer 35 (2008) 696–703. [14] A. Bejan, Entropy Generation Through Heat and Fluid Flow, Wiley, New York Inc, 1982.

513

[15] A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, 1996. [16] I.E. Sarris, I. Lekakis, N.S. Vlachos, Natural convection in a 2d enclosure with sinusoidal upper wall temperature, Numer. Heat Transfer Part A 42 (2002) 513–530. [17] A. De, A. Dalal, A numerical study of natural convection around a square, horizontal, heated cylinder placed in an enclosure, Int. J. Heat Mass Transfer 49 (2006) 4608–4623. [18] L. Adjlout, O. Imine, A. Azzi, M. Belkadi, Laminar natural convection in an inclined cavity with a wavy wall, Int. J. Heat Mass Transfer 45 (2002) 2141– 2152. [19] G. Lauriat, V. Prasad, Non-darcian effects on natural convection in a vertical porous enclosure, Int. J. Heat Mass Transfer 47 (1989) 2135–2148. [20] R. Kaluri, R. Anandalakshmi, T. Basak, Bejan’s heatline analysis of natural convection in right-angled triangular enclosures: effects of aspect-ratio and thermal boundary conditions, Int. J. Thermal Sci. 49 (2010) 1576–1592.