Entropy Generation During Natural Convection in a ... - CiteSeerX

Numerical Heat Transfer, Part A, 62: 336–364, 2012 Copyright # Taylor & Francis Group, LLC ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/10407782.2012.691059

ENTROPY GENERATION DURING NATURAL CONVECTION IN A POROUS CAVITY: EFFECT OF THERMAL BOUNDARY CONDITIONS Tanmay Basak, Ram Satish Kaluri, and A. R. Balakrishnan Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai, India Entropy generation plays a significant role in the overall efficiency of a given system, and a judicious choice of optimal boundary conditions can be made based on a knowledge of entropy generation. Five different boundary conditions are considered and their effect of the permeability of the porous medium, heat transfer regime (conduction and convection) on entropy generation due to heat transfer, and fluid friction irreversibilities are investigated in detail for molten metals (Pr ¼ 0.026) and aqueous solutions (Pr ¼ 10), with Darcy numbers (Da) between 10
5–10
3 and at a representative high Rayleigh number, Ra ¼ 5  105. It is observed that the entropy generation rates are reduced in sinusoidal heating (case 2) when compared to that for uniform heating (case 1), with a penalty on thermal mixing. Finally, the analysis of total entropy generation due to variation in Da and thermal mixing and temperature uniformity indicates that, there exists an intermediate Da for optimal values of entropy generation, thermal mixing, and temperature uniformity.

1. INTRODUCTION Ever since the pioneering work of Darcy [1] on flow through porous beds which resulted in the identification of permeability as the property of porous media, there has been tremendous interest in the study of flow through porous media which occur widely in nature and industry. The existence of convection currents in porous media was first pointed out by Horton and Rogers [2], who studied NaCl distribution in subterranean sand-layers. A pioneering research in convection in porous media was carried out by Wooding [3–7]. Convection in porous media is now an established concept, and there have been significant developments in modeling of porous media which include various nonlinear effects due to acceleration, inertia, drag, non-local thermal equilibrium between solid and fluid phase, temperature dependent viscosity effects, anisotropy, and others. An extensive review of literature on porous media may be found in earlier works [8]. Natural convection in fluid-saturated porous materials is encountered in many applications such as molten metals infiltration in porous media [9–12], drying and transport of gases in porous media [13–15], enhanced oil recovery by hot-water Received 27 January 2012; accepted 19 April 2012. Address correspondence to A. R. Balakrishnan, Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India. E-mail: [email protected]

336

ENTROPY GENERATION IN A POROUS CAVITY

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NOMENCLATURE Be Da g k L N p P Pr R Ra RMSD S T Th Tc To u U v V

Bejan number Darcy number acceleration due to gravity, m s
2 thermal conductivity, W m
1 K
1 side of the square cavity, m total number of nodes pressure, Pa dimensionless pressure Prandtl number residual of weak form Rayleigh number root-mean square deviation dimensionless entropy generation temperature of the fluid, K temperature of discrete heat sources, K temperature of cold portions of the cavity, K bulk temperature, K x component of velocity x component of dimensionless velocity y component of velocity y component of dimensionless velocity

b V x X y Y a b c C h, H m n q U w

dimensionless velocity distance along x coordinate dimensionless distance along x coordinate distance along y coordinate dimensionless distance along y coordinate thermal diffusivity, m2 s
1 volume expansion coefficient, K
1 penalty parameter boundary dimensionless temperature dynamic viscosity, kg m
1 s
1 kinematic viscosity, m2 s
1 density, kg m
3 basis functions streamfunction

Subscripts i residual number k node number cup cup-mixing avg, av spatial average

flooding in porous beds [16], combustion of heavy oils in porous reservoirs [17, 18], etc., where the thermal boundary conditions play a significant role in the overall process. Effects of thermal boundary conditions, such as sinusoidal heating and discrete heating, are reported in the literature [19–21]. Entropy generation minimization (EGM) is the emerging thermodynamic approach for the optimization of engineering systems. The main idea behind thermodynamic optimization is to relate degree of thermodynamic non-ideality of the design to the physical characteristics of the system, such as finite dimensions, shapes, materials, finite speeds, and finite-time of intervals of operation and vary one or more physical characteristics to optimize the design characterized by minimum entropy generation subject to finite-size and finite-constraints [22, 23]. The EGM approach has been applied to optimize natural convection systems [24, 25], pin-fin heat sinks [26], fuel cells [27], heat exchangers [28, 29], environmental control of aircraft [30, 31], combustion in porous media [32], etc. Analysis of EGM for natural convection in porous square cavities subject to various types of thermal boundary conditions may be helpful in identifying the proper boundary conditions for specific applications and, thus, the thermodynamic efficiency of the system may be enhanced. Further, with a knowledge of irreversibilities present in the system, the porous medium (such as metal foams for electronic cooling applications) can be designed with desirable characteristics. In a thermal convection system, the irreversibilities are due to heat transfer and fluid friction. Perusal of the literature reveals that only a few studies have been reported on entropy generation during natural convection in enclosures filled with porous media. Baytas [33] studied entropy generation in a differentially heated inclined square porous

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cavity and presented the effect of a Darcy modified Rayleigh number and Bejan number on entropy generation for various degrees of inclination. Mahmud et al. [34] also studied entropy generation characteristics in wavy enclosures filled with microstructures which are modeled as porous medium. They reported that high irreversibility is exhibited in cavities with lower aspect ratio in the phase-plus, while higher aspect ratio in the phase-minus mode exhibit low irreversibility. Further, Varol et al. [35] have numerically studied entropy generation due to natural convection in nonuniformly heated porous isosceles triangular enclosure positioned at various inclinations, and they concluded that the highest entropy generation due to heat transfer and fluid friction is observed at / ¼ 90 . Recently, Zahmatkesh [36] reported the entropy generation in a porous square cavity; however, only two boundary conditions (uniform and sinusoidal heating conditions) were considered. A comprehensive analysis on the effect of thermal boundary conditions on entropy generation due to natural convection in fluid-saturated porous media has yet to appear in the literature and forms the prime objective of the current study. In this study, five different thermal boundary conditions are considered based on the heating of various walls of the cavity. Entropy maps due to heat transfer irreversibility (Sh) and fluid friction irreversibility (Sw) are obtained for material processing applications with representative fluids such as molten metals (Pr ¼ 0.026) and aqueous solutions (Pr ¼ 10) within Darcy numbers (Da) of 10
5– 10
3 and at Rayleigh number Ra ¼ 5  105. Further, the relative dominance of Sh and Sw is analyzed in terms of average Bejan number and thermal mixing for temperature uniformity via cup-mixing temperature and root-mean square deviation. 2. MATHEMATICAL FORMULATION AND SIMULATION 2.1. Velocity and Temperature Distributions The physical domain of the square cavity is shown in Figure 1. The cavity is subjected to five different thermal boundary conditions, mentioned below: . Case 1. The bottom wall is maintained hot isothermal, and the side walls are maintained cold isothermal. . Case 2. The bottom wall is heated sinusoidally, and the side walls are maintained cold isothermal. . Case 3. The bottom wall is maintained hot isothermal, and the side walls are cooled linearly. . Case 4. The bottom wall is heated sinusoidally, the left wall is cooled linearly, and the right wall is maintained cold isothermal. . Case 5. The bottom and left walls are maintained hot isothermal, and the right wall is maintained cold isothermal. The top wall is maintained adiabatic in all cases. It may be noted that the boundary conditions in cases 1–3 are symmetric, while those in cases 4 and 5 are asymmetric. All the physical properties are assumed to be constant except the density in the buoyancy term. Change in density due to temperature variation is calculated using

ENTROPY GENERATION IN A POROUS CAVITY

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Figure 1. Schematic diagrams of the physical domain. The walls of the cavity are subjected to various thermal boundary conditions, as shown.

Boussinesq approximation. Another important assumption is that the local thermal equilibrium (LTE) is valid, i.e., the temperature of the fluid phase is equal to the temperature of the solid phase everywhere in the porous region. The momentum transfer in porous medium is based on generalized non-Darcy model proposed by Vafai and Tien [37]. However, the velocity square term or Forchheimer term which models the inertia effect is neglected here as in the present case only natural convection flow in porous medium is studied in an enclosed cavity. The inertia effect is more important for high velocity fluid flow in a high-porosity medium. Similar approximation of negligible inertial effects was considered by earlier researchers [38–40]. Under these assumptions and following Vafai and Tien [37] with Forchheimer inertia term being neglected, the governing equations for steady two-dimensional natural convection flow in a porous square cavity using conservation of mass, momentum, and energy may be written with the following dimensionless variables or numbers: X¼

P¼

x ; L

pL2 ; qa2

Y¼

y ; L

n Pr ¼ ; a

U¼

uL ; a

Da ¼

K ; L2

V¼

vL ; a

Ra ¼

h¼

T
Tc Th
Tc

gbðT h
T c ÞL3 Pr n2

ð1Þ

as qU qV þ ¼0 qX qY

ð2Þ

qU qU qP q2 U q2 U þ þV ¼
þ Pr U qX qY qX qX 2 qY 2 qV qV qP q2 V q2 V þV ¼
þ Pr U þ qX qY qY qX 2 qY 2

!


!

Pr U Da

ð3Þ

Pr V þ Ra Pr h Da

ð4Þ




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T. BASAK ET AL.

U

qh qh q2 h q2 h þV ¼ þ 2 qX qY qX qY 2

ð5Þ

The boundary conditions for velocities are as follows. UðX ; 0Þ ¼ UðX ; 1Þ ¼ Uð0; Y Þ ¼ Uð1; Y Þ ¼ 0 V ðX ; 0Þ ¼ V ðX ; 1Þ ¼ V ð0; Y Þ ¼ V ð1; Y Þ ¼ 0

ð6Þ

The boundary conditions for temperature in various cases are given below. Symmetric cases Case 1 h¼1

ðat bottom wallÞ

h¼0 qh ¼0 qY

ðat left and right wallsÞ

ð7aÞ

ðat top wallÞ

Case 2 h ¼ sin px h¼0 qh ¼0 qY

ðat bottom wallÞ ðat left and right wallsÞ

h¼1

ðat bottom wallÞ

h¼1
Y qh ¼0 qY

ðat left and right wallsÞ

ð7bÞ

ðat top wallÞ

Case 3

ð7cÞ

ðat top wallÞ

Asymmetric cases Case 4 h ¼ sin px

ðat bottom wallÞ

h¼1
Y h¼0 qh ¼0 qY

ðat left wallÞ ðat right wallÞ ðat top wallÞ

ð7dÞ

ENTROPY GENERATION IN A POROUS CAVITY

341

Case 5 h¼1 h¼1

ðat bottom wallÞ ðat left wallÞ

h¼0 qh ¼0 qY

ðat right wallÞ

ð7eÞ

ðat top wallÞ

Note that in Eqs. (1)–(7) (a–e), X and Y are dimensionless coordinates varying along horizontal and vertical directions, respectively; U and V are dimensionless velocity components in the X and Y directions, respectively; h is the dimensionless temperature; P is the dimensionless pressure; and Ra, Pr and Da are Rayleigh, Prandtl and Darcy numbers, respectively. The momentum and energy balance equations [Eqs. (3)—(5)] are solved using the Galerkin finite element method. The continuity equation (Eq. (2)) has been used as a constraint due to mass conservation, and this constraint may be used to obtain the pressure distribution. Eqs. (3) and (4) are solved using the penalty finite element method, where the pressure P is eliminated by a penalty parameter c and the incompressibility criteria given by Eq. (2) which results in the following.
 qU qV þ P ¼
c qX qY

ð8Þ

The continuity equation [Eq. (2)] is automatically satisfied for large values of c. Typical value of c that yield consistent solutions is 107. Using Eq. (8), the momentum balance equations [Eqs. (3) and (4)] reduce to the following. qU qU q þV ¼c U qX qY qX

!
 qU qV q2 U q2 U Pr þ þ Pr U þ
2 2 qX qY Da qX qY

ð9Þ

and qV qV q U þV ¼c qX qY qY

!
 qU qV q2 V q2 V Pr þ þ þ Pr V þ Ra Pr h
qX qY Da qX 2 qY 2

ð10Þ

The system of equations [Eqs. (5), (9), and (10)] with boundary conditions is solved by using the Galerkin finite element method [41]. Expanding the velocity components (U, V) and temperature (h) using basis set fUk gN k¼1 , as, follows. U

N X k¼1

U k Uk ðX ; Y Þ;

V

N X

V k Uk ðX ; Y Þ;

k¼1

h

N X k¼1

for 0  X;

and

Y 1

hk Uk ðX ; Y Þ ð11Þ

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The Galerkin finite element method yields the following nonlinear residual equations for Eqs. (9), (10), and (5), respectively, at nodes of internal domain X. ð1Þ Ri

Z " X N

X

þ Pr

N X

Z  Uk

k¼1

þ

Pr Da

X

Z

N X

qUk þ qX

!

N X

X

 qUi qUk qUi qUk þ dX dY qX qX qY qY !

ð12Þ

U k Uk Ui dX dY

k¼1

X

ð2Þ Ri

!

# qUk Ui dXdY ¼ Uk U k Uk V k Uk qY k¼1 k¼1 k¼1 X 2 3 Z Z N N X X qU qU qU qU i i k k dX dY þ dX dY 5 Uk Vk þ c4 qX qX qX qY k¼1 k¼1 N X

Z " X N

# qUk Ui dX dY ¼ Vk U k Uk V k Uk qY k¼1 k¼1 k¼1 X 2 3 Z Z N N X X qU qU qU qU i k i k dX dY þ dX dY 5 Uk Vk þ c4 qY qX qY qY k¼1 k¼1 N X

!

qUk þ qX

!

N X

X

X

 Z  qUi qUk qUi qUk þ dX dY Vk þ Pr qX qX qY qY k¼1 X ! Z N X Pr V k Uk Ui dX dY þ Da X k¼1 ! Z X N hk Uk Ui dX dY
RaPr N X

ð13Þ

k¼1

X

and ð3Þ Ri

¼

N X

hk

k¼1

þ

N X k¼1

Z " X N X

Z  hk

k¼1

! U k Uk

qUk þ qX

N X

! V k Uk

k¼1

 qUi @Uk qUi qUk þ dX dY qX qX qY qY

# qUk Ui dX dY qY ð14Þ

X

Bi-quadratic basis functions with three point Gaussian quadrature are used to evaluate the integrals in the residual equations, except the second term in Eqs. (12) and (13). In Eqs. (12) and (13), the second term containing the penalty parameter (c) are evaluated with two point Gaussian quadrature (reduced integration penalty formulation, [41]). The nonlinear residual equations [Eqs. (12)–(14)] are solved using the

ENTROPY GENERATION IN A POROUS CAVITY

343

Newton–Raphson method to determine the coefficients of the expansions in Eq. (11). The detailed solution procedure is given in an earlier work [42].

2.2. Streamfunction The fluid motion is displayed using the streamfunction (w) obtained from velocity components U and V. The relationships between streamfunction and velocity components for two-dimensional flows are as follows. U¼

qw qY

and

V ¼


qw qX

ð15Þ

which yield a single equation. q2 w q2 w qU qV þ ¼
qX 2 qY 2 qY qX

ð16Þ

Using the above definition of the streamfunction, the positive sign of w denotes anticlockwise circulation, and the clockwise circulation is represented by the negative sign of w. Expanding the streamfunction (w) using the basis set fUk gN k¼1 as PN w ¼ k¼1 wk Uk ðX ; Y Þ and the relationships for U and V from Eq. (11), the Galerkin finite element method yields the following linear residual equations for Eq. (16). Rsi

¼

N X

Z  wk

k¼1

þ

N X k¼1

 Z qUi qUk qUi qUk þ dX dY
Ui n  rw dC qX qX qY qY

X

Z

Uk

qUk dX dY
Ui qY

X

N X k¼1

Z Vk

C

ð17Þ

qUk dX dY Ui qX

X

The no-slip condition is valid at all boundaries, as there is no cross flow; hence, w ¼ 0 is used as residual equations at the nodes for the boundaries. The bi-quadratic basis function is used to evaluate the integrals in Eq. (17) and w’s are obtained by solving the N linear residual equations [Eq. (17)].

2.3. Entropy Generation In a natural convection system, the associated irreversibilities are due to heat transfer and fluid friction. According to local thermodynamic equilibrium of linear transport theory [22], the dimensionless form of local entropy generation rate due to heat transfer (Sh) and fluid friction (Sw) for a two-dimensional heat and fluid flow in porous media in Cartesian coordinates in explicit form is written as follows. "
Sh ¼

qh qX

2
2 # qh þ qY

ð18Þ

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(

"

!
 #)  2  qU 2 qV 2 qU qV 2 2 S w ¼ / U þ V þ Da 2 þ þ þ qX qY qY qX

ð19Þ

It may be noted that the viscous dissipation model as proposed by Al-Hadhrami et al. [43] is employed in Eq. (19). It may also be noted that there are three different models for viscous dissipation flow through porous media. In a recent study, Hooman and Gurgenci [44] compared three different viscous dissipation models and concluded that the three models are effectively the same at low values of Da, but for higher limits of Da only the model of Al-Hadhrami et al. [43] is valid. A detailed discussion on different viscous dissipation models, their limits of applicability, and other relevant issues on various aspects of modeling viscous dissipation in porous media may be found in earlier works [45, 46]. It may be noted that the effect of viscous dissipation is neglected in the energy equation (Eq. (5)), but is considered for estimation of Sw. The condition when the viscous dissipation may be neglected for natural convection systems is given by Ge