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Hydromagnetic Mixed Convective Transport in a Nonisothermally Heated Lid-Driven Square Enclosure Including a Heat-Conducting Circular Cylinder Dipankar Chatterjee*,† and Satish Kumar Gupta‡ †

Simulation & Modeling Laboratory, CSIR-Central Mechanical Engineering Research Institute, Durgapur 713209, India Department of Mechanical Engineering, National Institute of Technology Durgapur, Durgapur 713209, India

‡

ABSTRACT: A two-dimensional numerical study is performed in an effort to understand the fundamental characteristics of hydromagnetic mixed convective transport in a nonisothermally heated vertical lid-driven square enclosure filled with an electrically conducting fluid in the presence of a heat-conducting solid circular cylinder. Additionally, entropy generation due to heat transfer, fluid friction, and magnetic effect is also determined. Simulations are performed for various controlling parameters such as the Richardson number (1 ≤ Ri ≤ 10), Hartmann number (0 ≤ Ha ≤ 50), Prandtl number (0.02 ≤ Pr ≤ 7), Reynolds number based on the lid velocity (Re = 100, 150, and 200), and amplitude of the sinusoidal function (A = 0.25, 0.5, and 1), keeping the solid−fluid thermal conductivity ratio fixed as K = 5. The flow and thermal fields are analyzed through streamline and isotherm plots for various Ha, Ri, Re, and Pr. Furthermore, the drag coefficient on the moving lid and Nusselt numbers on heated surfaces are also computed to understand the effects of Ha, Ri, Re, Pr, and n on them. It is observed that the drag on the moving lid decreases with Re and increases with Ri and Ha, however, remains insensitive with Pr. The heat-transfer rate from the hot right wall increases as usual with Re, Pr, and Ri but decreases with Ha. The sinusoidally heated bottom wall shows a decrease in the heat-transfer rate with increasing Pr, and at higher Pr, it also decreases with Re. Furthermore, increasing magnetic field strength causes an increase in the heat-transfer rate from the bottom wall. It also decreases with decreasing value of the amplitude of the sinusoidal function.

1. INTRODUCTION Analysis of natural convection flow has received considerable attention among researchers and engineers because of its extensive practical applications such as food processing, solar collection, chemical processing, nuclear reaction, phase change applications, and thermal energy storage. The mixed convective transport in cavities subjected to an externally applied magnetic field can have potential applications in crystal growth processing and electronic cooling. The insertion of some additional active or passive elements within the cavity geometry offers substantial control and regulation of such transport. A wide gamut of fundamental issues evolving from the classical fluid flow and heat transfer in a cavity are discussed in this work. The first and most important issue is mixed convective transport in a lid-driven enclosure, which originates as a consequence of two competing mechanisms. Primarily, shearing action is offered by the translating lid, which results in forced flow, followed by buoyancy-driven transport originating as a result of the thermal nonhomogeneity between the cavity boundaries. The mixed convection problem in lid-driven enclosures finds wide applications in various fields of engineering and science such as float glass production, hydraulics of nuclear reactors, dynamics of lakes, food processing, and heat transfer in vertical tubes including thermal discharge in water bodies. There have been many investigations reported in the literature in the past on mixed convective flow in a lid-driven enclosure, and a comprehensive review in this regard can be found in Shankar and Deshpande.1 Furthermore, the lid-driven-cavity problem has been extensively used as a benchmark case for the evaluation of numerical solution algorithms.2,3 © XXXX American Chemical Society

The next important issue is the control and regulation of convective transport of heat and fluid in cavities through active and passive means, which are extremely important in the design of heat-exchanging systems. Many attempts have so far been reported in the literature that broadly include variations of both the techniques in a simple square enclosure addressing lid-driven flows, buoyancy-driven transport, forced convection in a cavity including inlet and outlet ports, etc. A recent trend is to use some additional passive elements such as triangular conductive fins,4 a circular body,5 or a heated block6 with the cavity geometry or using a fluid-saturated porous medium7,8 to control the heat and fluid flow without consumption of additional energy. When an electrically conducting fluid is acted upon by some external magnetic field, the electromagnetic field interacts with the fluid to produce magnetohydrodynamic (MHD) forces, which give rise to hydromagnetic convection. The applied magnetic field may have substantial influence on the flow as well as heat transfer in cavities subjected to mixed convective transport, as observed from contemporary studies.9−22 The effect of the magnetic field on convective transport is important in many chemical engineering science avenues such as applications in liquid metals, electrolytes, and ionized gases. Special Issue: Energy System Modeling and Optimization Conference 2013 Received: March 13, 2014 Revised: September 15, 2014 Accepted: September 17, 2014

A

dx.doi.org/10.1021/ie501080y | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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When there is variation of the temperature in both the solid and fluid due to thermal interaction, the heat-transfer process is commonly termed “conjugate heat transfer”. A typical example is the heating or cooling of a solid object by the flow of air in which it is immersed. Conjugate heat-transfer methodology finds extensive application in the solution of turbine heat-transfer problems. Rahman and co-workers23−25 studied the effects of the magnetic field with the Joule heating effect in a lid-driven cavity with a circular body insertion. They observed that heat transfer decreases with an increase in the Joule parameter. One technique for the active regulation of heat transfer in convection-dominated heat-exchanging systems is by the controlled movement of a wetted boundary. Chatterjee and co-workers26−29 in a series of articles demonstrated the effect of the magnetic field on mixed convective transport in a square enclosure with stationary and rotating heat-conducting objects. It was observed that rotation of the object causes significant variation in the heat-transfer rate. The majority of the studies mentioned above consider isothermal or isoflux thermal boundary conditions on the walls of the enclosures. However, nonisothermal boundary conditions can exist in many occasions such as in electrical arc furnaces, in rotary burners,30 and also in the case of cylindrical heaters.31 Some studies are available pertaining to the application of nonisothermal boundary conditions32−40 such as a sinusoidal temperature variation in the cavity walls. However, apart from Oztop et al.,40 no one else considered the magnetic effect in conjunction with mixed convection and conjugate heat transfer in a lid-driven cavity. Another important aspect with regard to the efficient thermal design of thermochemical equipment is exergy analysis. As such, efficiency analysis of any thermal system is an important issue to the thermal system designers because the optimal design criteria of such a system depends on the idea of minimizing entropy generation in the system. Works are available in the literature pertaining to entropy-generation analysis for convective transport in cavities41−44 frequently encountered in potential chemical engineering applications. Accordingly, we aim here to numerically analyze the hydromagnetic mixed convection flow and heat transfer along with entropy generation in a vertical lid-driven square enclosure involving a heat-conducting horizontal solid circular cylinder placed centrally within the enclosure. The side walls of the enclosure are kept isothermal, while the top is considered to be adiabatic and the bottom is heated and cooled with a sinusoidal function. Simulations are performed by deploying a finitevolume technique for various controlling parameters such as the Richardson number (1 ≤ Ri ≤ 10), Hartmann number (0 ≤ Ha ≤ 50), Prandtl number (0.02 ≤ Pr ≤ 7), Reynolds number based on the lid velocity (Re = 100, 150, and 200), and amplitude of the sinusoidal function (A = 0.25, 0.5, and 1).

Figure 1. Schematic diagram of the physical problem.

allowed to translate at a constant velocity V0 along the upward direction. The fluid inside the enclosure can be considered as electrically conducting. A heat-conducting horizontal solid circular cylinder of diameter d = 0.2L is placed centrally within the enclosure. An external magnetic field of amplitude B0 is applied along the horizontal direction normal to the vertical wall. All of the solid walls are assumed to be electrically insulated. Assuming an incompressible flow with constant fluid properties along with Boussinesq and low magnetic Reynolds number approximations, the governing inductionless differential equations consisting of mass, momentum, and energy in the dimensionless form can be expressed as For fluid: ∂U ∂V + =0 ∂X ∂Y U

(1)

∂U ∂U ∂P 1 ⎛ ∂ 2U ∂ 2U ⎞ + +V =− ⎟ ⎜ 2 + ∂X ∂Y ∂X Re ⎝ ∂X ∂Y 2 ⎠

(2a)

U

∂V ∂V 1 ⎛ ∂ 2V ∂P ∂ 2V ⎞ +V =− + ⎜ 2 + ⎟ − NV + Riθ ∂X ∂Y Re ⎝ ∂X ∂Y ∂Y 2 ⎠

U

∂θ ∂θ ∂ θ⎞ 1 ⎛∂ θ +V = ⎜ 2 + ⎟ ∂X ∂Y RePr ⎝ ∂X ∂Y 2 ⎠

(2b) 2

2. PHYSICAL MODEL AND MATHEMATICAL FORMULATION The physical model considered is shown in Figure 1 along with the important geometric parameters and boundary conditions. A Cartesian coordinate system is chosen, with the origin attached at the lower left corner of the simulation domain. A square enclosure of side L is considered as the computational domain of which the left and right walls are maintained at constant temperature Tc and Th(>Tc), respectively. The top wall is assumed to be adiabatic, whereas the bottom is heated and cooled with a sinusoidal function of the form T(x) = Tc + ΔTA sin(2πx/L), where ΔT = Th − Tc and A is the amplitude. The left wall is

2

(3)

For solid:

∂ 2θs

∂ 2θs

=0 (4) ∂X2 ∂Y 2 where U and V are the dimensionless velocity components along the X and Y directions, respectively, P is the dimensionless pressure, θ and θs are the dimensionless temperatures of the fluid and solid, Re (=V0L/υ) is the Reynolds number, N = Ha2/Re is the interaction parameter with Ha = B0L(σ/ρυ)1/2 being the Hartmann number, Ri = Gr/Re2 is the Richardson number with Gr = gβ(Th − Tc)L3/υ2 being the Grashof number and Pr = υ/α the Prandtl number. B

+



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momentum equations. The QUICK (Quadratic Upstream Interpolation Convective Kinetics) scheme is used for the spatial discretization of the convective terms, and a central difference scheme is used for the diffusive terms. The SIMPLEC algorithm is used as the pressure−velocity coupling scheme. Finally, the algebraic equations are solved by using the Gauss−Siedel pointby-point iterative method. The convergence criteria based on the absolute error are set as 10−10 for the discretized continuity and momentum equations and 10−12 for the discretized energy equation. As such, convergence is a big concern in numerical simulation particularly for buoyancy-driven transport. In general, this can be handled in Ansys Fluent in various ways. One of them is the volume adaptation technique. It is best for both accuracy and convergence to have a mesh in which the changes in the cell volume are gradual. If the mesh creation or adaption process has resulted in a mesh that does not have this property, the grid can be improved by using volume adaption with the option of refining based on either the cell volume or the change in the volume between the cell and its neighbors. The convergence may depend on the choice of the discretization schemes and pressurevelocity coupling scheme. Manipulation of the underrelaxation parameters and multigrid settings can significantly enhance the convergence. The buoyancy-driven flows need special attention as far as convergence is concerned. Apart from the above approaches, a body-force-weighted or PRESTO scheme for pressure interpolation can specifically be used for buoyancy-driven flows for better convergence. Detailed discussions in this regard can be found in the Fluent user’s manual45 (one can follow the special technique for high Rayleigh number flows, as outlined in the user’s manual45). One important aspect in implementing the present problem in Ansys Fluent is the strategy invoked for implementing the numerical scheme in a dimensionless form. The software solves the governing equations in the dimensional form having the following nature (because the present problem is steady, the governing equations are presented in steady forms):

The dimensionless variables are defined as y x u v , V= , X= , Y= , U= L L V0 V0 (p + ρgy)L2

P=

ρV0 2

θ=

,

T − Tc , Th − Tc

θs =

Ts − Tc Th − Tc (5)

The fluid properties are represented by density ρ, kinematic viscosity υ, electrical conductivity σ, and thermal diffusivity α = kf/ρcp, where kf and cp are the fluid thermal conductivity and specific heat, respectively. The work due to pressure and viscous dissipation and the Hall and Joule heating effects are neglected in the present computation. The following boundary conditions are imposed: (1) U = 0, V = 1, θ = 0: at the left vertical wall (2) U = 0, V = 0, θ = 1: at the right vertical wall (3) U = 0, V = 0, ∂θ/∂n = 0: at the top wall (4) U = 0, V = 0, θ = A sin (2πX): at the bottom wall (5) (∂θ/∂n)f = K(∂θ/∂n)s: at the fluid−solid interface K = ks/kf is the solid−fluid thermal conductivity ratio, and n is the unit outward normal. Pressure boundary conditions are not explicitly required because the solver extrapolates the pressure from the interior. 2.1. Global Parameters. The drag coefficient on the moving wall can be obtained from CD =

2FD 2

ρV0 L

=

2 ∂U Re ∂X

(6)

X=0

The local and average Nusselt numbers at the right hot wall of the enclosure are defined as Nu = −

∂θ ∂X

and

Nuav = −

∫0

1

∂θ dY ∂X

(7)

and the local and average Nusselt numbers on the bottom wall are computed from Nu = −

∂θ ∂Y

and

Nuav = −

∫0

1/2

∂θ dX ∂Y

∂u ∂v + =0 ∂x ∂y

(8)

The bulk average temperature in the enclosure is given as θav =

∫

θ dV ̅ V̅

u (9)

where V̅ is the enclosure volume (considering the unit thickness).

u

3. METHOD OF SOLUTION AND IMPLEMENTATION The finite-volume-based computational fluid dynamics package Ansys Fluent45 is deployed to solve the conservation equations subjected to the aforementioned boundary conditions. A userdefined function (UDF) written using Fluent macros is used to incorporate the Lorentz force source term (NV in eq 2b) in the momentum equation. Another UDF is used to prescribe the sinusoidal thermal boundary condition at the bottom wall of the cavity. The control-volume-based technique solves the governing system of partial differential equations in a collocated grid system by constructing a set of discrete algebraic equations. The numerical scheme solves the discretized governing equations sequentially. The sequence updates the velocity field through solution of the momentum equations using known values for the pressure and velocity. Then, it solves a “Poisson-type” pressure correction equation obtained by combining the continuity and

(10)

⎛ ∂ 2u ∂u ∂u ∂ 2u ⎞ 1 ∂p +v =− + ν⎜ 2 + 2 ⎟ ∂x ∂y ρ ∂x ∂y ⎠ ⎝ ∂x

(11a)

⎛ ∂ 2v 1 ∂p ∂v ∂v ∂ 2v ⎞ +v =− + ν⎜ 2 + 2 ⎟ + gβ(T − Tc) ∂x ∂y ρ ∂y ∂y ⎠ ⎝ ∂x (11b)

u

⎛ ∂ 2T ∂T ∂T ∂ 2T ⎞ +v = α⎜ 2 + 2 ⎟ ∂x ∂y ∂y ⎠ ⎝ ∂x

(12)

The magnetic field source term is not included in eq 3 because a separate UDF is used to model it, as mentioned earlier. Now comparing eqs 1 and 10, 2a and 11a, 2b and 11b, and 3 and 12, we could arrive at ρ = 1,

ν=

1 , Re

α=

1 , RePr

g = 1,

β = Ri (13)

Hence, in Ansys Fluent, the following parameters are fixed as ρ = 1, C

ν=

1 , Re

k f = 1,

c p = RePr ,

g = 1,

β = Ri

(14)



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Additionally, the length scale is considered as L = 1 while creating the geometry. The above strategy can effectively be utilized for implementing the numerical scheme in a dimensionless form in Ansys Fluent. A nonuniform grid distribution having a close clustering of grid points in the vicinity of the solid cylinder wall and other walls is used in the present computation. 240 grid points are used on the periphery of the circular obstacle. A comprehensive grid sensitivity analysis is also carried out. The following three different mesh sizes are chosen in order to check the grid independency: (M1) 52600 quadrilateral elements, 53242 nodes; (M2) 58500 quadrilateral elements, 59041 nodes; (M3) 62800 quadrilateral elements, 63401 nodes. Computations are performed for Re = 100, Ri = 10, Ha = 50, and Pr = 0.02. It is observed that the simulation results show a maximum difference of about 3.84% between M1 and M2, while it is 1.68% between M2 and M3 types, in terms of the average Nusselt number and bulk average fluid temperature. Accordingly, type M2 mesh is preferred, keeping in mind the accuracy of the results and computational convenience in the simulations.

4. RESULTS AND DISCUSSION Computations are performed for the following ranges of dimensionless parameters such as the Richardson number (1 ≤ Ri ≤ 10), Hartmann number (0 ≤ Ha ≤ 50), Prandtl number (0.02 ≤ Pr ≤ 7), Reynolds number based on the lid velocity (Re = 100, 150, and 200), and amplitude of the sinusoidal function (n = 0.25, 0.5, and 1), keeping the solid−fluid thermal conductivity ratio fixed as K = 5. The implications of varying Ha, Ri, Re, Pr, and n are emphasized through streamlines and isotherm patterns along with variations of the drag coefficient of the moving plate and average Nusselt number of the hot walls. The streamlines, representing the direction of flow, are basically isolines of constant stream functions. With the known velocity field, the stream function can be calculated from u = ∇·ψ, where u and ψ are the flow velocity and stream function, respectively, and ψ = (0, 0, ψ) if the velocity vector is u = (u, v, 0). In the Cartesian coordinate system, this is equivalent to u = ∂ψ/∂y and v = −∂ψ/∂x, where u and v are the velocity components along the x and y directions, respectively. The isotherms are the constant temperature lines. 4.1. Numerical Verification. In order to verify the present numerical scheme, first, the problem of hydromagnetic lid-driven cavity flow, as reported in Chamkha,14 is simulated and compared. The details of the physical problem and pertinent parameters are outlined in work by Chamkha.14 The comparisons are presented in Figure 2 for the average Nusselt numbers. Figure 2a shows variation of the average Nusselt number with the magnetic field strength for a representative Re = 1000 and Gr = 100. The heat-transfer rate decreases with increasing magnetic field, and the simulated results are in excellent agreement with that predicted by Chamkha.14 Figure 2b presents variation of the average Nusselt number with the natural convective strength for a fixed Re = 100 and no magnetic field, Ha = 0. With increasing Gr, the average Nusselt number expectedly rises very fast. Again, a very good agreement with the reported results can be observed. Next, the effect of a heat-conducting horizontal circular cylinder in a vertical lid-driven square enclosure is depicted in Figure 3 in terms of the average Nusselt number on the heated enclosure wall and bulk fluid temperature. The pertinent parameters are available in Rahman and Alim.25 Figure 3 shows that the average Nusselt number once again increases with increasing mixed convective strength (Ri). However, the bulk fluid temperature

Figure 2. Effect of (a) Ha and (b) Gr on Nu for Pr = 0.71.

Figure 3. Effect of a circular obstacle on heat transfer and bulk fluid temperature.

inside the domain increases first with Ri and then reaches a saturated state. The present computation shows satisfactory agreement with that reported in work by Rahman and Alim.25 The minor deviations can be attributable to the use of different grids and solution methods for the respective studies. 4.2. Flow and Thermal Fields. Figures 4−7 respectively show the streamlines and isotherms for different Hartmann, Richardson, and Reynolds numbers and the amplitude of the sinusoidal function. The fluid dynamics can be viewed as an outcome of the coupled effect of shear-driven flow as a consequence of the lid movement, buoyancy-driven flow due to differential heating of the two opposing sides of the enclosure, and the sinusoidally varying bottom boundary temperature and MHD flow due to the externally imposed magnetic field. Thermal transport is a consequence of the forced flow, thermal buoyancy, and conduction within the solid object. In Figure 4, the effect of the magnetic field on the flow and thermal fields is shown for fixed Re = 100, Pr = 0.02, Ri = 1, and A = 1. In general, when the lid is set into motion, a thin shear layer is formed adjacent to the surface of the lid. This shear layer grows and eventually affects the fluid that is farthest from the moving lid. D



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Figure 4. Streamlines and isotherms at Re = 100, Pr = 0.02, Ri = 1, and A = 1 for different Hartmann numbers.

magnetic field, heat transfer is dominated by the forced flow caused by the moving lid and the natural convection driven flow due to thermal buoyancy. However, as the magnetic field strength increases, the shearing effect increases and the buoyancy effect decreases, which causes heat transfer to be dominated by forced convection in the left portion of the enclosure and by conduction on the right half of the enclosure. Half of the bottom plate is heated whereas the rest is cooled owing to the sinusoidal temperature variation. As a result, positive and negative isotherms characterize the temperature distribution in the respective halves of the bottom plate. Figure 5 depicts the streamline and isotherm patterns for different Richardson numbers and at Re = 100, Ha = 20, Pr = 0.71, and A = 1. The flow and thermal fields, which are dominated almost equally by the shear and buoyancy forces at low-tomoderate Hartmann numbers at Ri = 1, gradually become thermal-buoyancy-dominated as the Richardson number increases. For higher Richardson numbers, the CW vortices adjacent to the left moving wall become narrower and the CCW vortices elongate to cover the majority of the enclosure. Additionally, with increasing Richardson number, heat transfer from the sinusoidally heated and cooled bottom wall becomes significant. The fluid from the heated half of the bottom plate rises upward and then strikes the strongly rotating fluids on the upper plate and finally comes down, resulting in the formation of a CW cell close to the bottom wall. This can particularly be visualized at larger Ri. Looking into the corresponding isotherms, it is revealed that at lower Ri a thermal plumelike structure develops and breaks down with increasing strength of buoyancy.

When there is no magnetic field (Ha = 0), two unequal counterrotating vortices are observed to form. These two vortices are basically an outcome of the forced flow as a result of the lid movement and buoyancy-driven flow due to differential heating. The vortices adjacent to the moving lid have a clockwise (CW) sense because the lid moves vertically upward (from bottom to top). Counterclockwise (CCW) vortices form adjacent to the right hot wall as a result of the thermal buoyancy effect. As the Hartmann number increases, the magnetic force, which is damping in nature, increases. This damping force actually reduces the effect due to thermal buoyancy (by observing that both the magnetic and buoyancy forces are featured in the y-momentum equation, eq 2b) on the flow field. The resulting flow field is predominantly influenced by the shear force offered by the translating lid. As a consequence of these effects, the CW vortices originating as a result of the lid movement increase in size toward the right and the CCW vortices are reduced. As the Hartmann number increases progressively, the CW vortices penetrate further to the right and occupy the major part of the enclosure. On the other hand, the CCW vortices adjacent to the hot wall become reduced in size and eventually turn into two small eddies at the top and bottom corners in the vicinity of the hot right wall. Additionally, because of the presence of the solid object, the shear flow is obstructed and a partial backward surge propagates toward the left wall along the horizontal centroidal axis. This eventually causes the elongated cell adjacent to the left wall to decompose into two smaller cells. This is particularly noticed at higher Hartmann number (Ha = 50). The corresponding isotherms reveal that when there is no E



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Figure 5. Streamlines and isotherms at Re = 100, Pr = 0.71, Ha = 20, and A = 1 for different Richardson numbers.

The convective strength is intensified at larger Ri, which causes an increase in the heat-transfer rate. Figure 6 illustrates the impact of Re on the variation of streamlines and isotherms for Ri = 1, Ha = 20, Pr = 7, and A = 1. At relatively small Reynolds number (say, Re = 100), there exist three recirculation cells. Among these cells, a CW cell occupying the major part of the enclosure and the remaining two small CCW cells developed near the right top and bottom corners of the enclosure. This implies that the fluid is well mixed within the enclosure. With increasing Re (Re = 150 and 200), the size of the CCW cell adjacent to the right vertical heated wall gradually increases and occupies almost the entire enclosure, pushing down the CW cell near the left vertical wall. This can be attributed to the fact that, as the Reynolds number increases while the Richardson number remains fixed, the Grashof number increases, which causes the natural convective strength to increase. From the corresponding temperature distributions, it can be observed that isotherms are nearly parallel to the hot wall, which is similar to a conduction-like distribution. However, at larger Re, convective distortion of the isotherms occurs throughout the enclosure because of the strong influence of the natural convective current. When the amplitude of the sinusoidal function is varied, with all other parameters fixed, the flow strength is observed to increase with increasing amplitude, as can be observed from the streamline plots in Figure 7. Significant variation in the temperature field can be seen for varying amplitude, as is evident from the corresponding isotherms in Figure 7. This may, in turn,

appreciably affect the heat-transfer rate from the bottom wall of the enclosure. The effect of the Prandtl number on the overall flow and heat-transfer characteristics can actually be realized by comparing similar plots of Figures 4−6 for Re = 100, Ri = 1, Ha = 20, and n = 1. Pr = 0.02 signifies liquid metal, which has a higher thermal diffusivity compared to the momentum diffusivity. Accordingly, the thermal boundary layer grows at a faster rate in comparison with the hydrodynamic boundary layer for such a fluid. Pr = 0.71 signifies air, which in a strict sense is not an electrically conducting fluid. Still, it is used in the computation for comparison purposes. For such a fluid, both the thermal and hydrodynamic boundary layers are of comparable thickness. Pr = 7 represents water for which the momentum diffusivity is higher, resulting in thicker hydrodynamic boundary layer development. The thermal buoyancy effect is observed to decrease progressively with increasing Prandtl number. The conduction mode heat transfer is predominant for lower Pr; however, convection dictates thermal transport at larger Pr. Figure 8 shows the velocity and temperature profiles for different Ha at Re = 100, Pr = 0.71, A = 1, and Ri = 1. The profiles are presented at two different locations along the x axis as x = 0.25 and 0.75. It is observed that negative velocities are formed on both sides of the enclosure. However, a slight positive value can be seen at Ha = 0. The left-side velocity originates as an outcome of the lid movement, whereas the right-side velocity is a manifestation of the natural convection. With increasing Hartmann number, the difference between these two velocities decreases. This establishes that the magnetic field is actually F



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Figure 6. Streamlines and isotherms at Ri = 1, Pr = 7, Ha = 20, and A = 1 for different Reynolds numbers.

signifying that the heat is transferred from the fluid to the wall. The average heat transfer from the bottom wall increases like the right wall with Ri. However, with increasing magnetic field strength, heat transfer from the bottom wall shows an increasing tendency, which is just the reverse of that observed in the right heated wall. Figure 11 shows the influence of the amplitude of the sinusoidal function on heat transfer from the right and bottom walls of the enclosure. The increasing amplitude suggests that heat transfer is more. This is quite obvious because, with more and more increasing amplitude, energy is imparted to the system, which causes an increase in the heat-transfer rate. However, with the Prandtl number, variations in the heat-transfer rate from the right and bottom walls show converse trends. 4.4. Entropy-Generation Analysis. One way of increasing the energy efficiency of the thermal system is to reduce exergy losses due to irreversibilities, measured as “entropy generation”. In the present problem, the boundary movement and thermal nonhomogeneity along with the existence of the external magnetic field set the fluid in a nonequilibrium state, which causes entropy generation in the system. According to the local thermodynamic equilibrium with linear transport theory, the local entropy generation can be given as46

damping in nature. With increasing magnetic field strength, the convective heat-transfer decreases and the conduction mode dominates, as established from the straightened temperature profiles at larger Ha. The maximum temperature also decreases with increasing Ha. 4.3. Global Transport Parameters. Figure 9 represents the effect of Ha, Re, Pr, and Ri on the drag coefficient of the moving lid. Because the lid is moving upward, the drag force is having a negative sense. As is evident from Figure 9a, the drag decreases with increasing Reynolds number, which is obvious. However, there is an insignificant effect of Pr on the drag coefficient. It increases with increasing Hartmann number because of the increase in shear stress, as depicted in Figure 9b. The drag also increases with increasing Richardson number, and the rate of increase becomes less at higher magnetic field strength. Parts a and b of Figure 10 suggest that the average Nusselt number on the right hot wall increases with increasing Re, Ri, and Pr. However, the increase of Nuav with Re is insignificant at low Pr. Furthermore, it is observed that heat transfer decreases with increasing magnetic field strength, and the difference increases as the thermal buoyancy effect becomes more. At higher Ha, heat transfer is predominantly caused by conduction at the hot wall, and convection has a small role to play. Consequently, heat transfer decreases at higher Ha. Parts b and c of Figure 10 show the average Nusselt number on the sinusoidally heated and cooled bottom wall. For smaller Pr, the average Nu shows no change with Re. However, as Pr increases, the average Nu shows a decreasing trend with Re, and for Pr = 7, it becomes negative,

̇ =− Sgen

μ σ |u × b|2 1 ·∇ + Φ + q T T T T2

(15)

where T is the local absolute temperature, q is the heat flux vector, μ is the dynamic viscosity, Φ is the viscous dissipation G



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Figure 7. Streamlines and isotherms at Re = 100, Pr = 0.71, Ri = 1, and Ha = 20 for different amplitudes of the sinusoidal temperature profile.

⎡⎛ ∂θ ⎞2 ⎛ ∂θ ⎞2 ⎤ ⎡⎛ ∂θ ⎞2 ⎛ ∂θ ⎞2 ⎤ Ns = ⎢⎜ ⎟ + ⎜ ⎟ ⎥ + K ⎢⎜ s ⎟ + ⎜ s ⎟ ⎥ ⎝ ∂Y ⎠ ⎦ ⎝ ∂Y ⎠ ⎥⎦ ⎢⎣⎝ ∂X ⎠ ⎣⎝ ∂X ⎠

function, u is the velocity vector, and b is the magnetic field. The first term on the right-hand side of eq 15 represents the irreversibility due to heat transfer within the fluid and solid, the second is due to fluid friction, and the third is due to magnetic force. According to Bejan, the characteristic entropy-transfer rate is given by

2 ⎧ ⎡⎛ ∂U ⎞2 ⎛ ∂V ⎞2 ⎤ ⎛ ∂U ∂V ⎞⎟ ⎫ ⎬ + λ 2V 2 + λ1⎨2⎢⎜ ⎟ + ⎜ ⎟ ⎥ + ⎜ + ⎝ ∂Y ⎠ ⎦ ⎝ ∂Y ∂X ⎠ ⎭ ⎩ ⎣⎝ ∂X ⎠ ⎪

⎪

⎪

⎪

= Nt + Nf + Nm

⎛ ΔT ⎞ σ0 = k ⎜ ⎟ ⎝ LT0 ⎠

̇ (L2T02/kfΔT2), K = ks/kf, λ1 = (μT0/kf)(V02/ΔT2), where Ns = Sgen and λ2 = λ1Ha2. Nt, Nf, and Nm are the contributions from heat transfer, fluid friction, and magnetic field on the total entropy generation. Total dimensionless entropy generation is obtained by a numerical integration of the dimensionless local entropy generation through the entire volume of the cavity as

(16)

where k, L, T0, ΔT = Th − Tc are the thermal conductivity, the characteristic length of the enclosure, a reference temperature, and a reference temperature difference, respectively. Following eq 16, the volumetric entropy-generation rate for a two-dimensional flow in Cartesian coordinates becomes

S=

2 ⎤⎫

⎧ ⎡ ⎡ 2 ⎛ ∂T ⎞ ⎪ ⎪ k k ⎛ ∂T ⎞ ⎛ ⎞2 ⎛ ⎞ ̇ = ⎨ f ⎢⎜ ∂T ⎟ + ⎜ ∂T ⎟ ⎥ + s ⎢⎜ s ⎟ + ⎜ s ⎟ ⎥⎬ Sgen 2 ⎢⎝ ⎪ T 2 ⎢⎝ ∂x ⎠ ⎠ ⎥ ∂ ∂ y x ⎝ ∂y ⎠ ⎥⎦⎪ ⎝ ⎠ ⎦ T0 ⎣ ⎭ ⎩ 0 ⎣ 2⎤ 2⎫ ⎧ ⎡ 2 2 ⎛ ∂u ⎛ ∂v ⎞ σv B0 2 μ ⎪ ⎛ ∂u ⎞ ∂v ⎞ ⎪ + + ⎨2⎢⎜ ⎟ + ⎜ ⎟ ⎥ + ⎜ ⎟⎬+ ⎢⎝ ⎠ ∂x ⎠ ⎪ T0 ⎪ T0 ⎝ ∂y ⎠ ⎥⎦ ⎝ ∂y ⎭ ⎩ ⎣ ∂x (17) 2⎤

(18)

∫Ω Ns dΩ

(19)

Figure 12 is plotted to show variation of the total dimensionless entropy generation with the Hartmann number for different Richardson (Figure 12a; Re = 100, Pr = 0.02, and A = 1) and Reynolds (Figure 12c; Ri = 1, Pr = 0.02, and A = 1) numbers. Additionally, to understand the contributions of various terms in the total entropy, Figure 12c is plotted to show variations of heat-transfer, fluid friction, and magnetic field contributions with the Hartmann number for various Richardson numbers, with the Reynolds number remaining fixed as Re = 100 with Pr = 0.02 and A = 1. It is observed that when natural and

By using the dimensionless scheme in eq 5, eq 17 takes the following form: H



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Figure 8. Velocity profiles (left) and temperature profiles (right) at different locations along the x axis for Re = 100, Pr = 0.71, A = 1, Ri = 1, and Ha = (a) 0, (b) 20, and (c) 50.

Figure 9. Variation of the drag coefficient with (a) the Reynolds number at Ha = 20 and Ri = 1 for different Prandtl numbers and (b) the Richardson number at Re = 100 and Pr = 0.71 for different Hartmann numbers.

fixed Reynolds number shows that entropy generation gradually starts, decreasing with the Hartmann number. When there is no magnetic field, entropy generation increases with both the Reynolds and Richardson numbers. Under such conditions, entropy production is due to heat transfer and fluid friction.

forced convective strengths are comparable (Ri = 1), the dimensionless total entropy increases with the Hartmann number irrespective of the Reynolds number. Additionally, as the Reynolds number increases, the entropy generation decreases. However, increasing natural convective strength for I



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Figure 10. Variation of the average Nusselt number with (a and c) the Reynolds number at Ha = 20 and Ri = 1 for different Prandtl numbers and (b and d) the Richardson number at Re = 100 and Pr = 0.71 for different Hartmann numbers.

Figure 11. Variation of the average Nusselt number on (a) the right and (b) bottom walls with amplitude, Re = 100, Ha = 20, and Ri = 1 for different Prandtl numbers.

and A = 1 (as a representative case) kept constant, total dimensional entropy generations are found as 109.3, 109.7, and 112.4; i.e., the entropy production increases with increasing Prandtl number. Increased Prandtl number is associated with increased momentum diffusivity and decreased thermal diffusivity. This causes the fluid velocity to increase and the temperature to reduce. Hence, the entropy production due to fluid friction increases, whereas that due to heat transfer decreases. Because fluid friction is the major source of entropy generation, it increases with increasing Pr. In order to understand the effect of the amplitude of the sinusoidal function used to heat the bottom surface, we simulate the case with Re = 100, Ri = 1, Ha = 20, and Pr = 0.025 for different amplitudes. The results show that Ns = 105.087, 106.07, and 109.3 for A = 0.25, 0.5, and 1, respectively. Hence, the entropy production increases with increasing amplitude. With increasing amplitude, the temperature increases, which causes more production of entropy.

With increasing Richardson number, heat transfer increases, and with increasing Reynolds number, fluid friction increases. Hence, entropy generation also increases. With the introduction of the magnetic field, the magnetic field itself contributes toward entropy production. Additionally, the magnetic field increases the shearing effect and reduces the buoyancy effect. Hence, with a larger magnetic field (say, at Ha = 50), the entropy increases faster with an increase in the Reynolds number, whereas the increase becomes narrower with increasing Richardson number. Upon inspection of the contributions of thermal, viscous, and magnetic effects on the total entropy generation, it is observed that generation due to heat transfer for increasing magnetic field strength remains insensitive to the Richardson number, as observed from Figure 12b. However, the fluid friction and magnetic field effects contribute significantly toward the total entropy production. With increasing Prandtl number from 0.02 to 0.71 and 7 with the parameters at Re = 100, Ri = 1, Ha = 20, J



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Figure 12. Total dimensionless entropy-generation variation with the Hartmann number: (a) for different Richardson numbers; (b) contributions of various terms of total entropy; (c) for different Reynolds numbers.

5. CONCLUSION Numerical computations are carried out to analyze the magnetoconvective transport in a lid-driven enclosure with a heat-conducting solid circular cylinder placed centrally within the enclosure. One of the enclosure walls is subjected to nonisothermal boundary conditions. The analysis is carried out for various controlling parameters such as the Richardson number (1 ≤ Ri ≤ 10), Hartmann number (0 ≤ Ha ≤ 50), Prandtl number (0.02 ≤ Pr ≤ 7), Reynolds number based on the lid velocity (Re = 100, 150, and 200), and amplitude of the sinusoidal function (A = 0.25, 0.5, and 1), keeping the solid−fluid thermal conductivity ratio fixed as K = 5. The following are some itemized observations from the present study: (a) The flow and thermal fields are basically an outcome of the coupled effect of shear, buoyancy, and magnetically driven transport. The magnetic field acts as a damping agent to the flow field. With increasing magnetic field strength, the convective heat transfer decreases and the conduction mode dominates. The thermal buoyancy effect is observed to decrease progressively with increasing Prandtl number. (b) The drag coefficient of the moving lid decreases with increasing Reynolds number. However, there is an insignificant effect of Pr on the drag coefficient. It increases with increasing Hartmann number and Richardson number. (c) The average Nusselt number on the heated right wall of the enclosure increases with increasing Re, Ri, and Pr. With an increase in the magnetic field strength, the heat-transfer rate from the heated right wall decreases. Finally, the heat-transfer rate is found to be low at higher magnetic field strength, and the difference increases as the thermal buoyancy effect increases. For the sinusoidally heated and cooled bottom wall, the average Nu

shows a decreasing trend with Re as Pr increases. With increasing magnetic field strength, heat transfer from the bottom wall shows an increasing tendency, which is the reverse to that of the right wall. (d) Heat transfer from both the right and bottom walls increases with increasing amplitude of the sinusoidal function. (e) The entropy generation increases with Ha for low Ri irrespective of Re. However, at higher Ri, it decreases with Ha. With increasing Re, the entropy production decreases. The entropy production increases with increasing Prandtl number and amplitude of the sinusoidal function used to heat the bottom surface.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +91-343-6510455. Fax: +91-343-2548204. Notes

The authors declare no competing financial interest.

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K

NOMENCLATURE A = amplitude of the sinusoidal function B0 = magnetic field strength, Wb m−2 CD = drag coefficient cp = specific heat at constant pressure, J kg−1 K−1 d = cylinder diameter, m FD = drag force, N g = gravitational acceleration, m s−2 Gr = Grashof number Ha = Hartmann number dx.doi.org/10.1021/ie501080y | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX


Article

kf = thermal conductivity of the fluid, W m−1 K−1 ks = thermal conductivity of the solid, W m−1 K−1 K = solid−fluid thermal conductivity ratio L = length of square enclosure, m n = normal direction N = Stuart number Ns = dimensionless entropy generation Nu = Nusselt number p = pressure, N m−2 P = dimensionless pressure Pr = Prandtl number Re = Reynolds number Ri = Richardson number S = total dimensionless entropy generation Ṡgen = local entropy generation, W m−3K−1 T = temperature, K u, v = velocity components, m s−1 U, V = dimensionless velocity components V0 = lid velocity, m s−1 V̅ = enclosure volume, m3 x, y = Cartesian coordinates, m X, Y = dimensionless Cartesian coordinates

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Greek Symbols

α thermal diffusivity, m2 s−1 β thermal expansion coefficient, K−1 θ dimensionless temperature ρ density of the fluid, kg m−3 σ electrical conductivity, Ω−1 m−1] μ dynamic viscosity, Pa s υ kinematic viscosity of the fluid, m2 s−1

Subscripts

av average c cold f fluid h hot s solid

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