Cmjv02i02p0336 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.2 (2), 336-351 (2011)

A Numerical Study on Developed MHD Laminar Mixed Convection in Vertical Channels G. ARUNA*, B. RAMA BHUPAL REDDY, S.V.K. VARMA and N. CH. S. N. IYENGAR *Dept. of Applied Mathematics, Y.V. University, Kadapa – 516 003, A.P. India Dept. of Mathematics, K.S.R.M. College of Engineering, Kadapa-516 003, A.P. - India. email: reddybrb@gmail.com

Dept. of Mathematics, S.V. University, Tirupati-517 502, A.P.- India. School of Computing Science and Engineering, VIT University, Vellore-632014, T. N., India ABSTRACT In this paper, a numerical study on developed MHD laminar mixed convection in vertical channels is considered. The flow problem is described by means of partial differential equations and the solutions are obtained by an implicit finite difference technique coupled with a marching procedure. The velocity, the temperature and the pressure profiles are obtained and their behaviour is discussed computationally for different values of governing parameters like buoyancy parameter Gr/Re, the wall temperature difference ratio rT, the magnetic field parameter ‘M’ and Prandtl number Pr. The analysis of the obtained results showed that the flow is significantly influenced by these governing parameters. Keywords: Mixed Convection, Vertical Channels, Buoyancy Effects and MHD.

1. INTRODUCTION Combined forced and free convection flows (or mixed convection flows) inside vertical channels, such as parallel-plate channels, circular tubes, concentric and eccentric annular ducts, are encountered in many industrial applications, engineering devices. Investigations have

shown considerable changes in heat transfer and friction coefficient due to mixed convection effects. This may provide the potential for optimizing some designs of heat transfer devices such as cooling systems for electronic components and reactors, cooling passages in turbine blades, combustion chambers, and many other heat exchanging surfaces.

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With the technological demand on heat transfer enhancement, most markedly related to compact heat exchangers, solar energy collection, as in the conventional flat plate collector and cooling of electronic equipment analysis, the parallel-plate channel geometry gained further attention from thermal engineering researchers. Using dimensionless parameters, Aung and Worku2 solved the problem of mixed convection between parallel plates and obtained closed form analytical solution. From the closed form solution for U, the criterion for the existence of reversed flow has been deduced under thermal boundary conditions of uniform heating on one wall while the other wall was thermally insulated. An analysis of the mixed convection in a channel provides information on the flow structure in the developing region and reveals the different length scales accompanying the different convective mechanisms operative in the developing flow region. Quantitative information on the temperature and velocity fields has been provided in a numerical study reported by Aung and Worku1. These authors noted that buoyancy effects dramatically increase the hydrodynamic development distance. With asymmetric heating, the bulk temperature is a function of Gr/Re and rT, and decreases as rT is reduced. Buoyancy effects are noticeable through a large segment of the channel, but not near the channel entrance or far downstream from it. Reddy8 studied on developed laminar mixed convection in vertical channels. Yih Nen Jeng et al.12 were studied on the Reynolds- number independence of mixed convection in a vertical channel subjected to asymmetric wall temperatures

with and without flow reversal. Evans and Greif6 were studied buoyant instabilities in downward flow in asymmetrically heated vertical channel. Barletta3 studied laminar and fully developed mixed convection in a vertical rectangular duct with one or more isothermal walls. The analysis refers to thermal boundary conditions such that atleast one of the four duct wall is kept isothermal. The in-depth literature cited above has revealed that, in spite of the huge amount of knowledge accumulated over the last few decades into the subject of laminar mixed convection in vertical channels, the mixed convection in such geometry is still not fully understood, especially the hydrodynamics of it. For instance, the term “aiding flow” is generally used in the literature as synonym for “up-flow in a heated vertical channel or down-flow for a cooled vertical channel” and vice versa for the term “opposing flow”. The general findings in the literature are that buoyancy effects enhance the pressure drop in buoyancy-aided flow (upward flow in a vertical heated channel), i.e., there is a monotonic pressure decrease of pressure in the axial direction of upward flow in a vertical heated channel. Chamka5, studied on laminar hydro magnetic mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions. Mazumber et al.7 studied hall effects on combined free and forced convective hydromagnetic flow through a channel. Umavathi and Malashetty11 analyzed combined free and forced convective magneto hydrodynamic flow in a vertical

Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-383)

G. Aruna, et al., J. Comp. & Math. Sci. Vol.2 (2), 336-351 (2011)

channel. The channel walls are maintained equal were are at different constant temperature. 2. b Cp g Gr Ho

J k M p p0 P

NOMENCLATURE Distance between the parallel plates Specific heat of the fluid Gravitational body force per unit mass (acceleration) Grashoff number, gβ (T1 – T0) b 3 /υ 2 Applied Magnetic Field Current Density vector Thermal conductivity of fluid Magnetic Parameter Local pressure at any cross section of the vertical channel Hydrostatic pressure Dimensionless pressure inside the channel at any cross section,(p – p0) /ρ u0 2 Prandtl Number, µ Cp / k

Velocity Vector Re Reynolds number, (b u0)/υ T Dimensional temperature at any point in the channel Ambient of fluid inlet temperature T0 Tw Isothermal temperatures of circular heated wall T1,T2 Isothermal temperatures of plate 1 and plate 2 of parallel plates u Axial velocity component Average axial velocity u u0 Uniform entrance axial velocity U Dimensionless axial velocity at any point, u / u0 v Transverse velocity component V Dimensionless transverse velocity, v / u0 x Axial coordinate (measured from the channel entrance)

X y Y θ ρ ρ0 µ µe σ

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Dimensionless axial coordinate in Cartesian, x / (b Re) Transverse coordinate of the vertical channel between parallel plates Dimensionless transverse coordinate, y/b Dimensionless temperature Density of the fluid Density of the fluid at the channel entrance Dynamic viscosity of the fluid Magnetic Permeability Electrical Conductivity Kinematic viscosity of the fluid, µ/ρ Volumetric coefficient of thermal expansion

3. FORMULATION OF THE PROBLEM We consider the laminar steady flow in a vertical channel. Both channel walls are assumed to be isothermal, one with temperature T1 and the other with temperature T2. The system under consideration as well as the choice of the coordinate axes is illustrated in Fig.1. The distance between the plates is ‘b’ i.e., the channel width. The Cartesian coordinate system is chosen such that the x-axis is in the vertical direction that is parallel to the flow direction and the gravitational force ‘g’ always acting downwards independent of flow direction. The y-axis is orthogonal to the channel walls, and the origin of the axes is such that the positions of the channel walls are y = 0 and y = b. The classic boundary approximation is invoked to model the buoyancy effect. A uniform transverse magnetic field of strength Ho is applied perpendicular to the plates.

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Continuity Equation

∂U ∂V + =0 ∂X ∂Y

→ →

(1)

X – Momentum Equation U

∂U ∂V dP Gr ∂ 2U +V =− + θ+ − M 2U ∂X ∂Y dX Re ∂Y 2

(2)

Energy Equation

→ → →

∂θ ∂θ 1 ∂ 2θ +V = ∂X ∂Y Pr ∂Y 2

(3)

Where M2 = σµ2e Ho2 b2/µ The form of continuity equation can be written in integral form as 1

∫ U dY = 1

(4)

Figure 1: Schematic view of the system and coordi coordinate axes corresponding to Up Up-flow

The governing equations for the steady viscous flow of an electrically conducting fluid in the presence of external magnetic field with the following assumptions are made: (i)

The flow is steady, viscous, incompressible and developed. (ii) The flow is assumed to be two twodimensional steady, and the fluid properties are constant except for the variation of density in the buoyancy term of the momentum equation. (iii) The electric field E and induced magnetic field are re neglected9, 10. (iv) Energy dissipation is neglected. After applying the above assumptions the boundary layer equations appropriate for this problem are

The boundary conditions are Entrance conditions At X = 0, 0< Y< 1 : U = 1, V = 0, θ = 0, P=0 (5a) No slip conditions : U = 0, V = 0 At X > 0, Y = 0 At X > 0, Y = 1 : U = 0, V = 0 (5b) Thermal boundary conditions At X > 0, Y = 0 : θ=1 At X > 0, Y = 1 : θ = rT (5c) In the above, dimensionless parameters have been defined as U = u / u0, V = vb /υ , X = x / (b Re), Y=y/b P = (p – p0 ) /ρ u0 2 , Pr = µ Cp / k , (6) Re = (b u0)/υ Gr = gβ (T1 – T0) b 3 /υ 2 , θ= (T – T0) / (T1 – T0) The system of non-linear linear equations (1) to (3) are solved by a numerical method based on finite difference approximations. An implicit difference technique is

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employed whereby the differential equations are transformed into a set of simultaneous linear algebraic equations. 4. NUMERICAL SOLUTION The solution of the governing

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equations for developing flow is discussed in this section. Considering the finite te difference grid net work of figure. 2, equations (2) and (3) are replaced by the following difference equations which were also used in4.

Figure 2: Mesh Network for Difference Representations

U (i, j ) =

U (i + 1, j ) − U (i, j ) U (i + 1, j + 1) − U (i + 1, j − 1) + V (i, j ) ∆X 2∆Y

U(i +1, j +1) − 2U(i +1, j) +U(i +1, j −1) P(i +1) − P(i) Gr − + θ(i +1, j) − M 2U(i +1, j) 2 ∆ X Re (∆Y)

U (i, j )

θ (i + 1, j ) − θ (i, j ) ∆X

+ V (i, j )

(7)

θ (i + 1, j + 1) − θ (i + 1, j − 1) 2∆Y

1 θ (i + 1, j + 1) − 2θ (i + 1, j ) + θ (i + 1, j − 1) Pr ( ∆Y ) 2

(8)

Numerical representation of the Integral Continuity Equation

trapezoidal rule of numerical integration and is as follows:

The integral continuity equation can represented by the employing a

 n  ∑ U ( i + 1, j ) + 0.5 (U ( i + 1, 0 ) + U ( i + 1, n + 1) )  ∆Y = 1  j =1 

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However, from the no slip boundary conditions

 n  ∑ U ( i + 1, j )  ∆Y = 1  j =1 

U(i+1, 0) = U(i+1, n+1) = 0

A set of finite-difference equations written about each mesh point in a column for the equation (7) as shown:

Therefore, the integral equation reduces to:

(9)

β 1U (i + 1,1) + γ 1U (i + 1,2 ) + ξP(i + 1) +

Gr θ (i + 1,1) = φ1 Re Gr α 2U (i + 1,1) + β 2U (i + 1,2 ) + γ 2U (i + 1,3) + ξP (i + 1) + θ (i + 1,2 ) = φ 2 Re - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- -

α nU (i + 1, n − 1) + β nU (i + 1, n ) + ξP (i + 1) + where

αk = γk =

1 2

−

( ∆Y ) 1

( ∆Y )

V ( i, j ) 2 ∆Y V ( i, j ) 2 ∆Y

Gr θ (i + 1, n ) = φ n Re



β k = −

ξ=

 (∆Y )

 U (i, j ) −M 2 ∆X   P ( i ) + U 2 ( i, j )   ∆X  

−1 , ∆X

φk = − 

for k = 1,2… n A set of finite-difference equations written about each mesh point in a column for the equation (8) as shown: β 1θ (i + 1,1) + γ 1θ (i + 1, 2) + − − − − − − − − − − − − −− = φ 1 − α 1 ,

α 2θ (i + 1,1) + β 2θ ( i + 1, 2 ) + γ 2θ (i + 1,3) + − − − − − − − − − − − = φ 2

- - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- -

α nθ (i + 1, n − 1) + β nθ ( i + 1, n ) = φ n − rT γ n

where

αk = γk =

1 2

−

Pr ( ∆Y ) 1 Pr ( ∆Y )

V ( i, j ) 2 ∆Y V ( i, j ) 2 ∆Y

U ( i, j )   2 ∆X   Pr ( ∆Y ) U ( i , j ) θ ( i, j )



βk = −

φk = −

∆X

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Equation (1) can be written as V (i + 1, j ) = V (i + 1, j − 1) −

∆Y (U (i + 1, j ) + U (i + 1, j − 1) − U (i, j ) −U (i, j − 1)), j = 1,2,.....n 2∆x

The numerical solution of the equations is obtained by first selecting the parameter that are involved such as Gr/Re, M, Pr and rT. Then by means of a marching procedure the variables U, V, θ and P for each row beginning at row (i+1) = 2 are obtained using the values at the previous row ‘i’. Thus, by applying equations (7), (8) and (9) to the points 1, 2, ...., n on row i, 2n+1 algebraic equations with the 2n+1 unknowns U(i+1,1), U(i+1,2),……., U(i+1,n), P(i+1), θ(i+1,1), θ(i+1,2), . . . . θ(i+1, n), θ(i+1, n) are obtained. This system of equations is then solved by Gauss – Jordan elimination method. Equations (10) are then used to calculate V(i+1,1), V(i+1,2), . . . ., V(i+1,n). 5. RESULTS AND DISCUSSION The numerical solution of the equation is obtained first selecting the parameters that are involved such as Gr/Re, rT, M and Pr. For fixed Pr = 0.7 and different values of Gr/Re, M and rT, the velocity profiles are shown in figures 3 to 6 and the temperature profiles are shown in figures 7 to 9. Figures 3 to 5 executive velocity profile fixed buoyancy parameter Gr/Re, rT and M. as can be seen, vary closed to the channel inlet, the heating effects are not yet and further down stain, the heating causes the fluid accelerate near the hot wall and decelerate near the and heated wall resulting

(10)

distortion of velocity profile to satisfy the continuity principle. However the velocity profile recovers and attains it asymptotic fully developed parabolic profile. A skewness in the velocity profile on the also appear as the fluid moves to ward hot wall.(y-=1 for fixed M). The smaller rT, the greater is the skewness. On then other hand increased buoyancy introduces a more severe distortion as illustrated in figure 6. The development of the temperature field is exemplified by figures 7 to 9. The effect of the buoyancy later parameter is felt in the developing regions, where the buoyancy decreases the temperature in the region adjacent to the hot wall. While increasing the temperature else where in the flow. Thus the buoyancy tense to equalize the temperature in the fluid. It is seen that increasing the magnetic field parameter M for fixed Gr/Re at rT for different X values the temperature is decreases. Figure 10 shows the variation of the dimensionless pressure parameter P. The figures indicate the steam wise variation of the parameter at different Gr/Re for fixed magnetic field parameter M. In the upper range of Gr/Re the maximum pressure occurs about the point where buoyancy effects begin to be felt and the centerline velocity starts to decrease for fixed magnetic field parameter M. In the same range it is also observed that P becomes positive when the center line velocity attains a value of less than that of the entry velocity.

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REFERENCES 1. Aung. W. Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, (1987). 2. Aung. W., and Worku, G. Developing flow and flow reversal in a vertical channel with asymmetric wall temperatures”, ASME. J. Heat Transfer, Vol. 108, pp.299- 304, (1986). 3. Barletta, A. Fully developed mixed convection and flow reversal in a vertical rectangular duct with uniform wall heat flux, Int. J. Heat Mass Transfer, Vol.45, pp.641-654, (2002). 4. Bodoia, J. R., and Osterle, J.F. The development of free convection between ed vertical plates, ASME. J. Heat Transfer, Vol. 84, pp.40-44, (1962). 5. Chamkha Ali, J. On laminar hydromagnetic mixed convection flow in a vertical channel with symmetric and asymmetric wall heating conditions, Int. J. Heat Mass Transfer, Vol.45, pp.25092525, (2002). 6. Evans, G., and Greif, R. Buoyant instabilities in downward flow in asymmetrically heated vertical channel, Int. J. Heat Mass Transfer, Vol.40,

No.10, pp.2419-2430, (1997). 7. Muzumder, B.S. Gupta, A.S., and Datta, N, Hall Effects on combined free and forced convective hydromagnetic flow through a channel, Int. J. Engg., Sci., Vol.14, pp.285-292, (1976). 8. Reddy B.R.B., HEAT TRANSFER IN FLUID FLOW PROBLEMS, Ph.D. Thesis, S.V. University, Tirupati, May (2008). 9. Rossow, V. J., “On flow of electrically conducting fluids over a Flat plate in the presence of a Transverse Magnetic Field;’, NASA-TN. 3971, (1957). 10. Sparrow E.M., and Cess, R.D., Effect of Magnetic Field on the Convection heat transfer, ASME J. Appl. Mech. Vol.29, 181, (1962). 11. Umavathi, J.C. and Malashetty, M.S., Magneto-hydrodynamics mixed convection in a vertical channel”, Int. J. Non Linear Mech. Vol.40, pp.91-101, (2005). 12. Yih Nen Jeng and Jiann Lin Chen., Yeong-Ysong Chen and Aung, W. Heat transfer enhancement in a vertical channel with asymmetric isothermal walls by local blowing are suction, Int. J. Heat and Fluid Flow, Vol.16, pp.2535, (1995).

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