Proceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT, Elounda, Greece, August 21-23, 2006 (pp357-362) Double Diffusive Convection from a Permeable Horizontal Cylinder of Elliptic Cross Section in a Saturated Porous Medium CHING-YANG CHENG Department of Mechanical Engineering Southern Taiwan University of Technology 1, Nantai Street, Yungkang 710 TAIWAN Abstract: - The double diffusive convection near a permeable horizontal cylinder of elliptic cross section with uniform wall temperature and concentration in a fluid-saturated porous medium are numerically studied. A coordinate transformation is employed to transform the governing equations into nondimensional nonsimilar boundary layer equations. The obtained boundary layer equations are then solved by the cubic spline collocation method. The influence of the transpiration parameter and the eccentricity on the heat and mass transfer characteristics near a permeable horizontal cylinder of elliptic cross section in a fluid-saturated porous medium is examined as the major axis of the elliptic cylinder is vertical (slender orientation) and horizontal (blunt orientation). Increasing the transpiration parameter tends to decrease the boundary layer thickness and thus enhances the heat and mass transfer rates between the fluid and the wall. Moreover, the heat and mass transfer rates of the cylinder with slender orientation are higher than those of the cylinder with blunt orientation. Key-Words: - Heat and mass transfer, Permeable, Natural convection, Porous medium, Elliptic cylinder, Cubic Spline collocation method, Coordinate transformation 1 Introduction characteristics by natural convection from a Coupled heat and mass transfer driven by combined horizontal cylinder embedded in porous media. thermal and solutal buoyancy forces in a Yücel  studied the heat and mass transfer about a fluid-saturated porous medium is of great importance vertical cylinder with constant wall temperature and in geophysical, geothermal and industrial concentration in a porous medium. Yih  examined applications, such as the extraction of geothermal the heat and mass transfer by natural convection from energy, the dispersion of chemical contaminants a permeable horizontal cylinder in a porous medium through water-saturated soil and the migration of with constant wall temperature and concentration. moisture through air contained in fibrous insulations. Merkin  studied the natural convection boundary Bejan and Khair  used Darcy’s law to study the layer flow on cylinders of elliptic cross section in a features of natural convection boundary layer flow porous medium. Pop et al.  examined the natural driven by temperature and concentration gradients. convection heat transfer about cylinders of elliptic Lai and Kulacki  studied the natural convection cross section in a porous medium. boundary layer along a vertical surface with constant Motivated by the works above, this article heat and mass flux including the effect of wall applied the coordinate transformation and the cubic injection. Yih  studied the heat and mass transfer spline collocation method to analyze the heat and characteristics in natural convection flow over a mass transfer by natural convection along a truncated cone subjected to variable wall temperature permeable horizontal cylinder of elliptic cross and concentration or variable heat and mass flux section embedded in fluid saturated porous media embedded in porous media. Cheng  uses integral with constant wall temperature and concentration. approach to study the magnetic effects on heat and The results obtained herein are compared with the mass transfer by natural convection from a vertical similarity solutions for horizontal cylinders obtained plate in a fluid-saturated porous medium. by Merkin  and by Yih  to check the accuracy. Similarity solutions for natural convection heat The influence of the transpiration parameter and the transfer on a horizontal cylinder in a saturated porous eccentricity on the heat and mass transfer medium have been presented by Merkin . Fand et characteristics near a permeable horizontal cylinder al.  examined experimentally the heat transfer of elliptic cross section in a fluid-saturated porous Proceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT, Elounda, Greece, August 21-23, 2006 (pp357-362) medium is examined when the major axis is horizontal (blunt orientation) and vertical (slender orientation). g b 2 Problem Formulation a Consider the effect of transpiration on the combined heat and mass transfer by free convection near a B buried horizontal cylinder of elliptic cross section with blunt orientation embedded in a homogeneous x fluid-saturated porous medium, as shown in Fig. 1, where a is the length of semi-major axis and b is the A length of semi-minor axis for the elliptical cylinder. y In this figure, A represents the angle made by the outward normal from the cylinder with the downward Fig. 1. Physical model and coordinates for an elliptic vertical and B is the eccentric angle. It should be cylinder of blunt orientation. noted that for cylinders of elliptic cross section there Here u and v are the volume-averaged velocity are two orientations to consider: the orientation is blunt when the major axis is horizontal, as shown in components in the x -direction and y -direction, Fig. 1, and the orientation is slender when the major respectively. T and C are the volume-averaged axis is vertical. temperature and concentration, respectively. The surface of the cylinder is held at a constant Property µ is the dynamic viscosity of the fluid, K temperature Tw which is higher than the ambient is the permeability of the porous medium, and ρ is porous medium temperature T∞ . In addition, the the fluid density. Furthermore, α and D are the concentration of a certain constituent in the solution equivalent thermal and mass diffusivity of the that saturates the porous medium varies from C w on saturated porous medium, respectively. β t and β c the fluid side of the surface of the cylinder to C∞ are the coefficients for thermal expansion and for concentration expansion of the saturated porous sufficiently far from the surface of the cylinder. The medium, respectively, and g is the gravitational transpiration velocity is uniform. The fluid properties are assumed to be constant except for density acceleration. Vw is the uniform transpiration variations in the buoyancy force term. velocity. With introducing the boundary layer and After introducing the stream function ψ to Boussinesq approximations, the equations governing satisfy the relations: u = ∂ψ ∂y and v = − ∂ψ ∂x , the steady-state conservation of mass, momentum, we then define the nondimensional variables: energy and constituent for Darcian flow through a homogeneous porous medium near the surface of the ξ = x / a , η = ( y a )Ra1 2 , ψ = ψ αRa1 2 ,( ) horizontal cylinder of elliptic cross section can be θ = (T − T∞ ) (Tw − T∞ ) , φ = (C − C ∞ ) (C w − C ∞ ) . written in two-dimensional Cartesian coordinates Equations (1)-(6) become the following equations: ( x, y ) as  ∂ψ = (θ + NC )sin A (7) ∂u ∂v ∂η + =0 (1) ∂x ∂y ∂ψ ∂θ ∂ψ ∂θ ∂ 2θ − = (8) u= K sin A [ρgβ t (T − T∞ ) + ρgβ c (C − C ∞ )] (2) ∂η ∂ξ ∂ξ ∂η ∂η 2 µ ∂ψ ∂φ ∂ψ ∂φ 1 ∂ 2φ ∂T ∂T ∂ T 2 − = (9) u +v =α 2 (3) ∂η ∂ξ ∂ξ ∂η Le ∂η 2 ∂x ∂y ∂y The associated boundary conditions are ∂C ∂C ∂ 2C ∂ψ V a u +v =D 2 (4) = − w 1 2 , θ = 1 , φ = 1 on η = 0 (10) ∂x ∂y ∂y ∂ξ αRa The appropriate boundary conditions are: ∂ψ = 0 , θ = 0 , φ = 0 as η → ∞ (11) v = Vw , T = Tw , C = C w on y = 0 (5) ∂η u = 0 , T = T∞ , C = C ∞ as y → ∞ (6) Proceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT, Elounda, Greece, August 21-23, 2006 (pp357-362) In the above equations, Ra = Kβ t ga (Tw − T∞ ) (αν ) Table 1. The value of Fi , j , Gi , j , and S i , j . is the Darcy-Rayleigh number, Le = α D is the Lewis number and N = β c (C w − C ∞ ) [β t (Tw − T∞ )] ⎡ ⎛ ∂θ ⎞ ⎤ n is the buoyancy ratio. Fi , j θ in j + ∆τ ⎢− ξ i l n ⎜ f ⎜ ⎟ ⎥ ⎟ ⎝ ∂ξ , ⎢ ⎠ i, j ⎥ A further transformation is needed for bodies ⎣ ⎦ with rounded lower ends because sin A ξ ⎡ ⎛ ∂f ⎞ ⎤ n +1 approaches a constant value as ξ approaches zero θ Gi , j ∆τ ⎢ f i ,nj + ξ i ⎜ ⎜ ⎟ ⎥ ⎟ . The new nondimensional variable is defined as ⎢ ⎣ ⎝ ∂ξ ⎠ i , j ⎥ ⎦ f (ξ ,η ) = ξ −1ψ (12) Si, j ∆τ Substituting Eq. (12) into Eqs. (7)-(9), we obtain the ⎡ ⎛ ∂φ ⎞ ⎤ n following boundary-layer governing equations: Fi , j φ n + ∆τ ⎢− ξ i l n +1 ⎜ ⎟ ⎥ f ⎜ ⎟ ⎝ ∂ξ ⎠ i , j ⎥ i, j ⎢ f′= sin A (θ + Nφ ) (13) ⎣ ⎦ ξ ⎡ ⎛ ∂f ⎞ ⎤ n ⎛ ∂θ ∂f ⎞ φ Gi , j ∆τ ⎢ f i ,nj + ξ i ⎜ ⎜ ⎟ ⎥ ⎟ θ ′′ + fθ ′ = ξ ⎜ f ′ ⎜ −θ ′ ⎟ (14) ⎢ ⎝ ∂ξ ⎠ i , j ⎥ ⎝ ∂ξ ∂ξ ⎟ ⎠ ⎣ ⎦ 1 ⎛ ∂φ ∂f ⎞ Si, j ∆τ φ ′′ + fφ ′ = ξ ⎜ f ′ − φ ′ ⎟ ⎜ ∂ξ (15) Le ⎝ ∂ξ ⎟ ⎠ Le The boundary conditions are Table 2. Comparison of values of Nu Ra for f = f w , θ = 1 , φ = 1 on η = 0 (16) N = 0 and b / a = 1 between the present results with f ′ = 0 , θ = 0 , φ = 0 as η → ∞ (17) the solutions reported by Merkin  and Yih . In terms of the new variables, the Darcian velocities in x- and y- directions can be expressed as ξ Merkin Yih  Present αξRa u= f′ (18)  results a 0 0.6276 0.6276 0.6276 αRa 1 2 ⎛ ∂f ⎞ v=− ⎜ f +ξ ⎜ ⎟ (19) 0.2 0.6245 0.6245 0.6245 a ⎝ ∂ξ ⎟ ⎠ 0.6 0.5996 0.5996 0.5997 Here primes denotes partial derivation with respect to η . As b / a = 1 , N = 0 and f w = 0 , Eqs. (13)-(14) 1.0 0.5508 0.5508 0.5510 become the equations for the Darcy natural 1.4 0.4800 0.4800 0.4804 convection heat transfer near a horizontal circular cylinder in a fluid-saturated porous medium 1.8 0.3901 0.3899 0.3904 presented by Merkin . f w = − (V w a ) αRa 1 2 is the 2.2 0.2847 0.2843 0.2849 transpiration parameter. Note that f w < 0 when V w > 0 (the case of blowing), and f w > 0 when 2.6 0.1679 0.1677 0.1680 V w < 0 (the case of suction). 3.0 0.0444 0.0446 0.0444 Here ξ and sin A can be given in terms of the eccentric angle B by the relations: (1) For blunt orientation: sin B sin A = (23) ( ) (1 − e ) B 12 ξ = ∫ 1 − e 2 sin 2 γ 12 dγ (20) cos 2 B 2 0 where e denotes the eccentricity expressed as ( ) b sin B sin A = (21) 12 2 ( a 1 − e sin 2 B 1 2 ) e = 1 − b2 a2 and b/a is the aspect ratio of the elliptic cylinder. When ξ approaches zero, as shown (2) For slender orientation: In Eqs. (20)-(23), the value of sin A ξ approaches 0 B ( ξ = ∫ 1 − e 2 cos 2 γ ) 12 dγ (22) the aspect ratio b / a for the elliptic cylinder with blunt orientation while the value of sin A ξ Proceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT, Elounda, Greece, August 21-23, 2006 (pp357-362) approaches the value of a 2 b 2 for the elliptic 1.6 slender cylinder with slender orientation. 1.4 fw=-0.3 The local Nusselt number can be written as fw=0 = −θ ′(ξ , 0 ) Nu 1.2 fw=0.3 (24) b/a=0.6 Ra1 2 1.0 Le=6 The local Sherwood number can be given by N=1 Sh blunt = −φ ′( ξ ,0 ) (25) Nu 0.8 Ra 1 2 Ra 0.5 In Eqs. (24)-(25), Nu = ha k and Sh = ja D where 0.6 h and j are the local heat transfer coefficient and 0.4 the local mass transfer coefficient, respectively. The average Nusselt number for the elliptic 0.2 cylinder can be derived as: (1) For blunt orientation 0.0 Nu m ∫ π ( θ ′( ξ ,0 ) 1 − e 2 sin 2 γ )12 dγ 0.0 0.4 0.8 1.2 1.6 B 2.0 2.4 2.8 3.2 =− 0 (26) ∫ (1 − e ) 12 π 12 Ra 2 sin 2 γ dγ Fig. 2. Effects of transpiration parameter on the local 0 Nusselt number. (2) For slender orientation Nu m ∫ π ( θ ′( ξ ,0 ) 1 − e 2 cos 2 γ ) 12 dγ 6.0 slender fw=-0.3 =− 0 (27) ∫ (1 − e ) 12 π 12 blunt fw=0 Ra 2 cos 2 γ dγ 5.0 fw=0.3 0 The average Sherwood number for the elliptic b/a=0.6 4.0 Le=6 cylinder can be given by: N=1 (1) For blunt oriention ( ) π Sh 3.0 ∫ 12 Shm φ ′( ξ ,0 ) 1 − e 2 sin 2 γ dγ Ra 0.5 =− 0 (28) ∫ (1 − e ) π Ra1 2 2 sin 2 γ 12 dγ 2.0 0 (2) For slender orientation 1.0 Shm ∫ π ( φ ′( ξ ,0 ) 1 − e 2 cos 2 γ ) 12 dγ =− 0 (29) ∫ (1 − e ) 12 π 12 0.0 Ra 2 cos 2 γ dγ 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 0 Note that Nu m = hm a k and Shm = j m a D where B hm and j m are the average heat transfer coefficient Fig. 3. Effects of transpiration parameter on the local and the average mass transfer coefficient for the Sherwood number. elliptic cylinder, respectively. layer η ∞ is about 12. Moreover, a grid with 150 grid 3 Problem Solution points is used in the ξ direction. At every grid point, The governing differential equations, Eqs. (14) and the iteration process continues until the convergence (15), and the appropriate boundary conditions, Eqs. criterion for all the variables, 10 −6 , is achieved. The (16) and (17), can be solved by the cubic spline present calculation for Eqs. (13)-(17) can be collocation method [11, 12]. The Simpson’s rule for performed from the bottom up to the top of the variable grids is used to calculate the value of f at elliptic cylinder without encountering a singularity. every position from Eq. (13) and boundary Here by using the cubic spline collocation method conditions (16) and (17). Variable grids with 200 grid [11, 12], Eqs. (14) and (15) can be discretized by points are used in the η direction. The minimum step using the false transient technique, as size is 0.01. The value of the edge of the boundary Proceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT, Elounda, Greece, August 21-23, 2006 (pp357-362) θ in, +1 − θ in, j n n 1.2 ⎛ ∂θ ⎞ ⎛∂ f ⎞ + ξil n ⎜ ⎟ − ξ i lθn +1 ⎜ ⎟ j f ⎜ ⎟ ⎜ ∂ξ ⎟ fw=0.3 ∆τ ⎝ ∂ ξ ⎠i, j ⎝ ⎠i, j Le=6 1.0 N=1 = Lθ +1 + f i n j lθn +1 n , (30) 0.8 φ in, +1 − φ in, j n⎛ ∂φ ⎞ n ⎛ ∂f ⎞ n +ξ il f ⎜ ⎟ − ξ i lφn +1 ⎜ ⎟ j ∆τ ⎜ ⎟ ⎜ ∂ξ ⎟ ⎝ ∂ξ ⎠ i , j ⎝ ⎠i, j Nu m 0.6 Ra 0.5 Lφ +1 n = + f i n j lφn +1 , (31) 0.4 slender Le blunt where ∆ξ = ξ i − ξ i −1 , ∆η = η i − η i −1 0.2 ⎛ ∂ϑ ⎞ ϑi , j − ϑi , j −1 ⎜ ⎟ = ⎜ ∂ξ ⎟ for i = 1 0.0 ⎝ ⎠ i, j ∆ξ i 0.25 0.40 0.55 0.70 0.85 1.00 b/a ⎛ ∂ϑ ⎞ − 3ϑi , j + 4ϑi , j −1 − ϑi , j − 2 ⎜ ⎟ = ⎜ ∂ξ ⎟ for i ≥ 2 Fig. 4. Effects of aspect ratio on the average Nusselt b ⎝ ⎠ i, j 2∆ξ i number. ∂ϑ ∂ 2ϑ lφ = , Lφ = (32) ∂η ∂η 2 4.0 Note that ϑ refers to θ and φ , and the quantity fw=0.3 Le=6 ∆τ = τ n +1 − τ represents the false time step. n 3.5 N=1 After some arrangement, Eqs. (30) and (31) can be written in the following spline approximation 3.0 form: Shm ϑin +1 = Fi , j + Gi , j lϑi+j1 + S i , j Lϑ+,1j ,j n n (33) 2.5 Ra 0.5 , i The quantities F, G, and S are known coefficients evaluated at previous time steps (Table 1). By using 2.0 the cubic spline relations [11, 12], Eq. (33) may be slender written in the following tridiagonal form as blunt 1.5 Ai , j ϑi , j −1 + Bi , j ϑi , j + C i , j ϑi , j +1 = Di , j (34) Here Eq. (34) can be easily solved by using the 1.0 Thomas algorithm. 0.25 0.40 0.55 0.70 0.85 1.00 In order to check the accuracy of the present method, the local Nusselt number Nu / Ra 0.5 for b/a b a = 1 and N = 0 obtained in the current study Fig. 5. Effects of aspect ratio on the average under Darcian assumptions for a horizontal circular Sherwood number. cylinder are compared with the solutions reported by Merkin  and Yih . As shown in Table 2, the and then decrease to zero at the top of the elliptical present results are found to be in excellent agreement cylinder. For an elliptical cylinder with slender with the results of Merkin  and Yih . orientation, the local Nusselt number and the local Figs. 2 and 3 plot the variation of the local Sherwood number decrease monotonically from the Nusselt number Nu / Ra 0.5 and the local Sherwood lower end of the cylinder to the upper end of the number Sherwood number Sh / Ra 0.5 as functions of cylinder; that is due to the increase in boundary layer the eccentric angle B of the elliptical cylinder for thickness. various transpiration parameters ( f w = -0.3, 0, 0.3), Comparing the curves in Figs. 2 and 3, we can deduce that increasing the transpiration parameter N=1, Le=6 and b/a=0.6. For the elliptical cylinder tends to decrease the boundary layer thickness and with blunt orientation, the local Nusselt number and thus increases the heat and mass transfer rates the local Sherwood number first increase with between the fluid and the wall. The results show that distance from the stagnation point, reach a maximum, Proceedings of the 4th WSEAS Int. Conf. on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT, Elounda, Greece, August 21-23, 2006 (pp357-362) the use of blowing ( f w < 0 and V w > 0 ) tends to References: decrease the heat and mass transfer rate while the use  A. Bejan, K.R. Khair, Heat and Mass Transfer by of suction ( f w > 0 and Vw < 0 ) increases the heat Natural Convection in a Porous Medium, International Journal of Heat and Mass Transfer, and mass transfer rate. Vol. 28, 1985, pp. 909-918. Figs. 4 and 5 show the average Nusselt number  F.C. Lai, F.A. Kulacki, Coupled Heat and Mass Nu m / Ra 0.5 and the average Sherwood number Transfer by Natural Convection from Vertical Shm / Ra 0.5 as a function of the aspect ratio b/a for Surfaces in Porous Media,” International Journal of Heat and Mass Transfer, Vol. 34, 1991, pp. f w = 0.3 , Le=6 and N=1. The average Nusselt 1189-1191. number and the average Sherwood number of the  K.A. Yih, Coupled Heat and Mass Transfer by elliptic cylinder with slender orientation are higher Free Convection over a Truncated Cone in Porous than those of the elliptical cylinder with blunt Media: VWT/VWC or VHF/VMF, Acta orientation for any aspect ratio b/a smaller than one. Mechanica, Vol. 137, 1999, pp. 83-97. When the aspect ratio b/a is increased (i.e., the  C.Y. 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The effects of the Medium, International Communications in Heat transpiration parameter and the aspect ratio on the and Mass Transfer, Vol. 26, 1999, pp. 431-440. Nusselt and Sherwood numbers for the permeable  J.H. Merkin, Free Convection Boundary Layers elliptical cylinders of blunt and slender orientations on Cylinders of Elliptic Cross Section, Journal of have been studied. The results show that increasing Heat Transfer, Vol. 99, 1977, pp. 453-457. the transpiration parameter tends to decrease the  I. Pop, M. Kumari, G. Nath, Free Convection boundary layer thickness and thus enhances the heat about Cylinders of Elliptic Cross Section and mass transfer rates between the fluid and the wall. Embedded in a Porous Medium, International Moreover, the heat and mass transfer rates of the Journal of Engineering Science, Vol. 30, 1992, cylinder with slender orientation are higher than pp. 35-45. those of the cylinder with blunt orientation.  S.G. Rubin, R.A. Graves, Viscous Flow Solution with a Cubic Spline Approximation, Computers and Fluids, Vol. 3, 1975, pp. 1-36. Acknowledements  P. Wang, R. Kahawita, Numerical Integration of This work was supported by National Science Partial Differential Equations Using Cubic Spline, Council of Republic of China under the grant no. International Journal of Computer Mathematics, NSC 94-2212-E-218-016. Vol. 13, 1983, pp. 271-286.
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