Double Diffusive Convection from a Permeable Horizontal Cylinder of by pharmphresh23

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									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 [7] 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 [8] 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 [9] studied the natural convection boundary
           Bejan and Khair [1] 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. [10] examined the natural
       driven by temperature and concentration gradients.                convection heat transfer about cylinders of elliptic
       Lai and Kulacki [2] 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 [3] 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 [4] 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 [5] and by Yih [8] 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 [5]. Fand et                 characteristics near a permeable horizontal cylinder
       al. [6] 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 [10]                                                   ∂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 ⎜
                                                                                                                                 ⎜       ⎟ ⎥
                                                                                                                                         ⎟
       [10]. 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 [5] and Yih [8].
       In terms of the new variables, the Darcian velocities
       in x- and y- directions can be expressed as
                                                                                     ξ                Merkin             Yih [8]        Present
             αξRa
        u=            f′                                  (18)                                          [5]                             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 [5]. 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 [5] and Yih [8]. 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 [5] and Yih [8].                            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            [1] 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
                                                                         [2] 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.
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       slender orientation equal to those of blunt orientation              of Arbitrary Shape in a Saturated Porous Medium,
       when the aspect ratio b/a equals to one. Therefore,                  International Journal of Heat and Mass Transfer,
       the elliptic cylinders of slender orientation are found              Vol. 22, 1979, pp. 1461-1462.
       to be superior to the elliptic cylinders of blunt                 [6] R.M. Fand, T.E. Steinberger, P. Cheng, Natural
       orientation from the viewpoint of the heat and mass                  Convection Heat Transfer from a Horizontal
       transfer rates in fluid-saturated porous media.                      Cylinder Embedded in a Porous Medium,
                                                                            International Journal of Heat and Mass Transfer,
                                                                            Vol. 29, 1986, pp. 119-133.
       4 Conclusion                                                      [7] A. Yücel, Natural Convection Heat and Mass
       The coupled heat and mass transfer by natural                        Transfer along a Vertical Cylinder in a Porous
       convection of a permeable horizontal cylinder with                   Medium, International Communications in Heat
       elliptic cross section has been studied. Here a                      and Mass Transfer, Vol. 33, 1990, pp.
       coordinate transformation is employed to transform                   2265-2274.
       the governing equations into nondimensional                       [8] K.A. Yih, Coupled Heat and Mass Transfer by
       nonsimilar boundary layer equations. The obtained                    Natural Convection Adjacent to a Permeable
       boundary layer equations are then solved by the cubic                Horizontal Cylinder in a Saturated Porous
       spline collocation method. 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                    [9] 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                 [10] 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.                     [11] S.G. Rubin, R.A. Graves, Viscous Flow
                                                                            Solution with a Cubic Spline Approximation,
                                                                            Computers and Fluids, Vol. 3, 1975, pp. 1-36.
       Acknowledements                                                   [12] 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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