CHEMICALLYREACTING NATURAL CONVECTIVE HEAT AND MASS TRANSFER FROM AN ISOTHERMAL VERTICAL CYLINDER IN A NONDARCY POROUS MEDIUM: NUMERICAL SOLUTIONS
INTRODUCTION
Unsteady free convection flow of a viscous incompressible fluid along a vertical or horizontal heated cylinder is an important problem relevant to many engineering applications such as geothermal power generation and drilling operations, where the freestreem and buoyancy induced fluid velocities are of roughly the same order of magnit
CHEMICALLYREACTING NATURAL CONVECTIVE HEAT AND MASSTRANSFER FROM AN ISOTHERMAL VERTICAL CYLINDER IN A NONDARCY POROUSMEDIUM: NUMERICAL SOLUTIONS
INTRODUCTION
Unsteady free convection flow of a viscous incompressible fluid along a vertical or horizontalheated cylinder is an important problem relevant to many engineering applications such asgeothermal power generation and drilling operations, where the freestreem and buoyancy inducedfluid velocities are of roughly the same order of magnitude. The exact solution for these type of nonlinear problems is still out of reach. Sparrow and Gregg [1] provided the first approximate solution for the laminar buoyant flow of air bathing a vertical cylinder heated with a prescribed surfacetemperature, by applying the similarity method and power series expansion. Minkowycz and Sparrow[2] obtained the solution for the same problem using the nonsimilarity method. Fujii and Uehara [3]analyzed the local heat transfer results for arbitrary prandtl numbers. Lee et all. [4] investigated theproblem of natural convection in laminar boundary layer flow along slender vertical cylinders andneedles for the powerlaw variation in wall temperature. Dring and Gebhart[5] presented the transientnatural convection results in association with the thin wires in liquids. Velusamy and Grag [6]presented the numerical solution for the transient natural convection over heatgenerating verticalcylinders of various thermal capacities and radii. The rate of propagation of the leading edge effectwas given special consideration by them. Recently, Rani [7] has investigated the unsteady naturalconvection flow over a vertical cylinder with variable heat and mass transfer using the finite differencemethod. Ganesan and Loganathan [8] solved the problem of unsteady natural convective flow past amoving vertical cylinder with heat and mass transfer.Chemicallyreacting boundary layer flows have received considerable attention in theengineering sciences owing to extensive applications in combustion processes (Cheng and Kovitz1958) [9], hypersonic aerodynamics (Inger 1966) [10], rocket nozzle heat transfer (Daily et al 1974)[11], metallurgical flows (Burke
et al.
1988) [12] and chemical engineering processes (Oliver and DeWitt 1995) [13]. Both homogenous and heterogenous reactions may occur for which respectively,chemical reaction occurs
uniformly throughout a given phase
or
in a restricted region
(or boundary) of the phase (Levenspiel 1999) [14]. Muthucumaraswamy and Ganesan (Muthucumaraswamy andGanesan 2001) [15] presented finitedifference solutions for the effect of firstorder chemical reactionon flow past an impulsively started vertical plate with uniform heat and mass flux, showing thatchemical reaction parameter reduces velocities. Ganesan and Loganathan (Ganesan andLoganathan 2002) [16] investigated the chemically reactive heat and mass transfer in boundary layer convection from a moving vertical cylinder.
Computational transport modelling of buoyancyinduced (i.e. natural convection) flows ingeomaterials is of considerable interest in environmental and civil engineering sciences.Environmental transport processes in soils [17], forest fire development through brush and dry soils[18], geothermal processes [19], radioactive waste storage [20] and thermal plumes inmagmaticgeosystems constitute just a few important applications of geological thermofluid transport modelling. A thorough discussion of these and other applications is available in the monographs by Ingham andPop [21] and Nield and Bejan [22]. The vast majority of porous media transport models haveemployed the Darcian model which for isotropic, homogenous materials utilizes a single permeabilityfor simulating the global effects of the porous medium on the flow. Effectively in the context of viscoushydrodynamic modelling, for example using boundarylayer theory, the momentum conservationequation (unidirectional Navier±Stokes equation) is supplemented by an additional body force, theDarcian bulk linear drag. Numerous studies in the context of transport modelling in soil mechanics,petroleum displacement in reservoirs, geothermics, geohydrology and filtration physics haveemployed such an approach. For example, Singh et al. [23] studied the free convection flow and heattransfer in a Darcian porous geomaterial using perturbation methods; this study also incorporatedpermeability variation via a transverse periodic function. Thomas and Li [24] analyzed numerically theunsteady coupled heat and mass transfer in unsaturated soil with a Darcy model. The Darcy modelassumes that the pressure drop across the geomaterial is proportional to the bulk drag force. Athigher velocities, however,
inertial effects
become important and the regime is no longer
viscousdominated
. Themost popular approach for simulating highvelocity transport in porous media, whichmay occur for example under strong buoyancy forces, through highly porous materials, etc., is theDarcy±Forchheimer drag force model. This adds a secondorder (quadratic) drag force tothemomentum transport equation. This term is related to the geometrical features of the porousmedium and is independent of viscosity, as has been shown rigorously by Dybbs and Edwards [25].In the context of coupled heat and mass transfer studies, Takhar et al. [26] studied the doublediffusive heat and species transport in porous media using the Darcy±Forchheimer model. Takhar and Bég [27] used the Keller±Box implicit difference method to analyze the viscosity and thermalconductivity effects in boundary layer thermal convection in nonDarcian porous media. Beg et al. [28]studied using NSM the unsteady hydrodynamic couette flow through a rotating porous mediumchannel using a Forchheimerextended Darcy model.This work however only considered the vertical cylinder scenario. Clearly much remains to beexplored in this topic, both in the way of geometrical bodies in the porous media (e.g. cones, spheres,ellipsoids, wedges, cylinders) and also regarding interactive effects of the thermophysical parameters.The objective of the present work is therefore to investigate the natural convection simultaneous heatand mass transfer from a vertical cylinder embedded in fluid saturated porous medium with Darcian
resistance, Forchheimer quadratic drag, chemical reaction, thermal Grashof number, species Grashof number, Prandtl number and Schmidt number effects. The effects of governing multiphysicalparameters on heat and mass transfer characteristics are analyzed. The equations of continuity,linear momentum, energy and diffusion, which governed flow field, are solved by using an implicitfinite difference method of Crank ± Nicolson type.The behavior of the velocity, temperature concentration skin friction, Nusselt and Sherwoodnumbers have been discoursed for variations in governing parameters, and benchmarked whereappropriate with previous studies.
2. Mathematical analysis
An unsteady twodimensional laminar natural convection boundary layer flow of a viscousincompressible fluid past a uniformly heated semiinfinite vertical cylinder of radius
0
r
is consideredas shown in Fig. 1.The xaxis is measured vertically upward along the axis of the cylinder. The srcin of x is taken to beat the leading edge of the cylinder, where the boundary layer thickness is zero. The radial coordinater is measured perpendicular to the axis of the cylinder. The surrounding stationary fluid temperature isassumed to be of ambient temperature (
w
T
¢
). Initially, i.e., at
t
¢
=
0 it is assumed that the cylinder andthe fluid are of the same temperature
T
¥
¢
. When
0
t
¢>
, the temperature of the cylinder is raised to
g
r
w
T
d
x
T
g
O
w
T
¢
( )
T
¥
¢
and maintained at the same level for all the time
t
¢
>
0. It is assumed that the effect of viscous dissipation is negligible in the energy equation. Under these assumptions, the boundary layer equations of mass, momentum and energy with Boussinesq¶s approximation are as follows:Continuity equation
( ) ( )0
ru rv x r
¶ ¶+ = ¶ ¶
(1)Momentum equation
* 2
( ) ( )
u u u u bu v g
T T
g C C r u u
t
x r r r r K K n nb b
¥ ¥
æ ö
¶ ¶ ¶ ¶ ¶÷ç
¢ ¢ ¢ ¢
+ + =  +  +  ÷ç÷ç
¢è ø
¶ ¶ ¶ ¶ ¶
(2)Energy equation
1
p
T T T k T u v r
t
x r c r r r r
æ
ö¢ ¢ ¢ ¢
¶ ¶ ¶ ¶ ¶ççç
¢
¶ ¶ ¶ ¶
è
¶
ø
(3)Mass diffusion equation
l
C C C D C u v r K C
t
x r r r r
æ ö¢ ¢ ¢ ¢
¶ ¶ ¶ ¶ ¶÷ç
¢
+ + = ÷ç÷÷ç
¢
¶ ¶ ¶ ¶
è
¶
ø
(4)The initial and boundary conditions are:
0: 0, 0, ,
t
u v
T T
C C
¥ ¥
¢ ¢ ¢ ¢ ¢£
= = = =
for all
0
x
³
and
0
r
³
0
0: , 0, ,
w w
t
u u v T T
¢ ¢ ¢ ¢ ¢= = = =
at
0
r r
=
0, 0, ,
u v
T T
C C
¥ ¥
¢ ¢ ¢ ¢
= = = =
at
0
0
x and r r
=
³
(5)
0, ,
u T T
¡¡
¥ ¥
¢ ¢ ¢ ¢® ® ®
as
r
® ¥
where
u
,
v
are the velocity components in
x
,
r
directions respectively,
t
¢
 the time,
g
 theacceleration due to gravity,
F

the volumetric coefficient of thermal expansion,
*
F

the volumetriccoefficient of expansion with
concentration,
T
¢
 the temperature of the fluid in the boundary layer
,
C
¢
 the species concentration in the boundary layer
,
w
T
¢
 the wall temperature,
w
¢
¢
 theconcentration at the wall,
T
¥
¢
 the free stream temperature of the fluid far away from the plate,
C
¥
¢

the species concentration in fluid far away from the cylinder,
R

the kinematic viscosity
,
V
 thedensity of the fluid,
p
c
 the specific heat at constant pressure, kpermeability, bForchheimer geometrical inertial parameter for the porous medium K
l
± Chemical reaction rate and
D

the speciesdiffusion coefficient. In the momentum equation (2) the first two terms on the righthand siderepresent the thermal buoyancy body force and the species buoyancy body force, respectively. Thepenultimate term is the Darcian linear drag and the final term is the Forchheimer quadratic drag force.