Studies of mass transfer in a porous medium are of interest to researchers as a result of its various uses in different fields of engineering practices. This work examined the steady flow of a reactive variable viscous fluid in a porous cylindrical pipe. Dimensionless variables were used to dimensionalize the governing equations. A regular perturbation technique was employed to obtain an approximate solution of the resulting dimensionless non-linear equations. Numerical simulation was done to get the threshold values for the flow parameters under consideration. The effects of viscous heating and permeability parameters on the steady flow were studied and reported.
Fluid is a substance that yields readily to any force that alters its shape. Thus, it conforms to the configuration of a containing vessel. In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress [
Pipe, in technology, is a tube (usually cylindrical) used to transport liquids, liquid-solid mixtures, or fragmented solids from one point to another. In petrochemical industries and petroleum refineries, studies related to thermal ignition and heat transfer in a reactive variable viscosity fluid are extremely useful in order to ensure safety of lives and properties [
The study of flow of viscous fluid with temperature dependent properties is of great importance in lubrication and tribology (study of interacting moving surfaces, i.e. the science and technology of interacting surfaces in relative motion, including the study of friction, lubrication and wear), food processing, instrumentation and viscometry. However, viscous heating is always a possible and frequently significant source of errors in viscometric measurement at high shear rates in instrumentation and viscometry [
Many fluids used in industrial and engineering processes like geological materials, liquid foams, polymeric fluids, hydrocarbon oils and grease do exhibit flow characteristics that cannot be adequately described by the classical linearly viscous fluid model [
Several researchers have worked on reactive variable viscosity flow with different non-porous channels [
The objective of this paper is to study the steady flow of a reactive variable viscous fluid in a porous cylindrical pipe with an isothermal wall under Arrhenius kinetics and to report the effect of porosity and heating parameters on velocity and temperature of the flow.
The work on steady flow of a reactive variable viscosity fluid in a cylindrical pipe with an isothermal wall under Arrhenius kinetics [
1 r ¯ d d r ¯ ( r ¯ d T d r ¯ ) + Q C 0 A k e − E R T + μ k ( d u ¯ d r ¯ ) 2 = 0 (1)
1 r ¯ d d r ¯ ( μ r ¯ d u ¯ d r ) + μ k 1 u ¯ = − G (2)
u ¯ = 0 , T = T 0 , on r ¯ = a d T d r ¯ = d u ¯ d r ¯ = 0 , on r ¯ = 0 (3)
where μ 0 , Q , C 0 , A , E , R , T , G are the fluid reference viscosity, heat of reaction, initial concentration of the reactant species, rate of constant, activation energy, universal gas constant, absolute temperature and constant axial
pressure gradient respectively, the last term μ k 1 u ¯ in the left hand side of Equation (2) is the porosity force, k 1 is the Darcy permeability [
μ = μ 0 e E R T (4)
where μ 0 , E , R , T are as defined.
Introducing the following dimensionless quantities into Equations (1)-(4),
r = r ¯ a , u = u ¯ b , θ = E R T 0 2 ( T − T 0 ) , ∈ = R T 0 E , e − E R T = e θ 1 + ∈ θ e − 1 ∈ , λ = a 2 Q C 0 A k ∈ T 0 e − 1 ∈ , β = G 2 a 2 μ 0 Q C 0 A , b = G a 2 μ 0 e 1 ∈
the dimensionless form of the governing equations are obtained as:
1 r d d r ( r d θ d r ) + λ e θ 1 + ∈ θ + λ β e − θ 1 + ∈ θ ( d u d r ) 2 = 0 (5)
1 r d d r ( r e − θ 1 + ∈ θ d u d r ) + δ e − θ 1 + ∈ θ u = − 1 (6)
with dimensionless boundary condition
u ( 1 ) = θ ( 1 ) = 0 , d u ( 0 ) d r = d θ ( 0 ) d r = 0 (7)
where δ = a 2 k 1 is the dimensionless porosity (permeability) parameter.
For all fuels of interest, the parameter ∈ is assumed small [
1 r d d r ( r d θ d r ) + λ e θ + λ β e − θ ( d u d r ) 2 = 0 (8)
1 r d d r ( r e − θ d u d r ) + δ e − θ u = − 1 (9)
where Equations (8) and (9) represent the temperature and velocity equations respectively.
Method of SolutionThe coupled nonlinear ordinary differential Equations (8) and (9), with the dimensionless boundary conditions (7), made it difficult to obtain exact solution. Since the cylindrical coordinate has singularity at r = 0 , singular perturbation technique was employed in order to obtain an approximate solution of the flow field and thermal distribution. It is convenient to take a power series expansion in the Frank-Kamenetskii parameter λ , and for easy computation, the solution to Equations (8) and (9) is assumed to be in the form
u ( r ) = ∑ i = 1 ∞ λ i u i = u 0 ( r ) + λ u 1 ( r ) + λ 2 u 2 ( r ) + λ 3 u 3 ( r ) + ⋯ (10)
and
θ ( r ) = ∑ i = 1 ∞ λ i θ i = θ 0 ( r ) + λ θ 1 ( ( r ) + λ 2 θ 2 ( r ) + λ 3 θ 3 ( r ) + ⋯ (11)
Substituting Equations (10) and (11) into (8) and (9) using the dimensionless boundary conditions (7), we have
θ ( r ) = λ ( α 4 + α 5 r 2 + α 6 r 4 + α 7 r 6 + α 8 r 8 ) + λ 2 ( α 12 + α 13 r 2 + α 14 r 4 ) + λ 3 ( α 18 + α 19 r 2 + α 20 r 4 ) (12)
and
u ( r ) = α 1 + α 2 r 2 + α 3 r 4 + λ ( α 9 + α 10 r 2 + α 11 r 4 ) + λ 2 ( α 15 + α 16 r 2 + α 17 r 4 ) (13)
where
α 1 = − − 16 + δ 64 − 16 δ + δ 2 ;
α 2 = ( − 16 + δ ) δ 4 ( 64 − 16 δ + δ 2 ) − 1 4 ;
α 3 = − ( − 16 + δ ) δ 2 64 ( 64 − 16 δ + δ 2 ) + 1 64 δ ;
α 4 = 1 4 β α 3 2 + 4 9 α 2 α 3 β + 1 4 β α 2 2 + 1 4 ;
α 5 = − 1 4 ;
α 6 = − 1 4 β α 2 2
α 7 = − 4 9 α 2 α 3 β ;
α 8 = − 1 4 α 3 2 β ;
α 9 = − 16 δ α 4 α 1 − 64 α 4 α 2 − 32 α 5 α 2 − 64 α 4 α 3 − 4 δ α 5 α 1 + δ 2 α 4 α 1 64 − 16 δ + δ 2 ;
α 10 = − ( − 16 δ α 4 α 1 − 64 α 4 α 2 − 32 α 5 α 2 − 64 α 4 α 3 − 4 δ α 5 α 1 + δ 2 α 4 α 1 ) δ 4 ( 64 − 16 δ + δ 2 ) + α 4 α 2 + 1 4 δ α 4 α 2 ;
α 11 = ( − 16 δ α 4 α 1 − 64 α 4 α 2 − 32 α 5 α 2 − 64 α 4 α 3 − 4 δ α 5 α 1 + δ 2 α 4 α 1 ) δ 2 64 ( 64 − 16 δ + δ 2 ) + 1 16 δ α 4 α 2 + 1 2 α 5 α 2 + α 4 α 3 + 1 16 δ α 5 α 1 − 1 64 ( 4 α 4 α 2 + δ α 4 α 1 ) δ ;
α 12 = 1 16 ( 64 − 16 δ + δ 2 ) ( 256 α 4 − 64 δ α 4 + 4 δ 2 α 4 + 8 β δ 2 α 2 α 1 α 5 + 64 α 5 − 16 δ α 5 + 4 β α 2 2 α 4 δ 2 + 256 β α 2 2 α 4 + δ 2 α 5 + 64 β δ α 2 2 α 4 + 128 β δ α 2 α 3 α 4 + 128 β δ α 1 α 2 α 4 + 64 β δ α 2 2 α 5 ) ;
α 13 = − 64 α 4 + 16 δ α 4 − δ 2 α 4 4 ( 64 − 16 δ + δ 2 ) ;
α 14 = 1 16 ( 64 − 16 δ + δ 2 ) ( − 8 β δ 2 α 1 α 2 α 5 − 64 α 5 + 16 δ α 5 − 4 β δ 2 α 2 2 α 4 − 256 β α 2 2 α 4 − δ 2 α 5 − 64 β δ α 2 2 α 4 − 128 β δ α 2 α 3 α 4 − 128 β δ α 1 α 2 α 4 − 64 β α 2 2 δ α 5 ) ;
α 15 = 1 2 ( 64 − 16 δ + δ 2 ) ( − δ 2 α 4 2 α 1 + 2 δ 2 α 4 α 9 + 2 δ 2 α 12 α 1 − 128 α 4 α 10 − 128 α 4 α 11 − 64 α 5 α 10 − 32 δ α 4 α 9 − 8 δ α 5 α 9 − 32 δ α 12 α 1 + 16 δ α 4 2 α 1 − 128 α 12 α 2 + 64 α 4 2 α 2 + 8 δ α 5 α 4 α 1 − 8 δ α 13 α 1 + 64 α 5 α 4 α 2 − 64 α 13 α 2 − 128 α 12 α 3 + 64 α 4 2 α 3 ) ;
α 16 = − δ 8 ( 64 − 16 δ + δ 2 ) ( − δ 2 α 4 2 α 1 + 2 δ 2 α 4 α 9 + 2 δ 2 α 12 α 1 − 128 α 4 α 10 − 128 α 4 α 11 − 64 α 5 α 10 − 32 δ α 4 α 9 − 8 δ α 5 α 9 − 32 δ α 12 α 1 + 16 δ α 4 2 α 1 − 128 α 12 α 2 + 64 α 4 2 α 2 + 8 δ α 5 α 4 α 1 − 8 δ α 13 α 1 + 64 α 5 α 4 α 2 − 64 α 13 α 2 − 128 α 12 α 3 + 64 α 4 2 α 3 ) + 1 4 δ α 4 α 9 + α 12 α 2 − 1 2 α 4 2 α 2 + α 4 α 10 + 1 4 δ α 12 α 1 − 1 8 δ α 4 2 α 1 ;
α 17 = δ 2 128 ( 64 − 16 δ + δ 2 ) ( − δ 2 α 4 2 α 1 + 2 δ 2 α 4 α 9 + 2 δ 2 α 12 α 1 − 128 α 4 α 10 − 128 α 4 α 11 − 64 α 5 α 10 − 32 δ α 4 α 9 − 8 δ α 5 α 9 − 32 δ α 12 α 1 + 16 δ α 4 2 α 1 − 128 α 12 α 2 + 64 α 4 2 α 2 + 8 δ α 5 α 4 α 1 − 8 δ α 13 α 1 + 64 α 5 α 4 α 2 − 64 α 13 α 2 − 128 α 12 α 3 + 64 α 4 2 α 3 ) + 1 2 α 10 α 5 − 1 16 δ α 5 α 4 α 1 − 1 2 α 5 α 4 α 2 − 1 32 δ α 4 2 α 2 + α 4 α 11 − 1 2 α 4 2 α 3 + 1 16 δ α 13 α 1 + 1 16 δ α 4 α 10 + 1 16 δ α 12 α 2 + 1 2 α 13 α 2 + 1 16 δ α 5 α 9 + α 12 α 3 − 1 64 ( δ α 4 α 9 + 4 α 12 α 2 − 2 α 4 2 α 2 + 4 α 4 α 10 + δ α 12 α 1 − 1 2 δ α 4 2 α 1 ) δ ;
α 18 = 1 8 α 4 2 + 1 4 α 12 + 1 16 α 5 α 4 + 1 8 β α 4 2 α 2 2 + 1 4 β α 10 2 + 1 16 α 13 + 1 2 β α 2 α 16 − 1 2 β α 2 α 4 α 10 − 1 4 β α 12 α 2 2 ;
α 19 = − 1 8 α 4 2 − 1 4 α 12 ;
α 20 = − 1 2 β α 2 α 16 − 1 4 β α 10 2 − 1 8 β α 4 2 α 2 2 + 1 2 β α 2 α 4 α 10 − 1 16 α 13 + 1 4 β α 12 α 2 2 − 1 16 α 14 α 5
parameter β, when permeability parameter δ = 0.0 and the Frank-Kamenetskii parameter λ = 0.5 .
The maximum temperature appeared at the centre of the cylindrical pipe and it decreases towards the boundary of the pipe. This result is in line with [
flow in the presence of permeability parameter. It is observed that velocity increases with increasing viscous heating parameter. It can be said that a combustible fluid at low viscosity flows faster than the one at high viscosity. A similar thing is observed in the velocity profile for various values of the porosity parameter. Velocity increases with increasing values of permeability parameter. Permeability is the measure of the material’s ability to permit liquid or gas through its pores or voids. Filters made of soil and earth dams are very much based upon the permeability of a saturated soil under load. Permeability is a part of the proportionality constant in Darcy’s law. Darcy’s law relates the flow rate and fluid properties to the pressure gradient applied to the porous medium. This supports that as permeability increases velocity should also increase [
From the findings of this work, it can be concluded that the viscous heating parameter has effect on the temperature and velocity profiles of the flow. The presence of pores on the geometry of the problem also has effect on the temperature and velocity of the fluid. Increase in permeability parameter increases the velocity of the fluid which is in agreement with the results of [
Drilling an oil well can be very expensive. Therefore an exploration of a site for oil well must be economically feasible before embarking on it. Some of the factors to be considered for effective oil exploration are the porosity and permeability of the reservoir rock. In deciding the suitability of drilling an oil well, it is necessary that the porosity be a minimum of 8 per cent. The distance between the pores in the sandstone should not be too far apart as the oil and gas would not be able to flow well thereby resulting in poor outputs. In this paper, it has been shown that when sandstone have low permeability as indicated in
I appreciate the Tertiary Education Trust Fund (TETFund), Abuja, Nigeria for sponsoring me to attend the 10th International Conference on Computational Heat, Mass and Momentum Transfer (ICCHM2T 2017) organized by Korean Society for Fluid Machinery (KSFM) held at Sejong Hotel, Seoul, South Korea where the paper was first presented.
Farayola, P.I. (2017) On Steady Flow of a Reactive Viscous Fluid in a Porous Cylindrical Pipe. Open Journal of Fluid Dynamics, 7, 359-370. https://doi.org/10.4236/ojfd.2017.73024