# Mathematical Analysis of a Reactive Viscous Flow through a Channel Filled with a Porous Medium

Authors: Samuel O. Adesanya; J. A. Falade; J. C. Ukaegbu; K. S. Adekeye
Journal: Journal of Mathematics
 (2016)

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## Abstract

An investigation has been carried out to study entropy generation in a viscous, incompressible, and reactive fluid flowing steadily through a channel with porous materials. Approximate solutions for both velocity and temperature fields are obtained by using a rapidly convergent Adomian decomposition method (ADM). These solutions are then used to determine the heat irreversibility and Bejan number of the problem. Variations of other important fluid parameters are conducted, presented graphically, and discussed.


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## Body

## 1. Introduction

Studies on heat irreversibility in moving fluid find its relevance in several geological, petrochemical, and industrial applications. In most flows at extremely high temperature, heat irreversibility is unavoidable. This usually leads to material waste due to reduced efficiency of the thermofluid machine. To conserve energy, Bejan [1] introduced an approach that is based on the thermodynamics second law to predict the performance of thermal systems so as to maximize scarce available energy for work and minimize wastages. Following his analysis, Al-Zaharnah and Yilbas [2] considered the irreversibility analysis in a viscous pipe flow. Haddad et al. [3] examined the heat irreversibility in forced convective flow in concentric cylindrical annulus under diverse flow conditions. Kahraman and Yürüsoy [4] applied the same approach to study the heat irreversibility in non-Newtonian fluid flow through pipes. Aksoy [5] considered the influence of couple stresses on the development of heat irreversibility in a channel with adiabatic surface and constant heat flux. Ting et al. [6] considered the irreversibility associated with nanofluids in a microchannel with porous materials using water-alumina. Moreover, Khan and Gorla [7] addressed the convective problem in non-Newtonian fluid flow through a channel with porous medium and heat flux. Revellin et al. [8] addressed the thermal performance of adiabatic two-phase flow using two different methods. Hedayati et al. [9] utilized the thermodynamics analysis to optimize flow on a nonstationary wedge. Butt and Ali [10] reported the irreversibility analysis of fluid slippage with convective boundaries. Other works that focused on the minimization of energy losses in a fluid flow can be found in references [11–19] and many more too numerous to be listed.From applications’ point of view, studies on transport reactive fluids in porous media are very important since they occur in many important areas like water treatment using fixed beds, agriculture, oil recovery, ground water flows, geothermal engineering, exhaust systems in combustion, material processing, and reservoir engineering. Recently, Rundora and his associates [20–22] documented several investigations on unsteady reactive fluid flow in porous medium and how the flow evolved to the steady state. Bég et al. [23] examined the flow of viscoelastic fluid through a non-Darcian porous medium. Makinde [24] studied the inherent heat irreversibility in reactive fluid through a channel filled with porous material.In all the studies above, the entropy productions in the flow of viscous incompressible fluid flow through porous medium have not been investigated. Therefore, the work done in [24] can be further extended to give more interesting results on the thermodynamics and heat transfer properties of the fluid flow. This is because huge amount of money and effort could be wasted if the inherent irreversibility in the fluid flow is not well addressed. Therefore, the specific objective of this article is to examine the rate at which entropy is produced in a viscous fluid flow system through a porous medium. The problem under consideration is nonlinear due to the exponential nature of the rate law in Arrhenius kinetics for combustible fluids. In view of this, exact solution for the temperature field may not be possible to get. To solve the problem, we seek Adomian series solution to avoid linearization of the exponential term. The Adomian decomposition method is a straightforward way of solving all kinds of differential equations arising from many physical scenarios. It has been used extensively in the last few decades as reported in the bibliography by Rach [25], and, more recently, the method has been used in [26–31]. The plan of the article is as follows: the problem is formulated and the mathematical analysis is presented in Section 2. Section 3 of the work gives the Adomian method of solution. Graphical results are presented and interpreted in Section 4 while, in Section 5, concluding remarks are given.


## 2. Mathematical Analysis

The steady flow of viscous incompressible reactive fluid through parallel-plate immersed in a porous medium is studied. The flow is assumed to be full-developed and driven by an applied pressure gradient. The channel wall temperatures are kept constant. Then, the balanced governing equations are [24](1)0=-dPdx+μd2u′dy′2-μu′K0=d2Tdy′2+QC0Ake-E/RT+μkdu′dy′2+μu′2kKwith the following boundary conditions:(2)T0=Th=T0u′0=0=u′h.Under these assumptions, entropy generation equation becomes(3)EG=kT02dTdy′2+μT0du′dy′2+μu′2T0K.To nondimensionalize (1)–(3), we need the following parameters and variables: (4)y=y′h,u=u′UM,θ=ET-T0RT02,Da=Kh2,Ns=h2E2EGkR2T02,ϵ=RT0Eλ=QEAC0h2e-E/RT0RT02k,M=-h2μUdPdx,δ=U2μM2eE/RT0QAC0h2,β2=1Dato get the dimensionless problems:(5)d2udy2-β2u=-1;u0=u1=0,(6)d2θdy2+λeθ/1+ϵθ+δdudy2+δβ2u2=0;θ0=θ1=0(7)Ns=dθdy′2+δλϵdudy2+β2u2.Setting(8)N1=dθdy2,N2=δλϵdudy2+β2u2,then, the irreversibility ratio becomes (9)Be=N1N1+N2=11+Φ,Φ=N2N1.From (9), it is evident that (10)Be=0,N2≫N10.5N1=N21N2≪N1.


## 3. Adomian Method of Solution

A direct integration of (5)-(6) leads to the integral equations(11)uy=∫0ydu0dYdY+∫0y∫0yβ2udYdYwith(12)θy=∫0ydθ0dYdY-∫0y∫0yλeθ/1+ϵθ+δdudY2+δβ2u2dYdY.Due to the exponential nonlinearity in (12), we now define a series of functions defined by(13)uy=∑n=0∞uny,θy=∑n=0∞θny.Substituting (13) into the integral equations (11)-(12), we obtain(14)∑n=0∞uny=∫0ydu0dYdY+∫0y∫0yβ2∑n=0∞θnydYdY,(15)∑n=0∞θny=∫0ydθ0dYdY-∫0y∫0yλExp∑n=0∞θny1+ϵ∑n=0∞θny+δddY∑n=0∞uny2+δβ2∑n=0∞uny2dYdY.The nonlinear term in (15) represented by(16)Bn=Exp∑n=0∞θny1+ϵ∑n=0∞θnyis expanded by Taylor’s series to get the following Adomian polynomials:(17)B0=eθ0/1+ϵθ0B1=θ11+ϵθ12eθ0/1+ϵθ0B2=1-2ϵ-2ϵ2θ0θ12+21+ϵθ02θ221+ϵθ14eθ0/1+ϵθ0⋮The zeroth-order components of the series solutions (14) and (15) are(18)u0y=a0y-y22θ0y=0.Since the integral of a continuous function is continuous, then each term of the series can be uniquely determined by(19)un+1y=β2∫0y∫0yundYdYn≥1θ1y=∫0ya1dY-λ∫0y∫0yB0+δdudY2+δβ2u2dYdYθn+1y=-λ∫0y∫0yBndYdYn≥1,where du0/dY=a0 and dθ0/dY=a1 are the parameters to be determined.Then, (17)–(19) are evaluated using MATHEMATICA and the solutions are obtained as finite series:(20)uy=∑n=0muny,θy=∑n=0mθny.The series solutions are shown to be convergent and twice differentiable (see Tables 1 and 2). Next, we establish the uniqueness solution of (20). It is well known that the Lipschitz condition is sufficient for the uniqueness of solution. Therefore, we first seek for a Lipschitz constant ϕ such that(21)Fθ-Fθ-≤ϕθy-θy-is satisfied. To do this, the boundary-valued problems (6) are converted to system first-order differential equations by introducing the following transformations:(22)y1=y,y2=uy,y3=θy,y4=u′y,y5=θ′y.With (22), (6) can now be written as(23)y1y2y3y4y5′=1y4y5β2y2-1-λey3/1+ϵy3+δy42+β2y22=f1y1,y2,…,y5f2y1,y2,…,y5f3y1,y2,…,y5f4y1,y2,…,y5f5y1,y2,…,y5;y10y20y30y40y50=000c1c2,where cii=1,2, the guess values that will ensure the boundary conditions, are satisfied. Then,(24)dfidyj=0,0,0,0,00,0,0,1,00,0,0,0,10,0,β21,10,-2λδβ2y2-λ1+ϵy41-ϵy31+ϵy3ey3/1+εy3-2δλy30since dfi/dyj, i,j=1,2,…,5, exist and are continuous in the domain [0,1]. Hence, the Lipschitz constant ϕ with the property(25)dfidyi≤ϕexists.Table 1
Uniqueness resultsm=3, β=1.
y u-exact u-Adomian Error 0 0 0 0 0.1 0.0412846 0.0412847 8.58689 × 10−8 0.2 0.072741 0.072742 1.72597 × 10−7 0.3 0.0953855 0.0953858 2.61030 × 10−7 0.4 0.108743 0.108744 3.51816 × 10−7 0.5 0.113181 0.113182 4.44491 × 10−7 0.6 0.108743 0.108744 5.34572 × 10−7 0.7 0.0953855 0.0953861 6.06498 × 10−7 0.8 0.0729741 0.0729747 6.1915 × 10−7 0. 9 0.0412846 0.0412851 4.79648 × 10−7 1 0 0 0Table 2
Convergence resultsβ=1=λ=δ, ϵ=0.1.
N b 1 0 0 1 0.537883 2 0.590362 3 0.589506 4 0.589078 5 0.589096 6 0.589100 7 0.589100 8 0.589100Uniqueness Analysis. The Adomian series solutions (20) of the nonlinear problem (6) converges if 0<α≪1 and θ0y<∞, where α=λKY2/2.Proof.
LetCJ,· be a Banach space for all continuous functions on J with the norm(26)FY=max∀yϵJ⁡FY.Let θy and θy- be any two solutions of the integral equation (12); then,(27)θy-θy-=-∫0y∫0yλFθ-Fθ-dYdY.This implies that(28)θy-θy-≤∫0y∫0yλFθ-Fθ-dYdY.In view of (21), we then have(29)θy-θy-≤∫0y∫0yθy-θy-dYdY≤θy-θy-∫0y∫0yλϕdYdY≤λϕY22θy-θy-.Let α=λϕY2/2; then,(30)θy-θy-≤αθy-θy-or(31)1-αθy-θy-≤0.Hence, the problem will have a unique solution whenever 0<α≪1 and as such(32)θy=θy-.Kindly see Tables 1 and 2 for the numerical results.


## 4. Results and Discussion

In this section, the effects of pertinent fluid parameters on the velocity and temperature profiles are shown graphically. Figure1 shows the effect of medium porosity on the entropy generation rate. Since an increase in the porous permeability parameter implies a decrease in the medium porosity, this caused a decreased flow and heat trapping strategy. The net effect is seen here; that is, entropy production increases only in the centerline of the channel while it decreases at the walls. Figure 2 represents the effect of activation energy parameter on the entropy generation rate. Since activation energy decreases the fluid temperature, it is therefore expected to decrease the entropy generated in the flow region. This is true since an increase in the Frank-Kamenetskii parameter is known to enhance the fluid temperature. Therefore, by increasing values of this parameter, entropy generated is expected to be on the increase as shown in Figure 3. In Figure 4, the effect of the viscous heating parameter is shown. The result shows that entropy generation rate increases with increasing values of the viscous heating parameter due to frictional interaction in the fluid layers. Figure 5 represents the Bejan number’s variations with the activation energy. From the plot, as the activation energy parameter increases, heat irreversibility due to heat transfer decreases. As a result, fluid friction irreversibility dominates over heat transfer irreversibility within the porous medium. Moreover, as the porous permeability parameter increases in Figure 6, there are reduced flow and fluid temperature rises. The net effect shows that heat transfer irreversibility dominates over irreversibility due to fluid friction. Similar behaviour is observed in Figure 7 as the exothermic Frank-Kamenetskii parameter increases. Finally, as viscous heating parameter increases, the viscous heat dissipation increases in the flow channel, and fluid friction irreversibility dominates over heat transfer irreversibility as observed in Figure 8.Figure 1
Entropy generation rate for different porosity permeability parameter.Figure 2
Entropy generation rate for different activation energy parameter.Figure 3
Entropy generation rate for different Frank-Kamenetskii parameter.Figure 4
Entropy generation rate for different viscous heating parameter.Figure 5
Bejan number at different activation energy parameter.Figure 6
Bejan number at different porosity permeability parameter.Figure 7
Bejan number at different Frank-Kamenetskii parameter.Figure 8
Bejan number at different viscous heating parameter.


## 5. Conclusion

The entropy generation rate in the flow of reactive fluid through a porous medium has been investigated. Adomian decomposition method is used to obtain approximate solution to the strongly nonlinear boundary-valued problem for the dimensionless energy equation. The main contributions to knowledge from the present analysis are as follows: a reactive fluid flowing through a porous medium, entropy generation is minimum at the centerline of the channel; as a result, fluid friction irreversibility dominates over heat transfer irreversibility in the centerline. Secondly, porous medium with low permeability is one major factor that depletes the useful available energy in a thermofluid.



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