# Influence of Chemical Reaction on Heat and Mass Transfer Flow of a Micropolar Fluid over a Permeable Channel with Radiation and Heat Generation

Authors: Khilap Singh; Manoj Kumar
Journal: Journal of Thermodynamics
 (2016)

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## Abstract

The effects of chemical reaction on heat and mass transfer flow of a micropolar fluid in a permeable channel with heat generation and thermal radiation is studied. The Rosseland approximations are used to describe the radiative heat flux in the energy equation. The model contains nonlinear coupled partial differential equations which have been transformed into ordinary differential equation by using the similarity variables. The relevant nonlinear equations have been solved by Runge-Kutta-Fehlberg fourth fifth-order method with shooting technique. The physical significance of interesting parameters on the flow and heat transfer characteristics as well as the local skin friction coefficient, wall couple stress, and the heat transfer rate are thoroughly examined.


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## Body

## 1. Introduction

The micropolar fluid theory is the one of the most important non-Newtonian fluid models described by Eringen [1]. This theory shows microrotation effects as well as microinertia and has many applications such as polymer fluids, liquid crystals, animal bloods, unusual lubricants, colloidal and suspension solutions, colloidal fluids, liquid crystals, and polymeric suspension. The extensive reviews of the micropolar fluid theory and its applications can be found in Eringen [2] and Lukaszewicz [3]. The unsteady mixed convection flow of a micropolar fluid from a vertical surface in the presence of viscous dissipation and the buoyancy force has been studied by El-Aziz [4]. Ashraf et al. [5] investigated the micropolar fluids flow through a porous channel. Sheikholeslami et al. [6] proposed the flow and heat transfer of micropolar fluid in a permeable channel. Prakash and Muthtamilselvan [7] investigated radiation effect on MHD micropolar fluid flow in porous vertical channel. Darvishi et al. [8] analyzed numerically the micropolar fluid flow from a porous channel. Sherief et al. [9] studied the motion along axis of a circular cylindrical pore in a micropolar fluid of a slip spherical particle. Mosayebidorcheh [10] analyzed the flow of micropolar fluid over a porous channel with changing walls. Recently, many authors [11–14] have studied the micropolar fluid flow for different fluid properties over different geometries. However, the effects of chemical reaction on micropolar fluid flow over a permeable channel in the presence of radiation and heat generation have not been considered in the above investigations.Combined heat and mass transfer flows in the presence of chemical reaction have numerous applications in engineering and geophysics such as drying, geothermal reservoirs, dehydration at the surface of a water body, drying of porous solids, geothermal pool, thermal insulation, enhanced oil recovery, cooling of nuclear reactors, fibrous insulation, evaporation at the surface of a water body, pollution studies, cooling the polymer production and manufacturing of ceramics, energy transfer in a wet cooling tower, the flow in a desert, and oxidation and synthesis materials. Mohamed and Abo-Dahab [15] studied the effects of chemical reaction and thermal radiation on hydromagnetic free convection heat and mass transfer for a micropolar fluid bounded by a semi-infinite vertical porous plate in the presence of heat generation. In recent years, many researchers have studied and reported the effect of first-order chemical reaction [16–25].Influence of thermal radiation on flow and heat transfer study has become more important industrially. The heat transfer and temperature profile of a micropolar fluid over different geometries can be affected significantly at high temperature. Bhattacharyya et al. [26] considered thermal radiation effect on micropolar fluid flow and heat transfer over a porous shrinking sheet. Hussain et al. [27] analyzed radiation effects on the thermal boundary layer flow of a micropolar fluid towards a permeable stretching sheet. Oahimire and Olajuwon [28] investigated the influence of Hall current and thermal radiation on heat and mass transfer of a chemically reacting MHD flow of a micropolar fluid through a porous medium. Mabood et al. [29] studied effects of nonuniform heat source/sink and Soret on MHD non-Darcian convective flow past a stretching sheet in a micropolar fluid with radiation.The effect of heat generation on heat transfer is an important issue in view of various physical problems. Ziabakhsh et al. [30] analyzed the micropolar fluid flow with heat generation. Singh and Kumar [31] considered the melting effect in stagnation-point flow of micropolar fluid towards a stretching/shrinking surface. Bakr [32] investigated the effects of chemical reaction and heat source magnetoconvection and mass transfer flow of a micropolar fluid in a rotating frame of reference. The heat generation/absorption effects on MHD flow and heat transfer of micropolar fluid through a stretching surface have been proposed by Mahmoud and Waheed [33]. Abbasi et al. [34] examined the flow of Maxwell nanofluid in the presence of heat generation/absorption. Mliki et al. [35] investigated the influence of nanoparticle Brownian motion and heat generation/absorption over linear/sinusoidally heated cavity in the presence of magnetohydrodynamic natural convection. Sheikholeslami and Ganji [36] studied three-dimensional heat and mass transfer flow of nanofluid over a rotating system. Thermal radiation effects on mixed convection flow and heat transfer of a micropolar fluid through an unsteady stretching surface with heat generation/absorption are presented by Singh and Kumar [37].Motivated by the above studies and applications, the present work explores effects of chemical reaction on heat and mass transfer flow of a micropolar fluid over a permeable channel in the presence of radiation and heat generation. The equations of continuity, momentum, angular momentum, energy, and concentration have been reduced to a system of nonlinear ordinary differential equations by similarity transforms which are solved by Runge-Kutta-Fehlberg method with shooting technique. It is expected that the results obtained from present paper will provide important information to the audience. To the best of our knowledge, such type of study is not investigated before in the scientific literature.


## 2. Mathematical Formulation

The heat and mass transfer flow of a micropolar fluid in a permeable channel with chemical reaction is considered in the present work. The thermal radiation and heat source are incorporated in the energy equation. The graphical model of the problem has been given along with flow configuration and coordinate system in Figure1. The assumptions of the problem in detail can be found in [6]. The governing equations of boundary layer are given in the following form:(1)∂u∂x+∂v∂y=0,(2)u∂u∂x+v∂u∂y=-1ρ∂P∂x+μ+κρ∂2u∂x2+∂2u∂y2+κρ∂N∂y,(3)u∂v∂x+v∂v∂y=-1ρ∂P∂y+μ+κρ∂2v∂x2+∂2v∂y2-κρ∂N∂y,(4)u∂N∂x+v∂N∂y=μsρj∂2N∂x2+∂2N∂y2-κρj2N+∂u∂y-∂v∂x,(5)u∂T∂x+v∂T∂y=kρCp∂2T∂y2-1ρCp∂qr∂y+μρCp∂u∂y2+Q0ρCpT-T2,(6)u∂C∂x+v∂C∂y=D∗∂2C∂y2-γ0C-C2,where u and v indicate the velocity components in the x and y directions, respectively, ρ is the fluid density, μ is the dynamic viscosity, κ is the material parameter, N is the angular or microrotation velocity, P is the fluid pressure, j is the microinertia density, μs=(μ+κ/2)j is the microrotation viscosity, T is the fluid temperature, Cp is the specific heat at constant pressure, C is the fluid concentration, k is the thermal conductivity, qr is the radiative heat flux, Q0 is the heat generation coefficient, D∗ is the molecular diffusivity, and γ0 is the chemical reaction rate coefficient.Figure 1
(a) Geometry of problem. (b)x-y view at z=W/2.
(a) (b)Using Rosseland’s approximation for radiation, we obtain(7)qr=-4σ3k0∂T4∂y,where σ is the Stefan–Boltzmann constant and k0 is the absorption coefficient. We consider that the temperature variation within the flow is such that T4 may be expanded in a Taylor’s series. Expanding T4 about T∞ and neglecting higher order terms, we get T4=4T∞3T-3T∞4. Now (5) reduces to(8)u∂T∂x+v∂T∂y=kρCp∂2T∂y2+16σT∞33k1ρCp∂2T∂y2+μρCp∂u∂y2+Q0ρCpT-T2.The appropriate boundary conditions for the flow are(9)u=v=0,N=-n∂u∂y,T=T1,C=C1,aty=-h,u=v0xh,v=0,N=v0xh2,T=T2,C=C2,aty=-h,where boundary parameters n(0≤n≤1) indicate the degree to which the microelements are free to rotate near the channel walls. The case when n=0 is called strong concentration of microelements, which implies N=0 near the wall surface. This represents concentrated particle flow where the microelements close to the wall surface are unable to rotate. In the case when n=1/2, this indicates vanishing of the antisymmetric part of the stress tensor and denoted weak concentration of microelement and n=1 is used for the modeling of turbulent boundary layer flow. In this paper, the authors considered n=1/2 for which the governing equations can be reduced to the classical problem of steady boundary layer flow of a viscous incompressible fluid near the channel wall.Equations (2), (3), (4), (6), and (8) can be transformed into a set of nonlinear ordinary differential equations by using the following similarity transformations:(10)η=yh,ψ=-υ0xfη,N=υ0xh2gη,θη=T-T2T1-T2,ϕη=C-C2C1-C2,where T2=T1-Ax and C2=C1-Bx, with A and B as constants. The stream function ψ is defined as(11)u=∂ψ∂y,v=-∂ψ∂x.The coupled system of transformed nonlinear ordinary differential equations is(12)1+N1fIV-N1g-Reff′′′-f′f′′=0,N2g′′+N1f′′-2g-N3Refg′-f′g=0,1+Rθ′′+PehEcf′′2+f′θ-fθ′+Hθ=0,ϕ′′+Pemf′ϕ-fϕ′-γϕ=0.Boundary conditions in nondimensional form are(13)f-1=f′-1=g-1=0,θ-1=ϕ-1=1,f1=θ1=ϕ1=0,f′1=-1,g1=1,where N1=κ/μ is the coupling number, N2=vs/μh2 is the spin-gradient viscosity parameter, N3=j/h2 is the micropolar material constant, Ec=v02xυ/h3cpA is the local Eckert number, H=θ0h/ρCpv0 is the heat generation parameter, Peh=PrRe and Pem=ScRe are the Peclet numbers for the diffusion of heat and the diffusion of mass, Re=(v0/υ)h is the Reynolds number, R=3k0k/16σT∞3 is the thermal radiation parameter, Gr=gβTAh4/υ2 is the Grashof number, Pr=υρcp/k is the Prandtl number, Sc=υ/D∗ is the Schmidt number, and γ=γ0h/v0 is the chemical reaction parameter.The other parameters of physical interest are the local NusseltNux and Sherwood Shx numbers, which are defined as follows:(14)Nux=xqwkT1-T2,Shx=xmwD∗C1-C2,where qw and mw are the local heat flux and mass flux, respectively, which are defined as(15)qw=-k∂T∂yy=-h,mw=D∗∂C∂yy=-h.Now using (10) and (15) in (14), we get(16)Nux=-θ′-1,Shx=-ϕ′-1.


## 3. Method of Solution

In this present paper, Runge-Kutta-Fehlberg fourth fifth-order method has been employed to solve the system of nonlinear ordinary differential equations (12) with the boundary conditions given by (13) for different values of governing parameters. The RKF45 method has a procedure to determine if the appropriate step size h is being used. The formula of fifth-order Runge-Kutta-Fehlberg method can be defined as follows:(17)zn+1=zn+16135k0+665612,825k2+28,56156,430k3-950k4+255k5h,where the coefficients k0 to k5 are given by(18)k0=fxn,yn,k1=fxn+14h,yn+14hk0,k2=fxn+38h,yn+332k0+932k1h,k3=fxn+1213h,yn+19322197k0-72002197k1+72962197k2h,k4=fxn+h,yn+439216k0-8k1+3680513k2-8454104k3h,k5=fxn+12h,yn+-827k0+2k1-35442565k2+18594104k3-1140k4h.The computation of the error can be achieved by subtracting the fifth-order from the fourth-order method:(19)yn+1=yn+25216k0+14082565k2+21974101k3-15k4h.If the error goes beyond a specified antechamber, the results can be recalculated using a smaller step size. The approach to computing the new step size is shown as follows:(20)hnew=holdεhold2zn+1-yn+11/4.The numerical computations were carried out with Δη=0.01. The variation of the dimensionless velocity, microrotation, temperature, and concentration is ensured to be less than 10-6 between any two successive iterations for the convergence criterion.


## 4. Results and Discussion

In order to study, the effects of various governing physical parameters on the flow, heat, and mass transfer numerical computations are carried out for0.1≤N1=N2=N3≤1, 0≤H≤10, -2≤Re≤5, 0≤R≤4, 0≤Peh≤2, 0≤Pem≤2, and 0≤γ≤10 while the Eckert number Ec=0.01 is fixed. A critical analysis with previously published work is done in Table 1 and these results are found to be in very good agreement.Table 1
Comparison of values ofϕ(η) for various values of η, Pem, and Re when Peh=0.2, N1=N2=N3=0.1, N1=N2=N3=0.1, andEc=R=γ=0.
η Sheikholeslami et al. [6] Present results P e m = 0.2, Re=0.5 P e m = 0.5, Re=1 P e m = 0.2, Re=0.5 P e m = 0.5, Re=1 −1.0 1 1 1 1 −0.6 0.814341 0.835986 0.814346 0.835992 −0.2 0.621160 0.653497 0.621165 0.653501 0.0 0.521109 0.553582 0.521114 0.553590 0.2 0.419033 0.448495 0.419039 0.448499 0.6 0.210678 0.227352 0.210684 0.227358 1.0 0 0 0 0The effects of the Reynolds numberRe on velocity f′(η) and microrotation g(η) are shown in Figures 2 and 3. It is seen from Figure 2 that velocity profile f′(η) decreases near the lower channel wall, while it increases near the upper channel wall when the Reynolds number Re increases. It is clear from Figure 3 that an increase in the magnitude of Re leads to decrease in microrotation profile g(η).Figure 2
Velocity profilef′(η) for various values of Re.Figure 3
Microrotation profileg(η) for various values of Re.The microrotation profileg(η) for the coupling number N1, the spin-gradient viscosity parameter N2, and the micropolar material constant N3 are displayed in Figures 4–6. It is noticed from Figure 4 that the microrotation profile increases with N1 at lower channel wall for 0≤η<-0.65, but reverse trend is found for the case η>-0.65. It is depicted from Figure 5 that the value of microrotation profile g(η) increases as the value of N2 increases. It is evident from Figure 6 that the values of microrotation are lower for higher values of N3.Figure 4
Microrotation profileg(η) for various values of N1.Figure 5
Microrotation profileg(η) for various values of N2.Figure 6
Microrotation profileg(η) for various values of N3.Figures7–9 show the temperature distribution θ(η) with collective variation in heat source parameter H, thermal radiation parameter R, and Peclet number Peh. Figure 7 exhibits that temperature θ(η) considerably increases with an increase in H. Figure 8 indicates that an increase in R induces a decrease in temperature θ(η). It is noticed from Figure 9 that a rise in Peh causes rapid increase in temperature.Figure 7
Temperature profileθ(η) for various values of H.Figure 8
Temperature profileθ(η) for various values of R.Figure 9
Temperature profileθ(η) for various values of Peh.The effects of the Peclet numberPem and chemical reaction parameter γ on the concentration profile ϕ(η) are shown in Figures 10 and 11. Figures 10 and 11 indicate that concentration decreases with increasing values of Pem and γ.Figure 10
Concentration profileϕ(η) for various values of Pem.Figure 11
Concentration profileϕ(η) for various values of γ.The variation of the Nusselt number (dimensionless heat transfer rate at the surface) is displayed for different parameters in Figures12–14. Figure 12 depicts the behavior of the Nusselt’s number against Reynolds number Re with various values of heat source parameter H. It is clear that, with the increasing of H, the heat transfer rate decreases. Figure 13 enlightens the variation of the heat transfer rates with Reynolds number Re for various values of thermal radiation parameter R. It is evident from Figure 13 that Nusselt number monotonically increases with a rise in R values. Figure 14 depicts the variation of Nusselt number as function of Reynolds number Re and Peclet number Peh. It is noted that heat transfer rate decreases with a rise in Peh values. Also, a negligible change in heat transfer rate with increasing values of Reynolds number Re is seen from Figures 12–14.Figure 12
Variation of Nusselt’s number withH and Re.Figure 13
Variation of Nusselt’s number withR and Re.Figure 14
Variation of Nusselt’s number withPeh and Re.Figures15 and 16 displayed the variation of the Sherwood number or mass transfer rate for different parameters. Figures 15 and 16 depict the behavior of the mass transfer rate against Reynolds number Re with different values of Peclet number Pem and chemical reaction parameter γ. The mass transfer rate rises with increasing values of the Peclet number Pem and chemical reaction parameter γ. It is also clear from these figures that there is no change in mass transfer rate with rise in Reynolds number Re values.Figure 15
Variation of Sherwood number withPem and Re.Figure 16
Variation of Sherwood number withγ and Re.


## 5. Conclusions

The present paper deals with numerical analysis of chemical reaction effects on heat and mass transfer flow of a micropolar fluid over a permeable channel in the presence of radiation and heat generation. The system of nonlinear partial differential equations was converted to a system of ordinary differential equations and then is solved numerically using the Runge-Kutta-Fehlberg method along with shooting method. From the above discussion, the important results are summarized as follows:(i)
Velocity and Reynolds number are inversely proportional to each other at lower channel wall, while velocity and Reynolds number are proportional to each other at upper channel wall.(ii)
Microrotation decreases with increase in the value of coupling number, micropolar material constant, and Reynolds number, but it increases with increase in the value of spin-gradient viscosity parameter.(iii)
Temperature increases with heat generation parameter andPeh and Pem are the Peclet numbers for the diffusion of heat and the diffusion of mass, and temperature decreases with thermal radiation parameter.(iv)
Concentration is inversely proportional to the Peclet number for the diffusion of mass and chemical reaction parameter.(v)
The rate of heat transfer increases with thermal radiation parameter and rate of heat transfer decreases with Peclet numberPeh and heat generation parameter.(vi)
The rate of mass transfer rises with increase Peclet numberPem and chemical reaction parameter.



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