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Effect of Heat Generation on Heat and Mass Transfer Over a Radially Stretching Surface Embedded in Porous Medium with Chemical Reaction and Activation Energy
Elsayed M. A. Elbashbeshy1*, Hamada Galal Asker2 and Tarek G. Emam3
1Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt.
2Department of Physics & Engineering Mathematics, Faculty of Engineering, Ain Shams University, Cairo, Egypt
3Department of Mathematics, College of Science and Arts Kuhliids, University of Jeddah, Jeddah, Saudi Arabia.
Submission:October 21, 2022;Published: December 19, 2022
*Corresponding author: Elsayed M. A. Elbashbeshy Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt
How to cite this article: Elsayed M. A. Elbashbeshy, Hamada Galal Asker, Tarek G. Emam. Effect of Heat Generation on Heat and Mass Transfer Over a Radially Stretching Surface Embedded in Porous Medium with Chemical Reaction and Activation Energy. Prospects Mech Eng Technol. 2022; 1(1):555555. 10.19080/PMET.2018.01.555555
Abstract
The effect of heat generation on heat and mass transfer over a radially stretching surface embedded in a porous medium with chemical reaction and activation energy are numerically discussed. The governing boundary layer equations are formulated and transformed into ordinary differential equations using a suitable similarity transformation. The resulting ordinary differential equations are solved numerically by applying the fourth order Runge-Kutta method with the shooting technique. The influence of the different parameters on the velocity, temperature, and concentration are discussed and analyzed. The skin friction coefficient, the Nusselt number, and Sherwood number are also computed and investigated for different embedded parameters in the problem statements.
Keywords:radially stretching surface, heat and mass transfer, porous medium, chemical reaction and activation energy, heat generation
Introduction
The investigation of flow and heat transfer occurred by stretching surface exist in a lot of industrial applications such as plastic, glass, and rubber sheets. The quality and the specification of the final product are controlled by the criteria of stretching and the mechanism of heat transfer. The radial stretching is one of the stretching methods takes place in different investigations.
The exact similarity solution for the governing equations deals the natural convection on a vertical radially stretching surface [1]. The study of mixed convection for Jeffery fluid show that the momentum boundary layer thickness enhanced while the thermal and concentration get worse [2]. The existence of magnetic field in the axisymmetric flow of Carneau nanofluid enhances the thermal and concentration boundary layer thickness [3] and over a radially stretching/shrinking disk [4]. The influence of thermal radiation and magnetic field is investigated numerically using Homotropy analysis for the Jeffrey nanofluid [5] and numerically [6]. The existence of Joule heating for steady radially permeable stretching/shrinking sheet takes place for MHD axisymmetric flow [7] and unsteady radially stretching surface [8]. The effect of nanoparticle shapes on the nanofluid over a radially stretching rotating disk takes place [4].
The present paper introduces the influence of heat generation, activation energy in the existence of porous media and chemical reaction on the flow over stretching surface radially.
Mathematical
The surface is stretched radially in two-dimensional inducing steady, and laminar flow. The surface is embedded in porous medium with influences of heat generation. The choice of coordinate system is made as shown in Figure1 such that r-axis is taken along the surface radially in the direction of motion with velocity Uw=Uo r, where Uo>0 is rate of stretching surface and z-axis is perpendicular to it by keeping the origin fixed. The plane z = 0 represents the surface. The flow in the region z >0.
Assuming a species chemical reaction with finite Arrhenius activation energy. The equations of flow analysis can be expressed as: (Figure 1)
where u and w are the velocity components in the r and z-direction respectively, μ is dynamic viscosity and α is thermal diffusivity of , ρ is the fluid density, K is the permeability, Q0 is the heat generation or absorption coefficient such that Q0 > 0 corresponds to heat generation while Q0 < 0 corresponds to heat absorption, T∞ is the ambient temperature, T is the temperature, D is the Solutal diffusivity,
Where Tw is surface temperature and Cw is solute concentration, T∞ and C∞ are the temperature and concentration of the ambient fluid.
The equation of continuity is satisfied if we choose a stream function ψ (r, z) such that
where the primes denote differentiation with respect to η ,M = μ a / k is the permeability parameter, Pr = v /α is the Prandtl number , γ = Q0 /U0 is the heat-source/sink parameter, Sc = v / D is the Schmidt number, δ = (Tw − T∞) / T∞ is the temperature relative parameter, and σ = kr / a is the dimensionless chemical reaction rate constant and E = Ea / kT∞ is the non-dimensional activation energy.
The non-dimensional quantities are the local skin friction coefficient Cfx ,the local Nusselt number Nux, and local Sherwood number Shx which are expressed as
Results and Discussions
The Mathematica program solves the equations from (7) to (10) are solved. The numerical method is fourth order Rung- Kutta. The shotting method predicts the missing boundary terms. In these results, Graphical representation shows the change in the flow profiles, and tabulates the non-dimensional parameters at different settings. Profile of the velocity is affected with M parameter shown in Figure 2, for any selected η increasing M parameter decreases the velocity (Figure 2) (Figure 3).
The profile of temperature is affected by M Pr, and γ parameters as shown in Figures 3, 4, and 5 respectively. Figure 3 shows that increasing M increases the temperature, while it decreases with increasing Pr number in Figure 4 & Figure 5 ensures that the heat source increases the temperature (Figure 4) (Figure 5).
Figure (6) to Figure (11) shows the variation of the profile of concentration against the presented parameters E, M, n,δ , Sc and σ respectively. Figure 6 increasing E increases the concentration while Figure 7 increasing M makes an inconsiderable increase. Figure (8) and Figure (9) show an inconsiderable decrease in the concentration with the n and δ parameters, respectively. While Figure (10) and Figure (11) Sc and σ parameters make a decrease in the concentration with, respectively (Figure 6) (Figure 7) (Figure 8) (Figure 9) (Figure10) (Figure 11) (Table 1).
Table 1 introduces tabulated results for the non-dimensional quantities local skin friction Cf Re , local Nusselt number Nu Re , and Sherwood number Sh / Re as − f "(0) , −θ '(0) , and −ϕ '(0) , respectively. E parameter decreases −ϕ '(0) , but it has no effect on − f "(0) and −θ '(0) . Prenumber has no effect on − f "(0) and −ϕ '(0) while it causes an increase in −θ '(0) . n parameter has no effect on − f "(0) and −θ '(0) , while it causes an increase in -φ’ (0). Sc parameter increases -φ’ (0) without no effect on − f "(0) and −θ '(0) . γ parameter increases the −θ '(0) and decreases -φ’ (0) without any effect on -f^’’ (0).δ and σ parameters increase -φ’ (0) and have no effect on − f "(0) and −θ '(0) . M parameter increases − f "(0) , while it decreases −θ '(0) and -φ^’ (0).
Conclusion
The Navier-stokes equations of the boundary layer flow over radial stretching surface is solved numerically using fourth order Runge-Kutta.
The velocity decreases with M parameter The temperature increases with M parameter, while it decreases with Pr and γ parameters.
The concentration increases with E and M parameter, while it deceases with n, Sc, δ, and σ parameters.
The skin friction coefficient increases with M parameter.
The local Nusselt number increases with Pr and γ parameters, while it decreases with M parameter.
The Sherwood number decreases with E, γ, and M parameters, while it increases with n, Sc, δ, and σ parameters (Table 2).
Declarations
Ethics Approval: This manuscript is original and has not been published elsewhere and is not submitted to another journal. All authors have contributed to the work.
References
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- Bilal Ashraf, M, T. Hayat, A. AL saeid, and S. A. Shehzad (2015) "Convective heat and mass transfer in MHD mixed convection flow of Jeffrey nanofluid over a radially stretching surface with thermal radiation" Journal of Central South University 22(3): 1114-1123.
- Azam, M, M. Khan, and A. S. Alshomrani (2017) "Effects of magnetic field and partial slip on unsteady axisymmetric flow of Carneau nanofluid over a radially stretching surface." Results in physics 7: 2671-2682.
- Soid Siti Khuzaimah, Anura Ishak, and Iona Pop (2018)"MHD flow and heat transfer over a radially stretching/shrinking disk." Chinese Journal of Physics 56(1): 58-66.
- Sreelakshmi, K, G. Sarojamma, and J. V. Murthy (2018) "Homotropy analysis of an unsteady flow heat transfer of a Jeffrey nanofluid over a radially stretching convective surface." Journal of Nanofluids 7(1): 62-71(10).
- Khashi'ie, Najiyah Safwa, Norihan Md Arifin, Rosalinda Nazra, Ezaz Hafiz Hafidzuddin et. al (2020) "Magnetohydrodynamics (MHD) axisymmetric flow and heat transfer of a hybrid nanofluid past a radially permeable stretching/shrinking sheet with Joule heating." Chinese Journal of Physics 64: 251-263.
- Shahzad, Azeem, Uzma Gulistan, Ramzan Ali, Azhar Iqbal, Muhammad Kamran et.al (2020) "Numerical study of axisymmetric flow and heat transfer in a liquid film over an unsteady radially stretching surface" Mathematical Problems in Engineering: 9.
- Saranya, S., and Qasem M. Al-Mdallal (2021)"Computational study on nanoparticle shape effects of Al2O3-silicon oil nanofluid flow over a radially stretching rotating disk." Case Studies in Thermal Engineering 25: 100943.
