Convective heat and mass transfer of chemically reacting fluids with activation energy with radiatio

Abstract

Keywords: Heat generation; Thermal radiation; Activation energy; Chemical reaction

Introduction

The main objective of this current study is to evaluate the fluid flow by rotating frame in a porous medium by radiation and heat generation. This kind of flow finds great

importance due to the vast application in industry and engineering fields [1–7]. The effect caused by activation energy has been studied. The smallest amount of energy required by atoms or molecules to bring about a chemical reaction is

*Corresponding author. *E-mail address: sd.siva@yahoo.com, sdsiva@gmail.com This paper was recommended for publication in revised form by ­Regional Editor Jovana Radulovic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

the activation energy. The increase in reaction rate with the increase in temperature is explained on the basis of collision theory. A certain minimum amount of energy possessed by some collisions alone results in a chemical reaction. In 1889 Svante Arrhenius initially introduced the term activation energy. Thus according to activation energy concept, the reactant molecules have to gain a certain amount of energy before they can react to yield products, and there is an energy barrier that must be crossed over by the reactants to be converted into the products. A chemical reaction will be ready to take place whenever this energy is crossed. The difference in rates of reactions is observed mainly due to the difference in activation energy because it stimulates molecules to take place in chemical reaction. Abbas et al. [8] conclude that within the boundary layer, the concentration profile is enhanced by the rising values of non-dimensional activation energy. The magnetohydrodynamic (MHD) peristaltic flow of Jeffrey matter which is chemically reactive along with activation energy was done by Hayat et al. [9]. The function of Arrhenius activation energy and the flow of MHD viscoelastic fluid with double chemical reaction were numerically computed by Mustafa et al. [10]. The mutual performance of activation energy and the double chemical reaction was scrutinized by Shafique et al. [11] in rotating fluid flow over a stretching surface. The latest field of research about activation energy can be learned in the study of Dhlamini et al. [12]. Eswaramoorthi et al. [13] investigated by the power of radiation and chemical effect, underflow on boundary layer the Soret and Dufour effects of irregular three-dimension. Sivasankaran et al. [14] analysed the consequence of slip, radiation, and chemical effect on a varied convective run of the viscous liquid in a permeable medium towards a perpendicular plate near a stagnation point. The result of nanofluid over a wedge on first order chemical effect, suction effects, and inner heat production/absorption was examined by Kasmani et al. [15]. Loganathan et al. [16] studied the appearance of radiation, chemical effect, convective heat, and cross-diffusion effects by means of CattaneoChristov heat up flux model to examine the changes of an Oldroyd-B fluid toward a stretching surface. The nth order chemical reaction, suction/ injection, thermal emission was examined by Makinde et al. [17] with buoyancy forces past a porous plate on uneven incompressible fluid flow. For a micropolar fluid that is enclosed by a semi-infinite permeable plate positioned vertically, Mohamed and Abo-Dahab [18] considered the special effects of chemical reaction besides thermal radiation in the occurrence of heat generation on free convection hydromagnetic heat and mass transport. Maleque [19] initiated that for an exothermic reaction, for rising values of the pre-exposure parameter a small reduction in temperature contour is found, but for endothermic reaction opposite effects are found. By increasing the energy of activation the chemical reaction rate

decreases. The extent of heat generated by a reaction can be quantified. The quantity that specifies the amount of heat is known as the “enthalpy” ∆H. When ∆H is positive the reaction is endothermic and if it is negative the reaction is exothermic. Kandasamy et al. [20] investigated alongside a wedge-shaped geometry, the collective special effects of chemical reaction and analysis of heat and mass transfer. Wahiduzzaman et al. [21] considered the impact of heat production, chemical effect, and thermal release by the flow of nanofluid on stagnation point. During the MHD flow of nanofluids, a flat plate with a chemical effect generated the radiative heat transport which was reported by Zhang et al. [22]. Natural convection heat transport alongside a perpendicular and inclined plate study by Yildiz [23] showed that local temperature values increase in distance along with the plate. Akinshilo [24] adopted the Adomain decomposition method for the heat transfer across convective straight fins taking into consideration temperature-dependent conductivity and interior generation of heat. In the existence of chemical reaction, collective heat and mass transport have plentiful application as a geothermal reservoir, drying, dryness at the exterior of a water body, geothermal pool, aeration of permeable solids, thermal filling, cooling action in nuclear reactors, improved oil recovery, toxic waste study, wet body surface evaporation, polymer fabrication cooling, and production of ceramics. By the action of thermal energy and chemical reaction, Rana et al. [25, 26] developed above a stretching sheet the mixed and free convection flow of Casson fluid. Zeeshan et al. [27] designate the effects of viscous dissipation parameter and thermal emission for multiphase magnetic fluid above a stretching surface. Darcy-Forchheimer flow was considered by means of Rashid et al. [28] in a rotating frame, it was seen that the flow as of the extended sheet filled the porous gap and the binary chemical process was entertained. An effort has not been made till now to study the behaviour of chemically reacting fluids with activation energy, radiation and heat generation. The authors have carried out work in this field for the major cause that this current article is to account for the outcome of radiation and heat generation of the chemically reacting fluid flow produced by rotating frame. The flow of fluid in the

rotating frame has been attracted for the reason of its extensive application in geophysics and production processes. Activation energy impact on the concentration of the fluid has also been applied.

Mathematical Analysis

We are interested to analyse the rotating flow by chemical reaction in a permeable space as shown in Figure 1. Effects of activation energy and radiation have been carried out in present flow situation. A stretching level surface induces the flow which coincides with the plane z≥0. Considering the temperature T = T(x,y,z) concentration C = C(x,y,z) and velocity V =[u(x,y,z), v(x,y,z), w(x, y, z)], we get: u v w    0 (1) x y z u

v v v 2v   v  w  2u   2  v  Fv 2, K x y z z

T T T Q  2T (T  T ) v w  2  c p x y z z 1 qr  ,  c p y

is the chemical reaction rate constant, k = 8.61×10–5 eV/K is the Boltzmann constant, n is a unitless constant exponent fitted rate constants typically lie in the range –1 < n < 1. We assume that the working medium is optically thick and the thermal radiation is local. When the medium is optically thick, the radiation can be approximated as an isotropic diffusion process by using the Rosseland approximation. By the means of the Rosseland approximation, we 4 T 4 attain the radiative heat flux qr  * where σ and k* 3k y are the Stefan-Boltzman constant and the mean absorption coefficient respectively. It is understood that within the flow field the difference in temperature is too small so that T4 can be extended in Taylor’s series. Expanding T4 about T∞ and neglect higher order terms we locate T  4T 3T  3T 4 The conditions for boundary are: u  ax , v  0, w  0,  k

Here T∞ and Tw is ambient and constant temperature, respectively. C∞ and Cw is ambient concentration and surface concentration and a is the stretching rate (>0 ). Utilising the following variables: 1 a , u  axf ( ), v  axg ( ), w    a  2 f    (7) C  C T  T     ,     Tw  T Cw  C 

where u, v and w are velocity components, respectively. Ω  is the angular velocity,     is kinematic viscosity, µ  dynamic viscosity and ρ the density of the flowing fluid. K represents porous medium permeability and D is the solute diffusivity, k denotes thermal conductivity, cp is at constant  k  pressure the specific heat,     is thermal diffusiv C p   

 C  ity, Cb is the drag co-efficient and F   1b/2  non-uniform K  n  T   Ea inertia co-efficient. The term kr2   e T  C  C  is the  T  modified Arrhenius function, Ea is the activation energy, kr2

Continuity Eq. (1) is automatically satisfied. In view of Eq. (7), Eqs. (2) – (6) become f   ff    f   2  g  1  Fr  f 2  0 (8) g   fg ’ f g  2  f    g  Fr g 2  0 (9)  1  4   Pr   1  3 Rd    f    Hg  0 (10)    E  n     Scf    Sc 1    exp      0 (11)  1    with boundary conditions being f  0, f   1, g  0,     Bi 1    ,   1, at   0 (12) f   0, g  0,   0,   0 as   

 is the Prandtl number,  4 *T3  C  Fr   1b/2  is the inertia coefficient, Rd  is the kk * K 

E  is Schmidt number, E  a is the  T D T  T non-dimensional activation energy,   w is the temT k2 perature difference parameter,   r is the dimensionless a h nu / a reaction rate and Bi  f is the Biot number. k The heat transfer, as a result of convection, through a fluid is given by the Nusselt number (Nu). The force and the mass transfer speed along the surface are defined in conditions of Sherwood number (Sh) and skin friction coefficient (Cf) as follows:

Since the leading system of equations are non-linear, to find the closed form of solutions it is not possible. Therefore, we seek a numerical solution of the governing model. With shooting technique the equations are solved by Runge-Kutta method.

Figure 11. Concentration profiles for different values of δ.

Results And Discussion

(E), Schmidt number (Sc) and fitted rate constant (n) on velocity, temperature and concentration have been studied. Figures 2–4 indicate the behaviour of (β) rotational parameter on velocity, concentration and temperature. Figure 2 shows that a velocity f’(η) decrease takes place for a percentage increase in rotational parameter (β). The reason for velocity decrease is that the stretching rate of the sheet reduces for larger β. In fact, when rotation in a framework

The present investigation has extended the variations of radiation parameters and heat generation parameters in the rotating framework. The impact of the rotational parameter(β), inertia coefficient (Fr), heat generation parameter (Hg), porosity parameter (λ), radiation parameter(Rd), Biot number(Bi), Prandtl number (Pr), temperature difference parameter (δ), the reaction rate (σ), the energy of activation

Figure 12. Concentration profiles for different values of E.

Figure 14. Concentration profiles for different values of n.

is increased, the stretching of the sheet will accordingly reduce and which will correspondingly slow down the flow of the fluid through it. From Figure 3, the temperature θ (η) increase with increasing β. According to Figure. 4, the concentration ϕ(η) is enhanced with increasing β. The major finding by this is that increase in porosity shows an opposite behaviour for temperature and concentration profile to the decrease in velocity. Qualitatively θ (η) and ϕ (η) through (β) show like results. Figure 5 shows the influence of inertia coefficient (Fr) on velocity f’(η). For rising values of Fr, there is a reduction in velocity. Figure 6 indicates the effect of the porosity parameter (λ) on velocity. Similar to the actions of β, the velocity is reduced for λ value rise. When the porosity of the fluid becomes large, the flow of fluid is restricted. This is caused because the fluid becomes more viscous and thus reduces the velocity. Impacts of heat generation parameter Hg and radiation parameter Rd are illustrated in Figures 7 and 8. Biot number Bi and Prandtl number Pr are illustrated in Figures 9 and 10. Here the temperature is a rising function for the Hg, Rd, and Bi variables and decreasing function for the Pr. In Figure 7 the temperature enhances for Hg. In the same manner, it can be examined from Figure 8 that temperature increases when Rd (Rd = 0, 2, 4, 6, 8)

Figure 13. Concentration profiles for different values of σ.

Figure 15. Concentration profiles for different values of Sc.

increases. It is obvious that when heat and radiation are increased in the fluid flow in a framework the temperature shows an increasing behaviour. The power of Bi is illustrated in Figure 9 on the temperature profile. θ(η) increase by means of an increase in Bi. In Figure 10 it is found that raising the value of Pr decreases the θ(η). From Figures11–15 it is to learn the influence of a range of parameters like activation energy, temperature difference, reaction rate, fitted rate constant, and Schmidt number on concentration. Figure 10 reveals that raising the values of δ leads to falling in ϕ(η). But from Figure 12, we understand that when activation energy (E) enhances ϕ(η) thickness also increases. In Figure 13, it is obvious that with raise in reaction rates σ values ϕ(η) come down rapidly. It can be examined from Figures 14 and 15 that ϕ(η) decreases for n and Sc increase. When n enhances, there is an increment in σ so automatically the liquid species terminate or dissolve and reduces ϕ (η). For larger Sc, mass diffusivity reduces which is in charge of the reduction in ϕ(η). Table 1 is prepared for dissimilar porosity parameter, rotational parameter and inertia coefficient. Temperature gradient –θ' (0) and concentration gradient –ϕ'(0) is decreased by β, λ and Fr but shows an increment in velocity gradient. Table 2 highlights the force of Biot number,

Table 1. The values of –f˝(0), –g˝(0), –θ´(0), –ϕ´(0) for ­different Fr, λ, β with Pr = 7, Sc = 1, Bi = 0.1, E = 0.1, n = 0.5, σ = 2, δ = 1, Hg = –0.5 and Rd = 0.2

Table 2. The values of –θ'(0), -ϕ'(0) for different Hg, Bi, Rd with Pr = 7, Sc = 1, β = 0.5, E = 0.1, n = 0.5, σ = 2, δ = 1, Fr = 0.2, λ = 0.2

Table 3. The values of –ϕ'(0) for different δ, σ, E, n, Pr with Sc = 1, β = 0.5, Bi = 0.1, Rd = 0.2, Hg = –0.5, Fr = 0.2, λ = 0.2 Pr

radiation parameter and heat generation parameter on local Nusselt and Sherwood numbers. Table 3 indicates power of Prandtl number, temperature difference parameter, activation energy, rate of reaction and fitted rate constant on concentration gradient –ϕ'(0). It is seen that there is an increment by δ, σ, n on concentration gradient –ϕ'(0).

Conclusion

The rotating flow in a permeable space in the presence of chemical reaction and effects of activation energy and thermal radiation has been carried out in the present flow situation. The modelled equations are solved numerically. In the present analysis, the result of relevant parameters on temperature, concentration, and velocity have been described and displayed by graphs. The major points of the current study are mentioned below:

• Velocity reduces but concentration and temperature enhance for elevated values of the rotational parameter (β). • Coefficient of inertia (Fr) and porosity parameter (λ) cause decay in velocity. • Behaviour of heat generation parameter (Hg), radiation parameter (Rd), and Biot number (Bi) are opposite on temperature to that of Prandtl number (Pr). • The concentration field shows a decrease with a rise in Schmidt number (Sc), temperature difference

parameter (δ), fitted rate constant (n), and reaction rate (σ). • With the increase in concentration, the activation energy increases. • The values λ and Fr show decrease in temperature gradient and concentration gradient. • Bi, Rd and Hg have similar results for concentration but in temperature gradient Bi is different from Rd and Hg.

Authorship Contributions

Concept: SS, MB; Design: MB, PY; Supervision: SS, KS; Data: MB, PY; Analysis: MB, PY; Literature search: PY; Writing: PY, MB; Critical revision: SS, KS.

Data Availability Statement

No new data were created in this study. The published publication includes all graphics collected or developed during the study.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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