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Thermal runaway and thermodynamic second law of a reactive couple stress hydromagnetic fluid with variable properties and Navier slips
Thermal runaway and thermodynamic second law of a reactive couple stress hydromagnetic fluid with variable properties and Navier slips
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Thermal runaway and thermodynamic second law of a reactive couple stress hydromagnetic fluid with variable properties and Navier slips S.O. Salawu, R.O. Oderinu, A.D. Ohaegbue PII: DOI: Reference:
S2468-2276(19)30822-1 https://doi.org/10.1016/j.sciaf.2019.e00261 SCIAF 261
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Scientific African
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22 June 2019 3 December 2019 23 December 2019
Please cite this article as: S.O. Salawu, R.O. Oderinu, A.D. Ohaegbue, Thermal runaway and thermodynamic second law of a reactive couple stress hydromagnetic fluid with variable properties and Navier slips, Scientific African (2019), doi: https://doi.org/10.1016/j.sciaf.2019.e00261
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Thermal runaway and thermodynamic second law of a reactive couple stress hydromagnetic fluid with variable properties and Navier slips 1
S.O. Salawu, 2 R.O. Oderinu, 2 A.D. Ohaegbue
1
2
Department of Mathematics, Landmark University, Omu-aran, Nigeria. Department of Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria. Corresponding author email: [email protected]
Abstract Theoretical solutions of a thermal runaway and irreversibility thermodynamic analysis of a couple stress hydromagnetic reactive fluid with Navier slip conditions and variable properties are examined. Ignoring the material assumption, the fluid is taken to be actively exothermic under Bimolecular kinetics. The dimensionless modeled equations are analytically solved using weighted residual method coupled with collocation techniques to obtain the bifurcation branch chain for criticality, irreversibility rate, flow rate and heat distribution solutions. It was noticed that the variable properties have no significant influence on the couple stress reactive liquid and heat distribution. Also, at high thermodynamic equilibrium, entropy generation in the system is minimized. This study results will help in understanding the relationship that exist between thermal explosion and thermal branched-chain in managing industrial engines in order to avoid system blow up. Keywords: Thermal runaway; Irreversibility; Non-Newtonian fluid; chemical kinetics; Hydromagnetic Nomenclature x¯, y¯, x, y: Cartesian coordinates [m] u¯, u: Fluid flow velocity [m/s] T : Dimensional fluid temperature [K] T0 : Fluid temperature at wall [K] P¯ : Dimensional pressure [P a] Cp : Heat capacity at constant pressure [JKg −1 K −1 ] Q: Heat reaction, [W ] h: Channel of width, [m] B0 : Magnetic field strength, [W bm−2 ] k(T ): Heat dependent thermal conductivity, [W m−1 K −1 ] P¯ : Dimensional fluid pressure, P a G: Pressure gradient v: Vibration frequency Ha: Hartmann number c: Injection/suction R: Universal gas constant Br: Brinkman number l: Planck number C: Reactant species P r: Prandtl number n: Chemical kinetics 1
r1 , r2 : Coefficient of Navier slips g1 , g2 : Navier slips A: constant reaction rate Greek symbol ρ: Fluid density, [kgm−3 ] θ: Dimensionless fluid temperature, [K] σ: Electrical conductivity, [Sm−1 ] µ(T ): Heat dependent dynamic viscosity [kgm−1 s−1 ] δ: Frank-Kamenetskii term α: Variable viscosity β: variable conductivity : Activation energy ν0 : Suction Λ: Couple stress η0 : Couple stress coefficient 1.0 Introduction The essential and usefulness of heat transfer in relation to non-Newtonian fluids cannot be over stress due to its applications in thermal technology lubricant, pharmaceutical mixtures, plastic, bitumen production and many more, (Nagaraju et al. (2019), Kareem et al. (2020). Among the non-Newtonian classical theory of liquids, the generalization of the couple stress fluid initiated by stokes’ have simple rheological properties without micro-structure. The fluid has an invariable material term responsible for the polar couple stress effects and viscosity, Ahmed et al. (2014). The motion of electrically conduction couple stress reactive fluid in porous motionless walls with heat dependent properties is essential in understanding the engineering hydrodynamic lubricants systems (Makinde (2018); Salawu et al. (2019)). Lubricant is a film viscous thin utilized to avoid contact of sliding solids during motion. In industrial and technology processes, reactive lubricants such as polyphenylethers, synthetic esters, hydrocarbon oils, etc., are mostly used, Ogunseye et al. (2019). The performance of the lubricants depends from time to time on the variation in the temperature; Rehman et al. (2017). Therefore, it is important to study the efficiency of the couple stress hydromagnetic reactive lubricant in order to enhance the performance of the industrial and engineering machines or devices, Srinivas et al. (2018). Engines effectiveness can be encouraged by increasing the thermodynamic equilibrium and reversibility process. The thermodynamic second law is reliable in determine the entropy generation as a result of irreversibility process, Salawu and Ogunseye (2020). Entropy generation measures availability of work destruction in a system. Hence, to upgrade system efficiency, entropy production needs to be minimized, (Kareem et al. (2016); Daniel et al. (2017)). Due to it necessity, Chinyoka and Makinde (2013) numerically investigated the entropy production in transient hydromagnetic heat dependent electrical conductivity of Couette flow. Salawu and Oke (2017) reported on the irreversibility of variable viscosity third-grade chemical reactive fluid with walls convective cooling. The problem was solved analytically by weighted residual techniques. Eegunjobi and Makinde (2017) considered analysis of hydromagnetic irreversibility of radiative couple stress fluid in a porous channel. It was reported that temperature transfer irreversibility dominates the reactive fluid also viscosity increases Bejan number field. In the work of (Nagaraju et al 2
(2016), Srinivas et al. (2017)), the irreversibility analysis of flow in a cylinder in the presence of magnetic field. Falade et al. (2016) studied irreversibility process in a couple stress viscous fluid with nonuniform wall heat. The model was solved by Adomian decomposition method, and it was reported that the inverse additive term enhances the flow liquid in permeable media. All the studies above was done without examined the thermal explosion of reactive fluids. Therefore, it is necessary to consider thermal runaway conditions for reactive viscous fluid performance as lubricants. The analysis of a thermal runaway plays an essential role in processing and handling of reactive non-Newtonian polar liquids. Thermal runaway exists where heat generation in a reactive system go beyond dissipating heat to the surroundings, (Salawu et al. (2019); Makinde (2009)). This is the basic conditions in a system for thermal explosion or runaway to occur, Frank-Kamenetskii (1969). The main objective of analyzing thermal runaways is to predict the unsafe or critical states of reactive flow fluid, Bowes (1984). Reactive entropy and thermal runaway analysis of Couette flow with variable viscosity was examined by Makinde and Maserumule (2008). From the analysis, it was reported that thermal explosion is useful in enhancing the operation and designing of several technology devices. Okoya (2011) discussed criticality disappearance for third grade reactive fluid flow in a flat channel with variable viscosity. Mathematically, heat transport and thermal explosion with ohmic heating in porous channel creates a nonlinear diffusion reaction and enduring solutions behavior in space gives more understanding to the complex fundamental physical process of systems thermal runaway. This study constructs an approximate solution for the diffusion reactive equation that formulates thermodynamic second law for irreversibility process and thermal runaway using collocation scheme along with weighted residual techniques. The exothermic reactive flow of entropy generation and critical areas splitting the regimes of ignition and non-ignition ways of couple stress fluid reactions is investigated. The graphical results for the momentum and energy solutions are also presented for some terms. 2.0 Problem formulation Consider the flow of chemical reactive couple stress liquid through a horizontal porous wall with unidirectional flow momentum field. The heat dependent fluid is stimulated by preexponential factor and pressure gradient acting on an absolutely enhanced magnetic field. The non-isothermal infinitely bounded configuration of the flow walls is positioned at y¯ = [0, h] with y¯ perpendicular to the non-Newtonian flow direction in x¯-axis. The impacts of z¯ is trivial as geometrically illustrated in figure 1. The incompressible and dynamically compatible hydromagnetic fluid momentum couple stress equation is expressed as Anwar et al. (2012). 1 ρU = curl(ρN) + µ∇2 A + ∇p + ρF − η0 ∇4 A − J × B. 2
(1)
The equation for temperature balance of an exothermic chemical reactive couple stress is expressed as dW ρ = −divq + η0 (∇2 A)2 + ρr + QCK0 . (2) dt where W defines internal heat, q denotes Fourier’s heat flux for which q = −K∇T represents thermal conductivity. The parameter r denotes radiation energy. According to (Okoya (2013); 3
Salawu et al. (2019)), the term K0 is written as; n E KT K0 = A e− RT . vl
(3)
The pre-exponential index i.e. for Bimolecular (n = 0.5), Arrhenius (n = 0) and for Sensitized (n = −2). Follow from above, the equations for the velocity and heat balance with variable conductivities and Navier slips boundary conditions are given as:
Figure 1. Flow geometry setup d¯ u dP¯ d4 u¯ u d2 u¯ dµ(T ) dT d¯ =− − η0 4 + µ(T ) 2 + − σB02 u¯, d¯ y d¯ x d¯ y d¯ y dT d¯ y d¯ y 2 2 2 dT d2 T dT dk(T ) d u¯ d¯ u = k(T ) 2 + + η0 + µ(T ) ρν0 Cp + QCK0 + σB02 u¯2 , d¯ y d¯ y d¯ y d¯ y d¯ y2 d¯ y ρν0
(4) (5)
Bounded by the conditions
d2 u¯(0) d¯ u(0) d2 u¯(1) d¯ u(1) = 0, u ¯ (0) = r , T (0) = T , = 0, u¯(1) = r2 , T (1) = T0 . 1 0 2 2 d¯ y d¯ y d¯ y d¯ y
(6)
The chosen heat dependent dynamic viscosity and thermal conductivity are assumed to be exponentially vary as defined by (Makinde and Franks (2014); Jangili et al. (2018); Salawu et al. (2019)), µ ¯(T ) = µ0 e−a(T −T0 ) ,
k(T ) = k0 eb(T −T0 ) .
(7)
By using equation (7) along with the consecutive dimensionless variables on equations (4)-(6), u¯ y¯ P¯ h E(T − T0 ) µ ¯(T ) k(T ) r1 x¯ , u= , y= , P = , θ= , µ= , k= , g1 = , 2 h ν0 h µ 0 ν0 RT0 µ0 k0 h 2 2 dP aRT02 bRT02 µ0 ν 2 E h σ B r h2 µ0 0 0 2 2 2 , H = G=− , α= , β= , Br = , g = , Λ = , (8) 2 dx E E RkT02 µ0 h η0 n 1 EQACh2 T0 K µ0 C p RT0 hν0 δ= e− , c = , Pr = , = . 2 RKT0 lv ν k E x=
The resulting non-dimensional equations are gotten as c
du d2 u du dθ 1 d4 u = G + e−αθ 2 − αe−αθ − 2 4 − H 2 u, dy dy dy dy Λ dy 4
(9)
d2 θ dθ cP r = eβθ 2 +βeβθ dy dy
dθ dy
2
−αθ
+Bre
Along with the boundary conditions
du dy
2
Br + 2 Λ
d2 u dy 2
2
θ
+BrH 2 u2 +δ(1+θ)n e 1+θ , (10)
d2 u(0) d2 u(1) du(0) du(1) , T (0) = 0, , T (1) = 0. = 0, u(0) = g = 0, u(1) = g2 1 2 2 dy dy dy dy
(11)
3.0 Thermodynamic second law for irreversibility analysis The analysis of irreversibility from second law for the couple stress hydromagnetic reactive liquid flow causes steady entropy generation. Thus, this is as a result of exchange of heat between reactive couple stress liquid and the channel surfaces. The local rate of entropy volumetric for the hydromagnetic fluid with variable properties and Navier slip conditions is according to (Salawu and Dada (2018); Hassan et al. (2018)): K(T ) EA = T02
¯ 2 2 2 dT η d2 u¯ u σB02 u¯2 µ(T ) d¯ + + . + d¯ y T0 d¯ y T0 d¯ y T0
(12)
The heat irreversibility is defined first on the right hand side of equation (12) while the irreversibility due to couple stress, fluid viscosity and joule heating are presented respectively in the remaining part of the equation. Introducing equation (8), the non-dimensional entropy generation in equation (12) is expressed as " # 2 2 2 E 2 EA h2 dθ Br 1 d2 u du βθ −αθ 2 2 =e Ns = 2 + +e +H u . (13) T0 KR2 dy Λ2 dy 2 dy Taken N1 = e
βθ
2 dθ dy
and N2 =
Br
1 Λ2
d2 u dy 2
The Bejan number (Be) is characterized as
Be = eβθ
2 dθ dy
+
Br
1 Λ2
2
+e
−αθ
2 du dy
+H u . 2 2
2 dθ eβθ dy N 1 = 1 = . 2 2 Ns 1+ξ d2 u du −αθ 2 2 +e +H u dy 2 dy
(14)
N2 where Ns = N2 + N1 , ξ = N , and N1 shows the heat transport irreversibility, N2 denotes the 1 couple stress, fluid viscosity and ohmic heating irreversibility and ξ is the irreversibility ratio. The range 0 ≤ Be ≤ 1 defined Bejan number (Be), irreversibility due to couple stress, fluid viscosity, and viscous heating prevails when Be = 0 but heat irreversibility overrules the reactive system when Be = 1 at different temperature.
4.0 Method of solution The solution procedure for the thermal criticality, irreversibility process, heat and velocity equations are done by applying weighted residual and collocation methods. The solutions assumed the sum n X v (y, a) = ψ0 (y) + ai ψi , (15) i=1
5
as given by (Odejide and Aregbesola (2011); Salawu (2018)), the ψi (y) is a prescribed function satisfying the boundary conditions. The trial function v (y, a) are separately define for the equations and is applied on the boundary condition. Thus, the residual equation is gotten as R(y, a) = Q(v(y, a)) − r(y)
(16)
v(y, a) satisfies the nonlinear derivative in successful approximation for the function ψi . The intention is to reduce R(y, a) errors in some sense. That is Z R(y, a)Wi dy = 0 i = 1, 2, .., n (17) Y
The weighted functions Wi must be the same with the unknown ai in v. The collocation scheme is employed for the solution where the weighted function is defined as Dirac delta i.e., Wi (y) = δ(y − yi ) such that R(y, a) = 0. Applying the methods on the boundary conditions and on the equations (9) and (10) to obtain redsidual equations as uR = c 10 y 9 a10 + 9 y 8 a9 + 8 y 7 a8 + 7 y 6 a7 + 6 y 5 a6 + 5 y 4 a5 + 4 y 3 a4 + 3 y 2 a3 + 2 ya2 + a1 − 10 9 8 7 6 5 4 3 2 G − e−α (b10 y +b9 y +b8 y +b7 y +b6 y +b5 y +b4 y +b3 y +b2 y +b1 y+b0 ) 90 y 8 a10 + 72 y 7 a9 + 56 y 6 a8 +42 y 5 a7 + 30 y 4 a6 + 20 y 3 a5 + 12 y 2 a4 + 6 ya3 + 2 a2 + ... (18) θR = cP r 10 y 9 b10 + 9 y 8 b9 + 8 y 7 b8 + 7 y 6 b7 + 6 y 5 b6 + 5 y 4 b5 + 4 y 3 b4 + 3 y 2 b3 + 2 yb2 + b1 − 10 9 8 7 6 5 4 3 2 eβ (b10 y +b9 y +b8 y +b7 y +b6 y +b5 y +b4 y +b3 y +b2 y +b1 y+b0 ) 90 y 8 b10 + 72 y 7 b9 + 56 y 6 b8 + 42 y 5 b7 +30 y 4 b6 + 20 y 3 b5 + 12 y 2 b4 + 6 yb3 + 2 b2 + ... (19) The residual equations are collocated within the domain and solve along with the boundary conditions to get the unknown constants. The solutions process is repeated for different values of terms. Maple software is adopted to get the unknown constant and completely solve the problem. 5.0 Discussion of results The solutions to the primary dimensionless equations for irreversibility, velocity and heat equations as well as thermal runaway are obtained by the adopted method and the results are illustrated graphically in figures 2 to 13. The consistence and accuracy of the method are confirmed and presented in the table. In table 1, the results comparison with exact and existing work is reported. The collocation weighted residual method (CWRM) gives good results while comparing with the exact and Adominan decomposition method (ADM) results. The CWRM agreed well with the other methods of solutions with the order of absolute error 10−7 . Table 1: Results comparison with the existing results and analytical for velocity profile when Λ = 1, δ = α = β = 0, H = 0.2, g1 = g2 = 0.1
6
w 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
wexact wADM , 0.00303054 0.00598827 0.00854151 0.01037320 0.01125880 0.01106440 0.00974517 0.00734506 0.00399756 -0.00007304 -0.00454921
Kareem et al. (2016) 0.00303070 0.00598858 0.00854196 0.01037370 0.01125950 0.01106510 0.00974577 0.00734554 0.00399783 -0.00007305 -0.00454955
wCW RM Absolute error 0.00303067 1.0012 × 10−7 0.00598851 1.6750 × 10−7 0.00854182 2.3164 × 10−7 0.01037361 2.7968 × 10−7 0.01125933 3.6802 × 10−7 0.01106497 4.2994 × 10−7 0.00974568 4.3776 × 10−7 0.00734550 4.4422 × 10−7 0.00399779 2.2145 × 10−7 -0.00007305 9.1365 × 10−8 -0.00454947 2.4321 × 10−7
5.1 Momentum and heat dependent parameters solutions In figures 2 and 3, the reaction of the flow momentum to rising in the couple stress term (Λ) and Hartmann number (H) are considered. The inverse of the term (Λ) that defined the ratio of relative rotation of viscous force to the viscous Newtonian force increases the flow momentum field. This is because the fluid forces resisted the produced forces by additive thereby induced couple stress. The diminishing in the created inverse couple stress leads to encouragement in the flow rate profile (figure 2). Magnetic field term declines the reactive liquid field due to the induced force. The Lorentz force enhances the liquid bonding forces which barred free fluid particles movement. Hence, the velocity field is suppressed as shown in diagram 3. Figures 4 and 5 illustrate the response of heat distribution to rising in the terms Brinkman number (Br) and Frank Kamenetskii (δ). Both terms are active sources of heat, therefore they both steadily and significantly boosts the heat distribution in an exothermic couple stress reaction. The non-Newtonian fluid temperature is increased as the heat source term in the temperature equation is enhanced.
Figure 2. Flow rate profile for rising Λ
Figure 3. Flow rate profile for rising H
7
Figure 4. Heat profile with rising Br
Figure 5. Heat profile with rising δ
5.2 Irreversibility and irreversibility ratio dependent parameters solutions Figures 6 and 7 in a channel, represent the entropy generation results for rising values of the terms Frank Kamenetskii (δ) and pressure gradient (G). Entropy generation is optimized at the lower static wall and rises continuously until its reaches the highest level at the center of the walls. Its then minimized regularly towards the upper stationary wall. Further minimization in the entropy generation is observed as the fluid viscosity is discouraged in an exothermic couple stress reaction fluid. From the perspective of engineering design, overall entropy production can be reduced if the fluid viscosity as a result of increase in the temperature can be exponentially decreased. In figures 8 and 9, the impact of rising in the values of the terms (δ) and (G) on the Bejan number are demonstrated. It is noticed that the liquid viscosity irreversibility dictates the couple stress reaction fluid at the upper stationary wall, while energy transfers irreversibility dictates at the lower static wall. The dictation effect at the lower wall is due to increase in the values of the terms Frank Kamenetskii and pressure gradient as reported in the figures.
Figure 6. Entropy generation with rising δ
Figure 7. Entropy generation with rising G
8
Figure 8. Bejan number with rising δ
Figure 9. Bejan number with rising G
5.3 Branch-chain bifurcation and ignition dependent parameters solutions Figures 10 and 11 show the bifurcation branch-chain plots for rising values of activation energy () and chemical kinetics (m) in the (δ, θmax ). The plots denote the maximum liquid heat (θmax ) and Frank Kamenetskii (δ). For every interval of the term () in [0,0.1], there is a critical branch chain solution (δcr ) for 0 ≤ δ ≤ δcr . Due to nonlinear form of the modeled equations, two solutions are recorded. The stable lower branch solution and the upper unstable branch solution, when δcr < δ represents the thermal runaway. A respective incline and decline in the thermal runaway is displayed with increasing values of () and (m) in figures 10 and 11. These results denote finite time of infinite temperature that resulted in blowup effect. The changes in the maximum energy (θmax,cr ) and thermal criticality (δcr ) for variation in the liquid variable viscosity (α) is presented in figures 12 and 13. The magnitude of (θmax,cr ) and (δcr ) is respectively enhanced and diminished for rising values of (α). The appearance of the figures depicts the effect of increasing and decreasing heat source terms in the exothermic couple stress reaction liquid.
Figure 10. Criticality bifurcation for rising
Figure 11. Criticality bifurcation for various m
9
Figure 12. Plot of θmax,cr against for rising α
Figure 13. Plot of δcr against for rising α
Conclusion The thermal runaway and irreversibility effects on an exothermic couple stress reaction with Navier slips and variable conductivities in a stationary wall using weighted residual approximation techniques along with collocation techniques is examined. The heat dependent variables have no significant effect on the couple stress hydromagnetic reactive fluid. It was observed from the perspective of engineering design, overall entropy production can be reduced if the fluid viscosity as a result of increase in the temperature can be exponentially decreased. Also, the thermal criticality region splitting the areas of implosion and non-implosion ways of exothermic couple stress reaction is obtained. The obtained results from this investigation are hoped not only to give essential applications information, but also to complement existing related studies. The present study can be extended to unsteady state in a cylinder in the presence of reaction initiation rate and branch chain order and porous medium. References Ahmed, S., Anwar Beg, O., Ghosh, S.K. (2014). A couple stress fluid modeling on free convection oscillatory hydromagnetic flow in an inclined rotating channel. Ain Shams Engineering Journal, 5, 1249-1265. Anwar, B.O., Ghosh, S.K., Ahmed, S, Be’g, T. (2012). Mathematical modelling of oscillatory magneto-convection of a couples-stress biofluid in an inclined rotating channel. J Mech Med Biol., 12(3), 1-35. Bowes, P. C. (1984). Self-Heating: Evaluating and Controlling the Hazard, Elsevier, Amsterdam. Chinyoka, T., Makinde, O.D. (2013). Numerical investigation of entropy generation in unsteady MHD generalized Couette flow with variable electrical conductivity. The Scientific World Journal, 13, 1-11. Daniel, Y.S., Aziz, Z.A., Ismail, Z., Salah, F. (2017). Entropy analysis in electrical magnetohydrodynamic (MHD) flow of nanofluid with effects of thermal radiation, viscous dissipation, and chemical reaction. Theoretical & Applied Mechanics Letters, 7, 235-242. 10
Eegunjobi, A.S., Makinde, O.D. (2017). Irreversibility analysis of hydromagnetic flow of couple stress fluid with radiative heat in a channel filled with a porous medium, Results Phys. 7, 459-469. Falade, J.A., Adesanya, S.O., Ukaegbu, J.C., Osinowo, M.O. (2016). Entropy generation analysis for variable viscous couple stress fluid flow through a channel with non-uniform wall temperature. Alexandria Engineering Journal, 55, 69-75 Frank-Kamenetskii, D. A. (1969). Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, New York. Hassan, A.R., Gbadeyan, J.A., Salawu, S.O. (2018). The effects of thermal radiation on a reactive hydromagnetic internal heat generating fluid flow through parallel porous plates. Springer Proceedings in Mathematics & Statistics, 259, 183-193. Kareem, R.A., Salawu, S.O., Yubin Yan (2020). Analysis of transient Rivlin-Ericksen fluid and irreversibility of exothermic reactive hydromagnetic variable viscosity. J. Appl. Comput. Mech., 6(1), 26-36. Kareem, S.O., Adesanya, S.O., Vincent, U.E. (2016). Second law analysis for hydromagnetic couple stress fluid flow through a porous channel. Alexandria Engineering Journal, 55, 925-931. Makinde, O.D. (2008). Thermal criticality in viscous reactive flows through channels with a sliding wall: An exploitation of the HermitePade approximation method. Mathematical and Computer Modelling, 47, 312-317. Makinde, O.D. (2009). Thermal criticality for a reactive gravity driven thin film flow of a third-grade fluid with adiabatic free surface down an inclined plane. Appl. Math. Mech. -Engl. Ed. 30(3), 373-380. Makinde, O.D., Franks, O. (2014). On MHD unsteady reactive Couette flow with heat transfer and variable properties. Cent. Eur. J. Eng., 4(1), 54-63. Makinde, O.D., Maserumule, R.L. (2008). Thermal criticality and entropy analysis for a variable viscosity Couette flow. Phys. Scr. 78, 1-6. Nagaraju, G., Srinivas, J., Ramana Murthy, J.V. Rashad, A.M. (2016). Entropy generation analysis of the MHD flow of couple stress fluid between two concentric rotating cylinders with porous lining. Heat TransferAsian Research, 46(4), 316-330. Nagaraju, G., Srinivas Jangili, Ramana Murthy, J.V., Bg, O.A., Kadir, A. (2019). Second law analysis of flow in a circular pipe with uniform suction and magnetic field fffects. J. Heat Transfer, 141(1): 012004 (9 pages). Odejide, S.A., Aregbesola, Y.A.S. (2011). Applications of method of weighted residuals to problems with semi-finite domain. Rom. Journ. Phys., 56(1-2), 14-24.
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Ogunseye, H.A., Salawu, S.O., Tijani, Y.O., Riliwan, M., Sibanda, P. (2019). Dynamical analysis of hydromagnetic Brownian and thermophoresis effects of squeezing EyringPowell nanofluid flow with variable thermal conductivity and chemical reaction. Multidiscipline Modeling in Materials and Structures, 15(6), 1100-1120. Okoya, S.S. (2013). On criticality and disappearance of criticality for a branched-chain thermal reaction with distributed temperature. Afr. Mat., 24, 465-476. Okoya, S.S. (2011). Disappearance of criticality for reactive third-grade fluid with Reynolds model viscosity in a flat channel. International Journal of Non-Linear Mechanics, 46, 1110-1115. Rehman, A.U., Mehmood, R., Nadeem, S. (2017). Entropy analysis of radioactive rotating nanofluid with thermal slip, Appl. Therm. Eng. 112, 832-840. Salawua, S.O., Dada, M.S., Fenuga, O.J. (2019). Thermal explosion and irreversibility of hydromagnetic reactive couple stress fluid with viscous dissipation and Navier slips. Theoretical & Applied Mechanics Letters, 9, 246-253. Salawu, S.O., Ogunseye, H.A. (2020). Entropy generation of a radiative hydromagnetic PowellEyring chemical reaction nanofluid with variable conductivity and electric field loading. Results in Engineering, 5, 100072. Salawu, S.O., Oke, S.I. (2017). Inherent irreversibility of exothermic Chemical Reactive thirdgrade poiseuille flow of a Variable Viscosity with Convective Cooling. J. Appl. Comput. Mech., 4(3), 167-174. Salawu, S.O., Oladejo, N.K., Dada, M.S. (2019). Analysis of unsteady viscous dissipative Poiseuille fluid flow of two-step exothermic chemical reaction through a porous channel with convective cooling. Ain Shams Engineering Journal, 10, 565-572. Salawu, S.O., Kareem, R.A., Shonola, S.A. (2019). Radiative thermal criticality and entropy generation of hydromagnetic reactive PowellEyring fluid in saturated porous media with variable conductivity. Energy Reports 5, 480-488. Salawu, S.O., Hassan, A.R., Abolarinwa, A. Oladejo, N.K. (2019). Thermal stability and entropy generation of unsteady reactive hydromagnetic Powell-Eyring fluid with variable electrical and thermal conductivities. Alexandria Engineering Journal. 58, 519-529. Salawu, S.O., Dada, M.S. (2018). Lie group analysis of soret and dufour effects on radiative inclined magnetic pressure-driven flow past a Darcy-forchheimer medium. Journal of the Serbian Society for Computational Mechanics, 12(1), 108-125. Salawu, S.O. (2018). Analysis of third-grade heat absorption hydromagnetic exothermic chemical reactive flow in a Darcy-forchheimer porous medium with convective cooling. WSEAS Transaction on mathematics, 17, 280-289. Srinivas, J., Ramana Murthy, J.V., Anwar Bg, O. (2017). Entropy generation analysis of radiative heat transfer effects on channel flow of two immiscible couple stress fluids. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 39(6), 2191-2202. 12
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Journal Pre-proof
Thermal runaway and thermodynamic second law of a reactive couple stress hydromagnetic fluid with variable properties and Navier slips S.O. Salawu, R.O. Oderinu, A.D. Ohaegbue PII: DOI: Reference:
S2468-2276(19)30822-1 https://doi.org/10.1016/j.sciaf.2019.e00261 SCIAF 261
To appear in:
Scientific African
Received date: Revised date: Accepted date:
22 June 2019 3 December 2019 23 December 2019
Please cite this article as: S.O. Salawu, R.O. Oderinu, A.D. Ohaegbue, Thermal runaway and thermodynamic second law of a reactive couple stress hydromagnetic fluid with variable properties and Navier slips, Scientific African (2019), doi: https://doi.org/10.1016/j.sciaf.2019.e00261
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Thermal runaway and thermodynamic second law of a reactive couple stress hydromagnetic fluid with variable properties and Navier slips 1
S.O. Salawu, 2 R.O. Oderinu, 2 A.D. Ohaegbue
1
2
Department of Mathematics, Landmark University, Omu-aran, Nigeria. Department of Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria. Corresponding author email: [email protected]
Abstract Theoretical solutions of a thermal runaway and irreversibility thermodynamic analysis of a couple stress hydromagnetic reactive fluid with Navier slip conditions and variable properties are examined. Ignoring the material assumption, the fluid is taken to be actively exothermic under Bimolecular kinetics. The dimensionless modeled equations are analytically solved using weighted residual method coupled with collocation techniques to obtain the bifurcation branch chain for criticality, irreversibility rate, flow rate and heat distribution solutions. It was noticed that the variable properties have no significant influence on the couple stress reactive liquid and heat distribution. Also, at high thermodynamic equilibrium, entropy generation in the system is minimized. This study results will help in understanding the relationship that exist between thermal explosion and thermal branched-chain in managing industrial engines in order to avoid system blow up. Keywords: Thermal runaway; Irreversibility; Non-Newtonian fluid; chemical kinetics; Hydromagnetic Nomenclature x¯, y¯, x, y: Cartesian coordinates [m] u¯, u: Fluid flow velocity [m/s] T : Dimensional fluid temperature [K] T0 : Fluid temperature at wall [K] P¯ : Dimensional pressure [P a] Cp : Heat capacity at constant pressure [JKg −1 K −1 ] Q: Heat reaction, [W ] h: Channel of width, [m] B0 : Magnetic field strength, [W bm−2 ] k(T ): Heat dependent thermal conductivity, [W m−1 K −1 ] P¯ : Dimensional fluid pressure, P a G: Pressure gradient v: Vibration frequency Ha: Hartmann number c: Injection/suction R: Universal gas constant Br: Brinkman number l: Planck number C: Reactant species P r: Prandtl number n: Chemical kinetics 1
r1 , r2 : Coefficient of Navier slips g1 , g2 : Navier slips A: constant reaction rate Greek symbol ρ: Fluid density, [kgm−3 ] θ: Dimensionless fluid temperature, [K] σ: Electrical conductivity, [Sm−1 ] µ(T ): Heat dependent dynamic viscosity [kgm−1 s−1 ] δ: Frank-Kamenetskii term α: Variable viscosity β: variable conductivity : Activation energy ν0 : Suction Λ: Couple stress η0 : Couple stress coefficient 1.0 Introduction The essential and usefulness of heat transfer in relation to non-Newtonian fluids cannot be over stress due to its applications in thermal technology lubricant, pharmaceutical mixtures, plastic, bitumen production and many more, (Nagaraju et al. (2019), Kareem et al. (2020). Among the non-Newtonian classical theory of liquids, the generalization of the couple stress fluid initiated by stokes’ have simple rheological properties without micro-structure. The fluid has an invariable material term responsible for the polar couple stress effects and viscosity, Ahmed et al. (2014). The motion of electrically conduction couple stress reactive fluid in porous motionless walls with heat dependent properties is essential in understanding the engineering hydrodynamic lubricants systems (Makinde (2018); Salawu et al. (2019)). Lubricant is a film viscous thin utilized to avoid contact of sliding solids during motion. In industrial and technology processes, reactive lubricants such as polyphenylethers, synthetic esters, hydrocarbon oils, etc., are mostly used, Ogunseye et al. (2019). The performance of the lubricants depends from time to time on the variation in the temperature; Rehman et al. (2017). Therefore, it is important to study the efficiency of the couple stress hydromagnetic reactive lubricant in order to enhance the performance of the industrial and engineering machines or devices, Srinivas et al. (2018). Engines effectiveness can be encouraged by increasing the thermodynamic equilibrium and reversibility process. The thermodynamic second law is reliable in determine the entropy generation as a result of irreversibility process, Salawu and Ogunseye (2020). Entropy generation measures availability of work destruction in a system. Hence, to upgrade system efficiency, entropy production needs to be minimized, (Kareem et al. (2016); Daniel et al. (2017)). Due to it necessity, Chinyoka and Makinde (2013) numerically investigated the entropy production in transient hydromagnetic heat dependent electrical conductivity of Couette flow. Salawu and Oke (2017) reported on the irreversibility of variable viscosity third-grade chemical reactive fluid with walls convective cooling. The problem was solved analytically by weighted residual techniques. Eegunjobi and Makinde (2017) considered analysis of hydromagnetic irreversibility of radiative couple stress fluid in a porous channel. It was reported that temperature transfer irreversibility dominates the reactive fluid also viscosity increases Bejan number field. In the work of (Nagaraju et al 2
(2016), Srinivas et al. (2017)), the irreversibility analysis of flow in a cylinder in the presence of magnetic field. Falade et al. (2016) studied irreversibility process in a couple stress viscous fluid with nonuniform wall heat. The model was solved by Adomian decomposition method, and it was reported that the inverse additive term enhances the flow liquid in permeable media. All the studies above was done without examined the thermal explosion of reactive fluids. Therefore, it is necessary to consider thermal runaway conditions for reactive viscous fluid performance as lubricants. The analysis of a thermal runaway plays an essential role in processing and handling of reactive non-Newtonian polar liquids. Thermal runaway exists where heat generation in a reactive system go beyond dissipating heat to the surroundings, (Salawu et al. (2019); Makinde (2009)). This is the basic conditions in a system for thermal explosion or runaway to occur, Frank-Kamenetskii (1969). The main objective of analyzing thermal runaways is to predict the unsafe or critical states of reactive flow fluid, Bowes (1984). Reactive entropy and thermal runaway analysis of Couette flow with variable viscosity was examined by Makinde and Maserumule (2008). From the analysis, it was reported that thermal explosion is useful in enhancing the operation and designing of several technology devices. Okoya (2011) discussed criticality disappearance for third grade reactive fluid flow in a flat channel with variable viscosity. Mathematically, heat transport and thermal explosion with ohmic heating in porous channel creates a nonlinear diffusion reaction and enduring solutions behavior in space gives more understanding to the complex fundamental physical process of systems thermal runaway. This study constructs an approximate solution for the diffusion reactive equation that formulates thermodynamic second law for irreversibility process and thermal runaway using collocation scheme along with weighted residual techniques. The exothermic reactive flow of entropy generation and critical areas splitting the regimes of ignition and non-ignition ways of couple stress fluid reactions is investigated. The graphical results for the momentum and energy solutions are also presented for some terms. 2.0 Problem formulation Consider the flow of chemical reactive couple stress liquid through a horizontal porous wall with unidirectional flow momentum field. The heat dependent fluid is stimulated by preexponential factor and pressure gradient acting on an absolutely enhanced magnetic field. The non-isothermal infinitely bounded configuration of the flow walls is positioned at y¯ = [0, h] with y¯ perpendicular to the non-Newtonian flow direction in x¯-axis. The impacts of z¯ is trivial as geometrically illustrated in figure 1. The incompressible and dynamically compatible hydromagnetic fluid momentum couple stress equation is expressed as Anwar et al. (2012). 1 ρU = curl(ρN) + µ∇2 A + ∇p + ρF − η0 ∇4 A − J × B. 2
(1)
The equation for temperature balance of an exothermic chemical reactive couple stress is expressed as dW ρ = −divq + η0 (∇2 A)2 + ρr + QCK0 . (2) dt where W defines internal heat, q denotes Fourier’s heat flux for which q = −K∇T represents thermal conductivity. The parameter r denotes radiation energy. According to (Okoya (2013); 3
Salawu et al. (2019)), the term K0 is written as; n E KT K0 = A e− RT . vl
(3)
The pre-exponential index i.e. for Bimolecular (n = 0.5), Arrhenius (n = 0) and for Sensitized (n = −2). Follow from above, the equations for the velocity and heat balance with variable conductivities and Navier slips boundary conditions are given as:
Figure 1. Flow geometry setup d¯ u dP¯ d4 u¯ u d2 u¯ dµ(T ) dT d¯ =− − η0 4 + µ(T ) 2 + − σB02 u¯, d¯ y d¯ x d¯ y d¯ y dT d¯ y d¯ y 2 2 2 dT d2 T dT dk(T ) d u¯ d¯ u = k(T ) 2 + + η0 + µ(T ) ρν0 Cp + QCK0 + σB02 u¯2 , d¯ y d¯ y d¯ y d¯ y d¯ y2 d¯ y ρν0
(4) (5)
Bounded by the conditions
d2 u¯(0) d¯ u(0) d2 u¯(1) d¯ u(1) = 0, u ¯ (0) = r , T (0) = T , = 0, u¯(1) = r2 , T (1) = T0 . 1 0 2 2 d¯ y d¯ y d¯ y d¯ y
(6)
The chosen heat dependent dynamic viscosity and thermal conductivity are assumed to be exponentially vary as defined by (Makinde and Franks (2014); Jangili et al. (2018); Salawu et al. (2019)), µ ¯(T ) = µ0 e−a(T −T0 ) ,
k(T ) = k0 eb(T −T0 ) .
(7)
By using equation (7) along with the consecutive dimensionless variables on equations (4)-(6), u¯ y¯ P¯ h E(T − T0 ) µ ¯(T ) k(T ) r1 x¯ , u= , y= , P = , θ= , µ= , k= , g1 = , 2 h ν0 h µ 0 ν0 RT0 µ0 k0 h 2 2 dP aRT02 bRT02 µ0 ν 2 E h σ B r h2 µ0 0 0 2 2 2 , H = G=− , α= , β= , Br = , g = , Λ = , (8) 2 dx E E RkT02 µ0 h η0 n 1 EQACh2 T0 K µ0 C p RT0 hν0 δ= e− , c = , Pr = , = . 2 RKT0 lv ν k E x=
The resulting non-dimensional equations are gotten as c
du d2 u du dθ 1 d4 u = G + e−αθ 2 − αe−αθ − 2 4 − H 2 u, dy dy dy dy Λ dy 4
(9)
d2 θ dθ cP r = eβθ 2 +βeβθ dy dy
dθ dy
2
−αθ
+Bre
Along with the boundary conditions
du dy
2
Br + 2 Λ
d2 u dy 2
2
θ
+BrH 2 u2 +δ(1+θ)n e 1+θ , (10)
d2 u(0) d2 u(1) du(0) du(1) , T (0) = 0, , T (1) = 0. = 0, u(0) = g = 0, u(1) = g2 1 2 2 dy dy dy dy
(11)
3.0 Thermodynamic second law for irreversibility analysis The analysis of irreversibility from second law for the couple stress hydromagnetic reactive liquid flow causes steady entropy generation. Thus, this is as a result of exchange of heat between reactive couple stress liquid and the channel surfaces. The local rate of entropy volumetric for the hydromagnetic fluid with variable properties and Navier slip conditions is according to (Salawu and Dada (2018); Hassan et al. (2018)): K(T ) EA = T02
¯ 2 2 2 dT η d2 u¯ u σB02 u¯2 µ(T ) d¯ + + . + d¯ y T0 d¯ y T0 d¯ y T0
(12)
The heat irreversibility is defined first on the right hand side of equation (12) while the irreversibility due to couple stress, fluid viscosity and joule heating are presented respectively in the remaining part of the equation. Introducing equation (8), the non-dimensional entropy generation in equation (12) is expressed as " # 2 2 2 E 2 EA h2 dθ Br 1 d2 u du βθ −αθ 2 2 =e Ns = 2 + +e +H u . (13) T0 KR2 dy Λ2 dy 2 dy Taken N1 = e
βθ
2 dθ dy
and N2 =
Br
1 Λ2
d2 u dy 2
The Bejan number (Be) is characterized as
Be = eβθ
2 dθ dy
+
Br
1 Λ2
2
+e
−αθ
2 du dy
+H u . 2 2
2 dθ eβθ dy N 1 = 1 = . 2 2 Ns 1+ξ d2 u du −αθ 2 2 +e +H u dy 2 dy
(14)
N2 where Ns = N2 + N1 , ξ = N , and N1 shows the heat transport irreversibility, N2 denotes the 1 couple stress, fluid viscosity and ohmic heating irreversibility and ξ is the irreversibility ratio. The range 0 ≤ Be ≤ 1 defined Bejan number (Be), irreversibility due to couple stress, fluid viscosity, and viscous heating prevails when Be = 0 but heat irreversibility overrules the reactive system when Be = 1 at different temperature.
4.0 Method of solution The solution procedure for the thermal criticality, irreversibility process, heat and velocity equations are done by applying weighted residual and collocation methods. The solutions assumed the sum n X v (y, a) = ψ0 (y) + ai ψi , (15) i=1
5
as given by (Odejide and Aregbesola (2011); Salawu (2018)), the ψi (y) is a prescribed function satisfying the boundary conditions. The trial function v (y, a) are separately define for the equations and is applied on the boundary condition. Thus, the residual equation is gotten as R(y, a) = Q(v(y, a)) − r(y)
(16)
v(y, a) satisfies the nonlinear derivative in successful approximation for the function ψi . The intention is to reduce R(y, a) errors in some sense. That is Z R(y, a)Wi dy = 0 i = 1, 2, .., n (17) Y
The weighted functions Wi must be the same with the unknown ai in v. The collocation scheme is employed for the solution where the weighted function is defined as Dirac delta i.e., Wi (y) = δ(y − yi ) such that R(y, a) = 0. Applying the methods on the boundary conditions and on the equations (9) and (10) to obtain redsidual equations as uR = c 10 y 9 a10 + 9 y 8 a9 + 8 y 7 a8 + 7 y 6 a7 + 6 y 5 a6 + 5 y 4 a5 + 4 y 3 a4 + 3 y 2 a3 + 2 ya2 + a1 − 10 9 8 7 6 5 4 3 2 G − e−α (b10 y +b9 y +b8 y +b7 y +b6 y +b5 y +b4 y +b3 y +b2 y +b1 y+b0 ) 90 y 8 a10 + 72 y 7 a9 + 56 y 6 a8 +42 y 5 a7 + 30 y 4 a6 + 20 y 3 a5 + 12 y 2 a4 + 6 ya3 + 2 a2 + ... (18) θR = cP r 10 y 9 b10 + 9 y 8 b9 + 8 y 7 b8 + 7 y 6 b7 + 6 y 5 b6 + 5 y 4 b5 + 4 y 3 b4 + 3 y 2 b3 + 2 yb2 + b1 − 10 9 8 7 6 5 4 3 2 eβ (b10 y +b9 y +b8 y +b7 y +b6 y +b5 y +b4 y +b3 y +b2 y +b1 y+b0 ) 90 y 8 b10 + 72 y 7 b9 + 56 y 6 b8 + 42 y 5 b7 +30 y 4 b6 + 20 y 3 b5 + 12 y 2 b4 + 6 yb3 + 2 b2 + ... (19) The residual equations are collocated within the domain and solve along with the boundary conditions to get the unknown constants. The solutions process is repeated for different values of terms. Maple software is adopted to get the unknown constant and completely solve the problem. 5.0 Discussion of results The solutions to the primary dimensionless equations for irreversibility, velocity and heat equations as well as thermal runaway are obtained by the adopted method and the results are illustrated graphically in figures 2 to 13. The consistence and accuracy of the method are confirmed and presented in the table. In table 1, the results comparison with exact and existing work is reported. The collocation weighted residual method (CWRM) gives good results while comparing with the exact and Adominan decomposition method (ADM) results. The CWRM agreed well with the other methods of solutions with the order of absolute error 10−7 . Table 1: Results comparison with the existing results and analytical for velocity profile when Λ = 1, δ = α = β = 0, H = 0.2, g1 = g2 = 0.1
6
w 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
wexact wADM , 0.00303054 0.00598827 0.00854151 0.01037320 0.01125880 0.01106440 0.00974517 0.00734506 0.00399756 -0.00007304 -0.00454921
Kareem et al. (2016) 0.00303070 0.00598858 0.00854196 0.01037370 0.01125950 0.01106510 0.00974577 0.00734554 0.00399783 -0.00007305 -0.00454955
wCW RM Absolute error 0.00303067 1.0012 × 10−7 0.00598851 1.6750 × 10−7 0.00854182 2.3164 × 10−7 0.01037361 2.7968 × 10−7 0.01125933 3.6802 × 10−7 0.01106497 4.2994 × 10−7 0.00974568 4.3776 × 10−7 0.00734550 4.4422 × 10−7 0.00399779 2.2145 × 10−7 -0.00007305 9.1365 × 10−8 -0.00454947 2.4321 × 10−7
5.1 Momentum and heat dependent parameters solutions In figures 2 and 3, the reaction of the flow momentum to rising in the couple stress term (Λ) and Hartmann number (H) are considered. The inverse of the term (Λ) that defined the ratio of relative rotation of viscous force to the viscous Newtonian force increases the flow momentum field. This is because the fluid forces resisted the produced forces by additive thereby induced couple stress. The diminishing in the created inverse couple stress leads to encouragement in the flow rate profile (figure 2). Magnetic field term declines the reactive liquid field due to the induced force. The Lorentz force enhances the liquid bonding forces which barred free fluid particles movement. Hence, the velocity field is suppressed as shown in diagram 3. Figures 4 and 5 illustrate the response of heat distribution to rising in the terms Brinkman number (Br) and Frank Kamenetskii (δ). Both terms are active sources of heat, therefore they both steadily and significantly boosts the heat distribution in an exothermic couple stress reaction. The non-Newtonian fluid temperature is increased as the heat source term in the temperature equation is enhanced.
Figure 2. Flow rate profile for rising Λ
Figure 3. Flow rate profile for rising H
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Figure 4. Heat profile with rising Br
Figure 5. Heat profile with rising δ
5.2 Irreversibility and irreversibility ratio dependent parameters solutions Figures 6 and 7 in a channel, represent the entropy generation results for rising values of the terms Frank Kamenetskii (δ) and pressure gradient (G). Entropy generation is optimized at the lower static wall and rises continuously until its reaches the highest level at the center of the walls. Its then minimized regularly towards the upper stationary wall. Further minimization in the entropy generation is observed as the fluid viscosity is discouraged in an exothermic couple stress reaction fluid. From the perspective of engineering design, overall entropy production can be reduced if the fluid viscosity as a result of increase in the temperature can be exponentially decreased. In figures 8 and 9, the impact of rising in the values of the terms (δ) and (G) on the Bejan number are demonstrated. It is noticed that the liquid viscosity irreversibility dictates the couple stress reaction fluid at the upper stationary wall, while energy transfers irreversibility dictates at the lower static wall. The dictation effect at the lower wall is due to increase in the values of the terms Frank Kamenetskii and pressure gradient as reported in the figures.
Figure 6. Entropy generation with rising δ
Figure 7. Entropy generation with rising G
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Figure 8. Bejan number with rising δ
Figure 9. Bejan number with rising G
5.3 Branch-chain bifurcation and ignition dependent parameters solutions Figures 10 and 11 show the bifurcation branch-chain plots for rising values of activation energy () and chemical kinetics (m) in the (δ, θmax ). The plots denote the maximum liquid heat (θmax ) and Frank Kamenetskii (δ). For every interval of the term () in [0,0.1], there is a critical branch chain solution (δcr ) for 0 ≤ δ ≤ δcr . Due to nonlinear form of the modeled equations, two solutions are recorded. The stable lower branch solution and the upper unstable branch solution, when δcr < δ represents the thermal runaway. A respective incline and decline in the thermal runaway is displayed with increasing values of () and (m) in figures 10 and 11. These results denote finite time of infinite temperature that resulted in blowup effect. The changes in the maximum energy (θmax,cr ) and thermal criticality (δcr ) for variation in the liquid variable viscosity (α) is presented in figures 12 and 13. The magnitude of (θmax,cr ) and (δcr ) is respectively enhanced and diminished for rising values of (α). The appearance of the figures depicts the effect of increasing and decreasing heat source terms in the exothermic couple stress reaction liquid.
Figure 10. Criticality bifurcation for rising
Figure 11. Criticality bifurcation for various m
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Figure 12. Plot of θmax,cr against for rising α
Figure 13. Plot of δcr against for rising α
Conclusion The thermal runaway and irreversibility effects on an exothermic couple stress reaction with Navier slips and variable conductivities in a stationary wall using weighted residual approximation techniques along with collocation techniques is examined. The heat dependent variables have no significant effect on the couple stress hydromagnetic reactive fluid. It was observed from the perspective of engineering design, overall entropy production can be reduced if the fluid viscosity as a result of increase in the temperature can be exponentially decreased. Also, the thermal criticality region splitting the areas of implosion and non-implosion ways of exothermic couple stress reaction is obtained. The obtained results from this investigation are hoped not only to give essential applications information, but also to complement existing related studies. The present study can be extended to unsteady state in a cylinder in the presence of reaction initiation rate and branch chain order and porous medium. References Ahmed, S., Anwar Beg, O., Ghosh, S.K. (2014). A couple stress fluid modeling on free convection oscillatory hydromagnetic flow in an inclined rotating channel. Ain Shams Engineering Journal, 5, 1249-1265. Anwar, B.O., Ghosh, S.K., Ahmed, S, Be’g, T. (2012). Mathematical modelling of oscillatory magneto-convection of a couples-stress biofluid in an inclined rotating channel. J Mech Med Biol., 12(3), 1-35. Bowes, P. C. (1984). Self-Heating: Evaluating and Controlling the Hazard, Elsevier, Amsterdam. Chinyoka, T., Makinde, O.D. (2013). Numerical investigation of entropy generation in unsteady MHD generalized Couette flow with variable electrical conductivity. The Scientific World Journal, 13, 1-11. Daniel, Y.S., Aziz, Z.A., Ismail, Z., Salah, F. (2017). Entropy analysis in electrical magnetohydrodynamic (MHD) flow of nanofluid with effects of thermal radiation, viscous dissipation, and chemical reaction. Theoretical & Applied Mechanics Letters, 7, 235-242. 10
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