EFFECTS OF CHEMICAL REACTION ON MHD BOUNDARY LAYER FLOW OVER AN EXPONENTIALLY STRETCHING SHEET WITH JOULE HEATING AND THERMAL RADIATION

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1 International Research Journal o Engineering and Technology (IRJET) e-issn: Volume: Issue: 9 Dec-5.irjet.net p-issn: EFFECTS OF CHEMICAL REACTION ON MHD BOUNDARY LAYER FLOW OVER AN EXPONENTIALLY STRETCHING SHEET WITH JOULE HEATING AND THERMAL RADIATION Mridul Kumar Gogoi Associate Proessor and Head, Department o Mathematics, Dhakuakhana College, Assam, India Abstract A numerical analysis has been carried out to study a steady to-dimensional lo o a viscous incompressible, electrically conducting dissipating luid past an exponentially stretching surace in presence o magnetic ield and thermal radiation. The present ork examines the eects o chemical reaction and magnetic ield on velocity, temperature and concentration ields. Eects o viscous dissipation and Joule heating are taken into consideration in the energy equation. Non-linear partial dierential equations governing the motion are reduced to a system o ordinary dierential equations using suitable similarity transormations. Resultant equations are then solved numerically using MATLAB s built in solver bvp4c. The eects o various lo governing parameters on velocity, temperature and species concentration proile are depicted graphically. Finally, numerical values o coeicient o skin-riction, Nusselt number and Sherood number are presented in tables or various physical parameters. Key Words: Chemical reaction, magnetic ield, Joule heating, heat transer, mass transer *** INTRODUCTION Heat and mass transer study on luids ith chemical reaction eect over an exponentially stretching sheet have important role in metallurgy and chemical engineering industries, such as ood processing and polymer production. Moreover, coupled heat and mass transer problems in the presence o chemical reaction are o importance in many processes, and have thereore received a considerable amount o attention in recent times. Possible applications can be ound in processes such as drying, distribution o temperature and moisture over agricultural ields and groves o ruit trees, damage o crops due to reezing, evaporation at the surace o a ater body and energy transer in a et cooling toer, and lo in a desert cooler. Chambre and Acrivos[] analyzed catalytic surace reactions in hydrodynamic los. The paper as concerned ith its counterpart, namely, an investigation o a certain special class o homogeneous volume reactions in lo systems. Chambre et al.[] had studied the diusion o a chemically reactive species in a laminar boundary layer lo. Goddard and Acrivos [3] analyzed the laminar orced convection mass transer ith homogeneous chemical reaction. A uniied boundary layer analysis as applied to the problem o steady state mass transer o a chemical species, diusing rom a surace and reacting isothermally in a linear luid stream. Yih [4] presented an analysis o the orced convection boundary layer lo over a edge ith uniorm suction/bloing, hereas Watanabe [5] investigated the behavior o the boundary layer over a edge ith suction or injection in orced lo. MHD laminar boundary layer lo over a edge ith suction or injection had been discussed by Kaoussias and Nanousis [6] and Kumari[7] discussed the eect o large bloing rates on the steady laminar incompressible electrically conducting luid over an ininite edge ith a magnetic ield applied parallel to the edge. Anjali Devi and Kandasamy [8] have studied the eects o heat and mass transer on nonlinear boundary layer lo over a edge ith suction or injection. The eect o induced magnetic ield is included in the analysis. Chamkha and Khaled [9] investigated the problem o coupled heat and mass transer by MHD ree convection rom an inclined plate in the presence o internal heat generation or absorption. For the problem o coupled heat and mass transer in MHD ree convection, the eects o viscous dissipation and Ohmic heating ith chemical reaction are not studied in the above investigation. Hossain et al.[] analyzed the eects o radiation on ree convection rom a porous plate. During chemical reaction beteen to species concentration, heat is also generated []. Here it has been assumed that the level o species concentration is very lo and that the heat generated during chemical reaction can be neglected. The eects o mass transer 5, IRJET ISO 9:8 Certiied Journal Page 768

2 International Research Journal o Engineering and Technology (IRJET) e-issn: Volume: Issue: 9 Dec-5.irjet.net p-issn: on lo past an impulsively started ininite vertical plate ith constant heat lux and chemical reaction ere studied []. Hence these results are ound useul in chemical technology here ammonia and ethyl alcohol are used. The present investigation deals ith the study o lo o a electrically conducting luid in the presence o chemical reaction ith a species concentration due to the MHD lo over an exponentially stretching sheet.. MATHEMATICAL FORMULATION Consider a steady to-dimensional lo o an incompressible viscous luid bounded by a stretching sheet in hich x-axis is taken along the stretching sheet in the direction o the motion and y-axis is perpendicular to it. A uniorm magnetic ield o strength B is assumed to be applied in the y-direction. Under the usual boundary layer approximations, the lo, heat and mass transer in the presence o a chemical reaction eects ith Joule heating and thermal radiation are governed by the olloing equations (Sajid and Hayat [3]): u v x y u u u B u v o u x y y () () T T T u J qr x y y y y c p u v C C C u v Dm k C x y y (3) (4) here u and v are the velocities in the x- and y- directions respectively, is the luid density, is the dynamic viscosity, T is the temperature, is the thermal conductivity, c p is the speciic heat and q r is the radiative heat lux, C is the species concentration o the luid ithin the boundary layer, Dm is the molecular diusivity o the species concentration. The boundary conditions are given by: u() U e x L, v(), T() T T e x L, C() C Ce x L (5) u, T, C as y. here U is the reerence velocity, T and C are the temperature and species concentration at the plate. and T and C are the temperature and species concentration ar aay rom the plate, L is a constant. Employing Rosseland approximation o radiation or an optically thick layer one has (Sajid and Hayat [3]) q r 4 T * 3k y * 4, (6) here k* is the mean absorption coeicient and is the Stean-Boltzmann constant, T 4 is expressed as a linear unction o temperature, hence T 4T T 3T (7) Let us introduce the olloing transormations: x x L, v e L ( ) ( ) u U e ( ) U x e L y T T, ( ) L T T x L ( ) T T T e, C C ( ) C C U, L, here x L C C C e ( )., here (8) here the prime ( ) denotes derivative ith respect to η. Using equations (6)-(8), ()-(4) reduces to M (9) 4 3 () Nr Pr Ec MEc Sc( ) () ith the boundary conditions (5) become 5, IRJET ISO 9:8 Certiied Journal Page 769

3 International Research Journal o Engineering and Technology (IRJET) e-issn: Volume: Issue: 9 Dec-5.irjet.net p-issn: (), (), (), (),,, as. here dimensionless parameters are: Pr c p is the Prandtl number, Ec () U Tc is the p * 3 4 T Eckert number, Nr is the radiation * k k parameter, is the chemical reaction parameter, U M c B is the magnetic ield parameter, u Ec is the Eckert number, Sc is cp( T T ) Dm the Schmidt number. For the present problem the coeicient o skinriction and Nusselt number are the parameters o engineering interest hich are given respectively as belo: Coeicient o skin-riction: Coeicient o skin-riction is given by u C, here is the shearing U y y stress. Using the non-dimensional variables, e get C / (Re) () number. Nusselt number:, here Re Ux The Nusselt number hich is deined as xq Nu T T sheet is given by is the Renolds, here q is the heat transer rom the q T y y. Using the non-dimensional variables, e get Nu / (Re) (). Sherood number: The Sherood number hich is deined as xm Sh, here Dm( C ) C the mass lux at the surace. Using the non-dimensional variables, e get / Sh(Re) (). 3. RESULTS AND DISCUSSION C m Dm y y The set o equations (9) to () under boundary conditions () are non-linear coupled ordinary dierential equations so their solutions cannot be obtained in closed orm thereore these equations are solved numerically ith MATLAB s built in solver bvp4c. Computations have been carried out or dierent values o magnetic parameter (M), Eckert number (Ec) and Prandtl number(pr), chemical reaction parameter( ), radiation parameter and Schimdt number(sc). Results are presented through graphs or velocity ( ), temperature ( ) and species concentration ( ) in Fig.-Fig.7 or various values o lo governing parameters. Values o the parameters are taken as: M=.5, Pr=.7, Nr=.5, Ec=.5, =. and Sc=. unless otherise stated. Fig. and Fig. sho the eect o magnetic parameter (M) on dimensionless velocity and temperature distributions, respectively. In the presence o a magnetic ield in an electrically conducting luid induces a orce called Lorentz orce, hich opposes the lo. This resistive orce tends to slo don the lo, so the eect o increase in M is to decrease the velocity and also causes increases in its temperature. Fig. 3 depicts the variation o concentration proiles ith chemical reaction parameter ( ). It is seen is 5, IRJET ISO 9:8 Certiied Journal Page 77

4 ' International Research Journal o Engineering and Technology (IRJET) e-issn: Volume: Issue: 9 Dec-5.irjet.net p-issn: that an increase in the value o chemical reaction parameter decreases the concentration o species in the boundary layer. This is due to the act that chemical reaction in this system results in consumption o the chemical and hence results in decrease o concentration proile. The variation o concentration proiles or dierent values o Schimdt number(sc) is shon in Fig.4. It is observed that concentration o species decreases ith increasing value o Sc. The eects o Prandtl number(pr) on the temperature ield is shon in Fig.5. It is seen that increase in the Pr leads to a all in the luid. This is due to the act that thermal boundary layer thickness decreases compared to velocity boundary layer thickness or increasing Pr. From Fig.6, it is observed that temperature proile increases or increasing value o radiation parameter(nr). This is because an increase in the radiation parameter Nr leads to increase in the boundary layer thickness and enhances the heat transer rate on the surace in the presence o chemical eect. Fig.7 shos that temperature o the luid increases ith the increasing value o Eckert number(ec) Fig.: Temperature proile or dierent M =.,.,., Fig.3: Concentration proile or dierent Sc =.,.3,.4, M=.5,.,.5, Fig.4: Concentration proile or dierent Sc Fig.: Velocity proile or dierent M Pr =.5,.8,., M=.5,.,.5, Fig.5: Temperature proile or dierent Pr 5, IRJET ISO 9:8 Certiied Journal Page 77

5 International Research Journal o Engineering and Technology (IRJET) e-issn: Volume: Issue: 9 Dec-5.irjet.net p-issn: Nr =.,.5,., Fig.6: Temperature proile or dierent Nr.9 Ec =.5,.3,., Fig.7: Temperature proile or dierent Ec TABLE Eects o M, and Sc on coeicient o skin-riction ( C ), Nusselt number( Nu ) and Sherood number(sh) M Sc C Nu Sh , IRJET ISO 9:8 Certiied Journal Page 77

6 International Research Journal o Engineering and Technology (IRJET) e-issn: Volume: Issue: 9 Dec-5.irjet.net p-issn: Table demonstrates the eects o M, Sc on coeicient o skin-riction and C, hich represent plate shearing stress, rate o heat transer rom the plate to the luid in terms o Nusselt number Nu and rate o mass transer in terms o Sherood number Sh. It is seen that the value o Nu and Sh decreases ith the increase o magnetic parameter(m). But magnitude o C increases ith M. Physically negative values o C mean that surace exerts a drag orce on the luid, so that stretching surace ill induce the lo. Also it is observed that Sherood number increases or increasing values o chemical reaction parameter and Schimdt number. 4. CONCLUSIONS On the basis o the above study olloing observations ere made: () Velocity o the luid decreases and temperature increases ith the increasing magnetic ield intensity. () The increase in the chemical reaction parameter and Schimdt number lead to a all in the species concentration. (3) Temperature o the luid increases or increasing radiation parameter and Eckert number, hile that o decreases ith the increase o Prandtl number. (4) Wall shear stress o luid increases ith the increase o magnetic ield strength. But rate o heat and mass transer rise due to the increase o magnetic ield strength. (5) Rate o mass transer increases ith the rise o chemical reaction parameter and Schimdt number. REFERENCES [] P.L.Chambre and J.D. Acrivos, Analysis o catalytic surace reactions in hydrodynamic los, Journal o applied physics, vol. 7, pp. 3, 956. [] P.L. Chambre and A. Acrivos, Diusion o a chemically reactive species in a laminar boundary layer lo, Indian Engng. Chem, vol. 49, pp. 5, 957. [3] A. Acrivos, Analysis o the laminar orced convection mass transer ith homogeneous chemical reaction, Amer. Inst. Chem. Engrs, vol. 6, pp. 4, 96. [4] K.A.Yih, Uniorm suction / bloing eect on orce convection about edge, Acta Mech., vol. 8, pp. 73, 998. [5] T.Watanabe, Thermal boundary layer over a edge ith uniorm suction or injection in Forced lo, Acta Mechanica, vol. 83,pp. 9, 99. [6] N. G. Kaoussias and N. D. Nanousis, Magnetohydrodynamic laminar boundary layer lo over a edge ith suction or injection, Canadian Journal o Physics, pp. 733, 997. [7] M.Kumari, Eect o large bloing rates on the steady laminar incompressible electrically conducting luid over an ininite edge ith a magnetic ield applied parallel to the edge, International Journal o Engng.Sci., vol. 36, pp. 99, 998. [8] S. P. Anjali Devi and R. Kandasamy, Eects o heat and mass transer on MHD laminar boundary layer lo over a edge ith suction or injection, Journal o Energy Heat and Mass Transer, vol. 3, pp. 67,. [9] A. J. Chamkha and A. R. A. Khaled, Similarity solutions or hydro magnetic simultaneous heat and mass transer, Heat Mass Transer, vol. 37, pp. 7,. [] M. A. Hossain, M. A. Alim and D. A. S. Rees, The eects o radiation on ree convection rom a porous plate, Int. J. Heat Mass Transer, vol. 4, pp. 8, 999. [] Byon Bird arrene R; Steart E, Transport phenomena, Johniley and sons, Neyork, 3 pp. 69-6, 99. [] U. N. Das, R. Deka and V. M. Soundalgekar, Eects o mass transer on lo past an impulsively started ininite vertical plate ith constant heat lux and chemical reaction, J Forschung Im Ingenieuresen-Engineering Research Bd 6: 84-87, 994. [3] M. Sajid and T. Hayat, Inluence o thermal radiation on the boundary layer lo due to an exponentially stretching sheet, International Communications in Heat and Mass Transer, vol. 35, pp , 8. 5, IRJET ISO 9:8 Certiied Journal Page 773