CHEMICAL REACTION EFFECTS ON FLOW PAST AN EXPONENTIALLY ACCELERATED VERTICAL PLATE WITH VARIABLE TEMPERATURE. R. Muthucumaraswamy and V.

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International Jornal of Atomotive and Mechanical Engineering (IJAME) ISSN: 9-8649 (int); ISSN: 18-166 (Online); Volme pp. 31-38 Jly-December 1 niversiti Malaysia Pahang DOI: http://dx.doi.org/1.

1 International Jornal of Atomotive and Mechanical Engineering (IJAME) ISSN: (int); ISSN: (Online); Volme pp Jly-December 1 niversiti Malaysia Pahang DOI: CHEMICAL REACTION EFFECTS ON FLOW PAST AN EXPONENTIALLY ACCELERATED VERTICAL PLATE WITH VARIABLE TEMPERATRE R. Mthcmarasamy and V. Valliammal Department of Applied Mathematics Sri Venkatesara College of Engineering Sripermbdr 6 15 India. msamy@svce.ac.in ABSTRACT An analysis is performed to stdy the nsteady flo past an exponentially accelerated infinite vertical plate ith variable temperatre and niform mass diffsion in the presence of a homogeneos chemical reaction of first-order. The plate temperatre is raised linearly ith time and the concentration level near the plate is raised niformly. The dimensionless governing eqations are solved sing the Laplace transform. The velocity profiles are stdied for different physical parameters sch as the chemical reaction parameter thermal Grashof nmber mass Grashof nmber a and time. It is observed that the velocity increases ith increasing vales of a or t. Bt the trend is jst the reverse in the chemical reaction parameter. Keyords: chemical reaction accelerated vertical plate exponential heat transfer mass diffsion. INTRODCTION Diffsion rates can be tremendosly altered by chemical reactions. Chemical reactions can be codified as either heterogeneos or homogeneos processes. This depends on hether they occr at an interface or as a single phase volme reaction. In ell-mixed systems the reaction is heterogeneos if it takes place at an interface and homogeneos if it takes place in soltion. In most chemical reactions the reaction rate depends on the concentration of the species itself. A reaction is said to be of first order if the rate of reaction is directly proportional to concentration. In many chemical engineering processes there is a chemical reaction beteen a foreign mass and a flid. These processes take place in nmeros indstrial applications sch as manfactring of ceramics food processing and polymer prodction. Chambre and Yong (1958) have analysed a first order chemical reaction in the neighborhood of a horizontal plate. Das et al. (1994) have stdied the effect of a homogeneos first order chemical reaction on the flo past an implsively started infinite vertical plate ith niform heat flx and mass transfer. The mass transfer effect on moving isothermal vertical plate in the presence of chemical reaction as stdied by Das et al. (1999). The dimensionless governing eqations ere solved by the sal Laplace transform techniqe and the soltions are valid only at loer time level. Gpta et al. (1979) stdied free convection in a flo past a linearly accelerated vertical plate in the presence of viscos dissipative heat sing a pertrbation method. Free convection effects on flo past an exponentially accelerated vertical plate as stdied by Singh and Kmar (1984). The skin friction for an accelerated vertical plate has been stdied 31

2 R. Mthcmarasamy et al./ International Jornal of Atomotive and Mechanical Engineering (1) analytically by Hossain and Shayo (1986). Jha et al. (1991) analysed mass transfer effects on an exponentially accelerated infinite vertical plate ith constant heat flx and niform mass diffsion. Mthcmarasamy et al. (9) stdied an exact soltion of flo past an accelerated infinite vertical plate ith prescribed heat and mass flx. It is here proposed to stdy the nsteady flo past an exponentially accelerated infinite vertical plate ith variable temperatre and niform mass diffsion in the presence of a first order chemical reaction. The dimensionless governing eqations are solved sing the Laplace transform. This stdy ill be sefl in chemical process indstries sch as ire draing fibre draing food processing and polymer prodction. The soltions are in terms of the exponential and the complementary error fnctions. ANALYSIS The effects of a first order chemical reaction on the nsteady flo of a viscos incompressible flid past an exponentially accelerated infinite vertical plate ith variable temperatre and niform mass diffsion is stdied. It is assmed that the effect of viscos dissipation is negligible in the energy eqation. Here the x-axis is taken along the plate in the vertically pard direction and the y-axis is taken normal to the plate. At time t the plate and flid are at the same temperatre T. At time t > the plate is exponentially accelerated ith a velocity = exp(at) in its on plane and the temperatre of the plate is raised linearly ith respect to time and the concentration level near the plate is raised to C. It is also assmed that there is a first order chemical reaction beteen the flid and the species concentration. The reaction is assmed to take place entirely in the stream. Then nder the sal Bossinesq approximation the nsteady flo is governed by the folloing eqations: * gβ(t T ) gβ (C C ) ν (1) t y ρc p T T k t y C C D KlC t y () (3) ith the folloing initial and bondary conditions: T T C C for all yt t : here A. ν exp(a t) T T T T (T T )At C C C C at at y y (4) On introdcing the non-dimensional qantities t t ν y Y ν T T θ T T 3

3 Chemical reaction effects on flo past an exponentially accelerated vertical plate ith variable temperatre * gβν(t T ) C C νgβ (C C ) Gr C Gc (5) 3 3 C c μc p aν νk a K k in (1) to (4) e obtain 1 Sc ν D Grθ GcC t Y θ 1 θ t Y C 1 C KC t Sc Y (6) (7) (8) The initial and bondary conditions in non-dimensional qantities are θ C for all Y t t : exp(at) θ t θ C 1 C at at Y Y (9) METHOD OF SOLTION nder the assmptions of this flo problem it is observed that the energy eqation (7) and the concentration eqation (8) are ncopled from the eqation of momentm (6). It is clear that the temperatre variable θ(yt) and concentration eqation C(Yt) hereas (Yt) can be expressed in terms of θ(yt) and C(Yt). The dimensionless governing eqations (6) to (8) ith the initial and bondary conditions (9) are tackled sing Laplace transform techniqe. (Abramoitz and Stegn 197) and the reslts are as follos: Y s e L( ) = s (1) Y Sc( s k) e L(C) = s (11) 1 Y s Gr 1 L( ) e e 3 s a 1 s Gc 1 Y Sc( sk ) Y e e 1 Sc s ( S b) Y s s e Y s (1) 33

4 R. Mthcmarasamy et al./ International Jornal of Atomotive and Mechanical Engineering (1) The inverse Laplace transform soltions of eqations (1) to (1) are as follos: θ t(1 η )erfc(n ) η exp( η π ) 1 C exp(η KtSc)erfc(η Sc Kt) exp( η KtSc)erfc(η Sc Kt) (14) exp(at) exp(η at )erfc(η at ) exp( η at )erfc(η at ) e erfc(η) dt 6 e (31η e exp(ct) exp(η exp(η 4 4η )erfc(η) 4 (31η 4η ct )erfc(η KtSc)erfc(η e exp(ct) exp(η exp( η Sc η (1 4η )exp( η ) π )erfc(η η ) ct ) exp( η Kt) exp( η Sc(K c)t )erfc(η Sc Sc(K c)t )erfc(η (1 4η )exp( η ) π ct )erfc(η KtSc)erfc(η (K c)t ) Sc ct ) (K c)t Sc ) Kt) (13) (15) KSc here c Gr d 1Sc 1 Gc Y e and η. c(1 Sc) t RESLTS AND DISCSSION In order to get some physical insight into the problem nmerical comptations ere carried ot for different physical parameters a Gr Gc Sc and t pon the natre of the flo and transport. The vale of the Schmidt nmber Sc is taken to be.6 hich corresponds to ater vapor. Also the vale of andtl nmber () is chosen sch that it represents air ( =.71). The nmerical vales of the velocity are compted for different physical parameters like a andtl nmber thermal Grashof nmber mass Grashof nmber Schmidt nmber and time. The effect of velocity for different times (t = ) k = a =.5 Gr = Gc = 5 are shon in Figre 1. In this case the velocity increases gradally ith respect to time t. Figre illstrates the effect of velocity for different vales of the chemical reaction parameter (k =. 5) a =.5 Gr = 5 Gc = 1 and t =.. The trend shos that the velocity increases ith decreasing chemical reaction parameter. It is observed that the relative variation of the velocity ith the magnitde of the chemical reaction parameter. The velocity profiles for different (a =..5.8) k = Gr = Gc = 5 at t =. are stdied and presented in Figre 3. It is observed that the velocity increases ith increasing vales of a. 34

5 Chemical reaction effects on flo past an exponentially accelerated vertical plate ith variable temperatre t=.8 t=.6 t=.4 t= Figre1. Velocity profiles for different vales of t k=. k= k= Figre. Velocity profiles for different vales of k 35

6 R. Mthcmarasamy et al./ International Jornal of Atomotive and Mechanical Engineering (1) a=.8 a=.5 a=. a= Figre 3. Velocity profiles for different vales of a Figre 4 represents the effect of concentration profiles at time t =. for different Schmidt nmber (S c = ). The profiles have the common featre that the concentration decreases in a monotone fashion from the srface to zero far aay in the free stream. It is observed that the all concentration increases ith decreasing Schmidt nmber S c =. S c =.3 S c =.6 S c = Figre 4. Velocity profiles for different vales of S c 36

7 Chemical reaction effects on flo past an exponentially accelerated vertical plate ith variable temperatre CONCLSION The theoretical soltion of flo past an exponentially accelerated infinite vertical plate in the presence of variable temperatre and niform mass diffsion has been stdied. The dimensionless governing eqations ere solved by the sal Laplace transform techniqe. The effect of different parameters sch as the thermal Grashof nmber mass Grashof nmber a and t ere stdied graphically. It as observed that the velocity increases ith increasing vales of Gr Gc a and t. Bt that trend is reversed ith respect to the Schmidt nmber. REFERENCES Jha B.K. asad R. and Rai S. (1991) Mass Transfer Effects on the Flo Past an Exponentially Accelerated Vertical Plate With Constant Heat Flx. Astrophysics and Space Science 181: Chambre P.L. and Yong J.D. (1958) On the Diffsion of a Chemically Reactive Species in a Laminar Bondary Layer Flo. The Physics of Flids l: Das.N. Deka R.K. and Sondalgekar V.M. (1994) Effects of Mass Transfer on Flo Past an Implsively Started Infinite Vertical Plate With Constant Heat Flx and Chemical Reaction. Forschng im Ingenieresen 6: Das.N. Deka R.K. and Sondalgekar V.M. (1999) Effects of Mass Transfer on Flo Past an Implsively Started Infinite Vertical Plate With Chemical Reaction. The Blletin of GMA 5: 13. Gpta A.S. Pop I. and Sondalgekar V.M. (1979) Free Convection Effects on the Flo Past an Accelerated Vertical Plate in an Incompressible Dissipative Flid. Rev. Rom. Sci. Techn.-Mec. Apl. 4: Hossain M.A. and Shayo L.K. (1986) The Skin Friction in the nsteady Free Convection Flo Past an Accelerated Plate. Astrophysics and Space Science 15: Mthcmarasamy R. Sndar Raj M. and Sbramanian V.S.A. (9) Exact Soltion of Flo Past an Accelerated Infinite Vertical Plate With Heat and Mass Flx International Jornal of Applied Mechanics and Engineering 14: Singh A.K. and Kmar N. (1984) Free Convection Flo Past an Exponentially Accelerated Vertical Plate Astrophysics and Space Science 98: NOMENCLATRE A a constants C species concentration in the flid molm 3 C dimensionless concentration C p specific heat at constant pressre Jkg 1 K 1 D mass diffsion coefficient m s 1 Gc mass Grashof nmber Gr thermal Grashof nmber g acceleration de to gravity ms k thermal condctivity Jm 1 K 1 K l chemical reaction parameter J K dimensionless chemical reaction parameter 37

8 R. Mthcmarasamy et al./ International Jornal of Atomotive and Mechanical Engineering (1) Sc T t t x y y andtl nmber Schmidt nmber temperatre of the flid near the plate K time s dimensionless time velocity of the flid in the x-direction ms 1 velocity of the plate ms 1 dimensionless velocity spatial coordinate along the plate coordinate axis normal to the plate m dimensionless coordinate axis normal to the plate Greek symbols * erfc volmetric coefficient of thermal expansion K 1 volmetric coefficient of expansion ith concentration K 1 coefficient of viscosity Pas kinematic viscosity m s 1 density of the flid kgm 3 dimensionless skin-friction dimensionless temperatre similarity parameter complementary error fnction Sbscripts conditions at the all conditions in the free stream 38

Radiation Effects on Heat and Mass Transfer over an Exponentially Accelerated Infinite Vertical Plate with Chemical Reaction

Radiation Effects on Heat and Mass Transfer over an Exponentially Accelerated Infinite Vertical Plate with Chemical Reaction A. Ahmed, M. N.Sarki, M. Ahmad Abstract In this paper the stdy of nsteady flow