RADIATION AND CHEMICAL REACTION EFFECTS ON ISOTHERMAL VERTICAL OSCILLATING PLATE WITH VARIABLE MASS DIFFUSION

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1 THERMAL SCIENCE: Vol. 13 (009), No., pp RADIATION AND CHEMICAL REACTION EFFECTS ON ISOTHERMAL VERTICAL OSCILLATING PLATE WITH VARIABLE MASS DIFFUSION by Kaliappan MANIVANNAN, Rajamanickam MUTHUCUMARASWAMY, and Venu THANGARAJ Orig i nal sci en tific pa per UDC: : /.4 BIBLID: , 13 (009),, DOI: 10.98/TSCI090155M The un steady flow of a vis cous in com press ible flow past an in fi nite iso ther mal ver ti - cal os cil lat ing plate, in the pres ence of ther mal ra di a tion and ho mo ge neous chem i - cal re ac tion of first or der has been stud ied. The fluid con sid ered here is a gray, ab - sorb ing-emit ting ra di a tion but a non-scat ter ing me dium. The plate tem per a ture is raised to T w and the con cen tra tion level near the plate is raised lin early with re spect to time. An ex act so lu tion to the dimensionless gov ern ing equa tions has been ob - tained by the Laplace trans form method, when the plate is os cil lat ing har mon i cally in its own plane. The ef fects of ve loc ity, tem per a ture, and con cen tra tion are stud ied for dif fer ent phys i cal pa ram e ters like phase an gle, ra di a tion pa ram e ter, chem i cal re ac tion pa ram e ter, Schmidt num ber, ther mal Grashof num ber, mass Grashof num ber, and time are stud ied graph i cally. It is ob served that the ve loc ity in creases with de creas ing phase an gle wt. Key words: chemical reaction, gray, oscillating, radiation, vertical plate, heat and mass transfer, radiation Introduction Ther mal ra di a tion ef fects on heat and mass trans fer play an im por tant role in man u fac - tur ing in dus tries for the de sign of fins, steel roll ing, nu clear power plants, gas tur bines, and var i - ous pro pul sion de vice for air craft, mis siles, sat el lites, and space ve hi cles are ex am ples of such en gi neer ing ap pli ca tions. If the tem per a ture of the sur round ing fluid is rather high, ra di a tion ef - fects play an im por tant role and this sit u a tion does ex ist in space tech nol ogy. Eng land et al. [1] have stud ied the ther mal ra di a tion ef fects of a op ti cally thin gray gas bounded by a sta tion ary ver ti cal plate. Ra di a tion ef fect on mixed con vec tion along a iso ther mal ver ti cal plate were stud - ied by Hossain et al. []. Raptis et al. [3] stud ied the ef fects of ther mal ra di a tion and free con vec - tion flow past a mov ing ver ti cal plate. The gov ern ing equa tions were solved an a lyt i cally. Das et al. [4] have an a lyzed ra di a tion ef fects on flow past an im pul sively started in fi nite iso ther mal ver ti cal plate. The ef fect of a chem i cal re ac tion de pend whether the re ac tion is ho mo ge neous or het - er o ge neous. This de pends on whether they oc cur at an in ter face or as a sin gle phase vol ume re - ac tion. In well-mixed sys tems, the re ac tion is het er o ge neous, if it takes place at an in ter face and ho mo ge neous, if it takes place in so lu tion. Chambre et al. [5] have an a lyzed a first or der chem i -

2 156 Manivannan, K., Muthucumaraswamy, R., Thangaraj, V.: Radiation and Chemical Reaction... cal re ac tion in the neigh bor hood of a sta tion ary hor i zon tal plate. Das et al. [6] have stud ied the ef fect of ho mo ge neous first or der chem i cal re ac tion on the flow past an im pul sively started in fi - nite ver ti cal plate with uni form heat flux and mass trans fer. Again, mass trans fer ef fects on mov - ing iso ther mal ver ti cal plate in the pres ence of chem i cal re ac tion stud ied by Das et al. [7]. The dimensionless gov ern ing equa tions were solved by the usual Laplace-trans form tech nique and the so lu tions are valid only at lower time level. The flow of a vis cous, in com press ible fluid past an in fi nite iso ther mal ver ti cal plate, os cil lat ing in its own plane, was solved by Soundalgekar [8]. The ef fect on the flow past a ver ti - cal os cil lat ing plate due to a com bi na tion of con cen tra tion and tem per a ture dif fer ences was stud ied ex ten sively by Soundalgekar et al. [9]. Ra di a tion ef fects on the os cil la tory flow past ver - ti cal in the pres ence of uni form tem per a ture an a lyzed by Mansour [10]. The gov ern ing were solved by per tur ba tion tech nique. The ef fect of mass trans fer on the flow past an in fi nite ver ti cal os cil lat ing plate in the pres ence of con stant heat flux has been stud ied by Soundalgekar et al. [11]. Muthucumaraswamy [1] stud ied ther mal ra di a tion ef fects on ver ti cal os cil lat ing plate in the pres ence of vari able tem per a ture and mass dif fu sion. It is pro posed to study chem i cal re ac tion and ther mal ra di a tion ef fects on un steady flow past in fi nite iso ther mal ver ti cal os cil lat ing plate with vari able mass dif fu sion. The dimensionless gov ern ing equa tions are solved us ing the Laplace trans form tech nique. The so lu - tions are in terms of ex po nen tial and com ple men tary er ror func tion. The pres ent study will be found use ful in the de sign of space ships. Mathematical formulation Ther mal ra di a tion ef fects on un steady flow of a vis cous in com press ible fluid past an in fi nite iso ther mal ver ti cal os cil lat ing plate with vari able mass dif fu sion is stud ied. Here the un - steady flow of a vis cous in com press ible fluid which is ini tially at rest and sur rounds an in fi nite ver ti cal plate with tem per a ture T and con cen tra tion C. Here, the x-axis is taken along the plate in the ver ti cally up ward di rec tion and the y-axis is taken nor mal to the plate. Ini tially, it is as - sumed that the plate and the fluid are of the same tem per a ture and con cen tra tion. At time t' > 0, the plate starts os cil lat ing in its own plane with fre quency w' and the tem per a ture of the plate is raised to T w and the con cen tra tion level near the plate is raised lin early with re spect to time. The fluid con sid ered here is a gray, ab sorb ing-emit ting ra di a tion but a non-scat ter ing me dium. It is also as sumed that there ex ists a ho mo ge neous first or der chem i cal re ac tion be tween the fluid and spe cies con cen tra tion. Then by usual Boussinesq s ap prox i ma tion, the un steady flow is gov erned by the fol low ing equa tions: u u T T C C t gb( ) gb* ( ) n y (1) rc p T k t T y q r y () C D t C K C y l (3) In most cases of chem i cal re ac tions, the re ac tion rate de pends on the con cen tra tion of the spe cies it self. A re ac tion is said to be of the or der n, if the re ac tion rate is pro por tional to the n th power of the con cen tra tion. In par tic u lar, a re ac tion is said to be first or der, if the rate of re ac - tion is di rectly pro por tional to con cen tra tion it self.

3 THERMAL SCIENCE: Vol. 13 (009), No., pp With the fol low ing ini tial and bound ary con di tions: t 0: u 0 T T, C C for all y t : u u cos wt, T T, C C ( C C ) At at y w u 0, T T, C C as y The lo cal ra di ant for the case of an op ti cally thin gray gas is ex pressed by: w q r 4a* s( T 4 T 4 ) (5) y It is as sume that the tem per a ture dif fer ences within the flow are suf fi ciently small such that T 4 may be ex pressed as a lin ear func tion of the tem per a ture. This is ac com plished by ex - pand ing T 4 in a Tay lor se ries about T and ne glect ing higher-or der terms, thus: By us ing eqs. (5) and (6), eq. () re duces to: rc The dimensionless quan ti ties are de fined as: g Gr bn (T w T u in eqs. (1) to (4), leads to: p (4) T 4 4T 3 T 3T 4 (6) T T k 16 a * st 3 ( T T ) t y (7) u t u yu T T U t 0 0,, Y, q u n n T T 0 ) C C gb* n( C C, C, w ) Gc C C u 0 3 w 0 3 mc * p n n s n Pr,,, w wn k D R 16a T 3 K l Sc K ku u u U t Grq GcC w U Y (8) (9) q 1 q R q t Pr Y Pr (10) C t 1 C Sc Y KC The ini tial and bound ary con di tions in non-di men sional form are: U 0, q 0, C 0, for all Y, t 0 t 0: U cos wt, q 1, C t, at Y 0 U = 0, q 0, C 0 as Y All the phys i cal vari ables are de fined in the no men cla ture. The so lu tions are ob tained for hy dro dy namic flow field in the pres ence of ther mal ra di a tion and chem i cal re ac tion. (11) (1)

4 158 Manivannan, K., Muthucumaraswamy, R., Thangaraj, V.: Radiation and Chemical Reaction... The eqs. (9) to (11), sub ject to the bound ary con di tions (1), are solved by the usual Laplace-trans form tech nique and the so lu tions are de rived as fol lows: where 1 q [exp( h Rt ) erfc( h Pr at ) exp( h Rt ) erfc ( h Pr at )] (13) C t [exp( h KtSc ) erfc( h Sc Kt ) exp( h KtSc ) erfc( h Sc Kt )] h Sct [exp( h KtSc ) erfc( h Sc Kt ) exp( h KtSc ) erfc( h Sc Kt )] (14) K i t U exp( w ) [exp( h iwt ) erfc ( h iwt ) exp( h iwt ) erfc ( h iwt )] 4 exp( iwt) [exp( h iwt ) erfc ( h iwt ) exp( h iwt ) erfc( h iwt )] 4 ( d ce) erfc( h) d exp( bt)[exp( h bt ) erfc( h bt ) exp( h bt ) erfc( h bt )] eexp( ct)[exp( h ct ) erfc( h ct ) exp( h ct ) erfc( h ct )] d[exp( h Rt ) erfc( h Pr at ) exp( h Rt ) erfc( h Pr at )] d exp( bt){exp[ h Pr( a b) t ] erfc[ h Pr ( a b) t ] exp[ h Pr( a b) t ]erfc[ h Pr ( a b) t ]} e( 1 ct)[exp( h KtSc ) erfc( h Sc Kt ) exp( h KtSc ) erfc( h Sc Kt )] ech Sct [exp( h KtSc ) erfc( h Sc Kt ) exp( h KtSc ) erfc( h Sc Kt )] K eexp( ct){exp[ h Sc( K c) t ] erfc[ h Sc ( K c) t ] exp[ h Sc( K c) t ] erfc[ h Sc ( K c) t ]} (15) R R K Gc a b c d e Pr, Pr, Sc, Gr 1 1 Sc b( 1 Pr), c ( 1 Sc ), and h Y t In or der to get the phys i cal in sight into the prob lem, the nu mer i cal val ues of U have been com puted from eq. (15). While eval u at ing this ex pres sion, it is ob served that the ar gu ment of the er ror func tion is com plex and, hence, we have sep a rated it into real and imag i nary parts by us ing the fol low ing for mula: where exp( a ) erf ( a ib ) erf ( a ) [ 1 cos( ab ) i sin( ab )] ap exp( a ) exp ( n/ 4 ) [ f ( a, b ) ig ( a, b )] ( a, n 4a n n e b) p n1 f a a cosh( nb)cos( ab) nsinh( nb)sin( ab) n g n a cosh( nb)sin( ab) nsinh( nb)cos( ab) e( a, b) erf ( a ib)

5 THERMAL SCIENCE: Vol. 13 (009), No., pp Re sults and dis cus sion In or der to get a phys i cal view of the prob lem the nu mer i cal val ues of the ve loc ity, tem per a ture and con cen tra tion for dif fer ent val ues of the phase an gle, ra di a tion pa ram e ter, mag - netic field pa ram e ter, Schmidt num ber, and time. The pur pose of the cal cu la tions given here is to asses the ef fect of dif fer ent wt, K, R, Sc, and t upon the na ture of the flow and trans port. The Laplace trans form so lu tions are in terms of ex po nen tial and com ple men tary er ror func tion. The tem per a ture pro files are cal cu lated for dif fer ent val ues of ther mal ra di a tion pa - ram e ter (R =, 5, 7, 10) from eq. (13) and these are shown in fig. 1 for air (Pr = 0.71) at time t = = 0.4. The ef fect of ther mal ra di a tion pa ram e ter is im por tant in tem per a ture pro files. It is ob - served that the tem per a ture in creases with de creas ing ra di a tion pa ram e ter. Fig ure dem on strates the ef fect of the con cen tra tion pro files for dif fer ent val ues of the chem i cal re ac tion pa ram e ter (K =, 5, 10), Sc = 0.6, and time t = 0.4. It is ob served that the con cen tra tion in creases with de creas ing chem i cal re ac tion pa ram e ter. Fig ure 3 rep re sents the ef - fect of con cen tra tion pro files at time t = 0.4 for dif fer ent Schmidt num ber (Sc = 0.16, 0.3, 0.6,.01), and K =. The ef fect of con cen tra tion is im por tant in con cen tra tion field. The pro files have the com mon fea ture that the con cen tra tion de creases in a mono tone fash ion from the sur - face to a zero value far away in the free stream. It is ob served that the wall con cen tra tion in - creases with de creas ing val ues of the Schmidt num ber. The con cen tra tion pro files for dif fer ent time (t = 0., 0.4, 0.6, 1), Sc = 0.6, and t = 0. are shown in fig. 4. The trend shows that the wall con cen tra tion in creases with in creas ing val ues of the time. Figure 1. Temperature profiles for different values of R Figure. Concentration profiles for different values of K Figure 3. Concentration profiles for different values of Sc The ve loc ity pro files for dif fer ent phase an - gles (wt = 0, p/4, p/3, p/), R = 10, K = 4, Gr = =, Gc =, Sc = 0.6, Pr = 0.71, and t = 0. are shown in fig. 5. It is ob served that the ve loc ity in creases with de creas ing phase an gle wt. Fig - ure 6 il lus trates the ef fect of the ve loc ity for dif fer ent val ues of the re ac tion pa ram e ter (K = = 0., 7, 0), wt = p/4, R = 5, Gr = 5, Gc = 5, Sc = 0.6, Pr = 0.71, and t = 0.4. The trend shows that the ve loc ity in creases with de creas ing chem i cal re ac tion pa ram e ter.

6 160 Manivannan, K., Muthucumaraswamy, R., Thangaraj, V.: Radiation and Chemical Reaction... Fig ure 4. Con cen tra tion pro files for dif fer ent val ues of t Fig ure 5. Ve loc ity pro files for dif fer ent val ues of wt The effect of velocity for different values of the radiation parameter (R = 0., 5, 0), wt = = p/4, K = 4, Gr = 5, Gc =, Pr = 0.71, Sc = 0.6, and t = 0. are shown in fig ure 7. The trend shows that the velocity increases with decreasing radiation parameter. It is observed that the velocity decreases in the pres ence of high ther mal ra di a tion. Fig ure 6. Ve loc ity pro files for dif fer ent val ues of K Fig ure 7. Ve loc ity pro files for dif fer ent val ues of R The ef fect of ve loc ity pro files for dif fer ent time (t = 0., 0.3, 0.4), R = 5, K =, wt = = p/4, Gr = 5, Gc = 5, Pr = 0.71, and Sc = 0.6 are shown in fig. 8. In this case, the ve loc ity in - Fig ure 8. Ve loc ity pro files for dif fer ent val ues of t Fig ure 9. Ve loc ity pro files for dif fer ent val ues of Gr and Gc

7 THERMAL SCIENCE: Vol. 13 (009), No., pp creases grad u ally with re spect to time t. The ve loc ity pro files for dif fer ent ther mal Grashof num ber (Gr =, 5), mass Grashof num ber (Gc =, 10), wt = p/4, K = 7, R =, Sc = 0.6, Pr = = 0.71, and time t = 0. are shown in fig. 9. It is clear that the ve loc ity in creases with in creas ing ther mal Grashof num ber or mass Grashof num ber. Con clu sions An ex act anal y sis is per formed to study ther mal ra di a tion ef fects on un steady flow past an in fi nite iso ther mal ver ti cal os cil lat ing plate, in the pres ence of vari able wall con cen tra tion. The gov ern ing equa tions are solved us ing Laplace trans form tech nique. The con clu sions of the study are: the velocity increases with decreasing phase angle wt and radiation parameter R, the wall concentration increases with decreasing Schmidt number, the temperature decreases due to high thermal radiation, and the leading edge effect is not affected by the oscillation of the plate. Nomenclature A constant, [ ] a* absorption coefficient, [m 1 ] C concentration, [kgm 3 ] C dimensionless concentration Cp specific heat at constant pressure, [Jkg 1 K] D mass diffusion coefficient, [m s 1 ] erfc complementary error function g acceleration due to gravity, [ms ] Gr thermal Grashof number, [ ] Gc mass Grashof number, [ ] K dimensionless chemical reaction parameter K l chemical reaction parameter, [ ] k thermal conductivity, [Wm 1 K 1 ] Pr Prandtl number, [ ] q r radiative heat flux in the y-direction, [Wm ] R radiation parameter Sc Schmidt number, [ ] T temperature of the plate, [K] t dimensionless time t 0 U time, [s] dimensionless velocity component in x-direction u velocity component in x-direction, [ms 1 ] u 0 amplitude of the oscillation, [ms 1 ] x Y y spatial coordinate along the plate, [m] dimensionless spatial coordinate normal to the plate spatial coordinate normal to the plate, [m] Greek let ters a thermal diffusivity, [m s 1 ] b volumetric coefficient of thermal expansion, [K 1 ] b * volumetric coefficient of expansion with concentration, [K 1 ] m coefficient of viscosity, [Ra s] wt phase angle n kinematic viscosity, [m s 1 ] r density of the fluid, [kgm 3 ] t dimensionless skin-friction, [kgm 1 s ] s Stefan-Boltzman constant (= ), [Wm K 4 ] q dimensionless temperature h similarity parameter Subscripts w conditions on the wall free stream conditions References [1] Eng land, W. G., Em ery, A. F., Ther mal Ra di a tion Ef fects on the Lam i nar Free Con vec tion Bound ary Layer of an Ab sorb ing Gas, J. Heat Trans fer, 91 (1969), 1, pp [] Hossain, M. A., Takhar, H. S., Ra di a tion Ef fect on Mixed Con vec tion Along a Ver ti cal Plate with Uni form Sur face Tem per a ture, Heat and Mass Trans fer, 31 (1996),, pp [3] Raptis, A., Perdikis, C., Ra di a tion and Free Con vec tion Flow Past a Mov ing Plate, Int. J. App. Mech Engg., 4 (1999), 4, pp

8 16 Manivannan, K., Muthucumaraswamy, R., Thangaraj, V.: Radiation and Chemical Reaction... [4] Das, U. N., Deka, R. K., Soundalgekar,V. M., Ra di a tion Ef fects on Flow Past an Im pul sively Started Ver - ti cal In fi nite Plate, J Theo. Mech., 1 (1996), 5, pp [5] Chambre, P. L., Young, J. D., On the Dif fu sion of a Chem i cally Re ac tive Spe cies in a Lam i nar Bound ary Layer Flow, The Phys ics of Flu ids, 1 (1958), 1, pp [6] Das, U. N., Deka, R. K., Soundalgekar,V. M., Ef fects of Mass Trans fer on Flow Past an Im pul sively Started In fi nite Ver ti cal Plate with Chem i cal Re ac tion, The Bul le tin, GUMA, 5 (1999), 1, pp [7] Das, U. N., Deka, R. K., Soundalgekar,V. M., Ef fects of Mass Trans fer on Flow Past an Im pul sively Started In fi nite Ver ti cal Plate with Con stant Heat Flux and Chem i cal Re ac tion, Forschung im Ingenieurwesen, 60 (1994), 10, pp [8] Soundalgekar, V. M., Free Con vec tion Ef fects on the Flow Past a Ver ti cal Os cil lat ing Plate, Astrophysics Space Sci ence, 64 (1979),, pp [9] Soundalgekar, V. M., Akolkar, S. P., Ef fects of Free Con vec tion Cur rents and Mass Trans fer on the Flow Past a Ver ti cal Os cil lat ing Plate, As trophysics Space Science, 89 (1983),, pp [10] Mansour, M. A., Ra di a tive and Free Con vec tion Ef fects on the Os cil la tory Flow Past a Ver ti cal Plate, As - tro phys ics and Space Sci ence, 166 (1990), 3, pp [11] Soundalgekar, V. M., et al., Ef fects of Mass Trans fer on the Flow Past an Os cil lat ing In fi nite Ver ti cal Plate with Con stant Heat Flux, Thermophysics and AeroMechanics, 1 (1994), 3, pp [1] Muthucumaraswamy, R.,The In ter ac tion of Ther mal Ra di a tion on Ver ti cal Os cil lat ing Plate with Vari able Tem per a ture and Mass Dif fu sion, Theoretical Applied Mechanics, 33 (006),, pp Authors' affiliations: K. Manivannan Department of Computer Applications Jayam College of Engineering & Technology Dharmapuri, India R. Muthucumaraswamy (corresponding author) Department of Applied Mathematics, Sri Venkateswara College of Engineering Sriperumbudur , India msamy@svce.ac.in V. Thangaraj Ramanujan Institute for Advanced Study in Mathematics University of Madras, Chepauk, Chennai, India Paper submitted: October 7, 007 Paper revised: June 1, 008 Paper accepted: August 1, 008

[Lakshmi* et al., 5.(6): June, 2016] ISSN: IC Value: 3.00 Impact Factor: 4.116

[Lakshmi* et al., 5.(6): June, 6] ISSN: 77-9655 IC Value: 3. Impact Factor: 4.6 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY THERMAL RADIATION AND CHEMICAL REACTION EFFECTS