Effect of transport ratio on source term in determination of surface emission coefficient

Transcription

1 Internationa Journa of heoretica & Appied Sciences, (): 74-78(9) ISSN : Effect of transport ratio on source term in determination of surface emission coefficient Sanjeev Kumar and Apna Mishra epartment of Mathematics r. B.R. Ambedkar University I.B.S., Khandari Campus, Agra (UP) INIA ABSRAC : A surface reaction may be regarded as occurring in foowing five consecutive steps which are, diffusion of the reactants moecues to the surface, adsorption of gases on the surface, reaction on the surface, desorption of the products and finay diffusion of the desorbed products into the main body of the gas. In genera, at the surface of the iquid or soid, the foreign moecues of the substance of the gas soid or iquid experiences resutant inward attraction. If the moecue is in the interior, it experiences equa force of attraction by the surrounding moecues and the resutant force is zero. hus, the moecue of the adsorbate on the surface of adsorbent experiences resutant inward pu by the moecue of the adsorbent. his way the process of adsorption hed on to the surfaces and these forces are Vanderwaas forces and then it is termed as physica adsorption. However, there may be some bond formation between the adsorption moecues and adsorbent moecues which is then known as chemica adsorption. his work anayzes the effect of transport ratio on source term by drawing some graphs using MAAB and this describes the variation of source term with respect to the dimensioness time for sorption and desorption process. Keywords : iffusion coefficient, surface emission coefficient, Vanderwaas forces, chemica bond, hydrogen bond, covaent bond INROUCION Adsorption is the process of attraction and accumuation of any substance on the surface of another. In fact adsorption (either physica or chemica) is a surface phenomena at which the concentration of the adsorbate is more at the surface than in the buk of adsorbent. he forces between adsorbate and adsorbent moecues are either week Vanderwaas forces or strong chemica bond forces. he rate of adsorption decreases with the passage of time ti the time of equiibrium is attained. On the other hand absorption is a buk phenomenon where the concentration is same throughout, that at the surface as we as in the buk of the sovent. he forces may be hydrogen bond or covaent bond with the constant rate of absorption. Sometimes absorption may be foowed by adsorption and such a process in which both adsorption and absorption take pace, normay the term sorption shoud be used, whie the reverse of it, is caed desorption. When diffusion is taking pace within the medium then movement of diffusing substance is controed by the diffusion coefficient whereas at the surface of the medium it is controed by the surface emission coefficient. Aso adsorption depends upon the surface area of the adsorbent and it increases as the surface area of adsorbent increases. Adsorption of gases in muti-moecuar ayers was shown by Brunauer et. a. [], which describe that the adsorption isotherm of gases at high temperatures not far removed from their condensation points, show two regions for most adsorbents: at ow pressure the isotherms are concave, at higher pressures convex toward the pressure axis. he higher pressure convex portion has been variousy interpreted. Sometimes it is assumed that it has been attributed to condensation in the capiaries of the adsorbent on the assumptions that in capiaries of moecuar dimensions condensation can occur at pressures far beow the vapor pressure of the iquid, whie in some other situations such isotherms are beieved to indicate the formation of muti-moecuar adsorbed ayers. Mathematica equations for the surface emission coefficient corresponding to the variabe diffusion coefficient obtained by iu [6], where surface emission coefficient in isotherma diffusion equation was derived aong with the assumption that the variabe diffusion coefficient can be taken to be exponentia function of the concentration of diffusing substances. he mathematica reationship between surface emission coefficient and diffusion coefficient was estabished by iu and Simpson [8] in capiary porous soids and aso presented a theoretica expanation for the phenomena of sorption and desorption. On the other hand, a study on the separation of diffusion and surface emission coefficient in wood was aso proposed by, Cai and Avvamidis [3], where a non-inear technique was used to obtain optimum pair of surface emission coefficient and diffusion coefficient from a singe sorption curve which was aso tested with the experimenta data. iffusion coefficient and surface emission coefficient can aso be evauated by optimization technique which was presented by Chen Youg et. a. [4]. he optimization technique was used to cacuate the foowing two coefficients where first is the diffusion coefficient as an indication of interna resistance to moisture remova and

2 Kumar and Mishra 75 another is surface emission coefficient, as an indication of externa resistant to moisture remova, where both coefficients express the rate at which wood dries. Yeo et. a. [] aso determined the interna moisture transport and surface emission coefficients for eastern white pine. hey evauate wood specimens during drying at specific temperature and RH conditions and buit an appropriate numeric anaysis mode for predicting moisture profie changes in wood. hey aso shows that the interna moisture transport coefficients were dependent on the temperature and average moisture content and the surface emission coefficients on the wood vapor pressure in air adjacent to the surface. he interna moisture transport coefficient decreased with decreasing temperature and interna moisture content and surface emission coefficients increased with increasing temperature and decreasing surface moisture content. A cosed form soution of the non-steady state diffusion equation with constant transport coefficient has aready been obtained iu and Simpson [9]. An inverse moisture diffusion agorithm for the determination of diffusion coefficient was presented by iu et. a. []. Furthermore this inverse determination of the diffusion coefficient in one dimensiona non-steady state diffusion equation which is based on desorption test data was aso expained and inverse probem is then defined in the resuting system of finite difference equations. iffusion coefficient of porous soids obtained from isotherma sorption tests again presented by iu [7], according to which a derivation of mathematica equation to separate the diffusion coefficient and the surface emission coefficient in Newmans soution of unsteady state diffusion equation from an experimenta sorption curve was, defined iu [7]. Cranks uses Newmans soution with the assumption that S the, surface emission coefficient was infinity to obtain the constant, the diffusion coefficient for each curve and then derived as a function of the concentration of choroform. However in many practica sorption or desorption tests, the vaue for S is finite. Much of this work was aso done by Kim and Kim [5] and Boznani et. a. []. MAHEMAICA MOE he governing equations which govern the mode are as foows. If is the haf of thickness of sheet of a considered porous medium and C is the concentration of the diffusing substance then a one-dimensiona form of diffusion equation with the diffusing substance moving in one direction is, C t X C X + Q ( < X <, t > )...() where is diffusion coefficient which assumed to be an exponentia function of C, t is time, X is space coordinate measured from the center of the sheet, and Q denotes the source term. Consider the initia condition is C C ( < X <, t )...() and the boundary conditions are C X (X, t > )...(3) C C C, ( X, t > )...(4) X where C is a constant concentration in the medium, S is S ( ) surface emission coefficient and C is equiibrium concentration. Aso the equation (4) prescribes the boundary condition just within the surface of the sheet but not at the surface. Now consider the foowing two cases in different dimensioness formuations with assuming that diffusion coefficient as a function of concentration in the form e kc, where and k are constants, Case : I Sorption ( C > C) et the foowing non-dimensiona parameters are introduced to reduce the system of equations ()-(4) in dimensioness form : c (C C )/( C C ), /, τ t/, x X/ and f(c) () Q. C C Now equations () to (4) in dimensioness form using the above dimensioness parameters aong with, transport ratio defined as S/, and ε : τ + f (c) ( < x <, τ > )...(5) x c ( < x <, τ )...(6a) (x, τ > )...(6b) + c, (x ε, τ > )...(6c) Case: II esorption ( C < C) he mathematica formuation for the desorption wi be differ ony in the initia and boundary conditions to sorption with c ( C C ) / ( C C ), herefore, τ + f(c) ( < x <, τ > )...(7) x c ( < x <, τ )...(8a) (x, τ > )...(8b)

3 76 Kumar and Mishra + c, (x ε, τ > )...(8c) It shoud be noted here that equation () to (4) wi be appicabe in the case of desorption when C e < C, that is equiibrium concentration wi be ess than the concentration of the diffusing substance. Considering the mathematica procedures for desorption process in detai, we proceed as foows. Suppose the tota concentration in the medium, M be defined as M(τ) ε c( τ,) x dx, where ε...(9) so that the fraction of tota concentration within the medium is E where M ( ) M () τ M (),...() From equations (7) and (8b) to (), we obtain (as E τ E τ ε c( τ,) x dx τ ε c + f () c dx ( τ,) ε + at x is zero and c(τ, ε) + ε f () c dx c at x ε) x ε f () c dx...() Now, from equations () and (), we may write E + E τ E d( τ + / n) E dτ E + c( τ,)() ε + ε [M(τ) c(τ, ε)] + Equation () may aso be written as f c dx ε d( τ + / n) E E dτ M [M(τ) c(τ, ε)] ε f () c dx...() f () c dx...(3) d( τ + / n) E or M(τ) dτ [M(τ) c(τ, ε)] ε f () c dx and make the use of equations (9) and () to obtain ε d( τ + / n) E c( τ,) x [( c,)( τ,)] x c τ ε dτ dx f () c dx...(4) ε aking derivative with respect to (τ : / n E) ε c ( τ,)( x / τ n) +.(,) E + c τ x τ + / n)( E / n) τ τ + E ε ε ε ( τ + / n) E [(,)( c τ, x)] c τ ε τ ( τ + / n) E dx ε c( τ,) x [( c,)( τ,)] x c τ ε τ ( τ + / n) E f () c ( τ + / n) E dx...(5) ε dx () f c ( τ + / n) E dx...(6) c( τ,) x [( c,)( τ,)] x c τ ε ( τ + / n) E ε dx () f c ( τ + / n) E dx...(7) [( c,)( τ x,)] c τ [( ε,)(,)] c τ x c τ ε ( τ + / n) E ε dx ε () f c ( τ + / n) E dx...(8) [ ] c( τ,)( x,) c τ ε [( c,)( τ x,)] c τ ε ( τ + / n) E dx () f c ( τ + / n) E dx...(9) et [c(τ, x) c(τ, ε)] c (τ, x) and τ τ + n E...() ε [( τ,) x () f c τ τ dx...()

4 Kumar and Mishra 77 or his is true when τ [( τ,) x () f c τ τ () f c τ...() When we consider f (c ) e τ then equation () wi turn into the origina form as : τ + f(c ) ( < x <, τ > )...(3) Equation (3) is as same as equation (7) in the case-ii of desorption, now with the hep of equation (9) and (), the initia and boundary conditions for equation (3) becomes [], i.e., c, ( < x <, τ > )...(4), (x, τ > )...(5) + c + c(τ, ε) c(τ, ε) (x ε, τ > )...(6) since, we must have to satisfy equation (6), however this resut does not give the needed boundary conditions. Comparing equation (3)-(5) with the equation (5)-(6b), we observe a one to one correspondence between (symmetry) these set of equations. he needed boundary must be obtained from equation from (6c). If we set in equation (6c), we have c at x ε and τ >. hus we obtain the boundary condition repacing equation (6) as : c (x ε, τ > )...(7) he justification of equation (7) as the needed boundary condition exists in the fact that the boundary condition is just within the surface but not at the surface. herefore if we consider equation (6c) to be true at x ε with, ε, we may consider equation (7) to be true at x ε with ε < ε, and a the boundary conditions are aso satisfied. reating k in being a constant quantity and aso consider and, then we obtain from equation () τ (, E ) τ(, E) + n E...(8) where τ and E are associated with the sorption whie τ, E and with desorption. et be associated with the sorption then in the same manner, the corresponding equation for sorption becomes τ (, E) τ (, E ) + n E...(9) he surface emission coefficients, S for desorption and S for sorption, which are contained in C and C, respectivey can now be expressed from equation (8) and (9) as foows. For desorption, S n E τ (,)( E,) τ E...(3) n E For sorption, S τ(,)( E,) τ E...(3) We note that in above equation, E + E...(3) τ(, E) t...(33) τ (, E ) t...(34) In equation (3) and (33) t and t are to be evauated from experimenta desorption and sorption time curves corresponding to satisfied vaues of E and E, respectivey. he τ (, E ) in equation (3) and τ (, E) in equation (3) are to be evauated using numerica techniques. RESUS AN ISCUSSION In connection with the mechanism of the surface reactions an important concept is the moecuarity, which is the number of moecues that come together during the course of reaction. he moecuarity of a surface reaction is deduced from the kinetics on the basis of the experimenta resuts and of theoretica considerations. Our resuts show the variation of source term with respect to dimensioness time for sorption and desorption process through the Fig.. Whie drawing graphs, different vaues of transport ratio in a desorption process is taken and as we are increase the vaues of transport ratio then f(c ) increases with a sight change, graphs aso indicates that as the vaue of the dimensioness τ time increases the vaue of the source term f(c ) increases, that is there is a inear reationship between τ and f(c ) through the Fig. 3. he corresponding resuts for the sorption process can aso be determined by the same process, which are shown through Fig. and Fig Fig.. Variation in f(c ) for different increasing vaues of.

5 78 Kumar and Mishra Fig.. Variation in f(c ) for different increasing vaues of Fig.3. Variation in f(c ) with respect to for different vaues of Fig.4. Variation in f(c ) with respect to for different vaues of. REFERENCES [] Boznani, E., Fytianos, K. and Voudrias, E., Sorption desorption of dyes from aqueous soution and wastewaters with different sorbent materias, Goba Nest: the Internationa Journa, Vo. 4, No., pp- 78-8(). [] Brunauer, S., Emmett, P.H. and eer, E., Adsorption of gases in muti-moecuar ayers, Journa of the American Chemica Society, Vo.6, No., pp-39-39(). [3] Cai, and Avvamidis, S., A study on the separation of diffusion and surface emission coefficient in wood, rying echnoogy, Vo. 5, pp (997). [4] Chen, Youg., Choong, E.. and Wetze,.M., Evauation of diffusion coefficient and surface emission coefficient by a optimization technique, Wood and Fiber Science, Vo. 7, No., pp-78-8(7). [5] Kim, Y.M. and Kim, S.C., Adsorption/desorption on isotherma of nitrogen in carbon micro pores, Journa of the Korean Physica Society, Vo. 4, No., pp (). [6] iu, J.Y., erivation of surface emission coefficient in isotherma diffusion anaysis, Proceedings of 9th Internationa rying Symposium, Vo. A, pp-4-48(994a). [7] iu, J.Y., iffusion coefficient of porous soid obtained from isotherma sorption test, Res. Pap. FP- RP-533, Madison, W: US epartment of Agricuture, Forest Service Products aboratory, pp--(994b). [8] iu, J.Y. and Simpson,., Mathematica reationship between surface emission coefficient and diffusion coefficients, rying echnoogy, Vo.4, pp (996). [9] iu, J.Y. and Simpson,., Soution of diffusion equation with constant diffusion and surface emission coefficients, rying echnoogy, Vo. 5, pp (997):. [] iu, J.Y., Simpson,. and Verri, S.P., An inverse moisture diffusion agorithm for the determination of diffusion coefficient, Proceedings of th Internationa rying Symposium, No. 4, pp-8-3 (). [] Yeo, H., Eom, C.., Han, Y., Kang, W. and Smith, W.B., etermination of interna moisture transport and surface emission coefficients for eastern white pine, Wood and Fiber Science, Vo. 4, No. 4, pp (8).