Mass transfer with complex chemical reaction in gas—liquid ... - ITM

132 R.D. Vas Bhat et al./Chemical Engineering Science 54 (1999) 121—136remains constant due to the equilibrium constraint(5d) as K , A and B are fixed. The value for E can be reduced in case of loaded solutions ifthere is an increase in the rate of the backwardreaction of reaction (1b). It shall be shown, by meansof the following analysis that this reduction inE does actually occur. The effect of solute loadingon the backward reaction of reaction (1b) can berepresented asr "k [C #x][D #x](13a) where C and D represent the initial concentrations and ‘x’ the addition to the concentration as a resultof production by the forward reaction of (1b).Expanding eq. (13a) givesrN "k [C D #x(C #D )#x]. (13b) The reaction equation can be written as three separateadditions to the reaction rate. The productC D is constant and if we can neglect the effect of solute loading on x then the addition to the backwardreaction due to solute loading can be described asrN "k [x(C #D )].(13c) Since C D "constant we can write eq. (13c) as f (C , D )"x(C #D )" const #DD x. (13d)Differentiating the right-hand side with respect toD gives f (D )(D ) " 1!const D x.(13e)This function equals zero at D "const. Thismeans that the backward reaction is minimum atD "C and is higher for unequal amounts ofC and D . Correspondingly, the addition to thebackward reaction rate of the first reaction increaseswith the difference D !C (higher K ), thus reducingthe value of E .z It is interesting to note that in Fig. 6(a), results arepresented for an α"0.5 (K "100) and α"0.45(K "0.01) as these correspond to cases where A isslightly less than A . For values of α above thosepresented, the initial concentration of A in the liquidphase is greater than the interfacial value. Consequently,desorption of A from the liquid phase occursfor values of α greater than those reported inFig. 6(a).6. Approximate technique to determine E of reversibleconsecutive reactionsA technique for calculating E of reversible consecutivereactions with equal diffusivities of all chemical specieshas been derived based on the method described byDeCoursey (1982). DeCoursey developed this methodfor single reversible chemical reactions based on theDanckwerts’ model of mass transfer. He used the approximationtechnique of van Krevelen and Hoftijzer forlinearising the reaction terms of the species conservationequations. However, since DeCoursey used an unsteadystate theory, the concentrations of the reactants from theliquid phase were approximated by a time—mean valueand assumed independent of time. Variation of theseconcentrations with distance from the interface was neglectedas well. The liquid bulk was assumed to be atequilibrium at all times. In addition, it was assumed thatat any point within the liquid phase, equilibrium could beexpressed in terms of the time—mean concentrations.For the situation described in this paper, time—mean concentrations are obtained from the instantaneousconcentrations by taking ‘s-multiplied’ Laplacetransforms of Eq. (2a) under initial condition (4a). Similarexercise is carried out for all the other species. Bridgingequations for each reaction species are then obtained byeliminating the reaction terms on manipulation of thecomponent mass balances.The bridging equations areD (BM !AM !EM )!s(BM !AM !EM )x#s(B !A !E )"0D (CM #AM #2EM )!s(CM #AM #2EM )x#s(C #A #2E )"0D (DM #AM #EM )!s(DM #AM #EM )x#s(D #A #E )"0D (EM #AM !BM )!s(EM #AM !BM )x#s(E #A !B )"0D (FM #AM !BM )!s(FM #AM !BM )x#s(F #A !B )"0(14a)(14b)(14c)(14d)(14e)where the overhead bars define time-averaged values.These equations are then solved along with the transformedspecies transport equation for A.D x AM !s(AM !A )!RM !RM "0.(14f)To do this, the equations are rewritten in a manner similarto that described by DeCoursey (1982) and implementing