Local stability of tubular reactors

Abstract

Some difficulties, which existed due to the nonsymmetric nature of matrix C, in a recent paper of the above title by Clough and Ramirez (1972) have been aptly noted by Liou et al. (1972a) who attacked these and presented suitably modified stability criteria. Unfortunately, however, there are still some problems in the case of the tubular reactor with axial dispersion (point 3 of Liou et al., 1972a) and the stability criterion reached for this case, inequality (6) of Liou et al. (1972a), is incomplete. We obtain new stability criteria in this note and show that this inequality is only one of the two that must actually hold to ensure asymptotic stability of the steady state in the general case of r1 # r2 and the other implies this one. This whole approach is, in fact, shown to yield more conservative stability criteria than those of Varma and Amundson (1972). Conclusions of Liou et al. (1972b), since they were based on their earlier work, are therefore also incorrect for rl z r2.In the notation of Clough and Ramirez (1972), the negative definiteness of the symmetric matrix -C corre- is assured if and only i f t h e following inequalities hold:where Pi(x) = exp( -K i x ) , with K i being positive constants, are the functions used in defining the Liapunov functional -and The idea of Clough and Ramirez (1972) was to choose the constants Ki in a manner such that the most liberal stability condition is reached. We will show here that the best choice is K1 = rlr K2 = r2 where rl and r2 are the Peclet numbers for heat and mass transfer, respectively.To arrive at this conclusion, we note that in the case of an In what follows (7). The necessary (though not sufficient) condition for inequality ( 3 ) to hold isbut under this restriction, A5 > 0 implies A4 > 0; so that we now only need to satisfy inequality (4) with K1 satisfying both ( 7 ) and (9). The difficulty of Liou et al.(1972a) is that they satisfy (9) but assume that A4 > 0 implies A5 > 0; this is not true.Since (9) may be rearranged asthe least restrictive way to satisfy (10) is to take K1 = rl and then satisfies both (7) and (9).We must now simply find conditions so that inequality (4) holds with K1 = rl and under the restriction (11).A rearrangement of (4) reveals that it cannot be satisfied unless B1R1,~/rlr2 < 1/4(11)With the choice K2 = r2, (12) is satisfied in the least restrictive manner; all our choices are now over but we must still satisfy (4) which now takes the form Note that (13) implies (11); so the final stability criteria is (13). In terms of parameters, the final stability condition is the satisfaction of inequality (13) over the entire range of the a priori bounds on concentration and temperature:Ody'l,A derivation of these (rate independent) a priori bounds for the adiabatic and the nonadiabatic tubular reactors is in preparation and will be published elsewhere.Let US now pause and see what has been done. The above analysis implies that if the choice of functions P i ( x ) = exp( -Kix) is made in defining the matrix P for the Liapunov functiona...