G E N E R A L I Z E D R E L A X A T I O N METHOD IN CHEMICAL KINETICS RICHARD W . ARBESMAN AND YOUNG GUL K I M Department of Chemicul Engineering, Xorthwestern University, Ezsanston, Ill. 60201
The relaxation method, originally developed by Eigen, applies only to reactions in a batch reactor. Furthermore, if the over-all reaction under study includes more than one independent reaction, the relaxation times of the component steps must be substantially separated from each other, usually an order of magnitude, if various relaxation times are to be determined from the experimental data. In this paper, the relaxation method is generalized so that it can be used in analyzing data obtained in a flow reactor with perturbed flow rate. It works well where the relaxation times are close to each other. The method involves obtaining zeroes and poles of the frequency response information directly from frequency response experiment, or from transient response to pulse disturbance via numerical Fourier transformation. Various rate constants can be obtained from the zeroes and poles and equilibrium constants. Computer-generated "experimental" data demonstrate the usefulness of this method.
A R E C E K T paper
(Hulburt and Kim, 1966) discussed the possibility of imposing oscillatory conditions on a reacting flow system in order to improve the extraction of kinetic information. I t shows that this can be accomplished if the st'eps of a mechanism have cliffering rates of response to a disturbance. It outlines t,he relaxation method stressing the transient response of a two-step reversible reaction in a batch reactor to a step change in thermodynamic variable as a means of obtaining kinet'ic information. This paper attempts to put these two ideas together, and to explore the possibility of generalizing the relaxation method. The relaxation method, developed most actively by Eigen and coworkers (Eigen, 1954; Eigen and DeMaeyer, 1963; Eigen et al., 1963), applies to reversible reactions taking place in a batch reactor. It consists of perturbing a thermodynamic state of a reacting system initially a t equilibrium and folloniiig the response of the system as it seeks a new equilibrium imposed by the perturbing influence. The perturbation must be small enough so that the differential equations describing the dynamic behavior of the system can be linearized with respect to the driving forces, the difference between the tirne-dependent concentrations and the new equilibrium concentrat,ions. I n this paper we develop a generalized relaxation method to extract rate constants from experimental data obtained in a f l o reactor. ~ To lay t'he ground work, let us briefly describe the salient features of the relaxation method. Suppose an elementary reaction ki
Ai
+ 14qF? A3
has reached an equilibrium in a batch reactor. If any parameter affecting the equilibrium is perturbed slightly, the system will seek a new equilibrium state with concentrations 6i (i = 1, 2 , 3 ) , and the time-dependent concentrations Ci's can be written as Ci = 6i Xi (i = 1, 2, 3) where Xi's are the concentration deviation variables. Incorporating these equations with the mass-balance and the rate equation, and making use of the following relations:
we obtain dX1
1 -+-xi=o at
+ c2)+
where T = kl[ (6, k~1-l is called the relaxation time. For complex reactions there will be a spectrum of relaxation times, and in general the various relaxation times do not have one-to-one correspondence with particular reaction steps. The number of relaxation times equals the number of independent reaction steps. For a two-step reaction
Xi = XZ= -X3 216
I h E C
k2C8
=0
(from stoichiometry)
FUNDAMENTALS
ka
A3e
A 4
kp
k2
there are two relaxation times, r1 and r2,r1 being the smaller. The deviation of relaxation times for the above reaction is given in Appendix I. Derivation of Generalized Relaxation Times
The basic principle of relaxation method can be applied t o reactions taking place in a flow reactor. I n this case the equilibrium state will be replaced by a steady state, and if the perturbation of the steady state is small, all the formal treatment of the relaxation method can be used to analyze the transient behavior of the system as it seeks the new steady state. N7e treat the two-step reversible reaction discussed earlier and derive generalized relaxation times by using a continuousflow stirred-tank reactor as the system. If the reaction has reached a steady state under a given operating conditioni.e., fixed inlet composition, flow rate, and temperature-a perturbation can be introduced by a number of means. We consider only perturbation of flow rate here. Suppose for a reaction Ai
+ A2*
ka
ki
A3F? A4 kz
k4
we write mass balances for A i and Ad, and incorporate the rate equations, r A i = --kiAiA2 kzAa
+
Xi
