Reactive barrier membranes: some theoretical observations regarding the time lag and breakthrough curves

In a recent paper, Yang et al. [AIChE J. 47 (2001) 295] demonstrated numerically that the classical time lag parameter in irreversibly reactive barrier membranes is independent of reactivity of a prescribed molar amount of uniformly dispersed additive. The approach to the final breakthrough asymptote, however, is dependent on additive reactivity. In this communication analytical formulas for the time lag in reactive membranes of various types are derived using variations of the method of Frisch [J. Phys. Chem. 61 (1957) 93]. These formulas, which are derived for arbitrary distributions of additive initially incorporated into the membrane, confirm independence of reactivity. It is further shown, in the fast-reaction case with uniformly dispersed additive, that the time lag is extremely close to, but slightly greater than, the time at which initial breakthrough of penetrant through the membrane occurs. Finally, it is argued that two discernable time lags are observable in principle when reaction occurs at the surfaces of additive particles.