Aggregation and gelation—II. Mixing effects in continuous flow vessels

Abstract

The state of mixing in a continuous flow vessel is shown to affect the extent of aggregation and the extent of aggregation at which mathematical gelation occurs. Three aspects of the state of mixing in vessels where aggregation alone occurs are considered: the degree of segregation, the residence-time distribution (RTD) and the earliness or lateness of mixing. The effect of the state of mixing is different for each kernel and depends primarily on the order of the moment rate law for the zeroth and sixth moments. For example, the sum kernel has a first-order decay moment rate law for its zeroth moment and is not affected either by the degree of segregation or the earliness or lateness of mixing. On the other hand, the product kernel (ω = 1) has a second-order growth moment rate law for its sixth moment and its gelling behaviour is strongly affected by the degree of segregation and the earliness or lateness of mixing. Our results follow directly from an analogy with reaction engineering based on the formal equivalence of our moment rate law for aggregating systems and a well-known reaction rate law. We show the following striking progression of the dependence of the gelling behaviour of the sum kernel on the RTD: it is a non-gelling kernel in a plug flow vessel, a gelling kernel in a well-mixed vessel and an instantaneously gelling kernel when a vessel contains a partially stagnant zone. We propose that these observations be used for predicting the effects of scale-up and also of departures from ideal mixing. For gelling kernels, any deviation from the ideal case of plug flow always leads to a reduction in th extent of aggregation at which mathematical gelation occurs. Finally, we recommend that batch experiments be used to obtain aggregation rate data because of the difficulty of interpreting unambiguously the data from continuous flow vessels where the state of mixing is not well defined.