Flow without transition to the solid state

There are some fundamental investigations devoted to analysis of the flow in tubular polymerization reactors where the viscosity of the final product has a limit (viscosity ) i.e., the reactive mass is fluid up to the end of the process. As a zero approximation, flow can be considered to be one-dimensional, for which it is assumed that the velocity is constant across the tube cross-section. This is a model of an ideal plug reactor, and it is very far from reality. A model with a Poiseuille velocity profile (parabolic for a Newtonian liquid) at each cross-section is a first approximation, but again this is a very rough model, which does not reflect the inherent interactions between the kinetics of the chemical reaction, the changes in viscosity of the reactive liquid, and the changes in temperature and velocity profiles along the reactor. 

If a liquid with constant viscosity r flows through a tube, the P(Q) function is described by the well known Poiseuille equation, which can be written as 

If the viscosity varies during flow for some reason (decreases with rising temperature or increases as a result of a chemical reaction such as polymerization), the linear Poiseuille P-vs-Q relation is violated and the pressure drop - flow rate curve may become nonmonotonic. This effect in polymerizing reactors can be explained by the fact that the most viscous products of a reaction are swept out of the reactor with increasing flow rate and are replaced. Instead, a reactor is refilled with a fresh reactive mixture of low viscosity. This leads to a decrease of the volume-averaged integral viscosity and therefore the pressure drop decreases. This can be illustrated by the following relationship  

Unstable branches on the P(Q) curve and the appearance of hysteresis loops can occur for various reasons usually connected with an increase in viscosity. Thus, a non-monotonic P(Q) curve was first encountered in an analysis of the flow of a hot inert (non-reactive) liquid in a cold tube when the viscosity of the liquid was strongly dependent on temperature.190 The intense dissipative heat output may have been the reason for the instability in the flow of an inert liquid.191 In both cases, the reason for the nonmonotonic in P(Q) dependence was the strong dependence of viscosity on temperature, which is equivalent here to time dependence for viscosity. Detailed investigations of the hysteresis transitions shown in Fig. 4.24 proved that they have a wave character 192 in this case, the transition occurs at a constant flow rate. 

An analogy can be seen between these phenomena and the decrease in volume-averaged viscosity in the flow of a polymerizing liquid through a tubular reactor as described in Eq. (4.14). In real situations, the flow of a polymerizing liquid is always non-isothermal, which leads to a very 

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