Unsteady Piston Flow

Dynamic analysis of piston flow reactors is fairly straightforward and rather unexciting for incompressible fluids. Piston flow causes the d5mamic response of the system to be especially simple. The form of response is a hmiting case of that found in real systems. We have seen that piston flow is usually a desirable regime from the viewpoint of reaction yields and selectivities. It turns out to be somewhat undesirable from a control viewpoint since there is no natural dampening of disturbances. 

FIGURE 14.6 Differential volume element in an unsteady piston flow reactor. 

If p is constant, Equation (14.12) shows Q to be constant as well. Then the component balance simplifies to 

This result is valid for variable but not for variable p. It governs a PER with a time-dependent inlet concentration but with other properties constant. The final simplification supposes that is constant so that u is constant. Then Equation (14.13) has a simple analytical solution  

Formal verification that this result actually satisfies Equation (14.13) is an exercise in partial differentiation, but a physical interpretation will confirm its validity. Consider a small group of molecules that are in the reactor at position z at time t. They entered the reactor at time i = t — (zju) and had initial composition a t, z) = ai (t ) = ai (t — z/u). Their composition has subsequently evolved according to batch reaction kinetics as indicated by the right-hand side of Equation (14.14). Molecules leaving the reactor at time t entered it at time t — t. Thus, 

If p is constant, Equation 14.12 shows Q to be constant every where within the tube at any instant of time. If (2 in varies with time, then Q z) will immediately adjust to this new value because the fluid is incompressible. Then Equation 14.11, the component balance, simplifies to 

Molecules leaving the reactor at time t entered it at time t -t. Thus, 

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This number is used to characterize the stationary and unsteady oscillatory flow when the oscillatory field frequency presents a significant value. This type of flow can be generated for example, when a fluid is transported by piston pumps. In this case, the frequency flow parameters could be described by a combination of Strouhal number and Reynolds number ... 

The physical significance of the various wave profiles discussed above can be appreciated more readily by considering their development under unsteady flow conditions. As a typical example, we now discuss wave generation in one-dimensional flow by an instantaneously accelerated piston in a frictionless pipe initially containing stationary wet steam. 

Unsteady flows with shock waves are difficult to compute even in single-phase flow. To demonstrate the accuracy of the numerical scheme, we therefore present results for the piston-in-a-pipe problem using a perfect gas rather than wet steam. Fig.5 shows the theoretical (t-x) characteristics diagram which consists, quite simply, of a shock wave of... 

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