Case A. Multiple Tanks in Series

Consider the case of three, constant-volume tanks in series, as represented in Fig. 2.11, in which the tanks have differing volumes V], V2, V3, respectively.. Assuming well-mixed tanks, the component balance equations are 

In this case, three time constants in series, X, %2 and X3, determine the form of the final outlet response C3. As the number of tanks is increased, the response curve increasingly approximates the original, step-change, input signal, as shown in Fig. 2.12. The response curves for three stirred tanks in series, combined with chemical reaction are shown in the simulation example CSTR. 

Response of a Second-Order Temperature Measuring Element 

The temperature response of the measurement element shown in Fig. 2.13 is strictly determined by four time constants, describing a) the response of the bulk liquid, b) the response of the thermometer pocket, c) the response of the heat conducting liquid between the wall of the bulb and the wall of the pocket and d) the response of the wall material of the actual thermometer bulb. The time constants c) and d) are usually very small and can be neglected. A realistic model should, however, take into account the thermal capacity of the pocket, which can sometimes be significant. 

Assuming the pocket to have a uniform temperature Tp, the heat balance for the bulb is now 

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In Section 8.3.1.4, equations relevant to the analysis of the transient behavior of an individual CSTR were developed and discussed. It is relatively simple to extend the most general of these relations to the case of multiple CSTRs in series. For example, equations (8.3.15) to (8.3.21) may all be applied to any individual reactor in the cascade of stirred-tank reactors, and these relations may be used to analyze the cascade in stepwise fashion. The difference in the analysis for the cascade, however, arises from the fact that more of the terms in the basic relations are likely to be time variant when applied to reactors beyond the first. For example, even though the feed to the first reactor may be time invariant during a period of non-steady-state behavior in the cascade, the feed to the second reactor will vary with time as the first reactor strives to reach its steady state condition. Similar considerations apply farther downstream. However, since there is no effect of variations downstream on the performance of upstream CSTRs, one may start at the reactor where the disturbance is introduced and work downstream from that point. In our generalized notation, equation (8.3.20) becomes... 

The other two methods are subject to both these errors, since both the form ofi the RTD and the extent of micromixing are assumed. Their advantage is that they permit analytical solution for the conversion. In the axial-dispersion model the reactor is represented by allowing for axial diffusion in an otherwise ideal tubular-flow reactor. In this case the RTD for the actual reactor is used to calculate the best axial dififusivity for the model (Sec. 6-5), and this diffusivity is then employed to predict the conversion (Sec. 6-9). This is a good approximation for most tubular reactors with turbulent flow, since the deviations from plug-flow performance are small. In the third model the reactor is represented by a series of ideal stirred tanks of equal volume. Response data from the actual reactor are used to determine the number of tanks in series (Sec. 6-6). Then the conversion can be evaluated by the method for multiple stirred tanks in series (Sec. 6-10). 

It is possible to extend this treatment to the case of multiple CSTR s operating in series by adapting the procedure outlined by Denbigh and Turner (2). Let (ACA)U (ACA)2, and (ACA)h represent the changes in the concentration of species A that take place in tanks one, two, and i, respectively. 

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