Some More Unsteady-State Problems

Great science nobly labored to increase the people s joys, But every invention seemed to add another noise 

For the most part we have labored over the analysis of steady-state problems, although there have been some important side trips into the unsteady state. Principal among these were the analysis of CSTR startup, visits to fixed-bed and CSTR dynamics arising from catalyst deactivation, and some discussion on adsorption variations. The purpose of this chapter is to pursue some of these topics in more detail the range of interests here is rather broad, but all can be linked through a common concern with fixed-bed dynamics. 

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We discuss one more unsteady-state problem after we have shown how some of the restrictions in Eq. 4.34 can be lifted. 

If dt and dx are always considered positive when increasing, you can use Eq. (6.1) without having any difficulties with signs. However, if you for some reason transfer an output to the left-hand side of the equation, or the equation is written in some other form, you should take great care in the use of signs (see Example 6.2 below). We are now going to examine some very simple unsteady-state problems which are susceptible to reasonably elementary mathematical analysis. You can (and will) find more complicated examples in texts dealing with all phases of mass transfer, heat transfer, and fluid dynamics. 

In our experience, on the final approach to the steady state in an Eulerian type of time-dependent reactive flow calculation (see Section 4), when the properties at a point in the flow field vary slowly with time and fresh calculations of the transport coefficients are only required at infrequent intervals, use of the detailed transport model presents little problem. A sound basis then exists for studies of the effect of variations in basic transport parameters on the steady-state flow. Unsteady flows, on the other hand, are most economically treated by less demanding, though more approximate, methods. Some approaches along these lines are discussed below and in Section 3.4. 

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