[Luyben] Process Mod.. - Student subdomain for University of Bath

EXAMPLES OF MATHEMATICAL MODELS OF CHEMICAL ENGINEERING SYSTEMS 59
The product that we want to make is component B. If we let the reaction go on
too long, too much of B will react to form undesired C; that is, the yield will be
low. If we stop the reaction too early, too little A will have reacted; i.e., the
conversion and yield will be low. Therefore there is an optimum batch time when
we should stop the reaction. This is often done by quenching it, i.e., cooling it
down quickly.
There may also be an optimum temperature profile. If the temperaturedependences
of the specific reaction rates kl and k2 are the same (if their activation
energies are equal), the reaction should be run at the highest possible
temperature to minimize the batch time. This maximum temperature would be a
limit imposed by some constraint: maximum working temperature or pressure of
the equipment, further undesirable degradation or polymerization of products or
reactants at very high temperatures, etc.
If k, is more temperature-dependent than k,, we again want to run at the
highest possible temperature to favor the reaction to B. In both cases we must be
sure to stop the reaction at the right time so that the maximum amount of B is
recovered.
If kl is less temperature-dependent that k,, the optimum temperature
profile is one that starts off at a high temperature to get the first reaction going
but then drops to prevent the loss of too much B. Figure 3.10 sketches typical
optimum temperature and concentration profiles. Also shown in Fig. 3.10 as the
dashed line is an example of an actual temperature that could be achieved in a
real reactor. The reaction mass must be heated up to T,,,. We will use the
optimum temperature profile as the setpoint signal.
With this background, let us now derive a mathematical model for this
process. We will assume that the density of the reaction liquid is constant. The
FIGURE 3.10
-I Batch pro!ilcs.