The reader will have noted our avoidance, up to this point, of any use of partial differential equations (PDEs). We limited ourselves to ODEs that arose from unsteady state “stirred tank” and from steady state “one-dimensional pipe” models, and from algebraic integral or cumulative balances. These equations served us well in a good many cases, yielding close approximations of the exact solutions or, at the very least, upper or lower bounds to them. Cases do arise, however, in which PDEs can no longer be avoided or circumvented by valid simplifications and assumptions. In particular, PDEs will have to be addressed in the following situations:
• Thermal conduction or diffusive processes in which temperature or concentration vary with time and distance or, if at steady state, vary in more than one dimension. Exceptions occur when transport coefficients are large or system dimensions small, so that the system may be approximated as a stirred tank at the ODE level. This was done in the case of the thermocouple response given in Illustration 4.18, and was valid there by virtue of the high thermal conductivity and small dimension of the device. When size is more substantial, as in the quenched steel billet shown in Figure 1.3, internal temperature gradients can no longer be ignored, and the full PDE model has to be applied. To distinguish between these two cases, we have provided a criterion, the so-called Biot number, which allows us to determine a priori whether it is appropriate to use the reduced ODE model (Illustration 1.7). When this is no longer possible, solutions to the full PDE model must be sought, and this task will be addressed in the subsequent chapters. Time-dependent temperature and concentration distributions may also arise in systems containing instantaneous or continuous heat or mass sources. This important subcase, which also calls for the use of the full PDE model, is taken up in Chapter 6.