Overview

The one-dimensional, transient PBE for a control point model is

or written slightly differently as

(4.1)

This is a special case of a first-order, homogeneous, linear partial differential equation, a type of equation of the general form

where x is a vector of free variables. Structured, transient control point PBEs turn out also to have this form, so the solution methods for solving linear first-order partial differential equations play a key role in the transient solution of PBMs. The complete PBM is obtained when the PBE, valid between the control points, is coupled to cell balances and substrate equations.

The solution method for linear first-order partial differential equations is known as the method of characteristics or Cauchy's method. The two names refer, not to different methods, but to two different ways of presenting or thinking about what is essentially the same underlying solution method. The method of characteristics presentation is based on a geometric interpretation of the differential equation and its solution, an interpretation that is very helpful for visualizing the solution method for two-dimensional problems but difficult to use in higher dimensions, i.e., for structured models. Cauchy's method, on the other hand, is more abstract and easily generalized to any number of dimensions. Both methods will be described in this chapter, and, to become fully comfortable with the methods, the reader is urged to study both to the point where it becomes apparent that they both represent the same, identical solution method. The material that...