TABLE OF CONTENTS We will, for illustrative purposes, continue using the partial differential equation for the plug-flow reactor, Eq. (4.2), from the first example. The solution method we will describe is called the method of characteristics. Central to the method, as it is presented here, is the concept of a directional derivative, the rate of change, not in the coordinate directions in phase space, but in some other direction, at an angle with the coordinate directions. In order to introduce this concept, we will digress briefly from the main topic.
Consider a curve C( s) in our phase plane ( t, x) parameterized by s. For instance, the upper trajectory in Fig. 4.2 can be parameterized the obvious way as ( t( s), x( s)) = ( s, vs + x initial). A parameterization of a curve in terms of arc length, starting from an arbitrary point on the curve, is called a natural parametric representation. We will indicate it as M: ( t( s) , x( s)) and assume in the following that s is always the arc length. [1]
Given a function c( t, x), one can certainly define a new function c( s) by c( s) = c( t( s), x( s)) and ask, What is the rate of change of c( s
