Overview

In the chapter on compartmental models, we dealt with systems or processes in which the state variables varied at most with time, but not at all with spatial distance. We now reverse the situation and examine time-invariant systems; that is, steady-state systems in which the state variables vary in one direction. We refer to these as one-dimensional distributed systems.

Physical systems that lead to such one-dimensional distributions include flow processes in which the state variables (e.g., temperature, concentration, pressure) vary in the direction of flow, diffusion, or conduction, usually taking place in a circular or spherical geometry, and strictly static processes in which the distributions are the result of static considerations. We examined two such static cases earlier (Example 4.2 and Practice Problem 4.2).

To model flow processes, we introduced the reader, to the concept of the one-dimensional pipe (Chapter 1) in which the principal variations took place in the direction of flow, and any radial or lateral variations were lumped at the conduit wall by means of appropriate rate equations and resistance (Figs. 1.3 and 1.4). An early example of the application of the one-dimensional pipe was given in example 1.7, Release of a Substance into a Flowing Fluid. In this chapter we will apply the concept to model a countercurrent heat exchanger (Example 6.6), to derive oxygen profiles in a polluted river (Example 6.10), and to compute the heat released by an electrically heated wire (Example 6.11). Diffusional processes make use of a modified version of the...