10.3 Generalized One-dimensional Compressible Flow Analysis

One-dimensional analyses of compressible flow with mass, momentum, and energy addition can be developed starting from the conservation equations for an elementary control volume in a duct or channel shown in Figure 10.2. Within the control volume there is the possibility for mass addition, frictional forces, body forces, shaft work, and heat exchange. The resulting differential equations can be numerically integrated, but the form of the equations themselves gives much information about the direction and nature of the solution path. In the development here we take both mainstream and injected fluid to be perfect gases with constant specific heat; the derivation for varying specific heats and the injection of liquid is given by Shapiro (1953). It is convenient (and simpler) to describe the general case in two parts. Channel flows with no shaft work or work done by body forces (but all other effects) are first dealt with, followed by examination of the effects of work production.

Figure 10.2: Control volume with addition of mass, momentum, and energy to a control volume.

10.3.1 Differential equations for one-dimensional flow

Referring to the control volume in Figure 10.2, conservation of mass is expressed as

In (10.3.1) d represents the incremental change in mass flow across the control volume. Expanding the differential, dividing through by ?uA and keeping terms which are first order in the quantities du/u, dA/A, d ?/ ?, gives

The convention for which variables are viewed as...