9 STEADY MICROSCOPIC BALANCES WITH GENERATION This chapter is the continuation of Chapter 8, with the addition of the generation term to the inventory rate equation. The breakdown of the chapter is the same as that of Chapter 8. Once the governing equations for the velocity, temperature, or concentration are developed, the physical significance of the terms appearing in these equations is explained and the solutions are given in detail. The obtaining of macroscopic level design equations by integrating the microscopic level equations over the volume of the system is also presented. 9.1 MOMENTUM TRANSPORT
For steady transfer of momentum, the inventory rate equation takes the form Rate of Rate of Rate of − + =0 momentum in momentum out momentum generation
(9.1-1)
In Section 5.1, it was shown that momentum is generated as a result of forces acting on a system, i.e., gravitational and pressure forces. Therefore, Eq. (9.1-1) may also be expressed as Rate of Rate of Forces acting − + =0 (9.1-2) momentum in momentum out on a system As in Chapter 8, our analysis will again be restricted to cases in which the following assumptions hold: 1. Incompressible Newtonian fluid, 2. One-dimensional, fully developed laminar flow, 3. Constant physical properties. 9.1.1 Flow Between Parallel Plates
Consider the flow of a Newtonian fluid between two parallel plates under steady conditions as shown in Figure 9.1. The pressure gradient is imposed in the z-direction while both plates are held stationary. Velocity components are simplified according to Figure 8.2. Since vz = vz (x) and vx = vy = 0, Table C.1 in Appendix C indicates that the only nonzero shear-stress component is 305