Page 493 - Modelling in Transport Phenomena A Conceptual Approach

Chapter 11

Unsteady-State Microscopic

Balances With Generation

This chapter briefly considers the cases in which all the terms in the inventory rate
equation are non-zero. The resulting governing equations for velocity, temperature
and concentration are obviously partial differential equations. Nonhomogeneity
introduced either by the governing equation itself or by the boundary conditions
further complicates the problem.

11.1 UNSTEADY LAMINAR FLOW IN A TUBE

A horizontal tube of radius R is filled with a stationary incompressible Newtonian
fluid as shown in Figure 11.1. At time t = 0, a constant pressure gradient is
imposed and the fluid begins to flow. It is required to determine the development
of velocity profile as a function of position and time.
Postulating v, = v,(t,r) and 'u, = ve = 0, Table C.2 in Appendix C indicates
that the only non-zero shear stress component is T,, and the components of the
total momentum flux are given by

8%
nrz = T,, + (p~,) V, = T,, = -pdr (1 1.1-1)
rez = 78, + (pv,) (1 1.1-2)
7rz.z = T,, + (pv,) v, (11.1-3)

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