4 - Unidirectional Transport: Cartesian Co-ordinates

Summary

Problems involving mass, momentum and energy transport in one spatial direction in a Cartesian co-ordinate system are considered in this chapter. The concentration, velocity or temperature fields, here denoted field variables, vary along one spatial direction and in time. The ‘forcing’ for the field variables could be due to internal sources of mass, momentum or energy, or due to the fluxes/stresses at boundaries which are planes perpendicular to the spatial co-ordinate. Though the dependence on one spatial co-ordinate and time appears a gross simplification of practical situations, the solution methods developed here are applicable for problems involving transport in multiple directions as well.

There are two steps in the solution procedure. The first step is a ‘shell balance’ to derive a differential equation for the field variables. The procedure, discussed in Section 4.1, is easily extended to multiple dimensions and more complex geometries. The second step is the solution of the differential equation subject to boundary and initial conditions. Steady problems are considered in Section 4.2, where the field variable does not depend on time, and the conservation equation is an ordinary differential equation. For unsteady problems, the equation is a partial differential equation involving one spatial dimension and time. There is no general procedure for solving a partial differential equation; the procedure depends on the configuration and the kind of forcing, and physical insight is necessary to solve the problem. The procedures for different geometries and kinds of forcing are explained in Sections 4.4–4.7.

The conservation equations in Sections 4.2 and 4.4–4.7 are linear differential equations in the field variable—that is, the equations contain the field variable to the first power in addition to inhomogeneous terms independent of the field variable. For the special case of multicomponent diffusion in Section 4.3, the equations are non-linear in the field variable. This is because the diffusion of a molecular species generates a flow velocity, which contributes to the flux of the species. The conservation equation for the simple case of diffusion in a binary mixture is derived in Section 4.3, and some simple applications are discussed.

In Section 4.8, correlations for the average fluxes presented in Chapter 2 are used in the spatial or time evolution equations for the field variables.