5 - Unidirectional Transport: Curvilinear Co-ordinates

Summary

In the previous chapter, a Cartesian co-ordinate system was used to analyse the transport between surfaces of constant co-ordinate, and the boundary conditions were specified at a fixed value of the co-ordinate z. For configurations with curved boundaries, such as a cylindrical pipe or a spherical particle, the boundaries are not surfaces of constant co-ordinate in a Cartesian system. It is necessary to apply boundary conditions at, for example, x² + y² + z² = R² for the diffusion around a spherical particle of radius R. It is simpler to use a co-ordinate system where one of the co-ordinates is a constant on the boundary, so that the boundary condition can be applied at a fixed value of the co-ordinate. Such co-ordinate systems, where one or more of the co-ordinates is a constant on a curved surface, are called curvilinear co-ordinate systems.

The procedure for deriving balance laws for a Cartesian co-ordinate system can be easily extended to a curvilinear co-ordinate system. First, we identify the differential volume or ‘shell’ between surfaces of constant co-ordinate separated by an infinitesimal distance along the co-ordinate. The balance equation is written for the change in mass/momentum/energy in this differential volume in a small time interval Δt. The balance equation is divided by the volume and Δt to derive the differential equation for the field variable. The balance equations for the cylindrical and spherical co-ordinate system are derived in this chapter, and the solution procedures discussed in Chapter 4 are applied to curvilinear co-ordinate systems.

Cylindrical Co-ordinates

Conservation Equation

A cylindrical surface is characterised by a constant distance from an axis, which is the x axis in Fig. 5.1. It is natural to define one of the co-ordinates r as the distance from the axis, and a second co-ordinate x as the distance along the axis. The third co-ordinate ϕ, which is the angle around the x axis, is considered later in Chapter 7. For unidirectional transport, we consider a variation of concentration, temperature, or velocity only in the r direction and in time, and there is no dependence on ϕ and x.