Question: Transport equation for contaminant in two-dimensional flow field
In many engineering applications one is interested in the transport of a contaminant by the fluid flow. The contaminant could be anything from a polluting chemical to particulate matter. To derive the governing equation one needs to recognize that, provided that the contaminant is not being created within the flow field, the mass of contaminant is conserved. The contaminant matter can be transported by two distinct physical mechanisms, convection and molecular diffusion. Let C be the concentration of contaminant (i.e., mass per unit volume of fluid); then the rate of transport of contamination per unit area is given by
where i and j are the unit vectors in the x and y directions respectively, and D is the diffusion coefficient (units m2/s, the same as kinematic viscosity). Note that diffusion transports the contaminant down the concentration gradient (i.e., the transport is from a higher to a lower concentration); hence the minus sign. It is analogous to thermal conduction.
a) Consider an infinitesimal rectangular control volume. Assume that no contaminant is produced within it and that the contaminant is sufficiently dilute to leave the fluid flow unchanged. By considering a mass balance for the control volume, show that the transport equation for a contaminant in a two-dimensional flow field is given by
(b) Why is it necessary to assume a dilute suspension of contaminant? What form would the transport equation take if this assumption were not made? Finally, how could the equation be modified to take account of the contaminant being produced by a chemical reaction at the rate of m0 per unit volume.