Consider the case of axial dispersion in steady, fully developed flow between stationary, parallel plates. The plates are of length, \(L\), and the separation distance between the plates is \(d\). Following the analysis of Section 13.6,

a. Develop a criterion to decide when diffusion is important.

b. By transforming the coordinate system to one that moves with the mean speed of the flow, solve the differential equation and determine a Fick's Law-type expression for the axial flux.

c. What is the dispersion constant for this situation?

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13.6 TAYLOR DISPERSION [23,24] We are going to discuss what happens when we have mass transfer occurring in a tube but where the solute is either present in the fluid initially or introduced into the flow stream and is being redis- tributed under the action of the flow field. This scenario is quite common and has many practical applications such as measuring the speed of flows, measuring blood and nutrient flow for pharmaco- logical studies, analyzing toxic waste spills in waterways or the atmosphere, and performing tracer experiments for diagnostic purposes in chemical plants. Consider a system where we have fully developed laminar flow in a tube and want to measure the speed of the flow. Usually we do a tracer experiment. We inject an amount of solute, the tracer, into the tube at the inlet or another position and then we time how long it takes to make it to a recording station some point further downstream. Experiments have shown that the solute slug does not trans- form into a paraboloid even though we know that the velocity profile in the tube is parabolic and that the centerline velocity is fully twice that of the mean flow. The slug is observed to move as a single unit with the mean speed of the flow and can therefore be used to determine the average speed of the stream. This is a remarkable result since it appears to say that the clear fluid in the center of the tube overtakes the slug of solute, and then passes through it as if the solute were not there. A second striking result is that the solute slug seems to spread out symmetrically from a point that moves with the mean speed of the flow, even though the flow field is not symmetric in that direction. If we were to inject solute into the tube for a period of 0.01 s for example, by the time the solute reached our next monitoring station, we might record its passage over a period of 0.1 s or more. The extent of the spread depends upon the mean speed of the flow, the physical properties of the fluid and the solute, and the dimensions of the pipe. The spreading out of the solute by the action of the flow field is termed dispersion.