The effect of flow profile

To take the effect of the flow profile properly into account we would have to solve partial differential equations in two or even three space variables. This is not only difficult, but the analysis of the results would be difficult also and there is some incentive to look for a simple situation in which it is possible to estimate the effect rather quickly. The ordinary laminar flow profile would seem to be a good candidate for study, for it is certainly far from uniform, with the central stream moving at twice the mean speed. If we assume that there is no diffusion, so that there is no transfer between different parts of the stream, we should be able to see the effect of velocity variations at their worst. In practice, there will be a certain amount of diffusion and reactants that would otherwise pass rapidly through the reactor will diffuse to slower streams and the distribution of residence times will be more uniform than when there is no diffusion at all. 

Consider a cylindrical tubular reactor of radius a through which reactants are flowing with mean linear velocity v. If the flow profile is parabolic, the linear velocity at a radius r from the axis of the tube is v(r) = 2v(l — and the volume flow rate through an annulus between the radii r and r + dr 

Since the central stream is moving with velocity 2v, the least possible residence time is Lllv = Ojl, hence a full specification of the residence time distribution 

Consider now a second order irreversible reaction for which 

Then the reactants with residence time t will react to extent 

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