In some cases it may be advantageous to operate R chemical reactor in the unsteady state. Douglas and Rippin (1 ) studied the cyclic operation of a back mix reactor with second-order kinetics and showed that for some values of system parameters the time average conversion of reactant was higher if the reactant concentration in the feed stream varied sinusoidally than if the concentration were held constant at the mean value of the fluctuations. We wish to consider the general problem of cyclic operation of isothermal tubular reactors with back mixing in contrast to their operation in the steady state.
THE UNSTEADY STATE AXIAL DISPERSION MODELIn this study we are concerned with the isothermal axial dispersion model. This model seems to have been proposed first by Hulburt (2) and Danckwerts ( 3 ) , but their analyses were limited to the steady state, although Danckwerts gave a solution to the unsteady state diffusion equation for the case of equimolar counterdiffusion and no reaction. Wehner and Wilhelm ( 4 ) , in their extension of Danckwerts's analysis of the steady state dispersed flow reactor, discussed the applicability of the boundary conditions to the unsteady state as well, but did not give sample solutions. Kramers and Alberda ( 5 ) reported on the frequency response of the axial diffusion model for plug and laminar flow cases giving input-output amplitude ratio and phase lag diagrams. An alternate formulation of the entrance boundary condition leading to the same approximate solution was published by McHenry and Wilhelm (6).Our formulation follows that of Danckwerts. By assuming constant overall fluid density and constant velocity, the equation of continuity in dimensionless form for a reactant species with second-order kinetics may be written as
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