The theoretical transient behavior of an isothermal packed bed or tubular reactor with direct recycle is investigated. It is shown that the recycle effect, coupled with the phenomenon of axial dispersion, causes waves in the reactant concentration to travel through the bed when the feed concentration undergoes a step change. The waves have a length almost equal to the bed length and they travel with the velocity of the fluid. The behavior of the waves is very insensitive to the value of the Peclet number.
The pertinent linear differential equation is solved by the method of generalized Fourier transforms and the boundary conditions are such that the Sturm-Liouville theorem cannot be used. The operator is nonself-adjoint and the eigenfunctions are not mutually orthogonal. The eigenvalues, which are complex, are found by means of the argument principle. Sample calculations are presented of the first one hundred terms of the Fourier expansion of a solution function and this is compared to a simplified approximate series which is developed.