Pulse-response analysis of a simple reaction-advection-diffusion equation

Abstract

We analyze a reaction-advection-diffusion (RAD) equation arising in pulse-response studies of transport and reaction in a narrow reactor tube. A short pulse of gas is injected near one end of the reactor, while unreacted gas and reaction products are detected at the outlet. Particular attention is given to the effect of a constant axial advection velocity as a minimal extension of the standard diffusion model. For a single pulse, we obtain analytical expressions for the exit-flow response and for experimentally accessible characteristics, including its moments and peak properties, as functions of the Péclet number and the reaction rate. The corresponding diffusion-advection model without reaction defines a standard transport curve that may be used as a baseline for identifying chemical activity. For a first-order irreversible reaction, the reactive and nonreactive exit-flow curves satisfy a simple exponential factorization, allowing the reaction rate to be extracted directly from their ratio. We also extend the single-pulse analysis to a uniform periodic train of pulses. Using Poisson summation, we derive explicit expressions for the Fourier coefficients of the asymptotic periodic exit flow and relate them to the Laplace transform and moments of the single-pulse response. Finally, a stochastic interpretation in terms of reflected diffusion, first-passage times, and exponential killing provides probabilistic meaning for several of the analytical results and unifies the single-pulse, periodic-response, and reaction aspects of the model.