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

Abstract

We undertake a detailed analysis of a reaction-advection-diffusion (RAD) equation from the viewpoint of pulse-response studies, with particular attention to effects due to the advection velocity. Our boundary-value problem is a mathematical model for a system consisting of a narrow reactor tube into which a short pulse of reactant gas is injected at one end and a mixture of reaction product and unreacted gas flows out at the opposite end. Exit flow properties such as moments and peak characteristics are obtained analytically as functions of the Péclet number. The description of a standard transport curve\ -- -including diffusion and advection but no reaction\ -- -can serve as the baseline for further characterization of chemical activity. This characterization is done here for a first order irreversible reaction. Among our main observations is that chemical activity is easily obtained from the ratio of the exit flow curve in the presence of reaction over the standard transport curve.

Figures (10)

• Figure 1: Rough scheme of a TAP-experiment. Small and short pulses of gas are injected into the micro-reactor tube, which is packed with solid material that is both permeable to gas diffusion and contains catalytic material promoting gas reaction. Exit flow gas mixture (amount of escaped gas per unit time for each gas species) is analyzed by a mass spectrometer and the time-function profiles of each gas species (on the right) is recorded.
• Figure 2: Ratio of $\mu_n$ over the asymptotic value $\mu^{\text{\tiny asymp}}_n=\left(n-\frac{1}{2}\right)\pi$.
• Figure 3: Profile of concentration $c(x,t)$ inside the reactor ($0\leq x\leq 1$) at different times. Parameters used: $x_0=0.01$, $k=1$, $\text{Pe}=4$. (Here we used $a=1$, $L=1$; note that $k$ and $t$ are used, not $\kappa$ and $\tau$), and $n=1500$.
• Figure 4: Exit flow as a function of $\text{Pe}$. Here, $k=0$ and $x_0=0.01$. A negative $\text{Pe}$ means that the advection velocity is negative. These are called the standard transport curves.
• Figure 5: Approximation of $j(L,t)$ by one and two terms of the infinite series. For not too small values of $t$, two terms already provide a very useful description of the exit flow. Here $k=0$ and $\text{Pe}=0$.