Equation (8.4) defines the average concentration, $a_{\text {out }}$, of material flowing from the reactor. Omit the $V_z(r)$ term inside the integral and normalize by the cross-sectional area, $A_c=\pi R^2$, rather than the volumetric flow rate, $Q$. The result is the spatial average concentration $a_{\text {spatial }}$, and is what you would measure if the contents of the tube were frozen and a small disk of the material was cut out and analyzed. In-line devices for measuring concentration may measure $a_{\text {spatial }}$ rather than $a_{\text {out }}$. Is the difference important?
(a) Calculate both averages for the case of a parabolic velocity profile and first-order reaction with $k \bar{t}=1.0$.
(b) Find the value of $k \bar{t}$ that maximizes the difference between these averages.