1. (15 points) In the short contact-time approximation, the differential equation below relates the spatial concentration gradients with the maximum flow velocity in the tube. 2 4. [Z)a_c=D\"6_E R)oz A A) (5 points) If we make the substitution 8 = c/cy, and n = y/, derive the following differential equation: 3n? do _ d'o dn dn' for = ()% 2Vmax B) (5 points) The solution to this differential equation for 8(n) is below: v OO 0 ~ 1.11{)3/ e dt 7 0(n) is a decaying function related to the Gamma function and its shape is depicted on page 19 of Book 3 in the course reader. If 8(n) decays to zero, roughly around n = 2.8, then derive a critical distance down the pipe, z. such that the short-contact time approximation breaks down. Show your work and explain your answer. Hint: When y=R and z=z., 8(n)=0 and the concentration at the middle of the pipe is still 0 so the short contact time approximation still holds. However, for z>z, and y=R, n becomes less than 2.8 and the concentration at the middle of the pipe is no longer zero. C) (3 points) Derive the convective flux, N, for solute at the inside surface of the pipe in the short-contact time limit. Show your work