1.

Molecular diffusion in a bounded medium [12 pts]

An instantaneous point of strength M is located at the position (0, 0, h)

above a horizontal plane (z = 0) that forms a reflective boundary. With a

uniform advective velocity field (U, 0, 0) and molecular diffusivity D, (1)

determine the solution for the resulting concentration distribution, and (2)

by differentiation normal to the boundary show that the normal diffusive

flux is indeed zero. [2 pts each]

Assuming an additional horizontal boundary, that behaves as a perfect

absorber for the substance under consideration, is located at z = H (where

H>h), determine the solution. [3 pts]

Assume a 1-D governing equation for vertical diffusion in a vertical tube:

a²c

Әс

at

= D azz assuming homogeneity in the x-y plane. Assume the tube has a

length of 100 cm and there is an injection of mass in the middle of the tube

(z=50 cm) of 10 g. Assume D is 0.001 m²/s and diameter of the tube is 10

cm. Perform 2 simulations with 2 different boundary conditions at z=0 cm

and z=100 cm: (1) a concentration of zero, and (2) a no flux boundary

condition. Use the numerical approach with a grid spacing of 5 cm to solve

for the vertical distribution of mass after 60 s (i.e., plot C vs z at t-60 s) for

(1) and (2). Choose your time step based on this criterion: At < 0.44² [5

pts]

a.

b.

C.