TRANSPORT PHENOMENA

Diffusion of a pigment through a polymer capsule. I want to do a visual experiment for the DTP course to show Fickian diffusion of a fluorescent pigment $($F$)$ through a polymer $($P$).$ For that, I filled up a hollow spherical capsule, made from a polymer, with a solution of the fluorescent pigment at a really high known concentration.

The polymer capsule loaded with the pigment solution ends up having the same density as water, so I can suspend it inside a $2$L Erlenmeyer flask filled with water. The water outside the capsule has no pigment at the beginning of the experiment. So$,$ the capsule will be fixed in the middle of the flask during the experiment $($using some wires to keep it stationary$)$ and surrounded by pure water. There is a magnetic stir bar at the bottom of the flask so that I can have proper mixing of the water to make the pigment move quickly by convection away from the capsule once it exists. Students will be able to see how the fluorescent pigment moves away from the capsule by looking at the fluorescent streamlines formed and maybe even visualize the gradients in pigment as a function of radius. The hollow polymer sphere has an inner radius ro and outer radius r$1.$ During the experiment, the capsule will slowly release a steady dose of the pigment due to the difference in molar concentrations of the pigment from the inside and the outside of the capsule. The molar concentration at the outer surface of the capsule can be assumed to remain at zero due to the very effective convective transport provided by the stirring $($agitation$).$ The flask is relatively very large compared to the capsule. The polymer wall of the capsule is permeable to the pigment and the water is not significantly diffusion into the capsule. The concentration of the pigment inside the capsule remains constant during the experiment because it is a relatively slow diffusion process. The diffusion coefficient of the pigment through the polymer Dfp is constant. The experimental is conducted under isothermal conditions. No chemical reaction occurs during the experiment.

a$.$Draw a $2$D and a $3$D sketches of the system in your problem. Note, they may look similar for certain geometries.

b$.$Include details of the complete experimental set$-$up$,$ these may be useful for the next points.

i$.$ What is the source of the species that is diffusing? What is the sink of the species that is diffusing? Explain your answer.

ii$.$ On at least one of your sketches, add the coordinate system that you will use to model the system. Think about where to place the origin and what are the appropriate axes.

iii. What is your control volume? Describe with words indicating to what area of space it corresponds in your drawing.

iv$.$ What is the range of space occupied by your control volume?

v$.$ Gradients are responsible for creating diffusional transfer processes. In this case, based on your previous answers, add to your drawing a vector arrow that indicates the direction of the diffusional mass transport $($flux$).$

c$.$ Propose a general partial differential equation $($PDE$)$ using a version of the $3-$dimensional form of the Continuity Equation in terms of moles containing all possible terms for a real$-$life system. Explain how this comes from a mass microbalance.

d$.$ Explain how you can use the experiment conditions mentioned in the problem statement to cancel out terms from the general PDE $($from b$)$ to simplify it to an ODE that you can integrate. Explain the cancellation and indicate your final GDE.