A colleague has proposed using a cylindrical polymer tube to transport hydrogen to a fuel cell. However, you are concerned that a significant amount of hydrogen may diffuse through the tube wall and be lost. To evaluate this we need to obtain an expression for the steady$-$state rate at which $H_2$ moves though the tube wall. We may assume diffusion of the hydrogen in the polymer is Fickian and that the tube is straight and long, so end effects are negligible. Let $R$ be the outer radius of the tube, and let $R$ be the tube's inner radius. We can assume that the concentration of hydrogen in the tube surroundings is effectively zero and that the concentration of hydrogen in the tube is constant at $c_0.$

$($a$)[20$ points$]$ Write the simplified balance equation describing the concentration of hydrogen in the polymer and the appropriate boundary conditions.

$($b$)[20$ points$]$ Solve the result of part $($a$)$ and obtain an expression for the concentration of hydrogen in the polymer.

$($c$)[10$ points$]$ Use the result of part $($b$)$ to obtain an expression for the total rate at which hydrogen is lost through the tube wall.

$[50$ points$]$ Your company is working with a producer of frozen vegetables on the design of a new process for freezing green peas. In the proposed process fresh peas, initially at room temperature, will be dropped through a vertical column while a stream of extremely cold air is blown upward through the column. The cold air will freeze the peas, and they will be collected for packaging at the bottom on the column.

In order to determine the necessary size of the column, you need to determine how long the falling peas must be exposed to the cold air to freeze completely. To do this, formulate a mathematical model capable of describing how long it will take for the maximum temperature in an individual pea to go below freezing. Your company has a mathematical geek in residence, so you do not need to solve the model equations that you develop. However, be sure to provide the appropriately simplified differential equations for the model, as well as any necessary boundary and initial conditions.