$[50$ 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.