Question

Consider a round potato being baked in an oven. Would you model the heat transfer to the potato as one-, two-, or three-dimensional? Would the heat transfer be steady or transient? Also, which coordinate system would you use to solve this problem, and where would you place the origin? Explain.

Short answer

Answer: To best model heat transfer to a round potato in an oven, it should be considered as three-dimensional, transient, using a spherical coordinate system with the origin placed at the center of the potato.

Step-by-step solution

Determine the dimensionality of heat transfer

In this problem, we are modeling heat transfer to a round potato in an oven. Since the potato is round, we can expect the heat to be transferred across its entirety uniformly. Therefore, it is safe to assume that the heat transfer is three-dimensional. So, the heat transfer is three-dimensional.

Determine if the heat transfer is steady or transient

Let's consider the heat transfer process. As the potato bakes in the oven, its temperature will change over time as it receives heat from the oven. The heat transfer is not constant throughout the entire process, as the potato will eventually get cooked and reach a point where its temperature no longer changes. Thus, the heat transfer would be transient, not steady.

Choose the coordinate system

Since the potato is round, it would be best to use a spherical coordinate system to model the heat transfer. Spherical coordinates are more suitable for dealing with round objects, and in this case, it will simplify the calculations.

Determine the origin for the coordinate system

In a spherical coordinate system, it is best to place the origin at the center of the object being studied. In this case, we place the origin at the center of the round potato. This will help in simplifying calculations and provide a more intuitive understanding of the heat transfer process within the potato. In conclusion, based on the analysis, the heat transfer to the round potato in an oven should be modeled as three-dimensional, transient, using a spherical coordinate system with the origin placed at the center of the potato.