Question

Consider an egg being cooked in boiling water in a pan. Would you model the heat transfer to the egg 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: An egg being cooked in boiling water can be modeled as a one-dimensional, transient heat transfer problem. The appropriate coordinate system to use is the polar (spherical) coordinate system, with the origin placed at the center of the egg.

Step-by-step solution

Determine the dimensionality of the heat transfer problem

Since heat is being conducted from the hot water to the egg, the heat transfer occurs in all three spatial dimensions; therefore, it is a three-dimensional problem. However, because the egg has a symmetrical shape, the temperature distribution would most likely be similar across its surface, and hence it can be simplified to a one-dimensional problem along the radial axis of the egg. In this case, a one-dimensional model is appropriate and practical.

Identify steady or transient heat transfer

Steady heat transfer implies that the temperature does not change with time at any point within the object. In the case of the egg being cooked in boiling water, the temperature changes throughout the heating process and depends on the time the egg is in contact with the hot water. That means the heat transfer is transient, not steady.

Choose the coordinate system and origin

Since we decided to consider this as a one-dimensional problem, the natural coordinate system to use would be the polar (spherical) coordinate system due to the shape of the egg. In this case, the radial coordinate (r) would represent the distance from center to surface. The origin should be placed at the center of the egg, as this gives the most straightforward way to calculate distances within the egg and simplifies the analysis. In conclusion: For cooking an egg in boiling water, the heat transfer can be modeled as a one-dimensional, transient problem using the polar (spherical) coordinate system with the origin placed at the center of the egg.