Question

Consider a hot dog being cooked in boiling water in a pan. Would you model the heat transfer to the hot dog 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: The heat transfer in a hot dog being cooked in boiling water can be modeled as one-dimensional and transient in the radial direction. The most appropriate coordinate system to use is the cylindrical coordinate system with the origin placed at the center of the hot dog.

Step-by-step solution

One-, Two-, or Three-Dimensional Heat Transfer

Firstly, we need to determine whether the heat transfer in the hot dog is one-, two-, or three-dimensional. Since hot dogs are generally cylindrical in shape, we can consider that heat transfers in the radial (r) and longitudinal (z) directions. As the boiling water temperature is constant and uniformly distributed around the hot dog, we can assume that the temperature will be the same along the entire length of the hot dog. Hence, we can model the heat transfer as one-dimensional in the radial direction (r).

Steady or Transient Heat Transfer

Next, we need to determine whether the heat transfer is steady or transient. A process is said to be steady if the temperature does not change with time at any point in the medium. While boiling the hot dog, the temperature will vary over time from the initial temperature until it reaches a uniform temperature close to the boiling water temperature. Therefore, we classify the heat transfer as transient.

Coordinate System and Placement of the Origin

Since we are dealing with a one-dimensional problem, the most appropriate coordinate system would be the cylindrical coordinate system. In this case, the origin (r = 0) can be placed at the center of the hot dog (cylinder axis), and the radial direction (r) will extend outward from the center to the outer surface of the hot dog. Thus, the heat transfer problem gets simplified to only considering the radial heat transfer.