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Chapter 2: Problem 10

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

Expert verified
Answer: The heat transfer for an egg in boiling water is three-dimensional transient heat transfer. A spherical coordinate system with the origin located at the center of the egg is the most appropriate for solving the problem.

Step by step solution

01

Determine the dimensionality of heat transfer

To decide whether the heat transfer should be modeled as one-, two-, or three-dimensional, we need to consider how the temperature changes with respect to different axes. Since the egg is submerged in boiling water, the heat transfer occurs uniformly throughout the egg. Therefore, it is reasonable to model the heat transfer as three-dimensional, as temperature variations can occur along all three axes (x, y, and z).
02

Assess the heat transfer as steady or transient

The heat transfer is considered steady if the temperature does not change with time, and transient if the temperature changes with time. In the case of an egg being cooked in boiling water, the temperature of the egg will change until it reaches a final, steady temperature. Thus, the heat transfer is transient in nature as the temperature of the egg changes with time.
03

Choose an appropriate coordinate system and origin

To solve the problem, we can use a spherical coordinate system, as it best fits the shape and symmetry of the egg and simplifies the analysis. In a spherical coordinate system, the origin (r = 0) would be placed at the center of the egg, while the polar and azimuthal angles (θ and φ) will describe the position of a point on the surface of the egg. In summary, we will model the heat transfer as three-dimensional transient heat transfer, and we will use a spherical coordinate system with the origin located at the center of the egg.

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Most popular questions from this chapter

Does heat generation in a solid violate the first law of thermodynamics, which states that energy cannot be created or destroyed? Explain.

Consider a cylindrical shell of length \(L\), inner radius \(r_{1}\), and outer radius \(r_{2}\) whose thermal conductivity varies in a specified temperature range as \(k(T)=k_{0}\left(1+\beta T^{2}\right)\) where \(k_{0}\) and \(\beta\) are two specified constants. The inner surface of the shell is maintained at a constant temperature of \(T_{1}\) while the outer surface is maintained at \(T_{2}\). Assuming steady one-dimensional heat transfer, obtain a relation for the heat transfer rate through the shell.

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