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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 (胃 and 蠁) 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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transient Heat Transfer
Understanding the concept of transient heat transfer is essential when analyzing real-life problems, such as the cooking of an egg in boiling water. Transient heat transfer happens when the temperature within a system varies with time.
For example, when you place the egg in boiling water, its temperature initially starts at a lower level and gradually rises until it reaches the temperature of the surrounding water.
  • This variation means that the heat transfer is not at equilibrium, as the rate of heat entering the egg will diminish over time until complete equilibrium is achieved and the egg is fully cooked.
  • Transient heat transfer is crucial in many operations where time-dependent temperature changes matter, such as cooling operations or heating processes in cooking.
In contrast, steady heat transfer occurs when the temperatures remain constant over time, which is not the case here.
The transient state captures this constant evolution of temperature until it steadies out.
Three-dimensional Analysis
When examining the heat transfer in the process of boiling an egg, a three-dimensional analysis is a crucial approach. In a 3D model, temperature changes occur along all three spatial dimensions鈥攛, y, and z鈥攑roviding a more accurate depiction of how heat flows.
This is important because:
  • The egg is a complex shape, not flat or purely cylindrical, making it necessary to consider how every part of the egg's surface is exposed to heat.
  • Heat travels through the egg in various directions, not solely from one side to another, illustrating the necessity of a comprehensive three-dimensional model.
By considering this, you get an accurate picture of how the heat conducts through the egg energetically and dynamically.
Analyzing the egg in this manner allows for a complete understanding of temperature distribution across its volume, thus ensuring precision in determining when the egg is perfectly cooked.
Spherical Coordinate System
Choosing the right coordinate system can make analyzing heat transfer much simpler, especially for axially symmetric objects like an egg. A spherical coordinate system is well-suited for this purpose due to its natural alignment with the egg's geometry.
  • With spherical coordinates, you describe points in space in terms of their distance from a center point, radial distance, and angles, instead of simply using cartesian coordinates (x, y, z).
  • This system is advantageous because it simplifies the mathematical modeling of heat flow within spherical objects, such as the egg.
For the egg, it's logical to place the origin at its center. The features of spherical coordinates will wrap around the egg's curved structure seamlessly.
This simplification helps in focusing on variables that are directly relevant to the heat transfer process. Thus, adopting spherical coordinates aligns well with the physical reality of the scenario, enabling clear and effective analysis of how heat pervades through the egg uniformly and radially.

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Most popular questions from this chapter

A spherical container, with an inner radius of \(1 \mathrm{~m}\) and a wall thickness of \(5 \mathrm{~mm}\), has its inner surface subjected to a uniform heat flux of \(7 \mathrm{~kW} / \mathrm{m}^{2}\). The outer surface of the container is maintained at \(20^{\circ} \mathrm{C}\). The container wall is made of a material with a thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where \(k_{0}=1.33 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.0023 \mathrm{~K}^{-1}\), and \(T\) is in \(\mathrm{K}\). Determine the temperature drop across the container wall thickness.

A large plane wall, with a thickness \(L\) and a thermal conductivity \(k\), has its left surface \((x=0)\) exposed to a uniform heat flux \(\dot{q}_{0}\). On the right surface \((x=L)\), convection and radiation heat transfer occur in a surrounding temperature of \(T_{\infty}\). The emissivity and the convection heat transfer coefficient on the right surface are \(\bar{\varepsilon}\) and \(h\), respectively. Express the houndary conditions and the differential equation of this heat conduction problem during steady operation.

Consider a plane wall of thickness \(L\) 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 wall surface at \(x=0\) is maintained at a constant temperature of \(T_{1}\), while the surface at \(x=L\) is maintained at \(T_{2}\). Assuming steady one-dimensional heat transfer, obtain a relation for the heat transfer rate through the wall.

Consider a short cylinder of radius \(r_{o}\) and height \(H\) in which heat is generated at a constant rate of \(\dot{e}_{\text {gen. }}\). Heat is lost from the cylindrical surface at \(r=r_{o}\) by convection to the surrounding medium at temperature \(T_{\infty}\) with a heat transfer coefficient of \(h\). The bottom surface of the cylinder at \(z=0\) is insulated, while the top surface at \(z=H\) is subjected to uniform heat flux \(\dot{q}_{H}\). Assuming constant thermal conductivity and steady two-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem. Do not solve.

Consider a spherical shell of inner radius \(r_{1}\), outer radius \(r_{2}\), thermal conductivity \(k\), and emissivity \(\varepsilon\). The outer surface of the shell is subjected to radiation to surrounding surfaces at \(T_{\text {surr }}\), but the direction of heat transfer is not known. Express the radiation boundary condition on the outer surface of the shell.
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