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Chapter 2: Problem 82

Consider the uniform heating of a plate in an environment at a constant temperature. Is it possible for part of the heat generated in the left half of the plate to leave the plate through the right surface? Explain.

Short Answer

Expert verified
Answer: Yes, it is possible for part of the heat generated in the left half of a uniformly heated plate to leave the plate through the right surface. This is due to the heat flow from higher temperature regions to lower temperature regions and the conduction process that transfers heat from the plate to the surrounding environment.

Step by step solution

01

Understanding heat transfer mechanisms

Heat can transfer through three main mechanisms: conduction, convection, and radiation. Conduction is the transfer of heat between two objects in direct contact, convection is the transfer of heat through a medium such as air, and radiation is the transfer of heat using electromagnetic waves like infrared radiation. In this case, we will focus on conduction, since the plate is in direct contact with the environment.
02

Analyzing uniform heating in the plate

In the given scenario, the plate is being heated uniformly, meaning that heat is being generated evenly throughout the plate. When heat is generated in the left half of the plate, it is transferred to the colder regions of the plate, including the right surface.
03

Understanding heat flow in the plate

Heat flows from higher temperature regions to lower temperature regions. Since the plate is uniformly heated, the temperature distribution inside the plate will also be uniform. Therefore, the heat generated on the left half of the plate will flow in all directions, including towards the right half of the plate.
04

Heat leaving through the right surface

As heat flows from the left half of the plate to the right half, it will eventually reach the right surface. At this point, if the right surface is in contact with a cooler environment, the heat will transfer from the plate to the environment through conduction. This means that it is indeed possible for part of the heat generated in the left half of the plate to leave the plate through the right surface.
05

Conclusion

In conclusion, it is possible for part of the heat generated in the left half of the uniformly heated plate to leave the plate through the right surface. This is due to the fact that heat flows from higher temperature regions to lower temperature regions, such as the environment surrounding the right surface of the plate, through conduction.

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Key Concepts

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

Conduction
Conduction is a primary heat transfer mechanism where thermal energy moves through materials that are in physical contact. This process occurs at the molecular level, where energetic, rapidly moving molecules or atoms collide with slower-moving ones, passing along energy in the form of heat.

Imagine a chain of people passing a basketball from one end to the other; in a similar manner, heat is transferred from the warm side to the cooler side of an object until a balanced temperature is achieved throughout.
Uniform Heating
Uniform heating refers to a scenario where heat is generated or applied evenly across an entire object or surface. For instance, think of a griddle pan that heats up at the same rate all over, ensuring that pancakes cook uniformly. In terms of a plate being heated, this means that every part of the plate reaches the same level of warmth without temperature gradients or 'hot spots'.
Temperature Distribution
The concept of temperature distribution is related to how heat is spread within a material or structure. A uniform temperature distribution signifies that every point within the object holds the same temperature. This is ideal in many manufacturing processes, where consistent quality and properties are crucial.

In contrast, non-uniform temperature distribution can lead to stresses and imperfections in materials as different parts expand or contract differently due to temperature variations.
Thermal Energy Flow
Thermal energy flow is the movement of heat from one part of a substance to another or from one substance to another. This flow is directed from regions of higher temperature to those of lower temperature, following the second law of thermodynamics. As such, the flow continues until thermal equilibrium is reached, meaning the involved areas reach the same temperature.
Heat Transfer Through Conduction
Heat transfer through conduction is particularly significant in solids, where atoms and molecules are tightly packed together and can efficiently transfer kinetic energy to their neighbors. Imagine touching a metal spoon that has been sitting in a pot of hot soup. The heat is conducted from the soup through the spoon to your hand, illustrating how effectively metals, due to their free electrons, can transfer heat.

An understanding of this process is not only fundamental to solving heat-related problems in physics but also crucial in engineering to design more efficient thermal systems, ranging from computer processors to large-scale heating installations.

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Most popular questions from this chapter

Consider a large 3 -cm-thick stainless steel plate \((k=\) \(15.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) in which heat is generated uniformly at a rate of \(5 \times 10^{5} \mathrm{~W} / \mathrm{m}^{3}\). Both sides of the plate are exposed to an environment at \(30^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Explain where in the plate the highest and the lowest temperatures will occur, and determine their values.

A spherical metal ball of radius \(r_{o}\) is heated in an oven to a temperature of \(T_{i}\) throughout and is then taken out of the oven and allowed to cool in ambient air at \(T_{\infty}\) by convection and radiation. The emissivity of the outer surface of the cylinder is \(\varepsilon\), and the temperature of the surrounding surfaces is \(T_{\text {surr }}\). The average convection heat transfer coefficient is estimated to be \(h\). Assuming variable thermal conductivity and transient one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve.

Consider a water pipe of length \(L=17 \mathrm{~m}\), inner radius \(r_{1}=15 \mathrm{~cm}\), outer radius \(r_{2}=20 \mathrm{~cm}\), and thermal conductivity \(k=14 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Heat is generated in the pipe material uniformly by a \(25-\mathrm{kW}\) electric resistance heater. The inner and outer surfaces of the pipe are at \(T_{1}=60^{\circ} \mathrm{C}\) and \(T_{2}=80^{\circ} \mathrm{C}\), respectively. Obtain a general relation for temperature distribution inside the pipe under steady conditions and determine the temperature at the center plane of the pipe.

A 2-kW resistance heater wire whose thermal conductivity is \(k=10.4 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot \mathrm{R}\) has a radius of \(r_{o}=0.06\) in and a length of \(L=15\) in, and is used for space heating. Assuming constant thermal conductivity and one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem during steady operation. Do not solve.

Consider a large plane wall of thickness \(L\) and constant thermal conductivity \(k\). The left side of the wall \((x=0)\) is maintained at a constant temperature \(T_{0}\), while the right surface at \(x=L\) is insulated. Heat is generated in the wall at the rate of \(\dot{e}_{\text {gen }}=a x^{2} \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{3}\). Assuming steady one-dimensional heat transfer, \((a)\) express the differential equation and the boundary conditions for heat conduction through the wall, \((b)\) by solving the differential equation, obtain a relation for the variation of temperature in the wall \(T(x)\) in terms of \(x, L, k, a\), and \(T_{0}\), and (c) what is the highest temperature \(\left({ }^{\circ} \mathrm{C}\right)\) in the plane wall when: \(L=1 \mathrm{ft}, k=5 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{\circ}{ }^{\circ} \mathrm{F}, a=1200 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{5}\), and \(T_{0}=700^{\circ} \mathrm{F}\).
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