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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 plane wall of thickness \(I_{\text {, }}\) thermal conductivity \(k\), and surface area \(A\). The left surface of the wall is exposed to the ambient air at \(T_{\infty}\) with a heat transfer coefficient of \(h\) while the right surface is insulated. The variation of temperature in the wall for steady one-dimensional heat conduction with no heat generation is (a) \(T(x)=\frac{h(L-x)}{k} T_{\infty}\) (b) \(T(x)=\frac{k}{h(x+0.5 L)} T_{\infty}\) (c) \(T(x)=\left(1-\frac{x h}{k}\right) T_{\infty}\) (d) \(T(x)=(L-x) T_{\infty}\) (e) \(T(x)=T_{\infty}\)

Consider a \(1.5\)-m-high and \(0.6-\mathrm{m}\)-wide plate whose thickness is \(0.15 \mathrm{~m}\). One side of the plate is maintained at a constant temperature of \(500 \mathrm{~K}\) while the other side is maintained at \(350 \mathrm{~K}\). The thermal conductivity of the plate can be assumed to vary linearly in that temperature range as \(k(T)=\) \(k_{0}(1+\beta T)\) where \(k_{0}=18 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\beta=8.7 \times 10^{-4} \mathrm{~K}^{-1}\). Disregarding the edge effects and assuming steady onedimensional heat transfer, determine the rate of heat conduction through the plate. Answer: \(22.2 \mathrm{~kW}\)

Consider a large plane wall of thickness \(L=0.8 \mathrm{ft}\) and thermal conductivity \(k=1.2 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\). The wall is covered with a material that has an emissivity of \(\varepsilon=0.80\) and a solar absorptivity of \(\alpha=0.60\). The inner surface of the wall is maintained at \(T_{1}=520 \mathrm{R}\) at all times, while the outer surface is exposed to solar radiation that is incident at a rate of \(\dot{q}_{\text {solar }}=300 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}\). The outer surface is also losing heat by radiation to deep space at \(0 \mathrm{~K}\). Determine the temperature of the outer surface of the wall and the rate of heat transfer through the wall when steady operating conditions are reached.

A long homogeneous resistance wire of radius \(r_{o}=\) \(0.6 \mathrm{~cm}\) and thermal conductivity \(k=15.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is being used to boil water at atmospheric pressure by the passage of electric current. Heat is generated in the wire uniformly as a result of resistance heating at a rate of \(16.4 \mathrm{~W} / \mathrm{cm}^{3}\). The heat generated is transferred to water at \(100^{\circ} \mathrm{C}\) by convection with an average heat transfer coefficient of \(h=3200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming steady one-dimensional heat transfer, \((a)\) express the differential equation and the boundary conditions for heat conduction through the wire, \((b)\) obtain a relation for the variation of temperature in the wire by solving the differential equation, and \((c)\) determine the temperature at the centerline of the wire.

A flat-plate solar collector is used to heat water by having water flow through tubes attached at the back of the thin solar absorber plate. The absorber plate has an emissivity and an absorptivity of \(0.9\). The top surface \((x=0)\) temperature of the absorber is \(T_{0}=35^{\circ} \mathrm{C}\), and solar radiation is incident on the absorber at \(500 \mathrm{~W} / \mathrm{m}^{2}\) with a surrounding temperature of \(0^{\circ} \mathrm{C}\). Convection heat transfer coefficient at the absorber surface is \(5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while the ambient temperature is \(25^{\circ} \mathrm{C}\). Show that the variation of temperature in the absorber plate can be expressed as \(T(x)=-\left(\dot{q}_{0} / k\right) x+T_{0}\), and determine net heat flux \(\dot{q}_{0}\) absorbed by the solar collector.
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