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Chapter 2: Problem 39

It is claimed that the temperature profile in a medium must be perpendicular to an insulated surface. Is this a valid claim? Explain.

Short Answer

Expert verified
Answer: Yes, the temperature profile must be perpendicular to an insulated surface to ensure the lowest possible heat transfer due to the temperature gradient driving the heat transfer in conduction and the purpose of insulation in minimizing heat transfer.

Step by step solution

01

Understanding the concept of temperature profile

A temperature profile is a graphical or mathematical representation of how the temperature varies within a medium or across a surface. In the context of this exercise, we are examining how the temperature distributes across an insulated surface.
02

Examining heat transfer across surfaces

Heat transfer across surfaces, and through media, occurs through three primary mechanisms: conduction, convection, and radiation. In this case, we are primarily concerned with conduction, as this is the dominant mode of heat transfer in contact between a medium and a solid surface.
03

Understanding conductive heat transfer

Conductive heat transfer occurs when there is a temperature gradient in a medium, which causes energy to flow from high-temperature regions to low-temperature regions. In most cases, this heat transfer is directly proportional to the temperature gradient (difference between temperatures), and inversely proportional to the distance of the medium.
04

Insulation and conductive heat transfer

The purpose of insulation is to reduce or prevent the flow of heat across a surface or within a medium. This can be achieved by using materials with low thermal conductivity, or by design features that minimize the surface area available for heat transfer or create barriers to heat flow.
05

Determining the validity of the claim

The temperature profile in a medium must be perpendicular to an insulated surface because the heat transfer via conduction is driven by the temperature gradient. Insulated surfaces are designed to minimize heat transfer so the maximum temperature difference occurs across the shortest possible distance, which would be perpendicular to the insulated surface. #Conclusion# The claim is valid because the temperature profile must indeed be perpendicular to an insulated surface to ensure the lowest possible heat transfer. This is due to the temperature gradient driving the heat transfer in conduction and the purpose of insulation in minimizing heat transfer.

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Most popular questions from this chapter

Consider a large plane wall of thickness \(L=0.4 \mathrm{~m}\), thermal conductivity \(k=1.8 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and surface area \(A=\) \(30 \mathrm{~m}^{2}\). The left side of the wall is maintained at a constant temperature of \(T_{1}=90^{\circ} \mathrm{C}\) while the right side loses heat by convection to the surrounding air at \(T_{\infty}=25^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(h=24 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming constant thermal conductivity and no heat generation in the wall, \((a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall, \((b)\) obtain a relation for the variation of temperature in the wall by solving the differential equation, and \((c)\) evaluate the rate of heat transfer through the wall. Answer: (c) \(7389 \mathrm{~W}\)

What is the difference between an ordinary differential equation and a partial differential equation?

In a food processing facility, a spherical container of inner radius \(r_{1}=40 \mathrm{~cm}\), outer radius \(r_{2}=41 \mathrm{~cm}\), and thermal conductivity \(k=1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is used to store hot water and to keep it at \(100^{\circ} \mathrm{C}\) at all times. To accomplish this, the outer surface of the container is wrapped with a 800 -W electric strip heater and then insulated. The temperature of the inner surface of the container is observed to be nearly \(120^{\circ} \mathrm{C}\) at all times. Assuming 10 percent of the heat generated in the heater is lost through the insulation, \((a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the container, \((b)\) obtain a relation for the variation of temperature in the container material by solving the differential equation, and \((c)\) evaluate the outer surface temperature of the container. Also determine how much water at \(100^{\circ} \mathrm{C}\) this tank can supply steadily if the cold water enters at \(20^{\circ} \mathrm{C}\).

Exhaust gases from a manufacturing plant are being discharged through a 10 - \(\mathrm{m}\) tall exhaust stack with outer diameter of \(1 \mathrm{~m}\), wall thickness of \(10 \mathrm{~cm}\), and thermal conductivity of \(40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The exhaust gases are discharged at a rate of \(1.2 \mathrm{~kg} / \mathrm{s}\), while temperature drop between inlet and exit of the exhaust stack is \(30^{\circ} \mathrm{C}\), and the constant pressure specific heat of the exhaust gasses is \(1600 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). On a particular day, the outer surface of the exhaust stack experiences radiation with the surrounding at \(27^{\circ} \mathrm{C}\), and convection with the ambient air at \(27^{\circ} \mathrm{C}\) also, with an average convection heat transfer coefficient of \(8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Solar radiation is incident on the exhaust stack outer surface at a rate of \(150 \mathrm{~W} / \mathrm{m}^{2}\), and both the emissivity and solar absorptivity of the outer surface are 0.9. Assuming steady one-dimensional heat transfer, (a) obtain the variation of temperature in the exhaust stack wall and (b) determine the inner surface temperature of the exhaust stack.

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.
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