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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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Key Concepts

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

Temperature Profile
A temperature profile is a crucial concept when analyzing how temperature changes over a surface or within a medium. Imagine slicing through a structure and looking at how temperature levels vary from one part of the side to another. This variation is what forms a temperature profile, which can be visualized as a graph or described mathematically.
  • It helps in understanding how heat is distributed.
  • It guides engineers in designing systems that manage heat efficiently.
In scenarios involving an insulated surface, the temperature profile tells you how effectively the insulation is working by showing you the steepness of the temperature change.
Conductive Heat Transfer
Conductive heat transfer is a major mechanism by which heat moves through a solid body or between tightly contacting bodies. In this process, heat is transferred from a region of higher temperature to a region of lower temperature.
  • This occurs due to the thermal motion of atoms and vibrations in the lattice of a solid.
  • It's essential for understanding heat flow in systems where substances are in direct contact.
  • The efficiency of this transfer is often described mathematically using heat equations.
Hence, understanding conductive heat transfer is vital in designing systems that either need to enhance or curtail heat transfer, such as in cooking utensils or building insulation.
Thermal Conductivity
Thermal conductivity is like the bridge that allows heat to travel through a material. It's a property that defines how easily heat is conducted through a given material. Materials with high thermal conductivity, like metals, transfer heat quickly.
  • It's measured in watts per meter-kelvin (W/m路K).
  • Determines how effective a material will be in applications needing either rapid or slow heat dispersion.
  • Materials with low thermal conductivity, such as foam or fiberglass, are good insulators.
This property is fundamental when selecting materials for thermal management in various engineering applications, ensuring that heat flow is controlled as desired.
Temperature Gradient
The temperature gradient is the rate at which temperature changes in a particular direction within a medium. Here, it serves as the driving force for conductive heat transfer, meaning that without a gradient, heat wouldn't naturally flow.
  • It is often described as the change in temperature per unit distance.
  • Significant in the context of thermal insulation as it dictates the rate of heat loss across surfaces.
  • Helps determine the efficiency of the thermal systems.
For example, in an insulated wall, a steep temperature gradient perpendicular to the wall indicates effective insulation, minimizing heat transfer across the surface.

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Most popular questions from this chapter

Is heat transfer a scalar or vector quantity? Explain. Answer the same question for temperature.

Heat is generated in a long wire of radius \(r_{o}\) at a constant rate of \(\dot{e}_{\text {gen }}\) per unit volume. The wire is covered with a plastic insulation layer. Express the heat flux boundary condition at the interface in terms of the heat generated.

Water flows through a pipe at an average temperature of \(T_{\infty}=90^{\circ} \mathrm{C}\). The inner and outer radii of the pipe are \(r_{1}=\) \(6 \mathrm{~cm}\) and \(r_{2}=6.5 \mathrm{~cm}\), respectively. The outer surface of the pipe is wrapped with a thin electric heater that consumes \(400 \mathrm{~W}\) per \(\mathrm{m}\) length of the pipe. The exposed surface of the heater is heavily insulated so that the entire heat generated in the heater is transferred to the pipe. Heat is transferred from the inner surface of the pipe to the water by convection with a heat transfer coefficient of \(h=85 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming constant thermal conductivity and one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of the heat conduction in the pipe during steady operation. Do not solve.

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.

Consider a 20-cm-thick large concrete plane wall \((k=0.77 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) subjected to convection on both sides with \(T_{\infty 1}=22^{\circ} \mathrm{C}\) and \(h_{1}=8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the inside, and \(T_{\infty 2}=8^{\circ} \mathrm{C}\) and \(h_{2}=12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, \((a)\) express the differential equations 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 temperatures at the inner and outer surfaces of the wall.
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