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Chapter 4: Problem 83

What is a semi-infinite medium? Give examples of solid bodies that can be treated as semi-infinite mediums for heat transfer purposes.

Short Answer

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Answer: A semi-infinite medium is a material with a large size compared to the characteristic length of the problem, which simplifies the analysis by considering only unidirectional heat transfer. Examples of solid bodies that can be treated as semi-infinite mediums for heat transfer include large metal slabs, Earth's surface, thick concrete walls, and large ice sheets.

Step by step solution

01

Definition of semi-infinite medium

A semi-infinite medium is a material with a large size (in one or more dimensions) compared to the characteristic length of the problem under consideration. One common application of the semi-infinite medium concept is in heat transfer analysis, where the semi-infinite medium assumption simplifies the analysis by considering only unidirectional heat transfer.
02

Semi-infinite medium examples in solid bodies for heat transfer

1. Large metal slabs: In industries, large metal slabs (e.g., steel or aluminum) are often used for various processes like casting, rolling, or welding. When heat is applied to the surface of these slabs, the heat transfer occurs mostly in one direction (perpendicular to the surface), and the slab can be considered a semi-infinite medium. 2. Earth's surface: For most practical purposes, the Earth's surface can be treated as a semi-infinite medium when analyzing heat transfer related to solar radiation, geothermal energy, or heat loss from buildings. The Earth's surface is much larger than the depth at which temperature variations are relevant, resulting in unidirectional heat transfer. 3. Thick concrete walls: In architectural or civil engineering applications, thick concrete walls can be considered semi-infinite mediums for heat transfer if the wall thickness is much larger than the heat penetration depth. The heat transfer in the wall will mostly occur in a single direction (perpendicular to the surfaces), simplifying the analysis. 4. Large ice sheets: In cryogenic and polar research, large ice sheets or glaciers often serve as examples of semi-infinite mediums for heat transfer. When a heat source is applied to the surface of an ice sheet, the heat transfer occurs predominantly in one direction (downward), simplifying the analysis and allowing for easier modeling of melting and freezing processes.

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Key Concepts

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

Heat Transfer Analysis
Heat transfer analysis is crucial in engineering and environmental studies as it allows us to understand how heat energy moves through materials. It involves the examination of thermal energy migration from hot to cold regions within a body. Key parameters in this analysis include thermal conductivity, which characterizes how well a material can conduct heat, and specific heat capacity, a measure of how much energy is needed to change the temperature of a material.

Through heat transfer analysis, engineers can predict the temperature distribution and the rate of heat flow within objects. The practical applications are vast, ranging from designing efficient cooling systems for electronic components to creating comfortable living environments in buildings. Heat transfer analysis makes use of mathematical models, typically differential equations, to describe how heat moves within materials. This is where simplified models, such as the semi-infinite medium assumption, often become beneficial, especially when dealing with large objects where heat moves in a predominant direction and the other dimensions are less significant.
Unidirectional Heat Transfer
Unidirectional heat transfer refers to a scenario where heat moves predominantly in one direction. This simplification is relevant when considering very large bodies, or semi-infinite mediums, where the heat transfer in the direction of thickness is significant, but negligible along the length and width. In practice, this means we can ignore heat flow in two out of the three spatial dimensions, making it easier to calculate and model the heat transfer within the material.

For example, when a large metal slab is heated from one side, the heat penetrates the material primarily perpendicularly to its heated surface. The dimensions along the slab's surface are so large that heat has no time to significantly spread sideways before it is conducted deeper into the slab. Therefore, the unidirectional model is adequate for predicting how the temperature varies with depth and time without the complexity of three-dimensional heat flow computation.
Thermal Penetration Depth
The thermal penetration depth indicates how far heat has penetrated or will penetrate into a material over time. It is a valuable concept when evaluating how quickly a material responds to temperature changes at its boundary. In the context of semi-infinite mediums, the thermal penetration depth helps define the depth at which the temperature within the material remains practically unaffected by surface temperature fluctuations.

The thermal penetration depth depends on the thermal diffusivity of the material - a measure combining the material's thermal conductivity, density, and specific heat capacity. High thermal diffusivity implies that heat spreads rapidly throughout the material, leading to a greater penetration depth for a given time. Conversely, low thermal diffusivity means heat moves slowly, so the penetration depth is smaller. Engineers and scientists use this concept to design materials and structures in which precise control over temperature distribution is critical.

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Most popular questions from this chapter

Under what conditions can a plane wall be treated as a semi-infinite medium?

Refractory bricks are used as linings for furnaces, and they generally have low thermal conductivity to minimize heat loss through the furnace walls. Consider a thick furnace wall lining with refractory bricks \(\left(k=1.0 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\) and \(\left.\alpha=5.08 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)\), where initially the wall has a uniform temperature of \(15^{\circ} \mathrm{C}\). If the wall surface is subjected to uniform heat flux of \(20 \mathrm{~kW} / \mathrm{m}^{2}\), determine the temperature at the depth of \(10 \mathrm{~cm}\) from the surface after an hour of heating time.

A potato may be approximated as a 5.7-cm-diameter solid sphere with the properties \(\rho=910 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4.25 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\), \(k=0.68 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\alpha=1.76 \times 10^{-1} \mathrm{~m}^{2} / \mathrm{s}\). Twelve such potatoes initially at \(25^{\circ} \mathrm{C}\) are to be cooked by placing them in an oven maintained at \(250^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(95 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The amount of heat transfer to the potatoes during a 30-min period is (a) \(77 \mathrm{~kJ}\) (b) \(483 \mathrm{~kJ}\) (c) \(927 \mathrm{~kJ}\) (d) \(970 \mathrm{~kJ}\) (e) \(1012 \mathrm{~kJ}\)

The walls of a furnace are made of \(1.2\)-ft-thick concrete \(\left(k=0.64 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right.\) and \(\left.\alpha=0.023 \mathrm{ft}^{2} / \mathrm{h}\right)\). Initially, the furnace and the surrounding air are in thermal equilibrium at \(70^{\circ} \mathrm{F}\). The furnace is then fired, and the inner surfaces of the furnace are subjected to hot gases at \(1800^{\circ} \mathrm{F}\) with a very large heat transfer coefficient. Determine how long it will take for the temperature of the outer surface of the furnace walls to rise to \(70.1^{\circ} \mathrm{F}\). Answer: \(116 \mathrm{~min}\)

For heat transfer purposes, an egg can be considered to be a \(5.5-\mathrm{cm}\)-diameter sphere having the properties of water. An egg that is initially at \(8^{\circ} \mathrm{C}\) is dropped into the boiling water at \(100^{\circ} \mathrm{C}\). The heat transfer coefficient at the surface of the egg is estimated to be \(800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the egg is considered cooked when its center temperature reaches \(60^{\circ} \mathrm{C}\), determine how long the egg should be kept in the boiling water. Solve this problem using analytical one-term approximation method (not the Heisler charts).
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