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Chapter 4: Problem 5

For which kind of bodies made of the same material is the lumped system analysis more likely to be applicable: slender ones or well-rounded ones of the same volume? Why?

Short Answer

Expert verified
Answer: Well-rounded bodies are more suitable for lumped system analysis due to their smaller Biot number, validating the assumption of uniform temperature throughout the body's volume.

Step by step solution

01

Understand the shape's effect on the lumped system analysis

The shape of an object plays a critical role in heat transfer and thus in the applicability of the lumped system analysis. A slender object has a higher surface area to volume ratio than a well-rounded object. Since heat transfer takes place on the surface, a larger surface area to volume ratio results in faster heat transfer rates from the surface of the object.
02

Analyze thermal resistance in slender and well-rounded bodies

In slender bodies, the larger surface area to volume ratio results in a smaller thermal resistance. This means that heat can flow rapidly from the surface into the body and vice versa. In contrast, well-rounded bodies have a larger thermal resistance due to their smaller surface area to volume ratio, resulting in slower rates of heat transfer.
03

Determine the effect on the Biot number

The Biot number (Bi) is a dimensionless number that measures the relativeness of heat conduction within the object to the heat transfer from the surface of the object. A smaller Biot number (Bi < 0.1) indicates that the lumped system analysis is more likely to be applicable. Since the slender bodies have a smaller thermal resistance, the heat transfer rate is faster, and therefore, the Biot number will likely be larger. On the other hand, well-rounded bodies will have a smaller Biot number due to their larger thermal resistances.
04

Conclude the type of body suitable for lumped system analysis

Based on the reasoning discussed above, the lumped system analysis is more likely to be applicable for well-rounded bodies made of the same material. This is because well-rounded bodies have a smaller Biot number due to their larger thermal resistances, making the assumption of uniform temperature throughout their volume more valid. In summary, the lumped system analysis is more suitable for well-rounded bodies as they have a smaller Biot number, which validates the assumption of uniform temperature throughout the body's volume.

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Most popular questions from this chapter

Chickens with an average mass of \(1.7 \mathrm{~kg}(k=\) \(0.45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\alpha=0.13 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ) initially at a uniform temperature of \(15^{\circ} \mathrm{C}\) are to be chilled in agitated brine at \(-7^{\circ} \mathrm{C}\). The average heat transfer coefficient between the chicken and the brine is determined experimentally to be \(440 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Taking the average density of the chicken to be \(0.95 \mathrm{~g} / \mathrm{cm}^{3}\) and treating the chicken as a spherical lump, determine the center and the surface temperatures of the chicken in \(2 \mathrm{~h}\) and \(45 \mathrm{~min}\). Also, determine if any part of the chicken will freeze during this process. Solve this problem using analytical one-term approximation method (not the Heisler charts).

A 10-cm-inner diameter, 30-cm-long can filled with water initially at \(25^{\circ} \mathrm{C}\) is put into a household refrigerator at \(3^{\circ} \mathrm{C}\). The heat transfer coefficient on the surface of the can is \(14 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming that the temperature of the water remains uniform during the cooling process, the time it takes for the water temperature to drop to \(5^{\circ} \mathrm{C}\) is (a) \(0.55 \mathrm{~h}\) (b) \(1.17 \mathrm{~h}\) (c) \(2.09 \mathrm{~h}\) (d) \(3.60 \mathrm{~h}\) (e) \(4.97 \mathrm{~h}\)

How can we use the transient temperature charts when the surface temperature of the geometry is specified instead of the temperature of the surrounding medium and the convection heat transfer coefficient?

How does the rate of freezing affect the tenderness, color, and the drip of meat during thawing?

It is claimed that beef can be stored for up to two years at \(-23^{\circ} \mathrm{C}\) but no more than one year at \(-12^{\circ} \mathrm{C}\). Is this claim reasonable? Explain.
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