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Chapter 4: Problem 1

What is the physical significance of the Biot number? Is the Biot number more likely to be larger for highly conducting solids or poorly conducting ones?

Short Answer

Expert verified
Answer: A large Biot number indicates that the internal resistance to heat conduction within the object is higher than the convective resistance at the surface, making convective heat transfer more dominant in the heat transfer process. This is typically observed in poorly conducting solids.

Step by step solution

01

Definition of Biot Number

The Biot number (Bi) is a dimensionless quantity that helps in determining the importance of conduction within an object compared to convection on its surface. It is defined as the ratio of the internal thermal resistance of the object to the convective thermal resistance at the surface. Mathematically, it can be expressed as: Bi = \(\frac{hL_c}{k}\) where: - h is the heat transfer coefficient due to convection at the surface (W/m虏路K) - Lc is the characteristic length of the object (m) - k is the thermal conductivity of the solid material (W/m路K)
02

Physical Significance

The physical significance of the Biot number lies in its ability to give information about the dominant mode of heat transfer within a system. A small Biot number (typically less than 0.1) indicates that the internal resistance to heat conduction within the object is much smaller than the convective resistance at the surface, which means that conduction is more dominant. In the case of large Biot numbers (typically greater than 10), the thermal resistance within the object is higher than the convective resistance at the surface, making convection more dominant in the heat transfer process.
03

Biot Number for Highly and Poorly Conducting Solids

Now, let's analyze the relationship between the Biot number and the thermal conductivity of solids. From the formula, we can see that the Biot number is inversely proportional to the thermal conductivity (k). As the thermal conductivity increases, the Biot number decreases and vice versa. For highly conducting solids, the thermal conductivity (k) is large, resulting in a smaller Biot number. This implies that heat conduction is more dominant in highly conducting solids as the internal resistance is lower compared to convective heat transfer at the surface. On the other hand, for poorly conducting solids, the thermal conductivity (k) is small, leading to a larger Biot number. This implies that convective heat transfer at the surface is more dominant in these materials, and the internal resistance to heat conduction is higher. To conclude, the Biot number is more likely to be larger for poorly conducting solids than for highly conducting ones.

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Key Concepts

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

Thermal Resistance
This resistance becomes particularly important when considering the insulation properties of materials. Higher thermal resistance implies better insulation, meaning less heat is lost through the material. In the context of the Biot number, it's the internal thermal resistance that we are concerned with鈥攚hich represents the resistance to heat conduction within a solid object. The comparison between this internal resistance and the resistance posed by convection at the surface of the object helps to predict which mode of heat transfer鈥攃onduction or convection鈥攚ill be more significant.
Convection Heat Transfer
The rate of convective heat transfer is characterized by the heat transfer coefficient, denoted by 'h'. A larger 'h' value means the surface is more effective at transferring heat through convection. This coefficient is affected by several factors, including fluid velocity, viscosity, and the temperature difference between the surface and the fluid.
Thermal Conductivity
In quantitative terms, thermal conductivity is the rate at which heat passes through a material with a given area and temperature gradient. In the context of our problem, understanding thermal conductivity is vital. It directly influences the Biot number and thus impacts whether convection or conduction is the predominant mode of heat transfer in a system.
Dimensionless Numbers in Heat Transfer
There are other important dimensionless numbers in heat transfer as well, such as the Reynolds number which indicates the flow regime of a fluid, and the Prandtl number, which relates the thickness of the thermal boundary layer to the velocity boundary layer. These numbers are used extensively in designing and analyzing systems where heat transfer is crucial, such as radiators, heat exchangers, and cooling systems in electronics.

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Most popular questions from this chapter

A potato that may be approximated as a \(5.7-\mathrm{cm}\) 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 \(\alpha=1.76 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\). 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 by the time the center temperature reaches \(100^{\circ} \mathrm{C}\) is (a) \(56 \mathrm{~kJ}\) (b) \(666 \mathrm{~kJ}\) (c) \(838 \mathrm{~kJ}\) (d) \(940 \mathrm{~kJ}\) (e) \(1088 \mathrm{~kJ}\)

What is an infinitely long cylinder? When is it proper to treat an actual cylinder as being infinitely long, and when is it not? For example, is it proper to use this model when finding the temperatures near the bottom or top surfaces of a cylinder? Explain.

Consider heat transfer between two identical hot solid bodies and their environments. The first solid is dropped in a large container filled with water, while the second one is allowed to cool naturally in the air. For which solid is the lumped system analysis more likely to be applicable? Why?

Layers of 23 -cm-thick meat slabs \((k=0.47 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\left.\alpha=0.13 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) initially at a uniform temperature of \(7^{\circ} \mathrm{C}\) are to be frozen by refrigerated air at \(-30^{\circ} \mathrm{C}\) flowing at a velocity of \(1.4 \mathrm{~m} / \mathrm{s}\). The average heat transfer coefficient between the meat and the air is \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming the size of the meat slabs to be large relative to their thickness, determine how long it will take for the center temperature of the slabs to drop to \(-18^{\circ} \mathrm{C}\). Also, determine the surface temperature of the meat slab at that time.

Consider two identical 4-kg pieces of roast beef. The first piece is baked as a whole, while the second is baked after being cut into two equal pieces in the same oven. Will there be any difference between the cooking times of the whole and cut roasts? Why?
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