Write a summary of chapter 3 from the SEVENTH EDITION Fundam

Write a summary of chapter 3 from the SEVENTH EDITION

Fundamentals

of Heat

and Mass

Transfer

THEODORE L. BERGMAN

Solution

Answer :

Summary
Despite its inherent mathematical simplicity, one-dimensional, steady-state heat transfer occurs in numerous engineering applications. Although one-dimensional, steady-state conditions may not apply exactly, the assumptions may often be made to obtain results of reasonable accuracy. You should therefore be thoroughly familiar with the means by which such problems are treated. In particular, you should be comfortable with the use of equivalent thermal circuits and with the expressions for the conduction resistances that pertain to each of the three common geometries. You should also be familiar with how the heat equation and Fourier’s law may be used to obtain temperature distributions and the corresponding fluxes. The implications of an internally distributed source of energy should also be clearly understood. In addition, you should appreciate the important role that extended surfaces can play in the design of thermal systems and should have the facility to effect design and performance calculations for such surfaces. Finally, you should understand how the preceding concepts can be applied to analyze heat transfer in the human body, thermoelectric power generation, and micro- and nanoscale conduction.

You may test your understanding of this chapter’s key concepts by addressing the following questions.
• Under what conditions may it be said that the heat flux is a constant, independent of the direction of heat flow? For each of these conditions, use physical considerations to convince yourself that the heat flux would not be independent of direction if the condition were not satisfied.
• For one-dimensional, steady-state conduction in a cylindrical or spherical shell without heat generation, is the radial heat flux independent of radius? Is the radial heat rate independent of radius?
• For one-dimensional, steady-state conduction without heat generation, what is the shape of the temperature distribution in a plane wall? In a cylindrical shell? In a spherical shell?
• What is the thermal resistance? How is it defined? What are its units?
• For conduction across a plane wall, can you write the expression for the thermal resistance from memory? Similarly, can you write expressions for the thermal resistance associated with conduction across cylindrical and spherical shells? From memory, can you express the thermal resistances associated with convection from a surface and net radiation exchange between the surface and large surroundings?
• What is the physical basis for existence of a critical insulation radius? How do the thermal conductivity and the convection coefficient affect its value?
• How is the conduction resistance of a solid affected by its thermal conductivity? How is the convection resistance at a surface affected by the convection coefficient? How is the radiation resistance affected by the surface emissivity?
• If heat is transferred from a surface by convection and radiation, how are the corresponding thermal resistances represented in a circuit?
• Consider steady-state conduction through a plane wall separating fluids of different temperatures, T,i and T,o, adjoining the inner and outer surfaces, respectively. If the convection coefficient at the outer surface is five times larger than that at the inner surface, ho 5hi, what can you say about relative proximity of the corresponding surface temperatures, Ts,o and Ts,i, to their adjoining fluid temperatures?
• Can a thermal conduction resistance be applied to a solid cylinder or sphere?
• What is a contact resistance? How is it defined? What are its units for an interface of prescribed area? What are they for a unit area?
• How is the contact resistance affected by the roughness of adjoining surfaces?
• If the air in the contact region between two surfaces is replaced by helium, how is the thermal contact resistance affected? How is it affected if the region is evacuated?
• What is the overall heat transfer coefficient ? How is it defined, and how is it related to the total thermal resistance? What are its units?
• In a solid circular cylinder experiencing uniform volumetric heating and convection heat transfer from its surface, how does the heat flux vary with radius? How does the heat rate vary with radius? In a solid circular sphere experiencing uniform volumetric heating and convection heat transfer from its surface, how does the heat flux vary with radius? How does the heat rate vary with radius?

• Is it possible to achieve steady-state conditions in a solid cylinder or sphere that is experiencing heat generation and whose surface is perfectly insulated? Explain.
• Can a material experiencing heat generation be represented by a thermal resistance and included in a circuit analysis? If so, why? If not, why not?
• What is the physical mechanism associated with cooking in a microwave oven? How do conditions differ from a conventional (convection or radiant) oven?
• If radiation is incident on the surface of a semitransparent medium and is absorbed as it propagates through the medium, will the corresponding volumetric rate of heat generation be distributed uniformly in the medium? If not, how will vary with distance from the surface?
• In what way is a plane wall that is of thickness 2L and experiences uniform volumetric heating and equivalent convection conditions at both surfaces similar to a plane wall that is of thickness L and experiences the same volumetric heating and convection conditions at one surface but whose opposite surface is well insulated?
• What purpose is served by attaching fins to a surface?
• In the derivation of the general form of the energy equation for an extended surface, why is the assumption of one-dimensional conduction an approximation? Under what conditions is it a good approximation?
• Consider a straight fin of uniform cross section (Figure 3.15a). For an x-location in the fin, sketch the temperature distribution in the transverse (y-) direction, placing the origin of the coordinate at the midplane of the fin (t/2 y t/2). What is the form of a surface energy balance applied at the location (x, t/2)?
• What is the fin effectiveness ? What is its range of possible values? Under what conditions are fins most effective?
• What is the fin efficiency ? What is its range of possible values? Under what conditions will the efficiency be large?
• What is the fin resistance ? What are its units?
• How are the effectiveness, efficiency, and thermal resistance of a fin affected if its thermal conductivity is increased? If the convection coefficient is increased? If the length of the fin is increased? If the thickness (or diameter) of the fin is increased?
• Heat is transferred from hot water flowing through a tube to air flowing over the tube. To enhance the rate of heat transfer, should fins be installed on the tube interior or exterior surface?
• A fin may be manufactured as an integral part of a surface by using a casting or extrusion process, or it may be separately brazed or adhered to the surface. From thermal considerations, which option is preferred?
• Describe the physical origins of the two heat source terms in the bioheat equation. Under what conditions is the perfusion term a heat sink?
• How do heat sinks increase the electric power generated by a thermoelectric device?
• Under what conditions do thermal resistances associated with molecule–wall interactions become important?