Steady Heat Conduction 3
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CHAPTER
STEADY HEAT CONDUCTION 3
n heat transfer analysis, we are often interested in the rate of heat transfer through a medium under steady conditions and surface temperatures. Such OBJECTIVES Iproblems can be solved easily without involving any differential equations When you finish studying this chapter, by the introduction of the thermal resistance concept in an analogous manner you should be able to: to electrical circuit problems. In this case, the thermal resistance corresponds ■ Understand the concept of thermal to electrical resistance, temperature difference corresponds to voltage, and the resistance and its limitations, and develop thermal resistance heat transfer rate corresponds to electric current. networks for practical heat We start this chapter with one-dimensional steady heat conduction in a conduction problems, thermal resis- plane wall, a cylinder, and a sphere, and develop relations for ■ Solve steady conduction problems tances in these geometries. We also develop thermal resistance relations for that involve multilayer rectangular, convection and radiation conditions at the boundaries. We apply this concept cylindrical, or spherical geome- to heat conduction problems in multilayer plane walls, cylinders, and spheres tries, and generalize it to systems that involve heat transfer in two or three dimen- ■ Develop an intuitive understand- sions. We also discuss the thermal contact resistance and the overall heat ing of thermal contact resistance, transfer coefficient and develop relations for the critical radius of insulation and circumstances under which it may be significant, for a cylinder and a sphere. Finally, we discuss steady heat transfer from ■ Identify applications in which finned surfaces and some complex geometrics commonly encountered in insulation may actually increase practice through the use of conduction shape factors. heat transfer, ■ Analyze finned surfaces, and as- sess how efficiently and effectively fins enhance heat transfer, and ■ Solve multidimensional practical heat conduction problems using conduction shape factors.
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136 STEADY HEAT CONDUCTION
11°C 3–1■ STEADY HEAT CONDUCTION IN PLANE WALLS 20°C 3°C Consider steady heat conduction through the walls of a house during a winter 11°C day. We know that heat is continuously lost to the outdoors through the wall. 20°C 3°C 3°C We intuitively feel that heat transfer through the wall is in the normal direc- tion to the wall surface, and no significant heat transfer takes place in the wall 11°C 20°C 3°C 3°C 3°C in other directions (Fig. 3–1). Recall that heat transfer in a certain direction is driven by the temperature 11°C gradient in that direction. There is no heat transfer in a direction in which 20°C 3°C 3°C 3°C · there is no change in temperature. Temperature measurements at several loca- Q tions on the inner or outer wall surface will confirm that a wall surface is 11°C 20°C 3°C 3°C 3°C nearly isothermal. That is, the temperatures at the top and bottom of a wall surface as well as at the right and left ends are almost the same. Therefore, 11°C there is no heat transfer through the wall from the top to the bottom, or from 20°C 3°C 3°C 3°C T(x) left to right, but there is considerable temperature difference between the in- A ner and the outer surfaces of the wall, and thus significant heat transfer in the 11°C 20°C 3°C 3°C direction from the inner surface to the outer one. y The small thickness of the wall causes the temperature gradient in that di- 11°C rection to be large. Further, if the air temperatures in and outside the house re- 20°C 3°C x main constant, then heat transfer through the wall of a house can be modeled z as steady and one-dimensional. The temperature of the wall in this case de- FIGURE 3–1 pends on one direction only (say the x-direction) and can be expressed as T(x). Heat transfer through a wall is one- Noting that heat transfer is the only energy interaction involved in this case dimensional when the temperature of and there is no heat generation, the energy balance for the wall can be ex- the wall varies in one direction only. pressed as
Rate of Rate of Rate of change heat transfer heat transfer of the energy into the wall out of the wall of the wall
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