2 02 02 where the constants a1 and 1 are functions of

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Unformatted text preview: T Ti T 2 A1e 1 J0( 1r/ro), T(r, t) T Ti T 2 sin( 1r /ro) , A1e 1 1r /ro 0.2 0.2 0.2 where the constants A1 and 1 are functions of the Bi number only, and their values are listed in Table 4–1 against the Bi number for all three geometries. The error involved in oneterm solutions is less than 2 percent when 0.2. Using the one-term solutions, the fractional heat transfers in different geometries are expressed as Plane wall: Cylinder: Sphere: Q Qmax wall Q Qmax cyl Q Qmax sin 1 0, wall 1 2 1 3 1 1 J1( 1) 0, cyl 1 cos 1 0, sph 1 3 1 sph erfc erfc x 2 exp t h x 2 t T(x, t) Ti Ts Ti erfc x 2 (Ts t constant) Using a clever superposition principle called the product solution these charts can also be used to construct solutions for the two-dimensional transient heat conduction problems encountered in geometries such as a short cylinder, a long rectangular bar, or a semi-infinite cylinder or plate, and even three-dimensional problems associated with geometries such as a rectangular prism or a semi-infinite rectangular bar, provided that all surfaces of the solid are subjected to convection to the same fluid at temperature T , with the same convection heat transfer coefficient h, and the body involves no heat generation. The solution in such multidimensional geometries can be expressed as the product of the solutions for the one-dimensional geometries whose intersection is the multidimensional geometry. The total heat transfer to or from a multidimensional geometry can also be determined by using the one-dimensional values. The transient heat transfer for a two-dimensional geometry formed by the intersection of two one-dimensional geometries 1 and 2 is 1 sin The analytic solution for one-dimensional transient heat conduction in a semi-infinite solid subjected to convection is given by T(x, t) Ti T Ti where the quantity erfc ( ) is the complementary error function. For the special case of h → , the surface temperature Ts becomes equal to the fluid temperature T , and the above equation reduces to hx k t h2 t k2 k Q Qmax total, 2D Q Qmax Q Qmax 1 12 Q Qmax 1 Transient heat transfer for a three-dimensional body formed by the intersection of three one-dimensional bodies 1, 2, and 3 is given by Q Qmax total, 3D Q Qmax 1 Q Qmax 3 Q Q 1Qmax 2 Qmax 1 Q Q 11Qmax 1 Qmax 2 REFERENCES AND SUGGESTED READING 1. ASHRAE. Handbook of Fundamentals. SI version. Atlanta, GA: American Society of Heating, Refrigerating, and Air-Conditioning Engineers, Inc., 1993. 2. ASHRAE. Handbook of Fundamentals. SI version. Atlanta, GA: American Society of Heating, Refrigerating, and Air-Conditioning Engineers, Inc., 1994. 3. H. S. Carslaw and J. C. Jaeger. Conduction of Heat in Solids. 2nd ed. London: Oxford University Press, 1959. 4. H. Gröber, S. Erk, and U. Grigull. Fundamentals of Heat Transfer. New York: McGraw-Hill, 1961. 5. M. P. Heisler. “Temperature Charts for Induction and Constant Temperature Heating.” ASME Transactions 69 (1947), pp. 227–36. 6. H. Hillman. Kitchen Science. Mount Vernon, NY: Consumers Union, 1981. 7. F. P. Incropera and D. P. DeWitt. Introduction to Heat Transfer. 4th ed. New York: John Wiley & Sons, 2002. cen58933_ch04.qxd 9/10/2002 9:13 AM Page 252 252 HEAT TRANSFER 8. L. S. Langston. “Heat Transfer from Multidimensional Objects Using One-Dimensional Solutions for Heat Loss.” International Journal of Heat and Mass Transfer 25 (1982), pp. 149–50. 9. M. N. Özisik, Heat Transfer—A Basic Approach. New , York: McGraw-Hill, 1985. 10. P. J. Schneider. Conduction Heat Transfer. Reading, MA: Addison-Wesley, 1955. 11. L. van der Berg and C. P. Lentz. “Factors Affecting Freezing Rate and Appearance of Eviscerated Poultry Frozen in Air.” Food Technology 12 (1958). PROBLEMS* Lumped System Analysis 4–1C What is lumped system analysis? When is it applicable? 4–2C Consider heat transfer between two identical hot solid bodies and the air surrounding them. The first solid is being cooled by a fan while the second one is allowed to cool naturally. For which solid is the lumped system analysis more likely to be applicable? Why? 4–3C 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? 4–4C Consider a hot baked potato on a plate. The temperature of the potato is observed to drop by 4°C during the first minute. Will the temperature drop during the second minute be less than, equal to, or more than 4°C? Why? Cool air Hot baked potato FIGURE P4–4C 4–5C Consider a potato being baked in an oven that is maintained at a constant temperature. The temperature of the potato is observed to rise by 5°C during the first minute. Will the temperature rise during the second minute be less than, equal to, or more than 5°C? Why? 4–6C What is the physical significance of the Biot number? Is the Biot number mor...
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This note was uploaded on 01/28/2010 for the course HEAT ENG taught by Professor Ghaz during the Spring '10 term at University of Guelph.

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