Document eZGQxRMdVXn91ORGpZeV7qG

106 CHAPTER 5 1958 Guide-' PERIODIC AND TRANSIENT HEAT FLOW -, The foregoing data and examples dealt with steady-state heat transfer , (not varying with time). In most practical heat transfer problems the heat /, flow depends upon time. Such cases can usually be divided into two i classes: periodic and transient. Periodic heat transfer repeats periodically.^ in'tinie.' Transient heat transfer-exhibits no periodicity. Graphical, ana-,-) lyrical and numerical methods are available for solving transient or periodic^ heat flow problems.4-6-10-11-12. Graphical and numerical methods are the : most versatile, and can be applied with minimum mathematical training. A large number of analytical solutions for the case of heat conduction in variously shaped solids are available in the literature. Table 7 gives a/- Example of a Graphical Solution to a Problem in Transient Heat Fiq. 9. Conduction. summary of the cases reported and tabulated. Many more analytical solutions are available in the form of infinite series,n-12-1MS but are not tabulated. Certain complex cases may be treated by combining the simple, analytical solutions as discussed in Reference 17. (See also Reference 20). Frequently, transient heat flow problems in one dimension have boundary') conditions which make the problem difficult to treat, analytically. In such, , cases, recourse may be made to a graphical method of solution sometimes j called the Schmidt method.4-10-18-21-?2 This method will be briefly outlined:; for the case of transient heat flow in a slab insulated on one face, and suddenly exposed on theiother face through-a fixed thermal resistance to ai higher temperature. The technique is general, however, and methods may? be devised-for any boundary conditions,6-21 and also; for one dimensional (radial) heat flow in spheres and cylinders.22 23 ' .- Heat-Transfer 107 Table 7. Analytical Solutions for Heat Conduction in Variously Shaped Solids Shape op Solid Semi-infinite Boundary Conditions Data Available m Graphs Surface temperature changed. suddenly Temperature distribution in solid as a function of time. "(W)^4P6.37'(5)PV-28^10) Heat flow from surface as a function of time. References: (10) pp.266,267; (13) p.47 A steady flow of heat is suddenly ap plied to the surface. Temperature distribution as a func tion of time. Reference: (10) p. 257. x-- The surface temperature has been . varying sinusoidally with tim^ for a long time. * Temperature distribution as a func tion of time.' Reference: - (10) p. 298. Semi-infinite with fluid 'aat free surface. The temperature of the fluid in con tact with the surface has a sudden change in temperature. (Thesurface conductance is constant). Heat flow from surface as a function of time. Reference: (10) p. 296. Temperature distribution as a func tion of time. References: (4) p. 37; (5) pp. V-45,46, % *-c Slab. The temperature of the fluid in con tact with the surface has been varying sinusoidally with for a long time. (Tim surface con ductance is oonstant.) The temperatures fi and tt are sud denly changed from the initial unifprm slab temperature to a new temperature. (The case where the surface on one side is insulated is treated by taking the case of a slab of twice the given thickness since the midplane has no heat flow due to symmetry.) Temperature distribution as a func tion of time. Reference: (10) p. 298. Heat flow from the surface as a func tion of time. Reference: (10) p. 298. Temperature distribution as a func tion of time. References: (6) p. V-12; (10) p. 265. The temperatures ti and ft suddenly begin to increase as linear func tions of time. The slab is in itially at uniform temperature. (The case where one surface is insulated against heat flow is treated as noted above.) Temperature distribution as a func tion of time. Reference: (10) p. 268. The temperature at both surfaces has been varying sinusoidally for a long tune. Temperature distribution as a func tion of time. Reference: (10) p. 300. Sl"b,,i,""er"d ip fluid with constant conductance between fluid and slab surfaSl % be. b%e. The temperature of the fluid is sud denly changed from the initial uniforaa slab temperature. (If one surface is insulated against heat flow, see above.) Heat flow from the surface. Reference: (10) p. 303. Temperature distribution as a func tion of time. References: (4) pp. 32, 33, 34, 35; (5) PP-, V-9. 10.35,42; (10) pp. 274, 284 Uo) p. 106. Heat flow from the surface as a func tion of time. References: (5) p. V-10; (10) p. 274: (13) p 107. * The temperature of the fluid at one surface varies as a periodic func- tion of time while the temperature Temperature distribution as a func - tion of time. Reference 14. of the fluid at the other surface is constant. The conductances need not be the same on both sides. (The variations in temperature are Heat flow at the surface as a function of time. Reference 14. expressible as a Fourier series.)