Document 7R549yb2VEXOmkvb985j4eyOR

106 CHAPTER 5 1953 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 classes: periodic and transient. Periodic heat transfer repeats periodically in time. Transient heat transfer exhibits no periodicity. Graphical, ana lytical and numerical methods are available for solving transient or periodic heat flow problems.4'6'9-10-11 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 Fio. 9. Example of a Graphical Solution to a Problem in Transient Heat Conduction. summary of the cases reported and tabulated. Many more analytical solutions are available in the form of infinite series,10-11-17-18 but are not tabulated. Certain complex cases may be treated by combining the simple analytical solutions as discussed in Reference 16. (See also Reference 19). 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 called the Schmidt method.4-9-18-20-21 This method will be briefly outlined for the case of transient heat flow in a slab insulated on one face, and suddenly exposed on the other face through a fixed thermal resistance to'a higher temperature: The technique is general, however, and methods may be devised for any boundary conditions,6-20 and also, for one Himpnsinmil (radial) heat flow in spheres and cylinders.21 -a .Heat Transfer 107 Table 7 Analytical Solutions for Heat Conduction in Variously Shaped AAB ` Solids Shape of Solid ` Boundary Conditions Data Available in G barbs Semi-infinite . Surface temperature changed. suddenly Temperature distribution in solid as a function of time. References: (4)p. 37, (6) p. V-28; (9) p. 264; (12) p. 46.. Heat flow from surface'as a function of time. ts- References: (9) pp.256, 267; (12) p.47. A steady flow of heat- is suddenly ap plied to the surface.' Temperature'distribution as a func tion of time. Reference: (9) p. 257. The surface temperature has been Temperature distribution as a func varying sinusoidally with time for . tion of time. a long time. .Reference: (9) p. 296.-. Heat flow from surface as a function of time. Reference: (9) p. 296. Semi-infinite' with fluid at free surface. The temperature of the fluid in con tact with the surface has a sudden change in temperature. (The sur face conductance is constant). Temperature distribution as a func tion of time. References: (4) p. 37; (5) pp. V-45,46, 47. The temperature of the fluid in con tact with the surface has been varyi ng sinusoidally with ti me for a long time. (The surface con ductance is constant). Temperature distribution as a func tion of time. Reference: (9) p. 298. Heat flow from the surface as a func tion of time. Reference: (9) p. 298. Slab. The temperatures fi and t* are sud -. denly changed from the initial uni-' form slab temperature to a new . temperature. (The case where the surface on one side is insulated is _ treated by taking the case of aslab of twice the given thickness since the midplane has no heat flow due to symmetry). Temperature distribution as a func tion of time. References: (5) p. V-12; (9) p. 265. The temperature 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: (9) p. 268. The temperature at both surfaces has been varying sinusoidally fora long time. Temperature distribution as a func tion of time. Reference: (9) p. 300. Heat flow from the surface. Reference: (9) p. 303. Slab immersed in a fluid with constant conductance be tween fluid and slab surface. The temperature of the fluid is sud denly changed from the initial uni form slab temperature. (If one surface is insulated 'against heat flow, see above). Temperature distribution as a func tion of time. References: (4) pp. 32, 33, 34, 35; (5) pp. V-9, 10, 35, 42; (9) pp. 274, 284; (12) p. 106. %% l>c 2 H^at flow from the surface as a func tion of time. References: (5) p. V-10; (9) p. 274; ri2) p 107. The temperature of the fluid at one surface varies as a periodic func tion of time while the temperature of the fluid at the other surface is constant. The conductances need not be the same on both sides. (The variations in temperature are expressible as a Fourier series). Temperature distribution as a func tion of time. Reference 13. Heat flow at the surface as a function of time. Reference 13.