Document Dvm9RX93nDVw3Xy1kb99DjJq5

Table 7. Analytical Solutions fob Heat Conduction in Variously Shaped iff Solids (Concluded) 3? Shape op Solid Boundary Conditions Data Available in Graphs Cylinder of infinite axial di mension The surface temperature ia suddenly changed from the initial (uniform) temperature. Temperature distribution as a func tion of time. Reference: (10) p. 265. i'filfU yl Cylinder of infinite axial di* mansion immersed in a fluid. nU !1J18 Kk` The surface temperature' suddenly begins to increase linearly with time. The surrounding fluid suddenly changes from the initial (uniform) temperature of the cylinder. i The temperature of the surrounding fluid changes sinusoidally. Heat flow from surface as a function of time. Multiply temperature difference be tween surface and fluid by surface conductance. Temperature distribution as a func tion of time.. Reference: (10) p. 269. Temperature distribution as a func tion of time. References: (4) p. 36; (5) pp. V-16, V-W, V-43, V-48; (10) pp. 278, 286; Heat flow from the surface as a func tion of time. References: (5) p. V-16; (10) p. 278. Temperature distribution as a func tion of time. Reference: (5) p. VI-34. Sphere Heat flow from the surface as a func tion of time. Reference: (5) p. VI-36. The temperature of the surface is suddenly changed from the initial uniform temperature. Temperature distribution as a func tion of time.' References: (5) p. V-23; (10) pp. 264, 265. The temperature at the surface sud Temperature distribution as a func- denly begins to change as a linear tion of time. function of time. Reference: (10) p. 269. Sphere immersed in fluid. The temperature of the surrounding fluid suddenly changes from the initial uniform sphere temperature. Temperature distribution as a func tion of time. References: (4) p. 36; (5) pp. V-21, V-35, V-44; (10) pp. 281, 282; (4). Heat flow as a function of time. References: (5) p. V-21; (10) p. 281. Rectangular bar of infinite length. P&rallelopiped (rectangular) Cylinder of finite length. Hollow cylinder of infinite ex terior radius. Any of the above noted boundary conditions for a slab. Temperature distribution as a func tion of time. Combine solutions as indicated in Refs. 16 and 17. Any of the above noted boundary conditions for a slab. Temperature distribution as a func tion of time. Combine solutions as indicated in Refs. 16 and 17. Any of the boundary conditions given above for a cylinder and a slab. Temperature distribution as a func tion of time. Combine solutions as indicated in Ref8. 16 and 17. The temperature of the surface sud denly changes from the initial (uniform) temperature. Temperature distribution as a func tion of time. Combine solutions as indicated in Refs. 16 and 17. Heat flow at the surface as a func tion of time. Reference: (10) p. 267. Consider the slab to be divided, as shown in Fig. 9, by n equidistant r planes parallel to the slab surface and a distance Ax apart. Let the tem- perature of the slab at any plane and any time (9) be denoted by Tx,j. ; Then the temperature of the slab at the two adjacent planes at the same f;: Heat Transfer 109 time will be denoted as T(x+&x,s) and Tix~4x,w. In a similar manner the temperature of the x plane at a time A9 later will be 71(,,,+a>. In accordance with this nomenclature, the temperature at any plane x and time 9 + A9 is given as Tlx, B+AS) T'x+Ax, 9) + T(s-A*,S) 2 (18) which may be interpreted as follows. The temperature of the slab at any plane, x, and any time, 9, is equal to the average temperature of the two adjacent planes obtained at the time (9 -- A9). The time interval A 9 is determined by the equation Ax' as = 2a (19) Omitting the graphical construction at the slab boundaries, reference to Fig. 9 demonstrates the graphical method by means of which the tempera ture at each plane is determined at successive intervals of time in accordance with Equation 18. For the problem stated, the boundary condition at the insulated surface, is specified by the equation 3T dx and at the uninsulated face by the equation MTM-T)=-fc -- ax In terms of finite differences these two equations (employing nomenclature established.by Fig. 9) become ----- =0 or Tr -- Te at x = L Ax and (Tb - Ta) A(T. - 2V) = -k at Ax or Ta - 7V = _ Ta - Te k/h Ax The details of the graphical construction are best obtained by inspection f Fig. 9. Note that the line (0,0,0') used to initiate the graphical con struction, is the only one drawn to the slab boundary A'. The numbered points indicate temperatures at the sub-slab boundaries at 1,2,3, etc., time intervals (A 9) after the slab is exposed to the high temperature. For transient heat flow in two dimensions, and also, for steady state conduction, numerical methods of solution are available in the litera ture.6-10-1219 These numerical methods are applicable to three dimensional problems, although the calculations involved normally become too tedious for most applications of the method. An additional technique of solution for one and two dimensional problems in transient conduction results from