Document 71BavazZ3QQG2qmw4REbpv6L8

108 4-t. CHAPTER S 1958 Guide ? ^gaaar-*" V""LI 1 Shape or Soup I Boundary Conditions Cylinder of infinite axial di mension The surface temperature is suddenly changed from the initial (uniform) temperature. Data Available in Graphs Temperature distribution as a func tion of time. Reference: (10) p. 265. Heat flow from surface as a function of time. Multiply temperature difference be tween surface and fluid by surface conductance. Cylinder of infinite axial di mension immersed in a fluid. The surfaoe temperature suddenly begins to increase bnearly with time. The surrounding fluid suddenly changes from the initial (uniform) temperature of the cylinder. Temperature distribution as a func tion of time. Reference: (10) p. 269. Temperature distribution as a func tion of time. References: (4) p. 56; (5) pp. V-16, V-35, V-43, V-48; (10) pp. 278, 286; (15). ___________ Heat flow from the surface as a funcnSenSsj p. V-16; (ID) p. 278. The temperature of the suiTOunding fluid changes sinusoidally. Temperature distribution as a func tion of time. Reference: (5) p. VI-34. _____ 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 denly begins to change aa a linear function of time. Temperature distribution as a func tion of time. Reference: (10) p. 269. 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-35r 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. {Temperature distribution as a func Any of the above noted boundary conditions for a slab. tion of time. Combine solutions os indicated in Refs. 16 and 17. Parallelopiped (rectangular) 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. Cylinder of finite length. Any of the boundary conditions given above for a cylinder and a Temperature distribution as a func tion of time. Combine solutions as indicated in slab. Refs. 16 and 17. Hollow cylinder of infinite ex terior radius- 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 surfaoe as a func tion of time. Reference: (10) p. 267. Consider the slab to be divided, as shown in Fig. 9, by n equidistant planes parallel to the slab surface and a distance Ax apart. Let the tem perature of the slab at any plane and any time (0) be denoted by Tx.sThen the temperature of the slab at the two adjacent planes at the same ' Heat Transfer 109 time will be denoted as T(x+Az.n an<3 T(x-&x,s). In a similar manner the temperature of the x plane at a time AS later will be Tlz, In accordance with this nomenclature, the temperature at any plane x and time 0 + A0 is given as m T(x+&x,d) + T(x--Az,$) T{x,6+A6) = -------------- ---------------- (18) which may be interpreted as follows. The temperature of the slab at any plane, x, and any time, 0, is equal to the average temperature of the two adjacent planes obtained at the time (0 -- A0). The time interval A 8 is determined, by the equation Ax* A9 = -- 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 and at the uninsulated face by the equation ST MT. - T) = -k -- OX In terms of finite differences these two equations (employing nomenclature established by Fig. 9) become and 7V -- 2V ------- -- = 0 or TV = TE at x = h Ax h{T,, - TV) = -k {T~ ^ -A> at x = 0 or T, - Ta-____ Ta - Tb k/h Ax The details of the graphical construction are best obtained by inspection of 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 6) 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 literature.M0-J2.i9 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