Document wgRDe9aE7LkYEgGy1zO9qD41D

CHAPTER 5 1954'Guide 108 Solutions fob Heat Conduction in Variously Shaped Table 7. Analytical c---------- Shape of Solid Cylinder of infinite axial di mension Boundary Conditions The surface temperature is suddenly changed from the initial (uniform; Data Available in Graphs Temperature distribution as a *unetion 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 surface temperature suddenly begins to increase linearly with time. The surrounding fluid suddenly changes from the initial (uniform) temperature of the cylinder. Temperature distribution as a function of time. --- ----Reference: (10) p. 269. Temperature distribution as a func tion of time. 4 ,T References: (4) p. 36; (5> pp. V-16, V-35, V-43, V-48; (10) pp. 278, 286; (15). 1 Heat flow from the surface as a func- tion of time. ,' References: (5) p. V-16; (10). p. 278. Temperature distribution as a func- The temperature of the surrounding fluid changes sinusoidally. tion of time. ,_ Reference: (5) p. Vi-34. . % i 5. A ' i. St ^ v a % sphere Sphere immersed in fluid. Qz 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 as a linear Temperature distribution as a func tion of time. Temperature distribution as a func- The temperature of the surrounding fluid suddenly changes from the ipitiftl uniform sphere temperature- tion of time. , References: (4) p. 36; (5) PP* Y."?'* V-35, V-44; (10) pp. 281, 282; (4). Heat flow as a function of time. References: (5) p- V-21; (10) p. 281. V Lv *' 5 ^ Rectangular bar of infinite length. Any of the above noted boundary conditions for a slab. Temperature distribution as a func- tion of time. . _. , , . Combine solutions as indicated m . Refs. 16 and 17. t > a Parallelepiped (rectangular) Any of the above noted boundary conditions for a slab. Temperature distribution as a tunc- tion of time. Combine solutions as indicated m Refs. 16 arid 17. ^ V? .y. Cylinder of finite length. Hollow cylinder of infinite ex terior radius. Any of the boundary conditions i given above for a cylinder and a slab. 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 m Refs-16 and 17. ---------------------------------:-------- r~ Temperature distribution as a func- tion of time. . ` ... Combine solutions as indicated m Refs. 16 and 17. , *S'v ^ f* Heat flow at the surface as a func- tion of time. ___ Reference: (10) p- 267. /;> ---------------- -------------------------------- % ilaCnoensspidaerraltlheel toslathbetoslabbe sduivrfiadceeda, nads ashdoiswtnanicneFAisg.a9p,arbt.y nLeeqt uthiaeistieamni-- ff ierature of. the slab at any plane and any time (0) be denoted by Txj. % rhen the temperature of the slab at the two adjacent planes at the same f Heat Transfer 109 time will be denoted as Tlx+Ax,o) and T(x^Ax,t). ' In a similar manner the temperature of the x plane at a time A0 later will be T(x,e+eA). In accordance with this nomenclature, the temperature at any plane x and time 8 + A0 is given as m T(x+Ax.e) Tix--Ax,o) id.s+AB) = --------------- --------------- (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 A0 is determined by the equation A* A0 = -- (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 =0 dx and at the uninsulated face by the equation - T) = -k^- ox In terms of finite differences these two equations (employing nomenclature established by Fig. 9) become Tf -------- - = 0 or Tr = Tt, at, x = L Air. and h(T,, - TV) = -k (Tb ~ Th- at x = 0 Ax or - 7V = _ T'a - 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 (A0) 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.6-10'12'!9 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