Document QgJgj0MEJ3D70NRdVDKeO85Q8

114 CHAPTER 6 1951 Guide! Table 7.. Analytical Solutions fob Heat Conduction in Vabiouslt.'Sbabk>1 . Solids (Concluded). Shape of Solid . Boundart Conditions Data Available m Graphs Cylinder of infinite axial dimension il > u Cylinder of mension iimnfmin.iiutBeadxiainl d: ia- fluid. The surfacetemperature is suddenly changed from the initial (uniform) temperature. 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 fm,.. tion of time. Reference: (9) p. 265. -Heat flow from surface as a funetifJ of time. Multiply temperature difference h*. tween surface and fluid by suifan Conductance. Temperature distribution as a txmZ tion of time Reference: (9) p. 269..!. Temperature distribution as a fuaZ tion of time. References: (4) p. 36: (5) pp. v.u V^5, V-43. -48; (9*) pb- &8,2$ y' Ri. a r< He Thfleutiedmcpnernapture of the surrounding Heat flow from the surface as a fm*J tion of time. References: (5) p. V-16; (9) p.278. Temperature distribution as a fun* tion of time. Reference: (5) p. VI-34. Sphere Sphere immeieed in fluid. Heat flow from the surface as a fun* tionof time. Reference: (5) p. VI-36. The. temperature of Hie surface is suddenly changed from the Temperature distribution as a func tion of time. uniform-temperature. References: (5) p. V-23; (9) pp. 264. 265.- Tie temperature at the surface sud denly begins to change as a Temperature distribution as a fun* tion of time. . function of time. Reference: (9) p. 269. The temperature of the surrounding Temperature distribution as a func fluid suddenly changes from the tion of time. -- initial uniform spheretemperature. References: (4) p. 86: (6) pp. V-21 V-35, V-44; (9) pp. 281. 2*7 (U) ' Heat flow as a function of time. References: (5) p. V-21; (9) p. 281. Rectangular bar of infinite , length. Any of the above noted boundary conditions for a slab. Parallelepiped (rectangular). ' Any of the above noted boundary conditions for a slab. Cylinder of finite length* Any of the boundary conditions given above for a cylinder and a slab. Hollow cylinder of infinite,ex terior radius. The temperature of the surfaoe sud denly changes from the initial (uniform) temperature. Temperature distribution as a func tion of time. CoRmefbBin. 1e5 sflo.nlHuti1o6n. s as indicated in Temperature distribution as a func tion oi time. Combine solutions as indicated in RefB. 15 and'16. Temperature distribution, as a fun* tion of time. Combine solutions as. indicted in Refs. 15 and 16, Temperature distribution as a fun* tion of time. Combine solutions as indicated In Refs. 15 and 16. Heat flow at the surface as a fun* * tion of time. Reference: (9) 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 At apart. Let the tem perature of the slab at any plane and any time (6) be denoted by Then the temperature of the slab at the two adjacent planes at the same /fHe't +a,&t*TransfeVr- - ;; :;k :iT5 itime will' be 'denoted:as'21(+'Ai,9)i:aiid Tci-ai.sj.' . Ih a similar manner the taeenmmIdnpptieeamrrcaaecttouuOrrrdeei+aoonAffcdetthni.seewgixxitvheppttnilaahhauniissseenna2SooUtmm- aOeretnnUimcciu.llaaeuttuuAurrvSee',,latthhteeer ttwfpem^millpnbpeerrnaTftu^'r7ei+I JaAtstt*.,i'a' n:y pl.ane x "T'-i, ^y^ Q****-*) +T(X-Ax.9 2 . (18) Wwhhiiccnh may bwe= interpr_e_t_e_d__a_s__fo_l_lo_ws. The temperature of the slab at any Diane 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 AS is determined 'bythe equation -6B > a (19) Omitting the graphical construction at the slab boundaries reference to Fig. 9 demonstrates the graphical method by means of which the temnera turn at each plane is determined at successive intervals of time in accordance with Equation 18. . r, For the^problem stated, the boundary condition at the insulated surface is specified by the equation ; a-*' = o 3x and at the uninsulated face by the equation ; dT In terms of finite differences these .two equations (employing nomenclature established by Fig. 9) become . >_ 0 or Ty ==. Ty, at x L Ax and Mr. - TaO = -k(rB . . A. at . x = 0 T,, - Ti.' _ k/h f Ta --Tb 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 l,2,3, etc., time intervals (A 0) 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.6'8-11.18 These numerical methods'are applicable to three dimensional problems, although the calculations, involved normally become too tediousfor most applications of the method. Ah additional technique of solution for one and two dimensional problems in transient conduction results from /7