Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 73
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The role of k and j in the 0~kj) is the same as in "0A(k), while m represents fluidbulk mean temperature.The other heat transfer results related to the four fundamental solutions can be obtainedusing the following equations in conjunction with Table 5.14 [1]:I)(1)oo"-_ .
( I ) ( l i -)-r , ( i ) l ~ )~----r, t ~ i o(1)0<~!m t - ~ 1 ~ oo)v mo(5.111)(5.112)¢It>¢I~>Nu'(°~)= Nu~) = ~1 - 0~ = ~0~)o(5.113)¢PI~)r*¢l~ )Nui(i) = Nu(°~ = 1 - o<') = o<'!(5.114)vmo0 ~ ) - vmtA(2!=r* [~2o) --(5.115)Jv Am<o 2 ) I1Nu~ ) - o I ? - o<2(5.116)1N_Uoo<2)_-- ~A(2) _(5.117)voo_ (2)A(2)vmo, (2)Nuoi = Nuio = 0(5.1~8)¢ I ? = -ooa'+= 0(5.119)(3)oivm!A(3):" ioA(3!--.'mrt~(3)--O mo --(5.120)11(I)(o4/)- ~ 4 o ) - - r *0(/4)1(5.121)r*.
<4)+ ~= r 0oo- Nu~ )Nu~olo~11Nu~°4°~= Nul4) = 014)---m,A(4"----~- r*Omo(4)1(5.123)r*Nu(°~ = Nu~) = (4------'-7A(4)- A - A<4! - Nu(°~voo(5.122)vmo(5.124)vmlThe direct use of these four fundamental solutions is rare in engineering applications. Thesolutions for practical problems must be developed. The following examples should be ofgreat interest with respect to the application of these fundamental solutions.Uniform Temperature at Both Walls. Wh e n Ti ¢ To, the problem is designated as la, andthe fully developed Nusselt numbers at the two walls are designated as Nu~la) and Nu~ a).
Wh e nTi = To, the problem is designated as lb, and the fully developed Nusselt numbers at the twowalls are designated as Nu~lb) and Nu~ b). These are presented in Fig. 5.13. Tabulated values forthese and the subsequent solutions are available in Shah and L o n d o n [1].A circumferentially averaged Nusselt n u m b e r in the case of Ti = To, designated as NuT, canbe obtained from Nu~1~) and Nu~ ~) by means of the following relation [1]:NUT =Numb) + r* Nu~'b)1 + r*(5.125)5.36C H A P T E R FIVE8.524fRe - - - - - - - - - " ' -8.0Nu H227.0fRe6.0 Nu20r.5.0184.01600.20.40.60.83.51.0r*FIGURE 5.14 Fully developed friction factor and Nusselt numbers for concentric annularducts [2].The NUT values for Eq. 5.125 are presented in Fig. 5.14, in which Nun, described later, andf R e , calculated from Eq.
5.107, are also displayed.Uniform Heat Fluxes at Both Walls. When qT= q~ the problem is designated as 2a, andthe fully developed Nusselt numbers at the two walls are designated as Nu~2a)and Nu(o~). WhenqT~ q~ the problem is designated as 2b and the fully developed Nusselt numbers at the twowalls are denoted Nu~TM and Nu(oTM. These Nusselt numbers are shown in Fig. 5.15. In the caseof qT-- q~, the circumferentially averaged Nusselt number Null can be obtained from Nu]~) andNu(oTMvia Eq. 5.125 by replacing subscript T with H and superscript 1 with 2.
The Null obtainedin this way is shown in Fig. 5.14.It should be noted that the heat flux is positive if the heat transfer occurs from the wall tothe fluid. Therefore, a negative Nusselt number like those shown in Fig. 5.15 signifies that heattransfer has taken place in the opposite direction (i.e., from the fluid to the wall). In bothaforementioned cases, Tw- Tm is considered positive. Therefore, the infinite Nusselt numbersin Figs.
5.13 and 5.15 indicate that Tw = Tm, not infinite heat flux.Uniform Temperature at One Wall and Uniform Heat Flux at the Other The subscripts 1and 2 refer to either the inside or the outside wall. When T1 ~ T2, the problem is known as 4a,and when 7'1 = T2 it is known as 4b. It has been shown by Shah and London [1] thatNu~) = Nu~4~)= Nu~1")(5.126)Nu(o~')= Nu(o4b)= Nu(o1")(5.127)Hydrodynamically Developing Flow.Shah and London [1] summarize the solutions forthe hydrodynamic development of laminar flow in concentric annuli. The apparent frictionfactor in the hydrodynamic entrance region, derived by Shah [103], is expressed as:fapp Re = 3.44(x+) -°5 +K(~)/(4x +) +f R e - 3.44(x+) -°51 + C(x+)-2(5.128)The values of K(oo), f R e , and C in Eq.
5.128 are given in Table 5.15. A very good agreement,within +_3 percent, with the various analytical predictions has been achieved using Eq. 5.128.FORCED CONVECTION,INTERNALFLOW IN DUCTS5.3738tItI30ritItI*='~o20%%~Nu!2a)Nu~2b) %%Nu10.'~Z'Z~"~"~.-_ . .....Nu(2a).0-10Case -_~(Nu~ 2a)~20 "--30....II ~0.00. lIII0.2J I0.3qi"C, qo (2b),,_ ,,q i - - q o (2a)I I I .l I I I I I I I l0.4 0.50.6 0.7 0.8 0.9 1.0r*FIGURE 5.15 Fully developed Nusselt numbers for uniform axial heatfluxes at both walls in concentric annular ducts [1].In addition, the hydrodynamic entrance lengthis given in Table 5.15.Lh~, recommendedby Shah and London [1],Thermally Developing Flow.
The solutions for thermally developing flow in concentricannular ducts under each of the four fundamental thermal boundary conditions are tabulatedin Tables 5.16, 5.17, 5.18, and 5.19. These results have been taken from Shah and London [1].Additional quantities can be determined from the correlations listed at the bottom of eachtable using the data presented.Hydrodynamically Developing FlowParameters and Constants to Use in Conjunctionwith Eq. 5.128 for Concentric Annular Ducts [103]TABLE 5.15r*L~,K(**)00.050.100.500.751.00.05410.02060.01750.01160.01090.01081.2500.8300.7840.6880.6780.674fRe16.00021.56722.34323.81323.96724.000C0.0002120.0000500.0000430.0000320.0000300.0000295.38CHAPTER FIVETABLE 5.16Fundamental Solutions of the First Kind for Thermally Developing Flow in ConcentricA n n u l a r Ducts (compiled from Shah and L o n d o n [1])r*x*(I) ....(1.)•0.020.000010.000050.00010.00050.0010.0050.010.050.10.5..78.557.550.8739.2835.47528.38126.12425.05125.051..0.000010.000050.00010.00050.0010.0050.010.050.10.5..52.035.430.4322.0319.39714.67113.26912.68512.685..0.000010.000050.00010.00050.0010.0050.010.050.10.50.000010.000050.00010.00050.0010.0050.010.050.10.50.050.100.25(1)O x, miNu!X)..
.Nu(1).t~ ....0 ....Nu .(1). . .~~~~0.0432.1332.7482.9482.94851.08129.35023.03312.9349.9935.2723.8811.5370.8350.5010.5010.003030.008760.013800.039300.061340.168480.256640.605400.757340.830060.8300651.23629.60923.35513.46310.6466.3405.2203.8963.4402.9482.9480.002970.008600.013550.038620.060310.165910.252470.593380.731910.789910.7899151.78129.93423.61613.62110.7746.4225.2863.9113.4133.0193.019. . . . .
.(1)(1)..0.00110.00310.005190.018740.033280.112940.151460.169930.16993..78.557.751.1440.0336.69731.99430.78730.17930.179..0.00140.00450.007590.026520.046060.146810.191400.210090.21009..52.135.630.6722.6320.33417.19516.40916.05816.058~~~-0.0542.2412.8413.0193.01951.62729.67623.29613.09510.1255.3563.9511.5900.9150.6340.63480.29049.63240.68226.24921.94913.83312.9189.2278.1997.8177.8170.000430.001290.002100.006620.010940.035420.061310.183820.233880.252560.2525680.32449.69640.76726.42422.19214.34113.76211.30510.70210.45910.459~~~-~~0.0642.3432.9333.0953.09552.18630.01923.57613.27510.2765.4614.0441.6701.0220.7820.7820.002870.008300.013080.037320.058320.160870.245300.574850.700580.747440.7474452.33630.27023.88813.78910.9126.5095.3593.9273.4133.0953.09566.50239.73331.94719.48215.8439.9758.2365.3154.5674.3284.3280.000790.002340.003750.011300.018260.056390.092290.253340.312310.331200.3312066.55539.82732.06719.70416.13810.5719.0737.1186.6416.4716.471m--m~~0.0832.5263.1203.2673.26753.27630.71024.15013.66510.6135.7174.2771.8851.2761.0821.0820.002570.007460.011760.033680.052730.146580.224730.528300.634740.668800.6688053.41430.94024.43814.14111.2046.6995.5173.9963.4943.2673.267Nu(xli)ommmmm0.56719.96227.50030.17930.179nmmmm0.16611.08514.85616.05816.058mmmm0.1557.4919.79210.45910.459mmm0.1304.8446.1416.4716.471FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.39TABLE 5.16 Fundamental Solutions of the First Kind for Thermally Developing Flow in ConcentricAnnular Ducts (compiled from Shah and London [1]) (Continued)x*0(1.....).0.500.000010.000050.00010.00050.0010.0050.010.050.10.5**60.47035.54128.29516.71113.3397.9306.3413.7133.0732.8852.8850.001210.003540.005630.016580.026420.078240.124880.322330.389950.409820.4098260.54335.66828.45516.99313.7018.6037.2465.4805.0374.8894.889------0.0922.7443.3743.5203.52054.61331.58324.88914.19011.0776.0864.6222.2041.6151.4431.4430.002200.006370.010070.028970.045490.127970.197730.471540.563250.590180.5901854.73331.78525.14214.61411.6056.9795.7614.1713.6983.5203.5200.1163.7514.6714.8894.8890.000010.000050.00010.00050.0010.0050.010.050.10.556.80433.04626.14115.10611.8956.7505.2352.7622.1682.0002.0000.001710.004980.007880.022880.036130.103710.162490.399260.477700.500000.5000056.90133.21126.34915.45912.3417.5316.2514.5974.1514.0004.000------0.0643.1123.8354.0004.00056.80433.04626.14115.10611.8956.7505.2352.7622.1682.0002.0000.001710.004980.007880.022880.036130.103710.162490.399260.477700.500000.5000056.90133.21126.34915.45912.3417.5316.2514.5974.1514.0004.0000.0643.1123.8354.0004.0001.0(1)miOx,NuO!.x,.
. . . .Nu(1).r*0 .(1). . . . . .0(1).(I)(')x, oi = -0(~1,)~, N U x o~0(")x,u = 10 (').X, Ol = 0~0).. . . . = _ 0 0x,) m o NuO~x,,oO(x,1) = 1• x, oi,N U(1)........Nu(I!a(l) = 0The thermal entrance lengths for thermally developing flow with these four f u n d a m e n t a lthermal b o u n d a r y conditions are given in Table 5.20.Using the four f u n d a m e n t a l solutions p r e s e n t e d in Tables 5.16--5.19, thermally developingflow with t h e r m a l b o u n d a r y conditions different from the f u n d a m e n t a l conditions p r e s e n t e din the section entitled "Four F u n d a m e n t a l T hermal B o u n d a r y Conditions" can be obtainedby the superposition method. Three examples are detailed in the following sections.U n i f o r m T e m p e r a t u r e s at B o t h Walls.As m e n t i o n e d in the section "Fully D e v e l o p e dFlow," when Ti ~ To, the p r o b l e m is designated as la, and its solution is expressed in terms ofthe following equations [1]:,, = 7', + (Ti - L ) O ~ m i + (To - T* e l~A(1)vx, moT ( x ,TMkqx~!la) -- - Dh[(Ti - T.
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