Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 87
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5.337 agree well (within +1 percent) with the tabulatedvalues in Table 5.56.Circular Sector DuctsFIGURE 5.54 A circular sector duct.A schematic drawing of a circular sector duct is presented inFig. 5.54. The fully developed f Re and Nu for circular sectorducts have been obtained by Eckert and Irvine [280], Sparrow and Haji-Sheikh [174], Hu and Chang [265], and BenAli et al.
[281]. The results are summarized in Table 5.57.Soliman et al. [282] numerically analyzed the problemof laminar flow development in circular sector ducts. Theirfapp Re and L~,y results for 2~ = 11.25, 22.5, 45, and 90 ° arepresented in Table 5.58. Furthermore, the hydrodynamicentrance lengths are 0.235, 0.144, 0.108, and 0.0786 forhydrodynamically developing flow in circular sector ductswith 2~ = 11.25, 22.5, 45, and 90 °, respectively.Circular Segment Ducts\\/I\\\\FIGURE 5.55 A circular segment duct.A circular segment duct is depicted in Fig.
5.55. The fullydeveloped flow and heat transfer characteristics obtained bySparrow and Haji-Sheikh [283] are given in Table 5.59.Hong and Bergles [284] have analyzed the thermalentrance solution of heat transfer for a circular segment ductwith 2~ = 180 ° (i.e., a semicircular duct). Two kinds of thermal boundary conditions are used: (1) a constant wall heatflux along the axial flow direction with a constant wall temperature along the duct circumference, and (2) a constantwall heat flux along the axial flow direction and a constantwall temperature along the semicircular arc, with zero heatFORCED CONVECTION, INTERNAL FLOW IN DUCTSTABLE 5.575.109Fully Developed f R e , K(oo), and Nu for Laminar Flow in a Circular Sector Duct2t~f ReK(oo)NUll1NUll2NUT2t~f ReK(oo)Null1NUll2NUT05810152012.00012.3312.41112.50412.72812.982.971-2.480~2.235~2.0592.2452.384~2.6192.742-0.018-0.0810.1950.354809010012015016014.59214.7914.92915.20015.5415.6111.5301.423~1.6921.9012.0733.6713.7303.8063.9063.9994.04-2.984-2.8982.995~m3.0603036404550607213.31013.51013.63513.78213.9514.17114.4351.855-~1.657~1.580~3.005~-3.273.3373.479--0.8381.1741.4001.6671.9902.4212.6082.341-2.543~2.702.822~18021024027030033035015.76715.9816.1516.2916.4216.5416.624.0894.1274.1714.2084.2444.2804.3042.9232.8712.8212.7812.7492.7232.708TABLE 5.581.5041.4881.4681.4633.1913.2683.3473.3703.3893.4073.4273.443Flow Parameters for Hydrodynamically Developing Flow in Circular Sector Ducts [282],,x÷2¢~ = 11.25 °22.5 °45 °90 °Lhyfapp ReK(x)fapp ReK(x)fapo ReK(x)fapp ReK(x)0.0010.0030.0060.0100.0200.0300.0400.0500.0700.1000.1500.2000.2500.3000.3500.4000.4500.5000.6000.7000.8000.9001.000109.366.0948.2739.5228.9325.0922.7021.1819.9017.3415.8214.9514.4714.0713.9313.5913.5113.3513.1913.0312.9512.9012.870.2070.3350.4560.5380.7580.8870.9911.0761.2111.3641.5401.6641.7561.8281.8851.9301.9701.9972.0512.0912.1182.1392.156147.981.6260.1848.3035.6430.2027.0224.9222.2019.9117.8116.6515.8916.3514.9514.6514.4114.2213.9513.7613.6313.5213.470.1770.2810.3770.4690.6280.7390.8300.9011.0191.1531.3131.4291.5171.5881.6461.6951.7351.7701.8251.8681.9021.9301.951180.498.2763.6953.9139.5933.4729.8927.5124.4121.7819.4018.0517.0916.5416.0615.6615.3415.1114.7914.5514.3914.2314.150.1710.2660.3520.4320.5670.6620.7380.8010.9031.0211.1651.2691.3501.4121.4641.5101.5461.5771.6251.6611.6891.7101.728226.0115.778.2960.2643.9337.3733.5030.8827.4424.4021.5819.9518.8718.0717.5117.0516.7016.4115.9615.6715.4615.3115.190.1540.2410.3140.3800.4920.5710.6350.6890.7780.8811.0071.1001.1711.2291.2751.3151.3471.3751.4181.4511.4751.4941.509TABLE 5.59The f R e , K(oo),NUll1, and NUll2for Fully Developed Laminar Flow in Circular Segment Ducts [283]2~f ReK(oo)NUll1NUll22~f ReK(oo)NUll1NUll20102040608015.55515.55815.56015.57515.59815.6271.7401.7391.7341.7151.6861.6503.5803.6083.6163.6483.6963.75600.0130.0520.2071.4560.78512018024030036015.69015.76715.84015.91516.0001.5711.4631.3851.3411.3333.8944.0894.2284.3284.3641.6082.9233.8824.2964.3645.110CHAPTER FIVETABLE 5.60 Local Nusselt Numbers in the Thermal Entrance Regionof a Semicircular Duct [284]NUx.H1NUx,H2x*BC1BC2x*BC1BC20.0004580.0009540.001490.002080.0027117.7113.7211.8010.559.60517.4313.4111.3710.089.1410.02790.03510.04420.05520.06864.7674.5624.4294.2764.2174.3394.0373.8303.6863.5430.003750.004930.006270.007777.4757.7237.1376.5568.1277.3756.7886.3120.08490.1050.1300.1594.1564.1244.1184.1083.4253.3303.2653.2080.009460.01280.01680.02176.3005.8215.3965.0775.9125.3684.9354.5790.1960.2410.261oomm3.1713.1613.1603.1604.089flux along the diameter.
The local Nusselt n u m b e r s for these two b o u n d a r y conditions arep r e s e n t e d in Table 5.60. The terms BC1 and BC2 in Table 5.60 refer to the previously mentioned first and second b o u n d a r y conditions.Annular Sector DuctsA n annular sector duct is displayed in the inset of Fig. 5.56.
Shah and L o n d o n [1] have calculated the f R e value to a high d e g r e e of accuracy using the analytical solution p r o p o s e d byS p a r r o w et al. [285]. The results of f Re are p r e s e n t e d in Fig. 5.56.24.023.0r*r,22.021.0ro - ~24'=35~20.019.0fRe18.017.016.015.014.00.00.10.20.30.40.50.60.70.80.91.0r*FIGURE 5.56 Fully developed friction factor for laminarflow in an annular sector duct [1].FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.111Schenkel [279] has developed the following approximate equation for f Re in an annularsecular duct:24[0.63 (1 - r*)][1 (l-r*)] 2(5.341)f R e = 1----~-- l + r .1+-~ l + r *This equation is valid for ~ > ~min(r*).
The values of ~min for r* = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7,0.8, and 0.9 are 60 °, 50 °, 42 °, 35 °, 28.5 °, 22.5 °, 17.5°, 13°, 8.5 °, and 4 °, respectively. For ~ >~)min(r*), the predictions of Eq. 5.341 are in excellent accord with the results presented inFig. 5.56.Ben-Ali et al. [281] and Soliman [286] have investigated fully developed flow and heattransfer in annular sector ducts with the 0), (~, and @ boundary conditions. The Nusselt numbers obtained by those investigators can be found in Table 5.61.Simultaneously developing flow in annular sector ducts for air (Pr = 0.7) has been analyzedby Renzoni and Prakash [287].
In their analysis, the outer curved wall is treated as adiabatic,and the ~ boundary condition is imposed on the inner curved wall as well as on the twostraight walls of the sector. The fully developed friction factors, incremental pressure dropnumbers, hydrodynamic entrance lengths, and thermal entrance lengths are presented inTable 5.62. The term L~y used in Table 5.62 is defined as the dimensionless axial distance atwhich fapp Re = 1.05f Re.
The fully developed Nusselt numbers are represented by NU/d inorder not to confuse the reader since the thermal boundary condition applied in Renzoni andPrakash [287] is different from those defined in the section.Stadium-Shaped DuctsA stadium-shaped duct and a modified stadium-shaped duct are displayed in the insets ofFig. 5.57. Zarling [288] has obtained the f Re and Null1 for fully developed laminar flow instadium-shaped ducts. Cheng and Jamil [289] have determined the f Re and Nu for the mod-24.023.022.0\', \',,.........-21.020.0•:'1-\ \\\',i12;.- 19.018.0 ./'Re5.517.05.016.0"4.5--NUj4.015.014.0,3.~!,0.lI ,.
I ,0.2 0.31 ~ t,0.4 0.5l,0.61 ~"T";"'-r--r--13.00.7 0.8 0.9 1.0a" =2M2bFIGURE 5.57 Fully developed f Re and Num for stadium-shapedand modified stadium-shaped ducts [1].5.112CHAPTER FIVETABLE 5.61 Fully Developed Nusselt Numbers for Annular Sector Duct [281,286]2~ = 5 °2~ = 10 °2~ = 15 °2~ = 20 °2~ = 30 °2~ = 40 °2~ = 50 °2~) = 60 °2~ = 90 °2~ = 120 °2~ = 150 °2~ = 180 °2~ = 210 °2~ = 240 °2~ = 270 °2~ = 300 °2~ = 330 °2~ = 350 °r* =0.10.20.30.40.50.60.70.80.9NumNUll2NUTNUll1NumNuxNumNumNUTNumNUH2NUTNUll1NUH2NUTNUHINUH2NUTNumNUH2NUTNUn1NumNUTNumNUH2NUTNumNumNUTNumNumNUT2.7230.03261.7072.8960.14271.9963.0410.31452.2083.1630.62532.3743.3541.3482.6153.4902.0412.7803.5892.5312.8933.6602.8212.9733.7823.0603.0973.8633.0283.1573.9562.9673.2224.0672.9153.3014.1922.8763.3924.3222.8473.4914.4532.8253.5944.5812.8083.6994.7042.7943.7984.7822.7873.8643.2540.06182.0453.3870.27462.3413.4880.62502.5413.5641.0832.6843.6562.0002.8683.6972.6032.9663.7112.9013.0153.7153.0253.0383.7613.0603.0863.8933.0033.1804.0792.9533.3224.2842.9163.4804.4902.8883.6414.6852.8683.8074.8672.8533.9615.0342.8424.1075.1862.8334.2415.2792.8274.3273.8500.12532.4403.8930.52982.7173.9031.1352.8763.8881.7602.9713.8212.6023.0453.7462.9423.0463.6903.0533.0323.6853.0793.0293.8153.0373.1284.0722.9823.3264.3602.9413.5534.6392.9123.7824.8942.8923.9995.1202.8784.1985.3202.8674.3795.4962.8604.5425.6522.8534.6885.7452.8494.7764.4600.27562.8864.3261.0453.0964.1771.8813.1664.0312.4563.1633.8022.9463.0903.6833.0653.0253.6583.0853.0103.6973.0723.0414.0063.0093.2804.3922.9603.5904.7572.9273.8925.0772.9054.1695.3512.8914.4155.5802.8824.6315.7772.8754.8185.9462.8704.9826.0932.8655.1246.1792.8615.2084.9760.64203.3554.5541.8493.4054.2072.5753.3163.9512.8873.1943.6863.0653.0293.6403.0862.9973.7123.0703.0533.8443.0463.1574.3512.9793.5624.8362.9373.9665.2392.9134.3225.5652.8994.6215.8352.8904.8746.0452.8825.0806.2252.8795.2636.3772.8755.4136.5102.8715.5406.5832.8705.6205.2431.4263.7734.4712.6033.5233.9972.9643.2493.7473.0513.0773.6273.0882.9883.7433.0663.0793.9483.0353.2414.1833.0053.4264.8552.9543.9875.3832.9164.4575.7782.9014.8286.0802.8915.1186.3342.8865.3526.5052.8815.5396.6602.8795.7006.7882.8755.8256.9042.8695.9266.9602.8695.9935.1012.4193.9624.0983.0153.3373.7093.1113.0513.6163.1292.9803.7853.0623.1134.1063.0693.3664.4463.0403.6424.7632.9573.9105.5102.9164.5766.0102.9005.0556.3572.8925.3966.6112.8855.6516.8452.8805.8436.9562.8745.9897.0782.8676.0997.1782.8666.2057.2792.8666.2957.3112.8676.3474.4662.9223.6343.6753.0773.0263.6413.0853.0003.8343.0563.1534.3712.9933.5824.8752.9534.0085.2902.9304.3375.6242.9164.6866.2992.8965.3426.7002.8875.7446.9642.8765.9977.1502.8686.1757.3682.8686.3237.3942.8696.4377.4792.8716.5287.5482.8746.6017.6382.8776.6647.6382.8786.6993.6483.0833.0053.8893.0493.1974.4622.9873.6624.9792.9474.0095.7262.9124.7826.2082.8985.2506.5372.8895.5736.7752.8835.8127.2082.8686.2387.4432.8696.4887.5902.8746.6467.6912.8816.7757.8842.8876.8347.8202.8946.8947.8642.9006.9417.8992.9056.9797.9762.9097.0107.9452.9117.705NUll1NUH2NUTNUHINUll2NUTNUll1NumNUTNumNUll2NUTNUll1NUH2NUTNumNumNUTNUll1NUH2NUTFORCED CONVECTION, INTERNAL FLOW IN DUCTS5.113TABLE 5.62 Fully Developed Fluid Flow and Heat TransferCharacteristics of Annular Sector Ducts with AdiabaticOuterCurved Wall [287]r*fReK(oo)0.20.50.815.6516.0114.211.771.321.420.20.50.815.3514.9015.031.641.371.330.20.50.814.7314.2917.581.461.421.07NUfdL Sth3.4334.3723.3400.15300.09240.08983.4933.9333.1130.13200.08380.10903.4613.2353.3270.10700.09720.1230t+hy2~ = 15°0.07750.05000.05292~ = 22.5°0.07030.05160.04762~ = 45°0.05740.05290.0303ified stadium-shaped ducts.
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