Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 80
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5.25.80i702ba* = 2b60-- 2ot50lapp Re002.0.514030150.0020.0050.010.020.050.1x+FIGURE 5.25 Apparent Fanning friction factors for hydrodynamically developingflow in rectangular ducts [158].5.70CHAPTER FIVET h e r m a l l y D e v e l o p i n g Flow. Wibulswas [160] and A p a r e c i d o and Cotta [161] have solvedthe thermal entrance p r o b l e m for rectangular ducts with the thermal b o u n d a r y condition ofuniform t e m p e r a t u r e and uniform heat flux at four walls. However, the effects of viscous dissipation, fluid axial conduction, and thermal energy sources in the fluid are neglected in theiranalyses. The local and m e a n Nusselt n u m b e r s NUx,T, NUm,T, and Nux, m and NUm,H1 obtained byWibulswas [160] are p r e s e n t e d in Tables 5.32 and 5.33.For square ducts, a* = 1, the Nusselt n u m b e r s for the ~ , ~ , and @ thermal b o u n d a r y conditions have b e e n obtained by C h a n d r u p a t l a and Sastri [162].
As r e c o m m e n d e d by Shah andL o n d o n [1], the results obtained by C h a n d r u p a t l a and Sastri [162], shown in Table 5.34, arem o r e accurate than those p r e s e n t e d by Wibulswas [160].The thermal entrance lengths for rectangular ducts in the 0) b o u n d a r y condition L *th,T ared e t e r m i n e d to be 0.008, 0.054, 0.049, and 0.041 for a* - 0 , 0.25, 0.5, and 1, respectively [1]. Thethermal entrance lengths in the ~ b o u n d a r y condition L *th,nl are found to be 0.0115, 0.042,TABLE 5.32 Local and Mean Nusselt Numbers in the Thermal Entrance Region of Rectangular DuctsWith the 03 Boundary Condition [160]1NUm,TNUx,Tx*a* = 1.00.51/30.250.2V61.00.51/30.250.201020304060801001201401601802002.652.863.083.243.433.784.104.354.624.855.035.245.413.393.433.543.703.854.164.464.724.925.155.345.545.723.964.024.174.294.424.674.945.175.425.625.805.996.174.514.534.654.764.875.085.325.555.775.986.186.376.574.924.945.055.135.225.405.625.836.066.266.456.636.805.225.255.345.415.485.645.866.076.276.476.666.867.022.653.504.034.474.855.506.036.466.867.227.567.878.153.393.954.464.865.245.856.376.847.247.627.978.298.583.964.545.005.395.746.356.897.337.748.118.458.779.074.515.005.445.816.166.737.247.718.138.508.869.179.474.825.365.776.136.457.037.537.998.398.779.149.469.795.225.666.046.376.707.267.778.178.639.009.359.6710.01TABLE 5.33 Local and Mean Nusselt Numbers in the Thermal Entrance Regionof Rectangular Ducts With the ~ Boundary Condition [160]1x*0102030406080100120140160180200NUx, H1a* = 1.03.603.713.904.184.454.915.335.696.026.326.606.867.10NUm, H10.5½0.251.00.5½0.254.114.224.384.614.845.285.706.056.376.686.967.237.464.774.855.005.175.395.826.216.586.927.227.507.768.025.355.455.625.775.876.266.637.007.327.637.928.188.443.604.485.195.766.247.027.668.228.699.099.509.8510.184.114.945.606.166.647.458.108.669.139.579.9610.3110.644.775.456.066.607.097.858.489.029.529.9310.3110.6710.975.356.036.577.077.518.258.879.399.8310.2410.6110.9211.23,,,FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.71TABLE 5.34 Local and Mean Nusselt Numbers in the Thermal Entrance Regionof a Square Duct With the 03, (~, and @ Boundary Conditions [162]1X*NUx,TNUm,TNRx,H1NUm,H1NUx,H2NRm,H20102025405080100133.31602002.9752.9763.0743.1573.4323.6114.0844.3574.755-5.4122.9753.5144.0244.2534.8415.1735.9896.4357.068-8.0843.6123.6863.9074.0484.4654.7205.3875.7696.3316.7307.2693.6124.5495.3015.6336.4766.9498.1118.7479.65310.27911.1033.0953.1603.3593.4713.8464.0674.6544.9935.4925.8486.3303.0953.9154.6024.8985.6566.0837.1387.7198.5519.1289.8910.048, 0.057, and 0.066 for o~* = 0, 0.25, 1/3, 0.5, and 1, respectively [1].
T h e r m a l l y developingflow in rectangular ducts with one wall or m o r e insulated is reviewed in Shah and L o n d o n [1].Simultaneously Developing Flow.Table 5.35 presents the results for simultaneously developing flow in rectangular ducts; these w e r e obtained by Wibulswas [160] for the 03 and (~b o u n d a r y conditions for air (Pr = 0.72). Transverse velocity is neglected in this analysis. H o w ever, C h a n d r u p a t l a and Sastri [163] include transverse velocity in their analysis for a squareduct with the ~ b o u n d a r y condition.The NUx,H1 and NUm,H1 obtained by C h a n d r u p a t l a and Sastri [163] are illustrated in Table5.36.
It should be n o t e d that in Table 5.36, Pr = 0 corresponds to slug flow, w h e r e a s Pr = oo corresponds to hydrodynamically d e v e l o p e d flow.TABLE 5.35 Local and Mean Nusselt Numbers for Simultaneously Developing Flow in Rectangular DuctsWith the 03 and ~ Boundary Conditions [160]1Nux,mx*0~* = 1.0510203040506080100120140160180200220.4.184.665.075.475.836.146.807.387.908.388.849.289.69...NUrn,H10.51A0.251.0.4.605.055.405.756.096.427.027.598.118.619.059.479.88..5.185.505.826.136.446.747.327.868.378.849.389.7010.06.5.665.926.176.436.707.007.558.088.589.059.599.8710.244.605.436.607.528.258.909.4910.5311.4312.1912.8713.5014.0514.5515.030.55.005.776.947.838.549.179.7710.7311.7012.4813.1513.7914.3514.8815.36NUm,T½0.251.05.586.277.318.138.859.4810.0711.1312.0012.7813.4714.1014.7015.2115.836.066.657.588.379.079.7010.3211.3512.2313.0313.7314.4814.9515.4916.02.3.754.394.885.285.635.956.577.107.618.068.508.919.309.700.5.½.4.204.795.235.615.956.276.887.427.918.378.809.209.6010.00.4.675.175.605.966.286.607.177.708.188.669.109.509.9110.300.25.5.115.565.936.276.616.907.477.988.488.939.369.7710.1810.585.726.136.476.787.077.357.908.388.859.289.7210.1210.5110.905.72CHAPTERFIVELocal and Mean Nusselt Numbers for Simultaneously Developing Flow in a Square Duct (ix* = 1)With the ~ Boundary Condition [163]TABLE 5.36NUx,H1Num,rtlx*Pr = 0.00.11.010.0~0.00.0050.00750.010.01250.020.0250.040.050.1**14.65312.54511.29710.4599.0318.5007.6757.4157.0517.01311.6599.5978.3917.6156.3535.8835.1084.8264.2433.6128.3737.1226.3795.8775.0114.6834.1523.9733.6873.6127.3296.3815.7165.4804.7594.5024.0803.9393.6863.6127.2696.3315.7695.3874.7204.4654.0483.9073.6863.6120.121.98619.09517.29016.00313.62212.64710.91310.2378.7017.01317.82315.39113.78112.62010.4759.6018.0437.4265.9483.6121.013.39011.48910.2979.4617.9347.3156.2145.7824.7833.61210.011.2009.7378.8238.1817.0106.5335.6825.3474.5803.61211.1039.6538.7478.1116.9496.4765.6335.3014.5493.612Turbulent FlowEntrance configuration is the key factor affecting flow transition in rectangular ducts.
Thelower limit of the critical Reynolds numbers Rent along with entrance configuration has beeninvestigated by Davies and White [164], Allen and Grunberg [165], Cornish [166], and Hartnett et al. [167]. The lower limits of the critical Reynolds numbers for a smooth rectangularduct with two entrance configurations are given in Table 5.37.For most engineering calculations of friction factors and Nusselt numbers for fully developed flow in rectangular ducts, it is sufficiently accurate to use the circular duct correlationsby replacing the circular duct diameter 2a with the hydraulic diameter Dh = 4ab/(a + b) or withDr, defined by the following equations to consider the shape effect [168]:2(Dr= -~ Dh(1 + 0~*)2 1 -192tx*~o1(2n + 1 ) u c t * ) ( 5 .
2 3 4 )n~ .= '--------~(2n + 1) tanh2An approximate expression for Dt is:2 Dh + "11Dt = "~~ o~*(2 - tx*)(5.235)which yields D~ values within +_2 percent of those given by Eq. 5.234.TABLE 5.37Lower Limits of the Critical Reynolds Numbers for Smooth Rectangular DuctsEntrance configurationsSmooth entranceSchematicst................_-,.gAbrupt entrancelm|~ " . . .
. . . . . "~"Aspect ratio ct*Critical Reynolds number Recnt00.10.20.33331.03400440070006000430000.10.20.25550.34251.0310029202500240023602200FORCED CONVECTION,INTERNALFLOW IN DUCTS5.73The fully developed friction factor and heat transfer coefficients for turbulent flow in anasymmetrically heated rectangular duct have been reported by Rao [59]. In this investigation,the experimental region of the Reynolds number was from 104 to 5 x 104.For fully developed Nusselt numbers for the turbulent flow of liquid metals in rectangularducts, a simple correlation has been derived for the 03 and ~ boundary conditions [169].
Thiscorrelation follows:Nu = ~Nus~ug + 0.015Pe °8(5.236)where Nus~ugis the Nusselt number corresponding to slug flow (Pr = 0) through rectangularducts, which is given in Fig. 5.26 as a function of or* for rectangular ducts under the 03 and (~boundary conditions.14'III'210NuautI\40FIGURE[169].\~/--- T ~boundary conditionL boundarycondi(~ tion~0.250.500.75t.O05.26Slug flow Nusselt numbers for rectangular ductsTRIANGULAR DUCTSThe flow and heat transfer characteristics of triangular ducts, as shown in Fig.
5.27, areexplained in this section. The coordinates shown in Fig. 5.27 are used in the presentation ofthe results.LaminarFlowIn this section, the laminar flow and heat transfer characteristics are explained for differenttriangular ducts.5.74CHAPTERFIVEI2ILzz2a),, f(a),2b(b)(c)(e)(0-I2bl(d)FIGURE 5.27 Triangularducts: (a) equilateral; (b) equilateral with rounded comers; (c) isosceles;(d) and (e) right; and (f) arbitrary.Fully Developed FlowEquilateral Triangular Ducts.
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