Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 81
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For equilateral triangle ducts as shown in Fig. 5.27a, thefully developed laminar flow velocity profile and friction factor have been obtained by Marcoand Han [155]:U15Um_yz 28 [ ( b ) 3- 3(-~-)(-~-)- 2(b) 2- 2(b)2-~ - -~~]Um=-- 1"~f R e = .40. = 13.333(5.237)(5.238)When the three walls of the equilateral triangles are subjected to the uniform wall temperature boundary condition @, the fully developed Nusselt number Nux is equal to 2.49[170]. However, when the uniform wall heat flux with the uniform circumferential wall temperature boundary condition ~ is applied, the Nusselt number Num is determined by the following equation [6]:28/9NUll1 = 1 + 1A2S*+ 4°AIBr'(5.239)The internal heat source and viscous dissipation effects are considered in Eq.
5.239.The Nusselt number for uniform wall heat flux in both the axial and circumferential directions under the ~) boundary condition Num is found to be 1.892 [171].Since sharp triangular ducts are rarely seen in practical use, triangular ducts with roundedcorners, such as that presented in Fig.
5.27b, have been investigated by Shah [172]. His resultsare presented in Table 5.38, in which the y and Ymaxrefer to the distances measured from theduct base to the centroid and to the point of maximum fluid velocity, respectively.FORCED CONVECTION,INTERNALFLOW IN DUCTS5.75TABLE 5.38 Fully Developed Laminar Flow and Heat Transfer Characteristicsof Equilateral Triangular Ducts With Rounded Comers [172]Dh/2ay/2aYmax[2aUmax~UrnK(oo)L~yf ReNumNUH2No roundedcornersOne roundedcomerTwo roundedcomersThree roundedcorners0.577350.288680.288682.2221.8180.039813.3333.1111.8920.597450.267780.286272.1721.6980.065914.0573.4012.1960.621150.309570.291172.1151.5670.031914.8993.7562.7150.649530.288680.288682.0641.4410.028415.9934.2053.780Isosceles Triangular Ducts.
For isosceles triangular ducts like those shown in Fig. 5.27c,the velocity distribution and friction factors for fully developed laminar flow are expressed bythe following set of equations suggested by Migay [173]:1 (dp)y2-z2tan2*[(z) B-2u =--~~1 _ tanZ ~2b2(dP)u~ = - - ~-~-]-1(B- 2) tan2 ~-d--x-x (B + 2)(1-tan2~)12(B + 2 ) ( 1 - tan 2 ¢)f R e = (B - 2)[tan ¢ + (1 + tan 2 ,)1/212B = [4 + 5/z(cot2 ¢ - 1)]1/2(5.240)(5.241)(5.242)(5.243)Apparently, when 2~ = 90 °, f Re is indeterminate from Eq.
5.242. Migay [173] obtainedanother relation for 2~ = 90 °, as follows:12(B + 2)(1 - 3 tan z ~)f R e = 13/2tan ~[4 tan z ~ + 5/2(1- tan 2 ~)]-1,1 _ 2}[tan ~ + (1 + tan 2 ,)1/212(5.244)The remaining flow and heat transfer characteristics, represented by K(oo), L+hy,NUT, NUll1,and NUll2, together with f R e , are given in Table 5.39 [1].The fully developed Nusselt numbers NUT and NUll1 for laminar flow in isosceles triangular ducts with one wall or more heated are given in Table 5.40.Right Triangular Ducts. For right triangular ducts, shown in Fig. 5.27d and e, the fullylaminar developed flow and heat transfer characteristics f R e , K(oo), Num, and Null2 are givenin Fig.
5.28 [2]. The data for this figure were taken from Haji-Sheikh et al. [170], Sparrow andHaji-Sheikh [174], and Iqbal et al. [175].Arbitrary TriangularDucts. For triangular ducts with arbitrary angles such as that shownin Fig. 5.27f, the fully developed friction factors and Nusselt numbers are presented for fullydeveloped laminar flow in Figs. 5.29 and 5.30 [2].Thermally and Simultaneously Developing Flows. Hydrodynamically developing laminarflow in triangular ducts has been solved by different investigators as is reviewed by Shah andLondon [1]. Wibulswas [160] obtained a numerical solution for the problem of simultaneouslyTABLE 5.39 Flow and Heat Transfer Characteristics for Fully Developed Laminar Flowin Isosceles Triangular Ducts [1]2b/2a2~K(oo)Lh~oo8.0005.7154.0002.8362.0001.8661.5001.3741.0721.0000.8660.7500.7140.5960.5000.2890.2500.1340.125007.1510.0014.2520.0028.0730.0036.8740.0050.0053.1360.0067.3870.0080.0090.00120.00126.87150.00151.93180.002.9712.5212.4092.2712.1281.9911.9661.8981.8761.8311.8241.8181.8241.8291.8601.9072.1652.2352.5432.5742.9710.10480.06480.05900.05330.04840.04430.04360.04180.04120.04010.03990.03980.03990.04000.04080.04210.04900.05150.06440.06590.1048fReNuTNumNum12.00012.35212.47412.63612.82213.02613.06513.18113.22213.30713.32113.33313.32113.31113.24813.15312.74412.62212.22612.19612.0000.9431.461.611.812.002.222.262.362.392.452.462.472.452.452.402.342.001.901.501.470.9432.0592.3482.4462.5752.7222.8802.9102.9983.0293.0923.1023.1113.1023.0953.0502.9822.6802.6032.3252.3022.05900.0390.0800.1730.3660.7470.8511.221.381.761.821.891.841.801.591.340.620.4900.1560.1360TABLE 5.40 Fully Developed N u t and Num for Heat Transfer in Isosceles Triangular Ducts With One Wall or MoreWalls Heated [1 ]NUTNUll1A A A2bA A A2a~ degrees5.0002.5001.6671.25005.7111.3116.7021.801.885m2.0582.2272.3120.0000.8221.2681.5251.6751.2151.4161.8492.0992.2371.2151.3121.5731.7241.8022.0592.4652.6832.7962.84501.0031.5151.8071.9781.3461.8242.2742.5412.6951.3461.7391.9462.0742.1411.0000.8330.7140.6250.55626.5630.9634.9938.6641.992.344m2.311~--1.758~1.812~~2.3012.3192.3062.2742.2321.8311.8221.7871.7351.6732.849w2.778~~2.076w2.146~~2.7732.8012.7922.7742.7382.1612.1462.1072.0531.9890.5000.4500.4000.3500.30045.0048.0151.3455.0159.042.162m~1.923~1.765--1.633--2.1832.1272.0551.9681.8611.6061.5291.4331.3151.1732.594w-2.332~2.111-~1.991w2.6962.6462.5832.5052.4121.9211.8431.7461.6281.4860.2500.2000.1500.1000.05063.4368.2073.3078.6984.291.6711.5121.3301.1260.8951.4711.3611.2291.0710.8781.7331.5811.4011.1820.8931.0040.8050.5780.3320.1062.0731.9171.7481.5761.4181.8431.7461.6351.5151.3982.3012.1742.0321.8811.7371.3161.1140.8740.5870.244090.000.60760.608~--1.3461.346~5.76FORCED CONVECTION, INTERNAL FLOW IN DUCTS3.05.773.0 ~ 13.2NUT2.5NUHI13.0-2.02.5 --" 12.8NUHIfRefReI--2.,.,I1.5a* =K ( = ) - 12.62b/2aK(=)1.0-Null2Null20.52.0~-12.4I-12.2~!o1.5 i-- 12.01.0o0.10.20.30.40.50.60.70.80.90l sFIGURE 5.28 Fully developed fRe and Nu for right triangular ducts [2].13.4a* = 1.013.20.513.012.8fRe12.612.42c > 2=a*'-12.22aI~- 2,= -.~112.0010203040506070809024,, d e gFIGURE 5.29 Fully developed friction factors for arbitrary triangular ducts [2].developing laminar flow for equilateral triangular and right-angled isosceles triangular ductsfor Pr = 0, 0.72, and ~.
His results for uniform wall temperature and axial uniform wall heatflux with circumferential uniform wall temperature boundary conditions are presented inTables 5.41 and 5.42. Since Pr = ~ implies that the flow is hydrodynamically developed, theresults for Pr = oo can be applied to all fluids in thermally developing laminar flow.5.78CHAPTER FIVE3.25a* = 1.03.00_ NUHI----- -NuT0.52..50~~~mm.,-- ~I.-..~I~0.7Nu2.00/'--/__L1.501-~/I/,//IxI/IS_/s Ss..--~-~.~ ~0.3/ssS s~,"sI.-.-/¢~,J/S~,~ S/"/0~2c>_2._~ " 2.-~"]1.00 F0.751q,¢, / IIIIII1020304050.... I60--II7080)02~, de8F I G U R E 5.30TurbulentFully developed Nusselt numbers for arbitrary triangular ducts [2].FlowT h e l o w e r limit of Recrit is c o n s i d e r e d to be a p p r o x i m a t e l y 2000 in t r i a n g u l a r ducts [45]. N oreliable results for the friction factor and Nusselt n u m b e r are available for transition flow int r i a n g u l a r ducts.
In this section, the t u r b u l e n t flow and h e a t t r a n s f e r characteristics for equilateral, isosceles, and right t r i a n g u l a r ducts are p r e s e n t e d .TABLE 5.41 Local and Mean Nusselt Numbers for Thermally and Simultaneously Developing Laminar Flowsand Equilateral Triangular Ducts [160]1Nux,'rNu,,,~NUx,H1Num,mx*Pr = oo0.720oo0.720oo0.720oo0.720102030405060801001201401601802002.572.732.903.083.263.443.734.004.244.474.674.855.032.803.113.403.673.934.154.504.764.985.205.405.605.803.273.934.464.895.255.566.106.607.037.477.888.208.543.103.664.074.434.755.025.495.936.296.616.927.187.423.524.274.885.355.736.086.687.217.688.098.508.889.214.655.796.647.327.898.369.239.9810.5911.1411.6612.1012.503.273.483.744.004.264.494.855.205.505.776.016.226.453.584.014.414.805.135.436.036.567.047.507.938.338.714.345.356.146.777.277.668.268.819.309.7410.1710.5310.874.024.765.325.826.256.637.277.878.388.849.259.6310.024.765.876.807.578.208.759.7310.6011.3812.0512.6813.2713.806.678.049.089.9610.6511.2712.3513.1513.8214.4615.0215.5016.00FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.79Local and Mean Nusselt Numbers for Thermally and Simultaneously Developing Laminar Flowsin Right-Angled Isosceles Triangular Ducts [160]TABLE 5.421Nux,TNUm,TNux,mNUm,H1x*Pr = oo0.720oo0.720oo0.720oo0.720102030405060801001201401601802002.402.532.702.903.053.203.503.774.014.214.404.574.742.522.762.983.183.373.543.854.154.434.704.965.225.493.754.414.825.175.485.776.306.757.137.517.848.108.382.873.333.704.014.284.524.915.235.525.786.006.176.333.123.734.204.584.904.175.696.106.506.827.107.337.574.815.856.486.977.387.738.318.809.189.479.709.9410.133.293.583.844.074.284.474.845.175.465.715.956.166.364.004.735.235.635.976.306.927.457.958.398.809.148.505.316.276.857.237.557.858.378.859.229.589.9010.1710.434.224.985.505.916.256.577.147.608.038.408.739.049.335.366.517.327.958.508.999.8010.4210.9011.3111.6712.0012.296.867.978.689.209.6710.0710.7511.3211.7712.1412.4712.7513.04Fully Developed FlowEquilateral Triangular Ducts.
The friction factor for an equilateral triangular duct hasbeen measured by Altemani and Sparrow [176]. Their data are fitted by the following equation in the region of 4000 < Re < 8 × 104:f=0.0425Re0. 2(5.245)These researchers have also obtained the fully developed Nusselt numbers for air (Pr = 0.7)in the range of 4000 < Re < 8 × 104 in an equilateral triangular duct with the ~ boundary condition on two walls and the third wall insulated as follows:Nun1 = 0.019Re °'781(5.246)Isosceles Triangular Ducts.
Bhatti and Shah [45] r e c o m m e n d e d that the friction factor forfully developed turbulent flow in isosceles triangular ducts can be determined using differentcorrelations. For 0 < 2~ < 60 °, the circular duct correlations in Table 5.8 can be used with Dhreplaced by D~, as defined by B a n d o p a d h a y a y and A m b r o s e [177]:D~ = ~3 In cot ~ -2 In tan ~ -In tan(5.247)where 0 = (90 ° - ~)/2. For 2~ = 60 °, the circular duct correlations in Table 5.8 should be usedwith Dh replaced by D~, which is equal to V3a.
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