Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 85
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It is noted that the fully developed Nusselt number canbe calculated using Gz ---) 0 (L ~ oo) in the corresponding equations.In the study of Miyatake and Iwashita [256], the relationship of local Nusselt number andGraetz number is formulated for developing longitudinal flow between a triangular array ofcylinders with a uniform heat flux and various pitch-to-diameter ratios. For P/D = 1.01-1.1:b Gzx1/3- aNUx.H2= 1 +451Gzx [154P/°-14937] + 1NUx.H2= a;For P/D = 1.1-4.0:wherewhen Gzx> (a/b) 3when Gzx < (a/b) 3(5.309)NU~.H2= (a 2 + b 2 Gz2/3)1/2a=(5.308)(5.310)3 . 1 ( P / D - 1)°1+ 3 2 4 ( P / D - 1) '.61 + 69.5(P/D - 1 )2.41.53611 + 8.24(P/D - 1)°39]b = [2V~(P/D - 1) 2 - rt]l/3[1 + 6.37(P/D-1)0"73](5.311)Square Array.
A square array is displayed in Fig. 5.39. The fully developed friction factorfor longitudinal flow in such a virtual channel has been investigated by Sparrow and Loeffler[240], Shih [243], Rehme [244, 245], Mal~ik et al. [247], Meyder [257], Kim [258], Ramachandra [238], and Ohnemus [259]. The f R e is given in Fig. 5.39.
It can be approximated by the following equation [237]:f R e = 40.70( P -1) 0.435(5.312)Equation 5.312 is valid in the range of 1.05 < P/D < 2.0.The fully developed Nusselt numbers for the ~ and @ boundary conditions in squarearrays have been analyzed by Kim [258], Ramachandra [238], Ohnemus [259], and Chen et al.[260]. The fully developed Null1 and NUll2 are shown in Fig. 5.39.Miyatake and Iwashita [255] also investigated the developing longitudinal laminar flowbetween a square array of cylinders with uniform wall temperature. The local and logarithmicNusselt number can be obtained using the following correlations:5.96CHAPTER FIVEP/D for f Re1.21.016 r,,1.41.61.82.02.22.42.62.83.0D1470126010508 I'--NUHI40fReNu~"_o"-rf6 I--4 -,pS- t 30Null2itfReI2 ~sO Chen et al.
[260], Null2•Ohnemus [259], NumARamachandra[238],Nu m0-- 20"- 1001.01,21.41.61.82.0P/D for NUHt, NUH2FIGURE 5.39 Fully developed f Re and Nusselt numbers for longitudinal laminar flow in asquare array [237].F o r P/D = 1.0-1.2:Nux, T = 4.08(1 + 0.0058Gz146) TM(5.313)NUtm,T = 4.08(1 + 0.0349Gzx]46) TM(5.314)For P/D = 1.2-4.0, the s a m e e q u a t i o n s as Eqs. 5.299 and 5.300 are used, but the a and b aredifferent:a=4.0011 + 0.509(P/D - 1)]l+0.765(P/D_l)5/3(5.315)1.6911 + 9.1(P/D- 1)]b = [1 -I- lO.8(e/o- 1)5/4][4(e/o- 1) 2 - 2] 1/3(5.316)F o r d e v e l o p i n g longitudinal l a m i n a r flow b e t w e e n a s q u a r e array of cylinders with a unif o r m wall heat flux, the local Nusselt n u m b e r c o r r e l a t i o n s w e r e m a d e by M i y a t a k e andI w a s h i t a [256] as follows:F o r P/D = 1.01-1.2:b Gzx]/3 - aNux, m = 1 + 94Gzx [7"66P/°-7-379] + aNux, m = aF o r P/D = 1.2-4.0:w h e n Gzx < (a/b) 3Nux, H2 = (a 2 + b 2 Gzff3) 1/2w h e n Gzx > (a/b) 3(5.317)(5.318)(5.319)FORCED CONVECTION, INTERNAL FLOW IN DUCTSwherea=bFully DevelopedTurbulent=5.973.6(P/D- 1) 0.2 4- 32.2(P/D- 1) 151 + 9.1(P/D- 1)2.2(5.320)1.224[ 1 + 4.40(P/D - 1 )0.39][4(P/D- 1) 2-/I;]1'311 + 2.66(P/D- 1) °"73](5.321)FlowFully developed turbulent flow and heat transfer in triangular and square arrays have beenanalyzed by Deissler and Taylor [261,262].
The friction factors for longitudinal flow betweenthe cylinders in a triangular and a square array are given in Fig. 5.40. Correspondingly, theNusselt numbers, in terms of the Stanton number, defined as Nu/(Re Pr), are given in Fig.5.41, where the cylinders are considered to be uniformly heated.I-eI---- -- - ' -F-IIiI IF'~iII|I }Expedmental data, Pld = 1.12Kays and Perkins [263]Ct~&~1~O.OtOF0 008P/d0.006~W,,, ~~..
_.0.004'-"F0.002ii.0.001o.o~o~0.008 ;.>...~=L__~,,J~i~'["'~r'..~1__-l...L,;~, , ; : ,.:, i:i;.l.I~......J.,,-'x~-~-;oooo......0.002~..L~-~ooI. 12 ~ Experimemldate, P/d1.20 J- - - , - , - Kays and Perldns [263]- - - - - - Circular tube0.001.. .104-I_-L-_,[II.~,~~--",~L--" '-...w... iiiIl I I I I .....IO sIO sReF I G U R E 5.40 Fully developed friction factor for longitudinal turbulent flow between a triangular and a rectangular array [261].5.98CHAPTER FIVE10-2" :'" ' "1'"'' "~~,'~ , ~ , . _ , , , j Re ,~" x r u"44==" '"]60~~\\110"3oStSto2.0.=.!10-310-4==Re =3 X lOS.ellO-S•,~' !iIO'~.
I/L•!! ,,Ilil,,,,1,,lOIi,,,,JIO0I ,.lt-.,,o-SI,O00PrIO'ZiI;;i-;I"l-I1"i' • 1!~110-3{~~Q 0=:t4~t~IJII! !,St10-zI "-"StP/dz.O1.0I0 "4-~-,~, 2.oI,= 10-3""~'!~~_., ~"10-510 -4"J!II!1 1 1 IIIIIlllII 1 I I111I0I00I I I=. 10-S1,000PrFIGURE 5.41 Fully developed heat transfer characteristics for longitudinal turbulent flow between a triangular array and a square array [261].FORCED CONVECTION, INTERNAL FLOW IN DUCTSIOOI'" r"'WI''I~'JII ......5.99I'°t60-40JNu30Pld20 - 2.201.75o.-~._._ooI100z2o.o° ooo~I_II357ti0 ~--I23,I,.,5tJ7~04PeFIGURE 5.42 Nusselt numbers for fully developed longitudinal flow between cylinders in a triangular array [263].Maresca and Dwyer [264] have analyzed the heat transfer of liquid metal flow in a triangular array with uniform longitudinal heat flux.
The Nusselt number resulting from their analysis is given in Fig. 5.42.INTERNALLY FINNED TUBESInternally finned tubes are ducts with internal longitudinal fins. These tubes are widely used incompact heat exchangers. The friction factor-Reynolds number product and the Nusseltnumber for such internally finned tubes, designated as ( f Re)d and NUb+,d, respectively, arecomputed from the following definitions:(5.322)aOh,finless=(Uml(Oh IRed =~\-~'-] \'~c-c ]finlessq"Dh,nnles~q"(D~]- ~=NUb~d- k(t~ - tin) 4k(t~- tm) \ Ac /f~nl~ss(5.323)(5.324)where Dh, finlessis the hydraulic diameter corresponding to finless ducts.
Based on actual geometry, the Dh, finned is used in the f R e and NUbc. The relationships between (fRe)d, NUbc,d, andf Re, NUbc are given in the following expressions:D hfinlessNUb¢d =(5.325)Zcfio+)NUbc( Dh'finned)2(Dh finless Acfinned(5.326)5.100CHAPTERFIVECircular Ducts With Thin Longitudinal FinsH u and Chang [265] have o b t a i n e d the friction factors and Nusselt n u m b e r s for fully develo p e d laminar flow t h r o u g h a circular duct having longitudinal rectangular fins equally spacedalong the wall.
The fin's efficiency was treated as 100 percent, while its thickness was t r e a t e das zero. The fully d e v e l o p e d ( f R e ) a and Num.a for laminar flow in a circular duct with longitudinal fins are given in Table 5.47, in which l* and n are relative fin length and the n u m b e r offins, respectively.Prakash and Liu [266] have numerically analyzed laminar flow and heat transfer in thee n t r a n c e region of an internally finned circular duct. In this study, the fully d e v e l o p e d f R e isc o m p a r e d with those r e p o r t e d by H u and C h a n g [265] and Masliyah and N a n d a k u m a r [267].The incremental pressure drop K(oo) and h y d r o d y n a m i c e n t r a n c e length Z+hy t o g e t h e r withf R e are given in Table 5.48, in which the term n refers to the n u m b e r of fins, while l* d e n o t e sthe relative length of the fins.TABLE 5.47 The Fully Developed (fRe)d and NuH2.afor Laminar Flow in a Circular DuctWith Longitudinal Fins [1](fRe)anl* =0.20.40.60.72812162017.2821.22.25.99.20.8342.8727.42101.10.219.54.31.89139.5522242832...30.43n..69.57.......348.860.79535.68161.0335.98162.03286.66439.37616.5236.64164.84~448.43632.11712.76813.671025.61251.6732.60838.231062.71298.71546.80.7950.80.96.1630.1053.6573.4883.606.2330.65~71.0680.416.9327.26u31.8586.8285.0075.3262.4384.0283.7078.0667.05--434.40607.72...0.79701.7591.65..372.37..773.69l* = 0.20.40.60.7281216204.254.27.4.12~4.324.674.888.66.7.29--5.3816.796.1129.4921.6572.6681.8922242832..3.841221.00.80.940.54172.70481.12NUH2,d..4.04~--TABLE 5.48n--.84.11..3.39.--..0.79..4.108.6255.7625.15Flow Characteristics for the Entrance Problem in an Internally Finned Circular Duct81"(fRe)K(oo)00.30.6115.9627.8897.37171.81.252.442.851.5816Lh+y0.04150.04430.03200.00524(fRe).39.18208.1477.424Lh+yK(oo)..4.1110.71.79.0.04380.05400.00235(fRe)K(oo)46.00293.0933.85.4023.51.93Lh+r.0.04170.06220.001365.101FORCED CONVECTION, INTERNAL FLOW IN DUCTSTABLE 5.49Heat Transfer Characteristics for Fully Developed Flow in a Finned Circular Ductn8l*NUT,d00.30.61.03.6584.1108.77933.25LT, d0.04210.03920.005490.0057416NUHI,d4.3715.24517.2641.58+LHI.dNUT,dLT, d24NUHI.d+LHI.dNUT,d+LT, d+NUHI,dLHI,d0.0571......0.06583.993 0.03015.107 0.06533.859 0.02584.830 0.06180.08486.545 0.041113.32 0.1215.313 0.01169.154 0.1160.00774 80.55 0.00336 106.5 0.00379 143.7 0.00246 2 0 0 .
0 0.00227The fully developed Nusselt numbers for the thermal boundary conditions of uniform walltemperature and axial uniform wall heat flux with circumferential uniform temperatureobtained by Prakash and Liu [266] are given in Table 5.49, along with the corresponding thermal entrance lengths. The term n in Table 5.49 denotes the number of fins, whereas l* represents the relative length of the fins.Square Ducts With Thin Longitudinal FinsGangal and Aggarwala [268] have analytically obtained the f R e and NUll1 for fully developedflow in a square duct with four equal internal fins, as that shown in Fig..5.43. The fins weretreated as having zero thickness and 100 percent efficiency. The results o f f Re and NUHI,dforfully developed flow are provided in Table 5.50.TABLE 5.50 Longitudinal Four Thin FinsWithin a Square Duct: (fRe)d and NUHI,dfor Fully Developed Laminar Flow [268][r2aLIFIGURE 5.43 A square duct with four equallongitudinal thin fins.l*(fRe)dNUm,d00.1250.2500.3750.5000.6250.750114.26115.28518.28123.63031.87742.52752.34156.9193.6093.7214.1605.1727.30911.09614.02514.431Rectangular Ducts With Longitudinal Thin Fins from Opposite WallsThe fully developed ( f Re)d and NUHI,d for rectangular ducts with two fins and four fins onopposite walls have beenobtained by Aggarwala and Gangal [269] and Gangal [270].
Theseare shown in Fig. 5.44.Circular Ducts With Longitudinal Triangular FinsNandakumar and Masliyah [271] and Masliyah and Nandakumar [267] have analyzed fullydeveloped laminar flow in a circular duct with equally spaced triangular fins, as shown in theinset in Fig. 5.45. The flow area and wetted perimeter for this type of duct are given byAc, finned "-/T,a2 -- n[aZ¢~ - a(a - l) sin ~]Pfinned -- 2ha + 2 n l ' -2n~a(5.327)(5.328)CHAPTERFIVE5.1026.0~'I~1~~~I~.o _ i _ ~ T"Nu""d6.0~where,~;,~/- 5.0k__d2,.o.H~oH,JiiL".'Y_ L L . . .
. . . ~2 03.0_o ii..LT ...........7/The hydraulic diameter can be calculated from Oh, finned =and Figs. 5.45 and 5.46, in which the case of 2~ = 0 ° representslongitudinal fins of zero thickness.-4.oA-/ 1t~Re);, - , -3.0"",,.,,_~-.:/ /(5.329)4(Ac/P)finned. T h e results of ( f Re)d and NUHI,d are providedK/ /r = [a 2 + (a - I) 2 - 2a(a - 1) cos ~]laCircular~o);/' / ..--.~t-2.o ~ . / ( r " o ) 7- 2.0DuctsWithTwistedTapeThe enhancement of heat transfer inside a circular duct isoften achieved by inserting a thin, metal tape in such a way1.0 ~ *t 1.0NuHl'dt / R e ) ; = (fRe)dthat the tape is twisted about its longitudinal axis, as indicated in Fig.
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