Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 43
Текст из файла (страница 43)
For higher accuracy, and for convenience since Ra* is used in place of Ra, use the following equation set:----6(Nur = He Ra .1/5Pr)1/5He = ~- 4 + 9~/-P-rr+ 10 Pr(4.36a)1.0Nue= In (1 + 1.0/Nu T)(4.36b)/( (c2pr)04)Nu,= (CV) 3/4 Ra *1/4 1 +Nu = (Nu~' + NU~n)1/mRa*C2 = 7 x 1012(4.36c)m = 6.0(4.36d)mHe is tabulated in Table 4.1.Vertical Plates of Various Planforms. There is a class of vertical plates, such as shown inFig.
4.9, for which any vertical line on the plate intersects the edge only twice. When suchplates are isothermal, the thin-layer laminar Nusselt number Nu r can be calculated [225] bydividing the surface into strips of width Ax and length S(x), applying Eq. 4.33a to compute theheat transfer from each strip, adding these to obtain the total (thin-layer) heat transfer, andusing this heat transfer to calculate Nur. The result can be expressed as~1/4 f0W S 3/4 d xN u T=GCt R a TMG=A(4.37a)where/? is any characteristic dimension of the surface.
This also results from the applicationof Eq. 4.14. Figure 4.9 shows examples of the constants G for three body shapes.BCFIGURE 4.9e=De=He=Btan0G=1.05G=8/7G=1+7cos0I_~B < C cos 0IwqeNu - AA~kgl3,~Te3Ra -wgl3q"e 4Ra* =v~kDefinition sketch for natural convection from plates of various planform.4.16CHAPTER FOURFor laminar heat transfer, the correlations for such disks have been given by Eq. 4.18 [227]:Nut = NUcoND + Nur(4.37b)Yovanovich and Jafarpur [294] have shown that, for the length scales in Fig.
4.9,eNUcoNo = X/A NUVXcOND(4.37C)If the plate is heated on one side only, but is immersed in a full space NUVXcOND= 3.55, if theplate is heated on one side and immersed in a half-space (i.e., the heated plate is embedded inan infinite adiabatic surface) NUvXcOND = 2.26, and if the plate is heated on both sides andimmersed in a full space NUVX.COND= 3.19. In all cases A is the surface area that is active. Fora general equation for Nu: substitute Eqs.
4.37a and c into Eq. 4.37b to obtain the equation forNut, use Eq. 4.33 for Nu,, and substitute these equations for Nut and Nut into Eq. 4.33 toobtain the equation for Nu.The isothermal surface correlation jus___ttprovided can also be used to estimate Nu for a uniform flu___xboundary by replaci_~ ATby AT. For higher accuracy, use Eq.
4.36a~to find the thinlayer ATfor each strip, find ATfor the entire surface by area weighting the ATfor each strip(Eq. 4.35), and calculate Nur from the area weighted AT. This results inA e 1/5mNur = C2Ht Ra *v5C2-- ~w(4.38)$6,5 dx%where e is again any characteristic dimension of the surface, and terms are defined in Fig. 4.9.To find Nu, substitute Eq. 4.38 for Nur and Eq. 4.37c for NUcoND into Eq. 4.37b to obtain anexpression for Nut. Use this Nut and Eq.
4.36c for Nut in Eq. 4.36d to obtain Nu.Horizontal Heated Upward-Facing Plates (~ = 0 °) With Uniform T. and TooCorrelation. For horizontal isothermal plates of various planforms with unrestricted inflowat the edges as shown in Fig. 4.10, the heat transfer is correlated for 1 < Ra < 10~°by the equation:Nur = 0.835Ce R a TM(4.39a)1.4Nut = In (1 + 1.4/Nu r)(4.39b)Nu, = C,v R a 1/3(4.39c)Nu = ((Nut) m + (Nut)m) l/m(4.39d)m=10Use of the length scale L* [116], defined in Fig. 4.10c, is intended to remove explicit dependence on the planform from the correlation.~- Perimeter pAL*= AlpqL*Nu= A A T kRe = cj/3 ~-T (L*) 3vaEdge view(a)Plon view(b)(c)FIGURE 4.10 Definition sketch for natural convection on a horizontal upwardfacing plate of arbitrary planform. Only the top heated surface of area A is heated.NATURAL CONVECTIONi•10 2,i'oAI-Arabi & EI-Riedy, circularnAI-Arabi & EI-Riedy, squareI'i•./.,.~"Clausing&Berton,square (Nr)o4.17J-Clausing & Berton, square (nitrogen)•Yousef,Tarasuk,&McKeen,square+Bovy&Woelkrectangul,arooi10 0-~Jm101f1lO*/,.I10 2,I,,10 4I10 6iI10 0,,10 ~°RaFIGURE 4.11 Comparisonof Eq.
4.39 to data for upward-facing heated plates of various planform in air.Comparison With Data. Figure 4.11 shows that Eq. (39) is in good agreement with measurements for gases (Pr --- 0.7). The Clausing and Berton [62] data shown in the figure havebeen extrapolated to Tw/To. = 1 using their correlation. The scatter in the data of Yousef et al.[290] is due to temporal changes in the heat transfer. Excellent agreement was also found withthe data of Goldstein et al. [116] for 1.9 < Pr < 2.5 for a variety of shapes, but the data ofSahraoui et al. [237] for a disk and flat annular ring, 1 < Ra < 103, fall below this equation. ForPr > 100, Eq. 4.39 is in excellent agreement with the measurements of Lloyd and Moran [297]for 107 < Ra < 109, and agrees, within the experimental scatter, with the measurements ofLewandowski et al.
[179] for 102 < Ra < 104.Horizontal Heated Upward-Facing Plates (~ = 0 °) With Uniform Heat FluxCorrelation. The nomenclature for this problem is also given in Fig. 4.10, where AT is thesurface average temperature difference. Equation 4.39 should also be used for this case,where the calculated Nu value provides the average temperature difference AT.Comparison With Data. Equation 4.39 agrees to within about 10 percent with the data ofFujii and Imura [103] and Kitamura and Kimura [161], for heat transfer in water (Pr = 6), for104 < Ra < 1011.
Both these experiments were performed using effectively infinite strips offinite width.Horizontal Isothermal Heated Downward-Facing Plates (~ = 180 °)Correlation. Definitions and a typical flow pattern for this problem are shown in Fig.4.12. The heat transfer relations given here assume that the downward-facing surface is substantially all heated; if the heated surface is set into a larger surface, the heat transfer will bereduced. Since the buoyancy force is mainly into the surface, laminar flow prevails up to veryhigh Rayleigh numbers. The following equation can be used for 103 < Ra < 101°:4.18CHAPTER FOURqL*Nu Heated plateAATkRa : g~ AT (L*)3VQ~=Ap_ _=heater areaheater perimeterFIGURE 4.12 Definition sketch for natural convection on adownward-facing plate.
Only the bottom surface of area A isheated.Nu r =0.527Ra 1/5(1 + (1.9/Pr)9/~°) ~9(4.40a)2.5Nue = In (l + 2.5/Nu r)(4.40b)The coefficient in Eq. 4.40a was obtained by fitting the results of the integral analysis of Fujiiet al. [102].Comparison With Data.
Measurements for isothermal plates in air are compared to Eq.4.40 in Fig. 4.13 for rectangular plates. Data lie within about +__20percent of the correlation.Measurements have also been done using water, but only with a uniform heat flux boundarycondition. For water, the data of Fujii and Imura [103] for a simulated 2D strip lie about 30percent below Eq.
4.40, but the data of Birkebak and Abdulkadir [20] lie about 3 percent102........,........,........,........oFaw & Dullforce, squareDHatfleld& Edwards, square,AHalfleld& Edwards, rectangular 3:1•Aihara,Yamada, & Endo, 2D strip........,........,".J•....if+Bevy& Woelk, rectangular 1:1,3:2, 3:1xRestrepo& Glicksman, s q u a r eJ~.~-.i,I--'~++.~+..I+Z10111~o ~0 o. . . . .
. . .10 si10 +|ii....1110 s. . . . . . . .!i. . . . . . .10 s!10 7. . . . . . . .i10 e. . . . . . . .!10 9. . . . . . .101°RaFIGURE 4.13 Comparison of Eq. 4.40 to data for downward-facing heated plates ofvarious planform in air.NATURAL CONVECTION4.19above the correlation. The importance of the heat transfer from the outer edge of the plate isbelieved to be the cause of such large discrepancies.Plates at Arbitrary Angle of Tilt.
The previous sections provide equations from which tocompute the total heat transfer from vertical plates (~ - 90°), horizontal upward-facing plates(~ = 0°), and horizontal downward-facing surfaces (~ = 180°). These equations are the basis forobtaining the heat transfer from tilted plates.For a wide isothermal plate at any angle of tilt, first compute the heat transfer from Eq.4.33 (vertical plate) but with g replaced by g sin ~, then compute the heat transfer from Eq.4.40 (downward-facing plate) with g replaced by g(0,-cos ~)max,then compute the heat transfer from Eq. 4.39 (upward-facing plate) with g replaced by g(0, cos ~)max,and take the maximum of the three heat transfer rates.
It is important to take maximum heat transfer ratherthan the maximum Nusselt number, because the Nusselt numbers are based on differentlength scales.For plates with small aspect ratio, such as shown in Fig. 4.9, follow the same procedureexcept use Eq. 4.37b in place of Eq. 4.33b.Vertical Isothermal Plate in Stably Stratified AmbientCorrelation. For an isothermal vertical plate (see Fig. 4.6) in an ambient fluid whose temperature Too increases linearly with height x, the heat transfer depends on the stratificationparameter S, defined byL dTooS - A---T dx(4.41)where the mean temperature difference AT is also the value of Tw- Too at the mid-height ofthe surface.
For 0 < S < 2, Tw - Too is positive over the entire plate; for S = 2, Tw - Too at x - L;and for S ~ oo the plate temperature is lower than Tooover the top half of the plate and greaterthan Tooover the bottom half.From laminar thin-layer analysis [41,225,226] the value of Nu T, corrected for stratificationeffects, isNuT= (1 + S/a)bCe Ra TM: cSl/4CfRa TMS<2(4.42a)S>2(4.42b)For gases: a = 1, b = 0.38, and c = 1.28; for water (Pr --- 6): a = 2, b = 0.5, and c = 1.19. For turbulent flow everywhere on the plate and for S _<2, Nut is given byNu,---7-~S-[(1 + -~)7/3- (1 - 5)7/3] (1 + 1.4Cyx Ral/3•109 Pr/Ra) 'S<2(4.43)To estimate Nu at intermediate Ra for S < 2, substitute Eq.
4.42 for Nur into Eq. 4.33b toobtain Nue and use this Nue and Nu, from Eq. 4.43 in Eq. 4.33d to find the Nusselt number.Confirmation of Procedure. This procedure leads to good agreement with the data ofChen and Eichhorn [41] for water, for 2 x 106 < Ra < 3 × 107, but it is unconfirmed outside thisrange. The accuracy of Eq. 4.43 is particularly questionable through transition since the transition to turbulence depends on S.CylindersLong Vertical Cylinders, Circular or NoncircularCorrelation. The objective is to calculate the heat transfer from the lateral surface of thelong vertical cylinder or wire, where heat transfer from the ends is ignored. See Fig. 4.14 forrelevant nomenclature.CHAPTERFOUR4.20For a vertical cylinder of length L and diameter D, firstcalculate the Nusselt number Nur and Nut for a vertical flatplate of height L, using Eq.
4.33 if the cylinder is isothermaland Eq. 4.36 if it has uniform heat flux. These Nusselt numbers are based on the length L of the plate (see Fig. 4.6) andare renamed NUplater and NUt,Plate here. The laminar Nusseltnumber for the vertical cylinder (defined in Fig. 4.14), Nut, isthen calculated fromTwqLNu-~AATkgI3ATL 3Ra-NutVGIn (1 + ~~~-N u t Plate;'1.8L/DNu rPlate(4.44)gl3q"L4( qL )For an isothermal boundary, calculate Nut from Eq. 4.33cNUpiate= . ~ k /Plateand substitute this N u t and Eq.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
