Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 88
Текст из файла (страница 88)
However, εM and εH are not properties of thefluid but functions of the flow. As such, they are not known a priori and experimentalresults or various turbulence models are used for approximating them.BOOKCOMP, Inc. — John Wiley & Sons / Page 559 / 2nd Proofs / Heat Transfer Handbook / Bejan[559], (35)Lines: 878 to 911———0.51814pt PgVar———Normal PagePgEnds: TEX[559], (35)560123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTIONSeveral turbulence models have been developed in recent years and employed forsolving various turbulent natural convection flows of practical interest.
Among these,the κ–ε model, where κ is the turbulence kinetic energy and ε the rate of dissipationof turbulence energy, has been employed extensively. Both these quantities are calculated from differential equations which describe their transport in the flow and whichare similar to the vorticity and energy transport equations. Launder and Spalding(1972) discuss this turbulence model, and similar models, in detail and also give therelevant constants and functions for these models.Various researchers have employed different turbulence models to simulate complex turbulent natural convection flows.
The algebraic eddy viscosity model has beena popular choice because of its simplicity and because the empirical constants inhigher-order models may not be available for a given flow circumstance. A considerable amount of work has been done on recirculating turbulent flows in enclosures,such as those due to fire in a room (Abib and Jaluria, 1995).
See, for instance, thereview paper on such flows by Yang and Lloyd (1985).Experimental work has also been done on turbulent natural convection flows andheat transfer. However, the data available for different flow configurations and circumstances are few. Cheesewright (1968) measured velocity and temperature profilesand provided heat transfer data from a heated surface in air.
In turbulent flow, the generalized temperature profiles were found to remain largely unchanged downstream.This aspect was employed by Cheesewright to determine the end of the transitionregime. The work of Vliet and Liu (1969) and of Vliet (1969) was directed at verticaland inclined uniform heat flux surfaces.
The turbulence level, which was defined asu /ū, was measured and found to be as high as 0.3. The velocity profile was found towiden with a decrease in the nondimensional velocity, as also observed by Jaluria andGebhart (1974). Both water and air were employed. Vliet and Ross (1975) consideredan inclined surface with a constant heat flux. In the Grashof number, g was replacedby g cos γ, where γ is the angle at which the surface is inclined with the vertical. Thisis in line with the earlier discussion on inclined surfaces.7.7EMPIRICAL CORRELATIONSIn several problems of practical interest, the heat transfer and flow processes are socomplicated that the analytical and numerical methods discussed earlier cannot beemployed easily and one has to depend on experimental data.
Over the years, a considerable amount of information on heat transfer rates for various flow configurationsand thermal conditions has been gathered. Some of this information has already beenpresented earlier. The present section gives some of the commonly used results fora few important cases. The results included here are only a small fraction of what isavailable in the literature, and the attempt is only to present useful results in a fewcommon circumstances and to indicate the general features of the empirical relationships. Unless mentioned otherwise, all fluid properties are to be evaluated at the filmtemperature Tf = (Tw + T∞ )/2.BOOKCOMP, Inc. — John Wiley & Sons / Page 560 / 2nd Proofs / Heat Transfer Handbook / Bejan[560], (36)Lines: 911 to 928———0.0pt PgVar———Short PagePgEnds: TEX[560], (36)EMPIRICAL CORRELATIONS1234567891011121314151617181920212223242526272829303132333435363738394041424344455617.7.1 Vertical Flat SurfacesTo cover the range of Rayleigh number Ra from laminar to turbulent flow, the following correlations have been employed over many years for isothermal surfaces(McAdams, 1954; Warner and Arpaci, 1968):Nu =0.59Ra1/40.10Ra1/3for104 < Ra < 109for10 < Ra < 109(laminar)13(turbulent)(7.70)(7.71)These equations are applicable for Pr values that are not very far from 1.0.
Churchilland Chu (1975a) have recommended the following correlation, which may be appliedover the entire range of Ra:20.387Ra1/6Nu = 0.825 + 1 + (0.492/Pr)9/168/27[561], (37)(7.72)For laminar flow, greater accuracy is obtained with the correlation [Churchill and Chu(1975a)]0.67Ra1/4Nu = 0.68 + 1 + (0.492/Pr)9/164/9for 0 < Ra 109(7.73)Here Nu and Ra are both based on the plate height L. The preceding correlations aregenerally preferred to the others, since they have the best agreement with experimental data.
The local Nusselt numbers Nux can be obtained from the results given in thepreceding references.For the uniform heat flux case, the heat transfer results given by Vliet and Liu(1969) in water yield the following relationships. For laminar flow,Nux,q = 0.60(Gr ∗x · Pr)1/5Nuq = 1.25NuL,qfor 105 < Gr ∗x · Pr < 1013(7.74)for 105 < Gr ∗x · Pr < 1011(7.75)For turbulent flow,Nux,q = 0.568(Gr ∗x · Pr)0.22Nuq = 1.136NuL,qfor 1013 < Gr ∗x · Pr < 1016(7.76)for 2 × 1013 < Gr ∗x · Pr < 1016(7.77)whereGr ∗x =BOOKCOMP, Inc.
— John Wiley & Sons / Page 561 / 2nd Proofs / Heat Transfer Handbook / Bejangβq x 4kν2(7.78)Lines: 928 to 978———9.5482pt PgVar———Short Page* PgEnds: Eject[561], (37)562123456789101112131415161718192021222324252627282930313233343536373839404142434445NATURAL CONVECTIONHere NuL,q represents the Nusselt number at x = L and Gr ∗ = gβq L4 /kν2 . In alater study, Vliet and Ross (1975) obtained a closer corroboration for data in air withthe following relationships:for laminar flow(7.79)0.55(Gr ∗x · Pr)0.2Nux,q =0.17(Gr ∗x · Pr)0.25for turbulent flow(7.80)Nuq can be obtained by computing the mean temperature difference and using theoverall heat transfer rate provided by Vliet and Ross (1975).7.7.2Inclined and Horizontal Flat SurfacesAs discussed earlier, the results obtained for vertical surfaces may be employed forsurfaces inclined at an angle γ up to about 45° with the vertical, by replacing gwith g cos γ in the Grashof number.
For inclined surfaces with constant heat flux,Vliet and Ross (1975) have suggested the use of eq. (7.74) for laminar flow, withthe replacement of Gr ∗x by Gr ∗x cos γ for both upward- and downward-facing heatedinclined surfaces. In the turbulent region also, eq. (7.76) is suggested, with Gr ∗xreplaced by Gr ∗x cos γ for an upward-facing heated surface and with Gr ∗x replacedby Gr ∗x cos2 γ for a downward-facing surface.Several correlations for inclined surfaces, under various thermal conditions, weregiven by Fujii and Imura (1972).
For an inclined plate with heated surface facing *upward with approximately constant heat flux, the correlation obtained is of the formNuq = 0.14[(Gr · Pr)1/3 − (Gr cr · Pr)1/3 ] + 0.56(Gr cr · Pr cos γ)1/4for 10 < Gr · Pr cos γ < 10511and 15° < γ < 75°for 105 < Gr · Pr cos γ < 1011 ,(7.81)γ < 88° (7.82)The fluid properties are evaluated at Tw −0.25(Tw −T∞ ), and β at T∞ +0.25(Tw −T∞ ).For horizontal surfaces, several classical expressions exist. For heated isothermalsurfaces facing downward (lower surface of heated plate), or cooled ones facingupward (upper surface of cooled plate), the correlation given by McAdams (1954)which has been used extensively, isNu = 0.27Ra1/4for 105 Ra 1010Fujii and Imura (1972) give the corresponding correlation asBOOKCOMP, Inc. — John Wiley & Sons / Page 562 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 978 to 1029———-0.52994pt PgVar———Normal PagePgEnds: Eject[562], (38)where Gr cr is the critical Grashof number at which the Nusselt number starts deviatingfrom the laminar relationship, which is the second expression on the right-hand sideof eq.
(7.81). This correlation applies for Gr > Gr cr . The value of Gr cr is also givenfor various inclination angles. For γ = 15, 30, 60, and 70°, Gr cr is given as 5 × 109,2 × 109, 108, and 106, respectively. For inclined heated surfaces facing downward,the expression given isNuq = 0.56(Gr · Pr cos γ)1/4[562], (38)(7.83)EMPIRICAL CORRELATIONS123456789101112131415161718192021222324252627282930313233343536373839404142434445Nu = 0.58Ra1/5for 106 ≤ Ra ≤ 1011563(7.84)Over the overlapping range of the two studies by Fujii and Imura (1972) and Rotemand Claassen (1969), the agreement between the two Nusselt numbers is fairly good.For the heated isothermal horizontal surface facing upward and cold surface facingdownward, the correlations for heat transfer are given by McAdams (1954) asfor 104 Ra 107(7.85)0.54Ra1/4Nu =0.15Ra1/3for 107 Ra 1011(7.86)The corresponding correlation given by Fujii and Imura (1972) for an approximatelyuniform heat flux condition isNuq = 0.14Ra1/37.7.3for Ra > 2 × 108(7.87)[563], (39)Lines: 1029 to 1081Cylinders and Spheres———A considerable amount of information exists on natural convection heat transfer5.2721ptPgVarfrom cylinders.
For vertical cylinders of large diameter, ascertained from eq. (7.47),———the correlations for vertical flat surfaces may be employed. For cylinders of smallNormal Pagediameter, correlations for Nu are suggested in terms of the Rayleigh number Ra,* PgEnds: Ejectwhere Ra and Nu are based on the diameter D of the cylinder.The horizontal cylinder has been of interest to several investigators.
McAdams(1954) gave correlation for isothermal cylinders as[563], (39)1/449for 10 < Ra < 10(7.88)0.53RaNu =1/39120.13Rafor 10 < Ra < 10(7.89)For smaller values of Ra, graphs are presented by McAdams (1954). A generalexpression of the form Nu = C · Ran is given by Morgan (1975), with C andn presented in tabular form. Churchill and Chu (1975b) have given a correlationcovering a wide range of Ra, Ra ≤ 1012 , for isothermal cylinders asRaNu = 0.60 + 0.387[1 + (0.559/Pr)9/16 ]16/91/6 2(7.90)This correlation is recommended for horizontal cylinders since it is convenient to useand agrees closely with experimental results.For natural convection from spheres, too, several experimental studies have provided heat transfer correlations.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
