Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 89
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It can be observed that as n ~ o% the value of f Reapproaches 6.222 for a* = 1 (annular duct); f R e approaches 16 for o~* = 0 (circular duct).Circular Ducts With Centered Regular Polygonal CoresThe product of fully developed friction factors and the Reynolds n u m b e r f Re obtained byH a g e n and R a t k o w s k y [298] for laminar flow in circular ducts with centered regular polygoTABLE 5.65The f R e and Num for fully Developed Laminar Flow in Confocal Elliptical Ducts [1](x* = 0.2r*0.020.10.20.30.40.50.60.70.80.90.950.980.40.6f ReNumf ReNUll1f Re19.41919.45219.47819.49519.50719.51619.52519.53419.54419.55519.56119.5655.12375.12525.12305.11855.11305.10725.10165.09655.09215.08855.0788m19.46819.62219.75919.87119.97320.07220.17120.26820.36520.46020.50620.5345.1231 20.2915.139520.6225.147920.9655.1541 21.2015.162621.4045.175121.5855.192221.7495.213721.8965.239022.0295.267622.1485.283622.20322.2340.80.90.95NUHIf ReNumf ReNumf ReNum5.47825.55345.61625.67705.74415.74415.89665.97796.05976.14046.1801--21.76622.38822.75022.97423.13523.25723.25723.42923.49023.53923.56023.5726.50836.73846.89737.02187.13257.13257.32247.40237.47247.53367.5606--22.43623.15123.45423.61023.70823.77323.81923.85123.87423.89023.89623.9007.19337.52737.69457.79617.86967.86967.96998.00468.03208.05368.0621m22.62023.36623.64323.77723.85523.90323.93423.95323.96623.97323.97523.9767.41007.79407.95748.04278.09558.09558.15468.17118.18258.19018.192824,'-'~--~~~ ~ ~~~~~,~~.~,~ ~-y,.,,,~ .!,,~..-- ._~.
~ ,. .~,,~,~',.~20t---- -....~.----....~,0 )'f Re"--,. !,'\,,L.\.,o-::TI-,860.200.40.60.81.0(IsF I G U R E 5.65 Fully developed friction factors for regularpolygonal ducts with centered circular cores and circular ductswith centered rectangular polygonal cores [2].:,,6~.f".'"-i///I". ,i/,0-0.1" ~..~."laro--~..J ..~-4 -~-~\,,-'= ~-, ~'=.0.20.3~.:~0.40.5FIGURE 5.66 Fully developed Nusselt numbers for regularpolygonal ducts with centered circular cores and circular ductswith centered rectangular polygonal cores [2].5.1195.120CHAPTERFIVEnal cores (see inset in Fig. 5.65) is shown in Fig. 5.65. Corresponding fully developed Null1 obtained by Cheng and Jamil[289] are depicted in Fig.
5.66. The f Re and NUll1 for concentric circular annular ducts are shown in Figs. 5.65 and 5.66for the purpose of comparison.FIGURE 5.67 An isosceles triangular duct with aninscribed circular core.Isosceles Triangular DuctsWith Inscribed Circular CoresAn isosceles triangular duct with an inscribed circular core is shown in Fig. 5.67. The f Reobtained for fully developed laminar flow in such a duct by Bowen [299] can be expressed interms of ~, as follows:f R e = 12.0000 - 0.1605~ + 4.2883 x 10-3t~2 - 1.0566 x 10-4t~3+ 1.6251 x 10-6t~4 - 1.04821 x 10-8~5 (5.358)where ~ is in degrees.Elliptical Ducts With Centered Circular CoresFor elliptical ducts with centered circular cores, fully developed laminar flow has been analyzed by Shivakumar [300].
The f Re values are given in Table 5.66, in which o~* denotes theratio of the length of the minor axis to the length of the major axis of the ellipse and r* is theratio of the diameter of the circular core to the length of the minor axis.TABLE 5.66 Fully Developed Friction Factorsfor Elliptical Ducts With CenteredCircular Cores [300]fReo~*r* = 0.50.50.70.919.32121.69423.5190.6-~23.4350.7-19.40223.1590.9516.816CONCLUDING REMARKSThis chapter discusses forced convection in various ducts.
The formulas, correlations, equations, tables, and figures included in this chapter are given for the purpose of practical calculations. However, the following effects are not considered: a detailed discussion of heat sourceand dissipation effects, non-newtonian fluids, varying thermal property effects, porous wallducts, unsteady-state effects, rotating ducts, combined radiation, and convection. The interested reader can consult Kays and Perkins [263] and Kakaq, Shah, and Aung [301] for furtherinformation regarding these effects.NOMENCLATUREAcflow cross-sectional area, m 2aradius of a circular duct, m; half-length of major axis of an elliptic duct, m;half-length of the width of a rectangular duct, mFORCED CONVECTION, INTERNAL FLOW IN DUCTSheDh/kBiBiot numberBrBrinkmann number for the 03 boundary condition, = ktUZm/k(Tw,m- Te)Brinkmann number for the (~ boundary condition, ktu2/q"DhBr'bCGDDeDe*DgDhDlE(m)e*Fflappf~f~c~( )GrGr'Gz=5.121=half-spacing of a parallel plate duct, m; coil spacing, m; half length of minoraxis of an elliptic duct, m; half length of height of a rectangular duct, mconstantspecific heat of the fluid at constant pressure, J/(kg.K)diameter of a circular cylinder, mDean number = Re k/--a/Rmodified Dean number = Re V'Dh/Rgeneral lengthhydraulic diameter of the duct = 4Ac/P, mlaminar equivalent diameter, mcomplete elliptic integral of the second kind with argument m, which isdefined by Eq.
2.252eccentricity of the eccentric annular duct = e/(ro - ri); amplitude of the circularduct with sinusoidal corrugation = e/aa multiplicative factor entering various expressionscircumferentially averaged fully developed friction factor = xw/(pu2/2)apparent Fanning friction factor = Ap*/(2x/Dh)friction factor for curved ducts = Xw/(puam/2)friction factor for straight ductseigenfunctionsGrashof number = ~ga3AT/v 2modified Grashof number = ~ga4q"/kv®Graetz number = mcp/kL = p/(4DhX*)uniform wall heat flux boundary conditions@thermal boundary condition referring to uniform axial wall heat flux withuniform peripheral wall temperaturethermal boundary condition referring to axially and circumferentially uniformwall heat fluxHeH~H~hheJi( )Kr(x)conductive thermal boundary conditionthermal boundary condition referring to exponential wall heat fluxhelical coil numberlength of the fin on the major axis in an elliptic duct, mlength of the fin on the minor axis in an elliptic duct, mconvective heat transfer coefficient, W/(m2.K)convective heat transfer coefficient for the duct exterior, W/(m2.K)Bessel functions of the first kind and orders 0 or 1 corresponding to i = 0 or 1wall conductivity parameter = ks/kw8~klincremental pressure drop number, defined by Eq.
5.5thermal conductivity, W/(m.K)length of fin, ml*relative length of fins =//a.,~og~8~'~B..~ ~ ~ ,,8=:~~-o~~" p,~~-~-~=••~o"--~*.,~- -~'~*T,*"-.*"~**~ ~~~~~I~-,~00~_ - - ~""~~~~~--~•r~~,,~°o0o~-""0=~= =~ =-~,=IZL~,i..1o~8 ~"~*oo~~~°..=,~-"~~E~,,=~~,.~=~0,Z.,'"I=. ,.-.~~.9.~~_~..~.
~oo-.=g- ~rD0~.~~bo" ~o00~~" ~ .~., ~ ~ , ~g-=~="~=~" = ' Z~E~II~,,E"~~>FORCED CONVECTION, INTERNAL FLOW IN DUCTS5.123dimensionless parameter for eccentric annular duct; thermal energy sourcefunction, rate of thermal energy generated per unit volume of the fluid, W / m3duct dimension, mSSD2/k(Tw, m- Te) for 03 boundary condition;S*thermal energy source number,Sk= SDh/q" for @ boundary conditionsStark number = ewGT3Dh/kTfluid temperature, KT,TmTwfluid bulk mean temperature, Kwall temperature at the inside duct periphery, KTw, mcircumferentially averaged wall temperature, KT*w,maxdimensionless maximum wall temperatureZ~w,mindimensionless minimum wall temperature®uniform wall temperature boundary condition=ambient fluid temperature, K,convection boundary condition@radiative boundary conditionufluid velocity, fluid axial velocity in x direction, m/sUmfluid mean axial velocity, m/sUmaxfluid maximum axial velocity for fully developed flow, rn/sUtturbulent friction or shear velocity = V~Xw/p,m/sU+wall coordinate = u/u,, dimensionlessI/fluid velocity component in y or r direction, m/sWfluid velocity component in the z or 0 direction, m/sXaxial (streamwise) coordinate in the Cartesian or cylindrical coordinatesystem, mX+dimensionless axial coordinate for the hydrodynamic entrance region, = x,/Dh ReX*dimensionless axial coordinate for the thermal entrance region, = X/Dh PeXLtwist ratioyCartesian coordinate across the flow cross section, m; distance measured fromthe duct wall, my+wall coordinate=yut/vYdistance of the centroid of the duct cross section measured from the base, mYmaxnormal distance from the base to the point where umax occurs in the ductcross section, mCartesian coordinates across the flow cross section, m; distance from the apexof a triangle, mGreek Symbolst~fluid thermal diffusivity = k/pcp, m2/stx*aspect ratio of a rectangular channel = 2b/2a; ratio of the minor axis to themajor axis of an elliptic duct, 2b/2a13coefficient of thermal expansion, 1/K~neigenvalues5.124CHAPTER FIVEr( )gamma functionYdimensionless parameter defined by Eq.
5.24; ratio of heat fluxes at two wallsof a parallel plate duct8hydrodynamic boundary layer thickness, m; thickness of a twisted tape, m8wduct wall thickness, mEdistance between centers of two circles of an eccentric annular duct, m;amplitude of a circular duct with sinusoidal corrugations, m; roughness of ductwall, m~wOemissivity of the duct wall material; eddy diffusivity, m2/sOmdimensionless fluid bulk mean temperature = (Tin- Tw)/(Te- Tw)dimensionless fluid temperature for the boundary condition of axiallyconstant wall heat flux, = ( T - Te)/q,'Dflkdimensionless fluid temperature for a doubly connected duct, defined in Shahand London [1]dimensionless circumferentially averaged wall temperature (l = i for the innerwall, l = o for outer wall) for the fundamental boundary condition of kind kwhen the inner or outer wall ( j = i or o) is heated or cooled; dimensionlessfluid bulk mean temperature if ! = mfluid bulk mean temperature for the fundamental boundary condition of kindk when the inner wall ( j = i) or outer wall ( j = o) is heated or cooledo~influence coefficients derived from the fundamental solutions of the secondkind, = (o(2) 0(2)'~/[0(2) 0(2)`\ " m o ~ ".'io l ' \ v i i -- mi )o~influence coefficients derived from the fundamental solutions of the secondkind, = (o(2) o" o(2>(2) O(2)'~\vmii )/(0oo....
lfluid dynamic viscosity coefficient, Pa-svfluid kinematic viscosity coefficient = la/p, m2/sPfluid density, kg/m 3Stefan-Boltzmann constant = 5.6697 × 10-8 W/(m2.K 4)wall shear stress, Pa(k)ljdimensionless heat flux at a point in the flow field for the jth wall of a doublyconnected duct, defined in Ref. 1,~(k) = q~,Dh/dimensionless wall heat flow defined in a manner similar to ,,tjk ( T j - Te) for k = 1, 3; = q"/q~ for k = 2, 4(ff~Jm, Tdimensionless mean wall heat flux for boundary condition of axially constantwall temperature, = q"Dh/k( 7",, - T~)~x,Tdimensionless local wall heat flux for boundary condition of axially constantwall temperature, = q.~'Dh/k( T~ - Te)07>, O(o'>dimensionless heat flux at a point in the flow field for the inner or outer wallof a concentric or eccentric annular ductapex angle or half-apex angle of a duct; angle of tube curvatured~coefficient, defined by Eq.
5.250Subscriptsbcthermal boundary conditioncenter, centroid, or curvedfinned ductinitial value at the entrance of the duct or where the heat transfer beginsFORCED CONVECTION, INTERNAL FLOW IN DUCTS/fdfluidfinlessfinless ductH® boundary condition5.125fully developed flowH1boundary conditionH2@ boundary conditionH4boundary conditionH5boundary conditionhyhydrodynamiciinner surface of a doubly connected ductininletJheated wall of a doubly connected duct, = i or ollaminar flowmmeanmaxmaximumminminimumoouter surface of a doubly connected ductPperipheral valueSsmooth, straight ductslugslug flowT03 boundary conditionT3@ boundary conditionT4@ boundary conditionstturbulentththermalXan arbitrary section along the duct length; a local value as opposed to a m e a nvalue; axialwwall or fluid at the walloofully developed value at x =ooREFERENCES1. R. K.
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