Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 97
Текст из файла (страница 97)
For Pr near unity, the growth factor is closer to Pr -°28.The adiabatic wall temperature (recovery temperature) is given by2UeTaw = Te + r(0) 2Cp(6.31)Figure 6.4 shows the dependence of the recovery factor r(0) on Pr as given by Refs. 2 and 5(solid line). In the region 0.5 < Pr < 2, the formular(0) = er 1/2(6.32)6.8CHAPTER SIXI001.92 P r l / 3 - ~ , , . ~,.- ~ "~ ~"r.,~ SI0I,.,.,..c',pr,,2f .-p r - Z 3_./._- % ,OJo""~'-AO0/IIO1001000F I G U R E 6.4 Influence of Prandtl number on the recovery factor and modifiedReynolds analogy for a laminar boundary layer on a flat plate.represents the calculated values to within 1 percent. For Pr = 7, Eq.
6.32 yields results high by5.4 percent. The dashed line labeled A in Fig. 6.4 represents an extrapolation of the exactnumerical results for Pr < 15 to approach asymptotically the limiting valuer(0) = 1.92Pr ~'3(6.33)resulting explicitly when a linear velocity distribution exists throughout the thermal boundarylayer.For a uniform-surface-temperature plate in high-speed flow, the temperature distributionwithin the boundary layer is expressed by a superposition of the two Pohlhausen solutions [3].This is permissible because the energy equation (Eq. 6.8) with constant properties is linear intemperature. The general solution of the energy equation is the sum of the general solution ofthe homogeneous equation (Eq.
6.18) and a particular solution of the inhomogeneous equation (Eq. 6.26):. e~T - Te = (Tw- Taw)I1o(11)+ ~r(rl)(6.34)Y0(rl) and r(rl) (Eqs. 6.19 and 6.28) are indicated in Figs. 6.2 and 6.3, respectively. The localheat flux is expressed asq " = Pe/peUeCp St (Tw - Taw)(6.35)The appropriate Stanton number is again represented by Eq. 6.23, Taw is given by Eq. 6.31,and r(0) is given by Eq. 6.32 or Fig. 6.4.Liquids With Variable Viscosity. When the temperature difference between a liquid anda surface becomes significant, it is necessary to consider the temperature dependence of theviscosity across the boundary layer. Calculations of convective heating were made [7] for aliquid whose viscosity varies as~wT+T~FORCED CONVECTION,EXTERNALFLOWS6.9where b and Tc are constants. The boundary layer equations are solved through a transformation of independent variables identical in form with Eq.
6.11, but with the kinematic viscosityv evaluated at the surface temperature. The resulting transformed momentum equation isf") '(~ww+-yl f f , , =0and the energy equation, where viscous dissipation has been neglected, is identical with Eq.6.20, but with Pr evaluated at the wall temperature. In Ref. 7 the form of the solution requiresa choice of the constant b (b = 3 in most of the examples) but avoids the necessity of predetermining an explicit value of the constant To. The skin friction and heat transfer areexpressed directly in terms of the viscosity ratio across the boundary layer ~w/~te and thePrandtl number at the surface.Figure 6.5 shows the velocity distributions in a boundary layer of a liquid with Prw = 100(e.g., sulfuric acid at room temperature). For this Prandtl number, the thermal boundary layerpenetration into the liquid is much less than the flow boundary layer, and the regions whereviscosity variations occur are confined close to the surface.
The curve corresponding to}.tw/l.te= 1 is the Blasius solution (see Fig. 6.1). The curve labeled ~w/l.te = 0.23 corresponds to aheated surface where the low viscosity near the surface requires steeper velocity gradients tomaintain a continuity of shear with the outer portion of the boundary layer. The heated freestream cases reveal the opposite effects. In general, the outer portions of the flow boundarylayers act similarly to the velocity distribution of the Blasius case except for their being displaced in or out by the effects that have taken place in the thermal boundary layer.0.80,6oOA0.2/¢/////23456y ~/Ue/(VEX)FIGURE 6.5 Velocityprofiles in the laminar boundary layer of a liquid,Prw = 100 [7].The temperature profiles for different Prw and ~w/~l, e a r e indicated in Fig.
6.6. Note that thecurves for Prw = 10 apply equally well for greater Prandtl numbers because of the use of thethermal boundary layer thickness parameter as the abscissa (see Fig. 6.2).The effects of the viscosity variation across the boundary layer on the surface shear stress andheat flux are shown in Fig.
6.7. The shear stress is normalized by the value obtained from the Bla-6.10CHAPTER SIX1.0~10"81: ~ ~P%->10I!°.6-iioii°'°0.4 I,i0.20I23456Pr1/a y Vue/vwxFIGURE 6.6 Temperature profiles in the laminar boundary layer of aliquid [7].sius solution with the same free-stream properties. The heat flux is normalized by the Pohlhausen value with the viscosity and Prandtl number evaluated at the wall temperature. Note thatat the higher Prandtl numbers the wall shear becomes less dependent on the fluid properties.Ideal Gases at High Temperatures. The speeds of m o d e m military aircraft, missiles, or reentry bodies are so high that the resulting recovery temperatures are several times to orders of2r::Lovxo0v~= •,~----~--c~z~<)~~0.80.60.06 O080.10.2040.6 0.8 I2P~5000IOO0IOOI0.Ix!I468 I0/.',w/~.eF I G U R E 6.7 Effects of viscosity changes across a laminar liquid boundary layer on surfaceshear and heat flux--reference shear from Blasius solution with free-stream properties, reference heat flux from Pohlhausen solution with wall properties [7].FORCED CONVECTION, EXTERNAL FLOWS6.11magnitude larger than the ambient atmospheric temperatures.
Under these conditions, thebehavior of atmospheric gases within the boundary layer changes from that of an ideal gas tothat of a real gas, including the physical effects of rotational and vibrational excitation, dissociation, and even ionization. The real-gas behavior is so complex that numerical analysis is the onlymeans of introducing it into boundary layer problems. Because the cases of simpler gas behavior are directly applicable to many aircraft problems and are guides for correlating the numerical results of the real-gas computations, examples of ideal-gas solutions will be presented first.Equations 6.6--6.8 are converted to a convenient form through the Howarth-Dorodnitsyntransformation of the independent variables [8] from x, y to ~, 11 as follows:--" Ue~[rPrX11=U~~ex~' PP, dy(6.36)The subscript r represents a reference condition for the properties, usually the conditions atthe edge of the boundary layer or at the surface.The transformed momentum and energy equations for a uniform-temperature flat plate are(Crf")" + ½ ff"= 0(6.37)]\--~r / +2if' +-)7 Cr 1---~-r fir' =0where_ u__f' - u~i=1ffeCr-~,l,pbt.trPr011 -- ((6.38))t(6.39)with the boundary conditions-1"1=0 f = f ' = 0I-11--+oo f ' + l ,I+1iwleThe fluid properties are introduced into these equations only through Cr and Pr, whichinvolve combinations of individual physical properties.
Certain values of Cr and Pr permitsimplifications that lead to useful general relationships.Prandtl Number Equal to Unity. If Pr = 1, considerable simplification results. Equation6.38 acquires a form identical with Eq. 6.37, I being analogous to f'. A solution of the energyequation, therefore, is directly expressible in terms of the velocity distribution as[= 1-iwre) f'+feiw(6.40)after the boundary conditions at the surface and boundary layer edge have_ been introduced.When the wall enthalpy equals the total enthalpy, Eq. 6.40 indicates that I = 1, i.e., the totalenthalpy is constant throughout the boundary layer. For other wall enthalpies, the local totalenthalpy is linearly dependent on the local velocity.
The corresponding static enthalpy distribution is_i __ 1 + iw--Ie (1 --f') + UZe (1 _f,2)iele(6.41)XThe Reynolds analogy is given bySt = cl2(6.42)6.12C H A P T E R SIXwith the requirement that the recovery enthalpy bei,w = Ieorr(0) = 1Note that cl/2 in Eq. 6.42 differs in magnitude from that given by Eq. 6.16 because of thedeparture from the Blasius solution by the existence of Cr(rl)in Eq. 6.37.Viscosity-Density Product Equal to a Constant.
If Cr = C and Pr are constant, a naturaltransformation suggests itself [9], where 11 is replaced by11(6.43)~ ' - V-~andF'(rl0 = f'(rl)(6.44)l(rl~) =/(rl)(6.45)F" + V2FF" = 0(6.46)Equations 6.37 and 6.38 becomeU~eI" + 1Apr FI' + - ; - ( P r - 1)(F'F")'= 0le(6.47)with boundary conditionsrio=0F=F'=0,rio ~ oo F' --->1,l=lwI ---> 1The assumption of constant Cr, therefore, permits separation of the momentum equationfrom its dependence on the energy equation and results in an energy equation that is linear in1Iso that general solutions can be obtained from a superposition of individual solutions.Equation 6.46 with its boundary conditions is the Blasius problem again. The energy equation is satisfied byiIS-iw- iawieU2eY°(1]c)+ ~r(rlc) + 1(6.48)where Y0 and r are obtainable from Figs.
6.2 and 6.3 when tin is adjusted according to Eq. 6.43.The recovery factor r(0) is independent of C and therefore is identical to the constantproperty value given in Fig. 6.4 and Eq. 6.32.cI0.332 ~twpw/~ePe {Cf~l, wPw/~ePe(6.49)The term (ci/2)~ represents the skin friction coefficient corresponding to a constant-propertyboundary layer at the same local length Reynolds number. Similarly, the Stanton number isgiven bySt - 0.332 gwPw/gePe pr_2/3=Sti gwP,,/gePe(6.50)where Sti is the corresponding constant-property Stanton number.For a gas that satisfies the perfect gas equation of state and whose viscosity obeys theequation~t~.LeT-Te(6.51)FORCED CONVECTION,EXTERNALFLOWS6.13mwhere Te is the reference condition, the constant C becomes-C: Ce- ~p - ~P~ = 1[-tePe ~tePe(6.52)ThUS, Eqs. 6.49 and 6.50 indicate that the skin friction coefficient and Stanton number remainequal to their constant-property values.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
