Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 98
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In terms of these dimensionless transfer coefficients,the effects of the linear dependence of viscosity on temperature just cancel those of the perfect gas variation of the density. It should be noted, however, that the density variation itselfstill affects the boundary layer thickness.For a constant-temperature plate, Chapman and Rubesin [9] modified Eq.
6.51['tl~e_Cew ~r(6.53)in order to approximate better the actual viscosity distribution near the surface. Equation6.53 with the perfect gas equation of state yieldsC= Cew-~twPw~ePeand Eqs. 6.49 and 6.50 becomec. (c.)2-"2"- i6,4.[.l,ep eS t = ~Q Pr -2/3(6.55)./~twpwSt = S t i ¥ ~(6.56)The above skin friction relationship was deduced intuitively many years earlier by vonK~irm~in [10], who assumed local wall properties would control the skin friction law whenproperty variations occur.
Thus, Eq. 6.54 is equivalent to(c,)5 w- V wu i.wwhere"t:w=pwu2(~£)~Similarly, since Pr has been assumed constant in Eq. 6.47, the modified Reynolds analogyalso applies under these conditions withStw = (ff)w pr-2/3whereq~' = Stw pwUe(iw-- iaw)Sutherland Law Viscosity.
Crocco [5] solved equations equivalent to Eqs. 6.37 and 6.38utilizing the rather accurate Sutherland viscosity relationship and a constant value of Pr otherthan unity. For a gas that satisfies the ideal equation of state, the quantity Cr, referred to freestream conditions, becomesCr_ ~tP - ~ (~ePe1+0° )T/Te + Oo(6.57)6.14CHAPTERSIXwhere O0= TsclTe and Tsc is the Sutherland constant corresponding to the specific gas.
Valuesof 00 are indicated in Table 6.1 for a variety of gases and boundary layer edge temperaturescharacteristic of those occurring in the stratosphere, under room conditions, and in productsof combustion. Crocco obtained numerical results for 00 = 0, ½, 1, and 3. Because enthalpy isemployed as the thermodynamic dependent variable in Eq. 6.38 to account for specific heatvariations, it is necessary to express Ce in terms of enthalpy rather than temperature as in Eq.6.57. This is no problem when attention is confined to a specific gas where the enthalpy andtemperature are uniquely related at specified pressures. Crocco, however, chose to avoid thisapproach because it would confine his results to specific gases and thermodynamic conditions.To retain generality, he made the assumption that the temperature ratio in Eq.
6.57 can bereplaced by the enthalpy ratio i/ie without introducing serious errors.TABLE 6.1 Prandtl Number and Sutherland Constant for Gases [5]00 = Tsc/TeSutherlandconstantT~c,KTeTeTeGasPr,T= 230 K218 K300 K3000 KH2CON2Air02CO2H200.7170.7650.7390.7250.7310.8051.08901041121161312666730.4130.4770.5140.5320.6011.2203.090.3000.3470.3730.3870.4370.8872.240.0300.0350.0370.0390.0440.0890.224A major result of Crocco's numerical solutions was the discovery that the functionaldependence of the local enthalpy on the local velocity is independent of the particular law ofviscosity employed.
Thus, i ( f ' ) found for the simplified case of Ce = 1 applies for all valuesof 0. The conclusions deduced from this discovery are that the modified Reynolds analogy ofEq. 6.24 or Fig. 6.4 and the recovery factor expression of Eq. 6.32 or Fig. 6.4 apply to all gases,regardless of their viscosity laws, as long as Prandtl number is constant.
Another significantconsequence of this discovery is that it simplifies the solution of Eqs. 6.37 and 6.38 by avoiding either a simultaneous solution of two differential equations or a sequential iteration process. The simpler process uses Eq. 6.48 and the Blasius solution to relate i and f'. Then Ce isevaluated in terms o f f ' through Eq. 6.57 with T/Te replaced by i/ie and with the proper 00, andEq.
6.37 is solved to yield the final velocity distribution. The local enthalpy distribution interms of the local velocity is given by2r(f'iiw- i,,,,,, Yo(f') + ue r(0) ~i--re= 1 + ie~r(0)(6.58)where the enthalpy profile functions Y0 and fir(0) are plotted in Figs. 6.8 and 6.9 for severalPrandtl numbers. Many authors have argued that Eq. 6.41 can be modified to account forPrandtl number deviations from unity by replacing Ie by iaw and multiplying the last term byr(0).
A comparison with Eq. 6.58 reveals that this suggestion is equivalent to retaining Y0 andr/r(0) characteristic of Pr = 1 for all Prandtl numbers. Reference to Figs. 6.8 and 6.9 indicatesthe errors introduced by this procedure. For example, on a plate at constant temperature, thelocal temperature at f ' = U/Ue = 0.5 is 10 percent higher at Pr = 0.725 from Fig. 6.8 than wouldbe given by the aforementioned rule.Correlation equations that fit these results within 0.015 of the ordinates areY0=1 - Pr ~'2 u _Ue// H \ 6"3Pr-1/2(1 - P r ~ / 3 ) / ~ /\ Ue ,](6.59)FORCEDCONVECTION,EXTERNALFLOWS6.15I.O0.8-Pr0.6"\ "q@,o,0.2 L00.20.40.60.81.0f ' , u/u eFIGURE 6.8 Laminar boundary layer enthalpy profilefunction on a uniform-temperature flat plate, Ce= 1 [5].0.80.60.4\\0.200.20.40.60.81.0f ' • U/UeFIGURE 6.9 Laminar boundary layer enthalpy profilefunction on an insulated flat plate, Ce= 1 [5].randr(O)-1-prl/2(u__u__121/2 /' u \7"3Pr-°38\Ue] -- (1 - Pr )~ Ue)(6.60)An example of the skin friction results obtained by Crocco for the case of 00 = 3 is shownin Fig.
6.10. The individual curves represent values for constant surface temperatures whereiw/ie = 0.25, 0.50, 1.0, 1.5, and 2.0, and for an insulated plate.For slender aircraft flying in the stratosphere, the temperature at the edge of the boundarylayer is 218 K (-67.6°F) and the value of 00 for air based on the Sutherland constant is 0.505.Van Driest [11] repeated Crocco's analysis for these conditions and Pr = 0.75. Graphs of the6.16CHAPTER SIX0.36IInsulated plateiw/iezo~0.34~"~~J04(Jv0.32JI0300 '0022345Mae0.81.8:5.25.0 ueZ/2ieFIGURE 6.10 Local laminar boundary layer skin friction coefficient on a flatplate at uniform temperature, Pr = 0.725 and 00 = 3 [5].local velocity, temperature, and Mach number profiles for an extensive range of conditionsare presented in Ref. 11.
The local skin friction coefficient is indicated in Fig. 6.11 for Mae upto 20, for wall temperatures 0.25 < iw/ie < 6, and for an insulated plate. Two examples of thesolutions based on a constant value of Ce-- Cew are indicated for comparison with the insulatedplate curve (circled points). Note that the use of wall properties underestimates the skin friction here by about 5 percent.Air as a Real Gas in Chemical Equilibrium. At reentry speeds, the high enthalpies introduce Prandtl number variations and the nonideal effects of dissociation and ionization in thebehavior of equilibrium air. Several studies [12-17] have determined the effects of these property variations on the behavior of the laminar boundary layer for successively increasingspeeds.
A characteristic common to these theories because of the complexity of the behaviorof air at elevated enthalpies is the reliance on completely numerical computation of a relatively limited number of examples. The results, however, are not markedly different from the0.34=-ll!iw/Je-.....030---..20.26~ 0.22Insulated plote J . ~040.18o Ce=Cew, Insulated plate0140.1002468I01214161820MaeFIGURE 6.11 Local skin friction coefficient for air flowing in a laminar boundary layer on aflat plate, Pr = 0.75, 00=0.505 [11].FORCED CONVECTION. EXTERNAL FLOWS6.17ideal-gas cases.
The variations in Prr cause the recovery factor to be dependent on the surfacetemperature and Mach number at the edge of the boundary layer. When this relatively smalleffect is taken into account, the skin friction and heat transfer coefficients exceed the vanDriest results of the previous section by less than 15 percent for enthalpies characteristic offlight speeds less than 25,000 ft/s (7620 m/s) and wall temperatures below the sublimationtemperature of carbon (6000°R or 3333 K). When ionization takes place, however, a markedincrease occurs in both the skin friction and heat transfer.Errors are inherent in the above solutions because of the uncertainties in the transport properties of air at very high temperatures.
The theories of Refs. 15 and 17 employed total properties from different sources, while Ref. 16 accounted for equilibrium air by using frozenproperties and Le = 1 in the diffusional heat flux contribution. A comparison of skin frictionand heat transfer coefficients reveals differences of less than 10 percent between the results ofRefs. 15 and 16 and only a few percent between the results of Refs. 16 and 17. Thus, prior to ionization the errors in convective heating predictions caused by property uncertainties are rathersmall.
With the onset of ionization, large errors may have been introduced because of the largeuncertainty of the thermal conductivity of ionized air as influenced primarily by the chargetransfer cross section of atomic nitrogen. Hence, the marked increase in heat transfer rate withthe presence of ionization [17] can only be considered qualitatively correct.A technique for correlating the results for the convective heating behavior of any ideal gasand real air is given in the following section.Reference Enthalpy Method. The behavior of the skin friction coefficient indicated inFigs. 6.10 and 6.11 can be correlated to a very good approximation by the modified incompressible formulacs0.332 / p ~ '(6.61)2 - RV-ff ex ~/P~ewhere the properties designated with the prime are evaluated at a reference enthalpy i' andthe boundary layer edge pressure.
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