Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 99
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This correlation technique was expressed originally interms of a reference temperature 7" [12, 18-20] and later as a reference enthalpy i' to accountfor variations in specific heat [14, 21].A convenient form of the reference enthalpy isi'~=iea * + b * iw~+C*teiawieOn evaluation of the coefficients a*, b*, and c* based on the van Driest values of skin frictionin Fig. 6.11, it is found that they differ from those given in the earlier references. Therefore,the following formula is adopted:i'ieiwiawie-- = 0.32 + 0.50 ~ + 0.18 ~le(6.62)A convenient form of the skin friction coefficient compatible with the Crocco and vanDriest formulations isc,,0.332I(i'V,+0o2 - ~x~ex, .///~\-~e] i'/ie+O0(6.63)Note that Eq.
6.62 differs slightly from the popular Eckert relationshipi"ieiwlei~wie-- = 0.28 + 0.50 - - + 0.22 ~(6.64)The relative accuracies of these formulations are indicated in Table 6.2 for the skin frictioncoefficients shown in Figs. 6.10 and 6.11.6.18CHAPTERSIXComparison of Skin Friction CoefficientsFrom Reference Enthalpy MethodsTABLE 6.2% RMS error [5]% RMS error [11]SourceAir00=000=½00=100=3Eq. 6.64Eq. 6.622.00.642.31.21.10.300.650.500.830.64The figures of merit indicated in this table represent the RMS errors of the formulas at thematrix of points Mae = 0 and 5, iw/ie = 0.25, 2.0, and iaw/ie in Fig. 6.10 for 00 = 3 and for similarpoints for the other values of 00; and at Mae = 0, 5, and 10, iw/ie = 0.25, 6, and iaw/ie in Fig.
6.11for 00 = 0.505.From Table 6.2 for air, it can be seen that Eq. 6.62 gives some improvement in comparisonwith the van Driest results over the older formulas. At the high speeds where air behaves as areal gas, Wilson [15] shows that equations equivalent to Eqs. 6.61 and 6.62 yield skin frictioncoefficients that agree with those found from numerical integrations of the boundary layerequations to within 5 percent for total enthalpies corresponding to free-stream speeds up to25,000 ft/s (7620 m/s).
Similar close agreement is achieved between the use of the Eckert reference enthalpy and results of Cohen [16].For real air, the total Prandtl number varies in an oscillating manner with enthalpy distribution across the boundary layer. In view of this behavior, it would not be expected a priorithat evaluation of Prr at the reference enthalpy would be appropriate for evaluating therecovery factor from Eq. 6.32 or the modified Reynolds analogy from Eq. 6.24. Comparisonwith the numerical results of Refs. 12, 13, and 15, however, reveals that this interpretation ofthe reference enthalpy technique yields results for the recovery factor correct to within 2.5percent, and for the Reynolds analogy correct to within 5 percent.
Note that Wilson [15] suggests the use of Prrw evaluated at wall enthalpy in the Reynolds analogy. Comparison of thismethod with the use of Pr~ evaluated at the reference enthalpy for surface temperaturesbelow the sublimation temperature of carbon reveals little difference. Because Pr~ ratherthan Prw or Pre yields better agreement with the modified Reynolds analogy of van Driest [13],the consistent use of Pr~ in both the recovery factor and Reynolds analogy as suggested byEckert is still appropriate.
Further, when the assumption that Le = 1 in Ref. 16 is interpretedas equivalent to the assumption that PrF = Prr, the use of Pr~ based on the reference enthalpyfor the recovery factor and Reynolds analogy is again borne out by these independent calculations to the accuracies quoted previously.With the onset of ionization, the reference enthalpy technique yields results that begin todepart from the exact calculations, and recourse to the latter [16, 17] is recommended foraccurate predictions.Nonuniform Surface Temperature.The previous section was devoted to uniformtemperature plates.
In practice, however, this ideal condition seldom occurs, and it is necessary to account for the effects of surface temperature variations along the plate on the localand average convective heat transfer rates. This is required especially in the regions directlydownstream of surface temperature discontinuities, e.g., at seams between dissimilar structural elements in poor thermal contact. These effects cannot be accounted for by merely utilizing heat transfer coefficients corresponding to a uniform surface temperature coupled withthe local enthalpy or temperature potentials. Such an approach not only leads to seriouserrors in magnitude of the local heat flux, but can yield the wrong direction, i.e., whether theheat flow is into or out of the surface.It has been shown that, for property variations for which superposition of solutions is permitted, a series of solutions corresponding to a step in surface temperature can be utilized torepresent an arbitrary surface temperature [22].
This approach is identical with the Duhamelmethod used in heat conduction problems to satisfy time-dependent boundary conditionsFORCED CONVECTION, EXTERNAL FLOWS6. ] 9with a series of solutions involving sudden changes in surface boundary conditions [23]. Theresulting convective heat flux distribution expressed in terms of the surface enthalpy distribution isq~'(x) =peUeSt (x, 0) [iw(0) - i~] +fox St (x, s)St (x, 0)d[i~(s)- iaw]ds+dsSt (x, sj)}[i~(s;) -i~(sT) ]j:l St (x, 0)(6.65)Here, St (x, s) represents the local Stanton number on a plate at a uniform temperature forx > s preceded by an unheated portion of length s, and St (x, 0) is the Stanton number on aplate with a uniform temperature over its entire length.
The first term in parentheses in theenthalpy potential arises from the difference between the leading-edge enthalpy of the plateand the recovery enthalpy. The integral term accounts for the portions where continuous surface enthalpy variations occur. The last term sums over a k number of discontinuous jumps insurface enthalpy that may occur downstream of the leading edge. The terms iw(sT) and iw(S;)represent the surface enthalpy just upstream and downstream of sj where the ]th jump inenthalpy occurs.The effect of a stepwise discontinuity in surface temperature on a flat plate can beexpressed ast,x s, E (s/3411,3St (x, 0) = 1 -(6.66)This closed-form equation was obtained through similarity solutions of the energy equation byinvestigators who assumed that the velocity profile in the boundary layer is linear in rio for thecase of constant Ce and Prr or in y for the case of constant fluid properties [24, 25].
Note thatthe right side does not contain terms involving the fluid properties, a direct consequence of Ceand Prr being assumed constant throughout the boundary layer. Again, an intuitive approachto include property variations is to use the local surface enthalpy in the reference enthalpytechnique for evaluating St (x, 0). The stepwise discontinuous-surface-temperature solutionsare used solely to define the functional form of the enthalpy potential appropriate to an arbitrary surface temperature.
A plot of Eq. 6.66 is given in Fig. 6.12 (13p= 0 for a flat plate).The preceding theory has been verified by several experiments. For example, in Ref. 26,local heat transfer rates were measured in the presence of ramplike temperature distributionsthat began downstream of the leading edge (see inset in Fig. 6.13). The data shown in Fig. 6.13agree with the theory (solid line) to within +10 percent, the estimated accuracy of the data.The dot-dashed line in the figure represents the use of a local temperature potential in estimating the heat flux and yields large errors for this particular form of the surface temperaturedistribution.
Had the leading edge been raised above the recovery temperature, the error inneglecting the variable-surface-temperature effects would have diminished. In general, if continuous variations in the surface temperature or enthalpy are large compared to the overalldriving potential, the variable-surface-temperature methods must be utilized. For discontinuous surface temperatures, much smaller variations are important.Stepwise and Arbitrary Heat Flux Distribution.
It is often necessary to evaluate the surface temperatures resulting from a prescribed heat flux distribution. The superposition ofsolutions yields the surface enthalpy distribution asiw(X) - iaw=Surface With Mass Transfer.peUe0.207foxq~'(x')dx"St (x, 0)x[1 (Xt]X)3'4] 213-(6.67)-An effective method of alleviating the intense convectiveheating of surfaces exposed to very-high-enthalpy streams is by means of mass transfercooling systems.
The coolant is introduced, usually in gaseous form, into the hot boundary6.20CHAPTERSIX2.4r2.22.0o/-1.8.,,....03tnO~0.5~~.61.4"1.000.2/0.40.8061.0s/xF I G U R E 6.12 Effect of a stepwise surface temperaturediscontinuity on the local Stanton numberNEq. 6.66 for aflat plate (13p- 0 ) and Eq. 6.124 for a surface with a pressuregradient (13p> 0)._,61I1I.,.Xo~.00.o~.0.8x~o.,•O.6'%O.40.200.10.2O.B0.40.50.60.7080.9i.oXo/XF I G U R E 6.13 Comparison of data and theory for a flat plate with a delayed ramp surface temperature distribution [26].FORCED CONVECTION,EXTERNALFLOWS6.21layer through the surface being protected. Mass transfer cooling is particularly applicableto leading edges of wings, fins, and nose tips of aircraft; turbine blades; and reentry capsules and missile nose cones.
The types of systems include transpiration, ablation, and filmcooling.A transpiration cooling system is characterized by a porous surface material that remainsintact while the coolant is being pumped through the pores toward the hot boundary layer.The coolant may be a gas or a liquid that changes phase within the porous surface or after itemerges from the surface. This system operates best when pore sizes are so small that thecoolant leaves at the surface temperature of the porous solid.
These systems are complicatedin that they require coolant storage vessels, pumps, controls, distribution ducts, and filters toavoid pore clogging. It is also difficult to fabricate strong, aerodynamically smooth poroussurfaces. Another drawback of these systems is that they are unstable because a clogged poreresulting from local overheating seals off the flow of coolant and causes local failure. Theadvantages of transpiration cooling systems are their versatility in the choice of coolant andcoolant distribution, the reusability of the system, and the retention of intact aerodynamicallycontoured surfaces.An ablation cooling system is one where the gas entering the boundary layer has beengenerated by the thermal destruction of a sacrificial solid thermally protecting an underlying structure.
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