Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 101
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This adiabatic condition is achieved inexperiments by setting Tic = Tw for a given mass flow rate so that the coolant gains no heat. Interms of the Stanton number, the heat balance indicated by Eq. 6.73 can be expressed asq" _qr~ opwVw ilw- ilcpeUeSt0 iZ~wo- i2~(6.74)6.24CHAPTER SIXThe subscript 0 in this equation signifies zero mass transfer at the surface (f(0) = 0). The subscript 2 indicates that the enthalpy potential contributing to the heat flux by conduction aloneis dependent only on the specific heat of the air and not that of the coolant.The effect of mass transfer on heat flux from slender bodies is shown in Fig.
6.14 for a variety of coolant gases. The corresponding mass transfer coefficients are shown in Fig. 6.15.1.0'-'IAir08xXN....0.6 ~~~~"--~J0.4 ~~lill00.10.20.30.40.50.60.70.8pwvw / (peueSt0 )FIGURE 6.15 Reduction of the local mass transfer coefficient by surface mass transfer of a foreign gas [31]. (Reprinted by permission of Pergamon Press.)An important result from these figures is that lighter gases are more effective in reducingthe transport coefficients and surface heat flux.
For a range of calculations including Machnumbers as high as 12 and surface temperatures from free-stream (392°R; 218 K) to recoverytemperatures, the maximum departure from these correlation lines of individual solutions is+15 percent for q" and +_25 percent for Cmi. T h e bulk of the discrete numerical results liewithin about one-third of these bandwidths. The maximum spread of results is obtained withthe mass transfer of hydrogen, and the spread is smaller with helium and much smaller withthe heavier gases. The differences in results from different calculations with the light gases aredue primarily to the use of different gas properties [32].Figures 6.14 and 6.15 are particularly useful for obtaining the mass transfer rate requiredin a transpiration cooling system.
Usually the following quantities are specified: the coolantand its reservoir conditions, the porous surface material and its maximum allowable surfacetemperature, and the inviscid flow conditions outside the boundary layer. For cases where thedifference between the surface temperature and the boundary layer edge temperature issmall compared to the temperature rise by frictional heating, Fig.
6.14 can be used directly.For the specified conditions, the factor (ilw - i l c)/ (i ~wo - i2w) can be readily established from thethermodynamic properties of the coolant and air and i2~w0= iaw. This factor is used as the slopeof a straight line that represents Eq. 6.74 and passes through the origin of Fig. 6.14. The intercept of this line with the appropriate heat blockage curve on the figure is the operating condition required to yield the specified surface temperature. The abscissa of this point yields therequired local mass transfer rate.The behavior of a coolant composed of mixtures of He, Ar, Xe, and N2 injected into anitrogen boundary layer has been analyzed in Ref.
33. Examination of the results reveals thatFORCED CONVECTION, EXTERNAL FLOWS6.25at a given mass transfer rate the skin friction coefficient is correlated for different coolantswithin +_5 percent using the mean molecular weight of the coolant. Thus, for a coolant composed of a mixture of n gases, the average molecular weight of the coolant at reservoir conditions defined as1M,a,, = Y.7 (K,c/M,)(6.75)can be used to interpolate between the curves of Figs. 6.14 and 6.15.Uniform Surface Injection.
Although a mass transfer distribution yielding a uniform surface temperature is most efficient, it is much easier to construct a porous surface with a uniform mass transfer distribution. Libby and Chen [34] have considered the effects of uniformforeign gas injection on the temperature distribution of a porous flat plate.
For these conditions, however, boundary layer similarity does not hold. Libby and Chen extended the workof Iglisch [35] and Lew and Fanucci [36], where direct numerical solutions of the partial differential equations were employed. An example of the nonuniform surface enthalpy andcoolant concentrations resulting from these calculations is shown in Fig.
6.16.tO,..-&'vu._/ J0.8._e/Ix0.6/_.._uto0.4.....,i0.20//I/0.10.2Pr F = Le = IAll Di} equal0.30.40.,50.60.7pwvwl(PeUe)"~peue x I/u.eFIGURE 6.16 Surface temperature and coolant concentration distribution along aplate with uniform foreign gas injection, Le = Pr = Ce= 1, all binary diffusion coefficientsequal [34]. (Reprinted by permission from The Physics of Fluids.)Film Cooling With Upstream Transpiration. Film cooling systems provide protection to asurface exposed to high-enthalpy streams by injecting a coolant into the hot boundary layerupstream of the surface. Injection can take place through local porous sections or slots at various angles to the surface.
The coolant may be the same gas as in the boundary layer, a foreigngas, or a liquid. In the upstream porous section, more coolant is provided than is required tomaintain safe surface temperatures. The excess coolant is entrained in the boundary layerclose to the surface and is carried downstream, providing an insulating layer between the hotfree-stream gas and the surface. This layer is dissipated by laminar mixing while flowing withthe boundary layer in such a way that protection is afforded over a limited distance.
Becauseof the discontinuous nature of the surface injection, the boundary layer downstream of the6.26CHAPTER SIXdiscontinuity is far from similar. Numerical solutions of higher-order integral equations [37]or of finite difference forms of the partial differential equations [38] have been used for evaluating the local convection to the surface.Examples of thermal protection offered by an upstream transpiration system are indicatedin Fig. 6.17 (see Ref. 37).
The figure shows the temperature rise that occurs on an insulatedsurface downstream of a transpiration-cooled section for two amounts of blowing. Also indicated are the corresponding surface temperatures for the case where the upstream sectionwas cooled internally, i.e., f(0) = 0.
The quantity TwL is the upstream surface temperature ineither case. The differences between the curves labeled f(0) = -0.5 and f(0) = -1 and the f(0)curve show how the presence of the transpired coolant within the downstream boundary layerdistorts the temperature profiles so as to afford greater downstream protection depending onthe amount of blowing.1.0O8f( 0 .
~-~'-o.5~~---~~ /0.6-"5'~, 0.4//;//r/0.20I2345x/LF I G U R E 6.17 Effect of upstream transpiration cooling on the temperature distribution of the impervious portion of a fiat plate.Cone in Supersonic Flow. The preceding solutions for a flat plate may be applied to acone in supersonic flow through the Mangler transformation [39], which in its most generalform relates the boundary layer flow over an arbitrary axisymmetric body to an equivalentflow over a two-dimensional body. This transformation is contained in Eq.
6.89, whichresults in transformed axisymmetric momentum and energy equations equivalent to thetwo-dimensional equations (Eqs. 6.95 and 6.97). Hence, solutions of these equations areapplicable to either a two-dimensional or an axisymmetric flow, the differences being contained solely in the coordinate transformations.For the case of an arbitrary pressure distribution, it is just as convenient to solve theaxisymmetric problem directly.
When the solution for the equivalent two-dimensional flowalready exists, however, as for a flat plate, then the results for the corresponding axisymmetric problem can be obtained by direct comparison. This correspondence exists for a cone insupersonic flow when the surface pressure is constant. Solutions of Eqs. 6.95 and 6.97 for a flatplate expressed in terms of ~ and rl may then be applied to a cone. Illustrative examples arepresented in the following subsections.FORCED CONVECTION,EXTERNALFLOWS6.27Uniform Surface Temperature.
Transformations (see Eq. 6.89) for a flat plate (k = 0) or acone (k = 1) become~- PwgwUe(sinOc)2a2 k l L2--k]x2k+I+(6.76)1"1=~/(2k + 1)Ue foe2pwg.,xP dywith Pw, gw, and Ueequal to constants, ~ = 0, and r = x sin 0c. The transformed momentum andenergy equations (Eqs. 6.95 and 6.97) are essentially the same as Eqs. 6.37 and 6.38 for a flatl~late.
For the same wall and boundary layer edge conditions, then, the solutions for f(rl) andI (rl) are the same for a flat plate or a cone. These results may be expressed in terms of physical variables as2pwla~fo~lY = ~ (2k~" 1)/./e-p- an(6.77)gw(3U~ =peUe2~(2k+ 1)ge g~Pw f"(O)Xw= \ by ]w2peUeX ktePe(6.78)• •kwle / ( 2 k + l ) k t e kLwPwP(o)q(~"= k {()T~-- \ ~Y ]w--- ~l.wCpw4 2peUeX ktePe(6.79)For a given value of x and the same flow properties, the boundary layer on a cone (k = 1) isthinner by a factor of l/X/3, and the surface shear stress and heat transfer are larger by afactor of V3. The local skin friction and heat transfer coefficients are related similarly:(Cf)cone_.(C/)flat plateStcone = X/~(6.80)Stflat plateThe local and average coefficients for a cone are related as follows:(~I)"~f cone --(St)4(6.81)"~- cone -" "~-These relationships may be used to obtain cone flow results from the fiat-plate results of thesection on uniform free-stream conditions.
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