Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 100
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The simplest ablation mechanism is the sublimation of a homogeneous material. More complex ablation involves the thermal degradation of composite materials suchas reinforced plastics where a heat-absorbing pyrolysis occurs below the surface. Thegaseous products of pyrolysis pass through a carbonaceous char, gaining additional sensibleheat or chemical heat through endothermic reactions, and then pass into the boundary layerto absorb additional heat.
Ablation cooling has the enormous advantage of being selfcontrolled and requiring no active elements. The disadvantages are that surface dimensionsare usually altered, part of the char is eroded mechanically by shear forces rather thanthrough heat-absorbing phase change, and often the heavy gaseous products are not aseffective in blocking the incoming convective heat as light gases. Furthermore, ablation systems are generally not reusable.
For short-time applications, dimensional stability has beenachieved in ablation systems by employing porous refractory metal surfaces that have beenimpregnated with lower-melting-temperature metals that absorb heat by melting, vaporizing, and transpiring through the porous refractory matrix.A film cooling system is one where a surface is protected by a film of coolant introducedinto the boundary layer from either a finite length of porous surface or a slot upstream. A liquid can be used as the coolant to absorb heat by vaporization as it is drawn along the surfaceby the main stream gas. A severe limitation on such systems is the requirement that gravity orinertial forces act in a direction that will keep the liquid film stable and against the surface. Inaddition, a film cooling system requires all the active elements of a transpiration cooling system.
Its main advantage over the latter is in the simpler construction of limited areas ofporous surfaces or slots and in localized ducting.The boundary layer behavior over a continuously transpiration-cooled surface and anablation-cooled surface is generally the same, differing only as a result of the specific chemical identity of the coolant. The effects of a porous surface when the pore size is below somethreshold dimension that is a small fraction of the local boundary layer thickness, and of theflow of liquid char over ablating surfaces, do not appreciably alter the behavior of the boundary layer and can be neglected in design considerations. Thus, boundary layer theory withcontinuous mass injection is applicable to both types of cooling systems.
Further, results ofexperiments involving transpiration systems can be utilized in the prediction of the behaviorof ablation systems. In film cooling systems, because of the discontinuities formed by slots orlimited porous regions, the boundary-layer profiles at various stations along the surface aredissimilar so that prediction methods are quite complex and rely on experimental data orrather complicated numerical analyses.Uniform Surface Temperature, Air as Coolant. When the boundary layer and coolantgases are the same, the equations controlling boundary layer behavior are Eqs.
6.6-6.8. Themass injection at the surface simply alters the boundary conditions (Eq. 6.9) at the wall to be6.22C H A P T E R SIXx>0, y=0u = 0 , V=Vw(X)~)i-~v = 0I = constant = iw or i,,,, i.e.,,wheref(0) = -t2pwV_______~Wpeue ~PeUeX~j~e.--(6.68)As boundary layer similarity requires f(0) to be independent of x, pwVw must be proportional tox -1/2. A simple heat balance on an element of a porous surface with a transpired gas, or of a subliming surface, reveals that this distribution of gaseous injection is uniquely compatible withthe requirements of a constant surface temperature. Thus, Pw = constant, and Vw - x -1~2. Thismass injection distribution has practical importance because the porous surface can operate atits maximum allowable temperature everywhere, thereby minimizing the coolant required.The boundary layer equations, together with these boundary conditions, were solved in aseries of similarity theories such as those described in the section on the two-dimensional laminar boundary layer, beginning with solutions for constant properties and progressing to idealgases with variable properties.
A rather complete bibliography of these theories and corroborating experiments is given in Ref. 27. The assumption of constant C e ~P/~ePe provedequally useful in this problem as with the impervious plate in extending constant-propertysolutions to high Mach number cases where air still behaves as an ideal gas. The results shownhere are predominantly from Refs. 28-30.It is found in these analyses that blowing, i.e., negative values of f(0), thickens both the flowand thermal boundary layers. In addition, the velocity profiles take on an S shape characteristicof boundary layers approaching separation (see Fig.
6.19). Separation, (3u/by)w = 0, occurs whenf(0) = - 1 . 2 3 8 [28]. These S-shaped profiles are less stable, thereby decreasing transitionReynolds numbers with increased blowing rates. Near the surface, blowing reduces the velocityand temperature gradients and the corresponding shear and heat transfer rates, respectively.The heat transfer does not drop as rapidly as the shear and, consequently, Reynolds analogybecomes dependent on the blowing rate. The recovery factor, r(0), also depends on the blowingrate when Pr does not equal unity.
For the case of a slender body where r(0)u2/2 >> I , , - le, thereduction in the heat flux qw is dependent on the product of the reduction in Stanton numberand the recovery factor. This is shown in Fig. 6.14 by the line labeled Air-Air.- -I . O-0.8L0.6:g.0.4_•I~/20-Air'~He-Air02,.-~,~-Air0O. I0.20.30.40.5PwV,,/(,%ue Sto)0.60.70.8FIGURE 6.14 Reductionof the local heat flux by surface mass transfer of a foreign gas,Tw- Te < r(O)U2e/2Cp,[31].FORCED CONVECTION,EXTERNALFLOWS6.23Uniform Surface Temperature, Foreign Gas as Coolant. The effectiveness of air injection inreducing convective heat flux stimulated investigations into the use of other coolants.
With theintroduction of a foreign species into the boundary layer, the boundary layer equations reduceto the continuity equation (Eq. 6.6), the momentum equation (Eq. 6.7), the energy equationpu-~-fxx+PV-~yy=by~-ffy-y-(il-i2)(1-Le)-~y j+g 1--~r Fbyand the diffusion equationbK1/)K1b ( g L e bK1)Pu--~--x +Pv by - by PrF by(6.70)Here, PrF/Le = Sc is the Schmidt number, K1 is the coolant mass concentration, and ia and i2are the coolant and air enthalpies, respectively. These equations are the bases for the bulk ofthe analyses involving foreign gas transpiration.The boundary conditions for Eqs. 6.69 and 6.70 arex=0, y>0K~=0x > O, y ---) oo K1---) Oy =0K1 = KlwThe value of Klw is dependent on the blowing rate, and some hypothesis is required to establish this dependency.
Most authors have assumed a zero net mass transfer of air into the surface. This requires that the air carried away from the surface by the mass motion normal to thesurface be balanced by the diffusion of air toward the surface. Since/(1 = 1 - K2 and J1 = -J2, thisbalance of air motion can be expressed directly in terms of the coolant properties asPw~)12 ( bgl I _]lw(6.71)Equation 6.71 is the required relationship between the blowing rate of the coolant and its concentration at the wall. When the diffusion flux is expressed in terms of the mass transfer coefficient, the concentration of the coolant at the wall is given by1K~w = 1 + (peUe/PwVw)Cml(6.72)Note that the mass transfer coefficient is defined differently in Ref.
31.Although the total heat flux at the surface in a binary gas is composed of the sum of a conduction term and a diffusion term, the results of analyses are expressed solely in terms of theheat conduction term. The reason is that this term is equal to the heat gained by the coolantin passing from its reservoir to the surface in contact with the boundary layer. This simpleheat balance is\-~y ]w= pwVw(ilw- ilc)(6.73)where the subscript c represents the initial coolant condition in the reservoir. Further, therecovery factor is defined in terms of the surface temperature that results in zero heat conduction at the surface for a given mass transfer rate.
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