o un U < RESEARCH MEMORANDUM SURVEY OF ADVANTAGES AND PROBLEMS ASSOCIATED WITH TRANSPIRATION COOLING AND FILM COOLING OF GAS-TURBINE BLADES By E. R. G. Eckert and Jack B. Esgar Lewis Flight Propulsion Laboratory- Cleveland, Ohio JlNlVERSITY OF FLORIDA I TV OF LIBRAE i Documents DEPAR-n^iENT a 20 MARSTON SCIENCE LIBRARY CONSOLIDATED VULTSE AIRCRAFT COffc. I >0. BOX 1 1 701 1 San Diego, California GAINESVILLE, FL 32611-7011 USA CLASSIFIED DOCUMENT This document contains classified information affecting the National Defense of the United States within the meaning of the Espionage Act, USC 50:31 and 32. Its transmission or the revelation of its contents in any manner to an unauthorized person is prohibited by law. Information so classified may be Imparted only to persons In the military and naval services of the United Stales, appropriate civilian officers and employees of the Federal Government who have a legitimate interest therein, and to United States citizens of known loyalty and discretion who of necessity must be informed thereof. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WASHINGTON February 12, 1951 RESTRICTED 2 * NACA RM E50K15 RESTRICTED NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS RESEARCH MEMORANDUM SURVEY OF ADVANTAGES AND PROBLEMS ASSOCIATED WITH TRANSPIRATION COOLING AND FILM COOLING OF GAS-TURBINE BLADES By E. R. G. Eckert and Jack B. Esgar SUMMARY Transpiration and film cooling promise to be effective methods of cooling gas-turbine blades; consequently, analytical and experimental investigations are being conducted to obtain a better understanding of these processes. This report serves as an introduction to these cool- ing methods, explains the physical processes, and surveys the informa- tion available for predicting blade temperatures and heat-transfer rates. In addition, the difficulties encountered in obtaining a uni- form blade temperature are discussed, and the possibilities of cor- recting these difficulties are indicated. Air is the only coolant considered in the application of these cooling methods. INTRODUCTION Gas-turbine blades are usually cooled by forcing air through the hollow interior of the blade. The possibilities of this cooling method are li mi ted by the heat -transfer rates from the blade to the coolant. Because of this limitation, other blade cooling methods are being investigated. Transpiration cooling, also called sweat or porous-wall cooling, promises to be a very effective means of cooling objects such as rocket nozzles (reference l) that must be in contact with high- temperature, high-veloc ity gas streams. This method of cooling in which the coolant is forced through a porous wall to form an insulating layer of fluid between the wall and the gas stream is now being con- sidered for use in gas-turbine stator and rotor blades. Film cooling is another means of cooling surfaces that is similar to transpiration cooling, although somewhat less effective. With film cooling the insulating fluid layer or film is formed by allowing a coolant to flow through slots or holes and then over the surface. References 2 and 3 have shown that film cooling with air as the coolant can be used effec- tively to decrease the temperatures of gas-turbine blades. RESTRICTED NACA RM E50K15 Since transpiration and film cooling show considerable promise as methods of turbine -"blade cooling, experimental and analytical investi- gations are being conducted at the NACA Lewis laboratory to gain more knowledge and to expand the application of these cooling methods. This report serves as an introduction to the basic principles of transpira- tion and film cooling and reviews briefly the status of methods for calculating heat transfer and temperatures in blades cooled by these means. It deals only with the thermodynamic aspect of the cooling methods. Neither stresses, a very important item for both methods, nor fabrication techniques are considered, but problems that have to be solved in the design and fabrication of transpiration and film- cooled blades are discussed. SYMBOLS The following symbols are used in this report: A area, sq ft Cj to C, constants c_ specific heat at constant pressure, Btu/(lb)(°F) d average width of pore in porous metal, ft F factor in equation (ll) g acceleration due to gravity, ft/sec^ H gas-to-surface heat-transfer coefficient, for porous or film- cooled surface, Btu/(°F)(sq ft) (sec) H' gas-to-surface heat-transfer coefficient for solid surface, Btu/(°F)(sq ft) (sec) k thermal conductivity, Btu/(°F) (ft) (sec) L length, ft m exponent Hx Nu Nusselt number, r— *g p pressure, lb/sq ft NACA RM E50K15 Pr Prandtl number, CpUgg Q heat flow rate, Btu/sec r ratio of velocities in "boundary layer (See equation (17)) V g x Re Reynolds number, ~r— g s slot width, ft Nu H St Stanton number, _ __ or ==- 7 RePr p c V g P g T temperature, °F v flow volume per unit area and time, ft/sec V velocity, ft/sec w flow rate per unit area, lb/(sec)(sq ft) W flow rate, lb/sec x distance, ft y distance from center line, ft (See fig. 3) Q Y distance from center line where -s =0.5 (See fig. 3.) a a max 5 Boundary- layer thickness, ft 9 temperature difference, °F \i viscosity, (lb) (sec)/sq ft Ug / 1? kinematic viscosity, — , sq ft/sec p density, lb/cu ft NACA RM E50K15 thickness, ft

^ e same temperature difference meas- ured on the center line I-I (for y = 0) . The investigations in refer- ence 5 show that Y increases linearly with the distance x from' the heat source. The area under all the temperature profiles at different distances has to be the same for continuity reasons when the loss in momentum of the fluid by the heat source can be neglected. In this case the temperature difference d max on the center line decreased inversely proportional to the distance x. Conditions in the flow on a film-cooled surface would be identical to the conditions considered in figure 3 when there are no frictional forces within the flowing gas and when the neighborhood of the slot is excluded. In this case the center line I-I represents the wall surface and the line source S represents the slot. In reality the flow is retarded by viscous forces in the neighborhood of the wall, which causes the convective heat flow and the intensity of turbulence to be somewhat changed. The temperature distribution meas- ured by Wieghardt (reference 6) on a film-cooled surface is also shown in figure 3 in order to give an indication of the order of magnitude of these changes. The same investigation shows that the difference between the wall temperature and the gas temperature decreases inversely propor- tional to x as opposed to the decrease of 9 ma x inversely propor- tional to x for the line heat source. The investigations of Wieghardt will be subsequently discussed in more detail. Transpiration Cooling It was previously stated that, in the method of film cooling, the air film is gradually destroyed by turbulent mixing with the hot gases. In this way, the effectiveness of the film decreases in the downstream direction from the point where it leaves the slot. This disadvantage is avoided in the transpiration- cooling method where the cooling film is continuously renewed along the blade surface. The blade is fabricated out of some porous material and the coolant is forced through the porous blade wall as shown in figure l(c) . A liquid coolant is again more effec- tive than a gas coolant because considerable heat is absorbed by the evaporation process. Cooling with a liquid may be distinguished from cooling with a gas by calling it evaporative-transpiration or sweat cool- ing. The most important coolant for aircraft power plants is air for the reasons mentioned previously and, therefore, only transpiration NACA RM E50K15 ' cooling with air will be considered in this report. Because the cooling film is continuously renewed in the transpiration-cooling method, the blade temperatures will be more uniform along the blade circumference than they will with film cooling. An additional advantage of transpira- tion cooling that tends to decrease the heat transfer from the hot gase? to the blade surface is the fact that there is a continuous slow movement of the cooling air away from the blade surface because new cooling air is continuously forced through the pores and then leaves the surface. A counterflow is thus created between the heat flowing from the hot gases towards the blade surface and the cooling air flowing away from the blade surface. The cooling air continuously carries heat away from the surface by convection and in this way decreases the over- all heat transfer from the hot gas to the surface. This principle is explained in more detail in the simple configura- tion in figure 4 for a duct with the cross-sectional area constant along its length and with air at a constant, uniform velocity flowing through it. Two screens separated by the distance L are placed in the duct. These screens are kept at two different temperatures Tj and T 2 . First, the air velocity is considered to be zero. In this case when free-convection currents are neglected, heat is conducted from the screen with the higher temperature Tg through the stagnant air between the screens to the screen with the temperature T^_. This heat- conduction process is described by the familiar equation Q=-Akg (1) in which Q is the heat flow per unit time, k is the thermal con- ductivity of the air that will be assumed independent of temperature, and T is the air temperature at the distance x from the screen having the temperature Tq_. In a steady state, the heat flow Q has to be constant along x when heat losses through the duct walls are neglected. An integration of equation (1) gives the well-known relation T - T-l = Cxx (2) stating that the temperature increases linearly from one screen to the other. The heat flow is then determined by the equation Q = -Ak - (3) ° NACA RM E50K15 If the air moves with a finite velocity' producing a weight flow w per unit time and cross-sectional area, the heat flow through any cross section in the duct is now composed of two parts: (l) the heat that is conducted in the direction opposite to the air stream and (2) the heat that is carried along "by the air stream. Therefore, the heat flow is represented by dT Q = -Ak ^ + c p w A (T - Ti) (4) where the specific heat of the air Cp will be assumed independent of temperature. The second term of this equation contains the temperature difference T - Tj_ because it is usual to disregard the heat that is carried along by an. unheated air stream (of temperature T3.) . Again, the heat flow Q has to be constant along x in a steady-state process. An integration of equation (4) together with the boundary condition T = T]_ when x = yields c p w T - T X = C 2 (e k - 1) (5) With the boundary condition T = Tg for x = L, the equation becomes ~F" x T - T X = (T 2 - T X ) - — (6) C P W T ~ L n e - 1 The resultant heat flow Q from the screen with the temperature T 2 to the screen with temperature T]_ is sLr-^-V^- (7) « = "A* IrbFj = - C P' 'x=0 -f- L - 1 NACA RM E50K15 For w = the numerator and denominator in this equation are zero. However, it can be determined that the value for this indeterminate form is the same as given by equation (3). In figure 4 the air tem- peratures in the space between the two screens are plotted with the value Cpw/k that characterizes the air velocity as the parameter. For zero air velocity a straight line represents the air temperatures. For finite velocity the temperature profiles are curved. A negative value of the parameter Cpw/k describes a flow of the air from the screen Tg towards the screen Tj_. The gradient of the lines at x = determines the amount of heat that flows from one screen to the other because, for x = 0, the second term on the right side of equation (4) is zero. It can be seen that the heat flow decreases with increasing velocities in the positive direction and increases for velocities in the negative direction. The decrease in heat flow for positive velocities is a consequence of the fact that part of the heat which leaves the screen with the temperature T2 by conduction in a direction opposite to the air velocity is picked up by the air stream and carried back towards the screen with temperature Tg and only part reaches the screen with the temperature T^_. This process by which the heat conduction flow is continuously decreased on its way by the count erf lowing air takes place within a transpiration- cooled wall as well as within the boundary layers that are built up on both surfaces of the wall. Figure 5(a) shows how the tem- perature varies throughout the wall and through the boundary layers. Figure 5(b) indicates the temperature variation within the wall in an enlarged scale. The transpiration- cooled wall is represented in cross section. The cooling air flows through the wall from left to right. Hot gases with a temperature of 1000° F flow along the right surface and build up a boundary layer that is usually turbulent. Within this boundary layer, the temperature drops to the value on the outside wall surface. Because the surface area in contact with the cooling air is very great in a porous material, there are practically no temperature differences between the cooling air on its flow through the pores and the wall. The temperature difference between the air and the wall shown in figure 5(b) is exaggerated. The cooling air therefore leaves the porous wall with the wall surface temperature T w g. It leaves the wall with a small velocity normal to the surface. In passing away from the surface, the cooling air picks up momentum from the gas flow until it finally reaches the outside gas velocity. At the same time its tem- perature increases either by conduction or by turbulent mixing until at some distance the gas temperature Tg is reached. The condition within the coolant-air film in the Imm ediate vicinity of the wall surface cor- responds principally to the condition in the air flow discussed in 10 NACA RM E50K15 connection with figure 4. The temperature Ti corresponds to the wall surface temperature T Wj g and the temperature T2 to the gas tempera- ture T g . The only difference is that the location of the plane in the cooling-air film that corresponds to the location of the screen with the temperature T2 is somewhat indeterminate because the turbulent mixing process occurs over some finite distance. Essentially, the same' process also takes place within the wall itself, the only difference being that the heat is conducted not only within the cooling air but also within the porous wall. The thermal conductivity k in equa- tions (4) to (7) must be adjusted to take this process into account. In addition, a heat-transfer process from the porous wall material to the cooling air takes place as the air flows through the pores. This process determines the temperature difference between the cooling air and the solid material within the wall. A mathematical investigation of this process by Weinbaum and Wheeler (reference 7) shows that this tem- perature difference is very small except within a very narrow range near that part of the wall surface where the cooling air enters. This temperature difference may therefore be neglected in calculating the wall temperatures. There is a boundary layer on the coolant entrance side of the wall within which the air temperature increases from the initial value T c to the temperature with which the air enters the pores. The thickness of this boundary layer and the temperature increase within it, however, are much smaller than on the hot side of the wall. The shape of the temperature curve within this boundary layer corresponds principally to the shape of the temperature profiles in figure 4 near the screen with the temperature T2. The temperatures T c , T w m , and Tg shown in figure 5 were measured on a transpiration-cooled turbine blade. The temperatures within the boundary layers are only approximate shapes deduced from boundary-layer calculations. The temperatures within the wall were determined using equation (6) and measured values for the thermal conductivity of porous steel (fig. 6). The notation &L in figure 5 will be explained later. A relation between the outside gas-to-surface heat-transfer coeffi- cient and the wall temperature on the outside of the turbine blade can be easily derived in connection with figure 5. If there is no heat flow along the wall within the boundary layer on the cooling-air side, which quite often is the case, then the total heat conducted from the inner wall surface into the cooling-air boundary layer is picked up by the cooling air and is carried back into the wall by convection. In other words the resultant heat flow Q in equation (4) at any place on the wall is zero. In a steady state, the heat flow is zero at any cross section throughout the wall. On the surface of the wall facing the gas stream, the temperature of the cooling air is Tw,g- The heat that is NACA RM E50K15 11 carried with the cooling air through the element dA of this surface is therefore wc p dA (T w , g - T c ) , (8) where w denotes the weight flow per unit area and time through dA and cp is the specific heaf of the cooling air at constant pressure. On the other hand, the heat that is transported through the gas boundary- layer to the surface may he defined with a heat-transfer coefficient by the expression H dA (T g - T^ g ) (9) At high velocities in the gas stream, the effective temperature has to he introduced as gas temperature into this expression. Because the total heat flow through the area dA is zero, expressions (8) and (9) have to he equal. Therefore, H (T g ,- T w , g ) = wc p (T w , g - T c ) (10) This relation determines the gas-to-surface heat-transfer coefficient as soon as the effective gas temperature, the coolant temperature, and the wall temperature on the surface facing the gas stream are known from measurements ; or inversely, it allows for the calculation of the blade- surface temperature from the gas-to-surface heat-transfer coefficient. GAS-TO-SURFACE HEAT -TRANSFER COEFFICIENT Extent of Laminar and Turbulent Boundary Layers It is well known that gases flowing through a turbine blade grid build boundary layers along the surface of the blades . The action of the viscous forces within the flowing gases is practically confined to this region and to the wake downstream of the blades . Sometimes the flow separates from the blade surface on the convex or suction side of the blade near the trailing edge. The boundary layers may be laminar or turbulent . 12 NACA RM E50K15 When the blade is cooled, a temperature boundary layer is built up around the blade, which, for gases, has a thickness of the same order of magnitude as the flow boundary layer. The gas temperature is influenced by the cooling process only within this boundary layer and within the wake behind the turbine blade. If separation occurs, then the region behind this separation has to be included. The remainder of the gas flow is not influenced by the cooling process. The heat transfer from the blade surface into this boundary layer depends very much on whether the flow within this boundary layer is laminar or turbulent. Before a calculation of the heat transfer can be started, it must therefore be known in which places along the blade surface the boundary layer is laminar or turbulent. Consequently, a short examination as to what kind of boundary layer is to be expected on a porous blade is required. At some point on the nose of the turbine blade, the gas flow divides into the part that flows along the suction side and the part that flows along the pressure side of the blade. This point is called the stagna- tion point and in this region the boundary layer is always laminar. At some distance downstream of this point, the flow within the boundary layer becomes turbulent. The location of this transition point mainly depends on the local pressure gradient and the thickness of the boundary layer at that point. The pressure distribution around a turbine blade on the pressure and suction side is determined by the outside gas flow. The pressure decreases at first with increasing distance from the stagna- tion point. Usually, it reaches a minimum somewhere along the blade on both sides and then increases. The transition to turbulent flow within the boundary layer probably occurs on turbine blades near the point where the pressure minimum is reached. The exact location of the point of transition cannot be predicted yet by calculation although great advances have been made in recent years in understanding the transition process. The factors influencing transition are now fairly well known. Some information also exists on their mutual importance. The investigations in this field that are of special importance for the prediction of the transition point on turbine blades will be mentioned briefly. Tollmien and Schlichting (references 8 and 9) show that on a solid surface the transition is caused by the fact that the boundary layer becomes unstable against small fluctuations under the influences of the outside pressure distribution along the surface and of the boundary-layer thickness. A pressure decrease in flow direction stabilizes, and a pressure increase destabilizes, the boundary layer. Outside disturbances that may be present to a considerable degree in the flow in a gas turbine move the transition point forward. A theory of this process is given by Taylor in reference 10. It was found that increasing curvature decreases the stability on a concave surface (reference ll) . Temperature differ- ences within the boundary layer increase the stability when the blade is NACA RM E50K15 13 cooler than the gas (reference 12). A comparison of average gas-to-blade heat-transfer coefficients calculated by Brown and Donoughe (refer- ence 13) with measured data indicates that, in internally- cooled turbine blades, the transition usually occurs near the point where the pressure minimum is reached along the surface. All these data referred to boundary layers on a solid blade. By the transpiration-cooling process, the stability of the boundary layer is decreased. Some information on this process is found in reference 14. No experimental data concerning the point of transition on turbine blades are available; however, it may be expected that laminar boundary layers will not be present downstream of the minimum-pressure point. Laminar boundary layers, therefore, will usually be confined to a region near the blade nose, and along the rest of the surface the boundary layer will be turbulent. It will be discussed later why it is well worth while to study the question of how the laminar part can be extended over a larger part of the blade surface. Heat Transfer in Transpiration-Cooled Laminar Boundary Layer The heat transfer in the laminar boundary layer can be analytically investigated. All the available information stems from calculations. The essential features of this type of heat transfer can be learned by a discussion of the conditions on a flat plate. Figure 7 shows the sketch of such a porous plate in a gas stream of uniform velocity V g . The plate is fabricated from porous material and cooling air is blown through it from a box attached to the back of the plate. The cooling air in this box has a temperature T c and the outside gas temperature is T g . The distribution of the cooling air along the plate surface depends on the local distribution of the porosity. Two important conditions may be considered. The first one is to find a distribution of the cooling air that keeps the outside surface tempera- ture of the. plate T w g constant over the whole plate length. The amount of coolant passing through the porous plate may be characterized by a velocity v c with which the cooling air moves away from the plate surface. This velocity v c is not the actual velocity of the cooling air coming out of the pores, but rather it is the volume flow through a unit surface of the plate. The cooling air has this velocity when all the jets coming out of the pores combine into a continuous flow and the velocity differences are equalized. It is assumed in all the calcula- tions for transpiration cooling that these conditions occur very close to the plate surface. t The distribution of the velocity v c along the plate that gives a constant wall temperature when equation (10) is valid 14 NACA RM E50K15 is v cfVx^ where x denotes the distance along the plate from the leading edge (reference 15) . In this case the partial differential equations describing the flow and the heat transfer within the boundary layer can be transformed into total differential equations and solved exactly. Such a calculation shows that the boundary- layer thickness along the wall increases proportional to the square root of x and the local heat-transfer coefficient H decreases inversely proportional to the square root of x. The heat- transfer coefficient is obtained from the dimensionless equation Nu = F/y/Re" (11) where The factor F is a function of the coolant-flow velocity v c and the Prandtl number Pr of the gas. The factor F is presented in fig- ure 8 for a gas with a Prandtl number of one with the assumptions that the property values are constant through the boundary layer and the Mach number is small. Negative values of the abscissa indicate suction of the gas into the wall. The flow boundary layer for this case was calculated by Schlichting (reference 16) . The temperature boundary layer and therefore the heat-transfer coefficient for Pr = 1 follow immediately from these calculations considering the similarity between velocity and temperature profile. For the case investigated, the porosity of the plate has to be varied in such a way that the velocity v decreases inversely proportional to the square root of x. The second important case is a plate where the velocity v c is constant. This case can be obtained with a constant wall thickness and porosity. Only, an approximate solution of the differential equations is possible for this case (reference 17) . The local behavior of the boundary- layer thickness, the wall temperature, and the heat -transfer coefficient is different from the case previously discussed. As in the other case, the boundary- layer thickness starts with the value zero at the leading edge of the plate but it increases linearly in the down- stream direction for great values of x. The local heat-transfer coefficient starts with the value infinity at the leading edge and approaches asymptotically the value zero with increasing x. The wall temperature is equal to the gas temperature at the leading edge and approaches the cooling-air temperature asymptotically with increasing x. It is interesting to note that the behavior of these values is NACA RM E50K15 15 different when the air is sucked into the plate instead of blown from it. In this case the boundary- layer thickness, the heat-transfer coefficient, and the wall temperature reach a constant value some distance from the leading edge. Conditions near the leading edge of a turbine blade where the boundary layer is expected to be laminar differ from those on a flat plate by the fact that the pressure and, therefore, the velocity V g outside the boundary layer are not constant but vary along the blade surface. Exact calculations are possible only for certain types of this velocity variation. Solutions were presented for constant property values, small Mach numbers, and for the case that V„ varies proportional to x (references 15 and 16). The solutions can be extended to include the effect of the variation of property values and a variation of the gas velocity Vg proportional to x m . The resulting formula for the heat-transfer coefficient will have the same form as equation (ll) . The factor F will then also depend on the ratio of the gas temperature T g to the wall temperature T w „ and on the exponent m. This exponent is related to the local pressure gradient dp/dx by the equation dp = m Pg V g dx x Normally, the variation of the velocity Vg proportional to x 111 does not correspond too well to the velocity variation encountered on turbine blades. For such a variation only approximate calculations of the heat- transfer coefficient are possible. A method using the integrated momentum and heat-flow equations of the boundary layer, similar to the procedure proposed by von Earman (reference 18), seems to be best aiiited for this purpose. Heat Transfer in Transpiration-Cooled Turbulent Boundary Layer No exact calculations are possible for the turbulent boundary layer with or without transpiration cooling. Experimental information obtained on transpiration- cooled turbulent boundary layers is also very meager. The present knowledge of the heat transfer in the transpiration- cooled turbulent boundary layer is therefore much less reliable than that of the laminar boundary layer. A first, approximate theory for the turbulent case was presented by Rannie (reference 19) . The essen- tial features of this theory maybe explained with the help of figure 5. The turbulent fluctuations within the boundary layer on the gas side of the wall decrease in a direction towards the wall. On the wall 16 NACA RM E50K15 itself they must be zero. Very little detailed information exists on this decrease of turbulence. However, to simplify the real conditions for heat-transfer investigations the boundary layer can be divided into two parts, a very thin sublayer near the wall (indicated by 5 T in fig. 5) within which the flow is assumed to be laminar, and a turbulent part where the heat transfer by turbulent mixing is assumed to exceed by far the heat transfer by conduction. This idealization, which gave useful results for the boundary layer on a solid wall, is used by Rannie in his theory. In addition he assumes that the transpiration- cooling process does not alter the value of 8-^ or the conditions in the turbulent part of the boundary layer; only the laminar sublayer is influenced. The temperature drop through the laminar sublayer on a solid wall is practically linear in accordance with equation (2) . For the porous wall it is non-linear and can be described by equation (6) . In this way a formula was derived in reference 19 for the heat-transfer coefficient and by use of equation (lO) the wall temperature could also be calculated. Comparisons of this formula with experimental data obtained in the Jet Propulsion Laboratory, California Institute of Technology showed fair agreement (reference 19) . The formula was simplified by Friedman (reference 20) in restricting it to fluids with a Prandtl number near one. The formula for the wall temperature T w „ derived by him is T w,g T g " T c e 1 ^ + r - (12) where r is the ratio of the velocity parallel to the surface at the border between the laminar sublayer and the turbulent part of the boundary layer (at the distance 6-r in fig. 5) to the stream velocity Vq. outside the boundary layer. The value Cp is given by p c v V-^ (13) with p , the density; c , the specific heat; v c , the velocity of c ( P the cooling air; and H , the heat-transfer coefficient that would be present on a solid surface under the same outside flow conditions. The formula was used by Friedman to study heat transfer in a tube, but it can be applied to the flat plate as well. Reference 21 gives the equation for heat transfer over a flat plate with turbulent flow as Nu = 0.0296 (Re) ' 8 (Fr) 1 / 35 (14) NACA RM E50K15 17 If both sides of equation (14) are divided by (Re)(Pr), the equation takes the form Nu H' 0-0296 . ^^ = " P g C p V g = (Re)V5 (Er) 2/3 < 15 > where the Reynolds number is based on the length x. Substituting the value of H from equation /( 15) into equation (13) results in (Re) 1 / 5 (Fr) 2 / 5 Pglc * 0.0296 p V ^ Xb ' g g The value cp therefore depends mainly on the ratio of density times velocity of the coolant to density times velocity of the outside gas flow and, to a smaller degree, on the Reynolds number for the outside flow and on the Prandtl number. The value 0.5 was used by Friedman for the velocity ratio r. For the flat plate it is better to use the relation in reference 22 2-11 #„„% r = TTTTTi ( 1? ) (Re) 0.1 Friedman also compared equation (12) with test data and found good agreement. For the specific values Re = 10 and Pr = 0.7, the tem- perature ratio given by equation (l2) is plotted in figure 9 as a function of the mass-flow ratio. The temperature ratio calculated from equations (10) and (ll) for the laminar boundary layer on a flat plate is also included. The figure reveals that the amount of coolant flow necessary to obtain the same wall temperature is very much lower for the laminar boundary layer than for the turbulent one. This fact means that very big gains can be obtained if the laminar portion of the boundary layer on the surface of the turbine blade can be increased. The conditions on a turbine blade in the region of the turbulent boundary layer differ again from the conditions on a flat plate by the fact that the velocity varies along the blade surface. Practically no information on the influence of this velocity variation is available, neither on the solid surface nor on a transpiration- cooled wall. It seems, however, that the influence of a pressure gradient is less on a turbulent boundary layer than on a laminar one. Nothing is known of the heat transfer in the separated region that often exists on the downstream part of the suction surface of the blade. 18 NACA RM E50K15 Eeat Transfer in Film Cooling It was pointed out that in film cooling the cooling-air film is diffused by turbulent mixing with the hot gases and, in this way, is gradually destroyed on its downstream path after leaving the slot. When the wall surface does not receive heat by radiation or by con- duction into or from the interior of the wall, it assumes a temperature that is called the adiabatic wall temperature. More information on the value of this adiabatic wall temperature T ^ may be obtained from an experimental investigation by Wieghardt (reference 6) , which is concerned primarily with the de-icing of air- plane wings. He therefore investigated a hot-air jet blown into a cold air stream by a slot in a flat plate. As a first approximation, however, his results should be applicable to the film-cooling process as well. He derived the expression 2** . a.e ft !SL£1 (i 8 ) T - T lx p V § c \ g g in which x is the distance in downstream direction from the slot and s, the slot width. The formula is applicable for x/s values greater than 100 and for a mass-flow ratio smaller than or equal to one . It gives a decreasing adiabatic wall temperature at a certain distance x with increasing mass-flow ratio. More research is required for x/s values smaller than 100, because this range will find most application in gas-turbine blades. The experiments (reference 6) showed that, for mass-flow ratios greater than one, the adiabatic wall temperature increased again with increasing mass-flow ratio. A value of one for the mass-flow ratio therefore gives the best cooling effectiveness. When there is heat flow through the wall by internal conduction from a hotter location in the wall, or when heat is radiated from the gas to the wall surface, a finite heat transfer by convection from the wall to the air film has to balance this heat exchange and the tempera- ture profile on a normal to the wall must have a finite gradient at the wall surface. A heat- transfer coefficient H describing this heat flow dQ has to be defined by the equation dQ= HdA (T e - T^ g ) (19) in analogy to the corresponding equation for the heat transfer in high- velocity flow. The effective temperature T has to be chosen in NACA RM E50K15 19 such a way that the temperature difference T - T is zero where e w, g the heat flow dQ is zero. Only this difference avoids a change of the heat-transfer coefficient to values infinite and zero as the temperature difference is varied. The exact evaluation of T is difficult owing to wall temperature gradients in the direction of flow, which influence the air-film temperature profile normal to the wall surface. The temperature T e affecting heat transfer will therefore probably be slightly different from the adiabatic wall temperature T -j in equation (18) . No information is presently available on the magnitude of the heat-transfer coefficients in film cooling or on an exact method of evaluating T in equation (l9) . The, information obtained by Wieghardt's investigations may be used to compare the effectiveness of film and transpiration cooling for the case when the heat flow dQ in equation (l9) is zero. If a wall is assumed to be cooled by a continuous succession of slots having the distance x from each other, equation (18) gives the highest values for the effective wall temperature. The amount of coolant flow per length x of the surface is sp V . In a transpiration-cooled wall of the same dimensions, the coolant flow through the length x is xp v c . In order to compare both walls at the same amount of coolant flow per unit area, the coolant velocity v on the transpiration- cooled wall has to be compared with a coolant velocity — V on the film-cooled wall. Therefore the expression p V — c c x p V K g g in equation (18) is equivalent to the expression P v K c c Pg V g in equation (16) . In this way the curve for film cooling is inserted in figure 9 . It can be seen that the transpiration cooling is more efficient than the film cooling. In this comparison it must be kept in mind that equation (l8) describing the temperatures obtained in film cooling is valid only for a ratio x/s greater than 100 and that the slot width is of the same order of magnitude as the boundary- layer thickness. 20 NACA RM E50KL5 FLOW THROUGH POROUS WALLS Required Flow Distribution The information in the previous section can be used to determine the wall-temperature distribution around the surface of a transpiration- cooled porous blade when the velocity of the cooling air v is prescribed, or it can be used to determine the distribution of the cooling-air velocity around the blade surface that is necessary to keep the wall temperature uniform. A first approximate answer to this question may be obtained with the equations presented and the accuracy of such a determination can be increased when more information on the heat-transfer coefficients under the circumstances prevailing on blade surfaces becomes available. The local distribution of the cooling-air velocity v necessary to keep the surface temperature of a plane wall at a constant value is shown in figure 10. The necessary coolant velocity begins with an infinite value at the leading edge and decreases rapidly in the region of the laminar boundary layer with increasing distance from the leading edge. It rises again when the boundary layer becomes turbulent in accordance with figure 9 and decreases in the turbulent region. This decrease, however, is much slower than the decrease in the laminar portion. Basically, the conditions around a turbine blade for constant surface temperature will be similar. They differ mostly in the region near the stagnation point where the finite thickness of the blade causes the cooling-air velocity to start out with a finite value. The required velocity v decreases from this value along both the suction and pressure side of the surface to the transition point, where a rise in the necessary velocity v c has to be anticipated followed by a more gradual decrease of its value in the downstream direction. The leading edge region where high coolant velocities v c are required probably extends a greater distance downstream on a blade than on a flat plate. The problem that has to be solved in the design of a turbine blade is providing the necessary distribution of the cooling air by proper choice of the local porosity of the blade wall and its thickness. Probably the variation of v determined theoretically along both sides of the blade can be approximated well enough by a constant value; however, the high v values necessary near the leading edge have to be provided by some means. Figure 11 shows the temperature distribution that was measured on a porous blade in a static cascade at the NACA RM E50K15 21 NACA Lewis laboratory. The manufacturer tried to maintain a uniform porosity around the blade perimeter. It can be seen that the tempera- tures are very lov on both sides of the blade. For comparison, blade temperatures are also given which were measured in a static cascade on two blades cooled internally with air; namely, a hollow blade and a blade where the internal area was increased by brazing 10 tubes into the hollow blade shell. It can be seen that the temperatures obtained in the central part of the porous transpiration- cooled blade are extremely low. Even with liquid cooling such low temperatures have not been obtained at comparable coolant flows . The reason for the high leading-edge temperatures is apparent from a comparison of the blade cross section in figure 11 and the velocity distribution necessary to maintain a constant wall tempera- ture as shown in figure 10. Figure 11 shows that the wall thickness near the leading edge was appreciably higher than at the sides of the blades. Owing to a high pressure drop through the thicker wall, the velocity was therefore small in the region where, according to fig- ure 10, a high value of this velocity is necessary to maintain a con- stant wall temperature. Near the trailing edge, no especially high values of the coolant velocity are necessary according to figure 10; however, the coolant velocity in this region in the turbine blade was extremely low because of the long path length of the coolant through the material in this region. This fact is evidenced by a measured local coolant-flow distribution shown in figure 12. Special attention must therefore be directed to the leading-edge and the trailing-edge regions of such a transpiration-cooled turbine blade. Further inves- tigations are necessary to determine whether a local change in the porosity can make the temperature sufficiently uniform around the periphery of the blade or whether other special means have to be used for cooling the leading- and trailing-edge regions. Another difficulty is present In turbine blades, which may be explained with the help of figure 13. This figure shows the outside pressure distribution around the turbine blade. The cooling-air pressure inside the turbine blade is practically constant because of the low coo ling -air velocities. This constant value is also indicated in the upper part of the figure. It can be seen that the pressure difference available to force the cooling air through the porous wall differs along the blade surface. Wear the leading edge where high coolant velocities v c are required, the pressure difference is smallest. The higher pressure difference and higher coolant flow on the suction side of the blade is responsible for the lower temperature 22 NACA RM E50KL5 seen in figure 11 on this side of the blade. The differences in the available pressure difference can be decreased by using a wall with low porosity, which makes the required inside pressure high. A high coolant pressure, however, should be avoided because this pressure may not be available in the compressor of the gas-turbine engine. It would be sufficient to provide a high coolant pressure only for the region near the stagnation point and possibly near the trailing edge. Another possibility is to let the variation of the local porosity take care of the differences in the available pressure as well as the dif- ferences in the necessary coolant flow. Porosity and Permeability The manufacturer of porous material is concerned with the fabri- cation of porous structures with a certain predetermined and repro- ducible porosity. On the other hand, the considerations in the previous paragraphs determine what the permeability of a wall should be. The porosity is defined as the value one minus the ratio of the density of the porous material to the density of a solid piece made of the same material. The permeability expresses the capacity of a porous material to pass liquids or air when pressure differences are present. In order to prescribe the required porosity to the manufacturer of transpiration- cooled walls, the law relating the porosity and the mass flow through a porous wall must be known. For low velocities Darcy's law Ap „ c c . . — - c 3 — (20) d gives the relation between the pressure difference Ap acting on the two surfaces of a plain porous wall with the thickness T , the viscosity p. , the velocity v of the coolant flowing through the porous wall, and the average width d of the pores. This law holds for Reynolds numbers smaller than one when the Reynolds number is based on the dimension d. It gives a linear relationship between pressure drop and velocity analogous to laminar flow in a tube. For large Reynolds numbers another law holds ¥ - c 4 2£- (21) according to which the pressure drop is proportional to the density p of the coolant and the square of the velocity v . Usually, there c *" NACA RM E50K15 23 is a large range of Reynolds numbers within which the law gradually changes from the linear to the quadratic relationship. Usually, this is the range that is of interest for the applications considered. In addition the constants in both equations depend on the structure of the porous wall and cannot be predicted exactly (reference 23) . The relation between the pressure drop and the velocity, therefore, has to be experimentally determined for each material. When the density of the coolant varies considerably on its way through the porous wall, it is more expedient to calculate with the mass flow p v per unit area. It was shown in references 23 and 24 that, when the change in density . is caused by pressure variation, the two foregoing equations become relations between the differences Ap^ of the square of the pressure and the mass flow rate P c v c - In turbine blades the surfaces cannot always be considered as plane walls in calculating the mass flow through the walls, especially near the leading and the trailing edge of the blade. The mass flow through curved walls may be determined by methods similar to the ones used to study the heat flow by conduction through curved walls. In the range, of low Reynolds numbers where the relations between the pressure _square and the mass flow rate is linear, there is a complete analogy between both physical processes. The lines of constant tempera- ture then correspond to the lines of constant pressure and the direction of the heat flow on any place within the wall coincides with the direction of the mass flow. In the range where the relation between pressure- square values and mass-flow rate is nonlinear this analogy ceases to exist. In "this case an adaptation of the relaxation method to the nonlinear character of the equations may be used to determine local flow rates through a porous wall. SUMMARY OF RESULTS The results of a survey on transpiration and film cooling have shown that : 1. Transpiration cooling is an extremely effective means of cooling objects in high- temperature, high-velocity gas streams such as gas-turbine blades; however, special care will be required to provide sufficient cooling at the leading and trailing edges. 2. The laminar boundary layer on most turbine blades is appar- ently limited to the portion of the blade near the leading edge. Because transpiration cooling is much more effective for laminar boundary layers than for turbulent layers, means of extending the laminar layer require investigation. 24 NACA EM E50KL5 3. Fabrication techniques must be developed for porous blades to provide high strength with controlled, variable permeability. 4. Film cooling is also an effective method for the cooling of turbine blades, although it is less effective than transpiration cooling. 5. Although considerable research has been conducted on transpi- ration and film cooling, additional research is required for a more thorough understanding of the heat transfer under the special con- ditions prevailing for turbine blades. Lewis Flight Propulsion Laboratory, National Advisory Committee for Aeronautics, Cleveland, Ohio. REFERENCES 1. Canright, Richard B. :• Preliminary Experiments of Gaseous Transpi- ration Cooling of Rocket Motors. Prog. Rep. No. 1-75, Power Plant Lab. Proj. No. MX801, Jet Prop. Lab., C.I.T., Nov. 24, 1948. (AMC Contract No. W-535-ac-20260, Ordnance Dept. Contract No. W-04-200-0RD-455.) 2. Kuepper, K. H.: Temperature Measurement on Two Stationary Bucket Profiles for Gas Turbines with Boundary- Layer Cooling. Trans. No. F-TS-1543-RE, Air Materiel Command, U.S. Air Force, Jan. 1948. (ATI No. 18576, CADO.) 3. Dempsey, W. W.: Turbine Blade Cooling (Final Hot Test Rep.). Rep. No. 2037, Stalker Development Co., June 29, 1949. 4. Kinney, George R., and Sloop, John L.: Internal Film Cooling Experiments in 4-Inch Duct with Gas Temperatures to 2000° F. NACA RM E50F19, 1950. 5. Corrsin, Stanley, and Uberoi, Mahinder S.: Spectrums and Diffusion in a Round Turbulent Jet. NACA TN 2124, 1950. 6. Wieghardt, K.: Hot-Air Discharge for De-icing. AAF Trans. No. F-TS-919-RE, Air Materiel Command, Dec. 1946. MCA RM E50K15 25 7. Weinbaum, S., and Wheeler, H. L., Jr.: Heat Transfer in Sweat- Cooled Porous Metals. Prog. Rep. No. 1-58, Air Lab. Pro j . No. MX121, Jet Prop. Lab., C.I.T., April 8, 1947. (AMC Contract No. W-535-ac-20260, Ordnance Dept. Contract No. W-04-200-0RL-455.) 8. Tollmien, W.: The Production of Turbulence. NACA TM 609, 1931. u 9. Schlichting, H. : Uber die theoretische Berechnung der kritischen Reynoldsschen Zahl einer Reibungsschicht in beschleunigter und verzogerter Stromung. Jahrb. d. D. Luftfahrtforschung, 1940, S. I 97-112. 10. Taylor, G. I.: Statistical Theory of Turbulence. V - Effect of Turbulence on Boundary Layer. Theoretical Discussion of Relation- ship between Scale of Turbulence and Critical Resistance of Spheres. Proc. Roy. Soc. London, vol. CLVI, no. A888, ser. A, Aug. 17, 1936, pp. 307-317. 11. Liepmann, Hans W.: Investigations on Laminar Boundary-Layer Stability and Transition on Curved Boundaries. ' NACA ACR 3H30, 1943. 12. Lees, Lester: The Stability of the Laminar Boundary Layer in a Compressible Fluid. NACA Rep. 876, 1947. (Formerly NACA TN 1360.) 13. Brown, W. Byron, and Donoughe, Patrick L.: Extension of Boundary- Layer Heat-Transfer Theory to Cooled Turbine Blades. NACA RM E50F02, 1950. 14. Lees, Lester: Stability of the Laminar Boundary Layer with Injection of Cool Gas at the Wall. Rep. No. 124, Aero. Eng. Lab., Princeton Univ., May 20, 1948. Tech. Rep. No. 11, Proj. Squid, under Navy Dept. Contract N6-0RI-105, T.0. Ill, Phase I, NR 220-038.) 15. Eckert, E. R. G. : Heat Transfer and Temperature Profiles in Laminar Boundary Layers on a Sweat-Cooled Wall. Tech. Rep. No. 5646, Air Materiel Command, Nov. 3, 1947. 16. Schlichting, Hermann, und Bussmann, Karl: Exacte Losungen fur die laminare Grenzschicht mit Absaugung und Ausblasen. Schriften d. D. Akad. Luftfahrtforschung, Bd. 7B, Heft 2, 1943. 26 NACA RM E50K15 17. Yuan, Shao Wen: Heat Transfer in Laminar Compressible' Boundary Layer on a Porous Flat Plate with Fluid Injection. Jour. Aero. Sci., vol. .16, no. 12, Dec. 1949, pp. 741-748. 18. von Karman, Th.: On Laminar and Turbulent Friction. MCA TM 1092, 1946. 19. Rannie, W. D. : A Simplified Theory of Porous Wall Cooling. Prog. Rep. No. 4-50 Power Plant Lab. Proj. No. MX801, Jet Prop. Lab., C.I.T., Nov. 24, 1947. (AMC Contract No. W-535-ac-20260, Ordnance Dept. Contract No. W-04-200-0RD-455. ) 20. Friedman, Joseph: A Theoretical and Experimental Investigation of Rocket-Motor Sweat Cooling. Jour. Am. Rocket Soc, no. 79, Dec. 1949, pp. 147-154. 21. Colburn, Allan P.: A Method of Correlating Forced Convection Heat Transfer Data and a Comparison with Fluid Friction. Trans. Am. Inst, of Chem. Eng., vol. XXIX, 1933, pp. 174-210. 22. Eckert, E. R. G. : Introduction to the Transfer of Heat and Mass. McGraw-Hill Book Co., Inc., 1950. 23. Green, Leon, Jr., and Duwez, Pol: The Permeability of Porous Iron. Prog. Rep. No. 4-85, Jet Prop. Lab., C.I.T., Feb. 9, 1949. (Ordnance Dept. Contract No. W-04-200-0RD-455. ) 24. Grootenhuis, P.: The Flow of Gases through Porous Metal Compacts. Engineering, vol. 167, April 1, 1949, pp. 291-292. NACA EM E50K15 27 ■d 8 ■H c H 03 co -d a) rH ■d rH o o o I C o •H -P o S o oling) ^ Laminar i (transpii •at ion cool ing) .002 .004 .006 .008 .010 Pc v c p g V g .012 Figure 9. - Effect of mass-flow ratio on wall temperature for transpiration and film cooling. Ee = 10 . 36 NACA EM E50K15 u •8 1 P C 0) r-l 3 •s \ hi £> H -d o O o CJ I o •H -P a) fn ■H P. 10 a p 3 P CO u & 0) p I +J fl (0 (0 c o o o > p ■H o o r-l A) t* ? O i-H Cw I P (3 O o u 4) s, •H pK o. < A 'Ji%\0O\3A AO"[J ^UBIOOQ KACA EM E50K15 37 Suction surface 1000 = 800 cf I EH 600 400 200 NACA Gas temperature Hollow "blade 10 -Tube "blade Pressure surface Suction surface Coolant temperature 20 40 60 Chord, percent 80 100 Figure 11. - Porous-blade temperature distribution for w c /v g = 0.026. 38 NACA EM E50E15 Suction surface ^NACA, CO o tt) O O O o 1N /I A~3 4 !1U ^^ s •*«, 3 s /-Prei ssure i 3urfac< i r/ \ / I \ / i ' N. > / / \ ' 1 Suction surface-' \\ 2 1 1 \\ 1 \\ 1 \ 1 20 40 60 Chord, percent 80 100 Figure 12. - Permeability variation around porous blade for pAP = 49.5 lb 2 /ft 5 . NACA EM E50EL5 Suction surface 39 U 3 ID ID 40 60 Chord, percent 80 100 Figure 13. - Pressure distribution around turbine blade. NACA-Langloy - 2-12-51 - 560 UNIVERSITY OF FLORIDA UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT , MARSTON SCIENCE LIBRARY P.O. 3OX117011 SVILLE, FL 32611-7011 USA 3k IT) CT> OS CQ