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,fA m'. 534 SI 1:DENTIAL cow RM E52H07 A.: ir :FOR CALCULATING TURBINE BLADE TEMPERATURES PRISON OF CALCULATED WITH OBSERVED VALUES FOR TWO STATIONARY AIRCOOLED BLADES :.Byrn Brown, Henry 0. Slone, and Hadley T. Richards Lewis Flight Propulsion Laboratory Cleveland, Ohio Z 'g iR. : I:... DATE AUG. 17, 1955 4Us V. 0N UNIVERSITY OF FLORIDA S"" DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE UBRA K:. ,.CIium DOCUm"Enr P.O. BOX 117011 w.::.,.....a .mm... anf tm ul malwegg* L 326117011 U .O. omagui lawip s 7adola, U. s.c3 MO70.U lm sen. m s mo si.sr t.. aipat laildvesa idtd by law. TIONAL ADVISORY COMMITTEE ',~~:'.i,.:..% .. 1., .. . 'R AERONAUTICS 51.. :.:W. ;. 4ASHIN.G.O 1K Od r^ i 1 : .:> ; i :'i '' * *if . ; ;" i.E * ... : ...E ".. . .. .* *..:..E:. " i mmm ";..: ] !: i~~a:' i.:' i 1* I I NACA RM E52H07 CONFIDEIITIAL NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS RESEARCH MEMORANDUM PROCEDURE FOR CALCULATING TURBINE BLADE TEMPERATURES AND COMPARISON OF CALCULATED WITH OBSERVED VALUES FOR TWO STATIONARY AIRCOOLED BLADES By W. Byron Brown, Henry 0. Slone, and Hadley T. Richards SUMMARY The accurate prediction of local turbine blade temperatures is necessary for the design of cooled turbines. When current methods of predicting blade temperatures are applied to cooled turbine blades, discrepancies between calculated and measured temperatures in some blade leading and trailing sections result. In an effort to reduce these discrepancies and hence to improve bladetemperature predictions, an investigation was conducted for stationary turbine blades with 10 tubes and 13 fins forming the internal heattransfer surfaces. Local blade temperatures were calculated using previously published NACA temperature distribution equations and the most recent theories for determining heat transfer coefficients, including for the first time the allowance for effects of variable wall temperature on gastoblade heattransfer coefficients at the leading and trailing sections of turbine blades. The calculated temperatures were compared with measured temperatures. Results indicate that calculated trailingsection temperatures can be greatly reduced and leadingsection temperatures increased in blades which have an appreciable temperature gradient in these sections when gastoblade heattransfer coefficients based oh variable wall tempera ture rather than coefficients based on constant wall temperature are used in the temperaturedistribution equations. Applications of gas toblade coefficients based on variable wall temperature and an average bladetocoolant coefficient for the coolant passage nearest the trail ing edge for the 10tube and 13fin blades resulted in probable errors for a point near the trailing edge of 60 and 40 F, respectively, for a 300 F gas temperature and of 360 and 300 F, respectively, for a 10000 F gas temperature. For a point near the leading edge, a similar procedure resulted in probable errors of 50 and 60 F, respectively, for a 3000 F gas temperature and of 8 and 18 F, respectively, for a 10000 F gas temperature. In the blade midchord region, where variable walltemperature effects are negligible, maximum probable errors of 80 and 13 F for gas temperatures of 3000 F and 10000 F were obtained for the two blades. CONFIDENTIAL NACA RM E52H07 niTR'.T'DUrT iii A knowledge of cooledblade temperatures for a turbine of known design and operatingg conditions is extremely important in the evaluation of cooled turbines. Reliably calculated blade temperatures enable the turbine designer to (1) determine accurately the required coolant flow necessary for the turbine design characteristics considered, (2) examine the thermal gradients in the blade, because large thermal gradients may cause the blade to fail, and (3) determine the strength characteristics of the turbine blade on the basis of stresstorupture data. Blade strength decreases rapidly as the blade temperature increases; thus an accurate calculation of blade tempieratLures is required. Also, experi mental and reliably calculated blade temperatures afford a check on the blade fabrication techniques; that is, a c.rmparison between experimental and calculated temperatures may indicate whether or not the thermal bond between the internal heattransfer surfaces and the blade shell is satisfactory. As early as 1945, equations for calculating blade temperature dis tributions were developed and published by the NACA. These investiga tions are summarized in references 1 and 2. Because the accuracy of the calculated temperatures depends primarily on the heattransfer coeffi cients inserted into the equations, it is quite important that appro priate values for these be determinable. For forcedconvection bladetocoolant heattransfer coefficients, hereinafter called inside coefficients, pipe correlations agree with experimental correlations for air in a stationary cascade (reference 3) and heated liquids in a rotating cascade (reference 4). The gastoblade heattransfer coefficients, hereinafter called outside coefficients, offer a more complex problem because turbine blade shapes differ widely among themselves and from round tubes, so that a great variety of pressure and velocity distributions occurs. For laminar flow, methods have been published for computing both average and local outside coefficients for a wedgetype flow which are applicable to tur bine blades (reference 5). The conditions covered by that investigation for a constant wall temperature include an Euler number range (a measure of the pressure gradient) from 0.09 to 2.0, Mach numbers approximating zero, a Prandtl number range from 0.6 to 1.0, and a temperature ratio (gas to wall) equal to 1.0. Additional analyses of the laminar region which can also be applied to cooled blades with impermeable walls were made for the case of transpiration cooling. These analyses are pre sented in references 6 to 9. An approximate method of solving the laminarboundarylayer equa tions for cylinders of arbitrary cross section is presented in refer ence 6. This method requires that the velocity and temperature profiles C .II IDEIITIAL CONFIDENTIAL NACA RM E52H07 in the boundary layer be assumed. The conditions covered by this anal ysis include a constant wall temperature, a temperature ratio equal to 1.0, and a range of Prandtl number. An exact method of solving the laminarboundarylayer equations for a wedgetype flow presented in ref erence 7 includes the conditions of a constant wall temperature, Euler number range from separation values to 1.0, Mach numbers approximating zero, a Prandtl number of 0.7, and temperature ratios from 1.0 to 4.0. Reference 8 presents a tabulation of these exact solutions of the laminarboundarylayer equations for most of the conditions of refer ence 7 and includes results from two additional temperature ratios of 1/4 and 1/2. Another approximate method which utilizes prepared charts to reduce calculation procedures is presented in reference 9 for the calculation of heat transfer in the laminar region around cylinders of arbitrary cross section. This method is based on the exact boundary layer solutions for wedgetype flow presented in references 7 and 8. The boundarylayer equations for cylinders of arbitrary cross section were used in reference 9 to compare the solutions obtained in references 7 and 8 for wedgetype flow. For impermeable blade walls, it can be shown (reference 9) that the effect of temperature ratios from 1.0 to 2.0 on heat transfer is negligible in the laminar region. At present, a tem perature ratio of 2.0 is probably the limit for aircooled blades having impermeable walls. Also, it is pointed out in reference 9 that heat transfer coefficients obtained for elliptical cylinders compared favorably with those determined from wedgetypeflow solutions. On the basis of the foregoing results, the simplified methods reported in reference 5 for the determination of heattransfer coefficients in the laminar region are currently adequate for application to aircooled turbine blades having impermeable walls. Local and average heattransfer coefficients for turbulent flow are computed in references 5, 10, and 11 for the cases of zero pressure gradient (flat plate), constant wall temperature, low subsonic Mach numbers, temperature ratio equal to 1.0, and a Prandtl number range from 0.5 to 10. An equation for the average outside heattransfer coefficient for turbine blades including both laminar and tL.rbu1,lent flow is derived in reference 5 for constant wall temperature, low subsonic Mach numbers, temperature ratio equal to 1.0, and including the Euler number and laminar to turbulent transitionratio effects. Calculated average outside coefficients using the equations of reference 5 compared favorably with observed data for stationary blades in references 12 and 13 and with data for a watercooled turbine in reference 14. The temperature equations reported in references 1 and 2 have been used recently to compute cooledblade temperatures in stationary and rotating turbine blade cascades in which temperatures were measured experimie..t.ali. Reference 15 gives a comparison between calculated and CONFIDENTIAL CONFIDEiNfrAL NACA RM E52H07 experimental blade temperatures for two stationary aircooled blade con figurations. One of the configurations investigated was a hollow alumi num blade with long leading and trailing sections. The other configura tion was an aircooled blade of low thermal conductivity with a long trailing section, a short leading section, and 10 tubes forming the internal heattransfer surfaces. Herei:lafter, the terms long leading and trailing sections will be used when referring to blades having a sufficiently large area of low thermal conductivity metal in the uncooled section to cause an appreciable temperature gradient. The methods reported in reference 5 were used to compute constant wall temperature outside coefficients for the calculations of reference 15; a stagnation point coefficient was used to calculate local leading section tempera tures, and average outside cefficients were employed to calculate an average midchord and local trailing section temperatures. Good agreement was obtained for both blades at low and high gas temperatures except at the trailing section of the 10tube blade where the calculated tempera ture was at least 1000 F higher than the experimental value at a gas temperature of 10000 F. A rotating cascade of aircooled thinshelled blades (blades having a mean wall thickness of approximately 0.040 in.) was used for a compar ison of average calculated and experimental blade temperatures in ref erence 16. An average constantwalltemperature outside coefficient was computed from reference 5 for the entire blade periphery, and the aver age calculated blade temperature was found to be approximately 300 F less than the experimental value for a gas temperature of approximately 15000 F. Reference 17 contains a comparison of calculated and experi mental blade temperatures for the leading and trailing sections and the midchord region of an aluminum watercooled turbine. Once again, the outside coefficients for constant wall temperature as computed from reference 5 were used in these calculations. Generally, good agreement resulted except at the leading section, where the maximum deviation was 470 F for a gas temperature range from 4000 to 16000 F. On the basis of the foregoing results, the methods reported in references 15 to 17 are considered accurate enough for calculating cooledblade temperatures at the midchord region of most blades and at the leading and trailing sections of thinshelled aircooled blades. Also, reliable temperatures can probably be calculated for blades of high thermal conductivity even if the leading and trailing sections are physically long. The methods are not accurate enough for long leading and trailing sections of low thermal conductivity as indicated by the results described for the trailing section of the 10tube aircooled blade (reference 15). In future cooledturbineblade applications, liquidcooled blades and possibly some cast aircooled blades may be made of steel (low thermal conductivity) and have long leading and trailing sections. Since low conductivity in a long blade section is CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 CONFIDENTIAL 5 conducive to appreciable temperature gradients in that section, a method of calculating blade temperatures more accurate than methods obtainable to date is desired. As was pointed out in the preceding discussion, local trailing section temperatures for a blade of low thermal conductivity and a long trailing section resulted in calculated temperatures at least 1000 F higher than the experimental values. Such trends were noticed as long ago as 1945 by E. Schmidt (reference 18) who observed a much lower trail ingsection temperature than he calculated. This result was attributed by E. R. G. Eckert to the shielding effect of the boundary layer, which is strongly cooled in the forward portion of the blade; that is, the metal temperature increases quite rapidly from the cooled midchord region to the edge of the long trailing section, especially where metals of low thermal conductivity are used. At the time, no method was given for evaluating this effect numerically. Recently, however, some attempts have been made to include these effects by consideration of variations in wall temperature on outside coefficients. References 19 to 21 con sider laminar flow along flat plates and wedges, and reference 22 con siders the case for turbulent flow of an incompressible fluid along a flat plate. Constant property values and pressure gradient effects are included in references 19 and 21, whereas reference 20 allows for certain variations in property values but does not include any pressure gradient effect. On the basis of the results of references 19 to 22, it can be concluded that the calculated trailingsection temperatures of the 10tube blade (reference 15), which were considerably higher than the experimental values, may be partially attributed to the effects of an appreciable temperature gradient in the long trailing section. Consequently, an investigation was conducted at the NACA Lewis laboratory on two aircooled turbine blade configurations (the 10tube blade of reference 15 and a 13fin blade which has long leading and trailing sections) in order to (1) obtain local experimental blade tem peratures around the blade periphery for two gas temperatures and a range of coolingair flow, and (2) compare the data with calculated blade temperatures, wherein an attempt was made to eliminate the out standing discrepancies obtained in previous investigations. The experi mental investigation was conducted in a static cascade because the instrumentation could be more complete and accurate than on a rotating cascade. The purposes of this report are to (1) apply variablewall temperature corrections to the outside coefficients for sections of the blade which require such corrections, and use these corrected coeffi cients to calculate blade temperatures, (2) compare the calculated tem peratures with experimental temperatures obtained in the investigation just described in order to determine the adequacy of the variablewall temperature correction in reducing the differences between experimental CONFIDENTIAL IJACA PR E52HOC' and calculated temperatures for long leading and trailing sections, and (3) present a detailed method for calculating local cooledblade tem peratures by use of the best available theories for obtaining outside and inside coefficients. The method of calculation is of such nature that it can be carried out entirely from design data without any test measurements. The results of this investigation are presented for gas temperatures of 3000 and 10000 F, a range of gas Mach number from 0.4 to 0.6, a mean temperature ratio (gas to average wall) of 1.40, a range of coolicgair flow for Reynolds numbers from 5000 to 40,000 for each gas temperature, and a range of Euler number in the laminar region from 1.0 to 0. APPARATUS AND FRPCEDURE Test Facility A sectional view of the blade test section used in this investiga tion is shown in figure 1. Combustion air passed successively through a flatplate orifice, a combustor, and a plenum chamber prior to entering the test section, and then into the exhaust system. The gasoline com bustor used in these investigations limited the gas temperature range to between 3000 and 10000 F. The inlet duct to the test section was equipped with a bellmouth to insure a uniform velocity profile at the entrance to the cascade. The setup was insulated against heat loss from just down stream of the combustor to just downstream of the test section. A cascade of seven blades was installed in the test section accord ing to the dimensions in figure 2. The test blade, installed as the center blade, was the only blade through which cooling air was passed. The other six blades had the same profile as the test blade. The cooling air that was supplied to the test blade was obtained from the laboratory refrigerated air system. The air passed successively riuLough a plenum chamber, the blade entrance extension, the test blade, the blade exit extension, another plenum chamber, a flatplate orifice, and then into the laboratory exhaust system (see fig. 1). Because it was impossible to connect the plenum chambers directly to the blade, blade entrance and exit extensions of the same shape as the blades were used to conduct the air from the entrance plenum chamber to the test blade and from the test blade to the exit plenum chamber. ITie blade entrance extension had a span of 6 inches and the blade exit extension, a span of 3 inches. Both extensions had an internal freeflow area of 0.043 square inch and a hydraulic diameter of 0.396 inch. In order to reduce the amount of heat conducted from the ends Li the hot test bide, metal was removed in the form of 1/8 inch wide chordwise slots cut thr.ugl the wills of these extensions near the ends a.ajacent to the test blade. These slots were then sealed with a low conductivity material to prevent coolant leakage. CiFAT IDEIITIA; ClIIFILErriTIAL NACA RM E5'27 Blade Description The aircooled turbine blade configurations used in this investi gation were a 10tube blade and a 13fin blade. The reason for selection, these two blade configurations is that the 13fin blade has both .1 .,r7 leading and trailing sections and the 10tube blade has a l cin truailini, section but a thinshell leading section. Figure 3 shows the end views of the two blades. The geometry factors pertinent to the two blade configuraticns are given in the following table: Geometry factor 10tube 13fin blade blade Blade chord, in. 2.00 2.00 Outside perimeter, in. 4.35 4.53 Span, in. 3.92 3.50 Portion of blade span exposed to gas stream, in. 3.00 3.00 T.tas freeflow area of internal .181 .0962 coolingair passage, sq in. Hydraulic diameter of internal .103 .0670 coolingair passage, in. 10tube blade. The 10tube blade used in this investigation was similar to the 10tube blades used in the investigations reported in references 15 and 23. The outside wall of the blade tapered linearly from the root to the tip for reduction of stresses during engine opera tion. The nominal thickness of the wall at the tip was 0.040 inch and at the base, 0.070 inch. The blade shell was cast of hightemperature alloy (X40) in such a manner that the core area was constant over the length of the blade. In order to increase the internal heattransfer surface, 10 tubes were inserted in the hollow blade (fig. 3). They extended through the blade from tip to base. These tubes were brazed to each other and to the inside surface of the hollow blade by Nicrobraze. Of the 10 tubes, four were made of stainlesssteel tubing having a 0.125 inch outside diameter and a wall thickness of 0.010 inch; and six were made of lowcarbon steel with a 0.156 inch outside diameter and a wall thickness of 0.0155 inch. Availability at the time of fabrication accounted for the difference in tubing materials. 13fin blade. The 13fin blade used in this investigation was designed for heattransfer investigations in a static cascade; therefore there was no taper in the blade wall, but the outside blade profile was essentially the same as for the 10tube blade. The blade was machined in two parts divided essentially at the mean camber line, and upon assembly the parts were welded together at the leading and trailing edges; therefore the 13 fins were not continuous (see fig. 3). The fins had an average thickness of 0.036 inch, and the average fin spaci:,n was 0.046 inch. The blade was machined from hightemperature alloy S816. C'ol'WILEilI .L CCITF IDEITr AL NACA RM E52H07 Instrumentation In order to calculate blade temperatures for the conditions of cascade operation, it was necessary that the coolingair weight flow, the blade entrance coolingair temperature, and the gas temperature be known. Also, for purposes of comparison, blade temperatures were meas ured for a range of coolingair weight flows. The coolingair weight flow was measured by means of a flatplate orifice downstream of the test section, and the coolingair temperatures were measured by means of thermocouples in the entrance and exit plenum chambers (fig. 1). The exit coolingair temperature was measured to check the calculated values of this temperature. The total temperature of the combustion gas was measured by means of thermocouples upstream of the test section, and the combustiongas weight flow was measured by means of a flatplate orifice upstream of the test section. The blade temperature distribution was measured by 11 thermocouples in the blade wall around the blade perimeter at the midspan (see fig. 3). In order to obtain the coolingair total temperatures at the blade entrance and exit, thermocouples were placed in the walls of the blade extensions to correct the values of coolingair total temperatures meas ured in the plenum chambers for any heat picked up in the blade entrance and exit extensions. The method of correcting the coolingair tempera tures for the heat picked up in the blade entrance and exit extensions is reported in reference 24. Cascade Operation Blade temperatures were measured for the 10tube and 13fin blades for the following conditions: Blade Gas Gas Gas Gas Mean Euler Cooling Cooling Average tem Prandtl Mach weight temper number, air air blade pera number number flow ature laminar weight Reynoildcs inlet ture Ib \ ratio region flow per number cooling (OF) sec) (gas to blade air ter average lsb perature wall) (OF) 300 0.672 0.40 3.5 1.32 1.0 0.010 5000 10tube and and to to and to to to 50 1000 0.654 0.60 4.5 1.61 0 0.070 40,000 300 0.672 0.40 5.0 1.28 1.0 0.005 5000 13fin and and to to and to to to 8 1000 0.654 0.60 6.0 1.38 0 0.022 17,000 CONFIDENTIAL COiFIDEriTLAL NACA RM E52H07 After the coolingair weight flow was set, frequent checks were made on the blade temperatures in order to ascertain when steadystate conditions were reached. When the blade temperatures and the combustiongas weight flow remained steady for approximately 15 minutes, the necessary readings were taken. METHODS OF CALCULATION The present investigation was limited for simplicity to one dimensional blade temperaturedistribution equations. In the central portion of the blade section, where the blade temperatures at different stations along the blade periphery do not differ very much, average values of chordwise blade temperatures were calculated for the entire region. At the two extreme sections, near the leading and trailing edges, large temperature gradients occurred between the surface swept by the coolant and the surface swept by the hot gas at a given spanwise distance from the blade root. Here, chordwise temperature distributions were calculated under the assumption that the heat conducted from the section in the spanwise direction was negligible when compared with that conducted chordwise from the outer blade surface to the coolant. Blade temperatures were calculated from a design basis; that is, it was assumed that only the blade geometry, effective gas temperature, coolingair weight flow, combustiongas weight flow, blade velocity distribution, and blade entrance effective coolingair temperature were known. The effective gas temperature was obtained from the definition of recovery coefficient, which is a function of the total, static, and effective gas temperatures. From the measured total gas temperature and the calculated velocity distribution, the static gas temperature was determined and used to calculate the effective gas temperature by assuming a recovery coefficient of 0.89 (reference 25). Inasmuch as the cal culated temperature distributions are for the blade midspan position so that a comparison could be made between calculated and experimental blade temperatures, an average effective coolingair temperature is required in the temperaturedistribution equations. It is sufficiently accurate, because of the linear distribution of the coolingair tem perature, to use an arithmetic average of the coolingair total tem peratures at the blade entrance and blade exit for the average effective coolingair temperature. Since the blade entrance coolingair total temperature was known, the blade exit coolingair temperature was cal culated from the equations presented in reference 26. For simplicity in presentation, the temperaturedistribution equa tions required in this investigation and the methods for determining the outside and inside heattransfer coefficients will be considered sepa rately in the following sections. CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 MidchordBladeTemperature Equation Average blade temperatures for the midchord section of the blade (see fig. 3) were obtained from equation (22) reported in reference 2. This equation, in the notation of this report, is Tge B 1 = (1) T T + A g, e a where A= (2) hf Zi (All symbols are defined in the appendix.) Blade temperatures were determined by inserting into the equations the appropriate perimeters of the blades in question and heattransfer coefficients determined by methods to be discussed in subsequent sections of this report. The methods for determining Tg e and T" which are required in equa tion (2) have been discussed in the preceding section. ChordwiseTemperatureDistribution Equation Through Blade Leading and Trailing Sections The leading and trailing sections of the 10tube and 15fin blades were approximated by trapezoids. The onedimensional blade chordwise temperaturedistribution equation for a trapezoidal approximation to the blade leading and trailingedge sections (equation (20), reference 1) is, in the notation of this report, T 2 [ (il1) Jo(it) + iJ1(il)iHo(i)] ge B 2K (3) T i 2t2 ge a i (it,) H,(i)2] [H(i(1)iJ1(i)] 2K kr The geometry factors and the effect of variable wall temperature on out side heattransfer coefficients incorporated in the terms K, 1, and t2 were evaluated from their definitions given in reference 1 for a wedge approximation of the blade leading and trailing sections. CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 The temperature distributions in the leading and trailing sections were calculated by insertion of appropriate blade dimensions and by use of heattransfer coefficients determined by the following methods. Outside HeatTransfer Coefficients The use of different equations for the calculation of blade temper atures in various parts of an aircooled turbine blade is accompanied by the use of different outside heattransfer coefficients in these equa tions. Consequently, outside coefficients will be discussed for the midchord region, the trailing section, and the leading section in that order. Midchord region of blade. The following procedure was used to determine an average outside coefficient for the blade midchord region (fig. 3 shows blade midchord region). Local outside coefficients at three points near the beginning, middle, and end of the pressure surface of the blade midchord region and local coefficients at three similar points on the suction surface were computed. The equation (equation (19), reference 5) 1 3 _ Nug/Prg = Flam Reg laminarr flow) (4) was used to calculate the local coefficients in the laminar region because, as pointed out in the INTRODUCTION, the simplified procedure of reference 5 is currently adequate for application to aircooled impermeablewall blades. For the calculation of local coefficients in the turbulent region, it is assumed that the effect of temperature ratios from 1.0 to 2.0 on heattransfer coefficients is negligible, as was the case in the laminar region (reference 9). Consequently, the equation (equation (26), reference 5) 1 Nug/Pr3 = 0.0296 Re 08 (turbulent flow) (5) was used to calculate the local coefficients in the turbulent region. An arithmetic average of the six local coefficients was used for the average outside coefficient for the blade midchord region. The fluid properties of equations (4) and (5) are based on an assumed wall tem perature. Therefore, the calculation of an average blade midchord tem perature is essentially an iteration process an average blade midchord temperature is assumed to calculate an average outside coefficient, this coefficient is used in equation (1) to calculate an average blade midchord CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 temperature, and then the two values are compared. It can be shown from equation (1) that a 25 percent change in the assumed blade temperature will result in a change of approximately 3 percent in the calculated blade temperature. The location of the transition point on each blade surface is required in order to determine whether laminar or turbulent flow pre vails at each of the six local midchord points in question, and hence whether equation (4) or equation (5) is applicable at each of these points. Moreover, if equation (4) is applicable, a knowledge of the local Euler number is necessary for the calculation of Flam. The value of the local velocity, the location of the transition point (considered herein as the minimum pressure point), and the calculation of the required local Euler number 2 Eu = (dp/dx)/(pgUg,/x) (6) were all determined from the velocity distribution about the blade in question; the velocity distribution was calculated by the method of reference 12 and checked by measurements made on a Lucite blade of sim ilar shape. From this calculated velocity distribution, the pressure distribution and hence the density were obtained. The Reynolds number, with properties based on wall temperature, was then determined. Trailing section of blade. The trailing sections of the blades considered in this investigation were approximated by trapezoids, to which equation (3) is applicable. In these blades, which have a long trailing section and are made of a hightemperature alloy (low thermal conductivity), the wall temperature varied rapidly beyond the coolant passage, and according to reference 22 the outside heattransfer coeffi cient could consequently be quite different from the coefficients for a constant wall temperature. This difference was provided for by use of a curve for the turbulent boundary layer (fig. 4) which is based on approximate analytical results of reference 22. The analysis of ref erence 22 is subject to the assumptions of a flat plate (no pressure gradient), constant property values, and no frictional dissipation of energy within the boundary layer. Since it can be assumed that trailing sections of turbine blades have a negligible pressure gradient and the effects of the other two assumptions are quitesmall, figure 4 should be applicable for trailing sections of turbine blades which have an appre ciable temperature gradient. Figure 4 presents the ratio of the outside heattransfer coefficient for variable wall temperature to that for con stant wall temperature as a function of the exponent n, where n is given by the relation * Tg,e (T) T T g,e B n (7) Tg,e TB CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 CONFIDENTIAL 13 and X is measured along the profile from a reference point where the midchord region with uniform temperature TB ends and the region of rapid change in wall temperature TB begins. In the investigations reported herein, this reference point was taken at the end of the tube or fin section (see fig. 3). Equation (67) of reference 22, derived for a flat plate and tur bulent flow, expresses the relation between the heattransfer coeffi cients based on variable and constant wall temperatures. In the nota tion employed herein, equation (67) is h An Yn (X/j)n S= (8) hoX An (X/j)n where n An (X/j)n is a power series expansion of the temperature ratio f(TB,X/Tge)I and Yn is a relation among certain gamma functions defined as (40n +2 r + r rY ( 9) Yn = 40n 32 r +  39 39 In the calculations necessary for the construction of figure 4, only one term in the series expansion was considered so that equation (8) sim plified to h S= Yn (10) Values of n were inserted into equation (9) to obtain a set of ratios of the variable to constantwalltemperature heattransfer coefficients. When an attempt is made to apply figure 4 to calculations for a blade trailing section, the value of n is unknown. However, a value of ho (constantwalltemperature outside coefficient) can be deter mined by the method employed in calculating the midchord coefficient. In this instance, two local outside coefficients were calculated on each surface of the trailing section by application of equation (5) (tur bulent flow prevails over the blade trailing section). The average of these four local coefficients To was used along with an assumed value of n, say 0.6, and ho (variablewalltemperature average outside coefficient) was then obtained by use of figure 4. The temperature distribution was then calculated by use of equation (3). Since the CONFIDENTIAL NACA RM E52H07 value of Tg was known, a logarithmic plot of Tg T against X determined n. If this value equaled the assumed n, the temperature distribution was the required one; if the two values of n differed, n was adjusted until the initial and final values agreed, and the desired value of ho was obtained from figure 4. Leading section of blade. The effect of variable wall temperature on outside heattransfer coefficient was also taken into consideration for the blade leading section. Here laminar flow prevails, and the analysis of reference 21 for wedge flows, flatplate flow, and stagna tion flow, assuming constant property values, was used to account for the effect of variable wall temperature. Figure 5 gives the relation between the outside heattransfer coefficients for variable and constant wall temperatures as a function of the exponent n (see equation (7) wherein X is replaced by x for this case). The values used in con structing figure 5 were obtained from table 1 (Euler number of zero) and table 2 (Euler number of 1.0) of reference 21 for a Prandti number of 0.7 by dividing the outside coefficients for various values of n (variable wall temperature) by the value for n equal to zero (constant wall temperature). Since the leading sections of turbine blades are in the laminar region and have Euler numbers ranging from 1.0 (stagnation point) to zero (transition from laminar to turbulent), figure 5 is applicable to blade leading sections. For this investigation, the constantwalltemperature outside coefficient was determined in a way similar to that for the midchord coefficient, that is, as an average of 12 local outside coefficients (six on each blade surface located between the coolant passage nearest the blade leading section and the stagnation point) determined from equation (4) for constant property values. Also, local Euler numbers were calculated for the 12 points and averaged. Therefore, figure 5 and values of ho and Eu were used to determine ho for the leading section in the manner used for the trailingsection turbulentflow outside coefficient. For the leading section, however, x was measured from the stagnation point, as in reference 21. Inside HeatTransfer Coefficient Different inside heattransfer coefficients were required for use in the two temperaturedistribution equations (equations (1) and (3)). In the midchord region of the blades, the heattransfer surface area was greatly augmented by the use of tubes and fins. In order to make allowance for this, a socalled effective inside coefficient was required. On the other hand, heat leaving either the leading or trail ing blade section is picked up by the cooling air adjacent to each sec tion, and the presence of tubes or fins in the midchord region had little or no effect. Consequently, an average inside coefficient for the coolant which swept the leading and trailing sections was required for CONF IDEINT IAL CONFIDENTIAL NACA RM E52H07 the calculation of temperatures in these regions. The methods of deter mining the effective and the average bladetocoolant heattransfer coefficients that were required follow. Effective coefficient. For the midchord region of the blades, where the tubes or fins substantially increased the heattransfer sur, face area, an effective coefficient was determined. An average heat transfer coefficient for fully developed turbulent flow in pipes was obtained from equation (90) of appendix F, reference 26, Nua = 0.019 Rea0.8 (11) The fluid property values in equation (ll) were based on wall temperature in references 16 and 26, which was considered adequate for these investi gations. It can be shown (reference 27) that for air flowing in a smooth round tube, a satisfactory correlation of average inside heattransfer :oefficients was obtained for a range of wall temperatures from approxi mately 1500 to 16000 F and an inletair temperature near 750 F when the fluid properties were based on wall temperatures. When the range of wall temperatures was extended to cover a more complete range from approxi mately 750 to 26000 F and the inletair temperature varied from 750 to 10000 F, a good correlation of the heattransfer data was obtained only when the fluid properties were based on film temperature (an average of the air and wall temperatures); it was assumed in reference 27 that the thermal conductivity of the air varied as the square root of the tempera ture. Since the best correlation of heattransfer data over an extended range of wall temperature and inletair temperature was obtained when the fluid properties were based on film temperature, the fluid property values in equation (11) were based on film temperature (an average of the cooling air and wall temperatures) in this investigation. The required effective inside coefficient, based on inner wall sur face area only, was determined from equation (6) of reference 16, which in the notation employed herein is h = hi m(2) tanh (LrcPr) + Z T (12) r= (mr + Tr) LrLpr r=l r=l r=l where ih is obtained from equation (11) and is based on film tem perature. .For application to aircooled turbine blades with internal surfaces other than fins, the 10tube blade in this instance required replacement of the internal surface by equivalent fins. The following general method of reference 16 was used: CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 (1) By inspection, the blade was divided into s sections of equi valent fins which appeared to have equal fin lengths, spacings, and thicknesses, and the number of fins a. in each section was determined. (2) For each section, the inside blade shell perimeter Ii.r li,r = a.(mrn + Tr and the total wetted perimeter li,W,r were determined. (3) The quantities required for equation (12) were determined by the following definitions: armr= ir aT (13) and L i,W,r .rmr Lr2 (14) The relation between bh and hi for the 10tube and 15fin blades is shown in figure 6. Examples showing the application of equa tion (12) to tube and fintype blades are presented in appendix B of reference 16. Average coefficient. Average inside coefficients for computing leading and trailingsection temperatures were calculated from equa tion (11). In each case, the coefficient was determined for the coolant which swept the one end of the blade section in question, and the dimen sions of this particular passage were used in equation (11). The cool ant flow for each of these passages was taken as the average coolant flow per unit of flow area. Since hi required for both leading and trailingsection temper ature calculations and for determining hf is based on film tempera ture, which necessitates the assumption of a wall temperature, an iterative process similar to that used in calculating midchord outside coefficients was used here. RESULTS AND DISCUSSION The comparisons of calculated and measured blade temperatures for the 10tube and 13fin blades for the conditions of this investigation (listed previously in APPARATUS AND PROCEDURE section) are shown in fig ures 7 to 9 for the midchord, trailing, and leading sections of the blades, respectively. In each case the calculated temperatures are plotted as the ordinates and the measured temperatures, as the abscissas, with the 450 line representing perfect agreement. CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 Comparison of Calculated and Measured Midchord Blade Temperatures Calculated midchord temperatures are compared with measured midchord temperatures in figure 7. In this region, temperatures were observed on each of the blade surfaces and the arithmetic average of these measure ments was used to represent the measured midchord temperature. For the calculation of the midchord temperatures, the average effective inside heattransfer coefficient was obtained by considering three groups of fins (or effective fins) for each of the blades. It must be remembered that in some cases more fin groups might be necessary since it is not advisable to average fins (or tubes) which differ very much in dimen sions. Figure 7 shows that the calculated and measured midchord tempera tures agree very closely for both the 10tube and 13fin blades; that is, the points for both blades are very close to the 450 perfect agree ment line. The quantity usually selected to compare the magnitudes of errors is the probable error (the number of errors greater than the probable error is the same as the number less than the probable error). For the midchord temperatures of the blades investigated, the probable error was found to be a maximum of 80 F for a gas temperature of 3000 F and a maximum of 130 F for the 10000 F gas temperature. The good agree ment between calculated and experimental temperatures clearly shows that departures from Mach numbers close to zero and a temperature ratio of 1.0 (the conditions used in obtaining equations (4) and (5)) are not enough to require corrections in the present investigation. Also, the assumption that the effect of temperature ratios from 1.0 to 2.0 on heattransfer coefficients in the turbulent region is negligible appears to be justified. The applicability of equations (4) and (5) to other blade configurations and conditions will be discussed in a subsequent section of this report. Comparison of Calculated and Measured Temperatures Near Blade Trailing Edge In figure 8, calculated blade temperatures are plotted against measured temperatures for the thermocouple located nearest the trailing edge of each blade (see fig. 3). For the 3000 F gas temperature the agreement is again very good; the probable error for the 10tube blade was found to be 60 F and for the 13fin blade, 40 F. For the 10000 F gas temperature, the probable errors were 360 F and 300 F for the 10tube and 13fin blades, respectively. Trailingsection temperatures were also calculated with no allowance for the effect of variation in wall temperature on the outside heattransfer coefficient, and the calculated temperatures were at least 1000 to 1500 F higher than CONFIDENTIAL CONFLDENTIAL NACA RM E52B07 the measured temperatures (see reference 15 for 10tube blade results). It thus appears that outside heattransfer coefficients which include walltemperaturevariation effects should be used in equation (3) for calculating accurate temperatures in cooled blades with long trailing sections. The relatively large probable errors obtained for the trailing section of the blades at the 10000 F gas temperature may be due to separation of flow along this section of each blade. Calculated tem peratures are dependent on the use of appropriate outside heattransfer coefficients; since no method for the calculation of such coefficients is as yet known for separated regions, special corrections for separa tion of flow cannot be made at the present time the separated region is simply included in the turbulent region. However, the conditions of equation (5) negligible pressure gradient, low subsonic Mach numbers, and a temperature ratio equal to 1.0 were fulfilled closely enough in the present investigation that the use of equation (5) precludes serious differences between experimental and calculated trailingsection tem peratures. The justification of the assumption that the temperature ratio has a negligible effect on heat transfer in the turbulent region is not so apparent from these results as for the midchord temperature calculations. Comparison of Calculated and Measured Temperatures Near Blade Leading Edge A plot of calculated against measured blade temperatures for the thermocouple located nearest the blade leading edge is shown in figure 9. Agreement at this point is also good. The maximum deviation between calculated and measured temperatures was found in the case of the 13fin blade when the gas temperature was 10000 F. The probable errors were found to be 50 and 80 F for the 10tube blade and 60 and 180 F for the 13fin blade for gas temperatures of 3000 and 10000 F, respectively. Temperatures were also calculated for this blade location by use of outside coefficients based on constant wall temperature for the 10000 F gas temperature. Calculated temperatures were found to be of the order of 500 to 700 F below the measured values for the 13fin blade, which has a long leading section. A negligible difference between calculated and measured temperatures for the 10tube blade which has a short leading section (thin shell) resulted. Thus, in order to be able to calculate leadingsection temperatures accurately, outside coefficients, which allow for variations in wall temperature, should be used in equation (3) for blades with long leading sections, whereas constantwalltemperature outside coefficients may be used in equa tion (3) for blades with short leading sections (thin shell). CONFIDENTIAL COITFIDEINTIAL NACA RM E52H07 For the present investigation the fact that equation (4) and fig ure 5, used to determine the outside coefficients, are based on constant property values has little effect on the coefficients, because for the laminar region a change from constant property values results in a change of less than 1.0 percent in the coefficient (references 7 to 9). Influence of Errors in Calculated Temperatures on Design Considerations The agreement between calculated and measured temperatures at all locations on the blades investigated is considered to be well within the accuracy required for turbine design. Errors in prediction of blade temperatures in the trailing or leading sections are less serious than in the midchord region because evidence from endurance tests (refer ence 28) and calculations based on reference 29 indicate that the midchord region of the blade is the main stress carrying member. The effect of errors in calculated average blade midchord temperatures on blade mate rial life will therefore be discussed. At gas temperatures higher than the 10000 F temperature investigated herein, the probable errors in the calculated midchord temperatures would be somewhat higher. An increase in gas temperature from 5000 to 10000 F resulted in an increase in the maximum probable error from 80 to 130 F for the midchordtemperature calculations. With stressrupture char acteristics as a basis, for most lowalloy steels suitable for non strategic turbine blades 250 F is about the maximum error permissible for design practice. For example, consider a turbine rotor blade made from Timken 1722A(S) steel, centrifugally stressed to 25,000 pounds per square inch at the critical section, and designed for a stressratio factor of 2.0. (The stressratio factor is an indication of the blade stress carrying capacity and is defined in reference 29 as the ratio of the integrated allowable stress to the integrated centrifugal stress at the blade critical section.) An error of 250 F in the average midchord temperature would result in stressratio factors of 2.36 and 1.56, respectively. On the basis of limited results, it is indicated in ref erences 28 and 29 that turbine design calculations should be based on stressratio factors above 1.5. It is believed that for the higher gas temperatures encountered in current turbine engines, approximately 17000 F, the probable error in the calculated average midchord tempera ture would not exceed the permissible value of 250 F. Modifications to this calculation method for other blade configurations and conditions are given in the following section. CONFIDENTIAL CONFIDENTIAL 2IACA RM E52H07 APPLICATION OF ANALYSIS TO OTHER BLADE CONFIGURATIONS AND COrDITIONS The 10tube and 13fin blades were selected for this investigation because they had long leading and trailing sections (sufficient amount of low thermal conductivity metal area in section to cause an appreciable temperature gradient), except the leading section of the '10tube blade, and large differences in coolant passage geometry. The results already presented indicate that calculated trailingsection temperatures are greatly reduced and leadingsection temperatures increased in blades which have an appreciable temperature gradient in these sections when outside heattransfer coefficients are corrected by newly published formulas for variable wall temperatures. Also, the results show that the same calculation procedures applied equally well to both blades and hence that passage configuration effects are negligible. However, the use of'equation (1) yielded only an average midchord temperature at the blade milsparn position. In order to calculate an average blade midchord temperature at a position other than midspan for a stationary blade, equation (20) of reference 2 must be applied. This equation requires that the effective coolingair temperature at the blade root be known, whereas equation (1) of this investigation utilizes an average cooling air temperature. For rotating blades, equation (18) of reference 2 is applicable; in this case, too, the blade root effective coolingair temperature must be known. However, for calculating local temperatures at the leading and trailing sections, equation (3) is still applicable. The procedure just described is applicable to other configurations with long leading and trailing sections (long metal heatconduction paths). On the other hand, blades such as those reported in refer ences 29 and 30 have thin blade shells so that the coolingair passages are extended well into the leading and trailing sections and the wall temperature variations in these regions are greatly reduced. For such blades the effect of walltemperature variation can often be neglected and the calculation of blade temperatures is simplified by using the outside heattransfer coefficients calculated by means of equations (4) and (5) without making a correction for walltemperature variation. This was verified in the present investigation by the leadingsection cal culations for the 10tube blade which has a thinshell leading section. That is, the correction in the outside heattransfer coefficient for the 10tube blade leading section was only about 2 percent, whereas that for the 13fin blade leading section was 20 percent. A comparison of calculated and measured average blade temperatures for two thinshell blades investigated in a turbojet engine using the simplified method (no variablewalltemperature correction) resulted in errors in calcu lated temperatures of about 300 F (reference 16). Some of this error may have been due to poor contact between the internal heattransfer surface and the shell of the blades, although it may be possible that the theories verified in this investigation are not entirely adequate for rotating aircooled blades. COIJFIDEIITIAL CONTFIDENTIAL NACA RM E52H07 CONFDErNTIAL 1i For forcedconvection liquidcooled turbine blades with long lead ing and trailing sections, exactly the procedure described herein can be used to account for the variablewalltemperature effect when cal culating outside heattransfer coefficients. The inside heattransfer coefficient should be calculated by means of an appropriate pipe cor relation, as shown in reference 4. The effect of temperature ratio may in some cases be an Irriprct.ant condition influencing outside heattransfer coefficients. Nevertheless, the effect seems to be negligible for temperature ratios from 1.0 to 2.0. As pointed out earlier, a temperature ratio (gastowall) of 2.0 is the probable limit for current aircooled blades with impermeable walls. Also, even though turbine inlet temperatures may exceed 17000 F in the near future, it is believed that the temperature ratio will still be close to 2.0 for aircooled blades with impermeable walls. Conse quently, the methods employed in this investigation may be utilized for such hightemperature applications. For future cooledturbine designs, probably those utilizing transpiration or liquid cooling, where tempera ture ratios higher than 2.0 may be encountered, the methods used herein must be modified. A detailed discussion of these modifications is beyond the scope of this report. SUMMARY OF RESULTS Temperatures calculated from previously published NACA equations and measured blade temperatures were compared for a 10tube and a 13fin aircooled turbine blade tested in a static cascade. The results of this investigation for gas temperatures of 3000 and 10000 F, subsonic gas'Mach numbers, mean temperature ratios (gastoaverage wall) of 1.3 to 1.6, and a range of coolingair weight flow are summarized as follows: 1. For those blades having long leading and trailing sections (sufficient amount of low thermal conductivity metal area in section to cause an appreciable temperature gradient), calculated trailingsection temperatures were greatly reduced and leadingsection temperatures increased when outside heattransfer coefficients were corrected by newly published formulas for variable wall temperatures. 2. For the midchord region of the blades investigated, the maximum probable error between calculated and measured blade temperatures was 80 F at a gas temperature of 3000 and 130 F at a 10000 F gas temperature. 3. For the trailing section of the blade, the probable errors were 60 and 40 F at the 3000 F gas temperature and 360 and 300 F at the 10000 F gas temperature for the 10tube and 13fin blades, respectively. CONFIDENTIAL NACA RM E52H07 4. For the leading section of the blade, the probable errors were 50 and 60 F at the 300 F gas temperature and 80 and 180 F at the 10000 F gas temperature for the 10tube and 13fin blades, respectively. 5. The short (thin shell) leading section of the 10tube blade required only a 2 percent correction of the outside heattransfer coefficient for the variablewalltemperature effect, whereas the long leading section of the 13fin blade required a 20 percent correction. Lewis Flight Propulsion Laboratory National Advisory Committee for Aeronautics Cleveland, Ohio CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 APPENDIX SYMBOLS The following symbols are used in this report: A flow area of coolant passage, sq ft An coefficients of power series cp specific heat at constant pressure, Btu/(lb)(oF) Dh hydraulic diameter, 4 times flow area, ft wetted perimeter Eu Euler nl.riber, (dpg/dx)/(pgcoUTg c/x) Flam variable, evaluated in figure 8, reference 5 h local heattransfer coefficient (constant wall temperature), Btu/(sec)(sq ft)(F) h average heattransfer coefficient, Btu/(sec)(sq ft)(oF) hf average effective inside heattransfer coefficient, Btu/(sec)(sq ft)(F) h local outside heattransfer coefficient (variable wall temperature), Btu/(sec)(sq ft)(F) ho* average outside heattransfer coefficient (variable wall temperature), Btu/(sec)(sq ft)(OF) iHo,H1 Bessel functions j chordwise distance from blade trailing or leading edge to coolant passage, ft Jo,iJ1 Bessel functions 1 K h _ kB sin k thermal conductivity, Btu/(sec)(ft)(0F) Lr fin length of fins in group r, ft (see equation (14)) CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 2 perimeter, ft mr spacing of fins in group r, ft U.u Nusselt number of cooling air, (hi Dh)/ka,F Nug Nusselt number of gas, hox/kg,w n exponent of X in relation (Tg,e Ti)/(Tg,e TB) Xn Prg Prandtl number of gas, (cp,g,wlg,w)/kg,w p static pressure, lb/sq ft absolute Reg Reynolds number for cooling air, (wa Dh)/(A 'a,F) Reg Reynolds number for gas, (Ug, a g,w x)/',w r index of summation s number of fin groups T temperature, OF T average temperature, OF T average coolingair total temperature, OF a T blade temperature for case with variable wall temperature, OF U velocity, ft/sec w air weight flow, lb/sec X distance along blade surface as shown in figure 3, ft x distance along blade surface from stagnation point, ft Yn function defined by equation (8) y distance from blade trailing or leading edge to blade element, ft (see reference 1) a number of fins in group r r CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 p Hl(i1l) Jo(iS2) + iJl(i~1) iHo(i2) except in.equation (9), where r denotes the usual gamma function 1 1 TI(l tan #) 2 TI(l tan 4) '1t2 2K y' + 2 tan 2 2K 2 tan 9 ' 1 2K +  2 tan f hfl i p viscosity, lb/(ft)(sec) p density, lb/(cu ft) T ,T2 trapezoidal thicknesses at leading or trailing edge and coolant passage, respectively, ft Tr thickness of fins in group r, ft 1 2 Ei 22 (r B2 1 where ih is based on film temperature (7B r1 tan1 T 2j 1 Subscripts: a air B blade e effective F evaluated at film temperature g gas i inside CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 o outside r any internal section of blade formed by fins or equivalent fins W wetted w evaluated at wall temperature X local point o0 free stream Superscripts: denotes linear dimension increased by T1/2 refers to variable wall temperature REFERENCES 1. Livingood, John N. B., and Brown, W. Byron: Analysis of Temperature Distribution in LiquidCooled Turbine Blades. NACA Rep. 1066, 1952. (Supersedes NACA TN 2321.) 2. Livingood, John N. B., and Brown, W. Byron: Analysis of Spanwise Temperature Distribution in Three Types of AirCooled Turbine Blade. NACA Rep. 994, 1950. (Supersedes NACA RM's E7Blle and E7G30.) 3. Cohen, H.: Heat Transfer in AirCooled GasTurbine Blades. Engineering, vol. 173, no. 4484, Jan. 4, 1952, pp. 2123. 4. Freche, John C., and Schum, Eugene F.: Determination of Bladeto Coolant HeatTransfer Coefficients on a ForcedConvection, WaterCooled, SingleStage Turbine. NACA RM E51E18, 1951. 5. Brown, W. Byron, and Donoughe, Patrick L.: Extension of Boundary Layer HeatTransfer Theory to Cooled Turbine Blades. NACA RM E50F02, 1950. 6. Eckert, E. R. G., and Livingood, John N. B.: Calculations of Laminar Heat Transfer Around Cylinders of Arbitrary Cross Section and TranspirationCooled Walls with Application to Turbine Blade Cooling. NACA RM E51F22, 1951. CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 7. Brown, W. Byron: Exact Solution of the Laminar Boundary Layer Equations for a Porous Plate with Variable Fluid Properties and a Pressure Gradient in the Main Stream. Paper presented at First U.S. National Congress of Applied Mechanics (Chicago), June 1116, 1951. 8. Brown, W. Byron, and Donoughe, Patrick L.: Tables of Exact Laminar BoundaryLayer Solutions When the Wall is Porous and Fluid Proper ties are Variable. NACA TN 2479, 1951. 9. Eckert, E. R. G., and Livingood, John N. B.: Method for Calculation of Heat Transfer in Laminar Region of Air Flow Around Cylinders Sof Arbitrary Cross Section (Including Large Temperature Differences and Transpiration Cooling). NACA TN 2733, 1952. 10. Boelter, L. M. K., Grossman, L. M., Martinelli, R. C., and Morrin, E. H.: An Investigation of Aircraft Heaters. XXIX Comparison of Several Methods of Calculating Heat Losses from Airfoils. NACA TN 1453, 1948. 11. Johnson, H. A., and Rubesin, M. W.: Aerodynamic Heating and Convective Heat Transfer Summary of Literature Survey. Trans. A.S.M.E., vol. 71, no. 5, July 1949, pp. 447456. 12. Hubbartt, James E., and Schum, Eugene F.: Average OutsideSurface HeatTransfer Coefficients and Velocity Distributions for Heated and Cooled Impulse Turbine Blades in Static Cascades. NACA RM E50L20, 1951. 13. Donoughe, Patrick L.: Outside Heat Transfer of Bodies in FlowA Comparison of Theory and Experiment. Thesis submitted to Case Inst. Tech., June 1951. 14. Freche, John C., and Schum, Eugene F.: Determination of Gasto Blade Convection HeatTransfer Coefficients on a ForcedConvection, WaterCooled SingleStage Aluminum Turbine. NACA RM E50J23, 1951. 15. Ellerbrock, Herman H., Jr.: Some NACA Investigations of Heat Transfer of Cooled GasTurbine Blades. Paper presented at the General Discussion on Heat Transfer. Inst. Mech. Eng. (London) and A.S.M.E. (New York) Conference (London), Sept. 1113, 1951. 16. Ziemer, Robert R., and Slone, Henry 0.: Analytical Procedures for Rapid Selection of Coolant Passage Configurations for AirCooled Turbine Rotor Blades and for Evaluation of HeatTransfer, Strength, and PressureLoss Characteristics. NACA RM E52G18, 1952. CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 17. Schum, Eugene F., Freche, John C., and Stelpflug, William J.: Comparison of Calculated and Experimental Temperatures of WaterCooled Turbine Blades. NACA RM E52D21, 1952. 18. Petrick, E. N.: A Survey of German Hollow Turbine Blade Develop ment. Pt. I Initial Investigations and Developments. Purdue Univ., Purdue Res. Foundation, pub. by USAFAMC, WrightPatterson Air Force Base, Dayton (Ohio) Oct. 1949. 19. Schuh, H.: Laminar Heat Transfer in Boundary Layers at High Velocities. Rep. and Trans. 810, British M.A.P., April 15, 1947. 20. Chapman, Dean R., and Rubesin, Morris W.: Temperature and Velocity Profiles in the Compressible Laminar Boundary Layer with Arbitrary Distribution of Surface Temperature. Jour. Aero. Sci., vol. 16, no. 9, Sept. 1949, pp. 547565. 21. Levy, Solomon: Heat Transfer to ConstantProperty Laminar Boundary Layer Flows with PowerFunction FreeStream Velocity and WallTemperature Variation. Jour. Aero. Sci., vol. 19, no. 5, May 1952, pp. 341348. 22. Rubesin, Morris W.: The Effect of an Arbitrary SurfaceTemperature Variation Along a Flat Plate on the Convective Heat Transfer in an Incompressible Turbulent Boundary Layer. NACA TN 2345, 1951. 23. Ellerbrock, Herman H. Jr., and Stepka, Francis S.: Experimental Investigation of AirCooled Turbine Blades in Turbojet Engine. I Rotor Blades with 10 Tubes in CoolingAir Passages. NACA RM E50104, 1950. 24. Brown, W. Byron, and Slone, Henry 0.: Pressure Drop in Coolant Passages of Two AirCooled TurbineBlade Configurations. NACA RM E52D01, 1952. 25. Esgar, Jack B., and Lea, Alfred L.: Determination and Use of the Recovery Factor for Calculating the Effective Gas Temperature for Turbine Blades. NACA RM E51G10, 1951. 26. Brown, W. Byron, and Rossbach, Richard J.: Numerical Solution of Equations for OneDimensional Gas Flow in Rotating Coolant Passages. NACA RM E50E04, 1950. CONFIDENTIAL CONFIDENTIAL NACA RM E52807 27. Humble, Leroy V., Lowdermilk, Warren H., and Desmon, Leland G.: Measurements of HeatTransfer and Friction Coefficients for Subsonic Flow of Air in Smooth Tubes at High Surface and Fluid Temperatures. NACA Rep. 1020, 1951. (Supersedes NACA RM's E7L31, E8L03, E50E23 and E50H23.) 28. Stepka, Francis S., and Hickel, Robert 0.: Experimental Investi gation of AirCooled Turbine Blades in Turbojet Engine. IX Evaluation of the Durability of Noncritical Rotor Blades in Engine Operation. NACA RM E51J10, 1951. 29. Esgar, Jack B., and Clure, John L.: Experimental Investigation of AirCooled Turbine Blades in Turbojet Engine. X Endurance Evaluation of Several TubeFilled Rotor Blades. NACA RM E52B13, 1952. 30. Bartoo, Edward R., and Clure, John L.: Experimental Investigation of AirCooled Turbine Blades in Turbojet Engine. XII Cooling Effectiveness of a Blade with an Insert and with Fins Made of a Continuous Corrugated Sheet. NACA RM E52F24, 1952. CONFIDENTIAL CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 O \ o rp 4, I: 0 fl 10 44 0 d 4r, 0) 0) ci Cl) H} 1U1 CONFIDENTIAL NACA RM E52H07 Figure 2. Cascade geometry. CONFIDENTIAL CONFIDENTIAL NACA RM E12HO7 o Thermocouple locations //// Midchord section (a) 10tube blade. Leading edge (b) 13fin blade. Figure 3. Sectional view of test blades showing blade thermocouple locations. CONFIDENTIAL Trailing edge CONFIDENTIAL NACA RM E52H07 1.2 1.1 1.0 .9 .8  .7 0 CONFIDENTIAL Figure 4. Correction factor for gastoblade heattransfer coefficients for variable wall temperature in turbulent boundary layer. CONFIDENTIAL NACA RM E52H07 Figure 5. Correction factor for gastoblade heattransfer coefficients for variable wall temperature in laminar boundary layer. CONFIDENTIAL OL 1.0 CONFIDENTIAL NACA RM E52H07 u0 Blade 07  10tube 13fin / .06 S/ / .05 / .04 // .03 II ,02 .01 0 .01 .02 .03 .O0 hi, Btu/(sec)(sq ft)(oF) Figure 6. Chart for determination of effective bladetocoolant heattransfer coefficients. CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 (a) 10tube blade. 600 o 400, i~f~i\ _____ _____ A .__ ____ _____ ___ ____ _____ j/  200 400 600 81 Experimental temperature, OF (b) 13fin blade. Figure 7. Comparison of calculated average and experimental average midchord blade temperatures for two aircooled blades in a static cascade. CONFIDENTIAL CLuu CONFIDENTIAL NACA RM E52H07 Combustion gas 0o 800 temperature  o (OF) c 1000  A 300 600 400 200_ (a) 10tube blade. Experimental temperature, OF (b) 13fin blade. Figure 8. Comparison of calculated and experimental local temperatures near blade trailing edge for two aircooled blades in a static cascade. CONFIDENTIAL CONFIDENTIAL CONFIDENTIAL NACA RM E52H07 400 60U Experimental temperature, 1UUU (b) 13fin blade. Figure 9. Comparison of calculated and experimental local temperatures near blade leading edge for two aircooled blades in a static cascade. CONFIDENTIAL NACAangleI 995 400 1 f,,f JUJUU     /      800 Combustion gas temperature ( F) o 1000 A 300 600 / 400  200 _ (a) 10tube blade. 1000  800 /o/ 8 0 0        __  600   400 / 200 __________ _____ ____ _____ _________I  Z 1 0 z/ 0 rU Id!0 Z Z~Z h 3 0 oW 0 Cd C', 00 C)1C g~p lw o S .n a0 0 0 = E ~ oa m U S " n~i S'S^Sf a s'S,, , (SW So Eoa c zap u P. cd cc)a bI "oa, (u *.B' cd : o 4141t o 't t eu 00' 31 U~ .04 C 4Q) 410. 410j 04 0. * Ocd cd, S ' ~U~f~CU... 2 0 C @2 C. > 0 .0000 4 e1 F4 CU l~il^^SlI a)C CD 0 t 0 .0 *4 . .4  s fci C.. @2 a C. 41 ''*2s S5S! *aor 'ES ^ S *SK h.2 a,3,3^ ,2= o.' c.Ucq ; Q B1 s (Y *0 0 raj aQ IV C o~* 4 ao E4a = u Ma 02 d .4 icU HS SzO < 41' "~Oo Oci B EI aci z 0a 0 z 0 0 >0 to u s 0 o E4 1 U (112c w E. p : ll::)gg P Ma uU M S Oz S"4.i~n^~ I ; E4 0r cd 1 Z  W < Z Ui 1 a: c r., uM >S E i MZu 4 CC ; m ..o Urt N CT I; H 4o 4 o c 4 1Z8 CI 1E2 p. CD 0 0' Si4 u iUSwio3, :13 1 bD k 4 mt u r CUTd 0 "^a 4 0 q > cl w c0 bp 0 0 Ck .0 c '0 Ur CUo 4 01. C. 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