DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN UNIVERS~ OF MICHIGAN Ann Arbor EXTERNAL MEIMRANDMM NO. 14 Project Ml-794 (AAF Contract 133-038 ac-14222) ProJect "Wizard" Aerodynamic Heating of a Supersonic Missile During Accelerated Flight Prepared By' dLa David W. Approved By Rutkowski October 9, 1947 This documnnt contains information affectinp the rat onal defense of the United Slates w1iin tk" '_.,':-',,:: tbt '''!,:;'.? 's e, t,,:, U. S. G., '1 and 32. its trals nissm-n. revni;,tson <:f!s coni ents in any '3ain-}cr '.o an unaulltllorizred nesr, n is prolii.;d by la 1

Printed and Lithoprinted in U. S. A. University Lithoprinters, Ypsilanti, Michigan 1947

DEPARTMENT OF ENGINEERING RESEARCH Report No. UW 14 UNIVERSITY OF MICHIGAN, Page 1 INTRODUCTION The air in the boundary layer of a high speed missile is heated by viscous retardation. The resulting high temperature of the missile skin is a controlling factor in the final choice of skin material and thickness. The importance of skin temperature problems in guided missile work wras realized and a preliminary study was initiated in the summer of 1946. As a result of that study, University of Michigan Memorandum #3 was issued in March 1947 to provide a conservative estimate of the order of magnitude of the skin equilibrium temperature to be expected. Inasmuch as the equilibrium temperature is usually not realized in the case of accelerated flight, the method of the present report has been expanded to include the temperature lag due to the heat capacity of the skin. The development is based upon quasi-steady methods, because the influence of acceleration upon heat transfer is unknown. Further, it has been noted that a different form of the Reynoldts number was used in the original German work. This change greatly affects the numerical results of all calculation methods based upon the German tests. The method of this report can be used to predict the maximsm temperatures and rates of heating to be used for tests instudies of the strength of structural materials at elevated temperatures.

DEPARTMENT OF ENGINEERING RESE ARCH Page 2 | UNIVERSITY OF MICHIGAN Report No. UI14 SUMIARY The skin heating investigation which was begun in UMM-3 has been continued to include the transient effects of the heat capacity of the missile skin. In this report, the method of computation has been reduced to a standard form and the accuracy of prediction improved by rewriting the expression given in UMM-3 for the average heat transfer coefficient. The method of computation is completely developed in the Appendix. An example of the technique of computation is included, with a comparison with flight test measurements made on the V-2 missile at White Sands, N. Il. The method gives good agreement with tests and, at this stage of progress, may be considered sufficient for the evaluation of skin temperatures. However, further experimental work is necessary before conclusions of satisfactory agreement and validity of assumptions can be derived.

DEPARTMENT OF ENGINEERING RESEARCH Report No UM- 14| UNIVERSITY OF MICHIGAN Page 3 SYMBOLS A = Surface area, sq ft a = Local speed of sound, ft per see cp = Specific heat at constant pressure, BTU per lb per degree F Cv - Specific heat at constant volume, BTU per lb per degree F Cakin Specific heat of skin material, BTU per lb per degree F g = Acceleration of gravity, ft per sec per sec h - Local heat transfer coefficient, BTU per sec per sq ft per degree F h - Average heat transfer coefficient, BTU per sec per sq ft per degree F H - Altitude, ft K Thermal conductivity, BTU per sec per ft per degree F ~ - Characteristic length, ft M - Mach number, V Nu - Nusselt number, -- h Pr = Prandtl number K Q m Quantity of heat, BTU Re Reynolds number, T - Temperature, degrees Rankine Tamb - Ambient temperature, degrees Rankine TBL = Boundary layer temperature, degrees Rankine Tskin Skin temperature, degrees Rankine t = Time, sec v = Velocity, ft per sec w - Weight of skin material, lb per sq ft

DEPARTMENT OF ENGINEERING RESEARCH Page 4 UNIVERSITY OF MICHIGAN Report No. UMM-14 3 Total cone angle degrees Ratio of specific heats E Ow e6 = airissivity: Ratio of emissive power of a surface to that of a black body / = Coefficient of viscosity, slug per ft see M ass density, slug per cu ft = Stefan-Boltmann constant, 4.8 x 10-13 BTU per seec per sq ft per degree R4 OF = Degrees Fahrenheit oR = Degrees Rankine

DEPARTMENT OF ENGINEERING RESEARCH Report No UI-,14 UNIVERSITY OF MICHIGAN Page 5 DISCUSSION In Report No. UIM-3 (Reference 1) the equilibriua temperature was defined as the temperature at which the heat radiated to space exactly equalled the heat received by the skin. For a missile in accelerated flight, the equilibrium temperature is usually not reached, due to the rather large heat capacity of the skin material. The transient temperature of the missile skin is then a function of the net heat added and the heat capacity of the skin material. An expression for the skin teaperature can befbund from the definition of the specific heat of the skin. caki- am.., (wA)-l Ai Tskin A To - ibm whore ATakin is the change of akin temperature when an amount of heat A Q is added. Purther, the, transient condition can be expressed in the differential form, dTskin (Acin1 dl (1) dt dtsin Now, for any flight path, an evaluation of the tern dQ will be dt needed for a solution. At least factors should be considered in the initial analysis. 1. Heat received through the boundary layer. 2. Heat lost by radiation to space. 3. beat received by radiation from the sun. 4. Heat received by radiation from the earth.

DEPARTMENT OF ENGIN'EERING RESEARCH Page 6 UNIVERSITY OF MICHIGAN Report NO. U-.14 50 Beat transmitted to the interior of the missile. 6. Heat received from the combustion chauber. It is to be noted that all of the above factors can be taken into account in ]quation (1). Factors (3) and (4) are, however, negligible by comparison with factors (1) and (2)*, and factors (5) and (6) must be oaitted because a means for their evaluation has not been determined. It is conservative to asume that no heat is transmitted to the interior, while the neglect of factor (6) implies a well insulated combustion chamber. It will be necessary, then, to evaluate the term C only for the heat received through the boundary layer and for the heat radiated to space. The Stefan-Boltzmann Law is used to find the amount of heat radiated. { bt}R = Aq fTin (2) It will be desirable to radiate as much heat as possible, therefore, the missile should be painted or coated with a substance having a high Bissivity. For this analysis, it has been assumed that -= 1. For the consideration of the heat received through the boundary layer, the heat transfer equation is written in the form, at~ s IAh(TL - Tsk) (3) * The,axisa amount of heat received from the sum is found to be 0.11 BTU per sq ft per sec, from Reference 7. The heat received by radiation froma the earth is found directly from the Stfeaa-Boltmann Law to be 0.02 BTU per sq ft per sec. (Effective radiating temperaturo of the earth 443~lR)

DEPARTMENT OF ENGINEERING RECH Report No. UIM-14 UNIVERSITY OF MICIGAN Page 7 The value of the heat transfer coefficient is found ifro the definition of the sleselt number. K (4) h 3.hL Eperimentst by Iber (Reference 2) indicate that for turbulent flow over a cone h = 0224 b (5) Re a i s understood to indicate the average value of the Nusselt number ahead of the place where the Reynold's number is computed. The expression for the Reynold's number in Equation 5 is based upon free stream values po and v. (ahead of the leading shook), and the value of )4L taken at the boundary laer** It was also reported in Reference 2 that if > 10~ Equation 5 can be replaced by the expression 0.0107 BRe where Re =2jR (6) Here the Reynold's number is based upon values of fp and v1behind the leading shock nd the ralue of,M aain based upon the conditions within the boundary layer. * Aecording to information received from Dr. Tber. In the past it has been generally assumed that the rvalu of a lso ras to be taken from the free stream condition.

DEPARTMENT OF ENGINEERING RESEARCH Page 8 lUNIVERSITY OF MICHIGAN Report No. Ut-14 When the average value of the Nusselt number is used in Equation 4, the heat transfer coefficient will be replaced by the average coefficient, h. Consequently, any skin temperature which is computed by means of Equation 6 will represent an average temperature ahead of the point where the Reynold's number is computedo Now the differential equation of skin heating can be written in the form of Equation 1 by the use of Equations 2 and 3. dT _ (IC)1 {(WC (TBL T ) T (7) dt '-skiin skin Upon integration, Equation 7 yields the value of the skin temperature as a function of the time. Unfortunately, Equation 7 cannot be integrated exactly, because TBL, h, and Takin are functions of time and the equation is non-linear.

DEPARTM[ENT OF ENGINEERING RESEARCH Report No UW 14 UNIVERSTY OF MICHIGAN Page 9 1. J. D. Schetzer and D. W. Lueck, "Curves for the Calculation of the Order of Magnitude of Skin Heating Due to Friction of a Missile in Steady Flight in the Atmosphere." UNM-3, University of Michigan, Ann Arbor, Michigan, March, 1947. 2. G. R. Eber, "kperimentelle Untersuch ng der Breastemperature und des Wliraberganges an Einfachen Kopern bei Ueberschall Geschwindigkeit", Peenande, Archiv Nr. 66/57g. 3. J. H. Keenan and J. Kaye, "Thermodynamic Properties of Air." John Wiley and Sons, Inc., New York, 1945. 4. H. J. Unger, "Thermal Effects on High Speed, Plastic, Ogival Missiles"o API-.TP-13. Applied Physics Laboratory, Johns Hopkins University, Silver Spring, Maryland. January 1945. 5. W. S. Aiken Jr., "Standard Nomenclature for Airspeeds with Tables and Charts for Use in Calculation of Airspeed". NACATN1120, Langley Field, Virginia, September 1946. 6. C. N. Warfield, "Tentative Tables for the Properties of the Upper Atmosphere." NACA-TN-.1200, Lngley Field, Virginia, January 1947. 7. Croft, "Therodynaiai cs", McGraw Hill Book Company, Inc., New York, 1938.

DEPARTMENT OF ENGINEERING RESEARCH Page 10 UNIVERSITY OF MICHIGAN Report No UW-14I APPIZDIX Heat Transferred Thro.uh the BodaMr, Lker The rate of heat transfer tbhrough the oudary layer is found from Eqution 3 ao: I A (Ta- I T,) (3) where the heat transufer oefficient is expresed by h a NUh (4) Using Ibers data, ohb is epressed in qution 6, Equation 4 is rewritten in the foer - 0.0107 i t 2 (8) where h represents the average value of the heat transfer coefficient ahead of the point where the Reynold's number is computed. Here it ie important to note that it was implied, in the originl German report, that the Reynold's nuber to be used with Equation 0 is to be based upon the values of p and v in the free stream behind the shock wave, and the value of M is to be found from condigtions within the boundar laFer. Inasuch as this fact was not expresslry stated in the German report, the free stream walue of the viscosity has been used almost exclusively in this country, wherever Eberls data has been used. The magnitde of the error introduced into the walue of h will be of the order of the ratio of the viscosities of the air in the boundary layer and in the free stream.

IDEPARTMENT OF ENGINEERING RESEARCH Report NO 1 UNIVESTY OF MICHIGAN PaP l Xquation 8 can be put into a more useful form by expanding the Reynold's number 0, 0107 K /,rs9,P W's 0 e } EL Re O1R (8a) Recognising the term in brackets to be the reciprocal of the Prandtl number, it is bund that h1.o P..1e (8b) Prom the data given in Reference 3 it is found that {Pr)B = 0.4 is a good approximation; therefore, Equation 8b can be written h = 0.138 t( (8c) The values of h obtained using Equation 8c approach niinity near the tip of the missile and the method breaks down. Experiaents reported by Unger (Reference 4) indicate that the actual temeperatures will fall below the calculated values in the stagnation region. This effect was attributed to the low thermal conductivity of stagnant air. Returning to the method of this analysis, it will be remembered that Equation 8 was intended to be used with free stream values of P and v behind the leading shock. These results can be generalised somewhat by assumingt the ehange in the product p v to be small through the shock ware. T, enly the abient density and the flight velocity are needed to solve

DEPARTMENT OF ENGINEERING RESEARCH Page 12 UNIVERSITY OF MICHIGAN Report No. UM-1 Equation Sc, This simplification leads to an unconservative error. The error is,however, blanketed by the difference between the average temperature and the actual temperature at a given point of the missile. The coefficient h was defined as the average value ahead of the point at which the Reynold's number was computed. Because the skin temperature decreases toward the rear of the missile as the Reynold's number increases, the temperature at a given point is less than the average temperature ahead of that point. The boundary layer temperature is needed for a complete solution of Equation 3. The expression is usually written in the form T, = Tab ( + c 2 M2) The value c = 0.9 is well established by experiment. The Heat Radiated to Space The amount of heat radiated to space can be computed using Equation 2. Upon substituting the value for 7 and assuming that 6 = 1, it is found that 4.8(l03~) AT4 (2a)?-t}R = 4.8(10-13) A in(2 The Practical Integration orf the Skin Heating Equation It has already been noted that the differential equation of skin heating, Equation 7, cannot be integrated exactly. A stepwise integration can be performed by putting the equation in the incremental form. skin wc~ski ha (TL Tskin 4) - 4.8(10h13)T4 (7a skin h T}T )

DEPARTMENT OF ENGINEERING RESEARCH Report No UW-I I UNIVERSITY OF MICHIGAN Page 13 Here the flight path is broken up into several finite intervals for which average values, ( )avs are comptted from the expressions previously determined. The change of temperature, ATs., is found for each interval and then added to the temperature at the end of the previous interval. The method will be demonstrated in the following example. Consider a V-2 missile flying in accordance with the fight path shown on Figure 1. The problem is to compute the transient skin temperture throughout the given path. The temperatures will be computed for a point 9.8 feet aft of the tip of the missile to conform with the position of the thermocouple used in the tests. At this point on the V-2, the steel skin is 0.5 mm thick. The quantities in Table I were computed according to the expressions already set up. Atmospheric data was taken from References 5 and 6, and the values of the thermodynamic coefficients where taken from Reference 3. In Table I the nflight path has been broken up into 10-second intervals. As an example of the technique of computation, Equation 7a will be used to compute the skin temperature at the end of 50 seconds of flight. It will be assumed that the temperature has beenfound to be 155~F (615~R) at the end of 40 seconds, from the computation of the previous interval. From a plot of the values in Table I, select average values of TBL, and h, for the 40-50 second interval. (TBL)MV = 7500R (h)av = 0.0064 * * If the error in the Reynold's number, which was noted in the development of the method, had been carried along; the value would be 0.0093.

DEPARTMENT OF ENGINEERING RESEARCH Page 14 UNIVERSITY OF MICHIGAN Repore t No * M-34 and further assum (T sdn)a 60 as a probable value, inasuch as the temperature at the end of 40 seconds was 6150R and the skin temperature must increase because the boundary layer temperature still exceeds the skin temperature. The coefficient At is computed from the physical properties of the skin and the length of the time interval. At _ Wskin Then Equation 7b becomes ATc - L12 (o0.0064 (750o650) - 4.8(10-13)(650)4 ATskin s 710 It is necessary to check the accuracy of the assumption for the average skin temperature, which was made before. (Tjkij)c y: (T5ko)40 + jATkin (Tsk)av - 651oR Having established the validity of the assumption, the temperature at the end of 50 seconds of flight is found directly from the expression (Tsn)50o (Tk)40 + 4 Tk. n (T5i)50 687"R (2270,)

"DEPARTMENT OF ENGINEERING RESEARCH Report No. UMIK-1 UNNERSITY OF MICHIGAN Page 15 Comparison with xperieat The process outlined in the previous section has been carried out for the complete flight path shown on Figure 1. The resulting skin temperatures are shown on Figure 2 as a solid line. Figure 1 represents the flight path of a V-2 rocket which was fired by the Naval Research Laboratory at White Sands, on 10 October, 1946. The creoses oa Figure 2 represent theocouple measurements which were read from the telemeter records of that flight. Telemeter accuracy ewas believed to be t 10%.

0 Time A1ltitude Velocity 6 170 116 >~ a [d piVelo city | a pl10 Xb|T ~ Ta |mb Rs~ Rs0O.18 34,000 1,90 __ _520 520 373.40 290 0 50 1 63,000 2,800 97112.7 9 1 92 5.e78 o 9.20 1. 0.0042 60 100,000 4,00oo 971 4.2 33 392 581 8.07. 614 1.70 13.20ooo004 20 137,000 5,.6 112 4.63 1 525 550 3.719 19.03 0.0026 1 0000 TA30 It Calculation 19000 S420ampl Probl 609 40 7A

DEPARTMENT OF ENGIGNEERING RESEARCH Report No. U-14 UNIVERSITY OF MICHIGAN Page 17 140 - - ___ BURN-OUT 130 6 V-2 FLIGHT PATH 10 OCTOBER 1946 120 - 100 90 - 0 _ U. 70_ — l.... TIME IN SECONDS I 50 IS THE PARAMETER W ON THE CURVE o 60....... 50 40 40 300 20 - I0: " ej/ REf: NAVAL RESEARCH LAB. R-3030 10 1/ 0 " 2 3 4 5 VELOCITY- FT/SEC x 10-3 FIGURE 1t Flight Path of V-2 Rocket

~~~~~~~~~400~~;0 ___,. -, i COMPUTED o200 0 r r I I TEMPERATURE 3 00 1 IJ I I I I I I I I I I I I1 W:1 ___ - ~~FLIGHT I TEST I20 I00 = x I 0 10 20 30 40 50 60 TIME - SEC..FIGU 2 Sdn Temperature o th V-2 During light Firing o 10 October,, 1946

DEPARTMENT OF GINEERING RESEARCH Report No. U1- 14 UNIVERSITYr OF MICHIGAN Page 19 DISTRIBUTION Distribution of this report is made in accordance with AN-GM Mailing List No. 4 dated October 1947, including Part A, Part C, and Part DA.

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