DEPARTMENT OF ENGINEERING RESEAIRCH....a_ |, UNIVERSITY OF MICHIGAN MM-?7 Copy No. 5... UNIVERSITY OF MICHIGAN Ann Arbor EXTERNAL MEMORANDUTM NO. 7 Project IZX-794 (AAF1 Contract W33-038 ac-14222) Project "Wizard" "A Simplified Ivlethod of Calculating Ram-Jet Performance Applicable To High Mach Numbers" Prepared by. Approved by vs. e rrJ E. T. Vincent Professor of Lechanical Engine e ring July 23, 1947

Pa.ge ii DEPARTMENT OF ENGINEERING RESEARCH iReport N. UMM-7 UNIVERSITY OF MICHIGAN. ACIGTOWLEDGEBMENT The author wishes to express sincere appreciation for the assistance given by Ir. John R. Sellars, whose suggestions and knowledge of the relationships previously developed and used herein have been most valuable. -

DEPARTMENT OF ENGINEERING RESEARCH Report l~o. UM..-7 UNIVERSITY OF MICHIGAN Page il CONTENTS - Introductio...... Summary.. Conclusions...... List of Symbois... Discussion...... Comousstion Chamber. Exit Nozzle.... Thrust Coefficient Diffuser.... Brief Outline of Method Sample Calculation I Sample Calculation II. Sample Calculation III References...... Appendix I....... Appendix II.... *.. 0 * 0 0 0 * 0 0 * 0 * *0 0 0 * 0 * 0 * a 0 0 * 0 0 * 0 * * * 0 0 * * 0.0 * 0 0 0 * 0 0 * * 0 0 0 * 0 0 0 * 0 * 0 0 0 *) 0 * 0 0 0 * * 0 0 0 0 0 * 0 0 ) 0 * * 0 0 0 *. 0 0 0 * 0 * 0 0 0 0 0 * 0 0 0 0 0 * 0 0 0 0 0 S * 0 0 0 0 0 0 * 0 0 * * 0 0 * 0 0) 0 * 0 0 * * 0 0 0 0 0 0 * 0 * 0 * 0 * 0 * 0 * 0 * 0 * 0 * 0 * 0 * 0 * 0 * 0 * kage 1 2, 4. 6.8.10.22 *.26.27.30.32.35.37.39.40.43 J

DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN Report No. M-7 LIST OF ILLUSTRATIONS Figure A 1 1A 2 2A Title Page Ram-Jet Station Designation........46 Solution for Velocity at Station 3 (Ram-Jet)47 Solution for V3 (Ram-Jet)..........48 Chart for Use in Calculating A/A2... 49 Total Temperature vs. Flight Mach Number..50 2B 3 3A 3B 3C 3D 3E 4 4A 4B 5 5A 6 7 Diffuser Efficiency vs.L'ach Number....51 F Specific Stream Thrust - W vs. T2 for a Various V2's............52 Heat Release - S vs. T2......... 53 Sa vs* T2...............54 S vs. T2.. ~.. ~ ~... 55 a vs T2. *...............56 Sa vs. Fuel-Air Ratio...........57 0(M) vs. Mach Number............58 0(M) vs. Iach Number...........59 0(M) vs, Mach Number...........60 0(M) vs /........61 Theoretical Total Pressure Recovery for a Constant Nozzle Efficiency Based on Energy,..............62 Maximum V2 for A = A2 and Various FuelAir Ratios..............63 Thrust Coefficient vs. Flight Mach Number..64 mg

DEPARTMENT OF ENGINEERING RESEARCH Report No. UJImd-7 UNIVERSITY OF MICHIGAN Page 1 INTRODUCTION The propulsion problems related to Project iMX-794, (Project Wizard) initially drew attention to the ramjet as a possible propulsion unit and performance estimates for various flight conditions were desired. A brief survey of t~he then existing methods of calculating ram-jet performance revealed that there were almost as many methods as there were groups investigating the ramjet. Lost of the methods were purposely developed holding constant certain important variables such as fuelair ratio, various efficiency curves, temperatures at the end of the combustion chamber, or general configuation, thus lacking the latitude of applications desired of a general method of calculation. Furthermore, Project Wizard is concerned with a wide range of Lach numbers (2-6) and the methods then in general use incorporated certain assumptions which are not valid at high Mach numbers. As a result, a method was devised (see Ref. 6) which is valid at high Mach numbers of flight, has the generality desired and is believed to give accurate results. However, this method is quite lengthy and tedious, and does not make readily apparent the important variables affecting performance. Certain subsequent investigations by Project Wizard using this method and the knowledge of some relationships previously developed by another group (see Ref. 3, 4, 5) led to the development of the simplified method presented in this report. i I

I DEPARTMENT OF ENGINEERING RESEARCH | Page 2 UNIVERSITY OF MICHIGAN IReport No,. Ud.-7 SUGARY The method developed in this report permits rapid calculation of ram-jet performance and still incorporates in a satisfactory manner the major effects of molecular dissociation in the combustion process and the variable specific heats of air as well as of the products of combustion. Variables such as the diffuser, combustion, and nozzle efficiencies; and operating factors, such as the velocity of the air at the entrance to the combustion chamber and the fuel-air ratio, can be assigned any value considered practical for the purpose at hand. The general configuration of the ram-jet can be varied as desired and the temperature at the combustion chamber exit can be selected, or allowed to assume whatever magnitude it will, depending on the particular conditions. The maximum possible velocity at the combustion chamber entrance determined by the fundamental equations of flow (that entrance velocity which produces sonic velocity or "choking" at the combustion chamber exit) is easily found for any Mach number, altitude, and fuel-air ratio. Also, at low Mach numbers of flight the existence of two solutions for the equations governing flow in the ram-jet combustion chamber is recognized and explained by the graphs and equations presented herein. Included in the last pages of this report is a condensed outline of the method, with three typical calcu

o No DEPARTMENT OF ENGINEERING RESEARCH Page 3 Report No. TJM-* UNIVERSITY OF MICHIGANage lations, each representing a common problem. Appendix I presents a discussion of the method for calculating ramjet performance as given in Reference 6 regarding its application to the condition where "choking" is present at the combustion chamber exit; and Appendix II is a discussion pertaining to the magnitude of the velocity at the combustion chamber entrance as determined by design considerations and the "choking" condition. I 1

Pagfe 4. l DEPARTMENT OF ENGINEERING RESEARCH eport No UlUNIVERSITY OF MICHIGAN Report o Ui CONCLUSIONS 1. The upper limit on the air velocity at the combustion chamber entrance, V2, for high Mach numbers of flight, imposed by the fundamental equations governing flow in a duct, is well above the velocities at which it is possible to sustain combustion at the present time, regardless of fuel-air ratio. Therefore, for flight at high Lach numbers, high velocities at the combustion chamber entrance are theoretically possible if "blow-out" and other practical difficulties can be overcome. However, since studies of the variation of the thrust coefficient, Ct, with changes in general ram-jet configuration at high Mach numbers of flight indicate that maximum Ct is obtained when the cross-sectional area of the diffuser inlet equals that of the combustion chamber; and, because at lower Mach numbers of flight, V2 is limited by the "choking" condition at the combustion chamber exit, it is not expected that combustion chamber entrance velocities over 500 ft per sec will be practical except in very special instances. 2. The stagnation temperature after combustion and the flow velocity at the combustion chamber entrance appear as important variables affecting ram-jet performance. 3. The variation of the specific heats and the de gree of molecular dissociation with temperature is appreciable under the conditions encountered in a ram-jet at Mach

w I DEPARTMENT OF ENGINEERING RESEARCH Report No. UIM-1 UNIVERSITY OF MICHIGAN Page 5 numbers of flight greater than 2 with fuel-air ratios which give maximum thrust and can be accounted for in a simple manner as shown in this report. The variation of the above effects due to different pressures at given temperatures can not be accounted for without use of the Thermodynamic Charts (see Reference 2) and their accompanying lengthy calculations, but the variation is negligibly small compared with that due to temperature. I

i DEPARTMENT OF ENGINEERING RESEARCH Page 6 "Rasv OF MICHEIGAN RTeport No. U&L-7 LIST OF SYBOLS A = area - sq ft a = acceleration - ft per sec per sec C = velocity of sound - ft per sec Ct = thrust coefficient Cs = velocity of sound at stagnation temperature - ft per sec s F = stream thrust - lb F - = specific stream thrust - sec a f = weight of fuel - lb g = gravitational acceleration - ft per sec per sec Y = ratio of specific heats (-) v h = enthalpy - Btu per lb h2 = fictitious enthalpy used to relate pressures in a 2MP non-isentropic adiabatic process h2 = actual enthalpy, assuming no losses through wall boundaries J = conversion factor, 778 ft lb = 1 Btu LI = Mach number mf = mass flow - slugs per sec 17 = combustion efficiency based on heating value of fuel?D = diffuser efficiency based on energy considerations 7n = nozzle efficiency based on energy considerations np = nozzle efficiency based on total pressure considera tions ~p = diffuser efficiency based on pressure recovery

b:~e por t No, XU~-7l V DEPARTMENT OF ENGINEERING RESEARCH Report No, _7 | UNIVERSITY OF MICHIGAN Page 7 ~Sa = combustion efficiency based on 100 %S P = pressure - lbs per unit area Pr = pressure ratio (adiabatic processes - see Reference 1) R = gas constant - 53.345 ft lb per lb per OR Sa =function of total temperature - sec T = static temperature - ~R T = gross thrust - lb g T = total temperature - OR V = flow velocity - ft per sec Wa = weight flow of air - lb per sec p = density - lb per cu ft Subscripts: 1, 2, 2' 3, 4, and 5 refer to stations (see Figure A)

i Page oDEPARTMENT OF ENGINEERING RESEARCH por.... T....-7 UNIVERSITY OF MICHIGAN Report No. UI-7 DISCUSSION A method for calculating ram-jet performance at high Mach numbers was presented in Reference 6. It successfully accounts for molecular dissociation and variable specific heats, both of which have an appreciable effect on performance at high velocities of flight with the fuelair ratios used (.0605,.0665 and.0782 by weight). Although the method presented in Reference 6 is quite lengthy and tedious, it is believed that the results obtained by its use will be accurate within the efficiency assumptions, calculation error, and chart (Reference 2) accuracy, in accounting for variable specific heats and dissociation in the burning and expansion process. Thus the accuracy of any other simplified method may be evaluated with reference to the results obtained therefrom. It is known that the conditions at the entrance to the combustion chamber and the amount of fuel added can uniquely determine the conditions after combustion. It remains to express this relationship in a simple manner which will still account for the effects of dissociation, variable specific heats, etc. Such a simplification would eliminate the use of the Thermodynamic Charts (Reference 2) and their attending complicated calculations. Therefore, if variable specific heats and dissociation in the combustion process are to be accounted for, which appears necessary at high Mach numbers, this relationship between the properties of the gases at stations 2 and 3 (see i snuOn -I -- I -

iI Re o. DEPARTMENT OF ENGINEERING RESEARCH Report INo. *Ul-7 UNIVERSITY OF MICHIGAN Page 9 Figure A) must in some manner incorporate previous calculations which have made use of either the Thermodynamic Charts or an equivalent method. Also the relationship should show the presence of two solutions at station 3 (one subsonic, one supersonic) at low Mach numbers and the reason for disappearance of the supersonic solution at higher L;ach numbers, as discussed in Appendix I. An analysis of flow through a ram-jet has been made by a group at JHU/APL, Silver Spring, Ld. (References 3, 4, 5) where it was shown to be possible to express the conditions after burning as a function of conditions before burning and heat added. Their analysis as presented in these reports is believed accurate for low flight LMach numbers (< 2) but is not adaptable to higher flight LMach numbers without revision, for the following reasons: investigations at the University of MLichigan have shown that at L.ach numbers greater than approximately 2.75, performance is impaired by the lack of a tail nozzle (the configuration used by the APL investigators); and the calculation of conditions through the diffuser by use of /Mach number relationships based on constant specific heats is sufficiently accurate at Liach numbers less than 2, but at higher iach numbers neither of the above are recommended if maxinum performance and accurate results are desired. For instance, if one were to use the -ach number relations in calculating total temperature or the temperature after the diffuser at a flight Iach number of 6 at sea level, 4260~R would be the result if a specific heat ratio (~) of a I.

I Page 10 Il DEPARTMENT OF ENGINEERING RESEARCH Page 10_ ___ UNIVERSITY OF MICHIGAN eport. ' 1.4 were used, and 2390~R if a Y of 1.2 were used. Thus the variation of Y over this possible range l1.4-1.2) is quite critical in such a calculation and the task of choosing the correct mean value is essentially one of trial and error. By using the Air Tables (Reference 1), as outlined in Reference 6, where the varying specific heats are accounted for, the temperature for the above conditions is found to be 3749~R. It is also generally known that a tail nozzle is a necessary item on the ram-jet at high Mach numbers, especially since its addition makes a considerable contribution to the thrust coefficient. The necessity of a tail nozzle at high Each numbers of flight arises from the fact that the velocity at the entrance to the combustion chamber would be excessive, accompanied by lowered pressure, if no nozzle were employed (mass flow constant for a given set of flight conditions), thus causing poor combustion efficiency and, probably, "blow-out". Also, for a ram-jet of given configuration, the lowest V2 possible without causing a spillover is desirable, because the thrust per unit mass flow in the duct decreases with an increase in V2. Thus the remainder of this report is an extension of the relationships already known to higher iLach numbers, with a development that permits a clear understanding of the important functions. Combustion Chamber In developing the relationship between stations 2 and 3 the following assumptions are made: J [ ----

,Report so.,3_ UNIVERSITY OF MICHIGAN 1 1. A cylindrical combustion chamber is employed; i.e., constant area burning. 2. No heat losses occur through the duct walls. 3. Heating value of fuel is constant, 19,500 Btu per lb fuel (CH2.25 ) x 4. Use of the arithmetical mean of specific heats in calculations between any two temperature limits sufficiently approximates the results obtained by the exact method for an adiabatic compression or expansion, 5. No frictional losses in the combustion chamber. See page 6 for list of symbols. The force-momentum equation makos it possible to express conditions at station 3 as a function of those at station 2 and the heat added, viz.92 2 P3 2 (1) A2(P2 + 2 + 2 ) A 3 + 3 ) P W where - V= g - mf = mass flow in slugs per sec The difference in the magnitude of expression (1) between two points in a duct can be shown to represent the net force on the duct when integrated over the interior surface as shown below. dv — //

I DEPARTMENT OF ENGINEERING RESEARCH P~ag~e }2 lUNIVERSITY OF MICHIGAN Rport No Uh-7 Consider a small element or region of the duct, dv. The force-momentum relation states that the net force exerted on the region is equal to the net out-flow of momentum per sec Force = mf u where mf is the mass flow undergoing the momentum change. Equating the forces acting on the region, dv, to the change of momentum of the mass flow, PA - (P + dP) (A + dA) + (P + P-) dA' -mr u+ mf(u + du) = + mf du Neglecting the products d dA' and dPdA and noting that PdA' is the reaction force on the wall, the integral of which is the thrust on the body, PA - PA - PdA - AdP + PdA' = m du PdA' = mf du + PdA + AdP The total force on the duct from station 1 to 2 then becomes: r2 12 Force (Ibs) = PdA = m.u + PA =F2 F1 /1 _1r Letting u = V, F represents the expression mfV + PA at any point and has been termed "stream thrust". Force = F2 - F1 = f2V2 + P2A2 - (mflV1 + PlA1) I I I - -

Report No UlI I-7 | DEPARTMENT OF ENGINEERING RESEARCH Report No. U12M-7, ^ ^S^ '-S'SSS. Page 13 Uo. VERSITY OF MICHIGAN.age =A2 (P + 2 V2)- A(1 + A1 V ) lbs Note that the expression F = A [p + V2i must be a constant from station 2 to 3 in the ram-jet, that is, any cylindrical section, since d.' = 0 and thus fPdAt =0 i.e., F2 = F Actually there is some loss of stream thrust, F,at the flame holders, which can be expressed as F - loss at 2' = F3 = F2,. Continuity of mass flow can be written as follows, using -= RT and C2 = TRTg; P W = VA. _ pVgYRTA C2 (2) W= P C Then, solving for PA and using the subscript o for ambient conditions: w(oC (3) POA = M0o0 Also from the conservation of energy and the state equations,

Page 14 DEPARTMENT OF ENGINEERING RESEARCH Report No T TJm.7 UNIVERSITY OF MICHIGAN | Report No. Ui-7 C (4) C. = -1 where Cs1 is the velocity 1 + - o21 of sound at the stagnation _- temperature. From (3) and (4) (5) P A = 0, 1 also noting that 1o 2 2 (6) ~ 0(Po + ) = APo(l + Y ) = P0 The following can be written from (5) and (6) (7) o= s 1 Qay Po~[o - -3 1 i 0 l IM0] 2 Writing F in lbs per lb of air per sec for station 3, rearranging, and multiplying by ___-tl_ for a reason explained later, also noting that at station 3 the flow is (1 + f) lb of gas per lb of air, - the following is obtained (8) CsF 42TM+l) (l+f) 1 + T"M32 (8) "r=- Yg A +i~ ~2- (( 2 + 2 Thus the following relationships are defined - c,/2( ) (1+f), (9 ) S =7a ( ~1 F A i I and the Mach number function --

I Re~ort No, U~Gt~-7 Page 15 UNIVERSITY OF MICHIGANge 15 (9A) 0(M}) = 1 Y-1 1 KM 12(Y+1) 1 +.] M2 (multiplying by _2,(T+) serves to normalize the Mach V2 (Y+1) number function so that 0(M) = 1 when M = 1). Then F3 (10) - = Sa 0(M3) lb per lb air per sec a Figures 4 and 4A show 0(M) vs Lach munber for three values of Y, two of which correspond to the extreme limits of temperature to be expected at station 3. The displacement of these curves due to the maximum variation in Y is appreciable but the actual effect on the performance calculation can not yet be evaluated as these curves are used only indirectly in calculating the thrust coefficient, Ct. Figures 4 and 4A are included in the report however so that M3 can be readily determined if desired. Figure 4B shows that for low values of 0(M), ((iL) = 1 to 0(M) ~ 1.7), there are two solutions for the Mach number at station 3, one subsonic, one supersonic. Only the subsonic solution is of practical interest, the supersonic solution being physically impossible. The supersonic solution disappears at values of 0(i) greater than approximately 1.7, as indicated above. It can be easily shown T 7I that 0(Ui) approaches the linit i for high Loach numbers. This limit is equal approximately to 1.67 and 1.91 for r = 1.25 and 1.175 respectively. Then for 0(L) greater than approximately 1.9 only one solution for the L

I I I i i Page 16 O DEPARTMENT OF ENGINEERING RESEARCH ort I T. 7 PageS6~~ 16 _ UNIVERSITY OF MICHIGAN __ Report No. U'.7 combustion chamber exit velocity is possible, the supersonic solution having disappeared.* In setting up the curves for fuel-air ratios.0605,.0665 and.0782, Li3 (and thus 0(m3) from Figure 4A) was *This is, after all, exactly what one would expect as the equation relating Mach numbers before and after a normal shock, M22= l 1 M) 1 + 2 (M 12 1) +-1 where M1 s the Mach number before shock, is derived from the same basic considerations, and it shows that for an increasing supersonic Mach number before shock there is a lower limit on the resulting subsonic Mach number after shock. So, conversely, one would reason that for certain subsonic Mach numbers there co-exist supersonic Mach numbers which would also satisfy the energy, force-momentum, and continuity equations even though they are of no practical significance for these flow conditions. The supersonic Mach number increases without bound as the subsonic Mach number decreases to the lower value,4, (let Ml become infinite in above equation and evaluate) below which the subsonic Mach number has no co-existent supersonic Mach number. This lower subsonic value for which the supersonic Mach number disappears is approximately M s.37 for T 1.4; and can be obtained by either the P(M) equation (Mach number corresponding to 0(M) = or the Mach number relation across a normal shock. Thus the existence of the two types of solutions given in Figures 1 and la and discussed in Appendix I is verified and explained. i --- -- I

Report No 7 DEPARTMENT OF ENGINEERING RESEARCH I.age 17 UNIVERSITY OF MICHIGAN calculated using the method given in Reference 6, which makes use of Thermodynaaic charts. The curves shown for fuel-air ratios.0499,.0333 and.01995 were obtained from calculations performed with the aid of Reference 7. The chart in Reference 7 does not account for molecular dissociation as the effect is unimportant at the lower temperatures encountered with the reduced fuel-air ratios. Because F2, = F3, S can be 2 found from Sa -= 2(;)W. Thus the important effects of a a dissociation and variable specific heats in the combustion process are retained and accounted for in the magnitude of Sa when found in this manner. From Equation 9, it can be seen that Sa is dependent only on total temperature at station 3. Thus, at a given combustion chamber inlet condition, Tt2, S3 is a function of the amount of heat added and has been termed the "heat release" or the "specific air impulse" by some investigators. The temperature at station 3 can be calculated from the following equatior which is obtained by solving for T3 in equation (9). ( gS 2 3gSa2 T3 aR(l+f)2T+l)(1 + Y-1 2 From this it can be seen that a constant Sa would Ieep T3 approximately constant, suggesting a convenient method of calculating performance at a constant combustion chamber exit temperature. For more accurate calculations, variation in M3 should be considered.

Page 18 DEPARTMENT OF ENGINEERING RESEARCH iport No. -7 Tg h e|ct wiUNIVERSITY OF MICHIGAN f The effect which a large variation of pressure t60 -5U00 psi) at station 3 has on Sa was investigated and seems to be quite small, almost insignificant as far as the results are concerned, but is large enough to w\arrant an explanation. As pressure is varied, keeping the temperature at station 2 and heat added constant, the position on the Thermodynamic chart of Reference 2 representing the state of the gas after combustion moves horizontally. As can be seen on the charts, the constant temperature lines are not independent of the pressure. The slope changes slightly as pressure varies, the main cause being the varying degrees of molecular dissociation at the same enthalpies and different pressures. The method as outlined does not account for this variation due to pressure, hence there is no change in Ct for different pressures with temperature constant at station 2. Therefore, for changes in altitude at altitudes where the temperature gradient is zero (above 35,332 - NACA Standard Air) there is no change in Ct at a constant iaach number, whereas, by che method presented in Reference 6, tltere is a slight increase in Ct above 40,000 ft due to the change in pressure. As previously stated, the variation of 0Ut3), and thus Sa due to the effects mentioned above, was investigated over a range of pressures up to 5000 psi and found to be approximately 3 %. However, the error that does arise in the practical example is neg ligible. As stated before, Sa is a function of the total temper

Report No. UM-7 UNIVERSITY OF MICHIGAN Page 19 ature at station 3 and therefore it can also be said that it is a function of total temperature at station 2 with a given heat addition. Sa was plotted against the total temperature at station 2 for different fuel-air ratios (see Figures 3A, 3B, 3C, 3D, and 3E). These curves are convenient, as the total temperature at station 2 is a function only of Mach number of flight and altitude, and thus easily determined (see Figure 2A). Sg has been plotted for six fuel-air ratios, however, the curve of Sa for any fuel-air ratio, Mach number and altitude can be obtained by cross-plotting from Figures 3D and 3E. The magnitudes of Sa were calculated and compared for the combustion efficiencies, c == 100% and 85 7, based on the heating value of the fuel; and it was found that the S for Ic = 857 was 95 % of the Sa forC = 1007. This fact could serve to re-define the corbustion efficiency; that is, in terms of Sa the new combustion efficiency,,Sa a ^a would be 95 % for the instance where 85 % of the heating value of the fuel was actually utilized in combustion (see Figure 3A). it should be stated here that applying an efficiency factor to the heating value and then entering the chart is not entirely valid, as each Thermodynamic chart of Reference 2 was made up for the full heating value of the particular kl+f) mixture. iowever, after considerable investigation and various calculations it was concluded that the:error thus introduced is negligibly small. As was explained in the derivation, the magnitude of

i I.. ~ ~DEPARTMENT OF ENGINEERING RESEARCH Page 20 UNIVERSITY OF MICHIGAN Report N1o. UiM-7 F = A(P + P V2) is constant from station 2 to station 3 except for flame holder losses. Some interesting relationships can be derived from this equation. When divided by the weight flow the following results: (1 )F A = P. V RT V (VA -- A V g V g For station 2 F2 RT2 V2 (lla) +Wa V2 g Thus F2 is dependent only on T2 and V2 and involves no 2 ~ ~ ~~~~2 2 F assumption regarding Y. The slope of the line- 2 vs T2 can ~~R ^~a be readily seen as V. 2 This relation presents a very convenient method for obF2 taining - for different flight Lach numbers and altitudes, a F F F 3 2' 2 but the magnitude of W or y-, which is less than - by the a a a loss due to the flame holders, is, in general, required. Previously, as given in Reference 6, the loss of pressure due to the flame holders was defined in terms of velocity head, P2-, which can be represented as a force AP-A. The I2g effect of the change in P2 and V2 across the flame-holders can be neglected, as calculations prove the error to be very small. Therefore S,12) F2 - A —A = F3 or npV2 13) F = F2 - - = F 2' 2 2g I I a.1

DEPARTMENT OF ENGINEERING RESEARCH Report io. U*Lahi-7I UNIVERSITY OF MICHIGAN Page 21 VWhere n = tne number of velocity heads loss. Dividing by tne weight flow at station 2, where Wa = weight of air a F2_ Ri2 V2 nV F3 (14) _-t a W a V2 g 2g = 1a iote that when n - 2 F3 RT2 F 2 t was calculated from Equation lla and has been a plotted versus static temperature for a rarLe of V2. s elieved practical at the present (see Figure 3). In order F3 nV to obtain an apnropriate value of: —should be suba F2 tracted from the - read off the graph, commensurate with a what one' s knowledge of, or experience with, flame holder losses dictates. The difference between the total and static temperatures at present day practical vaLue of V 2 is negligible, so that the same abscissa can be used for F2 as for Sa without introducing appreciable error. Another interesting application of this family of curves (Figure 3) is the instance where sonic velocity exists at the combustion chamber exit. From the equation F it can be seen that when it can be seen that when F3 aW S a 0(-) and thus Iv equals one. Because (M-3) can never be

i I I i 22^ ~ | DEPARTMENT OF ENGINEERING RESEARCH I-age 22;.R eport No. IRL-7 Page 22 l______ _UNIVERSITY OF MICHIGAN F 3 less than one, W- must always he equal to or greater than Sa. a Therefore, if it is required to find the maximum possible V2; that is, one which produces sonic velocity at station 3 for a given Sa and T2 (thus fuel-air ratio, Mach number of flight, and altitude) it is necessary only to enter Figure 3 with F2 = F3 = Sa, neglecting flame-holder losses, and read V2. Flame-holder losses can be taken into account by adding to F2 the amount of the loss, thus reading a slightly lower maximum V2. A more complete discussion related to the maximum velocities possible at the entrance to the combustion chamber is given in Appendix II. Thus, from Figure 3, F3 can be found when V2 and T2 are known. S can be found from Figures 3A-3E for the T2 and fuel-air ratio used. Then O(M3) can be obtained from F3 = Sa 0(M3) Wa, and M3 can be found from Figure 4A. Exit Nozzle Next, the relation between M3 and M5 must be established. This is easily done, as the Mach number at various points in a nozzle can be written as a function of the crosssectional area, assuming isentropic (no shock) flow. Y+l A 1 + l(16) A =l 2 J where subscript o denotes nozzle inlet conditions. Since the equation is "symmetrical" mathematically, subscript o may also denote nozzle outlet conditions. Thus, Mt being equal to 1, the ratio of the area at the inlet or outlet to the area at the throat may be written I'

Report INTo UIMI7 j DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN Page 23 Y+l 1 + M AO 1 o (17) 0t 1;+l Figure 5 shows a plot of,(M) (obtained from Figures 4, and 4A for each Mach number) vs A /At representing subsonic flow from station 3 to 4 and supersonic flow from station 4 to 5. Three curves were plotted, of which two represent the extreme limits of Y corresponding to the temperatures possible at station 3 and 5, and it can be seen that the effect of Y on the magnitude of ((M5) is not noticeable until a 0(M) greater than 3 is attained, where the curves separate. In this region it is believed that if the mean value of r were used the values obtained would be very nearly correct, inasmuch as the maximum possible error in,(M5) due to T is only 3 %. Using this curve, one need only know 0(M3). W(5) can then be obtained by going first to the subsonic curve, then vertically down to the abscissa, reading off the area ratio a3/At. Then move along the abscissa to the proper A5/At and up to the supersonic curve, then horizontally to the ordinate reading off P0(5). If A3 = A5, as would probably be true in the practical design, one can move vertically down from the subsonic to the supersonic curve and thence to the ordinate 0(M)5). It can be seen that the slope of the lower curve is very small at large area ratios, therefore, as will be seen from the equation for Ct, there is little loss in Ct due to making A5 = A2 at magnitudes of 0(3) greater than i

DEPARTMENT OF ENGINEERING RESEARCH. Page 24 UNIVERSITY OF MICHIGAN |Report No. UBM-7 2.5 or 3.0. This method of finding Mi5 assumes a 100 %o nozzle efficiency. In order to allow for friction and other losses during expansion the following development is given for nozzle efficiency based on total pressure. ~F3P3IvTA (18) p3VA3 = C = a constant (3V3A3 = - 3 Cs3 PT3 (19) Also C and P 1 a 2 2 I 3 (20) Substituting (19) in (18) and reducing, the following is obtained ~3PT313 r5P T5 5A5 ~3 + 1...~5 + 1 C-s 2 + ~1M 3 =2 2 ii5-1) CS3 1 +2la3 23Cs5 1 + 52 M252 Noting that, for an adiabatic process, Cs3 = Cs5; (21) A= - 2 1 ~5+1 + I ] 2 5) 5 T5 5 (22) Let 5 = 1 np and neglect the term 3 in view T3 np3 of the small effect it has on the magnitude of P(}5). I i

I I ReportNo. I DEPARTMENT OF ENGINEERING RESEARCH Repor t f~No. UME^i-7 UNIVERSITY OF MICHIGAN Page 25 r11 ~3'1 1 Y- + 2 T | A 3 2 (23) h3 ( 2 +1 A5 - 8 Y-1 A3 (24) Thus ----~np (A, isentropic compression or expansion. This is a convenient expression, as the curves for any efficiency can now be plotted directly from the?n = 100 % curve. Curves for np = 90 % and 80 % are shown on Figure 5 and the choice of the proper efficienc-y can be left to the user' s judgment. The nozzle efficiency defined in terms of energy losses, as shown in Reference 6, assumes a value of 95 %. The efficiency based on total pr ressure was calculated for a wide range of M3 and P3 and was found to vary from 80 % to 95 7% corresponding to a constant efficiency of 95 % based on energy. This data was plotted against E3 (see Figure 5A) and seems to show some lack of consistency, however, the indication of the general trend is somewhat significant. Here it might be stated that more information on nozzle I -J 16

PI ODEPARTMENT OF ENGINEERING RESEARCH i.age 26 UNIVERSITY OF MICHIGAN losses and efficiencies under the conditions of flow encountered in a ram-jet would be welcomed. Thrust Coefficient The equation used in calculating Ct is developed as follows: PS ( 21 ) v12) (25) Thrust = A5 (P5 + V ) (P + 1 - P1 (A5 - A1) which is similar to the equation given in Reference 6. Tg= M (V5 - V1) + A5(P5 - P1) From Equation 25, Tg= a5 P(M5) Wa - AP1 (1 + %2) P1(5 - A) Then, since Ct = - = -- 2 PV A PM2TA as Sa5 = Sa3 the subscripts can be dropped from the Sa term. a 1 11 Sa0(MI) Wa All 2 Pi A5 -t v~.... " YlA22 W = OVAl - g=l A From which (26) c = A2 ( s).^ (T 82 A V1 ' 0 2g5 2, 2 V) A2 2 As described before, the combustion chamber efficiency

DEPARTMENT OF ENGINEERING RESEARCH Report No UMM-7 UNIVERSITY OF MICHIGAN Page 27 can be applied to Sa and the product 7Sa x Sa actually used in the Ct equation. Diffuser Note that Equation 26, is directly affected by diffuser efficiency through the term A1/A2 which can be readily determined by a method similar to that given in Reference 6. The diffuser efficiency is still based on energy considerations h2p- h (27) ID = l which is the ratio of the isentropic change in enthalpy from PO to P2 through the diffuser to the actual change in enthalpy. V 2 V2 (28) 2v = g - where subscript 1 denotes conditions at the diffuser entrance and subscript o denotes T / v- -- c 7Actacld ambient conditions. l SubEtituting (28) in (27) and solving for hap the following is obtained _-...... S (29) h- 'j(ID 2P= gJ(V12 -V22) + h where?D is obtained from Figure 2B. Also from the continuity and state equations, the following results: I i

DEPARTMENT OF ENGINEERING RESEARCH Page 28 UNIVERSITY OF MICHIGAN Report No. UlTU-7 (a) p1VlA1= p2V2A2 (b) P RT P (c) P2 - ro and thus Al Pr2 T0 V2 (30) = 2 ro 2 1 where Pr is the Pressure Ratio corresponding to an enthalpy or temperature (see Reference 1) and Pr2 is the final Pressure Ratio after a non-isentropic change in pressure through the diffuser corresponding to the h2p from Equation 29. The diffuser efficiency is also left undetermined so that it can be chosen to suit the particular problem. (Figure 2B is included in this report as a basis for assumptions). At this point, it should be clearly understood that the above method also can be applied to a problem involving a fixed design ram-jet. In the fixed design calculations, the area ratio, A1/A2, would be known and thus V2 would be determined by the following: ro T2VlIA 2 - Pr2ToA2 r2 0 2 Inasmuch as some investigators prefer to define diffuser efficiency on the basis of pressure recovery, the following equation may be used for obtaining V2 directly; v2 = Pro 2V1k v2 =2p T

I DEPARTMENT OF ENGINEERING RESEARCH Report No. UlM$-71 UNIVERSITY OF MICHIGAN Page 29 where Pri is the total pressure ratio lisentropic) and corresponds to h2V. Neglecting V2 in Equation 28, V 2 h = 1 + h h2V Zg + ho Figure 2 shows h vs Pr from the Air Tables (Reference 1) and also has tabulated Pro, To, ho, and Co tsonic velocity) for various altitudes within the atmosphere, permitting rapid calculation of ram-jet performance without reference to material other than what is contained in this report. Some performance figures previously calculated by the method presented in Reference 6 were compared with those obtained for the same conditions by the method presented herein, and the maximum error was found to be 5 o. The Sa term is largely responsible for this discrepancy as it is quite critical. A small error in determining the exact magnitude of Sa for a particular fuel-air ratio and T2 can cause a greater error in Ct. This also serves to demonstrate the critical nature of the efficiency of combustion, as it directly affects Sa..-M I

DEPARTMENT OF ENGINEERING RESEARCH Page 30 UNIVERSITY OF MICHIGAN _ Report No. T3M-7 BRIEF OUTLINE OF METHOD (The following is intended as an aid and guide in calculating Ct for any Mach number, fuel-air ratio and altitude). For given Mach number of flight, fuel-air ratio, and altitude, read the following data from Figure 2; ho, Co, To, Pro, and calculate V1 from Mach number and Co. Select a diffuser efficiency (Figure 23) and an usable V2 and calculate h2 from hP - 2g (V - V )+ h From h2 find Prf, using Figure 2 or the Air Tables (Ref2 P2 erence 1), then calculate A/A2 from 00^ 22 A1 PrroToV2 A2 Pr oToV The A1/A2 may be known, as in a fixed design, and the V2 thus determined by the diffuser efficiency; this is a method of trial and error, as a V2 must be assumed to calculate h2p, from which Pr2 is found and thus V2. However, V2 is not very critical in computing h2p so that usually not more than one re-calculation is necessary (see Sample Calculation I). F2 From the T2 (Figure 2A) and V2, read W from Figure 3 nV2 a F and subtract 2- for flame holder loss to obtain.- Also 2g Wa from one of the following; Figures 3A, 3B, 3C, 3D, or 3E, read the Sa for the fuel-air ratio and T2. (If considerable I - __

Report NoI 7 DEPARTMENT OF ENGINEERING RESEARCH Report No. UMM-7 UNIVERSITY OF MICHIGAN Page 31 calculation is anticipated for a particular altitude and Mach number or fuel-air ratio, the curve can be obtained by cross-plotting from Figures 3D and 3E.) Calculate 0(M3) from F3 0(X3) W S a a F. If Sa is greater than Wa,either reduce S (the fuel-air a a a F ratio), or V2. (See Sample Calculation III). - must a always be equal to or greater than Sa as ((M3) can not be less than I. Knowing 0(M,), 0(M5) is obtained from Figure 5 for the nozzle efficiency expected, as indicated on the figure. If the fuel-air ratio is low, at low flight speeds, or at a high altitude, any combination which would produce relatively low temperatures at the combustion chamber exit, one should use the 0(M5) and 0((M5) curves corresponding to the larger Y and conversely for expected high temperatures. However, the error in 0(M5) due to using the mean Y would never be over 1.5 %. Then C can be calculated from I/2 ( aSao(,5) g ) A5 2~ = -$-Av -2

.......DEPARTMENT OF ENGINEERING RESEARCH Page 32 UNIVERSITY OF MICHIGAN Report No UM-7 Sample Calculation I PROBLEFI: To find Ct for the following conditions: A = 2 = 5 Altitude = 40,000 ft Flight iach No. = 3.0 Fuel-Air Ratio =.0333 lb fuel per lb air 050 7o stoichiometric) 7c =.85 7S =.95) a SOLUTION: ho = - 1.68 Btu per lb air lPr =.94 1. From { ~ I CO = 974; V1 = 3C = 2922 ft per sec Figure 2': o To = 393~R 2. From T2 = 10850R Figure 2A: 3. From =D 79 % Figure 2B: 4. Select a V2 for initial calculation = 400 ft per sec?7D 2 2.79 2 h2P 2g 1 - 2 o) + ho = 50,103 L2922 (400)2] - 1.68 130.4 5. From Reference 1:

1I DEPARTMENT OF ENGINEERING RESEARCH Report No...U... UNIVERSITY OF MICHIGAN Page 33 Pr2 - 20.23 6. Pr T2VA _ V - 2 Pr2ToL -.94 x 1085 x 2922 20.23 x 393 x 1 = 374.9 7. Re-calculating h2p = 50 [(2922)2 - (370)2 - 1.68 = = 130.9 From Reference 1, Pr2 = 20.39 v =.94 x 1085 1 2922 371.9 ft per se 2 - 20.39 x 393 79 8. From Figure 3 or calculate.P I -j F3 RT2 Wa V v2 nV2 g 2g Letting n = 1, F3 a 7;J 170 - 3769 164.2 64*5 4 9. From Figure 3D: S = 137 a 10. 7S Sa =.95 x 137 = 130.15 11 0(M3) =- 1644 =1262 130.15 162

PafPe 34 | DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN Report No. U3El-7 12. From Figure 5 VM5) = 1.05 for np = 88 3 C _- 2At 1 S[Sa0(M5) g 2 5 2 13. c * t ^- A2 l V1 - 1 - y A a - 2 130.15)(1.05)32.2 1 2. 22922 a 1 1.4(3)2 Ct =.85 for A, = A2 and 1:30 fuel-air ratio

DEPARTMENT OF ENGINEERING RESEARCH Report No. UMEM-. UNIVERSITY OF MICHIGAN Page 35 Sample Calculation II PROBLEM: To find Ct for the following conditions: Altitude = 20,000 ft Flight Llach No. = 4.0 V2 = 200 ft per sec Fuel-Air Ratio =.0665 lbs fuel per lb air (100 % stoichiometric) 71 = 85 %(1s = 95v) aL SOLUTION: ho 1. From Pro Figure 2 o T0 = 11.24 Btu per lb air -= 1.474 = 1037.5; = 446.8~R V1 = 4Co = 4150.0 ft per sec 2. From T2 2 = 17900R Figure 2A 3. From 7D tD =.70 /a Figure 2B 4. - D 2 h - j (V2 2P 2gJ 1" - V2) + h - 570 (41502 - 2002) - 22 +o =50,i03 + 11.24 = 251.3 Btu per lb 5. From Reference 1, (or Figure 2) Pr2 = 92.23 (corresponds to h2p = 251.3) 6. A1 Pr2 2- pr JL2 Pro o V2 92.23 200 446.8 T2 V1i = 474 4150 1790 =.753

I e 36 | DEPARTMENT OF ENGINEERING RESEARCH Report No, TU-7 Page 36 UNIVERSITY OF MICHIGAN '* 1 7. From Figure 3 F2 -r = 484 a Flame holder loss, 2V2 40021 =n 2) = -6 6.41 2 g 6 4 A thus F --- 484 - 6.21 - 477.79 a 8. From Figure 3A Sa = 177.5 a 9. F3 = Sa (M ) Wa ~ 3) 4 = =77.7892 - 177.5 2.692 10. From Figure 5, taking A5 = A3 I OM5) = 1.21 for?/np = 80 %i (low according to Figure 5A) I 11. 0c - 2A, St A2.7 = 2t.753) ( -X7.5x1. 21 4150 2A5 2' x 32.2,1 a J 2 1.4(4)2 ct =t.914 --

.....N. U ',7 DEPARTMENT OF ENGINEERING RESEARCH Page Report No., ULL7 UNIVERSITY OF MICHIGAN Pe Sarple Calculation 1II L PROBLE4iI: To calculate Ct for the following conditions: Altitude = Sea Level Flight Mach No. = 2.0 V2 = 350 ft per see Fuel-Air Ratio =.0605 (Lean) k91 '/ stoichiometric) lb fuel per lb air I c —.85 (/S =.95) 11 c sa~~~Q SOLUTION: I i / i I h = 28.56 Btu per lb air r~ = 2.49 CO = 1117.50; V1 = 2C = 2235 ft per sec 1. Froi Figure 2 2. From T = 519.0~R T2 = 920~R I Figure 2A 3. From D =.85 Figure 2B 2 2 2 285 4 h2= I2 (V1 - 2 ) + ho = 5 (2252 55010) + + 28.56 = 111.22 Btu per lb air 5. From Reference 1: Pr2 = 14.82 6. A1 Pr2 To V2 14.82 519 50 = 525 Pr T2 2.49 920 2235' A2 r02 v1 7. From Figure 3 or calculate - _

r DEPARTMENT OF ENGINEERING RESEARCH I or, T Page 38 UNIVERSITY OF MICHIGAN:I Report No. U1M-7! F3 w a _ ^ 53.34 x 920 1 - V _ 350 for two velocity heads, loss at flame holders V. n= 2 ( 2g) 8. From Figure 3B Sa = 160, which is larger than F3, thus pMt3) (1; as this is impossible either reduce Sa or decrease V2 and re-calculate. 9. Reduce S to 140; i.e., make mixture more lean - a 10, F3 (M3) =w aw aa (M5) = 1 11. c - 2A, tA2 Sa '1 - 1, V -1 n2 21 521 * 2 2.] -i 140.2 x 3 - 2( 525) 140. - ^ ~o~io; g9 - 1 2(1.4) C =.714 t ~- I

DEPARTMENT OF ENGINEERING RESEARCH Report No. w- UNIVERSITY OF MICHIGAN Page 39 REFERENCES 1. J. H. Keenan and J. Laye, "Thermodynamic Properties of Air." First Edition. New York, John Wiley and Sons, Inc. t1945) 2. R. L. Hershey, J. E. Eberhart, and i. C. Hottel, "Thermodynamic Properties of the Working Fluid in internal Combustion Engines." New York, S.A.E. Journal, Vol. 39, No. 4, Pages 409-424 (October 1936). 3. JHU/APL, CM-1. "Momentum of Combustion", Philip Rudnick (6 January 1945). 4. JHU/APL, CM-236. "Ram-jet Thrust Coefficients and Specific impulse." A. C. Beer. (April 1946). 5. JHU/APL Bumble-bee Report No. 32. "Analysis of Internal Flow in Ram-jets." Thomas Davis and J. R. Sellars. (March 1946) 6. University of iaichigan External Lemorandum No. 1, "An Analysis of the Performance Characteristics of the RamJet Engine." E. T. Vincent. (25 January 1947). 7. NACA Tech. Note No. 1026. "Charts of Thermodynamic Properties of Fluids Encountered in Calculations of internal Combustion Engine Cycles." H. C. Hottel and G. C. Williams. (M.I.T.)

I I i DEPARTMENT OF ENGINEERING RESEARCH Page 40 UNIVERSITY OF MICHIGAN I Report No. U.LE Appendix I Validity of method presented in UM-l (see Reference 6) for special cases. This portion of the report is the result of an investigation which had as its objective the presentation of evidence that the method presented in Reference 6 accounts for the particular condition of flow where a sach number of one exists at the combustion chamber exit and that no solution could be obtained if a higher Mach number would be required at this station. During the course of this investigation it was found that the method of solution given in Reference 6 gave two solutions at low flight velocities for the velocity at the end of the combustion chamber, one being supersonic and of no practical significance, while only one solution could be obtained at higher flight velocities. This was somewhat unexpected and warranted further investigation even though the supersonic solution was only of academic interest. The work that followed led to the development of the method of calculation given in the main body of this report and the explanation given there regarding the presence of two solutions for the flow conditions at the end of the combustion chamber is believed adequate. The following consists of a detailed description of the method of solution as given in Reference 6 and some representative graphical evidence which shows that the method is valid for the instances where sonic velocity exists at the combustion chamber exit and that the I 1.,i. -- ------

Report No. - DEPARTMENT OF ENGINEERING RESEARCH Report No. m'ai-71..UNIVERSITY OF MICHIGAN Page 41 method would give no solution if velocities greater than sonic were required. Figures I and 1A show the results of a series of calculations to determine the exit conditions from a cylindrical combustion chamber for various inlet air velocities kV2). See Figure A for the location of various sections referred to as station 2, 3, etc. The method is by trial and error; i.e., a value is assumed for V3, and by satisfying the energy, force-momentum and state equations, trial values of P3, p,' and T3 are determined. Then, satisfying the continuity of mass flow equation determines if the proper V3 was initially chosen. Figure 1 is a plot of the trial solutions for various combustion chamber inlet velocities at a flight oiach number of 1.75 at 40,000 feet with a fuel-air ratio of.0665. This diagram plots the initially assumed velocity at station 5 against the final velocity determined by the continuity equation, after having satisfied the energy, force-momentum, and state equations. It follows that the assumed V3 which gives a calculated V3 of the same magnitude is the correct solution for the flow at the conditions of the problem. Thus the curves of Figure I and 1A show the calculated V3 plotted against the assumed V3 for various values of combustion chamber inlet velocity (V2). The straight line in both figures having a slope of 1 is necessarily the locus of all solutions for V3; i.e., the assumed velocity equals the calculated velocity. Therefore, the intersections of each curve with the straight line are V3 solutions for the I 1 I

DEPARTMENT OF ENGINEERING RESEARCH Page 42 UNIVERSITY OF MICHIGAN Report No. UM-7 particular conditions represented by the curve. point worth noting is the small difference in the value of V necessary to give two widely spaced solutions, or no solution. For instance, in Figure L the V3 solutions for a V2 or 2U5 ft per sec are 2305 ft per sec and 3105 ft per sec while for a V2 > 207 no solution is obtained. Thus the method under discussion can be employed with a high degree of a accuracy to obtain the V2 that produces choking, but only after considerable calculation and work involving the Thermodynamic Charts. IReference 2). in the instance where no solution appeared (V2= 20b ft per sec in Figure i) the heat addition (fuel-air ratio) was decreased and a solution proved to be possible. Thus it is established that the method under discussion would offer no solution if conditions in the corabustion chamber entrance were such that for a given heat addition, velocities greater than the local sonic velocity would be required at the end of the combustion chamber. Consequently, it is concluded that the data presented in Reference 6 does represent possible solutions to the problem as there outlined despite some differences with the conclusions derived by other investigators.

Rep ort 4 Mo.U u11_7 DEPARTMENT OF ENGINEERING RESEARCH Pg.4. Rep ort No. UM-7 |UNIVERSITY OF MICHIGAN Page 43 Appendix II Probable Magnitude of the Air Velocity at the Entrance to the Combustion Chamber. it is of interest to note from Figure 3 that at those values of' iT corresponding to high leach numbers t5,6) the upper limits of the combustion chamber entrance velocity, V2, established by the fundamental equations governing flow in a duct are of the order 700-1000 ft per sec for a maximum Sa approximately stoichiometric fuel-air ratio). As a the fuel-air mixture is made more lean, reducing Sa, the maximum V2 increases even more. however, the applications of a particular ram-jet design that would use these excessively high velocities are believed very limited, for the following reasons: At low Aiacn numbers, say those less than 2.8to 3.0, the maximum Ct is obtained by using maximum Sa with the corresponding maximuml V2 sucn that sonic velocity exists at the combustion chamber exit; i.e., no nozzle required - open tail exhaust. At higher iach numbers the ram-jet configuration corresponding to maximum Ct (A1 = A2) serves to limit V2..igure 6 shows how this maximum V2 varies with lmach number at various fuel-air ratios, and also how it varies for A1 = A2' At low lkach numbers the ratio of the diffuser inlet area to the combustion chamber area, A1/A2, would necessarily be less than unity, and as the iach number increases

Page 44 Page 44 1 DEPARTMENT OF ENGINEERING RESEARCH Re t No *1IA 7 4 YUNIVERSITY OF MICHIGANepor at a constant altitude, for example, the diffuser inlet, A1, would be made larger, thus increasing V2, keeping F3 equal to S, and therefore, V3 equal to local sonic velocity, until a A1 equals A2. un 1igure 6 this would mean moving along the S. L. -.0665 fuel-air ratio curve to its intersection with the S. L. curve of "V2 for Al = A". At this point the best Ct possiole for that particular altitude of flight and fuelair ratio will be attained ksee Figure 7). As the ram-jet accelerates further to higher iach numbers, A1 should be kept equal to A2 and a tail nozzle should be used, whose area ratio varies as required by 3. If A1 is made greater than A2, the Ct Preferred to the maxiiluri frontal area) will decrease, because this will cause an increase in V2 and, as was establisned in reference 6, the thrust per Dl of air decreases with an increase in V2. it also follows that if Sa is reduced at constant flight conditions (by reducing fuelair ratio) the diffuser inlet area A, should be opened up increasing V2. Thus, ~3 is 'ept equal to the local sonic ve~ocity until Ak equals A2, at which point further reduction in S would necessitate a nozzle,. igure 6 shows that for a the.ub65 fue~-air ratio, which would be used to obtain the greatest Ct, the maximura V2 is approximately 430 ft per sec tintersection of S. L.u665 curve with S. L. kA = a2 curve). It also can be seen that the' diffuser efficiency afl'ects tnis peaKi, causing an increase in the maximum V2 for a less efficient diffuser. Rougn calculations seem to indicate that tne imaxmum specific impulse, Ig, occurs at a fuel-air ratio of ap roxi I 1

Report o. UWmi- DEPARTMENT OF ENGINEERING RESEARCH,,Report N~o. UiL/ __ UNIVERSITY OF MICHIGAN Page 45 mately.u4; for this reason, the curve corresponding to a fuel-air ratio of.u4 was also sKetched on rfigure 6, and tne maximum V2 for this curve appears to be 4oU ft per sec. Thus, the maximum V2 that would ever be considered in designs-where eitner miaxliua Ct or maximum specific ixapulse, ig, (reduced Sa) were the primary objectives would be something less than 500 ft per sec, unless these assumed diffuser efficiencies prove to be too optimistic. I

Page 46 Repo]t No. UIM-7 FUEL i FLAME HOLDER / - — 4 i - pO D DIFFUSER To Co I SECTION, - - iL D COMBUSTION r CHAMBER NOZZLE SECTION 2 I 2 3 4 5 FIG. A RAM-JET STATION DESIGNATION (REFERRED TO AS STN. 2, 3 ETC.)

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