DEPARTMENT OF ENGINEERING RESEARCH UNIVERSITY OF MICHIGAN UMM13 Copy No. _ UNIVERSITY OF MICHIGAN Ann Arbor (Api: cac'W33638 -ac-.14222 ) 1 * ", r*'. "',' * ~~*.' -,,....!,-.. t; S i Methol of Calc ulatin "A Simplified r:Methoiof Calculating Ram-Jet Performance Applicable To Low Mach Numbers" Prepared by James R. Gannett Approved by'.... E. T. Vincent Professor of Mechanical Engineering October 31, 1947

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' DEPARTMENT OF ENGINEERING RESEARCH Report No UMM-13 I UNIVERSITY OF MICHIGAN Page 1 LIST OF FIGURES Figure No. Page 1 P2/Po vs Flight Mach No. 7 2 P2/P3 s M3 For Various M2 8 3 F2/^a s T2 For Various V2 9 4 Sa v(M3) v M3 For Various Sa 10 5 V2 v Sa 11

DEPARTMENT OF ENGINEERING RESEARCH Page 2 __ |UNIVERSITY OF MICHIGAN Report No UM-13 This memorandum is published for use in conjunction with and as an extension of University of Michigan External Memorandum No. 7, "A Simplified Method of Calculating Ram-Jet Performance Applicable to High Mach Numbers", (UMM-7). The present memorandum may be considered as Appendix III of UMM-7, in which the theory is derived and the method developed.

DEPARTMENT OF ENGINEERING RESEARCH Report No UMM-13 NIVERSITY OF MICHIGAN Page 3 I. INTRODUCTION AND DISCUSSION The report for which this appendix was written, "Simplified Method of Calculating Ram-jet Performance Applicable to High Mach Numbers", (Reference 1) shows that the combustion chamber parameter, Sa, is not affected appreciably by variations in pressure in the combustion chamber. Consequently, a solution for conditions at the combustion chamber exit can be obtained independently of the pressure, at Mach numbers 2.0 to 6.0, mentioned in the report. However, for a particularly low range of Mach numbers, a solution for conditions at the combustion chamber exit (station 3, Figure A of Reference 1) is necessarily obtained in a less direct manner. In this low range of Mach numbers (1.0 - 1.25) the recovered pressure at the combustion chamber inlet, P2, is quite low, so that its ratio to the atmospheric pressure, P2/Po, is no longer sufficient to allow the particular nozzle design to determine M3 independently. That is, P2 has decreased to a point where its ratio to atmospheric pressure limits the available pressure drop in the combustion chamber and thus limits M3 (for a given M2) to some value less than one. It is assumed that P3 would be atmospheric if sonic velocity were not present at the combustion chamber "open tail" exit. As shown in Reference 1, for given conditions at station 2; namely, T2, V2, and the fuel-air ratio, M3 can be determined independently of the pressure, P2, when working with the higher Mach numbers. The pressure drop through the combustion chamber is given by P2 1+ 63M32 (1) P3"1 + 2M122

DEPARTMENT OF ENGINEERING RESEARCH~\~ Page 4 UNIVERSITY OF MICHIGAN Report No UMM-13 P3 can be any value above atmospheric when the exit velocity of the exhaust gases is sonic but, when PJ/PO is insufficient to accelerate the flow to M 1, P3 will necessarily be atmospheric. Figure 1 shows the ratio P2/PO plotted versus Mach number of flight for two values of V2. This curve obviously includes an assumption as to diffuser efficiency, ~ D, the values of which were taken from Figure 2B, Reference 1. The fact that V2 may be related closely to the diffuser efficiency has been sidestepped until more definite information is made available, so that the diffuser efficiency is considered only a function of flight Mach number. Assuming a specific heat ratio ( A) of 1.3, the maximum P2/P3 occurs for choking at station 3 and can be calculated from Equation 1, as shown on Figure 1. If P2/Po falls below this approximate "critical" value, sonic velocity cannot be maintained at the combustion chamber exit. A line is drawn on Figure 1 so that its intersection with the curves represents the approximate critical pressure ratio, P2/Po, and flight Mach number, below which P2/Po must be considered in calculating Sa and 0(M3). The method of solution is one of trial and error for these low Mach numbers (1.0 to approximately 1.25) and is outlined below with references to Figures 2, 3, and 4. II. METHOD OF SOLUTION Assume a V2 and calculate h2 from h2 2g (V12 - V22) + h(2) Find Pr2 corresponding to h2 from the Air Tables (Reference 2). P2/P3 is equal to Pr/Pr, since P3 would be atmospheric.

....... ~DEPARTMENT OF ENGINEERING RESEARCH Report No UMt.13 UNIVERSITY OF MICHIGAN Page 5 Calculate M2 from V2 and T2. Knowing P2/P3 and M2, read M3 from Figure 2. For T2 and V2, read F2/wa from Figure 3. Knowing M3 and the fact that Sa 0 (M3) F2/wa (for zero velocity heads (3) loss at the flame holders) read Sa from Figure 4. Probably Sa will be out of the possible range and the above steps will have to be repeated until the desired Sa is obtained. To account for flame-holder losses, subtract n2 from 2g a, where n is the number of velocity heads loss. $(M3) can be calwa culated and is equal to O(MO), so that CT can now be evaluated as outlined in Reference 1. CONCLUSIONS At a given flight velocity within this range of low Mach numbers, where the choking condition at the exit of the combustion chamber can not be obtained, V2 is determined by Sa and the diffuser efficiency. Figure 5 shows a curve drawn through several points obtained by trial and error solutions to obtain a maximm S^a From this curve it can be seen that as Sa increases, V2 decreases; this would necessarily be accompanied by a change in diffuser efficiency or mass flow, probably both. There is one V2 which will occur with the maximum possible Sa, as shown on the figure. Therefore, to obtain maximum performance, the inlet area ratio, A3/A2, will be determined by the V2 which corresponds to the maximum S and diffuser efficiency, from the relation A1 PrV2T (4) A2 PrVoT2

DEPARTMENT OF ENGINEERING RESEARCH ___ Page 6 ___ ___ ___ UNIVERSITY OF MICHIGAN I Report No U^H-13 where P2 corresponds to h2 of Equation 2. If \D, diffuser efficiency, is considered a function of V2, a curve similar to the one on Figure 5 would probably be plotted through points obtained by several trial and error solutions, each point calculated using a different D for each assumed V2. Then the V2 (corresponding to the maximum Sa available for the particular fuel-air ratio) and T2 would be the design point, thus determining A1/A2 and requiring a diffuser to obtain the assumed or predicted efficiency used in the calculation. An increase in A1/A2 will increase V2 but will penalize the Sa term in the equation for CT 2A1 (gsaA(l ) g 1) 2A5 T A2 V1 ) A @ The loss from the reduced Sa will be greater than the benefit fromi the A2/A2 term, resulting in a lower CT. Thus the V2 that gives the maximum Sa will also give the maximum CT.

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DEPARTMENT OF ENGINEBRING RESEARCH Report No U01-13 I UNIVSITY OF MICHIGAN Page 13 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 DP.