AFOSR TN 57-657 ASTIA Document No. AD 136-648 ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR Technical Note A PRELIMINARY STUDY OF THE APPLICATION OF STEADYSTATE DETONA,;VE COMBUSTION TO. A REACTION ENGINE *. r..-e... J. A. Nicholls Aircraft Propulsion Laboratory Aeronautical Engineering Department Project 2284 COMBUSTION DYNAMICS DIVISION AIR FORCE OFFICE OF SCIENTIFIC RESEARCH, ARDC CONTRACT NO. AF 18(600)-1199 WASHINGTON, D.C. September 1957

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The University of Michigan ~ Engineering Research Institute ACKNOWLEDGEMENT This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No. AF 18(600)-1199. Reproduction in whole or in part is permitted for any purpose of the United States Government. The assistance of this agency is gratefully acknowledged. ii

The University of Michigan ~ Engineering Research Institute TABLE OF CONTENTS Page LIST OF FIGURES iv ABSTRACT v OBJECTIVE v NOMENCLATURE vi DISTRIBUTION LIST viii I. INTRODUCTION 1 II. GENERAL CONSIDERATIONS 2 III. CONSIDERATIONS OF THE DETONATIVE PROCESS 3 IV. MIXING ANALYSIS 5 V. DETAILED EXAMINATION OF THE FLOW SYSTEM 8 VI. RESULTS AND CONCLUSIONS 12 REFERENCES 16 111

The University of Michigan ~ Engineering Research Institute LIST OF FIGURES Figure No. Title Page 1 Engine configuration 2 2 Normal detonation wave 3 3 Oblique detonation wave 4 4 A method of fuel injection 5 5 Mixing region 6 6 Total pressure ratio across the mixing zone 10 7 Specific thrust vs. flight Mach number 13 8 Specific thrust vs. flight Mach number 14 9 Specific fuel consumption vs. flight Mach number 15 iv

The University of Michigan ~ Engineering Research Institute ABSTRACT A rea-cti.on type engine employingsteady-state detonative combustion is considered. A simplified analysis treats the supersonic mixing of fuel and a,ir together with the requirements necessary to achieve steady-state detonal.ve combustion. Calculations of specific thrust and specific fuel consumpt:lon as functions of flight Mach number are made for hydrogen and acetylene fuels. The results of this study indicate that some supersonic diffusion of the air i.s necessary even though supersonic combustion exists. it is concluded.that the speed range of air-breathing engines may be materially extended. OB E C TIVE nhe purpose of this study is to determine the feasibility of a reaction engine employing a continuous detonation process at the combustion chamber. Vy

The University of Michigan ~ Engineering Research Institute NOMENCLATURE a speed of sound A area Cp specific heat at constant pressure F thrust g acceleration of gravity h enthalpy per unit mass CT specific thrust Ki defined by Equation (20) K2 defined by Equation (21) m molecular weight M Mach number P pressure Q heat release per unit mass Ro universal gas constant SFC specific fuel consumption T absolute temperature V velocity w mass flow rate x,y coordinate axes ratio of specific heats p density 0(M) defined by Equation (13) (M) defined by Equation (15) vi

The University of Michigan ~ Engineering Research Institute NOMENCLATUJRE (cont.) Subscripts a,f gases a and f (also a = air and f = fuel) D Chapman-Jouguet detonation e exit conditions ig ignition value n normal component s stagnation conditions t tangential component wo undisturbed air conditions 1 22,3,4 stations in the engine vii

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The University of Michigan ~ Engineering Research Institute I. INTRODUCTION To date there has been considerable experimental and theoretical work devoted to the study of detonation waves. It has been found that upon ignition of various combustible gas mixtures under certain conditions of pressure, volume, and composition, a flame front will propagate through the mixture at speeds ranging from 3000 to 12000 fps. This front (shock-initiated combustion) is called a detonation wave and the speed at which it propagates is the detonation velocity. So far this phenomenon has been observed mainly in shock tubes in which the wave is in a transient state. At present, however, attempts are being made to accelerate a combustible gas at high total temperature and pressure to the local detonation velocity and then cause it to ignite, thus establishing a standing (steady-state) detonation wave.1 One is led to wonder if such a burning process could be applied to a propulsive system. It seems that a jet device utilizing this mode of combustion, as opposed to deflagration burning, would offer several advantages. For example, the supersonic inlet diffuser could be greatly simplified since the burning would occur at supersonic speeds. Thus the incoming air need not be diffused subsonically and hence shock-swallowing problems, as well as large total pressure losses, could be eliminated. The static pressure rise, necessary for expelling the exhaust gases, would now be realized, at least in part, across the detonation wave rather than entirely through the diffuser. Furthermore, since the detonation process occurs at high velocities and total term peratures, it seems to be a natural means of extending the speed range of air breathing vehicles. Other advantages would include a shortened combustion chamber with no need for an ignition device. An obvious disadvantage is the lack of static thrust. In view of possible applications, a preliminary analysis was made in an effort to determine some of the performance characteristics of an engine in which heat is added by means of a standing detonation. Special attention is given to the supersonic mixing of fuel and air, and solutions are presented for the two cases when either the pressure or the area remain constant throughout the mixing zone. A general discussion of the detonation process together with the method in which it could be applied to a steady-flow engine is also given. Finally, problems associated with the matching of the various flow processes to produce an efficient thrust producing mechanism are discussed and some over-all characteristics, such as specific thrust and specific fuel cosumption,are calculated. _ _ __ __ 1

The University of Michigan ~ Engineering Research Institute It+ should be pointed out that the analysis contained herein is limited to the following general assumptions. The working fluid is assumed to be an inviscid perfect gas mixture. Average values of the specific heats, applicable to the temperature range and mixture considered, are used throughout. Heat and frictional losses are assumed absent, and any total pressure losses due to the formation of oblique shock waves while compressing a supersonic gas are neglected. Finally, it is supposed that there exists an "effective ignition temperature" below which the fuel-air mixture must be kept prior to detonating. In view of the above assumptions, it should be clear that the results presented in this report can only indicate approximate values of performance characteristics comparable to other idealized engines, which is all that was iintended. II. GENERAL CONSIDERATIONS The following is a brief discussion of the flow system assumed in the analysis of this engine. Figure 1 is a sketch of this flow system. 00) 2 3 4 e Fig. 1. Engine configuration. Tb.e desirability of an engine requiring no diffusion of the incoming air has already been mentioned. At first one might think that no diffusion would be required in this engine since both mixing and combustion are occurring at supersonic velocities. However, it will be shown that for maximum pressure recovery through the engine there must be supersonic diffusion to some extent previous to mixing the fuel and air. This diffusion occurs between stations (c) and (1).

The University of Michigan ~ Engineering Research Institute The fuel is injected into the supersonic stream at (1) and is assumed fully mixed with the air at station (2). Detonation occurs at station (3). Under certain conditions, namely, at low flight speeds, the tack pressure should be sufficient to initiate the detonation normal to the flow at some station (3). However, this type of initiation ray possibly be unstabl.e, and in practice some form of stabilization (such as a thin body placed in the stream) will probably be required. At high flight speeds, certain mixing requirements and efficiency considerations dictate that a body must definitely be placed in the stream. The resulting oblique shock wave initiates the chemical reaction and the body serves as the means of stabilization. For the oblique wave portion of this analysis, the detonation is assumed to be stabilized on a two-dimensional wedge. Chemical equiliLri.um conditions are assumed to exist at station (4) immediateely downstream of the detonation. Fi.nally, frozen equilibrium flow is assumed in the isentropic expansion to atmospheric pressure at the exit. Downstream reflections of the stabilized wave are neglected and of course variable geometry is implied. In the over-ail consideration of the analysis, two areas of especial interest and importance stand out: the supersonic mixing of fuel and air and the detonation process. Each will be considered separately in the following two sections. iT 1. CONSIDERATIONS OF THE DETONATIVE PROCESS For purposes of analysis a detonati-on wave A.s usually treated as a discontinuity with heat addition.* First, consider the one-dimensional model of a detonation in which the reference system is attached to the wave and the gases move relative to it (Fig. 2)~ Unreacted mixture Gaseous reaction of a comfLustible gas products state 3 state 4 Fig. 2. Normal detonation wave. The conservation equations for this system can be written: Mass: p3 V3 = p4 V4 or = 2M4P4 (i) a3 a4 *It may be thought of as a shock wave with combustion. 5

The University of Michigan ~ Engineering Research Institute Momentu' P3 + p3V32 = P4 + p4V42 or P3(1+Y3M32) = P4(ly+4M42) (2) Energyo CP3T3 + + r (l+Z3L M )+Q = ("4 ~2) (3) 2 2 Ys3-l 2 Y4-1 2 Next, consider an extension of the above model to the two-dimensional case wherein an oblique detonation is stabilized on a wedge (Fig. 3). VState3 State 4 v3n 3tn V4t Fig. 3. Oblique detonation wave. In this model the incoming velocity is resolved into components normal and tangential to the wave. Since there is no net pressure force in the tangential direction, there can be no change in the tangential velocity across the wave. The normal components may be treated as in the one-dimensional case. Hence the two-dimensional analysis reduces to the vector addition of a (tangential) velocity to the one-dimensional normal wave model. Note that the conservation equations for this model are identical to the previous onedimensional equations except that V3 and V4 are now understood to be the normal components of the total velocities. Note also that in the energy equation the tangential components of the velocity will cancel. One finds upon analyzing the above conservation equations that, for a given upstream temperature and heat release, solutions to the equations exist only for upstream velocities above a certain minimum. For all velocities above this minimum, the equations predict two possible solutions for the downstream conditions; however, one solution is ruled out by entropy considerations.* It is interesting to note that the minimum velocity is the velocity always measured in shock-tube investigations of detonation. Associated with this minimum velocity are downstream conditions such that the Mach number of the kburned gases relative to the wave is always unity. Also, the total pressure loss across such a detonation is a minimum. This type of wave is called a Chapman-Jouguet detonation. *See Reference 1 for a comprehensive discussion of the oblique wave solution. ___ ___ ___ ___ ___ ___ ___ ___ ___ ___4

The University of Michigan ~ Engineering Research Institute Since total pressure is of prime importance in the efficient conversion of thermal energy into kinetic energy, the total pressure loss across the detonation in this engine was minimized by imposing the conditions necessary for a Chapman-Jouguet type wave. This proves to be a fortunate choice from still another standpoint since the heat release computation can be eliminated. Thus, by using experimental values for the detonation velocity, the momentum and mass equations can be solved without considering the energy equation, and hence the laborious chemical equilibrium calculations behind the wave are avoided. IV. MIXING ANALYSIS An exact solution to the supersonic mixing of fuel and air would be prohibitively complicated. Consider, for example, a scheme for introducing gaseous fuel into a supersonic stream of air as shown in Fig. 4. Air, at a high stagnation temperature and pressure, would flow around the nozzles _X~g 7- i 7 Fuel Air IY Fuel Nozzles Fig. 4. A method of fuel injection. at supersonic speeds and proceed to mix with.the fuel through molecular and turbulent diffusion. The resulting shear flow would be nonuniform and nonsteady. The flow would be further complicated by the presence of shock waves generated in the nozzle region, as well as by possible local burning in the boundary layer in the vicinity of the nozzle exit. Since it would clearly be quite difficult to make a detailed investigation, the following analysis will be directed toward a simplified solution of the mixing problem. Consider the mixing of two inviscid perfect gases (denoted by subscripts a and f) as shown in Fig. 5. It is assumed that the rate of change of any flow parameter in the x or y direction is zero at stations 1 and 2 (except at the initial interface of the gases where a discontinuity may exist in the y direction in the temperature and velocity), and that the gases are fully mixed at station 2.

The University of Michigan ~ Engineering Research Institute =f x - a I 2 Fig. 5. Mixing region. With these assumptions the conservation equations may be written in difference form as followsMass Pla Via Ala + plf Vlf Alf = P2 V2 A2 (4) Energy: wla (kla + Via) + wlf(hlf + f) = w2 (h2 + V) (5) 2 2 2 Momentum in x-direction| P1 (Ala + Alf) - P2 A2 + PdA = P2A2V2 - PlaAlaVa - lfAlfVlf (6) Using the equation of state, Ro P = T and approximating the enthalpy by h = Cp T, where Cp is an average value of the specific heat between T=O and T=T1, the conservation equations are rewritten in the following form. Masss ilaMiaAlaPi ylfMlfAlfPl 72M2A2P2 (7) ala af a2 Energy CTs_ TS2Ener = T2 (1 + M2-1.M2) - + 2 Ts (8) LPzf wlf 1CPla wla 1 la Momentum: p L Wlf P1A1 (1+ l a Mid + 7 a M + 7f Al Mf) + PdA PA 1+M2 (9) where Al A1 = ALa + Alr Notice that the momentum equation contains an integral term that is dependent upon the variation of pressure during the mixing, while the other 6 _

The University of Michigan ~ Engineering Research Institute conservation equations express a relationship between the end points only. It is apparent that a solution can be found for constant pressure mixing, in which case A2 PA = P1(A2-A1) = P2(A2-A1), (1O) OA, or for constant area mixing where A2 PdA = 0. (11) In general the integral can be approximated by using an average constant value of the pressure during mixing although this case will not be considered. By combining Eqs. (7), (8), (9), and (10) or (11), the following solutions for the downstream Mach number are found for constant pressure or constant area mixing. Ca 2 Constant pressure mixing I1 + I- wa+f Y2 mia TsIf Cpla w1a L a)a wia / 1 m2 L- sa SCp~lf wf.where Wa (12) ~(M) =....M + M2 (13) 2 Constant area mixingerT" C 1 C mia Tsia f + (Mq.. - _ Cplf wlfJ,(Mla) (14) 1 + a wfS a) 7lf wia asia BN(Mzf) where M!4l"'+ M2 (T]) (1 + 7M2) It should be understood that the subscript for y in q(M) and 0(M) should correspond to the subscript for M. Of particular interest in the engine analysis is the downstream total temperature, given directly by Eq. (8), and the downstream total pressure. The total pressure ratios across the mixing zone for the two cases are given by 7

The University of Michigan ~ Engineering Research Institute 72 ]72-1 P _2. +2 1 for (P1 = P2), (16) Psla [1 + 2 M12 ]Y7a-1 2 and 721 2 72-1 A1 2 Alf 2 [ 2 + T[M21 2(+ 2 Ml a + 7 M +f) Psl C1. -e-Ya PS1a [1 7 M2] 7a-l (1 + 72 ) for (AA2 ) (17) Figure 6 shows a plot of the total pressure ratio as a function of M for a stoichiometric hydrogen-air mixture. For this case, and in the remainder of the report, the subscripts a and f will refer to the air and fuell respectively. The downstream boundary condition (T2 = 2000CR) was chosen so that the calculations would correspond to a practical case wherein the combustible mixture must be kept below an effective ignition temperature. Curves for two different fuel-inlet conditions are shown. It can be seen that as the Mach number of the air increases the total pressure ratio continually decreases and t-hat in the Mach number range between M=2 and M=4 the least loss in total pressure is obtained by injecting the fuel at the low energy leve'l. it is physically plausible that there should be a better total pressure recovery (referred to the air total pressure) when the fuel is at a low energy level since more thermal energy from the air must be transfered to the fuel.* As the Mach number of the air increases, the air total temperature -ecornes so large that the effect of the difference in fuel-inlet conditions becomes less significant. The results shown in Fig. 6 indicate the desirability of diffusing the incoming air somewhat before injecting the fuel because of the importance of total pressure recovery in engine performance. V. DETAILED EXAMINATION OF THE FLOW SYSTEM Th.e integration of the previous work into an over-all engine system will now be considered. *Recall that in a frictionless heat subtraction process the total pressure of the gas will increase. 8

The University of Michigan * Engineering Research Institute 0.9 -A =constant MI f P= constant TSf -0 0.8 0.7 0.6 Mlf =2.0 P=constont TSIf=2000*R A=constont a. u. 0.5 0.4 Stoichiometric H2-air mixture T2 =2000 OR 0.3 0.2 0. I 1 2 3 4 5 6 7 Mla Fig. 6. Total pressure ratio across the mixing zone. 9

The University of Michigan * Engineering Research Institute An important limitation peculiar to this type of engine is that the static temperature of the unreacted fuel-air mixture must be kept below an "effective ignition temperature." The "effective ignition temperature" is herein defined as that temperature at which a moving combustible gas mixture will spontaneously ignite. Clearly, this temperature will depend on the dynamic and thermodynamic state of the gas, composition, confining boundaries, the flow process involved, etc. To the authors' knowledge, no studies involving all these parameters have been made. To effect the calculations for this engine, it was necessary to assume some value of ignition temperature which appeared reasonable on the basis of known values found in the literature for stagnant mixtures. In applying the ignition-temperature condition, certain ramifications must be considered. The total pressure loss across the detonation can be minimized by the attainment of a Chapman-Jouguet detonation at the lowest possible detonation Mach number) Since this Mach number varies inversely as the square root of the temperature,* it is obvious that the detonation should occur at the highest possible temperature. This is advantageous from still another standpoint since (as can be seen from Fig. 6) the total pressure losses through the mixing zone are decreased when the Mach number of mixing is decreased. Hence, for best performance, the static temperature of the unreacted gas must be allowed to approach its limiting value, the ignition temperature. Also, from this reasoning, it is concluded that supersonic diffusion of the air prior to mixing with the fuel is necessary. In locating the detonation in this engine, the above discussion would indicate that it should coincide with the point of maximum static temperature; namely, at the end of mixing. Recall, however, that to achieve detonation, a certain minimum velocity is required. At low flight speeds this mirimuam velocity will not be realized at the end of mixing, and the flow muls hI+e expanded. The detonation will then occur at some downstream position (station3, Fig. 1), as a Chapman-Jouguet normal wave. As the flight speed is increased,the conditions sufficient to generate this type of detonation will occtur fur-ther upstream until finally they will exist just at the end of mixing. For any higher flight speeds, the wave must be stabilized on a wedge so that the normal component of velocity remains a minimum (Chapman-Jouguet oblique wave). With the general engine configuration thus determined, it is possible to compute over-all performance. By specifying the flight Mach number, ambient air temperature, fuel, fuel-air ratio, and fuel-inlet conditions,.the computations are made as follows: a) Setting T2 = Tig, determine M2 and TS2 from Eq. (8). b) Knowing M2 and TS2, compute Mla from either Eq. (12) or (14). c) Calculate Ps2/Ps-afrom Eq. (16) or (17). *The Chapman-Jouguet detonation velocities used in this report were assumed to be independent of initial temperature and pressure and were obtained from Ref.2 10

The University of Michigan ~ Engineering Research Institute d) Find V2 = M2 a2 and compare with VD. i. If V2 is less than VD, use the normal wave solution below [Eq. (18)] to compute Ve. 2. If V2 is greater than VD, use the oblique wave solution [Eq. (19)]. The following equations were derived by use of the conservation equations between the end of mixing and the exit. The solution across the detonation wave was determined so that a Chapman-Jouguet (normal or oblique) wave always occurred. 1/2 Normal detonation' r_ Ve = s.2 l - (Y4+1) Y/4 S2.I r + Y-t-1 V2 7i~l 2~4~~' r4-1 VD +- 2 Y2 - 7-1 I.4'4- 1 4-_1 | L 2 aS2 4 2 aS2 (18) Oblique detonation: Ve = K + V2 - K2 (P) 4 2+ 7Y4(Y2 ( -9) where 2 2 f4a 22- ) K (2 2 V.> (you )a2 + Ya VD )7 1 2 Y42-1V (21) Performance characteristics such as specific impulse and specific fuel consumption can now be computed by the usualformulas: wlf F Ve(l + w-a ) - V X CT = V wia g, sec (22) SFC = 3600 CT W HR (23) 11

The University of Michigan ~ Engineering Research Institute VI. RESULTS AND CONCLUSIONS Figures 7, 8, and 9 show the results of calculations of specific thrust and specific fuel consumption for hydrogen and acetylene fuels. in each case,the fuel-a.ir mixture was stoichiometric and the effective ignition temperature was assumed to be 2000~R. The Chapman-Jouguet detonation velocit:ies were approximately equal at a value of 5900 fps. The constant pressure mixing solution was used throughout. From. these graphs, it can 1be seen that there is no thrust below a flight Mach number of four and that the specific thrust reaches a mTax:ilrm between Mach numbers of six and seven, then decreases. The shape of the specificth;rust curve can be explained as follows. First, below a certain f.igh.t speed the total energy of the fuel-air mixture is insufficient to achieve steady detonation. At slightly higher flight speeds a detonation will occur -if the gases are expanded to a very high Mach number. However, for very high detonation Mach numbers, the total pressure loss is excessive and no thrust is realized. As the flight speed is increased, the Mach number of detonation decreases, and hence the specific thrust begins to increase. Fi.-nally the specific thrust reaches a maximum when the detonation is stabilized at its limiting position (the end of mixing). For still higher flight speeds the Chapman-Jouguet obtlique wave solution exists at a constant Mach number of detonation at Tig and hence the total pressure ratio across the wave remains constant. Thius the specific thrust decreases because of the:rising total pressure loss associated with the mixing process. The importance of fuel inlet conditions can be seen from Fig. 7. The upper curve corresponds to Mif = 2 and Tsif = 2%00RR, while the other curve represents the lower limiting va:lues for these parameters. The fuel-inlet conditions are seen to be guise important in determining the operating range for this type of engine. The carves in Figs. 8 and 9 compare two different fuels. Since the detonat.on velocities were the same for both mixtures and the effective ignition temperatures assumed equal, the differences in these curves are due to the unequal molecular weights, fuel-air ratios, and specific heat rat:ios. In deciding upon the feasibility of the proposed. engine, the specific thrust and specific fuel consumption may be compared to an ideal ramjet. The present engine offers comparable performance at much hilghe:r fl:.ght lMach numbers, and indicates a possible means of extending the speed range of air-breathing vehicles. Finally, it must be stressed that these performance calculations are highly dependent on the effective ignition temperature ass umed and are thus limited by the accuracy of this assumption. Furthermore, the results have not een optimi.zed from the standpoint of mixing process and fuel.-inlet conditions _____________________________ 12 ___________________

100......................;~~~~~~~ MIf= 2.0 LTs'f= 2000~R E L~~~~~~ 80...., F-<~ ~ ~ ~~ / H0 60 -. 0.... ~ (./W"W.. 40 Stoichiometric H2-air Tig 2000 ~R T= 520~R 20 - Yb I Iw0~~~~~~~~~~~~~~~~~~~~~~~ 3 4 5 6 7 8 9 I O Fig. 7- Specific thrust vs. flight Mach number.

100 -II Stoichiometric H2-air 80'"~~~~~~~~~~~~~80 ~~0.Stoichiometer 2 " -air cn w ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ LU N60 4 _J I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~MIf =2.0 20 Ts: =2000,0R 40' o' ~T i= 2000 oR 1_, "'A TS f:= 0' I i = 3' 4 5 6 7 8 9 10 -M n00 Fig. 8. Specific thrust vs. flight Macri number.

The University of Michigan ~ Engineering Research Institute All mixtures stoichiometric Tig = 2000 ~R To0=520~ R 6 C2H2aGir iMf= 2.0 Dv 4 wici LLI 03- i Ml2 H2 -air MIf, 2H_ - TSlf:_ =2000OR ___ 0 4 5 6 7 8 9 10 MOD Fig. 9- Specific fuel comsunption vs. flight Mach number. 15

The University of Michigan ~ Engineering Research Institute REFERENCES 1. Rutkowski, J., and Nicholls, J. A., "Considerations for the Attainment of a Standing Detonation Wave," Proceedings of the Gas Dynamics Symposium, Northwestern University, 1956. (Also issued as OSR-TN-55-216.) 2. Lewis, B., and Von Elbe, G., Combustion, Flames and Explosions of Gases, Academic Press, Inc., New York, 1951. 3. Morrison, R. B., "A Shock Tube Investigation of Detonative Combustion," The University of Michigan, Eng. Res. Inst. Report UMM-97, Ann Arbor,1952. 16

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