A Shock Tube Investigation of Detonative Combustion by Richard Boyd Morrison Contract AF 33 (038) 12657 Willow Run Research Center Engineering Research Institute University of Michigan UMM-97. January 1952

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 TABLE OF CONTENTS Section Title Page List of Figures iii List of Tables vii List of Plates viii List of Symbols and Definitions ix Acknowledgement xii Abstract xiii Introduction 1 I Theory of Detonative Processes 3 1.1 Hugoniot Relations 3 1. 2 Chapman-Jouguet Conditions 6 1.3 Hugoniot Relations for Ideal Gases 8 II Analysis of Detonation Velocities 11 2. 1 Relations for Chapman-Jouguet Detonation Velocities Derived from the Considerations of Mass Conservation, Momentum, and a State Equation 11 2.2 Relations Derived for Chapman-Jouguet Detonation Velocities from Consideration of Energy Conservation 17 2.3 Heat Addition to a Moving Gas in a OneDimensional Duct 21 2.4 Strong Detonations 31 2.5 Analysis of Shock Waves Considered as Degenerate Detonations 34 III Shock Tube Theory as Applied to Detonative Processes 38 3.1 Shock Tube Integral Relations 38 3.2 "Characteristics" Solution of Shock Tube Equations 42 3.3 Collision of a Detonation Wave with a Gaseous Inte rfac

rWILLOWN RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 TABLE OF CONTENTS (Cont.) Section Title Page 3.4 Detonations in Shock Tubes Which are Induced by Reservoir Gases of Insufficient Pressures to Create a Pressure Plateau Behind the Detonation 66 3. 5 Wave Shapes Behind Detonations 69 IV Experimental Equipment And Instrumentation 74 4. 1 Shock Tube or Detonation Tube and Charging Equipment 74 4. 2 Timing Apparatus 77 4. 2. 1 Timer 77 4. 2. 2 Calibration Equipment 79 4. 2. 3 Operation of Timing Equipment 82 4. 2.4 Probes 82 4. 3 Photographic Apparatus 85 4. 3. 1 Spark Source 85 4. 3. 2 Delay Circuit 87 4. 3. 3 Optical System 89 V Experimental Results 98 5. 1 Correlation of Detonation Velocities 104 5. 2 Photographic Results 108 VI Discussion 112 References 119 Bibliography 120 ii

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 LIST OF FIGURES Number Title Page 1 Flame Front in a Constant-Area Duct 3 2 Hugoniot Curve 5 3 Temperature Ratio, T /T1, Across a Chapman-Jouguet Detonation vs. Mach umber 14 4 Dimensionless Parameter, m1T2/m, vs Mach Number, MD, for a Chapman-Jouguet Detonation 16 5 Dimensionless Parameter, Q/C 1T1, vs. Detonation Mach Number, MD, for the Chapman-Jouguet Case 22 6 Dimensionless Parameter, ml/m2 x Q/CplT1 vs. Mach Number, MD, for a Chapman-Jouguet Detonation 23 7 Plot of Heat Addition in Btu/lb. vs. Velocity in Ft/Sec for a Series of Initial Temperatures 25 8 Temperature Along Path Lines in Degrees R vs. Velocity in Feet per Second 26 9 Heat Addition in One-Dimensional Duct 27 10 Detonative Processes 28 11 Heating Parameter, Q/CplT1, vs. Mach Number, MD 32 12 Shock Tube Representation 39 13 Time-Distance Plot Showing the Location of the Rarefaction Wave, Piston, and Shock Wave 40 14 Accelerating Piston in a One-Dimensional Duct 40 15 Use of a Detonation as a Reservoir Gas 50 16 Pressure Ratio, P2/P1, Across a Normal Shock vs. the Reservoir Pressure Ratio, P5/P1 51 17 Pressure Ratio, P2/P1i Across a Normal Shock vs. the Reservoir Pressure Ratio, P5/P1 51 Iiii

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 LIST OF FIGURES (Continued) Number Title Page 18 Pressure Ratio, P2/P1, Across a Normal Shock vs. Speed of Sound Ratio, CT/CR for P5/P1 = 10 52 19 Pressure Ratio, P2/P1, Across a Normal Shock vs. Speed of Sound Ratio, CT/CR for P5/P1 = 100 53 20 Pressure Ratio Across a Chapman-Jouguet Detonation, P2/P1, vs. the Reservoir Pressure Ratio, P5/P1 54 21 Pressure Ratio Across a Chapman-Jouguet Detonation, P2/P1, vs. the Reservoir Pressure Ratio, P5/P1 54 22 Pressure Ratio, P2/P1, Across a Chapman-Jouguet Detonation vs. the Speed of Sound Ratio, CT/CR, for P5/P1 = 10 55 23 Pressure Ratio, P2/P1, Across a Chapman-Jouguet Detonation vs. the Speed of Sound Ratio, CT/CR, for P5/P1 = 100 56 24 Reflected and Transmitted Waves 58 25 Reservoir Pressure Ratio, P5/P1, Necessary to Produce a Shock with no Reflected Waves vs. the Dimensionless Ratio, MC x CR/CT 59 26 Dimensionless Ratio, CR/CT x MT vs. the Reservoir Pressure Ratio, P5/P1, for Various Values of Pressure Ratio, P2/P1 61 27 Dimensionless Ratio, CR/CT x MT vs. the Reservoir Pressure Ratio, P5/P1, for Various Values of Pressure Ratio, P2/P1 62 28 Dimensionless Ratio, CR/CT X MT vs. the Reservoir Pressure Ratio, P5/P1, for Various Values of Pressure Ratio, P2/P 1 63 29 Reservoir Pressure Ratio, P5/P1, Necessary to Produce a Shock vs. Mach Number, MST 64 iv

WILLOVW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 LIST OF FIGURES (Continued) Number Title Page 30 Reservoir Pressure Ratio, P5/P1, Necessary to Produce a Chapman-Jouguet Detonation vs. Mach Number, MDT 65 31 Detonation Wave with a, Trailing Rarefaction 66 32 Characteristic Diagram for a Detonation Produced in a Shock Tube where the Reservoir Pressure was Insufficient to make P3 = P2 68 33 Characteristic Diagram for a Chapman-Jouguet Detonation Originating near the Closed End of a Flame Tube 71 34 Calculated Pressure Distribution for a 2H2 + 02 Detonation in a Closed End Flame Tube 72 35 Typical Wave Shapes in the Wake of Detonations 73 36 Shock Tube and Charging Equipment 74 37 Diaphragm Assembly 75 38 Timing Circuit 78 39 Pip Generator 80 40 Calibration Sequence Pulser 81 41 First Ionization Probe 83 42 Photoelectric Probe 84 43 Third Ionization Probe 85 44 First Spark Gap Assembly 86 45 Low Inductance Condenser 87 46 Time Delay Circuit 88 47 Diagram of Spark Schlieren 89 48 Detonation Velocity of Hydrocarbon Gases vs. Mixture Ratio 99

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 LIST OF FIGURES (Continued) Number Title Page 49 Detonation Velocity of Hydrocarbon Gases vs. Normalized Mixture Ratio 100 50 Detonation Velocity of Acetylene-Oxygen Mixtures vs. Mixture Ratio 101 51 Detonation Velocity of Acetylene-Oxygen Mixtures at Reduced Pressures 101 52 Detonation Velocity of Acetylene-Oxygen Mixtures at Reduced Pressures vs. Mixture Ratio 102 53 Detonation Velocity of Ether-Oxygen Mixtures vs. Mixture Ratio 102 54 Detonation Velocity of Hydrogen-Oxygen Mixtures vs. Mixture Ratio 103 55 Detonation Velocity of Hydrogen-Air Mixtures vs. Mixture Ratio 103 56 Detonation Velocity of a 51.5 per cent HydrogenOxygen Mixture vs. Distance from Diaphragm 105 57 Detonation Velocity of a 50 per cent AcetyleneOxygen Mixture vs. Pressure Ratio 105 58 Dimensionless Parameter, 4 = ml/m2 x Q/Cpl T1 for Various Fuel-Oxygen Ratios vs. Measured Mach Number, MD, of Detonation 107 vi

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 LIST OF TABLES Number Title Page 1 Values of (T2/T1) (M1/M2) for Various Mach Numbers forT1 = 1.4, Y2 = 1.2 17 2 Composition of Gases in the Burned Portion of a 10 per cent Acetylene-Oxygen Detonation 106 3 Comparison of Detonation Velocities and Temperatures Obtained from Various Sources 113 4 Comparison of Detonative Mach Numbers with Detonation Velocities 114 5 Comparison of Mach Numbers of Detonations from Theory with Observed Values 118 vii

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 LIST OF PLATES Number Title Page I Sphere Charging Apparatus 91 II Transparent Test Section Assembly 92 III Shock Tube Test Section 93 IV Shock Tube Diaphragm Assembly 94 V Timing Apparatus 95 VI Flash Unit 96 VII Schlieren Photographic Apparatus 97 VIII Spark Schlieren Photograph of a 50 per cent Hydrogen-Oxygen Detonation -- Vertical Knife Edge 110 IX Spark Schlieren Photograph of a 50 per cent Hydrogen-Oxygen Detonation -- Horizontal Knife Edge 110 X Spark Schlieren Photographs of a 25 per cent Hydrogen-Oxygen Detonation -- Horizontal Knife Edge 111 XI Spark Schlieren Photographs of a 25 per cent Hydrogen-Oxygen Detonation -- Vertical Knife Edge 111 viii

WVILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 LIST OF SYMBOLS AND DEFINITIONS Subscripts denote the following: C conditions of a convective gas C-J Chapman-Jouguet conditions D detonation process i conditions at an interface R reflected or reservoir conditions S shock process SD strong detonation process ST transmitted shock T transmitted or test chamber conditions 1, 2, 3, 4, 5 conditions as described in Figure 13, page 40. SYMBOLS a as defined by C =aC b as defined by Cp2 = bCp2 C speed of sound C specific heat at constant pressure p Cv specific heat at constant volume e internal energy per unit mass h enthalpy per'unit mass K as defined on page 20 M Mach number m molecular weight of gas P pressure ix

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 LIST OF SYMBOLS AND DEFINITIONS (Continued) Q heat added to the system.Qc heating value of a combustible mixture R gas constant = Ru/m R universal gas constant S entropy per unit mass T temperature (usually on the absolute scale) t time u velocity Z) specific volume ratio of specific heats = Cp/Cv p density heating parameter = - pl 1 DEFINITIONS Deflagration wave - a subsonic combustion wave characterized by a, pressure decrease and a specific volume increase on passing through the wave. See pages 4 - 6. Detonation wave -- a supersonic combustion wave characterized by a pressure increase and a specific volume decrease on passing through the wave. See pages 4 - 6. Chapman-Jouguet Conditions - the condition that exists for a deflagration or detonation wave when the gases in the burned portion of the wave move at a x

WILLOW RUN RESEARCH CENTER —UNIVERSITY OF MICHIGAN UMM-97 LIST OF SYMBOLS AND DEFINITIONS (Continued) velocity relative to the wave just equal to the local sonic velocity in the burned portion of the gas. Perfect gas -- one which conforms to the state equation P = pRT Ideal gas -- a perfect gas which has constant specific heats..,',. — ~~~Xi

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-97 ACKNOWLEDGEMENT Much credit is due to the author's co-workers James A. Nicholls and Robert E. Cullen whose understanding and experience in the field of gas dynamics proved invaluable. Henry H. Hicks aided in the construction of some of the electronic equipment. The author also wishes to express his deep appreciation to Professor John Luecht and other members of the faculty of the University of Michigan. xii

MWILLO OW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 ABSTRACT The combustion of inflammable gaseous mixtures may be divided into two categories: deflagration or subsonic combustion waves, and detonation or supersonic waves. Detonations are analytically and experimentally investigated to determine the influence of various thermodynamic parameters on the dynamic properties of a non-adiabatic, one-dimensional flow system. Unsteady and steady flows are analyzed in the case of detonations occuring in long, onedimensional ducts. The close connection between shock waves and detonation waves is discussed. From these analyses, a dimensionless heating parameter is derived which correlates all experimental detonation velocities when such velocities are expressed in terms of the Mach number of the detonation wave. The relation between the heating parameter and the Mach number of the detonation wave is substantiated by experimental investigations. Detonation waves were experimentally produced by means of shock ignition. The velocity of detonation waves was measured by means of a timing apparatus developed especially for this study. Spark schlieren photographs indicate the existence of flow instabilities for the case of lean detonating mixtures. xiii

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 INTRODUCTION The normal combustion of homogeneous gas mixtures usually occurs at a rate which is governed by the multiple diffusion processes within and in the neighborhood of the flame front. These processes are leisurely, and the propagation rates or "flame speeds" associated with such combustion are on the order of a few feet per second. In the latter part of the nineteenth century, however, the French physicists Vielle, Berthelot, Mallard, and Le Chatelier noted, in the course of their investigations with combustible mixtures, that under certain conditions combustion waves developed which possessed the extraordinary velocities of thousands of feet per second. About 1900 Chapman (Ref. 1) and Jouguet (Ref.2) independently advanced the explanation that such phenomena could be accounted for if this "detonation wave" were treated as a shock wave followed by combustion -- the combustion, in turn, being initiated by the high temperatures accompanying the shock rather than by the diffusion processes mentioned above. With the exception of minor alterations and elaborations this theory remains unchanged up to the present. Most of the theoretical investigations of detonation which are to be found in the literature treat the thermodynamic aspects of the process, with attention being focused on state changes across the wave. The dynamic aspects of the wave are de-emphasized with the elimination of velocity terms in the conservation equations. By the exclusive use of such analysis, many concepts concerning the dynamics of detonation are rendered indistinct. One purpose of this investigation is to find the functional relationship that exists between the thermodynamic properties and the dynamic properties of a detonating mixture. Detonations for experimental observation are usually produced in a long, one-dimensional duct called a "flame tube". This tube is charged with a combustible mixture and ignited at one end. The ensuing flame propagation, being unstable, continually accelerates down the tube, and with sufficient tube length, produces a detonation. Such experimental procedure affords few controls, and the flexibility of such a system is quite limited. However, the shock tube, which is considered primarily as a tool for the aerodynamist or physicist, possesses all the desirable qualities that are needed for a detailed study

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 of detonative processes. Hence the shock tube, which has received little attention in the past as a tool for combustion study, was employed to study the phenomenon of detonation.

WVILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 THEORY OF DETONATIVE PROCESSES 1. 1 Hugoniot Relations Analyses of flows involving deflagration and detonation fronts have been made in the past by such men as Chapman (1899) (Ref. 1), Jouguet (1905) (Ref. 2), and Becker (1922) (Ref. 3). These analyses, based on the laws of conservation of mass, momentum, and energy, predicted detonation velocities in good agreement with the experimental results obtained from detonation in flame tubes. Their results are briefly reviewed here. Consider as in Figure 1 a standing detonation or deflagration Gas Flow Wave Front FIG. 1 FLAME FRONT IN A CONSTANT-AREA DUCT wave where subscript (1) denotes conditions before the front (unburned gases) and subscript (2) conditions after the front (burned gases), then conservation of mass p u = p u (1. 1) 1 22 2 conservation of momentum P + p uZ = P + pu (1.2) 1 21 2 where P = pressure u = velocity p = density Combining Equations 1. 1 and 1. 2 gives P -p (p u ) = (p u ) = - (1.3) (PlU1 (PZ,2 1 uz)Z~~

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 where Z) = specific volume =- P Since (plu l) or (P2u2)2 must always be positive (P2 P1) and (V2 - V)1) are always of opposite sign, two types of combustion processes are indicated: Case Ir2 -P is - When 2IJ - ) i+s + Deflagration 2 - Case II P P is + Detonation When V Z-)1 is The above shows that cases of deflagration are characterized by a pressure decrease and a specific volume increase whereas detonations are accompanied by a pressure increase and a specific volume decrease. Introducing the energy equation uz U2 1 2 h + 1 =h +- -Q (1.4) and by use of Equation (1.3) to eliminate the velocity terms, Equation (1.5) results -h + Q 2 (P -P) (V+ V) (1. 5) 1 2 2 1 2 2 1 where h = enthalpy per unit mass Q= heat added to flow per unit mass or from h=e+PL)

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-97 where e = internal energy per unit mass 1 e -e +Q= - (P +P )(V) -) (1.6) 1 2 2 1 2 1 2 Equations (1.5) and (1.6) are the Hugoniot expressions for deflagrations and detonations. For a fixed value of Q (i. e., a particular chemical reaction) and fixed values of 1/L and P (i. e., particular initial conI 1 ditions), this becomes an equation in PZ and )Z. Graphs of such an equation are called Hugoniot curves. Along such curves lie the solutions for detonation or deflagration processes. There are two main branches to these curves, one branch for detonation processes and another for deflagration processes. A region exists between these two branches where imaginary solutions of the equation are obtained indicating that states along this portion of the curve cannot exist. A sketch of a Hugoniot curve with its deflagration and detonation branches is shown in Figure 2. K PD D Detonation Branch CP M O N Deflagration Branch _ 2 FIG. 2 HUGONIOT CURVE

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 Obviously there will be a Hugoniot curve for each assumed initial condition and each heat release, Q. Point O, which does not lie on the Hugoniot curve, is the point corresponding to the initial state, P1 and q1. The curve itself is the locus of points describing all possible final states of the burned gas. From Equation (1.3) it is seen that a propagation velocityul, is determined by the values of P and Z/2 This velocity is fixed by the slope 2 of a straight line through point O to some point C on the curve of Figure 2. Extension of a line such as OC to D indicates that for detonations or deflagrations, two different end states are possible, except for the case where the lines through O are tangent to the curve; namely, lines OA or OB. The waves corresponding to these slopes are called ChapmanJouguet detonations and deflagrations respectively. The other waves are classified as follows: M to A - weak detonations A to K - strong detonations N to B - weak deflagrations B to L - strong deflagrations The actual existence of the above waves is discussed later. 1. 2 Chapman - Jouguet Conditions The tangents through point O in Figure 2 to the Hugoniot curve may be expressed as dP P- P (1.7) 2 C-J 2 1 where the subscript C - J denotes Chapman-Jouguet conditions.

WILLOWX' RUN RESEARCH CENTER -LUNIVERSITY OF MICHIGAN UMM-97 The first law of thermodynamics gives the relation TdS = de + Pd z) (1.8) where S = entropy per unit mass T = temperature on the absolute scale but 1 1 de=-(V -v )dP 1 (P +P ) dZ) (1. 9) 2 de1 2 2 2 1 2 2 from Equation (1.6) By use of Equations (1.7) and (1.9) in Equation (1.8) it follows that dS = 0 (1. 10) which also implies dP z2 |= p d-7) where C = speed of sound The above relation shows that the entropy of the burned gas remains unchanged for a small change in state if Q is constant. In other words, along curve KAM the rate of change of entropy is 0 at point A. The same conditions hold for point B on curve LBN. By placing these Chapman-Jouguet conditions on Equation (1.3) it is found that P - P dP 22 d 2z! 2 2 d C-J or u2= C (1. 11)'' ~ ~ ~ 2...

WILLOWN RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 which states that the burned gases in a Chapman-Jouguet wave are moving at a Mach number of one relative to the wave. 1. 3 Hugoniot Relations for Ideal Gases In processes where C and C, the specific heats at constant pressure and constant volume, respectively, may be assumed to remain constant, Equation (1.5) may be written as P _- P _ + Q 2 ( -P ) ( V +) 7'-1 171 22 2 1 2 2 1 where C and h = - PZ C - v which simplifies to P7~ + 1 =Pl V f+1 2P Z 2 2 ~-1 i I 1 I V-P 7 (1. 12) It is of interest to examine this equation to find what types of propagation waves satisfy constant-volume or constant-pressure conditions. If =z) Z) 1 2 1 2 P P 1 + ( 7- 1)Q] (1. 13) Now for any case where Q is positive P>P 2 1...... -, 8

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 hence the situation of a constant volume process implies a detonation which, by Equation (1.3), propagates at an infinite speed. Equation (1. 12) may likewise be written as P P ['+1 +L1 =pV F~+1 z zQ Pz ] + - [ I + P P z) 22 [- 1 2 1 -11 P P1 1 (1. 14) For cases wherein P = P 1 2 Z.)2 Z.)1'1+ 7p1)Q (1. 15) It is required then,that when Q is positive Z>z) 2 > 1)l and hence the situation of a constant pressure process is a deflagration which, from Equation (1. 3), propagates itself at a zero speed. If a very strong detonation exists, P1/P2 may be regarded as approaching zero in value, Equation (1.12) reduces to 1 +i 7)1 z2 T- 1 (1. 16) for cases where Q is a finite quantity. From the above considerations it follows that there are, theoretically, three types of detonation waves that can travel at infinite velocity. The first type, described by Equation (1.13), is a very weak or constant volume detonation where Z l = Z)2 and Q is a finite quantity. 9

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 In this instance a definite finite pressure ratio, P2/P1, is required. Also the velocity of gases remains unchanged upon passing through the detonation front. The second type, in which Q is also a finite quantity, is a very strong detonation. The pressure ratio, P2/P, required to satisfy the conservation laws is equal to infinity. The gases on passing through the front suffer an extreme velocity decrease. A third type of detonation wave that may move at an infinite speed is that of a Chapman-Jouguet type accompanied by an infinite heat addition. 10

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 II ANALYSIS OF DETONATION VELOCITIES The last section very briefly reviews the salient features of existing detonation theory. It is the purpose of this section to augment the existing theory of detonative combustion with analyses based on flow parameters rather than state changes across the detonation front. As yet, such an approach has not been exploited to the extent that a clear concept might be obtained of the relationships between the thermodynamic and kinematic properties of a detonating mixture. 2. 1 Relations for Chapman-Jouguet Detonation Velocities Derived From the Considerations of Mass Conservation, Momentum Conversation, and a State Equation It was shown (1. 11) that the gases behind a Chapman-Jouguet detonation wave move relative to the front at a speed just equal to their own local sonic velocity. Equation (1.3) may then be written as P -P Yp Z2 I V 2 In cases where the gas is ideal and the molecular weight, m, does not change across the wave, Equation (2. 1) can be modified by the equation of State P = pRT to: P — 2 T P1 7+1 i/(7'+1)., (2.2) P2 T2 ~2 If the temperature ratio across the detonation is very large, i. e., T 11

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 then P 1 ~ -'+1,0 2 The ( + 1) root is not valid for detonation, because a detonation wave requires P 11 2 Similarly, from (2. 1) and the state equation it may be shown that T V1 T2 (2.3) + - T I 2 - 2 T wherein the minus sign is the only valid root for detonative phenomena. T 1 Z)I,, j+1 If T 0; T 2V2 l Equation (2.2) (with only the minus sign) may be combined with the momentum condition of Equation (1. 2) i. e., P1 1+/ i+I=.......(2.4) P2 1 + Mt MD = Mach number of detonation wave which relates conditions across a Chapman-Jouguet detonation front,to give Mz =( 21 (2.5) D i 2T1 2 T 1- 1 2 T2

WILLOWV RUN RESEAi.RCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 Further, if / = 1, it follows that M = -2 -1 (2.6) D T2 A graph of Equation (2. 5) is shown in Figure 3 for cases where / = 1 0 and I = 1.4. The difference between the two curves is insignificant: hence, the relation expressed by Equation (2. 6) adequately describes an ideal detonation taking place in media where C C, 7I, and molecular weights remain constant. Usually, however, a detonation is accompanied by changes in m, I, C, and Cv through the front. To take these changes into account the state equation is written as R P=p T (2.7) where: R = Universal gas constant Equation (1.3) takes the form of P2 1MZ D 1 (2.8) D VZ -l after suitable rearrangement of terms. Again, the momentum relation across the detonation zone (2.4) is used to determine the value of P2/P1 and it is noted from fhe state equation that T2 m2 2 Z_ 1 m 1 Z1 13

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 100 80 60 40 20 [ I 1.4 = 1.0 T2 Ti 10 6 2 2 MD - FIG. 3 TEMPERATURE RATIO, T2 / T1, ACROSS A CHAPMAN IJOUGUET Or: - i, DETONATION VERSUS MACH NUMBER, MD 14, ACROSS A CHAPMAN

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 hence it follows that T ml 1 + [1 M2 1 = + ] m1 + )2 T 2D 1 =D i2 2' + _ l('_2 1 rniMz1+2(2.9) T m 7/I +1 Ma (1+/)2 D 2 When 2 < < 1Mz (which is the case for high-velocity detonation waves) and 2 1. 0. 2= 1 1 [1+ I1Ma] +(2.10) T1 4 m or solving for MD 4 T2 ml 1 M = -(2. 11) D T1- m2 which simplifies still further' for large values of T2/T1 to 1 m_ T2 M = 2 a/ t - _ (2.12) D m2 T Equation (2.12) shows that detonation Mach numbers should vary approximately as the square root of the temperature ratio, T2/T1, and the molecular weight ratio, ml/m2. A comparison of values computed from Equations (2. 9) (exact equation i1 = 1.4; /2 = 1.2), (2.6) assuming ideal gas with V = 1 and T2/T1 >) 1), and (2.12) (assuming MD >> 1, V1 = 1.4, and T2/T1= >) 1) is shown in Table I and Figure 4. Equation (2. 6) predicts values of MD which are too high; however, it is apparent that Equation (2.12) yields values which are entirely adequate as long as the detonation Mach number is greater than 3. Hence, the simple form, (2.12), will be used in the following derivation. 15

WVILLOWU RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 100 80___/ 60 40 Equation 2.6 Equation 2.12 20 ml T2 Equation 2.9 m2 T1 10 8 2 1 2 4 6 8 10 20 MD FIG. 4 DIMENSIONLESS PARAMETER, m 1T2 / m 2T1, VERSUS MACH NUMBER, MD, FOR A CHAPMAN-JOUGUET DETONATION 16

WILLOWV RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 TABLE I T2 m Values of —T - for various Mach numbers, for = 1.4, / = 1.2. T1 m2 1 2 MD 3 5 7 10 15 Equation (2.9) 3.64 9.18 17.5 34.8 78.9 (2.12) 3.15 8.75 17.15 35 78.7 (2. 6) 2.78 6.76 12.76 25.5 57.1 1 2. 2 Relations Derived for Chapman-Jouguet Detonation Velocities from Consideration of Energy Conservation In the previous section use was made of a state equation and the laws of conservation of mass and momentum to relate detonation velocities to temperature ratios across the detonation fronts. In this section the concept of energy conservation is introduced in order to relate the detonation velocity, MD, to the heat addition, Q, of the combustible mixture. The Hugoniot relations (Equations 1.5 and 1. 6) may be used to obtain a theoretical detonation velocity. The possible end states in a Hugoniot curve are shown in Figure 2, and the slope of this curve for any end state determines the detonation velocity, as shown by Equation (1.3). The high temperatures* that exist in the burned gases behind the front force appreciable dissociations, with consequent lowering of Q; hence, it is necessary to take account of the variation of specific heat with temperature. The computations associated with this method are exceedingly laborious. Moreover, the Hugoniot curve is arrived at by elimination of the velocity terms in the conservation laws and hence any functional dependence of the detonation velocities upon the properties of the system are at least partially obscured. Indeed, the elegance of the Hugoniot relations lies in this very fact, which permits discussion and classification of detonative and deflagrative processes * These temperatures are much higher than the adiabatic flame temperatures because of the strong convective flow that is present in detonations. 17

WrILLOW~~ RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 without ever referring explicitly to their velocities of propagation. Therefore, in the work which follows, an attempt is made to find an explicit relationship between the heat release of the mixture and the Chapman-Jouguet detonation velocity in order that some correlative function may be indicated. Consider the standing detonation wave as shown in Figure 1. The energy equation for such a case is written as before 2 2 u1 2 hI + +Q h2 + (1.4) Defining Cp as the average specific heat from some base temperature, To to some other temperature, T1, or T0 C 2 as the average specific heat from the base temperature, TOl to temperature, T., or C (T2- T /) = C dT = h -h 0 1 p1 1 and'if hea o at T=O hi d piT1 and h2 = Cp T2 18

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 Equation (1. 4 may then be written as u u2 CpT1 + +Q =C T +- (2. 13) Also defining a and b as C = aC pl= pl C = bC pZ pZ where C and C 2 are the specific heats at T1 and T2. Then 1 2 aCp T + + Q=b Cp T +2 (2.14) noting that C: / C2 = /RT p 7/-1 M M 1 (From Eq. 1.11) C 2 universal gas constant where R = gas constant = molecUlar weight molecular weight C = speed of sound M: Mach number it follows that u2 2 1 Q T2 C 1 2C T + 2 p p1 1 + 1 2C T p2 2 19

WILLOWN RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 which along with Equation (2. 12) solves to 1 + C T Mp = l (2.15) ~4 L 2b' M 2a In cases where Q ~~m1 C Q 1 p1 C> > 1 and -p_ is not very Cp T m CPi 1 2 C p2 different from 1, Equation (2. 15) further simplifies to Cp m1 Q C m C T pZ 21 (2. 1 6 4 2b 2a providedfining K may be regardivided asby the denominator of Equation (2. 17) also, anpredicts athe dimensionlessg parameter in the form of MD/K vs. (the l 20 l The above relation predicts that the Mach number of detonation predicts anormalizig paramete in the frm of MD/Kvs. ~ (th

VWILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 of dissociation). Graphs of Equation (2.15) are shown in Figure 5 and of Equation (2. 17) in Figure 6. It may be concluded that the approximation (2. 17) is satisfactory. 2.3 Heat Addition to a Moving Gas in a One-dimensional Duct The problem of detonation may be regarded as analogous to the problem of heat addition to a moving gas in a one-dimensional duct. In many instances such a point of view greatly simplifies the concepts underlying detonation and makes possible more perspicuous explanations of certain phenomena. Generally, when considering heat addition in a duct, an ideal gas is assumed. In the real case, as pointed out in Section 2.2, the high temperatures associated with detonation rule out the use of such a simplifying assumption as that of constant specific heats. The equation of state, P = pRT, is a very good approximation to the actual physical conditions as long as the molecular weight of the gas remains constant and the density is sufficiently far below the critical density. In the case of the gases presently under consideration, the latter is true for the relatively high temperatures and low pressures being dealt with. It seems then that the use of the state equation in its elementary form for the analysis of a detonation wave is adequate and advantageous in the case where a gas cycle, such as for air, approximates the process. This applies to most cases of lean fuel-air mixtures, and to a lesser extent for rich fuel-air mixtures. The conservation laws give Mass: P1 1 = P2 U2 (1.1) Momentum: P1 + pu P + p2 u2 (1. 2) 2 2 1 2 2 (subscripts as in Figure 1). 21

WVILLOW RUN RESEARCH CENTER,-UNIVERSITY OF MICHIGAN UMM-97 100 80 60 Typical Lines for Q / 40 20 Q CP1 T1 10 4 2 1+ Q MD 1 1+ Y-'m i — - - a FIG.5DIENSINLES4 2bPm, CpA 2T 22'lu C Y 1.4 = 1.15 E'E b=.75 a- = 1 2 4 6 8 10 20 MD FIG. 5 DIMENSIONLESS PARAMETER, Q /?p1jT1, VERSUS 22

WILLOOW4 RUN RESEARCH CENTER " UNIVERSITY OF MICHIGAN UMM-97 1 00 80 60 40 20 10 mMD Q m2 CpI T1 1 2 4 6 8 10 20 M FIG. 6 DIMENSIONLESS PARAMETER, ml/m2 X Q/CP1 T1, VERSUS MACH NUMBER, MD, FOR A CHAPMAN-JOUGUET DETONATION c23

WILLOW RUN RESEARCH CENTER — UNIVERSITY OF MICHIGAN UMM-97 The equation of state may be written P = pRT R being the gas constant for the gas, equal to the universal gas constant divided by the molecular weight. By the use of the above equation, Equation (1.2) may be written as P RT1 + P1 U- = P2 RT2 + P U22 (2.18) 1 1 p1u~j pRT2 2u2 or in view of Equation (1.1) P1 U2 RT + u2 P2 2 2 2 - - - = (2.19) p2 u1 RT1+ U1 Solving Equation (2.19) for T2 u2/u1 (RT1 +u2) -u2 T = (2.20) 2 R Equation (2. 20), which includes the equation of state, momentum, and mass conservation, gives T2 as a function of u2 and conditions at (1). To introduce the conservation of energy, Equation (1.4) may be rewritten: U2. U 7U u2 Q = (h2 + 2-) - (h1 + (2.21) Since the gas has been regarded as a perfect gas, h is a function of T only, and Equations (2.20) and (2.21) are sufficient to eliminate u1 and thus obtain Q as a function of u2 and the initial conditions. The assumption of a perfect gas, i.e., a gas which conforms to the equation P = pRT, does not imply that the specific heats remain constant for a change in state of the gas. A graph of Equations (2.20) and (2.21) for air is shown in Figure 7. The values of the enthalpies, h1 and h2, were taken from 24

0 T~~~~~~5OOO ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ o'oo p =6000 T =5 000,_ _ /_ - \r 100 0Z cl)~~~~~~~~~~~~~~~~~~~~~~~~~~~~r 1 T3000 8',,L " \~~~~~ y~" 48.~,~..P( c: ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~iTp 20000 500 _____ -0 D N~~~ TlT=1000 500,dJ a50 Y5000 0 _ 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 (1 V- FT. PER SEC. FIG. 7 PLOT OF HEAT ADDITION IN BTU /LB. VERSUS VELOCITY IN FT. /SEC. FOR A SERIES OF INITIAL TEMPERATURES o

rWILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 7,000 6,000..... T = 6,000 o R 5,000 Z0 Ti~~~~~~~~~~= 5,000 0R C~4 uMr z 4,000 I-rLVLCT FT~. T= 3,000 R 0 0 1,000 2,000 3,000 4,000 5,000 6, 000 FIG. 8 TEMPERATURE ALONG PATH LINES IN DEGREES R VERSUS VELOCITY IN FEET PER SECOND 26

WILLOWV RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 Reference 4. For these curves T is the absolute temperature corresponding to u1 at M = 1. SincePheat cannot be added to a flow at Mach 1 without a readjustment of the flow, Q represents the amount of heat that would be subtracted from a stream having a velocity of Mach 1 before the extraction of heat, and a velocity u2 after extraction. During the extraction of heat, the path that would be followed is shown by the dashed curve. The portions of the dashed curves to the left of the minimum (M = 1) points (A, B, C, D, E, and F) are subsonic, and those portions to the right are supersonic. Isotherms are shown by solid lines. A plot of temperatures along the path lines is shown in Figure 8. As in cases with constant specific heats, maximum temperatures occur at Mach numbers somewhat less than one. A shock, being an adiabatic process, can be represented on a plot similar to Figure 7 by a horizontal line (Q = constant) from the supersonic branch of a path curve to its subsonic branch. Such a line is shown in Figure 9. Q;' T-~O FIG. 9 HEAT ADDITION IN ONE-DIMENSIONAL DUCT Shock Q = Const. I I Max. Temp. Pt. I -M<1P aQ> IM> M=1 Ua U In Figure 9 is sketched a single path line to illustrate the properties of Figure 7. The value of (Q2 - Q1), which is the difference in Q between M = 0, T = 0, and M = o~, T = 0, represents the energy of ordered motion, (streaming energy) contained in a super27

WILLOWVC RUN RESEARCH CENTER —UNIVERSITY OF MICHIGAN UMM-97 sonic stream whose temperature is zero (C = 0). It should be noted that at the point (ua, Q1 ) T = 0, P = 0 and p / 0. This follows directly from Equation (1.1) and the fact that when M = oo, u is a definite finite value. In fact, any arbitrary value may be assigned to p. Detonative processes may be traced on these curves. In Figure 10, two path lines and an isotherm are shown. e __A S > Qz A Q \~ i k As < o M Uh N' I FIG. 10 DETONATIVE PROCESSES Assume the initial mixture is at a temperature T = T1, and let the realized heat of combustion be Qc. In addition, suppose the gas mixture to be moving at a velocity of uh. The initial state is identified by point h in Figure'0. The occurrence of a normal shock jumps the state of the mixture from point h to point q. If the temperature at point q is sufficiently high to promote chemical reaction heat is added and the process continues down path line A-M-B to some point 28

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 j where Q1 - Q0 = Q. At point j the mixture velocity is subsonic relative to the front. Such a detonation is classified as a "strong detonation". If Q just equals Q1 the end state occurs at point M where the Mach number is one. As mentioned previously this detonation is classified as a "Chapman-Jouguet" detonation. It is interesting to note (see Fig. 8) that if the wave is assumed to have finite thickness, and is allowed to follow such a process as outlined, namely a shock followed by combustion, the maximum temperature occurs not after complete combustion but somewhere just before completion. The problem, which now presents itself, is the case of a combustible mixture with a heating value in excess of Q1, but with the same imposed shock velocity, uh. Inasmuch as heat cannot be added to a moving gas stream of M = 1 without a readjustment of flow, an unsteady state exists. To offer an explanation for this case, assume that the heating value of the mixture is Q2. Once the mixture burns, the full value of Q2 will be added to the gas. Therefore, no solution below the Q2 line is possible. Since the gas is undisturbed until the passage of the detonation wave, T1 remains the same, and hence, no solution is possible except along the isotherm T1. The intersection of T1 and Q2 indicates the minimum possible detonation velocity. The wave that is initiated at point h with a velocity of uh must then accelerate up to a velocity of at least uf. uf is then the Chapman-Jouguet detonation velocity that is compatible with the value of Q = Q2. A Chapman-Jouguet detonation is described then as the supersonic wave with heat addition that propagates itself at the minimum possible velocity which is consistent with the conservation laws. "Strong-detonation" solutions exist along the extension of the T1 isotherm. A process which starts at some subsonic point such as e and moves down a path line to some point k is called a deflagration. In 29

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 this case the initial flow is subsonic and the final flow is subsonic. Such a combustion is called a "weak deflagration". If the heat addition to a subsonic gas stream is sufficient to provide a final sonic velocity (points M and N) the wave is called a "Chapman-Jouguet Deflagration". A wave whose path can be described as e-k-e, i.e., a subsonic stream with heat addition having a final supersonic state, is called a "strong deflagration". To complete the description of possible waves, one other wave is indicated, namely, a supersonic wave with a supersonic end state. This wave is described in Figure 10 as a wave with an initial state f and a final state'R. This wave is referred to in literature as a "weak detonation". An element of doubt still exists as to the proper description of such a wave. In many cases this confusion results from the fact one name, "weak detonation", is used to describe two possible processes. The first process is described as a shock from f to e and a strong deflagration from e to k to F. To pass from k to 2 involves an entropy decrease under adiabatic conditions and hence, rules out the possibility of the supersonic end state. From this argument it is usually concluded that strong deflagrations and weak detonations are not possible. A process, however, is indicated which moves directly down the supersonic branch of the path curve without any shock process. This situation is identically that of heat addition to a supersonic stream which remains supersonic. Inasmuch as Jouguet's rule classifies deflagrations as having subsonic velocities ahead of the wave and detonations as having supersonic velocities ahead of the wave, a process directly from f to i is a detonation. On the other hand detonations are usually described as shocks followed by deflagrations. Since no shocks exist this process might well be viewed as a "supersonic deflagration". Such a name, however, is selfcontradictory. It is possible to imagine a set of circumstances under which such a process could be induced. For example, a supersonic stream of photo-sensitive material might be triggered into releasing energy upon passing a light beam. It seems probable that any such process would require some sort of artificial ignition. It seems very unlikely that any normal deflagrations could occur at supersonic speeds for 30

WILLOWN RUN RESEARCH CENTER I UNIVERSITY OF MICHIGAN UMM-97 their mechanism of propagation involves relatively slow-moving diffusion processes. In summary, only two types of detonations appear to be encountered normally: a strong detonation, and a Chapman-Jouguet detonation. The strong detonation, as shown in Section 2.5, occurs when the pressure behind the wave is in excess of that required to produce a Chapman-Jouguet detonation with the same heat release. Using the concept of heat addition to a flow in a one-dimensional duct, the phenomena discussed in Section 2.3 can be re-analyzed from a different point of view. A graph of Q/Cpl T vs. MD for ChapmanJouguet detonations is shown in Figure 11. This graph is to be compared with Figures 5 and 6. The slope of the curve of Figure 11 indicates that Q/ C T varies as approximately MD instead of p1 1 D MD as previously derived in Equation (2.17). This is explained by the fact that Equation (2. 17) was derived by assuming that. Q/C T1 >> 1. For low values of Q/C pT1, Equation (2. 15J should be used. Figure 5 shows that for low values of Q/Cpl T. the 2.5 exponent gives a good approximation to the curves. 2.4 Strong Detonations The analysis of strong detonation waves (i.e., those that move at speeds in excess of the Chapman-Jouguet velocities) are analyzed in substantially the same manner as outlined above. The use of the Chapman-Jouguet conditions that the gases behind the wave are moving at a sonic velocity can no longer be employed, hence, the relations are somewhat more complicated, but still solvable. Equation (2.8) may be combined with the momentum equation: P 1I + 1 MZD 2 1 D P11 +'M2D P ~31 31....

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 Values from Figure 7 T1 =500'R Cpl =.24 B.T. U. / Lb. ~F Chapman - Jouguet Case 10 6 4 ___/ Q Cp1 T1 1 2 4 6 8 10 MD FIG. 11 HEATING PARAMETER, Q/Cpi T1, VERSUS MACH NUMBER, MD 32

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 where MD =Mach number of gases behind a strong detonation wave relative to the wave. MD = Mach number of the detonation to give T 1 ~Ma2 1+ 1~ 2 / M m( L1D1~12 MZDJ 1+/ZM2Dm~ I Mz2 1 1 D 1+ y2 2D + 2D MI D which simplifies to T m M [ + M 2 (2.23)' 2 2D (. 23) T1 m 1 MD 2D When M2D = 1, Equation(2.23)reduces to the relation for the Chapman-Jouguet case or Tm2 2(1+ /1MD)2 T2)(. 24) 1 D ( rn )7 M2 (1 + /)2 (2.24) T 11 D which is identical with Equation 2.9. Dividing Equation (2. 23) by (2. 24) shows that the temperature ratio for a strong detonation is directly related to the temperature ratio of a Chapman-Jouguet detonation moving at the same Mach number by SD Ma [ 2 a' D] (2.25) T\ 2 M D i + le z" C-J 33

WILLOWV RUN RESEARCH CENTER l UNIVERSITY OF MICHIGAN UMM-97 SD = strong detonation; C-J = Chapman-Jouguet detonation. In writing Equation (2. 25) it is noted that the assumption is made that 7/ 7.' X, m2/mi1, etc., are identical for both waves. When -' 1 Equation (2. 25) reduces to T2 SD z T TT21 i M ZD I + z' Anaysi 2 MD 2 ]2(Z. 26) 2.5 Analysis of Shock Waves Considered as Degenerate Detonations It is interesting at this point to observe that a normal shock can be regarded as the degenerate case of a strong detonation with no heat addition. Such a consideration leads to relations that connect shocks with. detonations and provides some interesting concepts.. For simplicity consider an ideal gas. In this case the normal shock relations may be written, giving ( T2 ) 1 + 7.... 2 2 is (2.27) 11S: +...- 1 M2 5 2 2S and with MS = MS Is = MS 1+ /M2 2S = 2 M - +S1 7 _ (2.28) where S denotes normal shock, and 2S and IS denote conditions behind the wave and ahead of the wave, respectively. Then from Equations (2. 24) and (2. 27) 34

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 Substituting the value of M2S from Equation (2. 28), and noting that D =MS' Equation(2.29)becomes T 2 1 + AMD 2 _______ 2 ( + [ __ (___ (2.31) S I T1 3'-1 (1 + /M~ 7' +i 7' - Mz.3o) When MD > > 1 which for / = 1.4 gives 35

WILL OW RUN RESEARCH CENTER, UNIVERSITY OF MICHIGAN UMM-97 (TC-h 1.75 S and for 7 = 1.2 T2 C -J These equations indicate that high-velocity Chapman-Jouguet detonations, in most media, will possess temperatures that are in the order of magnitude of two to three times those of normal shocks of the same Mach number traversing the same media. In a like manner, using the momentum equation for the detonation wave and the normal shock equation P 2 ZI MZs 2 S= i P1 7+1 -+1 it follows immediately that P( ~ 1+ / M _____ 2 _M MD + I (2.32) 36

WILLOWCr RUN RESEARCH CENTER — UNIVERSITY OF MICHIGAN UMM-97 and when MD >> 1 P 1/2 P 2 i. e., the pressure behind a detonation wave having a high velocity is half of that which would exist behind a normal shock of the same velocity. 37

WILLOWV RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 III SHOCK TUBE THEORY AS APPLIED TO DETONATIVE PROCESSES The previous section was devoted to the development of analytical expressions for the various kinematic and thermodynamic properties of detonative combustion. In this section the characteristic expressions for a shock tube are reviewed and then applied to cases wherein detonations instead of shocks are produced in the test chamber of the shock tube. The use of a detonation wave (rather than a stagnant reservoir gas) to establish strong shocks and strong detonations is discussed. Consider the system shown in Figure 12; namely, that of a piston in a tube separating two gases of two different states. Assume that the gas to the left of the piston is at a higher pressure than the gas to the right of the piston. Let the piston be accelerated instantaneously up to some constant velocity, ui, to the right. A shock or detonation will develop on the front face of the piston and travel to the right at supersonic speed. A rarefaction wave will be formed on the rear face and move to the left at sonic speed. This situation is schematically shown in Figure 13. The flow problem may then be broken into two problems: (1) a study of the flow to the right of the piston, and (2) a study of the flow to the left of the piston. If the piston is replaced by a diaphragm and the diaphragm is ruptured, the problem is analogous; the only difference being that the pressures and velocities are continuous across the piston. 3.1 Shock Tube Integral Relations The detonative flow [problem (1) above] is most easily solved for the case of a shock, or for the case of a detonatiorl without a trailing rarefaction (Fig. 35d) by deriving the following integral relations, or so-called "jump relations" (Ref. 5). 38

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 Reservoir Piston Test Chamber P FIG. 12 SHOCK TUBE REPRESENTATION Assume, as in Figure 13, a piston starting from rest at x = 0 and immediately accelerated to a velocity, ui. Later at time, t, the piston is at some position, xi. Conservation relations may be written for the gas contained between this piston and some moving boundary at position xt, and traveling at a velocity ut (where the subscript, t, denotes transmitted conditions). The relations are: Conservation of Mass d dt pdx = P1Ut (3.1) X. Xt Conservation of Momentum d pudx = P3 - P (3.2) xi 39

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 Initial Piston Location ui Rarefaction r Shock or Detonation I I I 5 iA I i1B 4 C I. I [ 3 2 D I I I1I I 0 x I Expanded Compressed Test IICN ~ FIG. 13 TIME-DISTANCE PLOT SHOWING THE LOCATION OF THE RAREFACTION WAVE, PISTON, AND SHOCK WAVE ui ut P I 2 1

WILLOWX RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 Conservation of Energy xt d uZ P (e + ) dQ + P3u3 - PU2 +PlUt + (33) X. Where p = density u = velocity P = pressure e = internal energy per unit mass Q = heat added to the system contained between x. <x < Xt 1 and the subscripts are as noted in Figure 14. By use of Leibnitz' rule,Equations (3.1), (3. 2), and (3.3) may be reduced to x tdx +p2ut -P3u3 = p (3.4) x. 1 t a (pu) dx + P2 t 2 U P - P1 (3.5) Xi 1 -— p e +T dx+Pgut 2 e )-P3 ui 3+T) X. dt + P3 u- P u + P1 ut + (3.6) dt 3 3 2 2 respectively. 41

WILLOW, RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 For cases where the time derivatives of Equations(3.4), (3.5), and (3.6) vanish and where ut is regarded as a shock or detonation velocity and xtthe position of the shock or detonation wave at time t, these equations may be simplified. The conditions at 2 are the same as those at 3, and u and e do not change with time. Hence, Equations (3.4), (3.5), and (3.6) become P1 ut - ui P2 ut (3.7) P2 ui (Ut - Ui) = P2 P1 (3.8) P ut - ui 1-:_~ t i ~(3.9) P2 ut respectively. It is noted that Equations(3.7) and (3.9) are identical so nothing is gained by the introduction of the energy equation in this manner to the problem of shocks. Combining Equations (3.7) and (3.8) gives P2 - P1 = P t i (3.10) which is a very useful equation connecting the pressure difference across a discontinuity with the velocity of the discontinuity and the velocity of the gas behind it. Equation (3. 10) is applicable to detonations and deflagrations as well as shocks. 3. 2 "Characteristics" Solution of Shock Tube Equations This problem of a rarefaction resulting from a piston (Fig. 13) being withdrawn at constant speed (after an initial, infinite acceleration to that speed) may be conveniently handled by noting the "charac - terics" solution of the equations governing unsteady, one-dimensional flow. This handling is facilitated since the gas to the left and behind 42

WILLOWX RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 the piston may be regarded as undergoing an expansion of a reversible nature (neglecting viscosity effects), and all wavelets which compose the rarefaction wave, propagate into the reservoir at their own sonic speed. For this case conservation of mass requires t_. +~x /(pu) = 0 (3.11) (t' and the momentum relation is P u+ pau'+o (3.12) where p, u, and P are functions of x and t. Rearranging Equations (3. 1 1) and (3. 12) u ~x + p + p /au: 0 (3.13) ax at ) x C p2 + pu au + P = 0 (3.14) Ox 5x at where C = = velocity of sound and p, u, x, P. and t are (as before) the density, particle velocity,* distance, pressure, and time, respectively. Inasmuch as p = fl (x,t) and u = f2 (x,t) the identities du dx + dt (3.15) dx * u = velocity of fluid flow with respect to the shock tube. = wave velocity with respect to the shock tube. dt 43

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 and dp: dx + Z dt (3.16) a x C at may be written. Equations (3. 13) through (3. 16) contain four partial differential terms and there are four independent equations. It is possible then to solve for any of these partial terms. Solving for ax ax 0 1 p 0 O 0 pu p du 0 dx dt dp dt 0 0 ~P~ =..........._~~ -(3.17) ax u 1 p 0 C2 0 pu p O O dx dt dx dt 0 0 A solution for ap au au solution for or -) would likewise involve the same denominator as Equation (3. 17), differing only in one column of the numerator. Now Equation (3.17) will have a discontinuity in ~ at lx u 1 p 0 C2 0 pu p 0 0 dx dt 0 (3. 18) dx dt 0 0 and no solution is possible under these conditions unless 44

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 0 1 p 0 0 0 pu P du 0 dx dt 0 (3.19) dp dt 0 0 also. Equations (3. 18) and (3.19) then lead to the "characteristics" expressions dx = (u ~ C) dt (3.20) and dp dx + p du dt = u dp dt (3.21) respectively. Equation (3. 20) relates the fact that a disturbance in the flow will e dx move at a speed, dt, equal to the particle velocity, u, plus or minus the local sonic velocity, C. Substitution of Equation (3. 20) into Equation (3.21) gives du=: C d(3.22) P For the case presented here wherein the rarefaction wave moves in a negative direction the minus sign, (-), is used in Equation (3.22) and du p C (3. 23) u P P is the particle velocity at a point where the density is p where u is the particle velocity at a point where the density is p and up is the velocity at a point where the density is p[3' This equation applies to any two points in the region from A to C in Figure 13. Introducing the expression for the speed of sound!-1 C=C P 2 (3.24) 45

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 where C = speed of sound at state a 7 = Cp/C = the ratio of specific heats Equation (3. 23) becomes,P Pe X-3 c 2' du - dp (3.25) a p a which integrates immediately to 2C up =u a + 7 w[1 - 2 (3.26) a I'1 L or substituting for the isentropic relation connecting pressure and density 7(-1 2C u u + a 1 - ()(3.27) u a =a-i and likewise from the isentropic relation connecting pressure and temperature u| =u + I) P a [ (3.28) At position A in Figure 13 u =0 a= P aD 5| C = C = stagnation speed of sound. 46

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 Therefore Equation (3. 27) may be written as u C [ )- (3. 29) where u and P now take the place of up and Pp. It is to be noted from Equation (3. 29) that when 2CR u - (3.30) that the pressure ratio, P/P5 = 0. The velocity indicated in Equation (3.30) is often called the "escape speed" or "escape velocity" and is equivalent to the velocity obtained when a gas, initially at rest, is suddenly released into a vacuum (Ref. 5). If a piston is moved at a velocity greater than this escape velocity, 2CR u > a region exists between the piston and the first following gas particles that is completely void or "cavitated". This phenomenon is essentially analogous to case of cavitation in liquids. From mass and momentum considerations across a shock or detonating wave it was previously shown that P2- P1 = P1 uut i (3.10) But in the case of a steady state behind a shock or detonating'front, where the pressures and velocities are matched across the piston u2 = u3 = u4 = ui Pz = P3 = P4 Combining Equations (3. 29) and (3. 10) and noting that subscripts R, t, and T are reservoir, transmitted, and test chamber conditions, respectively, will give 47

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 C R F 2C P Z8R1 P -P Ca [ l 1] fR 21 - P1 T P 1 7R P5g Piut 7ITUt or CT R )(P1) Mt = 11] (3.31) 2PC2 Where Mt =Mach number of transmitted shock or detonation wave. Solving for P1/P5 2 / P1 P1 P 1/ P 2_K_( (MJ (3.32) L t R For the case where shocks are generated in the test chamber of the tube, Equation (3.32) may be reduced by the normal shock relation P 2/ Ma 1/ -1 2 T St T ~P2 1Mt T + 1 7+ I(3.33) P1 7T+ 1 MSt = Mach number of transmitted shock wave 2 to St P1 P1 CR RR - 1) 1 1 R (3.34) 5 2 P 2 T'" 11 1 2 5 2 1 )..., P2 (P YT+)_ When the shock tube is used to produce Chapman-Jouguet detonations with a pressure plateau after the detonation (as in Fig. 13 and 35d), Equation (3.32) may be combined with the momentum equation 48

WILLOWRUN RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 P = 1+ / MZ (3.35) 1 2 2 in which M2 = 1 to give 2R C -1 P C R 2.1 - P 1 1 (3.36) Pg P2 i} (Z + 1) p — - 7Z is used instead of IT in the denominator of Equation (3.35) because the gases behind a detonation wave possess a very high temperature, ca. 5, 000~ F. to 8,0000 F., hence,T' /2. It is noted from Equations (3.31), (3.34), and (3.36), that if the ratio of the speeds of sound, CT/CR, is small, high Mach numbers will occur even though the pressure in the reservoir section is low. This may be taken advantage of in practical use of shock tubes. 1Rand AT also exert a strong influence on the pressure ratios, but their effect is not so immediately obvious. It is to be anticipated that the best reservoir gases will be those with high speeds of sound such as hydrogen with C = 4160 ft/sec at 600 F. or helium with C = 3280 ft/sec at 600 F., and indeed this is the case. Figures 16 through 23 inclusive illustrate the effects of CT/CR and fR on the reservoir pressure ratio necessary to produce various test chamber Mach numbers, pressure ratios, etc. Figures 16, 17, 18, and 19 are for normal shocks while Figures 20, 21, 22, and 23 are for Chapman-Jouguet detonations with a pressure plateau behind them. As might be anticipated, detonations require less of a pressure ratio, P5/P1, than do shocks to establish a given plateau pressure ratio, PZ/P1. Most of the measured Chapman-Jouguet detonation Mach numbers for various hydrocarbon-oxygen mixtures are in the order of 49

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 magnitude of MD = 6 or 7. Hydrogen-oxygen mixtures have detonation speeds of about MD = 5. To produce a pressure plateau behind the detonating front, requires a reservoir pressure ratio equal to or in excess of 100 (see Fig. 20, 21, 22, and 23). Consequently, to produce detonations that have velocities in excess of ChapmanJouguet conditions it is necessary that the reservoir pressure ratio be even larger. 3.3 Collision of a Detonation Wave with a Gaseous Interface When a detonating mixture of strong intensity is used as a reservoir gas (Fig. 15) the convective velocity behind the detonating Detonation Diaphragm Separating Test Gas from Reservoir Gas. Burned Unburned! Test Mixture FIG. 15 USE OF A DETONATION AS A RESERVOIR GAS front, as well as the high temperatures existing there, serve to decrease the reservoir pressures necessary to produce a shock or detonating wave of given intensity. For this case Equation (3. 27) may be reduced in a manner analogous to that used to obtain Equation (3.32) to 2/ P5 - P — i- 2M (3.37) where: M = Mach number of the convective gases behind the detonating front relative to a fixed point on the shock tube. It is to be noted that Equation (3.37) is equally applicable to cases wherein shocks collide with an interface across which a discontinuity in pressure or in speed of sound exists. 50

WILLOW RUN RESEARCH CENTER ~ UNIVERSITY OF MICHIGAN UMM-97 P2 P1 100 T 1.4 =1.66 MST IG.17PRESSURERATIoN CH/K 0.6 43- io 2YR ~~~~~~~~1- PI 1 10 100 1,000 10,000 P5 P1 FIG. 16 PRESSURE RATIO, P /P1, ACROSS A NORMAL SHOCK VERSUS THE RESERVOIR PRESSURE RATIO, P5 /P1 P2 P1 100 7T = TR — 1.4 MST ol 5 - 0.6 4 - 3- 10 2-1 P RRP_ P_ C It,, ~~~1 10 100 1,000 10,000 P1 FIG. 17 PRESSURE RATIO, P2/ P1, ACROSS A NORMAL SHOCK VERSUS THE RESERVOIR PRESSURE RATIO, P5 /P1 51

WILLONW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 P1 TEST SECTION PRESSURE P5 RESERVOIR PRESSURE CT TEST SECTION SPEED OF SOUND CR RESERVOIR SPEED OF SOUND A Theoretical Point for Hydrogen in Reservoir and Air in Test Section B Theoretical Point for Helium in Reservoir and Air in Test Section C Theoretical Point for Hydrogen in Both Sections or Air in Both Sections 2.951 10 - m _..... 8 v 2.5- 0 O -j = 1.4 z z _ _ _ _ _ _ _ _ c2.0.- - 0 0.2 0.4 0.6 0.8 1.0 3: 52 U Ix 1.5 CA. FIG 18PESR AIO 2/1 COS OMLSO ~~~VRU SPE OFSUDRTIC/RFO 5P 1 Z~~~~~~~~5

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 P1 TEST SECTION PRESSURE P5 RESERVOIR PRESSURE CT TEST SECTION SPEED OF SOUND CR RESERVOIR SPEED OF SOUND A Theoretical Point for Hydrogen in Reservoir and Air in Test Section B Theoretical Point for Helium in Reservoir and Air in Test Section C Theoretical Point for Hydrogen in Reservoir, Stoichiometric Hydrogen-Oxygen Mixture in Test Chamber D Theoretical Point for Helium in Reservoir, Stoichiometric Hydrogen-Oxygen Mixture in Test Chamber E Theoretical Point for Hydrogen in Both Sections or Air in Both Sections 100 80 U 0 It) a 60 _ ___ - 0 <Z~~~ R 1.4 040 0 De' 5.08 5.o00 z 4.15- 20,~"' - -=1.66 2.95 Z 2.10- D 0:r ~0 0.2 0.4 0.6 0.8 1.0 CT/CR FIG. 19 PRESSURE RATIO, P2 /P1, ACROSS A NORMAL SHOCK VERSUS SPEED OF SOUND RATIO, CT/CR FOR P5/P1 =100 53

WVILLOWX/ RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 P2 P1 100 FIG. 2 PRESSURE RATIO ACROSS A CHAPMAN -JOUGUET DETONATION, 1.15 P2/P1, VERSUS THE RESERVOIR PRESSURE RATIO, P0/P6 P- 1 0 11 PI P2 1 10 100 1,000 10,000 P5 MD0 T oo R: = 1.4 MDT2 = 1.15 5-C 4- X PI PI CT IR A P P2 YR -1 T c(2-I/ Pi I 10 100 1,000 10,000 5i4

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 P1 TEST SECTION PRESSURE P 5 RESERVOIR PRESSURE CT TEST SECTION SPEED OF SOUND CR RESERVOIR SPEED OF SOUND A Theoretical Point for Hydrogen in Reservoir Stoichiometric Hydrogen -Oxygen Mixture in Test Chamber B Theoretical Point for Helium in Reservoir Stoichiometric Hydrogen -Oxygen Mixture in Test Chamber 10 Z 0 8 z 0 LtVI a=1l~~ = -L1.4 z'A z Z o Iu [ T' O rr< Z2 L LU 0 0.2 0.4 0.6 0.8 1.0 CT/ CR FIG. 22 PRESSURE RATIO, P2/P1, ACROSS A CHAPMAN-JOUGUET DETONATION VERSUS THE SPEED OF SOUND RATIO, CT/CR, FOR P5/P1 =10 55

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-97 P1 TEST SECTION PRESSURE P5 RESERVOIR PRESSURE CT TEST SECTION SPEED OF SOUND CR RESERVOIR SPEED OF SOUND A Theoretical Point for Hydrogen in Reservoir Stoichiometric Hydrogen-Oxygen Mixture in Test Chamber B Theoretical Point for Helium in Reservoir Stoichiometric Hydrogen-Oxygen Mixture in Test Chamber 100 z 0 80 z 0 I0 4- 60 40 o 7 0 o 6 -5 O3 ~ 20,4 - z 3 0 0: 0.2 0.4 0.6 0.8 1.0 CT/CR FIG. 23 PRESSURE RATIO, P2/ P1, ACROSS A CHAPMAN -JOUGUET DETONATION VERSUS THE SPEED OF SOUND RATIO, CT/CR, FOR P5/P1 =100 56

WILLOW' RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 When such a detonating reservoir gas is used to induce shocks in the test section, Equation (3.33) may be combined with Equation (3.37) to give 22R T 2 P P1 I R CR (R l) (P1 1 P1R R 1+ 2 MC L /Z/T ( YT + I) ( P + Ad+ 1 | (3.38) This relation is almost identical with Equation (3.34), differing from it only by the convective term, R M. The positive 2 C " sign associated with this term always results in an increase in the value of the bracket; and hence, a lowering of the value of P /P as compared to a conventional non-convective reservoir. When the value of the bracket becomes unity, P2 = P5 and the shock is transmitted at full pressure strength with no reflected shocks or rarefactions. If the bracket is less than unity a shock is transmitted and a rarefaction reflected. For this case P2 < P5' When the value of the bracket is greater than unity a shock is both transmitted and reflected with P2 > P5 For the case of the bracket reducing to unity M: (3.39) P rT + 1 A graph of this relation is shown in Figure 25. Equation (3.37) becomes invalid in cases where the bracket becomes greater than 57

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 unity, because this equation was developed for simple isentropic expansions or compressions, (Eq. 3.27), and a reflected shock departs from such a concept. A reflected wave, however, is readily analyzed in the following manner: Interface 4 3 2 P Reservoir5 UR t 1UtUi Test Chamber X FIG. 24 REFLECTED AND TRANSMITTED WAVES Across transmitted wave (3.10) P2 P1 = P1 it 3.10) Across reflected wave P2 - P5 = P5 (u5 - uR) (U - u) Solving for ui in each relation, equating, and reducing, PC 2 T R L = 1 + -- Mt [-R MC MR P1 71R CT where MT Mach number of transmitted wave MR =Mach number of reflected wave (relative to reservoir gas) 58

WILLOW RUN RESEARCH CENTER "-UNIVERSITY OF MICHIGAN UMM-97 ~~~~~~~80~~~~~~~~~~~~~~~ P, CR 2 (Ps/pl) - Mc - =_ 6CP5 T-1 2'T T+I + 40 20 Reflected 5 | Rarefaction'T 1.4 1.2 P1 When Actual Condition lies 10 Shock or 8 Reflected Shock / Detonation Interface When Actual Condition MC R 5 P CT T 6 Lies Reservoir Test Chamber 4 ___Mc = Convection Mach No. Lie/4 7of Reservoir Gas CT = Speed of Sound of Test Chamber Gas CR = Speed of Sound of Reservoir Gas 0/01" P5 = Reservoir Pressure p1 = Test Chamber Pressure 0.3 0.4 0.5 0.8 1 2 4 6 8 CR CT FIG. 25 RESERVOIR PRESSURE RATIO, P5 / PI, NECESSARY TO PRODUCE A SHOCK WITH NO REFLECTED WAVES VERSUS THE DIMENSIONLESS RATIO, MC XCR /CT 59

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 MC = Mach number of convective reservoir gas relative to a fixed point in the shock tube. For a reflected shock into the reservoir 2 2/R MSR ~R 1 5 OR +1 OR +1 and P2 RT CR t P I1 + M (3.40) RI1 C T R MC - 1 P ) Pi -RR CT2 Graphs of the above relation along with the relation expressed in Equation (3.37) appear in Figures 26, 27, and 28, and cross plots appear in Figures 29 and 30. Values of CT/CR,,T and RI are chosen to be representative of detonations passing into gases with = 1.4. As an example, it can be shown that a detonation passing through a stoichiometric mixture of hydrogen and oxygen requires that MC 0.83 and P5/P1 19. 0. This indicates that a transmitted shock of MST = 4.4 can be provided by means of such a detonating mixture in the reservoir, and a gas such as air in the test chamber (see Fig. 29). If, however, the reservoir were pressurized or the test chamber evacuated prior to the operation of the tube, the value of P5/P1 can be correspondingly increased so that Mach numbers in excess of 6 or 7 are possible. In the conventional use of the shock tube with a helium reservoir gas, shock Mach numbers of 6 or 7 require pressure ratios in excess of 1,000. The use, then, of detonation waves to create 60

a P2 P3 P4 5 4 3 2 1 Reflected as Transmitted McO0.6 YR=1.2 YT-1.4 Wave Interface Wave 100 (/I) P2 C) CRM Ref lected 2 15 C T 11I \1\ 1 50 C R Reflected r) 25~~~~Refec P2 5 c 20~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1 1.0 1.2 1.4 1.6 1.8 2. 0.1 I 10 100 1000 0 P 5~~~~~~~~~~P FIG. 26 DIMENSIONLESS RATIO, CR / CT X MT VERSUS THE RESERVOIR PRESSURE C) RATIO, P5 / P1, FOR VARIOUS VALUES OF PRESSURE RATIO, P2 / P1

4-~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -k -.+ P2=P3=P 4 3 2 1 McO.8 YR=1.2'YT=1.4 Ref lected Gas Transmitted Wave Interface Wave z 100 \ C,' P1 Pr 50 10 40 0 30 25 2 CR Reflected 4 20 MT hock 15 CT Reflec ed ~ I ~ 10 Rarefaction 1 5 z P21.0 1.2 1.4 1.6 1.8 2.0 C,' 0.1 1 10 100 1000 0 P5 1 PI FIG. 27 DIMENSIONLESS RATIO, CR / CT x MT VERSUS THE RESERVOIR PRESSURE 0 RATIO, P5 / PI, FOR VARIOUS VALUES OF PRESSURE RATIO, P2 / P1

P2P3P4 51 41 3 2 Mc = 1 R =1.2 YT=1.4 Reflected Gas Transmitted j Wave Interface Wave Z 100 P2i 500 10 ~I I _I40 C) 30 CR. 20 25 -MT 15,,!1 P1 CI 0.1 1 10 100 1000 0 P5 FIG. 28 DIMENSIONLESS RATIO, C R / CT x MT VERSUS THE RESERVOIR PRESSURE C) RATIO, P5 / P1, FOR VARIOUS VALUES OF PRESSURE RATIO, P2 / P1

WILLOWN RUN RESEARCH CENTER -- UNIVERSITY OF MICHIGAN UMM-97 100 60 30 f U,,.- 1. I 1 10 3 0 O 6 V =1.4 CR/CT=5 1 3 6 10 30 MST SHOCK 64

WILLONW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 100 60'I 30 i LO 0103 1 3 6 10 30 MDT DETONATION FIG. 30 RESERVOIR PRESSURE RATIO, P5 /P1, NECESSARY TO PRODUCE A CHAPMAN-JOUGUET DETONATION VERSUS MACH NUMBER, MDT 65 U,

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 convective reservoir gases with desirable properties seems to offer many advantages in producing very strong transmitted shocks. The same general comments as above hold in the case of detonations produced in a shock tube (see Fig. 30). As before, it is noticed that the reservoir pressure ratio, P5/P1, required for any particular Mach number is less for the case of a detonation than for a shock. The possibility immediately presents itself for the use of an intense detonation to produce a strong detonation, i. e., velocity in excess of the Chapman-Jouguet velocity. 3.4'Detonations in Shock Tubes Which are Induced by Reservoir Gases of Insufficient Pressures to Create a Pressure Plateau Behind the Detonation As shown in Section 2.3 (in the discussion of Fig. 10), the speed of a stable detonation wave cannot' fall below a minimum Chapman-Jouguet speed. For this Chapman-Jouguet condition the gases behind the front are moving at a sonic velocity relative to this front. A detonation with a trailing rarefaction would then move at a velocity just equal to that of the rarefaction wave. This situation is illustrated in Figure 31. The equations for Chapman-Jouguet Rarefaction Detonation 3 I I I I I ~ loDea'~gLg~'~~~~~ —-— I Rarefaction FIG. 31 DETONATION WAVE WITH A TRAILING RAREFACTION 66

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 detonations hold for the change from state (1) to state (2). In the region of change from state (2) to state (3), Equation (3. 22) with a plus (+) sign is valid. Equations (3. 26), (3. 27), and (3. 28) are altered by sign changes only, giving _'1 U a 7_ - 1 (3.41) u P u =u + - - ---- Pa (3.42) p a, - I ( ) ( * ) Using the notation of Figure 31, Equation (3.42) may be written in terms of conditions at state (2) and state (3) as aX-?-1 - 1 If the original detonation were induced by means of a stagnant "reservoir" gas bursting a diaphragm into the test chamber, Equa- 1 + 2 3 (3.44) tion (3. 23) can be written for u3 in terms of reservoir conditions. R u3 = 1R1 L Pi I

WILLOWN RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 Figure 32 illustrates this condition. Rarefaction Detonation 5 l Interface 2 4 3 P' I I X I I I I Expanded Expanding Reservoir Gas Expanded Equatins (344) aBurned Gas Expanding Reservoir Gas \ / Burned Gas ---— Y -t- - t Undisturbed / / / Undisturbed Reservoir Gas / Combustible Mixture FIG. 32 CHARACTERISTIC DIAGRAM FOR A DETONATION PRODUCED IN A SHOCK TUBE WHERE THE RESERVOIR PRESSURE WAS INSUFFICIENT TO MAKE P3=P2 Equations (3.44) and (3.45) are sufficient to solve for the central plateau states of (3) and (4), but they do not appear to be readily reducible to a simple relation unless OR = /2. In this case 68

WILL, OW' RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 21 C 1-1 R _ -1 I + 2- 3 __ Z MC C2 2 R 8 I M 1 (3.46) P1 1 1 2_ CR _ V l where MC = Mach number of gases immediately behind the detonation relative to the shock tube. In situations where the interface velocity, u3, is equal to 0, which correspond to cases where detonations are initiated very rapidly in the neighborhood of the closed end of a flame tube, Equation (3.44) reduces to 2 z P3 P2 F 1 2 ---. I.. (3.47) Pl PI21 When the reservoir pressure, P5/P1, is less than the plateau reservoir pressure, P3/P1, obtained from Equation (3.47), a shock is transmitted into the reservoir section and Equation (3.46) is invalidated. 3.5 Wave Shapes behind Detonations The wave shapes developed in a flame or shock tube are easily found for the ideal case wherein reflections, friction, variation of specific heats, etc., are neglected. The results obtained by such simplifying assumptions, though somewhat in error, serve to specify the order of magnitude of the variables involved. In general, to find the distribution of a pressure, density, or temperature behind a wave involves the solution of Equations (3. 20) and (3.22). For any given set of conditions Equation (3. 22) is used to find a velocity, u, at some pressure, density, or temperature. Any 69

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 of the equations developed from Equation (3. 22) may be used as an alternative in the cases where they apply. It is noted that isentropy is used to connect the variables of pressure, temperature, and density. The speed of sound corresponding to these conditions may also be found. Equation (3 20) then gives the slope of the characteristic line on a time-distance plot. The pressure distribution for the case of a detonation initiated in a closed-end flame tube is illustrated in Figures 33 and 34 for typical calculated values of the properties * associated with a stoichiometric hydrogen-oxygen detonation. This illustration applies equally well to a shock-induced detonation resulting from a reservoir pressure, P5, just equal to the plateau pressure, P3, calculated for the closed-end flame tube. In this instance it is presumed that ignition occurs immediately as the diaphragm bursts, and hence the reservoir gases remain stagnant while the ensuing detonation takes place. Let us denote this critical pressure ratio wherein the gas interface remains stationary as P ic When P5 < Pi a shock is reflected into the reservoir and the plateau gases will possess a convective velocity in the direction of the reservoir. When P5 > P'i l a rarefaction wave propagates into the reservoir and the plateau gases are convective in the direction of the transmitted detonation wave. This situation is the one illustrated in Figure 32. As the reservoir pressure, P5, is increased further, a point will be reached where P3 = P2'. Past this point P3 is always equal to P and the detonation moves at a rate in excess of Chapman-Jouguet. The above cases are schematically shown in Figure 35, but may be easily calculated in the same manner as outlined to obtain Figures 33 and 34. * Temperature values were obtained from Reference 2. 70

WILLOVW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 P (Atmospheres) ur It) ac t 0o 0 0 o- C C o'- 0 sO co.... to 06 1,000 P3 2 800 Plateau I/ / / I/ ///1/ Undisturbed Gas z 600 0 u u 2 H2 +02 Detonation mIx~~~~~~~~~~~~~~ ~Detonation U 2 400 W2 = 1.15 P1 = 1 ATM M2 =1 T2 = 7,110~ 200 20 40 60 80 100 X- INCHES FIG. 33 CHARACTERISTIC DIAGRAM FOR A CHAPMAN-JOUGUET DETONATION ORIGINATING NEAR THE CLOSED END OF A FLAME TUBE 71

WILLOWV RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 20 Time = 300 Microseconds I10 X —INCHES 20 0 20 40 60 80 115 X - INCHES 20 Time = 1000 Microseconds 10 0 20 40 60 80 115 X —INCHES 20~~~~ ~72....?Z,.~~~~~7

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIG UMM-97 ii':iii.i iiiii" Res r r. PD = Pressure Behind a Chapman - Jouguet Reservoir.:::::.: Test Chamber Detonation Diaphragm Chapman - Jouguet Detonation Interface p Pi d P PIC D 35 a P | Plc + _ _A P5 < PdIC IC 4 —— "- IC Chapman - Jouguet Detonation Inte rface PD Pi < Pd P P5=PIC + 35 b ~P -+D 35 c P PSP IC s P5= PIC,,Detonation Velocity Equal to - _5 ~ Chapman- Jouguet Detonation Velocity................:S > P IC Detonation Velocity Equal to;.. ji~i 5 gE t Chapman- Jouguet Detonation Velocity ~~~P: *Interface i35 d FIGC35 TYPIAL WAVEHAPESINHEWAKEFDETONA TIO PICP 5.. Chapman - Jouguet Detonation Velocity.ii.i.iiiiiiiiiii.ii.i Interface ei Pd iii iiiiiiiiiiiiiiii_.................,> c.... IC *....4.~~~i~P P.,IC._ FIG. 35 TYPICAL WAVE SHAPES IN THE WAKE OF DETONATIONS 73

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 IV EXPERIMENTAL EQUIPMENT AND INSTRUMENTATION 4.1 Shock Tube or Detonation Tube and Charging Equipment. The shock tube in which most of the velocity runs were made is made of 1/2 inch extra-strong commercial pipe which has an internal diameter of 0.546 inches. See Figure 36 for principal dimensions and arrangement. The reservoir and test chamber sections are connected by Pressure Gage Probe No. 1 Probe No. 2 Diaphragm Reservoir Test Chamber Jo 6' 7 1/2"7 9Lo 7' 9 2/3 "2't' 1/4"-j Helium or Hydrogeh I Cubic Foot Charging Nash Hytor Vessel Vacuum Pump Manometer FIG. 36 SHOCK TUBE AND CHARGING EQUIPMENT means of a pipe union which is machined to receive diaphragms, and also to provide a constant cross-sectional shape and area. The method..... 74..........74

WILLOYW RUN RESE.ARCH CENTER- UNIVERSITY OF MICHIGAN UMM- 97 of construction is illustrated in Figure 37. When the tube is operated as a shock tube (i.e., when the detonation is initiated from a shock), a diaphragm of cellophane or photographic film is inserted in the union along with an O-ring to seal the test chamber. To burst the diaphragm, compressed helium or hydrogen is bled into the reservoir section until — [R~~~ ASpark Plug Reservoir Test Chamber Diaphragm II O - Ring FIG. 37 DIAPHRAGM ASSEMBLY rupture occurs. The burst pressure is read from a standard Bourdon tube type pressure gage. This gage was calibrated against a master gage for which the manufacturer claims two percent accuracy. The master gage was not calibrated. For runs in which the shock tube is used as a flame tube (i. e., when the detonation is induced by a flame propagating down the tube rather than by a shock), a metal diaphragm is placed in the union. Ignition of the charge is secured from a miniature spark plug located just down-stream from the diaphragm. A commercial 6, 000-volt neon transformer is the current source for the plug. Fuels and oxidants are premixed in two one-cubic-foot spheres. Either or both spheres are used as needed. To charge the spheres, they are first evacuated to a pressure of about 1" Hg absolute, filled with fuel, evacuated again, and then filled with fuel to some predetermined pressure. Oxygen or air is then added to a new predetermined pressure, the mixture ratio being determined by the ratio of partial 75

WILLOW RUN RESEARCH CENTER~ UNIVERSITY OF MICHIGAN UMM-97 pressures (see Plate I). The pressures in the sphere were measured with a 100-inch, open mercury manometer. To change mixture ratios, fuel or oxygen is added to the known mixture in the sphere and the pressure change recorded. The sphere is provided with a three-inch blowout diaphragm as a safety precaution. The entire system was checked for leaks by evacuating it and noting any pressure rise on the spherecharging manometer. This procedure of checking leaks was repeated after each run. In all cases the leakage was less than 1/2" Hg for a 15-hour period, or less than 3 x 10-5 cfm for a one-atmosphere differential. To charge the tube itself, the same procedure as outlined above is followed. A piece of rubber tape is placed over the end of the shock tube, the tube evacuated, filled, evacuated, and filled. In instances where it was not necessary to conserve the charge in the spheres, about 0. 5 cubic feet of mixture was allowed to run through the tube and the end sealed immediately after charging. A second tube almost identical to the above was used to check the effect of distance on detonation velocity, the only difference being that the test section is 21 feet long. Holes are tapped along its length at intervals for the insertion of probes. The third tube, with transparent sections, is used when the waves are photographed (see Plates II and III). The test chamber consists of several interchangeable sections which may be bolted together to form any desired length. The reservoir chamber, similarly interchangeable, is a welded section of the same cross-sectional dimensions. These dimensions are 3/8" x 1/2". Separating the reservoir from the test section is a simple device for introducing diaphragms (see Plate IV). The windows of this tube are made from standard plate glass and are of sufficiently good optical quality for most needs. No structural difficulties were encountered as long as the windows were mounted between thin rubber gaskets to prevent them from resting directly against metal. 76

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 4. 2 Timing Apparatus 4. 2. 1 Timer Velocities of detonation are found by recording the time it takes for a wave to pass between two points in the tube at a known distance apart. Inasmuch as detonation waves travel at velocities up to 12, 000 ft. /sec., a timer that records intervals as small as 75 microseconds is necessary if distances between probes are to be kept between one and two feet. This spacing was selected arbitrarily as a compromise between unwieldy long lengths of tube and the necessity for a timer of extremely short time response. Also, if probe distances of one or two feet are employed, it is possible to investigate, at least partially, the effects of distance on detonation velocities. Since no known commercial timer of these capabilities was known, one was designed and fabricated, based on the familiar RC (resistor-condenser) bleed type of circuit (See Figure 38 and Plate V). The first pulse to the timer allows a current to start charging a condenser through a resistor and the second pulse stops the charging. A floating-grid-type vacuum-tube voltmeter was also constructed to record the charge on the condenser. The details of the circuit follow. The thyratrons, V3 and V6, are the basic controlling element for the charge to the condenser, C3. The grid bias of the thyratrons is controlled by R1 land R13, which are adjusted to maximum sensitivity. The first pulse, which is received at J3, fires V6. The current flow through V6 is controlled by R15, R16' or R23 in the cathode circuit. A negative potential then exists on the cathode of the diode V5, which allows V5 to conduct to C3. The second pulse, which is received at J2' fires V3. A positive potential then exists on the plate of V4 and hence on the cathode of V5. This stops the flow 77

F Sl T1 VI I Ch2 II V9 110 V.A.C. B+ XX - Heater Circuit Rl1- Potentiometer II x~~~~~~ R R12- 67,000 2 1 Watt X B- - 110v Pri. - 400v-Ct. R13- Pot. 2 Watt | 3 X 3+ (;) t 5.3 t 2 Amps R14- 67,000 Q 1 Watt B- R15- 180 2 5 Watt 2 - Fil. Trans. 110-6.3v R16- 500 5 Watt Ch1 C2 M2 Cu1- 8 Hry. 5OMA R17- 750,000 ~2 ~B+,o ~ 1H f I oB+ Ch 8 Hry. 5 MA Rl- 25,000 ~2 (1) CJ2' R19- 25,000 Q2 R9 C; 2'] 2 n v C R5 M 1 15 MA Meter R20- 750,000 ~2 6 4C 1 7 lM 2 1 MA Meter R21- 12,000 Q. — 6 Hit L Pilot Light R22- 12,000 Q2 ___L_.3~~~ — R23- 50 9 5 Watt B+ 0-4053 \ V8 |F - Fuse R22 2drR22 b XX X X S1 - SPST Switch - 20# f 450v oIF R1 0 S 2 - SPST Switch C 2-' 8, f 450v 2 nd Prob. _J2~~~~ S3 - DPST- Spring Loaded C 3 - 8 f Pyranol z 00~~~~~~~~ R ~~~~S4- SPST Switch C4- 40#f 450v R19''P | V4 1 V5 1 I = 3 RV7 6 Ss - Band Switch Electrolytic g S 6' DPST Switch C 5 - 40/ #f 450v,O R 20 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Electrolytic 20 - B20 f R 2 1 X J 1 Open Circuit Jack C6 50a/ / f 450v X X J 2 Jack C7 -.02 f 600v R13 R18 J3 Jack7 i 4 i 13 R18 \V 6 | | J3-Jack C 8-.02 f 600v 1 2 1 st Pro S5 RI' 56,000g 22Watt W 7 C8 | R -- 23,000 2 5 Watt V-1 80 Rectifier X X t__-L v J3 R3- 17,000 ~2 2 Watt V2- 117Z3 Rectifier _ R RR-' 47 2 1/2 Watt V 3' 884 Thyratron |j B.15 |16 23 R5- 60,000 ~ V4 - 6H6 Diode, fR6' 220,000 9 V5- 6H6 Diode |7 - Pot. 20 Q 10 Watt V'6- 884 Thyratron R 14 ICs| R 8 - 50,000 Q~ V7 - 6SJ7 Pentode O R9 - 25,000 ~2 V8 - VR105 Vol. Reg. | R10- 560 2 V9 - VR150 Vol. Reg. FIG. 38 TIMING CIRCUIT

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 of charge to C3, and all the current flow through V6 is by-passed to V and V. The part of the circuit making up the floating-grid vacuumtube voltmeter is constructed around a 6SJ7 pentode V7 operating at reduced heater voltage. The grid of V7 is placed at the same voltage as C3 by closing S3, and meter M1 or M2 measures the current flow through the tube, which in turn indicates the charge on C3. The power supply, which is conventional, supplies approximately 50 ma at +300 v and 50 ma at -300 v. Voltage-regulator tubes (V8 and V ) are placed at critical points in the circuit to help stabilize operation. 4. 2. 2 Calibration Equipment The calibration of the timer was secured in the following manner: a model 200c Hewlett Audio Oscillator, calibrated against the 440-cycle standard note of the Bureau of Standards station WWV, was used to control the frequency of a pip generator. The pip generator (Figure 39) is a thyratron oscillator in which a cathode-follower-type output is used to create a series of sharp pulses of approximately 5 microseconds duration and 15 volts amplitude. The frequency and wave shape is controlled through the potentiometers P1, P2, and P3. As a check on the frequency of the pip generator and the output wave form, a cathode ray oscilloscope was connected to the pip output. The pips are fed continuously into a calibration sequence pulser (Figure 40). The sequence pulser receives these pips on the grids of two thyratrons which are adjusted to sufficient grid bias to keep them from firing. When switch, S2, which is initially closed, is opened, the grid bias voltage is quickly reduced 79

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 P2 PIP MAKER C3 V P3 Audio.R6. Oscillator Input R5 R.1 - Indicator Light R2- 500 SQ R3 - 500 ~ AVAVV |VP |VA RA4 - 170,000 Q2 _ P1 R4 |R4 R5 - 170,000.Q R6 - 100,000 Q2 HII B- B+ iHl R7- 120,000, C1 C2 C1 - 40O#f, 450v C2 40 f, 450v C 3 -.005# f SR 1 SR2 T 1 - Fil. Trans. 6.3 v T 2- Fil. Trans. 6.3v T 3 - Fil. Trans. 6.3v T2 T3 S1 - SPST Switch SR1- Sel. Rect. 7.5 ma SR2- Sel. Redt. 7.5 ma._L- P 1 - Pot. 100,000 2 2 - Pot. 250,000 2 _________________ P3 - Pot. 20,000 Q T1,V - 884 Thyratron 110 AC FIG. 39 PIP GENERATOR 80

WILLOWV RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 BXX - 6v Heater Circuit X/,.S F1 Rwitch P1 R7 B- _ T - Power Transformer - tCS L.......J F X, F2,- 6 Filter Ckeircuit C.l P N2 - Pot- 500,000 R3 - R 3,300 Q F21 B+ R7, 500,000 R8 R Rg, l00,OOOQ C4 L RP1,7R 0 B0POUTPUT R Power T 16ransformer C13, F 2- Filter Choke P Pot. 500,000 CR 1, R 2, R5, R10,-10,000.g t R U1., U2- 884 Thyratron our$vr ~R5O - 11 7Z3 Rectifier C6, PIP I R 4, R6o, 120oo00 S2 B+7, I 0N - 80 Rectifier 4 Rl11' 2,700 S2 OUTTOUTPUT C2 - 16#f FIG. 40 CALIBRATION SEQUENCE PULSER 81

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 to a state wherein the next pip will fire it. The firing of the first thyratron simultaneously sends a pip to the timer and reduces the grid bias of the second thyratron. The next succeeding pip fires the second thyratron which sends a second pulse to the timer. A cathode-follower-type output is used, inasmuch as a positive pip output is needed at the timer. 4. 2. 3 Operation of Timing Equipment Before operation of the timing equipment, it was allowed a warm-up period of about one hour. During this time the heater current for the voltmeter tube was adjusted to a standardized value, and the thyratrons were adjusted to maximum sensitivity. These settings changed from day to day, probably due to changes in ambient temperature. The thyratrons were particularly sensitive to temperature, thus necessitating periodic checks during a run. After the above stabilization period, the timer was checked with the calibration equipment to be sure everything was functioning properly. The timer was then connected with the shock tube, a run made, and data recorded. The timer was next reconnected to the calibration equipment, and the audio oscillator adjusted to give the identical reading obtained from the previous run. In this manner, the error due to drift in the timing apparatus was eliminated. The audio oscillator was checked periodically against WWV as a standard, but no noticeable change in calibration occurred. A photograph of the timing equipment as it was used during tests appears in Plate V. 4. 2.4 Probes Several different probes were experimented with before a suitably sensitive and reliable one was found. The first kind, which is very simple in construction, consists of a short pipe nipple and a Plexiglas coupling, into which 82.

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 the probe points are mounted (see Figure 41). A positive voltage Plexiglas Adjustable Probe Points Pipe Nipple Shock Tube FIG. 41 FIRST IONIZATION PROBE is applied to one probe point and the other probe point is connected to the timer. These points are sufficiently gapped to prevent firing under ambient conditions. The passage of a detonation wave with its attending increased ionization level breaks down the gap and sends a pulse to the timer. The disadvantages of this system are many. The probes are quite insensitive unless gapped very closely, and the location of the "sensing" element of the probe is removed from the actual phenomenon. Positive potentials up to 6, 000 volts were applied across the points with some increase in sensitivity, but with the added disadvantages of inadvertent sparks igniting the mixture and the shock hazard to personnel. The sensitivity of these probes is not usually sufficient to respond to shock waves. The second kind of probe was fabricated around a type 921 photosensitive tube, and was designed to respond to the luminosity of the detonation wave. Figure 42 illustrates the method of mounting for the photocell. Since the grids of the timer thyratrons operate negatively at approximately 120 volts, the plate of the photocell is grounded and the cathode connected directly to the timer. This probe functioned very well and possessed many inherent advantages, such as good sensitivity, simplicity, and durability, but the response time was poor and not reproducible. 83

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 Light-tight Metal Cartridge To Timer Plexiglas Plexiglas Window A mp — 1 Insulator Shock Tube FIG. 42 PHOTOELECTRIC PROBE From the runs made with these probes it is estimated that their time lags are in excess of 40 microseconds. A type 923 tube was also used with no better results. These probes were used only for the runs where the detonation velocity was checked as a function of distance along the shock tube. It is felt that the scatter in the data for these runs can be attributed directly to the poor time response of the photocells. The third type and most successful type of probe is shown in Figure 43. In theory this probe works in the same manner as the first type; namely, by the ionization of the detonated gases. It possesses all the advantages of the first type and seemingly none of the disadvantages. The sensitivity is excellent, being sufficient to respond to shocks with a Mach number of 1. 5. The one difference between this probe and the first one is that it works on the low potential difference mentioned above under the photoelectric 84

WILLOW/ RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-97 I E To Timer Nut Plexiglas Washer Plexiglas Pipe Nipple Guide Brass Shock Tube FIG. 43 THIRD IONIZATION PROBE probe. The convergent section of this probe serves to intensify waves and hence increase the ionization level. These probes work very well on hydrogen and lean hydrocarbon mixtures, but foul badly when used with rich hydrocarbon mixtures. With rich acetylene mixtures it was necessary to clean the probes after each run. 4. 3 Photographic Apparatus 4. 3. 1 Spark Source The photographing of detonation waves that have velocities over 10, 000 ft/sec requires exposure times considerably under one microsecond. To obtain sufficient light during such short exposure times requires an intense source of light. These conditions lead immediately to the consideration of spark photography with shadowgraph or schlieren systems. Commercial condensers, when discharged in the proper manner, produce sparks of approximately 5 to 10 microseconds duration. To obtain shorter durations special effort must be 85

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 taken to insure that all inductive fields associated with the discharge cancel. The first system used accomplished this by arranging a dozen or so commercial plastic condensers axially between two circular plates with the spark gap itself centrally located (see Figure 44). Condenser Trigger Probe - Ground Spark Gap ghtNo. 60 Drill - Electrode + 10,000 Volts (Ground) Condenser FIG. 44 FIRST SPARK GAP ASSEMBLY The condensers are charged to 10, 000 volts and the negative electrode distance, d, adjusted to the point of breakdown. To spark, a pulse in the order of 5, 000 volts is applied to the trigger probe which sparks to the electrode. The ionization thus produced breaks down the main gap which supplies the light energy for photographing. The latter system functioned very well and a few excellent detonation shadow photographs were taken. The light obtained from this gap was marginal, so the voltage was intentionally increased above the rating of the condensers, The system worked very well photographically but the condensers eventually broke down. 86

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 To eliminate the shortcomings of the above assembly a second condenser and spark gap assembly was fabricated. The condenser (Figure 45) consists of built-up sections of copper sheet with polyethylene separations which serve to hold the plates apart and provide a high dielectric material. The copper plates are alternately connected in the center by heavy rods which also serve as the terminals to hold the electrodes. The whole assembly is placed in a Plexiglas case and the case filled with transformer oil. The same spark electrodes as those shown in Figure 44 are mounted on the terminals. This condenser is estimated to have a capacitance of approximately 0. 01 mfd. and is charged up to 50 kilovolts by means of a commercial R-F high voltage power supply. The above system was used for most of the pictures taken during this investigation. 4. 3. 2 Delay Circuit To photograph detonation waves at various stations in the test chamber, some means must be available to control the time at which the spark gap is fired. Therefore, a time delay apparatus was designed and fabricated (see Figure 46). The principle of operation follows. A pulse is received from the ionization probe on the grid of the first thyratron (No. 3 jack). The firing of the first thyratron from the above pulse starts a charge on one of the condensers Copper Sheets Transformer Oil' a Polythene Sheets Plexiglass Case FIG. 45 LOW INDUCTANCE CONDENSER 87

WILLOW RUN RESEARCH CENTER VUNIVERSITY OF MICHIGAN UMM-97 S1 F B+ 110 V R1 B+ MAIN SPARK | 2 XX - 6.3 Volts S~~ ~~~~~~~~~4 R5 Nos. 1,2,3,4 - Jack * 0 BB+ -F, F 2' Filter Choke 2 S"I S 2 S3, S 4, SPST Switch | 1S3 CIO 10 b |S5 Five Position Switch C1,C2 -20f 450v. R6 R7 m" R3 R2 /C3 C4 1 f 400v C5 - P.0001 f C I 1 1R8T2 -.001 f600v X 1 I IT r x C7-.R014-f 600v x \44L x+ l C8 -.lOOf 600v RiClo- 20, f 450v __O V- 000 QVoltm eter 2,' PL - Pilot Light C?" T1T 4 L - Auto. Ignition Coil B+ R C 7C R I 23,000.2 S''9R2, R R 7, R9, 25,000 RY, RY2 6X4 Rect00,000 ifier R 4FIG.- 3,900 TIME DELAY CIRCUIT R8' 68,000 88 R10- 10,000 S2 BP1,P2-100,000 S2Potentiometer T 1 ~ T2- 884 Thyratron 88....

WVILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 (C5, C6, C or C8) through the resistor R3. By means of the potentiometer, P2, the voltage on the appropriate condenser was previously set at some negative bias in excess of that required to hold the second thyratron from firing. When the voltage on the condenser (C5, C6, C7, or C8) drops to the critical bias voltage for the second thyratron, the tube fires, sending a pulse through the primary of an automotive-type spark coil. The pulse from the secondary of the spark coil triggers the spark of the main gap. S5 serves as a decade switch for selection of various time intervals. The voltmeter, V, indicates the bias on the second thyratron. S2 and S3 are restoring switches for the thyratrons. The time delay of this apparatus was calibrated against the timing apparatus to eliminate trial-and-error procedures of photographing the detonation waves. The entire flash unit is shown in Plate VI. 4. 3. 3 Optical System The optical equipment used to take the majority of the detonation photographs was a simple schlieren system using a pair of achromatic lenses (see Plates III and VII). The arrangement is illustrated in Figure 47. To prevent fogging of the negative Photographic PhotaFilm pc/ Knife Achromatic Test Achromatic Edge Lens Section Lens Spark Source e- - f.I. 2f 2f f = Focal Length of Lens FIG. 47 DIAGRAM OF SPARK SCHLIEREN 89

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 from the luminosity of the detonation, a pin hole is inserted at point A (Figure 47). The pin hole is sufficiently large so that the light cut-off at the knife edge is not interfered with. By removing the knife edge, the system functions as a shadowgraph. Inasmuch as shadowgraph sensitivity depends upon the distance of the photographic film from the test section, a system with a short focal length lens is necessary when operating in this manner to keep the sensitivity down; otherwise, all the detail of the picture is destroyed. The schlieren lens had a diameter of 2 1/4 inches with a focal length of 18 inches. The shadowgraph lens is an Eastman Kodak Aero-Ektar f. 5 seven-inch lens. 90

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 OXYGEN CYLINDER CYLINDER' - FUEL CONTROL: SPHERE VALVE CHARGING MANOMETER OXYGEN CONTROL VACUUM CONTROL VALVE VALVE PLATE I SPHERE CHARGING APPARATUS " 91..........~91

~~~~~~~~~~~~~~~~ l ~~", e0 9 9 9 3 0~~~~~~~~~~h bd 4u a i::4 jbjai:;C) mO mar-'~~~~~~~~~r 1\/~~~~~~~~~ _ r x~~~j~~;~,,~,,; _~~~~~~~~~~ h I 0 I 0 SE~~~~~~~~~~~ TEST SECTION A5ESS.Mo 12s~svr

0 (I) TEST SECTION n COLLIMATING LENS V PROBE C) dt!1 SPARK z TO TIME DELAY CIRCUIT PLATE MI SHOCK TUBE TEST SECTION

MIXTURE CHARGING VALVE COMBUSTIBLE MIXTURE CHAMBER -T1 C) HIGH PRESSURE e HELIUM RESERVOIR MIXTURE INLET z DIAPHRAGM PLATE IZ SHOCK TUBE DIAPHRAGM ASSEMBLY

SEQUENCE0 OSILSOPE AUDIO OSCILLATOR TIMER THYRATRON C) U't SAWTOOTH OSCILLATOR PT:P PLATE Y TIMING APPARA'TUS C)

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 TEST SECTION PROBE SPARK asd SPARK HIGH VOLTAGE GAP'in~'~ ~~~~ i s- ASPARK CONDENSER COLLIMATING LENS TIME DELAY CIRCUIT VARIAC 40 K. V. POWER.. SUPPLY PLATE "9 FLASH UNIT 96

PINHOLE ni PHOTOGRAPHIC KIEEDGE d PLATE FOCUSING LENS C) F~~~~~~~~~~~~1tZ1~~~~~~~~~~~~~~~~~~~~ ~~ ~C: PLATE YE~l SCHLIEREN PHOTOGRAPHIC APPARATUS

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 V EXPERIMENTAL RESULTS The initial shock tube experiments with detonation were carried out with acetylene-oxygen mixtures because of the ease with which such mixtures can be detonated. A series of runs were made at different mixture ratios and initial pressures. After this series of runs had been completed, the timing equipment was checked and a small amount of drift from the original calibration was noted. This drift was traced back to the operating conditions (line voltage, heat - er setting, etc.) on the day any one series of tests was made. It was assumed that during any one set of runs the error had remained constant; hence, a control run was made at atmospheric pressure, during which each point was checked by the calibration equipment immediately after taking data. The original data were then corrected to the control run. The results of the control run are shown in Figure 50, and that of the corrected data in Figures 51 and 5Z. After this experience all tests were made in the manner described under "Operation of Timing Equipment". The results from testing a paraffin series of fuels (viz., methane, ethane, propane, n-butane, and n-hexane) with oxygen are shown in Figure 48. These curves partially normalize when the velocity is plotted against the actual per cent fuel by volume divided by the per cent fuel under stoichiometric conditions, as is done in Figure 49. A graph of the detonation velocities for diethyl ether and oxygen detonations is shown in Figure 53. It is to be noted that in the case of hexane and ether the right hand limits of the curve are not the true detonation limits but only the limits imposed by the vapor pressures of these liquids at room temperature. The detonation velocities for hydrogen-oxygen mixtures are shown in Figure 54, together with a plot of velocities obtained from the work of Dixon (Ref. 6), Payman and Walls (Ref. 7). Those for hydrogenair detonations appear in Figure 55. 98 -

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 14000 12000 10000 z 6000 I1~~~~~~~~6000 LU 4000 - Methane A Ethane U.J ~ ~ ~ ~ ~ [oI Propane x Butane v Hexane 2000 0 10 20 30 40 PER CENT FUEL BY VOLUME FIG. 48 DETONATION VELOCITY OF HYDROCARBON GASES VERSUS MIXTURE RATIO 99

WILLOWV RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 14000 12000 10000 0 - l I LU 6000 0.40.80 1.20 1.60 4000 0

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 12000 U' 8000 LU )... 0 4000 101.2 00 20 40 60 80 PER CENT FUEL BY VOLUME FIG. 50 DETONATION VELOCITY OF ACETYLENE-OXYGEN MIXTURES VERSUS MIXTURE RATIO 12000 Per Cent Acetylene by Volume _ — -.- -..0% LU JL u 4000 0 LU 0 8 16 24 32 PR.ESSURE- INCHES OF MERCURY ABSOLUTE 101,,

WILLOWN RUN RESEARCH CENTER ~UNIVERSITY OF MICHIGAN UMM-97 12000 0 8000 >- 4000 I-,V> 4000 l l —-. —--— 30 in. Hg Abs. >. I ( I 1 20 in. Hg Abs. I I I-. —-10 in. Hg Abs. 0 I. 0 20 40 60 80 PER CENT FUEL BY VOLUME FIG. 52 DETONATION VELOCITY OF ACETYLENE-OXYGEN MIXTURES AT REDUCED PRESSURES VERSUS MIXTURE RATIO 12000 z V) 8000 4000 0*~102 LU LUi z LU 0.' 0 20 40 60 80 PER CENT FUEL BY VOLUME 102...

WILLOW' RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 12000 z 8 8000 000.... From Dixon, Payman, and Walls-E uJ o 2o 40 60 80 PER CENT FUEL BY VOLUME FIG. 54 DETONATION VELOCITY OF HYDROGEN-OXYGEN MIXTURES VERSUS MIXTURE RATIO Lu LL. 8000 4000 20 320 40 60 80 PER CENT FUEL BY VOLUME FIG. 55 DETONATION VELOCITY OF HYDROGEN-AIROXYGEN MIXTURES VERSUS MIXTURE RATIO 103

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 The velocity of detonation as a function of distance along the shock tube is shown in Figure 56. These runs were made in a round tube whose internal diameter is 0. 546 inches with the photoelectric probes described in Section 4.2.4. It is felt that the scatter in the data for this run can be directly attributed to the unsatisfactory response times of these probes. However, it is obvious from these data that the velocity of such a detonation does not change appreciably as the detonation wave proceeds down the tube. Figure 57 is a graph of reservoir pressure ratio or bursting pressure ratio vs. the velocity of detonation for a 50 per cent acetylene-oxygen mixture. The reservoir gas used in this instance was hydrogen which has a value for the speed of sound ratio, CT/CR, of about 0.25. From Figure 35, which is for P5/P1 = 100, it is seen that the maximum values of pressure ratio used would be just sufficient to establish a pressure plateau behind a detonation with a Mach number slightly under 7. The Mach number of a 50 per cent acetylene-oxygen detonation is 8.7. Such detonations, then, all had trailing rarefactions. There is a slight drop in the velocity of detonation as the pressure ratio increases. This is best explained (in view of Figures 51 and 52) as merely the drop in the actual detonation velocity that is associated with a decrease in the ambient pressure. These pressure ratios were established for the most part by controlling the test chamber pressures. 5. 1 Correlation of Detonation Velocities In Figure 58 is shown a plot of the dimensionless parameter, 4 = ml/m2 x Q/C 1T vs. the measured Mach number of detonation, MD, for the eight gases tested. 4 was evaluated for lean mixtures by assuming that all the products of combustion existed in the form of water vapor, carbon dioxide, and oxygen. This assumption, which avoided much laborious computation, is not justified a priori (particularly in view of the very high temperatures present in the wake of a detonation) but the values obtained for ml /m2 are quite close. Table 2 shows that for a 10 per cent acetylene-oxygen mixture under equilibrium conditions, the error introduced by this assumption is only 5 per cent. 104

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 12000 8000. 04000 LU LU LL 24000 0 LU 4000 0 4 8 12 16 DISTANCE FROM DIAPHRAGM IN FEET FIG. 56 DETONATION VELOCITY OF A 51.50% HYDROGENE-OXYGEN MIXTURE VERSUS DISTANCE FROM DIAPHRAGM 12000 U LU 8000 LU Uz >' 4000 Iu 0 20 40 60 80 100 RATIO OF BURSTING PRESSURE TO CHAMBER PRESSURE,, ~~~~ 105 -

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 TABLE 2 COMPOSITION OF GASES IN THE BURNED PORTION OF A 10 PER CENT ACETYLENE-OXYGEN DETONATION * Composition for Equilibrium Composition Conditions for Ideal Case Mole per cent Mole per cent O 5.40 H 0.29 C 2.48 x 10-8 OH 4.57 H2 0.'116 CO 1.35 CO2 18.68 21.0 02 62.07 68.5 H20 7.52 10.5 Note: The molecular weight ratio for the equilibrium case is 0. 998 as compared to 0.95 for the idealized case. For rich mixtures reactions were assumed which seemed reasonable and then a graph was drawn which faired in with the lean values above. This procedure could very well account for the scatter of the high Mach number points. The value used for the heat release, Q, should be the change in total enthalpy, i.e., the static enthalpy plus the streaming energy across the detonation front. For simplicity in this idealized case, the heat of combustion at 200 C. was used for Q. From a practical * These values for the equilibrium composition were obtained from the as yet unpublished work of Mr. Alexander Weir of the University of Michigan Combustion Group. 106

WVILLOW% RUN RESE.ARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 60 30 MD = 2.37 3.05 6! Methane Ethane ~ 3 Propane * N-Butane A Hexane 0 Acetylene A Ether Hydrogen O 1 3 6 10 30 D FIG. 58 DIMENSIONLESS PARAMETER, =ml /m2 x Q/Cpl Tj,FOR VARIOUS FUEL-OXYGEN RATIOS VERSUS MEASURED MACH NUMBER, MD, OF DETONATION 107

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 standpoint the use of heats of combustion at 20~ C. makes the prediction of detonative Mach numbers a simple matter. In spite of all the simplifying assumptions concerning the chemistry of the process, the data in Figure 58 correlates remarkably well. Assuming that the results can be represented as a simple power function, a curve is fitted to the points; the equation for this curve is MD 2.37 3. (5.1) Equation (2. 16) on neglecting minor terms and setting a = b = 1, 7= 2 = l and C = C = C may be put in the form of 2 1 p2 p1 p MZ - 8 1 Q D (+1) m C T (5.2) p1 when Y = 1.4 mi Q M~)= 2.38 m —C p1 or MD =1.54 < 1/2 (5.3) Plots of Equations (5.1) and (5.3) are included in Figure 58. It is to be remembered that the slope of the curve in Figure 11 predicts that Q/CpT1 varies as MD. 5.2 Photographic Results Plate VIII through Plate XI are characteristic spark schlieren photographs for hydrogen-oxygen detonations. Many schlieren photographs for acetylene-oxygen mixtures were also taken, but inasmuch as these photographs resemble Plate VIII very closely they are not included here. Plate VIII is a schlieren photograph for a 50 per cent hydrogen-oxygen mixture where the knife edge is vertical and 108

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 facing the wave. The over-all increase in lighting behind the wave indicates that a strong, adverse density gradient is present after the front. Plate IX is for the same conditions as Plate VIII except for the knife edge, which is horizontal in this case. The striated appearance of the burned gases in the vicinity of the front indicates a non-homogeniety in this region. This non-homogeniety seems to penetrate into the front itself, which has a globular appearance rather than the sharp well defined appearance of shock waves. As hydrogen-oxygen mixtures are leaned past 40 per cent, the relatively clean detonation front normally encountered (Plates VIII and IX) changes into an interacting and non-homogeneous region of shocks and combustion. Spark schlieren photographs of four such cases are shown in Plates X and XI, where the knife edge is located horizontally and vertically, respectively. 109

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-97 PLATE [ SPARK SCHLIEREN PHOTOGRAPH OF A 50% HYDROGENOXYGEN DETONATION — VERTICAL KNIFE EDGE PLATE IX SPARK SCHLIEREN PHOTOGRAPH OF, A 50% HYDROGENOXYGEN DETONATION - HORIZONTAL KNIFE EDGE 110

'VWILLtVOW I:RDN I::S..AI:Z(I - C(JK.N'FTIZI:.'- UNIV tS IT' Y OF MICIJIGAN 1UMM 97 PLATE X SPARK SCHLIEREN PHOTOGRAPHS OF A 25% HYDROGENOXYGEN DETONATION-.I HORIZONTAL KNIFE EDGE [i_[ 2t PLATE N X SPARK SCHLIEREN PHOTOGRAPHS OF A 25% HYDROGENOXYGEN DETONATION -...'.VERTICAL KNIFE EDGE.....................................1.............NN...... NNNN N11....... _ _ N'N' NNNN N~~~~~~~~~~~~~~~~~~~~t& N NNNN ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~:~~~~~' N:'_::: Nt N~~~NNN N N NNN~~~tNN~~ N N~~t~N' N N ~~~ NNN ~~ 4NN~~A4NNN>NNN%~i~ ii~~~~~~~NyN NNNN NNNNNN N N I~~~~~~~~5 ~ ~ Nr r N NttNN' NNNNNNN 4 N ~ N' N Nt Nt ~ N N N\NN NN NN NN N'~tNN N~NN N ~N NNNNNN NNN tNN~~~~~~tNN NNNNNN' N N N~~~~~~~~~~r PLAU X SPARK SCHLIEREN PHOOGRAPHS Oh A 25% HYDROGEN OXYGEN DETONATION HORIZONTAL KNIF I EDGE~~~~~~~~~~~":' N'' N \ N ki'NN~NNN'NtNNS~~~~?~~'::BS~~:'::~:~ N INNN~ 4<: ~NN &N4 NN <N~~~~~~~~~~~~~~~~~~~~~~~~~ NN<NN1NQ\T N\E< ~Z ANNRI SC~LINt N HTC;AW NF NNN N5 YB C~~XYG EN I1S&7~~NATIQ6)N ~~ VEBTICAN KNNNNNNNO~NG "<N>N"" \N NNNN ~ NNNN

WILLOWN RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-97 VI DISCUSSION The investigation of detonation waves is gratifying inasmuch as such waves lend themselves very well to both experimental and theoretical study. The velocities of these detonation waves can be predicted with a surprising degree of accuracy once the fundamentals of the process are understood. The results of some such calculations (Ref. 8) for hydrogen-oxygen mixtures, as reported in the literature, are compared to observed velocities in Table 3. The equilibrium temperatures calculated by Lewis and Friauf for these mixtures (column 1) are compared to those obtained from Equation (2. 12) (column 5), where the molecular weight ratio m1/m2 is based on the assumption that the reaction goes to completion and that no dissociation is present. The value for MD (for use in Equation 2.12) is calculated from the observed detonation velocities (column 4). The agreement between columns 1 and 5 is good considering the simplifying assumptions used in deriving Equation (2.12). As may be seen from Figure 54, the values of detonation velocities observed in this investigation are lower than those measured by Dixon, Payman and Walls. The discrepancy is slight and as yet unexplained. From the theoretical analysis of detonation waves (Section II) and from Figure 58 it is quite evident that the Mach number of detonation is a more fundamental parameter controlling detonative combustion than the velocity of detonation. Rich hydrogen-oxygen mixtures, for example, have detonation velocities in excess of 10,000 ft/sec while the Mach number of such detonations is below 5. The high velocity for these mixtures is due to the high speed of sound. Table 4 illustrates this case. Since the pressure ratio across the wave is governed to a large extent by its Mach number (as shown in Equation 2.4) it is apparent that detonations for mixtures such as hydrogen and oxygen are weak (i.e., have low pressure ratios across the front) as compared to mixtures such as butane or hexane and oxygen where the observed Mach numbers are 112

0 TABLE 3 COMPARISON OF DETONATION VELOCITIES AND TEMPERATURES z OBTAINED FROM VARIOUS SOURCES 1 2 3 4 5 6 7 Mixture T2 OK Detonation u Detonation T2 OK from.u1 Detonation u Detonation Calcu- Velocity Velocity ft/sec Equation Velocity Velocity lated ft/sec Observed 3. 13 and ft/sec Cal- ft/sec Cal — (Ref. 8) Calcu- Ob- (Sec. V) Observed culated from culated from lated served Velocity Equation 5.1 Equation 5.3 (Ref. 8) (Ref. 6 (Col. 4) and 7) _ _ _ w 4H2+02 3314 11,000 10,730 10,500 3100 10,500 10,100 2H 2+02 3583 9210 9250 8800 3900 9450 9550 H 2+02 3390 7550 7590 7440 3560 7560 7430 | 1 3 H +0 2620 5680 5580 5450 2360 5100 4620 3 2- 2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i

WVILLOWiR RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 TABLE 4 COMPARISON OF DETONATIVE MACH NUMBERS WITH DETONATION VELOCITIES Per cent H2 by vol. 25 per cent 50 per cent 66.6 per cent 80 per cent in H -O2 mixture Observed 5450 7440 8800 10, 500 detonation velocity ft/sec Speed of 1215 1455 1730 2120 sound of ambient mixture Mach num- 4.49 5.11 5.08 4.95 ber of detonation in excess of nine. As shown theoretically in Section 2. 1, the temperature ratio across a detonation is also governed by the Mach number of the wave. It seems then that detonation waves may be treated in an analogous sense to shock waves, where the Mach number is an important correlative parameter. The other important parameter to consider is A. The effect of a molecular weight decrease across a detonation is to increase the value of this parameter. In detonations where the molecular weight is large in the undisturbed gas and small in burned portion of the wave, large detonative Mach numbers can result even when the heat release is relatively small. Such a consideration might explain the observation of Breton (Ref. 9), in the investigation of the rich limit of acetylene-oxygen mixtures, that an increase in detonation velocity was observed as this rich limit was approached. 114

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 For the moment, as an example, suppose that a stoichiometric (28. 6 per cent) mixture of acetylene and oxygen reacts. Assuming that dissociation is negligible and that the reaction goes to completion, etc., it follows that 5 C H2 + 2O2 =2CO2 + H 20 The above reaction is accompanied by a heat release of 5,100 Btu. per pound of mixture and a molecular weight ratio, ml/m2, of 0.857 For a 50 per cent mixture Dalton and Bone have shown that the course of reaction is represented by C2H2 + 02 = 2CO + H2 This reaction evolves about 3,380 Btu. per pound of mixture and the molecular weight ratio is 1.5. From the above calculations, the numerical values of ml/m2 x Q are 4,370 and 5,080, respectively. Thus, it may be predicted that a 50 per cent mixture would have a higher detonative Mach number than a stoichiometric mixture, even though the heat release per pound of mixture is more than half again as large for the stoichiometric case. The difference in detonation velocity (measured in feet per second) would be even greater, because the speed of sound for a 50 per cent mixture is greater than that for the stoichiometric mixture. These conclusions are quantitatively corroborated by the experimental data shown in Figure 50. The statement of Breton to the effect that disintegration processes are at work in the vicinity of the rich limit are in the correct direction; for such disintegrations, if present, would make the value of m /m2 quite large. It would also follow that the higher and more comp 1ex hydrocarbons should also reflect the same tendencies. In Figures 48 and 53 it is observed that an inflection occurs in the detonation velocity curves at richer mixtures of hexane and ether, a fact that can be tentatively explained by such an increase of ml/m2. The above uncertainties could be at least partially removed by detailed equilibrium calculations of the products in the burned portion of the wave. Such calculations should be viewed 115

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 critically, however, with respect to times necessary to establish such equilibria. Inasmuch as the heat release and molecular weight ratio are so intimately associated with the chemical reaction, it becomes of interest to try to relate the parameter? to the actual chemical equation. To illustrate such a connection, consider the detonation of a paraffin hydrocarbon, C H2n+2 with oxygen. Further, make n 2n-i-2' the assumption as before, that the heat of combustion does not vary with initial temperature. For such a detonation in stoichiometric proportions 3n +1 C H +..0 =nCO +(n+ 1) HO (6.1) n 2n+2 2 2 2 2 The heat of combustion of this reaction can be estimated on the basis of the bond energies involved in the reaction; as an approximation, 52.8 K cal of heat are emitted per tnole for each C - C and C - H bond (Ref. 10) on the average. The value of the heat release for Equation 6.1 is then Q (K cal/g mole) = 52.8 x (3n + 1) In terms of Btu per pound of fuel 5 2.8 (3n + 1) x 453.6 x 3.97 12n + in + 2 = 47600 (3n + 1) (7n + 1) or in terms of Btu per pound of mixture 47,600 (3n + 1) 12n + 2n + 2 Q ='"(7n + 1..... (7n + 1) 12n + 2n + 2 + 16(3n + 1) = 47,600 (3n+ 1) (6. 2) 3In +9 The molecular weight change across the front is 116

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-97 m1 n+n +1 2 (2n + 1) m2 1 3n+1 3 (n+l) m 1+ Using a standard temperature of 5200 R ml Q 2 (Zn + 1) 47,600 (3n + 1) m2 C lT 3 (n +1) C x 520 (31n+9) 61 (2n + 1) (3n + 1) (6.4) C (n + 1)(31n + 9) Cp = specific heat of mixture (assumed constant).* Substitution of Equation (6.4) into Equations (5.1) and (5.2) gives MDJ 9-. l P (2n + 1) (3n + 1) 3.05 MD 9.1 C (n + i) (3In + 9) (6.5) 488 (2n + 1) (3n + 1) D'C + 1 (n +1)(31n 9) Equations (6.5) and (6.6), though highly idealized (the effects of dissociation, etc., are not rigorously treated), do point to a very close connection between the chemical composition of a mixture and the Mach number of detonation. Values of detonation Mach numbers from Equations (6.5) and (6.6) are compared to measured values for several paraffin hydrocarbons in Table 5. *The specific heat, C, may likewise be related to the number and character of the bondP but such a procedure makes the equation involved and does little to clarify any concepts. 117

WVILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 TABLE 5 COMPARISON OF MACH NUMBERS OF DETONATIONS FROM THEORY WITH OBSERVED VALUES - STOICHIOMETRIC MIXTURES OF HYDROCARBONS AND OXYGEN Mixture CH4 2 2H62 C3H8 -02 C4H10- 02 C6H14 0 MD Observed 6.82 7.45 7.9 7.83 7.81 MD Equation 7.43 7.85 8.0 8.05 8.1 (6.5) MDEquation 8.84 9.6 9.9 10 10.1 (6. 6) for =1.4 It is not surprising that the values from Equation (6.6) are high, due to the very idealized assumptions this equation contains. Figure 58 illustrates this discrepancy. The values from Equation (6.5) are quite close to those observed. This trend seems to indicate that nothing remarkable will take place with the higher hydrocarbons, the detonation Mach numbers remaining fairly constant at about eight as n gets larger. From the small amount of evidence presented by spark schlieren photographs, such as Plates X and XI, it is felt that the combustion behind the shocks of these lean detonations is quite unstable, continually lagging behind the shocks and catching up to them. In some of the photographs taken, the combustion zone was observed to lag behind the shock by nearly an inch. Such phenomena could bear some relationship to the "spinning detonations" observed by Bone, Frazer, and Wheeler (Ref. 11). The mixtures in Plates X and XI, however, are hydrogen and oxygen, while those of Bone, Frazer, and Wheeler were carbon monoxide and oxygen. All of the above results point to a single conclusion: that Mach number, a parameter which has been used almost exclusively by aerodynamicists, is fundamental to the understanding of chemical phenomena involving high-speed flow of gases.. 118

WILLOW RUN RESEeARCH CENTER - UNIVERSITY OF MICHIGAN UMM-97 REFERENCES 1. Chapman, D. L., Philosophical Magazine, (5), Vol. 47, p. 90, 1899. 2. Jouguet, E., J. Math, p.347, 1905; p.6, 1906; "Mecanique de explosifs", Paris, 1907; "La Theorie Thermodynamique de la Propagation des Explosions", Proceedings of the 2nd International Congress for Applied Mechanics, Zurich, 12-17 September 1926, pp. 12 to 22. 3. Becker, R., Z.Physik, Vol. 8, p.321, 1922. 4. Keenan, J. H., and Kaye, J., "Thermodynamic Properties of Air", John Wiley and Sons, Inc., 1945. 5. Courant and Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948. 6. Dixon, H. B., Philosophical Transactions of the Royal Society of London, A, pp. 184, 97, 1893; Transactions of the Chemical Society, 1896. 7. Payman, W., and Walls, Journal of American Chemical Society, p.420, 1923. 8. Lewis, B. and Friauf, Journal of American Chemical Society, Vol. 52, p.3905, 1930. 9. Breton, M. J., Annales de l'Office National des Combustieles Liquides, Vol. 11, p.487, 1936. 10. Millard, E. B., Physical Chemistry for Colleges, McGrawHill, p. 251, 1936 (gives 52.7 for each C - H bond and 52.9 for each C-C bond). 11. Bone, W. A., Frazer, R. P. and Wheeler, W. H., Philosophical Transactions of the Royal Society of London, A, Vol. 235,p. 29, 1936. 119

WILLO' RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-97 BIBLIOGRAPHY 1. Andrew, K. K., "Sbornik Statei Teorii Vzryvchatykh Veshchestv", Moscow, 1940. 2. Andrew, K. K., Journal of Applied Chemistry, U. S. S. R., Vol. 17, 1944. 3. Audibert, E., and Louis Delmas, Annales des Mines, Vol. 20, 1931. 4. Belyaev, A. F., Comptes Rendus de l'academie des Sciences, U.S.,S.R., Vol. 18, 1938. 5. Bone, W. A., and R. P. Frazer, Philosophical Transactions of Royal Society of London, A, p.228, 1929. 6. Bone, W. A., D. M. Newitt, and D. T. A. Townsend, Proceedings of the Royal Society of London, A, p.139, 1933. 7. Bone, W. A., R. P. Frazer, and W. H. Wheeler, Philosophical Transactions of the Royal Society of London, A, p.235, 1936. 8. Breton, M. J., Annales des l'office National des Combustieles Liquides, Vol. 11, 1936. 9. Brinkley, S. R., and J. G. Kirkwood, "Third Symposium on Combustion and Flame and Explosion Phenomena", Williams and Wilkins Company, p.586, 1949. 10. Chapman, D. L., Philosophical Magazine,(5), Vol. 47, 1899. 11. Dixon, H. B., Transactions of the Chemical Society, 1896. 12. Gordon, W. E., "Third Symposium on Combustion and Flame and Explosion Phenomena", Williams and Wilkins Company, p. 78, 1949. 13. Gawthrop, D., Journal Franklin Institute, Vol. 213, 1932...........l~~ ~120

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-97 BIBLIOGRAPHY (continued) 14. Jost, W., "Explosions and Combustion Processes in Gases", McGraw-Hill Book Company, 1st Edition, New York, 1946. 15. Jouguet, E., "Mecanique des explosifs", Paris, 1907. 16. Kistiakowsky, G. B., "Third Symposium on Combustion and Flame and Explosion Phenomena", Williams and Wilkins Company, p.560, 1949. 17. Laffitte, P., Comptes Rendus de l'academie des Sciences, Vol. 177, 1923. 18. Laffitte, P., and P. Dumanois, Comptes Rendus de l'academie des Sciences, Vol. 183, 1926. 19. Lewis, B., and von Elbe, G., "Combustion, Flames, and Explosions of Gases", Cambridge University Press, 1st Edition, Cambridge, 1938. 20. Lewis, B., and Friauf, Journal of the American Chemical Society, Vol. 52, 1930. 21. Payman, W., and W. C. F. Shepherd, Proceedings of the Royal Society of London, A, Vol. 158, 1937. 22. Rideal, E., and A. Robertson, "Third Symposium on Combustion and Flame and Explosion Phenomena", Williams and Wilkins Company, 536, 1949. 23. Shepherd, W. C. F., Bulletin of U. S. Bureau of Mines, 354, 1932. 24. Shepherd, W. C. F., "Third Symposium on Combustion and Flame and Explosion Phenomena", Williams and Wilkins Company, p. 301, 1949. 25. Ubbelohde, A. R., "Third Symposium on Combustion and Flame and Explosion Phenomena", Williams and Wilkins Company, p. 566, 1949. 121

UNIVERSITY OF MICHIGAN 3 901 5 03483 20901111111 3 9015 03483 2090