THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE INFLUENCE OF A COMPRESSIBLE BOUNDARY ON THE PROPAGATION OF GASEOUS DETONATIONS Eliahou K. Dab'ora A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Mechanical Engineering 1963 December, 1963 IP- 649

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ACKNOWLEDGMENT The author wishes to express his gratitude to the following members of the faculty of The University of Michigan who served on his doctoral committee: Associate Professor William Mirsky, Co-Chairman; Professor Richard B. Morrison, Co-Chairman; Professor Thomas CO Adamson, Jr.; Professor Stuart WO Churchill; Associate Professor James A. Nicholls. In particular the author wishes to acknowledge his indebtedness to Professor JO A, Nicholls, Supervisor of the Aircraft Propulsion Laboratory. His guidance, encouragement and understanding, throughout the course of this work, helped in a great measure in bringing it to a successful completion. Thanks are also due to several members of the Aircraft Propulsion Laboratory, where the work was performed, for their kind assistance at one time or other, Particular thanks are expressed to Mro Benjamin Kerkam, Assistant in Research, who was extremely helpful in conducting the experiments and reducing the data, and to Mrs. Eunice Sherbrook, Secretary, who did a fine job of typing the first draft. For the final preparation and typing, the author is grateful to the Industry Program of The University of Michigano Last but not least, the financial assistance of the Institute of Science and Technology of The University of Michigan to conduct feasibility studies, and that of the Army Research Office (Durham) under Grant DA-ARO(D)-31-124-G345 which made it possible to finish this work, are gratefully appreciated, ii

TABLE OF CONTENTS Page ACKNOWLEDGMENT................. ii LIST OF TABLES.. o................... o oV o...... v LIST OF FIGURES..ooo o o o o o.................o o o o o o o V o i o o o NOMENCLATURE. ix I INTRODUCTION o o o o o o o o o o o o o o o o...... o o o.... o o o o o o........o o o II GENERAL CHARACTERISTICS OF GASEOUS DETONATION WAVESo o..o.... o......................O o.. 2ol One Dimensional Waves.....o.............. o......o... 5 2.2 Classification of Detonation Waveso. o............... 9 2o3 Detonation Waves Bounded by an Inert Gas... o....o.ooo 11 a. Effect of an Area Change on the Propagation of a Detonation Wave....o.........o.............. 11 b. Degree of Confinement and Explosive Deflection Angle........ o....................... 20 co Shock Angle in the Inert Boundary Gaso......... 27 do Determination of the Detonation Wave Velocity Decrement.............. o..............o 33 III CHEMICAL ASPECTS AND QUENCHING CRITERIA........ o............ 36 3.1 ShchelkinYs Instability Criterion....................... 36 3o2 Belles Explosion Limit Criterion.................... o 39 303 Patch's Constant Temperature Criterion............... 43 IV EXPERIMENTAL REQUIREMENTS AND ARRANGEMENT................... 46 401 Preliminary Experiments OO.,,....... ooooo.......oo 46 4o2 Separation of the Explosive from the Inert by a Thin Membrane or Film.............. o...... o o o 51 403 Analytical Determination of the Critical Thickness of Thin Films.................... o..... o.o 54 4.4 Preparation of Thin Films and Estimation of Their Thickne s s.................................... 62 4.5 Diffusion Through Thin Films......................... 68 4.6 Experimental Arrangement and Procedure................ 74 iii

TABLE OF CONTENTS (CONT'D) Page V EXPERIMENTAL RESULTS AND DISCUSSION......................... 87 5o1 Critical Channel Widths of H2 - 02 Mixtures When Bounded by Nitrogen.............................. 87 5.2 Velocity Decrement of Stoichiometric H2 - 02 When Bounded by Nitrogen and Inferred Reaction Length...... 91 5~3 Behavior of Detonation Waves at Channel Widths Close to the Critical,................................. 99 5.4 Velocity Decrement of H2 - 02 Mixtures Bounded by Nitrogen Near Their Respective Critical Channel Widths......o.ooo...o..o.........................ooo 108 5.5 Estimation of the Reaction Length in the Stoichiometric Methane Oxygen Mixture....................... 113 5.6 Behavior of Detonation Waves When the Density of the Boundary Gas is Much Lower Than That of the Explosive.............................................. 117 5.7 Experimental Interface Angles and Inert Shock Angles, O..............o.. o..O...............O.O 124 VI CONCLUSIONS. o....................... o........................ 128 APPENDIX A - Error Analysis...................................... 130 BIBLIOGRAPHY..................................................... 135 iv

LIST OF TABLES Table Page I Hydrogen-Oxygen Composition Limits..........e 45 II Evaluation of the Functions h1 and 02................ 58 III Color-Thickness Correlation for Thin Films......... 66 IV Summary of the Experimental Results Obtained for the Stoichiometric H2 - 02 Bounded by N2............... 98 V Summary of the Experimental Results Obtained for H2 - 02 Mixtures Bounded by N2......................... 111

LIST OF FIGURES Figure Page 1 One Dimensional Detonation Wave........................ 6 2 Hugoniot Rayleigh Representation of Shock and Detonation Waves....................................... 6 3 Idealized Flow Model of Detonation Wave - Boundary Interaction.......................................... 13 4 Typical Photograph of the Interaction Process (Reference 31)..............O........O................ 14 5 Idealized Flow Model of Detonation Wave - Boundary Interaction with Finite Reaction Length................ 16 6 Shock Tube Analogy to the Interaction Process.......... 22 7 Variation of Non-Dimensional Pressure Ratio, P, with the Density Parameter............................ 25 8 Variation of Tangent of Interface Angle with the Density Parameter..................................... 28 9 Wave Refraction at the Interface of Two Media.......... 30 10 Relationship of Refracted to Incident Wave as a Function of the Density Parameter................... 32 11 Detonation Wave Velocity Decrement as a Function of the Area Increment.....,,........................... 35 12 Dependence of the Detonation and the Critical Mach Numbers on Hydrogen-Content in H2 - 02 Mixtures........ 40 13 Critical Mach Numbers According to Belles' Explosion Limit Criterion............................... 42 14 Schematic Drawing of the Test Section Used in the Preliminary Experiments..............................,, 48 15 Photograph of the Test Section Used in the Preliminary Experiments............................... 49 16 Detonation Wave Progress of Stoichiometric H2 - 02 Mixture in Channels of Two Different Widths, Boundary Gas = N2 o....... eo............... oo... o.. 50 vi

LIST OF FIGURES (CONTvD) Figure Page 17 Detonation Wave Appearance When Confined by a Soap Film...... o...o....o e o.oo ooo. o o... o. a. o o o o o a 0 0 0 0 53 18 Analogy of the Piston Path in a Shock Tube to the Film Position..... o......... o................. o o o.o. 55 19 Instantaneous Film Velocity vso Its Acceleration....... 59 20 Film Holder............. o...ooo...................... 64 21 Dependence of Film Thickness on Collodion Concentration in Amyl Acetate Times Number of Drops............. 67 22 Concentration Distribution Due to Diffusion..o......... 71 23 Schematic Diagram of Mixing and Charging System........ 75 24 Test Section Assembly............... o.. o............ o 77 25 Photograph of the Test Section........................... 78 26 Schlieren System and Block Diagram of Instrumentation for Spark Pictures............................... 81 27 Schlieren System and Block Diagram of Instrumentation for Streak Pictures..........o.....OO..O.......... 84 28 Typical Spark Photographs of a Quenched and an Unquenched Detonation Wave............................ 88 29 Critical Channel Widths of H2 - 02 Mixtures Bounded by Nitrogen o..................o.........o............. 90 30 Streak Photographs of a Detonation Wave in Stoichiometric H2 - 02 (b = 0o5 ino,),... o......... o... 93 31 Pertinent Dimensions Helpful for the Interpretation of Streak Photographs.o..oo....o.......o..o..o..... 94 32 Streak Photographs of a Detonation Wave in Stoichiometric H2 - 02 (b = 0.2 ino.)...........o...... 95 33 Variation of Detonation Velocity Decrement with Channel Widthoo...o......................o.......oo.oo. 100 34 Streak Photograph of a Quenching Detonation Wave in Stoichiometric H2 02 with Nitrogen Boundary...... 101 vii

LIST OF FIGURES (CONTwD) Figure Page 35 Streak Photograph of a Detonation Wave in Stoichiometric H2 - 02 and of Attendant Shock Wave in the Nitrogen Boundary (Double Slit)................... 103 36 A Guide for the Interpretation of "Double Slit" Streak Photographs................................. 10 4 37 Streak Photograph of a Detonation Wave in Stoichiometric H2 - 02 with Nitrogen Boundary Showing Spin Characteristics......................... 106 38 Streak Photograph of a Detonation Wave in 46S3% H2 - j;ll with Nitrogen Boundary....................... 109 39 Streak Photograph of a Detonation Wave in 75% H2 - 25% 02 with Nitrogen Boundary.......................... 110 40 Streak Photographs of a Detonation Wave in Stoichiometric CH4 - 02 and of Attendant Shock Wave in the Boundary Gas (Double Slit)............ 114 41 Spark Photographs of a Detonation Wave in 75% H2 - 25% 02 Mixture (b = 03 in. ) -(a) Hydrogen Boundary, (b) Helium Boundary................ O.o........... 119 42 Spark Photograph of a Detonation Wave in 33% CH4 - 67% 02 with Hydrogen Boundary (b = 0o5 in. )..,......... 120 43 Spark Photograph of a Detonation Wave in 75% H2 25% 02 Mixture with Hydrogen Boundary Showing an Oblique Shock in the Explosive Resulting from a Leading Shock in the Inert (b ~ 0o2 ino)............... 122 44 Detonation Wave Progress of 75% H2 - 25% 02 When Confined by Helium or Hydrogen......................... 123 45 Experimental Results of Interface Deflection Angles..... 125 46 Experimental Results of Shock Wave Angles in the Inert Gas............................................ o 126 47 Theoretical Detonation Wave Velocity of H2 - 02 Mixtures at One Atmosphere and 535R o.................... 133 V11i

NOMENCLATURE A area A Angstrom units 108 cm a speed of sound b channel width cp specific heat at constant pressure ca,Cb concentration of gas "a" or "b" mole fraction cc volumetric concentration of collodion in amyl acetate Cn mass of nitrocellulose/unit volume of collodion Dab binary diffusion coefficient d channel depth E activation energy e naperian base F function defined by Equation (2o15) f mole fraction h specific enthalpy k constant klk29etc reaction rate constants L permiter length of drum camera if film thickness M Mach number m molecular weight m,n integers in Equation (5 4) n number of drops P nondimensional pressure ratio defined in Equation (2~37)

P permeability p pressure q specific heat release Ro universal gas constant r film radius T temperature t time after passage of wave t non-dimensional time after passage of detonation wave, Equation (4010) tp drum period of drum camera u velocity in x direction umax maximum velocity in channel v specific volume v velocity in y direction v instantaneous non-dimensional velocity, Equation (4o9) vf acceleration of film in y direction v non-dimensional acceleration, Equation (4o8) [Xi third body concentration x reaction length. Xp pitch length z defined in Equation (4032) ea angle of slit with the direction of motion of camera P,"P angles of detonation streak with direction of motion of film F(x) "Complete" gamma function r(x,y) "Incomplete" gamma function defined as o tX-le-tdt Y ratio of specific heats

deflection angle of interface coefficient defined in Equation (2o21) reduced distance reduced reaction length Q wave angle 7/v0 magnificat ion v kinematic viscosity v number of drops/unit volume vum natural sound frequency fractional area increase defined by Equation (2 18) p density T induction time 01 function defined in Equation (2035),02 function defined in Equation (2~36) ~032R03vl functions defined in Equations (3~8) and (3o10) X see Equation (4o 44) function defined in Equation (2~23) Subscripts 1 condition ahead of wave 2 condition behind wave 20 condition immediately behind wave 3 condition behind wave after expansion c critical CJ Chapman Jouguet xi

e explosive i inert ion ionization probe max maximum th theoretical streak pictures xii

I. INTRODUCTION Research on detonation waves has received quite an impetus since the late nineteenth century when Berthelot and Vieille(l) and Mallard and LeChatelier(2) published their experimental works identifying the supersonic nature of these waves, and indicating that: constitute mode of combustion distinct from the slow mode or deflagration. Very shortly thereafter, Chapman(3) and Jouguet,(4) using a hydrodynamic approach, confirmed this observation. In addition, they recognized, independently, that the burned gas behind the wave travels at the local speed of sound relative to the wave. Waves possessing this unique property are now called Chapman-Jouguet waves in their honor. The present literature on detonation waves and allied subjects covering various aspects of the phenomenon is very extensive as can be seen from the reviews of Morrison, et al.,(5) Gross and Oppenheim,(6) Wagner,(7) and Evans and Ablow.(8) Experimentally, the study of detonation waves has, up to recent years, been conducted in shock tubes or flame tubes: for example, see the work of Morrison (9) In a tube it appears that the only stable type is thIe CJ wave as shown by Lewis and vonElbe (10) and Scorah. (11) Within the last decade some theoretical studies by Siestrunck, et al. (12) and Rutkowski and Nicholls(13) indicated the possibility of attaining standing detonation waves (ioeo, fixed with respect.to laboratory coordinates). These waves could be either the CJ type or the strong type where the burned gas travels at subsonic speed relative to the wave.

-2 -The first standing detonation wave was realized in 1958, in an underexpanded jet and was reported by Nicholls, Dabora and Oealero(14) Later on, other researchers such as Gross(l5p16) and Rhodes nd Chriss(l7) reported on experiments concerned with the establishment of standing detonation waveso Another aspect of the detonation phenomenon to which considerable attention is given at present is the structure of the wave itself. The pioneering works of Kirkwood and Wood(l8,19,20) and of Hirschfelder and colleagues(21,22,23) as well as the work of Adamson(24) are cited in this regard. White(25) used interferometric techniques to study the structure of the wave experimentally. Reviews on this subject can be found in the publications of Oppenheim and Stern(26) and Fayr(27) The chemical reaction behind the detonation front has been also a subject of study by many investigators. In particular, the hydrogen-oxygen-diluent reaction has been studied in detail: for example, the works of Duff,(28) Schott and Kinsey(29) and Nicholls.(30) The last two works were primarily concerned with the ignition time delayo Recently, in an effort to obtain a better understanding of. the propagation of detonation waves in condensed (liquid or solid) explosives, Sommers(3J-) studied, as an analogue, the interaction process between a gaseous detonation wave and an inert confining gas. The analogy stems from the realization that pressures behind detonation waves in condensed explosives are so high that any confining medium would behave much like a compressible material. Sommers succeeded- in showing that different confining gases can influence the behavior of

-3 -the detonation wave differently, and thus was able to confirm qualitatively what ha,-" been observed experimentally in condensed explosives by several investigators such as Campbell et a1.1(32) Mcdard(33) and Nahmani and Manheimer. (34) Apart from its usefulness as a tool in gaining an insight into the behavior of condensed explosives, the study of the interaction process between a gaseous detonation and a side inert gas is interesting by itself. The extensive experiments on detonations waves in H2 - 02 mixtures when confined by an inert gas reported here will show how such studies can lead to estimates of reaction lengths behind detonation waves and to a check on the reaction mechanism in the same region. An intriguing application of a detonation wave is its maintenance in an annulus with the front moving in the tangential direction. Detonation can be maintained continuously by exhausting the burned gas through one side and replenishing the annulus by the detonable gas from the other side. This type of detonation traveling in a circular track was reported by Voitsckhovskii (35) It has also been studied by Morrison and Cosens(36) as a prelude to the design of a rotating detonation wave rocket engine, the feasibility of which is under extensive study at the Aircraft Propulsion Laboratory of The University of Michigan(37,38,39) The physical design of such an engine is such that the unburned gas will in general be in contact with the burned gas from the previous cycle. Therefore, one of the problems relevant to the rotating detonation engine is the determination of conditions under which the detonation wave would continue to propagate. The results reported

-4 -herein will in part answer this question. The results are also considered relevant to the prevention of detonations that can arise in rocket launching sites due to leakage of fuels and oxidizers, as well as to the design of detonation traps.

II. GENERAL CHARACTERISTICS OF GASEOUS DETONATION WAVES 2.1 One Dimensional Waves The classical hydrodynamic formulation of the characteristics of a general, steady, one-dimensional wave will be presented briefly here. It is assumed that the wave can be considered as a planar discontinuity, with conditions along the wave being uniform, as represented in Figure 1o Thus the conservation equations can be written as follows: a. Conservation of mass: Plu = P2u2 (2.1) b. Conservation of momentum: (2.2) PlUl + Pl = P2u2 + P2 c, Conservation of energy 2 2 + hl + q - u2 + h2 (23) 2 2 where q is the heat release (if any) per unit mass due to chemical reaction. To keep the analysis as general as possible, one can assume that the gases ahead and behind the wave have different molecular weights and ratios of specific heats. However, for simplicity both gases are assumed thermally and calorically perfect. Thus the equation of state is p = RoT (2.4) -5 -

Pi P2 P, P2.. hi h2 Figure 1. One Dimensional Detonation Wave. Yr21 DETONATION HUGONIOT CURVES A C.N. RAYLEIGH LINES V2 / V Figure 2. Hugoniot Rayleigh Representation of Shock and Detonation Waves.

-7 -and the enthalpy can be written as h c AY Ro T P -P mp Y P L= Y pv (2.5) 7-1 p 7-1 Now if Equations (2 1) and (2,2) are combined, one can readily obtain P2-i 2 22 P1 U= ul M_ 2P2 (2 6) V2 1 Plvl PlPi Vl1 On a pv diagram this equation represents a straight line with a negatiye slope for any finite Mach number. This is called the Rayleigh lint-;. On the other hand, if Equations (2ol), (2,2), (2o3) and (2,5) are combined, the following can be obtained, after elimination of the velocity terms: 2q 7 + 1 72 1 72 1 P2 72 2 72 2 1 ( + ____ )(_=- ) (- ) (2e7) Plvl 7Y " 1 Y2 + 1 72 + 1 Pi 72 + 1 v1 Y2 + 1 This is called the Hugoniot relationand it can be seen that on a pv diagram it would plot as a family of hyperbolas with a function of q and y8s as a parameter. Figure 2 shows a schematic plot of Equation (2,6) (two straight lines) and Equation (2j7) for the adiabatic case and for two cases with heat release, The lower branch of the Hugoniot curves in conjunction with Rayleigh lines of absolute slope less than yl represent the deflagration mode of combustion and is not relevant to our work here. The upper branch and Rayleigh lines of absolute slope greater than y7

represent shock or detonation waves. In general, a Rayleigh line intersects the Hugoniot curve at two points which represent the simultaneous solution of Equations (2.6) and (2.7). As examples, point A represents an adiabatic shock wave with p2/P1 and P2/P1 greater than unity; point B is the solution for a strong detonation wave, and B' is that for a weak wave both of which occur for a value of q less than the limiting value. Points similar to B' are usually ruled out from entropy considerations, if the detonation process is considered a shock followed by heat release. Finally point CJ is the solution for the Chapman-Jouguet wave and is realized at the limiting value of heat release and is the type of wave observed in flame tubes. As can be seen from Figure 2, at point CJ, the Rayleigh line is tangent to the 'Hugoniot curve. If Equations (2.6) and (2.7) are differentiated, we obtain respectively: ( P2) 2 2 P1 2P2 (2.8) 6( V2 PlPl a( and a(P2) P2 2 - 1 P1 Pl 72+1 (2.9) 2 Pi Y2 - 1 vl P2 72 +1 Equating (2.8) and (2.9), substituting pl from (2.2) and ul from (2.1), the solution of u2 for the CJ point is obtained. U2 = -< = a"2 (2.10) VP2

This means that for a Chapman-Jouguet wave the velocity of the burned gas immediately behind the wave travels at the local sonic speed with respect to the wave. As has been mentioned earlier, this appears to be the most stable type of detonation waves in flame tubes. The question of whether the velocity of sound, a2, should be based on frozen or shifting equilibrium is raised in the literature, and for a' summary of its implications the reader is referred to the survey article by Gross and Oppenheimi(6) It is shown there that using either'!rrozen" or "equilibrium" speed of sound leads to an insignificant difference in the prediction of the detonation velocity; therefore, the "frozen"" speed of sound is used in this worko 202 Classification of Detonation Waves The conservation equations together with the perfect gas relation and the definition of the velocity of sound can be combined to give the changes in the thermodynamic properties across the wave in terms of the original Mach number M1, q and 7'so This was done by Adamson and Morrison(40) for the simple case where no changes in molecular weight and specific heat ratio were consideredo Later, Adamson(41) generalized the solutions to cases admitting these changeso These solutions are given here without derivation: a. Pressure ratio: P2 _ (MY 72 (2.11) Pi 72 +1

-10 -b. Density ratio: P2 = 1 (2.12) P1 1 F (N2 72 1 Y2 +1 1 1 M c. Temperature ratio: T2 m2 F (M2 X T1 m l M2 (72 + 1).1 + 71F (M 72) (213) 72 + 1 Y1 From the latter, one can easily obtain the speed of sound ratio, which will be needed in a later section of this work, as: (a) 7._2T2 F F(N1-M 2 721 F a(2 m2 2 7 1 1 71 F 2 727J a1 l1T1 71 M(72 + 1) 72 + 1 1\"~~~~~~~~~ ~(2.14) The definition of F in the above equations is: F 1+ 1.M. 2 ( q Y.2.) (2.15) ~ 2 _ 72)2C (71- 1) (M1 Y2 ) pl) Y11(72 - 1) and its implication is as follows: F = 1 represents cases with limiting q, ioeo, CJ waves. F = 2 represents cases with q = 0, ioeo, adiabatic shock waves, (wvhen Y1 = 72) 2 > F > 1 represents cases of strong detonation waves.

The solution for the Mach number behind the wave was also given in (41) as 2 1 (72 + 1 - F)(M1 _ -) + 7(72 1) Y2l ~ Y71F(M (2 +1) M1 It is to be noted that for the CJ case M2 1, and therefore, Equation (2o16) yields F 1. so that the propagation Mach number of a Chapman'-Jouguet wave can be found from Equation (2.15) by the relation 1- 7q~ _ 72 2(CT1 - 71(Y72 -1) 71- 1 Thus if the initial thermodynamic conditions as well as the value of heat release and 72 are known, the detonation velocity can be found. In general, however, finding 'q and y2 requires a trial and error solution that involves chemical equilibrium behind the wave as shown, for example, by Lewis and von Elbe,(10) Moyle, et alo,(42) Gealer and Churchill,(43) Dunn and Wolfson(44), Th'. et. -,..' (45):.: "'.>'c-.?.u.'.:.rtly _.....Jo..l...i.- and Gordono(46) 2.3 Detonation Waves Bounded by an Inert Gas a. Effect of an Area Change on the Propagation of a Detonation Wave When a plane detonation wave propagates in a gaseous explosive column surrounded by an inert gas, an oblique shock is induced in the latter due to the expansion of the burned gas behind the detonation front. This type of interaction is believed to have been first treated by

-12 -Doring and Burkhart.(47) The flow pattern, as it appears to an observer moving with the detonation front, is shown in Figure 3. In this frame of reference, both the detonation and the shock waves appear stationary, and the unburned explosive as well as the surrounding inert gas travel at a velocity, ul, into the waves. Recently Sommers(31) in his effort to simulate detonation of condensed explosives, devised a means of exposing a gaseous explosive column to an inert gas so that the interaction of the detonation wave with the latter could be observed photographically. Essentially the method consists of letting a stream of explosive gas flow either into a still medium or parallel to a side stream of an inert gas. A detonation is started in the explosive and when it comes in contact with the inert, the interaction process is photographed. Figure 4 is a typical schlieren picture of such an interaction, and it is seen that except for the curvature near the initial interface of both the shock and the detonation fronts, the interaction process follows closely the sketch in Figure 3o It will be shown later that the curvature is primarily due to diffusion between the two gases so that if it is minimized by appropriate means, the curvature will be reduced considerably. In order to find analytically what effect the expansion of the burned gas and the attendant induced shock in the inert have on the propagation velocity of the detonation wave a model of the interaction process must first be formulated. To begin with, it can be assumed after Zeldovich,(48) von Neumann(49) and Doering,(50) that the detonation wave consists of a shock followed by combustion, Further, since chemical reaction cannot be instantaneous, complete heat release is not immediate

P, P. INERT ORIGINAL INTERFACE EXPLOSIVE P \\ DETONATION Pe WAVE 20 Figure 5. Idealized Flow Model of Detonation Wave - Boundary Interaction.

SHOCK FINAL INTERFACE JUM R:i:4 DETONATION WAVE Figure 4. Typical Photograph of the Interaction Process. (Reference 31)

-15 so that the Chapman-Jouguet plane is at some distance from the shock front. Such a process is illustrated in the p-v plane (Figure 2) as going from C to A which is the shock, and then back to point CJ' through various stages of heat'release. It should be pointed out that recent evidence by Hornig(51) and White(52) indicates that the ZIJD model of detonation is essentially correct. The distance between the shock front and the CJ plane, denoted by x9 will be defined as the reaction length. In the case where the detonation takes place in a. solid tube, the CJ plane should essentially have - except for a boundary layer - the same area as the shock plane. However, when the detonation is exposed to a compressible inert medium the shocked' gas, being at higher pressure than the inert, expands so that the CJ plane in this case would have a larger area than the shock front. A detailed analysis of the reaction zone affected by the side relief is extremely difficult to make because the region of interaction would include subsonic and supersonic flow regimes which are further complicated by chemical reaction. Further, the adjustment of the subsonic flow behind the shock front of the detonation wave would most likely cause some curvature in the wave itself, the oblique shock in the inert and the interface. In order to formulate, then, a model that would be amenable to analysis, the region of curvature is assumed to be very small compared to the total reaction zone - an assumption which is justified only by the experimental evidence shown later. Thus, in the idealized model, shown in Figure 5, the detonation front, the oblique shock and the interface are all assumed to be planeo

Pi P INERT / \\\ EXPLOSIVE "* "CJ PLANE SHOCK PLANE Fg- X "-' Figure 5. Idealized Flow Model of Detonation Wave - Boundary Interaction with Finite Reaction Length. (Flow is assumed quasi one-dimensional within reaction zone but two dimensional behind CJ plane.)

-17 -The method for calculating the deflection angle of the: interface is presented in the next section, However, it ia. sufuficient to point out again, that due to this deflection, eacal strzcTsnmi-,lbe, originating at the front of the detonation wave, would experience in alreea increase at the CJ plane. The problem then is to find tle ei're1t of this area increase on the propagation velocityh The ana:ysi ti:;:.-t w.i! 'follow will make use of reasoning similar to tilet of Fay(53) >ho investigated the effect of boundary layer onl the detonaUion velci.-:y:n solid tubes, If the shock front area is A1 and til CJ plane re:a is A) which is different from A1, then A /A1 1 + e. -) where ~ is the average fractional change in the area.. of each streari tube. The conservation equations thus become: (1) Conservation of mass: plul = p2U2(1 + ~) (2ol,) (2) Conservation of n-omentumn: 2~ / 'PlUl + Pi = (P.2U2 + P29(1 + ) f p di (i2PO) 0 and the conservation of energy equation remains the scame as Equat-ion (2o3)o To evaluate the integral term in Equation (2''20) require, detailed knowledje of the variation of the pressure behind the shock& front along the interfare wi-thin the reaction zone. However, one can:, define the integral as: I P d = P2cE (\2~2) 0

-18 -and note from Equation (2.11) that P2/P1 for a shock (F - 2) is about twice that of a detonation wave (F = 1) when Ml is large. Thus, the value of C is between 2 and 1, It will be shown later that it is about 115. Using the same general assumptions as in Sections 2.1 - 2.2, Equations (2o3), (2.19), (2.20), (2.21) together with the definition of the speed of sound can be combined to give: 2(1 + q ) (2)(71 ) 1 2 (71M + 1) cTl1 71 2 1[1 - 1 (1 + y2) (l +) -(Y1 i 1) M12 (2.22) Introducing a new function 4 defined by 1 + /. (2.23) (1 + 72)(1 + )_ Equation (2.22) can be reduced to the following: 272 2 2 2 _1 (M1 2 1+) 2 [ --- 71 - 72 _2. 1= 271 + 2 71 (2,24) cpl~l Tj Y1 -1 Me Me 1 1 This equation is different from Equation (2.17) by only the last term which represents the effect of area change. For large M1, (M1 ~ 5), Equation (2.24) can be reduced further to: - 1 2 ]2 _ 1 (1 + r2) (2.25) CplT1 l72 -1) 71 M 2 If the heat release is assumed to remain the same whether there is an area change or not, the right hand side of (2,25) remains essentially

constant. It should be added that physically this is a reasonable assumption in that chemical equilibrium is expected to remain the same at the CJ plane whether there is an area change or not. This is so, because the influence of an area increase would be a decrease in both pressure and temperature at that plane, and if the heat release is controlled by dissociation at this point, the pressure and temperature effects would tend to cancel each other, As a check on this assumption, calculations on the stoichiometric H2 - 02 mixture at a pressure and a temperature of 20% less than atmospheric were performed using the method of Zeleznik and Gordon,(4') and it was found that the heat release changed only by 0o2%o Not.c-hg that f = 0 when, = 0, one can find, after making use of Equation (2.25)9 that the decrease in Mach number from the case where there is no area change to the case where ~ is finite can be written as: Ml~~) -- M1 -1- 2 (2r26) 2 ~-= ~ =l~ ~ ~ (2~26) M1(=0O) Ml(V3O) +' or in terms of Z'MI- X I 1+ )]2 2 2_ ( Ml(tD) [1 - (1)( 12 + 1y2 2L 1+721+ 1+ E72(e 1+) (2.27) which for << 1 reduces to AMY _, C.....

-20 -For 72 = 1l2 which is a reasonable value for the products of combustion in H2 - 02 detonations, the change in Mach number (or velocity) becomes MLa - o654 E (2o29) Ml( S_) Fay,(53) using numerical calculations for the stoichiometric hydrogen-oxygen, obtained a coefficient of 53 instead of the above ~654. In our applications, since 5 can be as high as o25, we shall make use of relation (2~27) rather than (2~28). bo Degree of Confinement and Explosive Deflection Angle The determination of the interface deflection angle is necessary for the calculation of the area change~ In this section the method of Sommers(31) for calculating this angle is described first and then a more general method is presented in detail. For his solution Sommers assumes the idealized model of Figure 3 wherein the detonation wave is considered a discontinuity. The burned gas behind it suffers a Prandtl-Meyer expansion from unity Mach number consistent with the CJ condition. It is possible then to find the pressure of the expanded gas for various Prandtl-Meyer angles. On the other hand, from a knowledge of the flow Mach number in the inert boundary, solutions for the pressure behind the wave for any wedge angle can be found, The correct solution to both flows is obtained when both the pressure behind the shock and the wedge angle match the pressure of the expanded gas behind the detonation wave and the Prandtl-Meyer angle respectively

\I2 k.~~~~-.,~~1 W]..~...e nomeniature of F~.&..re ~, t Me eauat~..on f o4 c' r t: r:-> No w, -an be writtten as Pe20 Pe20 Pel Pil Pel Pil a2O ail ael (il Pil" l,' ae2O ael ae2 7el1 Pel PI 1 ae2O rirther, from Equations (2o11) and (2.14) respectively Pel 01 M2 Pei el

PSEUDO SHOCK TUBE I -i INERT EXPLOSIVE e1~ ~ ~ ~~0 DETONATION C I I IPRESSURE ________________~ ~ ~ c L.1____ _ _,,, _,L__ X EQUIVALENT TO ult Figure 6. Shock Tube Analogy to the Interaction Process.

-23 - and ae20 02 Mel (2o34) ael where: YelF +(1.e20F ) (2~35) 7e20 + 1 Mel Ye20 + 1 and 02 + 1 {(Ye20 + 1 - F)Fye20 + [(e20 + 1)2(1 - F) 7e20 + Mel + 2F2? e20 F e20 e01/2 + ~2 e2 — 7e + — e — (7e2O + 1 - 7e20 F)} (2~36) 7el Me-l 7el It is to be noted that for large Mel, the functions 01 and 02 are essentially constants and depend primarily on the type of wave considered through Fo. If we define: 1 Pil) Pi2) (27) Mel Pel Pil and substitute Equations (2031)9 (2o32)9 (2033)9 (2034)3 and (2~37) into (2o30), the latter can be rearranged to the following form: Ye2O01 t2[ 7je2) p+(Yil- - - i) 1/2 PelYel Yil 1/2 il+l P Mel Pil7il 2(yil + 1) Ye201 P(il 1 Yel Pel Mel (2o38)

-24 -For large Mel this equation reduces to pilil) /2 2(7il+1) 1/2 ( e20-l) PelYel Yil 027el 7e2 -1 27Ye20 (2539) Although it is hard to conceive of practical situations where Pil is different from Pel, the ratio Pil/Pel was kept in Equation (2~36) for generality. It is seen that for a reasonably large Mel and a ratio of Pil/Pel moderately different from unity, the correlating parameter for P is essentially (Pilil/Pelel) /2, since [7il/2(7il + 1)] 1/2 changes very little with 7ilo The parameter P of course is nothing but the pressure ratio across the wave in the explosive after expansion due to side relief has taken place, normalized in terms of Mel9 since Pi2 Pe3~ Equations (2~38) and (2j39) are plotted in Figure 7 for a detonation condition in the explosive and for 7el = 7il = 1o4, and Y7e20 = 1.2. Further in Equation (2o38), Pil/Pel was taken as unity and Mel = 5~ Also plotted on the same figure are the results obtained using Sommers method under the same conditions. Good agreement between the two methods is apparent. The plot shows that the higher the density parameter, 1/2 1/2 (Pil7il) [2(il+ 1) 1/2 PelYel 7il the closer the pressure ratio P to the ideal pressure ratio of ~1

1.0.00 HOCK 0.9 0.8 0 rr 0.7 w: 0.6 (f) Y) w cr 5 00, TO NATION 0 0 0 z 0.I:: 0.5 -CL~~~~~~~~~ Figure2(. Variation of+Non-Dimensional)Pressure Ratio P., with the Density Parameter. -J 0.4- ~. 03.3 la 0q[.3-) w~~~~~~~~~~~~~~~6 61 i5?F X I DNSTYPAAMTE F 0. 2 z II. 0.5 5 10 50 I00 -I/z ~J ~l ~ DENSITY PA~RAM ETER p,, y~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.,, Figure 7- Variation of Non-Dimensional Pressure Ratio, P, with the Density Parameter.

-26 -indicating that the denser the confining medium is,the better the confinement. This is in agreement with the results of Sommers,(31) who found that the controlling parameter is the acoustic impedance ratio, Pilail/pelael, for it can be shown that in the special case where Pil T Pel' (Pi7il/Pe7ell)1/2 is indeed equivalent to Pilail/Pelael. In Figure 7, the plots labeled "Shock" are obtained again through Equations (2o38) and (2o39), except that the temperature and the pressure conditions behind the wave in the explosive are taken equivalent to what might be obtained if the detonation wave were a shock at the same Mel. In this c-ase e20 is taken as 1.4. This plot is not intended to imply an actual description of the pressure field, but it is used only as a means of calculating, as shall be done next, the interface angle under widely different conditionso In order to evaluate the deflection angle, 6, the shock tube analogy is retained. If vi is the interface velocity in direction normal to the detonation velocity, then tan = vi/ai vi/ail ail (2.40) uelil Mel ael But from Reference 54 F 2 1/2 2 -_(Pi_ _ ) (2.41) Pil p'il+ Therefore, writing a il ( 7ill _ (2.42) ael 7elPel Pil

and substituting it andEquations (2o41) and (2~37) into (2~40) we obtain: 1/2 1/2 - (-) 2 -tan Pelel 1/2 il el 1/2 tan 6 J 2(7i,+l) 1 Pel 1] (2o43) 7el Pil7i 2(7il+) + (l l / 7ii+1 Pel el If Equation (2~38) is substituted in the above equation, tan 6 becomes 7e20 - 1 2an 6 272re20 (~,2Y) t 6= e2O 02 L Since P is a function of the parameter, Pel~el 1/2 Yil 1/2 Phls isol g2( 1y + 1) it is possible to find the variation of tan 6 with this parameter. This is done in Figure 8 for the same conditions as those of Figure 7. It can be seen that under the various conditions used, tan 6 is nearly independent of Mel and as to whether a shock or a detonation is considered in the explosive. As a way of assessing the range of validity of the shock tube analogy9 the results obtained using Sommers' method for Mel = 5, 7il = Yel = 14 and Y7e20 = 12, are also shown in Figure 8. Excellent agreement is obtained for tan 6 up to about 0o5o The main conclusion, then, is that the deflection angle depends primarily on the original properties of the explosive and inert gases through the density parameter. co Shock Angle in the Inert Boundary Gas Although it will be seen that the shock angle in the confining gas does not appear explicitly in the evaluation of the propagation

1.0 i z 0.5....... w -J.. M DETONATION (3 z M 5 DETONATION M -.ooSHOCK ~~~~Z~~~~~~~~~~', M 5 SHOCK 0.05.,= 1.4 05 I 05 0 I I O M=5,~ 1.4 )Ref. 31 Method wD + M = 5,,= 1.667 LL U. 0 0.05 Z 0.05 0.1 0.5 1/2 1.0 5 10 DENSITY PARAMETER VPe,. ry i - 1 Figure 8. Variation of Tangent of Interface Angle with the Density Parameter.

-29 -velocity of the detonation in the explosive, it will nevertheless be solved for next for the sake of completeness of the flow configuration, and because it is one of the prime features that can be observed experimentallyo To this end, let us assume as in Figure 9 that an incident wave travels through the medium e at a velocity Alel/sin Gel in the direction parallel to the interface between two mediao The refracted wave travels, of course, at the same velocity but the normal components of the velocities are related by. sainl sin _ (2045) sin Gil sin Gel This is the general refraction law which applies for any type wave, linear or non-linear. It can be written as: Milail sin Gil (2.46) Melael sin Gel Now if we consider a shock wave only in the inert medium i without a change in yi, we see from Equation (2.11) that Pi2 =1 + 2 (1 -(M21 1) (2.47) P111 + Yil If we substitute (2.42), (2o37) and (2 47) into ( 2.46), we find that _______ _ I Pelyl 1/2 ~ 1/2 ~illi1 111/2 sin Gil _ ilt I 7il 0 (Pelel) [ il 1 [/2 + _ 14 sin Gel Yil Yel PilYil 2(Yil+l) Yil+1 Pel Mel (2048) so that from Equation (2.43) [P + 7ill(Pil) 1 ] sin Gil _ 7il + 1 il+l el M2el tan 6 (20 49) sin Gel 2 [p(Pil) 1]

INCIDENT WAVE _____ ~"'"MEDIUM e MEDIUM REFRACTED WAVE Uel ', Sin HeS NORMAL NORMAL Figure 9. Wave Refraction at the Interface of Two Media.

-31 -For the case where the incident wave is normal to the interface (Gel = 90'), Equation (2~49) reduces to: + Yil-l(Pi) sin i + + P M2 el ele which for large Mel reduces further to sin G 7il + 1 tan 6 (2~51) This is immediately recognized as the relation between wave angle and wedge angle for high Mach numbers and small wedge angles. Equation (2~48) is plotted in Figure 10 for the case where Y = 1o 4 and that where i 1o667. In both cases it is assumed that Mel is equal to 5 and a detonation condition prevails behind the wave in medium e, Also shown in the same figure is the solution of Sommers,(l31) for sin Gel = 1, and it is seen that the agreement is very good. An immediate application of Figure 10 is the experimental work of Gvozdeva,(55) where the refraction of detonation waves was studied. For the case of detonation of (CH4 + 202) refracted in air as boundary gas, we note that the parameter (Pe7e)l/2 [ _ i ] /2 = 519 (2~52) PiYi 2(7i + 1 Accordingly from Figure 10, sin Gil/sin Gel - 0498, Since Gel = 58, one finds Gil = 25'o Gvozdevals schileren photograph shows a shock wave angle Gil of about 260, which is in excellent agreement with the theoretical value~

1.0 ~ 0.1~ M =5, = 1.667 M =5, = 1.4 0.5 s Ref. 31 Ml Cn u)~~~+ M =5, = 1.6671 0.05 0.1 0.5 /21.0 5 10 / — J/2 DENSITY PARAMETER, Yie, r'' Figure 10. Relationship of Refracted to Incident Wave as a Function of Density Parameter.

-33 do Determination of the Detonation Wave Velocity Decrement With the detonation considered' as a discontinuity, one can calculate the deflection angle of the interface as is done in Section 2o3bo In order to determine the detonation wave velocity decrement, the assumption is now made-,. for the model of Figure 5, that the same deflection angle prevails along the reaction length~ This assumption is borne out by the experimental results presented in Chapter V. If one considers a two-dimensional channel of width b, exposed to the inert on one side normal to b, it is easily seen that x tan 6 (2053) From Equation (2027), it is seen that the only unknown left is E O An estimate of the value of E can be obtained from the work of Hirschfelder and Curtisso(21) In this work, the pressure distribution in the reaction zone of a detonation with an Irreversible unimolecular reaction is calculated for the case where the shock and reaction zones are coupledo In terms of their reduced distance which is proportional to x and which can be denoted by, it is found that the pressure distribution p could be represented with good fidelity by the following equation: (P 1) (Pma 1) e-k (2~ 54) P2 P2 where P2 is the CJ pressuree If we define the reduced reaction length as that point where p = p- is within one percent of the final value, one obtains 1 ln( mx (2 55) k P- P2

-34 -Further, since for a known channel width and a known angle 6, 5 is proportional to ~, definition (2.21) can be written in the alternative form f P d = e (2.56) o P2 and therefore, after substituting (2.54) into (2.56) and integrating the resulting equation, one gets = + 1 (Pmax - e k k5 P2 F -- Pmax [ 1 P2 P2 With Pmax/P2 = 1.53 from Reference 21 and P —/P2 = 1.01, it is found that E = 1.13 (2.59) With this value of E, a plot of the detonation velocity decrement as a function of the area increment can be made according to Equation (2.27). This is shown in Figure 11 for the case where Ye20 = 1.2. It should be added that the method of obtaining e is considered simply as a good estimate. However, if the exponential character of the pressure variation is retained, it can be shown that e is not very sensitive to either Pmax/P2 or p 2. For example, if P max/ = 2, E = 1.21 and if /P = 1.10, E = 1.20 that~~~~~~~~~~~~~

0.1 6 0.1 4 0.12 0 0.10 I-. I I i Y-,1.2 E z 1.13 z L 0.08 0 0.06 0 -J 0.04 z 0 z 0.02 0~~~ 0 0.1 0.2 0.3 ARA INCREMtENT Figure 11. Detonation Wave Velocity Decrement as a Function ~ of the Area Increment.

IIIo CHEMICAL ASPECTS AND QUENCHING CRITERIA Although Equation (2~27) can predict the velocity decrement in the propagation Mach number of a detonation wave, it does not give any information about whether the detonation would continue to propagate or quench. It is expected, however, that such information could be obtained by a consideration of the nature of the chemical reaction. The quenching criteria, to be presented here, are based on theories that are used to predict composition limits, and are confined to hydrogen-oxygen mixtures as explosives. 301 Shchelkin's Instability Criterion Shchelkin(56) examined the stability problem of a detonation wave in the light of an overall reaction rate to arrive at an analytical expression that predicts the onset of a spinning detonation. Since the latter is usually associated with composition limits (ioeo, explosive compositions that do not support detonation), it would seem that his solution could be extended to predict limits due to side relief. Briefly, ShchelkinYs theory is as follows. The detonation wave is considered a shock plane followed by a CJ plane after a suitable induction time, 3T This time can be determined from the overall reaction rate when pressure dependence is ignored, asE2 RoTe2o (301) where Te2o is the temperature in the shocked gas and E2 is the activation energy for the overall reaction2 If a disturbance starts in the f 6

-377 CJ plane,: the deformation becomes such that some of the gas in the post CJ.plane infringes on the unreacted shocked gas. Since the shocked gas is at higher pressure than the reacted gas, it tends to expand to a pressure close to that of the latter, resulting in a temperature decrease. This decrease, of course, tends to increase the induction time. Shchelkin postulated that if this increase is equal or greater than the induction time, instability sets in. Now E2 dT _ E2 RTe2o (52) d.Te2o..... (3.2) dge2o RTe2o If it is assumed that / 1. 1; i i v goes from Te2o to a temperature Te2o~ consistent with the expansion then the instability criterion becomes: E2 eo) RoTe2o -e2o If the detonation pressure in the CJ plane is P2 and if the ratio of specific heats in the shocked region is the same as that of the original gas, then,by assuming isentropic expansion, Equation (3~3) can be written aso rMe1 2 1el E2 K (P2) 7el - ---- > (3.4) RoTe2o l Pe2o - If the shocked gas has the same molecular weight as the original gas, substitution from Equations (2.11) and (213) with the appropriate value of F into (3.4) yields, after denoting the critical Mach number

-38 -satisfying the equality sign in (3.4) by Mlc: Yel - 1 FM2 7 Ye1 l (el i ) e2 + 7ellc Yel2 7e2 1) Yel + 1 + 27el (Mlc 1)J RoTe1 L +12(7el - 1)(Mlc - l)(7elMlc + 1) E2 (Yel+ 1)2 M2 The significance of the above analysis as related to this work is as follows. If the Mach number of propagation of a detonation wave is greater than Mlc, it is expected that it would be stable and thus continue to propagate. If, however, the propagation Mach number is smaller than Mlca instability would set in, leading to possible quenching. Thus quenching takes place when "el > Mel( —O) Mlc (3.6) Mel( (=) - Me-O) The critical Mach number, Mlc, is calculated for the case where the activation energy, E2 = 17 kcal/mole, 7el = 14, 7e2 = 1.2, and Tel = 5200Fo These values are consistent with hydrogen-oxygen mixtures. In particular the value of the activation energy corresponds to reaction II in the next section, which is generally considered to be the controlling reaction. A trial and error method used in Equation (3~5) yields Mlc = 4~o09 The theoretical detonation Mach numbers for the hydrogen-oxygen mixture without side relief have been recently calculated by Zeleznik and

-39 -Gordon, for the stoichiometric mixture(46) and for other mixtures that extend beyond both the rich and lean endso(57) Figure 12 shows a plot of the Mach number as a function of the hydrogen mole fraction with initial conditions of room temperature and pressure. On the same plot the line showing the critical Mach number according to Shchelkin s criterion is included, Thus, the intersections of this line with the detonation curve give the composition limit. Further, since the stoichiometric mixture has the highest Mach number, it would admit the largest decrease in Mach number before quenching. 3o2 Belles' Explosion Limit Criterion Belles(58) used the well known explosion criterion of hydrogenoxygen mixtures described by Lewis and vonElbe(59) to predict the composition limit. The important reactions considered in this scheme are the following o kl1 OH + H2 --- H20 + H, E1 2o 5 kcal/mole k2 H + 02 -_- OH + 0, E2 17 or 18 II O + H2 OH +H, E =6 III k4 X + H + 02 - HO2 + X, E4 - (2 - 5) IV The steady state approximation for this set of reactions yields the following relation between the chain propagating reaction II and the gas phase chain breaking reaction IVo 2k2 ) k4[X]

5.4 5,4 * I [ I!I 5.2 O AVERAGE QI RANGE 5.0 'r' 4.8 < CRITICAL L.. a: 4.6 MACH NUMBER z Ir~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 4.4 z 20 K CALLI2LE Ir E E4=~~i~~/"' 4.2- PATCH (REF.63) BE. Z 4.2 0REF(RE. 5.8' W..- SHCHELKIN ( REF.56 4.0. --- — -- 40 10 20 30 40 50 60 70 80 90 100 % HYDROGEN IN H2- 02 MIXTURE Figure 12. Dependence of the Detonation and the Critical Mach Numbers on Hydrogen-Content in H2 - 02 Mixtures.

-41 -The values for the rate constants as given by (59) are as follows:. 803 17,000 k2 = -1017 f e2o( - 1) (- cm3 k2= 10 830 e sec and k4 = 3.7 x 105 e80 sec If these are substituted into Equation (3o7)9 one obtains Pe2o e17,000/RTe2o 3o 11 0 (3.8) Te2o fx where fx is an effective mole fraction of third bodies given by (59) as: fx = fH2 + ~35f02 + 43fN2+ 2fAr + 1~47fC02 + 143fH20 (39) and where Te2o ( in ~K ) and Pe2o (in atmospheres) are the temperature and pressure respectively in the shocked gas behind the detonation front. The left hand, side of Equation (308) is a function of the Mach number, 7elD the initial pressure and temperature. This function which can be obtained from Equations (2o11) and (2o13) with F = 2, is denoted by 03o The right hand side of Equation (3.8) is a function of composition onlyo Plotting 03 against Mel as in Figure 13 (curve labeled E2 = 17 kcal/mole) where Yel is taken as 1.4, Tel = 300~K and p1 = 1 atm, one can obtain the value of Mel for each mixture ratio that satisfies Equation (308). This is the Mach number that satisfies the explosion criterion and is considered the critical Mach number below which detonation is not possible. It is plotted in Figure 12 as the curve labeled ' "Belles, E2 = 17 kcal/mole"o

-42 -100 15% H kn ~90% H _ —4 -C-~ 10 e z ell, 15% H (3.11 ) { 50% H2 5 |3.11 21 _/ _ __ 90% H2 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 MACH NUMBER Figure 15. Critical Mach Numbers According to Belles' Explosion Limit Criterion.

More recent data on reaction rate constants indicate that reaction IV has a negative activation energy with E4 = - (2 - 5) kcal/mole as reported by Clyne(60) and Baldwin (61) If the value of 2k2/k4 = 72 mm Hg at 8130K and E2 - E4 = 20 kcal/mole as quoted by the latter are assumed, then Equation (3 7) becomes Pe2o e20 000/RTe2o = 2553 (.10) Te2o fx The function 3 will be different from 03 as can be seen from the curve labeled "E2 - E4 = 20 kcal/mole" in Figure 13. Consequently, the critical Mach number variation corresponding to Equation (3.10) which is plotted in Figure 12 as the curve labeled "Belles, E2 - E4 = 20 kcal/mole" will also be different. It is apparent that this curve is much lower than that corresponding to Equation (3.8) where old reaction rates are used, and is very close to that of Shchelkin's criterion. Further, it should be mentioned at this point, that although Belles' analysis is criticized because it extends the explosion limit criterion to regions of pressures and temperatures far removed from static experiments, there is evidence of its validity in the standing detonation wave experiments by Nicholls and Dabora.(62) 3.3 Patches Constant Temperature Criterion Patch(63) examined the experimental works of Gordon, et al.,(64) and of Breton(65) near the rich and lean limits of hydrogen-oxygen-diluent mixtures for the purpose of deducing a limit criterion. He calculated the theoretical Mach number for mixtures with compositions at the limit corresponding to the values of (64) and (65), from which he deduced the

-44 -temperature, Te2o, behind the shock at various degrees of relaxation. From his results he concluded that the temperature Te2o gave the least standard deviation when rotational relaxation but no vibrational relaxation was considered. The average Te2o, he found, was 15140K when initial conditions were atmospheric. He then proceeded to calculate the theoretical compositions that have a detonation Mach number compatible with the average Te2oo Patch also calculated the theoretical limit composition using Belles'(58) explosion limit criterion outlined in the previous section. By comparing the deviation of the theoretical hydrogen mole fraction calculated by the two methods from the experimental mole fraction at the limit, he found that the average of the absolute value of the deviation is smaller in the case of the constant temperature calculation than that of the explosion limit calculation. His conclusion, then, is that a constant ignition temperature independent of composition can adequately predict the composition limits. On this basis, if one assumes that the same critical ignition temperature is required for all mixture ratios, then the same critical detonation Mach number is necessary. This can easily be found from Equation (2o13) to be Mlc = 4~23, and is shown on Figure 12 by the straight line labeled "Patch". It is interesting to note that it is close to the critical Mach number calculated by Shchelkin8s criterion. The composition limits as predicted by the above three criteria can now be compared with experimental observation. Table I, below, presents such a comparison. It indicates that all three criteria (except

-45 Belles with the old chemical kinetic data) predict limits very close to those observed. TABLE I HYDROGEN-OXYGEN COMPOSITION LIMITS Detonation Limits (Mixtures at Tel = 300K and Pel = 1 atm,) Lean Limit Rich Limit Criterion (% H2) (% H2) Shchelkin(E2 = 17 kcal/mole) 16o5 93~.0 Belles (E2 = 17 kcal/mole) 210 8708 (E2 - E4 = 20) 16o8 92.0 Patch (Te2o = 13140K) 180o 92~2 Observed: Gordon, et alo (Reference 64) 92.5 Breton (Reference 65) 1500 Lafitte (Reference 66) 15,0 90.0 However, we shall see from the experimental results that quenching due to side relief will follow trends similar to those predicted by Belles' criterion, when old kinetic data are used.

IV, EXPERIMENTAL REQUIREMENTS AND ARRANGEMENT 4.1 Preliminary Experiments In order to explore in at least a qualitative fashion the ideas embodied by Equations (2y27) and (3.6), it was decided to use a setup similar to that of Sommers(31) but with some modifications as shall be described presently. The main two ideas that were to be explored are, (1) that the decrease in the detonation velocity when the wave is confined by a compressible medium, should be approximately inversely proportional to the channel width as Figure 11 and Equation (2.53) indicate, and (2) that there should be a point where the decrease can become so high that the detonation would quench and degenerate into a pure shock. In Sommers' setup, an explosive mixture is realized by metering fuel gas and oxygen into the bottom of a rectangular flame tube of approximately 4 ft in length and.35 in. x.5 ino in crossection. The explosive mixture flows at very low velocity (order of 10 ft/sec) through the tube and into the test section. At the test section, the explosive jet is bounded by glass plates at the wider sides, and by a solid wall on one narrow side0 At the other narrow side the jet flows adjacent to either a quiescent or a flowing boundary inert gas0 The explosive mixture is ignited at the bottom of the flame tube, and by a suitable time delay arrangement a schlieren picture of the detonation wave and attendant phenomena at the test section can be taken. Figure 4 is a typical photograph obtained in this manner0 -46

-47 For the purpose of performing the preliminary experiments, the test section is modified as shown schematically in Figure 14, The explosive jet from the flame tube is divided into two columns, usually of different widths, Each column is bounded by a side jet of the inert to be used. The dividing wall is a.010 in, thick brass shim stock, the bottom of which is filed to a wedge. A photograph of the test section is shown in Figure 15. The idea now is that a detonation wave originating in the flame tube propagates into both channels at the same speed at the beginning of the test section where the explosive is bounded by solid walls. However, when the detonation wave reaches a point where the explosive is exposed to an inert, any effect on the propagation velocity could be noticed, Figure 16 shows a series of photographs taken at different time delays from the instant the detonation wave passes a fixed point in the flame tube. The explosive in this case is stoichiometric H2-02 mixture and the inert is nitrogen, In Figure 16a the detonation front is shown after it has been split by the dividing wall and it is apparent that the wall has a negligible effect, From Figure 16a-16d, one can clearly see that the detonation front travelled faster in the wide channel than in the narrow one, Further, in the wider channel the combustion zone seems to be attached to the front whereas in the narrow channel a "clean" front could be seen followed at an appreciable distance by a "turbulent" region. This is interpreted as a quenched detonation, The "turbulent" zone apparently corresponds to the region where the explosive either

DIVIDING WALL EXPLOSIVE - INERT CONTACT SURFACESE z 0rr I~_ INERT- -X INERT LL F W EXPLOSIVE Figure 14. Schematic Drawing of the Test Section Used in the Preliminary Experiments.

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-50 -(a) t = 72.5 tsec (b) t = 77.1 psec (c) t= 94. 6 jisec (d.) t 107.8 ~tsec Figure 16. Detonation Wave Progress of Stoichiometric H2 - 02 Mixture+li -"#= in ChanmnelsI -- of Two D)i -P-ffe-re-nt Wiat. -dths I~~~!~~~~~~~~~~~~ ~I 8 (c) t = 94.6 ~sec (d) t = 107.8 ~sec Figure 16. Detonation Wave Progress of Stoichiometric H2 - 020 Mixture in Channels of Two Different Widths. (Boundary Gas = N2) (Scale: Total width of both channels =.5 in.) (t = relative time picture was taken)

-51l ceases to burn or is deflagratingo This explanation is further corroborated by x-t plots of the detonations from these and similar photographs, in that the front appears to travel at continuously decreasing velocity in the narrow channel. On the other hand, in the wider channel the velocity seems to reach a new steady-state value somewhat lower than that in the solid tube, at about 10 Usec after the detonation passes the start of exposure to the inert. Experiments with mixture ratios other than the stoichiometric indicated the same features just described proving qualitatively the two ideas mentioned at the beginning of this section0 4.2 Separation of the Explosive from the Inert by a Thin Membrane or Film A close examination of Figure 16 reveals a mixing or diffusion zone between the inert and the explosive0 The question arises, then, as to what influence does the diffusion introduce into the problem. For example, could the slowing down of the detonation front be primarily due to diffusion or does the latter have only a minor effect? For his experiment Sommers calculated the diffusion width for the gases he used and found it to be of the order of,1 in, at two inches from the point where the two gases meet, He concluded that since his channel width is 05 in0 the effect of diffusion would be negligible, Unfortunately, such a diffusion width becomes too important in our experiments for the simple reason that channel widths less than 5 ino were contemplated, Therefore, methods for eliminating or at least minimizing the diffusion were sought~ Separation of the explosive from the inert gas by a thin wall appeared to be the

logical solution~ Sommers, early in his work, had used cellophane wrap.0005 in0 thick as the dividing wall but he found that such walls offered infinite confinement as far as the detonation wave was concerned, In other words, he found that the detonation wave in a cellophane tube propagated at the same speed as it would have in a tube having thick solid wallso His conclusion was that the presence of any solid wall no matter what its thickness is, would provide infinite confinement for detonations. Despite this discouraging observation, it was decided that it would be worthwhile to test a thinner membrane and therefore a soap film was tried, The film was obtained by dipping a wire frame into a soap solution and then retracting it, Because of the relatively large area required namely:.35 in, x 3 in,, it was found that a thick solution was necessary to obtain a film that would last about two minutes, which was the time necessary to carry out the experiment~ Thus, commercial liquid soap with one to one by volume dilution with water was found adequate. After obtaining the soap film on the frame, the frame was inserted into the test section and positioned so as to separate the explosive gas from the boundary inert gas. Figure 17 is typical of the results obtained. In Figure 17a the soap film was on the wider channel and in Figure 17b it was on the narrower channel. In either case, it is clear that the detonation front remained planar and travelled faster in the channel with the soap film, Further, from measurement of detonaton velocity in the solid tube ahead of the test section ae i and other necessary data, it appeared that the detonation wave did

FILM FRAME E ~ ilbii~iir~: Z alla iii'iiiasi~::::;:::::::::::::::::,:i Zi~~~~~iiaiiiiiiiiiiiiij':-i~~~~~~~~~~~~~~~c~~' iaii-iiiiiiijiii~~~~~~~~~~~~~~ii iii~~~~~iiiiiiiiiiii~~~~~~~~~iiiiiiiiii~~~~~~~~~i ~~%~~~.i~~~iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii ~ ~ ~ iii~ii~iiiiiiiiiii.~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~!!iiii!~~ii........ (a) Film on wider channel (b) Film on narrower channel Figure 17. Detonation Wave Appearance When Confined by a Soap Film. iiiii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~il~iiiiii~ iiiiiiiiiiii-iiii::~~~ii:~iii~ii~ -iiiii~ ~:' ~L:-: iii?~ -~ (a) Fi.~ on wi~~r o~~ne (b) Fi~~ on narowe, o~ }_ Figure 17. Detonation Wave Appearance When Confined by a Soap Film.

-54 -not change its velocity while travelling in the channel provided with the soap film. These results, then, supported Sommers' conclusion mentioned earlier, At this stage, the author's attention turned to the work of Gvozdeva(55) where a study of the refraction of detonation waves from one medium to another was made. The two gaseous media were separated by a thin nitrocellulose film which evidently introduced a negligible effect. From this work and the tests described above, it became apparent that an analytical determination of what constitutes a film of negligible effect is in order. This is presented in the following section0 4.3 Analytical Determination of the Critical Thickness of Thin Films For the determination of film thickness, one can resort back to the shock tube analogy. Again in this analogy, the driver gas is the gas behind the detonation wave and the driven gas is the inert medium. Further if there is a film between these two gases, it will be analogous to a piston of a gun tunnel. Because the film is thin and reasonably weak, it is assumed that the only resistance it offers to the expansion of the burned gas is that due to inertia. Figure 18 shows schematically the analogy with the film position assuming the "piston" path. 0 The acceleration, vf, of the film in the y direction is governed by the following equation: 0o p p A Ad (401) e3 i2 =fff (.1

I ~ F --- — --— 1 I — 1I I I.-I I I I I I I I, I I 1lLM I~I I I I I I I Q~. FILM INERT I PISTON EXPLOSIVE I I I I e~ I I EXPANSION -— r uI I I II I 01r I I I t ' I I I I g o ~I I I I I I I L_ L PRESSURE X Figure 18. Analogy of the Piston Path in a Shock Thbe to the Film Position.

The expansion of the gas behind the wave is assumed to take place isentropically so that: e20 Pe3 = Ye20 1ve( e20 V]e 2 ~ (4.2) P 22 Pe20 l 2 ae20J and finally the instantaneous velocity of the inert in the y direction is the same as that of Equation (2o41) and is rewritten here for convenience: 1/2 Vi Pi2 Yil (7il + 1) ail il i2 il Pil Yil+ 1 The velocity of the film and the velocities of the gases immediately in front or behind it are all the same and therefore can be denoted by v vf i e3 ( ) By using the definitions of a1l 02 and P of Equations (2233), (2.34) and (2o37) respectively and noting that: v ail ael -- 0 (4.5) ae20 ail ael ae20 one can write Equations (4ol) and (403) in the following forms respectively: P 0 eD | P 0v (4.6) Pe20 01

-57 -and 1 2 FPl Melel ~~v~~~~~~~ - Pa Me WW(4-7) Y el2 Lilil 2( il +1) 1 Pil 1 Z il- 2 Pe1 M 2 el ii _ el Mel Y il+ 1 where 0 o dav pfErfv -v= _,. (4 8) ~~dt P.e 2~ PelilMel ve2 v 1 v = v2 V 1 (4o9) ae20 ael 02Mel and t _ l9elMel t (4 10) 02Pf fael From Equation (2o435), Equation (4o7) is seen to be: tan (4 v 0 (4.11) ~2 However, tan 6 in Figure 8 is found to be a function of the density ratio parameter independent of whether a shock or a detonation condition behind the front of the detonation wave is assumed, and weakly dependent on the Mach number for Mel1 5. Hence, the high Mach number approximation can be used in Equation (4j7) to give:

-58 -1/2 v 2 P l i 1 (4.12) Yel 02 LPilyil 2(Yil + 1) Substituting this equation and Equation (4.2) into Equation (4o6), one obtains: 2ye20 Y _ e-_e20 - 1 Ye02 el 2(yil +1) 2 =2 e20 2 Pelyel Yi 2 At this point the values of 01 and 02 for the shock or the detonation conditions are calculated using the values of Yil Yel' Y7e20 and Pil/Pel shown in Table II below in Equations (2o33) and (2o34). TABLE II EVALUATION OF THE FUNCTIONS 01 AID 02 Pil Mel F Yil Yel Ye20 01 02 Pel 5 2(shock) 1 104 104 1.4 1.160.481 - o 2(shock) any 1.4 1.4 1.4 1.1667.441 5 l(deton) 1 1.4 1.4 1.2.654.560 -e o l(deton) any 1.4 1.4 1.2.636 o545 As is expected, the values of 01 and 02 depend primarily on whether shock or detonation conditions is assumed but not on the Mach number for Mel b 5~ Equation (4013) is plotted in Figure 19 for both shock and detonation conditions with ~1 and 02 corresponding to very large Mel

I! I I I Iif 1.0 Pr Y Lr I, ee ____ w\t Y 2(X4-1) <1 N 0. Ir --- ~~~~~~~~~~~~DETONATION ~~~~~n ~~~~~~~~~~o~HC. 0. I I -J~~~~~~~~~~~~~~~~~~~~ 2 z 0 0.2 P Y Y o I, Le. e. 0.4~~~~~~~~~~~~ =0.I, N NON- DIMENSIONAL VELOCITY V Figure 19. Instantaneous Film Velocity vs. Its Acceleration.

60 - and with y's corresponding to Table IIo Further, two values of the density ratio parameter appearing in the equation are considered, namely Pel'el Zil r2 0 271+ l)1 = ol or 20 L1w ~1 22(yil + 1) These values are chosen because they bracket the values for the combination of gases that are used experimentally. From Figure 19 it is seen that the variation of the nondimensional acceleration can reasonably be expressed as a linear function of the non-dimensional velocity, ioe., 0 v = 1 - - (4. 14) Voo 0O where v is the velocity at v = 0O 00 Equation (4o14) can be integrated by separation of variables to obtain the time tx, required for the velocity to reach a certain fraction, X, of the final velocity. Thus f f= af (dt15 O v 0 V 00 or -- 1 - v0 lIn (-) = tX (4o16) Setting this time to be a fraction, o, of the reaction time, one obtains after noting that the reaction time = x/uel and using Equation (4o10):

-61 -Qf ColpelMel 1 (4.17) x 2voPfaeluel in (lX) or -f = l 0 Pel (4.18) 1 (4.18) v Yel2 in () Pf With the values of X1 and 02 shown in Table II, it can be seen that the most stringent case is that for a detonation condition behind the wave. Further, when pe is smaller than Pi then v. is also small and when p e is larger than Pi, v. is large. By considering these effects and after setting a = ol and X =.99, one obtains for the most conservative case 'f.1 x.636 Pel x o185 x 1.4 x.545 x 4.6 pf 'l P1el.0978 -. el (4.19) Pf Pf The inertia of films with thickness given by this equation would be felt only within one-tenth of the reaction length. For the purpose of this study, this is considered the critical thickness beyond which the effect of the film is no longer deemed negligible. For film material, such as nitrocellulose, having a density of 1.58 gms/cm3 and for an explosive (80% H2 - 20% 0 2) density of 5357 x 104 gmns/cm3 at NoToPo and a reaction length of about 0.3 cmo,

-62o f 68 x cm = 680 A (4.20) This result explains why cellophane whose thickness is two orders of magnitude higher can act as infinite confinement0 It is to be noted that although Equation (4o19) predicts that the film thickness should be proportional to the operating pressure, the fact that the reaction length is inversely proportional to the pressure for a second order reaction cancels its dependence on pres sure 4)4 Preparation of Thin Films and Estimation of Their Thickness Thin films with thicknesses of the order required for our purposes have been used in electron microscopy as supporting membranes for the specimens being observed0 There are several materials that can be used, but apparently the two most common materials are Formvar (polyvinal formal) or Collodion (cellulose tetranitrate dissolved in ether and alcohol)0 The film is usually prepared by first obtaining a solution of the material in an appropriate volatile solvent and carefully placing a few drops of the solution on the surface of distilled water in a Petrie dish of about 20 cm in diameter. The solution spreads over the surface, the solvent evaporates, and one is left with a thin film floating on the water surface0 Various techniques for removing the film are used and the interested reader is referred to a book by Hall 7) which contains a chapter on the subject of films0

-63 -The material adopted for use in the experiments of this study is Collodion dissolved in amyl-acetate. It was found after some experimentation that three drops of 15% solution were adequate for they provided a film of acceptable thickness yet they allowed a reasonable time before the solvent completely evaporates -- a situation that renders the removal of the film subject to frequent breakage To prepare the film for use, three drops of the solution are placed on the water surface in rapid successiono These are easily seen to spread over most of the dish surface, As the solvent evaporates there appear broad diffraction color bands over the whole surface, As soon as they disappear a steel frame-film holder whose dimensions are shown in Figure 20, is introduced near the edge of the dish until all of the opening is submerged in the water under the film. The film holder is lifted gently at an angle until the edge of the film starts adhering to the surface of the holder near the top of the opening~ When the holder is lifted further its two outer edges usually start cutting the film. However, sometimes the cuts need to be started by a sharp razor or knife. The holder is now moved to the center of the dish, as it is lifted, until the film covers the opening adhering to both the flat and chamfered parts of the edges of the holder. During this operation care is taken that no fold back of film occurs and that no wrinkles are present. As soon as the frame is removed the flat part of the edge is blotted out so that the film rests essentially on the chamfered portion0 Sometimes a few water drops cling to the film and are allowed to

1/16 R ~+19 ~ 0 0 00 0 0 H 6o +I1 ~ 0 00 Ad 1 o do 0 Figure 20. Film Holder. +1 9 Ic tl~+ P, tb~O

evaporate before using the filmo This is done either by letting the water naturally evaporate or by passing dry nitrogen on either side of the film when the film holder is placed in the test section0 The film obtained in this manner is usually colorless or silvery grey, Its thickness can be estimated by two methods' The first method is based on calculating the residue volume and assuming it is evenly distributed over the water surface, Thus: cn cc n = v r2pf where cn = mass of nitrocellulose/unit volume of collodion = 004 gms/cm3 c = volumetric concentration of collodion in amyl acetate -= 15 n = number of drops used = 3 v = number of drops/unit volume = 30 cm-3 r = film radius = 7 5 cm pf = nitrocellulose density = 1l58 gms/cm5 With the above values one finds that -6 c f = 2o15 x 10 cm =215 A The second method of evaluating the film thickness is based on a study by Peachy(68) who gives a correlation between the interference color of thin sections and their thickness when observed under white light. For films with index of refraction = 1.5 the correlation is as shown in Table III belowg

-66 -TABLE III COLOR-THICKNESS CORRELATION FOR THIN FILMS (index of refraction = 1o5)(68) Thickness Range Interference Color 0o < 600 A Gray 600 - 900 Silver 900 - 1500 Gold 1500 - 1900 Purple 1900 - 2400 Blue 2400 - 2800 Green 2800 - 3200 Yellow The index of refraction of nitrocellulose is 1l514, and therefore the above table can be considered applicable, On this basis the color of films made from two different solutions and varying number of drops was observed and their thickness was estimated. Some averaging was necessary as apparently the films were not uniform in thickness0 The result is shown in Figure 21 from which the thickness of films used in this study can be estimated to be about 230 A, This result is in agreement with the result of the previous method0 In addition, it is apparent that such films satisfy the requirements set in the previous section0

c) 0 -0 0.50 COLLODION IN AMYL ACETATE LL 0 A.30 COLLODION IN AMYL ACETATE w 2 FILM DIAM.: 6" APPROX.:3 z x 0~~~~~~~~~~~~~ 3 z w L) z 2 __ __ __ 0 L) z 0 5 0 0 500 1000 1500 2000 0 FILM THICKNESS, A Figure 21. Dependence of Film Thickness on Collodion Concentration in Amyl Acetate Times Number of Drops.

-68 -4,5 Diffusion Through Thin Films A film with thickness of the order of that mentioned in the previous two sections is not expected to eliminate completely the diffusion between the gases it separates. However, it is plausible to assume that it represents a definite barrier to the diffusion, and therefore, the question arises as to how good a barrier it is. To answer this question, imagine two semi-infinite compartments one beside the other and each of which contains a different gas at the same temperature and pressure. The distribution of the two gases in the two compartments as a function of time is to be found for the following two cases: one, when at time zero the wall between the compartments is removed completely and two, when the wall is replaced by a thin film. If the two gases are denoted by "a" and "b" in compartments 1 and 2 respectively, use of Fick's second law gives, at t > 0 ~c 2 c Ca Ca (4.22) at ab ay2 For the first case, the boundary conditions are as follows: Cal = Ca2 for y = O, t > (4.23) and -Cal =aa2 for y = O, t > 0 (4.24) 6y by ca = 1 for y > 0 t = O (4.25) ca = O for y-< O t - O (4.26)

-69 -For the second case, the boundary conditions are the same except the condition (4023) is replaced by Dab +- (C -c c ) = 0, for y = 0, t > 0 (4.~27) Dab y ~ a2 al where P is the permeability of species "a" in the filmo This condition expresses the assumptions that no accumulation of species "a" in the film takes place and that its distribution is linear within the film. It can readily be seen that the diffusion problem above is analogous to the heat conduction problem between two semi-infinite solids of the same material but having two different temperatures initially0 The first case above is similar to the case where at time zero the two solids are brought together without contact resistance, and the second case is similar to that when there is a contact resistance such as a gaseous film, The solution for Equation (4~22) under the appropriate boundary conditions can be found in the treatise of Carslaw and Jeager (69) and is presented below0 For the first case: 1 y orf Y c = - 1 + erf - for y > 0 (4.28) a 2 and c = 1 er y for y < 0 (4~29) a 2 2 ID t ab

-70 -and -or the second case: Ca = ({ler2 y2ft + [ex(z2 a 2 DDabt [ab for y > 0 (4,30) and _ fJ L 2 ) c 1erf Y + exp erfc z Y aL 2 2 tT; -k(24Dabtl for y < 0 (431) where = + 2P (432) 2 Dabt Rf Dab To obtain the numerical solutions, one notes that for gases the diffusion coefficient Dab 1 cm /sec (4~33) and, that for permeability, the value for H2 through various cellulose compounds as quoted by Barrer(70) can be used. This value is P 10 7 cm2/sec (4 34) Equations (4~28) and (4~30) are plotted in Figure 22 for 0o various values of t. In the latter equation the value of ef = 200 A was used. It can be seen that for relatively short times a thin film offers an excellent barrier to diffusion. It appears that in order to take advantage of the film the gases should be exposed to the film

1.0..... t =0.01 SEC. /~- f = 0.25 SEC. /I I I~~~ 0.9 0 / —" o gI o i CO ) - // = o/ __ _ 0.8 Cl) ~/ //' U' 0 / / I/ ~c0.7 0NO FILM Eq. 4.28 Z 0.7 I/ ~~0 a IV L~ WITH FILM Eq. 4.30 Io.O,'/. WITH FILM Eq. 4.44, X= 3.25 IN., I-L / /U =60 IN/SEC. z /MAX. w "I / o z i o 0.6 / / / / 0.5 0 0.1 0.2 0.3 0.4 0.5 DISTANCE FROM INTERFACE OR FILM, INCHES Figure 22. Concentration Distribution Due to Diffusion.

-72 -for slo- ou time only. This can be accomplished by flowing the gases along the sides of the flim and choosing a flow velocity that allows a reasonably short residence time. It should be pointed out, however, that because of the noslip condition of the two streams of gases at the film surface, the solution of the diffusion problem is somewhat more involved than that given by Equation (4.30). For the appropriate solution, the following analysis is offered. Let both gases "a" and "b" be flowing in channels 1 and 2 respectively on either side of the film which is considered coincident with the x axis, By assuming incompressible steady flow and no change in density due to diffusion one can write the following conservation equations: a) Continuity: -u + -- = o (4035) ax by b) Momentum: u + v- - v aU 1 - (4 36) ax y ay2 op dx c) Diffusion of "b" u -- cb — (4 37) u -b+ v a = Dab For a fully developed, parallel flow in a channel, au/8x v = 0. Therefore, Equation (4036) reduces to that of a Poiseuille flow and Equation (4o37) becomes:

-73 -6cb 2c u b Dab b (4o38) 6x ab This equation is to be solved subject to the following boundary conditions: Cb = O, x = O, y > 0 (4o59) Cb = O x = O, y = o (4o40) and ( Cbl) + Dab b 0 x> 0, y = (4041) (bP - Cb ) + b2 If the diffusion through the film is assumed to be constant and equal to the maximum possible diffusion, this boundary condition can be written as: P _ acb Dab y x > 0, y= 0 (4 42) The system of Equations (4~38), (4~39), (4~40) and (4~42) is analogous to that obtained for the problem of heat transfer to a fluid flowing inside a tube when the heat flux at the wall is constant. If u is considered to change linearly with y, the solution is given in (71) the book by Bird et al. For a conservative answer one can write: u = u 2y (4o43) max b where Umax is the maximum velocity in a Poiseuille flow and b is the channel width. The solution involves a change of variables and is presented here without derivation:

-74 -cb c P b 36 Dab - X 4 a 2D 2 b 2 2 3 3 whe re > ~2y iax b2 - and F(2/3) is the "complete" gamma b 36x Dab function and r(2/3, X3) is the "incomplete gamma" function. Equation (4~44) is plotted in Figure 22 for a value of Umax = 60 in/sec, b = o5 in and x = 3.25 in. corresponding to the conditions used in the experiments described in this work. The incomplete gamma function is obtained from the tables by Pearson(72)0 It is apparent from the plot that the diffusion is negligible under the experimental conditions used. It should be added that taking b = o5 in. (the widest channel used) results in the most conservative answer, if the same average velocity is used in a narrower channel~ This is so, because the concentration depends on the slope of the velocity profile at the film. 4o6 Experimental Arrangement and Procedure The experimental apparatus is designed for obtaining two basic pieces of information: the critical explosive width and the velocity decrement of the detonation at explosive widths larger than the critical0 For both of these measurements, the basic equipment for the preparation of the explosive and for charging the test section is the same. Figure 23 is a schematic diagram of the system. All lines are 1/4 in. stainless steel tubes except the 40 in. long flame tube which has an inside cross-section of 0o36 in. x 0.50 in. The glow,lug is placed 8 ft ahead of the entrance to the flame tube to insure

INERT EXPLOSIVE \] TEST SECTION MIXING CHAMBER MANOMETER MtRIAM FLUID 0 2C VAC. PRES. BALLOON D /3 SWITCH 6 u4 r8 9 Sin s GLOW RI6b R 26 R36~ R4 b eb6 7 PLUG VENT RI-R4 REGULATORS VAC. PUMP NEEDLE VALVES VALVE INERT OXYGEN FUEL NITROGEN Figure 23. Schematic Diagram of Mixing and Charging System.

-76 -a fully developed wave before any measurements are taken, The flame tube is vertically mounted and is terminated by the test section shou.. - bo' i.u-es 2: a..d. 2>. 2 The test section is provided with two 1/2 in. glass plates constituting the front and back walls, and each plate has.035 in. wide by.135 in, deep grooves for the film holder (Figure 20) to slide through. The depth of the explosive channel in the test section matches its depth in the flame tube, but its width can be easily changed from.5 in, to.1 in, at.05 in, intervals by suitable spacers and a wedge piece. While the detonation propagation velocity may be affected by the wedge on the inclined side as evidenced by the work of Nicholls, et alo we shall see that this is not the case on the straight side, To prepare the explosive mixture, the mixing chamber (see Figure 23) is evacuated and then filled consecutively with oxygen and fuel, The partial pressure method is used to obtain the desired mixture, The pressure is monitored by a Marsh guage, type 200-C, which can be read to less than 0,5 psi, The mixing chamber, which is a piv,1ce one foot in diameter, is usually filled to about 60 psig total pressure. It is provided with a blow-off diaphragm for safety. Before a particular mixture is used a period of no less than two hours is allowed for complete mixing. To prepare for a run, the expendible toy balloon is evacuated and then filled with the explosive mixture. This process is repeated once more to insure purity of the mixture in the balloon. Valves 5 and 8 are closed and the plumbing between them is evacuated, This

-77 -0 0 /3/32D 10-32 SCREW — SECTION A-A 1/8 Da / —3/32 D 0o -32 sCREw IO 28-32 SECREW 3 Ir 24.u. 1t_ _ SetioAel I I I-15 I S GW I 5 I II I III I 4 Figure 2i. Test Section Assembly.

r7o `Ili-i,-i;;l:l-fi ~~ ~~r':-iil'I':' iiiiiiiii-i.: rii~~i~i~iiiiiii'~siil~iciiii~iii~ii:-.;.. iii~~i~i~iiiii~siiiiiiiiiii~iiiii........:::::~~~~~~~~~~~~~~~~~~~~~~~!~I XX:: liquefac i n.1::

provides a buffer zone that blocks the detonation wave from travelling back to the mixing chamber0 The film holder on which a thin film has previously been mounted according to Section 4o4, is inserted into the test section, after its edges have been covered by a thin layer of vacuum grease. The purpose of the grease is to prevent any mixing and/or diffusion between the inert and the explosive via the grooves in the glass plates. To make a run, valves 1 and 9 are opened slowly and simultaneously in such a way that pressures of the inert and the explosive are kept equal at all times until the balloon is nearly exhausted0 To insure that the flame tube is purged of air, the balloon is filled again and this process is repeated0 A simple but very sensitive check to insure equal pressures on both sides of the film at the test section is accomplished in the following manner. The filament of a small bulb is focused on a screen after reflection from the film, When the pressures are equal the film is plane and the image of the filament is sharp0 At a slight unbalance in pressure the film becomes curved and the image of the filament shifts and becomes diffused, The direction of the shift indicates which side is higher in pressure so that corrections could be made0 It was found that this method is sensitive to less than 1 mm of water of pressure differenceo It should be added that maintaining equal pressures on both sides of the film is advantageous by itself in that it insures constant explosive width and prevents gross mixing0 All runs are made with test section over-pressures not less than 4 cm of water0 The sizes of the exit orifices shown on the top of

s80 the test section in Figure 24 are so chosen that the overall velocity in the test section is about 40 in/sec, corresponding to a maximum velocity of 60 in/sec if a Poiseuille flow is assumed. For such a velocity, the diffusion at the top of the test section is found to be negligible in Section 405o When the balloon is nearly emptied the glow plug switch is closed momentarily thus starting two detonation waves, one going to the balloon and exploding it, and another travelling toward the test section. The sequence of events that take place as the latter travels down the flame tube depends on which of the two kinds of photographs is desired, as described next0 It should be mentioned that a glow plug is found more satisfactory than a spark plug because the latter seems to affect the electronic instruments used in the experiment0 ao Equipment and Procedure for Spark Photography The equipment for obtaining spark schlieren photographs of the detonation wave in the test section is shown schematically in Figure 26, Four ionization probes are used to detect the passage of the detonation wave in the tube resulting in the performance of certain functions which will be described shortly0 Each probe is made up of two 20 gauge enameled copper wires 6 in, in length0 The wires are twisted together for one inch and this part is inserted into a teflon tubing of equal length0 The wire and the tubing are then housed into a 1/8 in0 pipe thread Conax thermocouple fitting* resulting in a pressure tight assembly0 The protruding teflon tube *Manufactured by Conax Corporation, Buffalo, New York0

TEST SECTION - f -4 \ 1' 3f 1.5f TRIG. SPARKI L i 200/ Dr MULT. | VA RI BL E || TIMER X 2 L, ' |AL CH ROMATI LENSES POWER SUPPLY -L, P2 |oTIME DELA2 Po P 25 30 KV DC VARIABLE TIMER /2 LI,L2 ACHROMATIC LENSES POWER SUPPLY p 24" FOCAL LENGTH I. 5" DIAM. mr THYRATRON PONIZATION P I Po I I IIp PO. -P2i = 7 2 "....... UNIT GLOW PLUG -TIMER 00" TIME DELAY Po-Pm = 25" UNIT pip = 12.47,, UNIT EXPLOSIVE -~0 [ i _ GLOW PLUG TIMER I EXPLOSIVE Figure 26. Schlieren System and Block Diagram of Instrumentation for Spark Pictures.

82 -is cut and the wires are clipped to a length such that when the fitting is threaded into the tube, they will be flush with the tube wall. The clipped wires have pointed bare ends separated from each other by the two thin enamel layers thus constituting two conductors with a gap in between. One wire of each probe is grounded and the other, in the case of probe 1 and 3, is connected to the grid of an 884 thyratron in the thyratron unit. A unit similar to the one used is described in the work of Morrison.(9) The passage of the detonation wave, with attendant ionization, shorts the probe thus decreasing the grid bias of the thyratron causing it to fire and generate a voltage signal through the plate circuit. The voltage signal is fed to either the start or stop input of timer 1. In most runs a Berkley 1 megacycle counter, Model Noo 5120, was used as timer 1. It is accurate to + 1 Uisec and it is used to measure the wave velocity. In the case of probe 2 the ungrounded wire was connected to the grid of type 2050 thyratron in the time delay unit which is similar to the one described in Reference 9. The firing of this thyratron, in addition to providing a signal to the start input of timer 2, causes, after a predetermined time delay set by the potentiometer adjustment in the unit, a second 2050 thyratron to fire which in turn causes a spiked voltage signal to be induced in a coil, The output of the coil (about 15 kv) is used to trigger the spark source. A voltage signal due to the firing of the first thyratron is fed to the start of timer 2.

-83 - The spark source unit built by the Aircraft Propulsion Laboratory personnel consists essentially of two circular metal plates with 10 Sprague (type 20 DK-25) 500 [lifarad capacitors spaced evenly in a circle. They are charged to 25 kv and discharged by the coil pulse from the delay unit. An orifice in one plate acts as a point light source during the duration of the spark which is of the order of 0ol isec. The light strikes the collimating lens of the schlieren system as well as a photomultiplier whose circuit generates a voltage signal used as an input to timer 2. Model 524 B Hewlett-Packard 10 megacycle which is accurate to + 0.1 isec is usually used as this timer. Its function in this case is to measure the time delay between the passage of the.detonation of probe 2 and the spark discharge, i.e., when the spark picture is takeno It should be added thatoccasionally, Models 757 B and 727 BN timers made by Computer Measurement Company are used for checking purposes. They are accurate to + 0.o1 sec. b. Equipment and Procedure for Streak Photography The essential features of the equipment used for schlieren streak photography are the same as those for spark photography. As the block diagram shows in Figure 27, probes 1 and 3 are again used for the measurement of the wave velocity. Probe 2 in conjunction with the time delay unit provides a triggering spark to initiate a flash of about 1 msec duration in the flash tube shown. The flash tube is housed in a box-and again an orifice on one side acts as a point source. The optical system chosen was dictated by availability

TEST SECTION ~300 ~ft 24" 30"1 32" - 25" DRUM CAMERA 450 V LMAG. PICKUP SHIELD ~ EDGE FLASH TUBE |0.075" ORIFICE P3 (FT- 118) cn TIME DELAY _ P2 UNIT THYRATRON TIMER I UNIT o L, L2 ACHROMATIC LENSES 24" f. I. 5" DIAM. RELAY SWITCH L3 ACHROMATIC LENS 8.6" fl. L4 CAMERA LENS 7" fI- FREQUENCY GLOW PLUG OR PERIOD EXPLOSIVE METER Figure 27. Schlieren System and Block Diagram of Instrumentation for Streak Pictures.

of components and space limitation. At the right hand side of the test section all light is blocked off except for a O010 in, slit along the explosive channel close to the solid wall side0 One side of the slit has three notches providing increased exposure at the corresponding image points thus establishing the vertical spatial scale, The knife-edge was set horizontally in this schlieren system0 An Avco rotating drum camera was used to obtain the streak photographs0 The drum is 4 in0 in diameter and 3 ino long in inside dimensions, It is nitrogen or air driven with normal rotational speed being 600 rev/seco Thus, its rotational period is longer than the flash duration and therefore no double exposure can occur0 The camera is provided with a magnetic pick-up whose signal is fed to the Hewlett-Packard counter used as a period meter0 The signal is interrupted by a normally open relay switch (Potter and Brumfield type KCP 11) whose energizing coil is part of the plate circuit of one thyratron in the thyratron unit, The switch thus closes only after the passage of the detonation wave by probe O with a time delay of 5 msec, This means that the period measured is for a cycle somewhat later than the one at which the picture is taken0 However, the camera is usually brought to speed before a run is started and therefore the measurement is considered adequate0 Because of the comparatively slow linear speed of the film in the drum, it was found necessary to take pictures at o2 reduction. Such reduction results in a detonation streak at about 70 with the time axis0

Almost all film used for the streak pictures is Kodak plus-X o-. Es-tar base, which is a relatively curl resistant base. Further its dimensional changes due to processing, humidity or temperature variations are isotropic -- a quality which is essential for accurate determination of the velocity or the velocity change of the detonation wave.

V o EXPERIMENTAL RESULTS AND DISCUSSION 5o1 Critical Channel Widths of H2-02 Mixtures When Bounded by Nitrogen As was shown in Chapter III, the quenching criteria presented there are all based on the assumption that certain conditions have to be met in the post shock region of the detonation waves before the detonation could be considered self supporting. The different conditions could all be translated into a Mach number requirement which are shown in Figure 12. Before experimentally finding this critical Mach number for H2-02 mixtures, it is easier first to find the critical channel width for different mixture ratios when bounded by an inert gas and then determine the corresponding Mach numbero To do this, a series of go-no go type experiments were made where the mixture ratio as well 'as the channel width were varied. The boundary gas used was nitrogen in all cases. It was chosen because preliminary calculations indicated that quenching could be accomplished in a wide range of mixture ratios at channel widths compatible with the test section design. In the experiments, a mixture ratio is chosen and a run is made with the test section set at the widest possible channel width (5 in. )o Spark schlieren photographs are taken when the detonation wave is about two or three inches past the point where the explosive is exposed to the inert, Examination of the photograph revealed whether the detonation is quenched or not. Typical photographs of an unquenched and a quenched detonation wave are shown in Figure 28, -87 -

~ i 8ii 5, '8, '@@@.U M8: ' - s _... o...:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................... Figure 28. Typical Spark Photographs of a Quenched an: >.g.H IS ' D.. _ _.. swi.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~....... (a)Ru N. 88,3 b11 =lliE 0.25 in.Saii _... _....~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...... (b) Ru n N o. 84_, b_ =0. 5 ina. Quenched, 40% H2 - 60~ 02 with N2 Boundary~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........-....... __ R _ _ _w~~~~~~~~~~~~~~~~~~~~~w_...> B-E < - i M_8 sssE a Na _-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~................ (a) Ru No '. 881lE b =B 0:.25@ in. __uechd *o - __X@. 02 with N2 ESBound ary __ Run No.X _4o _ = 0aia58 in. — Qunhe__oH o 02 wit _2 EBounar

-89 -As was explaihed earlier, the quenched detonation wave is characterized by a "clean'" shock front followed, at an appreciable distance, by a "turbulent" zone. Further, from a knowledge of the detonation velocity inside the tube as measured through probes 1 and 3 (see Figure 26) and the time delay at which the picture is taken, an estimate of the decrease in velocity of the quenched detonation wave could be made0 It was found that the quenched detonation waves suffer an average of over 20% velocity decrement in 2-3 inches of exposure to the boundary gas0 To insure repeatability, four or five runs were made for each set of conditions (mixture ratio and channel width)~ If the detonation wave is unquenched, a smaller channel width is chosen and more runs are made0 If the detonation wave is still unquenched, a further decrease in channel width is made until quenching is observed0 Thus, a determination of the critical channel width could be made. Figure 29 represents the results obtained on such a series of experiments. Within the capabilities of the test setup, the leanest possible mixture ratio for which a critical channel width could be determined is 45% H2-55% 02~ Difficulties were encountered in starting a detonation wave in mixtures richer than 75% H2-25% 02~ Thus, the results of Figure 29 are subject to these two liminationso However, they are extrapolated up to the experimental lean and rich composition limits of Table I as shown by the broken lines0 In some runs, it was difficult to decide whether the detonation wave is quenched or not0 The reason was that in some cases only

X X X X QUENCHED O NOT QUENCHED * X DOUBTFUL I —.J w z 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 MOLE FRACTION OF HYDROGEN IN H 2- ~2 MIXTURE Figure 29. Critical Channel Widths of H2 - 02 Mixtures Bounded by Nitrogen. Figure 29. Critical Channel Widths of'H - 02 Mixtures Bounded by Nitrogen.

-91 -considerable thickening of the reaction zone could be observed; in others, the pictures revealed 50-50 chance of quenching and non-quenching. The points corresponding to these situations are plotted in Figure 29 as "doubtful" At this point, it is well to observe that Figure 28a reveals the same features of the assumed model of the interaction process shown in Figure 5. The absence of large curvatures in the detonation wave and the oblique shock as opposed to the curvature observed in Figures 4 and 16, is attributed to the reduction of diffusion offered by the separating film. Figure 29b shows that even in the quenched case, the interface is quite straight and thus adds more faith in the adopted model. 5.2 Velocity Decrement of Stoichiometric H2-02 When Bounded by Nigrogen and Inferred Reaction Length With the critical channel widths for various mixture ratios found, we now turn to the experimental determination of the effect of channel widths larger than the critical, on the propagation velocity of the detonation wave, In essence a check on the analytically calculated variation of the velocity with area increment, shown in Figure 11, is to be made. To carry out these experiments the stoichiometric hydrogenoxygen was chosen for two reasons. One, it allows the widest range of channel widths that could be used within our experimental limitations, as can be seen from the results of Figure 29. Two, this mixture has been investigated extensively in-the literature, and in particular, an

-92 -estimate of the reaction length which we shall need in our calculations, is available. Tests were made according to Section 4~6 using streak photography. Four different channel widths were investigated, namely.2ino,.25 in.,.3 ino, and o5 ino From five to ten runs were made for each width. A run with a solid wall, instead of a thin film, was also made for each channel width, This was intended to confirm that any observed changes in the detonation velocity is not due to some peculiarities of the test section but due to the side relief. Figure 30 shows typical streak photographs of the detonation wave, both with a solid wall and a thin film, for a channel width of.5 in. In the case of the solid wall, it can readily be seen that the slope of the detonation remains unchanged throughout the test section. A close examination of Figure 30b (the thin film case) shows that the detonation wave does slow down and indeed reaches a new steady state, indicating that it is not quenched. In all the streak pictures that will be presented henceforth, the detonation travels up and to the right, As a further guide to the identification of various streaks observed some pertinent dimensions are shown in Figure 31. Figure 32 shows a streak photograph for the same conditions as those of Figure 30, except in this case the channel width is only.2 in, It can again be seen that in the case of a solid wall the detonation speed remains unchanged even though the wedge piece was present. In all similar cases the results were the same and, therefore show that the geometric arrangements in the test section does

~ /,~ ~ ~~~~~~~~ ~ ~~~~~~ 'i: i'.i'~? ii'ii'~i # 2iii~iiiiiiiii~-~l~~iiiiiiii:.:..:._:::::,-:::.:;,:i::l~::: ii:iiiiiaiiiiii #1 (a) Run No. 797 Solid Boundary Vert. - x13 = 2.996 in. Horz. - dist. equiv. to x13 = 82.67 ptsec?,.. q:...~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.... (b) Run No. 796 Nitrogen Boundary Vert. - x13 = 2.996 in. Horz. - dist. equiv. to x13= 82.96 tsec Figure 30. Streak Photographs of a Detonation Wave in Stoichiometric H2 - 02 (b = 0.5 in.). (See Figure 51 for other dimensions. )

-94 -SLIT L4/ / / / / / / / / / TEST SECTION TOP END OF FILM i NOTCH/ 3 i 9 K 0 111,, 0) NO/TCH Il,' START OF FILM Ii 6C ~NOTCH/I Pi 0 Figure 31. Pertinent Dimensions Helpful for the Interpretation of Streak Photographs.

-95 -(a) Run No. 752 Solid Boundary vert. - x13= 2. 996 in. horz. - dist. equiv. to x13 = 81.67,i-sec. (b) Run No. 992 Nitrogen Bound~ary vert. - Xl3 = 2..996 in. horz. - d~ist. ecjuiv. to..78.0. sec. Figure 52. Streak Photograph of a Detonation Wave in.... Stoichiometric H - 0 (b = 0.2 in.). -:G.: —:..*,-,~:;,,a-zz-. - -- - --------------.-BRw.B............... 8k*..... 8a" BB ' 'aga ' l::*::; t:':::':: ':;': 0 lta04:::;i::00;: 0 0: 0 0;............; *A*Ba BuaBkin ' ', 77a"w,":::0X:;;e:X;;0EXg; gX::;:lj0:X::::::::, BaZB2>,za,,lg*g a: g g;; (ba) Run No. 992 Nitogen Boundary vert. -x l = 2.996 in. horz. - ist. equiv. to x13 _ 781.0 Utsec. FovE*igr 32. Strea Phtgrp of- ai Deoato Wav in iiiEii E ':E::i: -:BE:::E:::::::::::::i:::::::::::E~~~iL::iI - - * i 7

not affect the detonation wave being studied. In the case of exposure to the side relief (Figure 32b), a close examination reveals again that a new steady state velocity is reached in a similar fashion to that of Figure 30b. In addition to the incident detonation wave streak, some interesting patterns due to the reflected shock could be observed but these are of no consequence to our analysis. The striae observed in the case of channel widths less than.5 in. at the lower part of the photograph below the bottom white line is due to the reflection of the detonation wave in the small curved channel (see Figures 24 and 25). Each run is followed by taking a picture of the slit on the same film strip. This was done for the purpose of accounting for any deviation in its perpendicularity from the time axis, as shall presently be seen. If the slit is at an angle a from the ordinate and the detonation trace makes an angle P with the time axis, it can be shown that the velocity of the detonation wave is u r L tan (51) e i~t (1 - tan P tan a) where L is the perimeter length of the drum, t is the magnification and tp is the time for one revolution of the drum. For small (a and for a > 45~, Equation (5.1) can be simplified to ue = tan (P + a) (5.2) e tp Further, the change in velocity from ue to a new steady-state velocity ue' for which the corresponding slope angle is 5' and the slit angle

-97 -is a' on the streak photographs, is simply: Aue Ue - Ue tan (I'+ 2') = = - (513) u u tan ( + ) The original streak photographs obtained with about 5:1 reduction were examined in a tool-maker's microscope and the pertinent angles and distances were measured for the purpose of calculating the velocity ue at the entrance of the test section and the velocity decrement Aue/ue according to Equations (5.2) and (5.3) respectively. The velocity of the detonation wave also was calculated from knowledge of the distance between the ionization probes 1 and 3 and the time of passage of the detonation wave between them. Table IV shows a summary of the results. Strictly speaking, the comparison between theory and experiment should be made using in the latter the velocity decrease from the theoretical velocity. However, it is shown in the appendix that the error involved in Aue/ue is about.8% of the detonation velocity, whereas the error in the measurement of the detonation wave velocity by either the ionization probes or the streak photographs, plus the error due to composition would amount to about 2% of the detonation velocity. If either column (4) or column (6) in Table IV were added to column (7) to obtain the decrement of velocity from the theoretical, then the results would be subject to an error of 2.8% of the velocity. Since the greatest decrement is about 11% such an error becomes an appreciable fraction of the decrement. It is therefore felt that Aue/ue without correction would be more reliable.

-98 -TABLE IV SUMMARY OF THE EXPERIMENTAL RESULTS OBTAINED FOR THE STOICHIOMETRIC H2-02 BOUNDED BY N2 [(ue)th = 9320 ft/sec (Ref. 46] (1) (2) (3) (4) (5) (6) (7) (8) NTumber b (u ) [(uediol [& IUeQ FUe Range in of Runs e in (e)tio Range ine LL(ue)th h (e)t LUe (in) (ft/sec) (ft/sec) 9.2 9074.0264 9199.0121.070.051 -.097 8.25 9138.0195 9090.0245.087.063 -.111 5.3 9224.0103 9168.0160.072.054 -.o98 6.5 9186.0144 9209.0120.040.0235 -.053 (3) Average velocity obtained from ionization probes measurements. (4) Fractional average difference between (3) and the theoretical value. (5) Average velocity obtained from streak camera measurement, before wave reaches the side relief. (6) Fractional average difference between (5) and the theoretical value. (7) Average velocity decrement for wave with side relief obtained from streak pictures. (8) Range from which the average was obtained in (7).

-99 -On the other hand, the velocity decrement in the flame tube due to the effect of the boundary layer should be taken into account. This decrement, calculated by Fay's method(53), amounts to 1.8% for the tube size used in this investigation. It is added to columns (7) and (8) and the results plotted in Figure 330 Shown on the same plot is the theoretical curve deduced from Figure 11 where ~ is calculated according to Equation (2~53). The value of the reaction length x is taken as.138 in0 which is the value deduced by Fay(53) from the work of Kistiakowsky and Kyddo(74) The value of tan 5 was taken from Figure 8 to be.258 corresponding to the density parameter value of 0353. Figure 33 shows that the experimental results are in reasonable agreement with the theory and therefore confirm that the value of the reaction length is.14 in0 to within about 20%. 5o3 Behavior of Detonation Waves at Channel Widths Close to the Critical At channel widths below the critical, it was seen in Section 501, that the detonation wave quenches and degenerates into a shock. A typical streak photograph of a quenching detonation wave shown in Figure 34, beautifully reveals the dynamics of the quenching process0 At about two microseconds after the detonation meets the boundary gas, it starts slowing down0 Then there is some thickening of the reaction zone and finally, it appears that a pure continuously decelerating shock (black curve) travels into the explosive followed by the interface of the shocked but unburned gas with the last layer of the gas that suffered combustion (light, comparatively thick curve), at an ever widening distance~

.14 1 QUENCHED.12.......10 z 0,.08 o.06 o o 0.1 ~~~~~~~~> I I Y I I~0 AVERAGE =' Z.04 IRANGE.04 1= II8 1 j/ I I ~~~EXPLOSIVE STOICH H27-O 0.02 INERT NITROGEN 0 1 2 3 4 5 6 7 8 -I INVERSE OF CHANNEL WIDTH I/b IN. Figure 33. Variation of Detonation Velocity Decrement with Channel Width.

Figure 34. Streak Photograph of a Quenching Detonation Wave in Stoichiometric H2 - 02 with Nitrogen Boundary. Run No. 760 b = 0.1 in. Vert. - x 2.996 in. Horz. dist. equiv. to xl3 = 81.7,isec, approx.

-102 -At channel widths above the critical, it was maintained in the previous section that the detonation wave velocity reaches a new steady state value, as the various streak photographs revealed. To obtain a further proof of this, the velocity of the shock induced in the inert was checked, This was done by photographing the progress of the detonation wave through a slit close to the solid wall of the channel and the progress of the oblique shock in the inert through a slit at about 5 ino away from the separating filmo Such a "double slit" streak picture is shown in Figure 350 A guide for the interpretation of such photographs is presented in Figure 360 Slit 2 is partially blocked off near the top of the test section, for the purpose of interrupting traces due to it, so that it would be possible to identify which trace belongs to which slito The curved trace due to slit 2 near the bottom of the test section seen in Figure 35 is due to the cylindrical propagation of the shock from the point where the detonation wave first "sees" the boundary gas, After this initial curvature it can be seen that the trace of the oblique shock at points correspondihg to where the detonation wave reaches a new steady state has the same inclination as that of the detonation wave trace0 This is true of the interface trace also, although at first glance it does not appear to be so, However, this trace should be examined from the point where the initial curvature subsides to the point that corresponds to the detonation wave position when the latter hits the solid part of the film holder again0 The distance corresponding to these two points is rather short0

Figure 35. Streak Photograph of a Detonation Wave in Stoichiometric H2 - 02 and of Attendant Shock Wave in the Nitrogen Boundary (Double Slit). Run No. 801 b -.5 in. Vert. - x1 = 2.996 in. Horz. dist. equiv. to x13 = 91.86 psec

x, SLIT# I AT t TRACE ORIGINAL IN~~~~~~~~~~~~~~~TERACEDU ORIGINAL INTERFACE i 1/16" INTERFACE TRACE TO NOTCH SLIT # 2 SLIT # I SHOCK TRACE MASK ~, [ I ~~~d' DARKENING DUE TO MASKED D' REGION OF SLIT # 2 / x~ ~ SLIT * 2 SNAPSHOT AT t2 S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ At2 I/_ _ /' S' — ' — ~' - / I/DETONATION TRACE I.... 0 D ~~~I'S~LIT #1 SNAPSHOT /AT t ~ S L IT~ P.AT tI AT t I / AT tI SI t -m4 -- -- - - - - t --— S__ ~~~~I" -— "1TIME SNAPSHOT STREAK Figure 36. A Guide for the Interpretation of '"Double Slit" Streak Photographs.

-105 -At channel widths close to the critical, the trace of the detonation wave in the streak picture showed, at times, some periodic waviness with the detonation wave remaining unquenched. An example of such a situation is shown in Figure 37 in which tte appearance of the wave is very reminiscent of that of a "spinning detonation wave (see, for example, Ref. 7). In a way, this is not surprising, for "spin" is usually associated with composition limits which in turn must necessarily depend on limits in the chemical reaction involved. That side relief imposes some limitation on the chemical reaction is demonstrated by the quenching behavior of the detonation wave. The work of Fay(75) on spinning detonation wave showed that the waviness can reasonably be described as due to the transverse mode of vibration of the gas behind the shock front. It is, therefore, interesting to apply this theory to our work here. The sound frequencies of a column of gas in a rectangular channel with sides b and d, are: 1/2 v = -a + (5.4) nm 2 ~ with the fundamental frequency obtainable by setting n = m = 1. If the velocity of sound a, is taken as the average sound velocity behind the detonation wave then ae2 = aelMel2 (5.5) so that the theoretical fundamental frequency is aelMelO2 (b2 +' 1/2 th - 2b d

-lo6 -i i.i i E: T; 7 E..: g. s- ig: _ i i i.:_:i-i::-ii i_. ---. -........_. —: --:__ __::_:_: _ i-iii:i::.2 --: ':::- __iii~i -..-.-.-..: ----:-:::::::::::..........:. E Figure 37. Streak Photograph of a Detonation Wave in Stoichiometric H2 - 02 with Nitrogen Boundary Showing Spin Characteristics. Run No. 766 b = 0.2 in. Vert. -- =ix13 2. 996 in. Horz. - dist. equiv. to xl3 = 81.7 pisec, approx.

-107 -The observed frequency can be obtained from the observed pitch length, xp, as Uel aelMel v (5 7) xp xp and therefore the ratio of the theoretical frequency to the observed is vth 02 b2 1/2 - = - x (2 l) 1(5.8) From Figure 37 we find xp =.75 in,, and therefore, with b =.2 in., d =,36 in. and ~ =.52 obtained from Table II, application of this equation gives Vth =v.- 1.1 12 (5 9) which indicates that the observed frequency is about 12% smaller than the theoretical, The order of the variation is the same as that observed in spinning detonation in tubes without side relief, Fay's work does not delve into the origin of spin, nor has there been a satisfactory theory on this problem, Since our experimental work shows for the first time, -';hatspin could be created at will by proper choice of the boundary gas and channel widths in explosive mixtures far away from their composition limits, further experimental study along these lines may well prove helpful in solving the initiation (76) problem. As can be implied from the survey of Oppenheim et al. spin initiation may very well have some bearing on rocket motor instability.

5~4 Velocity Decrement of H2-02 Mixtures Bounded by Nitrogen Near Their Respective Critical Channel Widths In order to establish which of the quenching criteria presented in Chapter III is most adequate in predicting the quenching limit, a series of runs were made with various hydrogen-oxygen mixtures bounded by nitrogen, at channel widths very close to the critical. The idea was to determine the maximum velocity decrement possible before quenching takes place and to compare the results with each of the three criteria. Again streak schlieren photography was used and examples of the results for the leanest and richest mixtures tested are shown in Figures 38 and 39 respectively. Table V below gives a summary of the results obtained on the measurement of the velocity decrement using the same method as Section 5o2o To find the critical Mach number from the experimental data, the average velocity decrement column (9) was subtracted from the theoretical Mach number column (11) with the results shown in column (12)o Column (13) shows the range in the critical Mach number based on the range in the velocity decrement. These results which were corrected by a velocity decrement of o018 due to the boundary layer effect (see Section 5~2) are plotted in Figure 120 They show best agreement with Belles' explosion limit criterion (with the old chemical kinetics data) not only in magnitude but in trend as well0 It might be argued that either Patch's constant temperature criterion or Shchelkin's instability criterion could be made to give a reasonable fit to the experimental data by simply adjusting the critical temperature in the former and the value of the activation

Figure 3-8. Streak Photograph of a Detonation Wave in 46.3% H2 - 53.7% 02, with Nitrogen Boundary. Run No. 1014 b = 0.5 in. Vert. - x13 = 2.996 in. Horz. - dist. equiv. to x13 = 79.35 pUsec

-110 -Figure 59. Streak Photograph of a Detonation Wave in 75% H2 - 25% 02, with Nitrogen Boundary. Run No. 976 b =.25 in. Vert. - x13 = 2.996 in. Horz. - dist. equiv. to x13 = 78.60 Lsec

TABLE V SUMMARY OF THE EXPERIMENTAL RESULTS OBTAINED FOR H12-02 MIXTUEES IBOUNDED BY N2 (1) (2) (3) (4) (5) (6) (7) (8) (9) (0lo) (11) (12) (13) No. of (Aue)ion Au Au e Range in Range in % H2 Runs b (Ue)th (Ue)ion K l Aue (MUe)th Mec M Lue)th LethU ue (in.) ft/sec ft/sec 75 5.25 10,430 10,285.014 10,278.015.065 5 5.26 4.92 4.98-4.86.075.051 66.7 9.2 9,320 9,075.026 9,199.012.070 097 5.28 4.91 5.01-4.77.097.052 60 5.2 8,560 8,436.015 8,490.008.077 052 5.24 4.84 4.97-4.65.113 55 4.25 8,050 7,690.045 7,808.031.086.078 5.19 4.75 4.79-4.70.096 51.8 5.35 7,770 7,481.037 7,584.024.075 540 5.16 478 4.88-4.70.059 46.3 5.5 7,320 7,035.039 7,305.020.078 5.09 4.69 4.79-4.53.110' (4) Figure 47 (5), (6), (7), (8), (9) and (10): Similar to (3), (4), (5), (6), (7) and (8) respectively in Table IV. (12) = (Me)th - e e u e

-112l energy E2 used in the latter, since these are not known with considerable certainty. Doing this, however, would still give in both cases a constant critical Mach number and thereby the trend of the experimental data would not be followed. Further, the composition limits that would be predicted under these circumstances would be far removed from those observed. In experiments conducted at lower than atmospheric pressures such as those on the standing detonation wave reported by Nicholls and Dabora(62) it was found that ignition did not take place except under conditions compatible with Belles' criterion, A further support to plausibility of the explosion limit criterion is the observation made by Lewis and von Elbe(77) that composition limits in H2-02 can be widened if a special effort is made to dry the mixture~ They point out that this is indicative of the role of the chain breaking reaction IV and the high efficiency of water as third body (see Equation 3~9) in this reaction. In this regard, the experimental results reported here which show that quenching takes place at lower Mach number decrement than the theory predicts can be interpreted to be due, in part, to the water contamination, as no special precaution was taken to pre-dry the gases any further than as supplied. Neither Shchelkin's nor Patches criteria take into consideration any composition effect0 Further, Patch's criterion is probably applicable to mixtures which are initially at pressures near atmospheric, whereas Shchelkints criterion is independent of pressure0 Since the results presented here which show a definite composition

113 - effect, and since the results in (62) for initial pressures much lower than atmospheric are in good agreement with the explosion limit criterion, it is concluded that it is basically adequate in describing the conditions under which detonation waves in H202 mixtures could be self-maintained, 5~5 Estimation of the Reaction Length in the Stoichiometric Methane Oxygen Mixture The theory presented in Chapter II indicates that if data on the velocity decrement of an explosive when confined by a known inert medium are available, they can be used to infer the reaction length of the explosive, This is what was done in Section 5.2 for the stoichiometric hydrogen oxygen mixture with encouraging results. Stoichi-ometric methane oxygen was chosen as another explosive mixture for the purpose of calculating its reaction length. When the widest available channel width was used with nitrogen as boundary, it was found that the reaction zone thickened and the detonation wave tended to quench0 As the theory predicts, the situation was improved when carbon dioxide, a gas heavier than nitrogen, was used, More improvement was experienced when butane was used as the confining gas. Typical streak photographs for the above three conditions are shown in Figure 40[ In the butane case, the detonation wave still seemed to be thick but nevertheless it was considered unquenched because the velocity appeared constant near the top of the test section~ The velocity decrement calculated from several photographs ranged from o081 to ~093 with an average of ~088~ Thus the total velocity decrement becomes 106 after decrement due to boundary layer effect is added

(a) Run No. 1018 Boundary Gas: N2 b = 0.5 in. Vert. - x3 = 2.996 in. Horz. - dist. equiv. to x3 = 77.92 isec (b) Run No. 1020 Boundary Gas: C02 b = 0.5 in. Vert. - xl3 = 2.996 in. Horz. - dist. equiv. to x3 77.92 isec Figure 40. Streak Photograph of Detonation Wave in Stoichiometric CH4 - 02 and of AttendJant Shock Wave in the Boundary Gas (Double Slit).

-115-?I! -!i - * iiA -i (c) Run No. 1022 Boundary Gas: C4H10 Channel width = 0.5 in. Vert. - x = 2.996 in. Horz. - dist. equiv. to xl = 79.sec Figure 40. ( Continued)

-116 (as explained in Section 5~2)~ From Figure 11 such a velocity decrement corresponds to an incremental area increase ~ of o176o Using Equation (2o53), after finding tan 6 to be.272 from Figure 8 the value of 5325 in. is obtained for the reaction length of the stoichiometric methane-oxygen, which is about twice that of the stoichiometric hydrogen-oxygeno At this point, it is interesting to look at the results of Wagner(78) on detonations of mixtures of city gas* and oxygen conducted in paper tubes and examine them in the light of the findings on reaction lengths in this study, Wagner, by "specially treating" the paper tubes, was able to change the distance Xb, behind the detonation front at which they break. He finds that the detonation velocity in the paper tubes to be less than that in a solid tube with the decrement in velocity increasing with decreasing xb, and quenching setting in when xb =.4 in. His expression for the area increase due to the paper motion is = 10 3 t2 (5o10)** where t is the time (in isec) after the passage of the detonation wave, By assuming the time in which most of the reaction in detonation is completed is about 1 Usec, he concludes that the area increase, being thus very small, cannot alone explain the relatively large velocity decrements he obtains, and therefore the flow conditions behind the C-J plain must be influencing the propagation velocity0 *Composition not reported0 Main constituent is assumed to be methane~ **Expression (5o10) corresponds to a paper thickness of about o001 in,

-117 -An alternative explanation of Wagner's findings is offered as follows. Because of some impurities in city gas, it is reasonable to assume that its reaction length is higher than that of methane0 Such a reaction length would mean a reaction time of 6-9 Usec, which in turn gives ~ = ~04 - ~08~ Such an area increase corresponds to 2-5% velocity decrement which is what Wagner obtains at xb slightly larger than o4 ino This explanation of Wagner's results seems to be very plausible as it does away with the unlikely postulation that perturbations behind the C-J plane can influence the propagation velocity of the detonation wave. 5.6 Behavior of Detonation Waves When the Density of the Boundary Gas is Much Lower Than That of the Explosive In agreement with theory, it was demonstrated in the previous section that the denser the boundary gas the better the confinement of the detonation wave0 The limit of course is a solid boundary, inside which a detonation wave travels without loss in velocity0 The question arises as to what happenC to a detonation wave when a gas much lighter than the explosive is used as the confining medium0 Such a situation can usually arise in a rotating detonation wave rocket motor mentioned in the introduction0 After the first cycle, the burned gas is pushed by the fresh charge thereby acting as a confining medium for the detonation wave in the next cycle0 The burned gas is, of course, expanded from the Chapman-Jouguet pressure to some lower pressure; however, it would stillbe sta higher temperature than the injected charge and therefore would be much lighter than the charge~

-118 -Figure 9 which was derived using the shock tube analogy, indicates that when the density parameter is about 14, the shock angle in the inert becomes nearly 900o Such a shock angle with flow deflection downstream of it is not possible unless the shock is detached from the leading edge of the deflection plane. In actuality, detachment arises at shock angles ranging between 62-78~ depending on the detonation Mach number and the density parameter0 This situation was first recognized by Sommers(31) who was able to show photographically that indeed when the inert gas is light enough a detached shock can develop0 The question that needs to be answered here is what effect, if any, has the condition of shock detachment on the quenching criterion. Some experiments were carried out with H2-02 and CH4-02 detonations bounded by either He or H20 Typical spark photographs are shown in Figures 41 and 42~ In the case of 75% H2-25% 02 bounded by He (Figure 41b), one can see that the shock is still attached, whereas, in the case of the same explosive bounded by H2 (Figure 41a), the shock appears to lead the detonation wave. Finally, in the case of 33o3% CH4-66o7% 02 bounded by H2 (Figure 42) the shock is well ahead of the detonation wave and can be seen to be inducing an oblique shock into the explosive mixture0 For all the conditions represented by these photographs calculations would indicate that quenching of the detonation wave in the explosive should have taken place0 Why then do nore of these photographs show any quenching? A qualitative explanation can be offered as follows0 When the shock is detached it runs ahead of the detonation wave0 Now the

-119 -l >~ ~ ~~~~~~~~~b Fiur 1.SakPoorpsoaDentioWaei ~~~7 2 - 25 2 b=03i. (a) R_ o 18 - ydrgnbudr Ru o 19- Heimbudr

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-121 -density of the inert gas behind the shock is higher of course than its initial density, so that what the detonation wave "sees" essentially, is this higher density gas0 This in turn means a lower deflection angle and therefore a lower area increase and a lower Mach number decrement0 Therefore, it is possible that the steady-state detonation Mach number could still be higher than the critical0 Hence no quenching appears to take place. Despite these circumstances it is still reasonable to assume that quenching could be accomplished if the channel width is sufficiently reduced, This was done in the case of 75% H2-25% 02 bounded by hydrogen0 A typical photograph is shown in Figure 43 where it can be seen that the shock in the boundary gas leads by an appreciable distance the reaction zone in the explosive0 Further one can see the oblique shock induced by the shock into the explosive more clearly than could be seen in Figure 41, Also distinguishable is a new interface pushed into the explosive0 Needless to say, the flow pattern is very complicated0 Unfortunately the rotating drum camera was not available when tests of conditions similar to those of Figure 43 were being conducted0 However, tests where pictures at several time delays were taken when H2 and He were used, showed a considerable decrease in the velocity of the detonation wave0 The data is plotted in Figure 44, showing that the "reaction zone" propagation was at about 50P of the detonation velocity, Because the accuracy of the relative time delay is in question, it is not possible to say with certainty

-122 -Figure 43. Spark Photograph of a Detonation Wave in 75% H2 - 25% 02 Mixture with Hydrogen Boundary, Showing an Oblique Shock in the Explosive Resulting from a Leading Shock in the Inert. (b -.02 in.) (Run No. 1193)

3 () 0 0. x LL 0 2 2 |SLOPE = 5730 FT/SEC. CHANNEL WIDTH = 0.2 IN. 0c IL Z / Od HELIUM BOUNDARY wI o El0 HYDROGEN BOUNDARY za~~~~~~ 1, w X SLOPE 5210 FT./SEC. u) Li. W 0 70 80 90 100 110 RELATIVE TIME DELAY-/- SEC. Figure 44. Detonation Wave Progress of 75% H2 - 25% 02 When Confined by Helium or Hydrogen.

-124 -at this point, whether the detonation is continuously decreasing in velocity or whether it reached a steady state. However, it is interesting to mention that Voitsekhovskii(35) reports a steady detonation velocity in an annular channel at half the C-J velocity0 The situation in the annular channel is such that the burned gas is on the side of the fresh unburned gas acting as a low density boundaryo If this is more than a mere coincidence, it must mean that a rather new and interesting mode of detonation wave is possible, It should be added that such a mode is not completely inconsistent with the quenching criterion, for by inducing an oblique shock into the explosive, the detached shock of the inert essentially increases both pressure and temperature of the explosive. This can mean that the explosion limit criterion can be attained at somewhat lower Mach number than would be needed when the explosive is not so "prepared". Thus it would appear that a low density gas, when used as an inert boundary, can provide a mechanism by which some of the energy of the detonation wave could be transferred ahead of the detonation front0 5~7 Experimental Interface Deflection Angles and Inert Shock Angles As a check on the degree of accuracy with which the shock tube analogy can predict the interface deflection angles as well as the shock angles in the inert, measurements of these angles from spark schlieren photographs were made. The data for the deflection angles are plotted in Figure 45 where the theoretical deflection

z z z w ' W -.W C(,CD LLCD _ 0 w ZOZO WC0 0 w w 0 20 020 0r~1 0 %4%. - w0 0 w -- I:Z:: ~~~~~~~-~~~~~_::Z QI O O o ~ ~ 0. O e O e 0 to 00: 0 - 1i 0 Li.I I,,. I0 I. I w 40 I4 CIO I\ CD z 0 bz.5 IN w30 -— M=5 DETONATION CONDITION z 10 M=5 SHOCK CONDITION jTER 0 pi, 11~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 5 b=. 5 INg. DENSITY PARAMETER I~~~~~~~~~~~~~~~~~~~~~~~~ ~'..n' Elb=.3 IN. o 20 ~ --- ~, b=.25 ILN.Th 2~~' w M5DETONATION CONDITION IHER Z, M 5SHOCK CONDITION I0 ~ 0.5 1.0 1.5 2.0 DENSITY PARAMETER FPc'~1' '( / Figure 45. Experimental Results of Interface Deflection Angles.

-126 -40 ANGLE MEASURED NEAR X DETONATION WAVE I 30,, U) w C!, w z 20 z 'r (n L O~nr b.5 IN. o O b=.251N. A b.201IN. wZ 'Z. 0Z ozw Wm (0 D-( U. 0 ILLJL 0.I.2.3.4.5.6.7 ~~qDENSITY PARAMETER P: ai Inert Gas. Ole rZ ) U.) 0 0Le Figre 6. xpeimetalResltsof hoc Wae Agle inth

-127 -angle is also shown, Reasonable agreement is apparent throughout the range of conditions tested. Agreement appears, in general, to be excellent when the explosive channel width is large, i eo, when conditions are above and far away from the quenching limit. The data for the shock angle in the inert for limited test conditions, are plotted in Figure 460 They indicate good agreement with theory again. Cases in which the shock leads the detonation wave are not plotted because the shocks were curved and it was impossible to determine an "average angle." It is concluded on the basis of these measurements that estimation of area change near the detonation zone by use of the shock tube analogy results in a satisfactory agreement between theory and experiment.

VIo CONCLUSIONS 1. The shock tube analogy is successfully used to predict the shock angle induced by a gaseous detonation wave into the inert compressible medium that confines it, It can also predict the interface angle between the expanded burned gas behind the detonation wave and the shocked gas in the inert. 2o The parameter that controls the extent of the deflection angle contains essentially the ratio of the density of the explosive to that of the inert. The higher this ratio is, the larger the dew flection angle. 30 The effective area change of each stream tube between the shock plane of the detonation wave and the Chapman-Jouguet plane, can be calculated from a knowledge of the deflection angle, the explosive channel width and the reaction length. 41 It is possible to predict that an area increase in the reaction zone reduces the detonation velocity. Thus, if the velocity decrement of a detonation wave in a known explosive of known channel width, confined in a known inert can be determined experimentally, the rem action length. can be inferredo This is done in the case of a stoichiometric H2-02 mixture and a stoichiometric CH4-2 in which the reaction lengths are found to be o14 ino and ~32 in. respectively. The inferred reaction length of the H202 mixture agrees with the value reported in the literature. 50 Using Belles' explosion limit criterion it is possible to predict, for the H2-02 mixtures, a critical detonation velocity below which -1280

-129 -The detonation is expected to quench, Good agreement between this theory and the experimental results is obtained. It is found that a velocity decrement larger than 8-10% of the theoretical velocity can lead to the quenching of the detonation wave. 6. Near the quenching limit, the detonation wave propagates, at times, like a spinning detonation wave so common at composition limit. This is the first time that a "spin" could be induced in gas mixtures away from their composition limit and the method is therefore deemed convenient for the study of the origin of spin. 7. When the inert boundary gas is much lighter than the explosive, the induced shock is detached. In some instances, it appears that such a condition can provide a mechanism whereby energy from behind the detonation wave can be transferred ahead of it. In such situations, detonation waves traveling at about half the Chapman-Jouguet velocity have been observed experimentally. 8. A technique for minimizing diffusion between the boundary inert gas and the explosive gas by separating them by a thin film is developed. It is found that in order for the film itself to have a negligible effect on the interaction process being studied, the ratio of its thickness to that of the reaction length should be one-tenth the ratio of the explosive density to that of the filmo

APPENDIX A ERROR ANALYSIS The calculation of the detonation velocity and the velocity decrement are subject to errors due to measurements as well as explosive composition. It is therefore necessary to assess the accuracy of the reported calculations. According to Reference 79, the error in a quantity y, where y = F(xl, x2,.O, xn) (A-l) and be calculated by the following equation: n 2 (y)2 = (F ) (dxi)2 (A-2) which will be used belowo A.1 Error in Velocity Measurement from Streak Photographs The detonation velocity from streak photographs can be calculated according to Equation (5o1). This can be written, after assuming a = 0 and noting that the magnification is obtained by measurements of two distances, as ue = t tan B (A-3) where =o = distance between two fixed points on the test section Yi = distance between the corresponding points on the image. Applying Equation (A-2) to (A-3), one obtains due-2 dL2 deo)2 dei 2 dt 2 d tan 2/2 u() [I(~) +( ) + +( + tan-) I7 (A-4) -130 -

-131 -In using the tool maker's microscope for the measurement of angles and distances on the photographs, angles to within 0,1o and distances of,...5 ino to within.002 in. could be made. Since d tan P _ do (A-5) tan P sin B cos P and since P = 70o9 then d tan B/tan P =.0055. Further with, dL = ~002 (ability to measure drum perimeter of L 2.5 in, to within.025 in.) di o=.001 (measurement of 3 in. to within.003 in.) 'o =.004 (ability to define fixed points on the photographs when viewed in the toolmaker's microscope) and dt_ =.003 (measurement of the drum camera period by tp the Hewlett-Packard timer as reported by the manufacturer) one obtains du 2 (X =.775% (A-6) Ao2 Error in Velocity Measurement from Measurement by Ionization Probes The velocity in this case is calculated as ue = t3(A-7) tl3 where ~13 is the distance between two probes which can be estimated to within 1/16 in. in - 12 in., and tl3 is the time for the detonation wave to pass from probe 1 to probe 3 as measured by the Berkley

-132 -timer, to an accuracy of 1 Usec in 100 tsec, Therefore, d3 = o0052 and dtd.01 i13 t13 which gives: (Ue) = 113% (A-8) Ue A.3 Error Due to Composition The hydrogen oxygen mixture is prepared by the partial pressure method. The accuracy of the gauge used combined with the accuracy of reading it is about.5 psi. Usually a total pressure of 70 psia is attained during the preparation of each mixture. The mole fraction of H2 is fH2 = Ptot(A-9) For 40% H2 - 60% 02 mixture then dpH2.5 - 0178 (A-10) PH2 70 x.4 and for 75% H2 - 25% 02 mixture 2dPH _ o5 = 0095 (A-11) PH2 70 x.75 But, dPtt = ~ 5.0072 (A-l2) Ptot 70 so that fH2

-133 - 120 LLJ t) i0,, j 8 w z! REFERENCES (46,57) 0 O 6 5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 HYDROGEN MOLE FRACTION Figure 47. Theoretical Detonation Wave Velocity of H2 - ~2 Mixtures at One Atmosphere and 535~R.

The theoretical detonation wave velocity as calculated by Gordon(57) is shown in Figure 47 as a function of the hydrogen mole fraction. From this figure and the possible error in the mole fraction of H2 one can deduce that the possible variation in detonation velocity can be: du 2 (e)= -9% 1.3% (A-14) ue Ao4 Error in the Velocity Decrement as Measured from Streak Photographs When a = a 0, the velocity decrement in Equation (5-3) become s Aue 1 tan ~ (A-l) ue tan Applying Equation (A-2) and noting that tan D 2tan B', one obtains the following expression for the error in the velocity decrement: u- 2 2 X= e)= X 2( tan P) (A-16) e With the value of the error in tan X calculated to be.0055 in Section A.l, this becomes \d(u) =.78% (A-17)

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139 -55~ Gvozdeva, Lo G. "The Refraction of Detonation Waves Incident on the Boundary Between Two Gas Mixtures'" Soviet Physics, Techo Physo, 6, No. 6, December, 1961. 56. Shchelkin, K. I. "Two Cases of Unstable Combustion " Soviet Physo JETP, 36 (9), No. 8, August, 1959o 570 Gordon, S. Private Communication. 58. Belles, F. E. Detonability and Chemical Kinetics Prediction of Limits of Detonability of Hydrogen~ Seventh Sympo (Int'l) on Combustion, Butterworths Scientific Publications, London. (1959) 7450 59. Lewis, Bo and von Elbe, Go Combustion, Flames and Explosions of Gases. New York~ Academic Press Inc, 1951, Chap. I and Ii. 60o Clyne, M. A. A. Rates of Some Atomic Reactions Involving Hydrogen and Oxygen. Ninth Symposium on Combustion, Academic Press, New York, (1963), 211L 61. Baldwin, Ro Ro Comments on Reference 60, po 218. See also pc 667o 62. Nicholls, Jo A. and Dabora, Eo Recent Results on Standing Detonation Waveso Eighth Symposium on Combustion, The Williams and Wilkins Coo, Balt;imore, Mdo, 1962. 63. Patch, R. WO "Prediction of Composition Limits of Hydrogen-OxygenDiluent Mixtures~ T" ARS Journal (1961), 46-51. 64~ Gordon, Wo Eo, Mooridian, Ao Jo and Harper, So Ao Limit and Spin Effects in Hydrogen-Oxygen Detonationso Seventh Symposium on Combustion, Butterworths Scientific Publication, London (1959), 752-759o 65o Breton, Mo Jo "Recherches sur la Detonations des Melanges Gaseuxo" Annales de lOffice National des Combustibles Liquides, 11 (1936), 487-548. 66. Lafitte, P. F. "Flames of High Speed Detonation." Science of Petroleum, IV, Oxford Univo Press, London (1938), 2995-3003o 67. Hall, C. Eo Introduction to Electron Microscopyo McGraw-Hill Book Co., Inc.,, New York, 1953, Chap. 11. 68. Peachy, L. D. "Thin Sections I. A Study of Section Thickness and Physical Distortion Produced During Microtomyo" J. Biophysic and Biochemo Cytol., 4 (1958), 233. 69, Carslaw, H. So and Jeager, J. C. Conduction of Heat in Solids. 2nd Ed., Oxford Univo Press (1959) 8......

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