COPYRIGHT, 1950BY

JOHN WILEY & SONS, INC.

AU Righ~ Reserved

Thu book or any part thereof must notbe reproduced in any form withoutthe written permisswn 0/ the publisher.

PRINTED IN THE UNITED STATES OF AMERICA

To the memory ofthe founder of interior ballistics

BENJAMIN ROBINS, F.R.S.1707-1751

Preface

It was in 1944 that Dr. .J. W. Maccoll suggested to the writer thatthere was need for a modern textbook on the theory of the inte riorballistics of guns. The project could not be taken up at that t ime, butour discussion of the plan showed that the subject was ready for an up-to-date survey.

Probably the most st riking general result that has emerged from thewar time work on guns is the complexity of the physical and chemicalprocesses involved. It is, of course, not at all obvious that this needbe so. The system of gun, projectile, and charge appears at first sightto be surely one of the simplest of all heat engines. It was from thisidea that interior ballistics, about a hundred years ago, began to developtowards a mathematical st ructure based on a few simple physicalassumptions. Gradually the mathemati cs became more elegant, andmore subtle points were discussed. As there was relatively little studyof the physical assumptions, this mathematical work could not be saidto be helping greatly our understanding of the behav ior of real guns.

But now, a few years later, th eoretical interior ballistics has com-pletely changed its direction. This we must acknowledge to be duein the main to the use of modern instruments of research. Th ese haverevealed the complexity of the processes inside the gun and have reducedsome of the basic ideas previously held to the status of mere empiricalapproximations. Again, the study of gun behavior under extreme condi-tions has led, as could have been expected, to abnormal behavior ; themore important aspect is that these abn ormalities can be traced in modi-fied form right into the region of normal conditions. This explains whyso many aspects of inte rior ballistics have eluded theoretical analysis,and have been left to "past experience" acquired over many years andreliable only in a stagnant technique.

When thc physical basis of a subject becomes more complicated , themathematics becomes simpler. When four equations are thought tosum up the whole of interior ballistics, it is natural to spend almostany length of time on their study. If, however, the subject is seen asthe interrelation of a dozen different aspects, it is clearly best to keepas far as possible to simple mathematics.

In writing this book I have t ried to keep the best of the classicalvii

viii Preface

phase while introducing also some of the methods that promise to be mostvaluable in the future development of the subject. It is, fortunately,not yet possible to write of ballistics as a complete and perfect structure.The new complexity implies new possibilities of rational understandingand corresponding practical advances, and we can expect research togo ahead vigorously. I have indicated the lines on which theoreticalresearch is likely to be concentrated and have kept these in mind inchoosing topics. It must be added, also, that the best classical theory,with its tendency to oversimplification of the physics and chemistry,is for that reason a valuable introduction to the subject and would earna place in this book on that ground alone.

Permission for open publication has been kindly granted by the ChiefScientist, Ministry of Supply (England) . For permission to reproducematerial from my papers published in their journals I am: indebted tothe Royal Society (sections 2.33, 7.1-7.39, and 9.2-9.26, Figs. 7.1, 7.3,and 9.5-9.9), the Faraday Society (sections 2.41-2.44), the FranklinInstitute (sections 8.1-8 .16 and Fig. 8.1), the Physical Society of London(sections 3.24 and 3.32), and the Quarterly Journal of Mechanics andApplied Mathematics (sections 5.5-5.54). I am able to use Figs. 9.1and 9.2 by the courtesy of Mr. F . B. Pidduck and the Royal Society.I am indebted to Mr. E. P . Hicks and Mr. C. K. Thornhill for permissionto quote freely from their theory of heat transfer to gun barrels.

To my wife I am indebted not only for encouragement but also forsubstantial help in the preparation and editing.

J. CORNER

Contents

Chapter One: The Field of Interior Ballistics, 1

1.1 The scope of the present book . . . . . .1.2 "Practical" and "research" ballistics

1.21 The measurement of pressure in guns1.22 The measurement of muzzle velocity1.23 Other experimental research

1.3 The classical problem of interior ballistics .1.4 The future of interior ballistics . . . . . .1.5 The history of the classical problem of interior ballistics

Chapter Two: Gun Propellants, 24

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11141618

2.1 Propellant compositions . . . . . . . . . . . . . . 242.2 The arrangement of the propellant charge . . . . . . 26

2.21 Propellant shapes and the geometrical form function . 302.22 The true form function 352.23 The Charbonnier form of the equation of burning 41

2.3 Theories of the burning er gun propellants 422.31 Surface theories . . . . . . . 432.32 Vapor-phase theories 472.33 Theory of a flame zone in a gas . . 492.34 Application to the burning of propellant .632.35 The theoretical rate of burning at high preseures 662.36 The influence of the initial temperature of th e propellant . 692.37 Experimental knowledge of the rate of burning . . . . . 70

2.4 "Erosion" of propellant . . . . . . . . . . . . . . . . . . 732.41 A simple theory of the erosion of propellant at gun pressures 742.42 Hydrodynamic considerations . . . . . . . 762.43 Effective thermal conductivity in the flame. 812.44 Rate of burning with turbulence . . . .. 82

Chapter Three: The Thermochemistry of Propellants, 85

3.1 Sketch of closed-vessel technique . . . . . . . . . . . . . . . 873.11 Cooling corrections . . . . . . . . . . . . . . . . . . 85

3.2 Theory of the equilibrium state after burning without cooling or perform-ance of work . . . . . . . . . . . . . . . . . . . . 893.21 Theory without dissociation or pressure corrections 893.22 Dissociation . . . . . . . . . . . . . . . . . 973.23 Experimental results . . . . . . . . . . . . . . 1003.24 Theory of the covolume and other pressure corrections . 102

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3.3 The equilibrium state after burning under constant pressure withoutcooling . . . . . . . . . . . . . . . 1153.31 Theory without pressure corrections 1153.32 Pressure corrections . . . . . . . 1173.33 Specific impulse of a rocket fuel . . 120

3.4 Thermal behavior of the propellant gases in a gun 125

Chapter Four: Simple Ballistic Methods, 130

4.1 Introduction . . . . . .4.2 The "isothermal" solution .

4.21 Notation. . . . . .4.22 Equations of motion .4.23 Solution after "burn t"4.24 Summary of the working formulas .4.25 The efficiencies of gun and charge

4.3 Comparison with experiment . . . .4.31 Some typical ballistic solutions .

4.311 A typical AA gun . . . .4.312 Cord charges in a naval gun

4.32 Empirical corrections . . . . . .4.4 Ballistic effects of charge and design variables

4.41 The web size . . .4.42 The charge weight4.43 The total travel . .4.44 Chamber capacity .4.45 Shot weight4.46 Propellant shape4.47 Nature of propellant .

4.5 The history of the isothermal model .

Chapter Five: More Advanced Ballistic Methods, 174

130132132133139141142145145146154156159159162169169169170171171

5.1 The energy equation of interior ballistics 1755.2 Coppock's solution . . . . . . 177

5.21 Assumptions . . . . . . 1775.22 Equations of the problem 1775.23 Solution of the equations . 1785.24 Summary of the working formulas . 1825.25 The nature of the effects produced by covolume and kinetic-energy

terms . . . . . . . . . . 1845.26 Comparison with experiment . 1865.27 The effect of heat 10BBCs . . . . 189

5.3 A theory with a "shot-start pressure" 1895.31 Assumptions . . . . . . . . 1915.32 Goldie's solution . . . . . . 1925.33 Ballistic effects of a shot-start pressure . 204

5.4 Rate of burning not proportional to pressure . 2065.5 The ballistic effects of bore resistance . . . . 213

5.51 Disturbance of muzzle velocity by the standard bore resistance 215

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5.52 Accuracy of first-order theory . . . . . . . . . . . . . . 2205.53 Calculation of the effect of long stretches of bore resistance . 2225.54 Effect of thc covolume. . . 223

5.6 Numerical and mechanical methods . . . . . . . . . . . . . . 223

Chapter Six: Similarity Relations and Optimum Problems ofInterior Ballistics, 226

6.1 Introduction to ballistic similarity .. . . . . . . . . . . . . 2266.11 The similarity relations for a simple set of ballistic equations 2276.12 The tabulation of ballistic solutions ·235

6.2 Optimum ballistic solutions . . . . . . . . . . . . . . . . . 240

Chapter Seven: The Interior Ballistics of Leaking Guns, 243

7.1 Introduction . . . . . . . . 2437.2 The classical theory of nozzles 246

7.21 Covolume corrections . 2507.22 Thrust on a nozzle. . . 251

7.3 The equations of interior ballistics of a leaking gun 2537.31 Assumptions of the theory . . . . . . . . 2537.32 Notation. . . . . . . . . 2557.33 Pressure and density distributions in the gun 2557.34 Nozzle flow and energy relations . . . . . 2587.35 Summary of the equations . . . . . . 2607.36 The equivalent nonleaking ballistic problem 2617.37 Numerical integration . . . . . . 2667.38 Solution with linear rate of burning 2677.39 Solution after "burnt" . . . . 274

7.4 Gas leakage in a smooth-bore mortar . . 2767.5 The ballistics of a worn gun . . . . . . 279

7.51 Equations of interior ballistics of a worn gun 2817.511 Basic assumptions 2817.512 Notation 2827.513 Equations of the problem 2827.514 Solution after "all burnt" 2847.515 Approximate solution ncar the start . 285

7.52 Analysis of the experimental data on new and worn guns. 2877.521 Ballistics of the new gun 2887.522 Ballistics of the worn gun . . . . . . 289

7.53 Changes in ballistics during the life of the gun 2937.54 Rapid estimation of the effect of leakage 294

7.6 Ballistic properties of recoilless guns 2957.61 "Hecoillessness" 296

7.611 The calculation of recoil momentum 2977.62 The influence of design variables and loading conditions on the

interior ballistics of a typical recoilless gun . . . . . 2997.621 Effect of nozzle-start and shot-start pressures 3007.622 Effect of rate of burning or web size ... . 304

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7.623 Effect of charge weight on nozzle throat area for zero recoil 3057.624 Effect of shot weight on nozzle area for zero recoil 3067.625 Effect of changes of chamber capacity . 3067.626 Changes in total shot travel . 3077.627 Effect of nature of propellant . . . . 3077.628 Influence of nozzle design . . . . . . 308

7.63 The optimum pressure for opening of the nozzle . 3087.64 The value of a muzzle brake on a recoilless gun . 309

Chapter Eight: Some Special Types of Gun, 312

8.1 The high-low-pressure gun 312s.n Notation . . . . . . . . . . . . . . . . 3138.12 Assumptions . . . . . . . . . . . . . . 3148.13 Equations of interior ballistics up to "burn t" 3168.14 Interior ballistics after "burnt" . . . . . . 3238.15 Summary of the working formulas . . . . . 3248.16 The maximum possible piezometric efficiency 326

8.2 Small arms 3278.3 Composite charges . . . . . . . . . . . . . . 328

8.31 Reduction of a composite charge to a single effective web size . 3298.4 Tapered-bore guns . . . . . . . . . . . . . . . . . . . 334

8.41 The equations of interior ballistics for a tapered-bore gun. 336

Chapter Nine: The Hydrodynamic Problems ofInterior Ballistics, 339

9.1 Introduction to Lagrange's ballistic problem 3399.ll The conventional solution 3429.12 The results of Love and Pidduck . . 3479.13 Pidduck's limiting solution . . . . . 3519.14 Experimental results and future theoretical research 3579.15 The shock-wave equations . . . . . . . 3599.16 The maximum possible muzzle velocity 361

9.2 The interior ballistics of a gun after shot ejection 3649.21 The initial state of thc gas in the barrel 3659.22 Earlier work . . . . . . . . . . . . . 3689.23 Discussion of the problem . . . . . . . 3709.24 Solution before the arrival of the wave from thc muzzle 3719.25 Comparison with an experimental breech pressure record 3749.26 The rarefaction zone . . . . . . . . . 377

9.3 Muzzle brakes . . . . . . . . . . . . . . 3839.31 The calculation of muzzle-brake thrusts 3859.32 The "efficiency" of a muzzle brake 387

9.4 Conditions ahead of the shot . . . . . . . . 3929.41 Steady motion . . . . .. . . . . 3939.42 Riemann's method for the solution of the hydrodynamic equations 3959.43 Accelerated motion . . . . 397

9.5 The motion of propellant in a gun 4009.51 Ballistic consequences . . . 407

Contents

Chapter Ten: Heat Transfer to Gun Barrels, 409

10.1 Introduction . . . . . . . . . . .10.2 The heat-transmission coefficient . .10.3 The conduction of heat in the barrel10.4 Results of the heat transfer in guns .

10.41 Semiempiri cal formulas for heat loss

Appendix A: The Numerical Solution of tbe Equations ofInterior Ballistics, 427

A.1 Method of solution on a calculating machineA.2 Solution without a computing machine . . .

Appendix B : Some Constants Used in Ballistics, 432

Appendix C: Interpolation Coefficients, 433

Name Index, 435

Subject Index, 439

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