style="font-size:15px;"> The eighth chapter describes nonsteady propellant combustion regimes in the combustion chamber of a rocket engine. There are three time scales that are relevant for this problem: the thermal relaxation time of the heated layer of the condensed phase tc, the acoustic time ta, and the time of combustion products efflux from the chamber tch.
If the relaxation time of the condensed phase is close to the efflux time, tc∼tch (which occurs in small engines at low pressures), then such regimes may be called nonacoustic. Time scales in such problems are much larger than the acoustic time. This area of research may also be referred to as propellant combustion in semi‐enclosed volume.
On the other hand, over the last few decades a specific and dedicated area of research which may be termed ‘acoustics and combustion’ has taken shape. It deals with the case where acoustic time is close to the condensed phase thermal relaxation time ta∼tc, which leads to a possibility of sonic (in the general case nonlinear) oscillations development in the engine. The latter relation between the time scales applies to engines of large size with high pressure values in combustion chambers. The theory of such processes is still at a rudimentary stage. As an example, possible combustion regimes in a solid rocket engine with end burner grain geometry are investigated. A set of equations which allows the interaction between combustion and acoustic processes in a combustion chamber to be modelled is presented. The specific feature of the problem is the existence of the two distinctive time scales, namely the acoustic time and the time of pressure oscillation amplitude variation. These time scales differ by approximately three orders of magnitude, which demands high computational accuracy. A simpler solution method is developed in the quadratic approximation with respect to the amplitude of oscillations. This method accounts only for the effects related to the time scale of oscillation amplitude variation. Numerical results are obtained for the simplest propellant combustion model in the absence of entropic waves in combustion products. Stable and unstable combustion regimes are identified. In the latter regime, nonlinear effects may trigger shock waves in the combustion chamber.
The possibility of expanding the theory beyond the phenomenological framework is discussed in the final chapter. This development requires a more detailed combustion model that would adequately describe processes occurring in low‐inertia zones of a combustion wave. The influence of low‐inertia zones (the reacting layer of the condensed phase, preheat and reaction zones in the gas phase, the half‐space occupied by gaseous combustion products) on various nonsteady phenomena are investigated both analytically and numerically. The consideration is presented within the framework of the Belyaev model. It is demonstrated that under a weak dependence of surface temperature on initial temperature accounting for the above low‐inertia zones (even if their thermal inertia is small compared to the inertia of the preheat layer of the condensed phase) leads to significant corrections to the tc approximation.
Finally, it is our pleasure to acknowledge the significant contribution of the people who helped us in the preparation of this book.
We are very grateful to Professor Vladimir Marshakov, who discussed various topics throughout the book with us at great length.
Special thanks are given to Inga Novozhilov. It is certain that without her very careful and dedicated work the manuscript could not have been adequately prepared.
We are also incredibly thankful to Professor Vladimir Posvyanskii, Ludmila Novozhilova, and Natalia Golubnichaya for their help in preparing the manuscript.
The second author would like to thank his wife Natalia Golubnichaya again for her love and continuous support throughout the project.
Moscow – Belfast – Melbourne
2011–2019
Boris V. Novozhilov Vasily B. Novozhilov
Important Notation and Abbreviations
Abbreviations
| ADN | Ammonium dinitramide |
| BVP | Boundary value problem |
| c.c. | Complex conjugate |
| ZN | Zeldovich–Novozhilov |
| FM | Flame model |
| HMX | Cyclotetramethylene tetranitramine |
| ODE | Ordinary differential equation |
| PDE | Partial differential equation |
| PETN | Pentaerythritol tetranitrate |
| QSHOD | Quasi‐steady, homogeneous, one‐dimensional |
| RDX | Cyclotrimethylene trinitramine |
| SHS | Self‐propagating high–temperature synthesis |
| SRM | Solid rocket motor |
Mathematical Functions
| L n | Laguerre polynomials |
| lg | log10 |
| erfc | Complimentary error function |
| He n | Hermite polynomials |
| W | Whittaker function |
Notation
Over‐bar complex conjugate; Laplace–Carson transform
| prime | time derivative, case‐specific dimension; perturbed value, case‐specific dimension |
| Basic Physical Dimensions | M (mass), L (length), T (time), θ (temperature), N (amount of substance, e.g. mole) |
| a | speed of sound, LT−1; amplitude, case‐specific dimension |
| a f | amplitude of forced oscillations, case‐specific dimension |
| A | nozzle discharge coefficient, L−1T |
| b |
combustion temperature, nondimensional; correction
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