Within the framework of the theory presented in this monograph, the nonsteady process of propellant combustion is investigated by means of solving the heat transfer equation in the condensed phase with relevant initial and boundary conditions. Other necessary elements of the theory are the steady‐state dependencies of the burning rate and surface temperature on the pressure and initial temperature. These may be obtained experimentally or by considering a specific theoretical propellant combustion model. It is clear that all conclusions of the theory are applicable to real systems as the aforementioned dependencies are obtained from experiments with the exactly same systems.
Let us discuss briefly the assumptions which form the foundation of the theory. It should be noted that, with a few exceptions, these assumptions are adopted in all studies of the nonsteady combustion of solid rocket fuels. First of all, fuel is assumed to be homogeneous and isotropic. The scale of nonhomogeneity must be much smaller than the characteristic scale following from steady‐state theory, that is, the Michelson length. This requirement is undoubtedly fulfilled for ballistites. In the case of composite fuels, this assumption is valid for sizes of fuel and oxidizer particles much smaller than the thickness of the heated layer of the condensed phase. In the following discussion, one‐dimensional problem formulation assuming flat flame front and interface between the phases is considered nearly everywhere. Second, the basic assumption of the discussed theory is that thermal decomposition of the condensed phase and combustion in the gaseous phase occur much faster than the heating up of the condensed phase. This proposition may be justified by simple estimations, which are presented in the main body of the monograph.
The first review of the proposed theory was presented by Novozhilov (1968). It considered the major results obtained by that time: combustion stability at constant pressure, linear oscillating combustion regimes, acoustic admittance of the surface of burning propellant, combustion stability in a semi‐enclosed volume, nonlinear oscillations of burning rate, transitional combustion regimes, and propellant extinction. Later, the monographs (in Russian) by Novozhilov (1973a) and Zeldovich et al. (1975), as well as the more recent review by Novozhilov (1992a) appeared.
It seemed initially (to many, including the first author) that the phenomenological theory would be replaced soon by a more sophisticated approach to nonsteady propellant combustion. Such an approach could be built on the consideration (by numerical methods) of a set of differential equations, complete and consistent from the point of view of macroscopic chemical physics, describing the specific problem. This may eventually happen. However, progress beyond the framework of the ZN theory has been very slow owing to the significant complexity (even for homogeneous systems) of propellant combustion. This complexity, in the first place, is due to phase transition from condensed to gaseous, complicated even further by chemical reactions.
All the considerations below are applicable, strictly speaking, to homogeneous propellants only. The theory of composite systems is at a rudimentary stage since the processes in the combustion wave of such substances are much more complicated compared to homogeneous propellants. Apart from a nearly complete absence of data on the kinetics of chemical reactions, there are additional obstacles to a quantitative consideration of the combustion process of composite systems. Although during steady‐state conditions the mean burning rate is constant in time, the processes occurring in the vicinity of the surface are nonsteady. The geometry of the surface continuously changes in time as burnt particles are replaced by virgin ones at other locations at the surface. Temperature distribution in the vicinity of the interface between the phases, and on the interface itself, is a random function of time. It should be noted that attempts were made (Romanov 1976) to expand the theory to heterogeneous systems. It was proposed, in addition to steady‐state dependencies of the burning rate and surface temperature on pressure and initial temperature, to use dependencies of average values of these quantities on fuel (or oxidizer) mass fraction at the interface. Unfortunately, significant difficulties in obtaining such dependencies experimentally did not allow this approach to proceed.
Nevertheless, one may hope that some of the results obtained for homogeneous propellants would also be qualitatively applicable to composite systems. For example, the resonance response of the burning rate to periodically varying pressure may be expressed in the same terms as for homogenous systems. Naturally, the parameters characterizing such composite systems would have to be considered as adjustable values.
Let us discuss briefly the comparison of conclusions (which are discussed within the book) of the theory, with experimentation. As with any other theory, ZN theory requires, first of all, some experimental input data. These are steady‐state burning laws, that is, steady‐state dependencies of the burning rate and surface temperature (and, in some cases, of other properties of the combustion wave, e.g. combustion temperature) on external parameters, that is, on pressure and initial temperature. The theory demands a rather high accuracy of input experimental data as its conclusions follow from peculiarities of steady‐state burning laws. For example, the study of linear nonsteady phenomena is only possible if first derivatives of burning laws, with respect to external parameters, are known. Calculation of these derivatives obviously involves large errors as such a mathematical operation is ill‐posed.
The same applies to experimental data which is used for comparison with the theory outcomes. Observation of various unsteady combustion phenomena are associated with significant difficulties. At best, relative errors are of the order of tens of percent. The theory, however, predicts a number of quite distinctive qualitative effects,for example the existence of the natural frequency of propellant combustion and, as a consequence, a resonance response of the nonsteady burning rate to harmonically oscillating pressure. Such effects are actually being observed, and it is usually possible to reconcile the theory and experimental results quantitatively for the values of parameters within the experimental uncertainty region.
Before briefly outlining the monograph content, let us notice that the overwhelming majority of presented results are obtained using a universal approach. The following is its mathematical formulation.
A one‐dimensional unsteady heat transfer equation
is considered, along with the relevant boundary and initial conditions
Nonsteady relations between the burning rate v(τ) and the surface temperature ϑ(τ), on the one hand, and some external parameter η(τ) (most often pressure) and the temperature gradient ϕ(τ) = (∂θ/∂ξ)ξ = 0, on the other
are prescribed.
There must also be prescribed the function
or some auxiliary equation which determines this function.
A specification of the functions Φu(ϕ, η) and Φs(ϕ, η), and the external parameter dependence on time η(τ) lead to a class of problems that are related to rapid development in recent decades in the multidisciplinary area of synergetics (Mikhailov 2011).
The majority of results in this area are obtained by a numerical analysis of various model sets of differential equations. Most often, the systems with a finite number of degrees of freedom are considered.
The formulation discussed above is probably the simplest for distributed dynamical systems. Despite this simple form, however, the set of system behaviour scenarios is quite rich. For example (and this is demonstrated in the