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- Transient Burning Rate Model for Solid Rocket Motor Internal Ballistic Simulations
- Recent progress in modeling solid propellant combustion
- Transient Burning Rate Model for Solid Rocket Motor Internal Ballistic Simulations
- Solid Propellant Chemistry, Combustion, and Motor Interior Ballistics

*This volume brings together the world's most highly regarded scientists in the field of solid rocket propulsion.*

## Transient Burning Rate Model for Solid Rocket Motor Internal Ballistic Simulations

David R. A general numerical model based on the Zeldovich-Novozhilov solid-phase energy conservation result for unsteady solid-propellant burning is presented in this paper. Unlike past models, the integrated temperature distribution in the solid phase is utilized directly for estimating instantaneous burning rate rather than the thermal gradient at the burning surface.

The burning model is general in the sense that the model may be incorporated for various propellant burning-rate mechanisms. Given the availability of pressure-related experimental data in the open literature, varying static pressure is the principal mechanism of interest in this study. The example predicted results presented in this paper are to a substantial extent consistent with the corresponding experimental firing response data. An important aspect in the study of the internal ballistics of solid-propellant rocket motors SRMs is the ability to understand the behaviour of a given motor under transient conditions, that is, beyond what would be considered as quasisteady or quasiequilibrium conditions.

Transient combustion and flow conditions arise for example during the ignition and chamber filling phase [ 1 — 3 ] prior to nominal quasisteady operation, during the propellant burnout and chamber emptying phase [ 4 , 5 ] in the latter portion of a motor's firing, and on occasion when a motor experiences axial or transverse combustion instability symptoms [ 6 — 9 ] upon initiation by a disturbance.

The simulation of undesirable nonlinear axial combustion instability symptoms in SRMs employing cylindrical and noncylindrical propellant grains [ 10 — 13 ] has provided the motivation for the present study. SRM internal ballistic simulation models incorporate algorithms for describing the internal flow and the mass input to the core flow from the burning surface of the solid propellant.

More recent models may also incorporate the deflection of the surrounding structure, for example, propellant grain, casing, and heavyweight e. For numerical models, in the case of unsteady operation under transient flow conditions, one ideally would capture the dynamic characteristics of both the flow and combustion to a level of accuracy that would enable the prediction of inherent design limits for a given motor e.

An example head-end pressure-time profile from [ 13 ] illustrating classical axial combustion instability symptoms, of a limit-magnitude travelling axial shock wave moving back-and-forth within the motor chamber superimposed on a base dc pressure shift approaching 5? MPa above the normal operating pressure of approximately 10? MPa, is given in Figure 2. While in some instances the assumption of a quasisteady i. The Zeldovich-Novozhilov Z-N phenomenological approach [ 14 , 15 ], in its most general sense, was considered a good basis for the development of a numerical transient burning rate model that could function in an overall dynamic internal ballistic simulation environment.

The Z-N energy conservation criterion in this context requires a numerical heat conduction solution with time in the solid phase propellant beneath the burning surface , but conveniently, empirical or semiempirical steady-state burning rate information may be used in place of more complex dynamic flame-based reaction rate equations in tying in the gas phase above the propellant surface [ 15 ].

This is a distinct advantage for preliminary design and instability evaluation purposes, especially where quicker computational turnaround times are desirable. An approach of this kind can be easily adopted by motor developers, by fitting the model to observed response data for many kinds of propellants through a few parametric constants. In this paper, a working general numerical burning rate model is described.

Past transient burning rate models have typically been constructed in terms of the specific driving mechanism of interest, for example, with equations derived explicitly as a function of pressure. In the present approach, the equations are derived as a function of the quasisteady burning rate, which in turn may be a function of one or several driving mechanisms as noted earlier. Note that the work outlined in this paper is a continuation of the initial burning-rate model development that has been reported in [ 16 , 17 ] the reader may find some of the early model developments and evaluations thereof of interest.

Example results are presented in this paper in order to provide the reader with some background on the sensitivities of a number of pertinent parameters in the present study. Given the more ready availability of pressure-based burning experimental data in the open literature, comparisons are made to reported experimental pressure-based combustion response data for some composite and homogeneous solid propellants.

The Z-N solid-phase energy conservation result may be presented in the following time-dependent temperature-based relationship [ 15 ]:?? For the purposes of the development outlined in this paper, a more direct and general equation is sought relating?? In a finite difference format, energy conservation in the solid phase over a given time increment may be represented by the following equation:?? The quasisteady burning rate may be ascertained as a function of such parameters as static pressure of the local flow, for example through de St.

Robert's law [ 5 ]:?? In this respect, the present numerical model differs from past numerical models, for example, as reported by Kooker and Nelson [ 19 ].

In those past cases, the thermal gradient at the propellant surface?? In following this established trend, earlier versions of numerical Z-N models did not follow through on using 6 directly, but switched to a burning rate temperature sensitivity correlation such as [ 18 , 20 ]?? As reported by Nelson [ 18 ], the predicted??

Returning to the present model, in the solid phase, the transient heat conduction is governed by [ 14 , 21 ]?? The corresponding spatial step? Allowing for the propellant's surface regression of?? With respect to the burning surface temperature?? In the past, it was not uncommon to encounter estimation models assuming a mean or constant??

More recently, usage of an Arrhenius relation for solid pyrolysis dictating a variable?? The numerical model, as described above and with a preliminary assumption of a constant?? An additional equation limiting the transition of the instantaneous burning rate?? From the author's previous background in general numerical modeling where lagging a parameter's value is a desired objective, a simple empirical means for applying this constraint is as follows:??

The rate limiting coefficient?? In the numerical scheme, an incremental change in burning rate over a given time step would as a result be? In order to be consistent on input and output heat energy at the propellant surface, such that the converged solution is independent of time?

The empirical coefficient?? A limitation of the present approach would of course be the nonavailability of experimental response data, say from T-burner experiments, for the specific propellant in question, to allow for this alignment of?? An additional or complementary concern would be the strength of the assumption that 14 is in fact a practical means for describing the damped response, to a sufficient degree of accuracy for the purpose at hand, as compared to say, the usage of dynamic flame-based reaction rate equations that also intrinsically limits the movement of the propellant burning rate e.

Comparing the model's results to a number of different experimental results, for different propellants, would help establish the degree of confidence that might be warranted. Some comparisons are presented later in this paper, that do lend support to the present approach. The structure of composite propellants as opposed to homogeneous [double-base] propellants at the local microscale is physically more heterogeneous as a solid mixture of oxidizer crystals and polymer binder than the underlying assumption of solid homogeneity implied by the Z-N model.

Using bulk-average propellant properties and adjusting the rate limiting coefficient and net surface heat release to an appropriate setting may nevertheless produce a reasonable and pragmatic predictive capability for motor design and evaluation at the macroscale for both classes of propellants. In the present approach, the surface thermal gradient is free to find its own value at a given instant, via the numerical scheme for the regressing solid phase.

This contrasts with past approaches that dictated an analytical function tying the surface thermal gradient to surface regression rate. One can argue then that the use of 14 or some comparable damping function, while empirical, parallels the approach taken by past researchers in enforcing a stipulated surface thermal gradient a form of 7 or 8.

As part of the model development studies, a variable propellant surface temperature?? In practice,??

In this example, the effect of local static pressure?? As demonstrated by the numerically predicted temperature curves within the condensed phase Propellant A of Table 1 , where exponent?? The profiles in Figure 3 would be expected to conform to the following relationship for a homogeneous solid [ 14 ]:??

Analogous to constraining the burning rate to some degree via?? As reported in [ 17 ], the results for when?? The appearance of a low-frequency and a high-frequency peak in the frequency response graphs, or an initial peak dropping down to a prolonged plateau in magnitude at higher values for?? The more variable surface temperature in essence within the predictive model acts to suppress the degree of augmentation of the principal response peak as it moves the peak to a higher frequency for a given cyclic driving mechanism, and introduces a secondary resonant response peak at a low frequency.

From a modeling standpoint, one can appreciate that a moving?? This undoubtedly is playing a role in producing the odd results noted in [ 17 ] as?? Given the better overall comparisons to experimental profiles, to date, with an assumed constant?? In doing so, it is understood that this does not rule out the possibility of a future model development that would allow for a variable??

For example, application of a?? The nondimensional limit magnitude?? See Figure 5 for an example predicted result for Propellant A, for differing values of O. The limit magnitude profile may also be described in terms of the dimensionless frequency??

To allow for further comparison with frequency response data available in the literature, the simulation results can be presented in a form that relates to a specific flow parameter, in this case static pressure?? The real part of the pressure-based response function O is typically presented as a function of??

The response function is defined in terms of mean and fluctuating components of static pressure and incoming mass flow from the propellant surface [ 15 ]:? The base burn rate?? While there is some degree of variability in the experimental data [ 23 ] commonly encountered with T-burner results [ 24 ], in this particular case there is an appreciable level of agreement between the predicted curve of Figure 6 and the test data points. In [ 23 ], the authors found that setting the coefficient??

In Cohen's study of the application of the A-B response model to homogeneous propellants [ 25 ], he notes that an increase in value for coefficient A results in a higher peak response magnitude, at a higher dimensionless frequency. This corresponds to the effect of an increase in the value of??

Similarly, he notes that a decrease in the value of coefficient?? This corresponds to the effect of an increase in the positive value of?? As displayed in [ 24 ] Figure 3 of that paper , test data points produced from various institutions' experiments for that propellant range considerably for any given frequency.

For Figure 7 of the present paper, two institutions' data points are chosen they more or less reflect upper and lower bounds for the collected data; UTC refers to United Technologies Corp. Given the range of the experimental data, it was decided to also produce example upper and lower predictive model curves, rather than a single median curve. The assumed characteristics of this propellant are listed under C in Table 1.

The base burn rate O of this propellant at the test pressure is assumed to be 0. The log-linear format of the graph corresponds to that used in [ 24 ].

Given the range of variability of the experimental data, the agreement with the predictive model in terms of qualitative trends and quantitative magnitudes for some portions of the two predictive curves is encouraging.

Figure 8 provides an alternative linear-linear format for the above results, where the pressure-based response is in terms of the dimensionless frequency O output?? This presentation format conforms with that presented in [ 26 , Figures 3 and 4]. In [ 22 , Figure 4], the experimental T-burner pressure-based response for the A propellant is displayed for differing pressures and therefore, differing base burn rates.

For the present model, the base burning rates were estimated from available information, as follows: 0. MPa abs. As noted in [ 28 ], and as done as well in [ 29 ] for propellants with differing properties, it has been commonplace in the past to present output data in terms of??

This reference value differs substantially with the assumed actual value for the A propellant C in Table 1 , i. Conforming to the log-linear format of [ 22 ], Figure 9 presents the experimental data and the corresponding predicted curves for the three pressures using??

## Recent progress in modeling solid propellant combustion

The paper deals with mathematical simulation of dispersion of agglomerates formed in combustion of aluminized solid propellants. A substantial effect of the separation conditions of agglomerating metal particles from the surface of the burning propellant on the size of agglomerates is demonstrated. A mathematical model of agglomerate formation is constructed for propellants whose typical feature is active burning of the metallic fuel in the surface layer. Satisfactory quality of simulation is validated by the agreement of experimental and numerical data. This is a preview of subscription content, access via your institution. Rent this article via DeepDyve.

Tremendous progress has been achieved in the last ten years with respect to modeling the combustion of solid propellants. The vastly increased performance of computing capabilities has allowed utilization of calculation approaches that were previously only conceptual. The capability of modeling premixed combustion using detailed kinetic mechanisms has been evolving and successfully applied to solid propellant ingredients based on a one-dimensional approach. Much of the early work was performed at Novosibirsk. The approach allows calculating the burning rate as a function of pressure but also the temperature sensitivity and spatial distributions of temperature and species concentrations.

## Transient Burning Rate Model for Solid Rocket Motor Internal Ballistic Simulations

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The earliest rockets were solid-fuel rockets powered by gunpowder ; they were used in warfare by the Chinese , Indians , Mongols and Persians , as early as the 13th century. All rockets used some form of solid or powdered propellant up until the 20th century, when liquid-propellant rockets offered more efficient and controllable alternatives. Solid rockets are still used today in military armaments worldwide, model rockets , solid rocket boosters and on larger applications for their simplicity and reliability. Since solid-fuel rockets can remain in storage for a long time without much propellant degradation and because they almost always launch reliably, they have been frequently used in military applications such as missiles.

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### Solid Propellant Chemistry, Combustion, and Motor Interior Ballistics

Greatrix, greatrix ryerson. A general numerical model based on the Zeldovich-Novozhilov solid-phase energy conservation result for unsteady solid-propellant burning is presented in this paper. Unlike past models, the integrated temperature distribution in the solid phase is utilized directly for estimating instantaneous burning rate rather than the thermal gradient at the burning surface. The burning model is general in the sense that the model may be incorporated for various propellant burning-rate mechanisms.

The simulation used to predict solid rocket propellant performance is a safe and feasible alternative for developing rocket motors. An investigation into the combustion and performance of small solid-propellant rocket motors. Final Thesis Report. University of New South Wales. MORO, D. Experiments and Numerical.

Abstract: This volume brings together the world's most highly regarded scientists in the field of solid rocket propulsion. Thirty-nine papers present in-depth coverage on a wide range of topics including: advanced materials and nontraditional formulations; the chemical aspects of organic and inorganic components in relation to decomposition mechanisms, kinetics, combustion, and modeling; safety issues, hazards and explosive characteristics; and experimental and computational interior ballistics research, including chemical information and the physics of the complex flowfield. Skip to main content. No Access. Thomas B. Brill Search for more papers by this author.

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