The instantaneous burning rate of a propellant may be estimated from the pressure-time data obtained from a rocket motor static firing. This method is based on the knowledge that motor chamber pressure and burn rate are directly related in terms of Kn, c* and the propellant density. The all-important burn rate coefficient (a) and the pressure exponent (n) may also be estimated by this method.
The method described here was inspired by the treatise Non Parametric Burning Rate Estimation, authored by Henrik D. Nissen. This paper is available for downloading from the DARK web site.
Steady-state chamber pressure may be expressed in terms of the propellant properties, Kn, and burn rate:
where Kn is the klemmung (ratio of burning area to throat area), rp is the propellant mass density, c* is the propellant characteristic velocity, and r is the burn rate. Steady-state implies that the motor is operating under the condition of choked nozzle flow whereby any chamber pressure variation is due solely to grain geometry, and excludes the "pressure build-up" or "tail-off" phases of operation. Derivation of this equation may be found in the Chamber Pressure Theory web page.
The klemmung and burn rate may be expressed as
where Ab is the grain burning area and At is the nozzle throat cross-section area, Ds is the surface regression (depth burned) over the incremental time step Dt.
The equation for chamber pressure may be re-arranged as follows:
In this equation, it is important to note that the burning area,
Ab, is a function of surface regression,
s. The surface regression, as well as chamber pressure,
Po, are both functions of time. This may be explicitly written
The throat area is normally assumed constant, as are c* and rp. The equations for
calculating burning area as a function of surface regression, in terms of
grain initial geometry, is given in Appendix
A for hollow cylindrical grains.
The characteristic velocity, c*, is also obtained from the pressure-time data, given by time integral of chamber pressure over the burn, multiplied by the coefficient shown:
where mp is the propellant total mass (for english units, mass = weight divided by gravity constant, g). The pressure integral can be found simply by taking the sum of the pressure values factored by the time interval:
Acceptable systems of measurement units are provided in Appendix B
The pressure-time data from a static test of the APM-C.2 rocket motor will be used to obtain the propellant burn rate and burn rate parameters a and n. The APM-C.2 is a 51mm experimental rocket motor powered by the newly-developed KNPSB propellant.
After obtaining the burn rate parameters, these are inputted into SRM.XLS rocket motor design software, and the prediction for chamber pressure compared to the static test data.
The first step is to determine the delivered characteristic velocity, c*. This is accomplished by use of EQN.6 and EQN.7, whereby the pressure values are summed up, then multiplied by the time increment and nozzle throat area divided by propellant mass. The pressure-time curve from the static firing is shown in Figure 1, and the c* is calculated from the tabular form of the pressure time data.
The applicable parameters for this motor are shown below.
Characteristic velocity is calculated as such:
where the value of 145 is a conversion factor from psi to N/mm2.
The burn rate analysis is performed using an MS Excel spreadsheet, set up as shown in Figure 2.
The analysis, in this example, is done using metric units.
Proceeding row by row, the value in Column 8 is converged to zero by changing the value in Column 7, using the Data/What If Analysis/Goal Seek tool. It is recommended to write a macro to automate this process, as it can be quite tedious if there are a lot of data points.
The completed spreadsheet analysis is shown in Figure 3.
Notice that the value of total surface regression, 14.56 mm, is close
to the initial web thickness of 14.61 mm. As such,
the initial surface regression value of 0.00 is close enough and there is no need to
modify the value.
In Figure 4, the two different slopes (pressure rising/pressure falling) are a result of the computed burn rate being slightly different over the duration of pressure rise (from t = 0 to t = 0.37 in Figure 1) compared to pressure decline (t > 0.37). Ideally, the slopes should coincide. For the purpose of designing a motor utlizing this propellant, taking the average of the two curves is fully suitable. As well, the burn rate at the lower pressure regime may be neglected. For this example, the burn rate at pressure greater than 3.5MPa (500 psi) is solely of interest. To fit a curve through the data points, use Add Trendline/Power Series and select Display Equation on chart. The Power Series is chosen as this correlates to the St.Robert's Law burning rate relationship for a solid propellant. The result is shown in Figure 5, where the red dashed line is the trendline.
The derived burn rate parameters are therefore:
a = 6.4995
For purpose of design, the values may be rounded off to suitable engineering accuracy:
a = 6.50 (burn rate coefficient)
Note that the value of "a" is for pressure in megapascals (MPa) and gives burn rate in millimetres per second. The value can be readily converted for pressure in "psi" and burn rate in inches per second:
a = 0.0112 pressure = psi; r= in./sec
The values for "a" and "n" are now inserted into SRM.XLS that has been populated with the pertinent motor and propellant parameters. Specifically, these values are entered into the Other Propellant box in the BURNRATE tab.
The calculated value of characteristic velocity (c*) based on the static test data is also required. To accomplish this, a value of Combustion efficiency is entered to give c* = 973 m/sec. (using Goal Seek tool).
The resulting chamber pressure curve predicted by SRM with the derived a,n parameters is then plotted together with the chamber pressure curve obtained from the static test. The result is shown in Figure 6
The chamber pressure curve predicted by SRM using the derived values for the burning rate parameters is a close match to the static test data. The values should be considered tentative, albeit fully suitable for further experimental rocket motor design utilizing this propellant.
Excel files for this example: