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.

      EQN.1

where Kn is the klemmung, 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

                  EQN.2 and EQN.3

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:

      EQN.4

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 as

      EQN.5

The throat area is 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 chamber pressure as a function of time is the experimentally obtained pressure-time trace, where Po is known after each Dt time interval.

The characteristic velocity, c*, is also obtained from the pressure-time trace, given by time integral of chamber pressure over the burn, multiplied by the coefficient shown:

      EQN.6

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:

      EQN.7

Acceptable systems of measurement units are provided in Appendix B

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 / propellant mass. The pressure-time trace is shown in Figure 1, and the c* is calculated from the tabular form of the pressure time trace.

The delivered c* was found to be 2951 ft/s., which converts to 2951 ÷ 3.28 = 900 m/s. This compares well with a theoretical c* = 912 m/s., and to closed vessel testing measurements of 899 m/s. This is indicative of high combustion efficiency.

The next step is to estimate the region of the pressure-time curve where steady-state combustion exists. As mentioned, the method is only valid for this region, and not the pressure "build-up" or "tail-off" regions (hatched portions in Figure 1). In reality, this region is not at all distinct, especially during the tail-off phase where slivers of propellant are certainly present. In Figure 1, the "best guess" for the steady-state regime is between the two small circles (from data point 12 to 81). The first circle indicates "time zero" for this analysis.

The analysis is best performed using a spreadsheet software, such as MS Excel. A spreadsheet is set up as shown in Figure 2.

The analysis, in this example, is done using metric units.

• Col.4 -- The pressure units are converted to N/m² (1 psi = 6895 N/m² )
• Col.5 -- The formula for burning area is inserted, for a BATES grain, where the applicable grain geometry is shown in red text
• Col.6 -- This column is for Ds, which is the unknown quantity to be solved for
• Col.7 -- This column has the formula inserted for the left-side term of EQN.4. The constants are grouped together as beta, where b = At /(rp  c*)
• Col.8 -- The surface regression, s, which is simply the summed Ds values, plus an initial value, so, shown in blue. The initial value accounts for grain burning required to generate chamber pressure during the "build-up" phase. This initial value starts off with a guess, which can be later changed if required. The goal is to have the total surface regression equal to the initial web thickness, wo
• Col.9 -- The last column is the instantaneous burn rate, per EQN.3

Notice that the value of total surface regression, 18.69 mm, is close to the initial web thickness, 18.73 mm. Therefore, our "guess" of 2.50 mm as the initial surface regression was a good one, and there is no need to modify the value.
Obviously, the important values are in Column 9, the values for instantaneous burn rate, which correspond to the pressure values shown in Columns 3 and 4. A plot of the results is shown in Figure 4.

This plot clearly shows an increase in burn rate with increasing chamber pressure. The red curve is for the duration when steady-state chamber pressure is climbing, and the blue curve is for the duration when pressure is falling. One would expect the two curves to coincide. Almost certainly, this is a result of idealized grain geometry assumed in the analysis. In reality, the grain segments would not have all surfaces ignite simultaneously, resulting in a surface geometry that deviates from that idealized. Erosive burning likely plays a role, as well, in determining burn rate and geometry, although this may be a minor factor.

How this does estimation of burn rate compare with Strand Burner measurements? A plot showing this comparison is given in Figure 5.

Interestingly, the average burn rate over the pressure regime being investigated is close (approx. 13 mm/s) for both this method and the Strand Burner measurements. The trend differs, however. This regime falls within the plateau region based on Strand Burner results., but this method indicates a continuous rise in burn rate with pressure. It is known from the literature that burn rate measurements from static tests differs, to some extent, from strand burner measurements. The exact reasons are unclear.

The next step is to determine the burn rate coefficient and pressure exponent from the results. This is simply performed by fitting a power function curve through the burn rate v.s. pressure results. This may also be done in Excel using the TRENDLINE feature. This was found to give:

n = 0.61

a = 0.0073       (pressure in psi; burn rate inch/sec)

a = 3.85         (pressure in MPa; burn rate in mm/sec)

Valid for the range 800 - 1220 psi (5.5 - 8.5 MPa)

Interestingly, the modified prediction more closely fits the test results (only the shape of the curve is changed, overall performance is essentially identical).

It will be interesting to further investigate the results of this method with future pressure-time traces for this propellant (and others), to see if the trend holds.

For a hollow cylindrical grain with ends only inhibited, the burning area is constant, and is given by

For a hollow cylindrical grain with unrestricted burning, the burning area, Ab, as a function of surface regression, s, is given by: