Richard Nakka’s Experimental Rocketry Web Site

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Simplified Method to Estimate Burn Rate Parameters

Introduction

There are a number of methods commonly employed to obtain the burn rate parameters of a solid rocket propellant. Three methods are:

· Strand Burner

· Ballistic Evaluation Motor (BEM)

· Pressure versus time data from a motor firing

The Strand Burner method, which is detailed in the Strand Burner for Burn Rate Measurements web page, is arguably the most accurate method. The drawbacks with this method are the requirement for specialized apparatus, the method is expensive and time-consuming to perform, as multiple strand burns are needed at different pressure levels. If even a small modification is made to the propellant formulation, the series of measurements must be repeated. The validity of the results lessens under conditions of erosive burning.

The second method utilizes a specially designed and constructed rocket motor, as described in the burn rate webpage. The grain configuration is typically an end-burner or tubular (also known as hollow-cylindrical). If an end burner is employed, multiple firings are required, as an end-burner operates at a single pressure level. The tubular configuration has the advantage of producing a progressively increasing chamber pressure during its burn, as the Kn steadily increases throughout the burn. As such, only a single firing may be required to characterize the propellant burn rate. The drawback with an end-burner is the need to have a sufficiently large burning area, and small enough nozzle throat, to generate the desired pressure ranges. To date, I have not tried using an end-burner BEM for burn rate measurement. As such, I cannot comment on its real-life suitability. On the other hand, I have designed and built BEMs with the tubular configuration. For the propellents tested (APCP formulations), the resulting pressure-time curves did not correlate very well to the theoretical Kn versus time, especially near the end of the burn, where the pressure peak was rounded rather than sharp. I attributed this to web burn-through being unequal along the length of the grain due to pressure variation. This added difficulty to the process of interpreting the data and extracting burn rate parameters.

The third method, extraction of burn rate parameters using pressure versus time data from a motor firing, only requires a single motor firing. The method, detailed in the Burn Rate Determination from Static Test Pressure Measurement web page, is relatively straightforward and uses an iterative procedure to match the actual web regression to the theoretical Kn change over the course of the burn. I have used this method many times and have obtained good results. The drawbacks to this method are related to its inherent complexity of the calculations involved, combined with the need to suitably interpret the results. As well, the actual chamber pressure data should be free from erosive burning influence or other factors that would result in the data deviating significantly from the theortical burn profile.

Alternate Approach

The three methods described above are well-suited for deriving the burn rate parameters, specifically the burn rate coeffient, a, and the pressure exponent, n, for a new solid propellant that has reached a mature stage of development such that it is ready to put into effect for rocket motor design. Having done a lot of work recently in developing AP-based propellants, in particular, using different binder systems, I was somewhat hindered by not knowing the burn rate parameters of a new formulation. This makes it necessary to “guess” what value of Kn would be suitable for a given test motor during this development phase. This is particulary true when formulations are tweaked, which is often the case when new binders are being employed (for example, to obtain a suitable viscosity, be it pourable or packable). This being the case, I came up with a simple method to estimate the value of the burn rate parameters. The estimated values, obtained as such, are suitable for designing the development motors that are used for researching new propellants. The method described here utilizes the results of two (or more) motor test firings where pressure is measured during the burn

Method

As described in the Solid Rocket Motor Theory -- Chamber Pressure web page, the expression for solid rocket motor steady-state chamber pressure is given by equation 11:

Where the following terms are:

A_b/A* = Kn (burning area divided by throat cross-section area)

r_p = propellant mass density

k = ratio of specific heats of propellant combustion products

R = specific gas constant of propellant combustion products

To = combustion temperature

For a given propellant, the latter four parameters are all constants, and as such, can simply be denoted as a single constant C.

where MKS units of C are N-sec/m³. For convenience, the exponent of equation 11 can be denoted by m:

Equation 11 can therefore be re-written as:

As mentioned, a minimum of two chamber pressure measurements are needed for this method, which will be denoted by P₁ and P₂.

Applying equation 3 to these two pressure values:

Where K_n1 and K_n2 refer to the Kn values that correspond to the pressure values P₁ and P₂.

Pressure exponent, a, can be separated:

Solving for a in terms of P₁ and Kn₁:

where MKS units of a are m/sec/(N/m²)ⁿ. From equations 5a and 5b, the pressure ratio is given by:

Simplifying gives:

The exonent m can be solved for by applying the Power Rule for Logarithms:

Procedure:

1) Using equation 9, solve for parameter m by entering the values for measured chamber pressure P₁ and P₂ and the corresponding values for Kn₁ and Kn₂.

2) Using equation 2, solve for pressure exponent n where

3) Calculate the value of constant C using equation 1.

4) Use equation 6 to calculate the value of the burn rate coefficient a.

Important Notes:

1) The best grain configuration for utilizing this method is probably neutral BATES. The advantage of BATES is the Kn profile, which features a clear maximum value approximately one-half way through the burn, as illustrated in Figure 1. As such, the maximum design value of Kn would be used in conjuntion with the maximum steady-state non-erosive pressure measured during the motor firing.

2) A minimum of two static test pressure measurements are required. The peak pressures should be reasonably far apart and both pressures should be in the design range of interest.

3) The values of the burn rate parameters, a and n, obtained through this method should be considered to be approximate and tentative. Greater confidence can be obtained through applying this method over multiple firings.

4) Erosive burning tends to be greatest at the beginning of the motor burn, when the core diameter is at its minimum. Another advantage of the BATES grain is that the maximum Kn occurs around the middle of the burn time, when erosive burning effects are lessened.

5) The values for the four parameters (r_p , k, R and To) used for calculating the constant C can be obtained from ProPEP. Although ProPEP provides the ideal density of a propellant, it is best to use the measured density. A knock-down factor should be applied to the combustion temperature, To, based on the ratio of delivered c-star to ideal c-star. The examples that follow explain the details involved.

6) Consistent units of measure must be used in the calculation of C and a. For the calculation of m, any units of measure may be used for P₁ and P₂, as long as the same units are used for both pressure values (this is because m is dimensionless).

7) The values obtained for a and n are then checked by using a suitable application such as SRM to plot a pressure versus time curve for the two test motors from which P₁ and P₂ were obtained. Ideally, the ideal curve obtained from such should reasonably well match the actual (measured) pressure curve for both test motor firings.

8) The calculations involved in determining a and n are made easy by programming the equations into an Excel spreadsheet.

Figure 1 – Example neutral BATES Kn profile

Examples

Example 1

Use the results of static tests APM-J.1-ST1 and APM-J.1-ST2 to estimate the burn rate parameters for propellant AXP-AP7.2. This is an APCP with West System epoxy as binder.

The design Kn profiles, generated using SRM_1.3.xlsx, are shown below, together with the pressure-time curves from the static test firings.

APM-J.1-ST1 :

APM-J.1-ST2 :

From the two static test curves, it is clear that some degree of erosive burning occurs initially. It is assumed that the erosive burning peters out as the core grows in diameter during the burn. The maximum Kn is matched to the pressure during the mid-point of the burn, at which point burning is assumed to be non-erosive.

Static Test ID	Kn_max	Pressure@ Kn_max (psi)
APM-J.1-ST2	381	930
APM-J.2-ST1	345	790

Using these values, the exponent m is calculated, from which the pressure exponent, n, is obtained.

In order to calculate the constant, C, ProPep3 is used to obtain the specific heat ratio, k, the specific gas constant, R, and the ideal combustion temperature, To_i. The output for AXP-AP7.2 propellant is shown below.

Code AXP-AP7.2 WEIGHT D-H DENS COMPOSITION

0 AMMONIUM PERCHLORATE (AP) 68.000 -602 0.07040 1 CL 4 H 1 N 4 O

0 EPOXY 201 23.700 -661 0.04040 24 H 16 C 4 O

0 ALUMINUM (PURE CRYSTALINE) 8.000 0 0.09760 1 AL

0 CARBON BLACK 0.300 0 0.06370 1 C

THE PROPELLANT DENSITY IS 0.06100 LB/CU-IN OR 1.6886 GM/CC

THE TOTAL PROPELLANT WEIGHT IS 100.0000 GRAMS

NUMBER OF GRAM ATOMS OF EACH ELEMENT PRESENT IN INGREDIENTS

4.343715 H

1.377485 C

0.578738 N

2.653080 O

0.296516 AL

0.578738 CL

****************************CHAMBER RESULTS FOLLOW *****************************

T(K) T(F) P(ATM) P(PSI) ENTHALPY ENTROPY CP/CV GAS RT/V

2669 4346 68.02 1000.00 -56.60 239.05 1.2134 4.138 16.439

SPECIFIC HEAT (MOLAR) OF GAS AND TOTAL = 9.657 10.911

NUMBER MOLS GAS AND CONDENSED = 4.138 0.148

1.259959e+000 CO 1.165006e+000 H2 7.128118e-001 H2O 5.719621e-001 HCl

2.892328e-001 N2 1.476532e-001 Al2O3* 1.173632e-001 CO2 1.347614e-002 H

4.594527e-003 Cl 2.092405e-003 HO 3.509814e-004 AlCl2 3.143509e-004 AlCl3

2.807695e-004 AlCl 1.563391e-004 AlOCl 1.018212e-004 NH3 8.194096e-005 NO

5.068681e-005 AlHO2 3.362895e-005 CNH 3.090018e-005 CHO 2.329321e-005 AlHO

1.524328e-005 O 1.461942e-005 COCl 1.406953e-005 CH2O 1.359749e-005 Cl2

3.968182e-006 NH2 3.803548e-006 O2 3.068339e-006 CNHO 2.484860e-006 HOCl

1.17373E-06 CH4

THE MOLECULAR WEIGHT OF THE MIXTURE IS 23.333

**********PERFORMANCE: FROZEN ON FIRST LINE, SHIFTING ON SECOND LINE**********

IMPULSE IS EX T* P* C* ISP* OPT-EX D-ISP A*M EX-T

237.4 1.2291 2395 38.02 4784.4 8.38 400.8 0.14874 1216

241.0 1.2043 2425 38.34 4840.8 187.0 8.54 406.9 0.15049 1276

To_i = 2669 K.

k = 1.2134

In order to calculate the specific gas constant, the effective molecular mass is first calculated as the product mass divided by the number of gas moles:

M = 100/4.138 = 24.17 grams/mole

The specific gas constant is calculated as the universal gas constant, R¢, divided by the effective molecular mass

R¢ = 8314 J/kmole-K, giving

R = 8314/24.17 = 344 J/kg-K

The delivered c-star (characteristic velocity) for each of the two test firings can be determined using the method Measuring Chamber Pressure and Determining C-Star and Thrust Coefficient web page. Using this method, the values for delivered c-star are:

APM-J.1-ST1 4800 feet/second

APM-J.1-ST2 4735 feet/second

We’ll use the average value of 4768 feet/second. Using this we will calculate the actual combustion temperature of this propellant. Noting that the relationship between c-star (also denoted c*) and combustion temperature involves a square root, as shown in Equation 3 of the Solid Rocket Motor Theory -- Impulse and C-star web page, the actual combustion temperature is given by:

Therefore, To = 2669 × (4768/4784)² = 2651 K. The propellant density was measured and is:

r_p = 1.680 grams/cm³

The numerical value of the constant C may now be calculated using the formula given by Equation 1. Note that consistent units must be used, so density is converted to kg/m³ :

Equation 6 is used to calculate the numerical value of the burn rate coefficient. Again, consistent units must be employed so we convert pressure from psi to N/m²:

P₁ = 930 lbf/in² × 4.448 N/lbf × (39.37)² in²/m² = 6,411,779 N/m²

As stated earlier, the units of a are m/sec/(N/m²)ⁿ. As we usually measure chamber pressure in psi or MPa and desire burn rate to be inch/second or mm/second, the value of a can be readily converted to a the more useful form (note Pa, or Pascal, is a convenient notation for N/m² . MPa is megaPascal).

Pressure MPa and burn rate mm/sec:

0.0000147 m/sec/Paⁿ × 1000 mm/m × (1×10⁶)^0.392 Paⁿ /MPaⁿ = 3.31 mm/sec/MPaⁿ

Pressure psi and burn rate inch/sec:

0.0000147 m/sec/Paⁿ × 39.37 in/m × (6895)^0.392 Paⁿ /psiⁿ = 0.0185 in/sec/psiⁿ

Now that we’ve calculated the values for the burn rate parameters, the results are checked by plugging a and n into our design app (in this case SRM_1.3.xlsx) and see how the predicted pressure vs time curve compares to the measured test results.

Entered into SRM Burnrate sheet:

Predicted pressure curve for APM-J.1-ST1:

Comparison between pressure measured in static firing and SRM predicted pressure curve using derived burn rate parameters:

As expected, the initial portion of the predicted curve deviates from the measured pressure due to erosive burning. This aspect aside, the predicted curve is a reasonably good match, certainly validating the method for preliminary design of motors utilizing AXP-AP7.2 propellant.

Likewise, the predicted pressure curve for APM-J.1-ST2 is shown below, with the derived a and n values:

Comparison between pressure measured in static firing and SRM predicted pressure curve:

Similarly, and as expected, the initial portion of the predicted curve deviates from the measured pressure due to erosive burning. Otherwise, the predicted curve is a very good match.

Example 2

Use the results of static tests APM-E.11-ST1 and APM-E.14-ST1 to estimate the burn rate parameters for propellant AXP-AP4.7. This is an APCP with New Classic epoxy as binder and 10% aluminum content. Both the E.11 and E.14 are five-grain motors with a particularly long aspect (L/D) ratio. As such, both experience significant erosive burning.

The design Kn profiles, generated using SRM_1.3.xlsx, are shown below, together with the pressure-time curves from the static test firings.

APM-E.11-ST1 :

APM-E.14-ST1 :

From the two static test curves, it is clear that erosive burning occurs, apparent by the initial elevated pressure level and ramped tail-off. It is assumed that the erosive burning peters out as the core grows in diameter during the burn. The maximum Kn is matched to the pressure during the mid-point of the burn, at which point burning is assumed to be non-erosive.

Static Test ID	Kn_max	Pressure@ Kn_max (psi)
APM-E.11-ST1	385	650
APM-E.14-ST1	453	800

Using these values, the exponent m is calculated, from which the pressure exponent, n, is obtained.

AXP-AP4.7 WEIGHT D-H DENS COMPOSITION

0 AMMONIUM PERCHLORATE (AP) 65.800 -602 0.07040 1 CL 4 H 1 N 4 O

0 EPOXY 201 24.000 -661 0.04040 24 H 16 C 4 O

0 CARBON BLACK 0.200 0 0.06370 1 C

0 ALUMINUM (PURE CRYSTALINE) 10.000 0 0.09760 1 AL

THE PROPELLANT DENSITY IS 0.06119 LB/CU-IN OR 1.6937 GM/CC

THE TOTAL PROPELLANT WEIGHT IS 100.0000 GRAMS

NUMBER OF GRAM ATOMS OF EACH ELEMENT PRESENT IN INGREDIENTS

4.294500 H

1.386280 C

0.560014 N

2.582464 O

0.370645 AL

0.560014 CL

****************************CHAMBER RESULTS FOLLOW *****************************

T(K) T(F) P(ATM) P(PSI) ENTHALPY ENTROPY CP/CV GAS RT/V

2723 4443 68.02 1000.00 -55.48 237.72 1.2081 4.106 16.568

SPECIFIC HEAT (MOLAR) OF GAS AND TOTAL = 9.470 11.042

NUMBER MOLS GAS AND CONDENSED = 4.106 0.184

1.303868e+000 H2 1.302881e+000 CO 5.576438e-001 H2O 5.513949e-001 HCl

2.798577e-001 N2 1.842916e-001 Al2O3* 8.320438e-002 CO2 1.743026e-002 H

5.097948e-003 Cl 1.994683e-003 HO 6.314100e-004 AlCl2 6.207755e-004 AlCl

4.344841e-004 AlCl3 2.347277e-004 AlOCl 1.141001e-004 NH3 7.625845e-005 NO

6.395624e-005 AlHO2 5.225380e-005 CNH 4.278266e-005 AlHO 3.892884e-005 CHO

1.666849e-005 O 1.632010e-005 CH2O 1.578039e-005 COCl 1.341211e-005 Cl2

5.173023e-006 NH2 3.374803e-006 CNHO 2.897470e-006 O2 2.128502e-006 HOCl

1.8144E-06 CH4 1.8144E-06 CH4

THE MOLECULAR WEIGHT OF THE MIXTURE IS 23.309

**********PERFORMANCE: FROZEN ON FIRST LINE, SHIFTING ON SECOND LINE**********

IMPULSE IS EX T* P* C* ISP* OPT-EX D-ISP A*M EX-T

239.8 1.2215 2452 38.12 4827.9 8.51 406.1 0.15009 1267

243.9 1.1986 2481 38.42 4878.3 188.3 8.66 413.0 0.15166 1330

To_i = 2723 K.

k = 1.2081

In order to calculate the specific gas constant, the effective molecular mass is first calculated as the product mass divided by the number of gas moles:

M = 100/4.106 = 24.35 grams/mole

The specific gas constant is calculated as the universal gas constant, R¢, divided by the effective molecular mass

R¢ = 8314 J/kmole-K, giving

R = 8314/24.35 = 341 J/kg-K

APM-E.11-ST1 4443 feet/second

APM-E.14-ST1 4406 feet/second

We’ll use the average value of 4425feet/second. Using this we will calculate the actual combustion temperature of this propellant. Noting that the relationship between c-star (also denoted c*) and combustion temperature involves a square root, as shown in Equation 3 of the Solid Rocket Motor Theory -- Impulse and C-star web page, the actual combustion temperature is given by:

Therefore, To = 2723 × (4425/4828)² = 2287 K. The propellant density was measured and is:

r_p = 1.684 grams/cm³

The numerical value of the constant C may now be calculated using the formula given by Equation 1. Note that consistent units must be used, so density is converted to kg/m³ :

Equation 6 is used to calculate the numerical value of the burn rate coefficient. Again, consistent units must be employed so we convert pressure from psi to N/m²:

P₁ = 650 lbf/in² × 4.448 N/lbf × (39.37)² in²/m² = 4,481,750 N/m²

As stated earlier, the units of a are m/sec/(N/m²)ⁿ. As we usually measure chamber pressure in psi or MPa and desire burn rate to be inch/second or mm/second, the value of a can be readily converted to a the more useful form (note Pa, or Pascal, is a convenient notation for N/m² . MPa is megaPascal).

Pressure MPa and burn rate mm/sec:

0.000185 m/sec/Paⁿ × 1000 mm/m × (1×10⁶)^0.217 Paⁿ /MPaⁿ = 3.708 mm/sec/MPaⁿ

Pressure psi and burn rate inch/sec:

0.000185 m/sec/Paⁿ × 39.37 in/m × (6895)^0.217 Paⁿ /psiⁿ = 0.0497 in/sec/psiⁿ

Entered into SRM Burnrate sheet:

Predicted pressure curve for APM-E.11-ST1:

Comparison between pressure measured in static firing and SRM predicted pressure curve using derived burn rate parameters:

Likewise, the predicted pressure curve for APM-E.14-ST1 is shown below, with the derived a and n values:

Comparison between pressure measured in static firing and SRM predicted pressure curve:

Similarly, and as expected, the initial portion of the predicted curve deviates from the measured pressure due to erosive burning. Otherwise, the predicted curve is also a reasonably good match.

Example 3

Use the results of static tests APM-C.1-ST1 and APM-C.3-ST1 to estimate the burn rate parameters for propellant KNPSB. This is KP-enhanced sugar propellant. Compare the resulting burn rate parameters to the values obtained using Burn Rate Determination from Static Test Pressure Measurement method, published in the Burn Rate section of the KNPSB web page.

The design Kn profiles, generated using SRM_1.3.xlsx, are shown below, together with the pressure-time curves from the static test firings.

APM-C.1-ST1:

APM-C.3-ST1:

An interesting observation is the slow ramp-up of both curves. This is normal behaviour for propellants with a high pressure exponent. The maximum Kn is matched to the pressure during the mid-point of the burn.

Static Test ID	Kn_max	Pressure@ Kn_max (psi)
APM-C.1-ST1	169	1015
APM-C.3-ST1	131	517

Using these values, the exponent m is calculated, from which the pressure exponent, n, is obtained.

The specific heat ratio, k, the specific gas constant, R, and the ideal combustion temperature, To_i are obtained from the Technical Notepad – KNPSB Ideal Performance Calculation web page.

To_i = 1858 K.

k = 1.163

R = 228.5 J/kg-K

c*_i = 995×3.281 =3266 ft/sec. (@ 750 psi)

The delivered c-star (characteristic velocity) for each of the two test firings is taken from the KNPSB Propellant web page:

APM-C.1-ST1 964×3.281 = 3163 feet/second

APM-C.3-ST1 980 ×3.281 = 3215 feet/second

We’ll use the average value of 3189 feet/second. Using this we will calculate the actual combustion temperature of this propellant. Noting that the relationship between c-star (also denoted c*) and combustion temperature involves a square root, as shown in Equation 3 of the Solid Rocket Motor Theory -- Impulse and C-star web page, the actual combustion temperature is given by:

Therefore, To = 1858 × (3189/3266)² = 1771 K. The propellant density was measured and is:

r_p = 1.882 grams/cm³

The numerical value of the constant C may now be calculated using the formula given by Equation 1. Note that consistent units must be used, so density is converted to kg/m³ :

Equation 6 is used to calculate the numerical value of the burn rate coefficient. Again, consistent units must be employed so we convert pressure from psi to N/m²:

P₁ = 1015 lbf/in² × 4.448 N/lbf × (39.37)² in²/m² = 6,997,800 N/m²

As stated earlier, the units of a are m/sec/(N/m²)ⁿ. As we usually measure chamber pressure in psi or MPa and desire burn rate to be inch/second or mm/second, the value of a can be readily converted to the more useful form (note Pa, or Pascal, is a convenient notation for N/m² . MPa is megaPascal).

Pressure MPa and burn rate mm/sec:

a = 0.00000122 m/sec/Paⁿ × 1000 mm/m × (1×10⁶)^0.622 Paⁿ /MPaⁿ = 6.561 mm/sec/MPaⁿ

Pressure psi and burn rate inch/sec:

a = 0.00000122 m/sec/Paⁿ × 39.37 in/m × (6895)^0.622 Paⁿ /psiⁿ = 0.0117 in/sec/psiⁿ

These values are compared to those derived earlier, and presented in Figure 10 of the KNPSB Propellant web page, reproduced below:

The values for a and n using both methods are seen to be very close.

Originally posted October 26, 2023

Last updated October 26, 2023

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