**Richard Nakka’s Experimental Rocketry Web Site**

________________________________________________

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,

__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** =

*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/m*^{3}. 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 *P*_{1}
and *P*_{2}.

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 *P*_{1} and *P*_{2}.

Pressure
exponent, *a*, can be separated:

Solving
for *a* in terms of *P*_{1}
and *Kn*_{1}:

where
*MKS* units of *a*
are *m/sec/(N/m ^{2})^{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 *P*_{1}
and *P*_{2} and the corresponding
values for *K*n_{1} and *K*n_{2}.

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 P_{1} and P_{2}**,
as long as the same units are used for both pressure values (this is because

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 *P*_{1}
and *P*_{2} 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/cm^{3}

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/m*^{3} :

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/m^{2}:

P_{1} = 930 lbf/in^{2}
× 4.448 N/lbf × (39.37)^{2} in^{2}/m^{2} = 6,411,779
N/m^{2}

As stated earlier, the
units of *a* are *m/sec/(N/m ^{2})^{n}*.
As we usually measure chamber pressure in

Pressure *MPa* and burn rate *mm/sec*:

0.0000147 m/sec/Pa^{n}
× 1000 mm/m × (1×10^{6})^{0.392} Pa^{n} /MPa^{n}
= __3.31__*mm/sec/MPa ^{n}*

Pressure *psi* and burn rate *inch/sec*:

0.0000147 m/sec/Pa^{n}
× 39.37 in/m × (6895)^{0.392} Pa^{n} /psi^{n} = __0.0185__ *in/sec/psi ^{n}*

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.

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-AP4.7 propellant is shown below.

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

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-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/cm^{3}

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/m*^{3} :

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/m^{2}:

P_{1} = 650 lbf/in^{2}
× 4.448 N/lbf × (39.37)^{2} in^{2}/m^{2} = 4,481,750
N/m^{2}

As stated earlier, the
units of *a* are *m/sec/(N/m ^{2})^{n}*.
As we usually measure chamber pressure in

Pressure *MPa* and burn rate *mm/sec*:

0.000185 m/sec/Pa^{n}
× 1000 mm/m × (1×10^{6})^{0.217} Pa^{n} /MPa^{n}
= __3.708__
*mm/sec/MPa ^{n}*

Pressure *psi* and burn rate *inch/sec*:

** **0.000185 m/sec/Pa^{n}
× 39.37 in/m × (6895)^{0.217} Pa^{n} /psi^{n} = __0.0497__*in/sec/psi ^{n}*

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-E.11-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-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/cm^{3}

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/m*^{3} :

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/m^{2}:

P_{1} = 1015 lbf/in^{2}
× 4.448 N/lbf × (39.37)^{2} in^{2}/m^{2} = 6,997,800
N/m^{2}

As stated earlier, the
units of *a* are *m/sec/(N/m ^{2})^{n}*.
As we usually measure chamber pressure in

Pressure *MPa* and burn rate *mm/sec*:

*a* = 0.00000122 m/sec/Pa^{n} × 1000 mm/m × (1×10^{6})^{0.622}
Pa^{n} /MPa^{n} = __6.561__ *mm/sec/MPa ^{n}*

Pressure *psi* and burn rate *inch/sec*:

*a* = 0.00000122 m/sec/Pa^{n} × 39.37 in/m × (6895)^{0.622}
Pa^{n} /psi^{n} = __0.0117__*in/sec/psi ^{n}*

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**