Thermodynamic calculations.

Now to the fun part!

By now we have enough information to estimate the characteristic velocity c*, the theoretical Isp, the exhaust velocity ve and the theoretical upper limit of the total impulse Itot.

The characteristic velocity is important in the design of the nozzle. It allows us to solve for the throat area At from the equation:
We will come back to this equation later.

In the next section we will estimate the burn rate r for the KNDX 65/35 propellant for the given maximum chamber pressure.

By putting the established quantities mf, mo and Pmax from the previous sections we can now calculate what species will result from the combustion, steady state combustion temperature, number of moles of gas and condensed matter and other quantities related to combustion performance. By using PROPEP we retrieve the following output:

KNDX 65/35 Run using June 1988 Version of PEP,
Case 1 of 1 12 Apr 2009 at 1:47:49.89 am

CODE WEIGHT D-H DENS COMPOSITION
1102 DEXTROSE (GLUCOSE) 6.650 -1689 0.05670 6C 12H 6O
821 POTASSIUM NITRATE 12.350 -1167 0.07670 1N 3O 1K

THE PROPELLANT DENSITY IS 0.06827 LB/CU-IN OR 1.8897 GM/CC
THE TOTAL PROPELLANT WEIGHT IS 19.0000 GRAMS

NUMBER OF GRAM ATOMS OF EACH ELEMENT PRESENT IN INGREDIENTS

0.442935 H 0.221467 C 0.122147 N 0.587907 O
0.122147 K

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

T(K) T(F) P(ATM) P(PSI) ENTHALPY ENTROPY CP/CV GAS RT/V
1703. 2606. 49.93 734.00 -25.64 31.70 1.1317 0.450 111.021

SPECIFIC HEAT (MOLAR) OF GAS AND TOTAL= 10.683 15.132
NUMBER MOLS GAS AND CONDENSED= 0.4497 0.0577

0.16588 H2O 0.08500 CO 0.07873 CO2 0.06106 N2
0.05772 K2CO3* 0.05237 H2 0.00631 KHO 0.00032 K
3.47E-05 K2H2O2 1.21E-05 NH3 3.30E-06 H 2.20E-06 KH
1.06E-06 KCN 5.56E-07 HO 4.37E-07 CH4 3.64E-07 CH2O
2.96E-07 CNH

THE MOLECULAR WEIGHT OF THE MIXTURE IS 37.441

By following Nakka's instructions and calculation example we begin by calculating the effective molar mass of the exhaust matter:
Furthermore, we need to know the mass fraction of condensed matter versus total mass of propellant in the chamber. We note that the only product that is condensed in the liquid state (*) of the combustion species is K2CO3.

The molar mass of this molecule in the chamber is M(K2CO3) = 138.2055 g/mol. Thus the mass fraction is:
Now we need to calculate the specific heats for each of the products that is of significance!
Richard Nakka has compiled this nifty chart (hope you wont mind me borrowing it):

In order to find the proper values we perform linear interpolation over the temperature range marked with yellow. The interpolation parameter t is estimated to:
Here follows the resulting specific heats for the chamber temperature T0 from the interpolation:
Remaining species from the PROPEP output are negligible.

In order to calculate the ratios of specific heat for gas, mixed and two phase particles we use the following set of equations derived by Nakka:

and lastly the ratio of the specific heat for the 2 phase flow follows:

Now we can finally calculate some interesting combustion properties such as the characteristic velocity which tells us how efficient the combustion is:
If we assume that the exit pressure is matched with the ambient pressure such that Pe = Pa = 1 atm, and further note that the maximum chamber pressure is P0 = Pmax = 5.0600 MPa = 49.938 atm we can calculate the theoretical exhaust velocity which is due to the divergent portion of the nozzle:
Here we use the 2nd phase flow version of the ratio of specific heat since this is a dynamic condition.

The theoretical (ideal) specific impulse is then very easy to calculate:
We have a relationship between the actual specific impulse and the total impulse. If we assume that we have a perfect rocket motor then the total delivered impulse would be:
Which of course is the exact same thing as mg ve. I might use mp as a symbol for propellant mass further on which of course is the same thing as grain mass mg.

The total impulse indicates we got an E-class motor. The actual total impulse will be lower due to thermal losses, pressure buildups and nozzle friction etc.

In my next post I will try to find an estimate for the burn rate for the design pressure (i.e. maximum pressure). This will be the last piece of the puzzle for the rough design of the rocket motor.


Please read the safety guidelines before attempting anything like this on your own.