## Richard Nakka's Experimental Rocketry Web Site

### Rocket Theory Appendices

 Appx.A - Calculation of AFT for KN-Sucrose Appx. B- (Reserved) Appx. C - Flow Properties for Kappa-DX Nozzle Appx. D - Expression for Mass Flow rate through Nozzle Appx. E - Calculation of Max. Chamber Pressure for Kappa-DX Motor

C12H22O11 + 6.288 KNO3 -> 3.796 CO2 + 5.205 CO + 7.794 H2O + 3.065 H2 + 3.143 N2 + 2.998 K2CO3 + 0.274 KOH
The enthalpies of formation for the reactants are obtained from the CRC Handbook of Chemistry and Physics, and for the products, from the JANAF thermochemical tables:     (units are kJ/mole)

 C12H22O11 -2222.1 KNO3 -494.63 CO2 -393.52 CO -110.53 H2O -241.83 H2 0 N2 0 K2CO3 -1150.18 KOH -424.72

Using the energy balance equation (assuming no changes in K.E. or P.E.): Substituting in the values for hf , ni and ne gives :

1 (-2222.10 + 0) + 6.288(-494.63 + 0)    =    3.796(-393.52 + CO2) + 5.205(-110.53 + CO) + 7.794(-241.83 + H2O) + 3.065(0 + H2) + 3.143 (0 + N2) + 2.998 (-1150.18 + K2CO3) + 0.274 (-424.72 + KOH)

Expanding and gathering terms simplifies the equation to the following form:

2186.2    =    3.796 CO2 + 5.205 CO + 7.794 H2O + 3.065 H2 + 3.143 N2 + 2.998 K2CO3 + 0.274 KOH

Solution of the equation is obtained by simply substituting in values for at a certain temperature. This temperature is equal to the AFT when the the right hand side of the equation is equal to the left hand side (=2186.2).

Take a guess that the AFT lies somewhere between 1700 K and 1800 K (easy for me to guess, as I know the answer! But no matter what the guess, the answer will eventually converge).

From the JANAF tables, the values of are:     (units are kJ/mole)

 T CO2 CO H2O H2 N2 K2CO3 KOH 1700 K 73.480 45.945 57.758 42.835 45.429 280.275 116.505 1800 K 79.431 49.526 62.693 46.169 48.978 301.195 124.815

For the term on the right side of the equation, substituting in the values at T=1700K :

3.796 (73.480) + 5.205 (45.945) + 7.905 (57.758) + 3.065 (42.835) + 3.143 (45.429) + 2.998 (280.275) + 0.274 (116.505) = 2114.5 kJ/mole

Substituting in the values at T=1800 K:

3.796 (79.431) + 5.205 (49.526) + 7.794 (62.693) + 3.065 (46.169) + 3.143 (48.978) + 2.998 (301.195) + 0.274 (124.815) = 2280.6 kJ/mole

Clearly, the actual temperature lies in between 1700 and 1800 K. The actual value may be found by using linear interpolation: This is in close agreement with the combustion temperature predicted by GUIPEP (1720 K.), that being about 1% lower. The small deviation is a result of the simplified combustion equation assumed in this example. In reality, some trace products such as NH3 and monatomic K form, consuming energy in the process.

#### Appendix B

Reserved for future use.

#### Appendix C

The following are plots of the nozzle flow properties for the Kappa-DX rocket motor:      Time for flow to travel through nozzle = 430 microseconds.

#### Appendix D

The derivation of the expression for mass flow rate through the nozzle is presented here.
From Equation 9 of the
Nozzle Theory Web Page, the continuity equation for mass flow rate through the nozzle is given by: where * designates critical (throat) conditions. From Equation 7 of the referenced web page, the critical flow density may be written as: and from Equations 3 & 4, the critical (sonic) velocity may be given by: From the ideal gas law, the chamber density may be expressed as: Substitutionof this equation and those for critical density and velocity into the mass flow rate expression gives: which may be rearranged to the form of the expression shown as Equation 4 of the Chamber Pressure Theory Web Page: #### Appendix E

Example:   Calculate the maximum steady-state chamber pressure for the design of the Kappa-DX rocket motor.

Units of measure:
The most prudent (botch-proof) system of units is mks (metre : kilogram : second), however, for this example, appropriate English units will used, as well.

Equation 12 of the Chamber Pressure Theory Web Page: Burn/throat area Kn, max. = 378 Propellant density p = 1.806 g/cm3 = 1806 kg/m3 = 0.00203 slug/in3 Burn rate r = 12.65 mm/s = 0.01265 m/s = 0.50 in/s Propellant c-star c* = 912 m/s = 2992 ft/s

Therefore,
Po = 378 (1806) .01265 (926) = 7.9 x 106 N/m2 ( 7.9 MPa)
or
Po = 378 (0.00203) 0.50 (2992) = 1148 psi #### Last updated  August 19, 2001

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