
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)
Using the energy balance equation (assuming no changes in K.E. or P.E.):
Substituting in the values for h_{f} , n_{i }and n_{e} 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)
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 NH_{3} and monatomic K form, consuming energy in the process. Appendix BReserved for future use. Appendix CThe following are plots of the nozzle flow properties for the KappaDX rocket motor:
Time for flow to travel through nozzle = 430 microseconds. Appendix DThe 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 EExample: Calculate the maximum steadystate chamber pressure for the design of the KappaDX rocket motor.
Units of measure: Equation 12 of the Chamber Pressure Theory Web Page:
Therefore, 