## E

aijnj

propellants

Here the atomic coefficients atj are the number of kilogram atoms of element i per kg-mol of species j, and m and n are as defined above. The average molecular mass of the products in Eq. 5-5 would be jjj = 2«H2 + 32«Q2 + 18«H2Q + 16«p + nH + 17nOH ^

"H2 + «o2 + «H2O + "o + "H + "OH

Another way to determine the molar fractions for the equilibrium composition is to use a factor X that represents the degree of advancement of the chemical reaction. This factor X has the value of zero for the initial conditions before the reaction starts and 1.0 for the final conditions, when the reaction is completed and all the reaction gases are converted to product gases. For the reaction described by Eq. 5-24, X can be used in this way:

Number of moles of A: nA = aX (5—28) Number of moles of B: «B = bX

Number of moles of C: nc = c(l — X) (5—29) Number of moles of D: «D = d{\ — X)

By substituting these molar fractions into the Gibbs free energy equation (Eq. 5-12), then differentiating the expression with respect to X and setting the derivative dG/dX = 0, one can determine the value of X at which G is a minimum for the gas mixture. The degree of advancement X then determines the values of nA, nB, nc, and nD at equilibrium.

The approach used in Ref. 5-13 is commonly used today for thermochemi-cal analysis. It relies on the minimization of the Gibbs free energy and on mass balance and energy balance equations. As was explained in Eq. 5-12, the change in the Gibbs free energy function is zero at equilibrium (AG = 0): the chemical potential of the gaseous propellants has to equal that of the gaseous reaction products, which is Eq. 5-12:

AG - £(«,.AGy)products - £>yAG,)reactants - 0 (5-30)

To assist in solving this equation a Lagrangian multiplier or a factor of the degree of the completion of the reaction is often used. An alternative method for solving for the gas composition, temperature, and gas properties is to use the energy balance (Eq. 5-23) together with several mass balances (Eq. 5-26) and equilibrium relationships (Eq. 5-21).

After assuming a chamber pressure and setting up the energy balance, mass balances, and equilibrium relations, one method of solving all the equations is to estimate a combustion temperature and then solve for the various values of rij. Then a balance has to be achieved between the heat of reaction ArH° and the heat absorbed by the gases, Hj- — ff{j, to go from the reference temperature to the combustion temperature. If they do not balance, another value of the combustion temperature is chosen until there is convergence and the energy balances.

The energy release efficiency, sometimes called the combustion efficiency, can now be defined as the ratio of the actual change in enthalpy per unit propellant mixture to the calculated change in enthalpy necessary to transform the pro-pellants from the initial conditions to the products at the chamber temperature and pressure. The actual enthalpy change can be evaluated if the initial propellant condition and the actual composition and the temperature of the combustion gases are measured. Experimental measurements of combustion temperature and gas composition are difficult to perform accurately, and the combustion efficiency is therefore actually evaluated only in rare instances. The combustion efficiency in liquid propellant rocket thrust chambers depends on the method of injection and mixing and increases with increased combustion temperature. In solid propellants the combustion efficiency is a function of the grain design, the propellant, and the degree of mixing between the several solid constituents. Actual measurements on well designed rocket propulsion systems indicate efficiency values of 94 to 99%. These high values indicate that the combustion is essentially complete, that very little, if any, unreacted propellant remains, and that chemical equilibrium is indeed established.

The number of compounds or species in the exhaust can be 50 or more with solid propellants or with liquid propellants that have certain additives. The number of nearly simultaneous chemical reactions that have to be considered can easily exceed 150. Fortunately, many of these chemical species are present only in very small amounts and can usually be neglected. 