CHEMICAL ROCKET PROPELLANT PERFORMANCE ANALYSIS
5.2. ANALYSIS OF CHAMBER OR MOTOR CASE CONDITIONS
170 CHEMICAL ROCKET PROPELLANT PERFORMANCE ANALYSIS
a more general case of the reaction of Equation 5-8. In this case the reactants are not in stoichiometric proportion.
In the combustion of hydrogen with oxygen it is possible to form six pro- ducts: water, hydrogen, oxygen, hydroxyl, atomic oxygen, and atomic hydro- gen. In this case all the reactants and products are gaseous. Theoretically, there could be two additional products: ozone 03 and hydrogen peroxide H202;
however, these are unstable materials that do not readily exist at high tempera- ture, and they can be ignored. In chemical notation this can be stated by
The left side shows the condition before and the right side after the reaction.
Since H2 and 02 can be found on both sides, it means that not all of these species are consumed and a portion, namely nH2 and n02 , will remain unreacted. With chemical equilibrium at a particular temperature and pressure the molar concentrations on the right side will remain fixed. Here a, b, nH2o, nH2, no2, no, nH, and noH are the respective molar fractions or molar quantities of these substances before and after the reaction, and they can be expressed in kg-mo! per kilogram of propellant reactants or reaction products. The initial proportions of a and b are usually known. The number of kg-mo! per kilogram of mixture of each element can be established from this initial mix of oxidizer and fuel ingredients. For the hydrogen-oxygen relation above, the mass bal- ances would be
for hydrogen: 2a
=
2nH20+
2nH2+
nH+
noHl
for oxygen: 2b = nH2o
+
2n02+
no+
noH (5-25) The mass balance of Eq. 5-25 provides two more equations for this reaction (one for each atomic species) in addition to the energy balance equation. There are six unknown product percentages and an unknown combustion or equili- brium temperature. However, three equations provide a solution for only three unknowns, say the combustion temperature and the molar fractions of two of the species. If, for example, it is known that the initial mass mixture ratio of b/a is fuel rich, so that the combustion temperature will be relatively low, the percentage of remaining 02 and the percentage of the dissociation products (0, H, and OH) would all be very low and can be neglected. Thus n0 , nH, n0H, and n02 are set to be zero. The solution requires knowledge of the enthalpy change of each of the species, and that information can be obtained from existing tables, such as Table 5-2 or Refs. 5-8 and 5-9.In more general form, the mass for any given element must be the same before and after the reaction. The number of kg-mo! of a given element per kilogram of reactants and product is equal, or their difference is zero. For any one atomic species, such as the Hor the O in Eq. 5-25,
5.2. ANALYSIS OF CHAMBER OR MOTOR CASE CONDITIONS 171
[ m
LaunjJ
-[n
LaunjJ =
0j= 1 products j= 1 propellants
(5-26)
Here the atomic coefficients au are the number of kilogram atoms of element i per kg-mo! of species}, and m and n are as defined above. The average mole- cular mass of the products in Eq. 5-5 would be
fill = 2nH2
+
32n02+
l 8nH2o+
16n0+
nH+
17noHnH2
+
no2+
nH20+
no+
nH+
noH (5-27) Another way to determine the molar fractions for the equilibrium composition is to use a factor A that represents the degree of advancement of the chemical reaction. This factor A 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, A can be used in this way:Number of moles of A: nA = aA Number of moles of B: n8 = b..t Number of moles of C: nc = c(l - ..t) Number of moles of D: n0 = d(l - ..t)
(5-28) (5-29)
By substituting these molar fractions into the Gibbs free energy equation (Eq.
5-12), then differentiating the expression with respect to ), and setting the derivative dG/d..t = 0, one can determine the value of A at which G is a mini- mum for the gas mixture. The degree of advancement A then determines the values of nA, n8 , nc, and n0 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 (LlG = 0):
the chemical potential of the gaseous propellants has to equal that of the gaseous reaction products, which is Eq. 5-12:
(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
172 CHEMICAL ROCKET PROPELLANT PERFORMANCE ANALYSIS
to estimate a combustion temperature and then solve for the various values of n1. Then a balance has to be achieved between the heat of reaction t-.,H0 and the heat absorbed by the gases,
H~ - Hg,
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 pro- pellant condition and the actual composition and the temperature of the com- bustion 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.