## Background And Fundamentals

The principle of chemical reaction or combustion of one or more fuels with one or more oxidizing reactants is the basis of chemical rocket propulsion. The heat liberated in this reaction transforms the propellants (reactants) into hot gaseous reaction products, which in turn are thermodynamically expanded in a nozzle to produce thrust.

The chemical reactants or propellants can initially be either liquid or solid and occasionally also gaseous. The reaction products are usually gaseous, but with some propellants one or more reactant species remain in the solid or liquid phase. For example, with aluminized solid propellants, the chamber reaction gases contain liquid aluminum oxide and the colder gases in the nozzle exhaust contain solid, condensed aluminum oxide particles. For some of the chemical species, therefore, the analysis must consider as many as all three phases and the energy changes for the phase transitions must be included. If the amount of solid or liquid in the exhaust is small and the particles are small, then to assume a perfect gas introduces only small errors.

It is necessary to accurately know the chemical composition of the propellants and their relative proportion. In liquid propellant this means the mixture ratio and the major propellant impurities; in gelled or slurried liquid propellants it also includes suspended or dissolved solid materials; and in solid propellants it means all the ingredients, their proportions and impurities and phase (some ingredients, such as plasticizers, can be in a liquid state).

Dalton's law applies to the gas resulting from the combustion. It states that a mixture of gases at equilibrium exerts a pressure that is the sum of the partial pressures of the individual gases, all at the same temperature. The subscripts a, b, c, etc. refer to individual gas constituents.

The perfect gas equation pV = RT applies very closely to high temperature gases. Here V is the specific volume or the volume per unit mass of gas mixture, and the gas constant R for the mixture is obtained by dividing the universal gas constant R' (8314.3 J/kg-mol-K) by the average molecular mass 9K (often erroneously called the molecular weight) of the gas mixture. Using Dalton's law, Eq. 5-1 can be rewritten

The volumetric proportions of gas species in a gas mixture are determined from the molar concentration or molar fractions, nj, expressed as kg-mol for a particular species j per kg of mixture. If n is the total number of kg-mol of species j per kilogram of uniform gas mixture, then

p = RaT/Va + RbT/Vh + RCT/VC + ■•• = R'T/(mVmx) (5-3)

where rij is the kg-mol of species j per kilogram of mixture, m is the number of different gaseous species present in the equilibrium combustion gas products. The effective average molecular mass 9W of a gas mixture is then

There are n possible species which enter into the relationship and of these only m are gases, so n - m represents the number of condensed species. The molar specific heat for a gas mixture at constant pressure Cp can be determined from the individual gas molar fractions rij and their molar specific heats as shown by Eq. 5-6. The specific heat ratio k of the mixture can be determined by a similar summation or from Eq. 5-7.

When a chemical reaction goes to completion, that is, all of the reactants are consumed and transformed into reaction products, the reactants are in stoichiometric proportions. For example, consider this reaction:

All the hydrogen and oxygen are fully consumed to form the single product— water vapor—without any reactant residue of either hydrogen or oxygen. In this case it requires 1 mol of the H2 and \ mole of the 02 to obtain 1 mol of H20. On a mass basis this stoichiometric mixture requires half of 32.0 kg of 02 and 2 kg of H2, which are in the stoichiometric mixture mass ratio of 8:1. The release of energy per unit mass of propellant mixture and the combustion temperature are highest at or near the stoichiometric mixture.

Rocket propulsion systems usually do not operate with the proportion of their oxidizer and fuel in the stoichiometric mixture ratio. Instead, they usually operate fuel-rich because this allows lightweight molecules such as hydrogen to remain unreacted; this reduces the average molecular mass of the reaction products, which in turn increases the specific impulse (see Eq. 3-16). For rockets using H2 and 02 propellants the best operating mixture mass ratio for highperformance rocket engines is typically between 4.5 and 6.0, not at the stoichiometric value of 8.0.

Equation 5-8 is a reversible chemical reaction; by adding energy to the H20 the reaction can be made to go backward to create H2 and 02 and the arrow in the equation would be reversed. The decompositions of solid propellants into reaction product gases are irreversible chemical reactions, as is the reaction of liquid propellants burning to create gases. However, reactions among combustion product gases are usually reversible.

Chemical equilibrium exists in reversible chemical reactions when the rate of forming products is exactly equal to the reverse reaction of forming reactants from the products. Once this equilibrium is reached, no further changes in concentration can take place. In Equation 5-8 all three gases would be present and their relative proportions would depend on the pressure, temperature, and initial mixture.

The heat of formation AfH° is the energy released (or absorbed), or the value of enthalpy change, when 1 mole of a chemical compound is formed from its constituent atoms or elements at 1 bar (100,000 Pa) and isothermally at 298.15 K or 25°C. The A implies that it is an energy change. The subscript / refers to formation and the superscript 0 means that each product or reactant substance is at its thermodynamic standard state and at the reference pressure and temperature. By convention, the heat of formation of the gaseous elements (e.g., H2, 02, Ar, Xe, etc.) is set to zero at these standard conditions of temperature and pressure. Typical values of AjH° and other properties are given in Table 5-1 for selected species. When heat is absorbed in the formation of a product, then Ay//0 has a negative value. Earlier analyses have been made with the standard temperature at other values, such as 273.15 K and a slightly higher standard reference pressure of 1 atm (101,325 Pa).

The heat of reaction ArH° is the energy released or absorbed when products are formed from its reactants at standard reference conditions, namely at 1 bar and 25°C. The heat of reaction can be negative or positive, depending on whether the reaction is exothermic or endothermic. The heat of reaction at other temperatures or pressures has to be corrected in accordance with the change in enthalpy. When a species changes from one state to another (e.g., liquid becomes gas or vice versa), it may lose or gain energy. In most rocket propulsion the heat of reaction is determined for a constant-pressure combustion process. In general the heat of reaction can be determined from sums of the heats of formation of the products and the reactants, namely

Here rij is the molar fraction of each particular species j. In a typical rocket propellant there are a number of different chemical reactions going on simultaneously; Equation 5-9 provides the heat of reaction for all of these simultaneous reactions. For data on heats of formation and heats of reaction, see Refs. 5-7 to 5-13.

 TABLE 5-1. Chemical Thermodynamic Properties of Selected Substances at 298.15 K (25°C) and 0.1 MPa (1 bar)