The objectives here are to determine the theoretical combustion temperature and the theoretical composition of the resulting reaction products, which in turn will allow the determination of the physical properties of the combustion gases (Cp, k, or p). Before we can make this analysis, some basic data (e.g., propellants, their ingredients, desired chamber pressure, or all likely reaction products) have to be known or postulated. Although the combustion process really consists of a series of different chemical reactions that occur almost simultaneously and includes the breakdown of chemical compounds into intermediate and subsequently into final products, the analysis is only concerned with the initial and final conditions, before and after combustion. We will mention several approaches to the analysis of chamber conditions. In this section we will first give some definitions of key terms and explain some concepts and principles. The first principle concerns the conservation of energy. The heat created by the combustion is equal to the heat necessary to raise the resulting gases adiabatically to their final combustion temperature. The heat of reaction of the combustion ArH has to equal the enthalpy change AH of the gases.
The energy balance can be thought of as a two-step process. The chemical reaction occurs instantaneously but isothermally at the reference temperature, and the resulting energy release then heats the gases from this reference temperature to the final combustion temperature. The heat of reaction is
Here Ah is the increase in enthalpy for each species multiplied by its molar fraction, and Cp is the molar specific heat at constant pressure.
The second principle is the conservation of mass. The mass of any of the atomic species present in the reactants before the chemical reaction must be equal to the mass of the same species in the products. This can be illustrated by
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 products: water, hydrogen, oxygen, hydroxyl, atomic oxygen, and atomic hydrogen. 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 temperature, and they can be ignored. In chemical notation this can be stated by aH2 + b02 «h2oH20 + «h2H2 + n0l02 + n00 + «hH + «ohOH (5-24)
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 nHl and nQl, 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, nHlQ, «h2, nQ nQ, «H, and «OH are the respective molar fractions or molar quantities of these substances before and after the reaction, and they can be expressed in kg-mol per kilogram of propellant reactants or reaction products. The initial proportions of a and b are usually known. The number of kg-mol 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 balances would be for hydrogen: 2a = 2«h2o + 2«h2 + "h + "oh for oxygen: 2b = «h2o + 2no2 + no + «oh
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 equilibrium 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 (O, H, and OH) would all be very low and can be neglected. Thus n0, «h> moh> and nQl 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-mol 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 H or the O in Eq. 5-25,
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