All the analysis discussed in this chapter is done today by computer programs. Most are based on minimizing the free energy. This is a simpler approach than relying on equilibrium constants, which was used some years ago. Once the values of and Tx are determined, it is possible to calculate the molecular mass of the gases (Eq. 5-5), the average molar specific heats Cp by a similar formula, and the specific heat ratio k from Eqs. 3-6 and 5-7. This then characterizes the thermodynamic conditions in the combustion chamber. With these data we can calculate c*, R, and other parameters of the chamber combustion. The nozzle expansion process simulated by computer gives the performance (such as Is, c, or A2IAt) and the gas conditions in the nozzle; it usually includes several of the corrections mentioned in Chapter 3. Programs exist for one-, two-, and three-dimensional flow patterns.
More sophisticated solutions include a supplementary analysis of combustion chamber conditions where the chamber velocities are high (see Ref. 5-14), a boundary layer analysis, a heat transfer analysis, or a two-dimensional axi-symmetric flow with nonuniform flow properties across any cross section of the nozzle. Time-dependent chemical reactions in the chamber are usually neglected, but they can be analyzed by estimating the time rate at which the reaction occurs; one way is to calculate the time derivative of the degree of advancement dk/dt and then to set this derivative to zero. This is described in Ref. 5-3.
An example of a commonly used computer program, based on chemical equilibrium compositions, was developed at the NASA Lewis Laboratory. It is described in Ref. 5-13, Vols. 1 and 2. The key assumptions for this program are one-dimensional forms of the continuity, energy, and momentum equations, zero velocity at the forward end of the chamber, isentropic expansion in the nozzle, using ideal gas laws, and chemical equilibrium in the combustion chamber. It includes options to use frozen equilibrium and narrow chambers (for liquid propellant combustion) or port areas with small cross sections (for solid propellant grains), where the chamber flow velocities are high, causing an extra pressure loss and a slight loss in performance.
Table 5-4 shows calculated data for a liquid oxygen, liquid hydrogen thrust chamber taken from an example of this reference. It has shifting equilibrium in the nozzle flow. The narrow chamber has a cross section that is only a little larger than the throat area. The large pressure drop in the chamber (approximately 126 psi) is due to the energy needed to accelerate the gas, as discussed in Section 3.3 and Table 3-2.
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