This chapter is an introduction to the theory, performance, and description of the most useful gas parameters in chemical rocket propulsion systems. It identifies relevant chemical fundamentals, basic analytical approaches, and key equations. While not presenting complete and complex analytical results to actual propulsion systems, it does list references for the required detailed analyses. Also included are tables of calculated results for several common liquid propellant combinations and select solid propellants, including exhaust gas composition for some of these. Furthermore, it describes how various key parameters influence both performance and exhaust gas composition.

In Chapter 3, simplified one‐dimensional performance relations were developed that require knowledge of the composition of rocket propulsion gases and some properties of propellant reaction products, such as their chamber temperature T₁, average molecular mass , and specific heat ratios or enthalpy change across the nozzle (). This chapter presents several theoretical approaches to determine these relevant thermochemical properties for any given composition or propellant mixture, chamber pressure, and nozzle area ratio or exit pressure. Such information then allows the determination of performance parameters, including theoretical exhaust velocities and specific impulses in chemical rockets.

By knowing temperature, pressure, and composition, it is possible to calculate other combustion gas properties. This information also permits the analysis and selection of materials for chamber and nozzle structures. Heat transfer analyses require knowledge of specific heats, thermal conductivities, and viscosities for the gaseous mixture. The calculated exhaust gas composition also forms the basis for estimating environmental effects, such as the potential spreading of toxic clouds near a launch site, as discussed in Chapter 21. Exhaust gas parameters are needed for the analysis of exhaust plumes (Chapter 20), which include plume profiles and mixing effects external to the nozzle.

The advent of digital computers has made possible solving large sets of equations involving mass balances and energy balances together with the thermodynamics and chemical equilibria of complex systems for a variety of propellant ingredients. In this chapter, we discuss sufficient theoretical analysis background material so the reader can understand the thermodynamic and chemical basis for computer programs in use today. We do not introduce any specific computer programs but do identify relevant phenomena and chemical reactions.

The reader is referred to Refs. 5–1 through 5–5 (and 5–16) for much of the general chemical and thermodynamic background and principles. For detailed descriptions of the properties of each possible reactant and reaction products, consult Refs. 5–6 through 5–16.

All analytical work requires some simplifying assumptions and as more phenomena are understood and mathematically simulated analytical approaches and resulting computer codes become more realistic (but also more complex). The assumptions made in Section 3.1 for an ideal rocket are valid only for quasi‐one‐dimensional flows. However, more sophisticated approaches make several of these assumptions unnecessary. Analytical descriptions together with their assumptions are commonly divided into two separate parts:

1. The combustion process is the first part. It normally occurs at essentially constant pressure (isobaric) in the combustion chamber and the resulting gases follow Dalton's law, which is discussed in this chapter. Chemical reactions do occur very rapidly during propellant combustion. The chamber volume is assumed to be large enough and the residence time long enough for attaining chemical equilibrium within the chamber.
2. The nozzle gas expansion process constitutes the second set of calculations. The equilibrated gas combustion products from the chamber then enter a supersonic nozzle where they undergo an adiabatic expansion without further chemical reactions. The gas entropy is assumed constant during reversible nozzle gas expansions, although in real nozzles it increases slightly.

The principal chemical reactions commonly taken into account occur only inside the combustion chamber of a liquid propellant rocket engine or inside the grain cavity of a solid propellant rocket motor (usually within a short distance from the burning surface). Chamber combustion analyses are discussed further in Chapters 9 and 14. In reality, however, some chemical reactions also occur in the nozzle as the gases expand; with such shifting equilibrium the composition of the flowing reaction products may noticeably change inside the nozzle, as described later in this chapter. A further set of chemical reactions can occur in the exhaust plume outside the nozzle, as described in Chapter 20—many of the same basic thermochemical approaches described in this chapter may also be applied to exhaust plumes.

From Eqs. 2–6 and 3–33, we see that represent a rocket's performance. As discussed in Chapters 2 and 3, the thrust coefficient depends almost entirely on the nozzle gas expansion process and the characteristic velocity depends almost exclusively on the effects of combustion. In this chapter, we will focus on the key parameters that make up .

5.1 BACKGROUND AND FUNDAMENTALS

The description of chemical reactions between one or more fuels with one or more oxidizing reactants forms the basis for chemical rocket propulsion combustion analysis. The heat liberated in such reactions transforms the propellants into hot gaseous products, which are subsequently expanded in a nozzle to produce thrust.

The propellants or stored chemical reactants can initially be either liquid or solid and sometimes also gaseous (such as hydrogen heated in the engine's cooling jackets). Reaction products are predominantly gaseous, but for some propellants one or more reactant species may partly 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 may contain solid, condensed aluminum oxide particles. For some chemical species, therefore, analysis must consider all three phases and energy changes involved in their phase transitions. When the amount of solid or liquid species in the exhaust is negligibly small and the particles themselves are small, perfect gas descriptions introduce only minor errors.

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

Dalton's law may be applied to the resulting combustion gases. It states that a mixture of gases at equilibrium exerts a pressure that is the sum of all the partial pressures of its individual constituents, each acting at a common total chamber volume and temperature. The subscripts a, b, c, and so on below refer to individual gas constituents:

5–1

5–2

The perfect gas equation of state accurately represents high‐temperature gases. For the j‐th chemical species, V_j is the “specific volume” (or reaction‐chamber volume per unit component mass) and R_j is the gas constant for that species, which is obtained by dividing the universal gas constant R′ (8314.3 J/kg‐mol‐K) by the species molecular mass (called molecular weight in the earlier literature). Using Eq. 5–1 we may now write the total pressure in a mixture of chemical species as

5–3

The volumetric proportions for each gas species in a gas mixture are determined from their molar concentrations, n_j, expressed as kg‐mol for a particular species j per kg of mixture. If n is the total number of kg‐mol of all species per kilogram of uniform gas mixture, then the mol fraction X_j becomes

5–4

where n_j is the kg‐mol of species j per kilogram of mixture and the index m represents the total number of different gaseous species in the equilibrium combustion gas mixture. In Eq. 5–3, the effective average molecular mass for a gas mixture becomes

5–5

When there are ℓ possible species which chemically coexist in a mixture and of these m are gaseous, then represents the number of condensed species. The molar specific heat for a gas mixture at constant pressure C_p can be determined from the individual gas molar fractions n_j and their molar specific heats as shown by Eq. 5–6. The specific heat ratio k for the perfect gas mixture is shown in Eq. 5–7:

5–6

5–7

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

5–8

All the hydrogen and oxygen are fully consumed to form the single product—water vapor—without any reactant residue. It requires 1 mol of the H₂ and mol of the O₂ to obtain 1 mol of H₂O. On a mass basis, this stoichiometric mixture requires of 16.0 kg of O₂ 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 always highest at or near the stoichiometric condition.

It is usually not advantageous in rocket propulsion systems to operate with the oxidizer and fuel at their stoichiometric mixture ratio. Instead, they tend to operate fuel rich because this allows low molecular mass molecules such as hydrogen to remain unreacted; this reduces the average molecular mass of the reaction products, which in turn increases their exhaust velocity (see Eq. 3–16) provided other factors are comparable. For rockets using H₂ and O₂ propellants, the best operating mixture mass ratio for high‐performance rocket engines ranges between 4.5 and 6.0 (fuel rich) because here the drop in combustion temperature (T₀) remains small even though there is unreacted H₂(g) in the exhaust.

Equation 5–8 is actually a reversible chemical reaction; by adding energy to the H₂O(g) the reaction can be made to go backward to recreate H₂(g) and O₂(g) at high temperature and the arrow in the equation would be reversed. The decomposition of solid propellants, identified with an (s), into reaction product gases involves irreversible chemical reactions, as is the burning of liquid propellants, denoted with an (l), to create gases. However, reactions among gaseous combustion product may be reversible.

Chemical equilibrium occurs in reversible chemical reactions when the rate of product formation exactly equals the reverse reaction (one forming reactants from products). Once this equilibrium is reached, no further changes in concentration take place. In Equation 5–8 all three gases would be present in relative proportions that would depend on the pressure, temperature, and equilibrium state of the mixture.

The heat of formation Δ_fH⁰ (also called enthalpy of formation) is the energy released (or absorbed), or the value of enthalpy change, when 1 mol 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 symbol Δ implies an energy change. The subscript f 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, heats of formation of the elements in gaseous form (e.g., H₂, O₂, Ar, Xe, etc.) are set to zero at standard temperature and pressure. Typical values of Δ_fH⁰ and other properties are given in Table 5–1 for selected species. When heat is absorbed in the formation a chemical compound, the given Δ_fH⁰ has a positive value. Earlier published tables show values given at a temperature of 273.15 K and a slightly higher standard reference pressure of 1 atm (101, 325 Pa) than Table 5–1.

Substance	Phasea	Molar Mass (g/mol)	ΔfH0 (kJ/mol)	(kJ/mol)	S0 (J/mol‐K)	Cp (J/mol‐K)
Al (crystal)	s	29.9815	0	0	28.275	24.204
Al2O3	l	101.9612	−1620.567	−1532.025	67.298	79.015
C (graphite)	s	12.011	0	0	5.740	8.517
CH4	g	16.0476	−74.873	−50.768	186.251	35.639
CO	g	28.0106	−110.527	−137.163	197.653	29.142
CO2	g	44.010	−393.522	−394.389	213.795	37.129
H2	g	2.01583	0	0	130.680	28.836
HCl	g	36.4610	−92.312	−95.300	186.901	29.136
HF	g	20.0063	−272.546	−274.646	172.780	29.138
H2O	l	18.01528	−285.830	−237.141	69.950	75.351
H2O	g	18.01528	−241.826	−228.582	188.834	33.590
N2H4	l	32.0453	+50.434	149.440	121.544	98.666
N2H4	g	32.0453	+95.353	+159.232	238.719	50.813
NH4ClO4	s	117.485	−295.767	−88.607	184.180	128.072
N2O4	l	92.011	−19.564	+97.521	209.198	142.509
N2O4	g	92.011	9.079	97.787	304.376	77.256
NO2	g	46.0055	33.095	51.258	240.034	36.974
HNO3	g	63.0128	−134.306	−73.941	266.400	53.326
N2	g	28.0134	0	0	191.609	29.125
O2	g	31.9988	0	0	205.147	29.376
NH3	g	17.0305	−45.898	−16.367	192.774	35.652

^a. When species are listed twice, as liquid and gas, their existence is due to evaporation or condensation.

The molar mass can be in g/g‐mol or kg/kg‐mol; C_p is given in J/g‐mol‐K or kJ/kg‐mol‐K, for J/kg‐mol‐K multiply tabulated values by 10³.

The heat of reaction Δ_rH⁰ can be negative or positive, depending on whether the reaction is exothermic or endothermic. The heat of reaction at other than standard reference conditions has to be corrected in accordance with corresponding changes in the enthalpy. Also, when a species changes from one state to another (e.g., liquid becomes gas), it may lose or gain energy. In most rocket propulsion computations, 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 products and reactants, namely,

5–9

Here, n_j is the molar concentration of each particular species j. In a typical rocket propellant, there are a number of different chemical reactions going on simultaneously; Eq. 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 through 5–13 and 5–15.

Various thermodynamic criteria for representing necessary and sufficient conditions for stable equilibria were first advanced by J. W. Gibbs and are based on minimizing the system's energy. The Gibbs free energy G (often called the chemical potential) is a convenient function or “property of state” for the chemical system describing its thermodynamic potential, and it is directly related to the constituents' internal energy U, pressure p, molar specific volume V, enthalpy h, temperature T, and entropy S. For any single species j the free energy is defined below as G_j; it can be determined for specific thermodynamic conditions, for mixtures of gases as well as for individual gas species:

5–10

For most materials used as rocket propellants, the Gibbs free energy has been determined and tabulated as a function of temperature. It can then be corrected for pressure. Typical G_j's units are J/kg‐mol. For a series of different species the mixture or total free energy G is

5–11

For perfect gases, the free energy is only a function of temperature and pressure. It represents another defined property, just as the enthalpy or the density; only two such independent properties are required to characterize the state of a single‐species gas. The free energy may be thought of as a tendency or driving force for a chemical substance to enter into a chemical (or physical) change. Only differences in chemical potential can be measured. When the chemical potential of the reactants is higher than that of their likely products, a chemical reaction can occur and the gas composition may change. The change in free energy ΔG for reactions at constant temperature and pressure is the value of the chemical potential of the products less that of the reactants:

5–12

Here, the index m accounts for the number of gas species in the combustion products, the summation index r accounts for the number of gas species in the reactants, and the ΔG represents the maximum energy that can be “freed to do work on an open system” (i.e., one where mass enters and leaves the system). At equilibrium the free energy is a minimum—at its minimum any small change in mixture fractions causes negligible change in ΔG, and the free energies of the products and the reactants are essentially equal; here

5–13

and a curve of molar concentration n versus ΔG would display a minimum.

If the reacting propellants are liquid or solid materials, energy will be needed to change phase and/or vaporize them or to break them down into other (gaseous) species. This energy has to be subtracted from that available to heat the gases from the reference temperature to the combustion temperature. Therefore, values of ΔH⁰ and ΔG⁰ for liquid and solid species are considerably different from those for the same species initially in a gaseous state. The standard free energy of formation Δ_fG⁰ is the increment in free energy associated with the reaction forming a given compound or species from its elements at their reference state. Table 5–2 gives values of Δ_fH⁰ and Δ_fG⁰ and other properties for carbon monoxide as a function of temperature. Similar data for other species can be obtained from Refs. 5–7 and 5–13. The entropy is another thermodynamic property of matter that is relative, which means that it is determined only as a change in entropy. In the analysis of isentropic nozzle flow, for example, it is assumed that the entropy remains constant. For a perfect gas, the change of entropy is given by

5–14

and for constant C_p the corresponding integral becomes

5–15

where the subscript “zero” applies to the reference state. For mixtures, the total entropy becomes

5–16

Here, the entropy S_j is in J/kg‐mol‐K. The entropy for each gaseous species is

5–17

For solid and liquid species, the last two terms are zero. Here, refers to the standard state entropy at a temperature T. Typical values for entropy are listed in Tables 5–1 and 5–2.

Temp.
(K)	(J/mol‐K)		(kJ/mol)	(kJ/mol)	(kJ/mol)
0	0	0	−8.671	−113.805	−113.805
298.15	29.142	197.653	0	110.527	−137.163
500	29.794	212.831	5.931	−110.003	−155.414
1000	33.183	234.538	21.690	−111.983	−200.275
1500	35.217	248.426	38.850	−115.229	−243.740
2000	36.250	258.714	56.744	−118.896	−286.034
2500	36.838	266.854	74.985	−122.994	−327.356
3000	37.217	273.605	93.504	−127.457	−367.816
3500	37.493	279.364	112.185	−132.313	−407.497
4000	37.715	284.386	130.989	−137.537	−446.457

To change units to J/kg‐mol multiply tabulated values by 10³.

5.2 ANALYSIS OF CHAMBER OR MOTOR CASE CONDITIONS

The objective here is to determine the theoretical combustion temperature and the theoretical composition of the reaction products, which in turn will allow the determination of the physical properties of the combustion gases (C_p, k, ρ, or other). Before we can perform such analyses, some basic information has to be known or postulated (e.g., propellants, their ingredients and proportions, desired chamber pressure, and all likely reaction products). Although combustion processes really consist of a series of different chemical reactions that occur almost simultaneously and include the breakdown of chemical compounds into intermediate and subsequently into final products, here we are only concerned with initial and final conditions, before and after combustion. We will mention several approaches to analyzing chamber conditions; in this section we first give some definitions of key terms and introduce relevant principles.

The first principle concerns the conservation of energy. The heat created by the combustion is set equal to the heat necessary to adiabatically raise the resulting gases to their final combustion temperature. The heat of reaction of the combustion Δ_rH has to equal the enthalpy change ΔH of the reaction product gases. Energy balances may be thought of as a two‐step process: the chemical reaction process occurs instantaneously but isothermally at the reference temperature, and then the resulting energy release heats the gases from this reference temperature to the final combustion temperature. The heat of reaction, Equation 5–9, becomes

5–18

Here, the Δh_j, the increase in enthalpy for each species, is multiplied by its molar concentration n_j and C_pj is the species molar specific heat at constant pressure.

The second principle is the conservation of mass. The mass of any atomic species present in the reactants before the chemical reaction must equal that of the same species in the products. This can be better illustrated with a more general case of the reaction shown in Equation 5–8 when the reactants are not in stoichiometric proportion.

The combustion of hydrogen with oxygen is used below as an example. It may yield six possible products: water, hydrogen, oxygen, hydroxyl, atomic oxygen, and atomic hydrogen. Here, all reactants and products are gaseous. Theoretically, there could be two additional products: ozone O₃ and hydrogen peroxide H₂O₂; however, these are unstable compounds that do not exist for long at high temperatures and can be ignored. In chemical notation the mass balance may be stated as

5–19

The left side shows the condition before the reaction and the right side the condition after. Since H₂ and O₂ are found on both sides, it means that not all of these species are consumed and a portion, namely, and , will remain unreacted. At any particular temperature and pressure, the molar concentrations on the right side will remain fixed when chemical equilibrium prevails. Here, a, b, , , and n_OH are the respective molar concentrations of these substances before and after the reaction, these are expressed in kg‐mol per kilogram of propellant reaction products or of mixture; 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

5–20

The mass balances of Eq. 5–20 provide 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 can only solve for three unknowns, say the combustion temperature and the molar fractions of two of the species. When, 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 O₂ and the percentage of the dissociation products (O, H, and OH) would all be very low and may be neglected. Thus, n_O, n_H, n_OH, and 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 each atomic species, such as the H or the O in Eq. 5–20,

5–21

Here, the atomic coefficients a_ij are the number of kilogram atoms of element i per kg‐mol of species j, and m and r are indices as defined above. The average molecular mass for the products, using Eqs. 5–5 and 5–19, becomes

5–22

The computational approach used in Ref. 5–13 is the one commonly used today for thermochemical analyses. It relies on the minimization of the Gibbs free energy and on the mass balance and energy balance equations. As was indicated in Eqs. 5–12 and 5–13, the change in the Gibbs free energy function is zero at equilibrium; here, the chemical potential of the gaseous propellants has to equal that of the gaseous reaction products, which is Eq. 5–12:

5–23

To assist in solving this equation a “Lagrangian multiplier,” a factor representing the degree of the completion of the reaction, is often used. An alternative older method in solving for gas composition, temperature, and gas properties is to use the energy balance (Eq. 5–18) together with several mass balances (Eq. 5–21) and certain equilibrium constant relationships (see for example Ref. 5–16).

After assuming a chamber pressure and setting up the energy balance, mass balances, and equilibrium relations, another method of solving all the equations is to estimate a combustion temperature and then solve for the various values of n_j. Then, check to see if a balance has been achieved between the heat of reaction Δ_rH⁰ and the heat absorbed by the gases, , going from the reference temperature to the combustion temperature. If they do not balance, the value of the combustion temperature is iterated until there is convergence and the energy balances.

The energy release efficiency, sometimes called the combustion efficiency, can be defined here as the ratio of the actual change in enthalpy per unit propellant mixture to the calculated change in enthalpy necessary to transform the reactants from the initial conditions to the products at the chamber temperature and pressure. The actual enthalpy change is evaluated when the initial propellant conditions and the actual compositions and the temperatures of the combustion gases are measured. Measurements of combustion temperature and gas composition are difficult to perform accurately, and combustion efficiency is therefore only experimentally evaluated in rare instances (such as in some R & D programs). Combustion efficiencies in liquid propellant rocket thrust chambers also depend on the method of injection and mixing and increases with increasing combustion temperature. In solid propellants the combustion efficiency becomes a function of grain design, propellant composition, and degree of uniform mixing among the several solid constituents. In well‐designed rocket propulsion systems, actual measurements yield energy release efficiencies from 94 to 99%. These high efficiencies indicate that combustion is essentially complete, that is, that negligible amounts of unreacted propellant remain and that chemical equilibrium is indeed closely established.

The number of compounds or species in combustion exhausts can be large, up to 40 or more with solid propellants or with liquid propellants that have certain additives. The number of nearly simultaneous chemical reactions that take place may easily exceed 150. Fortunately, many of these chemical species are present only in relatively small amounts and may usually be neglected.

5.3 ANALYSIS OF NOZZLE EXPANSION PROCESSES

There are several methods for analyzing the nozzle flow, depending on the chemical equilibrium assumptions made, nozzle expansion particulates, and/or energy losses. Several are outlined in Table 5–3.

Step	Process	Method/Implication/Assumption
Nozzle inlet condition	Same as chamber exit; need to know T1, p1, , H, , ρ1, etc.	For simpler analyses assume the flow to be uniformly mixed and steady.
Nozzle expansion	An adiabatic process, where flow is accelerated and thermal energy is converted into kinetic energy. Temperature and pressure drop drastically. Several different analyses have been used with different specific effects. Can use one‐, two‐, or three‐dimensional flow pattern. The number of species can be small (Eq. 5–19 has 6) or large (Table 5–8 has 30).	Simplest method is inviscid isentropic expansion flow. Include internal weak shock waves; no longer a truly isentropic process. If solid particles are present, they will create drag, thermal lag, and a hotter exhaust gas. Must assume an average particle size and optical surface properties of the particulates. Flow is no longer isentropic. Include viscous boundary layer effects and/or nonuniform velocity profile.
Often a simple single correction factor is used with one‐dimensional analyses to modify nozzle exit condition for items 2, 3, and/or 4 above. Computational fluid dynamic codes with finite element analyses have been used with two‐ and three‐dimensional nozzle flow.
Chemical equilibrium during nozzle expansion	Due to rapid decrease in T and p, the equilibrium composition can change from that in the chamber. The four processes listed in the next column allow progressively more realistic simulation and require more sophisticated techniques.	Frozen equilibrium; no change in gas composition; usually gives low performance. Shifting equilibrium or instantaneous change in composition; usually overstates the performance slightly. Use reaction time rate analysis to estimate the time to reach equilibrium for each chemical reaction; some rate constants are not well known; analysis is more complex. Use different equilibrium analysis for boundary layer and main inviscid flow; will have nonuniform gas temperature, composition, and velocity profiles.
Heat release in nozzle	Recombination of dissociated molecules (e.g., ) and exothermic reactions due to changes in equilibrium composition cause internal heating of the expanding gases. Particulates release heat to the gas.	Heat released in subsonic portion of nozzle will increase the exit velocity. Heating in the supersonic flow portion of nozzle can increase the exit temperature but reduce the exit Mach number.
Nozzle shape and size	Can use straight cone, bell‐shaped, or other nozzle contour; bell can give slightly lower losses. Make correction for divergence losses and nonuniformity of velocity profile.	Must know or assume a particular nozzle configuration. Calculate bell contour by method of characteristics. Use Eq. 3–34 for divergence losses in conical nozzle. Most analysis programs are one‐ or two‐dimensional. Unsymmetrical nonround nozzles may need three‐dimensional analysis.
Gas properties	The relationships governing the behavior of the gases apply to both nozzle and chamber conditions. As gases cool in expansion, some species may condense.	Either use perfect gas laws or, if some of the gas species come close to being condensed, use real gas properties.
Nozzle exit conditions	Will depend on the assumptions made above for chemical equilibrium, nozzle expansion, and nozzle shape/contour. Assume no jet separation. Determine velocity profile and the pressure profile at the nozzle exit plane. If pressure is not uniform across a section it will have some cross flow.	Need to know the nozzle area ratio or nozzle pressure ratio. For quasi‐one‐dimensional and uniform nozzle flow, see Eqs. 3–25 and 3–26. If is not constant over the exit area, determine effective average values of and p2. Then calculate profiles of T, ρ, etc. For nonuniform velocity profile, the solution requires an iterative approach. Can calculate the gas conditions (T, p, etc.) at any point in the nozzle.
Calculate specific impulse	Can be determined for different altitudes, pressure ratios, mixture ratios, nozzle area ratios, etc.	Can be determined for average values of , p2, and p3 based on Eq. 2–6 or 2–13.

Once the gases reach a supersonic nozzle, they experience an adiabatic, reversible expansion process which is accompanied by substantial drops in temperature and pressure, reflecting the conversion of thermal energy into kinetic energy. Several increasingly more complicated methods have been used for analyzing nozzle processes. For the simplest case, frozen (composition) equilibrium and one‐dimensional flow, the state of the gas throughout expansion in the nozzle is fixed by the entropy of the system, which is considered to be invariant as the pressure is reduced. All assumptions listed in Chapter 3 for ideal rockets would be valid here. Again, effects of friction, divergence angle, heat losses, shock waves, and nonequilibrium are neglected in the simplest cases but are considered for the more sophisticated solutions. Any condensed (liquid or solid) phases present are similarly assumed to have zero volume and to be in kinetic as well as thermal equilibrium with the gas flow. This implies that particles and/or droplets are very small in size, move at the same velocity as the gas stream, and have the same temperature as the gas everywhere in the nozzle.

Chemical composition during nozzle expansion may be treated analytically in the following ways:

1. When the expansion is sufficiently rapid, composition may be assumed as invariant throughout the nozzle, that is, there are no chemical reactions or phase changes and the reaction products composition at the nozzle exit are identical to those of the chamber. Such composition results are known as frozen equilibrium rocket performance. This approach is the simplest, but tends to underestimate the system's performance typically by 1 to 4%.
2. Instantaneous chemical equilibrium among all molecular species may be significant in some cases under the continuously variable pressure and temperature conditions of the nozzle expansion process. Here, product compositions do shift because the chemical reactions and phase change equilibria taking place between gaseous and condensed phases in all exhaust gas species are fast compared to their nozzle transit time. The composition results so calculated are called shifting equilibrium performance. Here, gas composition mass fractions are different at the chamber and nozzle exits. This method usually overstates real performance values, such as or I_s, typically by 1 to 4%. Such analysis is more complex and many more equations are needed.
3. Even though the chemical reactions may occur rapidly, they do require some finite time. Reaction rates for specific reactions are often estimates; these rates are a function of temperature, the magnitude of deviation from the equilibrium molar composition, and the nature of the chemicals or reactions involved. Values of T, , or I_s in most types of actual flow analyses usually fall between those of frozen and instantaneously shifting equilibria. This approach is seldom used because of the lack of good data on reaction rates with multiple simultaneous chemical reactions.

The simplest nozzle flow analysis is also one dimensional, which means that all velocities and temperatures or pressures are equal at any normal cross section of an axisymmetric nozzle. This is often satisfactory for preliminary estimates. In two‐dimensional analyses, the resulting velocity, temperature, density, and/or Mach number do not have a flat profile varying somewhat over the cross sections. For nozzle shapes that are not bodies of revolution (e.g., rectangular, scarfed, or elliptic), three‐dimensional analyses need to be performed.

When solid particles or liquid droplets are present in the nozzle flow and when the particles are larger than about 0.1 µm average diameter, there will be both a thermal lag and a velocity lag. Solid particles or liquid droplets cannot expand as a gas; their temperature decrease depends on how they lose energy by convection and/or radiation, and their velocity depends on the drag forces exerted on the particle. Larger‐diameter droplets or particles are not accelerated as rapidly as smaller ones and their flow velocities are lower than that of those of the accelerating gases. Also, these particulates remain hotter than the gas and provide heat to it. While particles contribute to the momentum of the exhaust mass, they are not as efficient as an all‐gaseous flow. For composite solid propellants with aluminum oxide particles in the exhaust gas, losses due to particles could typically amount to 1 to 3%. Analyses of two‐ or three‐phase flows require assumptions about the nongaseous amounts from knowledge of the sizes (diameters), size distributions, shapes (usually assumed as spherical), optical surface properties (for determining the emission/absorption or scattering of radiant energy), and their condensation or freezing temperatures. Some of these parameters are seldom well known. Performance estimates of flows with particles are treated in Section 3.5.

The viscous boundary layers adjacent to nozzle walls have velocities substantially lower than those of the inviscid free stream. This viscous drag near the walls actually causes a conversion of kinetic energy into thermal energy, and thus some parts of the boundary layer can be hotter than the local free‐stream static temperature. A diagram of a two‐dimensional boundary layer is shown in Figure 3–15. With turbulent flows, this boundary layer can be relatively thick in small nozzles. Boundary layers also depend on the axial pressure gradient in the nozzle, nozzle geometry (particularly at the throat region), surface roughness, and/or the heat losses to the nozzle walls. The layers immediately adjacent to the nozzle walls always remain laminar and subsonic. Presently, boundary layer analyses with unsteady flow are only approximations, but are expected to improve as our understanding of relevant phenomena grows and as computational fluid dynamics (CFD) techniques improve. The net effect of such viscous layers appears as nonuniform velocity and temperature profiles, irreversible heating (and therefore increases in entropy), and minor reductions (usually less than 5%) of the kinetic exhaust energy for well‐designed systems.

At high combustion temperatures some portion of the gaseous molecules dissociate (splitting into simpler species); in this dissociation process, some energy is absorbed by the flow; this reduces the stagnation temperature of the flow within the nozzle even if some of this energy may be released back during reassociation (at the lower pressures and temperatures in the nozzle).

For propellants that yield only gaseous products, extra energy is released in the nozzle, primarily from the recombination of free‐radical and atomic species, which become unstable as the temperature decreases in the nozzle expansion process. Some propellant products include species that may actually condense as the temperature drops in the nozzle. If the heat release upon condensation is large, the difference between frozen and shifting equilibrium calculations can be substantial.

In the simplest approach, the exit temperature T₂ is calculated for an isentropic process (with frozen equilibrium). This determines the temperature at the exit and thus the gas conditions at the exit. From the corresponding change in enthalpy, it is then possible to obtain the exhaust velocity and the specific impulse. When nozzle flow is not really isentropic because the expansion process is only partly reversible, it is necessary to include losses due to friction, shock waves, turbulence, and so on. This results in a somewhat higher average nozzle exit temperature and a slight decrease in I_s. A possible set of steps used in such nozzle analysis is given in Table 5–3.

When the contraction between the combustion chamber (or port area) and the throat area is small (), acceleration of the gases in the chamber causes an appreciable drop in the effective chamber pressure at the nozzle entrance. This pressure loss in the chamber causes a slight reduction of the values of c and I_s. The analysis of this chamber configuration is treated in Ref. 5–14 and some data are shown in Table 3–2.

5.4 COMPUTER‐ASSISTED ANALYSIS

At the present time, all analyses discussed in this chapter are carried out with computer software. Most are based on minimizing the free energy. This is a simpler approach than relying on equilibrium constants, which was common some years ago. Once the values of n_j and T₁ are determined, it is possible to calculate the molecular mass of the gas mixture (Eq. 5–5), the average molar specific heats C_p by Eq. 5–6, and the specific heat ratio k from Eq. 5–7. This then characterizes the thermodynamic state conditions leaving the combustion chamber. With these data we may calculate , R, and other gas‐mixture parameters at the combustion chamber exit. From the process of nozzle expansion, as formulated in computer codes, we can then calculate performance (such as I_s, c, or A₂/A_t) and gas conditions in the nozzle; these calculations may include several of the correction factors mentioned in Chapter 3 for more realistic results. Programs exist for one‐, two‐, and three‐dimensional flow patterns.

More sophisticated solutions may include a supplementary analysis of combustion chamber conditions when the chamber velocities are high (see Ref. 5–14), boundary layer analyses, heat transfer analyses, and/or two‐dimensional axisymmetric flow models with nonuniform flow properties across nozzle cross sections. Time‐dependent chemical reactions in the chamber, which are usually neglected, may be analyzed by estimating the time rates at which the reactions occur. This is described in Ref. 5–3.

A commonly used computer program, based on equilibrium compositions, has been developed at the NASA Glenn Laboratories and is known as the NASA CEA code (Chemical Equilibrium with Applications). It is described in Ref. 5–13, Vols. 1 and 2, and is available for download (http://www.grc.nasa.gov/WWW/CEAWeb/ceaguiDownload‐win.htm). Key assumptions in this program are one‐dimensional forms of the continuity, energy and momentum equations, negligible velocity at the forward end of the combustion chamber, isentropic expansion in the nozzle, ideal gas behavior, and chemical equilibrium in the combustion chamber. It includes options for frozen flow and for narrow chambers (for liquid propellant combustion) or port areas with small cross sections (for solid propellant grains), where the chamber flow velocities are relatively high, resulting in noticeable pressure losses and slight losses in performance. NASA's CEA code has become part of a commercially available code named Cequel^TM, which also extends the code's original capabilities.

Other relatively common computer codes used in the United States for analyzing converging–diverging nozzle flows include:

• ODE (one‐dimensional equilibrium code), which features instantaneous chemical reactions (shifting equilibrium) and includes all gaseous constituents.
• ODK (one‐dimensional kinetics), which incorporates finite chemical reaction rates for temperature‐dependent composition changes in the flow direction with uniform flow properties across any nozzle section. It is used as a module in more complex codes but has no provision for embedded particles.
• TDK (two‐dimensional kinetic code), which incorporates finite kinetic chemical reaction rates and radial variation in flow properties. It has no provision for embedded particles.
• VIPERP (viscous interaction performance evaluation routine for two‐phased flows), a parabolized Navier–Stokes code for internal two‐phase nozzle flows with turbulent and nonequilibrium reacting gases. It can be used with embedded solid particles but requires data (or assumptions) on the amount of solids, particle size distribution, or their shape (see, e.g., pp. 503 to 505 in the Seventh Edition of this book).

More information on these computer codes may be obtained from the appropriate government offices and/or from private companies (who actually run the necessary codes for their customers). Many of the more sophisticated codes are proprietary to propulsion organizations or otherwise restricted and not publicly available.

5.5 RESULTS OF THERMOCHEMICAL CALCULATIONS

Extensive computer generated results are available in the literature and only a few samples are indicated here to illustrate effects typical to the variations of key parameters. In general, high specific impulse or high values of can be obtained when the average molecular mass of the reaction products is low (usually this implies formulations rich in hydrogen) and/or when the available chemical energy (heat of reaction) is large, which means high combustion temperatures (see Eqs. 3–16 and 3–32).

Table 5–4 shows computed results for a liquid oxygen, liquid hydrogen thrust chamber taken from Ref. 5–13. It shows shifting equilibrium results 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.

Chamber pressure at injector 773.3 psia or 53.317 bar; ; shifting equilibrium nozzle flow mixture ratio ; chamber to throat area ratio .
Parameters
Location	Injector face	Comb. end	Throat	Exit I	Exit II	Exit III	Exit IV
pinj/p	1.00	1.195	1.886	10.000	100.000	282.15	709.71
T (K)	3389	3346	3184	2569	1786	1468	1219
(molec. mass)	12.7	12.7	12.8	13.1	13.2	13.2	13.2
k (spec. heat ratio)	1.14	1.14	1.15	1.17	1.22	1.24	1.26
Cp (spec. heat, kJ/kg‐K)	8.284	8.250	7.530	4.986	3.457	3.224	3.042
M (Mach number)	0.00	0.413	1.000	2.105	3.289	3.848	4.379
A2/At	1.580a	1.580a	1.000	2.227	11.52	25.00	50.00
c (m/sec)	NA	NA	2879b	3485	4150	4348	4487
(m/sec)	NA	NA	1537b	2922	3859	4124	4309
Mol fractions of gas mixture
H	0.03390	0.03336	0.02747	0.00893	0.00024	0.00002	0.00000
HO2	0.00002	0.00001	0.00001	0.00000	0.00000	0.00000	0.00000
H2	0.29410	0.29384	0.29358	0.29659	0.30037	0.30050	0.30052
H2O	0.63643	0.63858	0.65337	0.68952	0.69935	0.69948	0.69948
H2O2	0.00001	0.00001	0.00000	0.00000	0.00000	0.00000	0.00000
O	0.00214	0.00204	0.00130	0.00009	0.00000	0.00000	0.00000
OH	0.03162	0.03045	0.02314	0.00477	0.00004	0.00000	0.00000
O2	0.00179	0.00172	0.00113	0.00009	0.00000	0.00000	0.00000

^a Chamber contraction ratio A₁/A_t.

^b If cut off at throat.

^c is the effective exhaust velocity in a vacuum.

is the nozzle exit velocity at optimum nozzle expansion.

NA means not applicable.

The above calculated values of specific impulse will be higher than those obtained from firing actual propellants in rocket units. In practice, it has been found that the experimental values can be lower than those calculated for shifting equilibrium by up to 12%. Because nozzle inefficiencies as explained in Chapter 3 must be considered, only a portion of this correction (perhaps 1 to 4%) is due to combustion inefficiencies.

Much input data for rocket‐propulsion‐system computer programs (such as the physical and chemical properties of various propellant species used in this chapter) are based on experiments that are more than 25 years old. A few of those have newly revised values, but the differences are believed to be relatively small.

Figures 5–1 through 5–6 indicate calculated results for the liquid propellant combination, liquid oxygen‐RP‐1 (Rocket Propellant #1). These data are taken from Refs. 5–7 and 5–8. RP‐1 is a narrow‐cut hydrocarbon similar to kerosene with an average of 1.953 g‐atoms of hydrogen for each g‐atom of carbon; thus, it has a nominal formula of CH_1.953. These calculations are for a chamber pressure of 1000 psia. Most of the curves are for optimum area ratio expansion to atmospheric pressure, namely 1 atm or 14.696 psia, and for a limited range of oxidizer‐to‐fuel mixture mass (not mol) ratios.

For maximum specific impulse, Figs. 5–1 and 5–4 show an optimum mixture ratio of approximately 2.3 (kg/sec of oxidizer flow divided by kg/sec of fuel flow) for frozen equilibrium expansion and 2.5 for shifting equilibrium, with the gases expanding to sea‐level pressure. The maximum values of occur at slightly different mixture ratios. These optimum mixture ratios are not at the value for highest temperature, which is usually fairly close to stoichiometric. The stoichiometric mixture ratio is more than 3.0 where much of the carbon is burned to CO₂ and almost all of the hydrogen to H₂O.

Because shifting equilibrium makes more enthalpy available for conversion to kinetic energy, it gives higher values of performance (higher I_s or ) and higher values of nozzle exit temperature for the same exit pressure (see Fig. 5–1). The influence of mixture ratio on chamber gas composition is evident from Fig. 5–2. A comparison with Fig. 5–3 indicates marked changes in the gas composition as the gases are expanded under shifting equilibrium conditions. The influence of the degree of expansion, or of nozzle exit pressure on gas composition is shown in Fig. 5–6 as well as in Table 5–4. As gases expand to higher area ratios and lower exit pressures (or higher pressure ratios) system performance increases; however, the relative increase diminishes as the pressure ratio is further increased (see Figs. 5–5 and 5–6).

The dissociation of gas molecules absorbs considerable energy and decreases the combustion temperature, which in turn reduces the specific impulse. Dissociation of reaction products increases as chamber temperature rises, and decreases with increasing chamber pressure. Atoms or radicals such as monatomic O or H and OH are formed, as can be seen from Fig. 5–2; some unreacted O₂ also remains at the higher mixture ratios and very high combustion temperatures. As gases cool in the nozzle expansion, the dissociated species tend to recombine and release heat into the flowing gases. As can be seen from Fig. 5–3, only a small percentage of dissociated species persists at the nozzle exit and only at the high mixture ratios, where the exit temperature is relatively high. (See Fig. 5–1 for exit temperatures with shifting equilibria). Heat release in supersonic flows actually reduces the Mach number.

Results of thermochemical calculations for several different liquid and solid propellant combinations are given in Tables 5–5 and 5–6. For the liquid propellant combinations, the listed mixture ratios are optimum and their performance is a maximum. For solid propellants, practical considerations (such as propellant physical properties, e.g., insufficient binder) do not always permit the development of satisfactory propellant grains where ingredients are mixed to optimum performance proportions; therefore values listed for solid propellants in Table 5–6 correspond in part to practical formulations with reasonable physical and ballistic properties.

		Mixture Ratio						Is (sec)
Oxidizer	Fuel	By Mass	By Volume	Average Specific Gravity	Chamber Temp.(K)	Chamber (m/sec)	, (kg/mol)	Shifting	Frozen	k
Oxygen	Methane	3.20	1.19	0.81	3526	1835	20.3		296	1.20
		3.00	1.11	0.80	3526	1853		311
	Hydrazine	0.74	0.66	1.06	3285	1871	18.3		301	1.25
		0.90	0.80	1.07	3404	1892	19.3	313
	Hydrogen	3.40	0.21	0.26	2959	2428	8.9		386	1.26
		4.02	0.25	0.28	2999	2432	10.0	389.5
	RP‐1	2.24	1.59	1.01	3571	1774	21.9	300	285.4	1.24
		2.56	1.82	1.02	3677	1800	23.3
	UDMH	1.39	0.96	0.96	3542	1835	19.8		295	1.25
		1.65	1.14	0.98	3594	1864	21.3	310
Fluorine	Hydrazine	1.83	1.22	1.29	4553	2128	18.5	334		1.33
		2.30	1.54	1.31	4713	2208	19.4		365
	Hydrogen	4.54	0.21	0.33	3080	2534	8.9		389	1.33
		7.60	0.35	0.45	3900	2549	11.8	410
Nitrogen tetroxide	Hydrazine	1.08	0.75	1.20	3258	1765	19.5		283	1.26
		1.34	0.93	1.22	3152	1782	20.9	292
	50% UDMH	1.62	1.01	1.18	3242	1652	21.0		278	1.24
	50% hydrazine	2.00	1.24	1.21	3372	1711	22.6	289
	RP‐1	3.4	1.05	1.23	3290		24.1		297	1.23
	MMH	2.15	1.30	1.20	3396	1747	22.3	289
		1.65	1.00	1.16	3200	1591	21.7		278	1.23
Red fuming nitric acid	RP‐1	4.1	2.12	1.35	3175	1594	24.6		258	1.22
		4.8	2.48	1.33	3230	1609	25.8	269
	50% UDMH	1.73	1.00	1.23	2997	1682	20.6		272	1.22
	50% hydrazine	2.20	1.26	1.27	3172	1701	22.4	279
Hydrogen peroxide (90%)	RP‐1	7.0	4.01	1.29	2760		21.7		297	1.19

Notes:

Combustion chamber pressure—1000 psia (6895 kN/m²); nozzle exit pressure—14.7 psia (1 atm); optimum expansion.

Adiabatic combustion and isentropic expansion of ideal gases.

The specific gravity at the boiling point has been used for those oxidizers or fuels that boil below 20 °C at 1 atm pressure, see Eq. 7–1.

Mixture ratios are for approximate maximum values of I_s.

Oxidizer	Fuel	ρb (g/cm3)a	T1 (K)	(m/sec)b	, (kg/mol)	Is (sec)b	k
Ammonium nitrate	11% binder and 7% additives	1.51	1282	1209	20.1	192	1.26
Ammonium perchlorate 78–66%	18% organic polymer binder and 4–20% aluminum	1.69	2816	1590	25.0	262	1.21
Ammonium perchlorate 84–68%	12% polymer binder and 4–20% aluminum	1.74	3371	1577	29.3	266	1.17

^a Density of solid propellant, see Eq. 12–1.

^b Conditions for I_s and : Combustion chamber pressure: 1000 psia; nozzle exit pressure: 14.7 psia; optimum nozzle expansion ratio; frozen equilibrium.

Calculated results obtained from Ref. 5–13 are presented in Tables 5–7 through 5–9 for a solid propellant to indicate typical variations in performance or gas composition. This particular propellant consists of 60% ammonium perchlorate (NH₄ClO₄), 20% pure aluminum powder, and 20% of an organic polymer of a given chemical composition, namely, C_3.1ON_0.84H_5.8. Table 5–7 shows the variation of several performance parameters with different chamber pressures expanding to atmospheric pressure. The area ratios listed are optimum for this expansion with shifting equilibrium. The exit enthalpy, exit entropy, thrust coefficient, and the specific impulse also reflect shifting equilibrium conditions. The characteristic velocity and the chamber molecular mass are functions of chamber conditions only. Table 5–8 shows the variation of gas composition with chamber pressure; here, some reaction products are in the liquid phase, such as Al₂O₃. Table 5–9 shows the variation of nozzle exit characteristics and composition for shifting equilibria as a function of exit pressure or pressure ratio for a fixed value of chamber pressure. Table 5–9 shows how composition shifts during expansion in the nozzle and indicates several species present in the chamber that do not appear at the nozzle exit. These three tables show computer results—some thermodynamic properties of the reactants and reaction products probably do not warrant the indicated high accuracy of five significant figures. In the analysis for chemical ingredients of this solid propellant, approximately 76 additional reaction products have been considered in addition to the major product species. These include, for example, CN, CH, CCl, Cl, NO, and so on. Their calculated mol fractions are very small, and therefore they may be neglected and are not included in Table 5–8 or 5–9.

Chamber pressure (psia)	1500	1000	750	500	200
Chamber pressure (atm) to sea‐level pressure ratio p1/p2	102.07	68.046	51.034	34.023	13.609
Chamber temperature (K)	3346.9	3322.7	3304.2	3276.6	3207.7
Nozzle exit temperature (K)	2007.7	2135.6	2226.8	2327.0	2433.6
Chamber enthalpy (cal/g)	–572.17	–572.17	–572.17	–572.17	–572.17
Exit enthalpy (cal/g)	–1382.19	–1325.15	–1282.42	–1219.8	–1071.2
Entropy (cal/g‐K)	2.1826	2.2101	2.2297	2.2574	2.320
Chamber molecular mass (kg/mol)	29.303	29.215	29.149	29.050	28.908
Exit molecular mass (kg/mol)	29.879	29.853	29.820	29.763	29.668
Exit Mach number	3.20	3.00	2.86	2.89	2.32
Specific heat ratio – chamber, k	1.1369	1.1351	1.1337	1.1318	1.1272
Specific impulse, vacuum (sec)	287.4	280.1	274.6	265.7	242.4
Specific impulse, sea‐level expansion (sec)	265.5	256.0	248.6	237.3	208.4
Characteristic velocity, (m/sec)	1532	1529	1527	1525	1517
Nozzle area ratioa, A2/At	14.297	10.541	8.507	8.531	6.300
Thrust coefficienta, CF	1.700	1.641	1.596	1.597	1.529

At optimum expansion.

Such calculated results are useful in estimating performance (I_s, , C_F, ε, etc.) for particular chamber and nozzle exit pressures, and knowledge of gas composition, as indicated in the previous figures and tables, permits more detailed estimates of other design parameters, such as convective properties for heat transfer determination, radiation characteristics of the flame inside and outside the thrust chambers, and acoustic characteristics of the gases. Some performance data relevant to hybrid propellants are presented in Chapter 16.

The thermochemical analyses found in this chapter can also be applied to gas generators; results (such as the gas temperature T₁, specific heat c_p, specific heat ratio k, or composition) are used for estimating turbine inlet conditions or turbine power. In gas generators and preburners for staged combustion cycle rocket engines (explained in Section 6.6) gas temperatures are much lower, to avoid damage to the turbine blades. Typically, combustion reaction gases are at 800 to 1200 K, which is lower than the gas temperature in the thrust chamber (2900 to 3600 K). Examples are listed in Table 5–10 for a chamber pressure of 1000 psia. Some gaseous species will not be present (such as atomic oxygen or hydroxyl), and often real gas properties will need to be used because some of these gases do not behave as a perfect gas at these lower temperatures.

SYMBOLS

Symbols referring to chemical elements, compounds, or mathematical operators are not included in this list.

a or b	number of kilogram atoms
At	throat area, m2
Ap	port area, m2
	characteristic velocity, m/sec
cp	specific heat per unit mass at constant pressure, J/kg‐K
Cp	molar specific heat at constant pressure of gas mixture, J/kg‐mol‐K
g0	acceleration of gravity at sea level, 9.8066 m/sec2
G	Gibbs free energy for a propellant combustion gas mixture, J/kg
ΔfG0	change in free energy of formation at 298.15 K and 1 bar
Gj	free energy for a particular species j, J/kg
ΔH	overall enthalpy change, J/kg or J/kg‐mol
ΔHj	enthalpy change for a particular species j, J/kg
ΔrH0	heat of reaction at reference 298.15 K and 1 bar, J
ΔfH0	heat of formation at reference 298.15 K and 1 bar, J/kg
hj	enthalpy for a particular species, J/kg or J/kg‐mol
Is	specific impulse, sec
k	specific heat ratio
ℓ	total number of given chemical species in a mixture
m	number of gaseous species, also total number of products
	mass flow rate, kg/sec
	molecular mass of gas mixture, kg/kg‐mol (lbm/lb‐mol)
n	total number of mols per unit mass (kg‐mol/kg or mol) of mixture
nj	mols of species j, kg‐mol/kg or mol
p	pressure of gas mixture, N/m2
r	total number of reactants
R	gas constant, J/kg‐K
R′	universal gas constant, 8314.3 J/kg mol‐K
S	entropy, J/kg mol‐K
T	absolute temperature, K
Tad	adiabatic temperature, K
U	internal energy, J/kg‐mol
	gas velocity, m/sec
V	specific volume, m3/kg
Xj	mol fraction of species j

PROBLEMS

1. 1. Explain the physical and/or chemical reasons for the maximum value of specific impulse at a particular flow mixture ratio of oxidizer to fuel.
2. 2. Explain why, in Table 5–8, the relative proportion of monatomic hydrogen and monatomic oxygen changes markedly with different chamber pressures and exit pressures.
3. 3. This chapter contains several charts for the performance of liquid oxygen and RP‐1 hydrocarbon fuel. If by mistake a new shipment of cryogenic oxidizer contains at least 15% liquid nitrogen discuss what general trends should be expected in results from its testing in performance values, likely composition of the exhaust gases under chamber and nozzle conditions, and find the new optimum mixture ratio.
4. 4. A mixture of perfect gases consists of 3 kg of carbon monoxide and 1.5 kg of nitrogen at a pressure of 0.1 MPa and a temperature of 298.15 K. Using Table 5–1, find (a) the effective molecular mass of the mixture, (b) its gas constant, (c) specific heat ratio, (d) partial pressures, and (e) density.Answer: (a) 28 kg/kg‐mol, (b) 297 J/kg‐K, (c) 1.40, (d) 0.0666 and 0.0333 MPa, (e) 1.13 kg/m³.
5. 5. Using information from Table 5–2, plot the value of the specific heat ratio for carbon monoxide (CO) as a function of temperature. Notice the trend of this curve, which is typical of the temperature behavior of other diatomic gases.Answer: at 3500 K, 1.30 at 2000 K, 1.39 at 500 K.
6. 6. Modify and tabulate two entries in Table 5–5 for operation in the vacuum of space, namely oxygen/hydrogen and nitrogen tetroxide/hydrazine. Assume the data in the table represent the design condition.
7. 7. Various experiments have been conducted with a liquid monopropellant called nitromethane (CH₃NO₂), which can be decomposed into gaseous reaction products. Determine the values of T, , k, , C_F, and I_s using the water–gas equilibrium conditions. Assume no dissociations and no O₂.Answer: 2470 K, 20.3 kg/kg‐mol, 1.25, 1527 m/sec, 1.57, 244 sec.
8. 8. The figures in this chapter show several parameters and gas compositions of liquid oxygen burning with RP‐1, which is a kerosene‐type material. For a mixture ratio of 2.0, use the given compositions to verify the molecular mass in the chamber and the specific impulse (frozen equilibrium flow in nozzle) in Fig. 5–1.

REFERENCES

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2. 5–2. S. S. Penner, Thermodynamics for Scientists and Engineers, Addison‐Wesley, Reading, MA, 1968.
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