There are several methods for analyzing the nozzle flow, depending on the assumptions made for chemical equilibrium, nozzle expansion, particulates, or energy losses. Several are outlined in Table 5-3.
Once the gases reach the nozzle, they experience an adiabatic, reversible expansion process which is accompanied by a drop in temperature and pressure and a conversion of thermal energy into kinetic energy. Several increasingly more complicated methods have been used for the analysis of the process. For the simple case of frozen 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 presumed to be invariant as the pressure is reduced to the value assigned to the nozzle exit plane. All the assumptions listed in Chapter 3 for an ideal rocket are also valid here. Again, the effects of friction, divergence angle, heat exchange, shock waves, or nonequilibrium are neglected in the simple cases, but are considered in the more sophisticated solutions. The condensed (liquid or solid) phases are again assumed to have zero volume and to be in kinetic as well as thermal equilibrium with the gas flow. This implies that particles or droplets are very small in size, move at the same velocity as the gas stream, and have the same temperature as the gas at all places in the nozzle.
The chemical equilibrium during expansion in the nozzle can be analytically regarded in the following ways:
1. When the composition is invariant throughout the nozzle, there are no chemical reactions or phase changes and the product composition at the nozzle exit is identical to that of its chamber condition. The results are known as frozen equilibrium rocket performance. This method usually is simple, but underestimates the performance, typically by 1 to 4%.
2. Instantaneous chemical equilibrium among all molecular species is maintained under the continuously variable pressure and temperature conditions of the nozzle expansion process. Thus the product composition shifts; similarly, instantaneous chemical reactions, phase changes or equilibria occur between gaseous and condensed phases of all species in the exhaust gas. The results so calculated are called shifting equilibrium performance. The gas composition mass percentages are different in the chamber and the nozzle exit. This method usually overstates the performance values, such as c* or Is, typically by 1 to 4%. Here the analysis is more complex.
3. The chemical reactions do not occur instantaneously, but even though the reactions occur rapidly they require a finite time. The reaction rates of specific reactions can be estimated; the rates are usually a function of temperature, the magnitude of deviation from the equilibrium molar composition, and the nature of the chemicals or reactions involved. The values of T, c*, or Is for these types of equilibrium analysis usually are between those of frozen and instantaneously shifting equilibria. This approach is almost never used, because of the lack of good data on reaction rates with multiple simultaneous chemical reactions.
For an axisymmetric nozzle, both one- and two-dimensional analyses can be used. The simplest nozzle flow analysis is one-dimensional, which means that all velocities and temperatures or pressures are equal at any normal cross section of an axisymmetric nozzle. It is often satisfactory for preliminary estimates. In a two-dimensional analysis the velocity, temperature, density, and/or Mach number do not have a flat profile and vary somewhat over the cross sections. For nozzle shapes that are not bodies of revolution (e.g., rectangular, scarfed, or elliptic) a three-dimensional analysis can be performed.
If solid particles or liquid droplets are present in the nozzle flow and if the particles are larger than about 0.1 urn average diameter, there will be a thermal lag and velocity lag. The solid particles or liquid droplets do not expand like a
TABLE 5-3. Typical Steps and Alternatives in the Analysis of Rocket Thermochemical Processes in Nozzles
Nozzle inlet condition
For simpler analyses assume the flow to be uniformly mixed and steady.
Simplest method is inviscid isentropic expansion flow with constant entropy.
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.
4. Include viscous boundary layer effects and/or non-uniform velocity profile.
Often a simple single correction factor is used with one-dimensional analyses to correct the 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.
Same as chamber exit; need to know Tupu Vi, H, c*, p\, etc.
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.
Chemical Due to rapid decrease in T and p, equilibrium the equilibrium composition can during nozzle change from that in the chamber, expansion The four processes listed in the next column allow progressively more realistic simulation and require more sophisticated techniques.
1. Frozen equilibrium; no change in gas composition; usually gives low performance.
2. Shifting equilibrium or instantaneous change in composition; usually overstates the performance slightly.
3. Use reaction time rate analysis to estimate the time to reach equilibrium for each of the several chemical reactions; some rate constants are not well known; analysis is more complex.
4. Use different equilibrium analysis for boundary layer and main inviscid flow; will have nonuniform gas temperature, composition, and velocity profiles.
TABLE 5-3. (Continued)
Heat release in nozzle
Nozzle shape and size
Nozzle exit conditions
Recombination of dissociated molecules (e.g., H + H = H2) and exothermic reactions due to changes in equilibrium composition cause an internal heating of the expanding gases. Particulates release heat to the gas.
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.
The relationships governing the behavior of the gases apply to both nozzle and chamber conditions. As gases cool in expansion, some species may condense.
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.
Can be determined for different altitudes, pressure ratios, mixture ratios, nozzle area ratios, etc.
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.
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 non-round nozzles may need three-dimensional analysis.
Either use perfect gas laws or, if some of the gas species come close to being condensed, use real gas properties.
Need to know the nozzle area ratio or nozzle pressure ratio. For quasi-one-dimensional and uniform nozzle flow, see Eqs. 3-25 and 326. If v2 is not constant over the exit area, determine effective average values of v2 and p2. Then calculate profiles of T, p, 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.
Can be determined for average values of v2, P2, and p} based on Eqs. 2-6, 3-35, and/or 2-14.
gas; their temperature decrease depends on losing energy by convection 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 the smaller ones and flow at a velocity lower than that of the adjacent accelerating gas. Also, the particulates are hotter than the gas and provide heat to the gas. While these particles contribute to the momentum of the exhaust mass, they are not as efficient as an all-gaseous exhaust flow. For composite solid propellants with aluminum oxide particles in the exhaust gas, the loss due to particles could typically be 1 to 3%. The analysis of a two- or three-phase flow requires knowledge of or an assumption about the nongaseous matter, the sizes (diameters), size distribution, shape (usually assumed to be 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 not well known. Performance estimates of flows with particles are explained in Section 3-5.
The viscous boundary layer next to the nozzle wall has velocities substantially lower than that of the inviscid free stream. The slowing down of the gas flow near the wall due to the viscous drag actually causes the 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 3-16. With turbulence this boundary layer can be relatively thick in large-diameter nozzles. The boundary layer is also dependent on the axial pressure gradient in the nozzle, the nozzle geometry, particularly in the throat region, the surface roughness, or the heat losses to the nozzle walls. Today, theoretical boundary layer analyses with unsteady flow are only approximations, but are expected to improve in the future as our understanding of the phenomena and computational fluid dynamics (CFD) techniques are validated. The net effect is a nonuniform velocity and temperature profile, an irreversible friction process in the viscous layers, and therefore an increase in entropy and a slight reduction (usually less than 5%) of the kinetic exhaust energy. The slower moving layers adjacent to the nozzle walls have laminar and subsonic flow.
At the high combustion temperatures a small portion of the combustion gas molecules dissociate (split into simpler species); in this dissociation process some energy is absorbed. When energy is released during reassociation (at lower pressures and temperatures in the nozzle), this reduces the kinetic energy of the exhaust gas at the nozzle exit. This is discussed further in the next section.
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 is decreased in the nozzle expansion process. Some propellant products include species that condense as the temperature drops in the nozzle expansion. If the heat release on condensation is large, the difference between frozen and shifting equilibrium performance can be substantial.
In the simplest method the exit temperature T2 is determined for an isen-tropic process (frozen equilibrium) by considering the entropy to be constant. The entropy at the exit is the same as the entropy in the chamber. This determines the temperature at the exit and thus the gas condition at the exit. From the corresponding change in enthalpy it is then possible to obtain the exhaust velocity and the specific impulse. For those analysis methods where the nozzle flow is not really isentropic and the expansion process is only partly reversible, it is necessary to include the losses due to friction, shock waves, turbulence, and so on. The result is a somewhat higher average nozzle exit temperature and a slight loss in Is. A possible set of steps used for the analysis of nozzle processes is given in Table 5-3.
When the contraction between the combustion chamber (or the port area) and the throat area is small (Ap/A, < 3), the acceleration of the gases in the chamber causes a 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 Is. The analysis of this chamber configuration is treated in Ref. 5-14 and some data are briefly shown in Tables 3-2 and 6-A.
Example 5-1. Various experiments have been conducted with a liquid monopropellant called nitromethane (CH3N02), which can be decomposed into gaseous reaction products. Determine the values of T, 9JÎ, k, c*, CF, and /, using the water-gas equilibrium conditions. Assume no dissociations and no 02.
SOLUTION. The chemical reaction for 1 mol of reactant can be described as
1.0 CH3N02 ncoCO + nCo2C02 + nH,H2 + "h,0H20 + «NîN2
Neglect other minor products. The mass balances are obtained for each atomic element.
The reaction commonly known as the water-gas reaction is
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