Multiphase Flow

In some rockets the gaseous working fluid contains many small liquid droplets and/or solid particles that must be accelerated by the gas. They give up heat to the gas during the expansion in a nozzle. This, for example, occurs with solid propellants (see Chapter 12) or some gelled liquid propellants (Chapter 7), which contain aluminum powder that forms small oxide particles in the exhaust. It can also occur with ion oxide catalysts, or propellants containing beryllium, boron, or zirconium.

In general, if the particles are very small (typically with diameters of 0.005 mm or less), they will have almost the same velocity as the gas and will be in thermal equilibrium with the nozzle gas flow. Thus, as the gases give up kinetic energy to accelerate the particles, they gain thermal energy from the particles. As the particle diameters become larger, the mass (and thus the inertia) of the particle increases as the cube of its diameter; however, the drag force increases only as the square of the diameter. Larger particles therefore do not move as fast as the gas and do not give heat to the gas as readily as do smaller particles. The larger particles have a lower momentum than an equivalent mass of smaller particles and they reach the nozzle exit at a higher temperature than the smaller particles, thus giving up less thermal energy.

It is possible to derive a simple theoretical approach for correcting the performance (Is,c, or c*) as shown below and as given in Refs. 3-13 and 314. It is based on the assumption that specific heats of the gases and the particles are constant throughout the nozzle flow, that the particles are small enough to move at the same velocity as the gas and are in thermal equilibrium with the gas, and that particles do not exchange mass with the gas (no vaporization or condensation). Expansion and acceleration occur only in the gas and the volume occupied by the particles is negligibly small compared to the gas volume. If the amount of particles is small, the energy needed to accelerate the particles can be neglected. There are no chemical reactions.

The enthalpy h, the specific volume V, and the gas constant R can be expressed as functions of the particle fraction f3, which is the mass of particles (liquid and/or solid) divided by the total mass. Using the subscripts g and s to refer to the gas or solid state, the following relationships then apply:

These relations are then used in the formulas for simple one-dimensional nozzle flow, such as Eq. 2-16, 3-15, or 3-32. The values of specific impulse or characteristic velocity will decrease as p, the percent of particles, is increased. For very small particles (less than 0.01 mm in diameter) and small values of P (less than 6%) the loss in specific impulse is often less than 2%. For larger particles (over 0.015 mm diameter) and larger values of P this theory is not helpful and the specific impulse can be 10 to 20% less than the Is value without flow lag. The actual particle sizes and distribution depend on the specific propellant, the combustion, the particular particle material, and the specific rocket propulsion system, and usually have to be measured (see Chapters 12 and 18). Thus adding a metal, such as aluminum, to a solid propellant will increase the performance only if the additional heat release can increae the combustion temperature T{ sufficiently so that it more than offsets the decrease caused by particles in the exhaust.

With very-high-area-ratio nozzles and a low nozzle exit pressure (high altitude or space vacuum) it is possible to condense some of the propellant ingredients that are normally gases. As the temperature drops sharply in the nozzle, it is possible to condense gaseous species such as H20, C02, or NH3 and form liquid droplets. This causes a decrease in the gas flow per unit area and the transfer of the latent heat of vaporization to the remaining gas. The overall effect on performance is small if the droplet size is small and the percent of condensed gas mass is moderate. It is also possible to form a solid phase and precipitate fine particles of snow (H20) or frozen fog of other species.

Other Phenomena and Losses

The combustion process is really not steady. Low- and high-frequency oscillations in chamber pressure of up to perhaps 5% of rated value are usually considered as smooth-burning and relatively steady flow. Gas properties (k, 97f, cp) and flow properties (v, V, T,p, etc.) will also oscillate with time and will not necessarily be uniform across the flow channel. These properties are therefore only "average" values, but it is not always clear what kind of an average they are. The energy loss due to nonuniform unsteady burning is difficult to assess theoretically. For smooth-burning rocket systems they are negligibly small, but they become significant for larger-amplitude oscillations.

The composition of the gas changes somewhat in the nozzle, chemical reactions occur in the flowing gas, and the assumption of a uniform or "frozen" equilibrium gas composition is not fully valid. A more sophisticated analysis for determining performance with changing composition and changing gas properties is described in Chapter 5. The thermal energy that is carried out of the nozzle (m cp Tf) is unavailable for conversion to useful propulsive (kinetic) energy, as is shown in Fig. 2-3. The only way to decrease this loss is to reduce the nozzle exit temperature T2 (larger nozzle area ratio), but even then it is a large loss.

When the operating durations are short (as, for example, with antitank rockets or pulsed attitude control rockets which start and stop repeatedly), the start and stop transients are a significant portion of the total operating time. During the transient periods of start and stop the average thrust, chamber pressure, or specific impulse will be lower in value than those same parameters at steady full operating conditions. This can be analyzed in a step-by-step process. For example, during startup the amount of propellant reacting in the chamber has to equal the flow of gas through the nozzle plus the amount of gas needed to fill the chamber to a higher pressure; alternatively, an empirical curve of chamber pressure versus time can be used as the basis of such a calculation. The transition time is very short in small, low-thrust propulsion systems, perhaps a few milliseconds, but it can be longer (several seconds) for large propulsion systems.

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