Nozzles

In rocket propulsion systems the mathematical tools needed to calculate performance
and to determine several key design parameters involve the principles from gas
dynamics and thermodynamics that describe processes inside a rocket nozzle
and its chamber. These relations are also used for evaluating and comparing the
performance between different rocket systems since with them we can predict
operating parameters for any system that uses the thermodynamic gas expansion
in a supersonic nozzle; they allow the determination of nozzle size and generic
shape for any given performance requirement. This theory applies to chemical
rocket propulsion systems (liquid and solid and hybrid propellant types), nuclear
rockets, solar-heated and resistance or arc-heated electrical rocket systems, and all
propulsion systems that use gas expansion as the mechanism for ejecting matter at
high velocities.

IDEAL ROCKET PROPULSION SYSTEMS

The concept of an ideal rocket propulsion system is useful because the relevant basic
thermodynamic principles can be expressed with relatively simple mathematical relationships, as shown in subsequent sections of this chapter. These equations
describe quasi-one-dimensional nozzle flows, which represent an idealization and
simplification of the full two- or three-dimensional equations of real aerothermochemical behavior. However, within the assumptions stated below, these descriptions
are very adequate for obtaining useful solutions to many rocket propulsion systems
and for preliminary design tasks. In chemical rocket propulsion, measured actual
performances turn out to be usually between 1 and 6% below the calculated ideal
values. In designing new rocket propulsion systems, it has become accepted practice
to use such ideal rocket parameters, which can then be modified by appropriate
corrections.

1. The working fluid (which usually consists of chemical reaction products) is
   homogeneous in composition.
2. All the species of the working fluid are treated as gaseous. Any condensed
   phases (liquid or solid) add a negligible amount to the total mass.
3. The working fluid obeys the perfect gas law.
4. There is no heat transfer across any and all gas-enclosure walls; therefore, the
   flow is adiabatic.
5. There is no appreciable wall friction and all boundary layer effects may be
   neglected.
6. There are no shock waves or other discontinuities within the nozzle flow.
7. The propellant flow rate is steady and constant. The expansion of the working
   fluid is uniform and steady, without gas pulsations or significant turbulence.
8. Transient effects (i.e., start-up and shutdown) are of such short duration that
   may they be neglected.
9. All exhaust gases leaving the rocket nozzles travel with a velocity parallel to
   the nozzle axis.
10. The gas velocity, pressure, temperature, and density are all uniform across any
    section normal to the nozzle axis.
11. Chemical equilibrium is established within the preceding combustion chamber and gas composition does not change in the nozzle (i.e., frozen composition flow).
12. Ordinary propellants are stored at ambient temperatures. Cryogenic propellants are at their boiling points.

These assumptions permit the derivation of the relatively compact, quasi-one dimensional set of equations shown in this chapter. Later in this book we present
more sophisticated theories and/or introduce correction factors for several of the
items on the above list, which then allow for more accurate determinations. The next
paragraph explains why the above assumptions normally cause only small errors.
For liquid bipropellant rockets, the idealized situation postulates an injection system in which the fuel and oxidizer mix perfectly so that a homogeneous working medium results; a good rocket injector can closely approach this condition. For solid
propellant rocket units, the propellant must essentially be homogeneous and uniform
and the burning rate must be steady. For solar-heated or arc-heated propulsion systems, it must be assumed that the hot gases can attain a uniform temperature at any
cross section and that the flow is steady. Because chamber temperatures are typically high (2500 to 3600 K for common propellants), all gases are well above their
respective saturation conditions and do follow closely the perfect gas law. Assumptions 4, 5, and 6 above allow the use of isentropic expansion relations within the
rocket nozzle, thereby describing the maximum conversion from heat and pressure to
kinetic energy of the jet (this also implies that the nozzle flow is thermodynamically
reversible). Wall friction losses are difficult to determine accurately, but they are usually negligible when the inside walls are smooth. Except for very small chambers,
the heat losses to the walls of the rocket are usually less than 1% (occasionally up
to 2%) of the total energy and can therefore be neglected. Short-term fluctuations in
propellant flow rates and pressures are typically less than 5% of their steady value,
small enough to be neglected. In well-designed supersonic nozzles, the conversion of
thermal and/or pressure energy into directed kinetic energy of the exhaust gases may
proceed smoothly and without normal shocks or discontinuities—thus, flow expansion losses are generally small.