Richard Nakkas Experimental Rocketry Web Site
Solid Rocket Motor Theory
Two-phase Flow
Introduction to Two-phase Flow
Most
solid rocket propellants produce combustion products that are a mixture of
gases and condensed-phase particles
(either liquid or solid) which is evident as visible smoke in the exhaust plume. Those
propellants containing metals, such as aluminum or magnesium, generate oxides
of the metals as condensed-phase combustion products. Metallic-compound
oxidizers, such as potassium nitrate (KN) or potassium perchlorate (KP),
generate condensed-phase products of particularly high molecular mass. As is
seen by Equation 12 of Nozzle Theory web page, a higher molecular mass (M)
of the products lowers the exhaust velocity and therefore overall performance. The
KN-Sugar propellants produce a dense white cloud of potassium
carbonate smoke. In fact, approximately 42% of the exhaust mass is condensed-phase
matter !
The
occurrence of solids or liquids in the exhaust leads to a reduction in
performance for a number of additional reasons:
· This portion of the combustion mass cannot perform any expansion
work and therefore does not contribute to acceleration of the exhaust flow.
· The higher effective molecular mass of these products lowers the
Characteristic Velocity (c-star).
· Due to thermal inertia, the heat of the condensed-phase is partly
ejected out of the nozzle before transferring this heat to the surrounding
gases, and is therefore not converted to kinetic energy. This is known as particle thermal lag.
· Likewise due to the relatively large mass of the particles
(compared to the gases), these cannot accelerate as rapidly as the surrounding
gases, especially in that portion of the nozzle where flow acceleration is
extremely high (throat region). Acceleration of the particles depends upon
frictional drag in the gasflow, which necessitates a differential velocity
between the particles and the gas. The net result is that the condensed-phase
particles exit the nozzle at a lower velocity than the gases. This is referred
to as particle velocity lag.
There
are a number of factors that determine how significant two-phase flow losses
have on a rocket motors performance. One important practical factor is nozzle contour, in particular at the
throat region. A more gradual acceleration of the flow in the neighbourhood of
the throat results in a reduction of thrust loss (Ref.1). Figure 1 illustrates
the flow acceleration for the Kappa nozzle.
The acceleration in the region of the throat (red dashed line) is extremely
high, especially just aft, where it is maximum. Most of the particle lag, which
is a strong function of acceleration, occurs in this region, thus the
importance of designing a nozzle with a well-rounded contour at the throat,
without any sharp changes in cross-section.
Figure 1 -- Gas/particle
acceleration for two-phase flow through Kappa nozzle
The size of the rocket motor as well
as condensed-phase particle size both
play an important role with regard to the influence of two-phase flow effects. This
is illustrated in Figure 2, which plots the fraction of Characteristic Velocity
loss with respect to:
· Motor size (thrust)
· Particle size
Note that the mass fraction of particles in the exhaust for
this study was 0.25. For the sugar propellants, the mass fraction can be as
high as 0.44.
Figure 5 -- Influence of
motor size and particle size on c-star
Excerpted
from Ref.2
For
example, for a 100 lb. (445 N.) thrust motor, the motor suffers a 6% loss in
Characteristic Velocity if the average particle size is 1.5 micron, as shown by
the orange dashed line.
It is clear from this plot that for amateur experimental motors, which are
typically of 1000 lb. thrust or less, that two-phase flow losses can be
significant, but are less significant for large "professional"
motors.
In the following treatment of two-phase flow and how it affects
the performance the various parameters related to rocket motor design, the
assumption of zero particle lag is assumed. In
other words, the particles are assumed to have the same
temperature and same velocity
as the surrounding gases as the
products flow through the nozzle. This assumption serves to greatly simplify
the two-phase problem in a practical way for rocket motor design. In reality,
this is largely true for exhaust particles of one micron size (Ref.1). This is
likely typical for sugar propellants, which produce a very fine smoke.
Rocket Motor Design Parameters for Two-Phase Flow
The
following section provides the means for calculating thermochemical and
performance parameters for a rocket motor that generates two-phase exhaust
products. The superscript ʺ is applied to properties denoting that is is
applicable to two-phase exhaust flow.
Effective Molecular Mass and Gas
Constant
When
the exhaust products consist of a mixture of gases and condensed-phase, the
effective molecular mass, designed Mʺ, is
given by the number of gas moles, Ng,
divided into the total mass of the exhaust products, mp:
The specific gas constant for
a mixture of gases and condensed-phase, Rʺ, is given by the universal gas constant, 
, divided by the effective molecular mass of the products, Mʺ :
The
specific gas constant for the gas-only products, R,
is related by the particle fraction, X, as shown in
Eqn.2b:
Ratio of Specific Heats
The
ratio of specific heats (also refered to as isentropic exponent)
for a mixture of gases and condensed phase, kmix,
is given by:
where Cp mix
is the specific heat (at constant pressure) of the mixture of gases and
condensed phase and is given by:
where
m1/mp, m2/mp, etc are the mass fraction of each of the gaseous and condensed-phase
products and Cp 1, Cp 2, etc. are the specific heats of
gaseous products. Note that for condensed-phase products, the specific heat is
denoted Cs. Specific heat is a function of
temperature, so the values must be determined at the combustion temperature.
The
specific heat for the two-phase mixture is related to that of the gas-only and
condensed-phase products and by the particle fraction as shown by Eqn.4b:
Eqn.4b
The
quantity kmix is valid for a static mixture of gases and condensed-phase products. What
about the dynamic case where the products are flowing through the nozzle? A
solution to this case is obtained from gasdynamics analysis, as given in
References 3 & 4, with the derivation shown in the Appendix. In this case,
the isentropic exponent for two-phase flow, kʺ, is
given by:
where
k is the gas-only products specific heat
ratio and Ψ (Greek letter Psi) is the
term X/(1-X), where X is the mass fraction of condensed-phase products. Note
that Cs is for the mixture of condensed-phase
products and Cp is for the mixture of gaseous
products.
An
alternate form of expression for kʺ is given by:
Eqn.3
and Eqn.5a can be shown (see Appendix) to be algebraically equivalent, such
that kʺ = kmix.
Critical (throat) Velocity
At
the nozzle throat, the velocity of the product flow is limited to mach one, or
sonic velocity. With the assumption of two-phase mixture behaving as an ideal
gas, the resulting equation is the same as for gas-only flow, except with the
specific heat ratio and gas constant being those applicable to the two-phase
mixture, and is given by Eqn.6a:
where
T* is the critical (or throat)
temperature of the flow. With the assumption of the two-phase mixture behaving
as an ideal gas, the critical temperature is found using Eqn.4 of the Nozzle Theory web page, yielding Eqn.6b:
The
critical velocity is of special interest to the rocket motor designer. As was
shown by Eqn.2, the higher the fraction of condensed-phase, the lower the value
of the gas constant, Rʺ. This
reduces the value of v* which has
the overall effect of lowering motor performance. The throat acts as a
restrictor, or limiter, of gas velocity.
Characteristic Exhaust Velocity
In
terms of the rocket performance parameters, the presence of condensed-phase
products is reflected in a reduced Characteristic Exhaust Velocity
(c-star or C*), due to the higher
effective molecular mass of the gas/particle mixture. As was explained earlier,
c-star is figure of thermochemical
merit for a particular propellant. With the assumption of two-phase mixture
behaving as an ideal gas, the Characteristic Exhaust Velocity for a two-phase mixture is given by Eqn.7:
Exhaust Velocity
The
exhaust velocity of a rocket nozzle, for two-phase flow, is given by Eqn.8:
where
kʺ and Rʺ
are for two-phase products. Equation 8 is of identical form to that of
gas-only flow, as seen by Eqn.12 of the Nozzle Theory web page.
Specific Impulse
The
ideal specific impulse of a rocket motor is given by Eqn.9:
where
c is the effective
exhaust velocity and g is the
acceleration of gravity ( sort of a conversion factor to give units of seconds). For a nozzle with optimum expansion ratio, the
effective exhaust velocity is equivalent to the ideal exhaust velocity given by
Eqn.8. As such, for two-phase flow, the ideal specific impulse is given by
Eqn.10:
Thrust
From
Equation 2 of Thrust Theory web page, the thrust of a rocket motor is given by:
..Eqn.11
with
the assumption of optimum nozzle expansion (Pe = Pa). The term ρ*A*v* represents
the mass flow rate through the nozzle (*
denotes critical or throat). Using Equation 7 of the Nozzle
Theory web page, and based on the assumption that a
two-phase mixture behaves as an ideal gas, the critical flow
density at the nozzle throat is given by:
.Eqn.12
A* is the throat cross-sectional area. The critical velocity, v*, is given by Eqn.6 and exhaust velocity ve is given by Eqn.8.
From
Eqn.6, Eqn.8, Eqn.11 and Eqn.12, the equation for thrust for two-phase flow is obtained
(for optimum nozzle expansion) and is given by Eqn.13:
It is seen that the
thrust equation for two-phase flow is of the same form as for gas-only flow
given by Equation 3 of the Thrust Theory web page except having the two-phase parameters Rʺ and kʺ.
Thrust Coefficient
As explained in the Thrust Theory web page, the Thrust Coefficient determines the amplification of thrust
due to gas expansion in the divergent portion of the nozzle as compared to the
thrust that would be exerted if the chamber pressure acted over the throat area
only. From equation 4 of the Thrust Theory web page, for a nozzle with optimum
expansion:
The Thrust Coefficient
for two-phase flow condition is therefore obtained from Eqn.13, and given by
Eqn.14:
Chamber Pressure
The equation for Chamber
Pressure for two-phase mixture is essentially the same as for the gas-only
condition, as given by Equation 11 of the Chamber Pressure Theory web page. The only difference is with the values for specific
heat ratio and gas constant, being those applicable for a two-phase mixture.
For a definition of the
terms introduced in this equation, refer to the Chamber
Pressure Theory web page.
Appendix
Derivation of Eqn.5a (2-phase isentropic exponent)
Derivation of Eqn.5b (2-phase isentropic exponent)
Derivation of Eqn.8 (2-phase exhaust velocity)
Derivation of Eqn.13 (2-phase thrust)
Derivation of Eqn.15 (2-phase chamber pressure)
Show that Eqn.3 and Eqn.5 are algebraically equivalent
Example 1: For KNSU propellant, calculate the various two-phase
thermochemical and performance parameters (ideal):
· Molecular mass, specific gas constant and ratio of specific heats
· Critical (throat) velocity, Characteristic Exhaust Velocity
· Exhaust velocity and Specific Impulse (ideal expansion from 1000
psi chamber pressure)
Example 2: Utilizing KNSU propellant, calculate the ideal chamber pressure,
thrust and thrust coefficient for the following motor conditions with optimum
expansion:
· Kn = 250
· Throat diameter 12.0 mm (0.472 inch)
Reference Sources
1.
Gas
Particle Flow in an Axisymmetric Nozzle,
W.S.Bailey, E.N.Nilson, R.A.Serra and T.F.Zupnik, ARS Journal, June 1961
2.
Dynamics
of Two-Phase Flow in Rocket Nozzles,
M.Gilbert, J.Allport and R.Dunlap, ARS Journal, December 1962
3.
Mechanics
and Thermodynamics of Propulsion,
Philip G. Hill, Carl R.Peterson
Addison-Wesley Publishing Co.
4.
Recent
Advances in Gas-Particle Nozzle Flows,
Richard F.Hoglund, ARS Journal, May 1962
5.
Perturbation
Analysis of One-dimensional Heterogeneous Flow in Rocket Nozzles,
W.D.Rannie, Detonation and Two-Phase Flow
edited by S.S.Penner & F.A.Williams, Academic Press, 1962
6.
Rocket
Propulsion Elements, George P. Sutton,
4th Ed., John Wiley and Sons
Originally
posted March 3, 2023
Last updated March 22, 2023