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,* N _{g}*,
divided into the total mass of the exhaust products,

The *specific gas constant *for
a mixture of gases and condensed-phase, *R*ʺ, 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, *k*_{mix},
is given by:

where C_{p mix}
is the specific heat (at constant pressure) of the mixture of gases and
condensed phase and is given by:

where
m_{1}/m_{p}, m_{2}/m_{p}, etc are the *mass fraction* of each of the gaseous and condensed-phase
products and C_{p 1}, C_{p 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 *k _{mix}* is valid for a

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 *= *k _{mix}*.

__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, *R*ʺ. 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 *v**e* 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 *R*ʺ 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,
4^{th} Ed., John Wiley and Sons

Originally
posted March 3, 2023

**Last updated March 22, 2023**