Shock Waves (1998) 8: 113 118
Real gas effects on the prediction of ram accelerator performance
P. Bauer1, C. Knowlen2, A. Bruckner2
1
Laboratoire de Combustion et Détonique (UPR 9028 du CNRS), ENSMA  University of Poitiers, BP 109, 86960 Futuroscope Cedex,
France
2
Aerospace and Energetics Research Program University of Washington, Box 352250 Seattle, WA 98195-2250, USA
Received 9 August 1996 / Accepted 23 May 1997
Abstract. The analysis of ram accelerator performance is readily applicable to any other mixture. The projectile ve-
based on one-dimensional modelling of the flow process that locity and acceleration histories determined by the Hugoniot
propels the projectile. The conservation equations are ap- analysis for the thermally choked ram accelerator mode, as-
plied to a control volume travelling with the projectile, and suming the Boltzmann EOS, turn out to be in much bet-
quasi-steady flow is assumed. To date the solution obtained, ter agreement with experimental observations up to the CJ
namely the generalized thrust equation, has been based on detonation velocity than that when based on the ideal gas
the ideal gas assumption. At the high level of pressure that assumption.
is encountered during the ram accelerator process, this as-
sumption cannot be regarded as adequate. Thus, a more ap- Key words: Ram accelerator, Ramjet-in-tube, Detonation,
propriate equation of state (EOS) should be used instead. Real gas, Equation of state
Depending upon the level of pressure, several equations of
state are available for dense gaseous energetic materials. The
virial type of EOS can be more or less sophisticated, de-
Nomenclature
pending upon the extent of complexity of the intermolecular
a: sound speed
modelling, and turns out to be totally appropriate for most
A: cross sectional area of the tube
gaseous explosive mixtures that have been investigated at
b: covolume
moderate initial pressures, i.e., less than 10MPa.
c: specific heat capacity
In the present case the Boltzmann EOS was applied. It
D: detonation velocity
is based on very simplified molecular interactions, which
F: axial force exerted on the flow
makes it relatively easy to use in calculations. Moreover,
h: specific enthalpy
the energetic EOS needs to be taken into account. This con-
H: enthalpy
cerns all the calorimetric coefficients, as well as the ther-
I: non-dimensional thrust
modynamic parameters, which can no longer be expressed
M: Mach number
as only a function of temperature. The higher the pressure
n: number of moles of the products
level, the more sophisticated these corrections become, but
N: finite difference operator
the main relationships that account for real gas effects are
p: pressure
basically the same. These include the use of a general form
P: pressure ratio of products to reactants
of analytical operators applied to correct the thermodynamic
Q: non-dimensional heat release
functions and coefficients. The equations governing the one-
R: gas constant
dimensional model were taken as a basis for the real gas cor-
T: temperature
rections and were solved analytically. The parameters which
u: velocity of the projectile
play the most crucial roles in this correction can thus be
U: internal energy
highlighted. A complete set of equations involving the real
v: specific volume
gas effects are presented in this paper. The QUARTET code
V: volume ratio of products to reactants
was used in this investigation, especially for determining
Å‚: heat capacity ratio
chemical equilibrium compositions.
“ : adiabatic heat capacity ratio
This more accurate model can better predict the projec-
·: caloric imperfection parameter
tile acceleration of the thermally choked propulsive mode.
Ã: compressibility parameter
Although the present analysis is applied to the fuel-rich
methane-oxygen-nitrogen mixture currently used in the ram
1: entry of the control volume
accelerator experiments, its general formulation makes it
2: exit of the control volume
Correspondence to: P. Bauer CJ: Chapman-Jouguet state
114
f: formation
p: constant pressure
S: constant entropy
T: constant temperature
v: constant volume
ig: ideal gas
Fig. 1. Control volume used for ram accelerator performance mod-
rg: real gas
elling
": correction term
1 Introduction
Successful prediction of the ram accelerator thrust-Mach
number relationship for the thermally choked propulsive
mode is accomplished in a straightforward manner that
yields the main parameters of acceleration process. This
analysis of subdetonative ram accelerator performance is
based on one-dimensional modelling of the flow process that
propels the projectile (Knowlen and Bruckner 1991; Bruck-
ner et al. 1991). The quasi-steady conservation equations are
applied to a control volume attached to the projectile. The
main assumption that oversimplifies the actual physical pro-
cess is to use the ideal gas equation of state (EOS) for the
combustion products. Since these products are at a substan-
tially elevated pressure, i.e., on the order of several tens of
Fig. 2. Non-dimensional thrust vs. flight Mach number at 10 and
MPa, the ideal EOS is no longer valid and a more realistic 50 bar
one should be used instead. This does not necessarily mean
an extremely sophisticated EOS is required, since there are
where the subscripts  1 and  2 refer to the flow properties
a number of EOS such as the virial type that, despite their
entering and leaving the control volume, respectively (as
simplicity, may be regarded as totally appropriate (Bauer et
shown in Fig. 1). Furthermore, the following dimensionless
al. 1985; 1994). The effects of increasing the initial pressure
parameters are defined:
beyond 8 MPa, thus yielding combustion products of the
order of 100 MPa, may still be sufficiently modelled with
hf1 - hf2 F p2 v2
a virial type EOS that includes more severe molecular in- Q = , I = , P = , V = , (2)
cp1T1 p1A p1 v1
teraction laws (Heuzé 1986). Such extremely elevated pres-
sures are beyond the scope of this report; however, it turns
which are the non-dimensional heat release, non-dimensional
out that the corrections to the aerothermodynamic equations
thrust, and the pressure and specific volume ratios between
presented here are totally general and can be applied with
the initial and final states, respectively.
any EOS.
The parameters · and à shown below denote a measure
of caloric imperfection and non ideal gas behavior.
h
2 Theoretical considerations
· = ; pv = ÃRT. (3)
cpT
2.1 General equations
After some algebraic combination of these relationships, the
following expression for the generalized Hugoniot results
The main idea of this modeling is to describe the aerother-
(Knowlen and Bruckner 1991):
modynamics of the flow around the projectile as a global
2cp1
(Q + ·1) - (V + 1)(I +1)
process between its state as it enters the control volume and
R1
P = . (4)
2cp2·2
the state of the exit flow. This modelling is based on the set
V ( - 1) - 1
ÃR2
of one-dimensional conservation equations for quasi-steady
flow:
At this stage, the ideal gas EOS will be taken to describe
u1 u2
the initial properties of the mixture, thus Ã1=1.
continuity : = ,
The sound velocity is expressed as follows:
v1 v2
a2 = “RT, (5)
u2 u2
1 2
energy : h1 + hf1 + = h2 + hf2 + , (1)
where “ , R, and T are the  adiabatic heat capacity ratio also
2 2
named adiabatic gamma (Byers Brown and Amaee 1992),
u2 F u2 the gas constant, and temperature, respectively. One may
1 2
momentum : p1 + + = p2 + ,
express “ in the form:
v1 A v2
115
Fig. 3. Sketch of the University of Washington ram accelerator facility
"h "Lnp
2.2 Real gas form of the thermodynamic parameters
“ = |s = - |s , (6)
"e "Lnv
The preceding equations require the knowledge of a series
where e and h are the specific internal energy and en-
of thermodynamic parameters, which can be provided by the
thalpy, respectively. This “ , specifically used in the case
classical thermodynamic functions. However, this requires
of combustion products undergoing rapid phase or composi-
some further numerical and analytical treatment which is de-
tion changes through compression or expansion processes, is
tailed below. Enthalpy and internal energy may be expressed
readily correlated to the usual Å‚ (Heuzé et al. 1986). Using
in the general following form:
the parameters described above, one can put the momentum
equation in terms of Mach number as follows:
¨rg = ¨ig + ¨", (13)
2 2
M2 “2R2T2 M1 Å‚1R1T1
where ¨ is either the enthalpy or internal energy and ¨" is
I = P - 1+ - . (7)
the correction term of the corresponding parameter. These
p1v2 p1v1
correction terms may be expressed in the following differ-
Note that when dealing with the initial state of the mixture,
ential forms (Byers Brown and Amaee 1992):
i.e., the unburned gases, the classical relationship was used
"v
for the sound speed, namely, that involving Å‚1.
dH" = v - T |p dp, (14)
Hence, the non-dimensional thrust may be expressed in the
"T
following form:
"p
2
dU" = T |v - p dv.
“2M2
2
"T
I = P (1 + ) - (1 + M1 Å‚1). (8)
Ã
We now introduce a series of operators involving partial
Using (3) and (5) together with the continuity equation
derivatives of à (Heuzé et al. 1986) that will be extensively
yields:
used in the following calculations:
M1 Å‚1R2T2
"Ã
P = Ã . (9)
Ãv = T |T (15)
M2 “2R1T1
"v
The temperature ratio may be derived from the energy bal-
"Ã
ÃT = v |v
ance in the form:
"T
·1 + M1 + Q
T2 cp1 Å‚1-1 2
Using these operators in (14) yields:
2
= , (10)
2
T1 cp2 ·2 + M2 “2R2
dv
2cp2
dH" = RT (Ãv + ÃT ) (16)
v
which yields:
dv
·1 + M1 + Q dU" = RT ÃT
M1 Å‚1R2 cp1 Å‚1-1 2
2
v
P = Ã , (11)
2
M2 “2R1 cp2 ·2 + M2 “2R2
2cp2
The real gas corrections should also be applied to the so-
called  adiabatic gamma. It can be shown (Heuzé et al.
This, in turn, leads to the final generalized relationship for
1987 a) that the generalized expression of this parameter is:
non-dimensional thrust:
Ãv
“ = Å‚ 1 - - nv , (17)
2
·1 + M1 + Q
M1 Å‚1R2 cp1 Å‚1-1 2 “2M2 Ã
2
I = Ã 1+
2
M2 “2R1 cp2 ·2 + M2 “2R2 Ã
where
2cp2
v "n
2
-(1 + M1 Å‚1). (12) nv = |T .
n "v
116
Fig. 4. Projectile configuration used in experiments
However, in the present form these parameters involve T
and v and, therefore, they cannot be readily calculated. In
order to perform such a calculation one can switch to p and T
variables which are more representative of the actual inputs
of the problem. On the basis of a set of finite difference
relationships which can be obtained by a simple equilibrium
calculation (Heuzé et al. 1987 b) for three distinct p, T sets
of values (Heuzé et al. 1987 a), the values of the dissociation
rates, nT and nv may thus be derived from the preceding
operators:
v "p
Fig. 5. a Experimental velocity vs. distance profile compared with
nv = Np |T , (18)
p "v theory at 25 bar. Projectile mass: 63 g (Mg). b Experimental veloc-
ity vs. distance profile compared with theory at 50 bar. Projectile
where
mass: 109 g (Ti)
p ´n
Np = |T .
n ´p
since analytical calculations can be easily handled and yet
It can be shown that the derivatives involved in these expres-
it is a very reliable EOS, as long as the pressure of the
sions can, in turn, be expressed as functions of the former
combustion products does not exceed 200 MPa (Bauer et al.
operators (15) as follows:
1991). In the present case, the EOS has the following form
Ãv Np
(Bauer and Brochet 1983):
nv = - 1 - . (19)
à 1 - Np
à =1 +x +0.625x2 +0.287x3 +0.193x4 + O(xn), (20)
A comment can be made at this point: since nv always has
b
where x = , and O(xn) indicates higher order terms which
v
a positive value and Ãv = 0 in the case where an ideal
are not taken into account here.
gas is concerned, “ for an ideal gas is always less than Å‚.
A simplified molecular interaction model was chosen
The general expressions for the thermodynamic parameters,
which assumes that only like molecules interact:
and more specifically those which appear in the generalized
thrust relationship, namely H2 and “2, can be expressed as
b = xibi. (21)
a function of the operators: nv, Ãv, and ÃT . It should be
noticed that all the preceding general formulation is not de-
This form of the EOS allows one to readily calculate all
pendent on any type of EOS, apart from a general parameter
the corrections expressed above. These corrections happen
à and its derivatives. A more detailed expression of this Ã
to take a simple form, since one gets a simple representation
used in the present case must now be given and its applica-
for both ÃT and Ãv (Heuzé et al. 1986) as follows:
tion to the thermodynamic parameters addressed.
ÃT =0, (22)
"Ã
Ãv = -x = -(x +1.25x2 +0.861x3 +0.772x4).
2.3 Equation of state
"x
The next step of this modelling requires the appropriate These rather straightforward expressions of the derivatives
choice of an EOS. The virial type EOS is very convenient, of Ã, as well as à itself, provide an easy way to calculate
117
the pertinent real gas corrections that should be accounted
for in the present situation:
H" =(Ã - 1)RT. (23)
This can be easily derived since, in the present case:
U" =0. (24)
The generalized form of the one-dimensional thrust equation
can now be readily calculated on the basis of these analyti-
cal expressions. The effect of the real gas corrections on the
non-dimensional thrust as a function of the inlet Mach num-
ber can be seen in Fig. 2 for the fuel-rich mixture: 2.8 CH4 +
2 O2 + 5.7 N2. The general profile of the curves is consistent
with the assumption of the thermally choked ram accelerator
mode (M2 = 1) which dictates decreasing thrust as the Mach
number increases up to the detonation velocity. Calculations
provided by the black box model of this propulsive mode
are presented for the assumptions of both the ideal and real
gas EOS. The initial pressures considered here were 10 and
50 bar in order to highlight the effect of real gas correc-
tions. These curves show that the real gas corrections tend
to shift upward the thrust value; thus the propellant mixture
is more effective at a higher initial pressure. In addition, the
enhancement of thrust tends to increase as the Mach number
increases.
3 Experimental facility
The ram accelerator facility at the University of Washington
consists of a helium gas gun, helium dump tank, ram acceler-
Fig. 6a,b. Non-dimensional thrust vs. flight Mach number for ex-
ator test section, and projectile decelerator as shown in Fig. 3
periment in Fig. 5
(Hertzberg et al. 1991). The 38 mm bore, 6-m-long, single-
stage light gas gun is used to accelerate the projectile to the
desired test section entrance velocity, usually around 1100
m/s. The 16-m-long ram accelerator test section consists of distance of the projectile in the tube are presented in Figs. 5a
eight 2-m-long, high-strength steel tubes having a bore of 38 and 5b. In the 25 bar experiment shown in Fig. 5a the pro-
mm and an outer diameter of 102 mm. The ram accelerator jectile was fabricated from magnesium alloy and had a mass
test section is designed to operate at propellant fill pressures of 63 g. It accelerated for approximately 12 m and then un-
of up to 50 bar. There are 40 equidistant multiple-port in- started. An unstart is the disgorging of a normal shock past
strument stations at 40 cm intervals along the test section, the projectile throat, often resulting in an overdriven detona-
which is typically instrumented with piezoelectric pressure tion wave which rapidly decelerates the projectile (Bruckner
transducers and electromagnetic (EM) sensors. Thin Mylar et al. 1991) at a velocity of 2100 m/s. The 50 bar experi-
diaphragms close off each end and separate different mix- ment shown in Fig. 5b was conducted with a titanium alloy
tures in multistage experiments. projectile having a mass of 109 g. It accelerated throughout
The experimental projectiles shown in Fig. 4 are fabri- the 16 m test section and exited the tube at 2000 m/s. The
cated from magnesium, aluminum or titanium alloys in two theoretical velocity-distance profiles for both ideal and real
hollow pieces (nose cone and body), which thread together gas assumptions are shown for these experiments. In addi-
at the throat (point of maximum projectile cross-sectional tion, the experimental detonation velocity (DCJ) for these
area). An annular magnet placed at the nose-body joint in- propellant mixtures are indicated in the figures.
duces signals in the EM sensors. The fins, in principle, serve The experimental velocity-distance profiles (Figs. 5a and
only to center the projectile in the tube. 5b) agree very well with the predictions for the thermally
choked ram accelerator mode propulsive mode using a real
gas EOS up to about 90% of the detonation velocity DCJ. In
4 Experimental results and discussion this velocity regime the projectiles are believed to undergo a
propulsive mode transition which enables them to accelerate
The results of two typical ram accelerator experiments using beyond the CJ speed (Hertzberg et al. 1991). Note that the
fuel-rich methane-oxygen-nitrogen mixtures at pressures of real gas EOS corrections predict a peak velocity that coin-
25 and 50 bar are presented in Figs. 5 and 6, respectively. In cides with the experimentally observed CJ speed, which is
each case the projectile entered the ram accelerator tube at in excellent agreement with the QUARTET thermochemical
approximately 1100 m/s. Records of the velocity vs. travel code (Heuzé et al. 1987 c). The real gas correction improve-
118
ment is even more obvious for the experiments having a References
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56-62
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5 Conclusion
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a
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