47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition
5 - 8 January 2009, Orlando, Florida
AIAA 2009-1601
A Modeling of Thermal Properties of Hydrogen/Oxygen
System Using Molecular Simulations
Shin-ichi Tsuda1 and Nobuhiro Yamanishi2
Japan Aerospace Exploration Agency, Tsukuba, Ibaraki, 305-8505, Japan
Takashi Tokumasu3
Tohoku University, Sendai, Miyagi, 980-8577, Japan
Nobuyuki Tsuboi4
Japan Aerospace Exploration Agency, Sagamihara, Kanagawa, 229-8510, Japan
and
Yoichiro Matsumoto5
The University of Tokyo, Bunkyo-ku, Tokyo, 113-8656, Japan
We have calculated some thermal properties of hydrogen and of oxygen using a classical
molecular dynamics (MD) method. To examine the applicability of a simplified
intermolecular interaction model, we employed the Lennard-Jones (12-6) potential with
ignoring the rotational freedom in both fluids and conducted MD simulations for the
Lennard-Jones fluid at a wide density-temperature range. Using the MD calculation data,
we made a polynomial function of Helmholtz free energy, which can derive every thermal
property, and determined the potential parameters of the Lennard-Jones model which
reproduce the thermal properties of hydrogen and of oxygen. In spite of the very simple
intermolecular interaction model, we found that it can reproduce the thermodynamic
properties of oxygen at a wide density-temperature range and the pressure-volumetemperature relationship of hydrogen below its critical density. Such simple model has a
potential to estimate reasonably an equation of state of hydrogen/oxygen mixture or the
interfacial tension which is very important in the analyses of the coaxial injection jet flows of
oxidizer with gaseous hydrogen in a rocket engine thrust chamber.
Nomenclature
e
A
Amn
Ep
h
kB
i
j
l
m
N
=
=
=
=
=
=
=
=
=
=
excess Helmholtz free energy from ideal gas
coefficients of the function for the excess Helmholtz free energy
excess internal energy from ideal gas
enthalpy
Boltzmann constant
molecular index i
molecular index j
liquid (as index)
mass of a molecule of interest
number of molecules
1
Aerospace Project Research Associate, JAXA’s Engineering Digital Innovation Center, Tsukuba, Ibaraki, Member
AIAA.
2
Associate Senior Engineer, JAXA’s Engineering Digital Innovation Center, Tsukuba, Ibaraki, Member AIAA.
3
Associate Professor, Institute of Fluid Science, Sendai, Miyagi.
4
Associate Professor, Institute of Space and Astronautical Science, Sagamihara, Kanagawa.
5
Professor, Department of Mechanical Engineering, Bunkyo-ku, Tokyo.
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Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
P
Psat
r
T
V
v
v
β
β0
ε
ρ
ρ0
σ
φ
=
=
=
=
=
=
=
=
=
=
=
=
=
=
pressure
saturation pressure
intermolecular distance
temperature
volume of the computational domain
specific volume
vapor (as index)
Boltzmann factor
standard Boltzmann factor
depth of the potential well
number density
standard number density
molecular diameter
intermolecular potential energy
I. Introduction
I
n a liquid rocket engine, various physical and chemical phenomena occurs such as cryogenic cavitation around
inducers in the turbopumps, subcritical or supercritical jet flows of oxidizer around injectors in a combustion
chamber, and the combustion reactions. Although various studies for each phenomenon have shown gradual
progress, many issues have to be clarified continuously by experiments and by numerical simulations.
To analyze each phenomenon mentioned above, we need an accurate prediction of the thermal and transport
properties of fluids such as the relationship of pressure-volume-temperature, specific heat, acoustic speed, latent heat,
and so on. Many data of one-component fluids at a standard pressure-temperature range have been obtained
experimentally, including hydrogen and oxygen which are the main working fluids of liquid rocket. On the other
hand, we do not well understand the fluid properties at extremely high pressure-temperature range or that of the
mixture due to the difficulty to approach by experiments. In a liquid rocket, temperature of the working fluids
changes from around 20K (in the tank of liquid hydrogen) to around 4,000K (at combustion chamber), and not all
thermal data have not been clarified including the data of hydrogen/oxygen mixture in spite that the thermal
properties are the basic information to analyze the phenomena mentioned above.
The present study aims to construct a reasonable method to derive thermodynamic data using molecular
simulations. Molecular simulations, e.g. Monte Carlo (MC) method and Molecular Dynamics (MD) method, are
often useful to understand thermal transport phenomena which are difficult to analyze by experiments or by
continuum mechanical calculations. In this paper, we describe a simple model to express the thermodynamic
properties of hydrogen and of oxygen, and its validity and the limitation is mainly discussed.
II. Calculation Methods
The process to construct the thermal properties and its validation in this study was as follows; 1) MD simulations
with a Lennard-Jones interaction model were performed at a very wide density-temperature range and we obtained
the pressure and internal energy at each density-temperature point. 2) Using MD data, we defined a function for
Helmholtz free energy. 3) Using the function, we estimated the liquid-vapor equilibria of the Lennard-Jones model.
4) Using the equilibria, we defined the Lennard-Jones potential parameters for both hydrogen and oxygen. 5) Some
characteristic thermodynamic properties were derived and compared with the experimental values1) or NIST
(National Institute of Standards and Technology) data2). The details are described below.
A. Molecular Dynamics Calculations
In this study, we conducted classical MD calculations where Newton’s momentum equation is applied to each
molecule to resolve its trajectory. We can obtain the thermodynamic properties statistically using all molecular
positions and velocities (∼momentums) at each time. To calculate the thermodynamic properties, we utilized a
microcanonical ensemble with the number of molecules, the volume of the computational domain, and the total
internal energy being kept constant. All molecular interactions were given by the Lennard-Jones (12-6) potential,
which fairly well reproduces the van der Waals force of rare gases, in a cubic computational domain on which
periodic boundary conditions were imposed on all (3 dimensional) directions. The Lennard-Jones (12-6) potential
φ(r), which is a function of intermolecular distance r, is given by
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⎧⎪⎛ σ ⎞12 ⎛ σ ⎞6 ⎫⎪
ϕ (r ) = 4ε ⎨⎜ ⎟ − ⎜ ⎟ ⎬,
⎪⎩⎝ r ⎠
⎝ r ⎠ ⎪⎭
(1)
where σ and ε are the potential parameters corresponding to the molecular diameter and the depth of the potential
well, respectively. In this paper, we mimicked oxygen and hydrogen as monatomic molecules although they are both
diatomic. This assumption is not widely accepted generally, but it is known that the bond length between the two
oxygen atoms can be set to around one-fifth of the diameter of the atom in the case that a diatomic rigid rotor model
(2-center Lennard-Jones potential) is employed2). On the other hand, for hydrogen, the validity of the assumption is
unapparent. Anyway, we can obtain a more vistaed view by employing a monatomic assumption because all
physical values can be non-dimensionalized by only m, σ, ε, which express material dependent constants (without
exaggeration and without omission). In this paper, all thermodynamic properties of the Lennard-Jones model are
shown in the non-dimensionalized unit by these constants, and various MD calculations at a wide densitytemperature range were conducted to construct an equation of state for the Lennard-Jones fluid.
As the conventional studies on the Lennard-Jones fluids, various Equations of State (EOS) have been already
proposed by many researchers such as Nicolas et al.3), Ree4), Johnson et al.5), and so on. Of these studies, Johnson et
al. 5) constructed a sophisticated EOS and we can utilize this EOS as a reference. However, as the next step, we will
also construct new EOSs for mixtures of Lennard-Jones fluids, or will employ more elaborated potentials such as
Buckingham, 2-center Lennard Jones, and so on, to examine the effect of repulsive force or of the rotational freedom.
Therefore, we would be better to prepare our original method to construct each EOS. As the first step in this paper,
we constructed a new EOS for the Lennard-Jones fluid using a somehow different method from Johnson et al5). or
Nicolas et al. 3) as described below. Our MD calculations are superior to the conventional EOSs in the statistical
accuracy because we employed more molecules to simulate and set more sampling points (density-temperature
points) than the others; 6912 molecules were used to calculate pressure and internal energy at around 281 given
density-temperature points (see Fig. 1). In each calculation, pressure is derived by the virial theorem6) as
P = ρk BT −
and the internal energy is as
1
3V
⎛
∑∑ ⎜⎜ r
i< j
j
⎝
ij
∂φij ⎞
⎟,
∂rij ⎟⎠
(2)
E p = ∑∑ φij .
i< j
(3)
j
In addition, we added a term for the long range correction6) to derive pressure and potential energy because we
employed a cut-off distance of 3.5σ in each MD calculation. Each detailed calculation method employed here is
described in the Reference No. 7.
Temperature [-]
2
1.5
1
0.5
0
0.2
0.4
0.6
0.8
Density [-]
Fig. 1. The density-temperature points where MD calculations were conducted (281 points).
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B. Kataoka’s Method to Express Equation of State
Kataoka8) proposed a calculation method of the Equation of State (EOS) by expressing the excess Helmholtz free
energy from an ideal gas, Ae, as the sum of the production of density ρ and temperature T,
⎛ ρ ⎞
A = ∑ ∑ Amn ⎜⎜ ⎟⎟
β n=1 m=−1 ⎝ ρ 0 ⎠
e
N
5
5
n
m
⎛β ⎞
⎜⎜ ⎟⎟ ,
⎝ β0 ⎠
(4)
where β = 1/(kBT), ρ0=1/σ3, β0 = 1/ε, and N is the number of molecules. The relationship between the potential
energy Ep, pressure p, and Ae is given by
( ) ⎞⎟
⎛ ∂ β Ae
E p = ⎜⎜
⎝ ∂β
⎟ ,
⎠ρ
(5)
(
⎛ ∂ β Ae / N
⎛ ρ ⎞⎧⎪
p = ⎜⎜ 0 ⎟⎟⎨1 + ρ ⎜⎜
∂ρ
⎝ β ⎠⎪⎩
⎝
) ⎞⎟
⎫⎪
⎟ ⎬.
⎠ β ⎪⎭
(6)
Once the pressure and the potential energy at each density-temperature point (Fig. 1.) is obtained by MD
calculations, 35 coefficients of Amn in Eq. (4) can be determined so that the calculated pressure and the potential
energy are reproduced well using least square method. Figure 2 shows some density-pressure curves as EOS (in
nondimensional unit) obtained using the present method. In addition, we also calculated the saturation curve using
Maxwell’s equal area rule9).
Figure 3 compares the calculated saturation curves with Nicolas et al.3) and Johnson et al.5) to confirm each
characteristics. As is shown in the figure, the calculated liquid-vapor equilibria shows a good correspondence to the
Johnson et al.’s result in the lower temperature region, and it matches to the Nicolas et al.’s result in the higher
temperature region. Although Nicolas’s EOS is the best known Lennard-Jones EOS, its statistical accuracy is
partially insufficient5). On the other hand, as mentioned above, Johnson’s EOS is sophisticated and has been
believed to have the highest accuracy of the conventional EOSs for Lennard-Jones fluid. From Fig.3, we could
confirm the following points; 1) the present EOS sufficiently expresses the liquid-vapor equilibria of the LennardJones fluid by Johnson et al. 2) Around the critical point, the present equilibria overestimates the critical temperature
of Johnson et al. by around 0.04.
One of the reasons why our EOS overestimates the critical temperature will be the limits of the use of a truncated
Lennard-Jones potential with a cut-off distance (3.5σ in this study). Actually, Johnson et al. reflected another
detailed calculation result around the critical point to their EOS while they employed their own MC or MD results
far from the critical point. The uncertainty around the critical point should be recognized in the application of our
EOS.
Fig. 2. Calculated density-pressure curves of the Lennard-Jones fluid.
(One of the Nicolas et al.’s data3) is also shown as a reference.)
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1.4
Temperature [-]
This study
Johnson
Nicolas
1.2
1
0.8
0
0.2
0.4
0.6
0.8
Density [-]
Fig. 3. Comparison of the calculated liquid-vapor equilibria with the other conventional studies.
C. The Values of the Potential Parameters for Hydrogen and for Oxygen
Thus far, we had not referred to the specific values of the potential parameters σ and ε. In this subsection, we
gave those values as follows. For oxygen, we determined the parameters so that the saturated liquid density and the
temperature at the atmospheric pressure correspond to the experimental values1). On the other hand, for hydrogen,
the potential parameters were determined so that the critical point corresponded to the experimental value1). The
reason why the oxygen’s potential parameters are fitted to the atmospheric saturation data is that it is sufficiently in
a high-density/low-temperature region where the non ideal gas effect (the intermolecular interaction effect) plays an
important role to the thermodynamic properties. For hydrogen, we did not employ the same method as oxygen,
because the so called quantum effect plays an important role in such dens/low-temperature region and the present
classical MD method cannot express such state. Therefore, we fitted the critical point to the experimental value so
that the lower-density/higher-temperature region could be mainly reproduced. (In this process, we did not employ
the least square method to determine the potential parameters because it would diminish the physical background of
the defined parameters.)
The given values are shown in Table 1. By giving the values of the potential parameters, we can compare each
thermal property with the experimental values and can discuss the validity of the method employed here.
Table 1. The values of the potential parameters given in this study.
----------------------------------------------------------------------σ [×10-10m]
molecule
ε [×10-22J]
----------------------------------------------------------------------hydrogen
3.37
3.290
oxygen
16.6
3.369
-----------------------------------------------------------------------
III. Results and Discussions
Both for hydrogen and for oxygen, the pressure-volume-temperature relationship and of the liquid-vapor
equilibria are shown from Fig. 4 to Fig. 6. As shown in each figure, it was confirmed that the calculated thermal
properties could well reproduce the experimental values for oxygen at a wide density-temperature range and for
hydrogen in its lower density region than the critical point. It indicates that a monatomic approximation shows a
certain amount of validity for hydrogen and for oxygen if we select the values of the Lennard-Jones potential
parameters as described in the previous subsection (II. C.). For oxygen, the reason why the prediction of the
thermodynamic properties failed near its critical point (around 450kg/m3 and 160K in Fig. 5) is in the overestimate
of the critical temperature as mentioned in the previous subsection (II. B.), although the limitations of the LennardJones potential itself as a two-body force will be also one of the reasons10). On the other hand, for hydrogen, the
reason of the prediction error is in the limitation of Lennard-Jones potential and a quantum effect which plays an
important role in the low temperature regions in the case of light molecules.
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Temperature [K]
35
calculation
NIST
30
25
20
15
0
10 20 30 40 50 60 70 80 90
Density [kg/m3]
Fig.4 Comparison of a density-pressure curve (left) and saturation curve (right) between the present calculation and
experimental values (hydrogen).
Fig. 5. Comparison of a density-pressure curve between the present calculation and experimental values
(oxygen; T = 100K).
Fig. 6. Comparison of the saturation curve between the present calculation and experimental values (oxygen).
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Specific Heat [kJ/kg-K]
7.5
●
calculation
NIST
7
6.5
0
10
Density
20
30
[kg/m3]
(a) oxygen (T=100K)
(b) hydrogen (T=40K)
Fig. 7. Comparison of specific heat between the present calculation and experimental values.
Latent Heat [kJ/kg]
800
calculation
NIST
600
400
200
20
25
30
35
[kg/m3]
Density
(a) oxygen
(b) hydrogen
Fig. 8. Comparison of latent heat between the present calculation and experimental values.
Next, we compared our calculation result of specific heat and that of the latent heat with the experimental value1)
or the NIST data2). Here, specific heat cv (at constant volume) and latent heat L were calculated from the
thermodynamic relationships below,
cv =
f k B ∂u p
∂u
,
=
+
∂T v 2 m ∂T v
(7)
and
L = hv − hl = Psat (vv − vl ) + (u p , v − u p ,l ),
(8)
where f is the intramolecular kinetic freedom (f=3 in monatomic molecules; f=5 in diatomic molecules), and u and up
are the specific internal energy and specific potential energy, respectively. In Fig. 7, the calculated specific heat of
oxygen and that of hydrogen are compared with the experimental values (oxygen; T=100K) or NIST data (parahydrogen; T=40K). In Fig. 8, the comparisons of the latent heat in both fluids are made although the large deviation
in hydrogen was predicted in advance from the disagreement of liquid-vapor equilibria in Fig.4 (right side).
For oxygen, the calculated specific heat showed good correspondence to the experimental values1) if an
appropriate correction of the kinetic rotational freedom (f=5 in Eq. (7)) was taken into account. The calculated latent
heat also well matched the experimental values except around the critical point as same as liquid-vapor equilibria
(Fig. 5). These two properties are the function of the potential energy as shown in Eq. (7) and Eq. (8), and the reason
why such good correspondence was obtained is the Lennard-Jones (12-6) model with the potential parameters of
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Table 1 well mimics the actual intermolecular interaction energy in a coarsened method. It was confirmed that our
EOS can reproduce the thermal properties of oxygen except around the critical point.
For hydrogen, the calculated specific heat did not show a good correspondence with the NIST data2). On the
whole, our EOS overestimated the specific heat. Here, note that the rotational kinetic term can be completely
ignored at the temperatures below 60K because the rotational characteristic temperature of hydrogen is 85.4K.
Hence, f=3 in Eq. (7) was applied in the right schematic of Fig. 7. The reason of the overestimation is in the strong
dependency on temperature of our potential energy function. Further, the large deviation of the calculated latent heat
is also in the misleading of potential energy in the liquid phase region. In the high-density/low-temperature region,
the so called quantum effect cannot be neglected in the case of hydrogen, and the simple application of the classical
molecular dynamics does not give good estimations for the pressure-volume-temperature relationship and the
internal energy11). Ultimately, our EOS is applicable for pressure-volume-temperature relationship in the lower
density than its critical point while its application should be deliberated for specific heat and for all properties of
dense hydrogen.
Finally, we note the application range of the present EOS on the basis of the Lennard-Jones (12-6) potential. The
most applicable flow field in a liquid rocket engine is the subcritical or supercritical jet flow of oxidizer with
gaseous hydrogen. This is because the present EOS does well express the actual thermodynamic properties of
oxygen in both subcritical and supercritical regions (except around its critical point), and also expresses the
pressure-volume-temperature relationship of hydrogen in its gas-like densities. To understand such jet flows or the
atomization process of the oxidizer, EOS of the mixtures and the interfacial tension between dense oxygen and
gaseous hydrogen become important. Here, the present EOS on the basis of the Lennard-Jones model can be simply
extended and applied. On the other hand, for high pressure combustions in a rocket engine chamber, this EOS may
cause some failures due to the stronger expression of the intermolecular repulsive forces. In addition, cryogenic
cavitation of hydrogen around the rocket pump inducers cannot be either well expressed due to the misleading of
potential energy function, which is affected by the quantum effect. To extend the application range of the present
EOS, other intermolecular potentials should be examined with some improvements of the classical molecular
simulation methods.
Conclusion
In this paper, we examined a simple and reasonable method to evaluate thermal properties of hydrogen/oxygen
system starting from the molecular dynamics calculations; i.e., we constructed an Equation of State (EOS) using
Kataoka’s method, based on a monatomic approximation for hydrogen and for oxygen. In spite of the simple
application of the Lennard-Jones (12-6) potential with a truncation of a cut-off distance (3.5σ), our EOS successfully
predicted the actual thermal properties of oxygen except around critical point. Also, pressure-volume-temperature
relationship of hydrogen is well reproduced in a wide temperature range if the density is below its critical point. On
the other hand, it was also shown that our present EOS cannot accurately predict the thermal properties of dense
hydrogen, including the specific heat in its gas region. Although the further improvement of the potential energy
function with a reasonable handling of quantum effect of hydrogen is necessary, the present EOS on the basis of
Lennard-Jones model can express the thermodynamic properties of the subcritical or supercritical jet flows of
oxidizer with gaseous hydrogen, which is very important to predict the rocket engine combustion characteristics.
Acknowledgments
This study has been supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for
Scientific Research (B) (No. 19360092). F. A. Author thanks Mr. Nagashima and Prof. Hayashi (Aoyamagakuin
University) for their strong supports to this study, and also thanks Prof. Koshi (The University of Tokyo) for his
many valuable advices.
References
1
Japan Society of Mechanical Engineers, “Thermophysical properties of fluids”, Trans. JSME Data Book, 1983.
http://webbook.nist.gov/chemistry/
3
Nicolas, J. J. et al., “Equation of state for the Lennard-Jones fluid”, Mol. Phys., Vol. 37, 1979, pp. 1429-1454.
4
Ree, F. H., “Analytic representation of thermodynamic data for the Lennard-Jones fluid”, J. Chem. Phys., Vol. 73, 1980, pp.
5401-5403.
5
Johnson, J. K. et al., “The Lennard-Jones equation of state revisited”, Mol. Phys., Vol. 37, 1993, pp. 591-618.
6
Allen, M. P. and Tildesley, D. J. “Computer Simulation of Liquids,” Clarendon Press Oxford, 1986.
2
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7
Tokumasu, T., Ohara, T., Kamijo, K. “Effect of molecular elongation on the thermal conductivity of diatomic liquids,” J. Chem.
Phys., Vol. 118, 2003, pp. 3677-3+685.
8
Kataoka, Y. “Studies of liquid water by computer simulations,” J. Chem. Phys., Vol. 87, 1987, pp. 589-598.
9
Debenedetti, P. G., “Metastable Liquids,” Princeton Univ. Press, 1996.
10
Anta, A. B. J. A. et al., “Influence of three-body forces on the gas-liquid coexistence of simple fluids: The phase equilibrium of
argon”, Phys. Rev. E., Vol. 55, 1997, pp.2707-2712.
11
Yonetani, Y. and Kinugawa, K, “Centroid molecular dynamics approach to the transport properties of liquid para-hydrogen
over the wide temperature range”, J. Chem. Phys., Vol. 120, 2004, pp. 10624-10633.
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