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Assessment of real-gas effects in high-pressure gas injection at engine-relevant conditions

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A numerical framework for fully consistent treatment of single-phase, real-gas thermodynamics has been established in the commercial, pressure-based CFD software STAR-CD. In this framework, additional options for the Equation of State (EoS) are introduced, high-pressure corrections in thermodynamic functions via the departure formalism are applied, molecular transport properties are calculated with semi-empirical models and appropriate mixing rules are account for. With the established tool, highpressure injection of methane, hydrogen and nitrogen into quiescent air at total pressure ratios of 2:5 and 5 is simulated in a RANS framework. The results are first validated against predictions of a high-fidelity hybrid solver with real-gas thermodynamics implemented in OpenFOAM. The significance of each aspect of realgas behavior is then investigated by examining global jet metrics as well as the distribution of local flow quantities. It is shown that at minimal cost, real-gas density can drastically increase the accuracy of mass flow rates and mixture formation. Activation of realgas enthalpy results in further improvements, modest with regard to mixing but considerable with respect to the temperature fields; yet it is accompanied by a massive computational overhead. Real-gas transport properties are found to exert a minimal influence on jet evolution. It is observed that deviations from the fully-fledged real-gas model are not consistent in their direction for all injection fluids.
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ASSESSMENT OF REAL-GAS EFFECTS IN HIGH-PRESSURE
GAS INJECTION AT ENGINE-RELEVANT CONDITIONS
V.D.Sakellarakis1,M.Banholzer2,D.Llugaliu1,Y.M.Wright1,M.Pfitzner2, and
K.Boulouchos1
1Aerothermochemistry and Combustion Systems Laboratory, ETH Zurich, Switzerland
2Institute for Thermodynamics, Bundeswehr University Munich, Neubiberg, Germany
sakdavid@lav.mavt.ethz.ch
Abstract
A numerical framework for fully consistent treat-
ment of single-phase, real-gas thermodynamics has
been established in the commercial, pressure-based
CFD software STAR-CD. In this framework, addi-
tional options for the Equation of State (EoS) are
introduced, high-pressure corrections in thermody-
namic functions via the departure formalism are ap-
plied, molecular transport properties are calculated
with semi-empirical models and appropriate mixing
rules are account for. With the established tool, high-
pressure injection of methane, hydrogen and nitrogen
into quiescent air at total pressure ratios of 2.5and 5is
simulated in a RANS framework. The results are first
validated against predictions of a high-fidelity hybrid
solver with real-gas thermodynamics implemented in
OpenFOAM. The significance of each aspect of real-
gas behavior is then investigated by examining global
jet metrics as well as the distribution of local flow
quantities. It is shown that at minimal cost, real-gas
density can drastically increase the accuracy of mass
flow rates and mixture formation. Activation of real-
gas enthalpy results in further improvements, modest
with regard to mixing but considerable with respect
to the temperature fields; yet it is accompanied by a
massive computational overhead. Real-gas transport
properties are found to exert a minimal influence on
jet evolution. It is observed that deviations from the
fully-fledged real-gas model are not consistent in their
direction for all injection fluids.
1 Introduction
Internal combustion engines (ICEs) fueled by typ-
ical liquid fuels, such as gasoline, diesel and heavy
oil, have served as the mainstay of energy conversion
powerplants in the transportation sector for decades,
while also finding extensive use in electricity genera-
tion and as power sources for smaller appliances. Nev-
ertheless, concerns about long-term availability and
stability of conventional fossil fuel supplies in con-
junction with concerted efforts for global reduction
of carbon-dioxide emissions have sparked interest in
the introduction of cleaner and more sustainable al-
ternative fuels, such as natural gas and gaseous hy-
drogen. Whether operating in spark ignition (SI) or
compression-ignition (CI) mode, in-cylinder direct in-
jection (DI) of the gaseous fuel is an attractive option,
as it eliminates power losses associated with low vol-
umetric efficiency, helps abate incomplete or abnor-
mal combustion and offers additional flexibility with
respect to the injection strategy Verhelst and Wall-
ner (2009), Korakianitis, Namasivayam and Crookes
(2011).
In order to counteract the lower density of the gas,
high pressure ratios are required to deliver the nec-
essary amount of fuel and to accomplish sufficient
in-cylinder mixing rapidly. Under such conditions,
the flow is usually chocked and moderately or highly
under-expanded jets emerge at the injectors outlet.
An extensive review of the main experimental stud-
ies dealing with free under-expanded jets can be found
in Franquet et al (2015). The structure and evolu-
tion of these jets as well as the influence of the DI
process on subsequent mixture formation and com-
bustion in the engine has also been the focus of sev-
eral numerical studies, both in a RANS Keskinen et
al (2016), Baratta and Rapetto (2014)and an LES
framework Hamzehloo and Aleiferis (2014), Schmitt
et al (2015), Vuorinen et al (2013), while others have
aimed at developing methodologies for simplifying the
injection process itself in the context of engine simu-
lations M¨
uller et al (2013).
At conditions of high injection pressures and cold
temperatures induced by the strong expansion of the
fluid, real-gas effects become important and a handful
of numerical investigations of engine-relevant config-
urations have been carried out with real-gas models
Traxinger, Banholzer and Pfitzner (2018), Banholzer,
M¨
uller and Pfitzner (2017), adding to the existing
body of literature associated with cryogenic or super-
critical injection, mixing and combustion in rocket en-
gines, where the importance of real-gas effects has
long been recognized Oefelein (2006), M¨
uller et al
(2016), Banuti et al (2016). Several dedicated com-
parisons between ideal-gas and real-gas models have
highlighted inadequacies of the former Pohl et al
(2013), Bonelli, Viggiano and Magi (2013), Hempert
et al (2017), Yue, Hessel and Reitz (2017), but a sys-
tematic assessment of the relative importance of the
high-pressure corrections introduced by real-gas mod-
elling is lacking. The aim of this work is, therefore,
first, to establish a comprehensive numerical frame-
work for modeling real-gas effects in single phase flu-
ids, and second, to simulate with it different high-
pressure gas injection events, with a view to quantify-
ing the significance of each aspect of real-gas behavior
as well as the computational cost imposed by the need
to account for it.
2 Numerical Framework
On the modeling side, inclusion of real-gas physics
translates into the need to introduce corrections into
the Equation of State (EoS), the caloric and the trans-
port properties as well as the mixing rules for multi-
component mixtures. The core unit of the numeri-
cal framework is provided by STAR-CD, a fully com-
pressible, finite-volume flow-field solver. The func-
tionalities of the solver were first expanded by inter-
facing it to Chemkin Real Gas Schmitt, R. G., But-
ler, P. B., and French, N. B. (1994), an extension
of Chemkin-II Kee, Rupley and Miller (1989). In
this way, on the one hand, apart from the ideal-gas
law (IGA) all the standard cubic EoS become available
Van der Waals (VdW), Redlich-Kwong (RK), Soave-
Redlich-Kwong (SRK), Peng-Robinson (PR)and, on
the other hand, thermodynamic properties such as en-
thalpy and heat capacities can be corrected for pres-
sure dependence via the departure formalism in ac-
cordance with the particular EoS chosen. In a second
step, Chung’s dense multi-component fluid models for
molecular viscosity and thermal conductivity Chung
et al (1988)were implemented in the code to account
for proper treatment of molecular transport properties.
Table 1: Test cases and operating conditions
Fuel/Ox. Pfuel /Pox ΠTf uel/Tox Case
CH4/Air 500/200 bar 2.5 294/294K CH4-p500200
CH4/Air 500/100 bar 5.0 294/294K CH4-p500100
H2/Air 500/200 bar 2.5 294/294K H2-p500200
H2/Air 500/100 bar 5.0 294/294K H2-p500100
N2/Air 500/200 bar 2.5 294/294K N2-p500200
N2/Air 500/100 bar 5.0 294/294K N2-p500100
Having established the numerical framework, a
subset of the operating conditions investigated nu-
merically by Banholzer et al (2018) was revisited,
as shown in Table 1. These correspond to highly
compressible flows extending from the subsonic to
the moderately under-expanded regime. Banholzer
employed a semi-implicit, hybrid PISO/Kurganov-
Tadmor scheme implemented in OpenFOAM (Kra-
poshin, Bovtrikova and Strijhak (2015)), which has
been extended to model real-gas thermodynamics with
and without phase separation. His simulations (Refer-
ence) are used as benchmarks for validating the newly
established framework in STAR-CD (Real Gas). The
latter utilizes the fully-implicit PISO algorithm Issa
et al (1991). Aside from the choice of solver and the
differencing schemes, the same computational mesh,
the same turbulence model (k-ωSST), the same EoS
(SRK) and the same models for transport properties
were employed, thus ensuring consistency between the
two codes to the extent that this is possible. In addition
to that, simulations based on ideal gas models were
conducted with STAR-CD (Ideal Gas) in order to dis-
entangle the impact of real-gas behavior from poten-
tial discrepancies owed to the application of different
numerics by the two solvers.
(a)
(b)
(c)
Figure 1: Normalized Jet Tip Penetrations as a Function of
Time
Although the deviation between ideal-gas and real-
gas based modelling is rather small, inclusion of real-
gas effects improves predictions of jet tip penetration
(Fig.1) for all injected fluids and at both pressure ra-
tios, leading to an excellent agreement with the refer-
ence simulations nearly throughout the whole injec-
tion event. More detailed investigation of the local
mixing state in the cases of methane injection at 3msec
after the start of injection, at which point the jet has
long reached a steady state, reveals differences of the
same order between between Real Gas and Ideal Gas
in mixture fraction (Fig.2 - first and third row) and a
clear improvement in temperature distribution in the
flow-field (Fig.2 - second and fourth row) with the
real-gas model, which matches predictions of the ref-
erence simulations very accurately. The same obser-
vations hold for hydrogen and nitrogen injections as
well, although real-gas effects are less pronounced in
those cases (and therefore not shown here).
The jet tip penetration, which alongside the numer-
ical aspects of the turbulence model and the disrceti-
zation scheme determines the mixing field, has been
found in previous studies to scale with momentum at
the nozzle outlet, which in turn depends on the density
and through velocity on the Mach number. Besides,
local temperature depends on enthalpy, for which a
transport equation is solved, through a two-way cou-
pling and on the local composition through the mix-
ing field. Given that the newly-established framework
shows very good agreement with the extensively vali-
dated hybrid OpenFOAM solver, it can be concluded
that it is able to reliably capture all physical quantities
influenced by real-gas effects for flows without phase
separation.
3 Assessment of Real-Gas Effects
To quantify the relative contribution of the vari-
ous aspects of real-gas behavior, the operating con-
ditions presented in Table 1 were simulated with dif-
ferent combinations of high-pressure corrections, pro-
gressively evolving from the fully ideal gas (M01)
to the fully real-gas model (M04) with two interme-
diate steps, as shown in Table 2. In here, obtain-
ing ideal-gas molecular transport properties refers to
employing built-in polynomial (in terms of tempera-
ture) approximations that match kinetic theory’s pre-
dictions at a considerable speed-up. On the other
hand, ideal-gas thermodynamic properties werel ob-
tained via Chemkin Real Gas rather than built-in look-
up tables. Besides, instead of the faster Upwind Dif-
ferencing scheme employed previously, the MARS
scheme with a blending factor of 0.75 and combined
with variable time-stepping for CFLmax 1.5has
been employed in this set of simulations, as it was
observed to be better suited at capturing shocks and
discontinuities in the near-nozzle area.
Results are first compared in terms of the global
metrics of jet tip penetration, jet volume, injected fluid
Table 2: Modelling Approaches
Model Equation of State Enthalpy Transport Properties
M01 IGA IGA IGA
M02 SRK IGA IGA
M03 SRK SRK IGA
M04 SRK SRK Chung
mass in the main chamber at 3msec after the start of
the injection event. Deviations from the fully real-gas
model M04 are summarized in Tables 3, 4 and 5. As
a consequence of the MARS scheme employing semi-
explicit flux correctors and therefore necessitating ad-
herence to the CFL criterion, due to the massive in-
crease in turn-around time brought about by the emer-
gence of very high velocities in the cases of hydro-
gen injection and because M03 and M04 were found
to consistently generate identical results, no sub-cases
with M04 were run for H2; M03 was used as the ref-
erence instead.
Table 3: CH4 - Deviations from M04
Operating Condition Π2.5 Π5
Metric M01 M02 M03 M01 M02 M03
Jet Penetration [%] -2.88 +0.29 +0.00 -1.42 +1.44 +0.00
Jet Volume [%] -12.6 +1.55 -0.02 -7.75 +2.88 +0.03
Injected Mass [%] -13.4 +2.15 -0.02 -14.0 +3.77 -0.01
CPU cost [%] -75.3 -59.3 -0.00 -73.8 -70.8 -22.6
Table 4: H2 - Deviations from M03
Operating Condition Π2.5 Π5
Metric M01 M02 M01 M02
Jet Penetration [%] -1.16 +0.59 +0.00 +0.29
Jet Volume [%] +3.89 +4.40 +8.96 +3.54
Injected Mass [%] +6.01 +2.87 +6.16 +2.78
CPU cost [%] -80.6 -75.9 -79.6 -66.7
Table 5: N2 - Deviations from M04
Operating Condition Π2.5 Π5
Metric M01 M02 M03 M01 M02 M03
Jet Penetration [%] -2.60 +0.00 +0.00 -1.14 +0.00 +0.00
Jet Volume [%] -7.90 +0.05 +0.00 -3.25 +0.22 +0.01
Injected Mass [%] -1.39 +0.21 +0.00 -1.23 +0.13 +0.01
CPU cost [%] -75.4 -70.1 +2.94 -53.9 -58.0 -6.51
The ideal-gas model (M01) always under-
estimates tip penetration by a small margin that
decreases with increasing pressure ratio. However,
this does not necessarily translate into smaller jet vol-
ume, as in the case of hydrogen a smaller jet is formed
when real-gas effects are accounted for. Similarly,
although the mass flow rate is notably under-estimated
by M01 for methane and to a lesser extend for
nitrogen at both pressure ratios, it is over-estimated
for hydrogen. Regardless, the deviations in injected
fuel mass are large enough to render ideal-gas based
modelling problematic. Activating real-gas density
(M02) leads to further improvements in capturing jet
penetration and jet volume and, more importantly, to
drastic improvement in predictions of mass flow rate,
at a small overhead of the order of 5to 15%.
Figure 2: Mixture fraction and Temperature distribution at 3msec for Cases CH4-p500200 and CH4-p500100
Figure 3: Mach Number and Temperature distribution at 3msec for Cases CH4-p500100 and N2-p500100
Activating real-gas enthalpy on top of that (M03)
leads to near-perfect agreement with the fully-real gas
model, which shows that the high pressure modifica-
tions for molecular transport processes exert a negli-
gible influence on global parameters. As can be ex-
pected from what was previously seen in Fig.2, differ-
ences in the mixture fraction field between models are
overall small and, with the unique exception of hydro-
gen, predictions follow the same trends as jet tip pen-
etration, meaning that a model predicting a longer jet
is at the same time predicting a larger radial spread for
the jet as well, in accordance with what the jet volume
metric suggests.
It still remains open to question, if the high-
pressure modifications for transport properties do af-
fect local jet structure and flow variables, as has
been sometimes reported in literature Hempert et al
(2017), and whether the activation of real-gas en-
thalpy offers any advantage that confidently justifies
the massive computational overhead associated with it
(60%). For this reason, the distribution of Mach
number and temperature at different cross-sections in
the domain are investigated for methane and nitrogen
injection at a pressure ratio of 5, as illustrated in Fig.3.
The former clearly reflects the “barel” structure in the
near-nozzle area, where a succession of Mach disks
forms. M01 predicts the strongest shocks of all mod-
els, however this difference in intensity does not carry
over to the height of the potential core of the jet, which
is identical for all models. Past that point, the axial
decay of the Mach number as well as its radial evolu-
tion in the domain also behave very similarly for all
models. In terms of temperature, however, M02 fol-
lows closely M01, whereas the M03 curves overlap
with M04 predictions, which always lag M01 by up to
a couple of tens of degrees, depending on the inten-
sity of real-gas effects, not just along the centerline,
but more importantly across the jet periphery, where
scalar dissipation rates are low enough to minimize
loss of heat and radicals. If reactive conditions were
to be established in the constant volume chamber, this
would have important repercussions from the perspec-
tive of ignition delay and ignition location, due to the
exponential dependence of kinetic rates on tempera-
ture, and this is the primary benefit from employing
real-gas enthalpy, alongside the modest gains in accu-
racy of calculation of the injection flow rate (3%).
Concerning M04, it is noted that activation of real-
gas adjustments for the molecular transport properties
does not appear to impact either temperature distribu-
tion in the chamber or shock strength and location in
the near-nozzle area, which suggests that for reasons
of expediency they can be dispensed with in the RANS
framework.
To interpret this, a term by term analysis of the en-
ergy equation budget was performed at several cross-
sections in the domain for case CH4-p500100 at
3msec. Results along the jet centerline and across a
representative axial station (z/Dnozzle = 60) are pre-
sented in Fig.4. The energy equation is cast in the form
of total enthalpy, including both the sensible and the
chemical part, consistent with the form of the equa-
tion that that is being solved for. The temporal term
and the temporal pressure change are neglected, as the
jet has reached steady-state at this point. The conduc-
tive heat flux is found to be 4-5 orders of magnitude
smaller than the leading terms of convection, turbu-
lent heat flux and rate of work by the stresses, which
explains why large differences in calculated values of
thermal conductivity may still lead to identical tem-
perature fields. Similar conclusions are expected for
the role of molecular viscosity in the momentum equa-
tions. The same may not necessarily apply in an LES
context, however.
(a)
(b)
Figure 4: Energy Equation Budget at 3msec for Case CH4-
p500100
4 Conclusions
A numerical framework with consistent real-gas
thermodynamics for single phase flows has been suc-
cessfully established. With it high pressure injections
of methane, nitrogen and hydrogen into quiescent air
in a constant volume chamber at weakly and moder-
ately under-expanded were investigated. It was found
that the most important high-pressure adjustment is re-
quired for the EoS, which predominantly affects the
mass flow rate and the emerging mixing field. The sec-
ond most important correction is required for enthalpy,
which leads to small improvements in the mixing field,
modest improvements in the injection flow rate and
jet structure and important improvements in temper-
ature distribution. Corrections for transport properties
have a negligible effect on all metrics considered in
this study due to the fact that in RANS they feature in
terms of relatively small order of magnitude.
Acknowledgments
Funding by the Research Association for Com-
bustion Engines eV (FVV, Frankfurt, Germany, nfvv-
net.de), by the Swiss Federal Office of Energy (grant
numbers SI/501584-01 and SI/501020-01) and by the
Swiss Competence Center for Energy Research (SC-
CER Mobility) is gratefully acknowledged. The au-
thors would also like to express their gratitude to Dr B.
Butler, President of Embry Riddle Aeronautical Uni-
versity, for providing them with Chemkin Real Gas.
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... It is evident that very large under-predictions of the injection rate and the jet momentum can be expected with ideal-gas modelling, which would carry over the error to the entire mixing process in the chamber. The impact of each of these real-gas corrections on both global (mass flow rate, jet tip penetration, jet volume) and local (temperature, mixture fraction) jet metrics was investigated in [18] and the main findings are summarized in Table 4. Calculations carried out with the ideal gas law (IG) and with Soave-Redlich-Kwong (SRK) as equation of state are compared against data from NIST. ...
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