A systematic investigation has been conducted to study the effects of ambient flow conditions (i.e. pressure and velocity) on supercritical droplet gasification in a forced-convective environment. The model is based on the time-dependent conservation equations in axisymmetric coordinates, and accommodates thermodynamic nonidealities and transport anomalies. In addition, an efficient scheme for evaluating thermophysical properties over the entire range of fluid thermodynamic states is established. The analysis allows a thorough examination of droplet behavior during its entire lifetime, including transient gasification, dynamic deformation, and shattering. A parametric study of droplet vaporization rate in terms of ambient pressure and Reynolds number is also conducted

Hsiao, Chia-Chun

Yang, Vigor

English

NASA Technical Reports Server

N95- 70887
LOX DROPLET VAPORIZATION IN A SUPERCRITICAL
FORCED CONVECTIVE ENVIRONMENT
Chia-chun Hsiao and Vigor Yang
Propulsion Engineering Research Center
Department of Mechanical Engineering
The Pennsylvania State University
University Park, PA 16802
SUMMARY:
A systematic investigation has been conducted to study the effects of ambient flow conditions (i.e.
pressure and velocity) on supercritical droplet gasification in a forced-convective environment. The model
is based on the time-dependent conservation equations in axisymmetric coordinates, and accommodates
thermodynamic nonidealities and transport anomalies. In addition, an efficient scheme for evaluating
thermophysical properties over the entire range of fluid thermodynamic states is established. The analysis
allows a thorough examination of droplet behavior during its entire lifetime, including transient gasification,
dynamic deformation, and shattering. A parametric study of droplet vaporisation rate in terms of ambient
pressure and Reynolds number is also conducted.
TECHNICAL DISCUSSION:
Many liquid-fueled combustion systems (such as diesel engines, ramjets, gas turbines and liquid
rockets) are designed to deliver fuel and oxidiser into a pressurized chamber through atomisers which break
the liquid fuel and form a large number of droplets. These droplets then undergo a series of vaporization,
breakup, and mixing process to form fuelloxidiser mixture for reaction. Modern combnstor designs even
extend the chamber operational condition into regimes well above the thermodynamic critical points of fuels
for better performance. Information regarding droplet trajectory, breakup criteria, and gasification rate in
a supercritical convective environments becomes crucial.
Although several studies have been conducted to investigate the characteristics of droplet
vaporization in supercritical conditions (Hsieh et al., 1991; Shuen el al., 1992; Yang el al., 1994), effects of
ambient convective flow on droplet behavior have not yet been addressed. The purpose of the present work
is to conduct a systematic investigation into supercritical droplet vaporization in a forced-convective stream.
The formulation starts with the time-dependent conservation equations of mass, momentum, energy, and
species concentration in axisymmetric coordinates for both droplet interior and ambient gases. Full account
is taken of thermodynamic non-ideality and transport anomaly during the droplet phase transition from
the subcriticai to supercritical state. In addition, a unified property evaluation scheme based on the BWR
equation of state and extended corresponding-state technique is established to predict fluid thermophysical
properties. The governing equations and associated boundary conditions are solved numerically using an
implicit finite-difference scheme with a dual time-stepping integration scheme.
In the first part of this study, a series of calculations has been carried out to study liquid oxygen
(LOX) droplet gasification in a hydrogen stream at different pressures and Reynolds numbers. Detailed
velocity and thermodynamic properties contours have been studied thoroughly. Secondly, a parametric
study is conducted to establish correlations for droplet lifetime, gasification rate, and drag coefficient as
functions of ambient pressure, temperature, droplet diameter and Reynolds number.
w
!
...
138
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w--r
- o
Governing Equation
The flow under consideration is laminar and _isymmetric. If body forces, viscous dissipation, mad
thermal radiation are ignored, the conservation laws can be written in the following form.
Mass:
_f//pdV+//pu, daj=o.
Momentum:
_///puidV+ffpu, ujdA_=/frijdAj.
(1)
(2)
Energy:
_///pedV+/fpeujdAj=//rijuidAj-/f(qr)jdAj. (3)
Species Concentration:
where
_ /ffpYtdV + //pYtujdAj =-/f(qM,s)jdAj. (4)
2
ro = -P_ij+ 2peij- ]t,(V •u)6ij.
Standardnotationsinfluidmechanicsand thermodynamicsareusedin(I)-(4).The specifictotalenergye
at a givenpressureisdefinedas
N IT Ui ui
e= EYz(h_"+ Jr C'"dT)-P-+-2-'
1=1 r,! P
where the index N represents the number of species considered, l_ the mass fraction of species I, and Trel
the reference temperature for enthalpy of formation. Fick's and Fourier's laws are used to approximate the
species and thermal diffusion in Eqs. (3) and (4), respectively.
Property Evaluation
An extended corresponding-state principle, which requires only critical constants and Pitser's
acentric factor for each component, is used to predict properties over the entire range of fluid l>-v-T states
for a mixture. The basic idea of the corresponding state model is relatively straightforward. It is assumed
that the configurational properties of a singie-phase mixture can be adequately represented by those of a
hypothetical pure fluid. The properties of this hypothetical pure fluid are then evaluated via corresponding
states with respect to a given reference fluid. The extended corresponding-state theory using shape factors
and mixture combining rules can satisfactorily predict the thermodynamic properties of mixtures from an
equation of state of the reference substance. Meanwhile, transport properties are expressed as a function of
density and temperature through conformal mapping to the reference fluid. For accurate property prediction,
itisdeslrabletohavean equationofsi;atethaia2curatelyrepresentsthep-V-Tbehaviorofa mixture.
Inthe presentstudyofmulticomponentdropletdynamicsatsupercriticalconditions,a generalised
BWR equationofstateisusedto predictboth gasand liquidphasebehaviorofthereferencefluid.
9 15
P = E a"(T)pn + E a"(T)P_n-lv e:'rp' (5)
n=l n=lO
139
where 7 is 0.04, and the functional coefficients an(T) vary with the reference fluid. This equation of state
must be solved iteratively to obtain the density of the reference fluid. The density of the mixture is then
obtain through an extended corresponding-state method.
p0[T/fm,0, p(hm,o/fm.o)] (6)
P = hm,o
Enthalpy, internal energy, entropy, fugacity, etc., are important thermodynamic properties, which
can be related to engine operating variables such as temperature and density. The heat capacity of a mixture
can be expressed as the sum of the ideal gas heat capacity at the same temperature and composition plus a
residual heat capacity (Leach, 1967)
C_ = C_ + AC_ (7)
where ACp is most conveniently determined by
/_" 2 T(Op/OT)_
Op
ACp= T ( p/OV)T R
Viscosity coefficient for a mixture fluid can be reproduced using a two-shape-factor corresponding-
state method with correction for mass and size difference.
p..,=(p, T, {z,=}, {m,=}) =/.=o(Po, To)F. + Ap eNsK°a (s)
where
[Mm '_ I/2¢I/2 h-213F.=V o) ..,o
Here Mm is the molecular weight of the mixture. A relatively simple correction taken from the Enskog hard
sphere theory is used with the predictions. The function corrects for deviations in mass mixing rules at low
densities, and for differences in molecular size.
Af_ENSKOG ENSKOG,t 3_ ENSKOG, if3 m _= {=.}, {m=})- ,, (9)
The Enskog model (1954) has been solved for a multicomponent mixture of hard spheres by Tham and
Gubbins (1971) and is used to calculate • ENSKOG and p_=NSKOGPmiz
Ely and Hanley proposed an expression for thermal conductivity which can be divided into three
contributions,
A = A'(p, T) + A"(T) + AAcr,,(p, T) (10)
The first term on the right-hand side is caused by the transfer of energy from purely collisional or translational
effects and can be calculated via the corresponding-state method. The second term in Eq. (10) is due to
the transfer of energy via internal degrees of freedom. This is independent of density and may be evaluated
by the modified Eucken correlation with an empirical mixing rule for polyatomic gases. Modern theory of
transport phenomena (Sengers,1971,1972) predicts an infinite thermal conductivity at the pure fluid critical
point and a large enhancement of thermal conductivity in the vicinity of the critical point. The third term
in Eq. (10) accounts for this phenomenon and is a function of temperature and density.
=
z
=
140
wv
RESULTS:
The theoretical model described in the preceding sections has been applied to LOX droplet
vaporization in a hydrogen stream at different pressures (100,200 and 400 atm) and Reynolds numbers
(60 - 250). A LOX droplet with initial temperature of 100 K and introductory diameter of 100 _rn is
injected into the 1000 K hydrogen stream. Part of the heat transferred from the gas phase goes into vaporizing
droplet, while the remainder goes into heating the droplet interior. Since the critical mixing temperature of
oxygen is 154.6 K at 1 atmosphere and decreases with increasing pressure, the droplet becomes supercritical
almost instantaneously upon introduction to the hydrogen gas at the supercritical pressure levels of interest
(100, 200 and 400 atm). Once this occurs, the enthalpy of vaporization and surface tension vanish, leaving
an essentially continuous medium with no abrupt phase change in the vaporization process. Although the
interior region of the droplet remains at the liquid state with a subcritical temperature distribution, the
medium attributes (p, p, T and thermophysical properties) vary continuously between the liquid core and
ambient gases with no distinct liquid surface as exists for a droplet in a subcritical environment. As a result,
a single-phase analysis is used to treat the droplet interior and ambient gases simultaneously as one fluid.
Since there is no discontinuity, the surface of the droplet is defined as the surface at which the temperature
attains the value of the critical mixing temperature.
Density contours with streamline patterns are shown in Fig. 1 for the 5 m/8 ambient flow case
considered here. A recirculating flow in the wake of the droplet occurs as a result of the shear force between
liquid oxygen and the surrounding gases. For the higher Reynolds number case (15 m/s), droplet shattering
is seen, as shown in Fig. 2. The convective flow penetrates through the liquid phase, breaking the droplet
into two parts - the core disk and the surrounding ring. The wake region, however, is convected away from
the droplet due to the diminishing shear flow between gas and liquid phases. In both of these supercritical
cases, no discernible flow circulation is found in the droplet interior, in contrast to the Hill's vortex found in
low-pressure cases.
Time variations of droplet residual mass at different pressures and velocities are shown in Figs. 3
and 4. When droplet surface reaches critical condition, no sharp boundary between gas and liquid phases
exists. The entire flow field becomes a continuous medium. Under this condition, droplet vaporisation is best
characterized by the droplet residual mass which is confined by a surface with critical mixing temperature.
Results show that high Reynolds number favors droplet evaporation. In a stronger convective flow (15 re s),
droplet tends to deform and as such increases contact surface with ambient flow, while the ambient stream
convects the gasified oxygen downstream and enhances both thermal and species diffusion. Figures 5 and
6 show the time variations of droplet moving velocity. The first derivative of these curves can be used to
calculate droplet acceleration. Drag force defined by Newton's second law can be obtained by multiplying
acceleration of center of gravity with its mass.
141
Figure I: Density Contours and Streamlines at t =
176ps, LOX/H_ System, D0 = 100#m, poo = 100 atm,
and Uoo = 5m/s.
LOX_I 2 System D O- 100 _.m
1.0 p. - 100 ,,tin
0.8
m/too 0.6 -=
0.4
0.2
O.C
0.00 0.05 0.10 0.15 020 0.25 0.30 0.35 0.40
Time, ms
Figure 4: Time Variations of Droplet Residual Mass
under Various Convective Velocities, LOX/H2 System,
Do = 100pro, and poo -- 100 atm.
Figure 2: Density Contours and Streamlines at t =
176ps, LOX/H_ System, Do = 100pro, poo = lOG arm,
and U_ = 15 m/s.
0.6 .... , .... , .... , .... , ....
LOX/_ System D O. 100 i_m
U.- 5 mPs
0.4
>_0.2
0.0
0.0 0.50.1 02 0.3 0.4
Time, ms
Figure 5: Time Variations of Velocity of Center of Grav-
ity for Oxygen at Various Pressures, LOX/H2 System,
Do = 100pro, and U¢o = 5 m/s.
LOX/I_ System O o . I00 _m
1.0 U.. Sm/s
0.8
mm'zo O.e
0.4
02
o.o
0.00 O.OS 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Time, ms
1.6
1.4
12
1.0
0.8
0.6
0.4
02
0.0
0.0
:tow .......... Ooioo,,m
p.- lo0 stm
lS
0.1 0_2 0.3 0.4 O.S
Time, ms
Figure 3: Time Variations of Droplets Residual Mass at
Various Pressures, LOX/H2 System, Do = 100#rn, and
Uoo = 5 m/s.
Figure 6: Time Variations of Velocity of Center of
Gravity for Oxygen under Various Convective Veloci-
ties, LOX/H2 System, Do = 100pm, and pcc = I00
arm.
142


Lox droplet vaporization in a supercritical forced convective environment

Modern liquid rocket engines often use liquid oxygen (LOX) and liquid hydrogen (LH2) as propellants to achieve high performance, with the engine operational conditions in the supercritical regimes of the propellants. Once the propellant exceeds its critical state, it essentially becomes a puff of dense fluid. The entire field becomes a continuous medium, and no distinct interfacial boundary between the liquid and gas exists. Although several studies have been undertaken to investigate the supercritical droplet behavior at quiescent conditions, very little effort has been made to address the fundamental mechanisms associated with LOX droplet vaporization in a supercritical, forced convective environment. The purpose is to establish a theoretical framework within which supercritical droplet dynamics and vaporization can be studied systematically by means of an efficient and robust numerical algorithm

N94 2 ')- C._
LOX DROPLET VAPORIZATION IN A SUPERCRITICAL
FORCED CONVECTIVE ENVIRONMENT
Chia-chun Hsiao and Vigor Yang
Propulsion Engineering Research Center
Department of Mechanical Engineering
The Pennsylvania State University
University Park, PA 16802
SUMMARY:
Modern liquid rocket engines often use liquid oxygen (LOX) and liquid hydrogen (LH2) as
propellants to achieve high performance, with the engine operational conditions in the supercriteal regimes
of the propellants. Once the propellant exceeds its critical state, it essentially becomes a puff of dense fluid.
The entire field becomes a continuous medium, and no distinct interfacial boundary between the liquid and
gas exists. Although several studies have been undertaken to investigate the supercritieal droplet behavior
at quiescent conditions, very little effort has been macle to address the fundamental mechanisms associated
with LOX droplet vaporization in a supercritical, forced convective environment. The purpo6e of this work
is to establish a theoretical framework within which supercritical droplet dynamics and vaporization can be
studied systematically by means of an efficient and robust numerical algorithm.
TECHNICAL DISCUSSION:
Several analyses based on the model of a spherical droplet in a quiescent environment were recently
developed to address the characteristics of supercritical droplet vaporization([1][2][3][4]). The model is
adequate only for small droplets which become equilibrium with the ambient flow. Contrarily, convective
effect is clearly observed for large droplets as a result of high momentum inertia of the liquid phase. Figure
1 depicts the situation examined here, showing an isolated vaporizing LOX droplet in a convective hydrogen
environment. The initial temperature of the droplet is subcritical, but the ambient temperature and pressure
are in the supercritical regime of oxygen. As a result of heat transfer from the surrounding gases, the droplet
surface temperature increases rapidly and reaches its critical mixing point. Under this condition, the sharp
distinction between gas and liquid phases disappears. The droplet regression can be best characterized by
tracing the location of the critical temperature.
Governing Equation
The analytical model is baaed on the complete time-dependent conservation equations of mass,
momentum, energy and species concentration for a multicomponent system with variable properties. If
body forces, viscous dissipation, radiation, and chemical reaction are ignored, these equation can be written
in the following vector form.
Mass:
Momentum:
(1)
(2)
Energy:
(3)
28
https://ntrs.nasa.gov/search.jsp?R=19940018561 2020-06-16T15:54:58+00:00Z
Species Concentration:
where
(4)
2
Standard notationsin fluidmechanics and thermodynamics are used in (I)-(4).The specifictotalenergy e
isdefinedas
1 fT UiUi
e = E Y_ JT Cp"dT- P +--
i=l ,e/' /0 2 '
where the index I represents the number of species considered, l_ the mass fraction of species 1, and Trey the
reference temperature for energy. Fick's and Fourier's laws are used to approximate the species and thermal
diffusion in (3) and (4), respectively.
Property Evaluation
Thermophysicai properties, including both thermodynamic and transport properties, play a decisive
role in determining droplet vaporization behavior. It is generally accepted that thermal conductivities and
heat capacities of pure fluids and fluid mixtures can be divided into three contributions, and correlated in
terms of density and temperature. For example, the thermal conductivity of fluid oxygen can be expressed
as [5]
A(p, T) = Ao(T) + AA,xc(p, T) + AAcri,(p, T). (5)
The first term on the right-hand side represents the value in the dilute-gas or sero-density limit, which is
independent of density and can be accurately predicted by kinetic-theory equations and an m-6-8 model
potential. The second is the excess thermal conductivity which, with the exclusion of unusual variations
near the critical point, characterizes the deviation from Ao for a dense fluid. The sum of Ao and AAezc is
sometimes called the "background" thermal conductivity. The third term refers to the critical enhancement
which accounts for the anomalous increase above the background thermal conductivity as the critical point
is approached. There are two separate regions where the critical enhancement of thermal properties is
important. The first region is located in the close vicinity of the critical point, and is the region where
a scaled equation of state must be used, as defined by Sengers et al[6]. For oxygen with Tc = 154.581 K
and Pc = 0.436 g/cm s, this region is bounded approximately by 150 _< T _< 160 K and 0.32 < p <_ 0.544
g/cm s. The second region, designated as the extended critical region, covers the p-T domain in which critical
enhancement is significant, but the fluid behavior can be described by an analytical equation of state such
as a modified Benedict-Webb-Rubin (BWR) type. This region can extend to quite high temperatures, up to
approximately 2To. The critical enhancement along various isotherms can be centered on a density, pc,,t,_,
which deviates slightly from the critical density and decreases with increasing temperature. More detail
description can be found in [4].
Numerical Algorithm
The model considered here is characterised by strong couplings among heat transfer, transient
diffusion, and phase transition, etc.. Because various time and length scales are involved in the entire process,
the numerical scheme for solving this serious stiffness problem becomes a challenge. Moreover, the singular
behavior of momentum balance in the limit of diffusion-dominant process adds further complications. In view
of these difficulties, a numerical algorithm capable of treating time-accurate low Mach-number compressible
-- 29
flow has been developed. The scheme is constructed in two steps. First, a rescaled pressure term is used in
the momentum equation to circumvent the pressure singularity at very low Much number.
u,t) = po+ t)
The second step is based on the establishment of a dual time-stepping integration procedure. With the
addition of a set of well-conditioned artificial time derivatives to the conservation laws, a new system of
governing equations is obtained whose converged solution in pseudo-time corresponds to a time-accurate
solution in the physical-time domain.
FaZ aQ O(E- E_)
+ -&- +
where r presents pseudo time, and Z : (p', u, v, h, Y/).
+ c3(F - F,_) _ S
coy
RESULTS:
As a first study of supercritical droplet behavior, A spherical droplet with an introductory
temperature of 300 K is first placed in a supercritical, forced convective environment(Too = 1000K). The
droplet Reynolds number based on the relative velocity is 60. The droplet is 100 pm in diameter and
contains 100% oxygen initially. As a result of large temperature gradient between gas and liquid phases,
surface temperature raises and reaches critical condition shortly. Due to the dimishment of surface tension,
droplet deformation and shattering can be clearly observed. Figure 2 shows the density contours, streamlines,
and velocity vectors at t = 0.02ms in a supercritical environment with H_ as the ambient gas. Figure 3
shows the oxygen concentration contours. Results indicate that the ambient gas mann diffusivity plays an
important role in determining the droplet behavior. A high mass diffusivity causes rapid penetration of the
ambient gas into the droplet, and consequently modifies the droplet dynamics (e.g., deformation, shattering,
etc.) caused by the density difference between the droplet and the gas.
REFERENCE:
[1] Litchford, R. J. and Jeng, S. M., "LOX Vaporisation in High-Pressure, Hydrogen-Rich Gas.", AIAA
Paper 90-2191, 1990.
[2] Delpanque, J. P. and Sirignano, W. A., "Transient Vaporization and Burning for an Oxygen Droplet
at Sub- and Near-Critical Conditions.", AIAA Paper 91-0075, 1991.
[3] Delpanque, J. P. and Sirignano, W. A., "Oscillatory and Transcritical Vaporisation of an Oxygen
Droplet.", AIAA Paper 92-0104, 1992.
[4] Yang, V., Lin, N. C., and Shuen, J. S., "Vaporisation of Liquid Oxygen (LOX) Droplets at
Supercritical Conditions", AIAA Paper 92-0103, presented at the AIAA 20th Aerospace Sciences
Meeting, January 1992.
[5] Roder, H. M., "The Thermal Conductivity of Oxygen", Journal of Research, National Bureau of
Standards, 87(4), 279.
[6] Sengers, J. V., Basu, R. S., and Levelt Sengers, "Representative Equations for the Thermodynamic
and Transport Properties of Fluids Near the Gas-Liqnid Critical Point.", NASA Contractor Report
3424.
30
STREAMLINES
ISOTHERMS
T-T$= 0.9
T=-Ts
Figure 1: Schematic of a vaporizing LOX droplet in hydrogen stream.
Figure 2: Density contours, streamlines and velocity vectors of LOX droplet vaporizing in a hydrogen
environment. Re = 60, t = 0.02ms.
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m E
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0
a uJ
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32


Lox droplet vaporization in a supercritical forced convective environment

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