Investigation on Deformation of an Evaporating Droplet in

Hindawi Publishing Corporation
Advances in Mechanical Engineering
Volume 2014, Article ID 326059, 12 pages
http://dx.doi.org/10.1155/2014/326059
Research Article
Investigation on Deformation of an Evaporating Droplet in
Convective Transcritical Environments
Shichun Yang, Yanfei Gao, Yaoguang Cao, Yue Gu, and Zhuoran Gong
School of Transportation Science and Engineering, Beihang University, Beijing 100191, China
Correspondence should be addressed to Yanfei Gao; [email protected]
Received 18 October 2013; Revised 4 March 2014; Accepted 4 March 2014; Published 22 April 2014
Academic Editor: Junwu Wang
Copyright © 2014 Shichun Yang et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An investigation of the dynamic deformation for an individual droplet under convective transcritical conditions is conducted with
a developed deformation model. This model takes account of gas solubility in liquid phase, variable thermophysical properties, the
effective conductivity accounting for convective heat and mass transfer and liquid phase internal circulation, and the disappearance
of surface tension at the critical mixing point. The results show that (1) at higher ambient pressures, surface tension decreases more
smoothly with the reduction of fuel mole fraction but more rapidly with the droplet surface temperature; (2) as time elapses, the drop
is continuously oscillating, together with the decreasing oscillation amplitude; (3) at high ambient pressures and temperatures the
maximum drop deformation occurs in the late stage of the evaporation process, while at low pressures and temperatures, it occurs
in the initial stage. In addition, the drop deformation and Weber number are initially quite insensitive to the ambient temperature.
1. Introduction
In the power systems such as aircraft jet engines, diesel
engines, and direct injection gasoline (GDI) engines, the
injected fuel enters the combustion chamber as a spray
of droplets, which then undergo a sequence of physical
processes, including deformation, breakup, evaporation, and
mixing with air. In the fuel-air mixing process, the drop
deformation induced by the aerodynamic force caused by
the relative motion between droplet and gas has a significant
effect on the drop evaporation and breakup yielding increased
surface area, which control the mixture preparation and thus
have a strong influence on combustion, performance, and
fuel consumption of engines. Generally, the temperatures
and pressures in the combustion chambers can reach nearor supercritical values of the fuels. Therefore, it is required
to fully understand the deformation characteristics of an
evaporating droplet at transcritical and convective conditions
[1, 2].
Several investigations on fuel injection and air-fuel mixing under subcritical or supercritical conditions have been
presented in published literature. As the chamber pressures
approach the critical pressures of the fuels, the atomization
and fuel-air mixture formation exhibit a great sensitivity to
pressures, temperatures, and local mixture concentrations.
Tavlarides and Anitescu [3] found that injecting fuels near
the supercritical region or supercritical fuel mixtures into the
engine cylinders has been suggested as one way to increase
engine efficiency and decrease emissions. Preheating the
diesel upon injection provides better fuel-air mixing when
this process is operated at the temperatures near the critical
region. Segal and Polikhov [4] investigated the effects of the
ambient pressure and temperature on the jet breakup. At
subcritical conditions, the gas inertia and surface tension
forces play dominant roles in the formation of ligaments
from which the materials break off and the drops form later,
but, at transcritical conditions, reduced surface tension has a
significant impact on the jet surface behavior. According to
Polikhov and Segal [5], in the transcritical and supercritical
regimes, the diffusion rate of oxygen into acetone is two
orders of magnitude faster than that at low ambient pressures. Yin and Lu [6] studied the influence of the injection
temperature on the flow evolution. It was concluded that, for
supercritical injection temperatures, the jet surface is more
unstable when the instability waves grow up, but, for subcritical injection temperatures, the jet surface is nearly straight
with the strong density stratification which suppresses the
instability wave development. Rachedi et al. [7] compared the
2
behaviors of a hydrocarbon and CO2 supercritical fluid jets.
They found that their behaviors are similar in most respects:
the jet swirl number has a significant effect on cone angle,
while the density ratio has a minor influence on it. Kim et
al. [8] conducted a computational analysis of four cryogen
nitrogen jets at near-critical and supercritical pressures with
their CFD code incorporated with two real fluid equations
of state and dense-fluid correction schemes. Roy et al. [9]
found that when a supercritical jet is injected into a subcritical
environment at relatively low chamber temperatures, the
droplet deformation occurs downstream beyond a 10 jet
diameters distance from the nozzle.
However, although Weckenmann [10] and Oldenhof et al.
[11] investigated free falling droplets experimentally at nearcritical conditions, which were injected in a heated and pressurized nitrogen atmosphere, most of the existing research
down to the level of the fundamental element of spray (i.e.,
an isolated droplet) concentrates on the droplet evaporation [12–14], and little work has been done on the droplet
deformation and breakup mechanisms at transcritical and
convective conditions. Several numerical methods, such as
the TAB and DDB models, and experimental investigations
have been presented in the published literature, involving
drop breakup in a high speed air stream, isolated droplet
deformation, droplet dynamics, and so forth, which are
studied for atmospheric conditions far below the critical
points of the fuels [15–17]. The TAB model is widely used
to describe the secondary breakup of hollow-cone gasoline
sprays in the case of low Weber number and the vibrational
regime (slower droplets, usually at the periphery of the spray)
[18–20]. In addition, in the case of high Weber number,
the droplets undergo high distortion in many applications
such as in diesel sprays and the drag coefficient changes as
a droplet departs from the spherical shape. The distortion
of a droplet can be calculated from the TAB model to
account for this [21]. Shim et al. utilized a modified TAB
model in comparison with experimental results to investigate
the atomization characteristics of high pressure swirl spray
[22]. Consequently, the TAB model was employed as part
of the numerical model in this work concerning the drop
deformation.
The main idea of this study is to develop a dynamic
deformation model for an isolated droplet in transcritical and
convective environments and to obtain a deeper understanding of the deformation and breakup physics under extreme
conditions. This model considers the behavior of real gases
and high pressure effects, including solubility of ambient gas
in the liquid phase and variable thermophysical properties
for both liquid and gas phases as functions of pressure,
composition, and temperature. It also takes account of the
disappearance of surface tension at the critical mixing point
and the spatial gradients of thermophysical properties inside
the liquid phase domain. In addition, the effective conductivity is used to account for heat and mass transfer in the gaseous
boundary layer around the droplet, which also considers
liquid phase internal circulation. The influences of ambient
conditions on surface tension and droplet deformation were
investigated with the developed model.
Advances in Mechanical Engineering
x
Evaporation (−dm/dt)
urel
Convection
Droplet
rs
Nitrogen dissolution
Effective conductivity
Gaseous film region
Figure 1: Schematic diagram of the droplet deformation and
evaporation model.
2. Mathematical Model
A schematic description of the developed numerical model
for the deformation of an evaporating droplet is shown in
Figure 1. In the figure, 𝑟𝑠 is droplet radius, 𝑢rel represents the
relative velocity between droplet and surrounding gas, 𝑥 is
the displacement of the droplet’s equator from its equilibrium
position, 𝑚 represents droplet mass, and 𝑡 is time. During
the deformation process, the droplet has a shape of an oblate
spheroid with ellipsoidal cross section, while it is assumed
to be spherically symmetric, having the same volume, during
evaporation. Pressure remains independent of time and space
due to low flow velocity in comparison with the acoustic
wave, and the radiation is negligible.
The physical problem described here is the deformation
of an isolated evaporating droplet which is initially at a
subcritical state and suddenly introduced into a warmer convective gaseous environment with its thermodynamics state
approaching the critical point. In the vicinity of the critical
point, the thermophysical properties of both gases and liquids
usually exhibit anomalous variations and are extremely sensitive to pressure and temperature, yielding a phenomenon
commonly referred to as near-critical enhancement. The
fluid characteristics are similar to those of both liquids and
gases, such as gas-like properties but liquid-like densities
[4]. As one is approaching the critical point, the densities
of both liquid and gas phases become approximately equal.
The molecule net attraction in the interface region toward the
dense phase is close to zero, namely, the metastable surface
[23]. In particular, the surface tension reaches zero and
droplets can no longer form at the critical point. In addition,
many phenomena coexist near-critical point, including rapid
decrease of vaporization enthalpy, thermophysical property
singularities, and strengthened dissolution of gas phase in
liquid phase. Due to the latter factor, the critical point of the
mixture can change dynamically depending upon the mixture properties, potentially influencing the interface between
liquid and gas phases. In addition to significant decrease
of liquid-gas density ratios, weakened surface tension at
Advances in Mechanical Engineering
3
the liquid interface can lead to further enhanced aerodynamic
interactions induced by the relative velocity between droplet
and surrounding gas. All the above factors can significantly
influence the drop deformation, breakup, and evaporation.
As a specific example, the deformation and breakup of an
isolated acetone droplet in a transcritical and convective
nitrogen environment were studied. Acetone is used as the
working fluid on account of existing data, which has a critical
temperature 𝑇𝑐 of 508 K and a critical pressure 𝑃𝑐 of 4.8 MPa.
2.1. Governing Equations. This model considers real fluid
behavior of both liquid and gas phases, the temporal variation
and nonuniform distribution of thermophysical properties,
high pressure effects, and solubility of inert species into
the liquid phase, where radiation heat transfer is neglected.
Ambient gas is assumed to be dissolved only in a very thin
layer of the gas-liquid interface and thus diffusion of gas into
the liquid interior is ignored. The effective conductivity is
used to deal with convective heat and mass transfer in the
gaseous boundary layer around the droplet, based on effective
film method, which takes account of liquid phase internal
circulation [24].
Energy conservation inside the droplet:
𝐶𝑝
𝜕𝑇
𝜕𝑇
1 𝜕
−
(𝑟2 𝜆 ) = 0.
𝜕𝑡 𝜌𝑟2 𝜕𝑟
𝜕𝑟
(1)
of ambient gas divided by the total mass rate of the liquid
component at droplet surface can be expressed as follows:
𝜌𝑔,sf ⋅ (1 − 𝑌F,sf ) ⋅ 𝑑𝑉𝑠 /𝑑𝑡
𝑊N 󵄨󵄨󵄨󵄨
.
󵄨󵄨 = −
𝑉
󵄨
𝑊F 󵄨sf
𝑑 (∫ 𝑠 𝜌 𝑑𝑉) /𝑑𝑡
The corrected mass and heat transfer coefficient in (3) and (4)
are calculated by
ℎ𝑔∗ = ℎ𝑔 ⋅
3.02514 × 10−3
{
{
,
2.685
+
{
{
(1.9868 × 10−3 + 𝐷FN 𝑡/𝑟𝑠2 )
{
{
{
{
𝐷 𝑡
𝑘𝑒 = 𝜆 𝑙 ⋅ {
< 1,
0.0004 ≤ FN
{
𝑟𝑠2
{
{
{
{
𝐷FN 𝑡
{2.685,
{
.
𝑟𝑠2 ≥ 1
{
(𝐻𝑔,sf − 𝐻𝑙,sf ) 𝑑𝑚
𝜕𝑇 󵄨󵄨󵄨󵄨
+ [ℎ𝑔∗ (𝑇∞ − 𝑇sf )] .
󵄨󵄨 =
𝜕𝑟 󵄨󵄨sf
4𝜋 ⋅ 𝑟𝑠2
𝑑𝑡
Nu = 2 +
𝑌F,sf − 𝑌F,∞
𝑑𝑚
∗
⋅
.
= 4𝜋 ⋅ 𝑟𝑠2 ⋅ 𝐾FN
𝑊F = −
󵄨
𝑑𝑡
1 − (1 + 𝑊N /𝑊F 󵄨󵄨󵄨sf ) 𝑌F,sf
𝜆 𝑔 Nu
2𝑟𝑠
,
0.555Re1/2 Pr1/3
√1 + 1.232/RePr4/3
2𝜌𝑔 𝑢rel 𝑟𝑠
Re =
𝜇𝑔
(3)
Pr =
Species conservation:
(7)
Heat transfer coefficient ℎ𝑔 for the gas mixture is expressed as
Liquid-gas phase energy balance:
𝑘𝑒 ⋅
(6)
and the effective thermal conductivity 𝑘𝑒 is estimated according to Jin and Borman [24]:
ℎ𝑔 =
(2)
𝑍𝑇
,
𝑒𝑍𝑇 − 1
𝑍
= 𝐾FN ⋅ 𝑍 𝑚 ,
𝑒 𝑚 −1
∗
𝐾FN
Continuity equation for liquid phase:
𝜕𝜌 1 𝜕 2
+
(𝑟 𝜌𝑢) = 0.
𝜕𝑡 𝑟2 𝜕𝑟
(5)
0
Cp𝑔 𝜇𝑔
𝜆𝑔
,
(8)
,
.
Mass transfer coefficient 𝐾FN for the gas mixture is calculated
by the following equations:
(4)
In the above expressions, 𝑇 stands for temperature, 𝑟
stands for the radial distance relative to droplet center, 𝑢
is the radial velocity inside the droplet relative to droplet
center, 𝜌 represents density, 𝐶𝑝 is the constant pressure heat
capacity, 𝜆 indicates thermal conductivity, 𝑘𝑒 indicates the
effective thermal conductivity, 𝐻 represents enthalpy, ℎ𝑔∗ and
∗
𝐾FN
are corrected heat transfer coefficient of gas mixture and
corrected mass transfer coefficient, 𝑊 is mass flow rate, 𝑌
is mass fraction in gas phase, subscripts F and N indicate
fuel and nitrogen components, subscripts 𝑔 and 𝑙 are gas
and liquid phases, and subscripts ∞ and sf represent infinite
boundary and drop surface, respectively.
Nitrogen dissolution in the liquid phase is assumed to
be negligible in the droplet mass calculation. The mass rate
𝐾FN =
Sh = 2 +
𝜌𝑔 𝐷FN Sh
2𝑟2
,
0.555Re1/2 Sc1/3
√1 + 1.232/ReSc4/3
Sc =
𝜇𝑔
(𝜌𝑔 𝐷FN )
(9)
,
(10)
.
(11)
Parameters 𝑍𝑚 and 𝑍𝑇 are determined by
𝑍𝑚 = ( 1 +
𝑍𝑇 =
𝑊N 󵄨󵄨󵄨󵄨
𝑊𝐹
,
󵄨 )⋅
𝑊F 󵄨󵄨󵄨sf 2𝜋 ⋅ 𝑟𝑠 ⋅ Sh ⋅ 𝜌𝑔 ⋅ 𝐷FN
󵄨
𝑊F (CpF,𝑔 + 𝑊N /𝑊F 󵄨󵄨󵄨sf ⋅ CpN,𝑔 )
4𝜋 ⋅ 𝑟𝑠2 ℎ𝑔
,
(12)
(13)
4
Advances in Mechanical Engineering
where 𝑉𝑠 represents droplet volume, 𝑉 represents volume,
ℎ is heat transfer coefficient, 𝐾FN indicates mass transfer
coefficient, 𝑍𝑇 and 𝑍𝑚 are correction factors of heat transfer
for high transfer rates, 𝐷FN is mass diffusivity, Nu indicates
Nusselt number, Re indicates Reynolds number, Pr represents
Prandtl number, 𝜇 represents viscosity, Sh is Sherwood
number, and Sc stands for Schmidt number.
As expressed by (12), the evaporation rate 𝑊F (𝑊F =
−𝑑𝑚/𝑑𝑡) is proportional to Sherwood number Sh, and
according to (10) Sh increases with Reynolds number Re.
Consequently, it is clear that the droplet evaporation is
enhanced as speed 𝑢rel of the drop increases, because Re is
directly proportional to 𝑢rel .
where 𝑇𝑐,mix is temperature of the mixture at the critical point
and Constant 𝐾 is taken as 57.193 to match experimental
results [26] with theory.
At the critical point, both liquid phase and vapor-phase
mass fractions of the fuel component at drop surface are
equal. Surface tension and enthalpy of vaporization decrease
to zero and the droplet surface disappears. For a binary
mixture composed of pure fuel and nitrogen, the critical
mixing point can be determined by [27]
(
(
Mass conservation of droplet:
4𝜋𝑟𝑠3 𝑑𝜌𝑙
𝑑𝑟
𝑑𝑚
= 4𝜋 ⋅ 𝑟𝑠2 ⋅ 𝜌𝑙 𝑠 +
⋅
.
𝑑𝑡
𝑑𝑡
3
𝑑𝑡
Droplet deformation.
The TAB model is widely adopted to predict the secondary breakup and drop deformation of hollow-cone gasoline sprays at low Weber numbers and those of the periphery
of engine sprays (i.e., low-speed droplets) [18–20]. Additionally, the TAB model can also be used to describe the droplet
deformation and drag coefficient [21] in the case of high
Weber number. Thus, the TAB model [25] is adopted in the
present investigation:
(16)
Model constant 𝐶𝑏 is 0.5 and 𝐶𝑑 is 5.0. Constants 𝐶F and
𝐶𝑘 will be given shortly in Section 3.1. In Figure 1, 𝑥 is the
displacement of the mass from the idle state and 𝑦 (𝑦 =
𝑥/𝐶𝑏 /𝑟𝑠 ) is the dimensionless droplet deformation. If (16) is
solved for the droplet, breakup is allowed to occur for 𝑦 > 1
[25]. The liquid-gas relative velocity 𝑢rel is calculated by [15]
𝜌𝑙 𝑉𝑠
2
𝜌𝑔 𝑢rel
𝑑𝑢rel
= 𝐶𝐷𝐴 𝑓
,
𝑑𝑡
2
(17)
where the initial velocity 𝑢rel |𝑡=0 is assumed to be zero.
In the above equations, 𝑃 represents pressure, 𝜌𝑙 is the
average droplet density, 𝐶𝐷 indicates the drop drag coefficient, 𝐴 𝑓 indicates the frontal area, and 𝜎𝑚 is the mixture
surface tension.
Thermodynamic Properties. In order to describe real fluid
behavior, high pressure effects, and phase equilibrium for
vapor and liquid mixtures, Peng-Robinson equation of state
(EOS) is adopted and can be expressed by
𝑃=
𝑎𝑚
𝑅𝑇
−
,
V − 𝑏𝑚 V (V + 𝑏𝑚 ) + 𝑏𝑚 (V − 𝑏𝑚 )
Surface Tension and Critical Mixing Point. The mixture
surface tension can be expressed as
𝑇
𝑇𝑐,mix
),
(18)
(20)
where 𝑎𝑚 and 𝑏𝑚 are the EOS parameters of the mixture and
can be calculated from the pure component parameters of the
EOS with the quadratic van der Waals one-fluid mixing rule.
V stands for molar volume and 𝑅 is universal gas constant. For
the acetone-nitrogen binary system, the binary interaction
coefficient of nitrogen and acetone is an adjustable binary
parameter and is taken as 0.2086 to correlate experimental
data in [28]. Peng-Robinson EOS above can be used to obtain
the specific enthalpy, constant pressure heat capacity, fugacity
coefficients, and so forth.
Other Physical Properties. The low pressure estimation model
proposed by Chung et al. for thermal conductivities of pure
components is modified with Chung et al. rules to calculate the thermal conductivity at high pressures. The liquid
viscosity is estimated via the corrected method of Chung
et al. accounting for high pressure and density effects [29].
The diffusion coefficient for binary gas systems is obtained
from Wilke and Lee method, which adopts high pressure
corrections of Takahashi method [30].
Thermodynamic properties of the gaseous mixture such
as 𝜇𝑔 and 𝑘𝑔 are evaluated with 1/3 rule [31] from the
average concentration and the average temperature, which
are expressed as follows:
𝑇𝑔 = 𝑇sf +
2.2. Thermophysical Properties
𝜎𝑚 = 𝐾 ⋅ (1 −
(19)
where 𝑓̃ represents fugacity, 𝜑̃ is fugacity coefficient, and 𝑋 is
molar fraction.
(15)
2
𝐶 𝜇 𝑑𝑦
𝑑2 𝑦 𝐶F 𝜌𝑔 𝑢rel 𝐶𝑘 𝜎𝑚
=
−
𝑦 − 𝑑 2𝑙 .
𝑑𝑡2
𝐶𝑏 𝜌𝑙 𝑟𝑠2
𝜌𝑙 𝑟𝑠3
𝜌𝑙 𝑟𝑠 𝑑𝑡
𝜕2 ln 𝑓̃F
) = 0,
𝜕𝑋F2
𝑃,𝑇
𝑓̃F = 𝑃𝑋F 𝜑̃F ,
(14)
Momentum conservation:
𝑃 = 𝑃∞ .
𝜕 ln 𝑓̃F
) = 0,
𝜕𝑋F 𝑃,𝑇
𝑇∞ − 𝑇sf
,
3
𝑋F,∞ − 𝑋F,sf 2
𝑋𝑔 = 𝑋F,sf +
= 𝑋F,sf ,
3
3
(21)
where 𝑇 is average temperature and 𝑋 is average molar
fraction.
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Table 1: Simulation conditions corresponding to experimental data.
Case
Case
Case
Case
0
1
2
3
𝑃ch (MPa)
2
4
6
6
𝑇ch (K)
293
293
293
376
𝑇inj (K)
293
293
293
376
𝐷0 (mm)
1.99
1.95
1.94
1.65
2.3. Boundary Conditions. All the variables are assumed to
remain at their initial values at the infinite boundary, and the
temperature gradient is assumed to be zero at the drop center.
The vapor-liquid equilibrium is satisfied at the drop
surface:
𝜑̃F,𝑔 𝑋F,𝑔 = 𝜑̃F,𝑙 𝑋F,𝑙 .
𝜑̃N,𝑔 𝑋N,𝑔 = 𝜑̃N,𝑙 𝑋N,𝑙 .
(22)
2.4. Numerical Scheme. An efficient numerical algorism
utilizing iterative method is needed in order to solve the
governing equations for liquid phase, liquid-gas interface,
and gas phase. A central difference scheme is used for liquid
phase energy conservation and a fully implicit scheme is
adopted for transient solution. A second-order one-sided
difference is used with a nonuniform grid system at the
liquid-gas interface, considering extremely high gradients
of thermophysical properties near the droplet surface. In
this work, the numerical solution method is illustrated as
follows: (1) at each time step, calculate the droplet evaporation firstly, where the drop is assumed to be spherically
symmetric concerning the combined calculation of heat
and mass transfer processes; (2) then resolve the droplet
deformation and oscillation in this time step, where the
spatial distribution of the physical properties for both liquid
and gas is assumed to be unchanged. By comparing numerical
results obtained at different time steps of integration, it was
found out that the temporal variations of droplet deformation
and evaporation rate and so forth exhibit very little difference
when dimensionless time steps are smaller than 1 × 10−4 .
Thus, the dimensionless time step is taken to be 1 × 10−4 .
3. Results and Discussion
3.1. Verification of the Model. The deformation of a free
falling acetone droplet in a nitrogen atmosphere at elevated
pressures and/or temperatures was predicted, where the
nitrogen gas is assumed to be stagnant and the initial 𝑦|𝑡=0 is
obtained from the experimental results. The variation curves
of droplet deformation 𝑦 and liquid-gas relative velocity for
cases in Table 1 are shown in Figure 2. The drop diameter is
obtained from that of a spherical droplet having same volume
as the distorting droplet. 𝑇inj is initial droplet temperature,
𝐷0 is initial droplet diameter, and 𝑃ch and 𝑇ch are ambient
pressure and temperature, respectively. Dimensionless constants 𝐶F and 𝐶𝑘 in (16) can be determined by comparing with
experimental and theoretical results [22, 25], which have been
chosen to be 2.0 and 1.4 according to the experimental data
for Case 0 (see Figure 2(a)). With the determined constants,
Figures 2(b)–2(d) show the comparisons between numerical
results calculated by the developed deformation model and
experimental data under the other three conditions (Case 1,
Case 2, and Case 3). From Figures 2(b)–2(d), it can be seen
that the simulation results agree well with the experimental
results, which convincingly proves that the proposed model
is effective for different extreme conditions.
3.2. Critical Mixing Point Calculated. A key fluid characteristic yielding the disappearance or appearance of liquid/gas
interface is the critical mixing point. Figure 3 presents the
critical molar fraction 𝑌cbs of acetone and the reduced critical
temperature 𝑇cr (𝑇cr = 𝑇cbs /𝑇𝑐 ) for an acetone-nitrogen
system with the reduced ambient pressure 𝑃𝑟 (𝑃𝑟 = 𝑃/𝑃𝑐 )
ranging from 0.02 to 2.8, where 𝑇cbs is the critical mixing
temperature and 𝑇𝑐 and 𝑃𝑐 are the critical temperature and
the critical pressure of acetone, respectively. The dotted line is
the reduced boiling point 𝑇br (𝑇br = 𝑇𝑏 /𝑇𝑐 ) for 𝑃𝑟 below 1.0,
where 𝑇𝑏 is the boiling point of acetone. It can be seen that
𝑌cbs decreases sensitively with 𝑃𝑟 except for 𝑃𝑟 less than 1.0.
In particular, 𝑌cbs attains a reduction of approximately 29%
at 𝑃𝑟 = 2.75. As shown in this figure, the variation curve of
𝑇cr is similar to that of 𝑌cbs , but 𝑇cr changes more smoothly,
with a decrease of about 3% at 𝑃𝑟 = 2.75. It shows that the
component fraction 𝑌cbs should have much more influence
on physical properties such as surface tension when elevating
ambient pressures. The predictions also show that, at 𝑃𝑟 less
than 1.0, 𝑌cbs reaches the value of approximately 1.0, which
implies that the component fraction has almost no influence
on the critical parameters. In this case, 𝑇br describes boiling
point of pure acetone instead of critical mixing temperature
𝑇cr , which increases with ambient pressure. Thus, 𝑇cr is taken
to be 1.0 for surface tension calculation.
3.3. Surface Tension Calculated. The effects of liquid phase
molar fraction 𝑋sf,𝑙 of acetone at droplet surface on surface tension, at various ambient pressures, are presented in
Figure 4. As shown in the figure, surface tension at low
pressures decreases with the reduction of 𝑋sf,𝑙 . This is because
more nitrogen which has lower surface tension compared
with acetone is dissolved in the droplet surface as 𝑋sf,𝑙
decreases, yielding reduction in surface tension of the liquid
mixture. The figure also shows that surface tension varies
more smoothly at higher ambient pressures. This is because
the critical mixing temperature decreases with pressure and
a droplet may reach the critical mixing point more easily
at higher pressures. Namely, the critical point can appear
at a lower temperature, which corresponds to a smaller
liquid phase molar fraction of acetone. However, surface
tension remains approximately independent of 𝑋sf,𝑙 at very
low pressures, especially when 𝑃𝑟 is lower than 0.2. This
is because 𝑋sf,𝑙 is approximately equal to 1.0 in this case,
indicating that the droplet behaves as a single component
droplet, which is different from results at high pressures (see
Figure 3).
Figure 5 shows the influence of ambient pressures and
droplet surface temperatures (𝑇sf /𝑇𝑐 ) on surface tension. Surface tension decreases with the droplet surface temperature,
6
Advances in Mechanical Engineering
0.4
0.6
0.6
0.3
0.5
0.2
0.4
0.1
0.3
0.0
0.2
0.1
−0.1
0.1
0.0
−0.2
0.2
0.5
0.1
0.4
0.0
0.3
0.2
−0.1
−0.2
20
40
60
80
100
Time (ms)
Adjusted y according
to experimental data
urel predicted
y measured
urel measured
0
20
40
0.4
0.3
0.5
0.3
0.2
0.4
0.2
0.3
0.1
0.2
0.1
0.0
0.0
−0.1
−0.1
60
Time (ms)
80
100
0.2
0.1
0.1
0.0
0.0
−0.1
−0.1
−0.2
−0.3
−0.2
0
5
10
15
20
25
30
35
Time (ms)
urel predicted
urel measured
y predicted
y measured
Droplet deformation
0.6
Relative velocity (m/s)
Droplet deformation
0.3
40
0.0
100
(b) Case 1
0.4
20
80
urel predicted
urel measured
y predicted
y measured
(a) Case 0
0
60
Time (ms)
Relative velocity (m/s)
0
Droplet deformation
0.7
Relative velocity (m/s)
Droplet deformation
0.3
Relative velocity (m/s)
0.4
(c) Case 2
urel predicted
urel measured
y predicted
y measured
(d) Case 3
Figure 2: Comparison of numerical and experimental results.
while the decrease in surface tension is more rapid in the case
of higher ambient pressures. This is due to the fact that, as
already mentioned above, it is easier for the drop to reach the
critical mixing point at higher pressures.
3.4. Dynamic Droplet Deformation and Weber Number. Fig2
𝜌𝑐 Cpref /𝜆 ref ))
ures 6, 7, 8, 9, and 10 show the time (𝑡/(𝑟𝑠0
histories of droplet deformation 𝑦 and Weber number We
2
(We = 𝜌𝑔 𝑢rel
𝑟𝑠 /𝜎𝑚 ) at various ambient pressures, where the
values of 𝑢rel are set at 2, 5, 8, 30, and 100 m/s, respectively.
𝑟𝑠0 is the initial radius of the droplet, 𝜌𝑐 is the critical
density of the fuel, and Cpref and 𝜆 ref are isobaric specific
heat capacity and thermal conductivity of the fuel at 293 K
and 0.1 MPa, respectively. 𝑇𝑟 (𝑇𝑟 = 𝑇amb /𝑇𝑐 ) stands for
the reduced ambient temperature, where 𝑇amb is ambient
temperature. The initial droplet diameter is 40 𝜇m, and
ambient temperature and initial drop temperature are 508.1 K
and 300 K, respectively. It is assumed that the droplet is
spherical and does not oscillate initially; that is, the initial
deformation 𝑦𝑡=0 and the initial oscillation (𝑑𝑦/𝑑𝑡)|𝑡 = 0 are
set at zero.
According to Figures 6–10, the variation curves of droplet
deformation 𝑦 appear to be fluctuant, indicating that the drop
is continuously oscillating as time elapses. It was found that
the amplitude of the oscillation is gradually decreasing with
time and becomes quite small during the late period. This is
because the viscous damping time scale 𝑡𝑑 (𝑡𝑑 = 2𝜌𝑙 𝑟𝑠2 /(5𝜇𝑙 ))
of the droplet decreases as time elapses, due to the reduction
of the droplet diameter and average density, which is caused
by heat and mass transfer between liquid and gas. As one
can see from these figures, We rises with time initially, due
to rapidly reduced surface tension caused by transient liquid
heating, although the drop becomes smaller. And then, We
falls rapidly with time, because the reduced droplet diameter
plays more important role and surface tension varies more
slowly due to approximatively steady heating. In addition, We
increases when ambient pressures and 𝑢rel are elevated. The
7
1.05
1.00
1.00
0.95
0.95
0.90
0.90
0.85
0.85
0.80
0.80
0.75
0.75
0.70
0.70
0.65
0.65
0.60
0
1
2
3
30
25
0.60
Pr
Surface tension (N/m)
1.05
Ycbs
Tcr and Tbr
Advances in Mechanical Engineering
20
15
10
5
0
0.5
Tbr
Tcr
Ycbs
Figure 3: Critical temperature, critical acetone molar fraction, and
boiling point as functions of pressure.
0.6
0.7
0.9
0.8
Dimensionless droplet surface temperature
Pr = 0.02
Pr = 1.00
Pr = 1.52
1.0
Pr = 2.12
Pr = 2.72
Figure 5: Influence of droplet surface temperature on surface
tension at different pressures.
Surface tension (N/m)
30
25
20
15
10
5
0
0.75
0.80
0.85
0.90
0.95
1.00
Molar fraction of liquid acetone at droplet surface
Pr = 0.02
Pr = 0.2
Pr = 1.0
Pr = 3.0
Figure 4: Influence of molar fraction of liquid acetone at droplet
surface on surface tension at different pressures.
peak deformation per oscillation cycle changes in a manner
similar to Weber number, but it declines slightly and then
rises in the initial stage. This implies that, during the early
stage, 𝑡𝑑 has a more important influence than We on the
drop deformation, while during the subsequent stage, We
dominates the drop deformation. The oscillation frequency
decreases slightly as ambient pressure is elevated, indicating
that elevating pressure has a negative influence on droplet
oscillation.
As shown in Figures 6–10, the predicted droplet deformation increases apparently with 𝑢rel and ambient pressure due
to Weber number enhancement. The droplet will be able to
breakup (𝑦 > 1) at 𝑃𝑟 higher than 0.7 when 𝑢rel is faster than
8 m/s but will not able to breakup even at a high 𝑃𝑟 (𝑃𝑟 = 1.8)
when 𝑢rel is slower than 2 m/s. However, at very high drop
speeds such as 𝑢rel = 100 m/s, the TAB model is only capable
of predicting the drop deformation instead of drop breakup as
is mentioned above. In this case, the droplet breakup occurs
at different levels of the droplet deformation. It was also found
that, the maximum deformation occurs in the later stage for
𝑃𝑟 exceeding 1.0 but occurs in the initial stage for 𝑃𝑟 lower
than 0.7. This result implies that thermophysical properties of
both liquid and gas are quite sensitive to pressure variations
at high pressures and temperatures, which yields a more
rapid increase of Weber number. Additionally, the maximum
deformation during the later stage appears approximately
at the same dimensionless time under different conditions.
This is due to the combined effects of the reduced droplet
diameter and the variation of physical properties, such as
surface tension and gas density. It can also be seen that the
droplet breakup occurs in the first oscillation cycle for 𝑢rel
faster than 8 m/s. In this case, the drop is rapidly broken up
at the beginning, and it is almost not affected by the heat and
mass transfer.
Figures 11, 12, 13, 14, and 15 show the time histories of 𝑦
and We at various ambient temperatures, where the values of
𝑢rel are set at 2, 5, 8, 30, and 100 m/s, respectively. The reduced
ambient pressure 𝑃𝑟 is 1.0, with other parameters having the
same values as used in Figures 6–10. It was found that 𝑦
and We at various ambient temperatures are quite similar to
those predicted at various pressures. 𝑦 and We seem to be
unaffected by the ambient temperature initially, due to the
fact that the physical properties of both liquid and gas are
hardly affected by high heat and mass transfer rates during
a very short period. With a constant pressure (𝑃𝑟 = 1.0), both
𝑦 and We vary more quickly at high ambient temperatures
(e.g., 𝑇𝑟 = 1.8), and the peak deformation per oscillation
cycle is rapidly increasing until its maximum is attained.
Therefore, the results indicate that, for very high ambient
8
Advances in Mechanical Engineering
4.5
1.4
0.6
0.2
0.02
0.04
0.2
0.4
0.6
0.8
9.1
3.5
7.8
3.0
6.5
2.5
2.0
5.2
1.5
3.9
1.0
2.6
0.2
0.5
1.3
0.0
0.0
0.00
0.01
0.02
Dimensionless time
0.4
Figure 6: Temporal variations of droplet deformation and Weber
number at different ambient pressures (𝑢rel = 2 m/s).
1.50
2.5
1.25
2.0
1.00
1.5
0.75
20
30
1.0
0.50
25
15
20
10
15
10
5
0.5
0.25
0.0
0.2
0.0
0.6
Figure 8: Temporal variations of droplet deformation and Weber
number at different ambient pressures (𝑢rel = 8 m/s).
Droplet deformation
3.0
We
1.75
0.00
0.00 0.01 0.02 0.03 0.04
0.5
Deformation y Pr = 0.7, Tr = 1.0
Deformation y Pr = 1.0, Tr = 1.0
Deformation y Pr = 1.8, Tr = 1.0
We Pr = 0.7, Tr = 1.0
We Pr = 1.0, Tr = 1.0
We Pr = 1.8, Tr = 1.0
Deformation y Pr = 0.7, Tr = 1.0
Deformation y Pr = 1.0, Tr = 1.0
Deformation y Pr = 1.8, Tr = 1.0
We Pr = 0.7, Tr = 1.0
We Pr = 1.0, Tr = 1.0
We Pr = 1.8, Tr = 1.0
Droplet deformation
0.1 0.2 0.3
Dimensionless time
0.4
0.6
0.8
Dimensionless time
Deformation y Pr = 0.7, Tr = 1.0
Deformation y Pr = 1.0, Tr = 1.0
Deformation y Pr = 1.8, Tr = 1.0
We Pr = 0.7, Tr = 1.0
We Pr = 1.0, Tr = 1.0
We Pr = 1.8, Tr = 1.0
Figure 7: Temporal variations of droplet deformation and Weber
number at different ambient pressures (𝑢rel = 5 m/s).
temperatures, thermophysical properties of both the droplet
and the gas exhibit a much greater sensitivity to changes in
ambient temperatures. The droplet can break up at 𝑇𝑟 higher
than 0.7 when 𝑢rel is faster than 8 m/s but cannot break up
even at a very high ambient temperature (𝑇𝑟 = 1.8) when
𝑢rel is slower than 2 m/s. Also, the drop breakup, if possible,
appears only in the first oscillation cycle.
3.5. Droplet Breakup. Considering that the TAB model is
mainly used to predict the secondary breakup for low Weber
We
0.0
0.00
0.4
Droplet deformation
0.8
We
Droplet deformation
1.0
0.4
10.4
4.0
1.2
We
0.6
5
0
0.00
0.01
0.02
0.1
Dimensionless time
0.2
0
0.3
Deformation y Pr = 0.7, Tr = 1.0
Deformation y Pr = 1.0, Tr = 1.0
Deformation y Pr = 1.8, Tr = 1.0
We Pr = 0.7, Tr = 1.0
We Pr = 1.0, Tr = 1.0
We Pr = 1.8, Tr = 1.0
Figure 9: Temporal variations of droplet deformation and Weber
number at different ambient pressures (𝑢rel = 30 m/s).
numbers (in the bag-type breakup regime), the characteristics
of droplet breakup for relatively low drop velocities are
studied in this section. The simulation results indicate that
the droplet breakup cannot occur if 𝑇𝑟 , 𝑃𝑟 , and/or 𝑢rel are
relatively low. Figure 16 shows the minimum 𝑇𝑟 required for
an acetone droplet to break up as a function of 𝑃𝑟 . 𝑢rel is set
at 5, 8, and 30 m/s, respectively, and initial drop temperature
is 300 K. With regard to the conditions below the curve,
the droplet breakup cannot occur during the evaporation
process at a specific 𝑢rel . On the contrary, the breakup can
appear under conditions above the curve. In order to enable
Advances in Mechanical Engineering
9
600
200
200
100
0.01
0.02
0.04
Dimensionless time
0
0.06
0.05
0.9
1.0
0.3
0.2
0.4
0.6
0.8
0.0
Dimensionless time
Deformation y Pr = 1.0, Tr = 0.7
Deformation y Pr = 1.0, Tr = 1.0
Deformation y Pr = 1.0, Tr = 1.8
We Pr = 1.0, Tr = 0.7
We Pr = 1.0, Tr = 1.0
We Pr = 1.0, Tr = 1.8
0.32
0.96
3.0
0.80
2.5
0.48
0.16
0.32
0.08
0.16
0.00
0.00
0.00
0.6
0.8
We
0.64
0.24
Figure 12: Time evolution of droplet deformation and Weber
number at different ambient temperatures (𝑢rel = 5 m/s).
Deformation y Pr = 1.0, Tr = 0.7
Deformation y Pr = 1.0, Tr = 1.0
Deformation y Pr = 1.0, Tr = 1.8
We Pr = 1.0, Tr = 0.7
We Pr = 1.0, Tr = 1.0
We Pr = 1.0, Tr = 1.8
Figure 11: Time evolution of droplet deformation and Weber
number at different ambient temperatures (𝑢rel = 2 m/s).
the droplet breakup, the minimum 𝑇𝑟 decreases with 𝑃𝑟 .
In addition, both the required 𝑇𝑟 and 𝑃𝑟 decrease when
improving the drop velocity 𝑢rel .
Ratios of the breakup time to droplet lifetime versus
ambient pressure 𝑃𝑟 at various ambient temperatures 𝑇𝑟 are
shown in Figure 17. 𝑢rel is equal to 5 m/s and initial drop temperature is 300 K. According to Figure 17, the ratio reduces
as 𝑇𝑟 and 𝑃𝑟 increase. It shows that, in comparison with
the evaporation process, the breakup occurs much earlier
when increasing 𝑇𝑟 and 𝑃𝑟 , due to enhanced aerodynamic
Droplet deformation
0.40
Droplet deformation
0.5
0.0
0.00 0.01 0.02 0.03
Figure 10: Temporal variations of droplet deformation and Weber
number at different ambient pressures (𝑢rel = 100 m/s).
0.2
0.4
0.04
Dimensionless time
1.5
0.6
Deformation y Pr = 0.7, Tr = 1.0
Deformation y Pr = 1.0, Tr = 1.0
Deformation y Pr = 1.8, Tr = 1.0
We Pr = 0.7, Tr = 1.0
We Pr = 1.0, Tr = 1.0
We Pr = 1.8, Tr = 1.0
0.02
2.0
9
8
7
2.0
6
5
1.5
4
1.0
We
0
0.00
100
2.5
We
300
We
400
300
Droplet deformation
500
400
Droplet deformation
3.0
1.2
3
2
0.5
1
0.0
0.00 0.01 0.02
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Dimensionless time
0
Deformation y Pr = 1.0, Tr = 0.7
Deformation y Pr = 1.0, Tr = 1.0
Deformation y Pr = 1.0, Tr = 1.8
We Pr = 1.0, Tr = 0.7
We Pr = 1.0, Tr = 1.0
We Pr = 1.0, Tr = 1.8
Figure 13: Time evolution of droplet deformation and Weber
number at different ambient temperatures (𝑢rel = 8 m/s).
interactions caused by anomalous variations of thermophysical properties. The drop will break up immediately at the
beginning, when 𝑇𝑟 and 𝑃𝑟 are high enough.
4. Conclusion
The dynamic deformation of an isolated droplet under
convective transcritical conditions was studied numerically.
The new deformation model was validated with the existing
10
Advances in Mechanical Engineering
12
15
8
10
5
0
0.00 0.01 0.02
0.1
0.2
Dimensionless time
0.3
0
0.4
350
500
300
400
250
300
We
Droplet deformation
400
200
100
50
1.2
1.0
0.8
0.6
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Figure 16: Minimum ambient temperature required for droplet
breakup versus ambient pressure at 5, 8, and 30 m/s 𝑢rel , respectively.
0.25
0.20
0.15
0.10
0.05
0.00
0
0.00 0.01 0.02
0.2
0.30
600
100
1.4
urel 5 m/s
urel 8 m/s
urel 30 m/s
Figure 14: Time evolution of droplet deformation and Weber
number at different ambient temperatures (𝑢rel = 30 m/s).
150
1.6
Pr
Deformation y Pr = 1.0, Tr = 0.7
Deformation y Pr = 1.0, Tr = 1.0
Deformation y Pr = 1.0, Tr = 1.8
We Pr = 1.0, Tr = 0.7
We Pr = 1.0, Tr = 1.0
We Pr = 1.0, Tr = 1.8
200
1.8
0.4
0.0
Ratio of breakup time to lifetime
4
Minimum Tr required for breakup
20
We
Droplet deformation
2.0
25
16
0
0.04 0.05 0.06 0.07 0.08 0.09 0.10
Dimensionless time
Deformation y Pr = 1.0, Tr = 0.7
Deformation y Pr = 1.0, Tr = 1.0
Deformation y Pr = 1.0, Tr = 1.8
We Pr = 1.0, Tr = 0.7
We Pr = 1.0, Tr = 1.0
We Pr = 1.0, Tr = 1.8
Figure 15: Time evolution of droplet deformation and Weber
number at different ambient temperatures (𝑢rel = 100 m/s).
experimental results. According to the numerical results, the
following conclusions are drawn.
(1) In the case of higher ambient pressures, the surface
tension reduces more smoothly with the fuel mole
fraction and more rapidly with the droplet surface
temperature.
(2) Weber number increases continuously in the initial
period and then decreases rapidly. Also, the drop
exhibits continuous oscillations as time elapses, with
the accompanying amplitude reduction.
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Pr
Tr = 1.0
Tr = 1.2
Tr = 1.5
Figure 17: Ratio of droplet breakup time to lifetime versus ambient
pressure at different ambient temperatures.
(3) The ambient temperature has almost no influence on
the drop deformation and Weber number initially.
The maximum deformation occurs in the later stage
for high pressure levels (e.g., 𝑃𝑟 exceeding 1.0) but
occurs in the initial stage for low pressures (e.g., 𝑃𝑟
lower than 0.7). When the ambient pressure (𝑃𝑟 =
1.0) keeps constant, both 𝑦 and We change very
rapidly at high temperatures (e.g., 𝑇𝑟 = 1.8), and the
peak deformation per oscillation cycle is increasing
continuously until its maximum is reached.
(4) The minimum ambient temperature required for drop
breakup decreases with the ambient pressure. In
addition, both the required ambient temperatures and
Advances in Mechanical Engineering
pressures for breakup decrease when increasing the
drop velocity.
Nomenclature
𝑎𝑚 , 𝑏𝑚 :
𝐴 𝑓:
𝐶𝑏 , 𝐶𝑑 , 𝐶F , 𝐶𝑘 :
𝐶𝐷:
Cp:
Cpref :
𝐷0 :
𝐷FN :
̃
𝑓:
ℎ:
ℎ𝑔∗ :
𝐻:
𝑘𝑒 :
𝐾:
𝐾FN :
∗
:
𝐾FN
𝑚:
Nu:
𝑃:
𝑃𝑐 :
𝑃ch :
𝑃𝑟 :
Pr:
𝑟:
𝑟𝑠 :
𝑟𝑠0 :
𝑅:
Re:
Sc:
Sh:
𝑡:
𝑡𝑑 :
𝑇:
𝑇amb :
𝑇𝑏 :
𝑇br :
𝑇𝑐 :
𝑇cbs :
𝑇ch :
𝑇𝑐,mix :
𝑇cr :
𝑇inj :
𝑇𝑟 :
𝑇:
𝑢:
𝑢rel :
EOS parameters
Frontal area of drop (m2 )
Model constants of the TAB model
Drop drag coefficient
Constant pressure heat capacity (J/(kg⋅K))
Constant pressure heat capacity of the fuel
at 293 K and 0.1 MPa (J/(kg⋅K))
Initial droplet diameter (m)
Mass diffusivity (m2 /s)
Fugacity (Pa)
Heat transfer coefficient (W/(m2 ⋅K))
Corrected heat transfer coefficient of gas
(W/(m2 ⋅K))
Enthalpy (J/kg)
Effective thermal conductivity (W/(m⋅K))
Constant
Mass transfer coefficient (kg/(m2 ⋅s))
Corrected mass transfer coefficient
(kg/(m2 ⋅s))
Droplet mass (kg)
Nusselt number
Pressure (Pa)
Critical pressure of the fuel (Pa)
Ambient pressure (Pa)
Reduced ambient pressure
Prandtl number
Radial distance relative to droplet center
(m)
Droplet radius (m)
Initial radius of droplet (m)
Universal gas constant (J/(moL⋅K))
Reynolds number
Schmidt number
Sherwood number
Time (s)
Viscous damping time scale
Temperature (K)
Ambient temperature (K)
Boiling point of acetone (K)
Reduced boiling point
Critical temperature of the fuel (K)
Critical mixing temperature (K)
Ambient temperature (K)
Mixture temperature at the critical point
(K)
Reduced critical temperature
Initial droplet temperature (K)
Reduced ambient temperature
Average temperature (K)
Radial velocity inside droplet relative to
droplet center (m/s)
Relative velocity between droplet and gas
(m/s)
11
Molar volume (m3 /mol)
Volume (m3 )
Droplet volume (m3 )
Mass flow rate (kg/s)
Weber number
Displacement of the mass from the idle
state (m)
𝑋:
Molar fraction
𝑋:
Average molar fraction
𝑦:
Dimensionless droplet deformation
𝑌:
Mass fraction in gas phase
Critical molar fraction of the fuel
𝑌cbs :
𝑍𝑇 , 𝑍𝑚 : Correction factors of heat transfer for high
transfer rates.
V:
𝑉:
𝑉𝑠 :
𝑊:
We:
𝑥:
Greek Letters
Density (kg/m3 )
Critical density of the fuel (kg/m3 )
Average droplet density (kg/m3 )
Thermal conductivity (W/(m⋅K))
Thermal conductivity of the fuel at 293 K
and 0.1 MPa (W/(m⋅K))
𝜇: Viscosity (Pa⋅s)
𝜎𝑚 : Mixture surface tension (N/m)
̃ Fugacity coefficient.
𝜑:
𝜌:
𝜌𝑐 :
𝜌𝑙 :
𝜆:
𝜆 ref :
Subscripts
F:
𝑔:
𝑙:
N:
sf:
∞:
Fuel component
Gas phase
Liquid phase
Nitrogen component
Drop surface
Infinite boundary.
Conflict of Interests
The authors declare that there is no conflict of interests
regarding the publication of this paper.
Acknowledgment
This work was supported by the Beijing Natural Science
Foundation (3132021) and the National Natural Science Foundation of China (50606014).
References
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[2] P. J. Kay, P. J. Bowen, M. Gold et al., “Studies of gasoline
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331, 2012.
12
[3] L. L. Tavlarides and G. Anitescu, “Supercritical diesel fuel
composition, combustion process, and fuel system,” US Patent
7488357, 2009.
[4] C. Segal and S. A. Polikhov, “Subcritical to supercritical mixing,”
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