Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 16.1.2 - Evaluation of a model for binary condensing flow
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 Material Information
Title: 16.1.2 - Evaluation of a model for binary condensing flow Droplet Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Sidin, R.
Hagmeijer, R.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: droplet size distribution
binary condensation
compressible flow
 Notes
Abstract: We evaluate a condensing flow model, which utilizes a kinetic equation to predict the evolution of the droplet size distribution for a two-component (binary) dispersion, generated by homogeneous nucleation. The binary kinetic equation (BKE) is based on the classic formulation by Reiss, with the monomer evaporation rates calculated by employing updated equilibrium distributions from Wilemski (1995). The evaluation of the condensing flow model is based on the simulation of nozzle flow experiments, for which measurement data are reported in Wyslouzil (2000). The mixture used in these experiments contains ethanol- and propanol vapor as the condensable components, with nitrogen as the inert carrier gas. In the simulation of the nozzle flows a full two-way coupling between the thermodynamic variables and the binary droplet size distribution is employed in order to account for the feedback effects of vapor depletion and latent heat release. For this purpose we have used a sectional approximation of the BKE, which we solve simultaneously with the fluid dynamics equations (FDE) for the mixture. We compare the predicted and measured vapor pressures and temperatures at condensation onset, which reveals a fair agreement between theory and experiment in case the backward rates are computed by using the fully self-consistent equilibrium distribution of Wilemski (1995). Finally, we show that the predicted thermodynamic variables are very sensitive to the type of equilibrium distribution used.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00392
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Resource Identifier: 1612-Sidin-ICMF2010.pdf

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ICMF 2010, Tampa, FL, May 30- June 4, 2010


Evaluation of a model for binary condensing flow


Ryan Sidin* and Rob Hagmeijer**

Department of Mechanical Engineering, University of Suriname, P.O.Box 9212, Paramaribo, Suriname
**Department of Mechanical Engineering, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
r.hagmeijer.ut@gmail.com
Keywords: droplet size distribution, binary condensation, compressible flow




Abstract

We evaluate a condensing flow model, which utilizes a kinetic equation to predict the evolution of the droplet size
distribution for a two-component (binary) dispersion, generated by homogeneous nucleation. The binary kinetic
equation (BKE) is based on the classic formulation by Reiss, with the monomer evaporation rates calculated by
employing updated equilibrium distributions from Wilemski (1995). The evaluation of the condensing flow model
is based on the simulation of nozzle flow experiments, for which measurement data are reported in Wyslouzil
(2000). The mixture used in these experiments contains ethanol- and propanol vapor as the condensable components,
with nitrogen as the inert carrier gas. In the simulation of the nozzle flows a full two-way coupling between the
thermodynamic variables and the binary droplet size distribution is employed in order to account for the feedback
effects of vapor depletion and latent heat release. For this purpose we have used a sectional approximation of the
BKE, which we solve simultaneously with the fluid dynamics equations (FDE) for the mixture. We compare the
predicted and measured vapor pressures and temperatures at condensation onset, which reveals a fair agreement
between theory and experiment in case the backward rates are computed by using the fully self-consistent equilibrium
distribution of Wilemski (1995). Finally, we show that the predicted thermodynamic variables are very sensitive to
the type of equilibrium distribution used.


Nomenclature


Roman symbols
A nozzle area (m2)
av, vapor phase activity (-)
b backward condensation rate (s 1)
Cn cluster number density (mi3)
Cp specific heat at constant pressure (Km2s
C, specific heat at constant pressure (Knm2s
fn forward condensation rate (s 1)
g mass fraction (-)
Jn condensation flux (m-3s 1)
Lb,k bulk latent heat of condensation (Jkg 1)
M Mach number (-)
n cluster composition vector (-)
p pressure (Nm 2)
R specific gas constant (m2s 1K 1)
Sn BKE source term (mi s 1)
T temperature (K)
u fluid velocity (nms1)
Xn,k average droplet molar fraction (-)
z nozzle axial coordinate (m)


Greek symbols
an sticking probability (-)
Kn shape factor (-)
p mass density (kgmi3)


Subscrip
b,k
v,k
0
SupersciF
eq
S


ts
component k bulk droplet phase
component k vapor phase
dry mixture state (before condensation)
its
equilibrium condition
saturation condition


Introduction

Multi-component condensation is an area of research
which is of fundamental interest to applications in both
nature and industry, e.g., in the formation of cloud con-
densation nuclei (see, e.g., Seinfeld (1986)), or the con-
ditioning of natural gas (see, e.g., Put (2003)). Bi-
nary condensation, which can be viewed as the most






elementary form of multi-component condensation, al-
ready poses a formidable research challenge, which is
reflected in the vast amount of investigations on this sub-
ject (Kashchiev (2000), Vehkamiki (2006)).
Most of the investigations in binary condensation are
primarily focused on steady-state nucleation, where the
aim is to predict or measure the steady-state nucleation
rate in the two-dimensional component space. Theo-
retical models for binary condensation describe the ki-
netics of cluster formation and destruction by means of
the so-called Binary Kinetic equation (BKE), which is
a conservation law in two-component space for the bi-
nary droplet size distribution (BDSD). Since the concep-
tion of the BKE in Reiss (1950), many improvements
to binary condensation theory have been suggested, no-
tably in Stauffer (1976), Kulmala (1992) and Wilemski
(1995). On the experimental front, measurement of the
steady state nucleation rate has been carried out in vari-
ous nucleation-pulse experiments, e.g. in Strey (1993).
More recently, new measurement techniques have
been developed in which condensation is achieved by
means of rapid expansion in nozzle flows, e.g. in
Wyslouzil (2000) and Heath (2003). Due to the rapid
variation in vapor supersaturation and temperature ex-
perienced in such experiments it is very likely that
quasisteady-state nucleation theory is no longer applica-
ble. Under such conditions, it is necessary to incorporate
the effects of vapor depletion and latent heat release on
the flow variables, which demands a fully two-way cou-
pled model between the gas phase and the liquid phase.
To this end it is necessary to solve the the fluid dynamics
equations (FDE) simultaneously with the BKE.
In this paper we present a fully two-way coupled
model for inviscid compressible flow with binary con-
densation. The accuracy of the model is evaluated by
comparing the predicted temperatures and pressures at
condensation onset with measurement data taken from
Wyslouzil (2000). By using a sectional approximation
of the BKE, the BDSD is resolved beyond the nanome-
ter size scale, in order to quantify the latent heat release
and vapor depletion effect with sufficient accuracy. The
simulations are carried out using various formulations
of the binary equilibrium size distribution to analyze its
impact on the flow field variables.

Physical-mathematical model

The Binary Kinetic Equation (BKE) Assuming the
Szilard model for condensation, i.e., droplets may only
capture or expel a single monomer of either type at any
instant, the BKE for an advected dispersion can be writ-
ten as:
09Cn a
dt i (Cnaj) J Jn-1 1 2 Jn, (1)
at 0 l-J+ e-


ICMF 2010, Tampa, FL, May 30- June 4, 2010


where n (ni, n2)T is the coordinate in binary space,
el (1, 0), e2 (0, 1)T, cn is the volumetric concen-
tration of droplets consisting of nl monomers of com-
ponent 1, and n2 monomers of component 2, and where
Jf denotes the nett rate at which clusters of size n grow
to size n + ek. Schematically, the BKE can be repre-
sented as a series of chemical reactions, such as depicted
in Figure 1. It is noted that creation of the binary dimer
n (1, 1)T from two monomers (1, O)T and (0, 1)T
constitutes a single reaction, which means that (i) the
monomer fluxes J2l,) and J ,) are equal, and (ii) that
only one of these should be used to compute the residu-
als for c(1,1).


t n +e


fn., Cn


n'' e J ,
i- ( J-el ",
O


n c .c,, > n+e,


n a-e,


Figure 1: Schematic representation of the Szilard model
for binary condensation.

The condensation flux Jn comprises the nett effect
of condensation and evaporation, via corresponding for-
ward rates fn and backward rates b', respectively:

Jn = fn ',, Cn+e,. (2)
Assuming Maxwellian velocity distributions for the
monomers and the clusters, the forward rate f for the
kh -component is given by (Vehkamaki (2006)):

f Pv,k f / 1 k1
fn = nn- V6 T -- + --
mv, k V n nv,


(v/3 + v113
n v,


where pv,k is the partial vapor mass density of compo-
nent k, and where an, Kn, rin, and Vn are the sticking
probability, shape factor, mass, and volume of a cluster
of size n, respectively. The mass and volume associated


47I3






with a monomer of component k are denoted by mr,k
and Vv,k, respectively. The sticking probability will be
set to constant value of an = 1, and a spherical droplet
shape will be assumed (Kn = 1), as is common practice
in the nucleation literature.
The backward rate b for component k is calculated
from the corresponding forward rate and the equilibrium
droplet size distribution crq, by virtue of the principle of
detailed balance, (Vehkamaki (2006)):


eq
, k n-efk
Cn-ek
en


The binary equilibrium size distribution The binary
equilibrium distribution ciq has been the subject of many
discussions in the nucleation literature, e.g., in Wilemski
(1995), and Kulmala (1992). The generic expression for
cq is given as (Vehkamaki (2006)):

c co exp (kT (5)

where AGn is the Gibbs free energy associated with a
droplet of size n, and co is a pre-factor which determines
the magnitude of the equilibrium number densities.
For AGn we adopt the following model (Wilemski
(1995), Vehkamaki (2006)):

AGn = -ksT In + n2 In a + Ann,
an, 1 an,2 /
(6)
where av,k is the vapor activity of component k in the
vapor phase, an, k is the liquid activity of vapor compo-
nent k in an n-droplet, and where An and an denote the
surface area and surface tension of an n-droplet, respec-
tively. The activities of the vapor and liquid phases are
defined as:

Pv,k k'
av, = pp ,and: akn,= ,p, (7)
Pvk Pv,k

respectively, where pv,k is the partial vapor pressure of
component k, p8P = p" (T) is the pure-component
saturation pressure of vapor component k over a flat liq-
uid surface, and where /. j. (n, T) is the partial
saturation pressure for vapor component k over the sur-
face of an n-droplet.
For the pre-factor co in Eq. (5) we consider the fol-
lowing variants:

1. co for the equilibrium distribution with limited self-
consistency, by Wyslouzil and Wilemski (Wilemski
(1995)):
C /^\ 3ni 1 hP\ n,12
co P,2 (8)
\m"y / \my2 /


ICMF 2010, Tampa, FL, May 30- June 4, 2010


2. co for the equilibrium distribution with full con-
sistency, by Wyslouzil and Wilemski (Wilemski
(1995)):
C = Co exp (n,IO,i + zn,2-,,2), (9)

where p"' is the mass density of pure vapor component
k at saturation, and where the dimensionless surface en-
ergy O,~, is given by:


v Av, v, k
kBT


(10)


with A4,k the effective surface area of a monomer of
component k, and cv,k the surface tension of a flat liquid
interface of pure component k. Xn,k denotes the average
molar fraction of vapor component k in the droplet,


(11)


nl + n2


The difference between c1 and cg is that, upon sub-
stitution in Eq. (5), the latter allows the unary size dis-
tribution to be retrieved, whereas the former does not.
The equilibrium distributions based on co and c2 will be
referred to as the WW1- and WW2-distributions, respec-
tively.
In calculating AGn, the spatial variation of the com-
position within the droplet should ideally be taken into
account when specifying the liquid phase activities and
surface tension. Fortunately, the binary system studied
in this investigation yields only a small error when the
mixture properties are evaluated at the bulk composition
Xn,k.
Mixture properties The binary mixtures employed in
this investigation can be approximated as ideal systems,
both in the gaseous and liquid states. The thermody-
namic model needs to be augmented with suitable ex-
pressions for the saturation vapor densities p, k, and
the surface tension on. Both the liquid mixture in
the droplets and the vapor mixture in the gas phase
are assumed to be ideal, which means that the liquid
phase activity coefficients are unity: 7n, = 1. Ne-
glecting droplet curvature, the partial saturation pressure
/ (n, T) of vapor component k over the surface of an
n-droplet is given by (Shavitt (1995)):


(12)


Using the thermal equation of state, the saturation den-
sity follows from:


Pvk(Xn,k) R,, T
Rv-,kcT


(13)


For the surface tension and latent heat of the liquid
mixture, a linear weighing of the pure-component sur-
face tensions in terms of the molar fractions will be em-
ployed. Further details regarding the fluid properties can
be found in Sidin (2009).


Pv, k (n, T) Xn, kcv",p (T).






The Fluid Dynamics Equation (FDE) We employ a
quasi-one dimensional approximation to simulate the
nozzle flow in the present investigation, since the area
variation of the nozzle investigated in this paper (see
Figure 2) implicates a slender nozzle geometry. The un-
known variables at each point along the nozzle include
the temperature and vapor densities, as well as the binary
number densities cn.
Since we consider a steady state flow, the mass flow
rate P, of the mixture is constant, which means that at
each axial position z:


puA = m,


ICMF 2010, Tampa, FL, May 30- June 4, 2010


number densities = -(c(2,o), C(o,2), .., C(N,N))T, with
p denoting the mixture mass density. Differentiation of
Eq. (17) with respect to z thus yields:


E o


( dLb,l Lb,2, dT
SdT lT ) ] dz


dgTbl dgT,2 d, u
Lb,1 ,2 + U- .
dz dz dz


(18)


Due to the inviscid approximation, the momentum
equation for the mixture is reduced to:


du 1 dp
dz p dz


(19)


where p is the mixture mass density, u the fluid velocity
and A = A(z) the cross-sectional area of the nozzle at
position z. By assuming choked flow, and that the onset
of condensation takes place downstream of the throat,
the mass flow rate can be obtained from isentropic flow
theory, thus:


P t m Atp /T,,oRn ,,oTo 1 + (-. ,o


S (1 2(1 ,o)
1)


(15)
where At is the flow area at the nozzle throat, To and po
are the total temperature and density of the dry mixture,
respectively, and where:

Ym,0 Cpo, Rm,o Cp,o C,,o,
Cv,o

are the Poisson constant, and specific gas constant of
the dry mixture, respectively. The Cp,O and C,,o are the
mixture specific heats at constant pressure and volume,
respectively, and also correspond with the dry state (i.e.,
before condensation takes place).
Differentiation of Eq. (14) with respect to z yields the
following quasi-one-dimensional and steady-state dif-
ferential form of the continuity equation for the mixture:


1 du 1 dp 1 dA
a+ + pd A
u dz p dz A dz


In the absence of viscous dissipation and heat conduc-
tion, the total enthalpy ht of the mixture remains invari-
ant along streamlines. Since the mixture is assumed to
behave ideally, the specific total enthalpy is composed
of the mass-weighted contributions of each phase:

12
ht Cp,oT- [gl,lLb,l(T)+gl,2Lb,2(T)] +_2, (17)

where gik is the liquid mass fraction of vapor compo-
nent k and Lb, Lb, (T) denotes the temperature de-
pendent latent heat of condensation for pure component
k. It is noted that gi9, is completely determined by the


for a quasi-one-dimensional system.
To calculate the liquid mass fractions, we require the
cluster number densities at each point along the noz-
zle. Instead of solving for Cn, the specific number den-
sity Cn Cn/p will be employed. By combining the
continuity equation for the mixture with the BKE (Eq.
(1)), the steady-state Lagrangian form of the BKE is ob-
tained:


deCn Sl


2(0n\


dx u
where Sn = Sn/p, with Sn the right-hand side of Eq.
(1).


Eqs. (20), (16), (18), and (19) constitute the govern-
ing system of equations which describe the condensing
flow in a quasi-one-dimensional geometry, in terms of
the state variables p, r, T, u, and p. To complete the
system, the thermal equation of state:


p =pRT


(21)


is also included, with the specific mixture gas constant
R = R(g1, gl,2) given by the mass-weighted sum of
the specific gas constants of each gas phase constituent:


2
R +ai 9 Rck z lma g l,k)Rv,k-
k=1


(22)


Here gj"" and Rc,k denote the mass fraction and spe-
cific gas constant associated with the carrier gas, respec-
tively, and gf"" represents the total mass fraction of
component k (in both liquid and vapor form) in the mix-
ture.
Differentiation of Eq. (21) with respect to z then
yields:


where:


1 p 1 dp 1 dR 1 dT
p dz pdz R dz T dz'


dR dR dgl,1 dR dg1,2
dz o9g,1 dz 9gl,2 dz


(23)


(24)


]




ICMF 2010, Tampa, FL, May 30- June 4, 2010


By combining Eqs. (16), (18), (19), and (23), the fol-
lowing expression for the spatial derivative of the tem-
perature can be derived:


dgi,i dgia
Lb,1 + Lb,2
dz dz


u2 1 dR 1 dA]
1 [R dz A dz
IRT


dz dLb, 1 Lb,2 u2/T
,T T 1
RT
(25)
The governing system of equations can now be reduced
to Eq. (20) for c, and Eq. (25) for T. Knowing c, the
liquid mass fractions gi, and g, 2 can be calculated. As
the total enthalpy ht is invariant, the fluid velocity can
be calculated from the temperature and the liquid mass
fractions, via Eq. (17). Finally, the fluid velocity and
flow area A(x) can be used to compute the mixture den-
sity via Eq. (14).


Test case description


A condensing nozzle flow is simulated for which mea-
surement data is reported in Wyslouzil (2000). The mix-
ture contains ethanol- and propanol vapor as the con-
densable components, with nitrogen as the carrier gas.
The nozzle area variation, depicted in Figure 2, is re-
constructed from the dry-flow pressure profiles given in
Wyslouzil (2000) by means of the isentropic relation-
ships for quasi-one-dimensional compressible flow. This
approach is adopted, because the measured pressure pro-
files account for the presence of boundary-layers in the
experiments. The total conditions at the nozzle inlet are
maintained at To = 286.15K, and po 59.1kPa, and
the mixture compositions are given in table 1, with the
index value k 1 assigned to ethanol. The relevant fluid
properties for the various components can be found in
Sidin (2009). Note that single-component condensation
takes place in case 1 and 5.


Yv,l (-)
1.00
0.75
0.50
0.25
0.00


yv,2 (-)
0.00
0.25
0.50
0.75
1.00


gv,i(x10-")
4.379
2.592
1.524
7.366
0.0


gv,2 (xl0-")
0.0
1.127
1.992
2.886
2.931


Table 1: Molar composition of the vapor phase and mix-
ture mass fractions for the ethanol-propanol
mixtures studied in the present investigation;
index 1 is assigned to ethanol.

In the simulation for each test case, a full two-
way coupling is employed. Unless specified otherwise,


\ ----- p/p(-) 05
< t \
0.4
0.0002 0.4
\ /

N 0.3

0.0001 0.2
-.. 0.2

0 0.02 0.04 0.06 0.08
distance from nozzle throat: x (m)


Figure 2: Reconstructed nozzle profile A(x) and di-
mensionless pressure profile for a dry flow of
N2 from Wyslouzil (2000), used for test cases
1 -5.


the backward rate is computed by using the WW2-
equilibrium distribution.

Numerical methods

To solve Eqs. (20) and (25) for the above test cases,
a space marching algorithm is used, which employs
Heun's predictor-corrector method to perform the nu-
merical integration. The calculation starts at a position
which is located slightly downstream of the throat, and
proceeds until the nozzle exit is reached. The binary
computational domain includes droplets with radii of
over ~ 10 ',,. in order to adequately capture the ef-
fects of vapor depletion and latent heat release. There-
fore, the BKE is solved on the domain 0 < nI < 106,
0 < n2 < 106, with I||n| > 1, which allows the binary
DSD to be captured in sufficient extent. To reduce the
computational effort, a sectional approximation is used,
for which the details are extensively described in Sidin
(2009).
In each of the simulations carried out, convergence
of the numerical integration is verified by performing
the solution using successively smaller integration steps
Az. A spatial resolution of Az = 10 ',, on a total
nozzle length of 0.08m was found to yield sufficiently
converged solutions for both the binary DSD and the
flow variables. With respect to the sectional approx-
imation of the BKE, it was found that a number of
NI x N2 400 x 400 bins yielded sufficiently con-
verged values for the liquid mass fractions. By using a
parallel implementation of the current numerical method
on 10 processors, we obtained the solution for the binary
DSD and flow variables in typical run times of 12 hours.




ICMF 2010, Tampa, FL, May 30- June 4, 2010


Results and discussion


Flow field- and BDSD-solutions The solution for the
flow field- and thermodynamic variables for test cases
1 to 5, are depicted in Figures 3.a-f. These solutions
have been obtained by solving Eq. (25) simultaneously
with Eq. (20), with the backward rates calculated using
the WW2-equilibrium distribution. In each case, the in-
fluence of condensation becomes noticeable at a certain
onset point, where the condensational release of latent
heat causes a slight rise in temperature, and a very slight
reduction in the Mach number, as shown in Figures 3.a
and b, respectively. Furthermore, the presence of nu-
cleation zones is observed in the regions where the va-
por activities attain extremal values in Figures 3.c and d.
Downstream of the nucleation zone the vapor activities
are significantly smaller, which leads to a reduced rate
of liquid production, as can be observed from the com-
ponent liquid mass fractions depicted in Figures 3.e and
f.
The solution for the BDSD at the nozzle exit is shown
in Figures 4.a, b, and c for the test cases 2, 3 and 4,
respectively. The influence of the initial vapor compo-
sition is reflected in the solution of the BDSD: As the
ethanol molar fraction is decreased from test case 2 to
4 (see 1, the highest cluster concentration moves away
from the ethanol-axis towards the propanol-axis.
Model validation To validate the condensation model,
the temperatures and partial vapor pressures at
condensation-onset are compared with the measurement
data reported in Wyslouzil (2000). Here, the onset-point
of condensation is defined as the position along the noz-
zle axis where the temperature of the condensing flow
deviates 0.5K from that of a dry flow under choked con-
ditions. The predicted and measured data are listed in
table 2, with corresponding plots given in Figures 5.a, b
and c. The agreement between both data sets is fairly
good, given that the flow model is inviscid and only
quasi-one-dimensional, and that the condensation model
is based on macroscopic theory. The latter is especially
remarkable, since at condensation onset, the liquid dis-
persion consists predominantly of droplets that typically
contain only a few tens of molecules, so that one would
expect the macroscopic theory to be invalid. The pre-
dictions obtained with the current model seem to sug-
gest, however, that the WW2-equilibrium distribution,
which was derived from pragmatic arguments in Wilem-
ski (1995), may still give a reliable description of the
average cluster dynamics at the microscopic level.
Influence of equilibrium distribution In this section,
the influence of the binary equilibrium distribution on
the flow field variables is investigated. The simula-
tions are carried out for test cases 2, 3, and 4, with ei-
ther the WW1- or WW2-equilibrium distribution used


220

S200 200
200

190
lOO%ohahol
180
002 0 04 0 06 008
distance from nozzle throat x (m)
(a)

,5 ,- ,,,----- ,- ,,, ,
25
-- 50 % ethanol
20 1075 %ehmol





-m
S-0 ---- h.-.
0 02 0 04 0 06 008
distance from nozzle throat x (m)
(c)

1 005
-- 25% Ohanol
50% ethanol
..... .l-- -- 75% ethanol

) 003

1002

) 001

0 002 0 04 0 06 0 08
distance from nozzle throat x (m)


i 000

0 001


0% thol
ah--
S75 % ethaol
100- OO% thanol
002 004 0 06 0 08
distance from nozzle throat x (m)
(b)

0 % e&hanol
---- 25%haol
50%aethmol
75 % thmol






002 004 0 06 0 08
distance from nozzle throat x (m)
(d)


25% tha1ol
-- 0 % ethanol
75 % thMol

/





002 004 0 06 0 08
distance from nozzle throat x (m)
(f


Figure 3: Solution of flow field and thermodynamic
variables for test cases 1 to 5 (ethanol percent-
ages are molar fractions, see table 1 for de-
tails): (a) temperature (with insert for the nu-
cleation zone), (b) Mach number, (c) ethanol
and (d) propanol vapor activities, (e) ethanol
and (f) propanol liquid mass fractions.

edxp Ep Texp Tnum
case: p, i P,2 /' T T

1 52.6 49.2 ... ... 210.2 210.9
2 30.9 30.1 10.3 9.5 209.5 212.8
3 18.5 17.5 18.5 17.2 210.7 212.8
4 9.3 8.4 27.8 25.4 212.9 213.2
5 ... ... 27.6 22.9 211.4 207.6


Table 2: Comparison of experimental (index 'exp',
from Wyslouzil (2000)) and simulation data
(index 'num') for vapor pressures pv,k (in Pa)
and temperature T (in K) at condensation on-
set. The uncertainty in the measured onset tem-
perature is 1 K, and the relative uncertainty in
the measured pressure is 5'




ICMF 2010, Tampa, FL, May 30- June 4, 2010


2.5E+05

a -7.9
.0 2.E+05 -18
S-28.9

1.5E+05 -50.0


2 1.0E+05


. 5.OE+04

1E t=n


O.OE+00


1.0E+05 2.OE+05
number of ethanol molecules: n2


6

C)


C)
C;


C)


2.5E+05


1.0E+05 2.0E+05
number of ethanol molecules: n,


60 ,1,-, ,

50 experiment E
50m
simulation

40

30-

20

10 -g

20 40 60 80 100
vapor molar fraction of ethanol (%)
(a)

5 0 , ,

experiment
40o simulation


30


20


10

0 20 40 60 80
vapor molar fraction of ethanol (%)
(b)


2.5E+05


2.0E+05 -18.
-28.
-39.
50.
1.5E+05 -


1.0E+05


5.0E+04


O.OE+OO
O.OE+00 1.0E+05 2.0E+05
number of ethanol molecules: n,


Figure 4: Binary droplet size distribution at the nozzle
exit (x 0.08m), for test cases (a) 2, (b)
3, and (c) 4. The contours correspond with
10 log Cn
c(1,o) + C(o,1)


J2 2 e ,


2220


215


210


'205


200


n experiment
simulation




- [ -

0




0 50 100
vapor molar fraction of ethanol (%)
(c)


Figure 5: Comparison of condensation-onset data (the
figures correspond with the data of table 2):
onset vapor pressures for ethanol (a) and
propanol (b); (c) onset temperatures.




ICMF 2010, Tampa, FL, May 30 June 4, 2010


for calculating the backward rates in the BKE. The re-
sults are shown in Figures 6.a f, where the temper-
ature and component liquid mass fractions have been
plotted for each of the cases mentioned. Clearly, both
the temperature and liquid mass fractions based on the
WW1-distribution are very much different from the ones
corresponding with the WW2-distribution. The use of
WW1 consistently delays the onset of condensation un-
til halfway the divergent section of the nozzle, where
the liquid mass fractions start to rise. In essence, the
fully consistent character of the WW2-equilibrium dis-
tribution causes c'~qWW > cq,ww' for small clusters.
As the number densities associated with the small clus-
ters are very near the equilibrium value, and because the
liquid dispersion is dominated by these small droplets at
condensation-onset, it is concluded that using the WW1-
equilibrium distribution to calculate the backward rate
should indeed delay the onset of condensation.
Given the reasonable agreement found between the
onset temperatures and vapor pressures from experiment
and simulation with the WW2-equilibrium distribution
(see table 2), it is thus concluded that using the WW1-
distribution yields a rather poor description of the aver-
age cluster dynamics at the microscopic scale.


Conclusions

We have validated the condensation model for five test
cases, with different inlet compositions of the vapor
phase. The predicted temperatures and pressures at
condensation-onset are found to be in good agreement
with measurement data, when the fully self-consistent
WW2-equilibrium distribution is used to calculate the
backward rates in the BKE.
Furthermore, we have quantified the sensitivity of the
flow field solution to the equilibrium distribution. The
results obtained by calculating the backward rates in
the BKE using either the partially self-consistent WW1-
equilibrium distribution, or the fully self-consistent
WW2-equilibrium distribution, show profound differ-
ences in terms of predicted temperatures and component
liquid mass fractions. The condensation-onset data pre-
dicted by using the WW1-equilibrium distribution are
shown to be much different from the reported measure-
ments.


Acknowledgements

This research was carried out as part of a PhD-project
within the framework of the Spearhead Program on Dis-
persed Multi-phase Flows of the Institute of Mechan-
ics, Processes and Control-Twente (IMPACT) at Twente
University, the Netherlands.


S ethanol, WW
0003 -- tha0l'V2
----- propanl,W 1
----- pWopol, WW2


0002



002 004 006 0
distance from nozzle throat x (m)
(b)


m nozzle throat x (m) distance from nozzle throat x (m)
(c) (d)


0004- ethanol, 1
230 --- 1 thano 2
Z2 --- propol, W1
220- dyflw 0003 ----- propnolW, W2
210 0002
200 0002
|200-
190 0001
,0o
18 o 0
002 004 006 0 0 002 004 006 008
distance from nozzle throat x (m) distance from nozzle throat x (m)
(e) (f)
Figure 6: Comparison of temperature and liquid mass
fraction profiles for simulations in which the
backward rates in the BKE are computed by
using either WW1 or WW2 for test cases 2 (a,
b), 3 (c, d), and 4 (e, f).



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