7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Direct Quadrature Method of Moments for Multicomponent Droplet Spray
Vaporization
Anne Bruyat*, Claire Laurent* Olivier Rouzaud*
ONERA, 2 avenue Edouard Belin, 31055 Toulouse, France
anne.bruyat@onera.fr, claire.laurent@onera.fr and olivier.rouzaud@onera.fr
Keywords: Multiphase flows, Multicomponentfuel droplet vaporization, Direct quadrature method of moments
Abstract
Complex composition of fuel oils makes modelling droplet and vapour composition during spray vaporization process
difficult. The classical Discret Component Model (DCM) should not be used in CFD codes when the number of species
is too large, because of its prohibitive computational time. An alternative is to model the droplet composition by a
probability density function (PDF) : this is called the Continuous Thermodynamic Model (CTM). Hallett (2000) first
assumed the PDF to be a Ffunction, but Harstad et al. (2003) demonstrated presuming the PDF mathematical form is
a weak point when vapour condenses on the droplet surface. The Quadrature Method of Moments (QMoM) does not
require any hypothesis on the PDF form, and only deals with moments of the PDF. Following Lage (2007) work on
phase equilibria, QMoM was implemented and studied by Laurent (2010). Even though the theorical robustness of the
method is mathematically proved (existence of a solution), this method may strike numerical difficulties. Indeed the
ProductDifference algorithm solving the problem is illconditionned and may lead to numerical errors in some cases
investigated in this work. Then an alternative method to link the evolution of moments to the evolution of descripting
variables of the problem is implemented to solve the droplet vaporization. The Direct Quadrature Method of Moments
(DQMoM), already used to study Polydisperse GasSolid fluidized beds (Marchisio and Fox, 2005) reveals as being
effective. Results and performances are analysed in this article, and show that DQMoM is a pertinent approach in
modelling droplet vaporization.
Nomenclature x molar fraction
Roman symbols
BM molar Spalding number (1)
c molar density (mol.m 3)
D Diffusion coefficient (m2.s 1)
f distribution function
F Function
I distribution parameter
Jp Jacobian matrix
lb Latent heat of vaporization
at normal boiling temperature(m2.s 2)
m moment
M moment vector
n molar flow rate (mol.s 1)
N number of pseudocomponents in the droplet
Av number of components in the droplet
P pressure (Pa)
R radius (m)
Sh Sherwood number
t time (s)
T Temperature (K)
U vector
Greek symbols
6 delta function
Subscripts
f fuel
9 gas
1 liquid
nb normal boiling point
s surface
sat saturation
Superscripts
a moment order
i relative to the component i
k relative to the pseudocomponent k
tot total
Introduction
The composition of fuel droplets has igniilik.ii effects
on the vapour flow rate computation, especially if nu
medical simulations investigate low temperature and low
pressure conditions. Indeed each fuel component has
its own boiling temperature. Important differences be
tween theses temperatures lead to a sequential vaporiza
tion, from light to heavy species. The classic Discrete
Component model (DCM) has a prohibitive computa
tional time since tens of components mixed in the fuel
are considered. Then it should not be used in CFD codes
for classical combustion applications, unless few repre
sentative species are able to correctly model the whole
composition. A less costly alternative, called the Con
tinuous Thermodynamics model (CTM) is to describe
the fuel composition with a probability density func
tion (PDF). Assuming the PDF is a Ffunction (Hal
lett, 2000) reduces the number of unknowns to few pa
rameters and improves the computational performances.
However, the drop composition PDF can be very dif
ferent from a Ffunction. Consequently, assuming PDF
shape does not guarantee the accuracy and the numeri
cal robustness of this model. As an example, the model
is inaccurate in the case of fuel vapour condensation,
because of a secondary peak appearing on the liquid
composition PDF. Modelling the PDF with a double F
function (Harstad et al., 2003) is in good agreement with
the classic Discrete Component model. However good
results are not ensured in all situations because of the
presuming the PDF shape. Another solution is to as
sume the fuel composition PDF f to be a sum of Dirac
functions : f(J) E= =XkAi, G (I) where I is the
distribution parameter and represents for example the
boiling temperature, and k the weight associated to lk,
corresponds to the molar fraction of the component k.
This model differs from the Discrete Component model
thanks to a method allowing the {I }k to change : the
weights and the corresponding abscissas are computed at
each time step from the moments. The number of pseu
docomponents N, is much smaller than the real number
of components. The key point of this method is to com
pute explicitly the {Ik } E [1, N] and the corresponding
weights from the first 2N moments of the PDF : this
corresponds to a Gaussian Quadrature. The Quadrature
Method of Moments (QMoM) has already been success
fully applied for population balance equations (McGraw,
1997) and thermodynamics equilibrium (Lage 2007). It
was then implemented to model multicomponent vapor
ization (Laurent, 2009 and 2010). This method enables
to reduce computational time and gives satisfactory re
sults, especially in case of vapor condensation. How
ever, the ProductDiffteren, e algorithm, used in QMoM
to compute weights and abscissas, fails when boiling
points of pseudocomponents in the gas and vapor phase
are the same, and if the weight of the pseudocompo
nent is very small. A derived method to solve Gaussian
Quadrature, called Direct Quadrature Method of Mo
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
ments (DQMoM), and which does not use this algorithm
is then implemented. At each time step, the evolution
of the weights and the abscissas are directly computed
without solving the Moment Problem. This method was
also applied to population balance equations (Marchisio
and Fox, 2005). Results are similar to the Quadrature
Method of Moments in casual cases, and they are im
proved in the difficult case described above. The numer
ical robustness of this method makes it interesting for
CFD codes.
Sample Section
1 Continuous modeling of the droplet
composition
This part describes the Continuous Thermodynamic
model (CTM), and especially how QMoM and DQMoM
are applied to solve the multicomponent droplet vapor
ization.
1.1 Description of the droplet
composition
In this study, the composition of the droplet is supposed
to be uniform, but it is not constant during the vaporiza
tion process : due to the volatility differences between
components, the mole fraction x' of each component i
evolves. The Discrete Component model (DCM) com
putes it each component evolution, whereas the CTM
approach models the droplet composition thanks to a
probability density function (PDF). This is an effec
tive way to reduce significantly the computational cost.
The chosen distribution parameter I of the PDF x4
fi(I)AI is the normal boiling point (I = Tb), since
this is a direct parameter for the heat of vaporization
and the saturation vapor pressure. The molar mass is
a common parameter (Hallett, 2000 and Harstad, 2003),
however it is less relevant to characterize the physical
behavior of the component during the vaporization pro
cess. In both QMoM and DQMoM methods, the droplet
composition is discrete and can be interpreted as a Dis
crete Component model using N pseudocomponents, to
stand for N real components (N < AN) of which nor
mal boiling point may change during the vaporization
process to best fit the real droplet composition. Hence
the PDF is the sum of N delta functions determined by
the mole fraction x and the normal boiling point 1 :
LI1 IIV~ I~lLIIIr~i LII LI~ 1VII~lIV~lll~jYVII 1
N
fi,N=(I) 6 (I)
k=1
Classical equations describing droplet composition dur
ing vaporization are integrated to be solved using the
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
PDF's moments of the droplet composition. The mo
ment of order a is given by
m0
f(I)Iidl
The integration, in the case of QMoM and DQMoM, can
be simply written as
N
k=l
Then an important I i p1 'llc, i, of the Continuous Ther
modynamic Model is to assume the physical properties
can be interpolated as functions of I Tb. Com
ponents are classified into homogeneous groups, such
as nalkanes, isoalkanes or cycloalkanes, according to
their physical properties. Modeling the droplet vaporiza
tion with QMoM and DQMoM gives satisfactory results
from 2 or 3 pseudocomponents for each homogeneous
group, instead of tens real components for the complex
mixture. Physical properties calculations for the Contin
uous Thermodynamic model are taken from the Reid's
book (Reid et al., 1977), and those implemented for the
kerosene composition in this study are presented in the
QMoM study applied to multicomponent spray vapor
ization (Laurent, 2010).
1.2 Droplet composition evolution
Since the multiPDF model uses a classical DCM ap
proach applied to groups instead of components, the fol
lowing section details only one group of components
equations.
Considering the composition to be uniform in the
droplet, the mass conservation equation, for each com
ponent i, can be written as:
dxL 3
t 4 (ht otx ) (4)
dt 47R3C
where c is the molar volume density and R is the radius
of the droplet. ntot is the total vapor molar flow rate, and
h represents the vapor molar flow rate of the component
i, and are related thanks to :
2dx'
h'i(r) = htj x (r) 472 C9gDShl c (5)
where D' is the diffusion coefficient. Introducing the
Sherwood number:
2R Ldxr
Sh' )} (6)
S r s\ dr( (6)
in Eq. (5) leads at the droplet surface (i.e r R) to :
nhi 2htotwi 2_ Rc, D Sh'(xr r x ,) (7)
2 7CggSrhgXg, s
A second Ilips' ll,'ii is done to perform the integra
tion of the continuous form and to obtain PDF moment
equations : the diffusion coefficients D' and Sh1 the
Sherwood number are assumed to be the same for all
the components (i.e D = Dg and Sh = Sh,). Ex
perience shows that this common assumption leads to a
little less good, but acceptable results, and so even in
the difficult case where the vapor composition is rather
different from the droplet composition.
The molar Spalding number BM is defined by :
tot tot
xBM ~  '0 (8)
1 tot
9 X,S
where' '.is obtained from the phase equilibrium at the
droplet surface and x,,,o, the total vapor mole fraction
in the cell, is known. Considering the previous hypoth
esis and the BM definition, the sum of Eq. (7) over all
components i gives :
ntot = 2W7RcgDShgBM (9)
When htot 0, a specific analysis of the model is
given (Laurent, 2010). We will only consider the case
nhtot 0 (i.e BM / 0). Hence, the term ShgD is
replaced in Eq (7) to introduce BM :
X'
htot 9' BM c9'00
Then, the equation (4) becomes :
3htotr
X (1 + BM)
BM
x,)o ) (11)
The molar fractions are multiplied by I" and, thanks
to the two Ii\hi.,Ilcci of the Continuous Thermody
namic Model, the sum is done over the components (Eq.
3) and leads to the PDF moment evolution equation :
"dt 4 3tt (1 + BM )
(12)
To numerically solve this equation, constituying the Mo
men Problem, the total vapor flow rate ntot and the PDF
moments at the droplet surface m", need to be known.
1.3 Vapor flow rate calculation
The total vapor flow rate ntot is derived from the inte
gration of Eq. (5) between the droplet radius R and the
limit of the boundary layer Rf of the droplet (Sirignano,
1999).
tot (1+ tot 1) p( tot 1
X,+( 4)c D R
91, 910 kV19
1;))
(13)
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Then the total vapor flow rate equation is derived from
this equation, using the BM definition:
ht't 2RcDgSt.*, (1 + BM) (14)
where Sh* is the ilihi, ., . ,/number.
Sh* 2R (15)
9 Rf R
The radius of the boundary layer Rf is given by Ranz
Marshall correlations depending on the Reynolds num
ber, the Schmidt number and the Spalding number
(Sirignano, 1999):
1.4 PDF moments at the droplet
surface
The moments of the vapor composition at the droplet
surface {ma}N1 are computed from the moments
of the droplet composition {ma }N 1, using the vapor
liquid equilibrium at the droplet interface. For ideal mix
tures, the Raoult's law governs the phase equilibrium at
the interface :
x9"PO x Psa(Ts)
9,S ,sat~s
ex
The
uous
where Psat(Ts) is the saturation vapor pressure
pressed at Ts, the temperature at the droplet surface.
Raoult's law should be integrated from its contain
form
fT usius o fe(I)Psat(I, TP)
The ClausiusClapeyron equation gives Psat :
1))
Hence the integration of Eq. (17) using the moment
method PDF fl,N gives:
SPsat(I TI ) (
k=1
1.5 Solving the moments equations
The keypoint of QMoM and DQMoM approaches is to
compute the normal boiling point {1kI} and the mole
fraction {xk}f for the N pseudocomponents. QMoM
and DQMoM are two different ways to solve this prob
lem.
1.5.1 QMoM
The QMoM approach consists in solving Eq. (12)
and searching the N nodes {Il}k and the N weights
{x" }I of a Gauss quadrature from the set of moments
{m }2N. The Moment Problem was studied and
the existence of a solution was proved (Laurent, 2010).
The ProductDiftereniie algorithm is used to perform the
computation (Gordon, 1968 and McGraw, 1997), but re
sults show that the algorithm may numerically fail in
some cases.
1.5.2 DQMoM
Unlikely the QMoM approach, integration of Eq. (12)
is not required for the DQMoM approach, hence it
does not use the ProductDiffterenc, algorithm. Indeed,
temporal evolution of the N nodes {Ik } and the N
weights {xz} are directly computed without using mo
ments.
Consider the F function :
F: U M
U \ / ,,,
SUN XN
UN+ = x111
U2N XNIN K2N
where
(17) f U)
mn x111a +  .+ aN f ,(U)
(18) m 2N1) 1 2N1 2N1 \f2N1 )
(18) (22)
(22)
Eq. (12) can be rewritten:
3&tot (
47wR3C ( I
"(1 B+M)
BM
This is equivalent to:
SR
dF(U)
dt
R2N 1/
(23)
t (24)
where R is a vector containing the 2N righthand side
of Eq. 23.
Psat(1, Ts) P"exp  (
SR G
The left hand side term is differentiated :
dF dU
dU R
dU dt
where JF is the Jacobian matrix. Differentiating the
functions {fk } with respect to the variables {ua } gives
(1 )( k
k=N+ 1,2N f (UN k)1
9Uk oa 1
(1 a)'I
(26)
= akV1
(27)
The Jacobian Matrix is similar to the Polydisperse
GasSolid fluidized beds study (Marchisio and Fox,
2005). The block matrix is :
JF (A B)
where the blocks A and B are :
1
0
I2
(1 a)I ?
\2(1 N)I2N1
0
1
211
a(2N 1
\(2N 1)2
1
0
2
2(1 N)I]f
0
1
21
2 1)
(2N 1) NN
(28)
E
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Table 1: Initial conditions
Variable Value
Composition of the droplet xkero 1.
Composition of the vapor x:60H14 = 0.3.
'ar = 0.7.
Diameter of the droplet (pm) 50.0.
Temperature of the droplet (K) 300.
Ambiant temperature (K) 500.
Ambiant pressure (bar) 5.0.
0.12
0.1
0.08
0.06
0.04
0.02
DCM
.... . ,, I I, . I .
300 350 400 450 500 550
Tnh (K)
1/
(29)
2/
(30)
Considering the boiling temperature {Ik} I 0 are dif
ferent, the Jacobian matrix Jp can be inversed, leading
to:
dt
An integration step leads to the temporal evolution of
{uk} and consequently of{xk} and {xklk}.
2 Results analysis
2.1 Test case description
A relevant test case is investigated for the QMoM and
DQMoM application to the multicomponent droplet va
porization modeling. Indeed this example is all the more
Figure 1: Initial composition of the droplet (DCM)
interesting since the GammaPDF model does not solve
properly the droplet vaporization if vapor condenses : in
this case the droplet composition PDF does not fit with
a Ffunction (Harstad et al, 2003). The Discret Com
ponent Model (DCM), which solves equations for each
component of the mixture, is considered as the reference
model in this study.
The test case presented in this part studies kerosene
droplets vaporization in air rich in vapor, composed with
isoC6H14, the most volatile component of the kerosene
(To ~6H14 = 331K). Initial conditions are presented
in Tab. 1. In this example, a oneway coupling ap
proach, numerically more difficult, is investigated : it
corresponds to a test case where the vapor composition
remains constant.
Initial composition of the droplet is plotted through
the mole fraction of each component for the Discrete
Component Model (DCM) in Fig. 1, and through the
nodes and the weights of the quadrature methods of mo
ments (QMoM and DQMoM) in Fig. 2.
kOfa
k =l1, N
Odu
.1
0.8
0.7
0,6
0.5
0.4
0,3
0.2
0.1
0
3
350 400 450 500 550
Tnh (K)
Figure 2: Initial composition of the droplet (QMoM and
DQMoM)
2.2 Global evolution
In order to understand the vaporization phenomenon, the
vapor flow rate evolutions of the droplet are first anal
ysed for the three different computing methods. The
moment methods (QMoM and DQMoM) are compared
to Discrete Component Model, which stands for the ref
erence solution. Fig. 3 represents the droplet vapor flow
rate evolution and only DQMoM results are plotted since
QMoM results are very close to DQMoM ones. The va
por flow rate is firstly negative due to the condensation of
isoC6H14, and becomes positive when the droplet com
position is rich enough in isoC6H14. Even if the droplet
lifetime is underestimated because of the Continuous
Thermodynamic model I.\pil.sIlcc, (Laurent, 2009), re
sults given by the moment methods are in good agree
ment with the reference model (DCM). As expected, in
creasing the number of pseudocomponents improves the
accuracy.
Fig. 4 illustrates the droplet mean boiling point evo
lution (i.e the first order moment ml). Moment methods
are compared to a DCM approach computed with the
two CTM Ipi illccw, described in parts 1.1 and 1.2 (the
physical properties can be interpolated as functions of
I Tnb, the diffusion coefficients and the Sherwood
number are assumed to be the same for all the compo
nents of a group). Indeed the approach would corre
spond to the converged results of moment methods (i.e.
the number of pseudocomponents is equal to the number
of components N = N). The evolution of the normal
boiling point follows the evolution of the vapor flow rate.
In particular, the temperature decreases during the con
densation of isoC6H14. At this step of the study, the
drawbacks of the PDalgorithm used in QMoM do not
clearly appear. However, the first order moment evo
lution (Fig. 4) points out a difference between QMoM
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
and DQMoM at the end of lifetime, when using 4 pseu
docomponents (i.e N 4). The following analysis of
nodes and weights evolution reveals QMoM difficulties.
Figure 3: Vapor flow rate evolution for different meth
ods
500 I I 
480 0 0 o0oo o
S460
..... .......................
440
S' DCM with CTM hypotheses
SDQMoM N=4
x DQMoM N=3
420 DQMoM N=2
o QMoM N=4
0 0.01 002 003 0.04 0.05 0
time (s)
Figure 4: First order moment (mean Tnb) evolution for
different methods
2.3 Node and weight evolutions and
convergence
Nodes and weights are displayed for the droplet's com
position, and for both QMoM and DQMoM. First Fig. 5
and Fig. 6 provide results for N = 3 (i.e three pseudo
components). The node and the weight k = 1 in Fig. 5
and Fig. 6 point out the C6H14 vapor condensation at
the beginning of the droplet vaporization : the C6H14
N=2
I N=3 .
SN=4 
I I I
I
I
I
I
I
I
II I
I ~I *
00
mole fraction in the droplet increases. It then decreases
in favour of an accumulation of heavy components, due
to the vaporization phenomenon.
08 ,
S   
"k I
a06 ki
304 *~ k2
=02  .
0 001 002 003
Lime (s)
700
600 
300 L
0 001 002 003
time (s)
Figure 5: Evolution of the QMoM
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
It was found out that the switch happens if one of
the pseudocomponent k has a low mole fraction xz and
if its boiling temperature I" remains constant (Laurent,
2010). It is due to the illconditionned calculation of
the ProductDiff'ereine algorithm employed to solve the
Moment Problem. As explained in the first part, DQ
MoM does not use this algorithm, and then avoids nu
merical difficulties (see Fig. 7 and Fig. 8).
004 005 006 0
   .kI
'04 *" *,i k2
S k=3
,2 k 02]
.. .;_ x %" I
0 001 002 003
time (s)
004 005 006 550
450 .
nodes and weights 4 0  J .....
nodes and weights 400oo
for the droplet composition, using 3 pseudocomponents
(N 3)
001 002 003
time (s)
004 005 006
0 01 002 003
time (s)
004 005
Figure 7: Evolution of the QMoM nodes and weights
for the droplet composition, using 4 pseudocomponents
(N 4)
600 
S500 ....................
400
S 001 002 003 004 005 006
time (s)
Figure 6: Evolution of the DQMoM nodes and weights
for the droplet composition, using 3 pseudocomponents
(N =3)
A noteworthy difference between QMoM and DQ
MoM is the evolution of the nodes and weights at the end
of the phenomenon : the nodes and weights k 2 and
k = 3 in Fig. 5 normally switch in the QMoM case. It
should not theorically happen since an obvious solution
of the Moment Problem is that once the nodes plateau,
they remain constant (Laurent, 2010). In this case, this
inversion has no repercussion on moments, and QMoM
and DQMoM results remain comparable (see for exam
ple the vapor flow rate).
time (s)
450
H 400
350 
001 002 003
time (s)
004 0 05 006
Figure 8: Evolution of the DQMoM nodes and weights
for the droplet composition, using 4 pseudocomponents
(N 4)
Fig. 7 displays the nodes and weights evolution for
QMoM N = 4, and shows numerical difficulties coming
with the increase of the number of pseudocomponents.
First weights and nodes are clearly unstable compared
to DQMoM results Fig. 8, and then the weight k = 4
becomes equal to 0. The computation is then finished
Sk=2
t3
k=37
x ~"^
;;;;;;
1 1 .3maa mrm~ow a
i ,, 
004 005 006

with only three pseudocomponent N = 3. Once again,
this does not have a NiInilik.iiii impact on the physical
results (Fig. 4), since the solution is quasiconverged
with 3 pseudocomponents. However, and because in an
other case QMoM may not be reliable, it is important
to underline that DQMoM is accurate, even for a large
number of pseudocomponents, since it avoids the use of
the ProductDiffere ce algorithm.
2.4 Computational efficiency
The computational cost of the Continuous Thermo
dynamics model (QMoM and DQMoM) make these
method to be really advantageous compared to DCM,
as long as a small number of pseudocomponents is used
(N = 2 or N 3). Indeed, using many pseudocom
ponents (N 4) does not improve the solution, and is
time consuming because it needs a smaller timestep, or
an iterative method. For this test case, the computation
time decreased at least of I .' for QMoM and DQMoM
(N = 2 or N 3) compared to the Discret Component
Model (DCM).
Conclusions
The Quadrature Method of Moments (QMoM) has been
implemented for the vaporliquid equilibrium (Lage,
2007) and then extended to the modeling of the droplet
vaporization phenomenon (Laurent, 2009 and 2010).
In the case of the Continuous Thermodynamic Model
resolution for the multicomponent droplet vaporization,
QMoM may fail because of the ProductDif'erelnce al
gorithm. Following Marchisio who applied DQMoM
to population balance equations (Marchisio and Fox,
2005), the article studies an alternative method to solve
the pseudocomponent evolution during the droplet va
porization. QMoM and DQMoM accuracy and robust
ness have been compared and analyzed in this study.
Both methods have been tested in the difficult case of
the oneway coupling vapor condensation. The improve
ments and reliability brought about by DQMoM have
been pointed out. The QMoM and DQMoM computa
tional time are satisfactory as long as few pseudocompo
nents are used (N < 3), and the convergence is reached
with only three pseudocomponents (N = 3). By way
of conclusion, DQMoM avoids the ProductDifferenne
algorithm failure and keeps the benefits of QMoM for
modeling droplet vaporization in case of vapor conden
sation. Consequently, DQMoM appears to be a relevant
method to compute multicomponent droplet vaporiza
tion in CFD code.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
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