7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Evaporation of Tetralin Spray with Direct Quadrature Method of Moments and
Eulerian Multi-Fluid Method


D. Choi*, L. Schneider*, N. Spyrou*, A. Sadiki* and J. Janicka*

Institute for Energy and Powerplant Technology, TU Darmstadt, Petersenstr. 30, 64287 Darmstadt, Germany
choi@ekt.tu-darmstadt.de, schneider@ekt.tu-darmstadt.de, spyrou@ekt.tu-darmstadt.de,
sadiki@ekt.tu-darmstadt.de and janicka@ekt.tu-darmstadt.de
Keywords: Evaporation, Saturation, DQMOM, Multi-Fluid




Abstract

The two classical approaches used to describe two-phase flows are Euler-Lagrange (EL) and Euler-Euler (EE) methods.
With EL methods a spray is treated as a cloud of discrete particles which are tracked in the computational domain.
Therefore, computational costs strongly depends on mass loading. In EE methods (also called multi-fluid method
(MF)) two or more phases are treated as interpenetrating continue. To apply MF to polydispers sprays a range of
droplet size are subsumed in one class or section and treated as one fluid. In this way the droplet size distribution
(DSD) can be resolved accurately but computational costs increase strongly. To overcome this aspect the DSD is
reduced with the direct quadrature method of moments (DQMOM) by approximating the DSD using N abscissas (e.g.
diameter, volume) and N weights. The coupling of DQMOM with the MF requires the transformation of the transport
equations for the weights and abscissas into balance equations for volume fraction and size of droplets. With this
model the evolution of the DSD is tracked by reconstructing the moments (of the DSD) with volume fraction and
droplet size. To account for complete evaporation Fox (2008) extended the procedure for DQMOM by introducing
additional assumptions called ratio constraints.
In this work both methods (MF and extended DQMOM) were coupled and implemented in PRECISE-UNS an
unstructured incompressible CFD Code to simulate the experiments of Wong and Chang (1992). To consider
decreasing or stopping evaporation due to saturation an additional transport equation for evaporated droplets was
implemented.


Nomenclature


Roman symbols
a, b, c DQMOM source term (-)
B Spalding's transfer number (-)
Cd drag coefficient (-)
d diameter (m)
D diffusion coefficient (-)
f distribution function (-)
g gravitational constant (nms1)
m moment (-)
Th mass flow rate (kgs-1)
p pressure (Nm 2)
Q momentum transfer source term (kgs 2rn 2)
r radius (m)
R, evaporation rate
t time (s)
u velocity (ms 1)
v droplet volume (m3)


V volume flow rate (ins 1)
w weight (-)
Y mass fraction (-)
Greek symbols
a Volume fraction (-)
3 factor (-)
6 dirac delta function (-)
p dynamic viscosity (kgs m1)
p density (kgm 3)
a standard deviation (-)
T stress tensor (kgs 2m 1)
Subscripts
g gas phase
k number of integer moment
1 liquid phase
m mean
n phase n
mod modified











particle
droplet surface
far away
initial


Introduction

Multiphase flows appear in natural and technical fluid
flows. In technical applications multiphase flows can
be found in e.g. painting or combustion processes. In
idustrialised societies combustion are used to generate
energy for electricity or passenger transportation. To
reduce the fossil fuel consumption renewable energy
sources are pushed forward but the greater part of en-
ergy sources are still fossil sources which need to be
combusted to generate energy. Therefore optimization
of combustion processes are of great interest. Additional
to experimental investigations the use of computer simu-
lation (computational fluid dynamics) increases to deve-
lope improved and optimized combustion technologies
in e.g. RQL (rich burn quick quench lean burn) or LPP
(lean premixed prevaporized) because of costs. Beside
that research and development are extended to the com-
plete combustion process involving spray generation and
evaporation because these multiphase processes strongly
influence the combustion. In the framework of Eulerian
multi-fluid method with the DQMOM Madsen (2005)
and Friedrich (2006) simulated sprays under considera-
tion of breakup, coalescence and collision. In this work
the Eulerian multi-fluid method and the extended ver-
sion of DQMOM will be implemented to simulate evap-
oration and compared to experimental investigations of
evaporating tetralin (CloH12) droplets.


Eulerian Multi-Fluid Method

In the Eulerian multi-fluid model each phase is treated as
a continuum. The gas phase is considered to be the main
phase. The dispersed phase can be described by a droplet
size distribution (DSD) which needs to be divided into
n different droplet phases with n different droplet sizes
and volume fractions an of phase n. By definition all
volume fractions of the phases sums to unity:

N
S- 1 (1)
n=l

Each phase is determined through balance equation for
mass and momentum. The balance equation for mass is


a(Pnan)+ (PnanUn) mn
a~t ax2


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


where pn is the density and u, the velocity of the phase
n. The source term on the right hand side m, considers
the masstransfer due to evaporation. The balance equa-
tions for momentum reads for each phase

a a
-(PnOnUn) + --(PnanUnUn)
ap ( ) (3)

+ Pnang + QF,n + Qm,n

where p is the pressure shared by all phases, g the grav-
itational vector, ij,n the stress tensor. QF,n and Qm,
are the source terms to account for momentum trans-
fer due to drag and evaporation. The source term QF,n
represents the momentum transfer between the gas and
dispersed phases. If the density of the gas phase is
much smaller than the density of the dispersed phase
(p, < pi) lift forces, virtual mass forces and basset
forces can be neglected. In this work only drag force is
considered and the source term for momentum transfer
reads
3 anpCd
QFn Urel,nr ,urel,n (4)
4 dn
Here dn is the diameter of the dispersed phase n. urel,n
is the relative velocity between the gas phase and the
dispersed phase and reads


Urel,n = Ug Un


Cd is the dimensionless drag coefficient and is given by

2 (1 +0.15Rec0687) Rp< 103,
Cd I
S0.44 Rep > 103,
(6)
p U P urel dn
Rep =-
[9
The multi-fluid model that is employed here can be
found in Hill (1998).

Direct Quadrature Method of Moments

The idea of DQMOM is to determine the evolution of
a DSD in time t and space x. Instead of transporting
the complete DSD f(d) first the DSD is approximated
with so called quadrature variables (weights w, and ab-
scissas dn) with the Product-Difference-alorithm (PD).
Therefore, the first k 0,..., 2N 1 moments mk of
the DSD f(d) needs to be calculated. The PD-algorithm
calculates the weights and abcissas in that way that the
moments m, of the original DSD can be reconstructed
with the quadrature variables


N
mk Wnd.
nl











The original DSD can be written as a sum of weighted
delta dirac-functions
N
f(d) wn,(d- dn) (8)
n-l
In this work the PD-algorithm is used to compute the
initial values for the weights and the abcissas. In Inves-
tigations of Fox (2008) or Schneider (2009) it was shown
that there are difficulties to describe vanishing droplets
due to evaporation with DQMOM. Fox (2008) intro-
duced an extended version of DQMOM to account for
vanishing droplets for continuous DSDs. Therefore Fox
used a DSD with the properties droplet volume v, and
droplet velocity u, and the approximated DSD f(v, u)
reads
N
f(v, u) wn6(v v)6(u n) (9)
n-l
where 6(u Un) = 6(u1 ul, n)6(U2 u2, n)6(U3 -
U3, n). The starting point to derive the DQMOM-
transport equations is the Williams-spray equation
which reads


a a.
f + (uif)
8 8f+


a (Rf)
8v


if only evaporation is considered. In this equation Rv
is the rate of evaporation. If (9) is substituted into (10)
three DQMOM-transport equations can be derived.

Wn + (Wnun) = an, (11)
at ax

a(PnvnWn) + (PnvnWnUn) = Pnbn (12)
at d0x
a a
at(pnvnwnun) + a9(Pnvnwnunun) PnCn (13)

where an, bn and c, are so called DQMOM-source
terms which can be evaluated by solving systems of lin-
ear equations. To determine these source terms moment
transformation is applied to (10) and moment constraints
are used. The resulting DQMOM-source terms can be
split into two parts
bn = b* + wRV ,
bn WnR(14)
Cn = Cn + WnUnRv.
Note that the first part does not account for vanishing
droplets. With respect to vanishing droplets Fox (2008)
used additional so called 'ratio constraints',

D wn) 0
Dt Wn+i)
D (n) 0 (15)
Dt \Vn+l
o-


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Using the 'ratio constraints' following expressions for
the second part of the source terms can be obtained ac-
counting for vanishing droplets
N
S an --,
n=l
N
b 0, (16)
n=l
N

n=l
where I is the evaporative flux. Further expressions to
compute the unknowns are
Wn+la, wnan+ = 0
(17)
for n 1=,...,N-1,

Wn+lVn+lbn-' ,*
WnWn+[Vn Rv(Vn + 1) Vn+lRv(Vn)] (18)
for n 1,...,N- 1,
Wn+lVn+lUn+lCn -- WnVnUnCn 1
UnUn+lWnWn+l[VnRv(Vn+l) Vn+nRv (Vn)] (19)
for n= 1,...,N-1


an = 3Wn,

N
n- nU~Wn


Details can be found in Fox (2008).

Coupling of Multi-Fluid and DQMOM


The fact that (12) and (13) are equivalent to the Eule-
rian multi-fluid model allow an easy way to couple both
methods. The quadrature variables and the source terms
used in DQMOM have to be modified to be compatible
with the Eulerian multi-fluid model. Using (2) and (3)
mass balance and momentum balance are determined.
The balance equations for the weights are not included in
the Eulerian multi-fluid model and additional equations
have to be implemented and used. Using the relationship

C = WnVn (21)
the missing DQMOM transport equation (11) can be ex-
pressed with the volumefraction a of the Eulerian multi-
fluid model.
(an) + --Un) anmo d, (22)
at \ v,n i \vx
where an,mod is the modified source term. All source
terms of the DQMOM procedure are modified due to
the fact that the variable Wn does not exist explicitly.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Evaporation and Saturation

An evaporation model of the form


d2(t) do E~t


is used in this work. Here, do is the initial droplet diam-
eter and E, the evaporation rate. The evaporation rate is
calculated with following equation.


E, = l8 n(1 + By)
Cppi


where A is the thermal conductivity and c, the specific
heat at constant pressure. By is Spalding's mass transfer
number. With the assumption Le 1 Spalding's trans-
fer number can be computed with mass fractions Y or
temperatures T


900
800
700
600
S500
" 400
Z 300
200
100
0 10 20 30 40 50
Diameter( im)


60 70


Figure 1: Number vs. diameter of droplets from experi-
ment Wong and Chang (1992)





I9


B


c. (TL,2
Ah,


where Yoo, Too is the mass fraction, temperature far
away from the droplet and Y,, T, the mass fraction, tem-
perature at the droplet surface. Ah, is the latent heat of
vaporization at temperature T,.
The effect of saturation can be modelled by account-
ing for the mass of the evaporated liquid. Therefore a
further transport equation is used which has the form


a (pY) + (pYu) (pD, aY) K, (26)
at Ox Ox 6xi

where D12 is the diffusion coefficient and K, the source
term accounting for evaporated liquid. The fact that
the information about evaporated liquid is provided by
DQMOM-source term bn the source term K, can be ex-
pressed by

N
Kv P= PIP bn,.mod (27)


With this procedure it is possible to modify the evapo-
ration law to account for saturation. The performance of
this method will be shown in following sections.


Experiments of Wong and Chang

Wong and Chang (1992) considered a tetralin spray that
undergoes acceleration, heating and evaporation in a
slowly moving nitrogen (N2) gas stream which moves
through a cylindrical tube. This tube is facing in the
direction of gravity. The measurements are conducted
with a phase doppler particle analyzer (PDPA) at the end
of the tube. The length of the tube is varied from 2 to 65
cm. The tube has a diameter of 2 cm and is insulated in


Ev


Condensing
chimney


Mixing
port pr
S, rit
aporation "." --- i :lili.:-.r


heated
rogen


tube


PDPA


Figure 2: Schematic of experimental apparatus


order to simulate an adiabatic evaporation environment
and to avoid condensation on the tube wall. It is pre-
heated to 75C. At the entrance the tetralin spray has a
mean velocity of approx. 0.9 ms 1, a mean temperature
of approx. 308 K and a mean diameter of approx. 30
pm. The measured DSD is depicted in Figure 1 and a
schematic of experimental apparatus is shown in Figure
2. The nitrogen gas surrounding the droplets has an ini-
tal velocity of approx. 1 ms 1, a mean temperature of
approx. 348 K and an approx. mass fraction of tetralin
vapour of 5 10-4. This experiment was conducted un-
der atmospheric pressure conditions. All measurements
were made on the axis of symmetry.


Numerical Investigation

The numerical investigation was performed using the
Eulerian multi-fluid model coupled with the extended
version of DQMOM. Forces on droplets (drag and grav-
ity), evaporation and saturation were considered. In this
work four numerical setups was calculated and com-
pared to the experiments of Wong and Chang (1992).








7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Table 1: Summary of gas and droplet properties used in
computations.

Variable Value Units

PN2 2.005 10-5 kg/sm
PN2 0.9676 kg/m3
PC1oH12 970 kg/m3
Re 869
dm 30 /rn
cr 2.5



For the calculations, the DSD was approximated with
the PD-algorithm with three weights and three abcis-
sas. The given DSD (see Figure 1) from the experi-
ment was represented by a normal distribution with a
mean diameter dm = 30 pm and a standard deviation of
- = 2.5. For this normal distribution the first moments
mk (k = 0,..., 5) were calculated. The PD-algorithm
provides initial values for droplet sizes and volume frac-
tions for each dispersed phase (N 3). The computa-
tional domain is modelled with a cylindrical tube with
a length of 1 m. The inlet velocities of the gas and the
droplets were generated using following equation


U(r) 2V
7-2


S)
2


where r is the radius of the tube. The volume flow rate
was adjusted to match 0.9 ms 1 on the axis of symmetry.
The properties of gas and dispersed phase are summer-
ized in Table 1.


Results and Discussion

In this section results of four different setups are com-
pared to the experiments of Wong and Chang (1992).
In the experiment the remaining volume fraction vs time
were measured for different numbers of tetralin droplets.
The following figures show the remaining droplet vol-
ume fraction vs time of numerical simulations compared
to the measurements. In Figure 3, 50 droplets per cm3
were evaporated. The measurements and the numerical
simulation of this case show that all tetralin droplets are
evaporated after approx. 0.45 s and the remaining vol-
ume fraction reaches zero. The numerical results are
in very good agreement with the measurements. Fig-
ure 4 shows the case with 1000 droplets per cm3. The
increased number of droplets results in an decreasing
evaporation rate that the measured time is to short for
complete evaporation. The numerical results shows sim-
ilar behaviour and a decreasing evaporation rate. The
case with 1500 droplets per cm3 is shown in Figure 5
and the case with 2100 droplets per cm3 in Figure 6. The


0.4 0.6
time (s)


Figure 3: Remaining droplet volume fraction vs time
for 50 droplets per cm3


.* 0.8

o 0.6
2 \

0.4

0.2
,02
0
0


Simulation
Experiment


0.2 0.4 0.6
time (s)


0.8 1


Figure 4: Remaining droplet volume fraction vs time
for 1000 droplets per cm3


1

. 0.8



S0.4
bO
" 0.2

0
0


Simulation
+Experiment



^-----______


0.2 0.4 0.6
time (s)


0.8 1


Figure 5: Remaining droplet volume fraction vs time
for 1500 droplets per cm3














.I 0.8

0.6
-5
S0.4

0.2

01
0 0.2 0.4 0.6
time (s)


Simulation
Experiment


Figure 6: Remaining droplet volume fraction vs time
for 2100 droplets per cm3


high number of droplets results into saturation of the gas
and the volume fraction of tetralin droplets remain con-
stant. The comparison between the experiments and the
numerical results are in very good agreement.


Conclusions

The results of the computations have shown that the
Eulerian multi-fluid model coupled with the extended
version of DQMOM can be used to simulate complete
evaporation of droplets. The used evaporation law was
extended to account for saturation using an additional
transport equation for mass fraction and the DQMOM-
source term. This model can capture the effect of satura-
tion and decreasing evaporation rates although tempera-
ture change is not included.
However, the application of DQMOM to real systems
requires further investigations.


Acknowledgements

This work is part of the Graduiertenkolleg 1344 at TU-
Darmstadt and financially supported by the DFG.


References

Fox R. O., Laurent F. and Massot M., Numerical simu-
lation of spray coalescence in an Eulerian framework:
direct quadrature method of moments and multi-fluid
method, J. Comput. Phys., Vol. 227, pp. 3058-3088,
2008

Friedrich and Weigand B., Eulerian Multi-Fluid Simu-
lation of Polydispers Dense Liquid Sprays by the Direct
Quadrature Method of Moments, ICLASS 2006.

Hill D. P., The Computer Simulation of Dispersed Two-
Phase Flows, Ph.D. Thesis, University of London 1998.


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Madsen J., Solberg T. and Hjertager B. H., Numerical
simulation of spray by the direct quadrature method of
moments, ILASS 2005.

Madsen J., Hjertager B. H., Solberg T., Norskov P. and
Rusas J., Application of the direct quadrature method of
moments to Y-Jet water sprays, ICLASS 2006.

Marchisio D. and Fox R., Solution of population bal-
ance equations using the direct quadrature method of
moments, J. of Aerosol Science, Vol. 36, pp 43-73, 2005

Schneider L., A Concise Moment Method for Unsteady
Sprays, Ph.D. Thesis, TU Darmstadt 2009.

Wong S-C. and Chang J.-C., Evaporation of non-dilute
and dilute monodisperse droplet clouds, Int. J. Heat
Mass Transfer, Vol. 35, pp. 2403-2411, 1992