Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 8.5.2 - Two-Phase Subgrid-Scale Stress Models for Two-Fluid Large-Eddy Simulation of Gas-Particle Flows
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 Material Information
Title: 8.5.2 - Two-Phase Subgrid-Scale Stress Models for Two-Fluid Large-Eddy Simulation of Gas-Particle Flows Particle-Laden Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Zhou, L.X.
Liu, Y.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: sub-grid scale model
LES
gas-particle flow
 Notes
Abstract: In present Eulerian-Lagrangian and Eulerian-Eulerian (two-fluid) large-eddy simulations (LES) of gas-particle flows most investigators use single-phase subgrid scale (SGS) stress models. The Interaction between the two-phase SGS stresses is totally not or at least not fully taken into account. In this paper, a unified second-order moment (USM) two-phase SGS stress model and a two-phase SGS energy equation (k-kp) model for the two-fluid LES of gas-particle flows are proposed, in which the interaction between the two-phase SGS stresses is fully taken into account. The proposed model is used in the LES of swirling and sudden-expansion gas-particle flows, together with RANS modeling using the USM two-phase turbulence model. The two-phase time-averaged velocities predicted by LES-USM/LES-k-kp and RANS-USM models are almost the same and are in good agreement with the experimental results. However, for two-phase RMS fluctuation velocities, the LES-USM and LES-k-kp results are better than the RANS-USM results.
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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Resource Identifier: 852-Zhou-ICMF2010.pdf

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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


Two-Phase Subgrid-Scale (SGS) Stress Models for
Two-Fluid Large-Eddy Simulation of Gas-Particle Flows



Lixing Zhou1, Yang Liu2

1 Department of Engineering Mechanics, Tsinghua University
Beijing, 10084 China, Zhoulximail.tsinghua.edu.cn
2 Marine Engineering College, Dalian Maritime University,
Daliani 16026, China, liuvaA@mail.tsinghua.edu.cn


Keywords: sub-grid scale model; large-eddy simulation; gas-particle flows


Abstract

In present Eulerian-Lagrangian and Eulerian-Eulerian (two-fluid) large-eddy simulations (LES) of gas-particle flows most
investigators use single-phase subgrid scale (SGS) stress models. The Interaction between the two-phase SGS stresses is
totally not or at least not fully taken into account. In this paper, a unified second-order moment (USM) two-phase SGS stress
model and a two-phase SGS energy equation (k-kp) model for the two-fluid LES of gas-particle flows are proposed, in which
the interaction between the two-phase SGS stresses is fully taken into account. The proposed model is used in the LES of
swirling and sudden-expansion gas-particle flows, together with RANS modeling using the USM two-phase turbulence model.
The two-phase time-averaged velocities predicted by LES-USM/LES-k-kp and RANS-USM models are almost the same and
are in good agreement with the experimental results. However, for two-phase RMS fluctuation velocities, the LES-USM and
LES-k-kp results are better than the RANS-USM results.


Introduction

In recent years, large-eddy simulation (LES) of
gas-particle flows attracts more and more attention (e,g.
Boivin et al., 2000; Riber et al., 2009). It is by and by
becoming an advanced CFD simulation tool. For LES of
gas-particle flows, one of the key problems is the selection
or development of the two-phase subgrid scale (SGS)
stress models. For LES of single-phase gas flows, the
widely used SGS stress models are Smagorinsky (1963)
eddy viscosity model, Germano (1991) dynamic eddy
viscosity model and Kim (1995) SGS energy equation
model. For Eulerian-Lagrangian LES of gas-particle flows,
many investigators use the single-phase SGS stress models,
for example, Apte et al i2 '" ). However, Zhou, H. et al.
(2007) proposed a gas SGS energy equation model,
accounting for the effect of particles on gas SGS stresses.
In the framework of two-fluid or Eulerian-Eulerian LES,
Xiang & Guo ("'ii4) proposed a particle Smagorinsky
SGS eddy viscosity model, which is a simple imitation of
the single gas phase SGS model. Similarly, Boileau et al.
(2008) adopt the Smagorinsky SGS eddy viscosity model
for both gas and droplet phases in their two-fluid-LES
modeling. In all these approaches the interaction between
the two-phase SGS stresses is not or at least not fully taken
into account.
In this paper, extending the idea of the two-phase
turbulence models in RANS modeling, a unified
second-order moment (USM) two-phase SGS stress model
and a two-phase SGS energy-equation stress model are
proposed for the two-fluid LES of gas-particle flows. The
proposed model may fully account for the interaction
between the gas and particle SGS stresses. The USM-SGS


and k-kp-SGS two-phase stress models are used in the
two-fluid LES of swirling and sudden-expansion
gas-particle flows respectively, and the statistical results
are validated by the measurement results reported in
references (Sommerfeld & Qiu, 1991; Xu & Zhou, 1999 ),
and also are compared with the RANS-USM modeling
results.

Nomenclature


D
G
k
P
p
R
t
V,v

a

A


V

p
T

i,j,k,l
g
1


Diffusion term
Source term
Subgrid scale kinetic energy
Production term
Pressure, Pa
Correlation term
time, s
velocity, m s-1

Volume fraction
Kronic-Delta unit tensor
Filtered space scale, m
Dissipation term
Dynamic viscosity, kg -m3 *s-1
Kinematic viscosity, m2Zs-'
Pressure-strain term
Density, kg -m3
Stress, kg -m' -s2

Coordinates directions

Gas
Laminar





Paper No


p
r Particle
Relaxation
Subgrid scale

LES Governing Equations

For a two-fluid LES, neglecting the forces other than
the drag force acting on the particles, the filtered continuity
and momentum equations for gas and particle phases can
be obtained as (k=g,p):


a a
-(akPk)+ (kPk uki) = 0


(aXgpgu g)+ -(agpgugiUg,)
t ax J


S gij
+ +


apg
Oxj


_gs ij+ agPg( pij)
C _X T Up


S pij
axj


+ + p (ugi upi)
Bxj T r
where the filtered gas and particle viscous forces are:
GgiB 2 ugi9j
Zg,ij [tgl( x + -xi -1gl -- ij

pUPi + pj)2. Upj
"p,ij = p xj i 3 P xj 3ij
The gas and particle subgrid scale (SGS) stresses are
defined as:

gsij = -pgRgs,ij = -Pg(ugigj -Ugiugj)

ps,ij = -PRps.i = -Pp (Upipj -UpiUpj)

The USM SGS Stress Model

Using the idea that used in modeling of two-phase
Reynolds stresses in RANS modeling (Zhou, L., 2002), the
SGS stresses of gas and particle phases can be given by the
following transport equations:

(aggpgRgs,ij) + (agPg gkRgsij) =
ct cgk (4)

Dsgs + psgs + Gsg+ 1sgs sgs
g g pg g g

(apppR ps ij) + (appppkRps,ij)
Ct Cxk (5)
Dsgs + sgs +sgs
p p p
where Dsgs, sgs ,1sgs sgs and Dsgs Psgs sgs
g g g hg d sn, pd ps n
are the diffusion, production, pressure-strain and


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

dissipation rate terms. Except the dissipation terms, all
other terms have the similar forms as those for the
corresponding terms in the two-phase Reynolds stress
equations in RANS modeling. For example, the closed
two-phase diffusion terms are:
a ksgs aR
Dsgs a g s Pg Rgskl gs
g axk c g gsk

a ksgs SS r
Dsgs (apCppy Rps,k )xl
p xk k e
The phase interaction term in the gas SGS stress equation


pg = Y (Rpgs,ij Rpgsji
P 'rp


2Rgs,ij)


where Rpgs,ij = (Upiug -UpiUgj) is the two-phase
SGS stress correlation.
The SGS kinetic energies for gas and particle phases kgs
s
and kp are defined as:
1 3 ksgs
ksgs= IRgs,ii ksgs Rps,
g 2 j~ gs,n lp Y Rj ps,u
2 Jj=l
The dissipation rates of SGS kinetic energies for gas and
particle phases e and Sp are given as:
=(ksgs 3/2/; p ( 3sgs 3/2
(=kl)g j /A; 8p = kp /A
where A is the sub-grid scale size. The transport equation
of the two-phase SGS stress correlation is:

(Rpgsij) + (Upk + Uk) (Rpgs,ij)
at kk
a IRpgs,ij
x [(vg + vp) ]
8xk axk

+ P- [ ppRps,ij + agPgRgs,ij
Pg'crp

(apPp + agpg)Rpgs,ij] (Rpgs,kj -k
cxk


+ Rpgs,ik )
Oxk


g
-sgs pgs,ij ij
ksg


(6)
where Vg = Cg(k gs)2/g, Vp = Cp(ksgs)2/p
So, the USM SGS stress model can fully account for
interaction between the two-phase SGS stresses and the
anisotropy of two-phase SGS stresses.

The Two-Phase SGS Energy Equation Model

For isotropic two-phase SGS stresses we have


ata Xpppp p)+ 0r~ p p ~i ,





Paper No


'Cgsij -PgK gs,ij = -2Pg kgs,ij


-2pg ( vgiVg1


V gi gj)


'ps,ij PpKpsij = -2pp kpsij


-2pp VpiVpj
2


VpiVpj)


The two-phase SGS energies are closed by


( gKgs,ij )+ (gvgjKgs,ij)
a OXj


a 8e Ogsij + sgs sgs
xj,, j xk gk + pg,

SjpKpsij)+ pvpjKpsij )


xa j p psiji + Gsgs + p
Ox j \Y x p P pk P
J y P J


pg8g


The terms on the right-hand side of Eqs.(7) and (8) have
the similar meanings as those in the two-phase turbulent
kinetic energy equations in RANS modeling. For
example,

sgs [np pi -ksgs
pg,


1
p =---[(VgiVpi
Tr


ugs p (v p)anp
p np Oxi
2k~gSX
CYP Pifi


The SGS two-phase velocity correlation v vp, i
determined by :

_(VgiVpi) + (Vk + Vpk) (Vgipi)


[(Ue + p) (VgiVpi)] +
Oxk Cxk
1 _
P-- [PgVpiVpi + PgVgigi -(Pg +Pp)VgiVpi]
Pg'rp


,-- pi -Vgk
pkgi k gkVpi X
C_ k SK k


s; -

g pig
kgs


(9)


Large-Eddy Simulation of Swirling Gas-Particle
Flows

Swirling gas-particle flows with s=0.47, measured by
Sommerfeld and Qiu (1991) was simulated using LES with
the proposed USM SGS stress model. The geometrical
configuration and sizes of the swirl chamber are given in


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

Fig.1.The inlet flow parameters are: central flow rate 9.9
g/s, annular flow rate 38.3 g/s, inlet Reynolds number
53256 and particle loading 0.034. The particles are glass
beads with a size of 60gum, the material density is 2500
kg/m3 and the mass flow rate is 0.34 g/s. The grid size is
taken as 1 mm and the grid number is about 100000 for a
half of the 2-D computation domain..
The numerical method is an extension of the
single-phase SIMPLEC algorithm (Patankar, 1980) to
two-phase flows. The LES-USM code is written in
Fortan-77 language, consisting of 12000 statements. The
time step is taken as 10-6s. The convergence criteria for gas
and particle phases are 10-5. For boundary conditions, the
inlet two-phase velocities, normal components of Reynolds
stresses and particle number density or volume fraction
are given by experiments, the shear stresses are given by
the eddy-viscosity assumption ; The fully-developed flow
condition of two phases are taken at the exit ; At the walls
no-slip condition is used for gas velocity, and the gas
Reynolds stress components as well as gas velocities are
determined via production term including the effect of wall
functions for near-wall grid nodes. Zero normal velocity,
zero gradients of longitudinal and tangential velocities and
normal components of Reynolds stresses, and zero mass
flux at the walls are used for the particle phase. At the axis
the symmetrical condition is adopted for both phases.
Periodic inlet fluctuations are superposed to the inlet
velocities
Figures 2 and 3 give the predicted gas and particle
time-averaged axial and tangential velocities by both
LES-USM and RANS-USM modeling and their
comparison with the experimental data respectively. It can
be seen that both modeling results are in good agreement
with the measured results and there is only a slight
difference between these two modeling results for axial
velocities. For the two-phase tangential velocities the
LES-USM results are somewhat better than the
RANS-USM results. Predictions can well give w-shaped
axial velocity profiles with an annular reverse-flow zone
and a typical Rankine-vortex structure with solid body
rotation plus free vortex in tangential velocity profiles.
Since only the time-averaged velocity field is of interest
for engineering design, one can conclude that the
RANS-USM modeling is good enough for engineering
applications.
Figures 4 and 5 give the gas and particle axial and
tangential RMS fluctuation velocities predicted by both
LES-USM and RANS-USM modeling and their
comparison with the experimental data respectively. It can
be seen that the LES-USM results are obviously better than
the RANS-USM results, in particular for the particle
fluctuation velocities.
The gas-particle fluctuation velocity correlation is an
important term in the RANS USM two-phase turbulence
model. It represents the turbulence interaction between the
gas and particle Reynolds stresses. Figure 6 gives the
correlations (U'u') and (w;w') predicted by both
LES-USM and RANS-USM modeling. In general, the LES
and RANS modeling give the same trend, however, the
LES predicted values are higher than the RANS modeling
values. Comparing with Figs 4 and 5, it is seen that both






Paper No


LES and RANS modeled gas-particle correlation
distribution is similar in shape to that of gas and particle
RMS fluctuation velocities, but its values are smaller than
the gas and particle RMS fluctuation velocities.

Large-eddy Simulation of Sudden-Expansion
Gas-Particle Flows

The k-kp SGS energy equation model is used for
large-eddy simulation of sudden-expansion gas-particle
flows measured by Xu and Zhou (1999). The geometrical
configuration of the sudden-expansion chamber is shown
in Fig.7. The heavy particles are glass beads with an
average size of 30nm and material density of 2500kg/m3.
The length of the chamber is 1000mm. The inlet gas
volumetric flow rate is 212.4m3/h, and the particle mass
loading is 0.005. For LES the grid size is taken as 1mm,
the grid number in a half 2-D computation domain is
60000. The time step is taken as 10-6s.
Figure 8 gives the LES-k-kp predicted two-phase
axial time-averaged velocities and their comparison with
the measurement results. It is seen that predictions are in
good agreement with the experimental results. Figure 9
shows the predicted two-phase axial RMS fluctuation
velocities using both LES-k-kp and RANS-USM and their
comparison with the measurement results. Both modeling
results are in good agreement with the experimental results,
and the LES results are somewhat better than the RANS
modeling results. Figure 10 gives the predicted axial and
radial components of gas-particle velocity correlation
using LES-k-kp and RANS-USM and their comparison
with the measurement results. Both modeling results give
the same tendency in agreement with the experimental
results, and the LES predicted values are greater than the
RANS predicted values. In most regions the LES results
are closer to the experimental results than the RANS
modeling results. Comparing with Fig.9 it is seen that the
gas-particle correlation distribution is similar in shape to
that of gas and particle RMS fluctuation velocities, but its
values are smaller than the gas and particle RMS
fluctuation velocities.

Conclusions

(1) The proposed USM two-phase SGS stress model and
two-phase SGS energy equation model can fully
account for interaction between the gas and particle
SGS stresses.
(2) These two-phase SGS models work well for LES of
gas-particle flows
(3) The LES-USM, LES-k-kp and RANS-USM modeling
can well predict time-averaged gas and particle
velocity fields with only a slight difference among
them.
(4) For gas and particle RMS fluctuation velocities the
LES-USM and LES-k-kp results are better than the
RANS-USM results.
(5) For the gas-particle fluctuation velocity correlation
distribution, the LES-USM, LES-k-kp and
RANS-USM modeling give the same trend: its shape is
similar to that of gas and particle RMS fluctuation
velocities, but the former is smaller in values than the
latter. The LES predicted results are closer to the


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

experimental results than the RANS-USM predicted
results.
(6) There is still some discrepancy between the LES
prediction results and the measurement results for gas
and particle RMS fluctuation velocities. This is due to
the shortcomings of 2-D LES and the errors of the
second-order difference scheme.

Acknowledgement

This study was sponsored by the Projects of National
Natural Science Foundation of China under the Grants
50606026 and 50736006.

References

Apte S V, Mahesh K, Moin P, Oefelein J C., Large-eddy
simulation of swirling particle-laden flows in a
coaxial-jet combustor, International Journal of
Multiphase Flow, 29:1311-1331 (2003)
Boileau, M., Pascaud, S., Riber, E., Cuenot, B., Gicquel, L.
Y. M., Poinsot, T. J., Cazalens, M., Investigation of
two-fluid methods for large eddy simulation of spray
combustion in gas turbines, Flow, Turbulence and
Combustion, 80:291-321 (2008)
Boivin M, Simonin O, Squires K., On the prediction of
gas-solid flows with two-way coupling using large
eddy simulation, Physics of Fluids, 12:2080-2090,
(2000)
Germano M, Piomelli U, Moin P Cabot W. H., A dynamic
subgrid-scale eddy viscosity model, Physics of Fluids.
A3:1760-1765 (1991)
Kim W W, Menon S S., A new dynamic one-equation
subgrid-scale model for large eddy simulation, AIAA
95-0356 (1995)
Patankar, S. V, Numerical Heat Transfer and Fluid Flow,
Hemisphere, New York (1980)
Riber E, Moureau V, Poinsot T, Simonin O., Evaluation of
numerical strategies for large eddy simulation of
particulate two-phase recirculating flows, Journal of
Computational Physics, 228:539-564 (2009)
Smagorinsky J. General circulation experiments with the
primitive equation (I) : The basic experiment. Monthly
Weather Review, 91 (3) : 99-164 (1963)
Sommerfeld M, Qiu H H., Detailed measurements in a
swirling particulate two-phase flow by a phase Doppler
anemometer, International Journal of Heat and Fluid
Flow, 12: 20-28 (1991)
Xiang P, Guo Y C., Modeling the hydrodynamics of dense
gas-particle flow in a riser, Journal of Engineering
Thermophysics (in Chinese), 25: 75-79 (2" 14)
Xu, Y, Zhou, L X, Experimental studies of two-phase
fluctuation velocity correlation in sudden-expansion
flows, 8th International Symposium on Gas-Particle
Flows, ASME-FED Summer Meeting, San Francisco,
CD-ROM, Paper FEDSM99-7909 (1999)
Zhou H S, Flamant G Gauthier D., Modelling of the
turbulent gas-particle flow structure in a
two-dimensional circulating fluidized bed riser,
Chemical Engineering Science, 62:269-280 (2007)
Zhou, L X, Dynamics of Multiphase Turbulent Reacting
Fluid Flows (in Chinese), Beijing, Defense Industry
Press, 105-110 (2002)






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


Figure 1 The Swirl Chamber (Sommerfeld & Qiu)
(1 primary gas-particle flow; 2 annular swirling flow,
Dl=32mm, D2=38mm, D3=64mm, D4=70mm,
D5=194mm)


X=3mm X=52mm X=155mm X=195mm
o Exp LES -----RANS-USM

(a) gas


X=315mm


0 - \

,, =! 1




5 0 5 10 15 0 5 10 0 5 0 5 0 5 10
X=3mm X=52mm X=155mm X=195mm X=315mm
a Exp LES RANS-USM

(b) particle
Figure 2 Two-phase time-averaged axial velocities


/ ,
\. /


I),
.I)"
ID,
I) i


X=3mm X=52mm X=155mm X=195mm X=315mm
o Exp -LES --RANS-USM

(b) particle
Figure 3 Two-phase tangential time-averaged
velocities


X=3mm X=52mm X=155mm X=195mm X=315mm
[ Exp LES ---RANS-USM

(a) gas


X=3mm X=52mm X=155mm X=195mm X=315mm
o Exp LES ----RANS-USM


(b) particle
Figure 4 Two-phase axial RMS fluctuation velocities


Paper No


2 12
1 1 1


9i c

I \I j 1 I I



0 5 10 0 4 80 2 4 0 2 4 0 2 4
X=3mm X=52mm X=155mm X=195mm X=315mm
Exp LES RANS-USM

(a) gas






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


X=3mm X=52mm X=155mm X=195mm X=315mm
o Exp LES ----RANS-USM



(a) gas


X=3mm X=52mm X=155mm X=195mm X=315mm
n Exp -LES -----RANS-USM

(b) particle
Figure 5 Two-phase tangential RMS fluctuation
velocities


0 05 10 00 05 10 00 05 10 00 05 10 00 05 10
X=3mm X=52mm X=155mm X=195mm X=315mm

-LES ----RANS-USM


(a) Axial


R (mm)
39 119 248 495


Di=120mm
Q=50mm 1



X(mm)


Figure 7 The Sudden-Expansion Chamber (Zhou & Xu)






1 -[
I \\

rv\ '


X=38.5mm X=119mm X=245mm X=494mm
Exp. LES


D 1I 2A 36 A 2 3D 0 1A 20 A 5
X=38.5mm X=119mm X=245mm X=494mm
Exp. LES
Figure 8 Two-phase time-averaged axial velocities

( Upper-gas; Lower-particle )


0 u
I ,,




0 I I I I I I

00 05 1000 05 10 00 05 10 00 05 1 0 00 05 10
X=3mm X=52mm X=155mm X=195mm X=315mm

-LES --RANS-USM


(b)Tangential
Figure 6 Gas-particle fluctuation velocity correlations


Paper No







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


X=38.6mm X=119mm X=246mm X=494mm
Exp. LES RANKS


X=38.5mm X=119mm X=245mm X=494mm
Exp. -LES ----RANS

Figure 9 Two-phase RMS axial fluctuation velocities
(Upper-gas; lower-particle)



SExp. LES --.--RANS


I 2 0 2002
X=38.5mm X=119mm X=245mm X=494mm

Exp. -LES -----RANS

Figure 10 Gas-particle velocity correlation
(Upper-axial; Lower-radial)


Paper No


1.i


DB




D2




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