PARTICLE TECHNOLOGY AND FLUIDIZATION
Dynamic Multiscale Method for Gas-Solid Flow via
Spatiotemporal Coupling of Two-Fluid Model and Discrete
Particle Model
Xizhong Chen and Junwu Wang
State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of
Sciences, P.O. Box 353, Beijing 100190, P.R. China
DOI 10.1002/aic.15723
Published online April 12, 2017 in Wiley Online Library (wileyonlinelibrary.com)
Various computational fluid dynamics methods have been developed to study the hydrodynamics of gas-solid flows, however, none of those methods is suitable for all the problems encountered due to the inherent multiscale characteristics
of gas-solid flows. Both discrete particle model (DPM) and two-fluid model (TFM) have been widely used to study gassolid flows, DPM is accurate but computationally expensive, whereas TFM is computationally efficient but its deficiency
is the lack of reliable constitutive relationships in many situations. Here, we propose a hybrid multiscale method
(HMM) or dynamic multiscale method to make full use of the advantages of both DPM and TFM, which is an extension
of our previous publication from rapid granular flow (Chen et al., Powder Technol. 2016;304:177–185) to gas-solid
two-phase flow. TFM is used in the regions where it is valid and DPM is used in the regions where continuum description fails, they are coupled via dynamical exchange of parameters in the overlap regions. Simulations of gas-solid channel flow and fluidized bed demonstrate the feasibility of the proposed HMM. The Knudsen number distributions are also
C 2017 American Institute of Chemical Engineers AIChE J, 63: 3681–
reported and analyzed to explain the differences. V
3691, 2017
Keywords: two-fluid model, discrete particle method, hybrid model, spatial coupling, dynamic coupling, gas-solid flow
Introduction
Gas-solid flows occur in many industries and thus have
attracted a lot of attention from engineers and scientists over
the past decades. Recently, computational fluid dynamics
(CFD) has already emerged as a popular tool in aiding the
design and scale-up of relevant processes. Various CFD methods, such as direct numerical simulation (DNS), discrete particle model (DPM) and two-fluid model (TFM), have been
developed to study the hydrodynamics of gas-solid flow.1,2
However, none of those methods is suitable for all the problems encountered due to the inherent multiscale characteristics
of gas-solid flows.3
Both TFM4–6 and DPM7–9 are very popular in fluidization
community. TFM treats both gas and solid as interpenetrating
continua, the model is commonly used in the study of largescale problems as it is computationally efficient, but the drawback lies at its continuum treatment of solid phase which
results in significant uncertainties of this model. The constitutive equations for particulate phase stresses is generally based
on the assumption that there is a clear scale separation in granular systems, or from thermodynamic viewpoint the local thermodynamic equilibrium postulate is valid. Unfortunately, this
is usually not the case.10–12 For example, Continuum method
Correspondence concerning this article should be addressed to J. Wang at
jwwang@home.ipe.ac.cn.
C 2017 American Institute of Chemical Engineers
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is inadequate due to the existence of Knudsen layer.13,14 Conversely, DPM treats the fluid as a continuum phase and treats
the solid phase as a collection of discrete particles that obeys
the Newton’s second law. The DPM is much more accurate
compared with TFM, however, the computational costs are
often very demanding, especially in case of dense suspensions
encountered in industrial reactors. Please note that highly
resolved TFM may offer insight of the physics of gas-solid
two-phase flow, but it meets difficult when there are big velocity, density and/or temperature gradients (or in general, the
magnitude of thermodynamic forces is large). More detailed
discussion on this topic could be referred to a recent publication of Fullmer and Hrenya.15
In present study, a hybrid multiscale method (HMM) or
dynamic multiscale method is developed for modeling the
complex gas-solid flow. The HMM aims to have both the efficiency of TFM and the accuracy of DPM for modeling the
hydrodynamics of gas-solid flows in large-scale reactors,
which is an extension of our previous works from rapid granular flow to gas-solid two-phase flow.16,17 Please note that only
numerical comparisons between all three models are carried
out to verify the developed HMM as its accuracy would not
surpass the DPM method theoretically. When compared with
the experimental results, some uncertainties about the particle
properties (shape, size distribution, moisture) and geometries
simplification may inevitably cause some ambiguities. The
advantage of the numerical comparison is that the initial condition of all the three models could be controlled to make sure
September 2017 Vol. 63, No. 9
3681
conditions for continuum description of solid phase in TFM.
The solid volume fraction, solid velocity, and granular temperature are calculated as follows
1 X
mi
(1)
eJ 5
VJ qs i J
Method
The TFM used in present study is a standard TFM with
kinetic theory of granular flow for closing particulate phase
stresses and Gidaspow’s model4 for closing the interphase
drag force term. The Johnson and Jackson model18 is used to
calculate the tangential velocity and granular temperature of
solid phase at the wall boundary. The governing equations of
TFM are listed in Appendix. The discrete particle model used
in present study is a linear spring-dashpot discrete element
method for particulate phase and Navier-Stokes equations for
gas phase. Gidaspow’s model is also used for closing the interphase drag force term. Please note that to consist with the
kinetic theory of granular flow used in TFM, the particles are
smooth in DPM. The governing equations of discrete particle
model are listed in Appendix.
A general schematic of the geometry of the hybrid multiscale method proposed in this work is shown in Figure 1.
DPM is used to solve the part of the channel near the wall,
while TFM is used to solve the remaining part. TFM and DPM
coincidently overlapped in overlap regions which are delicately designed to exchange solid phase boundary from each
side of the individual model to enforce the mass and momentum conservation in the total system. The overlap region contains three subregions: a discrete boundary subregion, a buffer
subregion and a continuum boundary. The buffer subregion
x1 x2 is used to minimize the disturbance caused by boundary data exchange. Detailed description of the handling solid
phase in overlap region could also refer to Chen, Wang, Li.17
The physical quantities of discrete particle in the subregion
x2 x3 are sampled and averaged to generate the boundary
1 X
mi vi
VJ eJ qs i J
(2)
1 X
mi ðvi 2 uJ Þ2
3VJ eJ qs i J
(3)
uJ 5
HJ 5
The subregion x0 x1 is used to provide boundary for solid
phase in discrete particle model from flow field of two-phase
model. The mass conservation is achieved through a flux monitor placed at the termination of the region of discrete particle
model (x0). The number of particles to be inserted or deleted is
calculated by
n 5 ðAJ uc Þ eJ qs 䉭tc = mi
(4)
where AJ uc is the flux of cell J calculated by TFM. The
velocities of newly inserted particle are generated from
Maxwell distribution with mean and standard deviation provided by the local information of TFM. A virtual wall is
placed at the termination of discrete region to prevent particle
from freely drifting to TFM region. After the compensation of
mass conservation, a constraint dynamics of particle velocities
in subregion x0 x1 is enforced to ensure the momentum conservation. The new velocity of particle i in this region is
relaxed to the local solid velocity of TFM uc calculated by
vi 0 5 vi 1 u c 2
NJ
1 X
Vi
NJ i51
(5)
After the solid phase flow field is resolved, the whole solid
volume fraction and drag force information are passed to the
gas phase. A unified gas phase solver based on the modified
SIMPLE algorithm is used to obtain the gas velocity and pressure of the whole simulated domain.
Figure 1. Schematic of the coupling strategy of hybrid multiscale model.
[Color figure can be viewed at wileyonlinelibrary.com]
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they are consistent. Simple geometry and flow condition are
likewise preferred and chosen here while more complex and
realistic industrial reactor simulation is the ultimate goal. The
instantaneous and time averaged Knudsen number distributions predicted by discrete particle model are reported and
analyzed to further clarify the differences between different
models.
The detailed flowchart of hybrid multiscale numerical algorithm for gas-solid flow is showed in Figure 2. In every time
step, the algorithm begins with solving the individual particle
movement in the regions of discrete particle model, the positions and velocities of the entire collection of particle are
known and subsequently the boundary conditions for the solid
phase in TFM could be determined. The governing equations
of solid phase in TFM is solved in the regions belong to TFM
given the boundary conditions, then the number of particles to
be added or deleted in the discrete particle region are calculated based on the mass flux monitor of the solid phase
Table 1. The Domain Decomposition Setting for HMM in
Horizontal Direction
Domain
TFM
DPM
Buffer
Grid No.
318
17 and 1420
46 and 1517
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information of TFM. If a particle is to be added, its velocity
will get from the Maxwell velocity distribution with the mean
velocity and standard deviation provided by velocity and
Table 2. Simulation Parameters of the Gas-Solid
Channel Flow
Parameter
Value
Gas viscosity (Pas)
Gas temperature (K)
Operation pressure (Pa)
Particle density (kg/m3)
Particle diameter (m)
Spring stiffness coefficient (N/m)
Particle–particle coefficient of restitution
Particle–wall coefficient of restitution
Height of channel (m)
Width of channel (m)
Time step (s)
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1.8 3 1025
298
1,01325
2000
1.2 3 1023
3000
0.95
0.95
0.8
0.12
1.0 3 1025
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Figure 2. Flowchart of hybrid multiscale model algorithm for gas-solid flows.
[Color figure can be viewed at wileyonlinelibrary.com]
Figure 4. Comparisons of the radial gas velocity distribution results between different models for
channel flow with uniform inlet velocity.
[Color figure can be viewed at wileyonlinelibrary.com]
Figure 6. The time series of Knudsen number in different monitor positions for channel flow with
uniform inlet velocity.
[Color figure can be viewed at wileyonlinelibrary.com]
Figure 5. The probability density distribution of Knudsen
number for channel flow with uniform inlet
velocity.
Figure 7. Radial distributions of solid volume fraction
predicted by all three methods.
[Color figure can be viewed at wileyonlinelibrary.com]
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[Color figure can be viewed at wileyonlinelibrary.com]
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Figure 3. Comparisons of the radial simulation results between different models for channel flow with uniform inlet
velocity: (a) Solid volume faction (b) Solid velocity.
Parameter
Value
Gas viscosity (Pas)
Gas temperature (K)
Operation pressure (Pa)
Particle density (kg/m3)
Particle diameter (m)
Spring stiffness coefficient (N/m)
Particle–particle coefficient of restitution
Particle–wall coefficient of restitution
Height of channel (m)
Width of channel (m)
Inlet gas velocity (m/s)
Inlet solid flux (kg/(m2s))
Time step (s)
Figure 8. The frequency density distribution of Knudsen
number inside the channel.
[Color figure can be viewed at wileyonlinelibrary.com]
1.8 3 1025
298
1,01325
2000
1.2 3 1023
24,000
0.95
0.95
1.6
0.12
7.0
60
5.0 3 1026
the momentum conservation. Once the solid phase volume
fraction and velocity distributions on the entire region are
known, the source term in gas phase governing equations
could be calculated and followed by a solving procedure to the
flow field of gas phase in the entire domain.
Results and Discussion
Figure 9. The 3D distribution of Knudsen number
inside the channel.
[Color figure can be viewed at wileyonlinelibrary.com]
granular temperature of local TFM correspondingly. A final
adjustment of particle velocities in the near boundary interface
is enforced according to constraint dynamics (Eq. 5) to ensure
To verify the implementation of the hybrid multiscale
numerical algorithm and further demonstrate the benefits of
hybrid multiscale model, three gas-solid channel flow cases
are designed and simulated. The boundary conditions at the
wall are said to play an important role in the accuracy of the
results of TFM for the gas-solid channel, due to the effect of
Knudsen Layer which cannot be captured by TFM. Therefore,
DPM is utilized in the near wall regions to simulate the gassolid flow inside the Knudsen Layer and the remaining part of
the channel is simulated by TFM. Meanwhile, each case is
also simulated using all the three methods, that is, pure TFM,
pure discrete particle model and hybrid multiscale model. The
domain decomposition setting in horizontal direction for
hybrid multiscale model is listed Table 1. All the cases have a
consistent grid setting, that is, the domain is discretized in 20
structural cells in the horizontal direction and 64 cells in the
vertical direction. In addition, the third direction in discrete
particle model is set as periodic boundary for solid phase and
has a thickness of 6.1 times particle diameter.
Figure 10. Radial distributions of gas and solid velocity predicted by all three methods.
[Color figure can be viewed at wileyonlinelibrary.com]
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Table 3. Simulation Parameters of the Gas-Solid Riser with
Coarse Particles
dp
Kn 5 pffiffiffi
6 2g0 es L
Figure 11. The snapshots of flow field predicted by all
three methods at t 5 15 s (Ug 5 7.0 m/s,
Gs 5 60 kg/(m2s)).
[Color figure can be viewed at wileyonlinelibrary.com]
(6)
The probability density distribution of Knudsen number inside
the channel predicted by DPM is showed in Figure 5. It can be
seen that the majority of Knudsen number inside the channel
are very small and the time averaged value is 0.00548, which
indicates that most of regions may be suitable for continuum
description of solid phase. Figure 6 presents the time series of
Knudsen number in different monitor positions. The fluctuation and average value of Knudsen number in the near wall
position are larger than other positions, which could be further
used to explain the reason why the simulation results of TFM
deviate from DPM.
Gas-solid channel flow with uniform inlet velocity
A uniform velocity-driven gas-solid channel flow without
considering the gravity is first studied. Both gas and solid are
flowed into the system from the bottom of the channel with
uniform velocity (2.0 m/s) and the inlet solid volume fraction
is 0.1. The gas density is calculated using idea gas law. Noslip boundary is imposed to gas phase at walls and pressure
outlet boundary is prescribed at the top of the channel. Johnson and Jackson boundary condition for solid phase is applied
at the wall in pure TFM, where the particle wall restitution
coefficient is the same as particle–particle restitution coefficient and the specularity coefficient is set to zero. Total simulation time of 3 s is completed for each simulation and mean
flow field information is extracted from the last 2 s numerical
results. Detailed parameters used in the numerical simulation
are listed in Table 2.
Figure 3 shows the comparisons of the radial distribution
of solid volume fraction and solid velocity simulations
results predicted by all the three models. It is found that
although both the inlet gas and solid velocities are uniform,
DPM predicts that there are a small oscillation of solid volume fraction and a decline of solid velocity in the near wall
region. The decline of solid velocity could be contributed to
the decline of gas velocity caused by the no-slip boundary,
which is presented in Figure 4. It is clear that the simulation
results of HMM are consistent with DPM while TFM failed
to capture the solid volume fraction oscillation near the wall.
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Figure 12. The predicted time averaged axial voidage
profiles.
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[Color figure can be viewed at wileyonlinelibrary.com]
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Noting that the gas velocity predicted by HMM is also corrected near the wall boundary, which indicated that both gas
and solid phase can be benefit from the design of hybrid multiscale method.
The Knudsen number (Kn) defined as the ratio of mean free
path (k) to the characteristic length scale (L) is often used in
microscale gas flows to characterize the validation of the continuum assumption. TFM treats solid phase as a continuum
media thus its application may be limited to the situation of
small Knudsen number. Conversely, DPM tracks individual
particle directly and thus do not rely on the magnitude of the
Knudsen number. Therefore, we could use the simulation
results of DPM to investigate of the distribution of Knudsen
number inside the channel. Here, we choose the width of channel as the characteristic length and calculate the mean free
path of particle using kinetic theory of granular flow.4 The formulation of local Knudsen number used in this case is given
as follows
[Color figure can be viewed at wileyonlinelibrary.com]
Gas-solid channel flow with parabolic inlet velocity
In this case, all simulation parameters are the same as the
previous one except that the inlet gas and solid velocities are
changed to be parabolic Uz 5Um ð12r 2 =R2 Þ, where Um is the
maximum velocity (2.0 m/s) in the center of the channel. Due
to the difference of the solid velocity across the channel, the
radial solid volume also became nonuniform across the channel. Figure 7 shows that all the three models predict a similar
flow pattern. Quantitatively, TFM has a relative large deviation from DPM in the center and the near wall region while
the simulation results of HMM is more consistent with DPM.
To disclose the reason, the Knudsen number (Eq. 6) based
on DPM simulation results is then calculated. Figure 8 shows
the frequency density distribution of Knudsen number inside
the channel. It can be seen that the average Knudsen number
is larger than the uniform inlet velocity case. Most of the large
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September 2017 Vol. 63, No. 9
value of Knudsen number are observed in the near wall region
as shown in Figure 9. As a result, it is expected that the simulation results of solid velocity in the near wall region between
TFM and DPM will have some differences. Figure 10 also
shows that the gas velocity predicted by HMM has a better
agreement with DPM simulation results.
Gas-solid fluidized bed
Different from the first two cases, the gravity is finally
added to the channel in this case. Detailed parameters used in
the numerical simulation are listed in Table 3. Due to the formation of heterogeneous structures, a longer simulation time
is needed. The flow field is found to arrive at the steady state
after 5 s. Total simulation time of 16 s is completed for each
simulation and mean flow field information is collected after
the initial 5 s.
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Figure 13. The predicted time averaged radial flow profiles.
[Color figure can be viewed at wileyonlinelibrary.com]
Figure 11 shows the snapshots of flow field predicted by
all the three models. The characteristic of a typical gas-solid
riser flow is the dynamic formation of heterogeneous structure like clusters. As it can be seen from Figure 11, all the
models predict the nonuniform flow field inside the channel,
which is qualitatively the same as the experimental observation.19 Figure 12 shows the quantitatively comparisons of
time averaged axial voidage profiles along the channel height
predicted by all the three models. Compared to TFM, the
overall simulation result of HMM is closer to DPM, especially in the middle and top parts of the channel. However,
both HMM and TFM show some differences in the near bottom parts of the channel.
Figure 13 presents the time averaged radial solid fraction,
gas velocity, and solid velocity predict by all the three models
with two different heights. Qualitatively, all the three models
predict dense solid volume fraction in the near wall region and
dilute solid volume fraction in the centre of the channel. Quantitatively, if we choose the simulation results of DPM as
benchmark, the overall performance of HMM is better than
TFM.
Due to clusters are very common in the predicted flow
field, we choose the local gradient of the solid volume fraction (L5es =jres j) as the characteristic length to calculate
the Knudsen number inside the channel in this case. The
solid volume fraction inside and outside the cluster is significantly different, thus this Knudsen number calculation
method takes into account for the structure of flow field.
Figure 14 shows the frequency distribution of Knudsen
number inside the channel. The majority of both the transient and time averaged value of the Knudsen number are
not large. However, there do exist some large values as
could be observed in Figure 15b. It is found that in the near
bottom and top region of the channel, the time averaged
Knudsen number is significant larger than the middle part.
This can qualitatively explain the reason why the simulation
results of TFM deviate more from DPM in those regions.
Conversely, the transient value of Knudsen number does not
have an evident regularity as shown in Figure 15a. Large
value of transient Knudsen number could appear in many
regions of the channel as the clusters are dynamically evolution inside the channel. However, the domain decomposition of DPM and TFM region in HMM simulation is always
fixed at present, which causes the simulation results of
HMM still have some difference with pure DPM simulation.
This indicates that the domain decomposition should be
automatically adjusted with a specific criteria such as given
Knudsen number in future research.
Figure 15. The 3D distributions of Knudsen number inside the channel: (a) Transient value (b) Time averaged value.
[Color figure can be viewed at wileyonlinelibrary.com]
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Figure 14. The distributions of Knudsen number inside the channel: (a) Transient value (b) Time averaged value.
Conclusions
This work presents a hybrid multiscale model with spatiotemporally coupled TFM and discrete particle model. In the
framework of hybrid multiscale model, the gas phase is solved
using a unified solver throughout the whole domain. An overlap region is constructed to facilitate the boundary communication of solid phase between TFM and discrete particle
model. Gas-solid channel flows without gravity were simulated by pure TFM, pure discrete particle model and hybrid
multiscale model. The simulation results indicated that the
hybrid multiscale model was consistent with pure discrete particle model while pure TFM had some differences with pure
discrete particle model in the near wall region. The distribution of Knudsen number based on the simulation results of
pure discrete particle model was analyzed to explain the reason of deviations. Further simulations of the gas-solid fluidized bed also showed the feasibility of the proposed hybrid
multiscale model. Large number of Knudsen number appeared
in many regions of the fluidized bed due to the dynamically
changed heterogeneous structures. Future works will involve
improving the model with the development of a criterion of
the validity of TFM and the implementation of adaptive
domain decomposition method for the cases of more complex
spatial heterogeneous flow structures.
Acknowledgments
We thank Professor Hans Kuipers at Eindhoven University of Technology for allowing the usage of his 3D-DPM
code, on which the HMM is based. This study is financially
supported by the National Natural Science Foundation of
China (21422608) and the “Strategic Priority Research Program” of the Chinese Academy of Sciences under the grant
No.XDA07080200.
Notation
TFM =
DPM =
HMM =
dp =
e=
ew =
F=
g=
g0 =
I=
kn =
Kn =
L=
m=
Np =
pg =
ps =
ra =
t=
Re =
ug =
us =
v=
two-fluid model
discrete particle model
hybrid multiscale model
particle diameter, m
coefficient of restitution between particle–particle interaction
coefficient of restitution between particle and wall interaction
force, N
gravitational acceleration, m/s2
radial distribution function
unit tensor
normal spring stiffness, N/m
Knudsen number
characteristic length scale, m
particle mass, kg
number of particles
gas pressure, Pa
particle pressure, Pa
position of particle a
time, s
Reynold number
gas velocity vectors, m/s
solid velocity vectors, m/s
particle velocity vectors, m/s
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September 2017 Vol. 63, No. 9
Vc = volume of cell, m3
Vp = volume of particle, m3
Greek letters
b=
eg =
es =
es,max =
lg, ls =
ks =
Hs =
s=
u=
d=
dn =
cs =
qg =
qs =
gn =
drag coefficient for a control volume, kg/m3s
voidage
solid volume fraction
solid volume fraction at packed condition
fluid and solid viscosity, Pas
solid bulk viscosity
granular temperature, m2/s2
stress tensor
specularity coefficient
dirac function
overlap between particles, m
energy dissipation, J/(m3s)
fluid density, kg/m3
solid density, kg/m3
normal damping coefficient
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Published on behalf of the AIChE
DOI 10.1002/aic
3689
15475905, 2017, 9, Downloaded from https://aiche.onlinelibrary.wiley.com/doi/10.1002/aic.15723 by Shanghai Jiaotong University, Wiley Online Library on [04/02/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
In this case, TFM simulation of the whole domain is fastest,
which takes 41 h of CPU clock time. Discrete particle model
simulation of the whole domain takes 320 h of CPU clock time
to complete the running, while hybrid multiscale model only
takes 243 h to complete. Therefore, hybrid multiscale model is
more efficient than pure discrete particle model even in this
demo and it is promising for larger scale simulation as well.
Table A1. Governing Equations of TFM
Description
Equation
@ ðeg qg Þ
1r ðeg qg ug Þ50
@t
@ ðes qs Þ
@t 1r ðes qs us Þ50
Mass balance equations
@ ðeg qg ug Þ
1r eg qg ug ug 52eg rpg 1r eg sg 1b us 2ug 1eg qg g
@t
@ ðes qs us Þ
1r ðes qs us us Þ52rps 2es rpg 1r ðes ss Þ1b ug 2us 1es qs g
@t
Momentum balance equations
h
i
ðes qs us Hs Þ 5ð2ps I1es ss Þ : rus 2r ðks rHs Þ2c23bHs
Granular temperature equation
3 @ðes qs Hs Þ
1r
@t
2
Gas stress
sg 5lg ðrug 1ruTg Þ1 23 lg ðr us ÞI
ss 5ls ðrus 1ruTs Þ1 ks 2 23 ls ðr us ÞI
Solid stress
Ps 5es qs Hs 12qs ð11eÞe2s g0 Hs
qffiffiffiffi
pffiffiffiffiffiffi
e qs dp pHs ls 5 45 qs dp e2s g0 ð11eÞ Hps 1 s 6ð32eÞ
11 25 ð11eÞð3e21Þes g0
qffiffiffiffi
ks 5 43 es qs dp g0 ð11eÞ Hps
qffiffiffiffi
pffiffiffiffiffiffi
2
150qs dp Hs p Hs
6
2
11
e
g
ð11eÞ
12q
e
d
ð11eÞg
ks 5 384ð11eÞg
s
0
p
0
s
s
5
p
0
qffiffiffiffi
Hs
2 2
4
c53ð12e Þes qs g0 Hs dp p 2ðr us Þ
Solid phase pressure
Solid shear viscosity
Solid bulk viscosity
Granular phase heat conductivity
Inelastic collision energy dissipation
1=3 21
es
g0 5 12 es;max
8
qg eg es jug 2us j 22:65
3
>
>
>
eg
> 4 CD
<
dp
b5
>
e2 l
q es jug 2us j
>
>
> 150 s 2g 11:75 g
:
eg dp
dp
(
ð24=Re Þð110:15Re 0:687 Þ;
CD 5
0:44;
Radial distribution function
Interphase drag force coefficient
Wall boundary condition of solid phase
es 0:2
es > 0:2
Re < 1000
Re 1000
e q d ju 2u j
Re 5 g g pl g s
g
6ls es;max
@us;w
us;w 52 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffi
3puqs es g0 Hs @n
pffiffi
3=2
3puqs es u2s;slip g0 Hs
@H
Hs;w 52 kcs Hs @ns;w 1
6es;max cs;w
s;w
Table A2. Governing Equations of Discrete Particle Model
Description
Equation
Mass balance equation for gas phase
@ ðeg qg Þ
1r
@t
Momentum balance equations for gas phase
@
Momentum exchange term
ðeg qg ug Þ50
ðeg qg ug Þ
1r ðeg qg ug ug Þ52eg rPg 1r
@t
ðX
Np
bVa
Sp 5 V1c
ðug 2va Þdðr2ra ÞdVc
es
a51
ðeg sg Þ2sp 1eg qg g
8
qg eg es jug 2us j 22:65
3
>
>
>
eg
es 0:2
> 4 CD
<
dp
b5
>
e2 l
q es jug 2us j
>
>
> 150 s 2g 11:75 g
es > 0:2
:
eg dp
dp
Particle movement equation
ma dvdta 52Va rpg 1
X
b2contactlist
dra
dt
3690
DOI 10.1002/aic
Fab;n 1
Va b
ðug 2va Þ1ma g
es
5va
Published on behalf of the AIChE
September 2017 Vol. 63, No. 9
AIChE Journal
15475905, 2017, 9, Downloaded from https://aiche.onlinelibrary.wiley.com/doi/10.1002/aic.15723 by Shanghai Jiaotong University, Wiley Online Library on [04/02/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Appendix
Description
Equation
Contact force
Fab;n 52kn dn nab 2gn vab;n
Overlap between particles
dn 5ðRa 1Rb Þ2jrb 2ra j
Normal unit vector
a
nab 5 jrrbb 2r
2ra j
Normal component of the relative velocity
vab;n 5ðvab nab Þnab
pffiffiffiffiffiffiffiffiffiffiffiffi
mab kn lnffi e
gn 5 22pffiffiffiffiffiffiffiffiffiffiffiffi
2
2
Normal damping coefficient
p 1ln e
Manuscript received Nov. 3, 2016, and revision received Feb. 12, 2017.
AIChE Journal
September 2017 Vol. 63, No. 9
Published on behalf of the AIChE
DOI 10.1002/aic
3691
15475905, 2017, 9, Downloaded from https://aiche.onlinelibrary.wiley.com/doi/10.1002/aic.15723 by Shanghai Jiaotong University, Wiley Online Library on [04/02/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
TABLE A2. Continued