DOI: 10.5545/sv-jme.2025.1365 349
© The Authors. CC BY 4.0 Int. Licencee: SV-JME  Strojniški vestnik - Journal of Mechanical Engineering   ▪   VOL 71   ▪   NO 9-10   ▪  Y 2025
Analysis of Gas Flow Distribution in a Fluidized Bed Using 
Two-Fluid Model with Kinetic Theory of Granular Flow and 
Coupled CFD-DEM: A Numerical Study
Matija Založnik - Matej Zadravec 
Faculty of Mechanical Engineering, University of Maribor, Slovenia
 matej.zadravec@um.si
Abstract Fluidized bed systems are widely used in chemical and process engineering due to their excellent heat and mass transfer properties. Numerical 
modeling plays a crucial role in understanding and optimizing these systems, with the two-fluid model enhanced by the kinetic theory of granular flow (TFM-KTGF) 
and the coupled computational fluid dynamics-discrete element method (CFD-DEM) emerging as leading techniques. This study employs both models to simulate 
gas-solid interactions and evaluates their performance using a benchmark single-spout fluidized bed case validated against experimental data. Subsequently, 
the influence of particle presence on gas flow distribution through a non-uniform distribution plate is analyzed. The results show that the common assumption 
of proportional flow distribution based on the opening area fraction is inaccurate, particularly in the presence of particles. Both numerical models capture this 
behavior, with TFM-KTGF showing trends comparable to the coupled CFD-DEM approach but at significantly reduced computational cost. The findings highlight 
the importance of accounting for particle dynamics in distribution plate design and promote the TFM-KTGF approach as a promising alternative for large-scale 
simulations.
Keywords fluidized bed, distribution plate, two-fluid model with kinetic theory of granular flow, coupled CFD-DEM, flow distribution
Highlights
 ▪ Two models (TFM-KTGF and CFD-DEM) simulate gas-solid flow in fluidized beds.
 ▪ Models validated against experiments, showing good particle behavior prediction.
 ▪ Gas flow depends on particles, not just plate geometry.
 ▪ CFD-DEM captures local effects; TFM-KTGF is faster and predicts overall trends.
1 INTRODUCTION
Fluidized bed systems are widely used in various industrial 
applications due to their excellent heat and mass transfer 
characteristics. Their applications range from chemical reactors and 
drying processes to coating technologies and catalytic cracking. 
Despite these advantages, fluidized beds remain inherently complex 
systems, where interactions between the gas and solid phases must be 
thoroughly understood to ensure efficient and stable operation [1,2].
With recent advances in computational modeling, the two-fluid 
model with added kinetic theory of granular flow (TFM-KTGF) 
and the coupled computational fluid dynamics-discrete element 
method (CFD-DEM) have emerged as powerful tools for simulating 
the complex behavior of fluidized bed systems. The TFM-KTGF 
approach treats both the gas and solid phases as interpenetrating 
continua within the Eulerian-Eulerian framework, with kinetic 
theory of granular flow (KTGF) playing a key role in characterizing 
particle behavior and inter-particle interactions. In contrast, the 
coupled CFD-DEM approach models the motion and interactions of 
individual particles in a Lagrangian framework, while the gas phase 
is treated using computational fluid dynamics (CFD) in the Eulerian 
framework. Although the coupled CFD-DEM approach provides 
a detailed resolution of particle dynamics, its complexity and high 
computational cost make it less practical for large-scale simulations 
compared to TFM-KTGF [3].
Esgandari et al. [4] conducted a direct comparison between these 
two modeling approaches in fluidized single- and multi-spout bed 
systems. Their study demonstrated that the TFM-KTGF approach 
could successfully replicate key hydrodynamic features observed in 
the more computationally intensive coupled CFD-DEM approach. 
Similarly, Ostermeier et al. [5] compared both numerical models for 
gas-solid fluidized beds and reported consistent global trends between 
them. These findings highlight why the TFM-KTGF approach is 
increasingly favored in both research and industry, offering reduced 
computational times while maintaining comparable predictive 
accuracy. Additional studies have examined the capabilities and 
limitations of both models through practical multiphase case studies 
of fluidized bed systems [6-9].
Flow distribution plays a crucial role in the proper functioning of 
fluidized bed systems, directly influencing particle mixing and the 
effectiveness of heat and mass transfer. One of the most essential 
components for ensuring optimal flow is the gas distribution plate 
(also referred to as the distributor), which governs the efficiency 
of gas introduction into the particle bed. Numerous designs for 
distribution plates have been proposed in the literature for various 
applications [10,11]. A numerical analysis of gas flow distribution 
across a distribution plate in a Wurster coater setup was performed 
by Kevorkijan et al. [12], using the coupled CFD-DEM approach. 
Their study revealed that both particle loading and inlet airflow rate 
significantly impact the uniformity of gas distribution across the 
distribution plate.
Recent studies have further examined distributor performance, 
pressure drop, and mixing efficiency in both industrial and laboratory 
systems, emphasizing that distributor geometry and particle 
properties critically influence hydrodynamic behavior inside the 
system. Gonzalez-Arango and Herrera [13] used CFD to study how 

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the geometries of different gas-phase distributors inside the fluidized 
bed affect the pressure drop and particle mixing. Their findings 
highlight that both the physical design and material selection of 
distribution plates can substantially impact system performance. The 
optimization of a uniform distributor inside a fluidized reactor was 
carried out by Singh et al. [14] using CFD, providing an example of 
the effective use of modeling tools for equipment optimization.
Although distribution plates are often designed based on open 
area fractions, this geometric assumption neglects particle effects that 
can significantly modify local gas flow through resistance, clustering, 
and particle-fluid interactions. While TFM-KTGF and coupled CFD-
DEM have been widely used to analyze fluidized bed hydrodynamics, 
few studies have investigated how particles influence gas distribution 
through non-uniform distribution plates. To address this gap, the 
present study employs both TFM-KTGF and coupled CFD-DEM 
modeling to evaluate deviations from theoretical, area-based flow 
distributions and to provide insights for more accurate distribution 
plate design. This approach improves our understanding of why 
simplified assumptions sometimes fail in real-world applications, 
especially when complex physical phenomena are involved. The 
analysis was conducted on a laboratory-scale fluidized bed equipped 
with a distribution plate featuring non-uniform opening sizes, as 
shown in Fig. 1.
Fig. 1. A distribution plate with non-uniform opening sizes in a laboratory-scale  
fluidized bed system was used for the numerical analysis
2 METHODS
2.1 Two-Fluid Model with Kinetic Theory of Granular Flow
In the TFM approach, both the gas and solid phases are treated as 
independent continua, each governed by its own set of conservation 
equations. For a non-reactive, transient, isothermal system composed 
of spherical particles, the governing equations for mass and 
momentum conservation are expressed as follows:






t
gg gg
 v
g
0, (1)


  
t
ss ss
 v
s
0, (2)






   

t
p
gg gg
gg g
 
 
vv v
gv v
gg g
gs g
  , (3)


  
    

t
pp
ss ss
ss ss
 
 
vv v
gv v
ss s
sg s
  , (4)
where the subscripts g and s denote the gas and solid phases, 
respectively. Here, α
i
 represents the volume fraction, ρ
i
 the density, 
v
i
 the velocity, p
s
 the solid pressure, τ
i
 the stress tensor, g the 
gravitational acceleration, and β the momentum exchange coefficient, 
which is computed using a drag model. The solid pressure and the 
stress tensor of the solid phase are calculated as follows:
pe g
ss ss ss ssss s
      21
2
0,
, (5)
 
is ss
vv v   




  
s
T
s
, (6)
where Θ
s
 is the granular temperature, e
ss
 is the restitution coefficient, 
g
0,ss
 is the radial distribution function, µ
s
 is the granular viscosity, and 
λ
s
 is the bulk viscosity. The radial distribution function is a correction 
factor that accounts for the increased probability of particle collisions 
as the solid phase becomes dense. It is calculated using the following 
equation:
g
ss
s
sm ax
0
3
1
1
,
,
, 











 (7)
where α
s,max
 represents the packing limit. The granular viscosity, 
which is related to the particle motion and interactions, is calculated 
using the following expressions [15,16]:
 
ss kins col

,,
, (8)

 

sk in
ssss
ss
ss ss ss s
d
e
ee g
,,
, 
 
   







63
1
2
5
13 1
0
 (9)
 

sc ol ssss ss s
s
dg e
,,
.  
4
5
1
0

 (10)
where d
s
 is the particle diameter. The bulk viscosity characterizes the 
material’s response to changes in pressure and stress and is calculated 
by the equation proposed by Lun et al. [17]:
 

ss ss ss ss
s
dg e  
4
3
1
0,
.

 (11)
The granular temperature Θ
s
 is a parameter introduced into the 
two-fluid model (TFM) through the KTGF. It quantifies the random 
fluctuations in particle velocity arising from collisions. The transport 
equation for the granular temperature is given as follows:
3
2


 







    


t
p
ss ss ss
ss
s
 




v
Iv
s
ss

 
 :
 
s
gs
 , (12)
where I is the identity tensor, κ
Θ
s
 is the diffusion coefficient, γ
Θ
s
 
represents the collisional dissipation of energy, and ϕ
gs
 denotes the 
interphase energy transfer due to particle-gas interactions. The first 
term on the right-hand side of the granular temperature equation 
corresponds to energy production; the second term represents the 
diffusion of granular temperature; the third accounts for energy 
dissipation due to particle collisions; and the final term describes 
the energy exchange between the gas and solid phases. The diffusion 
coefficient is calculated using the following expression [15]:

 






s
d
g
sss s
ss s

 
  
 
15
44 13 3
1
12
5
43
16
15
41 33
2
0
[
,
   
ss s
g
0,
], (13)
where η is a dimensionless parameter calculated as:
  
1
2
1 e
ss
. (14)

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The collisional dissipation of energy represents the rate at which 
energy is dissipated within the solid phase due to collisions between 
particles. It is calculated using the following equation [17]:





s
eg
d
ss ss
s
ss s



12 1
2
0
2
3
2
,
. (15)
Lastly, the interphase energy transfer is described by the following 
equation [18]:

gs s
 3  . (16)
In this work, the Syamlal-O’Brien drag model, which is based on 
the terminal velocity of particles, is employed [19]. The momentum 
exchange coefficient β is calculated using the following equation:












3
4
2
sgg
rs s
D
s
rs
vd
C
Re
v
,,
, vv
sg
 (17)
where v
r,s
 is the terminal particle velocity, d
s
 is the particle 
diameter, C
D
 is the drag coefficient, and Re
s
 is the Reynolds number 
for the solid phase. The drag coefficient, originally derived by Dalla 
Valle [20], is calculated as follows:
C
Re
v
D
s
rs















06 3
48
2
.
.
.
,
 (18)
The terminal particle velocity for the solid phase is calculated 
using the following expression [21]:
vA Re Re Re BA A
rs ss s ,
.. .. .,       05 00 30 50 06 01 22
2
2
(19)
with
A
g

41 4 .
,
 (20)
and
B
gg
gg








08 08 5
08 5
12 8
26 5
..
.
.
.
.


for
for
 (21)
2.2 Coupled CFD-DEM
In the coupled CFD-DEM approach, the hydrodynamic behavior 
of the gas within a gas-solid fluidized bed is modeled using CFD 
to solve the conservation equations. The particles in the system are 
modeled using the discrete element method (DEM), which governs 
their motion and interactions based on Newton’s second law of 
motion [22]. In the CFD-DEM coupling, the solid volume fraction 
field is computed using a volumetric diffusion Lagrangian-Eulerian 
mapping, which smoothly distributes each particle’s volume to the 
surrounding cells while conserving the total solid phase volume. For 
particle i with mass m
i
, the following set of equations is solved:
m
d
dt
i
j
n
v
FFF
i
ij
c
i
f
i
g
 


1
, (22)
I
d
dt
i
j
n
i
 
i
ij
M 


1
, (23)
where v
i
 is the translational velocity of the particle, ω
i
 is the angular 
velocity, F
ij
c
 and M
ij
 are the contact force and torque resulting from 
particle interactions with other particles and walls, F
i
f
 is the force 
due to particle-fluid interactions, F
i
g
 is the gravitational force, and I
i
 
is the moment of inertia. Particle-particle and particle-wall 
interactions are described using a soft-sphere model, where normal 
and tangential forces relative to the contact are modeled separately 
[23]. The normal contact force component is modeled using the 
Hysteretic linear spring model [24], as shown below:
F
F
F
ij
nt
ij
nt t
ij
nt t
,
,
,
min,
max





 
 
Ks Ks
Ks
l
t
u
u




if s0
,, .
,
0 001Ks
l
t






 if s0 
 (24)


ss s
tt t


, (25)
where F
ij
nt ,
 and F
ij
nt t ,( ) 
 are the normal forces acting on particle i at 
the current and previous time steps, ∆t is the time step size, s is the 
contact overlap, and K
l
 and K
u
 are the loading and unloading contact 
stiffnesses, determined by the particle properties as:
1
11
11
12
K
KK
KK
l
Lp Lp
Lp Lw



,,
,,
for particle-particle contact
fo or particle-wall contacts







, (26)
K
K
e
u
l
ss
=
2
, (27)
where subscripts 1 and 2 represent two contacting particles. The 
individual stiffnesses associated with a particle and a wall are 
calculated as:
KE L
lp p ,
, = (28)
KE L
lw w ,
, = (29)
where E is the Young’s modulus and L is the particle size. The 
tangential contact force is modeled using the linear spring Coulomb 
limit model. If the tangential force is assumed to be purely elastic, it 
can be calculated using the following equation:
FF
ij e
t
ij
tt
,
, ,
,
   

  
 Ks

 (30)
where F
ij
 ,( ) tt 
 is the tangential contact force at the previous time 
step, K
τ
 is the tangential stiffness, and ∆s
τ
 is the tangential overlap 
difference between two time steps. Since this model does not allow 
the tangential force to exceed Coulomb’s limit, the complete 
expression is given as follows:
FF F
F
F
ij
t
ij e
t
ij e
nt ij e
t
ij e
t
  
 
 
,
,
,
,
, ,
,
,
,
,, 

min 
 (31)
where µ is the friction coefficient. For a more detailed description 
of the model, the reader is encouraged to consult the literature by 
Walton and Braun [24] and Cundall and Strack [22].
The effect of fluid flow across the particle bed is modeled using 
a two-way coupled CFD-DEM approach. The force F
i
f
 consists of 
drag and pressure contributions, as shown below:
F
p
 Vp
p
 , (32)
Fv vvv
Dg si gs i
 
1
2

gp
A
,,
() , (33)
where V
p
 is the particle volume, ∆p is the local pressure gradient, A
p
 
is the projected particle area in the direction of the flow, and v
g
 −  v
s,i
 
is the relative velocity between particle i and the fluid. The Syamlal-
O’Brien drag model was again used to calculate the momentum 
exchange coefficient, as described in the equations shown above. 
In both numerical models, the k − ω SST turbulence model was 
employed [25].
2.3 Model Validation
The validation of both the TFM-KTGF and coupled CFD-DEM 
models was performed using a benchmark single-spout fluidized bed 
case. The simulation results were compared with experimental data 
reported by Van Buijtenen et al. [26]. In that study, particle velocities 

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were measured using particle image velocimetry (PIV) and positron 
emission particle tracking (PEPT) systems at two different heights: 
0.05 m and 0.10 m from the bottom, as indicated by the red dashed 
lines in Fig. 2.
Fig. 2. Schematic of the single-spout fluidized bed used for model validation,  
where the red dashed lines indicate the locations where particle velocities were measured
Simulations were performed using two software packages: ANSYS 
Fluent [27] for hydrodynamics and ANSYS Rocky [28] for DEM. A 
uniform numerical mesh consisting of 58,000 hexahedral elements 
was used for both models. The system under study contained 12,000 
spherical glass particles, each with a uniform diameter of 3 mm and a 
density of 2505 kg/m
3
. The restitution coefficients for all interactions 
were set to 0.97, while the friction coefficients for particle-particle 
and particle-wall interactions were set to 0.1 and 0.3, respectively, 
consistent with previous studies [26,29,30].
Table 1. Simulation parameters used for the validation study
Parameter Value
Material Glass
Number of particles, N 12000
Particle diameter, d
s
3 mm
Particle density, ρ
s
2505 kg/m
3
Restitution coefficient, e
ss
0.97
Particle-particle friction coefficient, μ
p‒P
0.1
Particle-wall friction coefficient, μ
p‒w
0.3
Spout velocity, v
sp
43.5 m/s
Background velocity, v
bg
2.4 s
Total simulation time, t 20 s
CFD time step, Δt
CFD
10
‒5
 s
The spout and background velocities at the inlet were set to 43.5 
m/s and 2.4 m/s, respectively, with the pressure outlet set to ambient 
pressure. All walls were assigned with no-slip boundary conditions. 
The total simulation time for both models was set to 20.0 s, with a 
CFD time step of 10
−5
 s, while the DEM time step was calculated 
automatically within ANSYS Rocky based on the hysteretic linear 
spring model [31]. A summary of all simulation parameters used in 
this study is presented in Table 1.
Figure 3 compares the particle velocity profiles in y direction at 
different simulation times. Figures 3a, b and c show results from the 
TFM-KTGF approach, while Figs. 3d, e and f present results from 
the coupled CFD-DEM approach at t = 6 s (Figs. 3a and d), t = 18 s 
(Figs. 3b and e), and as a time-averaged profile (Figs. 3c and f).
Fig. 3. a), b) and c) Average particle velocity in the y direction obtained using  
the TFM-KTGF approach, and d) e) and f) coupled CFD-DEM approach at:  
a) and d)  t = 6 s, b) and e)  t = 18 s, and c) and f) as a time-averaged result
It is evident that the particle velocities obtained using the TFM-
KTGF approach exhibit a very uniform profile throughout the 
simulation. This behavior arises from the nature of the TFM-KTGF 
model, in which particles are treated as a continuum phase. In this 
framework, there is no discrete mechanism driving the fluid to interact 
with particles in a way that would cause substantial variations in the 
velocity profile over time. In contrast, the velocity profiles obtained 
from the coupled CFD-DEM approach show a noticeable change 
as time progresses. This is because the direct interactions between 
particles and the airflow influence particle velocities, causing the 
profile to evolve dynamically over time.
The time-averaged particle velocity profiles in y direction, along 
the length of the fluidized bed at heights of 0.05 m and 0.10 m from 
the bottom, were compared with experimental data. The results are 
shown in Fig. 4. Good agreement between the numerical model 
predictions and the experimental data is observed, particularly at the 
height of 0.05 m from the bottom. Both numerical models produced 
similar velocity trends, demonstrating the validity of both approaches 
for simulating fluidized bed behavior.
Figure 5 shows the time-averaged particle velocity vectors 
obtained from PIV and PEPT measurements by Van Buijtenen et 
al. [26], along with the corresponding results from this study. Good 
agreement was observed between both the TFM-KTGF and coupled 
CFD-DEM approaches and the experimental data. In the coupled 
CFD-DEM approach, intensive circulation patterns are clearly 
visible, closely matching the experimental observations from PIV 
and PEPT. In contrast, the TFM-KTGF results show less pronounced 
circulation. The slight differences observed between the PIV and 
PEPT vector fields are attributed to challenges inherent in the 
experimental setup, as described by Van Buijtenen et al. [26].
In summary, both the TFM-KTGF and coupled CFD-DEM 
approaches for simulating single-spout fluidized beds provide 
satisfactory predictions of flow dynamics when compared with 
experimental results obtained using PIV and PEPT, despite 
slight deviations. Both models showed good agreement with the 
experimental particle velocity data, as shown in Fig. 3, confirming 

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their reliability of these models for further analyses of flow 
distribution through a non-uniform distribution plate.
2.4 Flow Distribution Analysis
The flow distribution analysis was conducted on the geometry of a 
laboratory-scale fluidized bed system with a non-uniform distribution 
plate, as shown in Fig. 6. The colored sections on the distribution 
plate (Fig. 6c) represent different groups of openings: cyan indicates 
4 mm, magenta 3.5 mm, red 3 mm, blue 1 mm, and green 2 mm. 
To reduce computational cost and simulation time, the geometry 
was symmetrically reduced to a quarter section, while preserving the 
essential flow characteristics.
As in the validation study, ANSYS Fluent [27] and ANSYS Rocky 
[28] were used to simulate multiphase flow using the TFM-KTGF and 
coupled CFD-DEM numerical models, respectively. Both approaches 
used the same numerical mesh, consisting of 1.5 million polyhedral 
elements. The simulations were performed with a total of 300 g of 
zeolite particles, with diameters ranging from 0.5 mm to 5 mm. The 
detailed particle size distribution is provided in Table 2, and the bulk 
particle density was set to 770 kg/m
3
. These values were selected 
based on the work of Zadravec et al. [32] to reflect realistic conditions 
in a laboratory-scale fluidized bed system. In the TFM-KTGF 
approach, where particles are represented as a continuous phase, the 
particle size distribution was first determined using the population 
balance model (PBM). The discrete method was applied, in which 
the overall particle size distribution is discretized into a finite number 
of size classes. From this distribution, the Sauter mean diameter was 
evaluated and used in the TFM-KTGF model. This ensures that the 
influence of the particle size distribution is captured in an averaged 
manner while maintaining the computational framework of the two-
fluid model. The interaction parameters, including restitution and 
friction coefficients for particle-particle and particle-wall contacts, 
were based on literature values [32] and are summarized in Table 3.
Fig. 6. a) Laboratory-scale fluidized bed system, b) the simplified geometry,  
and c) distribution plate geometry used in the analysis
Air was introduced into the system through the bottom inlet 
at volume flow rates ranging from 50 m
3
/h to 70 m
3
/h, increasing 
in increments of 5 m
3
/h to examine the effect of inlet velocity on 
flow distribution. Ambient pressure was applied at the outlet, and 
symmetry boundary conditions were imposed on the cut planes to 
represent the quarter geometry. All other walls were assigned no-slip 
boundary conditions to accurately capture near-wall interactions.
Fig. 4. Time-averaged particle velocity profiles in y direction along the length of the single-spout fluidized bed system at heights of a) 0.05 m, and b) 0.10 m from the bottom
Fig. 5. Time-averaged particle velocity vector fields in the single-spout fluidized bed system; a) velocity PIV [26], b) velocity PEPT [26], velocity TFM-KTGF, and d) velocity coupled CFD-DEM

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The total simulation time for each case was set to 5 s, with a CFD 
time step size of 10
−4
 s, while the DEM time step was calculated 
automatically, as in the validation case [31]. This ensured sufficient 
temporal resolution to capture flow evolution and particle behavior 
throughout the system.
Table 2. Zeolite particle size distribution
Particle size [mm] 0.5 1.0 2.0 3.15 5.0
Mass fraction [%] 0.5 0.8 3.5 77.3 17.9
Table 3. Restitution and friction coefficients used in this analysis
Particle-Particle Particle-Wall
Restitution coefficient 0.1 0.5
Friction coefficient 0.6 0.5
The complete set of parameters used in flow distribution 
simulations is summarized in Table 4.
Table 4. Simulation parameters used for the flow distribution study
Parameter Value
Material Zeolite
Total mass of particles, m
s
300 g
Particle diameter, d
s
Table 2
Particle density, ρ
s
770 kg/m
3
Restitution coefficient Table 3
Friction coefficient, μ Table 3
Inlet volume flow rate, V [50, 55, 60, 65, 70] m
3
/h
Total simulation time, t 5.0 s
CFD time step, Δt
CFD
10
−4
 s
3 RESULTS AND DISCUSSION
To thoroughly examine the effect of particsles on airflow distribution 
through the fluidized bed distribution plate, multiple simulation cases 
were performed. These included simulations of the system without 
particles, to establish the baseline flow distribution in an empty 
geometry, and simulations with particles using the TFM-KTGF and 
coupled CFD-DEM models to assess the influence of particles on 
flow distribution.
The objective was to compare the simulation results with the 
theoretical flow distribution, which assumes that air distributes 
proportionally according to sthe opening fraction of each hole size 
group on the distribution plate. In other words, the flow rate through 
each group of openings was assumed to correspond to its relative area 
fraction on the plate.
Figure 7 compares the theoretical flow distribution, based on 
the open-area fraction, with the simulation results under different 
operating conditions. The solid lines represent the theoretical flow 
fractions for each group of openings, while the symbols and dashed 
lines correspond to the simulation data at various inlet flow rates. 
Results are presented for both an empty system (without particles) and 
a fluidized system (with particles), evaluated using two multiphase 
modeling approaches: TFM-KTGF and coupled CFD-DEM. The 
figure illustrates that the actual flow distribution deviates from the 
theoretical prediction even in the absence of particles, with these 
deviations becoming more pronounced when particles are introduced. 
In particular, the system containing particles shows clear shifts in 
the flow fractions through each opening group (e.g. 3 mm, 3.5 mm 
and 4 mm opening groups experience increased flow relative to the 
geometric assumption, while the 1 mm and 2 mm groups exhibit 
reduced flow). Furthermore, increasing the inlet air flow rate slightly 
Fig. 7. Flow distribution through distribution plate openings of different sizes at various inlet air volume flow rates

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alters the distribution, indicating that operating conditions influence 
how gas is channeled through the distributor.
These findings highlight that particle interactions introduce 
additional resistance and non-uniformity in local flow paths that 
cannot be captured by geometric assumptions alone, emphasizing the 
importance of modeling approaches that explicitly account for the 
particle phase.
A broader overview of the flow distribution is presented in Fig. 
8, where the flow fractions for each opening size group are averaged 
across all inlet air flow rates. The results again confirm a significant 
mismatch between the theoretical distribution and the actual 
simulated distribution, particularly in the presence of particles.
Differences between the results obtained using the TFM-KTGF 
approach and those from the coupled CFD-DEM approach are also 
evident. These differences stem from the fundamental modeling 
approaches: TFM-KTGF treats the particle phase as a continuum, 
whereas the coupled CFD-DEM explicitly tracks individual particles. 
This distinction influences not only the flow predictions but also the 
computational performance of each model.
From a computational perspective, the TFM-KTGF approach 
proved to be significantly more efficient. Because it does not resolve 
individual particle trajectories, it requires far fewer computational 
resources than the coupled CFD-DEM model, which solves the 
equations of motion for each particle. In our simulations, the TFM-
KTGF model completed each run approximately five times faster 
than the coupled CFD-DEM model on the same hardware. This 
substantial difference in computational time underscores the appeal 
of TFM-KTGF for large-scale simulations involving high particle 
counts.
4 CONCLUSIONS
In this study, we investigated gas flow distribution through a non-
uniform distribution plate in a laboratory-scale fluidized bed system 
using two numerical approaches: TFM-KTGF and coupled CFD-
DEM. Both approaches were employed to examine their behavior 
and predictive capability in a laboratory-scale fluidized bed. CFD-
DEM provides detailed particle behavior, capturing particle-fluid 
interactions and heterogeneities, while TFM-KTGF efficiently 
predicts global flow trends. Using both methods allows us to evaluate 
how particle effects influence gas distribution and to assess the 
validity of the continuum assumptions in the TFM framework. Both 
models were validated against experimental data from a single-spout 
fluidized bed and demonstrated satisfactory agreement in predicting 
particle velocity profiles and overall flow behavior. These results 
confirm that both approaches can capture the essential features of 
fluidized bed dynamics.
This study demonstrates that particle effects can substantially alter 
the flow distribution in non-uniform distribution plates, despite the 
conventional assumption that gas flows proportionally to open area. 
By explicitly comparing the TFM-KTGF and coupled CFD-DEM 
approaches, we evaluated these deviations and provided evidence that 
particle-fluid interactions must be considered in distributor design. 
Incorporating the non-uniform plate geometry alongside realistic 
particle behavior bridges the gap between theoretical assumptions 
and actual flow patterns, offering a framework to guide improved 
design strategies for laboratory-scale fluidized beds.
Both numerical approaches captured the dynamic behavior of the 
particles, albeit with some differences. 
The coupled CFD-DEM approach provided more detailed 
results by tracking individual particles, their interactions, and their 
influence on the flow, highlighting particle-fluid interactions that 
are not captured by simple geometric assumptions. In contrast, the 
TFM-KTGF approach treats the particle phase as a continuum, which 
smooths out these details but still accurately represents the overall 
behavior on a global scale. A key advantage of the TFM-KTGF 
approach is its computational efficiency: in this study, it completed 
the simulations approximately five times faster than the coupled 
CFD-DEM approach on the same hardware, making it a practical 
choice for larger systems.
Fig. 8. Flow distribution through distribution plate openings of different sizes,  
averaged across all inlet air volume flow rate cases
While the coupled CFD-DEM approach captures particle-scale 
effects with high fidelity, its computational cost makes applications 
to large or industrial-scale fluidized beds impractical with current 
resources. Conversely, the TFM-KTGF model, though efficient, 
relies on a continuum treatment of the granular phase, which may 
smooth out local heterogeneities such as clustering or jet instabilities. 
These differences highlight that each model has inherent limitations, 
and their predictions should be interpreted within the context of 
laboratory-scale systems.
It should be emphasized that the present simulations were 
performed for laboratory-scale geometries, and direct extrapolation 
of the findings to full industrial-scale fluidized beds requires 
caution. Although the observed trends provide valuable guidance for 
distributor plate design, additional validation at pilot or industrial 
scale would be necessary to fully confirm the transferability of these 
results. Future work should therefore focus on bridging the gap 
between laboratory validation and industrial application.
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Acknowledgements The authors would like to express their sincere 
gratitude to the professors, assistants, and colleagues at the Faculty of 
Mechanical Engineering, University of Maribor for their valuable support 
and constructive discussions throughout this work. We also gratefully 
acknowledge the financial support provided by the Slovenian Research 
Agency (ARRS) under the framework of Research Program P2-0196: Power, 
Process, and Environmental Engineering.
Received: 2025-04-23, Revised: 2025-09-09, Accepted: 2025-09-24  
as Original Scientific Paper 1.01.
Data availability The data that support the findings of this study are 
available from the corresponding author upon reasonable request.
Authors contribution   Matija  Zalo žnik:  F o rmal  analysis,  In v es tigatio n, 
Writting - original draft, Writing – review & editing, Software, Visualization; 
Matej Zadravec: Methodology, Project Administration, Supervision.
Analiza distribucije toka plina v lebdečem sloju z uporabo modela 
dveh tekočin s kinetično teorijo granularnega toka in sklopljenega 
CFD-DEM: numerična študija
Povzetek   Sis t emi  z  lebdečim  slo jem  se  po go s t o   upo rabljajo   v  k emičnem 
in  pr o cesnem  inženir s tvu  zaradi  sv o jih  o dličnih  spo so bnos ti  preno sa 
t oplo t e  in  sno vi.  N umerično   mo deliranje  ima  ključno  vlo go   pri  razume v anju 
in  o p timizaciji  t eh  sis t emo v ,  pri  čemer  se  med  v o dilne  uv elja vljata  mo del 
dv eh  t ek o čin,  do polnjen  s  kine ti čno   t eo rijo   granularnega  t ok a  (TFM-KT GF) 
in  sklo pljena  računalnišk e  dinamik e  t ek o čin  z  me t o do   diskre tnih  element o v 
(CFD-DEM).  V  t ej  štud iji  s ta  upo rabljena  o ba  mo dela  za  simulacijo   int erakcij 
med  plino m  in  tr dnimi  delci  t er  o vredno t ena  njuna  učink o vit o s t  na  primeru 
eksperimentalnega primera iz literature. Analiziran je bil tudi vpliv delcev na 
po razdelit e v  t o k a    plina  sk o zi  ne-unif o rmno   dis tribucijsk o   plo ščo .  R ezultati 
k ažejo ,  da  je  po go s ta  predpo s ta vk a  o  so razmerni  po razdelitvi  t ok a  glede 
na  delež  o dpr tin  nepra vilna,  zlas ti  v  priso tno s ti  delce v .  Oba  numerična 
mo dela  zajame ta  t o   v edenje,  pri  čemer  TFM-KT GF  k aže  trende  primerljiv e 
s  sklo pljenim  CFD-DEM  pris t o pom  a  pri  bis tv eno   nižjih  računskih  časih. 
Ugo t o vitv e  po udarjajo   po men  upo št e v anja  dinamik e  delce v  pri  o blik o v anju 
dis tribucijskih  plo šč  t er  pr o mo virajo   upo rabo   TFM-KT GF  k o t  o be ta vno  
alternativo za simulacije na velikih sistemih.
Ključne besede lebdeči  slo j,  dis tribucijsk a  plo šča,  mo del  dv eh  t ek o čin  s 
kine ti čno  t eo rijo  granularnega t o k a, sklo pljen CFD-DEM, distribucija toka