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12th International Conference on CFD in Oil & Gas, Metallurgical and Process Industries
SINTEF, Trondheim, Norway
May 30th – June 1st 2017
CFD 2017
COMPARISON OF PARTICLE-RESOLVED DIRECT NUMERICAL SIMULATION AND
1D MODELLING OF CATALYTIC REACTIONS IN A PACKED BED
Arpit Singhal a, b*, Schalk Cloete c, Stefan Radl d, Rosa Quinta-Ferreira b and Shahriar Amini a, c
a NTNU (Norwegian University of Science and Technology), Department of Energy and Process Engineering,
Kolbjørn hejes v 1B, NO-7491, Trondheim, Norway
b University of Coimbra, Department of Chemical Engineering, Rua Sílvio Lima, Polo II, 3030-790 Coimbra, Portugal
c SINTEF Materials and Chemistry, Flow Technology Department, S. P. Andersens veg 15 B, NO-7031, Trondheim,
Norway
d TU Graz, Institute of Process and Particle Engineering, 8010 Graz, Inffeldgasse 13/III, Austria
Corresponding author’s e-mail: arpit.singhal@ntnu.no
ABSTRACT
The work presents a comparison of catalytic gas-solid reactions
in a packed bed as simulated on two widely different scales:
direct numerical simulation (capable of accurately predicting
transfer phenomena in and around a few particles) and 1D
modelling (capable of engineering simulations of industrial
scale reactors).
Particle-resolved direct numerical simulation (PR-DNS) is
performed on a small geometry containing ~100 realistically
packed monodisperse spherical particles generated via the
discrete element method (DEM). These results are compared to
a 1D packed bed reactor model using the effectiveness factor
approach to account for intra-particle mass transfer and a
suitable closure for gas-particle heat transfer.
The differences between the results from the two modelling
approaches are quantified over a range of Thiele moduli,
Prandtl numbers and reaction enthalpies. Results showed that
existing 1D-model closures perform well for a simple first order
catalytic reaction. Heat transfer completely dominates the
overall reaction system when large reaction enthalpies are
simulated, while mass transfer limitations dominate at low
reaction enthalpies. Future work will extend this comparative
approach to packings with more complex particle shapes and
complex reactions.
Keywords: Direct numerical simulation (DNS), CFD-DEM,
packed bed, catalytic gas-solid reaction, reaction rate, heat
transfer, multiscale.
NOMENCLATURE
Greek Symbols
Volume fraction
ԑ Void fraction
Thiele modulus (Th)
Effectiveness factor
Latin Symbols
Cp
Specific heat capacity of fluid [J/kg.K]
CA
Concentration of species A [mol/m3]
D
Molecular diffusivity [m2/s]
dp
Diameter of the cylindrical particle [m]
E
Activation energy [J/mol]
h
Heat transfer coefficient [W/m2K]
k0
Arrhenius constant [1/s]
Kf
Thermal Conductivity of fluid [W/m.K]
Nu
Nusselt number ()
Pr
Prandtl number ()
R
Gas constant [8.314 J/mol/K]
Catalytic reaction rate [mol/m3s]
r
Radius [m]
Re
Reynolds number ()
T
Temperature [K]
Superficial velocity of the fluid [m/s].
Sub/superscripts
f
Fluid
s
Solid.
p
Particle.
INTRODUCTION
Gas-solid reaction systems in packed beds are of great industrial
influence, with the application widespread from process to
metallurgical industries. The catalytic or non-catalytic role of
the solid defines the complexity involved in the gas-solid
reactions.
There are several advanced models available in literature for
gas-solid reaction systems. The non-catalytic reaction systems
are considered more complicated as they are transient in nature.
The detailed review of such systems is described by
(Ramachandran and Doraiswamy, 1982) and more recently by
(Nashtaee and Khoshandam, 2014). Meanwhile, (Ishida and
Wen, 1968) have described the effectiveness factor (η) in
catalytic reactions for gas-solid systems. The effectiveness
factor in heterogeneous catalyst reaction to obtain the intra
particle diffusion in porous particles is suggested in
(Levenspiel, 1999).
The recent work from (Yang et al., 2016) described an
effectiveness factor for general reaction forms. They presented
an analytical expression, which is applicable to wide range of
reaction rate forms and provides a direct and computationally
efficient approach of obtaining effectiveness factor in packed
bed reactors. The validity of such a simplified model when
added with heat transfer limitations motivates the current work.
Hence, the objective of the work is to obtain a comparison in
prediction of effectiveness factor for a catalytic gas-solid
reaction on two distinct scales. Firstly, a PR-DNS study of a
packed bed of ~100 spherical particles now involving a
catalytic reaction based on our previously published work
(Singhal et al., 2017) gives insight into a phenomenon of intra
particle diffusion along with heat transfer limitations. Secondly,
a 1D packed bed reactor model coupled with the effectiveness
Singhal et al.
factor model from (Yang et al., 2016) describes the intra-
particle heat and mass transfer. The results obtained from both
the approaches are compared and documented.
METHODOLOGY
Thiele Modulus and Effectiveness Factor
The effectiveness factor concept in heterogonous catalytic gas-
solid reactions can be explained as the effect of intra particle
diffusion on the reaction rate (Ishida and Wen, 1968;
Levenspiel, 1999).
Thus, the effectiveness factor in catalytic reactions is directly
linked with the Thiele modulus (Thiele, 1939). Thiele modulus
is explained as:
PR-DNS Simulation Setup
The spherical particle bed is generated using DEM (Discrete
Element Method) integrated in ANSYS FLUENT following the
procedure described in the paper (Singhal et al., 2017). The
geometry is meshed with fine body-fitted polyhedral elements
both inside and outside the particles with resolution of dp/30 on
the particle surfaces and the growth rate of 20% (Figure 1).
Figure 1: A section (y = 0) through the geometry meshed
with polyhedral elements.
ANSYS FLUENT is used to complete steady state DNS using
the SIMPLE algorithm for pressure-velocity coupling with 2nd
order spatial discretization of other equations. Steady state DNS
was found to be sufficient for this case since no transient
fluctuations occurred in the small spaces between particles
(Singhal et al., 2017). The geometry incorporates a velocity
inlet, a pressure outlet and a no-slip condition on the wall. The
reaction takes place in the porous solid particles (grain model
(Szekely, 1976)) modelled by the Eq. (1). The simulation
parameters used in the DNS simulations are describe in the
Table 1.
A (g) + B (s) → C (g) + B (s)
(1)
The reaction rate is described in the conventional way:
(2)
(3)
Simulations were completed at three different levels of mass
transfer resistance (Thiele modulus), heat transfer resistance
(Prandtl number) and reaction enthalpy as outlined in Table 1.
Mass and heat transfer was adjusted by setting the molecular
diffusivity and gas-phase thermal conductivity according to the
Th and Pr numbers specified in Table 1. No solids phase
thermal conductivity was included in order to accentuate heat
transfer resistances in the particle. For the reaction rate, the pre-
exponential factor in Eq. (3) was chosen to result in a reaction
rate constant of 10000 1/s at a temperature of 1000 K. A large
activation energy is selected to accentuate coupling between
heat and mass transfer.
Table 1: Simulation parameters for PR-DNS
Parameters
Value
Particle diameter (dp) (m)
0.001
Packed bed voidage
0.355
Particle void
fraction (internal)
0.3
Density (kg/m3)
Fluid :1
Particles :2500
Fluid velocity (m/s)
1
Inlet mole fraction (A)
0.1
Specific heat capacity (Cp)
(J/kg/k)
1000
Thiele moduli (Th)
5, 10, 20
Prandtl numbers (Pr)
0.4, 1.6, 6.4
Heat of reaction (kJ/mol)
100, 10, 0
1D Packed Bed Model
A detailed outline of the setup of the 1D model used in this work
can be viewed in a recent work by the authors (Cloete et al.,
2016). The model is solved in the commercial CFD code,
ANSYS FLUENT 16.2, on a domain with 100 cells arranged in
only one direction. In order to simulate a packed bed, the
Eulerian Two Fluid Model approach is followed and the
velocity of the solids phase is fixed to zero in all cells.
Conservation equations for mass, momentum, species and
energy are then solved in the conventional manner.
In the present study, the most important closures are the
effectiveness factor for modelling intra-particle mass transfer
limitations (Levenspiel, 1999) and the gas-particle heat transfer
coefficient for modelling external heat transfer limitations
(Gunn, 1978). The effectiveness factor for the simple first order
catalytic reaction considered in this study is written as follows:
(4)
(5)
(6)
The Thiele modulus represents the ratio of kinetic rate to
diffusion rate, so higher values represent greater mass transfer
limitation. The effective diffusivity is composed of the
molecular diffusivity , the void fraction of porous particles
and the tortuosity .
The classical Gunn correlation for gas-particle heat transfer is
written as follows:
(7)
PR-DNS and 1D Modelling of Catalytic Reactions in a Packed Bed
Inlet and outlet boundary conditions as well as the domain
length are set to identical values as the PR-DNS simulations.
The solids volume fraction in the bed is taken as the product of
the mean solids volume fraction in the PR-DNS domain (0.645)
and the solids volume fraction in the particles (0.7).
Figure 2: The PR-DNS results for the temperature variation in the packed bed of spherical particles for different Prandtl
numbers (Pr) and Thiele moduli (Th)
RESULTS AND DISCUSSIONS
Heat and Mass Transfer in Densely Packed Bed
PR-DNS results for simulations completed with different Thiele
moduli and Prandtl numbers for the highly endothermic
reaction kJ/mol are shown in Figure 2 and
Figure 3. The temperature variation in Figure 2 illustrates the
increasing effect of the heat transfer resistance as Pr is increased
by decreasing the gas-phase thermal conductivity. Even though
the thermal conductivity is also very low inside the particle, it
is clear that external gas-particle heat transfer still dominates.
This is most clearly visible in the Pr6.4 cases in Figure 2 where
the temperature gradient inside the particles is small relative to
the temperature gradient in the fluid film around the particles.
Figure 3 illustrates the mass transfer resistances. It is
immediately evident that mass transfer resistances are much
less influential in this case than heat transfer resistances because
the species concentration gradients are small relative to the
temperature gradients in Figure 2. The Pr0.4Th20 case shows
some intra-particle mass transfer resistance as a clear species
gradient within the particles. The importance of heat transfer
resistance relative to mass transfer resistance for this particular
case will be further discussed in the next sections.
Singhal et al.
Figure 3: The PR-DNS results for the reactant (A) mole fraction in the packed bed of spherical particles for different Prandtl
numbers (Pr) and Thiele moduli (Th).
Individual Particle Data
The PR-DNS approach allows for extraction of detailed data
from individual particles within the domain. In this way, the
effectiveness factor for individual particles can be extracted and
compared. This will be done for the case with the largest heat
and mass transfer limitations (Th20-Pr6.4). The definition of
the effectiveness factor becomes very important in this case.
Three different approaches will be followed (Figure 4):
Species: Comparing the species concentration on the
particle surface to the average concentration in the
particle (the effectiveness factor for an isothermal
first order reaction)
Surface: Comparing the average reaction rate in the
particle to the reaction rate that would occur using
species concentration and temperature on the particle
surface.
Volume: The same as the previous point, only using
data averaged over the volume of the particle.
The fact that the “species” effectiveness factor is close to unity
implies that mass transfer plays essentially no role in this
particular case (the reactant concentration on the particle
surface is essentially the same as the reactant concentration in
the particle volume). This case is therefore almost exclusively
controlled by heat transfer (as seen in the Th20-Pr6.4 case of
Figure 2 and Figure 3).
The heat transfer limitation becomes clear when looking at the
“surface” effectiveness factor. The temperature on the particle
surface is a lot higher than inside the particle volume where the
reaction takes place. Calculating the reaction based on the
particle surface temperature would therefore result in large
errors.
Interestingly, the “volume” effectiveness factor is larger than
unity. This implies that there is a significant amount of
temperature variation inside the particle, brought about by the
assumption of zero thermal conductivity by the solid material.
Naturally, this will not be the case in most catalyst particles, but
it presents an interesting phenomenon. Given the exponential
increase in reaction kinetics with temperature, any variation in
temperature around the mean will strongly increase the average
PR-DNS and 1D Modelling of Catalytic Reactions in a Packed Bed
kinetic rate inside the particle. This is what happened in this
case: the actual reaction rate inside the particle was higher than
the reaction rate calculated based on the average particle
temperature.
1D Model Predictions
Comparisons between PR-DNS and 1D model results are
discussed in this section. Firstly, the 1D model will be
compared to PR-DNS results over a range of Prandtl numbers
and Thiele moduli. Secondly, the reaction enthalpy will be
changed and the models will be compared again. Finally, an
important observation regarding the implementation of the 1D
model will be presented.
Variation of Prandtl number and Thiele modulus
A comparison of axial reactant concentration is given in Figure
5 for nine combinations Prandtl number and Thiele modulus. It
is clear that the 1D model successfully predicts the PR-DNS
results.
In addition, the dominance of heat transfer limitations is clear
in all cases because results for different Thiele moduli are
essentially identical, whereas results for different Prandtl
numbers differ substantially. As may be expected, the amount
of reaction in this endothermic system decreases as Pr is
increased by decreasing the gas phase thermal conductivity. A
lower thermal conductivity implies greater gas-particle heat
transfer resistance, thereby allowing less heat to enter and
sustain the highly endothermic reaction.
The continued dominance of heat transfer resistance at Pr = 0.4
is interesting given the clear intra-particle species gradients that
can be observed in the Th20-Pr0.4 case in Figure 2. This is
because the outer shell of the particles is slightly hotter than the
centre, implying that reduced species concentrations in the
centre of the particle (where the temperature is lower and the
kinetics is slower) does not have such a large impact on the
overall reaction rate.
Figure 6 shows the axial evolution of the difference between the
average gas temperature and the average particle temperature.
Again, it is clear that mass transfer limitations are essentially
negligible, while gas-particle heat transfer dominates the
system.
In this case, there is a clear deviation between the PR-DNS and
1D-simulation results: PR-DNS consistently predicts a larger
difference between the average gas and particle temperatures.
This implies that the PR-DNS predicts a lower particle
temperature than the 1D simulations (gas temperature reduces
with gas species concentration and is almost identical between
the PR-DNS and 1D simulations). As mentioned in the previous
section, the temperature variation inside the particle in the PR-
DNS allows the reaction rate to be higher than that implied by
the average particle temperature. On the other hand, the 1D
simulation inherently assumes constant temperature in all
particles. For this reason, the two models predict the same
overall reaction rate at different average particle temperatures.
Figure 4: Three different representations of effectiveness
factors for 20 particles from the Th20-P6.4 case.
0.9826
0.9828
0.983
0.9832
0.9834
0.9836
0.9838
0.984
0.9842
0.9844
0.9846
0 0.001 0.002 0.003 0.004 0.005
Effectiveness factor (species)
Height (m)
0.036
0.038
0.04
0.042
0.044
0.046
0.048
0 0.001 0.002 0.003 0.004 0.005
Effectiveness factor (surface)
Height (m)
3
3.5
4
4.5
5
5.5
6
0 0.001 0.002 0.003 0.004 0.005
Effectiveness factor (volume)
Height (m)
Singhal et al.
Figure 5: Comparison of axial species profiles between PR-
DNS (solid lines) and 1D simulations (dashed lines) for
different Prandtl numbers (Pr) and Thiele moduli (Th).
Figure 6: Comparison of axial gas-particle temperature
difference between PR-DNS (solid lines) and 1D
simulations (dashed lines) for different Prandtl numbers
(Pr) and Thiele moduli (Th).
0.07
0.075
0.08
0.085
0.09
0.095
0.1
00.001 0.002 0.003 0.004 0.005
Reactant mass fraction
Height (m)
Pr = 0.4
Th = 5 Th = 10 Th = 20
0.07
0.075
0.08
0.085
0.09
0.095
0.1
00.001 0.002 0.003 0.004 0.005
Reactant mass fraction
Height (m)
Pr = 1.6
Th = 5 Th = 10 Th = 20
0.07
0.075
0.08
0.085
0.09
0.095
0.1
00.001 0.002 0.003 0.004 0.005
Reactant mass fraction
Height (m)
Pr = 6.4
Th = 5 Th = 10 Th = 20
0
50
100
150
200
250
0 0.001 0.002 0.003 0.004 0.005
Temperature difference (K)
Height (m)
Pr = 0.4
Th = 5 Th = 10 Th = 20
0
50
100
150
200
250
300
0 0.001 0.002 0.003 0.004 0.005
Temperature difference (K)
Height (m)
Pr = 1.6
Th = 5 Th = 10 Th = 20
0
50
100
150
200
250
300
350
400
0 0.001 0.002 0.003 0.004 0.005
Temperature difference (K)
Height (m)
Pr = 6.4
Th = 5 Th = 10 Th = 20
T5 - 1D T10 - 1D T20 - 1D
PR-DNS and 1D Modelling of Catalytic Reactions in a Packed Bed
Figure 7: Comparison of axial species profiles between PR-
DNS (solid lines) and 1D simulations (dashed lines) for
different reaction enthalpies (dHrxn in kJ/mol). The
effectiveness factor predicted by the 1D model is also
shown for the different cases.
Variation of reaction enthalpy
Results in the previous section were generated with a strongly
endothermic reaction kJ/mol. This section will
investigate three additional reaction enthalpies on the case with
the greatest mass and heat transfer resistances (Th20-Pr6.4).
Figure 7 shows the effect of reaction enthalpy on the reactant
conversion. It is clear that a decrease in the reaction enthalpy
greatly increased reactant conversion and that the 1D model
accurately predicts the results from PR-DNS.
The increase in conversion with a decrease in the
endothermicity of the reaction is simply due to the large heat
transfer resistances included in this case. As the reaction
becomes less endothermic, the requirement for heat flow into
the particle reduces, thereby lessening the impact of this
limitation. As a result, mass transfer becomes the controlling
phenomenon, as can be seen from the reduction in the
effectiveness factor in Figure 7.
Figure 8: Comparison of the 1D simulations to the PR-
DNS results illustrating the importance of assigning the
reaction heat to the particle phase.
Importance of the reaction enthalpy source term
Finally, an important observation regarding the 1D-modelling
of gas-solid reaction systems with significant reaction
enthalpies can be shared. It is intuitive to add the energy source
term related to a reaction involving gas species to the gas phase,
but this results in large errors if significant gas-particle heat
transfer limitations exist. To get accurate predictions, all
reaction enthalpy must be assigned to the particle phase in the
1D simulation. This practice mimics the real case where all
reaction heat is released or consumed within the particle, even
if only gas species is involved in the reaction.
As an illustration of the importance of this observation, the axial
species profiles from the Th20-Pr6.4 case with
kJ/mol are presented in Figure 8. It is clear that assigning
the reaction heat to the gas phase completely over-predicts the
reaction. This is because the large gas-particle heat transfer
limitation observed in earlier sections is essentially eliminated
if the heat is not extracted in the particle phase.
CONCLUSION
This work presented a comparison of particle-resolved direct
numerical simulations (PR-DNS) results with 1D modelling of
a reactive gas-particle system with large heat and mass transfer
limitations. Existing 1D model closures for intra-particle mass
transfer and gas-particle heat transfer compared well to the PR-
DNS results. However, it was shown that it is vitally important
that all reaction heat must be assigned as a source term in the
particle phase, even if only gas species are reacting.
When a highly endothermic reaction is
simulated, gas-particle heat transfer completely dominates the
reaction phenomena in the particle assembly. Large heat
consumption in the particle requires large quantities of heat to
enter the particle from the gas phase. Mass transfer resistances
become increasingly important as the reaction enthalpy
becomes smaller until the system becomes exclusively mass
transfer controlled when no reaction heat is simulated.
It was also interesting to observe that the 1D model still
produced good results even though significant intra-particle
heat transfer limitations were included to generate some
temperature gradients inside the particles. This finding,
combined with the knowledge that a constant particle
temperature is normally a safe assumption, suggests that good
models for external gas-particle heat transfer and internal mass
0
0.02
0.04
0.06
0.08
0.1
0.12
0 0.001 0.002 0.003 0.004 0.005
Reactant mass fraction
Height (m)
dHrxn = 100 dHrxn = 10 dHrxn = 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.001 0.002 0.003 0.004 0.005
Effectiveness factor
Height (m)
dHrxn = 100 dHrxn = 10 dHrxn = 0
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0 0.001 0.002 0.003 0.004 0.005
Reactant mass fraction
Height (m)
PR-DNS dHrxn particle dHrxn gas
Singhal et al.
transfer are sufficient for accurate 1D model predictions of
packed bed reactors.
ACKNOWLEDGEMENT
This work is a part of a European Union project under Seventh
research framework program (FP7/2007-2013) under grant
agreement n° 604656 - A multi-scale Simulation based design
platform for Cost effective CO2 capture Processes using Nano-
structured materials (NanoSim). The authors are grateful to
European Research Council for its support. Additionally, the
computational resources at NTNU provided by NOTUR,
http://www.notur.no, were used during this project.
REFERENCES
Cloete, S., Gallucci, F., van Sint Annaland, M., Amini,
S., (2016). ''Gas Switching as a Practical Alternative for
Scaleup of Chemical Looping Combustion''. Energy
Technology 4, 1286-1298.
Gunn, D.J., (1978). ''Transfer of heat or mass to particles
in fixed and fluidised beds''. International Journal of
Heat and Mass Transfer 21, 467-476.
Ishida, M., Wen, C.Y., (1968). ''Comparison of kinetic
and diffusional models for solid-gas reactions''. AIChE
Journal 14, 311-317.
Levenspiel, O., (1999). ''Chemical Reaction Engineering,
3rd ed''. John Wiley & Sons, New York.
Nashtaee, P.S.b., Khoshandam, B., (2014). ''Noncatalytic
gas-solid reactions in packed bed reactors: a comparison
between numerical and approximate solution
techniques''. Chemical Engineering Communications
201, 120-152.
Ramachandran, P.A., Doraiswamy, L.K., (1982).
''Modeling of noncatalytic gas-solid reactions''. AIChE
Journal 28, 881-900.
Singhal, A., Cloete, S., Radl, S., Quinta-Ferreira, R.,
Amini, S., (2017). ''Heat transfer to a gas from densely
packed beds of monodisperse spherical particles''.
Chemical Engineering Journal 314, 27-37.
Szekely, J., Evans, J.W., Sohn, H.Y.,, (1976). ''Gas-solid
reactions''. Academic Press, New York.
Thiele, E.W., (1939). ''Relation between Catalytic
Activity and Size of Particle''. Industrial & Engineering
Chemistry 31, 916-920.
Yang, W., Cloete, S., Morud, J., Amini, S., (2016). ''An
Effective Reaction Rate Model for Gas-Solid Reactions
with High Intra-Particle Diffusion Resistance'',
International Journal of Chemical Reactor Engineering,
p. 331.
