PARSCALE -AN OPEN-SOURCE LIBRARY FOR THE SIMULATION OF INTRA-PARTICLE HEAT AND MASS TRANSPORT PROCESSES IN COUPLED SIMULATIONS

Abstract

We introduce the open-source library ParScale for the modeling of intra-particle transport processes in non-isothermal reactive fluid-particle flows. The underlying equations, the code architecture, as well as the coupling strategy to the widely-used DEM solver LIGGGHTS is presented. A set of verification cases, embedded into an automated test harness, is presented that proofs the functionality of ParScale. To demonstrate the capabilities of ParScale, we perform simulations of a non-isothermal granular shear flow including heat transfer to the surrounding fluid. We present results for the conductive heat flux through the particle bed for a wide range of dimensionless cooling rates and particle volume fractions. Our data suggests that intra-particle temperature gradients need to be considered for an accurate prediction of the conductive flux in case of (i) a dense particle bed and (ii) for large cooling rates characterized by a critical Biot number of ca Bi crit ≈ 0.1.

Full text

IV International Conference on Particle-based Methods – Fundamentals and Applications
PARTICLES 2015
E. O˜nate, M. Bischoff, D.R.J. Owen, P. Wriggers & T. Zohdi (Eds)
PARSCALE - AN OPEN-SOURCE LIBRARY FOR THE
SIMULATION OF INTRA-PARTICLE HEAT AND MASS
TRANSPORT PROCESSES IN COUPLED SIMULATIONS
Stefan Radl1, Thomas Forgber1, Andreas Aigner2and Christoph Kloss2
1Institute of Process and Particle Engineering
Graz University of Technology, Graz, Austria
Email: radl@tugraz.at, thomas.forgber@tugraz.at
Web page: http://ippt.tugraz.at
2DCS Computing GmbH
Altenbergerstr. 66a Science Park
4040 Linz, Austria
Email: andreas.aigner@dcs-computing.com, christoph.kloss@dcs-computing.com
Web page: http://www.dcs-computing.com
Key words: Granular Materials, DEM, LIGGGHTS , ParScale, Resolved Intra Particle
Profiles, Heat and Mass Transfer, Sheared Bed
Abstract. We introduce the open-source library ParScale for the modeling of intra-
particle transport processes in non-isothermal reactive fluid-particle flows. The underlying
equations, the code architecture, as well as the coupling strategy to the widely-used DEM
solver LIGGGHTS is presented. A set of verification cases, embedded into an automated
test harness, is presented that proofs the functionality of ParScale. To demonstrate the
capabilities of ParScale, we perform simulations of a non-isothermal granular shear flow
including heat transfer to the surrounding fluid. We present results for the conductive heat
flux through the particle bed for a wide range of dimensionless cooling rates and particle
volume fractions. Our data suggests that intra-particle temperature gradients need to be
considered for an accurate prediction of the conductive flux in case of (i) a dense particle
bed and (ii) for large cooling rates characterized by a critical Biot number of ca Bicr it ≈0.1.

Stefan Radl, Thomas Forgber, Andreas Aigner and Christoph Kloss
1 Introduction
Over the past ten years the coupling of the Discrete Element Method (DEM) and Com-
putational Fluid Dynamics (CFD) has been widely adopted by academia and industry to
predict fluid-particle flows [1]. Recently, the trend towards CFD-DEM has been fueled by
the introduction of open-source toolboxes [2]. These tools are able to model momentum
and thermal energy transport of the particles and the fluid with high computational effi-
ciency, hence provide a detailed understanding of granular flow behavior. However, these
tools typically do not take intra-particle transport phenomena into account, and hence
are unable to model intra-particle processes, e.g., heterogeneous reactions and diffusion
inside a porous particle. Unfortunately, in these reactive particulate systems intra-particle
processes often play a central role, and hence may dictate overall reactor performance.
Thus, spatially-resolved particle properties (e.g., the local gas concentration in the pores
of the particle) need to be resolved to account for all relevant phenomena in the system
[3].
ParScale, a newly developed open-source library implemented in a C/C++ environment
and publicly available through www.github.com [4], closes this gap. At the current devel-
opment state, ParScale contains a plurality of models that aim on predicting heat and
mass transfer, as well as homogeneous and heterogeneous reactions inside flowing porous
particles. Also, it is possible to account for a single or multiple-reactions, i.e., a whole
reaction network. Due to a modular class-based structure, and the integration into an
automated test harness, easy extendibility and a high software quality is ensured. Our
contribution outlines the governing equations for modeling various intra-particle phenom-
ena in Section 2. In section 3 a number of verification cases is presented in order to
demonstrate and verify the functionality of ParScale. Section 4 demonstrates the cou-
pling to LIGGGHTS and evaluates the need to account for intra-particle temperature
gradients in non-isothermal granular shear flows cooled by a surrounding fluid.
2 Simulation method and parallel coupling strategy
The key purpose of ParScale is to predict intra-particle target properties (e.g., the
temperature) as a function of time and space in a spherical particle and a fixed grid
consisting of equidistant grid points. The governing equations for the relevant transport
phenomena within each particle are outlined in the next section.
2.1 Transport within a particle
To illustrate a typical transport equation to be solved by ParScale within a single
particle, the Fourier equation in spherical coordinates with λef f =const. is considered:
ρ cp
∂T
∂t =div(λef f grad T ) + sT(1)
where ρis the density, cpis the heat capacity, Trepresents the target property profile, t
is the time, λef f is an effective conductivity (e.g., for heat), and sTis a volume-specific
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Stefan Radl, Thomas Forgber, Andreas Aigner and Christoph Kloss
source term (e.g., for thermal energy). By introducing the thermal diffusivity
a=λeff
ρ cp
(2)
and for constant thermal conductivity λeff , Eqn. (1) can be re-written as
∂T
∂t =a·∂2T
∂r2+2
r
∂T
∂r +sT
ρ cp
(3)
where ris the radial coordinate.
This partial differential equation (PDE) can be discretized in space, e.g., using a central-
differencing scheme, in the spirit of the so-called method of lines. The resulting system
of ordinary differential equations (ODEs) needs to be solved using a robust integration
approach, since the source sTmight lead to a stiff behavior of the PDE. Specifically, we
have chosen the flexible multi-step, variable-order solver CVODE, which uses a (modified)
Newton-iteration approach to robustly integrate stiff systems of ODEs. CVODE contains
a plurality of direct and iterative linear solvers for solving the resulting matrix-algebra
problems, and is actively maintained by the Lawrence Livermore National Laboratory
(LLNL, U.S.A) as part of the SUNDIALS package [5]. ParScale inherits the flexibility of
CVODE, and hence can handle quickly changing environmental conditions, or fast, strongly
exothermal reactions in porous particles.
For the second verification example (see Section 3.2), we will consider a single heteroge-
neous reaction in a porous particle. The corresponding transport equations are presented
in the next section.
2.2 Chemistry model
The following equations model the mole-based reactive species balance equations in a
particle with constant porosity ǫ. We have adopted the notation of Noorman et al. [6],
and from which we have extracted typical system parameters for a relevant application.
Here we focus on a single chemical reaction, which is considered to be irreversible and
involves a solid species A, as well as a gas species B educt that forms a solid product C
and a gaseous product D:
aA(s) + bB(g)→cC (s) + sD(g) (4)
Relevant real-world examples following this scheme are (i) the reduction of iron oxides
by H2, (ii) the oxidation of ZnS and FeS with O2to metal oxides (i.e., ZnO, Fe2O3), or
(iii) the combustion of coal with a high ash content. The mole balance equation for each
species in the porous particle is written as:
∂εci
∂t =−1
r2
∂
∂r (r2Ni) + siwith i=B, D (5)
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Stefan Radl, Thomas Forgber, Andreas Aigner and Christoph Kloss
Here εis the phase fraction, ciis the gas concentration (in kM ol per m3gas volume), Ni
the (convective and diffusive) flux, and siis the chemical source term. A typical model
for siis
si=
NR
X
j=1
νij ·
NG+NS
Y
k
cnk,j
k(6)
where NR is the number of reactions, N G is the number of gas components, νij is the
stoichiometric coefficient of species iin reaction j,N S is the number of solid species, and
nk,j is the reaction exponent. The mole balance equations for the solid phase are similar,
however, exclude the flux and are based on molar concentrations in kMol per m3total
volume:
∂ci
∂t =siwith i=A, C (7)
For spatial discretizing a second order central differencing scheme is used. For further
information about the underlying equations, available models and discretizion schemes
the interested reader may refer to the online documentation in the public repository of
ParScale [4]. The next section briefly outlines the run modes of ParScale including the
coupling to the open-source DEM-based solver LIGGGHTS .
2.3 Parallel coupling strategy
Besides a stand-alone mode the current development state of ParScale provides cou-
pling capabilities to LIGGGHTS and selected solvers of CFDEMcoupling in parallel. The
key idea is that ParScale acts as a slave to the master (i.e., LIGGGHTS ), and can ex-
change its data containers between individual processes as requested by the master. In this
paper we will only focus on the coupling to LIGGGHTS . The coupling to CFDEMcoupling
is handled via LIGGGHTS data structures, and hence is in fact unproblematic. Figure 1
illustrates the underlying coupling algorithm.
At every timestep tnLIGGGHTS advances the particle position and velocity (and other
integral particle quantities if desired) in the simulation domain. After this computation is
finished, the coupling is realized by updating the particle surface temperature (Tsurf ace)
directly, or (alternatively) a heat transfer coefficient αtogether with the fluid temperature
in the vicinity of the particle. In addition, a conductive heat flux due to particle-particle
collisions ( ˙qcond) can be imposed as well. ParScale initializes from the last time step tn−1,
and calculates all internal property fields according to the imposed boundary conditions.
Additional coupling options are available, e.g., LIGGGHTS is able to push a particle-
unique environment temperature (i.e., fluid temperature) to ParScale. This enables
ParScale to react due to changes in the environment temperature, e.g., if the particle
enters a region if a fixed temperature. Furthermore, the coupling provides the option to
reset the value of the target property to a certain value. Due to the automatic sub-time
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Stefan Radl, Thomas Forgber, Andreas Aigner and Christoph Kloss
Figure 1: Coupling between LIGGGHTS and ParScale for one timestep including required
coupling parameters and additional coupling options.
stepping of CVODE, the internal ParScale timestep does not necessarily has to correspond
to the DEM timestep from LIGGGHTS . After ParScale completed its calculation of the
surface temperatures of all particles, these temperatures are updated and taken into ac-
count by LIGGGHTS at timestep tn+1 when computing conductive fluxes. Furthermore,
source terms due to reactions, the core and the volume-averaged temperature and surface
fluxes handed over to LIGGGHTS .
Next, two verification cases, i.e., transient cooling of a sphere and a heterogeneous reac-
tion, is presented.
3 Verification cases
3.1 Cooled sphere
The first verification case considers a classical situation in which a spherical particle
(initially having the uniform temperature T0) is convectively cooled by an ambient fluid
with temperature Tenviro. Table 1 summarizes the parameters of this case. Figure 2
illustrates the comparison between the numerical solution by ParScale and the analytical
solution provided by [7] for a number of time coordinates.
As expected, excellent agreement (i.e., an average error of 10−6, and a maximum error
of 10−5) between analytical and numerical solution can be found.
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Stefan Radl, Thomas Forgber, Andreas Aigner and Christoph Kloss
Table 1: Parameters for the verification case ’cooling of a sphere’.
cp300 [J m−3K]
ρ1000 [kg m−3]
λs1 [W m−1K−1]
αp100 [W m−2K−1]
rp5·10−3[m]
T0800 [K]
Tenviro 300 [K]
t1, t2, t3, t42,5,8,10 [sec]
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
T / T0
r/R
t1
t2
t3
t4
analytical
Figure 2: Numerical (symbols) and analytical results (lines) for the temperature distri-
bution in a convectively cooled sphere at t1= 2 s, t2= 5 s, t3= 8 s, and t4= 10 s.
3.2 Heterogeneuous reation
This verification case considers a single reaction, and follows the analytical solution
provided by Wen [8] for a relative reaction speed (characterized by the Thiele Modulus)
of ≈3.16. All parameters are chosen following the copper oxidation case considered in
Noorman et al. [6]. The three basic assumpions are (i) an isothermal particle, (ii) a
reaction rate that is first-order with respect to the gas-phase species, (iii) and a reaction
of: 2 CO +O2→2C uO. Figure 3 shows a comparison of the solid and fluid concentration
inside the particle for two characteristic times.
Figure 3 (a) shows an excellent agreement for the early stage of the reaction. The small,
but noticable, differences in the later stage (see Figure 3 (b)) are due to the pseudo-steady
state assumption that needs to be adopted when deriving the analytical solution provided
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Stefan Radl, Thomas Forgber, Andreas Aigner and Christoph Kloss
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
r/R
Ci/Ci,0, i =A, S
(a)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
r/R
Ci/Ci,0, i =A, S
(b)
Figure 3: Solid and gas-phase concentration profiles inside a porous copper particle for
the early stage (a) and the late stage (b) of a heterogeneous reaction (lines: analytical
solution, symbols: predictions by ParScale).
by Wen [8]. Figure 4 shows the overall conversion, again indicating only minor differences
that can be explained by the shortcomings of the analytical solution. The comparison of
the analytical and numerical solution shows good agreement, and the mean difference is
below 2 %. We note here that previous work [6] came to a similar conclusion.
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9
t/treact
Xs
Figure 4: Conversion characteristics during a typical oxidation cycle of a porous copper
particle (line: analytical solution, dots: prediction by ParScale).
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Stefan Radl, Thomas Forgber, Andreas Aigner and Christoph Kloss
4 Simple shear flow
We now investigate the influence of ParScale under well-controlled flow conditions.
Therefore, particles are placed in a cubic periodic box (H/dp= 15) at various particle
volume fractions φp= (0.3−0.64). LeesEdwards boundary conditions are applied to
drive a homogeneous shear flow, which is typically used in studies of granular rheology
[9]. In the current contribution, the shear gradient is pointing in the y-direction, and we
analyze all quantities of interest (e.g., the conductive flux) only in this direction. Along
with the shear gradient, a temperature gradient is applied. We determine the Biot (Eqn.
8) and the Peclet (Eqn. 9) number as the two main non-dimensional influence parameters
as already mentioned by [10]. With the usage of ParScale at low Biot numbers, we
expect that our results agree with previous work [10]. However, for higher Biot numbers
the transferred flux to the ambient fluid is much larger than that sustained by conduction
inside the particle. Consequently, the influence on the heat transfer rate on the particles’
shell temperature, and hence the conductive flux becomes important. Therefore, we
expect that intra particle temperature profiles should be considered above some critical
Biot number Bicrit .
Bit=α dp
λp
(8)
where αis the heat transfer coefficient that characterized the rate of cooling by the
ambient fluid.
P e =(dp/2)2
λp/ ρccp
·˙γ(9)
where ˙γis the shear rate. The conductive reference flux is expressed as [9]
˙qcond,ref =−λp(∂yT)middle .(10)
where ˙qcond,ref is the reference conductive heat flux and ymiddle is the length of the region
over which the temperature gradient is applied. Table 2 shows the main non-dimensional
parameters of the sheared bed simulation. All other parameters, e.g., the particle stiffness,
the coefficient of restitution and time step are in agreement with our earlier simulations
[10].
Table 2: Dimensionless properties of the sheared bed particle case.
φp0.3...0.64
Bit10−6...10
P e 0.25
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Stefan Radl, Thomas Forgber, Andreas Aigner and Christoph Kloss
10-12
10-10
10-8
10-6
10-4
10-2
100
10-6 10-5 10-4 10-3 10-2 10-1 100101
Bit
˙q/ ˙qcond
φp= 0.35
φp= 0.45
φp= 0.54
φp= 0.60
φp= 0.64
Figure 5: Effect of the Biot number on the conductive heat flux in the gradient direction
for various particle volume fractions.
It can be seen in Figure 5 that the influence of the Biot number on the overall conduc-
tive flux needs to be considered above a certain value for the Biot number, which depends
on φp. Even at low Biot number regimes (i.e., Bit≈10−3) the conductive flux is under-
predicted by up to 10 %. For the highest particle volume fractions considered, for which
conductive fluxes are most relevant since they are comparable to the particle-convective
flux, the critical Biot number is Bicrit ≈10−2. Above this Biot number intra particle
profiles should be taken into account for the moderately fast sheared particle bed that we
considered. It is also shown that the influence of the Biot number on the conductive heat
flux is becoming more important in case the particle volume fraction is decreasing. Thus,
the slope of the curve indicating the relationship between Bi and qcond is becoming larger
with decreasing φpfor the high Bi regime. However, in the rather dilute flow at which
these extreme dependency is observed, the conductive flux is negligibly small compared
to the particle-convective flux.
5 Conclusions
We presented a novel open-source simulation tool ParScale which is published under
LGPL licence and can be linked to any particle-based solver. We outlined the coupling to
the open-source DEM solver LIGGGHTS and demonstrated the usage of ParScale with
selected verification cases. A good agreement is found between numerical results produced
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Stefan Radl, Thomas Forgber, Andreas Aigner and Christoph Kloss
by ParScale and analytical solutions available in literature. We extended the analysis of
the well-known sheared particle bed by taking intra-particle temperature gradients into
account. A key result is that even at comparably low Biot numbers the intra-particle
property profiles have a substantial influence on the conductive flux. The physical reason
is that the particle surface temperature is lower than the particle-average temperature
in case cooling by the ambient fluid is taken into account. This leads to smaller surface
temperature differences in the event of a particle-particle collisions. Thus, the transferred
heat flux to the environment should be considered when predicting the particle-particle
conductive fluxes. This is especially true for high particle concentrations and fast cooling
conditions, since the conductive flux for low particle concentrations is anyhow very low.
The current study was limited to selected particle volume fractions and Peclet numbers.
Future work will consider wider ranges of these parameters.
6 Aknowledgement
The authors acknowledge support by the European Commission through FP7 Grant
agreement no. 604656 (NanoSim). T.F. and S.R. acknowledge support from ”NAWI
Graz” by providing access to dcluster.tugraz.at. LIGGGHTS is a registered trademark of
DCS Computing GmbH.
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Stefan Radl, Thomas Forgber, Andreas Aigner and Christoph Kloss
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