Particle distributions
MFIX-Exa can generate constant
(monodisperse or single-sized) particles or
polydisperse particles that follow a normal
, log-normal
, or uniform
distribution. A user-defined custom
distribution can be specified by providing
discrete probabilities via a text input file.
In addition to the distribution type, the distribution weighting must be specified; MFIX-Exa supports two distribution weightings:
- Number-weighted diameter distribution
The number-weighted diameter density function, \(f_X^N(x)\,dx\), can be interpreted as defining the number of particles with diameters between \(x\) and \(x+dx\), divided by the total number of particles.
- Volume-weighted diameter distribution
The volume-weighted diameter density function, \(f_X^V(x)\,dx\), similarly can be interpreted as defining the volume of particles with diameters between \(x\) and \(x+dx\), divided by the total volume of particles.
Warning
Number-weighted and volume-weighted distributions are not interchangeable. Failing to specify the correct distribution weighting can result in unexpected behavior.
The code uses both distribution weightings, computing (or approximating) whichever distribution weighting that was not provided. For example, if a number-weighted normal distribution is provided, the corresponding volume-weighted normal distribution is computed. The volume-weighted distribution is used to calculate the number of particles (or parcels) to generate in an initial condition region. The number-weighted distribution is sampled when DEM particles are created, while the volume-weighted distribution is sampled when creating PIC parcels. Sampling the volume-weighted distribution for PIC parcels allows for fewer parcels to represent very small particles while using more parcels to represent large particles in the distribution. The statistical weight of the parcels is adjusted so that the number-distribution is represented accurately.
The following sections overview the distributions supported by MFIX-Exa and the input parameters that control their behavior.
constant
distribution
The constant
distribution defines a monodisperse (single-valued) distribution and uses the following inputs:
Description |
Type |
Default |
|
---|---|---|---|
constant |
size of the mono-disperse distribution |
Real |
N/A |
In the following example, a cuboid is filled with 1 mm diameter monodisperse particles with 2500 kg·m-3 density. The initial condition region volume is 0.001 m3, and it has a solids volume fraction of 0.05 (5% solids).
# Regions for defining ICs and BCs
# -----------------------------------------------------------------------
mfix.regions = full-domain
regions.full-domain.lo = 0.0 0.0 0.0
regions.full-domain.hi = 0.1 0.1 0.1
# Initial Conditions
# -----------------------------------------------------------------------
ic.regions = full-domain
mfix.particle_init_type = Auto
# Full domain
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>
ic.full-domain.fluid.volfrac = 0.95
ic.full-domain.fluid.density = 1.0
ic.full-domain.fluid.pressure = 0.0
ic.full-domain.fluid.velocity = 0.0 0.0 0.0
ic.full-domain.solids = solid0
ic.full-domain.packing = random
ic.full-domain.solid0.volfrac = 0.05
ic.full-domain.solid0.velocity = 0.00 0.00 0.00
ic.full-domain.solid0.diameter = constant
ic.full-domain.solid0.diameter.constant = 1000.e-6 # (m)
ic.full-domain.solid0.density = constant
ic.full-domain.solid0.density.constant = 2500.0 # (kg/m^3)
An MFIX-Exa DEM simulation generates approximately 95,500 particles, while a PIC simulation with
pic.close_pack = 0.64
, pic.parcels_per_cell_at_pack = 64
, and an Eulerian mesh spacing
\(\Delta x\) of 3.125 mm generates approximately 164,000 parcels.
normal
distribution
The normal
or Gaussian distribution is given by
where the parameter \(\mu\) is the distribution mean, and \(\sigma\) is the standard deviation. The inputs used to specify the distribution are provided in the following table.
Description |
Type |
Default |
|
---|---|---|---|
type |
distribution weighting: |
string |
N/A |
mean |
mean of the normal random variable, \(X\) |
Real |
N/A |
std |
standard deviation of the normal random variable, \(X\) |
Real |
N/A |
min |
Minimum particle diameter. Drawn samples below |
Real |
N/A |
max |
Maximum particle diameter. Drawn samples above |
Real |
N/A |
bins |
Number of bins used when discretizing the distribution to approximate the number of particles within an initial condition region. |
int |
64 |
In the following example, a cuboid with volume 0.001 m3 is filled with a number-weighted normal distribution with 1 mm mean diameter and 0.25 mm standard deviation. The minimum and maximum particle diameters are 0.25 mm and 1.75 mm, respectively.
# Regions for defining ICs and BCs
# -----------------------------------------------------------------------
mfix.regions = full-domain
regions.full-domain.lo = 0.0 0.0 0.0
regions.full-domain.hi = 0.1 0.1 0.1
# Initial Conditions
# -----------------------------------------------------------------------
ic.regions = full-domain
mfix.particle_init_type = Auto
# Full domain
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>
ic.full-domain.fluid.volfrac = 0.95
ic.full-domain.fluid.density = 1.0
ic.full-domain.fluid.pressure = 0.0
ic.full-domain.fluid.velocity = 0.0 0.0 0.0
ic.full-domain.solids = solid0
ic.full-domain.packing = random
ic.full-domain.solid0.volfrac = 0.05
ic.full-domain.solid0.velocity = 0.00 0.00 0.00
ic.full-domain.solid0.diameter = normal
ic.full-domain.solid0.diameter.type = number-weighted
ic.full-domain.solid0.diameter.mean = 1000.e-6 # (m)
ic.full-domain.solid0.diameter.std = 250.e-6 # (m)
ic.full-domain.solid0.diameter.min = 250.e-6 # (m)
ic.full-domain.solid0.diameter.max = 1750.e-6 # (m)
ic.full-domain.solid0.density = constant
ic.full-domain.solid0.density.constant = 2500.0 # (kg/m^3)
An MFIX-Exa DEM simulation with these settings generates approximately 81,700 particles. The particle size distribution is shown in Fig. 9 (open circles) in addition to the number-weighted (orange) and volume-weighted (blue) density functions. As noted by Weiner [Wei11], the volume-weighted probability density function (PDF) is shifted to the right of the number-weighted PDF.
An MFIX-Exa PIC simulation with the same settings generates approximately 82,000 parcels when pic.close_pack = 0.64
,
pic.parcels_per_cell_at_pack = 32
, and the Eulerian mesh spacing \(\Delta x\) is 3.125 mm. The parcel size distribution
shown in Fig. 10 (open triangles) follows the volume-weighted density function.
Small closed circles illustrate the effective particle size distribution, computed by multiplying each observation
by the parcel statistical weight. For example, if a parcel has statistical weight of 10, then 10 observations are
recorded in the bin capturing the parcel’s diameter.
The following sections outline how MFIX-Exa computes volume-weighted normal density function parameters from number-weighted normal density function parameters, and conversely, how number-weighted normal density function parameters are computed from volume-weighted density function parameters.
Computing volume-weighted normal density function parameters, \(\mu_V\) and \(\sigma_V\), from number-weighted normal density function parameters, \(\mu_N\) and \(\sigma_N\)
The volume-weighted density function, \(f_X^V(x)\), computed from the normally distributed number-weighted density function, \(f_X^N(x)\), is normally distributed with mean, \(\mu_V\), and variance, \(\sigma_V^2\). An expression for the volume-weighted density function is obtained by multiplying Eq.1 by \(x^3\) and substituting in the mean and standard deviation of the number-weighted distribution.
The mean of this distribution is computed as
and the variance is given by
where \(b = (\mu_N - \mu_V)\). Because the volume-weighted distribution is normally distributed, the computed mean and variance can be substituted directly into Eq.1.
Computing number-weighted normal density function parameters, \(\mu_N\) and \(\sigma_N\), from volume-weighted normal density function parameters, \(\mu_V\) and \(\sigma_V\)
The number-weighted density function mean and standard deviation, \(\mu_N\) and \(\sigma_N\), are computed from the volume-weighted mean and standard deviation, \(\mu_V\) and \(\sigma_V\), by solving the nonlinear system of equations constructed from Eq.2 and Eq.3 for \(\mu_N\) and \(\sigma_N\).
This system is solved using the homotopy method outlined in Burden and Faires [BF10]. To ensure rapid convergence, initial guesses for \(\mu_N\) and \(\sigma_N\) are calculated from the density function created by dividing Eq.1 by \(x^3\) and substituting in the mean and standard deviation of the volume-weighted distribution, \(\mu_V\) and \(\sigma_V\).
\(\tilde{f}_X^N(x)\), shown in Fig. 11, is not the number-weighted density function, but an approximation to it. The mean and variance are calculated by numerically integrating \(\tilde{f}_X^N(x)\) over the interval \(x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right]\) where \(x_{min}\) and \(x_{mid}\) are found by setting the derivative of Eq.4 to zero and computing the roots. \(x_\mathrm{max}\) is chosen to mirror \(x_\mathrm{min}\) across the midpoint.
Sampling a normal distribution
MFIX-Exa samples the normal distribution with mean m_mean
and standard deviation m_stddev
to return an
distribution observation.
amrex::Real observation;
do { observation = amrex::RandomNormal(m_mean, m_stddev, a_engine); }
while (!(m_min <= observation && observation <= m_max));
return observation;
The function amrex::RandomNormal
returns one pseudo-random real number (double). The sample is discarded and the distribution
is sampled again until the value satisfies \(\mathrm{xmin} \le x\) and \(x \le \mathrm{xmax}\).
log-normal
distribution
The log-normal
distribution is given by
where \(\mu\) and \(\sigma\) are parameters defining the distribution.
Note
\(\mu\) and \(\sigma\) are not the mean and standard deviation of \(x\). They are the mean and standard deviation of \(\mathrm{ln}(x)\). The mean (expectation), \(E(x)\), and variance, \(V(x)\), of the log-normal random variable are given by
and
respectively. See William Navidi [WN14] for an overview of log-normal distributions.
Description |
Type |
Default |
|
---|---|---|---|
type |
distribution weighting: |
string |
N/A |
mean |
mean of of the normal random variable, \(\mathrm{ln}(X)\) |
Real |
N/A |
stddev |
standard deviation of the normal random variable, \(\mathrm{ln}(X)\) |
Real |
N/A |
min |
Minimum particle diameter. Drawn samples below |
Real |
N/A |
max |
Maximum particle diameter. Drawn samples above |
Real |
N/A |
bins |
Number of bins used when discretizing the distribution to approximate the number of particles within an initial condition region. |
int |
64 |
In the following example, a cuboid with volume 0.001 m3 is filled with a volume-weighted log-normal distribution with the listed parameters. The minimum and maximum particle diameters are 0.23 mm and 3.00 mm, respectively.
# Regions for defining ICs and BCs
# -----------------------------------------------------------------------
mfix.regions = full-domain
regions.full-domain.lo = 0.0 0.0 0.0
regions.full-domain.hi = 0.1 0.1 0.1
# Initial Conditions
# -----------------------------------------------------------------------
ic.regions = full-domain
mfix.particle_init_type = Auto
# Full domain
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>
ic.full-domain.fluid.volfrac = 0.75
ic.full-domain.fluid.density = 1.0
ic.full-domain.fluid.pressure = 0.0
ic.full-domain.fluid.velocity = 0.0 0.0 0.0
ic.full-domain.solids = solid0
ic.full-domain.packing = random
ic.full-domain.solid0.volfrac = 0.25
ic.full-domain.solid0.velocity = 0.00 0.00 0.00
ic.full-domain.solid0.diameter = log-normal
ic.full-domain.solid0.diameter.type = volume-weighted
ic.full-domain.solid0.diameter.mean = -3.662955279
ic.full-domain.solid0.diameter.std = 1.04
ic.full-domain.solid0.diameter.min = 30.e-6 # (m)
ic.full-domain.solid0.diameter.max = 3000.e-6 # (m)
ic.full-domain.solid0.density = constant
ic.full-domain.solid0.density.constant = 2500.0 # (kg/m^3)
An MFIX-Exa DEM simulation with these settings generates approximately 162,000 particles. The particle size distribution (open circles), and number-weighted (orange), and volume-weighted (blue) density functions are shown in Fig. 12.
An MFIX-Exa PIC simulation with the same settings generates approximately 410,000 parcels when pic.close_pack = 0.64
,
pic.parcels_per_cell_at_pack = 32
, and the Eulerian mesh spacing \(\Delta x\) is 3.125 mm.
Fig. 13 shows the parcel size distribution (open triangles) and number-weighted (orange)
and volume-weighted (blue) density functions. Again, small closed circles indicate the effective particle distribution
captured by the PIC parcels.
Converting between number-weighted and volume-weighted parameters for log-normal distributions
As reported by El-Hilo [EH12], El-Hilo and Chantrell [EHC12], number-weighted and volume-weighted log-normal distributions use the same \(\sigma\) parameter
while the parameter \(\mu\) is computed directly from the other distribution.
Sampling a log-normal distribution
Rather than sample the log-normal distribution, MFIX-Exa samples the normal distribution with
mean m_mean
and standard deviation m_stddev
.
amrex::Real observation;
do { observation = amrex::RandomNormal(m_mean, m_stddev, a_engine); }
while (!(m_log_min <= observation && observation <= m_log_max));
return std::exp(observation);
The function amrex::RandomNormal
returns one pseudo-random real number (double). The sample is discarded and the distribution
is sampled again until the value satisfies \(\log \mathrm{xmin} \le \log x\) and \(\log x \le \log \mathrm{xmax}\).
The log-normal variable is returned by applying the exponential function, \(x = \exp ( \log x )\).
uniform
distribution
The uniform
distribution is given by
where \(x_\mathrm{min}\) and \(x_\mathrm{max}\) are the minimum and maximum particle diameters. The inputs used to specify the distribution are provided in the following table.
Description |
Type |
Default |
|
---|---|---|---|
type |
distribution weighting: |
string |
N/A |
min |
Minimum particle diameter. Drawn samples below |
Real |
N/A |
max |
Maximum particle diameter. Drawn samples above |
Real |
N/A |
bins |
Number of bins used when discretizing the distribution to approximate the number of particles within an initial condition region. |
int |
64 |
In the following example, a cuboid with volume 0.001 m3 is filled with a number-weighted uniform distribution. The minimum and maximum particle diameters are 0.25 mm and 1.75 mm, respectively.
# Regions for defining ICs and BCs
# -----------------------------------------------------------------------
mfix.regions = full-domain
regions.full-domain.lo = 0.0 0.0 0.0
regions.full-domain.hi = 0.1 0.1 0.1
# Initial Conditions
# -----------------------------------------------------------------------
ic.regions = full-domain
mfix.particle_init_type = Auto
# Full domain
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>
ic.full-domain.fluid.volfrac = 0.95
ic.full-domain.fluid.density = 1.0
ic.full-domain.fluid.pressure = 0.0
ic.full-domain.fluid.velocity = 0.0 0.0 0.0
ic.full-domain.solids = solid0
ic.full-domain.packing = random
ic.full-domain.solid0.volfrac = 0.05
ic.full-domain.solid0.velocity = 0.00 0.00 0.00
ic.full-domain.solid0.diameter = uniform
ic.full-domain.solid0.diameter.type = number-weighted
ic.full-domain.solid0.diameter.min = 250.e-6 # (m)
ic.full-domain.solid0.diameter.max = 1750.e-6 # (m)
ic.full-domain.solid0.density = constant
ic.full-domain.solid0.density.constant = 2500.0 # (kg/m^3)
An MFIX-Exa DEM simulation with these settings generates approximately 110,000 particles. The particle size distribution (open circles), and number-weighted (orange), and volume-weighted (blue) density functions are shown in Fig. 14.
An MFIX-Exa PIC simulation with the same settings generates approximately 410,000 parcels when pic.close_pack = 0.64
,
pic.parcels_per_cell_at_pack = 32
, and the Eulerian mesh spacing \(\Delta x\) is 3.125 mm.
Fig. 15 shows the parcel size distribution (open triangles) and number-weighted (orange)
and volume-weighted (blue) density functions. Again, small closed circles indicate the effective particle distribution
captured by the PIC parcels.
Sampling a uniform distribution
DEM simulations where a number-weighted distribution is specified and PIC simulations where a volume-weighted distribution is specified compute a uniformly distributed variable \(rand \in \left( 0, 1 \right)\), then scale and translate the result such that \(x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right)\).
amrex::Real rand = amrex::Random(a_engine);
return m_min + (m_max-m_min)*rand;
The function amrex::Random
returns one pseudo-random real number (double) from a
uniform distribution between 0.0 and 1.0 (0.0 included, 1.0 excluded).
Inverse sampling is used when a volume-weighted distribution is specified for a DEM simulation and when a number-weighted distribution is specified for a PIC simulation. This approach generates random variates from the continuous distribution function \(F\) using the inverse function \(F^{-1}\).
If \(\mathcal{U}\) is a uniform random variable on the interval \([0,1]\) and \(F\) is a continuous cumulative distribution function on \([a,b]\) with inverse
\[F^{-1}(u) = \inf \left\{ x: F(x) = u \right\}\]then \(F^{-1}(\mathcal{U})\) has cumulative distribution function \(F\) (adapted from Devroye [Dev86]).
The number-weighted density function, \(f_X^N(x)\), is computed from a specified volume-weighted density function by dividing Eq.6 by \(x^3\). The brace notation indicating that the density function is zero for \(x \notin \left[ x_\mathrm{min}, x_\mathrm{max} \right]\) is dropped for conciseness.
The cumulative distribution function and its inverse are
and
where \(a = x_{\mathrm{min}}\) and \(b = x_{\mathrm{max}}\). To sample \(f_X^V(x)\), a uniformly distributed variable \(u \in \mathcal{U}(0,1)\) is generated and substituted into the inverse cumulative distribution function.
amrex::Real rand = amrex::Random(a_engine);
amrex::Real const b(m_max);
amrex::Real const a(m_min);
amrex::Real const a2(a*a), b2(b*b);
return (a*b)/std::sqrt(b2 - rand*(b2-a2));
Similarly, a volume-weighted density function, \(f_X^V(x)\), is computed from a number-weighted distribution by multiplying Eq.6 by \(x^3\). Again, the brace notation indicating that the density function is zero for \(x \notin \left[ x_\mathrm{min}, x_\mathrm{max} \right]\) is dropped for conciseness.
The cumulative distribution function and its inverse are
and
where \(a = x_{\mathrm{min}}\) and \(b = x_{\mathrm{max}}\). To sample \(f_X^V(x)\), a uniformly distributed variable \(u \in \mathcal{U}(0,1)\) is generated and substituted into the inverse cumulative distribution function.
amrex::Real rand = amrex::Random(a_engine);
amrex::Real const b(m_max);
amrex::Real const a(m_min);
amrex::Real const a4(a*a*a*a), b4(b*b*b*b);
return std::pow(a4 + rand*(b4-a4),0.25);
custom
distribution
A user-defined custom
distribution can be specified by providing discrete probabilities via a text file.
Description |
Type |
Default |
|
---|---|---|---|
custom |
File name of user-defined distribution |
string |
N/A |
min |
Minimum particle diameter. Drawn samples below |
Real |
N/A |
max |
Maximum particle diameter. Drawn samples above |
Real |
N/A |
interpolate |
Enable linear interpolation between discrete bins. This option is only available when the initial distribution probability is zero. |
bool |
false |
The distribution input file contains four components:
line 1: number of distribution entries (integer)
line 2: specifies the distribution as
CDF
orPDF
(string)line 3: comment or blank line (unused by solver)
remaining lines define the bin and probability (Real Real)
The following sections provide a few examples of custom distribution configurations.
Bidisperse mixutre without interpoaltion
In the following example, a cuboid with volume 0.001 m3 is filled with a number-weighted distribution. The distribution is a 50/50 mixture of two particle sizes.
# Regions for defining ICs and BCs
# -----------------------------------------------------------------------
mfix.regions = full-domain
regions.full-domain.lo = 0.0 0.0 0.0
regions.full-domain.hi = 0.1 0.1 0.1
# Initial Conditions
# -----------------------------------------------------------------------
ic.regions = full-domain
mfix.particle_init_type = Auto
# Full domain
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>
ic.full-domain.fluid.volfrac = 0.95
ic.full-domain.fluid.density = 1.0
ic.full-domain.fluid.pressure = 0.0
ic.full-domain.fluid.velocity = 0.0 0.0 0.0
ic.full-domain.solids = solid0
ic.full-domain.packing = random
ic.full-domain.solid0.volfrac = 0.05
ic.full-domain.solid0.velocity = 0.00 0.00 0.00
ic.full-domain.solid0.diameter = custom
ic.full-domain.solid0.diameter.custom = bidisperse-pdf.dist
ic.full-domain.solid0.diameter.type = number-weighted
ic.full-domain.solid0.density = constant
ic.full-domain.solid0.density.constant = 2500.0 # (kg/m^3)
ic.full-domain.solid0.density = constant
ic.full-domain.solid0.density.constant = 2500.0 # (kg/m^3)
12
2PDF
3# bidisperse mixture
4 500.e-6 0.5
51000.e-6 0.5
An MFIX-Exa DEM simulation with these settings generates approximately 180,000 particles with half
of the particles having diameter 0.5 mm and half have diameter 1.0 mm.
An MFIX-Exa PIC simulation with the same settings generates approximately 82,000 parcels when pic.close_pack = 0.64
,
pic.parcels_per_cell_at_pack = 32
, and the Eulerian mesh spacing \(\Delta x\) is 3.125 mm. Approximately
10% of the parcels have diameter 0.5 mm and 90% have diameter 1.0 mm.
Bimodal mixutre
The following example uses the same layout as the previous example, however the custom distribution is of a CDF describing a bimodal distribution created by combining two normal distributions.
ic.full-domain.solid0.volfrac = 0.05
ic.full-domain.solid0.velocity = 0.00 0.00 0.00
ic.full-domain.solid0.diameter = custom
ic.full-domain.solid0.diameter.custom = bimodal-cdf.dist
ic.full-domain.solid0.diameter.type = number-weighted
125
2CDF
3# bimodal distribution created from two normal distributions
40.000144 0.0000
50.000240 0.0063
60.000336 0.0360
70.000384 0.0818
80.000416 0.1356
90.000432 0.1919
100.000456 0.2495
110.000480 0.3053
120.000496 0.3587
130.000528 0.4055
140.000568 0.4447
150.000584 0.4822
160.000600 0.5191
170.000616 0.5566
180.000632 0.5958
190.000664 0.6408
200.000704 0.6942
210.000720 0.7501
220.000752 0.8075
230.000768 0.8638
240.000784 0.9177
250.000816 0.9635
260.000864 0.9932
270.000960 0.9995
280.001056 1.0000
An MFIX-Exa DEM simulation with these settings generates approximately 2,285,000 particles. The particle size distribution (open circles), and number-weighted (orange), and volume-weighted (blue) density functions are shown in Fig. 16.
An MFIX-Exa PIC simulation with the same settings generates approximately 82,000 parcels when pic.close_pack = 0.64
,
pic.parcels_per_cell_at_pack = 32
, and the Eulerian mesh spacing \(\Delta x\) is 3.125 mm. The parcel size distribution
shown in Fig. 17 (open triangles) follows the volume-weighted density function.
Small closed circles illustrate the effective particle size distribution, computed by multiplying each observation
by the parcel statistical weight.
Measured size distribution
The final example uses the same layout as the previous examples, however the custom distribution
is a CDF describing an experimentally measured size distribution and the distribution type is volume-weighted
.
ic.full-domain.solid0.volfrac = 0.05
ic.full-domain.solid0.velocity = 0.00 0.00 0.00
ic.full-domain.solid0.diameter = custom
ic.full-domain.solid0.diameter.custom = qicpic-cdf.dist
ic.full-domain.solid0.diameter.type = volume-weighted
132
2CDF
3
40.00025 0.0013499
50.000298387 0.00250452
60.000346774 0.00448884
70.000395161 0.00777403
80.000443548 0.0130136
90.000491935 0.0210638
100.000540323 0.0329789
110.00058871 0.0499683
120.000637097 0.0733046
130.000685484 0.104184
140.000733871 0.143547
150.000782258 0.191886
160.000830645 0.24907
170.000879032 0.314239
180.000927419 0.385785
190.000975806 0.461453
200.00102419 0.538547
210.00107258 0.614215
220.00112097 0.685761
230.00116935 0.75093
240.00121774 0.808114
250.00126613 0.856453
260.00131452 0.895816
270.0013629 0.926695
280.00141129 0.950032
290.00145968 0.967021
300.00150806 0.978936
310.00155645 0.986986
320.00160484 0.992226
330.00165323 0.995511
340.00170161 0.997495
350.00175 0.99865
An MFIX-Exa DEM simulation with these settings generates approximately 963,000 particles. The particle size distribution (open circles), and number-weighted (orange plus), and volume-weighted (blue cross) probabilities are shown in Fig. 18.
An MFIX-Exa PIC simulation with the same settings generates approximately 164,000 parcels when pic.close_pack = 0.64
,
pic.parcels_per_cell_at_pack = 64
, and the Eulerian mesh spacing \(\Delta x\) is 3.125 mm. The parcel size distribution
shown in Fig. 19 (open triangles) follows the volume-weighted probabilities.
Small closed circles illustrate the effective particle size distribution, computed by multiplying each observation
by the parcel statistical weight.