From 580a44edcef206fc31861a8273097fc71ac29e42 Mon Sep 17 00:00:00 2001 From: Jordan Musser Date: Mon, 13 May 2024 08:21:15 -0400 Subject: [PATCH 1/6] Adding recent updates --- docs/source_docs/_static/theme_overrides.css | 7 + docs/source_docs/conf.py | 6 +- docs/source_docs/index.rst | 4 +- .../images/distribution-DEMvsPIC.png | Bin 0 -> 125018 bytes .../references/size_distributions.rst | 268 ++++++++++++++++++ .../user_guide/inputs/initial_conditions.rst | 1 + .../user_guide/inputs/model_options.rst | 60 +--- 7 files changed, 292 insertions(+), 54 deletions(-) create mode 100644 docs/source_docs/references/images/distribution-DEMvsPIC.png create mode 100644 docs/source_docs/references/size_distributions.rst diff --git a/docs/source_docs/_static/theme_overrides.css b/docs/source_docs/_static/theme_overrides.css index 7c1a520..75a87ae 100644 --- a/docs/source_docs/_static/theme_overrides.css +++ b/docs/source_docs/_static/theme_overrides.css @@ -8,3 +8,10 @@ max-width: 100%; overflow: visible; } + +.math { + text-align: left; +} +.eqno { + float: right; +} diff --git a/docs/source_docs/conf.py b/docs/source_docs/conf.py index 23ddcc2..82aa9ba 100644 --- a/docs/source_docs/conf.py +++ b/docs/source_docs/conf.py @@ -35,7 +35,8 @@ extensions = ['sphinx.ext.mathjax', 'sphinx.ext.githubpages', 'sphinx.ext.viewcode', 'sphinx_tabs.tabs', - 'sphinx.ext.todo'] + 'sphinx.ext.todo', + 'sphinxcontrib.bibtex' ] # Add any paths that contain templates here, relative to this directory. templates_path = ['ytemplates'] @@ -84,6 +85,9 @@ todo_include_todos = False # This is necessary to reference numbered figures numfig = True +math_numfig = True + +bibtex_bibfiles = ['refs.bib'] # -- Options for HTML output ---------------------------------------------- diff --git a/docs/source_docs/index.rst b/docs/source_docs/index.rst index 7d9dc38..9dc8059 100644 --- a/docs/source_docs/index.rst +++ b/docs/source_docs/index.rst @@ -54,11 +54,14 @@ To learn more about the implementation, follow 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zz|8tNm5}94AEngX3cN=*6Bh|TtKhQ-``cA;WS>`$q|ZpWj9< zqUT4o!Rc=}|I3phu?hXJCRi1NkkJ21r(crzzgi}?JCqrKSAn3Hx&FU@$ycq2sn#xZ z^TpZg{`;8V%-k1Q&K?wM3P1k8u3Wv3m2b$;i4-!6zjkT6Y=K{a4S1PFNXW#<+z_wt HeE + 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) + + + +``normal`` distribution +^^^^^^^^^^^^^^^^^^^^^^^ + +The ``normal`` or Gaussian distribution is given by + + +.. math:: + :label: eq_normal_dist + + F(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu)^2 }{ 2 \sigma^2 }\right)} + +where the parameter :math:`\mu` is the distribution mean, and :math:`\sigma` is the standard deviation. The inputs +used to specify the distribution are provided in the following table. + ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| | Description | Type | Default | ++=====================+=======================================================================+=============+===========+ +| type | distribution weighting: ``number-weighted`` or ``volume-weighted`` | string | N/A | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| mean | mean of the normal random variable, :math:`x` | Real | N/A | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| stddev | standard deviation of the normal random variable, :math:`x` | Real | N/A | +| | | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| min | Minimum particle diameter. Drawn samples below ``min`` are discarded | Real | N/A | +| | and a new sample is drawn. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| max | Maximum particle diameter. Drawn samples above ``max`` are discarded | Real | N/A | +| | and a new sample is drawn. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| bins | Number of bins used when discretizing the distribution to approximate | int | 64 | +| | the number of particles within an initial condition region. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ + + +.. rubric:: Converting a number-weighted normal distribution to volume-weighted: :math:`F_N(x) \rightarrow F_V(x)` + +The volume-weighted distribution, :math:`F_V(x)`, computed from the normal-weighted distribution, :math:`F_N(x)` +is normally distributed with mean, :math:`\mu_V`, and variance, :math:`\sigma_V^2`. An expression for the +volume-weighted probability density function (PDF) is obtained by multiplying :math:`F_N(x)` by :math:`x^3`. + +.. math:: + + f(x) = \frac{x^3}{\sigma \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu)^2 }{ 2 \sigma^2 }\right)} + +The mean is computed as + +.. math:: + + \mu_V = \frac{ \int_{-\infty}^{\infty} x f(x) dx }{\int_{-\infty}^{\infty} f(x) dx } + = \frac{3\sigma_N^4 + 6\sigma_N^2\mu_N^2 + \mu_N^4}{3\sigma_N^2 \mu_N + \mu_N^3} + +The variance is given by + +.. math:: + + \sigma^2_V = \frac{ \int_{-\infty}^{\infty} (x - \mu_V)^2 f(x) dx }{\int_{-\infty}^{\infty} f(x) dx } + = \frac{3\sigma_N^4 (3\mu_N + 2b) + \sigma^2(\mu_N^3 + 6\mu_N^2 b + 3\mu_Nb^2)+\mu_N^3 b^2}{3\sigma_N^2 \mu_N + \mu_N^3} + +where :math:`b = (\mu_N - \mu_V)`. The compute mean and variance, :math:`\mu_V` and :math:`sigma^2_V` can +be substituted into :eq:`eq_normal_dist`. The probability density and cumulative density functions +of a typical number-weighted normal distribution and its companion volume-weighted distribution are +shown in :numref:`fig_normal_dist_pdf_cdf`. As noted by :cite:t:`weiner11`, the volume-weighted +PDF and CDF are always shifted to right of the normal weighted distribution. + + +.. _fig_normal_dist_pdf_cdf: + +.. figure:: ./images/normal-dist-pdf_cdf.png + :width: 95% + :align: center + :alt: domain used with box embedded boundary + + Comparison of the number-weighted and volume-weighted normal probability density (solid lines) and cumulative + density (dashed lines) functions. The number-weighted distribution is shown in orange and has mean + :math:`\mu_N = 1000.0\mathrm{e-}6` and standard deviation :math:`\sigma_N = 250.0\mathrm{e-}6`. The volume-weighted + distribution is shown in green and has mean :math:`\mu_V \approx 1167.7\mathrm{e-}6` and standard deviation + :math:`\sigma_N \approx 232.6\mathrm{e-}6`. + + +.. rubric:: Converting a volume-weighted normal distribution to number-weighted: :math:`F_V(x) \rightarrow F_N(x)` + + +The number-weighted distribution, :math:`F_V(x)`, cannot be computed directly from the volume-weighted +distribution, :math:`F_N(x)`. + +.. is normally distributed with mean, :math:`\mu_V`, and variance, :math:`\sigma_V^2`. An expression for the + + +``log-normal`` distribution +^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The ``log-normal`` distribution is given by + +.. math:: + + F(d_p) = \frac{1}{d_p \sigma \sqrt{2\pi}} \exp{\left(- \frac{(\mathrm{ln}(d_p) - \mu)^2 }{ 2 \sigma^2 }\right)} + +where :math:`\mu` and :math:`\sigma` are parameters defining the distribution. + +.. note:: + :math:`\mu` and :math:`\sigma` **are not** the mean and standard deviation of :math:`d_p`. They are the mean and + standard deviation of :math:`\mathrm{ln}(d_p)`. The mean, :math:`E(d_p)`, and variance, :math:`V(d_p)`, of the + log-normal random variable are given by + + .. math:: + + E(d_p) = \exp{\left(\mu + \sigma^s/2\right)} + + and + + .. math:: + + V(d_p) = \exp{\left(2\mu + 2\sigma^2\right)} - \exp{\left(2\mu + \sigma^2\right)} + + respectively. See See :cite:t:`william2014` for an overview of log-normal distributions. + + ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| | Description | Type | Default | ++=====================+=======================================================================+=============+===========+ +| type | distribution weighting: ``number-weighted`` or ``volume-weighted`` | string | N/A | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| mean | mean of of the normal random variable, | Real | N/A | +| | :math:`\mathrm{ln}(d_p)` | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| stddev | standard deviation of the normal random variable, | Real | N/A | +| | :math:`\mathrm{ln}(d_p)` | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| min | Minimum particle diameter. Drawn samples below ``min`` are discarded | Real | N/A | +| | and a new sample is drawn. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| max | Maximum particle diameter. Drawn samples above ``max`` are discarded | Real | N/A | +| | and a new sample is drawn. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| bins | Number of bins used when discretizing the distribution to approximate | int | 64 | +| | the number of particles within an initial condition region. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ + + + + +The uniform, normal, and log-normal distributions + diff --git a/docs/source_docs/user_guide/inputs/initial_conditions.rst b/docs/source_docs/user_guide/inputs/initial_conditions.rst index 8293348..0af28ed 100644 --- a/docs/source_docs/user_guide/inputs/initial_conditions.rst +++ b/docs/source_docs/user_guide/inputs/initial_conditions.rst @@ -42,6 +42,7 @@ using the ``ic.[region].[fluid]`` compound prefix. +------------------------+------------------------------------------------------------------------+-------------+-----------+ +.. _InputsICSolids: Solids settings ~~~~~~~~~~~~~~~ diff --git a/docs/source_docs/user_guide/inputs/model_options.rst b/docs/source_docs/user_guide/inputs/model_options.rst index f647c83..d00510b 100644 --- a/docs/source_docs/user_guide/inputs/model_options.rst +++ b/docs/source_docs/user_guide/inputs/model_options.rst @@ -60,8 +60,8 @@ The following inputs are defined using the ``mfix`` prefix. | redistribution_type | Algorithms to address the '*small cell problem*' associated with | String | StateRedist | | | explicit cut-cell algorithms. | | | | | | | | -| | * ``StateRedist`` [Berger_Giulini_2021]_ [Giulini_etal_2022]_ | | | -| | * ``FluxRedist`` [Chern_Colella_1987]_ [Colella_etal_2006]_ | | | +| | * ``StateRedist`` :cite:p:`berger21,giuliani22` | | | +| | * ``FluxRedist`` :cite:p:`chern87,colella06` | | | | | * ``NoRedist``: Do not redistribute fluxes. | | | | | | | | +---------------------------------+-----------------------------------------------------------------------+-------------+--------------+ @@ -78,18 +78,6 @@ The following inputs are defined using the ``mfix`` prefix. +---------------------------------+-----------------------------------------------------------------------+-------------+--------------+ -.. [Berger_Giulini_2021] Berger, M., & Giuliani, A. (2021). - A state redistribution algorithm for finite volume schemes on cut cell meshes. Journal of Computational Physics, 428, 109820. - -.. [Giulini_etal_2022] Giuliani, A., Almgren, A. S., Bell, J. B., Berger, M. J., de Frahan, M. H., & Rangarajan, D. (2022). - A weighted state redistribution algorithm for embedded boundary grids. Journal of Computational Physics, 464, 111305. - -.. [Chern_Colella_1987] Chern, I.-L., and Colella, P. (1987). - A conservative front tracking method for hyperbolic conservation laws,” LLNL Rep. No. UCRL-97200, Lawrence Livermore National Laboratory. - -.. [Colella_etal_2006] Colella, P., Graves, D. T., Keen, B. J., & Modiano, D. (2006). - A Cartesian grid embedded boundary method for hyperbolic conservation laws, J. Comput. Phys., vol. 211, no. 1, pp. 347–366. - Additional available constraints -------------------------------- @@ -272,7 +260,7 @@ The following inputs are defined using the ``mfix`` prefix. | | is applied to all transferred properties. | | | | | | | | | | * ``StateRedist`` Use the state redistribution algorithm of | | | -| | [Berger_Giulini_2021]_ and [Giulini_etal_2022]_ to reallocate | | | +| | :cite:p:`berger21,giuliani22` to reallocate | | | | | transferred quantities in small cells. | | | | | | | | +---------------------------------+-----------------------------------------------------------------------+-------------+--------------+ @@ -298,9 +286,9 @@ The following input is defined using the ``mfix`` prefix. +===================+=======================================================================+=============+===========+ | drag_type | Fluid-particle drag model. | String | None | | | | | | -| | * ``WenYu`` [Wen_Yu_1966]_ | | | -| | * ``Gidaspow`` [Ding_Gidaspow_1990]_, [Lathouwers_Bellan_2000]_ | | | -| | * ``BVK2`` [BVK_2007]_, [Tenneti_et_2011]_, [Tang_etal_2015]_ | | | +| | * ``WenYu`` :cite:p:`Wen66` | | | +| | * ``Gidaspow`` :cite:p:`ding90,Lat00` | | | +| | * ``BVK2`` :cite:p:`Bee07,Bee071,tenneti11,tang15` | | | | | * ``UserDrag`` | | | | | | | | +-------------------+-----------------------------------------------------------------------+-------------+-----------+ @@ -312,30 +300,6 @@ The following input is defined using the ``mfix`` prefix. drag model, and recompiling the code. An example can be found in ``tests/DEM06-x``. -.. [Wen_Yu_1966] Wen, C.Y. & Yu, Y.H. (1966). - Mechanics of Fluidization. The Chemical Engineering Progress Symposium Series, 162, 100-111. - -.. [Ding_Gidaspow_1990] Ding, J., & Gidaspow, D. (1990). - A bubbling fluidization model using kinetic theory of granular flow. AIChE journal, 36(4), 523-538. - -.. [Lathouwers_Bellan_2000] Lathouwers D. & Bellan J.(2000). - Proceedings of the 2000 U.S. DOE Hydrogen Program Review NREL/CP-570-28890. - Available from http://www.eere.energy.gov/hydrogenandfuelcells/pdfs/28890k.pdf. - -.. [BVK_2007] Beetstra, R., van der Hoef, M. A., & Kuipers, J. A. M. (2007). - Numerical study of segregation using a new drag force correlation for polydisperse systems derived - from lattice-Boltzmann simulations. Chemical Engineering Science, 62(1-2), 246-255. - -.. [Tenneti_et_2011] Tenneti, S., Garg, R., & Subramaniam, S. (2011). - Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of - flow past fixed assemblies of spheres. International journal of multiphase flow, 37(9), 1072-1092. - -.. [Tang_etal_2015] Tang, Y., Peters, E. A. J. F., Kuipers, J. A. M., Kriebitzsch, S. H. L., & van der Hoef, M. A. (2015). - A new drag correlation from fully resolved simulations of flow past monodisperse static arrays of spheres. - AIChE journal, 61(2), 688-698. - - - Heat transfer coefficients -------------------------- @@ -346,15 +310,7 @@ The following input is defined using the ``mfix`` prefix. +===================+=======================================================================+=============+===========+ | convection_type | Fluid-particle heat transfer coefficient model. | String | None | | | | | | -| | * ``RanzMarshall`` [RanzMarshall_1952]_ | | | -| | * ``Gunn`` [Gunn_1978]_ | | | +| | * ``RanzMarshall`` :cite:p:`ranz52` | | | +| | * ``Gunn`` :cite:p:`gunn78` | | | | | | | | +-------------------+-----------------------------------------------------------------------+-------------+-----------+ - -.. [RanzMarshall_1952] Ranz, W.E. & Marshall, W.R. (1952). - Friction and transfer coefficients for single particles and packed beds. - Chemical Engineering Science, 48(5), 247-253. - -.. [Gunn_1978] Gunn, D. J. (1978). - Transfer of heat or mass to particles in fixed and fluidised beds. - International Journal of Heat and Mass Transfer, 21(4), 467-476. -- GitLab From 8cd58f8cf6e1c67cd3dd9130e4d8f42c5d950cdd Mon Sep 17 00:00:00 2001 From: Jordan Musser Date: Mon, 13 May 2024 08:29:36 -0400 Subject: [PATCH 2/6] Add some items missed in previous commit --- .gitlab-ci.yml | 2 +- .../references/images/normal-dist-cdf.png | Bin 0 -> 71783 bytes .../references/images/normal-dist-pdf.png | Bin 0 -> 80462 bytes .../references/images/normal-dist-pdf_cdf.png | Bin 0 -> 102187 bytes docs/source_docs/refs.bib | 436 ++++++++++++++++++ docs/source_docs/works_cited.rst | 4 + 6 files changed, 441 insertions(+), 1 deletion(-) create mode 100644 docs/source_docs/references/images/normal-dist-cdf.png create mode 100644 docs/source_docs/references/images/normal-dist-pdf.png create mode 100644 docs/source_docs/references/images/normal-dist-pdf_cdf.png create mode 100644 docs/source_docs/refs.bib create mode 100644 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--quiet-level 2 docs/source_docs/ docs/picvva/ + - codespell --ignore-words-list strack --enable-colors --quiet-level 2 docs/source_docs/ docs/picvva/ stage: build-docs tags: - docker diff --git a/docs/source_docs/references/size_distributions.rst b/docs/source_docs/references/size_distributions.rst index 0a981b7..a522cc2 100644 --- a/docs/source_docs/references/size_distributions.rst +++ b/docs/source_docs/references/size_distributions.rst @@ -32,7 +32,7 @@ The number-weighted distribution is *sampled* when DEM particles are created, wh the volume-weighted distribution is *sampled* when creating PIC parcels. .. warning:: - Number-weighted and volume-weighted distributions are not interchangable. Failing + Number-weighted and volume-weighted distributions are not interchangeable. Failing to specify the correct distribution weighting can result in unexpected behavior. Number-weighted and volume-weighted log-normal distributions are illustrated in -- GitLab From 3aa6f9fdd3c11d1fb519dc3d851dc449b59b1624 Mon Sep 17 00:00:00 2001 From: Jordan Musser Date: Wed, 22 May 2024 07:25:30 -0400 Subject: [PATCH 4/6] Update normal dist and add log-normal --- docs/source_docs/definitions.rst | 3 + .../references/images/log-normal_dem_FVw.png | Bin 0 -> 29624 bytes .../references/images/log-normal_pic_FVw.png | Bin 0 -> 29642 bytes .../references/images/normal-dist-cdf.png | Bin 71783 -> 0 bytes .../images/normal-dist-pdf-approx_FN.png | Bin 0 -> 25849 bytes .../references/images/normal-dist-pdf_cdf.png | Bin 102187 -> 0 bytes .../references/images/normal_dem_FNw.png | Bin 0 -> 27613 bytes .../references/images/normal_pic_FNw.png | Bin 0 -> 29012 bytes .../references/size_distributions.rst | 302 +++++++++++++++--- docs/source_docs/refs.bib | 32 +- 10 files changed, 295 insertions(+), 42 deletions(-) create mode 100644 docs/source_docs/definitions.rst create mode 100644 docs/source_docs/references/images/log-normal_dem_FVw.png create mode 100644 docs/source_docs/references/images/log-normal_pic_FVw.png delete mode 100644 docs/source_docs/references/images/normal-dist-cdf.png create mode 100644 docs/source_docs/references/images/normal-dist-pdf-approx_FN.png delete mode 100644 docs/source_docs/references/images/normal-dist-pdf_cdf.png create mode 100644 docs/source_docs/references/images/normal_dem_FNw.png create mode 100644 docs/source_docs/references/images/normal_pic_FNw.png diff --git a/docs/source_docs/definitions.rst b/docs/source_docs/definitions.rst new file mode 100644 index 0000000..2aed9ff --- /dev/null +++ b/docs/source_docs/definitions.rst @@ -0,0 +1,3 @@ + +.. |cdot| unicode:: 0xB7 .. middle dot (https://www.compart.com/en/unicode/U+00B7) + diff --git a/docs/source_docs/references/images/log-normal_dem_FVw.png b/docs/source_docs/references/images/log-normal_dem_FVw.png new file mode 100644 index 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z%ks|1V;W0fS|%SR<`tCVpUnShh%CV=*bi>4P24Bs!bM4LN(kg7_drf(<8&NuTQyB+ z*T&v7y-1{Z{-iiou5{7k5~gwkHGm|!(!)AZEVuT@q!vuXWd`AGbKP+U$j_+#LvyKo ztcX`X-Nc93rq4ZDLRC|}xFVPjkEqdOLZ!!DeLsIuFd1%e>Ok1_jOo$xt17A0kuYRo zJQV@`TmlB9#5x?tz79 zbH|3HoxE#cY#lu@yKXQC4>UKT_zirlRP%TGNWp$3X6ONd1TOB!*beM>_lLxebN=2E zJx7pNk%777{+6L<#DVgbiUMV541NMz|C$3hBV0cQ6TI%gGeu6S<5I|cML|){c^#B3jZ|*hW!k`%2VpstJDER^)Zk<~NwG{c7 zGy5FW_L}8fN$R*SDU!OSGqAVHjuVbb0~PU08fSy!3vzbp*P(40ruw4GgzyI^0)?4d z;9H&bU`WO#L8BIS4>?BHwmD>1IqQX9Xjv%Vmhlo$%zX#&(ct-N^Xzs=uJpgoKs4Uh zQ(Xm}mzRnD_BqCRGe#tYr zhY0S&57c&_Y9{sjk#2&4Oc=Q_3jr;VN@Ht{s_huubx`GsMDmX`>DVuHe{OHWHwXhNheTt6N z#C#JvD*+bv@P(4Iw&jpQxfr_xJ%H@Y4;}%8=gS|Th0gAld4R`$o|!uA*CPM9F|_wn z+0HrGf4_bS&-p${$3@(;`-Q3XRf}8xIq?C_51vVb6dKf*ayezI_%hc*3CInB816f2 z`D)Bt_fr3R^<#S*pL~Z$c<4$@W;Ig2d4fl?ba~ucJX$h81?<{%IOu=JNcNjc0qIbL z!}Lyq3nQkX#&`~g@HA%7k5nJ7t{{-g8m z%}|lEGyqQohb-p*&yU?Z=KlR4pA>ct?|elBsU3``SZ NY-IL#!TH + 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 for a solids +volume fraction of 0.0507, modestly overshooting the specified value of 0.05. The particle size distribution +is shown in :numref:`fig_normal_dem_FNw_MFIX_Np` (open circles) in addition to the number-weighted (orange) +and volume-weighted (blue) density functions. As noted by :cite:t:`weiner11`, the volume-weighted PDF is +shifted to the right of the number-weighted distribution. + +.. _fig_normal_dem_FNw_MFIX_Np: + +.. figure:: ./images/normal_dem_FNw.png + :width: 95% + :align: center + :alt: DEM FN and Np + + MFIX-Exa DEM particle size distribution (open circles) generated from example inputs. The number-weighted + (orange) with 1 mm mean and 0.25 mm standard deviation, and volume-weighted distribution with + ~1.17 mm mean and ~0.23 mm standard deviation are also plotted. + +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 :math:`\Delta x` is 1/320 m. The total solids volume +fraction of the region is ~0.0498, slightly undershooting the target value of 0.05. The parcel size distribution shown in +:numref:`fig_normal_pic_FNw_MFIX_Np` (open triangles) to follows the volume-weighted density function. +Small closed circles illustrate the effective particle distribution, computed by multiplying a size observation +by the parcel statistical weight as previously mentioned. + +.. _fig_normal_pic_FNw_MFIX_Np: + +.. figure:: ./images/normal_pic_FNw.png + :width: 95% + :align: center + :alt: PIC FN and Np + + MFIX-Exa PIC parcel size distribution (open triangles) generated from example inputs. The number-weighted + (orange) with 1 mm mean and 0.25 mm standard deviation, and volume-weighted distribution with + ~1.17 mm mean and ~0.23 mm standard deviation are also plotted. + +The following sections outline how MFIX-Exa computes a volume-weighted normal distribution from a number-weighted +normal distribution, and conversely, how a number-weighted normal distribution is computed from a volume-weighted +normal distribution. .. rubric:: Converting a number-weighted normal distribution to volume-weighted: :math:`F_N(x) \rightarrow F_V(x)` -The volume-weighted distribution, :math:`F_V(x)`, computed from the normal-weighted distribution, :math:`F_N(x)` +The volume-weighted distribution, :math:`F_V(x)`, computed from the normal-weighted distribution, :math:`F_N(x)`, is normally distributed with mean, :math:`\mu_V`, and variance, :math:`\sigma_V^2`. An expression for the -volume-weighted probability density function (PDF) is obtained by multiplying :math:`F_N(x)` by :math:`x^3`. +volume-weighted probability density function (PDF) is obtained by multiplying :eq:`eq_normal_dist` by :math:`x^3`. .. math:: - f(x) = \frac{x^3}{\sigma \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu)^2 }{ 2 \sigma^2 }\right)} + \tilde{F}_V(x) = \frac{x^3}{\sigma_N \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu_N)^2 }{ 2 \sigma_N^2 }\right)} The mean is computed as .. math:: + :label: eq_normal_dist_FN_to_FV_mean - \mu_V = \frac{ \int_{-\infty}^{\infty} x f(x) dx }{\int_{-\infty}^{\infty} f(x) dx } - = \frac{3\sigma_N^4 + 6\sigma_N^2\mu_N^2 + \mu_N^4}{3\sigma_N^2 \mu_N + \mu_N^3} + \begin{align} + \mu_V =& \frac{ \int_{-\infty}^{\infty} x \tilde{F}_V(x) dx }{\int_{-\infty}^{\infty} \tilde{F}_V(x) dx } \\[10pt] + =& \frac{3\sigma_N^4 + 6\sigma_N^2\mu_N^2 + \mu_N^4}{3\sigma_N^2 \mu_N + \mu_N^3} + \end{align} The variance is given by .. math:: + :label: eq_normal_dist_FN_to_FV_variance - \sigma^2_V = \frac{ \int_{-\infty}^{\infty} (x - \mu_V)^2 f(x) dx }{\int_{-\infty}^{\infty} f(x) dx } - = \frac{3\sigma_N^4 (3\mu_N + 2b) + \sigma^2(\mu_N^3 + 6\mu_N^2 b + 3\mu_Nb^2)+\mu_N^3 b^2}{3\sigma_N^2 \mu_N + \mu_N^3} + \begin{align} + \sigma^2_V =& \frac{ \int_{-\infty}^{\infty} (x - \mu_V)^2 \tilde{F}_V(x) dx }{\int_{-\infty}^{\infty} \tilde{F}_V(x) dx } \\[10pt] + =& \frac{3\sigma_N^4 (3\mu_N + 2b) + \sigma^2(\mu_N^3 + 6\mu_N^2 b + 3\mu_Nb^2)+\mu_N^3 b^2}{3\sigma_N^2 \mu_N + \mu_N^3} + \end{align} -where :math:`b = (\mu_N - \mu_V)`. The compute mean and variance, :math:`\mu_V` and :math:`sigma^2_V` can -be substituted into :eq:`eq_normal_dist`. The probability density and cumulative density functions -of a typical number-weighted normal distribution and its companion volume-weighted distribution are -shown in :numref:`fig_normal_dist_pdf_cdf`. As noted by :cite:t:`weiner11`, the volume-weighted -PDF and CDF are always shifted to right of the normal weighted distribution. +where :math:`b = (\mu_N - \mu_V)`. The mean and variance can be substituted directly into :eq:`eq_normal_dist` +because the volume-weighted distribution is normally distributed. +.. math:: -.. _fig_normal_dist_pdf_cdf: - -.. figure:: ./images/normal-dist-pdf_cdf.png - :width: 95% - :align: center - :alt: domain used with box embedded boundary + F_V(x) = \frac{1}{\sigma_V \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu_V)^2 }{ 2 \sigma_V^2 }\right)} - Comparison of the number-weighted and volume-weighted normal probability density (solid lines) and cumulative - density (dashed lines) functions. The number-weighted distribution is shown in orange and has mean - :math:`\mu_N = 1000.0\mathrm{e-}6` and standard deviation :math:`\sigma_N = 250.0\mathrm{e-}6`. The volume-weighted - distribution is shown in green and has mean :math:`\mu_V \approx 1167.7\mathrm{e-}6` and standard deviation - :math:`\sigma_N \approx 232.6\mathrm{e-}6`. .. rubric:: Converting a volume-weighted normal distribution to number-weighted: :math:`F_V(x) \rightarrow F_N(x)` +The number-weighted distribution, :math:`F_V(x)` is computed from the volume-weighted distribution +by solving solving the nonlinear system of equations constructed from :eq:`eq_normal_dist_FN_to_FV_mean` and +:eq:`eq_normal_dist_FN_to_FV_variance` for :math:`\mu_N` and :math:`\sigma_N`. + +.. math:: + + \begin{align} + f_1(\mu_N,\sigma_N) &= \left( 3\sigma_N^4 + 6\sigma_N^2\mu_N^2 + \mu_N^4 \right) - \mu_V \left( 3\sigma_N^2 \mu_N + \mu_N^3 \right) = 0 \\ + f_2(\mu_N,\sigma_N) &= \left(3\sigma_N^4 (3\mu_N + 2b) + \sigma^2(\mu_N^3 + 6\mu_N^2 b + 3\mu_Nb^2)+\mu_N^3 b^2\right) - \sigma^2_V \left( 3\sigma_N^2 \mu_N + \mu_N^3 \right) = 0 + \end{align} + +This system is solved using the homotopy method outlined in :cite:t:`Burden10`. The initial guesses for +:math:`\mu_N` and :math:`\sigma_N` are generated by approximating the number-weighted distribution function +by dividing :eq:`eq_normal_dist` by :math:`x^3`. + +.. math:: + :label: eq_normal_dist_FV_to_FN_initial_expression + + \tilde{F}_N(x) = \frac{1}{x^3\sigma \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu)^2 }{ 2 \sigma^2 }\right)} + -The number-weighted distribution, :math:`F_V(x)`, cannot be computed directly from the volume-weighted -distribution, :math:`F_N(x)`. +:math:`\tilde{F}_N(x)` is not the PDF of the number-weighted distribution, but an approximation to it. +Observe that :math:`\lim_{x \to 0}\tilde{F}_N(x) = \infty`, therefore the mean and variance are calculated by +numerically integrating :math:`\tilde{F}_N(x)` over :math:`x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right]`. +The lower bound, :math:`x_{min}`, and midpoint, :math:`x_{mid}` are inflection points of :math:`\tilde{F}_N(x)` +and are found by setting the derivative of :eq:`eq_normal_dist_FV_to_FN_initial_expression` to +zero and computing the roots. :math:`x_\mathrm{max}` is chosen to mirror :math:`x_\mathrm{min}` across the midpoint. + +.. math:: + :label: eq_normal_dist_FV_to_FN_discrete_values + + \begin{align} + x_{\mathrm{min}} \; =& \; \left( \mu_V - \sqrt{\mu_V^2 -12\sigma_V^2}\right)/2 \\ + x_{\mathrm{mid}} \; =& \; \left(\mu_V + \sqrt{\mu_V^2 -12\sigma_V^2}\right)/2 \\ + x_{\mathrm{max}} \; =& \; 2x_{\mathrm{mid}} - x_{\mathrm{min}} + \end{align} + + +.. figure:: ./images/normal-dist-pdf-approx_FN.png + :width: 95% + :align: center + :alt: approximate number-weighted pdf + + The number-weighted normal distribution function approximated by dividing the volume-weighted + distribution function by x\ :sup:`3`. The an approximate mean and variance are computed by numerically + integrating this function from x\ :sub:`min` to x\ :sub:`max`. -.. is normally distributed with mean, :math:`\mu_V`, and variance, :math:`\sigma_V^2`. An expression for the ``log-normal`` distribution @@ -218,24 +341,24 @@ The ``log-normal`` distribution is given by .. math:: - F(d_p) = \frac{1}{d_p \sigma \sqrt{2\pi}} \exp{\left(- \frac{(\mathrm{ln}(d_p) - \mu)^2 }{ 2 \sigma^2 }\right)} + F(x) = \frac{1}{x \sigma \sqrt{2\pi}} \exp{\left(- \frac{(\mathrm{ln}(x) - \mu)^2 }{ 2 \sigma^2 }\right)} where :math:`\mu` and :math:`\sigma` are parameters defining the distribution. .. note:: - :math:`\mu` and :math:`\sigma` **are not** the mean and standard deviation of :math:`d_p`. They are the mean and - standard deviation of :math:`\mathrm{ln}(d_p)`. The mean, :math:`E(d_p)`, and variance, :math:`V(d_p)`, of the + :math:`\mu` and :math:`\sigma` **are not** the mean and standard deviation of :math:`x`. They are the mean and + standard deviation of :math:`\mathrm{ln}(x)`. The mean, :math:`E(x)`, and variance, :math:`V(x)`, of the log-normal random variable are given by .. math:: - E(d_p) = \exp{\left(\mu + \sigma^s/2\right)} + E(x) = \exp{\left(\mu + \sigma^s/2\right)} and .. math:: - V(d_p) = \exp{\left(2\mu + 2\sigma^2\right)} - \exp{\left(2\mu + \sigma^2\right)} + V(x) = \exp{\left(2\mu + 2\sigma^2\right)} - \exp{\left(2\mu + \sigma^2\right)} respectively. See See :cite:t:`william2014` for an overview of log-normal distributions. @@ -246,10 +369,10 @@ where :math:`\mu` and :math:`\sigma` are parameters defining the distribution. | type | distribution weighting: ``number-weighted`` or ``volume-weighted`` | string | N/A | +---------------------+-----------------------------------------------------------------------+-------------+-----------+ | mean | mean of of the normal random variable, | Real | N/A | -| | :math:`\mathrm{ln}(d_p)` | | | +| | :math:`\mathrm{ln}(x)` | | | +---------------------+-----------------------------------------------------------------------+-------------+-----------+ | stddev | standard deviation of the normal random variable, | Real | N/A | -| | :math:`\mathrm{ln}(d_p)` | | | +| | :math:`\mathrm{ln}(x)` | | | +---------------------+-----------------------------------------------------------------------+-------------+-----------+ | min | Minimum particle diameter. Drawn samples below ``min`` are discarded | Real | N/A | | | and a new sample is drawn. | | | @@ -262,7 +385,104 @@ where :math:`\mu` and :math:`\sigma` are parameters defining the distribution. +---------------------+-----------------------------------------------------------------------+-------------+-----------+ +In the following example, a cuboid with volume 0.001 m\ :sup:`3` 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. + +.. code-block:: bash + :caption: Snippet of inputs defining a number-weighted ``log-normal`` distribution. **This is not a complete input file.** + + # 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 -The uniform, normal, and log-normal distributions + 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 = 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 for a solids +volume fraction of 0.7506. The particle size distribution (open circles), and number-weighted (orange), +and volume-weighted (blue) density functions are shown in :numref:`fig_log_normal_dem_FVw_MFIX_Np`. + +.. _fig_log_normal_dem_FVw_MFIX_Np: + +.. figure:: ./images/log-normal_dem_FVw.png + :width: 95% + :align: center + :alt: DEM FN and Np + + MFIX-Exa DEM particle size distribution (open circles) and number-weighted (orange) and volume-weighted (blue) + density functions. + +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 :math:`\Delta x` is 1/320 m. The total solids volume +fraction of the region is ~0.2490. :numref:`fig_log_normal_pic_FVw_MFIX_Np` 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. + +.. _fig_log_normal_pic_FVw_MFIX_Np: + +.. figure:: ./images/log-normal_pic_FVw.png + :width: 95% + :align: center + :alt: PIC FN and Np + + MFIX-Exa PIC parcel size distribution (open triangles), number-weighted density function (orange), + and volume-weighted density function (blue). Small closed circles indicated the particle size + distribution modeled by the parcels. + + +.. rubric:: Converting between number-weighted and volume-weighted log-normal distributions + +The conversion between number-weighted and volume-weighted log-normal distributions is shown +by :cite:t:`elhilo12,elhilo12a`. The distributions employ the same standard deviation, + +.. math:: + + \sigma = \sigma_{N} = \sigma_{V} + +while the parameter :math:`\mu` is computed directly from the initial distribution parameters. + +.. math:: + + \begin{align} + \mu_V =& \mu_N + 3.0\sigma^2 \\[10pt] + \mu_N =& \mu_V - 3.0\sigma^2 + \end{align} + + +``uniform`` distribution +^^^^^^^^^^^^^^^^^^^^^^^^^^^ +The ``uniform`` distribution is given by diff --git a/docs/source_docs/refs.bib b/docs/source_docs/refs.bib index 0a59849..f65d072 100644 --- a/docs/source_docs/refs.bib +++ b/docs/source_docs/refs.bib @@ -83,6 +83,15 @@ pages = {379-402} url = {https://www.sciencedirect.com/science/article/pii/S0021999120305945}, } +@book{Burden10, + title={Numerical Analysis}, + author={Burden, R.L. and Faires, J.D.}, + isbn={9780538733519}, + lccn={2010922639}, + year={2010}, + publisher={Cengage Learning} +} + @article{cap13, author = {Capecelatro, J. and Desjardins, O. }, @@ -160,7 +169,28 @@ pages = {379-402} pages = {12304} } - +@article{elhilo12, +author = {M. El-Hilo and R.W. Chantrell}, +title = {Rationalisation of distribution functions for models of nanoparticle magnetism}, +journal = {Journal of Magnetism and Magnetic Materials}, +volume = {324}, +number = {16}, +pages = {2593-2595}, +year = {2012}, +issn = {0304-8853}, +doi = {https://doi.org/10.1016/j.jmmm.2012.02.108}, +url = {https://www.sciencedirect.com/science/article/pii/S0304885312002132}, +} + +@article{elhilo12a, + author={El-Hilo, M}, + title={Nano-particle magnetism with a dispersion of particle sizes}, + journal={Journal of Applied Physics}, + volume={112}, + number={10}, + year={2012}, + publisher={AIP Publishing} +} @article{flad03, author = {Alexander Fladerer and Reinhard Strey}, -- GitLab From ded6c345889f2ae24431e04c50bca71f7815641f Mon Sep 17 00:00:00 2001 From: Jordan Musser Date: Wed, 5 Jun 2024 14:51:29 -0400 Subject: [PATCH 5/6] Add uniform distribution notes --- .../references/images/uniform_dem_FNw.png | Bin 0 -> 21098 bytes .../references/images/uniform_pic_FNw.png | Bin 0 -> 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zRy~V)yL>GhgBm9&rvU|cfG4dm7<7WQtntXNy{v0+DeQ0#EO`Ml;h;DTdefN`!qAhf u6V;ELsMp^;0elj4lS0Rg$Kw0{{b&56xW}sb&E9RGV#Cwb&t;ucLK6TESLZ$e literal 0 HcmV?d00001 diff --git a/docs/source_docs/references/size_distributions.rst b/docs/source_docs/references/size_distributions.rst index 153077d..a583eee 100644 --- a/docs/source_docs/references/size_distributions.rst +++ b/docs/source_docs/references/size_distributions.rst @@ -14,13 +14,13 @@ In addition to the distribution type, the `distribution weighting` must be specified; MFIX-Exa supports two distribution weightings: Number-weighted diameter distribution - A number-weighted diameter distribution, :math:`F_{N}(d_p)`, defines the + A number-weighted diameter density function, :math:`f^N(d_p)`, defines the fraction of particles between diameter :math:`d_p` and :math:`d_p+\Delta d_p`. Specifically, the *number of particles* with diameters between :math:`d_p` and :math:`d_p+\Delta d_p`, divided by the total number of particles. Volume-weighted diameter distribution - A volume-weighted diameter distribution, :math:`F_{V}(d_p)`, defines the + A volume-weighted diameter density function, :math:`f^V(d_p)`, defines the fraction of total particle volume between diameter :math:`d_p` and :math:`d_p+\Delta d_p`. Specifically, the *volume of particles* with diameters between :math:`d_p` and :math:`d_p+\Delta d_p`, divided by total particle volume. @@ -42,8 +42,8 @@ Number-weighted and volume-weighted log-normal distributions are illustrated in distribution (left vertical axis) and the dashed gray curve is the volume-weighted diameter distribution (right vertical axis). The symbols represent the observed probabilities computed by binning particle data from equivalent DEM and PIC simulations, respectively. -The DEM particle distribution tracks the number-weighted distribution, :math:`F_{N}(d_p)`. -The PIC parcel distribution follows the volume-weighted distribution, :math:`F_{V}(d_p)`. +The DEM particle distribution tracks the number-weighted distribution, :math:`f^{N}(d_p)`. +The PIC parcel distribution follows the volume-weighted distribution, :math:`f^{V}(d_p)`. .. figure:: ./images/distribution-DEMvsPIC.png :name: distribution-DEMvsPIC @@ -51,12 +51,12 @@ The PIC parcel distribution follows the volume-weighted distribution, :math:`F_{ :align: center :alt: domain used with box embedded boundary - Comparison of the volume-weighted diameter distribution, :math:`F(V_p)`, and - number-weighted diameter distribution, :math:`F(N_p)`, with the generation of + Comparison of the volume-weighted diameter distribution, :math:`f^V(d_p)`, and + number-weighted diameter distribution, :math:`f^N(d_p)`, with the generation of DEM and PIC distributions. -The number of particles represented by PIC parcels follows :math:`F_{N}(d_p)`. This distribution +The number of particles represented by PIC parcels follows :math:`f^{N}(d_p)`. This distribution is obtained by multiplying each observation (each parcel) by its 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. Sampling the volume-weighted distribution for PIC parcels allows for @@ -80,7 +80,7 @@ The ``constant`` distribution defines a monodisperse (single-valued) distributio | constant | size of the mono-disperse distribution | Real | N/A | +---------------------+-----------------------------------------------------------------------+-------------+-----------+ -In the following example, a cuboid is filled with monodisperse particles with a 1 mm diameter and 2500 kg\ |cdot|\ m\ :sup:`-3` +In the following example, a cuboid is filled with 1 mm diameter monodisperse particles with 2500 kg\ |cdot|\ m\ :sup:`-3` density. The initial condition region volume is 0.001 m\ :sup:`3`, and it has a solids volume fraction of 0.05 (5% solids). .. code-block:: bash @@ -134,7 +134,7 @@ The ``normal`` or Gaussian distribution is given by .. math:: :label: eq_normal_dist - F(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu)^2 }{ 2 \sigma^2 }\right)} + f_X(x) = \frac{d}{dx}F_X(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu)^2 }{ 2 \sigma^2 }\right)} where the parameter :math:`\mu` is the distribution mean, and :math:`\sigma` is the standard deviation. The inputs used to specify the distribution are provided in the following table. @@ -205,9 +205,8 @@ and 1.75 mm, respectively. ic.full-domain.solid0.density.constant = 2500.0 # (kg/m^3) -An MFIX-Exa DEM simulation with these settings generates approximately 81,700 particles for a solids -volume fraction of 0.0507, modestly overshooting the specified value of 0.05. The particle size distribution -is shown in :numref:`fig_normal_dem_FNw_MFIX_Np` (open circles) in addition to the number-weighted (orange) +An MFIX-Exa DEM simulation with these settings generates approximately 81,700 particles. The particle size +distribution is shown in :numref:`fig_normal_dem_FNw_MFIX_Np` (open circles) in addition to the number-weighted (orange) and volume-weighted (blue) density functions. As noted by :cite:t:`weiner11`, the volume-weighted PDF is shifted to the right of the number-weighted distribution. @@ -223,9 +222,8 @@ shifted to the right of the number-weighted distribution. ~1.17 mm mean and ~0.23 mm standard deviation are also plotted. 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 :math:`\Delta x` is 1/320 m. The total solids volume -fraction of the region is ~0.0498, slightly undershooting the target value of 0.05. The parcel size distribution shown in -:numref:`fig_normal_pic_FNw_MFIX_Np` (open triangles) to follows the volume-weighted density function. +``pic.parcels_per_cell_at_pack = 32``, and the Eulerian mesh spacing :math:`\Delta x` is 1/320 m. The parcel size distribution +shown in :numref:`fig_normal_pic_FNw_MFIX_Np` (open triangles) follows the volume-weighted density function. Small closed circles illustrate the effective particle distribution, computed by multiplying a size observation by the parcel statistical weight as previously mentioned. @@ -244,15 +242,15 @@ The following sections outline how MFIX-Exa computes a volume-weighted normal di normal distribution, and conversely, how a number-weighted normal distribution is computed from a volume-weighted normal distribution. -.. rubric:: Converting a number-weighted normal distribution to volume-weighted: :math:`F_N(x) \rightarrow F_V(x)` +.. rubric:: Converting a number-weighted normal distribution to volume-weighted: :math:`f_X^N(x) \rightarrow f_X^V(x)` -The volume-weighted distribution, :math:`F_V(x)`, computed from the normal-weighted distribution, :math:`F_N(x)`, +The volume-weighted distribution, :math:`f_X^V(x)`, computed from the number-weighted distribution, :math:`f_X^N(x)`, is normally distributed with mean, :math:`\mu_V`, and variance, :math:`\sigma_V^2`. An expression for the volume-weighted probability density function (PDF) is obtained by multiplying :eq:`eq_normal_dist` by :math:`x^3`. .. math:: - \tilde{F}_V(x) = \frac{x^3}{\sigma_N \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu_N)^2 }{ 2 \sigma_N^2 }\right)} + \tilde{f}_X^V(x) = \frac{x^3}{\sigma_N \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu_N)^2 }{ 2 \sigma_N^2 }\right)} The mean is computed as @@ -260,7 +258,7 @@ The mean is computed as :label: eq_normal_dist_FN_to_FV_mean \begin{align} - \mu_V =& \frac{ \int_{-\infty}^{\infty} x \tilde{F}_V(x) dx }{\int_{-\infty}^{\infty} \tilde{F}_V(x) dx } \\[10pt] + \mu_V =& \frac{ \int_{-\infty}^{\infty} x \tilde{f}_X^V(x) dx }{\int_{-\infty}^{\infty} \tilde{f}_X^V(x) dx } \\[10pt] =& \frac{3\sigma_N^4 + 6\sigma_N^2\mu_N^2 + \mu_N^4}{3\sigma_N^2 \mu_N + \mu_N^3} \end{align} @@ -270,7 +268,7 @@ The variance is given by :label: eq_normal_dist_FN_to_FV_variance \begin{align} - \sigma^2_V =& \frac{ \int_{-\infty}^{\infty} (x - \mu_V)^2 \tilde{F}_V(x) dx }{\int_{-\infty}^{\infty} \tilde{F}_V(x) dx } \\[10pt] + \sigma^2_V =& \frac{ \int_{-\infty}^{\infty} (x - \mu_V)^2 \tilde{f}_X^V(x) dx }{\int_{-\infty}^{\infty} \tilde{f}_X^V(x) dx } \\[10pt] =& \frac{3\sigma_N^4 (3\mu_N + 2b) + \sigma^2(\mu_N^3 + 6\mu_N^2 b + 3\mu_Nb^2)+\mu_N^3 b^2}{3\sigma_N^2 \mu_N + \mu_N^3} \end{align} @@ -279,14 +277,12 @@ because the volume-weighted distribution is normally distributed. .. math:: - F_V(x) = \frac{1}{\sigma_V \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu_V)^2 }{ 2 \sigma_V^2 }\right)} + f_X^V(x) = \frac{1}{\sigma_V \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu_V)^2 }{ 2 \sigma_V^2 }\right)} +.. rubric:: Converting a volume-weighted normal distribution to number-weighted: :math:`F_X^V(x) \rightarrow F_X^N(x)` - -.. rubric:: Converting a volume-weighted normal distribution to number-weighted: :math:`F_V(x) \rightarrow F_N(x)` - -The number-weighted distribution, :math:`F_V(x)` is computed from the volume-weighted distribution -by solving solving the nonlinear system of equations constructed from :eq:`eq_normal_dist_FN_to_FV_mean` and +The number-weighted distribution, :math:`f_X^V(x)` is computed from the volume-weighted distribution +by solving the nonlinear system of equations constructed from :eq:`eq_normal_dist_FN_to_FV_mean` and :eq:`eq_normal_dist_FN_to_FV_variance` for :math:`\mu_N` and :math:`\sigma_N`. .. math:: @@ -296,22 +292,20 @@ by solving solving the nonlinear system of equations constructed from :eq:`eq_no f_2(\mu_N,\sigma_N) &= \left(3\sigma_N^4 (3\mu_N + 2b) + \sigma^2(\mu_N^3 + 6\mu_N^2 b + 3\mu_Nb^2)+\mu_N^3 b^2\right) - \sigma^2_V \left( 3\sigma_N^2 \mu_N + \mu_N^3 \right) = 0 \end{align} -This system is solved using the homotopy method outlined in :cite:t:`Burden10`. The initial guesses for -:math:`\mu_N` and :math:`\sigma_N` are generated by approximating the number-weighted distribution function -by dividing :eq:`eq_normal_dist` by :math:`x^3`. +This system is solved using the homotopy method outlined in :cite:t:`Burden10`. Initial guesses for +:math:`\mu_N` and :math:`\sigma_N` are calculated from the distribution function created by dividing +:eq:`eq_normal_dist` by :math:`x^3`. .. math:: :label: eq_normal_dist_FV_to_FN_initial_expression - \tilde{F}_N(x) = \frac{1}{x^3\sigma \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu)^2 }{ 2 \sigma^2 }\right)} + \tilde{f}_X^N(x) = \frac{1}{x^3\sigma \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu)^2 }{ 2 \sigma^2 }\right)} -:math:`\tilde{F}_N(x)` is not the PDF of the number-weighted distribution, but an approximation to it. -Observe that :math:`\lim_{x \to 0}\tilde{F}_N(x) = \infty`, therefore the mean and variance are calculated by -numerically integrating :math:`\tilde{F}_N(x)` over :math:`x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right]`. -The lower bound, :math:`x_{min}`, and midpoint, :math:`x_{mid}` are inflection points of :math:`\tilde{F}_N(x)` -and are found by setting the derivative of :eq:`eq_normal_dist_FV_to_FN_initial_expression` to -zero and computing the roots. :math:`x_\mathrm{max}` is chosen to mirror :math:`x_\mathrm{min}` across the midpoint. +:math:`\tilde{f}_X^N(x)` is not the number-weighted distribution, but an approximation to it. The mean and variance +are calculated by numerically integrating :math:`\tilde{f}_X^N(x)` over the interval :math:`x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right]` +where :math:`x_{min}` and :math:`x_{mid}` are found by setting the derivative of :eq:`eq_normal_dist_FV_to_FN_initial_expression` +to zero and computing the roots. :math:`x_\mathrm{max}` is chosen to mirror :math:`x_\mathrm{min}` across the midpoint. .. math:: :label: eq_normal_dist_FV_to_FN_discrete_values @@ -329,9 +323,24 @@ zero and computing the roots. :math:`x_\mathrm{max}` is chosen to mirror :math:` :alt: approximate number-weighted pdf The number-weighted normal distribution function approximated by dividing the volume-weighted - distribution function by x\ :sup:`3`. The an approximate mean and variance are computed by numerically + distribution function by x\ :sup:`3`. The approximate mean and variance are computed by numerically integrating this function from x\ :sub:`min` to x\ :sub:`max`. +.. rubric:: Sampling a normal distribution + +MFIX-Exa samples the normal distribution with mean ``m_mean`` and standard deviation ``m_stddev`` to return :math:`x`. + +.. code:: + + 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 :math:`\mathrm{xmin} \le x` and :math:`x \le \mathrm{xmax}`. ``log-normal`` distribution @@ -341,7 +350,7 @@ The ``log-normal`` distribution is given by .. math:: - F(x) = \frac{1}{x \sigma \sqrt{2\pi}} \exp{\left(- \frac{(\mathrm{ln}(x) - \mu)^2 }{ 2 \sigma^2 }\right)} + f_X(x) = \frac{d}{dx}F_X(x) = \frac{1}{x \sigma \sqrt{2\pi}} \exp{\left(- \frac{(\mathrm{ln}(x) - \mu)^2 }{ 2 \sigma^2 }\right)} where :math:`\mu` and :math:`\sigma` are parameters defining the distribution. @@ -360,7 +369,7 @@ where :math:`\mu` and :math:`\sigma` are parameters defining the distribution. V(x) = \exp{\left(2\mu + 2\sigma^2\right)} - \exp{\left(2\mu + \sigma^2\right)} - respectively. See See :cite:t:`william2014` for an overview of log-normal distributions. + respectively. See :cite:t:`william2014` for an overview of log-normal distributions. +---------------------+-----------------------------------------------------------------------+-------------+-----------+ @@ -420,7 +429,7 @@ and 3.00 mm, respectively. ic.full-domain.solid0.volfrac = 0.25 ic.full-domain.solid0.velocity = 0.00 0.00 0.00 - ic.full-domain.solid0.diameter = normal + 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 @@ -431,9 +440,9 @@ and 3.00 mm, respectively. ic.full-domain.solid0.density.constant = 2500.0 # (kg/m^3) -An MFIX-Exa DEM simulation with these settings generates approximately 162,000 particles for a solids -volume fraction of 0.7506. The particle size distribution (open circles), and number-weighted (orange), -and volume-weighted (blue) density functions are shown in :numref:`fig_log_normal_dem_FVw_MFIX_Np`. +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 :numref:`fig_log_normal_dem_FVw_MFIX_Np`. .. _fig_log_normal_dem_FVw_MFIX_Np: @@ -446,10 +455,10 @@ and volume-weighted (blue) density functions are shown in :numref:`fig_log_norma density functions. 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 :math:`\Delta x` is 1/320 m. The total solids volume -fraction of the region is ~0.2490. :numref:`fig_log_normal_pic_FVw_MFIX_Np` 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. +``pic.parcels_per_cell_at_pack = 32``, and the Eulerian mesh spacing :math:`\Delta x` is 1/320 m. +:numref:`fig_log_normal_pic_FVw_MFIX_Np` 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. .. _fig_log_normal_pic_FVw_MFIX_Np: @@ -462,17 +471,16 @@ circles indicate the effective particle distribution captured by the PIC parcels and volume-weighted density function (blue). Small closed circles indicated the particle size distribution modeled by the parcels. - .. rubric:: Converting between number-weighted and volume-weighted log-normal distributions -The conversion between number-weighted and volume-weighted log-normal distributions is shown -by :cite:t:`elhilo12,elhilo12a`. The distributions employ the same standard deviation, +As reported by :cite:t:`elhilo12,elhilo12a`, number-weighted and volume-weighted log-normal distributions +use the same standard deviation, .. math:: \sigma = \sigma_{N} = \sigma_{V} -while the parameter :math:`\mu` is computed directly from the initial distribution parameters. +while the parameter :math:`\mu` is computed from the other distribution. .. math:: @@ -481,8 +489,219 @@ while the parameter :math:`\mu` is computed directly from the initial distributi \mu_N =& \mu_V - 3.0\sigma^2 \end{align} +.. rubric:: Sampling a log-normal distribution + +Rather than sample a log-normal distribution, MFIX-Exa samples the normal distribution with +mean ``m_mean`` and standard deviation ``m_stddev`` to return :math:`\log (x)`. + +.. code:: + + 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 :math:`\log \mathrm{xmin} \le \log x` and :math:`\log x \le \log \mathrm{xmax}`. +The log-normal variable is returned by applying the exponential function, :math:`x = \exp ( \log x )`. + ``uniform`` distribution ^^^^^^^^^^^^^^^^^^^^^^^^^^^ The ``uniform`` distribution is given by + +.. math:: + :label: eq_uniform_dist + + f_X(x) = \frac{d}{dx}F_X(x) = + \begin{cases} + \left(x_\mathrm{max} - x_\mathrm{min}\right)^{-1}, & \text{for} \; x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right] \\ + 0, & \text{otherwise} + \end{cases} + +where :math:`x_\mathrm{min}` and :math:`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: ``number-weighted`` or ``volume-weighted`` | string | N/A | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| min | Minimum particle diameter. Drawn samples below ``min`` are discarded | Real | N/A | +| | and a new sample is drawn. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| max | Maximum particle diameter. Drawn samples above ``max`` are discarded | Real | N/A | +| | and a new sample is drawn. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| bins | Number of bins used when discretizing the distribution to approximate | int | 64 | +| | the number of particles within an initial condition region. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ + +In the following example, a cuboid with volume 0.001 m\ :sup:`3` is filled with a number-weighted uniform distribution. +The minimum and maximum particle diameters are 0.25 mm and 1.75 mm, respectively. + +.. code-block:: bash + :caption: Snippet of inputs defining a number-weighted ``uniform`` distribution. **This is not a complete input file.** + + # 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 :numref:`fig_uniform_dem_FNw_MFIX_Np`. + +.. _fig_uniform_dem_FNw_MFIX_Np: + +.. figure:: ./images/uniform_dem_FNw.png + :width: 95% + :align: center + :alt: DEM FN and Np + + MFIX-Exa DEM uniform particle size distribution (open circles) and number-weighted (orange) + and volume-weighted (blue) uniform density functions. + +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 :math:`\Delta x` is 1/320 m. +:numref:`fig_uniform_pic_FNw_MFIX_Np` 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. + +.. _fig_uniform_pic_FNw_MFIX_Np: + +.. figure:: ./images/uniform_pic_FNw.png + :width: 95% + :align: center + :alt: PIC FN and Np + + MFIX-Exa PIC uniform parcel size distribution (open triangles), number-weighted (orange) and + volume-weighted (blue) uniform density functions. Small closed circles indicated the particle size + distribution modeled by the parcels. + + +.. rubric:: 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 :math:`rand \in \left( 0, 1 \right)`, then scale and translate the +result such that :math:`x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right]`. + +.. code:: + + 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 for DEM simulations where a volume-weighted distribution is specified and +PIC simulations where a number-weighted distribution is specified. + +The number-weighted distribution, :math:`f_X^N(x)`, computed from the volume-weighted distribution is obtained +by dividing :eq:`eq_uniform_dist` by :math:`x^3`. The brace notation indicating that the density +function is zero for :math:`x \notin \left[ x_\mathrm{max}, x_\mathrm{min} \right]` is dropped for conciseness. + +.. math:: + + f_X^N(x) = \frac{1}{x^3 (x_{\mathrm{min}} - x_{\mathrm{max}})} + +The cumulative distribution function and its inverse are + +.. math:: + + \mathrm{Pr}\left(X \le x\right) = F_X^N(x) + = \frac{ \int_{a}^{x} f_X^N(y) dy }{\int_{a}^{b} f_X^N(y) dy } + = \frac{ b^2(x^2-a^2)}{x^2(b^2 - a^2)} + +and + +.. math:: + + {F_X^N}^{-1}(y) = \frac{ab}{\sqrt{b^2 - y(b^2 -a^2)}} + +where :math:`a = x_{\mathrm{min}}` and :math:`b = x_{\mathrm{max}}`. To sample :math:`f_X^V(x)`, a uniformly distributed +variable :math:`rand \in \left( 0, 1 \right)` is obtained and substituted into the inverse cumulative distribution +function. + +.. code:: + + 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, the volume-weighted distribution, :math:`f_X^V(x)`, computed from a number-weighted distribution is obtained +by multiplying :eq:`eq_uniform_dist` by :math:`x^3`. Again, the brace notation indicating that the density +function is zero for :math:`x \notin \left[ x_\mathrm{max}, x_\mathrm{min} \right]` is dropped for conciseness. + +.. math:: + + f_X^V(x) = \frac{x^3}{(x_{\mathrm{min}} - x_{\mathrm{max}})} + +The cumulative distribution function and its inverse are + +.. math:: + + \mathrm{Pr}\left(X \le x\right) = F_X^V(x) + = \frac{ \int_{a}^{x} f_X^V(y) dy }{\int_{a}^{b} f_X^V(y) dy } + = \frac{ x^4 - a^4 }{ b^4 - a^4 } + +and + +.. math:: + + {F_X^V}^{-1}(y) = \left(a^4 + y(b^4 - a^4)\right)^{1/4} + +where :math:`a = x_{\mathrm{min}}` and :math:`b = x_{\mathrm{max}}`. To sample :math:`f_X^V(x)`, a uniformly distributed +variable :math:`rand \in \left( 0, 1 \right)` is obtained and substituted into the inverse cumulative distribution +function. + +.. code:: + + 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); -- GitLab From b2a4708167a7f112427b049005b208901af83a75 Mon Sep 17 00:00:00 2001 From: Jordan Musser Date: Mon, 24 Jun 2024 08:57:41 -0400 Subject: [PATCH 6/6] Updates --- .../images/distribution-DEMvsPIC.png | Bin 125018 -> 0 bytes .../references/images/log-normal_dem_FVw.png | Bin 29624 -> 30734 bytes .../references/images/log-normal_pic_FVw.png | Bin 29642 -> 30741 bytes .../references/images/normal-dist-pdf.png | Bin 80462 -> 0 bytes .../references/images/normal_dem_FNw.png | Bin 27613 -> 28999 bytes .../references/images/normal_pic_FNw.png | Bin 29012 -> 29871 bytes .../references/images/uniform_dem_FNw.png | Bin 21098 -> 22834 bytes 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zRy~V)yL>GhgBm9&rvU|cfG4dm7<7WQtntXNy{v0+DeQ0#EO`Ml;h;DTdefN`!qAhf u6V;ELsMp^;0elj4lS0Rg$Kw0{{b&56xW}sb&E9RGV#Cwb&t;ucLK6TESLZ$e diff --git a/docs/source_docs/references/size_distributions.rst b/docs/source_docs/references/size_distributions.rst index a583eee..b16dede 100644 --- a/docs/source_docs/references/size_distributions.rst +++ b/docs/source_docs/references/size_distributions.rst @@ -14,16 +14,18 @@ In addition to the distribution type, the `distribution weighting` must be specified; MFIX-Exa supports two distribution weightings: Number-weighted diameter distribution - A number-weighted diameter density function, :math:`f^N(d_p)`, defines the - fraction of particles between diameter :math:`d_p` and :math:`d_p+\Delta d_p`. - Specifically, the *number of particles* with diameters between :math:`d_p` - and :math:`d_p+\Delta d_p`, divided by the total number of particles. + The number-weighted diameter density function, :math:`f_X^N(x)\,dx`, can be interpreted + as defining the *number of particles* with diameters between :math:`x` and :math:`x+dx`, + divided by the total number of particles. Volume-weighted diameter distribution - A volume-weighted diameter density function, :math:`f^V(d_p)`, defines the - fraction of total particle volume between diameter :math:`d_p` and - :math:`d_p+\Delta d_p`. Specifically, the *volume of particles* with diameters - between :math:`d_p` and :math:`d_p+\Delta d_p`, divided by total particle volume. + The volume-weighted diameter density function, :math:`f_X^V(x)\,dx`, similarly can be interpreted + as defining the *volume of particles* with diameters between :math:`x` and :math:`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 @@ -31,39 +33,11 @@ number-weighted normal distribution is provided, the corresponding volume-weight 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. - -.. warning:: - Number-weighted and volume-weighted distributions are not interchangeable. Failing - to specify the correct distribution weighting can result in unexpected behavior. - -Number-weighted and volume-weighted log-normal distributions are illustrated in -:numref:`distribution-DEMvsPIC`. The solid orange curve is the number-weighted diameter -distribution (left vertical axis) and the dashed gray curve is the volume-weighted diameter -distribution (right vertical axis). The symbols represent the observed probabilities computed -by binning particle data from equivalent DEM and PIC simulations, respectively. -The DEM particle distribution tracks the number-weighted distribution, :math:`f^{N}(d_p)`. -The PIC parcel distribution follows the volume-weighted distribution, :math:`f^{V}(d_p)`. - -.. figure:: ./images/distribution-DEMvsPIC.png - :name: distribution-DEMvsPIC - :width: 100% - :align: center - :alt: domain used with box embedded boundary - - Comparison of the volume-weighted diameter distribution, :math:`f^V(d_p)`, and - number-weighted diameter distribution, :math:`f^N(d_p)`, with the generation of - DEM and PIC distributions. - - -The number of particles represented by PIC parcels follows :math:`f^{N}(d_p)`. This distribution -is obtained by multiplying each observation (each parcel) by its 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. Sampling the volume-weighted distribution for PIC parcels allows for -fewer parcels to represent very small particles while using more parcels to represent larger -particles in the distribution. The statistical weight of the parcels is adjusted so that -the number-distribution is represented accurately. - +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. @@ -122,8 +96,7 @@ density. The initial condition region volume is 0.001 m\ :sup:`3`, and it has a 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 -:math:`\Delta x` of 1/320 m generates approximately 164,000 parcels. - +:math:`\Delta x` of 3.125 mm generates approximately 164,000 parcels. ``normal`` distribution @@ -144,9 +117,9 @@ used to specify the distribution are provided in the following table. +=====================+=======================================================================+=============+===========+ | type | distribution weighting: ``number-weighted`` or ``volume-weighted`` | string | N/A | +---------------------+-----------------------------------------------------------------------+-------------+-----------+ -| mean | mean of the normal random variable, :math:`x` | Real | N/A | +| mean | mean of the normal random variable, :math:`X` | Real | N/A | +---------------------+-----------------------------------------------------------------------+-------------+-----------+ -| std | standard deviation of the normal random variable, :math:`x` | Real | N/A | +| std | standard deviation of the normal random variable, :math:`X` | Real | N/A | | | | | | +---------------------+-----------------------------------------------------------------------+-------------+-----------+ | min | Minimum particle diameter. Drawn samples below ``min`` are discarded | Real | N/A | @@ -207,8 +180,8 @@ and 1.75 mm, respectively. An MFIX-Exa DEM simulation with these settings generates approximately 81,700 particles. The particle size distribution is shown in :numref:`fig_normal_dem_FNw_MFIX_Np` (open circles) in addition to the number-weighted (orange) -and volume-weighted (blue) density functions. As noted by :cite:t:`weiner11`, the volume-weighted PDF is -shifted to the right of the number-weighted distribution. +and volume-weighted (blue) density functions. As noted by :cite:t:`weiner11`, the volume-weighted +probability density function (PDF) is shifted to the right of the number-weighted PDF. .. _fig_normal_dem_FNw_MFIX_Np: @@ -218,14 +191,15 @@ shifted to the right of the number-weighted distribution. :alt: DEM FN and Np MFIX-Exa DEM particle size distribution (open circles) generated from example inputs. The number-weighted - (orange) with 1 mm mean and 0.25 mm standard deviation, and volume-weighted distribution with - ~1.17 mm mean and ~0.23 mm standard deviation are also plotted. + density function (orange) with 1 mm mean and 0.25 mm standard deviation, and volume-weighted + density function with ~1.17 mm mean and ~0.23 mm standard deviation are also plotted. 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 :math:`\Delta x` is 1/320 m. The parcel size distribution +``pic.parcels_per_cell_at_pack = 32``, and the Eulerian mesh spacing :math:`\Delta x` is 3.125 mm. The parcel size distribution shown in :numref:`fig_normal_pic_FNw_MFIX_Np` (open triangles) follows the volume-weighted density function. -Small closed circles illustrate the effective particle distribution, computed by multiplying a size observation -by the parcel statistical weight as previously mentioned. +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. .. _fig_normal_pic_FNw_MFIX_Np: @@ -235,24 +209,26 @@ by the parcel statistical weight as previously mentioned. :alt: PIC FN and Np MFIX-Exa PIC parcel size distribution (open triangles) generated from example inputs. The number-weighted - (orange) with 1 mm mean and 0.25 mm standard deviation, and volume-weighted distribution with - ~1.17 mm mean and ~0.23 mm standard deviation are also plotted. + density function (orange) with 1 mm mean and 0.25 mm standard deviation, and volume-weighted density function + (blue) with ~1.17 mm mean and ~0.23 mm standard deviation are also plotted. -The following sections outline how MFIX-Exa computes a volume-weighted normal distribution from a number-weighted -normal distribution, and conversely, how a number-weighted normal distribution is computed from a volume-weighted -normal distribution. +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. -.. rubric:: Converting a number-weighted normal distribution to volume-weighted: :math:`f_X^N(x) \rightarrow f_X^V(x)` +.. rubric:: Computing volume-weighted normal density function parameters, :math:`\mu_V` and :math:`\sigma_V`, from + number-weighted normal density function parameters, :math:`\mu_N` and :math:`\sigma_N` -The volume-weighted distribution, :math:`f_X^V(x)`, computed from the number-weighted distribution, :math:`f_X^N(x)`, -is normally distributed with mean, :math:`\mu_V`, and variance, :math:`\sigma_V^2`. An expression for the -volume-weighted probability density function (PDF) is obtained by multiplying :eq:`eq_normal_dist` by :math:`x^3`. +The volume-weighted density function, :math:`f_X^V(x)`, computed from the normally distributed number-weighted density +function, :math:`f_X^N(x)`, is normally distributed with mean, :math:`\mu_V`, and variance, :math:`\sigma_V^2`. An +expression for the volume-weighted density function is obtained by multiplying :eq:`eq_normal_dist` by :math:`x^3` +and substituting in the mean and standard deviation of the number-weighted distribution. .. math:: \tilde{f}_X^V(x) = \frac{x^3}{\sigma_N \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu_N)^2 }{ 2 \sigma_N^2 }\right)} -The mean is computed as +The mean of this distribution is computed as .. math:: :label: eq_normal_dist_FN_to_FV_mean @@ -262,7 +238,7 @@ The mean is computed as =& \frac{3\sigma_N^4 + 6\sigma_N^2\mu_N^2 + \mu_N^4}{3\sigma_N^2 \mu_N + \mu_N^3} \end{align} -The variance is given by +and the variance is given by .. math:: :label: eq_normal_dist_FN_to_FV_variance @@ -272,18 +248,19 @@ The variance is given by =& \frac{3\sigma_N^4 (3\mu_N + 2b) + \sigma^2(\mu_N^3 + 6\mu_N^2 b + 3\mu_Nb^2)+\mu_N^3 b^2}{3\sigma_N^2 \mu_N + \mu_N^3} \end{align} -where :math:`b = (\mu_N - \mu_V)`. The mean and variance can be substituted directly into :eq:`eq_normal_dist` -because the volume-weighted distribution is normally distributed. +where :math:`b = (\mu_N - \mu_V)`. Because the volume-weighted distribution is normally distributed, the computed +mean and variance can be substituted directly into :eq:`eq_normal_dist`. .. math:: f_X^V(x) = \frac{1}{\sigma_V \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu_V)^2 }{ 2 \sigma_V^2 }\right)} -.. rubric:: Converting a volume-weighted normal distribution to number-weighted: :math:`F_X^V(x) \rightarrow F_X^N(x)` +.. rubric:: Computing number-weighted normal density function parameters, :math:`\mu_N` and :math:`\sigma_N`, from + volume-weighted normal density function parameters, :math:`\mu_V` and :math:`\sigma_V` -The number-weighted distribution, :math:`f_X^V(x)` is computed from the volume-weighted distribution -by solving the nonlinear system of equations constructed from :eq:`eq_normal_dist_FN_to_FV_mean` and -:eq:`eq_normal_dist_FN_to_FV_variance` for :math:`\mu_N` and :math:`\sigma_N`. +The number-weighted density function mean and standard deviation, :math:`\mu_N` and :math:`\sigma_N`, are computed from the volume-weighted +mean and standard deviation, :math:`\mu_V` and :math:`\sigma_V`, by solving the nonlinear system of equations constructed from +:eq:`eq_normal_dist_FN_to_FV_mean` and :eq:`eq_normal_dist_FN_to_FV_variance` for :math:`\mu_N` and :math:`\sigma_N`. .. math:: @@ -292,20 +269,21 @@ by solving the nonlinear system of equations constructed from :eq:`eq_normal_dis f_2(\mu_N,\sigma_N) &= \left(3\sigma_N^4 (3\mu_N + 2b) + \sigma^2(\mu_N^3 + 6\mu_N^2 b + 3\mu_Nb^2)+\mu_N^3 b^2\right) - \sigma^2_V \left( 3\sigma_N^2 \mu_N + \mu_N^3 \right) = 0 \end{align} -This system is solved using the homotopy method outlined in :cite:t:`Burden10`. Initial guesses for -:math:`\mu_N` and :math:`\sigma_N` are calculated from the distribution function created by dividing -:eq:`eq_normal_dist` by :math:`x^3`. +This system is solved using the homotopy method outlined in :cite:t:`Burden10`. To ensure rapid convergence, initial guesses for +:math:`\mu_N` and :math:`\sigma_N` are calculated from the density function created by dividing :eq:`eq_normal_dist` by :math:`x^3` +and substituting in the mean and standard deviation of the volume-weighted distribution, :math:`\mu_V` and :math:`\sigma_V`. .. math:: :label: eq_normal_dist_FV_to_FN_initial_expression - \tilde{f}_X^N(x) = \frac{1}{x^3\sigma \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu)^2 }{ 2 \sigma^2 }\right)} + \tilde{f}_X^N(x) = \frac{1}{x^3\sigma_V \sqrt{2\pi}} \exp{\left(- \frac{(x - \mu_V)^2 }{ 2 \sigma_V^2 }\right)} -:math:`\tilde{f}_X^N(x)` is not the number-weighted distribution, but an approximation to it. The mean and variance -are calculated by numerically integrating :math:`\tilde{f}_X^N(x)` over the interval :math:`x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right]` -where :math:`x_{min}` and :math:`x_{mid}` are found by setting the derivative of :eq:`eq_normal_dist_FV_to_FN_initial_expression` -to zero and computing the roots. :math:`x_\mathrm{max}` is chosen to mirror :math:`x_\mathrm{min}` across the midpoint. +:math:`\tilde{f}_X^N(x)`, shown in :numref:`fig_normal-dist-pdf-approx_FN`, is not the number-weighted density function, +but an approximation to it. The mean and variance are calculated by numerically integrating :math:`\tilde{f}_X^N(x)` +over the interval :math:`x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right]` where :math:`x_{min}` and :math:`x_{mid}` +are found by setting the derivative of :eq:`eq_normal_dist_FV_to_FN_initial_expression` to zero and computing the roots. +:math:`x_\mathrm{max}` is chosen to mirror :math:`x_\mathrm{min}` across the midpoint. .. math:: :label: eq_normal_dist_FV_to_FN_discrete_values @@ -316,6 +294,7 @@ to zero and computing the roots. :math:`x_\mathrm{max}` is chosen to mirror :mat x_{\mathrm{max}} \; =& \; 2x_{\mathrm{mid}} - x_{\mathrm{min}} \end{align} +.. _fig_normal-dist-pdf-approx_FN: .. figure:: ./images/normal-dist-pdf-approx_FN.png :width: 95% @@ -328,7 +307,8 @@ to zero and computing the roots. :math:`x_\mathrm{max}` is chosen to mirror :mat .. rubric:: Sampling a normal distribution -MFIX-Exa samples the normal distribution with mean ``m_mean`` and standard deviation ``m_stddev`` to return :math:`x`. +MFIX-Exa samples the normal distribution with mean ``m_mean`` and standard deviation ``m_stddev`` to return an +distribution observation. .. code:: @@ -356,8 +336,8 @@ where :math:`\mu` and :math:`\sigma` are parameters defining the distribution. .. note:: :math:`\mu` and :math:`\sigma` **are not** the mean and standard deviation of :math:`x`. They are the mean and - standard deviation of :math:`\mathrm{ln}(x)`. The mean, :math:`E(x)`, and variance, :math:`V(x)`, of the - log-normal random variable are given by + standard deviation of :math:`\mathrm{ln}(x)`. The mean (expectation), :math:`E(x)`, and variance, :math:`V(x)`, + of the log-normal random variable are given by .. math:: @@ -378,10 +358,10 @@ where :math:`\mu` and :math:`\sigma` are parameters defining the distribution. | type | distribution weighting: ``number-weighted`` or ``volume-weighted`` | string | N/A | +---------------------+-----------------------------------------------------------------------+-------------+-----------+ | mean | mean of of the normal random variable, | Real | N/A | -| | :math:`\mathrm{ln}(x)` | | | +| | :math:`\mathrm{ln}(X)` | | | +---------------------+-----------------------------------------------------------------------+-------------+-----------+ | stddev | standard deviation of the normal random variable, | Real | N/A | -| | :math:`\mathrm{ln}(x)` | | | +| | :math:`\mathrm{ln}(X)` | | | +---------------------+-----------------------------------------------------------------------+-------------+-----------+ | min | Minimum particle diameter. Drawn samples below ``min`` are discarded | Real | N/A | | | and a new sample is drawn. | | | @@ -455,7 +435,7 @@ shown in :numref:`fig_log_normal_dem_FVw_MFIX_Np`. density functions. 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 :math:`\Delta x` is 1/320 m. +``pic.parcels_per_cell_at_pack = 32``, and the Eulerian mesh spacing :math:`\Delta x` is 3.125 mm. :numref:`fig_log_normal_pic_FVw_MFIX_Np` 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. @@ -471,16 +451,16 @@ captured by the PIC parcels. and volume-weighted density function (blue). Small closed circles indicated the particle size distribution modeled by the parcels. -.. rubric:: Converting between number-weighted and volume-weighted log-normal distributions +.. rubric:: Converting between number-weighted and volume-weighted parameters for log-normal distributions As reported by :cite:t:`elhilo12,elhilo12a`, number-weighted and volume-weighted log-normal distributions -use the same standard deviation, +use the same :math:`\sigma` parameter .. math:: \sigma = \sigma_{N} = \sigma_{V} -while the parameter :math:`\mu` is computed from the other distribution. +while the parameter :math:`\mu` is computed directly from the other distribution. .. math:: @@ -491,8 +471,8 @@ while the parameter :math:`\mu` is computed from the other distribution. .. rubric:: Sampling a log-normal distribution -Rather than sample a log-normal distribution, MFIX-Exa samples the normal distribution with -mean ``m_mean`` and standard deviation ``m_stddev`` to return :math:`\log (x)`. +Rather than sample the log-normal distribution, MFIX-Exa samples the normal distribution with +mean ``m_mean`` and standard deviation ``m_stddev``. .. code:: @@ -598,7 +578,7 @@ shown in :numref:`fig_uniform_dem_FNw_MFIX_Np`. and volume-weighted (blue) uniform density functions. 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 :math:`\Delta x` is 1/320 m. +``pic.parcels_per_cell_at_pack = 32``, and the Eulerian mesh spacing :math:`\Delta x` is 3.125 mm. :numref:`fig_uniform_pic_FNw_MFIX_Np` 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. @@ -620,7 +600,7 @@ captured by the PIC parcels. DEM simulations where a number-weighted distribution is specified and PIC simulations where a volume-weighted distribution is specified compute a uniformly distributed variable :math:`rand \in \left( 0, 1 \right)`, then scale and translate the -result such that :math:`x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right]`. +result such that :math:`x \in \left[ x_\mathrm{min}, x_\mathrm{max} \right)`. .. code:: @@ -632,12 +612,22 @@ The function ``amrex::Random`` returns one pseudo-random real number (double) fr uniform distribution between 0.0 and 1.0 (0.0 included, 1.0 excluded). -Inverse sampling is for DEM simulations where a volume-weighted distribution is specified and -PIC simulations where a number-weighted distribution is specified. +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 :math:`F` using the inverse function :math:`F^{-1}`. + + If :math:`\mathcal{U}` is a uniform random variable on the interval :math:`[0,1]` and + :math:`F` is a continuous cumulative distribution function on :math:`[a,b]` with inverse + + .. math:: + + F^{-1}(u) = \inf \left\{ x: F(x) = u \right\} -The number-weighted distribution, :math:`f_X^N(x)`, computed from the volume-weighted distribution is obtained + then :math:`F^{-1}(\mathcal{U})` has cumulative distribution function :math:`F` (adapted from :cite:t:`devroye86`). + +The number-weighted density function, :math:`f_X^N(x)`, is computed from a specified volume-weighted density function by dividing :eq:`eq_uniform_dist` by :math:`x^3`. The brace notation indicating that the density -function is zero for :math:`x \notin \left[ x_\mathrm{max}, x_\mathrm{min} \right]` is dropped for conciseness. +function is zero for :math:`x \notin \left[ x_\mathrm{min}, x_\mathrm{max} \right]` is dropped for conciseness. .. math:: @@ -655,14 +645,15 @@ and .. math:: - {F_X^N}^{-1}(y) = \frac{ab}{\sqrt{b^2 - y(b^2 -a^2)}} + {F_X^N}^{-1}(u) = \frac{ab}{\sqrt{b^2 - u(b^2 -a^2)}} where :math:`a = x_{\mathrm{min}}` and :math:`b = x_{\mathrm{max}}`. To sample :math:`f_X^V(x)`, a uniformly distributed -variable :math:`rand \in \left( 0, 1 \right)` is obtained and substituted into the inverse cumulative distribution -function. +variable :math:`u \in \mathcal{U}(0,1)` is generated and substituted into the inverse cumulative distribution function. .. code:: + amrex::Real rand = amrex::Random(a_engine); + amrex::Real const b(m_max); amrex::Real const a(m_min); @@ -671,9 +662,9 @@ function. return (a*b)/std::sqrt(b2 - rand*(b2-a2)); -Similarly, the volume-weighted distribution, :math:`f_X^V(x)`, computed from a number-weighted distribution is obtained +Similarly, a volume-weighted density function, :math:`f_X^V(x)`, is computed from a number-weighted distribution by multiplying :eq:`eq_uniform_dist` by :math:`x^3`. Again, the brace notation indicating that the density -function is zero for :math:`x \notin \left[ x_\mathrm{max}, x_\mathrm{min} \right]` is dropped for conciseness. +function is zero for :math:`x \notin \left[ x_\mathrm{min}, x_\mathrm{max} \right]` is dropped for conciseness. .. math:: @@ -691,17 +682,84 @@ and .. math:: - {F_X^V}^{-1}(y) = \left(a^4 + y(b^4 - a^4)\right)^{1/4} + {F_X^V}^{-1}(u) = \left(a^4 + u(b^4 - a^4)\right)^{1/4} where :math:`a = x_{\mathrm{min}}` and :math:`b = x_{\mathrm{max}}`. To sample :math:`f_X^V(x)`, a uniformly distributed -variable :math:`rand \in \left( 0, 1 \right)` is obtained and substituted into the inverse cumulative distribution -function. +variable :math:`u \in \mathcal{U}(0,1)` is generated and substituted into the inverse cumulative distribution function. .. code:: + 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 ``min`` are discarded | Real | N/A | +| | and a new sample is drawn. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| max | Maximum particle diameter. Drawn samples above ``max`` are discarded | Real | N/A | +| | and a new sample is drawn. | | | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| interpolate | Enable linear interpolation between discrete bins | bool | true | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| resample | Resample the distribution to obtain uniform bin widths | bool | true | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ +| bins | Number of bins used when resampleing the distribution. | int | 64 | ++---------------------+-----------------------------------------------------------------------+-------------+-----------+ + + + +.. code-block:: bash + :caption: Snippet of inputs defining a ``constant`` distribution. **This is not a complete input file.** + :linenos: + + 32 + CDF + # comment about the distribution + 0.00025 0.0013499 + 0.000298387 0.00250452 + 0.000346774 0.00448884 + 0.000395161 0.00777403 + 0.000443548 0.0130136 + 0.000491935 0.0210638 + 0.000540323 0.0329789 + 0.00058871 0.0499683 + 0.000637097 0.0733046 + 0.000685484 0.104184 + 0.000733871 0.143547 + 0.000782258 0.191886 + 0.000830645 0.24907 + 0.000879032 0.314239 + 0.000927419 0.385785 + 0.000975806 0.461453 + 0.00102419 0.538547 + 0.00107258 0.614215 + 0.00112097 0.685761 + 0.00116935 0.75093 + 0.00121774 0.808114 + 0.00126613 0.856453 + 0.00131452 0.895816 + 0.0013629 0.926695 + 0.00141129 0.950032 + 0.00145968 0.967021 + 0.00150806 0.978936 + 0.00155645 0.986986 + 0.00160484 0.992226 + 0.00165323 0.995511 + 0.00170161 0.997495 + 0.00175 0.99865 diff --git a/docs/source_docs/refs.bib b/docs/source_docs/refs.bib index f65d072..5bec8d1 100644 --- a/docs/source_docs/refs.bib +++ b/docs/source_docs/refs.bib @@ -25,7 +25,6 @@ pages = {358–369}, numpages = {12}, } - @article{And67, author = {Anderson, T. B. and Jackson, R. }, title = {A Fluid Mechanical Description of Fluidized Beds}, @@ -138,8 +137,8 @@ pages = {379-402} } @article{deen07, - title={Review of discrete particle modeling of fluidized beds}, author={Deen, NG and Annaland, M Van Sint and Van der Hoef, Martin Anton and Kuipers, JAM}, + title={Review of discrete particle modeling of fluidized beds}, journal={Chemical Engineering Science}, volume={62}, number={1-2}, @@ -148,6 +147,18 @@ pages = {379-402} publisher={Elsevier} } +@Inbook{devroye86, + author="Devroye, Luc", + title="General Principles in Random Variate Generation", + bookTitle="Non-Uniform Random Variate Generation", + year="1986", + publisher="Springer New York", + address="New York, NY", + pages="27--82", + isbn="978-1-4613-8643-8", + doi="10.1007/978-1-4613-8643-8_2" +} + @article{ding90, author = {Ding, Jianmin and Gidaspow, Dimitri}, title = {A bubbling fluidization model using kinetic theory of granular flow}, -- GitLab

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