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docs/source_docs/_static/theme_overrides.css

+4 −0
Changes for docs/source_docs/_static/theme_overrides.css: 4 added lines, 0 removed lines.
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@@ -15,3 +15,7 @@ 
.eqno {

    float: right;

}



.wy-nav-content {

    max-width: none;

}

docs/source_docs/references/size_distributions.rst

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Changes for docs/source_docs/references/size_distributions.rst: 10 added lines, 10 removed lines.
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@@ -233,20 +233,20 @@ The mean of this distribution is computed as 
.. math::

   :label: eq_normal_dist_FN_to_FV_mean



   \begin{align}

   \begin{aligned}

       \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}

   \end{aligned}



and the variance is given by



.. math::

   :label: eq_normal_dist_FN_to_FV_variance



   \begin{align}

   \begin{aligned}

       \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}

   \end{aligned}



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`.
@@ -264,10 +264,10 @@ mean and standard deviation, :math:`\mu_V` and :math:`\sigma_V`, by solving the 


.. math::



   \begin{align}

   \begin{aligned}

     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}

   \end{aligned}



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`
@@ -288,11 +288,11 @@ are found by setting the derivative of :eq:`eq_normal_dist_FV_to_FN_initial_expr 
.. math::

   :label: eq_normal_dist_FV_to_FN_discrete_values



   \begin{align}

   \begin{aligned}

     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}

   \end{aligned}



.. _fig_normal-dist-pdf-approx_FN:
@@ -464,10 +464,10 @@ while the parameter :math:`\mu` is computed directly from the other distribution 


.. math::



   \begin{align}

   \begin{aligned}

       \mu_V =& \mu_N + 3.0\sigma^2 \\[10pt]

       \mu_N =& \mu_V - 3.0\sigma^2

   \end{align}

   \end{aligned}



.. rubric:: Sampling a log-normal distribution

docs/source_docs/user_guide/inputs/chemical_reactions.rst

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Changes for docs/source_docs/user_guide/inputs/chemical_reactions.rst: 1 added line, 1 removed line.
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@@ -4,7 +4,7 @@ 
Defining chemical reactions

===========================



.. |tol_eq| replace:: :math:`\left|\sum\text{products - \sum\text{reactants}\right| < \text{tolerance}`

.. |tol_eq| replace:: :math:`\left|\sum \mathrm{products} - \sum \mathrm{reactants}\right| < \mathrm{tolerance}`



The following inputs are defined using the prefix ``chemistry``:

docs/source_docs/user_guide/inputs/mesh_and_gridding.rst

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Changes for docs/source_docs/user_guide/inputs/mesh_and_gridding.rst: 3 added lines, 3 removed lines.
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@@ -27,9 +27,9 @@ number of cells. The mesh spacing is required to be the same in all directions: 


.. math::



   \frac{\text{prob_hi[0] - prob_lo[0]}}{\text{n_cell[0]}}

   = \frac{\text{prob_hi[1] - prob_lo[1]}}{\text{n_cell[1]}}

   = \frac{\text{prob_hi[2] - prob_lo[2]}}{\text{n_cell[2]}}

   \frac{\text{prob\_hi[0] - prob\_lo[0]}}{\text{n\_cell[0]}}

   = \frac{\text{prob\_hi[1] - prob\_lo[1]}}{\text{n\_cell[1]}}

   = \frac{\text{prob\_hi[2] - prob\_lo[2]}}{\text{n\_cell[2]}}





The inputs for defining the mesh for a single-level simulation are demonstrated in the

docs/source_docs/user_guide/inputs/model_options.rst

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@@ -307,7 +307,7 @@ The following input is defined using the prefix ``mfix.drag``: 


.. |VM_Zuber_Eq| replace:: :math:`C_{p,vm} = (\frac{1}{2})\frac{(1+2\varepsilon_p)}{(1-\varepsilon_p)}`



.. |VM_Nijssen_Eq| replace:: :math:`\begin{split}C_{p,vm} &= C_{0,vm} \\ &+ \sum_{k=1}^3 \left\{ \left[  a_{k,0} + a_{k,1} \ln \left( \frac{\rho_p}{\rho_f} \right) \right] \varepsilon_p^k \right\}\end{split}`

.. |VM_Nijssen_Eq| replace:: :math:`C_{p,vm} = C_{0,vm} + \sum_{k=1}^3 \left\{ \left[ a_{k,0} + a_{k,1} \ln \left( \frac{\rho_p}{\rho_f} \right) \right] \varepsilon_p^k \right\}`



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