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<section id="solid-diffusion-models">
<h1>Solid Diffusion Models<a class="headerlink" href="#solid-diffusion-models" title="Link to this heading"></a></h1>
<p>We use the Doyler, Fuller and Newman model, see <span id="id1">[<a class="reference internal" href="bibliography.html#id3" title="Marc Doyle, Thomas F. Fuller, and John Newman. Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell. Journal of The Electrochemical Society, 140(6):1526–1533, June 1993. URL: http://dx.doi.org/10.1149/1.2221597, doi:10.1149/1.2221597.">DFN93</a>]</span>. For each given spatial location, the concentration
profile in the solid is given for a representative spherical particle located there. The diffusion in the solid
electrode is then modeled as a diffusion equation in thee radial dimension of the pseudo particle. We use a standard
control volume method to solve the diffusion equation after discretization in the radial direction.</p>
<p>Reduced models for the solid diffusion have been considered to reduced the computation time. We have implemented the
<em>polynomial approximation method</em> as described in <span id="id2">[<a class="reference internal" href="bibliography.html#id2" title="Qi Zhang and Ralph E. White. Comparison of approximate solution methods for the solid phase diffusion equation in a porous electrode model. Journal of Power Sources, 165(2):880–886, March 2007. URL: http://dx.doi.org/10.1016/j.jpowsour.2006.12.056, doi:10.1016/j.jpowsour.2006.12.056.">ZW07b</a>]</span>.</p>
<p>The option <code class="code docutils literal notranslate"><span class="pre">diffusionModelType</span></code> in the Active Material part of the json schema (see <a class="reference external" href="https://github.com/BattMoTeam/BattMo/blob/dev/Utilities/JsonSchemas/ActiveMaterial.schema.json#L10">here</a>) is used to choose the type of diffusion model.</p>
<p>In the script <a class="reference external" href="https://github.com/BattMoTeam/BattMo/blob/dev/Examples/Advanced/runChen2020.m">runChen2020</a>, we compare the two diffusion models in addition to the solution produced by <a class="reference external" href="https://pybamm.org/">PyBaMM</a> for the Chen dataset <span id="id3">[<a class="reference internal" href="bibliography.html#id10" title="Chang-Hui Chen, Ferran Planella, Kieran O’Regan, Dominika Gastol, Dhammika Widanagea, and Emma Kendrick. Development of Experimental Techniques for Parameterization of Multi-scale Lithium-ion Battery Models. Journal of The Electrochemical Society, 167:080534, 2020. doi:10.1149/1945-7111/ab9050.">CPORegan+20</a>]</span>.</p>
<p>We reproduce the last plot produced in this script here. We observe a good overall match between the full and reduced
model for solid diffusion. The discripancies are large only at the beginning of the discharge. We observe also a good
agreement between BattMo and PyBaMM.</p>
<figure class="align-center" id="id4">
<a class="reference external image-reference" href="_images/chencomp.png"><img alt="_images/chencomp.png" src="_images/chencomp.png" style="width: 100%;" /></a>
<figcaption>
<p><span class="caption-text">Discharge curves for Chen’s dataset with the full and reduced model for diffusion.</span><a class="headerlink" href="#id4" title="Link to this image"></a></p>
</figcaption>
</figure>
<figure class="align-center" id="id5">
<a class="reference external image-reference" href="_images/chencompzoom.png"><img alt="_images/chencompzoom.png" src="_images/chencompzoom.png" style="width: 100%;" /></a>
<figcaption>
<p><span class="caption-text">Zoom on the initial phase of the discharge.</span><a class="headerlink" href="#id5" title="Link to this image"></a></p>
</figcaption>
</figure>
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