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<title>Air Propulsion Simulation</title>
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<meta name="citation_reference" content="citation_title=Orbital mechanics for engineering students;citation_publication_date=2020;citation_publisher=Butterworth-Heinemann Publications;citation_author=Howard D Curtis"/>
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{"title":"Air Propulsion Simulation","description":"Simulating the performace of an air propulsion system as an alternative to solid rocket motors.","authors":[{"author":"Anson Biggs","authorURL":"https://ansonbiggs.com","affiliation":" ","affiliationURL":"#","orcidID":""}],"publishedDate":"2021-04-01T00:00:00.000-07:00","citationText":"Biggs, 2021"}
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<h1>Air Propulsion Simulation</h1>
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<div class="dt=tag">Julia</div>
<div class="dt=tag">Capstone</div>
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<p><p>Simulating the performace of an air propulsion system as an alternative to solid rocket motors.</p></p>
</div>
<div class="d-byline">
Anson Biggs <a href="https://ansonbiggs.com" class="uri">https://ansonbiggs.com</a>
<br/>04-01-2021
</div>
<div class="d-article">
<p>Boilerplate intro about why all of this was done</p>
<div class="layout-chunk" data-layout="l-body">
<details>
<summary>
Show code
</summary>
<div class="sourceCode" id="cb1"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> Plots</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>plotly()</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>theme(<span class="op">:</span>ggplot2)<span class="op">;</span> <span class="co"># In true R spirit</span></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> Unitful</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> DataFrames</span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> Measurements</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> Measurements<span class="op">:</span> value<span class="op">,</span> uncertainty</span></code></pre></div>
</details>
</div>
<p>This code is just the setup, using values scraped from various parts of the world wide web.</p>
<div class="layout-chunk" data-layout="l-body">
<div class="sourceCode" id="cb2"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Tank https://www.amazon.com/Empire-Paintball-BASICS-Pressure-Compressed/dp/B07B6M48SR/</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>V <span class="op">=</span> (<span class="fl">85</span> ± <span class="fl">5</span>)u<span class="st">"inch^3"</span></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>P0 <span class="op">=</span> (<span class="fl">4200.0</span> ± <span class="fl">300</span>)u<span class="st">"psi"</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a>Wtank <span class="op">=</span> (<span class="fl">2.3</span> ± <span class="fl">0.2</span>)u<span class="st">"lb"</span></span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a>Pmax <span class="op">=</span> (<span class="fl">250</span> ± <span class="fl">50</span>)u<span class="st">"psi"</span> <span class="co"># Max Pressure that can come out the nozzle</span></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a>Wsolenoid <span class="op">=</span> <span class="fl">1.5</span>u<span class="st">"kg"</span></span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a><span class="co"># Params</span></span>
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a>d_nozzle <span class="op">=</span> ((<span class="fl">1</span> <span class="op">//</span> <span class="fl">18</span>) ± <span class="fl">0.001</span>)u<span class="st">"inch"</span></span>
<span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a>a_nozzle <span class="op">=</span> (<span class="cn">pi</span> <span class="op">/</span> <span class="fl">4</span>) <span class="op">*</span> d_nozzle<span class="op">^</span><span class="fl">2</span></span>
<span id="cb2-12"><a href="#cb2-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-13"><a href="#cb2-13" aria-hidden="true" tabindex="-1"></a><span class="co"># Universal Stuff</span></span>
<span id="cb2-14"><a href="#cb2-14" aria-hidden="true" tabindex="-1"></a>P_amb <span class="op">=</span> (<span class="fl">1</span> ± <span class="fl">0.2</span>)u<span class="st">"atm"</span></span>
<span id="cb2-15"><a href="#cb2-15" aria-hidden="true" tabindex="-1"></a>γ <span class="op">=</span> <span class="fl">1.4</span> ± <span class="fl">0.05</span></span>
<span id="cb2-16"><a href="#cb2-16" aria-hidden="true" tabindex="-1"></a>R <span class="op">=</span> <span class="fl">287.05</span>u<span class="st">"J/(kg * K)"</span></span>
<span id="cb2-17"><a href="#cb2-17" aria-hidden="true" tabindex="-1"></a>T <span class="op">=</span> (<span class="fl">300</span> ± <span class="fl">20</span>)u<span class="st">"K"</span></span></code></pre></div>
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<p>This is the actual simulation. Maybe throw some references in and explain some equations.</p>
<p>The rocket equation is pretty sick:</p>
<p><span class="math display">\[T = \dot{m} \cdot v_\text{Exit} + A_\text{Nozzle} \cdot (P - P_\text{Ambient}) \]</span> And thats about all you need to get to the moon<span class="citation" data-cites="curtis_orbital">(<a href="#ref-curtis_orbital" role="doc-biblioref">Curtis 2020</a>)</span>.</p>
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<div class="sourceCode" id="cb3"><pre class="sourceCode julia"><code class="sourceCode julia"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="kw">let</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>t <span class="op">=</span> <span class="fl">0.0</span>u<span class="st">"s"</span></span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a>P <span class="op">=</span> P0 <span class="op">|></span> u<span class="st">"Pa"</span></span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a>M <span class="op">=</span> V <span class="op">*</span> (P <span class="op">/</span> (R <span class="op">*</span> T)) <span class="op">|></span> u<span class="st">"kg"</span></span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a>ts <span class="op">=</span> <span class="fl">1</span>u<span class="st">"ms"</span></span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a><span class="kw">global</span> df <span class="op">=</span> DataFrame(Thrust<span class="op">=</span>(<span class="fl">0</span> ± <span class="fl">0</span>)u<span class="st">"N"</span><span class="op">,</span> Pressure<span class="op">=</span>P0<span class="op">,</span> <span class="dt">Time</span><span class="op">=</span><span class="fl">0.0</span>u<span class="st">"s"</span><span class="op">,</span> Mass<span class="op">=</span>M)</span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a> <span class="kw">while</span> M <span class="op">></span> <span class="fl">0.005</span>u<span class="st">"kg"</span></span>
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a> <span class="co"># while t < 30u"s"</span></span>
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a> <span class="co"># Calculate what is leaving tank</span></span>
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a> P <span class="op">=</span> minimum([P<span class="op">,</span> Pmax])</span>
<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a> ve <span class="op">=</span> sqrt((<span class="fl">2</span> <span class="op">*</span> γ <span class="op">/</span> (γ <span class="op">-</span> <span class="fl">1</span>)) <span class="op">*</span> R <span class="op">*</span> T <span class="op">*</span> (<span class="fl">1</span> <span class="op">-</span> P_amb <span class="op">/</span> P)<span class="op">^</span>((γ <span class="op">-</span> <span class="fl">1</span>) <span class="op">/</span> γ)) <span class="op">|></span> u<span class="st">"m/s"</span></span>
<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a> ρ <span class="op">=</span> P <span class="op">/</span> (R <span class="op">*</span> T) <span class="op">|></span> u<span class="st">"kg/m^3"</span></span>
<span id="cb3-13"><a href="#cb3-13" aria-hidden="true" tabindex="-1"></a> ṁ <span class="op">=</span> ρ <span class="op">*</span> a_nozzle <span class="op">*</span> ve <span class="op">|></span> u<span class="st">"kg/s"</span></span>
<span id="cb3-14"><a href="#cb3-14" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb3-15"><a href="#cb3-15" aria-hidden="true" tabindex="-1"></a> Thrust <span class="op">=</span> ṁ <span class="op">*</span> ve <span class="op">+</span> a_nozzle <span class="op">*</span> (P <span class="op">-</span> P_amb) <span class="op">|></span> u<span class="st">"N"</span></span>
<span id="cb3-16"><a href="#cb3-16" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb3-17"><a href="#cb3-17" aria-hidden="true" tabindex="-1"></a> <span class="co"># Calculate what is still in the tank</span></span>
<span id="cb3-18"><a href="#cb3-18" aria-hidden="true" tabindex="-1"></a> M <span class="op">=</span> M <span class="op">-</span> ṁ <span class="op">*</span> ts <span class="op">|></span> u<span class="st">"kg"</span></span>
<span id="cb3-19"><a href="#cb3-19" aria-hidden="true" tabindex="-1"></a> P <span class="op">=</span> (M <span class="op">*</span> R <span class="op">*</span> T) <span class="op">/</span> V <span class="op">|></span> u<span class="st">"Pa"</span></span>
<span id="cb3-20"><a href="#cb3-20" aria-hidden="true" tabindex="-1"></a> t <span class="op">=</span> t <span class="op">+</span> ts</span>
<span id="cb3-21"><a href="#cb3-21" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb3-22"><a href="#cb3-22" aria-hidden="true" tabindex="-1"></a> df_step <span class="op">=</span> DataFrame(Thrust<span class="op">=</span>Thrust<span class="op">,</span> Pressure<span class="op">=</span>P<span class="op">,</span> <span class="dt">Time</span><span class="op">=</span>t<span class="op">,</span> Mass<span class="op">=</span>M)</span>
<span id="cb3-23"><a href="#cb3-23" aria-hidden="true" tabindex="-1"></a> <span class="cn">append</span><span class="op">!</span>(df<span class="op">,</span> df_step)</span>
<span id="cb3-24"><a href="#cb3-24" aria-hidden="true" tabindex="-1"></a> <span class="kw">end</span></span>
<span id="cb3-25"><a href="#cb3-25" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span></code></pre></div>
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<p>Heres the results plotted. Notice the massive error once the tank starts running low.</p>
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<p><img src="air-propulsion-simulation_files/figure-html5/unnamed-chunk-4-J1.png" width="350" /></p>
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<p>Big conclusion about things.</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r distill-force-highlighting-css"><code class="sourceCode r"></code></pre></div>
<div id="refs" class="references csl-bib-body hanging-indent" role="doc-bibliography">
<div id="ref-curtis_orbital" class="csl-entry" role="doc-biblioentry">
Curtis, Howard D. 2020. <em>Orbital Mechanics for Engineering Students</em>. Fourth edition. Butterworth-Heinemann Publications.
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