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<p class="caption"><span class="caption-text">Contents:</span></p>
<ul class="current">
<li class="toctree-l1"><a class="reference internal" href="chap1_balanceEquations_Chap.html">1. Balance equations</a></li>
<li class="toctree-l1"><a class="reference internal" href="chap2_thermMachinesBasics_Chap.html">2. Thermal machines: Basics</a></li>
<li class="toctree-l1 current"><a class="reference internal" href="chap3_CompExpGas_Chap.html">3. Compression / Expansion of Gas and vapors</a><ul class="current">
<li class="toctree-l2 current"><a class="current reference internal" href="#">3.1. Compressions/Expansions</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#expression-of-technical-work">3.1.1. Expression of technical work</a></li>
<li class="toctree-l3"><a class="reference internal" href="#reversible-compression-expansion">3.1.2. Reversible compression/expansion</a><ul>
<li class="toctree-l4"><a class="reference internal" href="#isothermal-transformation">3.1.2.1. Isothermal transformation</a></li>
<li class="toctree-l4"><a class="reference internal" href="#isentropic-transformation">3.1.2.2. Isentropic transformation</a></li>
<li class="toctree-l4"><a class="reference internal" href="#polytropic-transformation">3.1.2.3. Polytropic transformation</a></li>
</ul>
</li>
<li class="toctree-l3"><a class="reference internal" href="#real-compressions-expansions">3.1.3. Real Compressions/Expansions</a><ul>
<li class="toctree-l4"><a class="reference internal" href="#adiabatic-transformations-isentropic-efficiencies">3.1.3.1. Adiabatic transformations/ isentropic efficiencies</a></li>
<li class="toctree-l4"><a class="reference internal" href="#non-adibatic-transformations-polytropic-efficiencies">3.1.3.2. Non adibatic transformations/ polytropic efficiencies</a></li>
</ul>
</li>
</ul>
</li>
<li class="toctree-l2"><a class="reference internal" href="chap3_2Optimisation.html">3.2. Optimisation of Compression/Expansion</a></li>
<li class="toctree-l2"><a class="reference internal" href="chap3_3Overview.html">3.3. Overview of comp/exp systems</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="chap4_ThermalEngines_Chap.html">4. Heat engines</a></li>
<li class="toctree-l1"><a class="reference internal" href="chap5_ThermalGenerators_Chap.html">5. Heat pumps and refrigerators</a></li>
<li class="toctree-l1"><a class="reference internal" href="zBibliography.html">6. References</a></li>
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<div class="section" id="compressions-expansions">
<h1><span class="section-number">3.1. </span>Compressions/Expansions<a class="headerlink" href="#compressions-expansions" title="Permalink to this headline">¶</a></h1>
<div class="section" id="expression-of-technical-work">
<span id="sec-chap3-technicalwork"></span><h2><span class="section-number">3.1.1. </span>Expression of technical work<a class="headerlink" href="#expression-of-technical-work" title="Permalink to this headline">¶</a></h2>
<p>For a <strong>closed system</strong> (cf. thermodynamics course), the work of reversible pressure forces for a transformation between state 1 and 2 expresses as:</p>
<div class="math" id="equation-workclosedsystem">
<p><span class="eqno">(3.1)<a class="headerlink" href="#equation-workclosedsystem" title="Permalink to this equation">¶</a></span><img src="_images/math/0095e37c97c8fb01d4e4a80c2f62529aea19ddd9.svg" alt="w_{12} = -\int_1^2 pdv"/></p>
</div><p>For an <strong>open system</strong> (crossed by a fluid flow), the balance energy equation has been obtained in <a class="reference internal" href="chap1_2BalanceForControlVolume.html#sec-chap1-balanceequations"><span class="std std-numref">Section 1.1: </span></a> and reads per unit of mass crossing the system:</p>
<div class="math" id="equation-balanceenergyspecific">
<p><span class="eqno">(3.2)<a class="headerlink" href="#equation-balanceenergyspecific" title="Permalink to this equation">¶</a></span><img src="_images/math/4ee79cef38c2ab7fd0e5ee11f9b93eaff7995861.svg" alt="h_2 - h_1 = w_{t,12} + q_{12}"/></p>
</div><p>During the transformation and thanks to the <em>Gibbs equation</em>, we also have:</p>
<div class="math" id="equation-gibbs">
<p><span class="eqno">(3.3)<a class="headerlink" href="#equation-gibbs" title="Permalink to this equation">¶</a></span><img src="_images/math/ed0a41b13035a2687672c7e8729b87b4d753eab9.svg" alt="dh = Tds + vdp"/></p>
</div><p>Such that between state 1 and 2:</p>
<div class="math" id="equation-gibbsint">
<p><span class="eqno">(3.4)<a class="headerlink" href="#equation-gibbsint" title="Permalink to this equation">¶</a></span><img src="_images/math/c08ef41b3c22de9dabf0ffcc7c0efe1077ee3857.svg" alt="h_2 - h_1 = \int_1^2 Tds + \int_1^2 vdp"/></p>
</div><p>Relations <a class="reference internal" href="#equation-balanceenergyspecific">Eq.3.2</a> and <a class="reference internal" href="#equation-gibbsint">Eq.3.4</a> applied to a reversible transformation allows to express the technical work <img class="math" src="_images/math/32680814b8ddaa418cebd6978ae352d4f941e71a.svg" alt="w_t" style="vertical-align: -2px"/>:</p>
<div class="math" id="equation-techworkint">
<p><span class="eqno">(3.5)<a class="headerlink" href="#equation-techworkint" title="Permalink to this equation">¶</a></span><img src="_images/math/0fbf85ae30cefb3b3f84b8720d6104d4ec01568b.svg" alt="w_{t,12} = \int_1^2 vdp"/></p>
</div><div class="admonition important">
<p class="admonition-title">Important</p>
<p>For a <strong>reversible transformation</strong>:</p>
<ul>
<li><p>The specific pressure work in a <em>closed system</em> expresses as:</p>
<blockquote>
<div><div class="math">
<p><img src="_images/math/7a837d01e5aaacbb2b726318b80f2e581e9dc023.svg" alt="w_{12} = - \int_1^2 pdv"/></p>
</div></div></blockquote>
</li>
<li><p>The specific technical work in an <em>open system</em> expresses as:</p>
<blockquote>
<div><div class="math">
<p><img src="_images/math/0fbf85ae30cefb3b3f84b8720d6104d4ec01568b.svg" alt="w_{t,12} = \int_1^2 vdp"/></p>
</div></div></blockquote>
</li>
</ul>
</div>
<p>According to the system under study, the work will be:</p>
<blockquote>
<div><ul class="simple">
<li><p>The opposite of the aera under the curve in (p,v) plane for <em>closed systems</em></p></li>
<li><p>The aera on the left of the curve in the (p,v) plane for <em>open systems</em></p></li>
</ul>
</div></blockquote>
<div class="figure align-center" id="id1">
<span id="fig-chap3-workinclapeyron"></span><a class="reference internal image-reference" href="_images/workInClapeyron.png"><img alt="_images/workInClapeyron.png" src="_images/workInClapeyron.png" style="width: 387.3px; height: 294.3px;" /></a>
<p class="caption"><span class="caption-number">Figure 3.1: </span><span class="caption-text">work for a closed system transformation between state 1 and 2 (pink aera) and for an open system between same states (green stripes aera).</span><a class="headerlink" href="#id1" title="Permalink to this image">¶</a></p>
</div>
</div>
<div class="section" id="reversible-compression-expansion">
<h2><span class="section-number">3.1.2. </span>Reversible compression/expansion<a class="headerlink" href="#reversible-compression-expansion" title="Permalink to this headline">¶</a></h2>
<p>Reversible compressions (or expansions) can be:</p>
<blockquote>
<div><ul class="simple">
<li><p>Isotherme: <img class="math" src="_images/math/b70afd77a8f68e386335856460636b043cfec3eb.svg" alt="T=cte" style="vertical-align: 0px"/></p></li>
<li><p>Polytropic: <img class="math" src="_images/math/fdef488edcf46af337908b2fac0c1c2e560b7034.svg" alt="pv^k=cte" style="vertical-align: -3px"/></p></li>
<li><p>Isentropic: <img class="math" src="_images/math/80db19aa8e31d7c3c81a109ee6d1a61785096d47.svg" alt="s=cte" style="vertical-align: 0px"/></p></li>
</ul>
</div></blockquote>
<p>A representation of these transformations in the <em>Clapeyron</em> diagram is presented in figure <a class="reference internal" href="#fig-chap3-compclapeyron"><span class="std std-numref">Figure 3.2: </span></a></p>
<div class="figure align-center" id="id2">
<span id="fig-chap3-compclapeyron"></span><a class="reference internal image-reference" href="_images/compClapeyron.png"><img alt="_images/compClapeyron.png" src="_images/compClapeyron.png" style="width: 559.5px; height: 282.9px;" /></a>
<p class="caption"><span class="caption-number">Figure 3.2: </span><span class="caption-text">Isotherme, Polytropic and Isentropic reversible compressions in (p,v).</span><a class="headerlink" href="#id2" title="Permalink to this image">¶</a></p>
</div>
<p>For <strong>ideal gas</strong> equation of state, the expressions of specific works are recalled hereafter.</p>
<div class="section" id="isothermal-transformation">
<h3><span class="section-number">3.1.2.1. </span>Isothermal transformation<a class="headerlink" href="#isothermal-transformation" title="Permalink to this headline">¶</a></h3>
<div class="math" id="equation-workigisothermal">
<p><span class="eqno">(3.6)<a class="headerlink" href="#equation-workigisothermal" title="Permalink to this equation">¶</a></span><img src="_images/math/ded46bb017a8b9b1b5ddd7e850572bb76c4d84b7.svg" alt="w_{12}^T = w_{t,12}^T = rT ln \left( \frac{p_2}{p_1} \right)"/></p>
</div></div>
<div class="section" id="isentropic-transformation">
<h3><span class="section-number">3.1.2.2. </span>Isentropic transformation<a class="headerlink" href="#isentropic-transformation" title="Permalink to this headline">¶</a></h3>
<div class="math" id="equation-workigisent">
<p><span class="eqno">(3.7)<a class="headerlink" href="#equation-workigisent" title="Permalink to this equation">¶</a></span><img src="_images/math/a64d78ee2ca435f219536d4e9dd3b03ecb399e94.svg" alt="w_{12}^s = \frac{rT_1}{\gamma-1} \left[ \left( \frac{p_2}{p_1} \right)^{\frac{\gamma-1}{\gamma}} -1 \right]"/></p>
</div><p>and</p>
<div class="math" id="equation-techworkigisent">
<p><span class="eqno">(3.8)<a class="headerlink" href="#equation-techworkigisent" title="Permalink to this equation">¶</a></span><img src="_images/math/0728b23c094366a9b1d3cc8f2f820e9618aab39f.svg" alt="w_{t,12}^s = \frac{\gamma rT_1}{\gamma-1} \left[ \left( \frac{p_2}{p_1} \right)^{\frac{\gamma-1}{\gamma}} -1 \right]"/></p>
</div></div>
<div class="section" id="polytropic-transformation">
<h3><span class="section-number">3.1.2.3. </span>Polytropic transformation<a class="headerlink" href="#polytropic-transformation" title="Permalink to this headline">¶</a></h3>
<div class="math" id="equation-workigpolyp">
<p><span class="eqno">(3.9)<a class="headerlink" href="#equation-workigpolyp" title="Permalink to this equation">¶</a></span><img src="_images/math/3f13cf950f86440625f90c526531b25dc7f19560.svg" alt="w_{12}^p = \frac{rT_1}{k-1} \left[ \left( \frac{p_2}{p_1} \right)^{\frac{k-1}{k}} -1 \right]"/></p>
</div><p>and</p>
<div class="math" id="equation-techworkigpolyp">
<p><span class="eqno">(3.10)<a class="headerlink" href="#equation-techworkigpolyp" title="Permalink to this equation">¶</a></span><img src="_images/math/79086b044ed3eca4f7211f86f1fa4f1d9b490cad.svg" alt="w_{t,12}^p = \frac{krT_1}{k-1} \left[ \left( \frac{p_2}{p_1} \right)^{\frac{k-1}{k}} -1 \right]"/></p>
</div></div>
</div>
<div class="section" id="real-compressions-expansions">
<span id="sec-chap3-efficiencies"></span><h2><span class="section-number">3.1.3. </span>Real Compressions/Expansions<a class="headerlink" href="#real-compressions-expansions" title="Permalink to this headline">¶</a></h2>
<p>If during a real compression (or expansion) the transformation remains <em>irreversible</em>, calculs are performed thanks to associated reversible transformations and the knowledge of corresponding <em>efficiencies</em> that are known as manufacturer’s data.</p>
<p>For a <em>compression</em>, real work (positive) is always greater than the corresponding theoretical reversible transformation, such that we define the <strong>compression efficiency</strong> of a compressor by:</p>
<div class="math" id="equation-compefficiency">
<p><span class="eqno">(3.11)<a class="headerlink" href="#equation-compefficiency" title="Permalink to this equation">¶</a></span><img src="_images/math/117b51f2b046be5fe26c71f5c82ccbb149f6265b.svg" alt="\eta_{C} = \frac{w_{t,rev}}{w_{t,real}} \leq 1"/></p>
</div><p>For an <em>expansion</em>, the reversed situation appears as the technical work provided by the machine is always lower than the corresponding theoretical reversible transformation such that we define the <strong>expansion efficiency</strong> of a turbine by:</p>
<div class="math" id="equation-turbefficiency">
<p><span class="eqno">(3.12)<a class="headerlink" href="#equation-turbefficiency" title="Permalink to this equation">¶</a></span><img src="_images/math/d4259dc3ca4b6bb42bb80faf4b6b7c434f2043ea.svg" alt="\eta_{T} = \frac{w_{t,real}}{w_{t,rev}} \leq 1"/></p>
</div><p>Thus depending on insulating of compressor/turbine, one will talk about isentropic, isothermic or polytropic efficiency.</p>
<div class="section" id="adiabatic-transformations-isentropic-efficiencies">
<h3><span class="section-number">3.1.3.1. </span>Adiabatic transformations/ isentropic efficiencies<a class="headerlink" href="#adiabatic-transformations-isentropic-efficiencies" title="Permalink to this headline">¶</a></h3>
<p>When the compression/expansion machine is <em>thermically insulated</em>, we use the <strong>isentropic efficiency</strong> that provides the ratio between the real work and those associated to the corresponding reversbile transformation (isentropic). The real transformation induces an increase of the entropy.</p>
<div class="figure align-center" id="id3">
<span id="fig-chap3-isentropicefficiency"></span><a class="reference internal image-reference" href="_images/isentropicEfficiency.png"><img alt="_images/isentropicEfficiency.png" src="_images/isentropicEfficiency.png" style="width: 284.7px; height: 320.7px;" /></a>
<p class="caption"><span class="caption-number">Figure 3.3: </span><span class="caption-text">Adiabatic compression and expansion. Comparaison between isentropic path and real path.</span><a class="headerlink" href="#id3" title="Permalink to this image">¶</a></p>
</div>
<p>Because the transformations are adiabatics, the <em>isentropic efficiencies</em> for compressors or turbines become:</p>
<div class="math" id="equation-isentropicefficiencies">
<p><span class="eqno">(3.13)<a class="headerlink" href="#equation-isentropicefficiencies" title="Permalink to this equation">¶</a></span><img src="_images/math/8effe82804aa69d7e8a046af83d6651a1f60baea.svg" alt="\eta_{C}^S = \frac{h_{t,1}^S-h_{t,0}}{h_{t,1}-h_{t,0}} \qquad \text{and} \qquad \eta_{T}^S = \frac{h_{t,2}-h_{t,0}}{h_{t,2}^S-h_{t,0}}"/></p>
</div><div class="admonition-simplifications admonition">
<p class="admonition-title">Simplifications</p>
<ul class="simple">
<li><p>When Kinetic energy variations are negligible</p></li>
<li><p>When the working fluid is consiered as an <em>ideal gas</em></p></li>
</ul>
<p><em>Isentropic efficiencies</em> become:</p>
<div class="math">
<p><img src="_images/math/746a68975b8afcc413686fee79394fe05bf5bf1f.svg" alt="\eta_{C}^S = \frac{T_{1}^S-T_{0}}{T_{1}-T_{0}} \qquad \text{and} \qquad \eta_{T}^S = \frac{T_{2}-T_{0}}{T_{2}^S-T_{0}}"/></p>
</div><p>Using <em>Laplace</em> relations for isentropic transformation, we get:</p>
<div class="math" id="equation-isentropicefficienciesig">
<p><span class="eqno">(3.14)<a class="headerlink" href="#equation-isentropicefficienciesig" title="Permalink to this equation">¶</a></span><img src="_images/math/c81ace0f43b5d7da0fe0e08fd4f02cf8532a6576.svg" alt="\eta_{C}^S = \frac{T_{0} r_p^{\frac{\gamma-1}{\gamma}}-T_{0}}{T_{1}-T_{0}} \qquad \text{and} \qquad \eta_{T}^S = \frac{T_{2}-T_{0}}{T_{0} r_p^{\frac{\gamma-1}{\gamma}}-T_{0}}"/></p>
</div><p>Then, it is possible to determine easily the final temperature of the working fluid by knowledge of the pressure ratio (<img class="math" src="_images/math/18e8d474c0d3668b9f91f811411028fed8353ad4.svg" alt="r_p = p_{final}/p_{init}" style="vertical-align: -5px"/>) and isentropic efficiencies.</p>
</div>
</div>
<div class="section" id="non-adibatic-transformations-polytropic-efficiencies">
<h3><span class="section-number">3.1.3.2. </span>Non adibatic transformations/ polytropic efficiencies<a class="headerlink" href="#non-adibatic-transformations-polytropic-efficiencies" title="Permalink to this headline">¶</a></h3>
<p>When the compression/expansion is not <em>thermically insulated</em>, isentropic efficiencies are no longer valid and we use the <strong>polytropic efficiency</strong> providing the ratio between the real work and those associated to the corresponding reversible polytropic transformation leading to the same thermodynamic state at the end of compression/expansion.</p>
<div class="figure align-center" id="id4">
<span id="fig-chap3-polytropicefficiency"></span><a class="reference internal image-reference" href="_images/polytropicEfficiency.png"><img alt="_images/polytropicEfficiency.png" src="_images/polytropicEfficiency.png" style="width: 606.3px; height: 383.09999999999997px;" /></a>
<p class="caption"><span class="caption-number">Figure 3.4: </span><span class="caption-text">Non adiabatic compressions and expansions. Comparaison between polytropic path (lines) and real path (dash lines).</span><a class="headerlink" href="#id4" title="Permalink to this image">¶</a></p>
</div>
<p>Possible transformation with heat exchanges during compression/expansions can lead to different possibilities summarized in <a class="reference internal" href="#fig-chap3-polytropicefficiency"><span class="std std-numref">Figure 3.4: </span></a>. Whatever the case is, we have for the polytropic (<img class="math" src="_images/math/2b449adf8e7e9062b514cc5038a5f2a6b73d4952.svg" alt="p" style="vertical-align: -3px"/>) and real transformation:</p>
<div class="math">
<p><img src="_images/math/cc55923c243a2ffd15549333a9db55c852f8076a.svg" alt="w_{t}^P = (h_{t,i}-h_{t,0}) - q_P \qquad \text{and} \qquad w_{t} = (h_{t,i}-h_{t,0}) - q"/></p>
</div><p>Such that the <em>polytropic efficiencies</em> express as:</p>
<div class="math" id="equation-polytropicefficiencies">
<p><span class="eqno">(3.15)<a class="headerlink" href="#equation-polytropicefficiencies" title="Permalink to this equation">¶</a></span><img src="_images/math/5f6820e80afbd4ea2bcf06f4ba879df76ccb66cb.svg" alt="\eta_{C}^P = \frac{(h_{t,1}-h_{t,0})-q_P}{(h_{t,1}-h_{t,0})-q} \qquad \text{and} \qquad \eta_{T}^P = \frac{(h_{t,2}-h_{t,0}) - q}{(h_{t,2}-h_{t,0}) -q_P}"/></p>
</div><div class="admonition-simplifications admonition">
<p class="admonition-title">Simplifications</p>
<ul class="simple">
<li><p>When Kinetic energy variations are negligible</p></li>
<li><p>When the working fluid is consiered as an <em>ideal gas</em></p></li>
</ul>
<p>The technical work for a polytropic transformation is given by relation <a class="reference internal" href="#equation-techworkigpolyp">Eq.3.10</a>, and the real technical work expresses as:</p>
<div class="math">
<p><img src="_images/math/584eddc48a838ce2a64882f9ed0505e4d4f5581a.svg" alt="w_t = (h_{t,i}-h_{t,0}) - q = \frac{\gamma r}{\gamma -1} T_0 (r_p^{\frac{k-1}{k}}-1) - q"/></p>
</div><p><em>Polytropic efficiencies</em> thus become:</p>
<div class="math" id="equation-polytropicefficienciesig">
<p><span class="eqno">(3.16)<a class="headerlink" href="#equation-polytropicefficienciesig" title="Permalink to this equation">¶</a></span><img src="_images/math/100ff6c494b783fedb565c8444db26b1d471f318.svg" alt="\eta_{C}^P = \frac{ \frac{k}{k-1} (r_p^{\frac{k-1}{k}}-1)} { \frac{\gamma }{\gamma -1} (r_p^{\frac{k-1}{k}}-1) - \frac{q (\gamma-1)}{r T_0} } \qquad \text{and} \qquad \eta_{T}^P = \frac{ \frac{\gamma }{\gamma -1} (r_p^{\frac{k-1}{k}}-1) - \frac{q (\gamma-1)}{r T_0} } {\frac{k}{k-1} (r_p^{\frac{k-1}{k}}-1)}"/></p>
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