Flusso di Rayleigh

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vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inizio</div> </a> </li> <li id="toc-Curva_di_Rayleigh" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Curva_di_Rayleigh"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Curva di Rayleigh</span> </div> </a> <button aria-controls="toc-Curva_di_Rayleigh-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Curva di Rayleigh</span> </button> <ul id="toc-Curva_di_Rayleigh-sublist" class="vector-toc-list"> <li id="toc-Approssimazione_della_curva_di_Rayleigh_mediante_trasformazioni_politropiche" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Approssimazione_della_curva_di_Rayleigh_mediante_trasformazioni_politropiche"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Approssimazione della curva di Rayleigh mediante trasformazioni politropiche</span> </div> </a> <ul id="toc-Approssimazione_della_curva_di_Rayleigh_mediante_trasformazioni_politropiche-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proprietà_termodinamiche_nel_flusso_di_Rayleigh" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proprietà_termodinamiche_nel_flusso_di_Rayleigh"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Proprietà termodinamiche nel flusso di Rayleigh</span> </div> </a> <button aria-controls="toc-Proprietà_termodinamiche_nel_flusso_di_Rayleigh-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Proprietà termodinamiche nel flusso di Rayleigh</span> </button> <ul id="toc-Proprietà_termodinamiche_nel_flusso_di_Rayleigh-sublist" class="vector-toc-list"> <li id="toc-Spiegazione_dell'andamento_non_univoco_della_temperatura" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spiegazione_dell'andamento_non_univoco_della_temperatura"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Spiegazione dell'andamento non univoco della temperatura</span> </div> </a> <ul id="toc-Spiegazione_dell'andamento_non_univoco_della_temperatura-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Choking_nel_flusso_di_Rayleigh" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Choking_nel_flusso_di_Rayleigh"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Choking nel flusso di Rayleigh</span> </div> </a> <ul id="toc-Choking_nel_flusso_di_Rayleigh-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" 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href="https://it.wikipedia.org/w/index.php?title=Flusso_di_Rayleigh&action=edit">Contribuisci</a> a migliorarla secondo le <a href="/wiki/Aiuto:Manuale_di_stile" title="Aiuto:Manuale di stile">convenzioni di Wikipedia</a>. Segui i suggerimenti del <a href="/wiki/Progetto:Fisica" title="Progetto:Fisica">progetto di riferimento</a>.</div> <div class="hide-when-compact"> </div> </div> </div> </div> <p>Il <b>flusso di Rayleigh</b> è un modello matematico di flusso unidimensionale in condizioni stazionarie con scambio di <a href="/wiki/Calore" title="Calore">calore</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta q\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>q</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta q\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b49b6b8c8f695d4d85f145c190314c162dc507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.379ex; height:2.843ex;" alt="{\displaystyle \delta q\neq 0}"></span>) con l'esterno, in condotti a sezione costante, dove le irreversibilità dovute all'<a href="/wiki/Attrito" title="Attrito">attrito</a> sono trascurate (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta l_{w}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta l_{w}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e36616da2fdfa0c741479ea7bbdf11d614c087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.412ex; height:2.676ex;" alt="{\displaystyle \delta l_{w}=0}"></span>)<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>. </p><p>Il flusso deve il suo nome a <a href="/wiki/John_William_Strutt_Rayleigh" title="John William Strutt Rayleigh">John Strutt, III barone Rayleigh</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>. </p><p>Analogamente al <a href="/wiki/Flusso_di_Fanno" title="Flusso di Fanno">flusso di Fanno</a>, il flusso di Rayleigh è non-isoentropico, in quanto reversibile ma non adiabatico a causa della presenza di calore scambiato con l'ambiente esterno. Infatti, scrivendo l'equazione del <a href="/wiki/Secondo_principio_della_termodinamica" title="Secondo principio della termodinamica">secondo principio della termodinamica</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Tds=\delta q+\delta l_{w}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mi>δ</mi> <mi>q</mi> <mo>+</mo> <mi>δ</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Tds=\delta q+\delta l_{w}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82b3602f289b7461f79c93778ad09bcf8d2b2243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.151ex; height:2.676ex;" alt="{\displaystyle Tds=\delta q+\delta l_{w}}"></span> </p><p>e, tenendo conto delle condizioni sopra ipotizzate, si evince che la variazione di entropia non è nulla <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\biggl (}ds={\frac {\delta q}{T}}\neq 0{\biggr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>≠</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\biggl (}ds={\frac {\delta q}{T}}\neq 0{\biggr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3abbdae481d08f6895f59279709736e90b03e31d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.041ex; height:6.176ex;" alt="{\displaystyle {\biggl (}ds={\frac {\delta q}{T}}\neq 0{\biggr )}}"></span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Curva_di_Rayleigh">Curva di Rayleigh</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flusso_di_Rayleigh&veaction=edit&section=1" title="Modifica la sezione Curva di Rayleigh" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Flusso_di_Rayleigh&action=edit&section=1" title="Edit section's source code: Curva di Rayleigh"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Partendo dall'equazione di continuità: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\dot {M}}{A}}=\rho c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>˙</mo> </mover> </mrow> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mi>ρ</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\dot {M}}{A}}=\rho c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a184d3f052e7aeaf9c03934dd77eb7c4713078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.586ex; height:6.009ex;" alt="{\displaystyle {\frac {\dot {M}}{A}}=\rho c}"></span> </p><p>(dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e17d7f361e94093d5346b9d04a5d4b8c7f4d22f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.676ex;" alt="{\displaystyle {\dot {M}}}"></span>è la portata di massa espressa in kg/s, A è l'area della sezione del condotto, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> e c sono rispettivamente la densità e la velocità del fluido all'interno del condotto) </p><p>e dal bilancio della quantità di moto in caso di flussi unidimensionali e stazionari: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+\rho c^{2}={\frac {I}{A}}=cost.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>+</mo> <mi>ρ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>I</mi> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p+\rho c^{2}={\frac {I}{A}}=cost.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad587a7b763b1138c0d2ebef2c213db0ab7cb5aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:20.85ex; height:5.343ex;" alt="{\displaystyle p+\rho c^{2}={\frac {I}{A}}=cost.}"></span> </p><p>(dove p è la pressione, I/A è definito come impulso specifico che è costante in condizioni stazionarie) </p><p>si ricava una relazione dove la pressione è espressa in funzione della densità: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+{\biggl (}{\frac {\dot {M}}{A}}{\biggr )}^{2}{\frac {1}{\rho }}={\frac {I}{A}}=cost.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>˙</mo> </mover> </mrow> <mi>A</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>I</mi> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p+{\biggl (}{\frac {\dot {M}}{A}}{\biggr )}^{2}{\frac {1}{\rho }}={\frac {I}{A}}=cost.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af6fc3b2e7b84b6c15792770cc2b765117339c7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:27.379ex; height:6.509ex;" alt="{\displaystyle p+{\biggl (}{\frac {\dot {M}}{A}}{\biggr )}^{2}{\frac {1}{\rho }}={\frac {I}{A}}=cost.}"></span> </p><p> Tenuto conto che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\frac {1}{\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {1}{\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ccf21e664429563c03bd0d2c29e8001093ce231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.264ex; height:5.676ex;" alt="{\displaystyle v={\frac {1}{\rho }}}"></span>si può rappresentare tale relazione sul piano di Clapeyron <i>p-v.</i></p><p><small><span class="error">Questo grafico non è disponibile a causa di un problema tecnico.<br /><strong>Si prega di <span style="text-decoration-line: underline;">non</span> rimuoverlo.</strong></span></small> </p><p>È possibile valutare anche l'entalpia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=h(p,\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=h(p,\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af981bcdc89fcbee1e70cba4ae242e8dd4ce6726" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.991ex; height:2.843ex;" alt="{\displaystyle h=h(p,\rho )}"></span> e l'entropia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=s(p,\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=s(p,\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b2de89382495dae1788ee24561c54ec969ae201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.494ex; height:2.843ex;" alt="{\displaystyle s=s(p,\rho )}"></span>e considerando la relazione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ddb9197516e9bc39cd438324349b3bb55b470ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.27ex; height:2.843ex;" alt="{\displaystyle p(\rho )}"></span>precedentemente ricavata, si ottiene la curva di Rayleigh sul piano di Gibbs <i>h-s</i>. </p><p><br /> </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Rayleigh_Line.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Rayleigh_Line.PNG/368px-Rayleigh_Line.PNG" decoding="async" width="368" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Rayleigh_Line.PNG/552px-Rayleigh_Line.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Rayleigh_Line.PNG/736px-Rayleigh_Line.PNG 2x" data-file-width="912" data-file-height="621" /></a><figcaption><b>Figura 1.</b> Rappresentazione della curva di Rayleigh sul diagramma h-s.</figcaption></figure> <p>Differenziando la funzione impulso specifico, si ottiene: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dp-{\biggl (}{\frac {\dot {M}}{A}}{\biggr )}^{2}{\frac {1}{\rho ^{2}}}d\rho =0\longrightarrow dp-{\biggl (}{\frac {\dot {M}}{\rho A}}{\biggr )}^{2}d\rho =0{\xrightarrow[{}]{eq.continuit{\grave {a}}}}dp-c^{2}d\rho =0\longrightarrow c={\sqrt {\frac {dp}{d\rho }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>p</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>˙</mo> </mover> </mrow> <mi>A</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>d</mi> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⟶</mo> <mi>d</mi> <mi>p</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>˙</mo> </mover> </mrow> <mrow> <mi>ρ</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset="-.24em"> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mpadded> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mi>e</mi> <mi>q</mi> <mo>.</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>i</mi> <mi>n</mi> <mi>u</mi> <mi>i</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>`</mo> </mover> </mrow> </mrow> </mpadded> </munderover> </mrow> <mi>d</mi> <mi>p</mi> <mo>−</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⟶</mo> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>ρ</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dp-{\biggl (}{\frac {\dot {M}}{A}}{\biggr )}^{2}{\frac {1}{\rho ^{2}}}d\rho =0\longrightarrow dp-{\biggl (}{\frac {\dot {M}}{\rho A}}{\biggr )}^{2}d\rho =0{\xrightarrow[{}]{eq.continuit{\grave {a}}}}dp-c^{2}d\rho =0\longrightarrow c={\sqrt {\frac {dp}{d\rho }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe9fbcd48217b89043a728b03a4b91ae9e92b29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:87.669ex; height:7.509ex;" alt="{\displaystyle dp-{\biggl (}{\frac {\dot {M}}{A}}{\biggr )}^{2}{\frac {1}{\rho ^{2}}}d\rho =0\longrightarrow dp-{\biggl (}{\frac {\dot {M}}{\rho A}}{\biggr )}^{2}d\rho =0{\xrightarrow[{}]{eq.continuit{\grave {a}}}}dp-c^{2}d\rho =0\longrightarrow c={\sqrt {\frac {dp}{d\rho }}}}"></span> </p><p>Nel punto M l'entropia è massima, pertanto assumerà un valore costante e ds = 0 di conseguenza. Da ciò discende che la velocità nel punto M è </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{M}={\sqrt {{\biggl (}{\frac {dp}{d\rho }}{\biggr )}}}_{s=cost}={\sqrt {\gamma RT}}=a_{S,M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>γ</mi> <mi>R</mi> <mi>T</mi> </msqrt> </mrow> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mo>,</mo> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{M}={\sqrt {{\biggl (}{\frac {dp}{d\rho }}{\biggr )}}}_{s=cost}={\sqrt {\gamma RT}}=a_{S,M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05137d4d5409be6b4c04a33c0bc973a8a81dcfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.109ex; height:7.676ex;" alt="{\displaystyle c_{M}={\sqrt {{\biggl (}{\frac {dp}{d\rho }}{\biggr )}}}_{s=cost}={\sqrt {\gamma RT}}=a_{S,M}}"></span>ossia la <a href="/wiki/Velocit%C3%A0_del_suono" title="Velocità del suono">velocità locale del suono</a> e pertanto ci troveremo in condizioni critiche (o soniche) con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ma_{M}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ma_{M}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed847115d0afb368a40cc0e87af21682476af92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.892ex; height:2.509ex;" alt="{\displaystyle Ma_{M}=1}"></span>. </p><p>Nel ramo superiore il flusso è in <a href="/wiki/Regime_subsonico" title="Regime subsonico">regime subsonico</a>, percorso verso entropie crescenti, se il sistema è in riscaldamento, mentre verso entropie decrescenti, se il sistema è in raffreddamento. </p><p>Nel ramo inferiore, invece, il flusso è in <a href="/wiki/Regime_supersonico" title="Regime supersonico">regime supersonico</a>, percorso analogamente al ramo superiore. </p><p>A differenza della curva di Fanno, la curva di Rayleigh può essere percorsa passando dal tratto subsonico al supersonico o viceversa in quanto l'entropia non è strettamente maggiore di zero, pertanto non si ha la necessità di muoversi esclusivamente verso le entropie crescenti come accade in Fanno. Nel flusso di Fanno infatti, le ipotesi di partenza sono <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta q=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta q=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b8e9cbd5b7e15d5d6027ef8915376042eeb6a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.379ex; height:2.676ex;" alt="{\displaystyle \delta q=0}"></span>e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta l_{w}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta l_{w}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b15df7b539753a5adde8372503e31a71ce350f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.412ex; height:2.843ex;" alt="{\displaystyle \delta l_{w}\neq 0}"></span> ma, essendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta l_{w}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta l_{w}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a36ff106c9c3d19349fe01a58057af0b60c64a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.412ex; height:2.676ex;" alt="{\displaystyle \delta l_{w}>0}"></span> per il secondo principio della termodinamica discende necessariamente che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>s</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aaa2f13574d6279677763051d00a44fd0eba3c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.287ex; height:2.176ex;" alt="{\displaystyle \Delta s>0}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Approssimazione_della_curva_di_Rayleigh_mediante_trasformazioni_politropiche">Approssimazione della curva di Rayleigh mediante trasformazioni politropiche</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flusso_di_Rayleigh&veaction=edit&section=2" title="Modifica la sezione Approssimazione della curva di Rayleigh mediante trasformazioni politropiche" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Flusso_di_Rayleigh&action=edit&section=2" title="Edit section's source code: Approssimazione della curva di Rayleigh mediante trasformazioni politropiche"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><br /></p><figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Rayleigh_2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Rayleigh_2.jpg/334px-Rayleigh_2.jpg" decoding="async" width="334" height="294" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Rayleigh_2.jpg/501px-Rayleigh_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Rayleigh_2.jpg/668px-Rayleigh_2.jpg 2x" data-file-width="1600" data-file-height="1409" /></a><figcaption><b>Figura 2</b>. Curve politropiche approssimano la curva di Rayleigh nei punti critici A, B, O e M.</figcaption></figure> <p>Fissato un piano di Gibbs <i>h-s</i> una curva di Rayleigh può essere approssimata da una generica politropica di equazione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pv^{m}=cost.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pv^{m}=cost.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b69e409ebd42fdd2d840a94e638dce5ad99566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.871ex; height:2.676ex;" alt="{\displaystyle pv^{m}=cost.}"></span>o equivalentemente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {p}=cost\cdot \rho ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>t</mi> <mo>⋅</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {p}=cost\cdot \rho ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19c02ac10e242c1643dcf62c0cb71ee673b560ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.978ex; height:2.843ex;" alt="{\displaystyle {p}=cost\cdot \rho ^{m}}"></span> </p><p>È possibile conoscere attraverso una semplice relazione il <a href="/wiki/Numero_di_Mach" title="Numero di Mach">numero di Mach</a> a seconda del punto della curva considerato. Il punto A ad esempio è approssimato da un'isobara, politropica con esponente m = 0. Il punto B è appartenente a un'isocora ossia una politropica avente m tendente a infinito. Infine i punti O e M sono comuni alla curva e, rispettivamente, a un'isoterma (esponente della politropica m = 1) e un'isoentropica (ossia adiabatica reversibile con esponente m = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span>1,4 per l'aria). </p><p>L'equazione della politropica viene dunque differenziata ottenendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dp}{d\rho }}=m\cdot cost\cdot \rho ^{m-1}\rightarrow {\frac {dp}{d\rho }}=m\cdot cost\cdot {\frac {\rho ^{m}}{\rho }}\rightarrow {\frac {dp}{d\rho }}=m{\frac {p}{\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>t</mi> <mo>⋅</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>t</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>ρ</mi> </mfrac> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>ρ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dp}{d\rho }}=m\cdot cost\cdot \rho ^{m-1}\rightarrow {\frac {dp}{d\rho }}=m\cdot cost\cdot {\frac {\rho ^{m}}{\rho }}\rightarrow {\frac {dp}{d\rho }}=m{\frac {p}{\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10a8b3538a5d64c27beb69a6224cc2bb8740291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:57.981ex; height:5.843ex;" alt="{\displaystyle {\frac {dp}{d\rho }}=m\cdot cost\cdot \rho ^{m-1}\rightarrow {\frac {dp}{d\rho }}=m\cdot cost\cdot {\frac {\rho ^{m}}{\rho }}\rightarrow {\frac {dp}{d\rho }}=m{\frac {p}{\rho }}}"></span> </p><p>Se il fluido considerato è un <a href="/wiki/Gas_ideale" title="Gas ideale">gas ideale</a>, la relazione si può riscrivere come: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dp}{d\rho }}=mRT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dp}{d\rho }}=mRT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1fdeedcf2b6f71fa7b9c6c7ae3c1ed397bae0f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.793ex; height:5.843ex;" alt="{\displaystyle {\frac {dp}{d\rho }}=mRT}"></span> </p><p>Pertanto, sulla curva di Rayleigh, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\sqrt {\frac {dp}{d\rho }}}={\sqrt {mRT}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>ρ</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>m</mi> <mi>R</mi> <mi>T</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\sqrt {\frac {dp}{d\rho }}}={\sqrt {mRT}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e6b75f4c9e4d90a1cbec0a9cf876e5e3280cdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.158ex; height:7.509ex;" alt="{\displaystyle c={\sqrt {\frac {dp}{d\rho }}}={\sqrt {mRT}}}"></span>. </p><p>La relazione che permette di calcolare il numero di Mach è pertanto: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ma={\frac {c}{a_{S}}}={\frac {\sqrt {mRT}}{\sqrt {\gamma RT}}}={\sqrt {\frac {m}{\gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>m</mi> <mi>R</mi> <mi>T</mi> </msqrt> <msqrt> <mi>γ</mi> <mi>R</mi> <mi>T</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>m</mi> <mi>γ</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ma={\frac {c}{a_{S}}}={\frac {\sqrt {mRT}}{\sqrt {\gamma RT}}}={\sqrt {\frac {m}{\gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73e76f7c94d2f8d6d2913073d3575a7aa7d2ccfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.739ex; height:7.176ex;" alt="{\displaystyle Ma={\frac {c}{a_{S}}}={\frac {\sqrt {mRT}}{\sqrt {\gamma RT}}}={\sqrt {\frac {m}{\gamma }}}}"></span>. </p><p>Notiamo che: </p> <ul><li>Nel punto A, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ma=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ma=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17e52a64c44956f0c232baad8c6346e2c937f3c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.933ex; height:2.176ex;" alt="{\displaystyle Ma=0}"></span> in quanto m = 0;</li> <li>Nel punto B, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ma\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>a</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ma\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3010a0fc1532d703ede7fa05fb86bf7565d1ef24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.61ex; height:2.176ex;" alt="{\displaystyle Ma\rightarrow \infty }"></span> in quanto m tende a infinito;</li> <li>Nel punto O, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ma={\frac {1}{\sqrt {\gamma }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>γ</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ma={\frac {1}{\sqrt {\gamma }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1401a8bb5c3893df19eac5f3f65a53f93d0b8c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:10.805ex; height:6.176ex;" alt="{\displaystyle Ma={\frac {1}{\sqrt {\gamma }}}}"></span> in quanto m = 1;</li> <li>Nel punto M, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ma=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ma=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/092cb0ae9c2cac5a1f88ea586be2b0a99fc0f52d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.933ex; height:2.176ex;" alt="{\displaystyle Ma=1}"></span> perché m = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>;</li></ul> <p>I risultati ottenuti trovano un'importante applicazione negli <a href="/wiki/Scambiatore_di_calore" title="Scambiatore di calore">scambiatori di calore</a>, per i quali consideriamo velocità del fluido molto basse ossia con numero di Mach tendente a zero. Infatti, la trattazione finora vista porta all'ipotesi forte di considerare uno scambiatore come isobaro, proprio perché nel tratto di curva approssimato dall'isobara vale Ma = 0. </p> <div class="mw-heading mw-heading2"><h2 id="Proprietà_termodinamiche_nel_flusso_di_Rayleigh"><span id="Propriet.C3.A0_termodinamiche_nel_flusso_di_Rayleigh"></span>Proprietà termodinamiche nel flusso di Rayleigh</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flusso_di_Rayleigh&veaction=edit&section=3" title="Modifica la sezione Proprietà termodinamiche nel flusso di Rayleigh" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Flusso_di_Rayleigh&action=edit&section=3" title="Edit section's source code: Proprietà termodinamiche nel flusso di Rayleigh"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nella tabella seguente sono illustrati gli andamenti qualitativi delle principali proprietà lungo la curva di Rayleigh, a seconda che si stia riscaldando o raffreddando il sistema. </p> <table class="wikitable"> <caption> </caption><tbody><tr><td></td></tr></tbody></table> <table class="wikitable"> <caption> </caption> <tbody><tr> <th>Grandezza </th> <th>Riscaldamento subsonico </th> <th>Riscaldamento supersonico </th> <th>Raffreddamento subsonico </th> <th>Raffreddamento supersonico </th></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad87bd7009e2a5c52bd0fb5a9bda9d8c1c23a79b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\textstyle p}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span></b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span></b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ma}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ma}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb92c88283886ac47fae5b0cf7001bba4192f264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.672ex; height:2.176ex;" alt="{\displaystyle Ma}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span></b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4618f22b0f780805eb94bb407578d9bc9487947a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \downarrow }"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ff2ad177893308b831fbb51780b878bd696659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.676ex;" alt="{\displaystyle T^{0}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \downarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">↓</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \downarrow }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26fd164d7cc33ea1d92946165c57e478ca2277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\textstyle \downarrow }"></span> </td></tr></tbody></table> <ul><li><b>Pressione:</b> aumenta nel senso del riscaldamento (sia subsonico che supersonico) e diminuisce nel senso del raffreddamento. L'andamento è deducibile tracciando le isobare nel diagramma <i>h-s</i>;</li> <li><b>Densità:</b> per stabilire l'andamento di tale grandezza va considerata la relazione rappresentabile sul diagramma di Clapeyron: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+{\biggl (}{\frac {\dot {M}}{A}}{\biggr )}^{2}{\frac {1}{\rho }}={\frac {I}{A}}=cost}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>˙</mo> </mover> </mrow> <mi>A</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>I</mi> <mi>A</mi> </mfrac> </mrow> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p+{\biggl (}{\frac {\dot {M}}{A}}{\biggr )}^{2}{\frac {1}{\rho }}={\frac {I}{A}}=cost}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a05e6eb29489414e212442875bc5655339a58f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:26.732ex; height:6.509ex;" alt="{\displaystyle p+{\biggl (}{\frac {\dot {M}}{A}}{\biggr )}^{2}{\frac {1}{\rho }}={\frac {I}{A}}=cost}"></span>. Noto l'andamento della pressione, è possibile dedurre quello della densità;</li> <li><b>Velocità:</b> Può essere espressa come <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\frac {\dot {M}}{A}}{\frac {1}{\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>˙</mo> </mover> </mrow> <mi>A</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\frac {\dot {M}}{A}}{\frac {1}{\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/020abca8d18f5d4b2ea1b23616c414221d48e24f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.422ex; height:6.343ex;" alt="{\displaystyle c={\frac {\dot {M}}{A}}{\frac {1}{\rho }}}"></span>ed è quindi inversamente proporzionale alla densità e alla pressione;</li> <li><b>Numero di Mach:</b> è direttamente proporzionale alla velocità c per definizione;</li> <li><b>Temperatura:</b> varia in maniera analoga all'entalpia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (dh=c_{p}dT)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mi>h</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>d</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (dh=c_{p}dT)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16aa0113585ea40ff2fcc271e9ee4f7016b0d98b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.381ex; height:3.009ex;" alt="{\displaystyle (dh=c_{p}dT)}"></span>, ma nel riscaldamento e nel raffreddamento subsonici non ha un trend univoco: è prima crescente e poi decrescente (paradossale);</li> <li><b>Entropia:</b> per il secondo principio della termodinamica <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Bigl (}ds={\frac {\delta q}{T}}{\Bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl (}ds={\frac {\delta q}{T}}{\Bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48fe68d10fd0eefd1e55f9cf86eeeb656f8c2a97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.135ex; height:5.343ex;" alt="{\displaystyle {\Bigl (}ds={\frac {\delta q}{T}}{\Bigr )}}"></span>varia in maniera direttamente proporzionale con il calore, pertanto crescente nel riscaldamento e decrescente nel raffreddamento;</li> <li><b>Temperatura totale:</b> detta anche <a href="/wiki/Temperatura_di_ristagno" title="Temperatura di ristagno">temperatura di ristagno</a>, dal primo principio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta q=dh+d{\Bigl (}{\frac {c^{2}}{2}}{\Bigr )}=dh^{0}=c_{p}dT^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>q</mi> <mo>=</mo> <mi>d</mi> <mi>h</mi> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>d</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta q=dh+d{\Bigl (}{\frac {c^{2}}{2}}{\Bigr )}=dh^{0}=c_{p}dT^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42f39b92c76dd46595ac926c821457b170e7dad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.363ex; height:5.676ex;" alt="{\displaystyle \delta q=dh+d{\Bigl (}{\frac {c^{2}}{2}}{\Bigr )}=dh^{0}=c_{p}dT^{0}}"></span>si nota il suo andamento direttamente proporzionale alla quantità di calore fornita o ceduta.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Spiegazione_dell'andamento_non_univoco_della_temperatura"><span id="Spiegazione_dell.27andamento_non_univoco_della_temperatura"></span>Spiegazione dell'andamento non univoco della temperatura</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flusso_di_Rayleigh&veaction=edit&section=4" title="Modifica la sezione Spiegazione dell'andamento non univoco della temperatura" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Flusso_di_Rayleigh&action=edit&section=4" title="Edit section's source code: Spiegazione dell'andamento non univoco della temperatura"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>È stato visto nel paragrafo precedente che nel caso di riscaldamento o raffreddamento con Ma < 1 il trend della temperatura non è univoco. Ciò costituisce un paradosso, perché non è logicamente prevedibile una diminuzione della temperatura quando viene fornito del calore a un sistema o viceversa un aumento della temperatura se il calore viene ceduto verso l'esterno. Per capire meglio il fenomeno, è sufficiente osservare la curva di Rayleigh: nel tratto da O a N (vedasi figura 2) l'entalpia decresce e, con essa, la temperatura. </p><p>Il paradosso può essere spiegato tenendo conto che è più corretto parlare di aumento di temperatura totale in caso di calore fornito al sistema o di diminuzione di temperatura totale in caso di calore ceduto all'esterno. Considerando infatti il primo principio: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta q=dh+d{\Bigl (}{\frac {c^{2}}{2}}{\Bigr )}=dh+cdc=c_{p}dT+cdc=c_{p}dT^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>q</mi> <mo>=</mo> <mi>d</mi> <mi>h</mi> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>d</mi> <mi>h</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mi>c</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>d</mi> <mi>T</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mi>c</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta q=dh+d{\Bigl (}{\frac {c^{2}}{2}}{\Bigr )}=dh+cdc=c_{p}dT+cdc=c_{p}dT^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f1e56de0bd5efef7588c4542e7e8ec701e1d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.465ex; height:5.676ex;" alt="{\displaystyle \delta q=dh+d{\Bigl (}{\frac {c^{2}}{2}}{\Bigr )}=dh+cdc=c_{p}dT+cdc=c_{p}dT^{0}}"></span> </p><p>e immaginando di somministrare calore al sistema <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\delta q>0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>δ</mi> <mi>q</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\delta q>0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18b4d7559be98cc04b6e6b6bdc2b851daea0bc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.188ex; height:2.843ex;" alt="{\displaystyle (\delta q>0)}"></span> nel tratto da O a N, affinché si soddisfi necessariamente la condizione <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dT^{0}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dT^{0}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9836adba85b41cc3d5c24ce74f4ba98f8030960f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.251ex; height:2.676ex;" alt="{\displaystyle dT^{0}>0}"></span>, per poter compensare il fatto che <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dT<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>T</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dT<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c197ed4f28f3335773f76e895c9f807f14436fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.113ex; height:2.176ex;" alt="{\displaystyle dT<0}"></span>, il termine cinetico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cdc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>d</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cdc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e38fedf5807485a86a0e3924b1098af7b3b61eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.23ex; height:2.176ex;" alt="{\displaystyle cdc}"></span>sarà il termine maggiormente influente e crescerà con un tasso di variazione maggiore di quanto aumenti la temperatura totale. Dunque il termine statico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd04a4416fcdeca1fd2399cebc31ad284d9fa7e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.176ex;" alt="{\displaystyle dT}"></span> diminuisce ma il termine dinamico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cdc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>d</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cdc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e38fedf5807485a86a0e3924b1098af7b3b61eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.23ex; height:2.176ex;" alt="{\displaystyle cdc}"></span>riesce a compensare il "crollo" del termine statico aumentando più rapidamente e pertanto ogni effetto anomalo sulla temperatura di ristagno, che è direttamente collegata al calore secondo il primo principio, non verrà "avvertito". </p> <div class="mw-heading mw-heading2"><h2 id="Choking_nel_flusso_di_Rayleigh">Choking nel flusso di Rayleigh</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flusso_di_Rayleigh&veaction=edit&section=5" title="Modifica la sezione Choking nel flusso di Rayleigh" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Flusso_di_Rayleigh&action=edit&section=5" title="Edit section's source code: Choking nel flusso di Rayleigh"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il fenomeno del <i>choking</i> è essenzialmente definibile come la violazione delle condizioni stazionarie o di principi inviolabili come conservazione della portata a seguito della riduzione della pressione all'uscita di una tubazione o di un <a href="/wiki/Ugello_di_scarico" title="Ugello di scarico">ugello</a> oppure a seguito dell'aumento di un cosiddetto "<i><b>driving factor</b></i>" al di sopra di una certa soglia, come ad esempio il termine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {4fL}{D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>f</mi> <mi>L</mi> </mrow> <mi>D</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {4fL}{D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1f61fcadb0b126e67440d042c076b1e0349056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.86ex; height:5.343ex;" alt="{\displaystyle {\frac {4fL}{D}}}"></span>nel flusso di Fanno. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Nuovo_documento_2019-04-24_11.25.22.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Nuovo_documento_2019-04-24_11.25.22.jpg/296px-Nuovo_documento_2019-04-24_11.25.22.jpg" decoding="async" width="296" height="268" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Nuovo_documento_2019-04-24_11.25.22.jpg/444px-Nuovo_documento_2019-04-24_11.25.22.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Nuovo_documento_2019-04-24_11.25.22.jpg/592px-Nuovo_documento_2019-04-24_11.25.22.jpg 2x" data-file-width="2212" data-file-height="2000" /></a><figcaption><b>Figura 3</b>. Rappresentazione grafica della massima quantità di calore somministrabile per non avere choking.</figcaption></figure> <p>Si può osservare che esiste una quantità massima di calore somministrabile al sistema affinché vengano mantenute le condizioni stazionarie. </p><p>Graficamente, osservando la figura 3, la quantità di calore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{max}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{max}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94f21f2bd39918f43b76e720a807ed672af1acd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.522ex; height:2.009ex;" alt="{\displaystyle q_{max}}"></span>corrisponde all'area sottesa dalla curva nel tratto da P a N, cioè dalle condizioni iniziali di Ma = 0 fino alle condizioni critiche. Si può scrivere dunque: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{max}\cong {\widehat {P_{0}PONN_{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>P</mi> <mi>O</mi> <mi>N</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{max}\cong {\widehat {P_{0}PONN_{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e7ead6824cf944bbca5d7b9329980bc5625c235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.67ex; height:3.509ex;" alt="{\displaystyle q_{max}\cong {\widehat {P_{0}PONN_{0}}}}"></span> </p><p>Immaginando di aumentare la portata <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e17d7f361e94093d5346b9d04a5d4b8c7f4d22f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.676ex;" alt="{\displaystyle {\dot {M}}}"></span>ci si sposterebbe verso una curva di Rayleigh più interna, come è possibile notare dal grafico. Ciò determina una diminuzione della quantità massima di calore che il sistema può ricevere. </p><p>Se tale quantità di calore viene superata, viene a rompersi l'ipotesi di flusso stazionario unidimensionale, portando a una diminuzione della <a href="/wiki/Portata" title="Portata">portata</a> nel condotto<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>. </p><p>Può altresì essere effettuato un altro tipo di analisi, considerando che le condizioni finali sono sempre quelle critiche. Si può dire, cioè, che assegnata una <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{finale}^{0}>T_{iniziale}^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mi>i</mi> <mi>n</mi> <mi>a</mi> <mi>l</mi> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo>></mo> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>i</mi> <mi>z</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{finale}^{0}>T_{iniziale}^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0394976c297f67325364c2020d7a2f0168f948d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.446ex; height:3.509ex;" alt="{\displaystyle T_{finale}^{0}>T_{iniziale}^{0}}"></span>, esiste un <a href="/wiki/Numero_di_Mach_critico" title="Numero di Mach critico">numero di Mach</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ma_{1}<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ma_{1}<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a41e1b0bc21016465a03521c458c675682933e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.987ex; height:2.509ex;" alt="{\displaystyle Ma_{1}<1}"></span> critico massimo in caso di regime subsonico (oppure un <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ma_{1}>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ma_{1}>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff39cd83bfe9539eae8cd001ae0683d979a5005" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.987ex; height:2.509ex;" alt="{\displaystyle Ma_{1}>1}"></span> minimo in caso di regime supersonico) per cui si verificano le condizioni stazionarie con numero di Mach unitario all'uscita dal condotto. Spingendosi oltre tale valore critico del numero di Mach, la portata in ingresso teorica sarebbe oltre il valore massimo e pertanto non riuscirebbe più a confluire interamente nel condotto, disperdendosi in parte e provocando quindi una diminuzione della portata in uscita, manifestando il <i>choking</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flusso_di_Rayleigh&veaction=edit&section=6" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Flusso_di_Rayleigh&action=edit&section=6" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation pubblicazione" style="font-style:normal"> Alessandro Ferrari, <span style="font-style:italic;">Fondamenti di termofluidodinamica per le macchine</span>.</cite></span> </li> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.ob-ultrasound.net/rayleigh.html"><span style="font-style:italic;">Lord Rayleigh, John William Strutt</span></a>, su <span style="font-style:italic;">www.ob-ultrasound.net</span>. <small>URL consultato il 23 aprile 2019</small>.</cite></span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190424102808/https://nptel.ac.in/courses/112103021/13"><span style="font-style:italic;">NPTEL :: Mechanical Engineering - Gas Dynamics</span></a>, su <span style="font-style:italic;">nptel.ac.in</span>. <small>URL consultato il 24 aprile 2019</small> <small>(archiviato dall'<abbr title="https://nptel.ac.in/courses/112103021/13">url originale</abbr> il 24 aprile 2019)</small>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flusso_di_Rayleigh&veaction=edit&section=7" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Flusso_di_Rayleigh&action=edit&section=7" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A. Ferrari, “Fondamenti di termofluidodinamica per le macchine”, Città Studi, De Agostini, 2018. <a href="/wiki/Speciale:RicercaISBN/9788825174236" class="internal mw-magiclink-isbn">ISBN 978-8825174236</a></li></ul> <ul><li>E. Catania, "Complementi di Macchine", Levrotto & Bella, 1979.</li> <li>A Mittica, "Turbomacchine idrauliche operatrici", Appunti dai corsi seminariali di Vercelli, 1991</li> <li>A. Capetti, "Motori termici", UTET, 1967.</li> <li>G. Lozza, “Turbine a gas e cicli combinati”, Terza Edizione, Società Editrice Esculapio, 2016. <a href="/wiki/Speciale:RicercaISBN/9788874889341" class="internal mw-magiclink-isbn">ISBN 978-8874889341</a></li> <li>V. Dossena, G. Ferrari, P. Gaetani, G. Montenegro, A. Onorati, G. Persico, “Macchine a fluido”, Città Studi Edizioni, 2015.</li></ul> <ul><li>N. Nervegna, “Oleodinamica e Pneumatica”, Politeko, ed. 2003.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flusso_di_Rayleigh&veaction=edit&section=8" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Flusso_di_Rayleigh&action=edit&section=8" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Flusso_quasi-unidimensionale" title="Flusso quasi-unidimensionale">Flusso quasi-unidimensionale</a></li> <li><a href="/wiki/Flusso_di_Fanno" title="Flusso di Fanno">Flusso di Fanno</a></li> <li><a href="/wiki/Trasformazione_isoentropica" title="Trasformazione isoentropica">Trasformazione isoentropica</a></li> <li><a href="/wiki/Gasdinamica" title="Gasdinamica">Gasdinamica</a></li> <li><a href="/wiki/Flusso_compressibile" title="Flusso compressibile">Flusso compressibile</a></li> <li><a href="/w/index.php?title=Flusso_saturato&action=edit&redlink=1" class="new" title="Flusso saturato (la pagina non esiste)">Flusso saturato</a></li> <li><a href="/wiki/Entalpia" title="Entalpia">Entalpia</a></li> <li><a href="/wiki/Entropia" title="Entropia">Entropia</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flusso_di_Rayleigh&veaction=edit&section=9" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Flusso_di_Rayleigh&action=edit&section=9" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><span class="plainlinks" title="commons:Category:Rayleigh flow"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Rayleigh_flow?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file su <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Rayleigh_flow?uselang=it">Flusso di Rayleigh</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flusso_di_Rayleigh&veaction=edit&section=10" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Flusso_di_Rayleigh&action=edit&section=10" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080308105311/http://meweb.ecn.purdue.edu/~meapplet/java/comp_calculator/rayleigh_calculator.html"><span style="font-style:italic;">Purdue University Rayleigh flow calculator</span></a>, su <span style="font-style:italic;">meweb.ecn.purdue.edu</span>. <small>URL consultato il 28 aprile 2009</small> <small>(archiviato dall'<abbr title="https://meweb.ecn.purdue.edu/~meapplet/java/comp_calculator/rayleigh_calculator.html">url originale</abbr> l'8 marzo 2008)</small>.</cite></li> <li><cite class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://web.archive.org/web/20071129223024/http://www.engr.uky.edu/~jdjacob/me530/rayleigh.html"><span style="font-style:italic;">University of Kentucky Rayleigh flow Webcalculator</span></a>, su <span style="font-style:italic;">engr.uky.edu</span>. <small>URL consultato il 28 aprile 2009</small> <small>(archiviato dall'<abbr title="http://www.engr.uky.edu/~jdjacob/me530/rayleigh.html">url originale</abbr> il 29 novembre 2007)</small>.</cite></li></ul> <div class="noprint" style="width:100%; 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Flusso di Rayleigh - Wikipedia