View source for Thermodynamic and Mechanical Study/fr

<span id="1._Scope_and_method"></span>
== 1. Périmètre et méthode ==

Cette étude décrit le cycle récepteur complet de la machine Dada sous forme symbolique. Le modèle couple une cinématique imposée des deux pistons, des volumes de gaz bien mélangés, des échangeurs thermiques 0D, des clapets passifs et une fermeture hydraulique compressible générique.

L'origine du cycle est fixée au volume maximal du grand cylindre, avec :

<math>t=0,\qquad \theta(0)=0,\qquad V_L(0)=V_{L,\max}.</math>

Les quatre volumes de gaz sont :

* <code>S</code> : petit cylindre, directement associé à l'échangeur froid <code>C</code> ;
* <code>L</code> : grand cylindre, directement associé à l'échangeur chaud <code>H</code> ;
* <code>C</code> : échangeur thermique froid, qui prélève de la chaleur au réservoir froid ;
* <code>H</code> : échangeur thermique chaud, qui rejette de la chaleur vers le réservoir chaud.

Les clapets autorisent <code>H → S</code> pendant la Phase II et <code>C → L</code> pendant la Phase IV.

Le cycle comporte quatre phases :

* '''Phase I''' — clapets fermés, nominalement adiabatique : compression côté <code>L+H</code> ;
* '''Phase II''' — échange thermique, nominalement isotherme : transfert <code>L → H → S</code>, avec rejet de chaleur côté chaud ;
* '''Phase III''' — clapets fermés, nominalement adiabatique : détente côté <code>S+C</code> ;
* '''Phase IV''' — échange thermique, nominalement isotherme : transfert <code>S → C → L</code>, avec prélèvement de chaleur côté froid.

Les qualificatifs ''nominalement adiabatique'' et ''nominalement isotherme'' décrivent l'objectif du cycle. Ils ne constituent pas des contraintes thermodynamiques exactes : les températures sont calculées par les bilans, les échangeurs restent couplés à leurs réservoirs, et les zones de faible déplacement des pistons ne sont pas supposées parfaitement immobiles.

<span id="1.1_First-level_assumptions"></span>
=== 1.1 Hypothèses de premier niveau ===

Le modèle repose sur les hypothèses suivantes :

* gaz monophasique, parfait et calorifiquement parfait ;
* propriétés constantes <math display="inline">R</math>, <math display="inline">C_p</math>, <math display="inline">C_v</math>, <math display="inline">\gamma</math> avec <math display="inline">R=C_p-C_v</math> et <math display="inline">\gamma=C_p/C_v</math> ;
* chaque volume de gaz est uniforme et bien mélangé ;
* énergie cinétique et énergie potentielle du gaz négligées dans les bilans 0D ;
* volumes des échangeurs fixes ;
* cinématique des pistons imposée ;
* pas de frottement mécanique dans le modèle thermodynamique ;
* échanges thermiques représentés par une conductance globale <math display="inline">UA</math> ;
* clapets passifs commandés par la différence de pression.

<span id="2._Notation_and_conventions"></span>
== 2. Notations et conventions ==

<span id="2.1_Geometry_and_kinematics"></span>
=== 2.1 Géométrie et cinématique ===

Pour <math display="inline">i\in\{S,L\}</math> :

<math>V_{i,\min}>0,\qquad V_{i,\max}>V_{i,\min}.</math>

Le volume balayé est :

<math>V_{i,\mathrm{swept}}=V_{i,\max}-V_{i,\min}.</math>

Les volumes thermodynamiques imposés par la mécanique sont :

<math>V_S=V_S(t),\qquad V_L=V_L(t),</math>

avec leurs dérivées '''signées''' :

<math>\dot V_S=\frac{dV_S}{dt},\qquad \dot V_L=\frac{dV_L}{dt}.</math>

Si la manivelle tourne à vitesse constante <math display="inline">\omega</math> :

<math>\theta(t)=\omega t,\qquad \dot V_i=\omega\frac{dV_i}{d\theta}.</math>

Lorsque seule la valeur absolue de la vitesse volumique est utile :

<math>\nu_{V,i}=|\dot V_i|.</math>

Une '''zone quasi immobile''' désigne un intervalle où le déplacement ou <math display="inline">|\dot V|</math> reste faible par rapport aux phases de transfert. Cette zone correspond au « plateau » de l'optimisation cinématique, sans supposer <math display="inline">\dot V=0</math> exactement.

<span id="2.2_Kinematic_closure_fraction"></span>
=== 2.2 Fraction de fermeture cinématique ===

Pour décrire la fermeture normalisée d'un cylindre :

<math>\Lambda_i(t)=\frac{V_{i,\max}-V_i(t)}{V_{i,\max}-V_{i,\min}},\qquad i\in\{S,L\}.</math>

Ainsi <math display="inline">\Lambda_i=0</math> correspond au volume maximal et <math display="inline">\Lambda_i=1</math> au volume minimal. Les valeurs <math display="inline">\Lambda_L^*</math> et <math display="inline">\Lambda_S^*</math> sont des cibles cinématiques nominales aux transitions. La valeur réellement atteinte lors d'un événement de clapet est :

<math>\Lambda_{\mathrm{real}}=\Lambda(t_{\mathrm{event}}).</math>

Les transitions physiques restent déterminées par les pressions ; <math display="inline">\Lambda^*</math> n'est donc pas une condition d'ouverture imposée.

<span id="2.3_Thermodynamic_variables"></span>
=== 2.3 Variables thermodynamiques ===

Pour chaque volume <math display="inline">i\in\{S,L,C,H\}</math> :

<math>m_i,\qquad U_i,\qquad T_i,\qquad P_i,\qquad V_i.</math>

Le vecteur d'état complet est choisi sous la forme :

<math>\mathbf X=(m_S,U_S,m_L,U_L,m_C,U_C,m_H,U_H).</math>

Les températures et pressions en sont déduites :

<math>T_i=\frac{U_i}{m_iC_v},\qquad P_i=\frac{m_iRT_i}{V_i}.</math>

Les volumes <math display="inline">V_S(t)</math> et <math display="inline">V_L(t)</math> sont imposés par la cinématique et ne sont pas des variables d'état thermodynamiques indépendantes. Les volumes <math display="inline">V_C</math> et <math display="inline">V_H</math> sont constants.

<span id="2.4_Energy_sign_convention"></span>
=== 2.4 Convention de signe énergétique ===

La chaleur est positive lorsqu'elle est '''reçue par le gaz'''. Le travail est positif lorsque le '''gaz fournit du travail''' :

<math>\dot W=P\dot V.</math>

En fonctionnement frigorifique recherché :

<math>Q_C>0,\qquad Q_H<0,\qquad W_{\mathrm{cycle}}<0.</math>

<span id="3._Thermal_closure_and_validity_domain"></span>
== 3. Fermeture thermique et domaine de validité ==

<span id="3.1_Exchange_with_the_thermal_reservoirs"></span>
=== 3.1 Échange avec les réservoirs thermiques ===

Pour un échangeur <math display="inline">HX\in\{C,H\}</math> :

<math>\dot Q_{HX}=(UA)_{HX}(T_{HX,\mathrm{res}}-T_{HX}).</math>

<math display="inline">T_{HX,\mathrm{res}}</math> est la température du réservoir thermique externe, imposée et constante dans le modèle de base. <math display="inline">T_{HX}</math> est la température moyenne 0D du gaz dans l'échangeur. <math display="inline">(UA)_{HX}</math> représente la conductance thermique globale, qui peut regrouper convection, conduction de paroi et résistances de contact.

Côté froid :

<math>\dot Q_C=(UA)_C(T_{C,\mathrm{res}}-T_C).</math>

En fonctionnement frigorifique, <math display="inline">T_C<T_{C,\mathrm{res}}</math> donne <math display="inline">\dot Q_C>0</math>.

Côté chaud :

<math>\dot Q_H=(UA)_H(T_{H,\mathrm{res}}-T_H).</math>

En fonctionnement frigorifique, <math display="inline">T_H>T_{H,\mathrm{res}}</math> donne <math display="inline">\dot Q_H<0</math>.

Une phase nominalement isotherme ne signifie donc pas <math display="inline">T_{HX}=T_{HX,\mathrm{res}}</math> : un écart de température fini est nécessaire pour transférer une puissance thermique finie lorsque <math display="inline">UA</math> est fini.

Un indicateur de qualité d'isothermie peut être défini sur une phase donnée par :

<math>\varepsilon_T=\frac{T_{\max}-T_{\min}}{T_{\mathrm{ref}}}.</math>

<span id="3.2_Thermophysical_validity_domain"></span>
=== 3.2 Domaine de validité thermophysique ===

Le modèle de base suppose :

<math>PV=mRT,\qquad Z=1,</math>

<math>C_p=\mathrm{const},\qquad C_v=\mathrm{const},\qquad \gamma=\mathrm{const}.</math>

La validité doit être contrôlée a posteriori sur tout le cycle, notamment par :

<math>|Z-1|\ll1,</math>

et par une faible variation des propriétés thermophysiques, par exemple :

<math>\varepsilon_{C_p}=\frac{C_{p,\max}-C_{p,\min}}{C_{p,\mathrm{ref}}}\ll1.</math>

Le fluide doit rester monophasique gazeux et suffisamment éloigné de toute condensation ou transition de phase dans l'ensemble du domaine <math display="inline">(P,T)</math> parcouru.

Si ces critères deviennent insuffisants, une extension pourra employer <math display="inline">Z(P,T)</math>, <math display="inline">C_p(T)</math>, <math display="inline">C_v(T)</math> ou une équation d'état réelle sans modifier l'architecture générale des bilans de masse et d'énergie.

<span id="4._Reduced_formulation_of_a_quasi-pressure-equalized_pair"></span>
== 4. Formulation réduite d'une paire quasi égalisée en pression ==

Lorsqu'un cylindre et son échangeur sont reliés par une liaison interne très peu résistive, on peut utiliser l'approximation :

<math>P_{\mathrm{cyl}}\approx P_{HX}=P_{\mathrm{pair}}.</math>

Elle est acceptable si :

<math>\varepsilon_P=\frac{|\Delta P_{\mathrm{int}}|}{P_{\mathrm{pair}}}\ll1,\qquad \Delta P_{\mathrm{int}}=P_{\mathrm{cyl}}-P_{HX}.</math>

Un faible nombre de Mach interne constitue un contrôle complémentaire :

<math>Ma_{\mathrm{int}}=\frac{|u_{\mathrm{int}}|}{a}\ll1,</math>

mais il ne suffit pas, à lui seul, à garantir l'égalisation des pressions.

<span id="4.1_Closed-pair_case"></span>
=== 4.1 Cas fermé de la paire ===

Pour une paire fermée, avec <math display="inline">V_{HX}</math> fixe et <math display="inline">V=V_{\mathrm{cyl}}+V_{HX}</math> :

<math>
\frac{dP}{dt}=
\frac{(\gamma-1)(UA)_{HX}(T_{HX,\mathrm{res}}-T_{HX})-\gamma P\dot V_{\mathrm{cyl}}}
{V_{\mathrm{cyl}}+V_{HX}}
.</math>

Lorsque <math display="inline">(UA)_{HX}=0</math> :

<math>P(V_{\mathrm{cyl}}+V_{HX})^\gamma=\mathrm{const}.</math>

<span id="4.2_Internal_redistribution_flow_rate"></span>
=== 4.2 Débit interne de redistribution ===

On définit positif le débit interne cylindre → échangeur. Posons :

<math>N=V_{HX}\dot P+(\gamma-1)(UA)_{HX}(T_{HX}-T_{HX,\mathrm{res}}).</math>

Le débit transporte l'enthalpie de l'état amont :

<math>
\dot m_{\mathrm{int}}=
\begin{cases}
\dfrac{N}{\gamma R T_{\mathrm{cyl}}}, & N\ge0 \quad (\mathrm{cyl}\to HX),\\[6pt]
\dfrac{N}{\gamma R T_{HX}}, & N<0 \quad (HX\to\mathrm{cyl}).
\end{cases}
</math>

La température de l'échangeur évolue selon :

<math>
\dot T_{HX}=\frac{T_{HX}}{P}\dot P-
\frac{RT_{HX}^2}{PV_{HX}}\dot m_{\mathrm{int}}
.</math>

<span id="4.3_Open_receiving_pair"></span>
=== 4.3 Paire réceptrice ouverte ===

Lors d'une phase active, le débit externe entre physiquement dans le '''cylindre récepteur''', et non directement dans son échangeur associé. Si <math display="inline">\dot m_{\mathrm{ext}}>0</math> entre dans le cylindre à la température <math display="inline">T_{\mathrm{ext}}</math> :

<math>
\dot P=
\frac{\gamma RT_{\mathrm{ext}}\dot m_{\mathrm{ext}}
+(\gamma-1)(UA)_{HX}(T_{HX,\mathrm{res}}-T_{HX})
-\gamma P\dot V_{\mathrm{cyl}}}
{V_{\mathrm{cyl}}+V_{HX}}
.</math>

Les masses vérifient :

<math>\dot m_{\mathrm{cyl}}=\dot m_{\mathrm{ext}}-\dot m_{\mathrm{int}},\qquad
\dot m_{HX}=\dot m_{\mathrm{int}},</math>

et donc :

<math>\dot m_{\mathrm{pair}}=\dot m_{\mathrm{ext}}.</math>

Le cas fermé est obtenu immédiatement avec <math display="inline">\dot m_{\mathrm{ext}}=0</math>.

<span id="5._Hydraulic_closure"></span>
== 5. Fermeture hydraulique ==

<span id="5.1_Generic_formulation"></span>
=== 5.1 Formulation générique ===

Toute liaison hydraulique est décrite par une loi générique :

<math>\dot m=\Phi(P_u,P_d,T_u,\mathcal G,\mathcal F)</math>

<math display="inline">u</math> et <math display="inline">d</math> désignent respectivement l'amont (''upstream'') et l'aval (''downstream''). L'enthalpie transportée est celle de l'amont :

<math>\dot H_{\mathrm{mass}}=\dot m C_pT_u.</math>

<div lang="en" dir="ltr" class="mw-content-ltr">
For a bidirectional connection, the upstream state is determined by the actual direction of the pressure gradient. For a check valve, reverse flow is prohibited.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== 5.2 First-level closure using a compressible orifice ===
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
A first approximation consists in using an effective hydraulic area:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>(C_dA)_{\mathrm{eff}},</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
which represents the overall ease of gas flow through the actual connection.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
With:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>r=\frac{P_d}{P_u},\qquad
r_{\mathrm{crit}}=\left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)},</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
the unchoked flow rate, for <math display="inline">r>r_{\mathrm{crit}}</math>, is:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\dot m=(C_dA)_{\mathrm{eff}}P_u
\sqrt{\frac{2\gamma}{RT_u(\gamma-1)}
\left(r^{2/\gamma}-r^{(\gamma+1)/\gamma}\right)}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For <math display="inline">r\le r_{\mathrm{crit}}</math>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\dot m=(C_dA)_{\mathrm{eff}}P_u
\sqrt{\frac{\gamma}{RT_u}}
\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
This closure is not essential to the model: it may later be replaced by a law that more faithfully represents the pressure losses of a real heat exchanger, pipe, or check valve.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
== 6. Complete thermodynamic cycle ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== 6.1 Phase I — check valves closed, nominally adiabatic: compression on the L side ===
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Both check valves are closed. The <code>L+H</code> and <code>S+C</code> pairs are closed. The <code>L+H</code> side is nominally compressed; the motion of the small piston remains that provided by the actual kinematics.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For <code>L+H</code>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot P_{LH}=
\frac{(\gamma-1)(UA)_H(T_{H,\mathrm{res}}-T_H)-\gamma P_{LH}\dot V_L}
{V_L+V_H}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For <code>S+C</code>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot P_{SC}=
\frac{(\gamma-1)(UA)_C(T_{C,\mathrm{res}}-T_C)-\gamma P_{SC}\dot V_S}
{V_S+V_C}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The internal redistribution and temperature equations of §4 apply to both pairs.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The transition to Phase II occurs when the <code>H → S</code> check valve satisfies its opening condition.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== 6.2 Phase II — heat exchange, nominally isothermal: L → H → S ===
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The gas leaves <code>L</code>, passes through the hot heat exchanger <code>H</code>, where it rejects heat, crosses the <code>H → S</code> check valve, and then enters the receiving cylinder <code>S</code>. The <code>S+C</code> pair remains quasi-pressure-equalized if the criterion <math display="inline">\varepsilon_P\ll1</math> is satisfied.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
==== 6.2.1 Donor cylinder L ====
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The fundamental balance is:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\frac{d(m_LC_vT_L)}{dt}
=-P_L\dot V_L-\dot m_{L,\mathrm{out}}C_pT_L
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The kinematics are designed to maintain an outflow from the donor cylinder. In this case, the analytical solution is:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\frac{T_L}{T_{L,\mathrm{ref}}}=
\left[\frac{m_L}{m_{L,\mathrm{ref}}}\frac{V_{L,\mathrm{ref}}}{V_L}\right]^{\gamma-1}
</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
and:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
P_L=P_{L,\mathrm{ref}}
\left[\frac{m_L}{m_{L,\mathrm{ref}}}\frac{V_{L,\mathrm{ref}}}{V_L}\right]^\gamma
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
One also obtains:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{T_L}{T_{L,\mathrm{ref}}}=
\left(\frac{P_L}{P_{L,\mathrm{ref}}}\right)^{(\gamma-1)/\gamma},</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
and the specific entropy of the remaining gas satisfies <math display="inline">ds=0</math> under these assumptions. If reverse flow occurs despite the kinematic design, the analytical solution is no longer applicable and the complete open-system balance in <math display="inline">(m_L,U_L)</math> must be used.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
==== 6.2.2 Hot heat exchanger H ====
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Mass conservation:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m_H=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Fundamental energy balance:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\frac{d(m_HC_vT_H)}{dt}
=\dot m_{\mathrm{in}}C_pT_L
-\dot m_{\mathrm{out}}C_pT_H
+(UA)_H(T_{H,\mathrm{res}}-T_H)
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
In expanded form:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\dot T_H=
\frac{\dot m_{\mathrm{in}}(C_pT_L-C_vT_H)
-\dot m_{\mathrm{out}}RT_H
+(UA)_H(T_{H,\mathrm{res}}-T_H)}
{C_vm_H}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The flow rates are determined by the hydraulic laws:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,H}}(P_L,P_H,T_L,\ldots),</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,H}}(P_H,P_{SC},T_H,\ldots).</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
==== 6.2.3 Receiving pair S+C ====
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The flow from <code>H</code> enters '''S'''. The pair pressure satisfies:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\dot P_{SC}=
\frac{\gamma R\dot m_{\mathrm{out}}T_H
+(\gamma-1)(UA)_C(T_{C,\mathrm{res}}-T_C)
-\gamma P_{SC}\dot V_S}
{V_S+V_C}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The internal redistribution equations of §4 remain unchanged: the external flow acts on heat exchanger <code>C</code> indirectly through the evolution of the pair pressure.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The transition to Phase III is the closing event of the <code>H → S</code> check valve.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== 6.3 Phase III — check valves closed, nominally adiabatic: expansion on the S side ===
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Both check valves are closed. The <code>S+C</code> pair expands nominally; the <code>L+H</code> pair also remains closed. Neither piston is assumed to be strictly stationary.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For <code>S+C</code>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot P_{SC}=
\frac{(\gamma-1)(UA)_C(T_{C,\mathrm{res}}-T_C)-\gamma P_{SC}\dot V_S}
{V_S+V_C}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For <code>L+H</code>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot P_{LH}=
\frac{(\gamma-1)(UA)_H(T_{H,\mathrm{res}}-T_H)-\gamma P_{LH}\dot V_L}
{V_L+V_H}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The transition to Phase IV occurs when the <code>C → L</code> check valve satisfies its opening condition.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== 6.4 Phase IV — heat exchange, nominally isothermal: S → C → L ===
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The gas leaves <code>S</code>, passes through the cold heat exchanger <code>C</code>, where it receives heat from the cold reservoir, crosses the <code>C → L</code> check valve, and then enters the receiving cylinder <code>L</code>. The <code>L+H</code> pair remains quasi-pressure-equalized if the criterion <math display="inline">\varepsilon_P\ll1</math> is satisfied.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
==== 6.4.1 Donor cylinder S ====
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The fundamental balance is:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\frac{d(m_SC_vT_S)}{dt}
=-P_S\dot V_S-\dot m_{S,\mathrm{out}}C_pT_S
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For outflow guaranteed by the kinematic design:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\frac{T_S}{T_{S,\mathrm{ref}}}=
\left[\frac{m_S}{m_{S,\mathrm{ref}}}\frac{V_{S,\mathrm{ref}}}{V_S}\right]^{\gamma-1}
</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
and:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
P_S=P_{S,\mathrm{ref}}
\left[\frac{m_S}{m_{S,\mathrm{ref}}}\frac{V_{S,\mathrm{ref}}}{V_S}\right]^\gamma
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Reverse flow requires returning to the complete open-system balance in <math display="inline">(m_S,U_S)</math>.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
==== 6.4.2 Cold heat exchanger C ====
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m_C=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The fundamental energy balance is:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\frac{d(m_CC_vT_C)}{dt}
=\dot m_{\mathrm{in}}C_pT_S
-\dot m_{\mathrm{out}}C_pT_C
+(UA)_C(T_{C,\mathrm{res}}-T_C)
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
In expanded form:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\dot T_C=
\frac{\dot m_{\mathrm{in}}(C_pT_S-C_vT_C)
-\dot m_{\mathrm{out}}RT_C
+(UA)_C(T_{C,\mathrm{res}}-T_C)}
{C_vm_C}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The flow rates are determined by:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,C}}(P_S,P_C,T_S,\ldots),</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,C}}(P_C,P_{LH},T_C,\ldots).</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
==== 6.4.3 Receiving pair L+H ====
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The flow from <code>C</code> enters '''L'''. The pair pressure satisfies:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\dot P_{LH}=
\frac{\gamma R\dot m_{\mathrm{out}}T_C
+(\gamma-1)(UA)_H(T_{H,\mathrm{res}}-T_H)
-\gamma P_{LH}\dot V_L}
{V_L+V_H}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Closing the <code>C → L</code> check valve returns the system to Phase I. Geometric closure of the pistons alone is not sufficient to guarantee thermodynamic closure of the cycle.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
== 7. Physical transitions of the check valves ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For a check valve oriented from upstream <math display="inline">u</math> to downstream <math display="inline">d</math>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>P_u-P_d\ge\Delta P_{\mathrm{open}}
\quad\Rightarrow\quad \text{opening},</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>P_u-P_d\le\Delta P_{\mathrm{close}}
\quad\Rightarrow\quad \text{closing},</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
with hysteresis, if present:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\Delta P_{\mathrm{close}}\le\Delta P_{\mathrm{open}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
An opening or closing event changes the hydraulic topology and therefore the active equations; it does not cause any instantaneous jump in the thermodynamic state. For each volume:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>m_i^+=m_i^-,\qquad U_i^+=U_i^-,\qquad V_i^+=V_i^-.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For an ideal gas:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>T_i^+=T_i^-,\qquad P_i^+=P_i^-.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
There is therefore no instantaneous pressure equalization when a check valve opens.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The values <math display="inline">\Lambda^*</math> serve as kinematic design targets; the values actually observed at the transitions are <math display="inline">\Lambda_{\mathrm{real}}=\Lambda(t_{\mathrm{event}})</math>.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
== 8. Work, heat, and coefficient of performance ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The instantaneous work delivered by the gas on the two pistons is calculated during all phases:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\dot W=P_S^\star\dot V_S+P_L^\star\dot V_L
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math display="inline">P_S^\star</math> and <math display="inline">P_L^\star</math> denote the thermodynamic pressure effectively applied to the gas in each cylinder according to the phase topology: pair pressure when the cylinder belongs to a quasi-pressure-equalized pair, and its own pressure when it is a hydraulically isolated donor.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The net work over the cycle is:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
W_{\mathrm{cycle}}=
\int_0^\tau
\left(P_S^\star\dot V_S+P_L^\star\dot V_L\right)dt
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The exchanged heats are:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>Q_C=\int_0^\tau (UA)_C(T_{C,\mathrm{res}}-T_C)\,dt,</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>Q_H=\int_0^\tau (UA)_H(T_{H,\mathrm{res}}-T_H)\,dt.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
In periodic steady operation:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\Delta U_{\mathrm{cycle}}=0,</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
and the first law gives:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>Q_C+Q_H=W_{\mathrm{cycle}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The refrigeration COP is:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>COP_c=\frac{Q_C}{-W_{\mathrm{cycle}}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The heat-pump COP is:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>COP_h=\frac{-Q_H}{-W_{\mathrm{cycle}}}=COP_c+1.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The signs <math display="inline">Q_C>0</math>, <math display="inline">Q_H<0</math>, and <math display="inline">W_{\mathrm{cycle}}<0</math> provide checks of the intended refrigeration regime.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The thermodynamic force exerted by the gas on a piston face may be written <math display="inline">F_{\mathrm{gas}}=PS</math>. Net mechanical force, inertia, and friction belong to the subsequent mechanical sizing stage.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
== 9. Gas charge and periodic regime ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The total amount of enclosed gas is a physical parameter:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>M_{\mathrm{tot}}=m_S+m_L+m_C+m_H=\mathrm{const}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
It may be imposed directly or defined experimentally by a charging pressure and temperature. The chosen reference configuration is that at <math display="inline">t=0</math>, with the large cylinder at maximum volume. If all volumes communicate and are in uniform equilibrium during charging:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
M_{\mathrm{tot}}=
\frac{P_{\mathrm{charge}}
\left[V_S(0)+V_{L,\max}+V_C+V_H\right]}
{RT_{\mathrm{charge}}}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math display="inline">P_{\mathrm{charge}}</math> and <math display="inline">T_{\mathrm{charge}}</math> define the amount of gas charged; they are not conditions that the periodic cycle must recover.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The established periodic regime is a solution of the system such that, between two successive passages through the maximum volume of the large cylinder with the same kinematic direction:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\mathbf X(t+\tau)=\mathbf X(t).</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Geometric periodicity alone:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>V_i(t+\tau)=V_i(t)</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
is not sufficient to guarantee thermodynamic periodicity.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The numerical state used to initialize a calculation may be approximate; it must not be confused with a physical parameter of the machine. The future solver may search for the periodic fixed point by successive cycles, a shooting method, or a Newton method.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
== 10. Parameters, design data, and results ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== 10.1 Prescribed data ===
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
* working fluid and reference properties <math display="inline">R</math>, <math display="inline">C_p</math>, <math display="inline">C_v</math>, <math display="inline">\gamma</math>;
* reservoir temperatures <math display="inline">T_{C,\mathrm{res}}</math>, <math display="inline">T_{H,\mathrm{res}}</math>;
* total charge <math display="inline">M_{\mathrm{tot}}</math>, or equivalently <math display="inline">(P_{\mathrm{charge}},T_{\mathrm{charge}})</math> in the charging configuration defined in §9;
* kinematics <math display="inline">V_S(t)</math>, <math display="inline">V_L(t)</math>, and, where relevant, <math display="inline">\omega</math>.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== 10.2 Design parameters ===
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
* <math display="inline">V_{S,\min}</math>, <math display="inline">V_{S,\max}</math>, <math display="inline">V_{L,\min}</math>, <math display="inline">V_{L,\max}</math>;
* <math display="inline">V_C</math>, <math display="inline">V_H</math>;
* <math display="inline">(UA)_C</math>, <math display="inline">(UA)_H</math>;
* hydraulic parameters of the heat exchangers, pipes, and check valves, represented at first level by <math display="inline">(C_dA)_{\mathrm{eff}}</math>;
* thresholds <math display="inline">\Delta P_{\mathrm{open}}</math>, <math display="inline">\Delta P_{\mathrm{close}}</math>;
* kinematic targets <math display="inline">\Lambda_L^*</math>, <math display="inline">\Lambda_S^*</math>.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== 10.3 Calculated variables and results ===
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
* <math display="inline">m_i</math>, <math display="inline">U_i</math>, <math display="inline">T_i</math>, <math display="inline">P_i</math>;
* internal and external mass flow rates;
* <math display="inline">\dot Q_C</math>, <math display="inline">\dot Q_H</math>, <math display="inline">Q_C</math>, <math display="inline">Q_H</math>;
* <math display="inline">W_{\mathrm{cycle}}</math>, <math display="inline">COP_c</math>, <math display="inline">COP_h</math>;
* pressure, temperature, and flow-rate extrema;
* actual check-valve events and <math display="inline">\Lambda_{\mathrm{real}}</math>;
* isothermal quality <math display="inline">\varepsilon_T</math>;
* validity criteria <math display="inline">\varepsilon_P</math>, <math display="inline">Ma</math>, <math display="inline">Z</math>, and property variations.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
== 11. Global conservation checks ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== 11.1 Mass conservation ===
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The solver must satisfy:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{dM_{\mathrm{tot}}}{dt}=0.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
A useful numerical residual is:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\varepsilon_M(t)=M_{\mathrm{tot}}(t)-M_{\mathrm{tot}}(0).</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== 11.2 Global energy conservation ===
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Whatever the phase, the internal mass and enthalpy fluxes must cancel when the balances of all volumes are summed. The global balance must reduce to:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\frac{dU_{\mathrm{tot}}}{dt}
=\dot Q_C+\dot Q_H
-P_S^\star\dot V_S
-P_L^\star\dot V_L
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
A cumulative energy residual may be defined by:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\varepsilon_E(t)=
U_{\mathrm{tot}}(t)-U_{\mathrm{tot}}(0)
-Q_C(0,t)-Q_H(0,t)+W(0,t)
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The solver must keep <math display="inline">\varepsilon_M</math> and <math display="inline">\varepsilon_E</math> close to zero to the expected numerical accuracy.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
----
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
== Appendix A — Symbolic derivations and validated checks =
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== A.1 Pressure equation for a closed pair ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For a cylinder + heat-exchanger pair at quasi-uniform pressure:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>U=\frac{P(V_{\mathrm{cyl}}+V_{HX})}{\gamma-1}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The first law gives:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{dU}{dt}=\dot Q-P\dot V_{\mathrm{cyl}},</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
with:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot Q=(UA)_{HX}(T_{HX,\mathrm{res}}-T_{HX}).</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Differentiating <math display="inline">U</math>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{1}{\gamma-1}
\left[(V_{\mathrm{cyl}}+V_{HX})\dot P+P\dot V_{\mathrm{cyl}}\right]
=\dot Q-P\dot V_{\mathrm{cyl}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Hence:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\dot P=
\frac{(\gamma-1)\dot Q-\gamma P\dot V_{\mathrm{cyl}}}
{V_{\mathrm{cyl}}+V_{HX}}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
If <math display="inline">\dot Q=0</math>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{\dot P}{P}=-\gamma\frac{\dot V}{V},</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
then:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>PV^\gamma=\mathrm{const}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== A.2 Internal flow rate of the pair ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For the heat exchanger alone, at fixed volume:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>U_{HX}=\frac{PV_{HX}}{\gamma-1}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Therefore:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{V_{HX}}{\gamma-1}\dot P
=(UA)_{HX}(T_{HX,\mathrm{res}}-T_{HX})
+\dot m_{\mathrm{int}}C_pT_{\mathrm{up}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Using <math display="inline">C_p=\gamma R/(\gamma-1)</math>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>V_{HX}\dot P+(\gamma-1)(UA)_{HX}(T_{HX}-T_{HX,\mathrm{res}})
=\gamma R T_{\mathrm{up}}\dot m_{\mathrm{int}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
This recovers the definition of the numerator <math display="inline">N</math> and the selection of the upstream temperature according to the sign of the flow rate.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== A.3 Evolution of the heat-exchanger temperature within a pair ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For a fixed volume:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>m_{HX}=\frac{PV_{HX}}{RT_{HX}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Differentiating:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{\dot m_{HX}}{m_{HX}}
=\frac{\dot P}{P}-\frac{\dot T_{HX}}{T_{HX}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
With <math display="inline">\dot m_{HX}=\dot m_{\mathrm{int}}</math>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\dot T_{HX}=\frac{T_{HX}}P\dot P-
\frac{RT_{HX}^2}{PV_{HX}}\dot m_{\mathrm{int}}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
This relation follows solely from mass conservation and the equation of state; it is valid for both flow directions.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== A.4 Open receiving pair ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For a pair receiving <math display="inline">\dot m_{\mathrm{ext}}</math> into its cylinder:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{dU}{dt}=\dot m_{\mathrm{ext}}C_pT_{\mathrm{ext}}
+\dot Q-P\dot V_{\mathrm{cyl}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
With <math display="inline">U=P(V_{\mathrm{cyl}}+V_{HX})/(\gamma-1)</math>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\dot P=
\frac{\gamma RT_{\mathrm{ext}}\dot m_{\mathrm{ext}}
+(\gamma-1)\dot Q
-\gamma P\dot V_{\mathrm{cyl}}}
{V_{\mathrm{cyl}}+V_{HX}}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
The local mass balance:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m_{\mathrm{cyl}}=\dot m_{\mathrm{ext}}-\dot m_{\mathrm{int}},</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m_{HX}=\dot m_{\mathrm{int}},</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
immediately gives:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m_{\mathrm{pair}}=\dot m_{\mathrm{ext}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== A.5 Analytical solution for the adiabatic donor cylinder ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
For an adiabatic, well-mixed cylinder with outflow only:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{d(mC_vT)}{dt}=-P\dot V-\dot m_{\mathrm{out}}C_pT,</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
and:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m=-\dot m_{\mathrm{out}}.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Expanding:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>C_vm\dot T+C_vT\dot m=-P\dot V+C_pT\dot m,</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
thus:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>C_vm\dot T=-P\dot V+RT\dot m.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
With <math display="inline">P=mRT/V</math> and <math display="inline">R/C_v=\gamma-1</math>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{dT}{T}=(\gamma-1)
\left(\frac{dm}{m}-\frac{dV}{V}\right).</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
After integration:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\frac{T}{T_0}=
\left[\frac{m}{m_0}\frac{V_0}{V}\right]^{\gamma-1}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Then, using <math display="inline">PV=mRT</math>:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
P=P_0\left[\frac{m}{m_0}\frac{V_0}{V}\right]^\gamma
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
and:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>
\frac{T}{T_0}=
\left(\frac{P}{P_0}\right)^{(\gamma-1)/\gamma}
.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Under these assumptions, the specific entropy of the remaining gas is constant: <math display="inline">ds=0</math>. The total entropy of the gas contained in the cylinder is not constant because its mass varies.
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
=== A.6 Active heat exchanger: expanded balance ==
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Fundamental balance:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\frac{d(mC_vT)}{dt}
=\dot m_{\mathrm{in}}C_pT_{\mathrm{in}}
-\dot m_{\mathrm{out}}C_pT
+\dot Q.</math>
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
Expanding the left-hand side and using:
</div>

<div lang="en" dir="ltr" class="mw-content-ltr">
<math>\dot m=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}},</math>
</div>

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one obtains:
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<math>
\dot T=
\frac{\dot m_{\mathrm{in}}(C_pT_{\mathrm{in}}-C_vT)
-\dot m_{\mathrm{out}}RT+\dot Q}
{C_vm}
.</math>
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Limiting checks:
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* with no flow, the equation recovers the thermal relaxation of a closed volume;
* with equal steady inlet/outlet flow rates, it recovers <math display="inline">mC_v\dot T=\dot mC_p(T_{\mathrm{in}}-T)+\dot Q</math>;
* if <math display="inline">T_{\mathrm{in}}=T</math> and <math display="inline">\dot m_{\mathrm{in}}=\dot m_{\mathrm{out}}</math>, the net contribution of the flow to <math display="inline">\dot T</math> vanishes.
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=== A.7 Global mass test during an active phase ==
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For Phase II:
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<math>\dot m_L=-\dot m_{\mathrm{in}},</math>
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<math>\dot m_H=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}},</math>
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<math>\dot m_{SC}=\dot m_{\mathrm{out}}.</math>
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Summing:
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<math>\dot M_{\mathrm{tot}}=0.</math>
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Phase IV gives exactly the same result by symmetry.
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=== A.8 Global energy test during an active phase ==
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For Phase II:
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<math>\dot U_L=-P_L\dot V_L-\dot m_{\mathrm{in}}h_L,</math>
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<math>\dot U_H=\dot m_{\mathrm{in}}h_L-\dot m_{\mathrm{out}}h_H+\dot Q_H,</math>
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<math>\dot U_{SC}=\dot m_{\mathrm{out}}h_H+\dot Q_C-P_{SC}\dot V_S.</math>
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The internal enthalpy fluxes cancel exactly:
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<math>-\dot m_{\mathrm{in}}h_L+\dot m_{\mathrm{in}}h_L=0,</math>
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<math>-\dot m_{\mathrm{out}}h_H+\dot m_{\mathrm{out}}h_H=0.</math>
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What remains is:
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<math>
\dot U_{\mathrm{tot}}
=\dot Q_C+\dot Q_H
-P_L\dot V_L-P_{SC}\dot V_S
.</math>
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Phase IV provides the symmetric relation. In the closed phases, the same structure follows directly from summing the balances of the two pairs. Hence, for any phase:
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<math>
\dot U_{\mathrm{tot}}
=\dot Q_C+\dot Q_H
-P_S^\star\dot V_S-P_L^\star\dot V_L
.</math>
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Integrated over a periodic cycle, this relation gives:
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<math>Q_C+Q_H=W_{\mathrm{cycle}}.</math>
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=== A.9 Continuity at transitions ==
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At the instant of a check-valve event, no finite mass or energy can be transferred in zero time. The conserved variables and geometry are therefore continuous:
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<math>m_i^+=m_i^-,\qquad U_i^+=U_i^-,\qquad V_i^+=V_i^-.</math>
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For an ideal gas:
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<math>T_i=\frac{U_i}{m_iC_v},\qquad P_i=\frac{m_iRT_i}{V_i},</math>
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which implies:
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<math>T_i^+=T_i^-,\qquad P_i^+=P_i^-.</math>
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The event only creates a change in hydraulic topology and in the active system of equations.
</div>