Thermodynamic and Mechanical Study/fr: Difference between revisions

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1. Périmètre et méthode

Cette étude décrit le cycle thermodynamique complet de la machine Dada sous forme symbolique, en mode réfrigération ainsi qu’en mode moteur. 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 :

$t = 0, θ (0) = 0, V_{L} (0) = V_{L, max} .$

Les quatre volumes de gaz sont :

$S$ : petit cylindre ;
$L$ : grand cylindre ;
$H_{i}$ : échangeur thermique "Heat-in", situé sur le chemin hydraulique $S \to L$ et qui apporte de la chaleur au gaz ;
$H_{o}$ : échangeur thermique "Heat-out", situé sur le chemin hydraulique $L \to S$ et qui enlève de la chaleur au gaz.

The two hydraulic branches have fixed circulation orientations:

$S \to H_{i} \to L,$

$L \to H_{o} \to S .$

Each branch contains one passive check valve. The check valve may be installed on either side of its heat exchanger. The two admissible arrangements for the heat-in branch are:

$S \to {C V}_{i} \to H_{i} \to L,$

or:

$S \to H_{i} \to {C V}_{i} \to L .$

Likewise, the heat-out branch may be arranged as:

$L \to {C V}_{o} \to H_{o} \to S,$

or:

$L \to H_{o} \to {C V}_{o} \to S .$

In all cases the check valve enforces the same overall circulation direction of its branch. Its position relative to the heat exchanger is a design parameter because, when the valve is closed, it determines which cylinder remains hydraulically connected to the exchanger volume.

The physical function of each heat exchanger is independent of the operating mode. What changes between refrigeration and motor operation is the external thermal reservoir connected to each exchanger.

The cycle is described by four hydraulic/thermodynamic phases: compression, transfer through $H_{o}$ from $L$ to $S$ , expansion, and transfer through $H_{i}$ from $S$ to $L$ . In refrigeration operation, starting from the reference origin used in this study, these phases are traversed in that order. Motor operation reverses the crank kinematics. The resulting thermodynamic chronology must be recalculated with the unchanged check-valve orientations and the reversed reservoir assignment; it is detailed in §6.6.

The phase boundaries describe the kinematics and the dominant thermodynamic regime. They are not check-valve events. Check-valve opening and closing are determined independently by the local pressure difference across each valve; mass transfer and heat transfer may therefore continue during compression or expansion, and a valve event may occur inside a kinematic phase.

The two transfer phases are intended to be quasi-isobaric: one cylinder empties while the other fills, and the pressure variation is intended to remain small compared with the pressure change during compression and expansion. Compression and expansion may involve simultaneous motion of both pistons in the same volumetric direction, so both cylinders may contribute to the pressure-changing phase.

For comparison with an ideal thermodynamic cycle, compression and expansion may be idealized as adiabatic transformations; if they are also reversible, they are isentropic. These are reference transformations only. The real machine does not impose zero heat transfer, zero mass transfer, or closed check valves during compression and expansion, and no construction capable of enforcing perfectly adiabatic phases is assumed here.

1.1 Hypothèses de premier niveau

Le modèle repose sur les hypothèses suivantes :

gaz monophasique, parfait et calorifiquement parfait ;
propriétés constantes $R$ , $C_{p}$ , $C_{v}$ , $γ$ avec $R = C_{p} - C_{v}$ et $γ = C_{p} / C_{v}$ ;
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 $U A$ ;
clapets passifs commandés par la différence de pression.

2. Notations et conventions

2.1 Géométrie et cinématique

Pour $k \in {S, L}$ :

$V_{k, min} > 0, V_{k, max} > V_{k, min} .$

Le volume balayé est :

$V_{k, s w e p t} = V_{k, max} - V_{k, min} .$

Les volumes thermodynamiques imposés par la mécanique sont :

$V_{S} = V_{S} (t), V_{L} = V_{L} (t),$

avec leurs dérivées signées :

${\dot{V}}_{S} = \frac{d V_{S}}{d t}, {\dot{V}}_{L} = \frac{d V_{L}}{d t} .$

La vitesse angulaire de la manivelle est signée :

$θ (t) = ω t, {\dot{V}}_{k} = ω \frac{d V_{k}}{d θ}, k \in {S, L} .$

Le sens récepteur en réfrigération est défini comme positif :

$ω > 0.$

Le fonctionnement moteur utilise la direction de manivelle opposée :

$ω < 0.$

L’origine géométrique est identique dans les deux modes :

$t = 0, θ (0) = 0, V_{L} (0) = V_{L, max} .$

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

$ν_{V, k} = | {\dot{V}}_{k} | .$

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

2.2 Fraction de fermeture cinématique

Pour décrire la fermeture normalisée d'un cylindre :

$Λ_{k} (t) = \frac{V_{k, max} - V_{k} (t)}{V_{k, max} - V_{k, min}}, k \in {S, L} .$

Ainsi $Λ_{k} = 0$ correspond au volume maximal et $Λ_{k} = 1$ au volume minimal. Les valeurs $Λ_{L}^{*}$ et $Λ_{S}^{*}$ sont des cibles cinématiques nominales aux transitions. La valeur réellement atteinte lors d'un événement de clapet est :

$Λ_{r e a l} = Λ (t_{e v e n t}) .$

Les transitions physiques restent déterminées par les pressions ; $Λ^{*}$ n'est donc pas une condition d'ouverture imposée.

2.3 Variables thermodynamiques

Pour chaque volume $j \in {S, L, C, H}$ :

$m_{j}, U_{j}, T_{j}, P_{j}, V_{j} .$

Les indices $i$ et $o$ dénotent respectivement le gaz contenu dans $H_{i}$ et $H_{o}$ .

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

$𝐗 = (m_{S}, U_{S}, m_{L}, U_{L}, m_{i}, U_{i}, m_{o}, U_{o}) .$

Les températures et pressions en sont déduites :

$T_{j} = \frac{U_{j}}{m_{j} C_{v}}, P_{j} = \frac{m_{j} R T_{j}}{V_{j}} .$

Les volumes $V_{S} (t)$ et $V_{L} (t)$ sont imposés par la cinématique et ne sont pas des variables d'état thermodynamiques indépendantes. Les volumes $V_{i}$ et $V_{o}$ sont constants.

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 :

$\dot{W} = P \dot{V} .$

Par définition des deux échangeurs thermiques physiques :

$Q_{i} > 0, Q_{o} < 0$

dans les conditions de fonctionnement prévues, tant pour le système de réfrigération que pour le moteur.

Le travail net du cycle distingue les deux modes :

$W_{c y c l e} < 0$

pour le mode récepteur en réfrigération, alors que :

$W_{c y c l e} > 0$

pour le mode moteur.

3. Fermeture thermique et domaine de validité

3.1 Échange avec les réservoirs thermiques

Pour l’échangeur "heat-in" :

${\dot{Q}}_{i} = (U A)_{i} (T_{i, r e s} - T_{i}) .$

Dans les conditions de fonctionnement prévues :

$T_{i} < T_{i, r e s} \Rightarrow {\dot{Q}}_{i} > 0.$

Pour l’échangeur "heat-out" :

${\dot{Q}}_{o} = (U A)_{o} (T_{o, r e s} - T_{o}) .$

Dans les conditions de fonctionnement prévues :

$T_{o} > T_{o, r e s} \Rightarrow {\dot{Q}}_{o} < 0.$

Les températures des réservoirs dépendent du mode de fonctionnement

En mode réfrigération :

$T_{i, r e s} = T_{c o l d}, T_{o, r e s} = T_{h o t} .$

En mode moteur :

$T_{i, r e s} = T_{h o t}, T_{o, r e s} = T_{c o l d} .$

Ainsi les équations de transfert thermique sont identiques dans les deux modes.

3.2 Domaine de validité thermophysique

Le modèle de base suppose :

$P V = m R T, Z = 1,$

$C_{p} = c o n s t, C_{v} = c o n s t, γ = c o n s t .$

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

$| Z - 1 | ≪ 1,$

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

$ε_{C_{p}} = \frac{C_{p, max} - C_{p, min}}{C_{p, r e f}} ≪ 1.$

Le fluide doit rester monophasique gazeux et suffisamment éloigné de toute condensation ou transition de phase dans l'ensemble du domaine $(P, T)$ parcouru.

Si ces critères deviennent insuffisants, une extension pourra employer $Z (P, T)$ , $C_{p} (T)$ , $C_{v} (T)$ ou une équation d'état réelle sans modifier l'architecture générale des bilans de masse et d'énergie.

4. Reduced formulations for quasi-pressure-equalized connected volumes

The complete model treats the four gas volumes independently. In some operating conditions, however, a set of volumes connected through sufficiently low hydraulic resistance may remain close to a common pressure. Such a set can then be treated by a reduced analytical formulation.

Let $𝒞$ denote any connected set of gas volumes for which:

$P_{j} \approx P_{𝒞}, j \in 𝒞 .$

The composition of $𝒞$ is determined by the actual hydraulic connectivity and by the position and state of the check valves.

A useful pressure-equalization criterion is:

$ε_{P, 𝒞} = max_{(a, b) \in 𝒞} \frac{| P_{a} - P_{b} |}{P_{𝒞}} ≪ 1.$

A low internal Mach number provides an additional check:

$M a_{i n t} = \frac{| u_{i n t} |}{a} ≪ 1,$

but is not sufficient by itself to guarantee pressure equalization.

4.1 Pressure equation for a connected set

Define the total volume:

$V_{𝒞} = \sum_{j \in 𝒞} V_{j} .$

Only cylinder volumes vary, so:

${\dot{V}}_{𝒞} = \sum_{k \in 𝒞 \cap {S, L}} {\dot{V}}_{k} .$

For a calorically perfect ideal gas at common pressure:

$U_{𝒞} = \sum_{j \in 𝒞} m_{j} C_{v} T_{j} = \frac{P_{𝒞} V_{𝒞}}{γ - 1} .$

Let the total heat received by the gas in the set be:

${\dot{Q}}_{𝒞} = \sum_{j \in 𝒞} {\dot{Q}}_{j} .$

Mass crossing the boundary of the set transports the enthalpy of its upstream state. Define the net external enthalpy flow into the set as:

${\dot{H}}_{𝒞}^{e x t} = \sum_{i n} \dot{m} C_{p} T_{u} - \sum_{o u t} \dot{m} C_{p} T_{u} .$

The first law for the complete connected set is then:

$\frac{d U_{𝒞}}{d t} = {\dot{Q}}_{𝒞} + {\dot{H}}_{𝒞}^{e x t} - P_{𝒞} {\dot{V}}_{𝒞} .$

Therefore:

${\dot{P}}_{𝒞} = \frac{(γ - 1) ({\dot{Q}}_{𝒞} + {\dot{H}}_{𝒞}^{e x t}) - γ P_{𝒞} {\dot{V}}_{𝒞}}{V_{𝒞}} .$

The total mass of the set satisfies:

${\dot{M}}_{𝒞} = \sum_{i n} \dot{m} - \sum_{o u t} \dot{m} .$

Internal mass and enthalpy transfers between members of $𝒞$ cancel from these global balances.

4.2 Closed connected set

If no mass crosses the boundary of $𝒞$ :

${\dot{H}}_{𝒞}^{e x t} = 0, {\dot{M}}_{𝒞} = 0.$

The pressure equation becomes:

${\dot{P}}_{𝒞} = \frac{(γ - 1) {\dot{Q}}_{𝒞} - γ P_{𝒞} {\dot{V}}_{𝒞}}{V_{𝒞}} .$

If the set is also adiabatic:

${\dot{Q}}_{𝒞} = 0,$

then:

$P_{𝒞} V_{𝒞}^{γ} = c o n s t .$

This is a limiting analytical case. A compression or expansion phase of the complete machine does not require the corresponding connected set to be closed or adiabatic.

4.3 Fixed-volume heat exchanger within a pressure-equalized set

For a heat exchanger $H_{j}$ of fixed volume $V_{j}$ belonging to $𝒞$ :

$m_{j} = \frac{P_{𝒞} V_{j}}{R T_{j}} .$

Differentiation gives:

$\frac{{\dot{m}}_{j}}{m_{j}} = \frac{{\dot{P}}_{𝒞}}{P_{𝒞}} - \frac{{\dot{T}}_{j}}{T_{j}},$

hence:

${\dot{T}}_{j} = \frac{T_{j}}{P_{𝒞}} {\dot{P}}_{𝒞} - \frac{R T_{j}^{2}}{P_{𝒞} V_{j}} {\dot{m}}_{j} .$

The exchanger mass rate ${\dot{m}}_{j}$ is the algebraic sum of the actual flows through all links connected to it. This relation is therefore independent of whether the check valve lies upstream or downstream of the exchanger.

Its energy balance may equivalently be written:

$\frac{V_{j}}{γ - 1} {\dot{P}}_{𝒞} = {\dot{Q}}_{j} + \sum_{i n} \dot{m} C_{p} T_{u} - \sum_{o u t} \dot{m} C_{p} T_{j} .$

These equations determine the local mass redistribution and temperature evolution once the hydraulic flow rates are known.

4.4 Two-volume cylinder–exchanger special case

For the particular case of one cylinder and one heat exchanger connected at quasi-uniform pressure, with no other flow entering or leaving the exchanger directly, define the internal mass flow as positive from cylinder to heat exchanger.

Let:

$N = V_{H X} \dot{P} + (γ - 1) (U A)_{H X} (T_{H X} - T_{H X, r e s}) .$

The internal flow rate is:

${\dot{m}}_{i n t} = {\begin{matrix} \frac{N}{γ R T_{c y l}}, & N \geq 0 (c y l \to H X), \\ \frac{N}{γ R T_{H X}}, & N < 0 (H X \to c y l) . \end{matrix}$

The heat-exchanger temperature then satisfies:

${\dot{T}}_{H X} = \frac{T_{H X}}{P} \dot{P} - \frac{R T_{H X}^{2}}{P V_{H X}} {\dot{m}}_{i n t} .$

This special reduction must not be used when an additional external flow enters or leaves the heat exchanger directly; in that case the general balances of §4.1 and §4.3 apply.

4.5 Single external-flow special cases

If a quasi-pressure-equalized connected set receives a single external flow ${\dot{m}}_{e x t} > 0$ at upstream temperature $T_{e x t}$ :

${\dot{P}}_{𝒞} = \frac{γ R T_{e x t} {\dot{m}}_{e x t} + (γ - 1) {\dot{Q}}_{𝒞} - γ P_{𝒞} {\dot{V}}_{𝒞}}{V_{𝒞}},$

with:

${\dot{M}}_{𝒞} = {\dot{m}}_{e x t} .$

If instead the set delivers a single external outflow ${\dot{m}}_{e x t} > 0$ from a boundary volume at temperature $T_{o u t}$ :

${\dot{P}}_{𝒞} = \frac{- γ R T_{o u t} {\dot{m}}_{e x t} + (γ - 1) {\dot{Q}}_{𝒞} - γ P_{𝒞} {\dot{V}}_{𝒞}}{V_{𝒞}},$

with:

${\dot{M}}_{𝒞} = - {\dot{m}}_{e x t} .$

The location at which the external flow crosses the boundary of the connected set affects the local masses and temperatures, but not the summed pressure equation once the set $𝒞$ , the boundary enthalpy flow, and its total heat and volume rates are specified.

5. Fermeture hydraulique

5.1 Formulation générique

Toute liaison hydraulique est décrite par une loi générique :

$\dot{m} = Φ (P_{u}, P_{d}, T_{u}, 𝒢, ℱ)$

$u$ et $d$ désignent respectivement l'amont (upstream) et l'aval (downstream). L'enthalpie transportée est celle de l'amont :

${\dot{H}}_{m a s s} = \dot{m} C_{p} T_{u} .$

Pour une liaison bidirectionnelle, l'état amont est déterminé par le sens réel du gradient de pression. Pour un clapet, le débit inverse est interdit.

5.2 Fermeture de premier niveau par orifice compressible

Une première approximation consiste à employer une aire hydraulique effective :

$(C_{d} A)_{e f f},$

qui représente la facilité globale de passage du gaz dans la liaison réelle.

Avec :

$r = \frac{P_{d}}{P_{u}}, r_{c r i t} = {(\frac{2}{γ + 1})}^{γ / (γ - 1)},$

le débit non étranglé, pour $r > r_{c r i t}$ , est :

$\dot{m} = (C_{d} A)_{e f f} P_{u} \sqrt{\frac{2 γ}{R T_{u} (γ - 1)} (r^{2 / γ} - r^{(γ + 1) / γ})} .$

Pour $r \leq r_{c r i t}$ :

$\dot{m} = (C_{d} A)_{e f f} P_{u} \sqrt{\frac{γ}{R T_{u}}} {(\frac{2}{γ + 1})}^{\frac{γ + 1}{2 (γ - 1)}} .$

Cette fermeture n'est pas l'essence du modèle : elle pourra être remplacée par une loi plus fidèle aux pertes de charge d'un échangeur réel, d'une conduite ou d'un clapet.

6. Complete thermodynamic cycle

The cycle is described by four successive kinematic and thermodynamic regimes: compression, transfer through $H_{o}$ , expansion, and transfer through $H_{i}$ . These phases describe the dominant evolution of the machine; they are not defined by the state of the check valves. Valve opening and closing remain determined independently by the instantaneous pressure differences and may occur within a phase rather than exactly at a phase boundary.

The numbering below follows the refrigeration direction $ω > 0$ . Motor operation uses the same physical machine with reversed crank direction and is described in §6.6.

6.1 Thermodynamic rationale of the four phases

6.1.1 Compression and expansion

Compression and expansion are primarily pressure-changing phases. In the intended kinematics, both cylinder volumes may decrease simultaneously during compression and increase simultaneously during expansion.

The instantaneous work delivered by the gas is:

$\dot{W} = P_{S} {\dot{V}}_{S} + P_{L} {\dot{V}}_{L} .$

When the two cylinder pressures are of the same order, both piston contributions therefore add during a simultaneous expansion and both contribute to the work required during a simultaneous compression. This allows the swept volumes of both cylinders to participate in the pressure-changing parts of the cycle.

In an ideal reversible reference cycle, compression and expansion may be considered adiabatic and reversible. This is not imposed on the real machine. The heat exchangers remain thermally coupled to the gas, mass redistribution may continue, and neither check valve is required to be closed during the whole compression or expansion phase.

6.1.2 Exchange phases

During an exchange phase, gas is transferred from one cylinder to the other through one of the heat exchangers. Hydraulic resistance requires a finite pressure difference to produce a finite mass flow. This pressure difference is intrinsically irreversible.

For the hydraulic-loss contribution idealized locally as an adiabatic, isenthalpic throttling process of a calorically perfect ideal gas:

$h_{u} = h_{d} \Rightarrow T_{u} = T_{d},$

and therefore:

$Δ s_{h y d} = R \ln (\frac{P_{u}}{P_{d}}) > 0 for P_{u} > P_{d} .$

The reversible limit is consequently:

$P_{u} - P_{d} \to 0.$

The exchange phases therefore tend ideally toward quasi-pressure-equalized operation. A finite real machine retains a finite pressure difference because a finite flow must cross the hydraulic resistances.

Pressure equalization between communicating volumes at a given instant does not, by itself, imply that their common pressure remains constant throughout the exchange. Let $𝒞$ denote a closed set of communicating gas volumes that are approximately at a common pressure $P$ . For an ideal gas:

$M_{𝒞} = \frac{P}{R} \sum_{j \in 𝒞} \frac{V_{j}}{T_{j}},$

and therefore:

$P = \frac{M_{𝒞} R}{\sum_{j \in 𝒞} V_{j} / T_{j}} .$

At constant mass, an exactly isobaric evolution requires:

$\frac{d}{d t} (\sum_{j \in 𝒞} \frac{V_{j}}{T_{j}}) = 0.$

Thus the piston motions and the temperature evolution must compensate each other. Equal cylinder-volume changes are neither required nor generally expected.

The same condition can be expressed through the energy balance. For a quasi-pressure-equalized set:

$U_{𝒞} = \frac{P V_{𝒞}}{γ - 1}, V_{𝒞} = \sum_{j \in 𝒞} V_{j} .$

Its first-law balance gives:

$V_{𝒞} \dot{P} = (γ - 1) {\dot{Q}}_{𝒞} - γ P {\dot{V}}_{𝒞} .$

An approximately isobaric exchange therefore satisfies:

$γ P {\dot{V}}_{𝒞} \approx (γ - 1) {\dot{Q}}_{𝒞} .$

The volume evolution imposed by the pistons can consequently compensate the thermal expansion or contraction produced by heat transfer, allowing substantial mass transfer while the common pressure remains nearly constant.

A useful first-order cylinder-sizing relation follows from the same condition. Over a sufficiently small part of an isobaric exchange, if the donor and receiver temperatures may be treated as locally constant and the temperature-storage terms of the fixed exchanger volumes are secondary, then:

$\frac{d V_{r}}{T_{r}} \approx - \frac{d V_{d}}{T_{d}},$

hence:

$\frac{d V_{r}}{- d V_{d}} \approx \frac{T_{r}}{T_{d}} .$

The hotter side therefore requires a larger volume change for the same transferred gas mass at the same pressure. When the exchange uses comparable fractions of the available cylinder strokes and the gas temperatures remain close to characteristic working temperatures $T_{h}$ and $T_{c}$ , this gives the first-order geometric scaling:

Failed to parse (syntax error): {\displaystyle </div> <div lang="en" dir="ltr" class="mw-content-ltr"> \frac{V_{\mathrm{swept},h}} {V_{\mathrm{swept},c}} \sim \frac{T_h}{T_c} . }

Temperatures must be expressed in kelvin. This relation is a sizing guide, not an exact design constraint: exchanger hold-up, clearance volumes, temperature evolution during the exchange, finite pressure losses, and the compression and expansion phases can all shift the optimum. When these effects are significant, the complete condition involving $\sum V_{j} / T_{j}$ must be used instead.

These relations explain why a low-loss exchange naturally tends toward both small pressure differences along the hydraulic path and, with suitable piston kinematics, a nearly constant pressure throughout the exchange. Exact isobaricity is not imposed as a thermodynamic constraint.

6.1.3 Balance equations valid throughout the cycle

The four control volumes remain $S$ , $H_{i}$ , $L$ , and $H_{o}$ . Define the signed mass flow rates:

${\dot{m}}_{S i} : S \to H_{i}, {\dot{m}}_{i L} : H_{i} \to L,$

${\dot{m}}_{L o} : L \to H_{o}, {\dot{m}}_{o S} : H_{o} \to S .$

The positive directions correspond to the permanent circulation directions of the two branches:

$S \to H_{i} \to L, L \to H_{o} \to S .$

Each branch contains one passive check valve. Its position relative to the heat exchanger is a design choice:

$S \to {C V}_{i} \to H_{i} \to L or S \to H_{i} \to {C V}_{i} \to L,$

$L \to {C V}_{o} \to H_{o} \to S or L \to H_{o} \to {C V}_{o} \to S .$

The check valve constrains the link on which it is installed; the other link may be bidirectional according to its hydraulic law.

Let ${\dot{H}}_{a b}$ denote the signed enthalpy transport from volume $a$ toward volume $b$ . For a calorically perfect gas:

${\dot{H}}_{a b} = {\begin{matrix} {\dot{m}}_{a b} C_{p} T_{a}, & {\dot{m}}_{a b} \geq 0, \\ {\dot{m}}_{a b} C_{p} T_{b}, & {\dot{m}}_{a b} < 0. \end{matrix}$

The mass balances are then:

${\dot{m}}_{S} = {\dot{m}}_{o S} - {\dot{m}}_{S i},$

${\dot{m}}_{i} = {\dot{m}}_{S i} - {\dot{m}}_{i L},$

${\dot{m}}_{L} = {\dot{m}}_{i L} - {\dot{m}}_{L o},$

${\dot{m}}_{o} = {\dot{m}}_{L o} - {\dot{m}}_{o S} .$

The corresponding energy balances are:

${\dot{U}}_{S} = {\dot{H}}_{o S} - {\dot{H}}_{S i} - P_{S} {\dot{V}}_{S},$

${\dot{U}}_{i} = {\dot{H}}_{S i} - {\dot{H}}_{i L} + {\dot{Q}}_{i},$

${\dot{U}}_{L} = {\dot{H}}_{i L} - {\dot{H}}_{L o} - P_{L} {\dot{V}}_{L},$

${\dot{U}}_{o} = {\dot{H}}_{L o} - {\dot{H}}_{o S} + {\dot{Q}}_{o} .$

These equations are valid during all four phases. The phase determines the prescribed piston motion and the dominant thermodynamic process; the hydraulic laws and pressure differences determine the actual flow rates and check-valve states.

6.2 Phase I — compression

During compression, the two cylinder volumes may decrease simultaneously:

${\dot{V}}_{S} < 0, {\dot{V}}_{L} < 0$

over the principal part of the phase.

The pressure rises from the lower exchange-pressure region toward the higher one. Both pistons may contribute to the compression work.

No closed-pair topology is imposed. A check valve may remain open during part of the compression, and gas may continue to move through the hydraulic network. In particular, a cylinder approaching its minimum volume may transfer its remaining gas toward the other cylinder. Heat transfer through $H_{i}$ and $H_{o}$ also remains active.

The actual evolution is therefore calculated from the complete balances of §6.1.3. Adiabatic compression is only the reversible reference limit described in §6.1.1.

The end of the compression phase is defined by the prescribed kinematic law, not by a check-valve event.

6.3 Phase II — exchange through Ho: L → Ho → S

The dominant circulation is:

$L \to H_{o} \to S .$

The gas leaves the large-cylinder side, passes through the heat-out branch, and reaches the small-cylinder side. The passive check valve ${C V}_{o}$ may be located either before or after $H_{o}$ ; in both cases it enforces the same net branch direction.

During the exchange, $L$ acts predominantly as donor and $S$ as receiver. Their volume changes need not have equal magnitudes. In the quasi-isobaric limit their first-order ratio follows the temperature relation derived in §6.1.2.

The heat-out exchanger removes heat from the gas:

${\dot{Q}}_{o} < 0$

in the intended operating regime.

The ideal exchange tends toward small pressure differences along the active path and an approximately constant pressure over the phase. The finite real pressure differences required to drive the flow are determined by the hydraulic closure of §5.

The state of the other check valve and any secondary redistribution flow are determined by the instantaneous pressures; they are not prescribed by the phase definition.

6.4 Phase III — expansion

During expansion, the two cylinder volumes may increase simultaneously:

${\dot{V}}_{S} > 0, {\dot{V}}_{L} > 0$

over the principal part of the phase.

The pressure decreases from the higher exchange-pressure region toward the lower one. Both pistons may then contribute simultaneously to the work delivered by the gas.

As during compression, no zero-flow or closed-valve condition is imposed. Mass redistribution may continue and both heat exchangers remain thermally active. The complete balances of §6.1.3 therefore remain applicable.

Adiabatic expansion is the reversible reference limit, not a required operating condition of the real machine.

The end of the expansion phase is determined by the prescribed kinematic law independently of the check-valve events.

6.5 Phase IV — exchange through Hi: S → Hi → L

The dominant circulation is:

$S \to H_{i} \to L .$

The gas leaves the small-cylinder side, passes through the heat-in branch, and reaches the large-cylinder side. The passive check valve ${C V}_{i}$ may be located either before or after $H_{i}$ ; both arrangements impose the same net branch direction.

During the exchange, $S$ acts predominantly as donor and $L$ as receiver. Their required volume changes depend on the temperatures of the gas on the two sides according to the relations of §6.1.2.

The heat-in exchanger supplies heat to the gas:

${\dot{Q}}_{i} > 0$

in the intended operating regime.

As in Phase II, the ideal exchange tends toward quasi-pressure-equalized and approximately isobaric operation, while the real mass flow requires finite hydraulic pressure differences.

After this phase, the prescribed kinematics return to the compression region and the four-phase sequence repeats. Thermodynamic closure nevertheless requires the complete state vector, and not only the piston geometry, to be periodic.

6.6 Motor operation

No second set of mass, energy, heat-transfer, or hydraulic equations is required for motor operation.

The physical branch directions remain:

$S \to H_{i} \to L, L \to H_{o} \to S,$

and the positions and orientations of the two passive check valves remain unchanged.

Motor operation is obtained by reversing the crank direction:

$ω < 0,$

and exchanging the external reservoirs connected to the two heat exchangers:

$T_{i, r e s} = T_{h o t}, T_{o, r e s} = T_{c o l d} .$

The geometric cycle is therefore traversed in the opposite direction. Pressure histories, mass flow rates, check-valve events, and the periodic thermodynamic state must all be recalculated; the refrigeration valve chronology is not assumed simply to carry over.

In the intended refrigeration operation, the $L \to H_{o} \to S$ transfer occurs on the high-pressure side of the cycle, whereas the $S \to H_{i} \to L$ transfer occurs on the low-pressure side.

In the intended motor operation, reversal of the crank direction reverses these pressure roles: $L \to H_{o} \to S$ becomes the low-pressure exchange, whereas $S \to H_{i} \to L$ becomes the high-pressure exchange.

The physical thermal functions of the exchangers do not change: $H_{i}$ always transfers heat into the gas and $H_{o}$ always transfers heat out of the gas.

The heat-in exchanger absorbs heat from the hot reservoir:

$Q_{i} > 0,$

the heat-out exchanger rejects heat to the cold reservoir:

$Q_{o} < 0,$

and motor operation is obtained when:

$W_{c y c l e} > 0.$

7. Transitions physiques des clapets

Pour un clapet orienté de l'amont $u$ vers l'aval $d$ :

$P_{u} - P_{d} \geq Δ P_{o p e n} \Rightarrow opening,$

$P_{u} - P_{d} \leq Δ P_{c l o s e} \Rightarrow closing,$

avec, en présence d'hystérésis :

$Δ P_{c l o s e} \leq Δ P_{o p e n} .$

Une ouverture ou une fermeture modifie la topologie hydraulique et donc les équations actives ; elle ne provoque aucun saut instantané de l'état thermodynamique. Pour chaque volume :

$m_{j}^{+} = m_{j}^{-}, U_{j}^{+} = U_{j}^{-}, V_{j}^{+} = V_{j}^{-} .$

Pour le gaz parfait :

$T_{j}^{+} = T_{j}^{-}, P_{j}^{+} = P_{j}^{-} .$

Il n'y a donc pas d'égalisation instantanée des pressions à l'ouverture d'un clapet.

Les valeurs $Λ^{*}$ servent de cibles de conception cinématique ; les valeurs réellement observées lors des transitions sont $Λ_{r e a l} = Λ (t_{e v e n t})$ .

Les deux clapets passifs ont une orientation fixe :

$H_{o} \to S, H_{i} \to L .$

Leurs conditions d’ouverture et de fermeture sont identiques dans les deux modes de fonctionnement. Seule l’historique des pressions change en raison de la cinématique inversée de la manivelle en fonctionnement moteur.

8. Travail, chaleur et performance

Le travail instantané du gaz sur les deux pistons est calculé pendant toutes les phases :

$\dot{W} = P_{S}^{⋆} {\dot{V}}_{S} + P_{L}^{⋆} {\dot{V}}_{L} .$

Le travail net du cycle est :

$W_{c y c l e} = \int_{0}^{τ} (P_{S}^{⋆} {\dot{V}}_{S} + P_{L}^{⋆} {\dot{V}}_{L}) d t .$

When the two cylinder pressures are comparable during a pressure-changing phase,

$\dot{W} \approx P ({\dot{V}}_{S} + {\dot{V}}_{L}) .$

If both cylinders contract during compression, or both expand during expansion, their work contributions therefore add instead of partly cancelling. This allows a larger fraction of the available swept volume to participate in compression and expansion for a comparable pressure history. It does not by itself prove a higher thermal efficiency, because the heat transfers and the resulting pressure history change at the same time.

Les chaleurs échangées sont :

$Q_{i} = \int_{0}^{τ} (U A)_{i} (T_{i, r e s} - T_{i}) d t,$

$Q_{o} = \int_{0}^{τ} (U A)_{o} (T_{o, r e s} - T_{o}) d t .$

En régime périodique :

$Δ U_{c y c l e} = 0,$

et le premier principe donne :

$Q_{i} + Q_{o} = W_{c y c l e} .$

Le COP frigorifique est :

$C O P_{c} = \frac{Q_{i}}{- W_{c y c l e}} .$

Le COP en pompe à chaleur est :

$C O P_{h} = \frac{- Q_{o}}{- W_{c y c l e}} = C O P_{c} + 1.$

Les signes $Q_{i} > 0$ , $Q_{o} < 0$ et $W_{c y c l e} < 0$ constituent des contrôles du régime frigorifique recherché.

Pour le cycle moteur :

$Q_{i} > 0, Q_{o} < 0, W_{c y c l e} > 0.$

Le rendement thermique est :

$η_{t h} = \frac{W_{c y c l e}}{Q_{i}} = 1 + \frac{Q_{o}}{Q_{i}} .$

La puissance thermodynamique moyenne du moteur est :

$\overline{\dot{W}} = \frac{W_{c y c l e}}{τ} .$

La force thermodynamique exercée par le gaz sur une face de piston peut être écrite $F_{g a s} = P S$ . La force mécanique nette, les inerties et les frottements relèvent du dimensionnement mécanique ultérieur.

9. Charge de gaz et régime périodique

La quantité totale de gaz enfermée est un paramètre physique :

$M_{t o t} = m_{S} + m_{L} + m_{i} + m_{o} = c o n s t .$

Elle peut être imposée directement ou définie expérimentalement par une pression et une température de charge. La configuration de référence choisie est celle de $t = 0$ , avec le grand cylindre au volume maximal. Si tous les volumes communiquent et sont en équilibre uniforme pendant le remplissage :

$M_{t o t} = \frac{P_{c h a r g e} [V_{S} (0) + V_{L, max} + V_{i} + V_{o}]}{R T_{c h a r g e}} .$

$P_{c h a r g e}$ et $T_{c h a r g e}$ définissent la quantité de gaz chargée ; ils ne constituent pas une condition que le cycle périodique doit retrouver.

Le régime périodique établi est une solution du système telle que, entre deux passages successifs au volume maximal du grand cylindre avec la même direction cinématique :

$𝐗 (t + τ) = 𝐗 (t) .$

La périodicité géométrique seule :

$V_{k} (t + τ) = V_{k} (t), k \in {S, L} .$

ne suffit pas à garantir la périodicité thermodynamique.

L'état numérique utilisé pour lancer un calcul peut être approximatif ; il ne doit pas être confondu avec un paramètre physique de la machine. Le futur solveur pourra rechercher le point fixe périodique par cycles successifs, méthode de tir ou méthode de Newton.

10. Paramètres, données de conception et résultats

10.1 Données imposées

mode de fonctionnement ;
températures des réservoirs $T_{c o l d}$ et $T_{h o t}$ ;
vitesses angulaire signée de la manivelle $ω$ .
fluide de travail et propriétés de référence $R$ , $C_{p}$ , $C_{v}$ , $γ$ ;
charge totale $M_{t o t}$ , ou équivalent $(P_{c h a r g e}, T_{c h a r g e})$ dans la configuration de remplissage définie au §9 ;
cinématique $V_{S} (t)$ , $V_{L} (t)$ .

10.2 Paramètres de conception

$V_{S, min}$ , $V_{S, max}$ , $V_{L, min}$ , $V_{L, max}$ ;
$V_{i}$ , $V_{o}$ ;
$(U A)_{i}$ , $(U A)_{o}$ ;
paramètres hydrauliques des échangeurs, conduites et clapets, représentés au premier niveau par $(C_{d} A)_{e f f}$ ;
seuils $Δ P_{o p e n}$ , $Δ P_{c l o s e}$ ;
cibles cinématiques $Λ_{L}^{*}$ , $Λ_{S}^{*}$ .

10.3 Variables et résultats calculés

$m_{j}, U_{j}, T_{j}, P_{j}, j \in {S, L, i, o}$ ;
débits massiques internes et externes ;
${\dot{Q}}_{i}$ , ${\dot{Q}}_{o}$ , $Q_{i}$ , $Q_{o}$ ;
$W_{c y c l e}$ , $C O P_{c}$ , $C O P_{h}$ ;
extrema de pression, température et débit ;
événements réels des clapets et $Λ_{r e a l}$ ;
qualité d'isothermie $ε_{T}$ ;
critères de validité $ε_{P}$ , $M a$ , $Z$ et variation des propriétés.

11. Contrôles globaux de conservation

11.1 Conservation de la masse

Le solveur doit satisfaire :

$\frac{d M_{t o t}}{d t} = 0.$

Un résidu numérique utile est :

$ε_{M} (t) = M_{t o t} (t) - M_{t o t} (0) .$

11.2 Conservation globale de l'énergie

Quelle que soit la phase, les flux massiques et enthalpiques internes doivent s'annuler lorsque les bilans de tous les volumes sont additionnés. Le bilan global doit se réduire à :

$\frac{d U_{t o t}}{d t} = {\dot{Q}}_{i} + {\dot{Q}}_{o} - P_{S}^{⋆} {\dot{V}}_{S} - P_{L}^{⋆} {\dot{V}}_{L} .$

Un résidu énergétique cumulatif peut être défini par :

$ε_{E} (t) = U_{t o t} (t) - U_{t o t} (0) - Q_{i} (0, t) - Q_{o} (0, t) + W (0, t) .$

Sur un cycle périodique :

$Q_{i} + Q_{o} = W_{c y c l e} .$

Le solveur doit maintenir $ε_{M}$ et $ε_{E}$ proches de zéro à la précision numérique attendue.

Annexe A — Dérivations symboliques et contrôles validés

A.1 Équation de pression d'une paire fermée

Pour une paire cylindre + échangeur à pression quasi uniforme :

$U = \frac{P (V_{c y l} + V_{H X})}{γ - 1} .$

Le premier principe donne :

$\frac{d U}{d t} = \dot{Q} - P {\dot{V}}_{c y l},$

avec :

$\dot{Q} = (U A)_{H X} (T_{H X, r e s} - T_{H X}) .$

En dérivant $U$ :

$\frac{1}{γ - 1} [(V_{c y l} + V_{H X}) \dot{P} + P {\dot{V}}_{c y l}] = \dot{Q} - P {\dot{V}}_{c y l} .$

D'où :

$\dot{P} = \frac{(γ - 1) \dot{Q} - γ P {\dot{V}}_{c y l}}{V_{c y l} + V_{H X}} .$

Si $\dot{Q} = 0$ :

$\frac{\dot{P}}{P} = - γ \frac{\dot{V}}{V},$

puis :

$P V^{γ} = c o n s t .$

A.2 Débit interne de la paire

Pour l'échangeur seul, volume fixe :

$U_{H X} = \frac{P V_{H X}}{γ - 1} .$

Donc :

$\frac{V_{H X}}{γ - 1} \dot{P} = (U A)_{H X} (T_{H X, r e s} - T_{H X}) + {\dot{m}}_{i n t} C_{p} T_{u p} .$

En utilisant $C_{p} = γ R / (γ - 1)$ :

$V_{H X} \dot{P} + (γ - 1) (U A)_{H X} (T_{H X} - T_{H X, r e s}) = γ R T_{u p} {\dot{m}}_{i n t} .$

On retrouve la définition du numérateur $N$ et la sélection de la température amont selon le signe du débit.

A.3 Évolution de la température de l'échangeur dans une paire

Pour un volume fixe :

$m_{H X} = \frac{P V_{H X}}{R T_{H X}} .$

En dérivant :

$\frac{{\dot{m}}_{H X}}{m_{H X}} = \frac{\dot{P}}{P} - \frac{{\dot{T}}_{H X}}{T_{H X}} .$

Avec ${\dot{m}}_{H X} = {\dot{m}}_{i n t}$ :

${\dot{T}}_{H X} = \frac{T_{H X}}{P} \dot{P} - \frac{R T_{H X}^{2}}{P V_{H X}} {\dot{m}}_{i n t} .$

Cette relation découle uniquement de la conservation de masse et de l'équation d'état ; elle est valable pour les deux sens de débit.

A.4 Paire réceptrice ouverte

Pour une paire recevant ${\dot{m}}_{e x t}$ dans son cylindre :

$\frac{d U}{d t} = {\dot{m}}_{e x t} C_{p} T_{e x t} + \dot{Q} - P {\dot{V}}_{c y l} .$

Avec $U = P (V_{c y l} + V_{H X}) / (γ - 1)$ :

$\dot{P} = \frac{γ R T_{e x t} {\dot{m}}_{e x t} + (γ - 1) \dot{Q} - γ P {\dot{V}}_{c y l}}{V_{c y l} + V_{H X}} .$

Le bilan de masse local :

${\dot{m}}_{c y l} = {\dot{m}}_{e x t} - {\dot{m}}_{i n t},$

${\dot{m}}_{H X} = {\dot{m}}_{i n t},$

redonne immédiatement :

${\dot{m}}_{c y l} = - {\dot{m}}_{i n t}, {\dot{m}}_{H X} = {\dot{m}}_{e x t} + {\dot{m}}_{i n t} .$

${\dot{m}}_{p a i r} = {\dot{m}}_{e x t} .$

A.5 Solution analytique du cylindre donneur adiabatique

Pour un cylindre adiabatique, bien mélangé, avec débit uniquement sortant :

$\frac{d (m C_{v} T)}{d t} = - P \dot{V} - {\dot{m}}_{o u t} C_{p} T,$

et :

$\dot{m} = - {\dot{m}}_{o u t} .$

En développant :

$C_{v} m \dot{T} + C_{v} T \dot{m} = - P \dot{V} + C_{p} T \dot{m},$

soit :

$C_{v} m \dot{T} = - P \dot{V} + R T \dot{m} .$

Avec $P = m R T / V$ et $R / C_{v} = γ - 1$ :

$\frac{d T}{T} = (γ - 1) (\frac{d m}{m} - \frac{d V}{V}) .$

Après intégration :

$\frac{T}{T_{0}} = {[\frac{m}{m_{0}} \frac{V_{0}}{V}]}^{γ - 1} .$

Puis, avec $P V = m R T$ :

$P = P_{0} {[\frac{m}{m_{0}} \frac{V_{0}}{V}]}^{γ} .$

et :

$\frac{T}{T_{0}} = {(\frac{P}{P_{0}})}^{(γ - 1) / γ} .$

Sous ces hypothèses, l'entropie spécifique du gaz restant est constante : $d s = 0$ . L'entropie totale du contenu du cylindre n'est pas constante puisque sa masse varie.

A.6 Échangeur actif : bilan développé

Bilan fondamental :

$\frac{d (m C_{v} T)}{d t} = {\dot{m}}_{i n} C_{p} T_{i n} - {\dot{m}}_{o u t} C_{p} T + \dot{Q} .$

En développant le membre de gauche et en utilisant :

$\dot{m} = {\dot{m}}_{i n} - {\dot{m}}_{o u t},$

on obtient :

$\dot{T} = \frac{{\dot{m}}_{i n} (C_{p} T_{i n} - C_{v} T) - {\dot{m}}_{o u t} R T + \dot{Q}}{C_{v} m} .$

Contrôles limites :

sans débit, l'équation redonne la relaxation thermique d'un volume fermé ;
à débit stationnaire égal entrant/sortant, elle redonne $m C_{v} \dot{T} = \dot{m} C_{p} (T_{i n} - T) + \dot{Q}$ ;
si $T_{i n} = T$ et ${\dot{m}}_{i n} = {\dot{m}}_{o u t}$ , la contribution nette du débit à $\dot{T}$ s'annule.

A.7 Global mass test

Using the signed link flows defined in §6:

${\dot{m}}_{S} = - {\dot{m}}_{S i} + {\dot{m}}_{o S},$

${\dot{m}}_{L} = {\dot{m}}_{i L} - {\dot{m}}_{L o},$

${\dot{m}}_{i} = {\dot{m}}_{S i} - {\dot{m}}_{i L},$

${\dot{m}}_{o} = {\dot{m}}_{L o} - {\dot{m}}_{o S} .$

Summing the four equations cancels every internal mass flow exactly:

${\dot{M}}_{t o t} = 0.$

This result is independent of the active phase, the flow directions on bidirectional links, and the positions of the two check valves.

A.8 Global energy test

Using the signed enthalpy fluxes defined in §6:

${\dot{U}}_{S} = - P_{S} {\dot{V}}_{S} - {\dot{H}}_{S i} + {\dot{H}}_{o S},$

${\dot{U}}_{L} = - P_{L} {\dot{V}}_{L} + {\dot{H}}_{i L} - {\dot{H}}_{L o},$

${\dot{U}}_{i} = {\dot{H}}_{S i} - {\dot{H}}_{i L} + {\dot{Q}}_{i},$

${\dot{U}}_{o} = {\dot{H}}_{L o} - {\dot{H}}_{o S} + {\dot{Q}}_{o} .$

All internal enthalpy fluxes cancel exactly when the four balances are summed:

${\dot{U}}_{t o t} = {\dot{Q}}_{i} + {\dot{Q}}_{o} - P_{S} {\dot{V}}_{S} - P_{L} {\dot{V}}_{L} .$

Integrated over a periodic cycle:

$Q_{i} + Q_{o} = W_{c y c l e} .$

A.9 Continuité aux transitions

À l'instant d'un événement de clapet, aucune masse ni énergie finie ne peut être transférée en un temps nul. Les variables conservées et la géométrie sont donc continues :

$j \in {S, L, i, o}$

$m_{j}^{+} = m_{j}^{-}, U_{j}^{+} = U_{j}^{-}, V_{j}^{+} = V_{j}^{-} .$

Avec le gaz parfait :

$T_{j} = \frac{U_{j}}{m_{j} C_{v}}, P_{j} = \frac{m_{j} R T_{j}}{V_{j}},$

ce qui implique :

$T_{j}^{+} = T_{j}^{-}, P_{j}^{+} = P_{j}^{-} .$

L'événement ne crée qu'un changement de topologie hydraulique et de système d'équations actif.

@@ Line 15: / Line 15: @@
 * <math display="inline">H_o</math> : échangeur thermique "Heat-out", situé sur le chemin hydraulique <math display="inline">L \to S</math> et qui enlève de la chaleur au gaz.
-La topologie hydraulique est fixe :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The two hydraulic branches have fixed circulation orientations:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>S\to H_i\to L,</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>L\to H_o\to S.</math>
+</div>
-Les clapets passifs permettent donc toujours :
-<math>H_i\to L,\qquad H_o\to S.</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Each branch contains one passive check valve. The check valve may be installed on either side of its heat exchanger. The two admissible arrangements for the heat-in branch are:
+</div>
-La fonction physique de chaque échangeur thermique est indépendante du mode de fonctionnement. Ce qui change entre les cycles récepteur et moteur est le réservoir thermique externe connecté à chaque échangeur.
+<div lang="en" dir="ltr" class="mw-content-ltr">
-Le cycle comprend quatre phases hydrauliques/thermodynamiques :
+<math>S\to \mathrm{CV}_i\to H_i\to L,</math>
+</div>
-* '''Phase I''' — clapets fermés, nominalement adiabatique : compression du côté <math display="inline">L+H_o</math> ;
+<div lang="en" dir="ltr" class="mw-content-ltr">
-* '''Phase II''' — échange thermique, nominalement isotherme : transfert <math display="inline">L \to H_o \to S</math>, avec extraction de chaleur du gaz ;
+or:
-* '''Phase III''' — clapets fermés, nominalement adiabatique : expansion du côté <math display="inline">S+H_i</math> ;
+</div>
-* '''Phase IV''' — échange thermique, nominalement isotherme : transfert <math display="inline">S \to H_i \to L</math>, avec de la chaleur ajoutée au gaz.
-Ces définitions de phases décrivent la topologie physique permanente de la machine. Le réservoir externe associé à chaque échangeur de chaleur dépend du mode de fonctionnement. Les limites de phase et leurs positions angulaires sont déterminées indépendamment dans chaque mode de fonctionnement par les événements réels liés aux clapets anti-retour.
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>S\to H_i\to \mathrm{CV}_i\to L.</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Likewise, the heat-out branch may be arranged as:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>L\to \mathrm{CV}_o\to H_o\to S,</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+or:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>L\to H_o\to \mathrm{CV}_o\to S.</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+In all cases the check valve enforces the same overall circulation direction of its branch. Its position relative to the heat exchanger is a design parameter because, when the valve is closed, it determines which cylinder remains hydraulically connected to the exchanger volume.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The physical function of each heat exchanger is independent of the operating mode. What changes between refrigeration and motor operation is the external thermal reservoir connected to each exchanger.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The cycle is described by four hydraulic/thermodynamic phases: compression, transfer through <math display="inline">H_o</math> from <math display="inline">L</math> to <math display="inline">S</math>, expansion, and transfer through <math display="inline">H_i</math> from <math display="inline">S</math> to <math display="inline">L</math>. In refrigeration operation, starting from the reference origin used in this study, these phases are traversed in that order. Motor operation reverses the crank kinematics. The resulting thermodynamic chronology must be recalculated with the unchanged check-valve orientations and the reversed reservoir assignment; it is detailed in §6.6.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The phase boundaries describe the kinematics and the dominant thermodynamic regime. They are not check-valve events. Check-valve opening and closing are determined independently by the local pressure difference across each valve; mass transfer and heat transfer may therefore continue during compression or expansion, and a valve event may occur inside a kinematic phase.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The two transfer phases are intended to be quasi-isobaric: one cylinder empties while the other fills, and the pressure variation is intended to remain small compared with the pressure change during compression and expansion. Compression and expansion may involve simultaneous motion of both pistons in the same volumetric direction, so both cylinders may contribute to the pressure-changing phase.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+For comparison with an ideal thermodynamic cycle, compression and expansion may be idealized as adiabatic transformations; if they are also reversible, they are isentropic. These are reference transformations only. The real machine does not impose zero heat transfer, zero mass transfer, or closed check valves during compression and expansion, and no construction capable of enforcing perfectly adiabatic phases is assumed here.
+</div>
-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>
@@ Line 96: / Line 151: @@
 <math>\nu_{V,k}=|\dot V_k|.</math>
+<div class="mw-translate-fuzzy">
 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.
+</div>
 <span id="2.2_Kinematic_closure_fraction"></span>
@@ Line 105: / Line 162: @@
 <math>\Lambda_k(t)=\frac{V_{k,\max}-V_k(t)}{V_{k,\max}-V_{k,\min}},\qquad k\in\{S,L\}.</math>
+<div class="mw-translate-fuzzy">
 Ainsi <math display="inline">\Lambda_k=0</math> correspond au volume maximal et <math display="inline">\Lambda_k=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 :
+</div>
 <math>\Lambda_{\mathrm{real}}=\Lambda(t_{\mathrm{event}}).</math>
+<div class="mw-translate-fuzzy">
 Les transitions physiques restent déterminées par les pressions ; <math display="inline">\Lambda^*</math> n'est donc pas une condition d'ouverture imposée.
+</div>
 <span id="2.3_Thermodynamic_variables"></span>
@@ Line 194: / Line 255: @@
 Ainsi les équations de transfert thermique sont identiques dans les deux modes.
-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>
@@ Line 222: / Line 277: @@
 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>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-== 4. Formulation réduite d'une paire quasi égalisée en pression ==
+== 4. Reduced formulations for quasi-pressure-equalized connected volumes ==
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The complete model treats the four gas volumes independently. In some operating conditions, however, a set of volumes connected through sufficiently low hydraulic resistance may remain close to a common pressure. Such a set can then be treated by a reduced analytical formulation.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Let <math display="inline">\mathcal C</math> denote any connected set of gas volumes for which:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+P_j\approx P_{\mathcal C},
+\qquad j\in\mathcal C.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The composition of <math display="inline">\mathcal C</math> is determined by the actual hydraulic connectivity and by the position and state of the check valves.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+A useful pressure-equalization criterion is:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\varepsilon_{P,\mathcal C}
+=
+\max_{(a,b)\in\mathcal C}
+\frac{|P_a-P_b|}{P_{\mathcal C}}
+\ll1.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+A low internal Mach number provides an additional check:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+Ma_{\mathrm{int}}=\frac{|u_{\mathrm{int}}|}{a}\ll1,
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+but is not sufficient by itself to guarantee pressure equalization.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== 4.1 Pressure equation for a connected set ===
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Define the total volume:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+V_{\mathcal C}=\sum_{j\in\mathcal C}V_j.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Only cylinder volumes vary, so:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot V_{\mathcal C}
+=
+\sum_{k\in\mathcal C\cap\{S,L\}}\dot V_k.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+For a calorically perfect ideal gas at common pressure:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+U_{\mathcal C}
+=
+\sum_{j\in\mathcal C}m_jC_vT_j
+=
+\frac{P_{\mathcal C}V_{\mathcal C}}{\gamma-1}.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Let the total heat received by the gas in the set be:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot Q_{\mathcal C}
+=
+\sum_{j\in\mathcal C}\dot Q_j.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Mass crossing the boundary of the set transports the enthalpy of its upstream state. Define the net external enthalpy flow into the set as:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot H_{\mathcal C}^{\mathrm{ext}}
+=
+\sum_{\mathrm{in}}\dot m\,C_pT_u
+-
+\sum_{\mathrm{out}}\dot m\,C_pT_u.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The first law for the complete connected set is then:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\frac{dU_{\mathcal C}}{dt}
+=
+\dot Q_{\mathcal C}
++
+\dot H_{\mathcal C}^{\mathrm{ext}}
+-
+P_{\mathcal C}\dot V_{\mathcal C}.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Therefore:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot P_{\mathcal C}
+=
+\frac{
+(\gamma-1)
+\left(
+\dot Q_{\mathcal C}
++
+\dot H_{\mathcal C}^{\mathrm{ext}}
+\right)
+-
+\gamma P_{\mathcal C}\dot V_{\mathcal C}
+}
+{V_{\mathcal C}}
+.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The total mass of the set satisfies:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot M_{\mathcal C}
+=
+\sum_{\mathrm{in}}\dot m
+-
+\sum_{\mathrm{out}}\dot m.
+</math>
+</div>
-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>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Internal mass and enthalpy transfers between members of <math display="inline">\mathcal C</math> cancel from these global balances.
+</div>
-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>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== 4.2 Closed connected set ===
+</div>
-Un faible nombre de Mach interne constitue un contrôle complémentaire :
-<math>Ma_{\mathrm{int}}=\frac{|u_{\mathrm{int}}|}{a}\ll1,</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+If no mass crosses the boundary of <math display="inline">\mathcal C</math>:
+</div>
-mais il ne suffit pas, à lui seul, à garantir l'égalisation des pressions.
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot H_{\mathcal C}^{\mathrm{ext}}=0,
+\qquad
+\dot M_{\mathcal C}=0.
+</math>
+</div>
-<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> :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The pressure equation becomes:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\frac{dP}{dt}=
+\dot P_{\mathcal C}
-\frac{(\gamma-1)(UA)_{HX}(T_{HX,\mathrm{res}}-T_{HX})-\gamma P\dot V_{\mathrm{cyl}}}
+=
-{V_{\mathrm{cyl}}+V_{HX}}
+\frac{
-.</math>
+(\gamma-1)\dot Q_{\mathcal C}
+-
+\gamma P_{\mathcal C}\dot V_{\mathcal C}
+}
+{V_{\mathcal C}}.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+If the set is also adiabatic:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot Q_{\mathcal C}=0,
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+then:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+P_{\mathcal C}V_{\mathcal C}^{\gamma}
+=
+\mathrm{const}.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+This is a limiting analytical case. A compression or expansion phase of the complete machine does not require the corresponding connected set to be closed or adiabatic.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== 4.3 Fixed-volume heat exchanger within a pressure-equalized set ===
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+For a heat exchanger <math display="inline">H_j</math> of fixed volume <math display="inline">V_j</math> belonging to <math display="inline">\mathcal C</math>:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+m_j=\frac{P_{\mathcal C}V_j}{RT_j}.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Differentiation gives:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\frac{\dot m_j}{m_j}
+=
+\frac{\dot P_{\mathcal C}}{P_{\mathcal C}}
+-
+\frac{\dot T_j}{T_j},
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+hence:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot T_j
+=
+\frac{T_j}{P_{\mathcal C}}\dot P_{\mathcal C}
+-
+\frac{RT_j^2}{P_{\mathcal C}V_j}\dot m_j
+.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The exchanger mass rate <math display="inline">\dot m_j</math> is the algebraic sum of the actual flows through all links connected to it. This relation is therefore independent of whether the check valve lies upstream or downstream of the exchanger.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Its energy balance may equivalently be written:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\frac{V_j}{\gamma-1}\dot P_{\mathcal C}
+=
+\dot Q_j
++
+\sum_{\mathrm{in}}\dot m C_pT_u
+-
+\sum_{\mathrm{out}}\dot m C_pT_j.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+These equations determine the local mass redistribution and temperature evolution once the hydraulic flow rates are known.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== 4.4 Two-volume cylinder–exchanger special case ===
+</div>
-Lorsque <math display="inline">(UA)_{HX}=0</math> :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+For the particular case of one cylinder and one heat exchanger connected at quasi-uniform pressure, with no other flow entering or leaving the exchanger directly, define the internal mass flow as positive from cylinder to heat exchanger.
+</div>
-<math>P(V_{\mathrm{cyl}}+V_{HX})^\gamma=\mathrm{const}.</math>
-<span id="4.2_Internal_redistribution_flow_rate"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-=== 4.2 Débit interne de redistribution ===
+Let:
+</div>
-On définit positif le débit interne cylindre → échangeur. Posons :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+N=
+V_{HX}\dot P
++
+(\gamma-1)(UA)_{HX}
+(T_{HX}-T_{HX,\mathrm{res}}).
+</math>
+</div>
-<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 :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The internal flow rate is:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\dot m_{\mathrm{int}}=
+\dot m_{\mathrm{int}}
+=
 \begin{cases}
-\dfrac{N}{\gamma R T_{\mathrm{cyl}}}, & N\ge0 \quad (\mathrm{cyl}\to HX),\\[6pt]
+\dfrac{N}{\gamma RT_{\mathrm{cyl}}},
-\dfrac{N}{\gamma R T_{HX}}, & N<0 \quad (HX\to\mathrm{cyl}).
+& N\ge0
+\quad(\mathrm{cyl}\to HX),\\[6pt]
+\dfrac{N}{\gamma RT_{HX}},
+& N<0
+\quad(HX\to\mathrm{cyl}).
 \end{cases}
 </math>
+</div>
-La température de l'échangeur évolue selon :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The heat-exchanger temperature then satisfies:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\dot T_{HX}=\frac{T_{HX}}{P}\dot P-
+\dot T_{HX}
-\frac{RT_{HX}^2}{PV_{HX}}\dot m_{\mathrm{int}}
+=
-.</math>
+\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 special reduction must not be used when an additional external flow enters or leaves the heat exchanger directly; in that case the general balances of §4.1 and §4.3 apply.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== 4.5 Single external-flow special cases ===
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+If a quasi-pressure-equalized connected set receives a single external flow <math display="inline">\dot m_{\mathrm{ext}}>0</math> at upstream temperature <math display="inline">T_{\mathrm{ext}}</math>:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot P_{\mathcal C}
+=
+\frac{
+\gamma RT_{\mathrm{ext}}\dot m_{\mathrm{ext}}
++
+(\gamma-1)\dot Q_{\mathcal C}
+-
+\gamma P_{\mathcal C}\dot V_{\mathcal C}
+}
+{V_{\mathcal C}},
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+with:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot M_{\mathcal C}
+=
+\dot m_{\mathrm{ext}}.
+</math>
+</div>
-<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> :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+If instead the set delivers a single external outflow <math display="inline">\dot m_{\mathrm{ext}}>0</math> from a boundary volume at temperature <math display="inline">T_{\mathrm{out}}</math>:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\dot P=
+\dot P_{\mathcal C}
-\frac{\gamma RT_{\mathrm{ext}}\dot m_{\mathrm{ext}}
+=
-+(\gamma-1)(UA)_{HX}(T_{HX,\mathrm{res}}-T_{HX})
+\frac{
--\gamma P\dot V_{\mathrm{cyl}}}
+-\gamma RT_{\mathrm{out}}\dot m_{\mathrm{ext}}
-{V_{\mathrm{cyl}}+V_{HX}}
++
-.</math>
+(\gamma-1)\dot Q_{\mathcal C}
+-
+\gamma P_{\mathcal C}\dot V_{\mathcal C}
+}
+{V_{\mathcal C}},
+</math>
+</div>
-Les masses vérifient :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+with:
+</div>
-<math>\dot m_{\mathrm{cyl}}=\dot m_{\mathrm{ext}}-\dot m_{\mathrm{int}},\qquad
+<div lang="en" dir="ltr" class="mw-content-ltr">
-\dot m_{HX}=\dot m_{\mathrm{int}},</math>
+<math>
+\dot M_{\mathcal C}
+=
+-\dot m_{\mathrm{ext}}.
+</math>
+</div>
-et donc :
-<math>\dot m_{\mathrm{pair}}=\dot m_{\mathrm{ext}}.</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The location at which the external flow crosses the boundary of the connected set affects the local masses and temperatures, but not the summed pressure equation once the set <math display="inline">\mathcal C</math>, the boundary enthalpy flow, and its total heat and volume rates are specified.
+</div>
-Le cas fermé est obtenu immédiatement avec <math display="inline">\dot m_{\mathrm{ext}}=0</math>.
 <span id="5._Hydraulic_closure"></span>
@@ Line 312: / Line 754: @@
 <math>\dot m=\Phi(P_u,P_d,T_u,\mathcal G,\mathcal F)</math>
+<div class="mw-translate-fuzzy">
 <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 :
+</div>
 <math>\dot H_{\mathrm{mass}}=\dot m C_pT_u.</math>
+<div class="mw-translate-fuzzy">
 Pour une liaison bidirectionnelle, l'état amont est déterminé par le sens réel du gradient de pression. Pour un clapet, le débit inverse est interdit.
+</div>
 <span id="5.2_First-level_closure_using_a_compressible_orifice"></span>
@@ Line 350: / Line 796: @@
 Cette fermeture n'est pas l'essence du modèle : elle pourra être remplacée par une loi plus fidèle aux pertes de charge d'un échangeur réel, d'une conduite ou d'un clapet.
-<span id="6._Complete_thermodynamic_cycle"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-== 6. Cycle thermodynamique complet ==
+== 6. Complete thermodynamic cycle ==
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The cycle is described by four successive kinematic and thermodynamic regimes: compression, transfer through <math display="inline">H_o</math>, expansion, and transfer through <math display="inline">H_i</math>. These phases describe the dominant evolution of the machine; they are not defined by the state of the check valves. Valve opening and closing remain determined independently by the instantaneous pressure differences and may occur within a phase rather than exactly at a phase boundary.
+</div>
-<span id="6.1_Phase_I_—_check_valves_closed,_nominally_adiabatic:_compression_on_the_L_side"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-=== 6.1 Phase I — clapets fermés, nominalement adiabatique : compression côté L ===
+The numbering below follows the refrigeration direction <math display="inline">\omega>0</math>. Motor operation uses the same physical machine with reversed crank direction and is described in §6.6.
+</div>
-Les deux clapets sont fermés. Les paires <math display="inline">L+H_o</math> et <math display="inline">S+H_i</math> sont fermées. Le côté <math display="inline">L+H_o</math> est nominalement comprimé ; le mouvement du petit piston reste celui fourni par la cinématique réelle.
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== 6.1 Thermodynamic rationale of the four phases ===
+</div>
-Pour <math display="inline">L+H_o</math> :
-<math>\dot P_{Lo}=
+<div lang="en" dir="ltr" class="mw-content-ltr">
-\frac{(\gamma-1)(UA)_o(T_{o,\mathrm{res}}-T_o)-\gamma P_{Lo}\dot V_L}
+==== 6.1.1 Compression and expansion ====
-{V_L+V_o}.</math>
+</div>
-Pour <math display="inline">S+H_i</math> :
-<math>\dot P_{Si}=
+<div lang="en" dir="ltr" class="mw-content-ltr">
-\frac{(\gamma-1)(UA)_i(T_{i,\mathrm{res}}-T_i)-\gamma P_{Si}\dot V_S}
+Compression and expansion are primarily pressure-changing phases. In the intended kinematics, both cylinder volumes may decrease simultaneously during compression and increase simultaneously during expansion.
-{V_S+V_i}.</math>
+</div>
-Les équations de redistribution interne et de température du §4 s'appliquent aux deux paires.
-La transition vers la Phase II survient lorsque le clapet <math display="inline">H_o \to S</math> satisfait sa condition d'ouverture.
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The instantaneous work delivered by the gas is:
+</div>
-<span id="6.2_Phase_II_—_heat_exchange,_nominally_isothermal:_L_→_Ho_→_S"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-=== 6.2 Phase II — échange thermique, nominalement isotherme : L → Ho → S ===
+<math>
+\dot W=P_S\dot V_S+P_L\dot V_L.
+</math>
+</div>
-Le gaz quitte <math display="inline">L</math>, traverse l'échangeur chaud <math display="inline">H_o</math>, y rejette de la chaleur, franchit le clapet <math display="inline">H_o \to S</math>, puis entre dans le cylindre récepteur <math display="inline">S</math>. La paire <math display="inline">S+H_i</math> reste quasi égalisée si le critère <math display="inline">\varepsilon_P\ll1</math> est satisfait.
-<span id="6.2.1_Donor_cylinder_L"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-==== 6.2.1 Cylindre donneur L ====
+When the two cylinder pressures are of the same order, both piston contributions therefore add during a simultaneous expansion and both contribute to the work required during a simultaneous compression. This allows the swept volumes of both cylinders to participate in the pressure-changing parts of the cycle.
+</div>
-Le bilan fondamental est :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+In an ideal reversible reference cycle, compression and expansion may be considered adiabatic and reversible. This is not imposed on the real machine. The heat exchangers remain thermally coupled to the gas, mass redistribution may continue, and neither check valve is required to be closed during the whole compression or expansion phase.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+==== 6.1.2 Exchange phases ====
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+During an exchange phase, gas is transferred from one cylinder to the other through one of the heat exchangers. Hydraulic resistance requires a finite pressure difference to produce a finite mass flow. This pressure difference is intrinsically irreversible.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+For the hydraulic-loss contribution idealized locally as an adiabatic, isenthalpic throttling process of a calorically perfect ideal gas:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\frac{d(m_LC_vT_L)}{dt}
+h_u=h_d
-=-P_L\dot V_L-\dot m_{L,\mathrm{out}}C_pT_L
+\quad\Rightarrow\quad
-.</math>
+T_u=T_d,
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+and therefore:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\Delta s_{\mathrm{hyd}}
+=
+R\ln\left(\frac{P_u}{P_d}\right)>0
+\qquad\text{for}\qquad P_u>P_d.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The reversible limit is consequently:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+P_u-P_d\to0.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The exchange phases therefore tend ideally toward quasi-pressure-equalized operation. A finite real machine retains a finite pressure difference because a finite flow must cross the hydraulic resistances.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Pressure equalization between communicating volumes at a given instant does not, by itself, imply that their common pressure remains constant throughout the exchange. Let <math display="inline">\mathcal C</math> denote a closed set of communicating gas volumes that are approximately at a common pressure <math display="inline">P</math>. For an ideal gas:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+M_{\mathcal C}
+=
+\frac{P}{R}
+\sum_{j\in\mathcal C}\frac{V_j}{T_j},
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+and therefore:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+P=
+\frac{M_{\mathcal C}R}
+{\displaystyle\sum_{j\in\mathcal C}V_j/T_j}.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+At constant mass, an exactly isobaric evolution requires:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\frac{d}{dt}
+\left(
+\sum_{j\in\mathcal C}\frac{V_j}{T_j}
+\right)=0.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Thus the piston motions and the temperature evolution must compensate each other. Equal cylinder-volume changes are neither required nor generally expected.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The same condition can be expressed through the energy balance. For a quasi-pressure-equalized set:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+U_{\mathcal C}
+=
+\frac{P V_{\mathcal C}}{\gamma-1},
+\qquad
+V_{\mathcal C}=\sum_{j\in\mathcal C}V_j.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Its first-law balance gives:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+V_{\mathcal C}\dot P
+=
+(\gamma-1)\dot Q_{\mathcal C}
+-\gamma P\dot V_{\mathcal C}.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+An approximately isobaric exchange therefore satisfies:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\gamma P\dot V_{\mathcal C}
+\approx
+(\gamma-1)\dot Q_{\mathcal C}.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The volume evolution imposed by the pistons can consequently compensate the thermal expansion or contraction produced by heat transfer, allowing substantial mass transfer while the common pressure remains nearly constant.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+A useful first-order cylinder-sizing relation follows from the same condition. Over a sufficiently small part of an isobaric exchange, if the donor and receiver temperatures may be treated as locally constant and the temperature-storage terms of the fixed exchanger volumes are secondary, then:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\frac{dV_r}{T_r}
+\approx
+-\frac{dV_d}{T_d},
+</math>
+</div>
-La cinématique est conçue pour maintenir un débit sortant du cylindre donneur. Dans ce cas, la solution analytique est :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+hence:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\frac{T_L}{T_{L,\mathrm{ref}}}=
+\frac{dV_r}{-dV_d}
-\left[\frac{m_L}{m_{L,\mathrm{ref}}}\frac{V_{L,\mathrm{ref}}}{V_L}\right]^{\gamma-1}
+\approx
+\frac{T_r}{T_d}
+.
 </math>
+</div>
-et :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The hotter side therefore requires a larger volume change for the same transferred gas mass at the same pressure. When the exchange uses comparable fractions of the available cylinder strokes and the gas temperatures remain close to characteristic working temperatures <math display="inline">T_h</math> and <math display="inline">T_c</math>, this gives the first-order geometric scaling:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-P_L=P_{L,\mathrm{ref}}
+</div>
-\left[\frac{m_L}{m_{L,\mathrm{ref}}}\frac{V_{L,\mathrm{ref}}}{V_L}\right]^\gamma
-.</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+\frac{V_{\mathrm{swept},h}}
+{V_{\mathrm{swept},c}}
+\sim
+\frac{T_h}{T_c}
+.
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Temperatures must be expressed in kelvin. This relation is a sizing guide, not an exact design constraint: exchanger hold-up, clearance volumes, temperature evolution during the exchange, finite pressure losses, and the compression and expansion phases can all shift the optimum. When these effects are significant, the complete condition involving <math display="inline">\sum V_j/T_j</math> must be used instead.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+These relations explain why a low-loss exchange naturally tends toward both small pressure differences along the hydraulic path and, with suitable piston kinematics, a nearly constant pressure throughout the exchange. Exact isobaricity is not imposed as a thermodynamic constraint.
+</div>
-On obtient également :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+==== 6.1.3 Balance equations valid throughout the cycle ====
+</div>
-<math>\frac{T_L}{T_{L,\mathrm{ref}}}=
-\left(\frac{P_L}{P_{L,\mathrm{ref}}}\right)^{(\gamma-1)/\gamma},</math>
-et l'entropie spécifique du gaz restant vérifie <math display="inline">ds=0</math> sous ces hypothèses. Si un débit rétrograde apparaît malgré la conception cinématique, la solution analytique n'est plus applicable et il faut utiliser le bilan ouvert complet en <math display="inline">(m_L,U_L)</math>.
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The four control volumes remain <math display="inline">S</math>, <math display="inline">H_i</math>, <math display="inline">L</math>, and <math display="inline">H_o</math>. Define the signed mass flow rates:
+</div>
-<span id="6.2.2_Heat-out_exchanger_Ho"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-==== 6.2.2 Échangeur Heat-out Ho ====
+<math>
+\dot m_{Si}: S\to H_i,\qquad
+\dot m_{iL}: H_i\to L,
+</math>
+</div>
-Conservation de la masse :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot m_{Lo}: L\to H_o,\qquad
+\dot m_{oS}: H_o\to S.
+</math>
+</div>
-<math>\dot m_o=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}}.</math>
-Bilan énergétique fondamental :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The positive directions correspond to the permanent circulation directions of the two branches:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\frac{d(m_oC_vT_o)}{dt}
+S\to H_i\to L,
-=\dot m_{\mathrm{in}}C_pT_L
+\qquad
--\dot m_{\mathrm{out}}C_pT_o
+L\to H_o\to S.
-+(UA)_o(T_{o,\mathrm{res}}-T_o)
+</math>
-.</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Each branch contains one passive check valve. Its position relative to the heat exchanger is a design choice:
+</div>
-Sous forme développée :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+S\to \mathrm{CV}_i\to H_i\to L
+\quad\text{or}\quad
+S\to H_i\to \mathrm{CV}_i\to L,
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\dot T_o=
+L\to \mathrm{CV}_o\to H_o\to S
-\frac{\dot m_{\mathrm{in}}(C_pT_L-C_vT_o)
+\quad\text{or}\quad
--\dot m_{\mathrm{out}}RT_o
+L\to H_o\to \mathrm{CV}_o\to S.
-+(UA)_o(T_{o,\mathrm{res}}-T_o)}
+</math>
-{C_vm_o}
+</div>
-.</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The check valve constrains the link on which it is installed; the other link may be bidirectional according to its hydraulic law.
+</div>
-Les débits sont déterminés par les lois hydrauliques :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Let <math display="inline">\dot H_{ab}</math> denote the signed enthalpy transport from volume <math display="inline">a</math> toward volume <math display="inline">b</math>. For a calorically perfect gas:
+</div>
-<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,o}}(P_L,P_o,T_L,\ldots),</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot H_{ab}=
+\begin{cases}
+\dot m_{ab}C_pT_a, & \dot m_{ab}\ge0,\\[4pt]
+\dot m_{ab}C_pT_b, & \dot m_{ab}<0.
+\end{cases}
+</math>
+</div>
-<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,o}}(P_o,P_{Si},T_o,\ldots).</math>
-<span id="6.2.3_Receiving_pair_S+Hi"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-==== 6.2.3 Paire réceptrice S+Hi ====
+The mass balances are then:
+</div>
-Le débit issu de <math display="inline">H_o</math> entre dans '''S'''. La pression de la paire vérifie :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot m_S=\dot m_{oS}-\dot m_{Si},
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\dot P_{Si}=
+\dot m_i=\dot m_{Si}-\dot m_{iL},
-\frac{\gamma R\dot m_{\mathrm{out}}T_o
+</math>
-+(\gamma-1)(UA)_i(T_{i,\mathrm{res}}-T_i)
+</div>
--\gamma P_{Si}\dot V_S}
-{V_S+V_i}
+<div lang="en" dir="ltr" class="mw-content-ltr">
-.</math>
+<math>
+\dot m_L=\dot m_{iL}-\dot m_{Lo},
+</math>
+</div>
-Les équations de redistribution interne du §4 restent inchangées : le débit externe agit sur l'échangeur <math display="inline">H_i</math> indirectement par l'évolution de la pression de paire.
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot m_o=\dot m_{Lo}-\dot m_{oS}.
+</math>
+</div>
-La transition vers la Phase III est l'événement de fermeture du clapet <math display="inline">H_o \to S</math>.
-<span id="6.3_Phase_III_—_check_valves_closed,_nominally_adiabatic:_expansion_on_the_S_side"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-=== 6.3 Phase III — clapets fermés, nominalement adiabatique : expansion du côté S ===
+The corresponding energy balances are:
+</div>
-Les deux clapets sont fermés. La paire <math display="inline">S+H_i</math> se détend nominalement ; la paire <math display="inline">L+H_o</math> reste également fermée. Aucun piston n'est supposé rigoureusement immobile.
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot U_S
+=
+\dot H_{oS}-\dot H_{Si}
+-P_S\dot V_S,
+</math>
+</div>
-Pour <math display="inline">S+H_i</math> :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot U_i
+=
+\dot H_{Si}-\dot H_{iL}
++\dot Q_i,
+</math>
+</div>
-<math>\dot P_{Si}=
+<div lang="en" dir="ltr" class="mw-content-ltr">
-\frac{(\gamma-1)(UA)_i(T_{i,\mathrm{res}}-T_i)-\gamma P_{Si}\dot V_S}
+<math>
-{V_S+V_i}.</math>
+\dot U_L
+=
+\dot H_{iL}-\dot H_{Lo}
+-P_L\dot V_L,
+</math>
+</div>
-Pour <math display="inline">L+H_o</math> :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot U_o
+=
+\dot H_{Lo}-\dot H_{oS}
++\dot Q_o.
+</math>
+</div>
-<math>\dot P_{Lo}=
-\frac{(\gamma-1)(UA)_o(T_{o,\mathrm{res}}-T_o)-\gamma P_{Lo}\dot V_L}
-{V_L+V_o}.</math>
-La transition vers la Phase IV survient lorsque le clapet <math display="inline">H_i \to L</math> satisfait sa condition d'ouverture.
+<div lang="en" dir="ltr" class="mw-content-ltr">
+These equations are valid during all four phases. The phase determines the prescribed piston motion and the dominant thermodynamic process; the hydraulic laws and pressure differences determine the actual flow rates and check-valve states.
+</div>
-<span id="6.4_Phase_IV_—_heat_exchange,_nominally_isothermal:_S_→_Hi_→_L"></span>
-=== 6.4 Phase IV — échange thermique, nominalement isotherme : S → Hi → L ===
-Le gaz quitte <math display="inline">S</math>, traverse l'échangeur froid <math display="inline">H_i</math>, y reçoit de la chaleur du réservoir froid, franchit le clapet <math display="inline">H_i \to L</math>, puis entre dans le cylindre récepteur <math display="inline">L</math>. La paire <math display="inline">L+H_o</math> reste quasi égalisée si le critère <math display="inline">\varepsilon_P\ll1</math> est satisfait.
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== 6.2 Phase I — compression ===
+</div>
-<span id="6.4.1_Donor_cylinder_S"></span>
-==== 6.4.1 Cylindre donneur S ====
-Le bilan fondamental est :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+During compression, the two cylinder volumes may decrease simultaneously:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\frac{d(m_SC_vT_S)}{dt}
+\dot V_S<0,\qquad \dot V_L<0
-=-P_S\dot V_S-\dot m_{S,\mathrm{out}}C_pT_S
+</math>
-.</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+over the principal part of the phase.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The pressure rises from the lower exchange-pressure region toward the higher one. Both pistons may contribute to the compression work.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+No closed-pair topology is imposed. A check valve may remain open during part of the compression, and gas may continue to move through the hydraulic network. In particular, a cylinder approaching its minimum volume may transfer its remaining gas toward the other cylinder. Heat transfer through <math display="inline">H_i</math> and <math display="inline">H_o</math> also remains active.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The actual evolution is therefore calculated from the complete balances of §6.1.3. Adiabatic compression is only the reversible reference limit described in §6.1.1.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The end of the compression phase is defined by the prescribed kinematic law, not by a check-valve event.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== 6.3 Phase II — exchange through Ho: L → Ho → S ===
+</div>
-Sous débit sortant garanti par la conception cinématique :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The dominant circulation is:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\frac{T_S}{T_{S,\mathrm{ref}}}=
+L\to H_o\to S.
-\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">
+The gas leaves the large-cylinder side, passes through the heat-out branch, and reaches the small-cylinder side. The passive check valve <math display="inline">\mathrm{CV}_o</math> may be located either before or after <math display="inline">H_o</math>; in both cases it enforces the same net branch direction.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+During the exchange, <math display="inline">L</math> acts predominantly as donor and <math display="inline">S</math> as receiver. Their volume changes need not have equal magnitudes. In the quasi-isobaric limit their first-order ratio follows the temperature relation derived in §6.1.2.
+</div>
-et :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The heat-out exchanger removes heat from the gas:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-P_S=P_{S,\mathrm{ref}}
+\dot Q_o<0
-\left[\frac{m_S}{m_{S,\mathrm{ref}}}\frac{V_{S,\mathrm{ref}}}{V_S}\right]^\gamma
+</math>
-.</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+in the intended operating regime.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The ideal exchange tends toward small pressure differences along the active path and an approximately constant pressure over the phase. The finite real pressure differences required to drive the flow are determined by the hydraulic closure of §5.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The state of the other check valve and any secondary redistribution flow are determined by the instantaneous pressures; they are not prescribed by the phase definition.
+</div>
-Un débit rétrograde impose de revenir au bilan ouvert complet en <math display="inline">(m_S,U_S)</math>.
-<span id="6.4.2_Heat-in_exchanger_Hi"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-==== 6.4.2 Échangeur Heat-in Hi ====
+=== 6.4 Phase III — expansion ===
+</div>
-<math>\dot m_i=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}}.</math>
-Le bilan énergétique fondamental est :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+During expansion, the two cylinder volumes may increase simultaneously:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\frac{d(m_iC_vT_i)}{dt}
+\dot V_S>0,\qquad \dot V_L>0
-=\dot m_{\mathrm{in}}C_pT_S
+</math>
--\dot m_{\mathrm{out}}C_pT_i
+</div>
-+(UA)_i(T_{i,\mathrm{res}}-T_i)
-.</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+over the principal part of the phase.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The pressure decreases from the higher exchange-pressure region toward the lower one. Both pistons may then contribute simultaneously to the work delivered by the gas.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+As during compression, no zero-flow or closed-valve condition is imposed. Mass redistribution may continue and both heat exchangers remain thermally active. The complete balances of §6.1.3 therefore remain applicable.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Adiabatic expansion is the reversible reference limit, not a required operating condition of the real machine.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The end of the expansion phase is determined by the prescribed kinematic law independently of the check-valve events.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== 6.5 Phase IV — exchange through Hi: S → Hi → L ===
+</div>
-Sous forme développée :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The dominant circulation is:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\dot T_i=
+S\to H_i\to L.
-\frac{\dot m_{\mathrm{in}}(C_pT_S-C_vT_i)
+</math>
--\dot m_{\mathrm{out}}RT_i
+</div>
-+(UA)_i(T_{i,\mathrm{res}}-T_i)}
-{C_vm_i}
-.</math>
-Les débits sont déterminés par :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The gas leaves the small-cylinder side, passes through the heat-in branch, and reaches the large-cylinder side. The passive check valve <math display="inline">\mathrm{CV}_i</math> may be located either before or after <math display="inline">H_i</math>; both arrangements impose the same net branch direction.
+</div>
-<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,i}}(P_S,P_i,T_S,\ldots),</math>
-<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,i}}(P_i,P_{Lo},T_i,\ldots).</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+During the exchange, <math display="inline">S</math> acts predominantly as donor and <math display="inline">L</math> as receiver. Their required volume changes depend on the temperatures of the gas on the two sides according to the relations of §6.1.2.
+</div>
-<span id="6.4.3_Receiving_pair_L+Ho"></span>
-==== 6.4.3 Paire réceptrice L+Ho ====
-Le débit issu de <math display="inline">H_i</math> entre dans '''L'''. La pression de la paire vérifie :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The heat-in exchanger supplies heat to the gas:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
-\dot P_{Lo}=
+\dot Q_i>0
-\frac{\gamma R\dot m_{\mathrm{out}}T_i
+</math>
-+(\gamma-1)(UA)_o(T_{o,\mathrm{res}}-T_o)
+</div>
--\gamma P_{Lo}\dot V_L}
-{V_L+V_o}
+<div lang="en" dir="ltr" class="mw-content-ltr">
-.</math>
+in the intended operating regime.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+As in Phase II, the ideal exchange tends toward quasi-pressure-equalized and approximately isobaric operation, while the real mass flow requires finite hydraulic pressure differences.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+After this phase, the prescribed kinematics return to the compression region and the four-phase sequence repeats. Thermodynamic closure nevertheless requires the complete state vector, and not only the piston geometry, to be periodic.
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== 6.6 Motor operation ===
+</div>
-La fermeture du clapet <math display="inline">H_i \to L</math> ramène à la Phase I. La fermeture géométrique des pistons ne suffit pas à garantir la fermeture thermodynamique du cycle.
-<span id="6.5_Motor_operation"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-=== 6.5 Fonctionnement moteur ===
+No second set of mass, energy, heat-transfer, or hydraulic equations is required for motor operation.
+</div>
-Un deuxième ensemble d’équations d’équilibre thermodynamique n’est pas nécessaire pour le cycle moteur.
-La topologie hydraulique physique reste :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The physical branch directions remain:
+</div>
-<math>L\to H_o\to S,\qquad
+<div lang="en" dir="ltr" class="mw-content-ltr">
-S\to H_i\to L.</math>
+<math>
+S\to H_i\to L,
+\qquad
+L\to H_o\to S,
+</math>
+</div>
-La direction des clapets reste :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+and the positions and orientations of the two passive check valves remain unchanged.
+</div>
-<math>H_o\to S,\qquad H_i\to L.</math>
-Le fonctionnement moteur est obtenu par :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Motor operation is obtained by reversing the crank direction:
+</div>
-# inverser le sens de la manivelle,
+<div lang="en" dir="ltr" class="mw-content-ltr">
-# échanger les réservoirs thermiques externes connectés à <math display="inline">H_i</math> et <math display="inline">H_o</math>.
+<math>
+\omega<0,
+</math>
+</div>
-ainsi :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+and exchanging the external reservoirs connected to the two heat exchangers:
+</div>
-<math>\omega<0,</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+T_{i,\mathrm{res}}=T_{\mathrm{hot}},
+\qquad
+T_{o,\mathrm{res}}=T_{\mathrm{cold}}.
+</math>
+</div>
-avec la même origine du cycle :
-<math>V_L(0)=V_{L,\max}.</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The geometric cycle is therefore traversed in the opposite direction. Pressure histories, mass flow rates, check-valve events, and the periodic thermodynamic state must all be recalculated; the refrigeration valve chronology is not assumed simply to carry over.
+</div>
-L’affectation des réservoirs thermiques devient :
-<math>T_{i,\mathrm{res}}=T_{\mathrm{hot}},\qquad
+<div lang="en" dir="ltr" class="mw-content-ltr">
-T_{o,\mathrm{res}}=T_{\mathrm{cold}}.</math>
+In the intended refrigeration operation, the <math display="inline">L\to H_o\to S</math> transfer occurs on the high-pressure side of the cycle, whereas the <math display="inline">S\to H_i\to L</math> transfer occurs on the low-pressure side.
+</div>
-Ainsi l’échangeur Heat-in absorbe de la chaleur de la source chaude :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+In the intended motor operation, reversal of the crank direction reverses these pressure roles: <math display="inline">L\to H_o\to S</math> becomes the low-pressure exchange, whereas <math display="inline">S\to H_i\to L</math> becomes the high-pressure exchange.
+</div>
-<math>Q_i>0,</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The physical thermal functions of the exchangers do not change: <math display="inline">H_i</math> always transfers heat into the gas and <math display="inline">H_o</math> always transfers heat out of the gas.
+</div>
-pendant que l’échangeur Heat-out rejette de la chaleur à la source froide :
-<math>Q_o<0.</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+The heat-in exchanger absorbs heat from the hot reservoir:
+</div>
-Les variations de pression, débits massiques, évènements de clapets, et les états thermodynamiques périodiques doivent être recalculés avec la cinématique inversée.
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+Q_i>0,
+</math>
+</div>
-Le cycle moteur est obtenu quand :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+the heat-out exchanger rejects heat to the cold reservoir:
+</div>
-<math>W_{\mathrm{cycle}}>0.</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+Q_o<0,
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+and motor operation is obtained when:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+W_{\mathrm{cycle}}>0.
+</math>
+</div>
 <span id="7._Physical_transitions_of_the_check_valves"></span>
@@ Line 616: / Line 1,493: @@
 <math>\Delta P_{\mathrm{close}}\le\Delta P_{\mathrm{open}}.</math>
+<div class="mw-translate-fuzzy">
 Une ouverture ou une fermeture modifie la topologie hydraulique et donc les équations actives ; elle ne provoque aucun saut instantané de l'état thermodynamique. Pour chaque volume :
+</div>
 <math>m_j^+=m_j^-,\qquad U_j^+=U_j^-,\qquad V_j^+=V_j^-.</math>
@@ Line 626: / Line 1,505: @@
 Il n'y a donc pas d'égalisation instantanée des pressions à l'ouverture d'un clapet.
+<div class="mw-translate-fuzzy">
 Les valeurs <math display="inline">\Lambda^*</math> servent de cibles de conception cinématique ; les valeurs réellement observées lors des transitions sont <math display="inline">\Lambda_{\mathrm{real}}=\Lambda(t_{\mathrm{event}})</math>.
+</div>
+<div class="mw-translate-fuzzy">
 Les deux clapets passifs ont une orientation fixe :
+</div>
+<div class="mw-translate-fuzzy">
 <math>H_o\to S,\qquad H_i\to L.</math>
+</div>
+<div class="mw-translate-fuzzy">
 Leurs conditions d’ouverture et de fermeture sont identiques dans les deux modes de fonctionnement. Seule l’historique des pressions change en raison de la cinématique inversée de la manivelle en fonctionnement moteur.
+</div>
 <span id="8._Work,_heat,_and_performance"></span>
 == 8. Travail, chaleur et performance ==
+<div class="mw-translate-fuzzy">
 Le travail instantané du gaz sur les deux pistons est calculé pendant toutes les phases :
+</div>
+<div class="mw-translate-fuzzy">
 <math>
 \dot W=P_S^\star\dot V_S+P_L^\star\dot V_L
 .</math>
+</div>
-<math display="inline">P_S^\star</math> et <math display="inline">P_L^\star</math> désignent la pression thermodynamique effectivement appliquée au gaz dans chaque cylindre selon la topologie de la phase : pression de paire lorsque le cylindre appartient à une paire quasi égalisée, pression propre lorsqu'il est donneur isolé hydrauliquement.
 Le travail net du cycle est :
+<div class="mw-translate-fuzzy">
 <math>
 W_{\mathrm{cycle}}=
@@ Line 652: / Line 1,542: @@
 \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">
+When the two cylinder pressures are comparable during a pressure-changing phase,
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>
+\dot W\approx P(\dot V_S+\dot V_L).
+</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+If both cylinders contract during compression, or both expand during expansion, their work contributions therefore add instead of partly cancelling. This allows a larger fraction of the available swept volume to participate in compression and expansion for a comparable pressure history. It does not by itself prove a higher thermal efficiency, because the heat transfers and the resulting pressure history change at the same time.
+</div>
 Les chaleurs échangées sont :
@@ Line 700: / Line 1,606: @@
+<div class="mw-translate-fuzzy">
 La force thermodynamique exercée par le gaz sur une face de piston peut être écrite <math display="inline">F_{\mathrm{gas}}=PS</math>. La force mécanique nette, les inerties et les frottements relèvent du dimensionnement mécanique ultérieur.
+</div>
 <span id="9._Gas_charge_and_periodic_regime"></span>
@@ Line 748: / Line 1,656: @@
 === 10.2 Paramètres de conception ===
+<div class="mw-translate-fuzzy">
 * <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_i</math>, <math display="inline">V_o</math> ;
@@ Line 754: / Line 1,663: @@
 * seuils <math display="inline">\Delta P_{\mathrm{open}}</math>, <math display="inline">\Delta P_{\mathrm{close}}</math> ;
 * cibles cinématiques <math display="inline">\Lambda_L^*</math>, <math display="inline">\Lambda_S^*</math>.
+</div>
 <span id="10.3_Calculated_variables_and_results"></span>
 === 10.3 Variables et résultats calculés ===
+<div class="mw-translate-fuzzy">
 * <math display="inline">m_j,U_j,T_j,P_j,\qquad j\in\{S,L,i,o\}</math> ;
 * débits massiques internes et externes ;
@@ Line 766: / Line 1,677: @@
 * qualité d'isothermie <math display="inline">\varepsilon_T</math> ;
 * critères de validité <math display="inline">\varepsilon_P</math>, <math display="inline">Ma</math>, <math display="inline">Z</math> et variation des propriétés.
+</div>
 <span id="11._Global_conservation_checks"></span>
@@ Line 786: / Line 1,698: @@
 Quelle que soit la phase, les flux massiques et enthalpiques internes doivent s'annuler lorsque les bilans de tous les volumes sont additionnés. Le bilan global doit se réduire à :
+<div class="mw-translate-fuzzy">
 <math>
 \frac{dU_{\mathrm{tot}}}{dt}
@@ Line 792: / Line 1,705: @@
 -P_L^\star\dot V_L
 .</math>
+</div>
 Un résidu énergétique cumulatif peut être défini par :
@@ Line 896: / Line 1,810: @@
 Cette relation découle uniquement de la conservation de masse et de l'équation d'état ; elle est valable pour les deux sens de débit.
-<span id="A.4_Open_receiving_pair"></span>
+<span id="A.4_Open_quasi-pressure-equalized_pair"></span>
+<div class="mw-translate-fuzzy">
 === A.4 Paire réceptrice ouverte ===
+</div>
+<div class="mw-translate-fuzzy">
 Pour une paire recevant <math display="inline">\dot m_{\mathrm{ext}}</math> dans son cylindre :
+</div>
 <math>\frac{dU}{dt}=\dot m_{\mathrm{ext}}C_pT_{\mathrm{ext}}
@@ Line 916: / Line 1,834: @@
 .</math>
+<div class="mw-translate-fuzzy">
 Le bilan de masse local :
+</div>
 <math>\dot m_{\mathrm{cyl}}=\dot m_{\mathrm{ext}}-\dot m_{\mathrm{int}},</math>
+<div class="mw-translate-fuzzy">
 <math>\dot m_{HX}=\dot m_{\mathrm{int}},</math>
+</div>
+<div class="mw-translate-fuzzy">
 redonne immédiatement :
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>\dot m_{\mathrm{cyl}}=-\dot m_{\mathrm{int}},\qquad
+\dot m_{HX}=\dot m_{\mathrm{ext}}+\dot m_{\mathrm{int}}.</math>
+</div>
+<div class="mw-translate-fuzzy">
 <math>\dot m_{\mathrm{pair}}=\dot m_{\mathrm{ext}}.</math>
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>\dot m_{\mathrm{pair}}=\dot m_{\mathrm{ext}}.</math>
+</div>
 <span id="A.5_Analytical_solution_for_the_adiabatic_donor_cylinder"></span>
@@ Line 1,005: / Line 1,940: @@
 * si <math display="inline">T_{\mathrm{in}}=T</math> et <math display="inline">\dot m_{\mathrm{in}}=\dot m_{\mathrm{out}}</math>, la contribution nette du débit à <math display="inline">\dot T</math> s'annule.
-<span id="A.7_Global_mass_test_during_an_active_phase"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-=== A.7 Test global de masse en phase active ===
+=== A.7 Global mass test ===
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Using the signed link flows defined in §6:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>\dot m_S=-\dot m_{Si}+\dot m_{oS},</math>
+</div>
-Pour la Phase II :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>\dot m_L=\dot m_{iL}-\dot m_{Lo},</math>
+</div>
-<math>\dot m_L=-\dot m_{\mathrm{in}},</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>\dot m_i=\dot m_{Si}-\dot m_{iL},</math>
+</div>
-<math>\dot m_o=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}},</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>\dot m_o=\dot m_{Lo}-\dot m_{oS}.</math>
+</div>
-<math>\dot m_{Si}=\dot m_{\mathrm{out}}.</math>
-En sommant :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Summing the four equations cancels every internal mass flow exactly:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>\dot M_{\mathrm{tot}}=0.</math>
+</div>
-La Phase IV donne exactement le même résultat par symétrie.
-<span id="A.8_Global_energy_test_during_an_active_phase"></span>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-=== A.8 Test global d'énergie en phase active ===
+This result is independent of the active phase, the flow directions on bidirectional links, and the positions of the two check valves.
+</div>
-Pour la Phase II :
-<math>\dot U_L=-P_L\dot V_L-\dot m_{\mathrm{in}}h_L,</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+=== A.8 Global energy test ===
+</div>
-<math>\dot U_o=\dot m_{\mathrm{in}}h_L-\dot m_{\mathrm{out}}h_o+\dot Q_o,</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+Using the signed enthalpy fluxes defined in §6:
+</div>
-<math>\dot U_{Si}=\dot m_{\mathrm{out}}h_o+\dot Q_i-P_{Si}\dot V_S.</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>\dot U_S=-P_S\dot V_S-\dot H_{Si}+\dot H_{oS},</math>
+</div>
-Les flux enthalpiques internes s'annulent exactement :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>\dot U_L=-P_L\dot V_L+\dot H_{iL}-\dot H_{Lo},</math>
+</div>
-<math>-\dot m_{\mathrm{in}}h_L+\dot m_{\mathrm{in}}h_L=0,</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>\dot U_i=\dot H_{Si}-\dot H_{iL}+\dot Q_i,</math>
+</div>
-<math>-\dot m_{\mathrm{out}}h_o+\dot m_{\mathrm{out}}h_o=0.</math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>\dot U_o=\dot H_{Lo}-\dot H_{oS}+\dot Q_o.</math>
+</div>
-Il reste :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+All internal enthalpy fluxes cancel exactly when the four balances are summed:
+</div>
+<div lang="en" dir="ltr" class="mw-content-ltr">
 <math>
 \dot U_{\mathrm{tot}}
 =\dot Q_i+\dot Q_o
--P_L\dot V_L-P_{Si}\dot V_S
+-P_S\dot V_S-P_L\dot V_L
 .</math>
+</div>
-La Phase IV fournit la relation symétrique. Dans les phases fermées, la même structure résulte directement de la somme des bilans des deux paires. D'où, pour toute phase :
-<math>
+<div lang="en" dir="ltr" class="mw-content-ltr">
-\dot U_{\mathrm{tot}}
+Integrated over a periodic cycle:
-=\dot Q_i+\dot Q_o
+</div>
--P_S^\star\dot V_S-P_L^\star\dot V_L
-.</math>
-Intégrée sur un cycle périodique, cette relation redonne :
+<div lang="en" dir="ltr" class="mw-content-ltr">
+<math>Q_i+Q_o=W_{\mathrm{cycle}}.</math>
+</div>
-<math>Q_i+Q_o=W_{\mathrm{cycle}}.</math>
 <span id="A.9_Continuity_at_transitions"></span>
@@ Line 1,077: / Line 2,044: @@
 <math>T_j^+=T_j^-,\qquad P_j^+=P_j^-.</math>
+<div class="mw-translate-fuzzy">
 L'événement ne crée qu'un changement de topologie hydraulique et de système d'équations actif.
+</div>

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Thermodynamic and Mechanical Study/fr: Difference between revisions

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Revision as of 19:03, 20 September 2026

1. Périmètre et méthode

1.1 Hypothèses de premier niveau

2. Notations et conventions

2.1 Géométrie et cinématique

2.2 Fraction de fermeture cinématique

2.3 Variables thermodynamiques

2.4 Convention de signe énergétique

3. Fermeture thermique et domaine de validité

3.1 Échange avec les réservoirs thermiques

3.2 Domaine de validité thermophysique

4. Reduced formulations for quasi-pressure-equalized connected volumes

4.1 Pressure equation for a connected set

4.2 Closed connected set

4.3 Fixed-volume heat exchanger within a pressure-equalized set

4.4 Two-volume cylinder–exchanger special case

4.5 Single external-flow special cases

5. Fermeture hydraulique

5.1 Formulation générique

5.2 Fermeture de premier niveau par orifice compressible

6. Complete thermodynamic cycle

6.1 Thermodynamic rationale of the four phases

6.1.1 Compression and expansion

6.1.2 Exchange phases

6.1.3 Balance equations valid throughout the cycle

6.2 Phase I — compression

6.3 Phase II — exchange through Ho: L → Ho → S

6.4 Phase III — expansion

6.5 Phase IV — exchange through Hi: S → Hi → L

6.6 Motor operation

7. Transitions physiques des clapets

8. Travail, chaleur et performance

9. Charge de gaz et régime périodique

10. Paramètres, données de conception et résultats

10.1 Données imposées

10.2 Paramètres de conception

10.3 Variables et résultats calculés

11. Contrôles globaux de conservation

11.1 Conservation de la masse

11.2 Conservation globale de l'énergie

Annexe A — Dérivations symboliques et contrôles validés

A.1 Équation de pression d'une paire fermée

A.2 Débit interne de la paire

A.3 Évolution de la température de l'échangeur dans une paire

A.4 Paire réceptrice ouverte

A.5 Solution analytique du cylindre donneur adiabatique

A.6 Échangeur actif : bilan développé

A.7 Global mass test

A.8 Global energy test

A.9 Continuité aux transitions

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