Thermodynamic and Mechanical Study

1. Scope and method

This study describes the complete thermodynamic cycle of the Dada engine in symbolic form, in both refrigeration and motor operation. The model couples prescribed kinematics of the two pistons, well-mixed gas volumes, 0D heat exchangers, passive check valves, and a generic compressible hydraulic closure model.

The cycle origin is set with the large cylinder at maximum volume, with:

$t=0,\theta (0)=0,V_L(0)=V_{L,max}.$

The four gas volumes are:

• $S$: small cylinder;
• $L$: large cylinder;
• $H_i$: heat-in exchanger, located on the $S\to L$ hydraulic path and transferring heat into the gas;
• $H_o$: heat-out exchanger, located on the $L\to S$ hydraulic path and transferring heat out of the gas.

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:

$S\to {\mathrm{C}\mathrm{V}}_i\to H_i\to L,$

or:

$S\to H_i\to {\mathrm{C}\mathrm{V}}_i\to L.$

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

$L\to {\mathrm{C}\mathrm{V}}_o\to H_o\to S,$

or:

$L\to H_o\to {\mathrm{C}\mathrm{V}}_o\to S.$

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 First-level assumptions

The model is based on the following assumptions:

• single-phase, ideal and calorically perfect gas;
• constant properties $R$, $C_p$, $C_v$, $\gamma$, with $R=C_p-C_v$ and $\gamma =C_p/C_v$;
• each gas volume is uniform and well mixed;
• gas kinetic and potential energies are neglected in the 0D balances;
• fixed heat-exchanger volumes;
• prescribed piston kinematics;
• no mechanical friction in the thermodynamic model;
• heat exchange represented by an overall conductance $UA$;
• passive check valves controlled by the pressure difference.

2. Notation and conventions

2.1 Geometry and kinematics

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

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

The swept volume is:

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

The thermodynamic volumes prescribed by the mechanism are:

$V_S=V_S(t),V_L=V_L(t),$

with their signed derivatives:

${\dot{V}}_S=\frac{d{V}_S}{dt},{\dot{V}}_L=\frac{d{V}_L}{dt}.$

The crank angular velocity is signed:

$\theta (t)=\omega t,{\dot{V}}_k=\omega \frac{d{V}_k}{d\theta },k\in \{S,L\}.$

The driven refrigeration direction is chosen as positive:

$\omega >0.$

Motor operation uses the opposite crank direction:

$\omega <0.$

The geometric origin is identical in both modes:

$t=0,\theta (0)=0,V_L(0)=V_{L,max}.$

When only the absolute value of the volumetric speed is useful:

${\nu }_{V,k}=|{\dot{V}}_k|.$

A quasi-stationary region denotes an interval in which the displacement or $|\dot{V}|$ remains small compared with the characteristic piston motion over the cycle. This region corresponds to the “plateau” of the kinematic optimization, without assuming $\dot{V}=0$ exactly.

2.2 Kinematic closure fraction

To describe the normalized closure of a cylinder:

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

Thus ${\Lambda }_k=0$ corresponds to maximum volume and ${\Lambda }_k=1$ to minimum volume. The values ${\Lambda }_L^{\ast }$ and ${\Lambda }_S^{\ast }$ may be used as nominal kinematic reference levels at selected phase boundaries. The value actually observed at a check-valve event may be recorded as:

${\Lambda }_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}=\Lambda (t_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}}).$

Check-valve events are determined by the local pressure difference across the valve and are independent of the selected kinematic phase boundaries; ${\Lambda }^{\ast }$ is therefore not an imposed opening condition.

2.3 Thermodynamic variables

For each volume $j\in \{S,L,i,o\}$:

$m_j,U_j,T_j,P_j,V_j.$

The subscripts $i$ and $o$ denote respectively the gas contained in $H_i$ and $H_o$.

The complete state vector is chosen as:

$𝐗=(m_S,U_S,m_L,U_L,m_i,U_i,m_o,U_o).$

Temperatures and pressures are derived from:

$T_j=\frac{U_j}{m_j{C}_v},P_j=\frac{m_{j}R{T}_j}{V_j}.$

The volumes $V_S(t)$ and $V_L(t)$ are prescribed by the kinematics and are not independent thermodynamic state variables. The volumes $V_i$ and $V_o$ are constant.

2.4 Energy sign convention

Heat is positive when it is received by the gas. Work is positive when the gas delivers work:

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

By definition of the two physical heat exchangers:

$Q_i>0,Q_o<0$

in the intended operating regime of both refrigeration and motor operation.

The net cycle work distinguishes the two modes:

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

for driven refrigeration operation, whereas:

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

for motor operation.

3. Thermal closure and validity domain

3.1 Exchange with the thermal reservoirs

For the heat-in exchanger:

${\dot{Q}}_i=(UA{)}_i(T_{i,\mathrm{r}\mathrm{e}\mathrm{s}}-T_i).$

In the intended operating regime:

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

For the heat-out exchanger:

${\dot{Q}}_o=(UA{)}_o(T_{o,\mathrm{r}\mathrm{e}\mathrm{s}}-T_o).$

In the intended operating regime:

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

The reservoir temperatures depend on the operating mode.

For refrigeration operation:

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

For motor operation:

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

Thus the heat-transfer equations themselves are identical in both modes.

3.2 Thermophysical validity domain

The base model assumes:

$PV=mRT,Z=1,$

$C_p=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t},C_v=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t},\gamma =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$

Validity must be checked a posteriori over the entire cycle, notably through:

$|Z-1|\ll 1,$

and through small variations of the thermophysical properties, for example:

${\epsilon }_{C_p}=\frac{C_{p,max}-C_{p,min}}{C_{p,\mathrm{r}\mathrm{e}\mathrm{f}}}\ll 1.$

The working fluid must remain single-phase and gaseous, and sufficiently far from any condensation or phase transition throughout the $(P,T)$ domain traversed.

If these criteria become insufficient, an extension may use $Z(P,T)$, $C_p(T)$, $C_v(T)$, or a real-gas equation of state without changing the general architecture of the mass and energy balances.

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:

${\epsilon }_{P,𝒞}=\underset{(a,b)\in 𝒞}{max}\frac{|P_a-P_b|}{P_{𝒞}}\ll 1.$

A low internal Mach number provides an additional check:

$M{a}_{\mathrm{i}\mathrm{n}\mathrm{t}}=\frac{|u_{\mathrm{i}\mathrm{n}\mathrm{t}}|}{a}\ll 1,$

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:

$U_{𝒞}=\sum _{j\in 𝒞}{m}_j{C}_v{T}_j=\frac{P_{𝒞}{V}_{𝒞}}{\gamma -1}.$

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:

${\dot{H}}_{𝒞}^{\mathrm{e}\mathrm{x}\mathrm{t}}=\sum _{\mathrm{i}\mathrm{n}}\dot{m}{C}_p{T}_u-\sum _{\mathrm{o}\mathrm{u}\mathrm{t}}\dot{m}{C}_p{T}_u.$

The first law for the complete connected set is then:

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

Therefore:

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

The total mass of the set satisfies:

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

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 $𝒞$:

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

The pressure equation becomes:

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

If the set is also adiabatic:

${\dot{Q}}_{𝒞}=0,$

then:

$P_{𝒞}{V}_{𝒞}^{\gamma }=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$

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:

$\frac{V_j}{\gamma -1}{\dot{P}}_{𝒞}={\dot{Q}}_j+\sum _{\mathrm{i}\mathrm{n}}\dot{m}{C}_p{T}_u-\sum _{\mathrm{o}\mathrm{u}\mathrm{t}}\dot{m}{C}_p{T}_j.$

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:

$N=V_{HX}\dot{P}+(\gamma -1)(UA{)}_{HX}(T_{HX}-T_{HX,\mathrm{r}\mathrm{e}\mathrm{s}}).$

The internal flow rate is:

${\dot{m}}_{\mathrm{i}\mathrm{n}\mathrm{t}}=\{\begin{array}{cc}\frac{N}{\gamma R{T}_{\mathrm{c}\mathrm{y}\mathrm{l}}},& N\ge 0(\mathrm{c}\mathrm{y}\mathrm{l}\to HX),\\ \frac{N}{\gamma R{T}_{HX}},& N<0(HX\to \mathrm{c}\mathrm{y}\mathrm{l}).\end{array}$

The heat-exchanger temperature then satisfies:

${\dot{T}}_{HX}=\frac{T_{HX}}{P}\dot{P}-\frac{R{T}_{HX}^2}{P{V}_{HX}}{\dot{m}}_{\mathrm{i}\mathrm{n}\mathrm{t}}.$

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 ${\dot{m}}_{\mathrm{e}\mathrm{x}\mathrm{t}}>0$ at upstream temperature $T_{\mathrm{e}\mathrm{x}\mathrm{t}}$:

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

with:

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

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

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

with:

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

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. Hydraulic closure

5.1 Generic formulation

Any hydraulic connection is described by a generic law:

$\dot{m}=\Phi (P_u,P_d,T_u,𝒢,ℱ)$

$u$ and $d$ respectively denote the instantaneous upstream and downstream states. The transported enthalpy is that of the upstream state:

${\dot{H}}_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}=\dot{m}{C}_p{T}_u.$

For a bidirectional connection, the upstream state is determined by the actual direction of the pressure gradient. For a check valve, reverse flow is prohibited. On each of the two hydraulic branches, the check valve may be placed on either side of the heat exchanger; the hydraulic law must therefore use the pressures immediately adjacent to the actual valve position.

5.2 First-level closure using a compressible orifice

A first approximation consists in using an effective hydraulic area:

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

which represents the overall ease of gas flow through the actual connection.

With:

$r=\frac{P_d}{P_u},r_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}={\left(\frac{2}{\gamma +1}\right)}^{\gamma /(\gamma -1)},$

the unchoked flow rate, for $r>r_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$, is:

$\dot{m}=(C_{d}A{)}_{\mathrm{e}\mathrm{f}\mathrm{f}}{P}_u\sqrt{\frac{2\gamma }{R{T}_u(\gamma -1)}(r^{2/\gamma }-r^{(\gamma +1)/\gamma })}.$

For $r\le r_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$:

$\dot{m}=(C_{d}A{)}_{\mathrm{e}\mathrm{f}\mathrm{f}}{P}_u\sqrt{\frac{\gamma }{R{T}_u}}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}.$

This closure is not essential to the model: it may later be replaced by a law that more faithfully represents the pressure losses of a real heat exchanger, pipe, or check valve.

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 $\omega >0$. 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:

$\Delta s_{\mathrm{h}\mathrm{y}\mathrm{d}}=R\ln \left(\frac{P_u}{P_d}\right)>0\text{for}{P}_u>P_d.$

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:

$\frac{d}{dt}\left(\sum _{j\in 𝒞}\frac{V_j}{T_j}\right)=0.$

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:

$U_{𝒞}=\frac{P{V}_{𝒞}}{\gamma -1},V_{𝒞}=\sum _{j\in 𝒞}{V}_j.$

Its first-law balance gives:

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

An approximately isobaric exchange therefore satisfies:

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

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:

$\frac{V_{\mathrm{s}\mathrm{w}\mathrm{e}\mathrm{p}\mathrm{t},h}}{V_{\mathrm{s}\mathrm{w}\mathrm{e}\mathrm{p}\mathrm{t},c}}\sim \frac{T_h}{T_c}.$

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:

${\dot{m}}_{Si}:S\to H_i,{\dot{m}}_{iL}:H_i\to L,$

${\dot{m}}_{Lo}:L\to H_o,{\dot{m}}_{oS}:H_o\to S.$

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:

$S\to {\mathrm{C}\mathrm{V}}_i\to H_i\to L\text{or}S\to H_i\to {\mathrm{C}\mathrm{V}}_i\to L,$

$L\to {\mathrm{C}\mathrm{V}}_o\to H_o\to S\text{or}L\to H_o\to {\mathrm{C}\mathrm{V}}_o\to S.$

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

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

${\dot{H}}_{ab}=\{\begin{array}{cc}{\dot{m}}_{ab}{C}_p{T}_a,& {\dot{m}}_{ab}\ge 0,\\ {\dot{m}}_{ab}{C}_p{T}_b,& {\dot{m}}_{ab}<0.\end{array}$

The mass balances are then:

${\dot{m}}_S={\dot{m}}_{oS}-{\dot{m}}_{Si},$

${\dot{m}}_i={\dot{m}}_{Si}-{\dot{m}}_{iL},$

${\dot{m}}_L={\dot{m}}_{iL}-{\dot{m}}_{Lo},$

${\dot{m}}_o={\dot{m}}_{Lo}-{\dot{m}}_{oS}.$

The corresponding energy balances are:

${\dot{U}}_S={\dot{H}}_{oS}-{\dot{H}}_{Si}-P_S{\dot{V}}_S,$

${\dot{U}}_i={\dot{H}}_{Si}-{\dot{H}}_{iL}+{\dot{Q}}_i,$

${\dot{U}}_L={\dot{H}}_{iL}-{\dot{H}}_{Lo}-P_L{\dot{V}}_L,$

${\dot{U}}_o={\dot{H}}_{Lo}-{\dot{H}}_{oS}+{\dot{Q}}_o.$

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 ${\mathrm{C}\mathrm{V}}_o$ 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 ${\mathrm{C}\mathrm{V}}_i$ 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:

$\omega <0,$

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

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

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:

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

7. Physical transitions of the check valves

For a check valve oriented from upstream $u$ to downstream $d$:

$P_u-P_d\ge \Delta P_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}}\Rightarrow \text{opening},$

$P_u-P_d\le \Delta P_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}}\Rightarrow \text{closing},$

with hysteresis, if present:

$\Delta P_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}}\le \Delta P_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}}.$

An opening or closing event changes the admissible hydraulic flow on the valve link; it does not define a kinematic phase boundary and it does not cause any instantaneous jump in the thermodynamic state. For each volume:

$m_j^{+}=m_j^{-},U_j^{+}=U_j^{-},V_j^{+}=V_j^{-}.$

For an ideal gas:

$T_j^{+}=T_j^{-},P_j^{+}=P_j^{-}.$

There is therefore no instantaneous pressure equalization when a check valve opens.

The values ${\Lambda }^{\ast }$ may serve as kinematic reference levels at selected phase boundaries; the values actually observed at check-valve events are ${\Lambda }_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}=\Lambda (t_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}})$.

The two passive check valves have permanent branch orientations: one permits circulation along $L\to H_o\to S$, the other along $S\to H_i\to L$.

Each valve may be installed upstream or downstream of its heat exchanger. Its opening and closing conditions therefore use the pressures immediately adjacent to its actual position.

The opening and closing laws are identical in both operating modes. Only the pressure histories change because the crank kinematics are reversed in motor operation.

8. Work, heat, and performance

The instantaneous work delivered by the gas on the two pistons is calculated during all phases from the actual cylinder pressures:

$\dot{W}=P_S{\dot{V}}_S+P_L{\dot{V}}_L.$

The net work over the cycle is:

$W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}={\int }_0^{\tau }(P_S{\dot{V}}_S+P_L{\dot{V}}_L)dt.$

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.

The exchanged heats are:

$Q_i={\int }_0^{\tau }(UA{)}_i(T_{i,\mathrm{r}\mathrm{e}\mathrm{s}}-T_i)dt,$

$Q_o={\int }_0^{\tau }(UA{)}_o(T_{o,\mathrm{r}\mathrm{e}\mathrm{s}}-T_o)dt.$

In periodic steady operation:

$\Delta U_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}=0,$

and the first law gives:

$Q_i+Q_o=W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}.$

The refrigeration COP is:

$CO{P}_c=\frac{Q_i}{-W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}}.$

The heat-pump COP is:

$CO{P}_h=\frac{-Q_o}{-W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}}=CO{P}_c+1.$

The signs $Q_i>0$, $Q_o<0$, and $W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}<0$ provide checks of the intended refrigeration regime.

For motor operation:

$Q_i>0,Q_o<0,W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}>0.$

The thermal efficiency is:

${\eta }_{\mathrm{t}\mathrm{h}}=\frac{W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}}{Q_i}=1+\frac{Q_o}{Q_i}.$

The mean thermodynamic motor power is:

$\bar{\dot{W}}=\frac{W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}}{\tau }.$

The thermodynamic force exerted by the gas on a piston face may be written $F_{\mathrm{g}\mathrm{a}\mathrm{s}}=PS$. Net mechanical force, inertia, and friction belong to the subsequent mechanical sizing phase.

9. Gas charge and periodic regime

The total amount of enclosed gas is a physical parameter:

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}=m_S+m_L+m_i+m_o=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$

It may be imposed directly or defined experimentally by a charging pressure and temperature. The chosen reference configuration is that at $t=0$, with the large cylinder at maximum volume. If all volumes communicate and are in uniform equilibrium during charging:

$M_{\mathrm{t}\mathrm{o}\mathrm{t}}=\frac{P_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}}[V_S(0)+V_{L,max}+V_i+V_o]}{R{T}_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}}}.$

$P_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}}$ and $T_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}}$ define the amount of gas charged; they are not conditions that the periodic cycle must recover.

The established periodic regime is a solution of the system such that, between two successive passages through the maximum volume of the large cylinder with the same kinematic direction:

$𝐗(t+\tau )=𝐗(t).$

Geometric periodicity alone:

$V_k(t+\tau )=V_k(t),k\in \{S,L\}.$

is not sufficient to guarantee thermodynamic periodicity.

The numerical state used to initialize a calculation may be approximate; it must not be confused with a physical parameter of the machine. The future solver may search for the periodic fixed point by successive cycles, a shooting method, or a Newton method.

10. Parameters, design data, and results

10.1 Prescribed data

• operating mode;
• reservoir temperatures $T_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d}}$ and $T_{\mathrm{h}\mathrm{o}\mathrm{t}}$;
• signed crank angular velocity $\omega$.
• working fluid and reference properties $R$, $C_p$, $C_v$, $\gamma$;
• total charge $M_{\mathrm{t}\mathrm{o}\mathrm{t}}$, or equivalently $(P_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}},T_{\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}})$ in the charging configuration defined in §9;
• kinematics $V_S(t)$, $V_L(t)$.

10.2 Design parameters

• $V_{S,min}$, $V_{S,max}$, $V_{L,min}$, $V_{L,max}$;
• $V_i$, $V_o$;
• $(UA{)}_i$, $(UA{)}_o$;
• hydraulic parameters of the heat exchangers, pipes, and check valves, represented at first level by $(C_{d}A{)}_{\mathrm{e}\mathrm{f}\mathrm{f}}$;
• thresholds $\Delta P_{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}}$, $\Delta P_{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}}$;
• position of each check valve upstream or downstream of its heat exchanger;
• kinematic reference levels ${\Lambda }_L^{\ast }$, ${\Lambda }_S^{\ast }$, when used.

10.3 Calculated variables and results

• $m_j,U_j,T_j,P_j,j\in \{S,L,i,o\}$;
• internal and external mass flow rates;
• ${\dot{Q}}_i$, ${\dot{Q}}_o$, $Q_i$, $Q_o$;
• $W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}$, $CO{P}_c$, $CO{P}_h$;
• ${\eta }_{\mathrm{t}\mathrm{h}}$ in motor operation.
• pressure, temperature, and flow-rate extrema;
• actual check-valve events and ${\Lambda }_{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$;
• validity criteria ${\epsilon }_P$, $Ma$, $Z$, and property variations.

11. Global conservation checks

11.1 Mass conservation

The solver must satisfy:

$\frac{d{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{dt}=0.$

A useful numerical residual is:

${\epsilon }_M(t)=M_{\mathrm{t}\mathrm{o}\mathrm{t}}(t)-M_{\mathrm{t}\mathrm{o}\mathrm{t}}(0).$

11.2 Global energy conservation

Whatever the phase, the internal mass and enthalpy fluxes must cancel when the balances of all volumes are summed. The global balance must reduce to:

$\frac{d{U}_{\mathrm{t}\mathrm{o}\mathrm{t}}}{dt}={\dot{Q}}_i+{\dot{Q}}_o-P_S{\dot{V}}_S-P_L{\dot{V}}_L.$

A cumulative energy residual may be defined by:

${\epsilon }_E(t)=U_{\mathrm{t}\mathrm{o}\mathrm{t}}(t)-U_{\mathrm{t}\mathrm{o}\mathrm{t}}(0)-Q_i(0,t)-Q_o(0,t)+W(0,t).$

Over a periodic cycle:

$Q_i+Q_o=W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}.$

The solver must keep ${\epsilon }_M$ and ${\epsilon }_E$ close to zero to the expected numerical accuracy.

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Appendix A — Symbolic derivations and validated checks

A.1 Pressure equation for a closed pair

For a cylinder + heat-exchanger pair at quasi-uniform pressure:

$U=\frac{P(V_{\mathrm{c}\mathrm{y}\mathrm{l}}+V_{HX})}{\gamma -1}.$

The first law gives:

$\frac{dU}{dt}=\dot{Q}-P{\dot{V}}_{\mathrm{c}\mathrm{y}\mathrm{l}},$

with:

$\dot{Q}=(UA{)}_{HX}(T_{HX,\mathrm{r}\mathrm{e}\mathrm{s}}-T_{HX}).$

Differentiating $U$:

$\frac{1}{\gamma -1}[(V_{\mathrm{c}\mathrm{y}\mathrm{l}}+V_{HX})\dot{P}+P{\dot{V}}_{\mathrm{c}\mathrm{y}\mathrm{l}}]=\dot{Q}-P{\dot{V}}_{\mathrm{c}\mathrm{y}\mathrm{l}}.$

Hence:

$\dot{P}=\frac{(\gamma -1)\dot{Q}-\gamma P{\dot{V}}_{\mathrm{c}\mathrm{y}\mathrm{l}}}{V_{\mathrm{c}\mathrm{y}\mathrm{l}}+V_{HX}}.$

If $\dot{Q}=0$:

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

then:

$P{V}^{\gamma }=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$

A.2 Internal flow rate of the pair

For the heat exchanger alone, at fixed volume:

$U_{HX}=\frac{P{V}_{HX}}{\gamma -1}.$

Therefore:

$\frac{V_{HX}}{\gamma -1}\dot{P}=(UA{)}_{HX}(T_{HX,\mathrm{r}\mathrm{e}\mathrm{s}}-T_{HX})+{\dot{m}}_{\mathrm{i}\mathrm{n}\mathrm{t}}{C}_p{T}_{\mathrm{u}\mathrm{p}}.$

Using $C_p=\gamma R/(\gamma -1)$:

$V_{HX}\dot{P}+(\gamma -1)(UA{)}_{HX}(T_{HX}-T_{HX,\mathrm{r}\mathrm{e}\mathrm{s}})=\gamma R{T}_{\mathrm{u}\mathrm{p}}{\dot{m}}_{\mathrm{i}\mathrm{n}\mathrm{t}}.$

This recovers the definition of the numerator $N$ and the selection of the upstream temperature according to the sign of the flow rate.

A.3 Evolution of the heat-exchanger temperature within a pair

For a fixed volume:

$m_{HX}=\frac{P{V}_{HX}}{R{T}_{HX}}.$

Differentiating:

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

With ${\dot{m}}_{HX}={\dot{m}}_{\mathrm{i}\mathrm{n}\mathrm{t}}$:

${\dot{T}}_{HX}=\frac{T_{HX}}{P}\dot{P}-\frac{R{T}_{HX}^2}{P{V}_{HX}}{\dot{m}}_{\mathrm{i}\mathrm{n}\mathrm{t}}.$

This relation follows solely from mass conservation and the equation of state; it is valid for both flow directions.

A.4 Open quasi-pressure-equalized pair

For a pair receiving ${\dot{m}}_{\mathrm{e}\mathrm{x}\mathrm{t}}$ across its external boundary:

$\frac{dU}{dt}={\dot{m}}_{\mathrm{e}\mathrm{x}\mathrm{t}}{C}_p{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}+\dot{Q}-P{\dot{V}}_{\mathrm{c}\mathrm{y}\mathrm{l}}.$

With $U=P(V_{\mathrm{c}\mathrm{y}\mathrm{l}}+V_{HX})/(\gamma -1)$:

$\dot{P}=\frac{\gamma R{T}_{\mathrm{e}\mathrm{x}\mathrm{t}}{\dot{m}}_{\mathrm{e}\mathrm{x}\mathrm{t}}+(\gamma -1)\dot{Q}-\gamma P{\dot{V}}_{\mathrm{c}\mathrm{y}\mathrm{l}}}{V_{\mathrm{c}\mathrm{y}\mathrm{l}}+V_{HX}}.$

If the external stream enters the cylinder and ${\dot{m}}_{\mathrm{i}\mathrm{n}\mathrm{t}}$ is positive from cylinder to heat exchanger:

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

${\dot{m}}_{HX}={\dot{m}}_{\mathrm{i}\mathrm{n}\mathrm{t}}.$

If instead the external stream enters the heat exchanger first:

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

In both cases:

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

A.5 Analytical solution for the adiabatic donor cylinder

For an adiabatic, well-mixed cylinder with outflow only:

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

and:

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

Expanding:

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

thus:

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

With $P=mRT/V$ and $R/C_v=\gamma -1$:

$\frac{dT}{T}=(\gamma -1)(\frac{dm}{m}-\frac{dV}{V}).$

After integration:

$\frac{T}{T_0}={\left[\frac{m}{m_0}\frac{V_0}{V}\right]}^{\gamma -1}.$

Then, using $PV=mRT$:

$P=P_0{\left[\frac{m}{m_0}\frac{V_0}{V}\right]}^{\gamma }.$

and:

$\frac{T}{T_0}={\left(\frac{P}{P_0}\right)}^{(\gamma -1)/\gamma }.$

Under these assumptions, the specific entropy of the remaining gas is constant: $ds=0$. The total entropy of the gas contained in the cylinder is not constant because its mass varies.

A.6 Active heat exchanger: expanded balance

Fundamental balance:

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

Expanding the left-hand side and using:

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

one obtains:

$\dot{T}=\frac{{\dot{m}}_{\mathrm{i}\mathrm{n}}(C_p{T}_{\mathrm{i}\mathrm{n}}-C_{v}T)-{\dot{m}}_{\mathrm{o}\mathrm{u}\mathrm{t}}RT+\dot{Q}}{C_{v}m}.$

Limiting checks:

• with no flow, the equation recovers the thermal relaxation of a closed volume;
• with equal steady inlet/outlet flow rates, it recovers $m{C}_v\dot{T}=\dot{m}{C}_p(T_{\mathrm{i}\mathrm{n}}-T)+\dot{Q}$;
• if $T_{\mathrm{i}\mathrm{n}}=T$ and ${\dot{m}}_{\mathrm{i}\mathrm{n}}={\dot{m}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$, the net contribution of the flow to $\dot{T}$ vanishes.

A.7 Global mass test

Using the signed link flows defined in §6:

${\dot{m}}_S=-{\dot{m}}_{Si}+{\dot{m}}_{oS},$

${\dot{m}}_L={\dot{m}}_{iL}-{\dot{m}}_{Lo},$

${\dot{m}}_i={\dot{m}}_{Si}-{\dot{m}}_{iL},$

${\dot{m}}_o={\dot{m}}_{Lo}-{\dot{m}}_{oS}.$

Summing the four equations cancels every internal mass flow exactly:

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

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:

${\dot{U}}_S=-P_S{\dot{V}}_S-{\dot{H}}_{Si}+{\dot{H}}_{oS},$

${\dot{U}}_L=-P_L{\dot{V}}_L+{\dot{H}}_{iL}-{\dot{H}}_{Lo},$

${\dot{U}}_i={\dot{H}}_{Si}-{\dot{H}}_{iL}+{\dot{Q}}_i,$

${\dot{U}}_o={\dot{H}}_{Lo}-{\dot{H}}_{oS}+{\dot{Q}}_o.$

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

${\dot{U}}_{\mathrm{t}\mathrm{o}\mathrm{t}}={\dot{Q}}_i+{\dot{Q}}_o-P_S{\dot{V}}_S-P_L{\dot{V}}_L.$

Integrated over a periodic cycle:

$Q_i+Q_o=W_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{e}}.$

A.9 Continuity at transitions

At the instant of a check-valve event, no finite mass or energy can be transferred in zero time. The conserved variables and geometry are therefore continuous:

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

$m_j^{+}=m_j^{-},U_j^{+}=U_j^{-},V_j^{+}=V_j^{-}.$

For an ideal gas:

$T_j=\frac{U_j}{m_j{C}_v},P_j=\frac{m_{j}R{T}_j}{V_j},$

which implies:

$T_j^{+}=T_j^{-},P_j^{+}=P_j^{-}.$

The event only changes the admissible hydraulic flow on the valve link; the thermodynamic state remains continuous.