Thermodynamic and Mechanical Study/en - Revision history

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	== 1. Scope and method ==		== 1. Scope and method ==

	This study describes the complete driven cycle of the Dada ~~machine~~ in symbolic form. 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.		This study describes the complete driven cycle of the Dada engine in symbolic form. 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 ~~at the '''top position of~~ the large cylinder~~'''~~, with:		The cycle origin is set with the large cylinder at maximum volume, with:

	<math>t=0,\qquad \theta(0)=0,\qquad V_L(0)=V_{L,\max}.</math>		<math>t=0,\qquad \theta(0)=0,\qquad V_L(0)=V_{L,\max}.</math>
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	If these criteria become insufficient, an extension may use <math display="inline">Z(P,T)</math>, <math display="inline">C_p(T)</math>, <math display="inline">C_v(T)</math>, or a real-gas equation of state without changing the general architecture of the mass and energy balances.		If these criteria become insufficient, an extension may use <math display="inline">Z(P,T)</math>, <math display="inline">C_p(T)</math>, <math display="inline">C_v(T)</math>, or a real-gas equation of state without changing the general architecture of the mass and energy balances.

	== 4. Reduced formulation of a quasi pressure-equalized pair ==		== 4. Reduced formulation of a quasi-pressure-equalized pair ==

	When a cylinder and its heat exchanger are connected by a very low-resistance internal path, the approximation		When a cylinder and its heat exchanger are connected by a very low-resistance internal path, the approximation
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	=== 6.2 Phase II — heat exchange, nominally isothermal: L → H → S ===		=== 6.2 Phase II — heat exchange, nominally isothermal: L → H → S ===

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

	==== 6.2.1 Donor cylinder L ====		==== 6.2.1 Donor cylinder L ====
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	=== 6.4 Phase IV — heat exchange, nominally isothermal: S → C → L ===		=== 6.4 Phase IV — heat exchange, nominally isothermal: S → C → L ===

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

	==== 6.4.1 Donor cylinder S ====		==== 6.4.1 Donor cylinder S ====
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	.</math>		.</math>

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

	The net work over the cycle is:		The net work over the cycle is:
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	<math display="inline">P_{\mathrm{charge}}</math> and <math display="inline">T_{\mathrm{charge}}</math> define the amount of gas charged; they are not conditions that the periodic cycle must recover.		<math display="inline">P_{\mathrm{charge}}</math> and <math display="inline">T_{\mathrm{charge}}</math> define the amount of gas charged; they are not conditions that the periodic cycle must recover.

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

	<math>\mathbf X(t+\tau)=\mathbf X(t).</math>		<math>\mathbf X(t+\tau)=\mathbf X(t).</math>
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	----		----

	= Appendix A — Symbolic derivations and validated checks =		== Appendix A — Symbolic derivations and validated checks =

	== A.1 Pressure equation for a closed pair ==		=== A.1 Pressure equation for a closed pair ==

	For a cylinder + heat-exchanger pair at quasi-uniform pressure:		For a cylinder + heat-exchanger pair at quasi-uniform pressure:
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	<math>PV^\gamma=\mathrm{const}.</math>		<math>PV^\gamma=\mathrm{const}.</math>

	== A.2 Internal flow rate of the pair ==		=== A.2 Internal flow rate of the pair ==

	For the heat exchanger alone, at fixed volume:		For the heat exchanger alone, at fixed volume:
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	This recovers the definition of the numerator <math display="inline">N</math> and the selection of the upstream temperature according to the sign of the flow rate.		This recovers the definition of the numerator <math display="inline">N</math> and the selection of the upstream temperature according to the sign of the flow rate.

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

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

	== A.4 Open receiving pair ==		=== A.4 Open receiving pair ==

	For a pair receiving <math display="inline">\dot m_{\mathrm{ext}}</math> into its cylinder:		For a pair receiving <math display="inline">\dot m_{\mathrm{ext}}</math> into its cylinder:
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	<math>\dot m_{\mathrm{pair}}=\dot m_{\mathrm{ext}}.</math>		<math>\dot m_{\mathrm{pair}}=\dot m_{\mathrm{ext}}.</math>

	== A.5 Analytical solution for the adiabatic donor cylinder ==		=== A.5 Analytical solution for the adiabatic donor cylinder ==

	For an adiabatic, well-mixed cylinder with outflow only:		For an adiabatic, well-mixed cylinder with outflow only:
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	Under these assumptions, the specific entropy of the remaining gas is constant: <math display="inline">ds=0</math>. The total entropy of the gas contained in the cylinder is not constant because its mass varies.		Under these assumptions, the specific entropy of the remaining gas is constant: <math display="inline">ds=0</math>. The total entropy of the gas contained in the cylinder is not constant because its mass varies.

	== A.6 Active heat exchanger: expanded balance ==		=== A.6 Active heat exchanger: expanded balance ==

	Fundamental balance:		Fundamental balance:
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	* if <math display="inline">T_{\mathrm{in}}=T</math> and <math display="inline">\dot m_{\mathrm{in}}=\dot m_{\mathrm{out}}</math>, the net contribution of the flow to <math display="inline">\dot T</math> vanishes.		* if <math display="inline">T_{\mathrm{in}}=T</math> and <math display="inline">\dot m_{\mathrm{in}}=\dot m_{\mathrm{out}}</math>, the net contribution of the flow to <math display="inline">\dot T</math> vanishes.

	== A.7 Global mass test during an active phase ==		=== A.7 Global mass test during an active phase ==

	For Phase II:		For Phase II:
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	Phase IV gives exactly the same result by symmetry.		Phase IV gives exactly the same result by symmetry.

	== A.8 Global energy test during an active phase ==		=== A.8 Global energy test during an active phase ==

	For Phase II:		For Phase II:
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	<math>Q_C+Q_H=W_{\mathrm{cycle}}.</math>		<math>Q_C+Q_H=W_{\mathrm{cycle}}.</math>

	== A.9 Continuity at transitions ==		=== 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:		At the instant of a check-valve event, no finite mass or energy can be transferred in zero time. The conserved variables and geometry are therefore continuous:

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== 1. Scope and method ==

This study describes the complete driven cycle of the Dada machine in symbolic form. 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 at the '''top position of the large cylinder''', with:

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

The four gas volumes are:

* <code>S</code>: small cylinder, directly associated with the cold heat exchanger <code>C</code>;
* <code>L</code>: large cylinder, directly associated with the hot heat exchanger <code>H</code>;
* <code>C</code>: cold heat exchanger, which absorbs heat from the cold reservoir;
* <code>H</code>: hot heat exchanger, which rejects heat to the hot reservoir.

The check valves allow <code>H → S</code> during Phase II and <code>C → L</code> during Phase IV.

The cycle comprises four phases:

* '''Phase I''' — check valves closed, nominally adiabatic: compression on the <code>L+H</code> side;
* '''Phase II''' — heat exchange, nominally isothermal: transfer <code>L → H → S</code>, with heat rejection on the hot side;
* '''Phase III''' — check valves closed, nominally adiabatic: expansion on the <code>S+C</code> side;
* '''Phase IV''' — heat exchange, nominally isothermal: transfer <code>S → C → L</code>, with heat absorption on the cold side.

The terms ''nominally adiabatic'' and ''nominally isothermal'' describe the objective of the cycle. They do not constitute exact thermodynamic constraints: temperatures are calculated from the balances, the heat exchangers remain coupled to their reservoirs, and the low-displacement regions of the pistons are not assumed to be perfectly stationary.

=== 1.1 First-level assumptions ===

The model is based on the following assumptions:

* single-phase, ideal and calorically perfect gas;
* constant properties <math display="inline">R</math>, <math display="inline">C_p</math>, <math display="inline">C_v</math>, <math display="inline">\gamma</math>, with <math display="inline">R=C_p-C_v</math> and <math display="inline">\gamma=C_p/C_v</math>;
* 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 <math display="inline">UA</math>;
* passive check valves controlled by the pressure difference.

== 2. Notation and conventions ==

=== 2.1 Geometry and kinematics ===

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

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

The swept volume is:

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

The thermodynamic volumes prescribed by the mechanism are:

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

with their '''signed''' derivatives:

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

If the crank rotates at constant angular speed <math display="inline">\omega</math>:

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

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

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

A '''quasi-stationary region''' denotes an interval in which the displacement or <math display="inline">|\dot V|</math> remains small compared with the transfer phases. This region corresponds to the “plateau” of the kinematic optimization, without assuming <math display="inline">\dot V=0</math> exactly.

=== 2.2 Kinematic closure fraction ===

To describe the normalized closure of a cylinder:

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

Thus <math display="inline">\Lambda_i=0</math> corresponds to maximum volume and <math display="inline">\Lambda_i=1</math> to minimum volume. The values <math display="inline">\Lambda_L^*</math> and <math display="inline">\Lambda_S^*</math> are nominal kinematic targets at the transitions. The value actually reached at a check-valve event is:

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

The physical transitions remain determined by the pressures; <math display="inline">\Lambda^*</math> is therefore not an imposed opening condition.

=== 2.3 Thermodynamic variables ===

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

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

The complete state vector is chosen as:

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

Temperatures and pressures are derived from it:

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

The volumes <math display="inline">V_S(t)</math> and <math display="inline">V_L(t)</math> are prescribed by the kinematics and are not independent thermodynamic state variables. The volumes <math display="inline">V_C</math> and <math display="inline">V_H</math> 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''':

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

For the intended refrigeration operation:

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

== 3. Thermal closure and validity domain ==

=== 3.1 Exchange with the thermal reservoirs ===

For a heat exchanger <math display="inline">HX\in\{C,H\}</math>:

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

<math display="inline">T_{HX,\mathrm{res}}</math> is the temperature of the external thermal reservoir, prescribed and constant in the base model. <math display="inline">T_{HX}</math> is the mean 0D temperature of the gas in the heat exchanger. <math display="inline">(UA)_{HX}</math> represents the overall thermal conductance, which may combine convection, wall conduction, and contact resistances.

Cold side:

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

In refrigeration operation, <math display="inline">T_C<T_{C,\mathrm{res}}</math> gives <math display="inline">\dot Q_C>0</math>.

Hot side:

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

In refrigeration operation, <math display="inline">T_H>T_{H,\mathrm{res}}</math> gives <math display="inline">\dot Q_H<0</math>.

A nominally isothermal phase therefore does not mean <math display="inline">T_{HX}=T_{HX,\mathrm{res}}</math>: a finite temperature difference is required to transfer finite thermal power when <math display="inline">UA</math> is finite.

An indicator of isothermal quality may be defined over a given phase by:

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

=== 3.2 Thermophysical validity domain ===

The base model assumes:

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

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

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

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

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

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

The working fluid must remain single-phase and gaseous, and sufficiently far from any condensation or phase transition throughout the <math display="inline">(P,T)</math> domain traversed.

If these criteria become insufficient, an extension may use <math display="inline">Z(P,T)</math>, <math display="inline">C_p(T)</math>, <math display="inline">C_v(T)</math>, or a real-gas equation of state without changing the general architecture of the mass and energy balances.

== 4. Reduced formulation of a quasi pressure-equalized pair ==

When a cylinder and its heat exchanger are connected by a very low-resistance internal path, the approximation

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

may be used. It is acceptable if:

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

A low internal Mach number provides an additional check:

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

but it is not sufficient on its own to guarantee pressure equalization.

=== 4.1 Closed-pair case ===

For a closed pair, with fixed <math display="inline">V_{HX}</math> and <math display="inline">V=V_{\mathrm{cyl}}+V_{HX}</math>:

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

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

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

=== 4.2 Internal redistribution flow rate ===

The internal flow rate is defined as positive from cylinder → heat exchanger. Let:

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

The flow carries the enthalpy of the upstream state:

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

The heat-exchanger temperature evolves according to:

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

=== 4.3 Open receiving pair ===

During an active phase, the external flow physically enters the '''receiving cylinder''', not directly its associated heat exchanger. If <math display="inline">\dot m_{\mathrm{ext}}>0</math> enters the cylinder at temperature <math display="inline">T_{\mathrm{ext}}</math>:

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

The masses satisfy:

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

and therefore:

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

The closed case is obtained immediately with <math display="inline">\dot m_{\mathrm{ext}}=0</math>.

== 5. Hydraulic closure ==

=== 5.1 Generic formulation ===

Any hydraulic connection is described by a generic law:

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

<math display="inline">u</math> and <math display="inline">d</math> respectively denote the upstream and downstream states. The transported enthalpy is that of the upstream state:

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

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.

=== 5.2 First-level closure using a compressible orifice ===

A first approximation consists in using an effective hydraulic area:

<math>(C_dA)_{\mathrm{eff}},</math>

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

With:

<math>r=\frac{P_d}{P_u},\qquad
r_{\mathrm{crit}}=\left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)},</math>

the unchoked flow rate, for <math display="inline">r>r_{\mathrm{crit}}</math>, is:

<math>
\dot m=(C_dA)_{\mathrm{eff}}P_u
\sqrt{\frac{2\gamma}{RT_u(\gamma-1)}
\left(r^{2/\gamma}-r^{(\gamma+1)/\gamma}\right)}
.</math>

For <math display="inline">r\le r_{\mathrm{crit}}</math>:

<math>
\dot m=(C_dA)_{\mathrm{eff}}P_u
\sqrt{\frac{\gamma}{RT_u}}
\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}
.</math>

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 ==

=== 6.1 Phase I — check valves closed, nominally adiabatic: compression on the L side ===

Both check valves are closed. The <code>L+H</code> and <code>S+C</code> pairs are closed. The <code>L+H</code> side is nominally compressed; the motion of the small piston remains that provided by the actual kinematics.

For <code>L+H</code>:

<math>\dot P_{LH}=
\frac{(\gamma-1)(UA)_H(T_{H,\mathrm{res}}-T_H)-\gamma P_{LH}\dot V_L}
{V_L+V_H}.</math>

For <code>S+C</code>:

<math>\dot P_{SC}=
\frac{(\gamma-1)(UA)_C(T_{C,\mathrm{res}}-T_C)-\gamma P_{SC}\dot V_S}
{V_S+V_C}.</math>

The internal redistribution and temperature equations of §4 apply to both pairs.

The transition to Phase II occurs when the <code>H → S</code> check valve satisfies its opening condition.

=== 6.2 Phase II — heat exchange, nominally isothermal: L → H → S ===

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

==== 6.2.1 Donor cylinder L ====

The fundamental balance is:

<math>
\frac{d(m_LC_vT_L)}{dt}
=-P_L\dot V_L-\dot m_{L,\mathrm{out}}C_pT_L
.</math>

The kinematics are designed to maintain an outflow from the donor cylinder. In this case, the analytical solution is:

<math>
\frac{T_L}{T_{L,\mathrm{ref}}}=
\left[\frac{m_L}{m_{L,\mathrm{ref}}}\frac{V_{L,\mathrm{ref}}}{V_L}\right]^{\gamma-1}
</math>

and:

<math>
P_L=P_{L,\mathrm{ref}}
\left[\frac{m_L}{m_{L,\mathrm{ref}}}\frac{V_{L,\mathrm{ref}}}{V_L}\right]^\gamma
.</math>

One also obtains:

<math>\frac{T_L}{T_{L,\mathrm{ref}}}=
\left(\frac{P_L}{P_{L,\mathrm{ref}}}\right)^{(\gamma-1)/\gamma},</math>

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

==== 6.2.2 Hot heat exchanger H ====

Mass conservation:

<math>\dot m_H=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}}.</math>

Fundamental energy balance:

<math>
\frac{d(m_HC_vT_H)}{dt}
=\dot m_{\mathrm{in}}C_pT_L
-\dot m_{\mathrm{out}}C_pT_H
+(UA)_H(T_{H,\mathrm{res}}-T_H)
.</math>

In expanded form:

<math>
\dot T_H=
\frac{\dot m_{\mathrm{in}}(C_pT_L-C_vT_H)
-\dot m_{\mathrm{out}}RT_H
+(UA)_H(T_{H,\mathrm{res}}-T_H)}
{C_vm_H}
.</math>

The flow rates are determined by the hydraulic laws:

<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,H}}(P_L,P_H,T_L,\ldots),</math>

<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,H}}(P_H,P_{SC},T_H,\ldots).</math>

==== 6.2.3 Receiving pair S+C ====

The flow from <code>H</code> enters '''S'''. The pair pressure satisfies:

<math>
\dot P_{SC}=
\frac{\gamma R\dot m_{\mathrm{out}}T_H
+(\gamma-1)(UA)_C(T_{C,\mathrm{res}}-T_C)
-\gamma P_{SC}\dot V_S}
{V_S+V_C}
.</math>

The internal redistribution equations of §4 remain unchanged: the external flow acts on heat exchanger <code>C</code> indirectly through the evolution of the pair pressure.

The transition to Phase III is the closing event of the <code>H → S</code> check valve.

=== 6.3 Phase III — check valves closed, nominally adiabatic: expansion on the S side ===

Both check valves are closed. The <code>S+C</code> pair expands nominally; the <code>L+H</code> pair also remains closed. Neither piston is assumed to be strictly stationary.

For <code>S+C</code>:

<math>\dot P_{SC}=
\frac{(\gamma-1)(UA)_C(T_{C,\mathrm{res}}-T_C)-\gamma P_{SC}\dot V_S}
{V_S+V_C}.</math>

For <code>L+H</code>:

<math>\dot P_{LH}=
\frac{(\gamma-1)(UA)_H(T_{H,\mathrm{res}}-T_H)-\gamma P_{LH}\dot V_L}
{V_L+V_H}.</math>

The transition to Phase IV occurs when the <code>C → L</code> check valve satisfies its opening condition.

=== 6.4 Phase IV — heat exchange, nominally isothermal: S → C → L ===

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

==== 6.4.1 Donor cylinder S ====

The fundamental balance is:

<math>
\frac{d(m_SC_vT_S)}{dt}
=-P_S\dot V_S-\dot m_{S,\mathrm{out}}C_pT_S
.</math>

For outflow guaranteed by the kinematic design:

<math>
\frac{T_S}{T_{S,\mathrm{ref}}}=
\left[\frac{m_S}{m_{S,\mathrm{ref}}}\frac{V_{S,\mathrm{ref}}}{V_S}\right]^{\gamma-1}
</math>

and:

<math>
P_S=P_{S,\mathrm{ref}}
\left[\frac{m_S}{m_{S,\mathrm{ref}}}\frac{V_{S,\mathrm{ref}}}{V_S}\right]^\gamma
.</math>

Reverse flow requires returning to the complete open-system balance in <math display="inline">(m_S,U_S)</math>.

==== 6.4.2 Cold heat exchanger C ====

<math>\dot m_C=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}}.</math>

The fundamental energy balance is:

<math>
\frac{d(m_CC_vT_C)}{dt}
=\dot m_{\mathrm{in}}C_pT_S
-\dot m_{\mathrm{out}}C_pT_C
+(UA)_C(T_{C,\mathrm{res}}-T_C)
.</math>

In expanded form:

<math>
\dot T_C=
\frac{\dot m_{\mathrm{in}}(C_pT_S-C_vT_C)
-\dot m_{\mathrm{out}}RT_C
+(UA)_C(T_{C,\mathrm{res}}-T_C)}
{C_vm_C}
.</math>

The flow rates are determined by:

<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,C}}(P_S,P_C,T_S,\ldots),</math>

<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,C}}(P_C,P_{LH},T_C,\ldots).</math>

==== 6.4.3 Receiving pair L+H ====

The flow from <code>C</code> enters '''L'''. The pair pressure satisfies:

<math>
\dot P_{LH}=
\frac{\gamma R\dot m_{\mathrm{out}}T_C
+(\gamma-1)(UA)_H(T_{H,\mathrm{res}}-T_H)
-\gamma P_{LH}\dot V_L}
{V_L+V_H}
.</math>

Closing the <code>C → L</code> check valve returns the system to Phase I. Geometric closure of the pistons alone is not sufficient to guarantee thermodynamic closure of the cycle.

== 7. Physical transitions of the check valves ==

For a check valve oriented from upstream <math display="inline">u</math> to downstream <math display="inline">d</math>:

<math>P_u-P_d\ge\Delta P_{\mathrm{open}}
\quad\Rightarrow\quad \text{opening},</math>

<math>P_u-P_d\le\Delta P_{\mathrm{close}}
\quad\Rightarrow\quad \text{closing},</math>

with hysteresis, if present:

<math>\Delta P_{\mathrm{close}}\le\Delta P_{\mathrm{open}}.</math>

An opening or closing event changes the hydraulic topology and therefore the active equations; it does not cause any instantaneous jump in the thermodynamic state. For each volume:

<math>m_i^+=m_i^-,\qquad U_i^+=U_i^-,\qquad V_i^+=V_i^-.</math>

For an ideal gas:

<math>T_i^+=T_i^-,\qquad P_i^+=P_i^-.</math>

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

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

== 8. Work, heat, and coefficient of performance ==

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

<math>
\dot W=P_S^\star\dot V_S+P_L^\star\dot V_L
.</math>

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

The net work over the cycle is:

<math>
W_{\mathrm{cycle}}=
\int_0^\tau
\left(P_S^\star\dot V_S+P_L^\star\dot V_L\right)dt
.</math>

The exchanged heats are:

<math>Q_C=\int_0^\tau (UA)_C(T_{C,\mathrm{res}}-T_C)\,dt,</math>

<math>Q_H=\int_0^\tau (UA)_H(T_{H,\mathrm{res}}-T_H)\,dt.</math>

In periodic steady operation:

<math>\Delta U_{\mathrm{cycle}}=0,</math>

and the first law gives:

<math>Q_C+Q_H=W_{\mathrm{cycle}}.</math>

The refrigeration COP is:

<math>COP_c=\frac{Q_C}{-W_{\mathrm{cycle}}}.</math>

The heat-pump COP is:

<math>COP_h=\frac{-Q_H}{-W_{\mathrm{cycle}}}=COP_c+1.</math>

The signs <math display="inline">Q_C>0</math>, <math display="inline">Q_H<0</math>, and <math display="inline">W_{\mathrm{cycle}}<0</math> provide checks of the intended refrigeration regime.

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

== 9. Gas charge and periodic regime ==

The total amount of enclosed gas is a physical parameter:

<math>M_{\mathrm{tot}}=m_S+m_L+m_C+m_H=\mathrm{const}.</math>

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

<math>
M_{\mathrm{tot}}=
\frac{P_{\mathrm{charge}}
\left[V_S(0)+V_{L,\max}+V_C+V_H\right]}
{RT_{\mathrm{charge}}}
.</math>

<math display="inline">P_{\mathrm{charge}}</math> and <math display="inline">T_{\mathrm{charge}}</math> define the amount of gas charged; they are not conditions that the periodic cycle must recover.

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

<math>\mathbf X(t+\tau)=\mathbf X(t).</math>

Geometric periodicity alone:

<math>V_i(t+\tau)=V_i(t)</math>

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 ===

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

=== 10.2 Design parameters ===

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

=== 10.3 Calculated variables and results ===

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

== 11. Global conservation checks ==

=== 11.1 Mass conservation ===

The solver must satisfy:

<math>\frac{dM_{\mathrm{tot}}}{dt}=0.</math>

A useful numerical residual is:

<math>\varepsilon_M(t)=M_{\mathrm{tot}}(t)-M_{\mathrm{tot}}(0).</math>

=== 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:

<math>
\frac{dU_{\mathrm{tot}}}{dt}
=\dot Q_C+\dot Q_H
-P_S^\star\dot V_S
-P_L^\star\dot V_L
.</math>

A cumulative energy residual may be defined by:

<math>
\varepsilon_E(t)=
U_{\mathrm{tot}}(t)-U_{\mathrm{tot}}(0)
-Q_C(0,t)-Q_H(0,t)+W(0,t)
.</math>

The solver must keep <math display="inline">\varepsilon_M</math> and <math display="inline">\varepsilon_E</math> close to zero to the expected numerical accuracy.

----

= 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:

<math>U=\frac{P(V_{\mathrm{cyl}}+V_{HX})}{\gamma-1}.</math>

The first law gives:

<math>\frac{dU}{dt}=\dot Q-P\dot V_{\mathrm{cyl}},</math>

with:

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

Differentiating <math display="inline">U</math>:

<math>\frac{1}{\gamma-1}
\left[(V_{\mathrm{cyl}}+V_{HX})\dot P+P\dot V_{\mathrm{cyl}}\right]
=\dot Q-P\dot V_{\mathrm{cyl}}.</math>

Hence:

<math>
\dot P=
\frac{(\gamma-1)\dot Q-\gamma P\dot V_{\mathrm{cyl}}}
{V_{\mathrm{cyl}}+V_{HX}}
.</math>

If <math display="inline">\dot Q=0</math>:

<math>\frac{\dot P}{P}=-\gamma\frac{\dot V}{V},</math>

then:

<math>PV^\gamma=\mathrm{const}.</math>

== A.2 Internal flow rate of the pair ==

For the heat exchanger alone, at fixed volume:

<math>U_{HX}=\frac{PV_{HX}}{\gamma-1}.</math>

Therefore:

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

Using <math display="inline">C_p=\gamma R/(\gamma-1)</math>:

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

This recovers the definition of the numerator <math display="inline">N</math> and the selection of the upstream temperature according to the sign of the flow rate.

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

For a fixed volume:

<math>m_{HX}=\frac{PV_{HX}}{RT_{HX}}.</math>

Differentiating:

<math>\frac{\dot m_{HX}}{m_{HX}}
=\frac{\dot P}{P}-\frac{\dot T_{HX}}{T_{HX}}.</math>

With <math display="inline">\dot m_{HX}=\dot m_{\mathrm{int}}</math>:

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

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

== A.4 Open receiving pair ==

For a pair receiving <math display="inline">\dot m_{\mathrm{ext}}</math> into its cylinder:

<math>\frac{dU}{dt}=\dot m_{\mathrm{ext}}C_pT_{\mathrm{ext}}
+\dot Q-P\dot V_{\mathrm{cyl}}.</math>

With <math display="inline">U=P(V_{\mathrm{cyl}}+V_{HX})/(\gamma-1)</math>:

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

The local mass balance:

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

<math>\dot m_{HX}=\dot m_{\mathrm{int}},</math>

immediately gives:

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

== A.5 Analytical solution for the adiabatic donor cylinder ==

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

<math>\frac{d(mC_vT)}{dt}=-P\dot V-\dot m_{\mathrm{out}}C_pT,</math>

and:

<math>\dot m=-\dot m_{\mathrm{out}}.</math>

Expanding:

<math>C_vm\dot T+C_vT\dot m=-P\dot V+C_pT\dot m,</math>

thus:

<math>C_vm\dot T=-P\dot V+RT\dot m.</math>

With <math display="inline">P=mRT/V</math> and <math display="inline">R/C_v=\gamma-1</math>:

<math>\frac{dT}{T}=(\gamma-1)
\left(\frac{dm}{m}-\frac{dV}{V}\right).</math>

After integration:

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

Then, using <math display="inline">PV=mRT</math>:

<math>
P=P_0\left[\frac{m}{m_0}\frac{V_0}{V}\right]^\gamma
.</math>

and:

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

Under these assumptions, the specific entropy of the remaining gas is constant: <math display="inline">ds=0</math>. The total entropy of the gas contained in the cylinder is not constant because its mass varies.

== A.6 Active heat exchanger: expanded balance ==

Fundamental balance:

<math>\frac{d(mC_vT)}{dt}
=\dot m_{\mathrm{in}}C_pT_{\mathrm{in}}
-\dot m_{\mathrm{out}}C_pT
+\dot Q.</math>

Expanding the left-hand side and using:

<math>\dot m=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}},</math>

one obtains:

<math>
\dot T=
\frac{\dot m_{\mathrm{in}}(C_pT_{\mathrm{in}}-C_vT)
-\dot m_{\mathrm{out}}RT+\dot Q}
{C_vm}
.</math>

Limiting checks:

* with no flow, the equation recovers the thermal relaxation of a closed volume;
* with equal steady inlet/outlet flow rates, it recovers <math display="inline">mC_v\dot T=\dot mC_p(T_{\mathrm{in}}-T)+\dot Q</math>;
* if <math display="inline">T_{\mathrm{in}}=T</math> and <math display="inline">\dot m_{\mathrm{in}}=\dot m_{\mathrm{out}}</math>, the net contribution of the flow to <math display="inline">\dot T</math> vanishes.

== A.7 Global mass test during an active phase ==

For Phase II:

<math>\dot m_L=-\dot m_{\mathrm{in}},</math>

<math>\dot m_H=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}},</math>

<math>\dot m_{SC}=\dot m_{\mathrm{out}}.</math>

Summing:

<math>\dot M_{\mathrm{tot}}=0.</math>

Phase IV gives exactly the same result by symmetry.

== A.8 Global energy test during an active phase ==

For Phase II:

<math>\dot U_L=-P_L\dot V_L-\dot m_{\mathrm{in}}h_L,</math>

<math>\dot U_H=\dot m_{\mathrm{in}}h_L-\dot m_{\mathrm{out}}h_H+\dot Q_H,</math>

<math>\dot U_{SC}=\dot m_{\mathrm{out}}h_H+\dot Q_C-P_{SC}\dot V_S.</math>

The internal enthalpy fluxes cancel exactly:

<math>-\dot m_{\mathrm{in}}h_L+\dot m_{\mathrm{in}}h_L=0,</math>

<math>-\dot m_{\mathrm{out}}h_H+\dot m_{\mathrm{out}}h_H=0.</math>

What remains is:

<math>
\dot U_{\mathrm{tot}}
=\dot Q_C+\dot Q_H
-P_L\dot V_L-P_{SC}\dot V_S
.</math>

Phase IV provides the symmetric relation. In the closed phases, the same structure follows directly from summing the balances of the two pairs. Hence, for any phase:

<math>
\dot U_{\mathrm{tot}}
=\dot Q_C+\dot Q_H
-P_S^\star\dot V_S-P_L^\star\dot V_L
.</math>

Integrated over a periodic cycle, this relation gives:

<math>Q_C+Q_H=W_{\mathrm{cycle}}.</math>

== 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:

<math>m_i^+=m_i^-,\qquad U_i^+=U_i^-,\qquad V_i^+=V_i^-.</math>

For an ideal gas:

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

which implies:

<math>T_i^+=T_i^-,\qquad P_i^+=P_i^-.</math>

The event only creates a change in hydraulic topology and in the active system of equations.