View source for Thermodynamic and Mechanical Study

<translate>
== 1. Scope and method == <!--T:1-->

<!--T:2-->
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.

<!--T:3-->
The cycle origin is set with the large cylinder at  maximum volume, with:

<!--T:4-->
<math>t=0,\qquad \theta(0)=0,\qquad V_L(0)=V_{L,\max}.</math>

<!--T:5-->
The four gas volumes are:

<!--T:6-->
* <math display="inline">S</math>: small cylinder;
* <math display="inline">L</math>: large cylinder;
* <math display="inline">H_i</math>: heat-in exchanger, located on the <math display="inline">S \to L</math> hydraulic path and transferring heat into the gas;
* <math display="inline">H_o</math>: heat-out exchanger, located on the <math display="inline">L \to S</math> hydraulic path and transferring heat out of the gas.

<!--T:346-->
The hydraulic topology is fixed:

<!--T:347-->
<math>S\to H_i\to L,</math>

<!--T:348-->
<math>L\to H_o\to S.</math>

<!--T:349-->
The passive check valves therefore always allow:

<!--T:350-->
<math>H_i\to L,\qquad H_o\to S.</math>

<!--T:351-->
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 comprises four hydraulic/thermodynamic phases:

<!--T:352-->
* '''Phase I''' — check valves closed, nominally adiabatic: compression on the <math display="inline">L+H_o</math> side;
* '''Phase II''' — heat exchange, nominally isothermal: transfer <math display="inline">L \to H_o \to S</math>, with heat removed from the gas;
* '''Phase III''' — check valves closed, nominally adiabatic: expansion on the <math display="inline">S+H_i</math> side;
* '''Phase IV''' — heat exchange, nominally isothermal: transfer <math display="inline">S \to H_i \to L</math>, with heat supplied to the gas.

<!--T:353-->
These phase definitions describe the permanent physical topology of the machine. The external reservoir associated with each heat exchanger depends on the operating mode. The phase boundaries and their angular positions are determined independently in each operating mode by the actual check-valve events.

<!--T:10-->
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 === <!--T:11-->

<!--T:12-->
The model is based on the following assumptions:

<!--T:13-->
* 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 == <!--T:14-->

=== 2.1 Geometry and kinematics === <!--T:15-->

<!--T:16-->
For <math display="inline">k\in\{S,L\}</math>:

<!--T:17-->
<math>V_{k,\min}>0,\qquad V_{k,\max}>V_{k,\min}.</math>

<!--T:18-->
The swept volume is:

<!--T:19-->
<math>V_{k,\mathrm{swept}}=V_{k,\max}-V_{k,\min}.</math>

<!--T:20-->
The thermodynamic volumes prescribed by the mechanism are:

<!--T:21-->
<math>V_S=V_S(t),\qquad V_L=V_L(t),</math>

<!--T:22-->
with their '''signed''' derivatives:

<!--T:23-->
<math>\dot V_S=\frac{dV_S}{dt},\qquad \dot V_L=\frac{dV_L}{dt}.</math>

<!--T:354-->
The crank angular velocity is signed:

<!--T:355-->
<math>\theta(t)=\omega t,\qquad
\dot V_k=\omega\frac{dV_k}{d\theta},
\qquad k\in\{S,L\}.</math>

<!--T:356-->
The driven refrigeration direction is chosen as positive:

<!--T:357-->
<math>\omega>0.</math>

<!--T:358-->
Motor operation uses the opposite crank direction:

<!--T:359-->
<math>\omega<0.</math>

<!--T:360-->
The geometric origin is identical in both modes:

<!--T:361-->
<math>t=0,\qquad \theta(0)=0,\qquad V_L(0)=V_{L,\max}.</math>

<!--T:26-->
When only the absolute value of the volumetric speed is useful:

<!--T:27-->
<math>\nu_{V,k}=|\dot V_k|.</math>

<!--T:28-->
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 === <!--T:29-->

<!--T:30-->
To describe the normalized closure of a cylinder:

<!--T:31-->
<math>\Lambda_k(t)=\frac{V_{k,\max}-V_k(t)}{V_{k,\max}-V_{k,\min}},\qquad k\in\{S,L\}.</math>

<!--T:32-->
Thus <math display="inline">\Lambda_k=0</math> corresponds to maximum volume and <math display="inline">\Lambda_k=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:

<!--T:33-->
<math>\Lambda_{\mathrm{real}}=\Lambda(t_{\mathrm{event}}).</math>

<!--T:34-->
The physical transitions remain determined by the pressures; <math display="inline">\Lambda^*</math> is therefore not an imposed opening condition.

=== 2.3 Thermodynamic variables === <!--T:35-->

<!--T:36-->
For each volume <math display="inline">j\in\{S,L,i,o\}</math>:

<!--T:37-->
<math>m_j,\qquad U_j,\qquad T_j,\qquad P_j,\qquad V_j.</math>

<!--T:362-->
The subscripts <math display="inline">i</math> and <math display="inline">o</math> denote respectively the gas contained in <math display="inline">H_i</math> and <math display="inline">H_o</math>.

<!--T:38-->
The complete state vector is chosen as:

<!--T:39-->
<math>\mathbf X=(m_S,U_S,m_L,U_L,m_i,U_i,m_o,U_o).</math>

<!--T:40-->
Temperatures and pressures are derived from:

<!--T:41-->
<math>T_j=\frac{U_j}{m_jC_v},\qquad
P_j=\frac{m_jRT_j}{V_j}.</math>

<!--T:42-->
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_i</math> and <math display="inline">V_o</math> are constant.

=== 2.4 Energy sign convention === <!--T:43-->

<!--T:44-->
Heat is positive when it is '''received by the gas'''. Work is positive when the '''gas delivers work''':

<!--T:45-->
<math>\dot W=P\dot V.</math>

<!--T:363-->
By definition of the two physical heat exchangers:

<!--T:364-->
<math>Q_i>0,\qquad Q_o<0</math>

<!--T:365-->
in the intended operating regime of both refrigeration and motor operation.

<!--T:366-->
The net cycle work distinguishes the two modes:

<!--T:367-->
<math>W_{\mathrm{cycle}}<0</math>

<!--T:368-->
for driven refrigeration operation, whereas:

<!--T:369-->
<math>W_{\mathrm{cycle}}>0</math>

<!--T:370-->
for motor operation.

== 3. Thermal closure and validity domain == <!--T:48-->

=== 3.1 Exchange with the thermal reservoirs === <!--T:49-->

<!--T:371-->
For the heat-in exchanger:

<!--T:372-->
<math>\dot Q_i=(UA)_i(T_{i,\mathrm{res}}-T_i).</math>

<!--T:373-->
In the intended operating regime:

<!--T:374-->
<math>T_i<T_{i,\mathrm{res}}
\quad\Rightarrow\quad
\dot Q_i>0.</math>

<!--T:375-->
For the heat-out exchanger:

<!--T:376-->
<math>\dot Q_o=(UA)_o(T_{o,\mathrm{res}}-T_o).</math>

<!--T:377-->
In the intended operating regime:

<!--T:378-->
<math>T_o>T_{o,\mathrm{res}}
\quad\Rightarrow\quad
\dot Q_o<0.</math>

<!--T:379-->
The reservoir temperatures depend on the operating mode.

<!--T:380-->
For refrigeration operation:

<!--T:381-->
<math>T_{i,\mathrm{res}}=T_{\mathrm{cold}},\qquad
T_{o,\mathrm{res}}=T_{\mathrm{hot}}.</math>

<!--T:382-->
For motor operation:

<!--T:383-->
<math>T_{i,\mathrm{res}}=T_{\mathrm{hot}},\qquad
T_{o,\mathrm{res}}=T_{\mathrm{cold}}.</math>

<!--T:384-->
Thus the heat-transfer equations themselves are identical in both modes.


<!--T:59-->
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.

<!--T:60-->
An indicator of isothermal quality may be defined over a given phase by:

<!--T:61-->
<math>\varepsilon_T=\frac{T_{\max}-T_{\min}}{T_{\mathrm{ref}}}.</math>

=== 3.2 Thermophysical validity domain === <!--T:62-->

<!--T:63-->
The base model assumes:

<!--T:64-->
<math>PV=mRT,\qquad Z=1,</math>

<!--T:65-->
<math>C_p=\mathrm{const},\qquad C_v=\mathrm{const},\qquad \gamma=\mathrm{const}.</math>

<!--T:66-->
Validity must be checked a posteriori over the entire cycle, notably through:

<!--T:67-->
<math>|Z-1|\ll1,</math>

<!--T:68-->
and through small variations of the thermophysical properties, for example:

<!--T:69-->
<math>\varepsilon_{C_p}=\frac{C_{p,\max}-C_{p,\min}}{C_{p,\mathrm{ref}}}\ll1.</math>

<!--T:70-->
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.

<!--T:71-->
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 == <!--T:72-->

<!--T:73-->
When a cylinder and its heat exchanger are connected by a very low-resistance internal path, the approximation

<!--T:74-->
<math>P_{\mathrm{cyl}}\approx P_{HX}=P_{\mathrm{pair}}.</math>

<!--T:75-->
may be used. It is acceptable if:

<!--T:76-->
<math>\varepsilon_P=\frac{|\Delta P_{\mathrm{int}}|}{P_{\mathrm{pair}}}\ll1,\qquad \Delta P_{\mathrm{int}}=P_{\mathrm{cyl}}-P_{HX}.</math>

<!--T:77-->
A low internal Mach number provides an additional check:

<!--T:78-->
<math>Ma_{\mathrm{int}}=\frac{|u_{\mathrm{int}}|}{a}\ll1,</math>

<!--T:79-->
but it is not sufficient on its own to guarantee pressure equalization.

=== 4.1 Closed-pair case === <!--T:80-->

<!--T:81-->
For a closed pair, with fixed <math display="inline">V_{HX}</math> and <math display="inline">V=V_{\mathrm{cyl}}+V_{HX}</math>:

<!--T:82-->
<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>

<!--T:83-->
When <math display="inline">(UA)_{HX}=0</math>:

<!--T:84-->
<math>P(V_{\mathrm{cyl}}+V_{HX})^\gamma=\mathrm{const}.</math>

=== 4.2 Internal redistribution flow rate === <!--T:85-->

<!--T:86-->
The internal flow rate is defined as positive from cylinder \to heat exchanger. Let:

<!--T:87-->
<math>N=V_{HX}\dot P+(\gamma-1)(UA)_{HX}(T_{HX}-T_{HX,\mathrm{res}}).</math>

<!--T:88-->
The flow carries the enthalpy of the upstream state:

<!--T:89-->
<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>

<!--T:90-->
The heat-exchanger temperature evolves according to:

<!--T:91-->
<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 === <!--T:92-->

<!--T:93-->
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>:

<!--T:94-->
<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>

<!--T:95-->
The masses satisfy:

<!--T:96-->
<math>\dot m_{\mathrm{cyl}}=\dot m_{\mathrm{ext}}-\dot m_{\mathrm{int}},\qquad
\dot m_{HX}=\dot m_{\mathrm{int}},</math>

<!--T:97-->
and therefore:

<!--T:98-->
<math>\dot m_{\mathrm{pair}}=\dot m_{\mathrm{ext}}.</math>

<!--T:99-->
The closed case is obtained immediately with <math display="inline">\dot m_{\mathrm{ext}}=0</math>.

== 5. Hydraulic closure == <!--T:100-->

=== 5.1 Generic formulation === <!--T:101-->

<!--T:102-->
Any hydraulic connection is described by a generic law:

<!--T:103-->
<math>\dot m=\Phi(P_u,P_d,T_u,\mathcal G,\mathcal F)</math>

<!--T:104-->
<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:

<!--T:105-->
<math>\dot H_{\mathrm{mass}}=\dot m C_pT_u.</math>

<!--T:106-->
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 === <!--T:107-->

<!--T:108-->
A first approximation consists in using an effective hydraulic area:

<!--T:109-->
<math>(C_dA)_{\mathrm{eff}},</math>

<!--T:110-->
which represents the overall ease of gas flow through the actual connection.

<!--T:111-->
With:

<!--T:112-->
<math>r=\frac{P_d}{P_u},\qquad
r_{\mathrm{crit}}=\left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)},</math>

<!--T:113-->
the unchoked flow rate, for <math display="inline">r>r_{\mathrm{crit}}</math>, is:

<!--T:114-->
<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>

<!--T:115-->
For <math display="inline">r\le r_{\mathrm{crit}}</math>:

<!--T:116-->
<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>

<!--T:117-->
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 == <!--T:118-->

=== 6.1 Phase I — check valves closed, nominally adiabatic: compression on the L side === <!--T:119-->

<!--T:120-->
Both check valves are closed. The <math display="inline">L+H_o</math> and <math display="inline">S+H_i</math> pairs are closed. The <math display="inline">L+H_o</math> side is nominally compressed; the motion of the small piston remains that provided by the actual kinematics.

<!--T:121-->
For <math display="inline">L+H_o</math>:

<!--T:122-->
<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>

<!--T:123-->
For <math display="inline">S+H_i</math>:

<!--T:124-->
<math>\dot P_{Si}=
\frac{(\gamma-1)(UA)_i(T_{i,\mathrm{res}}-T_i)-\gamma P_{Si}\dot V_S}
{V_S+V_i}.</math>

<!--T:125-->
The internal redistribution and temperature equations of §4 apply to both pairs.

<!--T:126-->
The transition to Phase II occurs when the <math display="inline">H_o \to S</math> check valve satisfies its opening condition.

=== 6.2 Phase II — heat exchange, nominally isothermal: L → Ho → S === <!--T:127-->

<!--T:128-->
The gas leaves <math display="inline">L</math>, passes through the heat-out exchanger <math display="inline">H_o</math>, where it rejects heat, crosses the <math display="inline">H_o \to S</math> check valve, and then enters the receiving cylinder <math display="inline">S</math>. The <math display="inline">S+H_i</math> pair remains quasi-pressure-equalized if the criterion <math display="inline">\varepsilon_P\ll1</math> is satisfied.

==== 6.2.1 Donor cylinder L ==== <!--T:129-->

<!--T:130-->
The fundamental balance is:

<!--T:131-->
<math>
\frac{d(m_LC_vT_L)}{dt}
=-P_L\dot V_L-\dot m_{L,\mathrm{out}}C_pT_L
.</math>

<!--T:132-->
The kinematics are designed to maintain an outflow from the donor cylinder. In this case, the analytical solution is:

<!--T:133-->
<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>

<!--T:134-->
and:

<!--T:135-->
<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>

<!--T:136-->
One also obtains:

<!--T:137-->
<math>\frac{T_L}{T_{L,\mathrm{ref}}}=
\left(\frac{P_L}{P_{L,\mathrm{ref}}}\right)^{(\gamma-1)/\gamma},</math>

<!--T:138-->
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 Heat-out exchanger Ho ==== <!--T:139-->

<!--T:140-->
Mass conservation:

<!--T:141-->
<math>\dot m_o=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}}.</math>

<!--T:142-->
Fundamental energy balance:

<!--T:143-->
<math>
\frac{d(m_oC_vT_o)}{dt}
=\dot m_{\mathrm{in}}C_pT_L
-\dot m_{\mathrm{out}}C_pT_o
+(UA)_o(T_{o,\mathrm{res}}-T_o)
.</math>

<!--T:144-->
In expanded form:

<!--T:145-->
<math>
\dot T_o=
\frac{\dot m_{\mathrm{in}}(C_pT_L-C_vT_o)
-\dot m_{\mathrm{out}}RT_o
+(UA)_o(T_{o,\mathrm{res}}-T_o)}
{C_vm_o}
.</math>

<!--T:146-->
The flow rates are determined by the hydraulic laws:

<!--T:147-->
<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,o}}(P_L,P_o,T_L,\ldots),</math>

<!--T:148-->
<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,o}}(P_o,P_{Si},T_o,\ldots).</math>

==== 6.2.3 Receiving pair S+Hi ==== <!--T:149-->

<!--T:150-->
The flow from <math display="inline">H_o</math> enters '''S'''. The pair pressure satisfies:

<!--T:151-->
<math>
\dot P_{Si}=
\frac{\gamma R\dot m_{\mathrm{out}}T_o
+(\gamma-1)(UA)_i(T_{i,\mathrm{res}}-T_i)
-\gamma P_{Si}\dot V_S}
{V_S+V_i}
.</math>

<!--T:152-->
The internal redistribution equations of §4 remain unchanged: the external flow acts on heat exchanger <math display="inline">H_i</math> indirectly through the evolution of the pair pressure.

<!--T:153-->
The transition to Phase III is the closing event of the <math display="inline">H_o \to S</math> check valve.

=== 6.3 Phase III — check valves closed, nominally adiabatic: expansion on the S side === <!--T:385-->

<!--T:155-->
Both check valves are closed. The <math display="inline">S+H_i</math> pair expands nominally; the <math display="inline">L+H_o</math> pair also remains closed. Neither piston is assumed to be strictly stationary.

<!--T:156-->
For <math display="inline">S+H_i</math>:

<!--T:157-->
<math>\dot P_{Si}=
\frac{(\gamma-1)(UA)_i(T_{i,\mathrm{res}}-T_i)-\gamma P_{Si}\dot V_S}
{V_S+V_i}.</math>

<!--T:158-->
For <math display="inline">L+H_o</math>:

<!--T:159-->
<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>

<!--T:160-->
The transition to Phase IV occurs when the <math display="inline">H_i \to L</math> check valve satisfies its opening condition.

=== 6.4 Phase IV — heat exchange, nominally isothermal: S → Hi → L === <!--T:161-->

<!--T:162-->
The gas leaves <math display="inline">S</math>, passes through the heat-in exchanger <math display="inline">H_i</math>, where it receives heat from its external reservoir, crosses the <math display="inline">H_i \to L</math> check valve, and then enters the receiving cylinder <math display="inline">L</math>. The <math display="inline">L+H_o</math> pair remains quasi-pressure-equalized if the criterion <math display="inline">\varepsilon_P\ll1</math> is satisfied.

==== 6.4.1 Donor cylinder S ==== <!--T:163-->

<!--T:164-->
The fundamental balance is:

<!--T:165-->
<math>
\frac{d(m_SC_vT_S)}{dt}
=-P_S\dot V_S-\dot m_{S,\mathrm{out}}C_pT_S
.</math>

<!--T:166-->
For outflow guaranteed by the kinematic design:

<!--T:167-->
<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>

<!--T:168-->
and:

<!--T:169-->
<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>

<!--T:170-->
Reverse flow requires returning to the complete open-system balance in <math display="inline">(m_S,U_S)</math>.

==== 6.4.2 Heat-in exchanger Hi ==== <!--T:386--> 

<!--T:172-->
<math>\dot m_i=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}}.</math>

<!--T:173-->
The fundamental energy balance is:

<!--T:174-->
<math>
\frac{d(m_iC_vT_i)}{dt}
=\dot m_{\mathrm{in}}C_pT_S
-\dot m_{\mathrm{out}}C_pT_i
+(UA)_i(T_{i,\mathrm{res}}-T_i)
.</math>

<!--T:175-->
In expanded form:

<!--T:176-->
<math>
\dot T_i=
\frac{\dot m_{\mathrm{in}}(C_pT_S-C_vT_i)
-\dot m_{\mathrm{out}}RT_i
+(UA)_i(T_{i,\mathrm{res}}-T_i)}
{C_vm_i}
.</math>

<!--T:177-->
The flow rates are determined by:

<!--T:178-->
<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,i}}(P_S,P_i,T_S,\ldots),</math>

<!--T:179-->
<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,i}}(P_i,P_{Lo},T_i,\ldots).</math>

==== 6.4.3 Receiving pair L+Ho ==== <!--T:180-->

<!--T:181-->
The flow from <math display="inline">H_i</math> enters '''L'''. The pair pressure satisfies:

<!--T:182-->
<math>
\dot P_{Lo}=
\frac{\gamma R\dot m_{\mathrm{out}}T_i
+(\gamma-1)(UA)_o(T_{o,\mathrm{res}}-T_o)
-\gamma P_{Lo}\dot V_L}
{V_L+V_o}
.</math>

<!--T:183-->
Closing the <math display="inline">H_i \to L</math> check valve returns the system to Phase I. Geometric closure of the pistons alone is not sufficient to guarantee thermodynamic closure of the cycle.

=== 6.5 Motor operation === <!--T:387-->

<!--T:388-->
No second set of thermodynamic balance equations is required for motor operation.

<!--T:389-->
The physical hydraulic topology remains:

<!--T:390-->
<math>L\to H_o\to S,\qquad
S\to H_i\to L.</math>

<!--T:391-->
The check-valve directions remain:

<!--T:392-->
<math>H_o\to S,\qquad H_i\to L.</math>

<!--T:393-->
Motor operation is obtained by:

<!--T:394-->
# reversing the crank direction,
# exchanging the external reservoirs connected to <math display="inline">H_i</math> and <math display="inline">H_o</math>.

<!--T:395-->
Thus:

<!--T:396-->
<math>\omega<0,</math>

<!--T:397-->
with the same cycle origin:

<!--T:398-->
<math>V_L(0)=V_{L,\max}.</math>

<!--T:399-->
The thermal-reservoir assignment becomes:

<!--T:400-->
<math>T_{i,\mathrm{res}}=T_{\mathrm{hot}},\qquad
T_{o,\mathrm{res}}=T_{\mathrm{cold}}.</math>

<!--T:401-->
The heat-in exchanger therefore absorbs heat from the hot reservoir:

<!--T:402-->
<math>Q_i>0,</math>

<!--T:403-->
while the heat-out exchanger rejects heat to the cold reservoir:

<!--T:404-->
<math>Q_o<0.</math>

<!--T:405-->
The pressure histories, mass flow rates, check-valve events, and periodic thermodynamic state must be recalculated with the reversed kinematics.

<!--T:406-->
The motor regime is obtained when:

<!--T:407-->
<math>W_{\mathrm{cycle}}>0.</math>

== 7. Physical transitions of the check valves == <!--T:184-->

<!--T:185-->
For a check valve oriented from upstream <math display="inline">u</math> to downstream <math display="inline">d</math>:

<!--T:186-->
<math>P_u-P_d\ge\Delta P_{\mathrm{open}}
\quad\Rightarrow\quad \text{opening},</math>

<!--T:187-->
<math>P_u-P_d\le\Delta P_{\mathrm{close}}
\quad\Rightarrow\quad \text{closing},</math>

<!--T:188-->
with hysteresis, if present:

<!--T:189-->
<math>\Delta P_{\mathrm{close}}\le\Delta P_{\mathrm{open}}.</math>

<!--T:190-->
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:

<!--T:191-->
<math>m_j^+=m_j^-,\qquad U_j^+=U_j^-,\qquad V_j^+=V_j^-.</math>

<!--T:192-->
For an ideal gas:

<!--T:193-->
<math>T_j^+=T_j^-,\qquad P_j^+=P_j^-.</math>

<!--T:194-->
There is therefore no instantaneous pressure equalization when a check valve opens.

<!--T:195-->
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>.

<!--T:408-->
The two passive check valves have permanent physical orientations:

<!--T:409-->
<math>H_o\to S,\qquad H_i\to L.</math>

<!--T:410-->
Their opening and closing conditions 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 == <!--T:196-->

<!--T:197-->
The instantaneous work delivered by the gas on the two pistons is calculated during all phases:

<!--T:198-->
<math>
\dot W=P_S^\star\dot V_S+P_L^\star\dot V_L
.</math>

<!--T:199-->
<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.

<!--T:200-->
The net work over the cycle is:

<!--T:201-->
<math>
W_{\mathrm{cycle}}=
\int_0^\tau
\left(P_S^\star\dot V_S+P_L^\star\dot V_L\right)dt
.</math>

<!--T:202-->
The exchanged heats are:

<!--T:203-->
<math>Q_i=\int_0^\tau (UA)_i(T_{i,\mathrm{res}}-T_i)\,dt,</math>

<!--T:204-->
<math>Q_o=\int_0^\tau (UA)_o(T_{o,\mathrm{res}}-T_o)\,dt.</math>

<!--T:205-->
In periodic steady operation:

<!--T:206-->
<math>\Delta U_{\mathrm{cycle}}=0,</math>

<!--T:207-->
and the first law gives:

<!--T:208-->
<math>Q_i+Q_o=W_{\mathrm{cycle}}.</math>

<!--T:209-->
The refrigeration COP is:

<!--T:210-->
<math>COP_c=\frac{Q_i}{-W_{\mathrm{cycle}}}.</math>

<!--T:211-->
The heat-pump COP is:

<!--T:212-->
<math>COP_h=\frac{-Q_o}{-W_{\mathrm{cycle}}}=COP_c+1.</math>

<!--T:213-->
The signs <math display="inline">Q_i>0</math>, <math display="inline">Q_o<0</math>, and <math display="inline">W_{\mathrm{cycle}}<0</math> provide checks of the intended refrigeration regime.

<!--T:411-->
For motor operation:

<!--T:412-->
<math>Q_i>0,\qquad Q_o<0,\qquad W_{\mathrm{cycle}}>0.</math>

<!--T:413-->
The thermal efficiency is:

<!--T:414-->
<math>
\eta_{\mathrm{th}}
=
\frac{W_{\mathrm{cycle}}}{Q_i}
=
1+\frac{Q_o}{Q_i}.
</math>

<!--T:415-->
The mean thermodynamic motor power is:

<!--T:416-->
<math>
\overline{\dot W}
=
\frac{W_{\mathrm{cycle}}}{\tau}.
</math>


<!--T:214-->
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 == <!--T:215-->

<!--T:216-->
The total amount of enclosed gas is a physical parameter:

<!--T:217-->
<math>M_{\mathrm{tot}}=m_S+m_L+m_i+m_o=\mathrm{const}.</math>

<!--T:218-->
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:

<!--T:219-->
<math>
M_{\mathrm{tot}}=
\frac{P_{\mathrm{charge}}
\left[V_S(0)+V_{L,\max}+V_i+V_o\right]}
{RT_{\mathrm{charge}}}
.</math>

<!--T:220-->
<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.

<!--T:221-->
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:

<!--T:222-->
<math>\mathbf X(t+\tau)=\mathbf X(t).</math>

<!--T:223-->
Geometric periodicity alone:

<!--T:224-->
<math>V_k(t+\tau)=V_k(t),\qquad k\in\{S,L\}.</math>

<!--T:225-->
is not sufficient to guarantee thermodynamic periodicity.

<!--T:226-->
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 == <!--T:227-->

=== 10.1 Prescribed data === <!--T:228-->

<!--T:229-->
* operating mode;
* reservoir temperatures <math display="inline">T_{\mathrm{cold}}</math> and  <math display="inline">T_{\mathrm{hot}}</math>;
* signed crank angular velocity <math display="inline">\omega</math>.
* 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>;
* 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>.

=== 10.2 Design parameters === <!--T:230-->

<!--T:231-->
* <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>;
* <math display="inline">(UA)_i</math>, <math display="inline">(UA)_o</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 === <!--T:232-->

<!--T:233-->
* <math display="inline">m_j,U_j,T_j,P_j,\qquad j\in\{S,L,i,o\}</math>;
* internal and external mass flow rates;
* <math display="inline">\dot Q_i</math>, <math display="inline">\dot Q_o</math>, <math display="inline">Q_i</math>, <math display="inline">Q_o</math>;
* <math display="inline">W_{\mathrm{cycle}}</math>, <math display="inline">COP_c</math>, <math display="inline">COP_h</math>;
* <math display="inline">\eta_{\mathrm{th}}</math> in motor operation.
* 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 == <!--T:234-->

=== 11.1 Mass conservation === <!--T:235-->

<!--T:236-->
The solver must satisfy:

<!--T:237-->
<math>\frac{dM_{\mathrm{tot}}}{dt}=0.</math>

<!--T:238-->
A useful numerical residual is:

<!--T:239-->
<math>\varepsilon_M(t)=M_{\mathrm{tot}}(t)-M_{\mathrm{tot}}(0).</math>

=== 11.2 Global energy conservation === <!--T:240-->

<!--T:241-->
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:

<!--T:242-->
<math>
\frac{dU_{\mathrm{tot}}}{dt}
=\dot Q_i+\dot Q_o
-P_S^\star\dot V_S
-P_L^\star\dot V_L
.</math>

<!--T:243-->
A cumulative energy residual may be defined by:

<!--T:244-->
<math>
\varepsilon_E(t)=
U_{\mathrm{tot}}(t)-U_{\mathrm{tot}}(0)
-Q_i(0,t)-Q_o(0,t)+W(0,t)
.</math>

<!--T:417-->
Over a periodic cycle:

<!--T:418-->
<math>Q_i+Q_o=W_{\mathrm{cycle}}.</math>

<!--T:245-->
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.

<!--T:246-->
----

== Appendix A — Symbolic derivations and validated checks == <!--T:247-->

=== A.1 Pressure equation for a closed pair === <!--T:248-->

<!--T:249-->
For a cylinder + heat-exchanger pair at quasi-uniform pressure:

<!--T:250-->
<math>U=\frac{P(V_{\mathrm{cyl}}+V_{HX})}{\gamma-1}.</math>

<!--T:251-->
The first law gives:

<!--T:252-->
<math>\frac{dU}{dt}=\dot Q-P\dot V_{\mathrm{cyl}},</math>

<!--T:253-->
with:

<!--T:254-->
<math>\dot Q=(UA)_{HX}(T_{HX,\mathrm{res}}-T_{HX}).</math>

<!--T:255-->
Differentiating <math display="inline">U</math>:

<!--T:256-->
<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>

<!--T:257-->
Hence:

<!--T:258-->
<math>
\dot P=
\frac{(\gamma-1)\dot Q-\gamma P\dot V_{\mathrm{cyl}}}
{V_{\mathrm{cyl}}+V_{HX}}
.</math>

<!--T:259-->
If <math display="inline">\dot Q=0</math>:

<!--T:260-->
<math>\frac{\dot P}{P}=-\gamma\frac{\dot V}{V},</math>

<!--T:261-->
then:

<!--T:262-->
<math>PV^\gamma=\mathrm{const}.</math>

=== A.2 Internal flow rate of the pair === <!--T:263-->

<!--T:264-->
For the heat exchanger alone, at fixed volume:

<!--T:265-->
<math>U_{HX}=\frac{PV_{HX}}{\gamma-1}.</math>

<!--T:266-->
Therefore:

<!--T:267-->
<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>

<!--T:268-->
Using <math display="inline">C_p=\gamma R/(\gamma-1)</math>:

<!--T:269-->
<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>

<!--T:270-->
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 === <!--T:271-->

<!--T:272-->
For a fixed volume:

<!--T:273-->
<math>m_{HX}=\frac{PV_{HX}}{RT_{HX}}.</math>

<!--T:274-->
Differentiating:

<!--T:275-->
<math>\frac{\dot m_{HX}}{m_{HX}}
=\frac{\dot P}{P}-\frac{\dot T_{HX}}{T_{HX}}.</math>

<!--T:276-->
With <math display="inline">\dot m_{HX}=\dot m_{\mathrm{int}}</math>:

<!--T:277-->
<math>
\dot T_{HX}=\frac{T_{HX}}P\dot P-
\frac{RT_{HX}^2}{PV_{HX}}\dot m_{\mathrm{int}}
.</math>

<!--T:278-->
This relation follows solely from mass conservation and the equation of state; it is valid for both flow directions.

=== A.4 Open receiving pair === <!--T:279-->

<!--T:280-->
For a pair receiving <math display="inline">\dot m_{\mathrm{ext}}</math> into its cylinder:

<!--T:281-->
<math>\frac{dU}{dt}=\dot m_{\mathrm{ext}}C_pT_{\mathrm{ext}}
+\dot Q-P\dot V_{\mathrm{cyl}}.</math>

<!--T:282-->
With <math display="inline">U=P(V_{\mathrm{cyl}}+V_{HX})/(\gamma-1)</math>:

<!--T:283-->
<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>

<!--T:284-->
The local mass balance:

<!--T:285-->
<math>\dot m_{\mathrm{cyl}}=\dot m_{\mathrm{ext}}-\dot m_{\mathrm{int}},</math>

<!--T:286-->
<math>\dot m_{HX}=\dot m_{\mathrm{int}},</math>

<!--T:287-->
immediately gives:

<!--T:288-->
<math>\dot m_{\mathrm{pair}}=\dot m_{\mathrm{ext}}.</math>

=== A.5 Analytical solution for the adiabatic donor cylinder === <!--T:289-->

<!--T:290-->
For an adiabatic, well-mixed cylinder with outflow only:

<!--T:291-->
<math>\frac{d(mC_vT)}{dt}=-P\dot V-\dot m_{\mathrm{out}}C_pT,</math>

<!--T:292-->
and:

<!--T:293-->
<math>\dot m=-\dot m_{\mathrm{out}}.</math>

<!--T:294-->
Expanding:

<!--T:295-->
<math>C_vm\dot T+C_vT\dot m=-P\dot V+C_pT\dot m,</math>

<!--T:296-->
thus:

<!--T:297-->
<math>C_vm\dot T=-P\dot V+RT\dot m.</math>

<!--T:298-->
With <math display="inline">P=mRT/V</math> and <math display="inline">R/C_v=\gamma-1</math>:

<!--T:299-->
<math>\frac{dT}{T}=(\gamma-1)
\left(\frac{dm}{m}-\frac{dV}{V}\right).</math>

<!--T:300-->
After integration:

<!--T:301-->
<math>
\frac{T}{T_0}=
\left[\frac{m}{m_0}\frac{V_0}{V}\right]^{\gamma-1}
.</math>

<!--T:302-->
Then, using <math display="inline">PV=mRT</math>:

<!--T:303-->
<math>
P=P_0\left[\frac{m}{m_0}\frac{V_0}{V}\right]^\gamma
.</math>

<!--T:304-->
and:

<!--T:305-->
<math>
\frac{T}{T_0}=
\left(\frac{P}{P_0}\right)^{(\gamma-1)/\gamma}
.</math>

<!--T:306-->
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 === <!--T:307-->

<!--T:308-->
Fundamental balance:

<!--T:309-->
<math>\frac{d(mC_vT)}{dt}
=\dot m_{\mathrm{in}}C_pT_{\mathrm{in}}
-\dot m_{\mathrm{out}}C_pT
+\dot Q.</math>

<!--T:310-->
Expanding the left-hand side and using:

<!--T:311-->
<math>\dot m=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}},</math>

<!--T:312-->
one obtains:

<!--T:313-->
<math>
\dot T=
\frac{\dot m_{\mathrm{in}}(C_pT_{\mathrm{in}}-C_vT)
-\dot m_{\mathrm{out}}RT+\dot Q}
{C_vm}
.</math>

<!--T:314-->
Limiting checks:

<!--T:315-->
* 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 === <!--T:316-->

<!--T:317-->
For Phase II:

<!--T:318-->
<math>\dot m_L=-\dot m_{\mathrm{in}},</math>

<!--T:319-->
<math>\dot m_o=\dot m_{\mathrm{in}}-\dot m_{\mathrm{out}},</math>

<!--T:320-->
<math>\dot m_{Si}=\dot m_{\mathrm{out}}.</math>

<!--T:321-->
Summing:

<!--T:322-->
<math>\dot M_{\mathrm{tot}}=0.</math>

<!--T:323-->
Phase IV gives exactly the same result by symmetry.

=== A.8 Global energy test during an active phase === <!--T:324-->

<!--T:325-->
For Phase II:

<!--T:326-->
<math>\dot U_L=-P_L\dot V_L-\dot m_{\mathrm{in}}h_L,</math>

<!--T:327-->
<math>\dot U_o=\dot m_{\mathrm{in}}h_L-\dot m_{\mathrm{out}}h_o+\dot Q_o,</math>

<!--T:328-->
<math>\dot U_{Si}=\dot m_{\mathrm{out}}h_o+\dot Q_i-P_{Si}\dot V_S.</math>

<!--T:329-->
The internal enthalpy fluxes cancel exactly:

<!--T:330-->
<math>-\dot m_{\mathrm{in}}h_L+\dot m_{\mathrm{in}}h_L=0,</math>

<!--T:331-->
<math>-\dot m_{\mathrm{out}}h_o+\dot m_{\mathrm{out}}h_o=0.</math>

<!--T:332-->
What remains is:

<!--T:333-->
<math>
\dot U_{\mathrm{tot}}
=\dot Q_i+\dot Q_o
-P_L\dot V_L-P_{Si}\dot V_S
.</math>

<!--T:334-->
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:

<!--T:335-->
<math>
\dot U_{\mathrm{tot}}
=\dot Q_i+\dot Q_o
-P_S^\star\dot V_S-P_L^\star\dot V_L
.</math>

<!--T:336-->
Integrated over a periodic cycle, this relation gives:

<!--T:337-->
<math>Q_i+Q_o=W_{\mathrm{cycle}}.</math>

=== A.9 Continuity at transitions === <!--T:338-->

<!--T:339-->
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:

<!--T:340-->
<math>\qquad j\in\{S,L,i,o\}</math>

<!--T:419-->
<math>m_j^+=m_j^-,\qquad U_j^+=U_j^-,\qquad V_j^+=V_j^-.</math>

<!--T:341-->
For an ideal gas:

<!--T:342-->
<math>T_j=\frac{U_j}{m_jC_v},\qquad P_j=\frac{m_jRT_j}{V_j},</math>

<!--T:343-->
which implies:

<!--T:344-->
<math>T_j^+=T_j^-,\qquad P_j^+=P_j^-.</math>

<!--T:345-->
The event only creates a change in hydraulic topology and in the active system of equations.
</translate>