Thermodynamic and Mechanical Study - Revision history

E.dada at 12:42, 10 October 2026

2026-10-10T12:42:49Z

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

E.dada: summary numbers and newlines removal

2026-10-10T12:35:59Z

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E.dada at 00:01, 21 September 2026

2026-09-21T00:01:24Z

← Older revision		Revision as of 01:01, 21 September 2026
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	<!--T:27-->		<!--T:27-->
	<math>\nu_{V,k}=\|\dot V_k\|.</math>		<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 characteristic piston motion over the cycle. 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-->		=== 2.2 Kinematic closure fraction === <!--T:29-->

E.dada: Marked this version for translation

2026-09-20T17:02:14Z

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E.dada at 16:54, 20 September 2026

2026-09-20T16:54:45Z

← Older revision		Revision as of 17:54, 20 September 2026
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	The cycle is described by four hydraulic/thermodynamic ~~stages~~: compression, transfer through <math display="inline">H_o</math> from <math display="inline">L</math> to <math display="inline">S</math>, expansion, and transfer through <math display="inline">H_i</math> from <math display="inline">S</math> to <math display="inline">L</math>. In refrigeration operation, starting from the reference origin used in this study, these ~~stages~~ are traversed in that order. Motor operation reverses the kinematics and ~~traverses the thermodynamic cycle in~~ the ~~opposite direction~~; ~~its chronology~~ is detailed in §6.5.		The cycle is described by four hydraulic/thermodynamic phases: compression, transfer through <math display="inline">H_o</math> from <math display="inline">L</math> to <math display="inline">S</math>, expansion, and transfer through <math display="inline">H_i</math> from <math display="inline">S</math> to <math display="inline">L</math>. In refrigeration operation, starting from the reference origin used in this study, these phases are traversed in that order. Motor operation reverses the crank kinematics. The resulting thermodynamic chronology must be recalculated with the unchanged check-valve orientations and the reversed reservoir assignment; it is detailed in §6.6.


	The ~~stage~~ 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 ~~stage~~.		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 ~~stages~~ 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 ~~stage~~.		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 ~~stages~~ is assumed here.		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.


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	<!--T:32-->		<!--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> may be used as nominal kinematic reference levels at selected ~~stage~~ boundaries. The value actually observed at a check-valve event may be recorded as:		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> 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:

	<!--T:33-->		<!--T:33-->
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	<!--T:34-->		<!--T:34-->
	Check-valve events are determined by the local pressure difference across the valve and are independent of the selected kinematic ~~stage~~ boundaries; <math display="inline">\Lambda^*</math> is therefore not an imposed opening condition.		Check-valve events are determined by the local pressure difference across the valve and are independent of the selected kinematic phase boundaries; <math display="inline">\Lambda^*</math> is therefore not an imposed opening condition.

	=== 2.3 Thermodynamic variables === <!--T:35-->		=== 2.3 Thermodynamic variables === <!--T:35-->
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	For the hydraulic-loss contribution ~~considered~~ locally as an adiabatic throttling process of a calorically perfect ideal gas:		For the hydraulic-loss contribution idealized locally as an adiabatic, isenthalpic throttling process of a calorically perfect ideal gas:

	<math>		<math>
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	In refrigeration operation, the <math display="inline">L\to H_o\to S</math> transfer occurs on the high-pressure side of the cycle, whereas the <math display="inline">S\to H_i\to L</math> transfer occurs on the low-pressure side.		In the intended refrigeration operation, the <math display="inline">L\to H_o\to S</math> transfer occurs on the high-pressure side of the cycle, whereas the <math display="inline">S\to H_i\to L</math> transfer occurs on the low-pressure side.

	In motor operation, reversal of the crank direction reverses these pressure roles: <math display="inline">L\to H_o\to S</math> becomes the low-pressure exchange, whereas <math display="inline">S\to H_i\to L</math> becomes the high-pressure exchange.		In the intended motor operation, reversal of the crank direction reverses these pressure roles: <math display="inline">L\to H_o\to S</math> becomes the low-pressure exchange, whereas <math display="inline">S\to H_i\to L</math> becomes the high-pressure exchange.

	The physical thermal functions of the exchangers do not change: <math display="inline">H_i</math> always transfers heat into the gas and <math display="inline">H_o</math> always transfers heat out of the gas.		The physical thermal functions of the exchangers do not change: <math display="inline">H_i</math> always transfers heat into the gas and <math display="inline">H_o</math> always transfers heat out of the gas.
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	<!--T:190-->		<!--T:190-->
	An opening or closing event changes the admissible hydraulic flow on the valve link; it does not define a kinematic ~~stage~~ boundary and it does not cause any instantaneous jump in the thermodynamic state. For each volume:		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:

	<!--T:191-->		<!--T:191-->
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	<!--T:195-->		<!--T:195-->
	The values <math display="inline">\Lambda^*</math> may serve as kinematic reference levels at selected ~~stage~~ boundaries; the values actually observed at check-valve events are <math display="inline">\Lambda_{\mathrm{real}}=\Lambda(t_{\mathrm{event}})</math>.		The values <math display="inline">\Lambda^*</math> may serve as kinematic reference levels at selected phase boundaries; the values actually observed at check-valve events are <math display="inline">\Lambda_{\mathrm{real}}=\Lambda(t_{\mathrm{event}})</math>.

	<!--T:408-->		<!--T:408-->
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	<!--T:197-->		<!--T:197-->
	The instantaneous work delivered by the gas on the two pistons is calculated during all ~~stages~~ from the actual cylinder pressures:		The instantaneous work delivered by the gas on the two pistons is calculated during all phases from the actual cylinder pressures:

	<!--T:198-->		<!--T:198-->
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	When the two cylinder pressures are comparable during a pressure-changing ~~stage~~,		When the two cylinder pressures are comparable during a pressure-changing phase,

	<math>		<math>
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	<!--T:214-->		<!--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~~.		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 phase.

	== 9. Gas charge and periodic regime == <!--T:215-->		== 9. Gas charge and periodic regime == <!--T:215-->
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	This result is independent of the active ~~stage~~, the flow directions on bidirectional links, and the positions of the two check valves.		This result is independent of the active phase, the flow directions on bidirectional links, and the positions of the two check valves.

E.dada at 15:56, 20 September 2026

2026-09-20T15:56:02Z

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E.dada at 02:30, 20 September 2026

2026-09-20T02:30:05Z

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E.dada: Marked this version for translation

2026-09-07T14:59:12Z

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E.dada at 13:54, 7 September 2026

2026-09-07T13:54:12Z

← Older revision		Revision as of 14:54, 7 September 2026
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	The transition to Phase II occurs when the <math display="inline">H_o \to S</math> check valve satisfies its opening condition.		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 ~~\to~~ Ho ~~\to~~ S === <!--T:127-->		=== 6.2 Phase II — heat exchange, nominally isothermal: L → Ho → S === <!--T:127-->

	<!--T:128-->		<!--T:128-->
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	The transition to Phase IV occurs when the <math display="inline">H_i \to L</math> check valve satisfies its opening condition.		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 ~~\to~~ Hi ~~\to~~ L === <!--T:161-->		=== 6.4 Phase IV — heat exchange, nominally isothermal: S → Hi → L === <!--T:161-->

	<!--T:162-->		<!--T:162-->

E.dada at 13:52, 7 September 2026

2026-09-07T13:52:16Z

← Older revision		Revision as of 14:52, 7 September 2026
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	The transition to Phase II occurs when the <math display="inline">H_o \to S</math> check valve satisfies its opening condition.		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: ~~<math display="inline">~~L \to ~~H_o~~ \to S~~</math>~~ === <!--T:127-->		=== 6.2 Phase II — heat exchange, nominally isothermal: L \to Ho \to S === <!--T:127-->

	<!--T:128-->		<!--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.		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 ~~<math display="inline">~~L~~</math>~~ ==== <!--T:129-->		==== 6.2.1 Donor cylinder L ==== <!--T:129-->

	<!--T:130-->		<!--T:130-->
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	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.		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 ~~<math display="inline">H_o</math>~~ ==== <!--T:139-->		==== 6.2.2 Heat-out exchanger Ho ==== <!--T:139-->

	<!--T:140-->		<!--T:140-->
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	<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,o}}(P_o,P_{Si},T_o,\ldots).</math>		<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,o}}(P_o,P_{Si},T_o,\ldots).</math>

	==== 6.2.3 Receiving pair ~~<math display="inline">~~S+~~H_i</math>~~ ==== <!--T:149-->		==== 6.2.3 Receiving pair S+Hi ==== <!--T:149-->

	<!--T:150-->		<!--T:150-->
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	The transition to Phase III is the closing event of the <math display="inline">H_o \to S</math> check valve.		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 ~~<math display="inline">~~S~~</math>~~ side ===		=== 6.3 Phase III — check valves closed, nominally adiabatic: expansion on the S side ===

	<!--T:155-->		<!--T:155-->
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	The transition to Phase IV occurs when the <math display="inline">H_i \to L</math> check valve satisfies its opening condition.		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: ~~<math display="inline">~~S \to ~~H_i~~ \to L~~</math>~~ === <!--T:161-->		=== 6.4 Phase IV — heat exchange, nominally isothermal: S \to Hi \to L === <!--T:161-->

	<!--T:162-->		<!--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.		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 ~~<math display="inline">~~S~~</math>~~ ==== <!--T:163-->		==== 6.4.1 Donor cylinder S ==== <!--T:163-->

	<!--T:164-->		<!--T:164-->
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	Reverse flow requires returning to the complete open-system balance in <math display="inline">(m_S,U_S)</math>.		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 ~~<math display="inline">H_i</math>~~ ====		==== 6.4.2 Heat-in exchanger Hi ====

	<!--T:172-->		<!--T:172-->
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	<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,i}}(P_i,P_{Lo},T_i,\ldots).</math>		<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,i}}(P_i,P_{Lo},T_i,\ldots).</math>

	==== 6.4.3 Receiving pair ~~<math display="inline">~~L+~~H_o</math>~~ ==== <!--T:180-->		==== 6.4.3 Receiving pair L+Ho ==== <!--T:180-->

	<!--T:181-->		<!--T:181-->