Thermodynamic and Mechanical Study - Revision history

E.dada: Marked this version for translation

2026-09-07T14:59:12Z

Marked this version for translation

E.dada at 13:54, 7 September 2026

2026-09-07T13:54:12Z

← Older revision		Revision as of 15:54, 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 15:52, 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-->

E.dada at 13:45, 7 September 2026

2026-09-07T13:45:14Z

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E.dada at 13:27, 7 September 2026

2026-09-07T13:27:12Z

← Older revision		Revision as of 15:27, 7 September 2026
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	<!--T:126-->		<!--T:126-->
	The transition to Phase II occurs when the <code>~~H_o→~~ S</code> check valve satisfies its opening condition.		The transition to Phase II occurs when the <code>H_o → S</code> check valve satisfies its opening condition.

	=== 6.2 Phase II — heat exchange, nominally isothermal: L → H_o → S === <!--T:127-->		=== 6.2 Phase II — heat exchange, nominally isothermal: L → H_o → S === <!--T:127-->
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	<!--T:153-->		<!--T:153-->
	The transition to Phase III is the closing event of the <code>~~H_o→~~ S</code> check valve.		The transition to Phase III is the closing event of the <code>H_o → S</code> check valve.

	=== 6.3 Phase III — check valves closed, nominally adiabatic: expansion on the S side === <!--T:154-->		=== 6.3 Phase III — check valves closed, nominally adiabatic: expansion on the S side === <!--T:154-->
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	<!--T:160-->		<!--T:160-->
	The transition to Phase IV occurs when the <code>~~H_i→~~ L</code> check valve satisfies its opening condition.		The transition to Phase IV occurs when the <code>H_i → L</code> check valve satisfies its opening condition.

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

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

E.dada at 13:19, 7 September 2026

2026-09-07T13:19:14Z

← Older revision		Revision as of 15:19, 7 September 2026
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	<!--T:36-->		<!--T:36-->
	For each volume <math display="inline">k\in\{S,L,i,o\}</math>:		For each volume <math display="inline">j\in\{S,L,i,o\}</math>:

	<!--T:37-->		<!--T:37-->
	<math>~~m_k~~,\qquad ~~U_k~~,\qquad ~~T_k~~,\qquad ~~P_k~~,\qquad ~~V_k~~.</math>		<math>m_j,\qquad U_j,\qquad T_j,\qquad P_j,\qquad V_j.</math>

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

	<!--T:192-->		<!--T:192-->
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	<!--T:193-->		<!--T:193-->
	<math>~~T_k~~^+=~~T_k~~^-,\qquad ~~P_k~~^+=~~P_k~~^-.</math>		<math>T_j^+=T_j^-,\qquad P_j^+=P_j^-.</math>

	<!--T:194-->		<!--T:194-->
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	* 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>;		* 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;		* 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>.		* kinematics <math display="inline">V_S(t)</math>, <math display="inline">V_L(t)</math>.

	=== 10.2 Design parameters === <!--T:230-->		=== 10.2 Design parameters === <!--T:230-->
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	<!--T:233-->		<!--T:233-->
	* <math display="inline">~~m_k~~,~~U_k~~,~~T_k~~,~~P_k~~,\qquad k\in\{S,L,i,o\}</math>;		* <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;		* 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">\dot Q_i</math>, <math display="inline">\dot Q_o</math>, <math display="inline">Q_i</math>, <math display="inline">Q_o</math>;
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	<!--T:340-->		<!--T:340-->
	<math>\qquad k\in\{S,L\}</math>		<math>\qquad j\in\{S,L,i,o\}</math>

	<math>~~m_k~~^+=~~m_k~~^-,\qquad ~~U_k~~^+=~~U_k~~^-,\qquad ~~V_k~~^+=~~V_k~~^-.</math>		<math>m_j^+=m_j^-,\qquad U_j^+=U_j^-,\qquad V_j^+=V_j^-.</math>

	<!--T:341-->		<!--T:341-->
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	<!--T:342-->		<!--T:342-->
	<math>~~T_k~~=\frac{~~U_k~~}{~~m_kC_v~~},\qquad ~~P_k~~=\frac{~~m_kRT_k~~}{~~V_k~~},</math>		<math>T_j=\frac{U_j}{m_jC_v},\qquad P_j=\frac{m_jRT_j}{V_j},</math>

	<!--T:343-->		<!--T:343-->
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	<!--T:344-->		<!--T:344-->
	<math>~~T_k~~^+=~~T_k~~^-,\qquad ~~P_k~~^+=~~P_k~~^-.</math>		<math>T_j^+=T_j^-,\qquad P_j^+=P_j^-.</math>

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

E.dada at 12:45, 7 September 2026

2026-09-07T12:45:43Z

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E.dada at 12:15, 7 September 2026

2026-09-07T12:15:37Z

← Older revision		Revision as of 14:15, 7 September 2026
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	<!--T:320-->		<!--T:320-->
	<math>\dot m_{SC}=\dot m_{\mathrm{out}}.</math>		<math>\dot m_{Si}=\dot m_{\mathrm{out}}.</math>

	<!--T:321-->		<!--T:321-->
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	<!--T:327-->		<!--T:327-->
	<math>\dot U_o=\dot m_{\mathrm{in}}h_L-\dot m_{\mathrm{out}}~~h_H~~+\dot Q_o,</math>		<math>\dot U_o=\dot m_{\mathrm{in}}h_L-\dot m_{\mathrm{out}}h_o+\dot Q_o,</math>

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

	<!--T:329-->		<!--T:329-->
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	<!--T:331-->		<!--T:331-->
	<math>-\dot m_{\mathrm{out}}~~h_H~~+\dot m_{\mathrm{out}}~~h_H~~=0.</math>		<math>-\dot m_{\mathrm{out}}h_o+\dot m_{\mathrm{out}}h_o=0.</math>

	<!--T:332-->		<!--T:332-->

E.dada at 12:09, 7 September 2026

2026-09-07T12:09:34Z

← Older revision		Revision as of 14:09, 7 September 2026
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	<!--T:2-->		<!--T:2-->
	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.		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-->		<!--T:3-->
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	<!--T:147-->		<!--T:147-->
	<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,H}}(P_L,P_o,T_L,\ldots),</math>		<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,o}}(P_L,P_o,T_L,\ldots),</math>

	<!--T:148-->		<!--T:148-->
	<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,H}}(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 S+H_i==== <!--T:149-->		==== 6.2.3 Receiving pair S+H_i==== <!--T:149-->

	<!--T:150-->		<!--T:150-->
	The flow from <code>H</code> enters '''S'''. The pair pressure satisfies:		The flow from <code>H_o</code> enters '''S'''. The pair pressure satisfies:

	<!--T:151-->		<!--T:151-->
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	<!--T:152-->		<!--T:152-->
	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 internal redistribution equations of §4 remain unchanged: the external flow acts on heat exchanger <code>H_i</code> indirectly through the evolution of the pair pressure.

	<!--T:153-->		<!--T:153-->
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	<!--T:162-->		<!--T:162-->
	The gas leaves <code>S</code>, passes through the heat-in exchanger <code>C</code>, where it receives heat from ~~the cold~~ reservoir, crosses the <code>~~H_i→~~ L</code> check valve, and then enters the receiving cylinder <code>L</code>. The <code>L+H_o</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 heat-in exchanger <code>H_i</code>, where it receives heat from its external reservoir, crosses the <code>H_i → L</code> check valve, and then enters the receiving cylinder <code>L</code>. The <code>L+H_o</code> 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-->		==== 6.4.1 Donor cylinder S ==== <!--T:163-->
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	<!--T:178-->		<!--T:178-->
	<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,C}}(P_S,P_i,T_S,\ldots),</math>		<math>\dot m_{\mathrm{in}}=\Phi_{\mathrm{HX,i}}(P_S,P_i,T_S,\ldots),</math>

	<!--T:179-->		<!--T:179-->
	<math>\dot m_{\mathrm{out}}=\Phi_{\mathrm{valve,C}}(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 L+H_o==== <!--T:180-->		==== 6.4.3 Receiving pair L+H_o==== <!--T:180-->

	<!--T:181-->		<!--T:181-->
	The flow from <code>C</code> enters '''L'''. The pair pressure satisfies:		The flow from <code>H_i</code> enters '''L'''. The pair pressure satisfies:

	<!--T:182-->		<!--T:182-->
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	<!--T:183-->		<!--T:183-->
	Closing the <code>~~H_i→~~ 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.		Closing the <code>H_i → 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.

	=== 6.5 Motor operation ===		=== 6.5 Motor operation ===

E.dada at 11:49, 7 September 2026

2026-09-07T11:49:54Z

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