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| {{thermodynamics|cTopic=Processes and Cycles}}
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| [[File:Lenoir gas engine 1860.jpg|thumb|235px|Lenoir [[gas engine]] 1860]]
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| The '''Lenoir cycle''' is an idealized [[thermodynamic cycle]] often used to model a [[pulse jet engine]]. It is based on the operation of an engine patented by [[Jean Joseph Etienne Lenoir]] in 1860. This engine is often thought of as the first commercially produced [[internal combustion engine]]. The absence of any compression process in the design leads to lower [[thermal efficiency]] than the more well known [[Otto cycle]] and [[Diesel cycle]].
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| ==The cycle==
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| In the cycle, an [[ideal gas]] undergoes<ref>{{cite web|url=http://books.google.co.uk/books?id=WLdcxNvWB2wC&pg=PA93&lpg=PA93&dq=lenoir+cycle&source=bl&ots=HThE5gCr4F&sig=BNhcHOwCShVa2uvJYGdbX0ghtrc&hl=en&sa=X&ei=8addUf2cHMi_0QX8r4CgDg&sqi=2&ved=0CF8Q6AEwCA#v=onepage&q=lenoir%20cycle&f=false |title=Ic Engines - V. Ganesan - Google Books |publisher=Books.google.co.uk |date= |accessdate=2013-04-04}}</ref>
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| :1-2: Constant volume ([[isochoric]]) heat addition;
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| :2-3: [[Isentropic process|Isentropic]] expansion;
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| :3-1: Constant pressure ([[isobaric process|isobaric]]) heat rejection—compression to the volume at the start of the cycle.
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| The expansion process is [[isentropic process|isentropic]] and hence involves no heat interaction. Energy is absorbed as heat during the isochoric heating and rejected as work during the isentropic expansion.
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| Waste heat is rejected during the isobaric cooling which consumes some work.
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| ===Constant volume heat addition (1-2)===
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| In the ideal gas version of the traditional Lenoir cycle, the first stage (1-2) involves the addition of heat in a constant volume manner. This results in the following for the first law of thermodynamics:
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| <math>{}_1Q_2 = mc_v \left( {T_2 - T_1 } \right)</math>
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| There is no work during the process because the volume is held constant:
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| <math>{}_1W_2 = \int\limits_1^2 {pdV} = 0</math>
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| and from the definition of constant volume specific heats for an ideal gas:
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| <math>c_v = \frac{R}{{\gamma - 1}}</math>
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| Where ''R'' is the ideal gas constant and ''γ'' is the ratio of specific heats (approximately 287 J/(kg·K) and 1.4 for air respectively). The pressure after the heat addition can be calculated from the ideal gas law: <math>p_2 V_2 = RT_2 </math>
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| ===Isentropic expansion (2-3)===
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| The second stage (2-3) involves a reversible adiabatic expansion of the fluid back to its original pressure. It can be determined for an isentropic process that the second law of thermodynamics results in the following:
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| <math>\frac{{T_2 }}{{T_3 }} = \left( {\frac{{p_2 }}{{p_3 }}} \right)^{{\textstyle{{\gamma - 1} \over \gamma }}} = \left( {\frac{{V_3 }}{{V_2 }}} \right)^{\gamma - 1} </math>
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| Where <math>p_3 = p_1</math> for this specific cycle. The first law of thermodynamics results in the following for this expansion process: <math> {}_2W_3 = mc_v \left( {T_2 - T_3 } \right)</math> because for an adiabatic process:
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| <math>{}_2 Q_3 = 0
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| </math>
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| ===Constant pressure heat rejection (3-1)===
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| The final stage (3-1) involves a constant pressure heat rejection back to the original state. From the first law of thermodynamics we find: <math>{}_3Q_1 - _3 W_1 = U_1 - U_3 </math>.
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| From the definition of work: <math>{}_3W_1 = \int\limits_3^1 {pdV} = p_1 \left( {V_1 - V_3 } \right)</math>, we recover the following for the heat rejected during this process: <math>{}_3Q_1 = \left( {U_1 + p_1 V_1 } \right) - \left( {U_3 + p_3 V_3 } \right) = H_1 - H_3 </math>.
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| As a result, we can determine the heat rejected as follows: <math>
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| {}_3Q_1 = mc_p \left( {T_1 - T_3 } \right)
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| </math>
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| from the definition of constant pressure specific heats for an ideal gas: <math>c_p = \frac{{\gamma R}}
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| {{\gamma - 1}}</math>.
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| The overall efficiency of the cycle is determined by the total work over the heat input, which for a Lenoir cycle equals <math>\eta _{th} = \frac{{{}_2W_3 + {}_3W_1 }}
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| {{{}_1Q_2 }}</math>. Note that we gain work during the expansion process but lose some during the heat rejection process.
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| ==Cycle diagrams==
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| {|
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| |- valign=bottom
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| |[[Image:lenoir pv.jpg|400px|thumb|[[PV diagram]] of the Lenoir cycle]]
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| |[[Image:lenoir Ts.jpg|400px|thumb|[[TS diagram]] of the Lenoir cycle]]
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| |}
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| ==References==
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| <references />
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| {{Thermodynamic cycles|state=uncollapsed}}
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| [[Category:Thermodynamic cycles]]
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| [[Category:Belgian inventions]]
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