Lenoir cycle

Thermodynamics
The classical Carnot heat engine
Branches Classical Statistical Chemical Quantum thermodynamics Equilibrium / Non-equilibrium
Laws Zeroth First Second Third
Systems Closed system Open system Isolated system State Equation of state Ideal gas Real gas State of matter Phase (matter) Equilibrium Control volume Instruments Processes Isobaric Isochoric Isothermal Adiabatic Isentropic Isenthalpic Quasistatic Polytropic Free expansion Reversibility Irreversibility Endoreversibility Cycles Heat engines Heat pumps Thermal efficiency
System properties Note: Conjugate variables in italics Property diagrams Intensive and extensive properties Process functions Work Heat Functions of state Temperature / Entropy (introduction) Pressure / Volume Chemical potential / Particle number Vapor quality Reduced properties
Material properties Property databases Specific heat capacity  ${\displaystyle c=}$ ${\displaystyle T}$ ${\displaystyle \partial S}$ ${\displaystyle N}$ ${\displaystyle \partial T}$ Compressibility  ${\displaystyle \beta =-}$ ${\displaystyle 1}$ ${\displaystyle \partial V}$ ${\displaystyle V}$ ${\displaystyle \partial p}$ Thermal expansion  ${\displaystyle \alpha =}$ ${\displaystyle 1}$ ${\displaystyle \partial V}$ ${\displaystyle V}$ ${\displaystyle \partial T}$
Equations Carnot's theorem Clausius theorem Fundamental relation Ideal gas law Maxwell relations Onsager reciprocal relations Bridgman's equations Table of thermodynamic equations
Potentials Free energy Free entropy Internal energy ${\displaystyle U(S,V)}$ Enthalpy ${\displaystyle H(S,p)=U+pV}$ Helmholtz free energy ${\displaystyle A(T,V)=U-TS}$ Gibbs free energy ${\displaystyle G(T,p)=H-TS}$
History Culture History General Entropy Gas laws "Perpetual motion" machines Philosophy Entropy and time Entropy and life Brownian ratchet Maxwell's demon Heat death paradox Loschmidt's paradox Synergetics Theories Caloric theory Vis viva ("living force") Mechanical equivalent of heat Motive power Key publications An Experimental Enquiry Concerning ... Heat On the Equilibrium of Heterogeneous Substances Reflections on the Motive Power of Fire Timelines Thermodynamics Heat engines Art Education Maxwell's thermodynamic surface Entropy as energy dispersal
Scientists Bernoulli Boltzmann Bridgman Carathéodory Carnot Clapeyron Clausius de Donder Duhem Gibbs von Helmholtz Joule Kelvin Lewis Massieu Maxwell von Mayer Nernst Onsager Planck Rankine Smeaton Stahl Tait Thompson van der Waals Waterston
Other Nucleation Self-assembly Self-organization Order and disorder
Category
v t e

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.

The cycle

In the cycle, an ideal gas undergoes^[1][2]

1–2: Constant volume (isochoric) heat addition;

2–3: Isentropic expansion;

3–1: Constant pressure (isobaric) heat rejection.

The expansion process is isentropic and hence involves no heat interaction. Energy is absorbed as heat during the isochoric heating and rejected as work during the isentropic expansion. Waste heat is rejected during the isobaric cooling which consumes some work.

Constant volume heat addition (1–2)

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: ${\displaystyle {}_{1}Q_{2}=mc_{v}\left({T_{2}-T_{1}}\right)}$

There is no work during the process because the volume is held constant: ${\displaystyle {}_{1}W_{2}=\int _{1}^{2}{p\,dV}=0}$

and from the definition of constant volume specific heats for an ideal gas: ${\displaystyle c_{v}={\frac {R}{\gamma -1}}}$

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: ${\displaystyle p_{2}v_{2}=RT_{2}}$

Isentropic expansion (2–3)

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: ${\displaystyle {\frac {T_{2}}{T_{3}}}=\left({\frac {p_{2}}{p_{3}}}\right)^{\textstyle {{\gamma -1} \over \gamma }}=\left({\frac {V_{3}}{V_{2}}}\right)^{\gamma -1}}$

Where ${\displaystyle p_{3}=p_{1}}$ for this specific cycle. The first law of thermodynamics results in the following for this expansion process: ${\displaystyle {}_{2}W_{3}=\int _{2}^{3}{p\,dV}}$ because for an adiabatic process: ${\displaystyle {}_{2}Q_{3}=0}$

Constant pressure heat rejection (3–1)

The final stage (3–1) involves a constant pressure heat rejection back to the original state. From the first law of thermodynamics we find: ${\displaystyle {}_{3}Q_{1}-{}_{3}W_{1}=U_{1}-U_{3}}$.

From the definition of work: ${\displaystyle {}_{3}W_{1}=\int _{3}^{1}{p\,dV}=p_{1}\left({V_{1}-V_{3}}\right)}$, we recover the following for the heat rejected during this process: ${\displaystyle {}_{3}Q_{1}=\left({U_{1}+p_{1}V_{1}}\right)-\left({U_{3}+p_{3}V_{3}}\right)=H_{1}-H_{3}}$.

As a result, we can determine the heat rejected as follows: ${\displaystyle {}_{3}Q_{1}=mc_{p}\left({T_{1}-T_{3}}\right)}$. For an ideal gas, ${\displaystyle c_{p}={\frac {\gamma R}{\gamma -1}}}$.

Efficiency

The overall efficiency of the cycle is determined by the total work over the heat input, which for a Lenoir cycle equals

${\displaystyle \eta _{\rm {th}}={\frac {{}_{2}W_{3}+{}_{3}W_{1}}{{}_{1}Q_{2}}}.}$

Note that we gain work during the expansion process but lose some during the heat rejection process. Alternatively, the first law of thermodynamics can be used to put the efficiency in terms of the heat absorbed and heat rejected,

${\displaystyle \eta _{\rm {th}}=1-{\frac {{}_{3}Q_{1}}{{}_{1}Q_{2}}}=1-\gamma \left({\frac {T_{3}-T_{1}}{T_{2}-T_{1}}}\right).}$

Utilizing that, for the isobaric process, T₃/T₁ = V₃/V₁, and for the adiabatic process, T₂/T₃ = (V₃/V₁)^γ−1, the efficiency can be put in terms of the compression ratio,

${\displaystyle \eta _{\rm {th}}=1-\gamma \left({\frac {r-1}{r^{\gamma }-1}}\right),}$

where r = V₃/V₁ is defined to be > 1. Comparing this to the Otto cycle's efficiency graphically, it can be seen that the Otto cycle is more efficient at a given compression ratio. Alternatively, using the relationship given by process 2–3, the efficiency can be put in terms of r_p = p₂/p₃, the pressure ratio,^[2]

${\displaystyle \eta _{\rm {th}}=1-\gamma \left({\frac {r_{p}^{1/\gamma }-1}{r_{p}-1}}\right).}$

Cycle diagrams

References

This page was last edited on 2 March 2023, at 20:58