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eISSN: 2576-4500

Aeronautics and Aerospace Open Access Journal

Review Article Volume 2 Issue 3

Effective temperature and performance characteristics of heat engines

Mahmoud Huleihil

The Arab Academic Institute of Education, Beit-Berl Academic College, Israel

Correspondence: Mahmoud Huleihil, The Arab Academic Institute of Education, The Academic Institute Beit-Berl, Kfar Saba 44905, Israel

Received: April 12, 2018 | Published: May 25, 2018

Citation: Huleihil M. Effective temperature and performance characteristics of heat engines. Aeron Aero Open Access J. 2018;2(3):147-153. DOI: 10.15406/aaoaj.2018.02.00045

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Abstract

For a heat engine working between two heat reservoirs, a hot reservoir at high temperature TH and a cold reservoir at low temperature TL, an effective temperature and an effective efficiency are introduced. The effective temperature is defined as the square root of the ratio between the net work output of the heat engine wnet and the SOT function (a new function introduced in this article). The SOT function is defined as the negative of the cyclic integral of the heat transfer change divided by the square of the temperature. The effective efficiency of the heat engine is defined (a novel definition) as one minus the ratio between the low temperature and the effective temperature. The effective temperature and the effective efficiency are worked out in details for the Carnot heat engine and the air standard cycles (Otto, Brayton, Stirling, and Ericsson). It was found for the considered cycles that the effective temperature is given by the expression and the effective efficiency is given by the expression n eff =1 T L T H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaeaajugWaiaadwgacaWGMbGaamOzaaqcfayabaqcLbsa cqGH9aqpcaaIXaGaeyOeI0scfa4aaOaaaeaadaWcaaqaaKqzGeGaam ivaKqbaoaaBaaabaqcLbmacaWGmbaajuaGbeaaaeaajugibiaadsfa juaGdaWgaaqaaKqzadGaamisaaqcfayabaaaaaqabaaaaa@49D2@ for all the considered cycles. The importance of these two proposed measures is twofold: educational and they could be used as a quick tool by the designer.

Introduction

For a heat engine working between two heat reservoirs, a hot reservoir at high temperature TH and a cold temperature at low temperature TL, what is the maximal achievable efficiency by the heat engine? This question was answered by Sadi Carnot at the beginning of the 19th century by using methods of thermodynamics.1-2 It was found that the maximal efficiency is limited by the Carnot efficiency ( n C =1 T L T H ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaju gibiaad6gajuaGdaWgaaqaaKqzadGaam4qaaqcfayabaqcLbsacqGH 9aqpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaamivaKqbaoaaBa aaleaajugWaiaadYeaaSqabaaakeaajugibiaadsfajuaGdaWgaaWc baqcLbmacaWGibaaleqaaaaaaOGaayjkaiaawMcaaaaa@4881@ . It is well known that the Carnot cycle is the most efficient cycle operating between these specified temperature limits. The Carnot cycle includes four branches: isentropic compression, isothermal heat addition, isentropic expansion and isothermal heat rejection.3 The Carnot cycle is an idealized thermodynamic cycle and is not appropriate to describe real heat engines such as air standard cycles. The Otto cycle, named after Nikolaus A. Otto, is the ideal cycle for spark-ignition reciprocating engines. The Otto cycle describes the ideal behavior of spark-ignition engines, in which the piston traces four strokes (four-stroke internal combustion engine). The internal combustion engine has intake and exhaust valves. These valves are closed during compression and expansion and they are open while exchanging intake (air mixture) and exhaust (combustion gases).3 The Brayton cycle, named after George Brayton, is used to describe the behavior of the reciprocating oil-burning engine that was developed by Brayton around 1870. The Brayton cycle is used today to describe the behavior of gas turbines which includes four processes: isentropic compression and expansion, and constant pressure heat addition and rejection. The working fluid is ideal gas.3

Stirling and the Ericsson engines are considered external combustion engines. That is, the energy flows to the cylinder from outside. External combustion has some advantages compared to internal combustion. Among these is the flexibility of choosing thermal energy sources, less air pollution due to complete combustion, more efficient use of energy sources, closed cycles operation which enables choosing best working fluids such as Hydrogen and Helium. The Stirling engine includes four processes: isothermal compression, isochoric heat addition, isothermal expansion and isochoric heat rejection. The Ericsson cycle differs from the Stirling cycle by the heat addition and heat rejection processes. While in the Stirling cycle these are isochoric processes, in the Ericsson cycle they are isobaric processes.3 Curzon & Ahlborn4 investigated the efficiency of a heat engine at maximum power operation in finite time.5-10 They considered in 1975 a model of heat engine (usually called the Curzon-Ahlborn engine) with finite heat transfer rates. The heat engine produces zero power output in the extremes of very slow operation and very fast operation. It was found that the heat engine attains a maximum power point with the efficiency at this point being one minus the square root of the ratio between the temperature of the cold reservoir and the temperature of the hot reservoir ( n CA =1 T L T H ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaju gibiaad6gajuaGdaWgaaqaaKqzadGaam4qaiaadgeaaKqbagqaaKqz GeGaeyypa0JaaGymaiabgkHiTKqbaoaakaaabaWaaSaaaeaajugibi aadsfajuaGdaWgaaqaaKqzadGaamitaaqcfayabaaabaqcLbsacaWG ubqcfa4aaSbaaeaajugWaiaadIeaaKqbagqaaaaaaeqaaaGccaGLOa Gaayzkaaaaaa@4A33@ . Bejan11 showed in 1994 that this result of heat engine efficiency at maximum power operation was previously derived by Novikov12 in 1957 when analyzing the performance of nuclear power plants.

Harvey S Leff13 considered four air standard cycles (Otto, Disel, Brayton, Atkinson) and found that the efficiency of these cycles at maximum power operation is similar to the achieved result by Curzon-Ahlborn. The methods of irreversible thermodynamics and finite thermodynamics have been used to analyze thermodynamic systems.14-32 Analysis of the air standard cycles (Otto and others) could be found elsewhere.33-60 In this article we ask: what are the effective temperature and the effective efficiency of a heat engine working between two heat reservoirs for the cycles described above? The answers to these questions are given in the following paragraphs. The following sections describe: the SOT function in section II, the model of the heat engine in section III; the Carnot cycle is considered in section IV, the air standard cycles are considered in section V, numerical examples are given in section VI, and finally summary and conclusions are given in section VII.

The SOT function

For a thermal process that goes from state A to state B, the negative integral of the heat transfer change divided by the square of the temperature from state A to state B is given by (a new function introduced in this article and it is called by the author the SOT function):

SOT= A B δq T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb Gaam4taiaadsfacqGH9aqpcqGHsisljuaGdaWdXbGcbaqcfa4aaSaa aOqaaKqzGeGaeqiTdqMaamyCaaGcbaqcLbsacaWGubqcfa4aaWbaaS qabeaajugWaiaaikdaaaaaaaWcbaqcLbmacaWGbbaaleaajugWaiaa dkeaaKqzGeGaey4kIipaaaa@4A5D@         (1)

For isothermal process the SOT function is given by:

SOT= 1 T 2 A B δq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb Gaam4taiaadsfacqGH9aqpcqGHsisljuaGdaWcaaGcbaqcLbsacaaI XaaakeaajugibiaadsfajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaa qcfa4aa8qCaOqaaKqzGeGaeqiTdqMaamyCaaWcbaqcLbmacaWGbbaa leaajugWaiaadkeaaKqzGeGaey4kIipaaaa@4BA7@    (2)

The first law of thermodynamics for a process with ideal gas working fluid is given by:

δq=dE+pdV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azcaWGXbGaeyypa0JaamizaiaadweacqGHRaWkcaWGWbGaamizaiaa dAfaaaa@3F74@    (3)

By considering the various process types and using the first law of thermodynamics (eq. 3), makes it easy to write explicit forms of the SOT function. In the following subsections some typical processes are considered.

Constant volume process

For the constant volume process (v=const.), the heat transfer change is proportional to the temperature change (dT) with constant volume heat capacity proportionality factor (cv). In this case the SOT function is given by:

SOT= c v ( 1 T A 1 T B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb Gaam4taiaadsfacqGH9aqpcaWGJbqcfa4aaSbaaSqaaiaadAhaaeqa aKqbaoaabmaabaWaaSaaaeaacaaIXaaabaGaamivamaaBaaabaqcLb macaWGbbaajuaGbeaaaaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamiv amaaBaaabaqcLbmacaWGcbaajuaGbeaaaaaacaGLOaGaayzkaaaaaa@4840@    (4)

Constant pressure process

For the constant pressure process (p=const.), the heat transfer change is proportional to the temperature change (dT) with constant pressure heat capacity proportionality factor (cp). In this case the SOT function is given by:

SOT= c p ( 1 T A 1 T B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb Gaam4taiaadsfacqGH9aqpcaWGJbqcfa4aaSbaaSqaaiaadchaaeqa aKqbaoaabmaabaWaaSaaaeaacaaIXaaabaGaamivamaaBaaabaqcLb macaWGbbaajuaGbeaaaaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamiv amaaBaaabaqcLbmacaWGcbaajuaGbeaaaaaacaGLOaGaayzkaaaaaa@483A@     (5)

 Constant temperature process

For the constant temperature process (T=const.), the heat transfer change is proportional to the volume change (dv) with pressure proportionality factor (p). In this case the SOT function is given by:

SOT= R T ln( v A v B ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aad+eacaWGubGaeyypa0ZaaSaaaeaacaWGsbaabaGaamivaaaaciGG SbGaaiOBamaabmaabaWaaSaaaeaacaWG2bWaaSbaaeaajugWaiaadg eaaKqbagqaaaqaaiaadAhadaWgaaqaaKqzadGaamOqaaqcfayabaaa aaGaayjkaiaawMcaaaaa@4689@    (6)

Isentropic process

For the isentropic process-constant entropy process (s=const.), the heat transfer change is zero. In this case the SOT function is given by:

SOT=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aad+eacaWGubGaeyypa0JaaGimaaaa@3AC9@     (7)

For a heat engine going in a cycle the SOT function is given by:

SOT= δq T 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aad+eacaWGubGaeyypa0JaeyOeI0Yaa8qbaeaadaWcaaqaaiabes7a KjaadghaaeaacaWGubWcdaahaaqcfayabeaajugWaiaaikdaaaaaaa qcfayabeqacqWIr4E0cqGHRiI8aaaa@45FB@     (8)

Effective temperature and efficiency of heat engines

Consider a heat engine working between two heat reservoirs, a hot reservoir at high temperature (TH) and a cold reservoir at low temperature (TL). The schematics of the heat engine are depicted in Figure 1.

The net work output wnet is calculated by means of the first law of thermodynamics (eq. 3) and is given by:

w net = Q H Q L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b qcfa4aaSbaaeaajugWaiaad6gacaWGLbGaamiDaaqcfayabaqcLbsa cqGH9aqpcaWGrbqcfa4aaSbaaeaajugWaiaadIeaaKqbagqaaKqzGe GaeyOeI0IaamyuaKqbaoaaBaaabaqcLbmacaWGmbaajuaGbeaaaaa@47F3@      (9)

The SOT function is given by equation (8).

The effective temperature is defined as the square root of the ratio between the net work output and the SOT function, and is given by:

w net = Q H Q L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b qcfa4aaSbaaeaajugWaiaad6gacaWGLbGaamiDaaqcfayabaqcLbsa cqGH9aqpcaWGrbqcfa4aaSbaaeaajugWaiaadIeaaKqbagqaaKqzGe GaeyOeI0IaamyuaKqbaoaaBaaabaqcLbmacaWGmbaajuaGbeaaaaa@47F3@      (10)

Finally, the effective efficiency is defined as the difference between unity minus the ratio between the low temperature and the effective temperature, and is given by:

n eff =1 T L T eff MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaeaajugWaiaadwgacaWGMbGaamOzaaqcfayabaqcLbsa cqGH9aqpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaamivaKqbao aaBaaaleaajugWaiaadYeaaSqabaaakeaajugibiaadsfajuaGdaWg aaWcbaGaamyzaiaadAgacaWGMbaabeaaaaaaaa@49A0@     (11)

The Carnot heat engine

The schematic of the Carnot heat engine is depicted in Figure 1. The net work output is given by (eq. 9). The Carnot cycle includes two processes: heat addition and heat rejection at constant temperatures (high and low) connected with two isentropic (reversible adiabatic) processes. The entropy of the heat engine is calculated by means of the second law of thermodynamics and is given by:

δq T =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qbae aadaWcaaqaaiabes7aKjaadghaaeaacaWGubaaaaqabeqacqWIr4E0 cqGHRiI8aiabg2da9iaaicdaaaa@4010@     (12)

By applying the assumptions of the Carnot cycle, equation (12) could be written explicitly as follows:

Q H T H = Q L T L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aajugibiaadgfajuaGdaWgaaqaaKqzadGaamisaaqcfayabaaabaqc LbsacaWGubqcfa4aaSbaaeaajugWaiaadIeaaKqbagqaaaaajugibi abg2da9KqbaoaalaaabaqcLbsacaWGrbqcfa4aaSbaaeaajugWaiaa dYeaaKqbagqaaaqaaKqzGeGaamivaKqbaoaaBaaabaqcLbmacaWGmb aajuaGbeaaaaaaaa@4B49@     (13)

The SOT function for the Carnot cycle is derived by means of equation 1 and is given by:

Q H T H = Q L T L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aajugibiaadgfajuaGdaWgaaqaaKqzadGaamisaaqcfayabaaabaqc LbsacaWGubqcfa4aaSbaaeaajugWaiaadIeaaKqbagqaaaaajugibi abg2da9KqbaoaalaaabaqcLbsacaWGrbqcfa4aaSbaaeaajugWaiaa dYeaaKqbagqaaaqaaKqzGeGaamivaKqbaoaaBaaabaqcLbmacaWGmb aajuaGbeaaaaaaaa@4B49@     (14)

Applying equation (13) to equation (4) aids to simplify the expression for the SOT function. After algebraic manipulation and using equation (9), the SOT function of the Carnot heat engine simplifies to the following:

SOT= w net T L T H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uai aad+eacaWGubGaeyypa0ZaaSaaaeaacaWG3bWaaSbaaeaajugWaiaa d6gacaWGLbGaamiDaaqcfayabaaabaqcLbsacaWGubqcfa4aaSbaae aajugWaiaadYeaaKqbagqaaKqzGeGaamivaKqbaoaaBaaabaqcLbma caWGibaajuaGbeaaaaaaaa@4912@     (15)

Then, the effective temperature is defined as the geometric mean of the high temperature and the low temperature, a result suggested by (eq. 15) and is given by:

n eff = T L T H = w net SOT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaeaajugWaiaadwgacaWGMbGaamOzaaqcfayabaqcLbsa cqGH9aqpjuaGdaGcaaqaaKqzGeGaamivaKqbaoaaBaaabaqcLbmaca WGmbaajuaGbeaajugibiaadsfajuaGdaWgaaqaaKqzadGaamisaaqc fayabaaabeaacqGH9aqpdaGcaaqaamaalaaabaGaam4DamaaBaaaba qcLbmacaWGUbGaamyzaiaadshaaKqbagqaaaqaaiaadofacaWGpbGa amivaaaaaeqaaaaa@5174@    (16)

Finally, the effective efficiency for the Carnot heat engine is given by:

n eff =1 T L T eff =1 T L T H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb qcfa4aaSbaaeaajugWaiaadwgacaWGMbGaamOzaaqcfayabaqcLbsa cqGH9aqpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaamivaKqbao aaBaaaleaajugWaiaadYeaaSqabaaakeaajugibiaadsfajuaGdaWg aaWcbaGaamyzaiaadAgacaWGMbaabeaaaaqcfaOaeyypa0JaaGymai abgkHiTmaakaaabaWaaSaaaeaajugibiaadsfajuaGdaWgaaqaaKqz adGaamitaaqcfayabaaabaqcLbsacaWGubqcfa4aaSbaaeaajugWai aadIeaaKqbagqaaaaaaeqaaaaa@5640@     (17)

Figure 1 Schematics of the engine working between two heat reservoirs, a hot reservoir at high temperature TH and a cold reservoir at low temperature TL. The heat input to the heat engine is QH and the heat rejection from the heat engine is QL. The net work output wnet is given by the QH-QL.

Air standard cycles

Ideal Otto cycle

The ideal Otto cycle is used to estimate the efficiency of spark ignition (SI) engine (Otto engine). The schematic of the Pressure–Volume (P-V) diagram of the ideal Otto cycle is shown in Figure 2. The cycle includes four processes: 1à2 isentropic compression, 2à3 constant volume heat addition, 3à4 isentropic expansion, and 4à1 constant volume heat rejection.

For a given initial state (pressure P1, volume V1, temperature T1) with known highest temperature T3, the application of the thermodynamic state relations and the first law of thermodynamics of the different branches lead to the following results:

The heat input to the engine QH is given by:

Q H = c v ( T 3 T 2 )= c v T 3 (1τa) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaajugibiabg2da9iaadoga juaGdaWgaaWcbaqcLbmacaWG2baaleqaaKqzGeGaaiikaiaadsfaju aGdaWgaaWcbaqcLbmacaaIZaaaleqaaKqzGeGaeyOeI0IaamivaKqb aoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacaGGPaGaeyypa0Jaam 4yaKqbaoaaBaaaleaajugWaiaadAhaaSqabaqcLbsacaWGubqcfa4a aSbaaSqaaKqzadGaaG4maaWcbeaajugibiaacIcacaaIXaGaeyOeI0 IaeqiXdqNaamyyaiaacMcaaaa@59B9@     (18)

Where cv is the constant volume heat capacity,  is the ratio between T1 and T3 and a () is the compression ratio r raised to the power (k-1) with k equals the ratio between constant pressure heat capacity (cp) and constant volume heat capacity (k=cp/cv).1

The heat rejection from the engine is given by:

Q L = c v ( T 4 T 1 )= c v T 3 (1τa) a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaajugibiabg2da9iaadoga juaGdaWgaaWcbaqcLbmacaWG2baaleqaaKqzGeGaaiikaiaadsfaju aGdaWgaaWcbaqcLbmacaaI0aaaleqaaKqzGeGaeyOeI0IaamivaKqb aoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacaGGPaGaeyypa0Jaam 4yaKqbaoaaBaaaleaajugWaiaadAhaaSqabaqcLbsacaWGubqcfa4a aSbaaSqaaKqzadGaaG4maaWcbeaajuaGdaWcaaqaaKqzGeGaaiikai aaigdacqGHsislcqaHepaDcaWGHbGaaiykaaqcfayaaKqzGeGaamyy aaaaaaa@5C5E@     (19)

Where T4 is the temperature at state 4.

The net work output (wnet) extracted by the engine is given by:

w net = Q H Q L = c v T 3 (1τa)( 1 1 a ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b qcfa4aaSbaaeaajugWaiaad6gacaWGLbGaamiDaaqcfayabaqcLbsa cqGH9aqpcaWGrbqcfa4aaSbaaeaajugWaiaadIeaaKqbagqaaKqzGe GaeyOeI0IaamyuaKqbaoaaBaaabaqcLbmacaWGmbaajuaGbeaajugi biabg2da9iaadogajuaGdaWgaaWcbaqcLbmacaWG2baaleqaaKqzGe GaamivaKqbaoaaBaaaleaajugWaiaaiodaaSqabaqcLbsacaGGOaGa aGymaiabgkHiTiabes8a0jaadggacaGGPaqcfa4aaeWaaOqaaKqzGe GaaGymaiabgkHiTKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGa amyyaaaaaOGaayjkaiaawMcaaaaa@5F84@    (20)

The SOT function is calculated based on equation (8). Noting that the contributions of the isentropic branches to the SOT function are zero, the resulting SOT function for the ideal OTTO cycle is given by:

SO T Otto = c v T 1 (1τa)( 1 1 a ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGtb Gaai4taiaacsfajuaGdaWgaaqaaKqzadGaam4taiaadshacaWG0bGa am4BaaqcfayabaqcLbsacqGH9aqpjuaGdaWcaaqaaKqzGeGaam4yaK qbaoaaBaaakeaajugWaiaadAhaaKqbagqaaaqaaKqzGeGaamivaKqb aoaaBaaabaqcLbmacaaIXaaajuaGbeaaaaqcLbsacaGGOaGaaGymai abgkHiTiabes8a0jaadggacaGGPaqcfa4aaeWaaOqaaKqzGeGaaGym aiabgkHiTKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamyyaa aaaOGaayjkaiaawMcaaaaa@58E2@      (21)

By comparing equations (20) and (21) it is observed that the effective temperature could be deduced and is given by:

T eff = w net SO T Otto = T 1 T 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamyzaiaadAgacaWGMbaaleqaaKqzGeGa eyypa0tcfa4aaOaaaOqaaKqbaoaalaaakeaajugibiaadEhajuaGda WgaaWcbaqcLbmacaWGUbGaamyzaiaadshaaSqabaaakeaajugibiaa dofacaWGpbGaamivaKqbaoaaBaaaleaajugWaiaad+eacaWG0bGaam iDaiaad+gaaSqabaaaaaqabaqcLbsacqGH9aqpjuaGdaGcaaGcbaqc LbsacaWGubqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiaads fajuaGdaWgaaWcbaqcLbmacaaIZaaaleqaaaqabaaaaa@5880@     (22)

The effective efficiency for the ideal OTTO cycle is given by equation (17), where T eff = w net SO T Otto = T 1 T 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamyzaiaadAgacaWGMbaaleqaaKqzGeGa eyypa0tcfa4aaOaaaOqaaKqbaoaalaaakeaajugibiaadEhajuaGda WgaaWcbaqcLbmacaWGUbGaamyzaiaadshaaSqabaaakeaajugibiaa dofacaWGpbGaamivaKqbaoaaBaaaleaajugWaiaad+eacaWG0bGaam iDaiaad+gaaSqabaaaaaqabaqcLbsacqGH9aqpjuaGdaGcaaGcbaqc LbsacaWGubqcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiaads fajuaGdaWgaaWcbaqcLbmacaaIZaaaleqaaaqabaaaaa@5880@ is replaced by T L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaaaaa@3A22@ and T 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaaG4maaWcbeaaaaa@3A0E@ is replaced by T H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaaaaa@3A1E@ .

Ideal Brayton cycle

The ideal Brayton cycle is used to estimate the efficiency of gas turbines. The schematic of the Pressure–Volume (P-V) diagram of the ideal Brayton cycle is shown in Figure 3. The Brayton cycle includes four processes: 1à2 isentropic compression, 2à3 constant pressure heat addition, 3à4 isentropic expansion, and 4à1 constant pressure heat rejection.

For a given initial state (pressure P1, volume V1, temperature T1) with known highest temperature T3, the application of the thermodynamic state relations and the first law of thermodynamics of the different branches lead to the following results:

The heat input to the engine QH is given by:

T H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaaaaa@3A1E@    (23)

Where cp is the constant pressure heat capacity, τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHep aDaaa@384A@ is the ratio between T1 and T3 and a ( a= r k1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGHb Gaeyypa0JaamOCaKqbaoaaCaaaleqabaqcLbmacaWGRbGaeyOeI0Ia aGymaaaaaaa@3DE9@ ) is the compression ratio r raised to the power (k-1) with k equals the ratio between constant pressure heat capacity (cp) and constant volume heat capacity (k=cp/cv).

The heat rejection from the Brayton cycle is given by:

Q L = c p ( T 4 T 1 )= c p T 3 (1τa) a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaajugibiabg2da9iaadoga juaGdaWgaaWcbaqcLbmacaWGWbaaleqaaKqzGeGaaiikaiaadsfaju aGdaWgaaWcbaqcLbmacaaI0aaaleqaaKqzGeGaeyOeI0IaamivaKqb aoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacaGGPaGaeyypa0Jaam 4yaKqbaoaaBaaaleaajugWaiaadchaaSqabaqcLbsacaWGubqcfa4a aSbaaSqaaKqzadGaaG4maaWcbeaajuaGdaWcaaqaaiaacIcacaaIXa GaeyOeI0IaeqiXdqNaaiyyaiaacMcaaeaacaWGHbaaaaaa@5AA5@    (24)

Where T4 is the temperature at state 4.

The net work output (wnet) extracted by the Brayton cycle is given by:

w net = Q H Q L = c p T 3 (1τa)( 1 1 a ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b qcfa4aaSbaaeaajugWaiaad6gacaWGLbGaamiDaaqcfayabaqcLbsa cqGH9aqpcaWGrbqcfa4aaSbaaeaajugWaiaadIeaaKqbagqaaKqzGe GaeyOeI0IaamyuaKqbaoaaBaaabaqcLbmacaWGmbaajuaGbeaajugi biabg2da9iaadogajuaGdaWgaaWcbaqcLbmacaWGWbaaleqaaKqzGe GaamivaKqbaoaaBaaaleaajugWaiaaiodaaSqabaqcLbsacaGGOaGa aGymaiabgkHiTiabes8a0jaadggacaGGPaqcfa4aaeWaaOqaaKqzGe GaaGymaiabgkHiTKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGa amyyaaaaaOGaayjkaiaawMcaaaaa@5F7E@     (25)

The SOT function is calculated based on equation (8). Noting that the contributions of the isentropic branches to the SOT function are zero, the resulting SOT function for the ideal Brayton cycle is given by:

SO T Brayton = c p T 1 (1τa)( 1 1 a ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaGGtb Gaai4taiaacsfajuaGdaWgaaqaaKqzadGaamOqaiaadkhacaWGHbGa amyEaiaadshacaWGVbGaamOBaaqcfayabaqcLbsacqGH9aqpjuaGda WcaaqaaKqzGeGaam4yaKqbaoaaBaaakeaajugWaiaadchaaKqbagqa aaqaaKqzGeGaamivaKqbaoaaBaaabaqcLbmacaaIXaaajuaGbeaaaa qcLbsacaGGOaGaaGymaiabgkHiTiabes8a0jaadggacaGGPaqcfa4a aeWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaalaaakeaajugibiaaig daaOqaaKqzGeGaamyyaaaaaOGaayjkaiaawMcaaaaa@5BA4@     (26)

By comparing equations (25) and (26) it is observed that the effective temperature could be deduced and is given by:

T eff = w net SO T Brayton = T 1 T 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamyzaiaadAgacaWGMbaaleqaaKqzGeGa eyypa0tcfa4aaOaaaOqaaKqbaoaalaaakeaajugibiaadEhajuaGda WgaaWcbaqcLbmacaWGUbGaamyzaiaadshaaSqabaaakeaajugibiaa dofacaWGpbGaamivaKqbaoaaBaaaleaajugWaiaadkeacaWGYbGaam yyaiaadMhacaWG0bGaam4Baiaad6gaaSqabaaaaaqabaqcLbsacqGH 9aqpjuaGdaGcaaGcbaqcLbsacaWGubqcfa4aaSbaaSqaaKqzadGaaG ymaaWcbeaajugibiaadsfajuaGdaWgaaWcbaqcLbmacaaIZaaaleqa aaqabaaaaa@5B48@    (27)

The effective efficiency the ideal Brayton cycle is given by equation (17), where T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaaa@3A0C@ is replaced by T L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaaaaa@3A22@ and T 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaaG4maaWcbeaaaaa@3A0E@ is replaced by T H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaaaaa@3A1E@ .

Ideal Stirling cycle

The ideal Stirling cycle is used to estimate the efficiency of Stirling engine. The schematic of the Pressure–Volume (P-V) diagram of the ideal Stirling cycle is shown in Figure 4. The cycle includes four processes: 1à2 isothermal compression, 2à3 constant volume heat addition, 3à4 isothermal expansion, and 4à1 constant volume heat rejection.

For a given initial state (pressure P1, volume V1, temperature T1) with known highest temperature T3, the application of the thermodynamic state relations and the first law of thermodynamics of the different branches lead to the following results:

The heat input to the Stirling cycle QH is given by:

Q H = c v ( T 3 T 2 )+R T 3 ln(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaajugibiabg2da9iaadoga juaGdaWgaaWcbaqcLbmacaWG2baaleqaaKqzGeGaaiikaiaacsfaju aGdaWgaaWcbaqcLbmacaaIZaaaleqaaKqzGeGaeyOeI0IaaiivaKqb aoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacaGGPaGaey4kaSIaam OuaiaadsfajuaGdaWgaaWcbaqcLbmacaaIZaaaleqaaKqzGeGaciiB aiaac6gacaGGOaGaaiOCaiaacMcaaaa@548C@    (28)

Where cv is the constant volume heat capacity, τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHep aDaaa@384A@ is the ratio between T1 and T3 and a ( a= r k1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGHb Gaeyypa0JaamOCaKqbaoaaCaaaleqabaqcLbmacaWGRbGaeyOeI0Ia aGymaaaaaaa@3DE9@ ) is the compression ratio r raised to the power (k-1) with k equals the ratio between constant pressure heat capacity (cp) and constant volume heat capacity (k=cp/cv).

The heat rejection from the Stirling cycle is given by:

Q L = c v ( T 4 T 1 )+R T 1 ln(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb qcfa4aaSbaaSqaaiaadYeaaeqaaKqzGeGaeyypa0Jaam4yaKqbaoaa BaaaleaajugWaiaadAhaaSqabaqcLbsacaGGOaGaaiivaKqbaoaaBa aaleaacaaI0aaabeaajugibiabgkHiTiaacsfajuaGdaWgaaWcbaqc LbmacaaIXaaaleqaaKqzGeGaaiykaiabgUcaRiaadkfacaWGubqcfa 4aaSbaaSqaaiaaigdaaeqaaKqzGeGaciiBaiaac6gacaGGOaGaaiOC aiaacMcaaaa@50E3@    (29)

Where T4 is the temperature at state 4 and R is the ideal gas constant.

The net work output (wnet) extracted by the Stirling cycle is given by:

w net = Q H Q L = c v R T 3 (1τ)ln(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b qcfa4aaSbaaeaajugWaiaad6gacaWGLbGaamiDaaqcfayabaqcLbsa cqGH9aqpcaWGrbqcfa4aaSbaaeaajugWaiaadIeaaKqbagqaaKqzGe GaeyOeI0IaamyuaKqbaoaaBaaabaqcLbmacaWGmbaajuaGbeaajugi biabg2da9iaadogajuaGdaWgaaWcbaGaamODaaqabaqcfaOaamOuaK qzGeGaamivaKqbaoaaBaaaleaajugWaiaaiodaaSqabaqcLbsacaGG OaGaaGymaiabgkHiTiabes8a0jaacMcaciGGSbGaaiOBaiaacIcaca GGYbGaaiykaaaa@5B2A@     (30)

The SOT function is calculated based on equation (8). Noting that the contributions of the isothermal branches to the SOT function cancel each other, the resulting SOT function for the ideal Stirling cycle is given by:

SO T Stirling =Rln(r)( 1 T 1 1 T 3 )= Rln(r) T 1 ( 1τ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb Gaam4taiaadsfajuaGdaWgaaWcbaqcLbmacaWGtbGaamiDaiaadMga caWGYbGaamiBaiaadMgacaWGUbGaam4zaaWcbeaajuaGcqGH9aqpju gibiaadkfaciGGSbGaaiOBaiaacIcacaGGYbGaaiykaKqbaoaabmaa baWaaSaaaeaacaaIXaaabaGaamivamaaBaaabaqcLbmacaaIXaaaju aGbeaaaaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamivamaaBaaabaqc LbmacaaIZaaajuaGbeaaaaaacaGLOaGaayzkaaGaeyypa0ZaaSaaae aacaWGsbGaciiBaiaac6gacaGGOaGaaiOCaiaacMcaaeaacaWGubWa aSbaaeaajugWaiaaigdaaKqbagqaaaaadaqadaqaaiaaigdacqGHsi slcqaHepaDaiaawIcacaGLPaaaaaa@63A3@     (31)

By comparing equations (30) and (31) it is observed that the effective temperature could be deduced and is given by:

T eff = w net SO T Stirling = T 1 T 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamyzaiaadAgacaWGMbaaleqaaKqzGeGa eyypa0tcfa4aaOaaaOqaaKqbaoaalaaakeaajugibiaadEhajuaGda WgaaWcbaqcLbmacaWGUbGaamyzaiaadshaaSqabaaakeaajugibiaa dofacaWGpbGaamivaKqbaoaaBaaaleaajugWaiaadofacaWG0bGaam yAaiaadkhacaWGSbGaamyAaiaad6gacaWGNbaaleqaaaaaaeqaaKqz GeGaeyypa0tcfa4aaOaaaOqaaKqzGeGaamivaKqbaoaaBaaaleaaju gWaiaaigdaaSqabaqcLbsacaWGubqcfa4aaSbaaSqaaKqzadGaaG4m aaWcbeaaaeqaaaaa@5C3A@     (32)

The effective efficiency the ideal OTTO cycle is given by equation (17), where is replaced by  and  is replaced by.

Ideal Ericsson cycle

The ideal Ericsson cycle is used to estimate the efficiency of the Ericsson engine. The schematics of the Pressure–Volume (P-V) diagram of the ideal Ericsson cycle is shown in Figure 5. The cycle includes four processes: 1à2 isothermal compression, 2à3 constant pressure heat addition, 3à4 isothermal expansion, and 4à1 constant pressure heat rejection.

For a given initial state (pressure P1, volume V1, temperature T1) with known highest temperature T3, the application of the thermodynamic state relations and the first law of thermodynamics of the different branches lead to the following results:

The heat input to the engine QH is given by:

Q H = c p ( T 3 T 2 )+R T 3 ln(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb qcfa4aaSbaaSqaaiaadIeaaeqaaKqzGeGaeyypa0Jaam4yaKqbaoaa BaaaleaacaWGWbaabeaajugibiaacIcacaGGubqcfa4aaSbaaSqaai aaiodaaeqaaKqzGeGaeyOeI0IaaiivaKqbaoaaBaaaleaacaaIYaaa beaajugibiaacMcacqGHRaWkcaWGsbGaamivaKqbaoaaBaaaleaaca aIZaaabeaajugibiGacYgacaGGUbGaaiikaiaackhacaGGPaaaaa@4E69@    (33)

Where cv is the constant volume heat capacity, τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHep aDaaa@384A@ is the ratio between T1 and T3 and a ( a= r k1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGHb Gaeyypa0JaamOCaKqbaoaaCaaaleqabaqcLbmacaWGRbGaeyOeI0Ia aGymaaaaaaa@3DE9@ ) is the compression ratio r raised to the power (k-1) with k equals the ratio between constant pressure heat capacity (cp) and constant volume heat capacity (k=cp/cv).

The heat rejection from the engine is given by:

Q L = c p ( T 4 T 1 )+R T 1 ln(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGrb qcfa4aaSbaaSqaaiaadYeaaeqaaKqzGeGaeyypa0Jaam4yaKqbaoaa BaaaleaajugWaiaadchaaSqabaqcLbsacaGGOaGaaiivaKqbaoaaBa aaleaacaaI0aaabeaajugibiabgkHiTiaacsfajuaGdaWgaaWcbaqc LbmacaaIXaaaleqaaKqzGeGaaiykaiabgUcaRiaadkfacaWGubqcfa 4aaSbaaSqaaiaaigdaaeqaaKqzGeGaciiBaiaac6gacaGGOaGaaiOC aiaacMcaaaa@50DD@      (34)

Where T4 is the temperature at state 4.

The net work output (wnet) extracted by the engine is given by:

w net = Q H Q L =R T 3 (1τ)ln(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b qcfa4aaSbaaeaajugWaiaad6gacaWGLbGaamiDaaqcfayabaqcLbsa cqGH9aqpcaWGrbqcfa4aaSbaaeaajugWaiaadIeaaKqbagqaaKqzGe GaeyOeI0IaamyuaKqbaoaaBaaabaqcLbmacaWGmbaajuaGbeaajugi biabg2da9KqbakaadkfajugibiaadsfajuaGdaWgaaWcbaqcLbmaca aIZaaaleqaaKqzGeGaaiikaiaaigdacqGHsislcqaHepaDcaGGPaGa ciiBaiaac6gacaGGOaGaaiOCaiaacMcaaaa@588D@     (35)

The SOT function is calculated based on equation (8). Noting that the contributions of the isobaric branches to the SOT function cancel each other, the resulting SOT function for the ideal Ericsson cycle is given by:

SO T Ericsson =Rln(r)( 1 T 1 1 T 3 )= Rln(r) T 1 ( 1τ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb Gaam4taiaadsfajuaGdaWgaaWcbaqcLbmacaWGfbGaamOCaiaadMga caWGJbGaam4CaiaadohacaWGVbGaamOBaaWcbeaajuaGcqGH9aqpju gibiaadkfaciGGSbGaaiOBaiaacIcacaGGYbGaaiykaKqbaoaabmaa baWaaSaaaeaacaaIXaaabaGaamivamaaBaaabaqcLbmacaaIXaaaju aGbeaaaaGaeyOeI0YaaSaaaeaacaaIXaaabaGaamivamaaBaaabaqc LbmacaaIZaaajuaGbeaaaaaacaGLOaGaayzkaaGaeyypa0ZaaSaaae aacaWGsbGaciiBaiaac6gacaGGOaGaaiOCaiaacMcaaeaacaWGubWa aSbaaeaajugWaiaaigdaaKqbagqaaaaadaqadaqaaiaaigdacqGHsi slcqaHepaDaiaawIcacaGLPaaaaaa@639D@    (36)

By comparing equations (35) and (36) it is observed that the effective temperature could be deduced and is given by:

T eff = w net SO T Ericsson = T 1 T 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamyzaiaadAgacaWGMbaaleqaaKqzGeGa eyypa0tcfa4aaOaaaOqaaKqbaoaalaaakeaajugibiaadEhajuaGda WgaaWcbaqcLbmacaWGUbGaamyzaiaadshaaSqabaaakeaajugibiaa dofacaWGpbGaamivaKqbaoaaBaaaleaajugWaiaadweacaWGYbGaam yAaiaadogacaWGZbGaam4Caiaad+gacaWGUbaaleqaaaaaaeqaaKqz GeGaeyypa0tcfa4aaOaaaOqaaKqzGeGaamivaKqbaoaaBaaaleaaju gWaiaaigdaaSqabaqcLbsacaWGubqcfa4aaSbaaSqaaKqzadGaaG4m aaWcbeaaaeqaaaaa@5C34@     (37)

The effective efficiency the ideal OTTO cycle is given by equation (17). where T 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaaa@3A0C@ is replaced by T L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamitaaWcbeaaaaa@3A22@ and T 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaaG4maaWcbeaaaaa@3A0E@ is replaced by T H MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamisaaWcbeaaaaa@3A1E@ .

Figure 2 Schematics if the ideal Otto cycle.

Figure 3 Schematics if the ideal Brayton cycle.

Figure 4 Schematics if the ideal Stirling cycle.

Figure 5 Schematics if the ideal Ericsson cycle.

Numerical results

The following table compares the effective efficiency with Carnot efficiency and with the observed efficiency for some real plants. As can be seen, the effective efficiency is much closer to the observed data.60

Power Source

TL ºC

TH ºC

Carnot efficiency

Effective efficiency

Observed efficiency

Effective temperature K

West Thurrock (UK) coal fired power plant

25

565

64.40%

40.40%

36%

500

CANDU (Canada) nuclear power plant

25

300

48%

27.90%

30%

413

Larderello (Italy) geothermal power plant

80

250

32.50%

17.80%

16%

430

Stirling/Ericsson

27

600

41.4

512

Otto

27

1800

85.5

62

789

Brayton

27

1100

78.1

53.2

 

642

Summary and conclusion

For a heat engine working between two heat reservoirs, a hot reservoir at high temperature TH and a cold reservoir at a low temperature TL the Carnot cycle and Curzon-Ahlborn heat engine were shortly reviewed and their performance efficiencies are given via the Carnot efficiency and the Curzon-Ahlborn respectively. The new terms SOT function, effective temperature and the effective efficiency were introduced. The SOT function was defined as minus the cyclic integral of the heat change divided by the temperature squared. The effective temperature was defined as the square root of the ratio between the net work output and the SOT function, and finally the effective efficiency was defined as one minus the ratio between the cold reservoir temperature and the effective temperature. The SOT function was calculated for different thermodynamic processes: isochoric, isobaric, isothermal and isentropic. A model of heat engine was considered and general expressions of the SOT function, effective temperature and effective efficiency were given as useful tools for the designer. These expressions were applied to different thermodynamic cycles: The Carnot cycle and air standard cycles (Otto, Brayton, Stirling and Ericsson). The effective temperature for the considered heat engines was given as the square root of the product of the reservoirs' temperatures and the efficiency was given as one minus the square root of the ratio between the cold reservoir temperature and the hot reservoir temperature. The derived expressions can serve two important purposes: in education and for a quick estimation tool for heat engine designers.

Acknowledgements

None.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this paper.

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