Research Article Work Criteria Function Of Irreversible Heat . - Hindawi
Hindawi Publishing CorporationPhysics Research InternationalVolume 2014, Article ID 890713, 7 pageshttp://dx.doi.org/10.1155/2014/890713Research ArticleWork Criteria Function of Irreversible Heat EnginesMahmoud HuleihilAcademic Institute for Arab Teacher Training, Beit-Berl College, 44905 Doar Beit-Berl, IsraelCorrespondence should be addressed to Mahmoud Huleihil; mahmud.ana@gmail.comReceived 19 May 2014; Accepted 17 July 2014; Published 5 August 2014Academic Editor: Ashok ChatterjeeCopyright 2014 Mahmoud Huleihil. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.The irreversible heat engine is reconsidered with a general heat transfer law. Three criteria known in the literatureβpower, powerdensity, and efficient powerβare redefined in terms of the work criteria function (WCF), a concept introduced in this study. Theformulation enabled the suggestion and analysis of a unique criterionβthe efficient power density (which accounts for the efficiencyand power density). Practically speaking, the efficient power and the efficient power density could be defined on any order basedon the WCF. The applicability of the WCF is illustrated for the Newtonian heat transfer law (π 1) and for the radiative law (π 4).The importance of WCF is twofold: it gives an explicit design and educational tool to analyze and to display graphically the differentcriteria side by side and thus helps in design process. Finally, the criteria were compared and some conclusions were drawn.1. IntroductionFinite-time thermodynamics [1, 2] has been used extensivelyin different fields of research [3β24]. A recent review [25]reported several key advances in theoretical investigationsof efficiency at maximum power operation of heat engines.In the review, a presentation of the analytical results ofefficiency at maximum power was given for the CurzonAhlborn heat engine, for the stochastic heat engine constructed from a Brownian particle, and for Feynmanβs ratchetas a heat engine [25]. The endoreversible heat engine as presented by Curzon and Ahlborn [1] considered an internallyreversible heat engine working between two heat reservoirs.A Newtonian-type heat transfer was assumed. The objectivewas to maximize power output. Different heat transfer lawsand different types of irreversibility were also considered [4β24]. In order to find the maximum power and efficiencyat maximum power output, relation between the designparameters of internally and externally radiative heat engineswas presented [26]. Another objective function in finite-timethermodynamics with an ecological criterion was appliedto an irreversible Carnot heat engine interacting with finitethermal capacitance rates of the heat reservoirs and finitetotal conductance of the heat exchangers [27]. The reverseheat engine (refrigerator) was considered also for economicoptimization of endoreversible operation and was carried outin [28]. The results obtained involve the following commonheat transfer laws: Newtonβs law (π 1), the linear phenomenological law in irreversible thermodynamics (π 1),and the radiative heat transfer law (π 4). In a different study[29], the thermoeconomic optimization of an irreversiblesolar-driven heat engine model was carried out using finitetime/finite-size thermodynamic theory. In this study losseswere taken into account due to heat transfer across finite timetemperature differences, heat leakage between thermal reservoirs and internal irreversibilities in terms of a parameter thatcomes from the Clausius inequality [29]. A more generalizedradiative heat transfer law was introduced into a modelexternal combustion engine with a movable piston, andeffects of heat transfer laws on the optimizations of the enginefor maximum work output were investigated [30]. Numericalexamples for the optimizations with linear phenomenological(π 1), Newtonβs (π 1), square (π 2), cubic (π 3), and radiative (π 4) heat transfer laws were provided,respectively, and the obtained results were compared witheach other. A power analysis was conducted on a reversibleJoule-Brayton cycle [31]. Although many studies have beencarried out using different performance criteria, resulting infamous efficiencies (Carnot, Curzon-Ahlborn), most do notconsider the sizes of the engines [31], but Gordon [24] wasthe first to observe that finite size of a system shows upexactly like finite time (duration). In the studies of Curzon
2and Ahlborn and others, researchers utilized the thermalefficiency at maximum power as a performance measure thatdictates an efficiency standard for practical heat engines. Asdescribed in [31], instead of maximizing power for certaincycle parameters, the power density defined as the ratio ofpower to the maximum specific volume in the cycle wasmaximized, taking into account the effects of the enginesizes. The obtained results showed an efficiency value atthe maximum power density that was always greater thanthat at the maximum power as given by Curzon-Ahlbornefficiency. Evaluations showed that design parameters atthe maximum power density resulted in smaller and moreefficient Joule-Brayton engines [31]. The maximum powerdensity criteria and the efficient power of an engine (definedas the product of power output and efficiency) were takenas the objectives for performance analysis and optimizationof an internally and externally irreversible radiative Carnotheat engine model. These objectives were approached fromthe standpoint of finite-time thermodynamics or entropygeneration minimization. The consequences of a maximumefficient power (MEP) design were analyzed. The obtainedresults showed that engines designed at MEP conditionshad an advantage of smaller size and were more efficientthan those designed at MP and MPD conditions [32]. Thesefindings were in agreement with those given by [31]. Otherconsiderations were studied for a class of irreversible Carnotengines that resulted from combining the characteristics oftwo models found in the literatureβthe model in finite timeand the model in finite size [33]. In a different study [34],the efficient power criterion was reconsidered analyticallyand finite-time thermodynamic optimization was carried outfor an irreversible Carnot heat engine. The obtained resultswere compared with those obtained by using the maximumpower and maximum power density criteria similar to theanalysis done in [32]. The optimal design parameters havebeen derived analytically, and the effect of the irreversibilities on general and optimal performances was investigated.Maximizing the efficient power gave a compromise betweenpower and efficiency. The derived results showed that thedesign parameter at the maximum efficient power conditionresulted in more efficient engines than at the MP conditions,and that the MEP criterion may have a significant poweradvantage with respect to the MPD criterion [34], a resultsupporting what had been reported in [32]. In summary,different performance criteria were employed to optimizethe performance of heat engines and heat pumps, usingthe methods of finite-time thermodynamics and taking intoaccount endoreversible and internally irreversible conditions.The maximization of the power density is defined as the ratioof the maximum power to the maximum specific volume inthe cycle. It takes into consideration the engine size instead ofjust maximizing its power output. The inclusion of the enginesize in the calculation of its performance is a very importantfactor from an economical point of view. In this study, knowncriteria are reconsidered and the work criterion function isintroduced. The formulation of this function enabled thesuggestion of a performance criterion to be consideredβthe efficient power density (EPD), defined as the efficientpower divided by the maximum volume of the working fluid,Physics Research InternationalTHqhThIrreversibleheat engineWTcqcTLFigure 1: Schematics of irreversible heat engines showing thecomponents and variables involved.following the definitions given in [32], and its maximumvalue (MEPD). Details of the criteria function are given inthe next section.2. The Work Criteria FunctionThe irreversible heat engine, as considered earlier, in [32],was assumed to follow law of radiative heat transfer. Itwas analyzed to determine the maximum efficient power.In this section, the irreversible heat engine is reconsideredbased on laws of general heat transfer, and criteria knownin the literature (maximum power, maximum power density,and maximum efficient power) are reviewed and recast interms of the work criteria function (WCF). The WCF allowsthe definition of a new criterion, the maximum efficientpower density, as suggested from the result obtained. Theanalysis relies upon applying the first and second laws ofthermodynamics written as equality, which is a commonexercise that is well documented in the literature available infinite-time thermodynamics research. The schematics of theengine are given in Figure 1, and the schematics of its temperature entropy π-π diagram are depicted by Figure 2. Theirreversible heat engine under consideration works betweentwo heat reservoirs at high and low temperatures. The rate ofheat input to the engine, πβ , is given byπ πβπ ) .πβ πβ (ππ»(1)In this equation, exponent π represents well-known heattransfer rates: the Newtonian heat transfer rate (π 1), theradiative heat transfer rate (π 4), and so on; πβ is thethermal conductance at the hot side of the engine; ππ» is thetemperature of the hot reservoir; and πβ is the temperature ofthe working fluid at the hot side of the engine. Similarly, the
Physics Research Internationalwork criteria function formulation. Equation (1) is rearrangedto give the temperature at the hot side of the engine byTHTh32s2qhπ πβπ ππ»3πβ.πβ(6)Similarly, based on (2), the temperature at the cold side of theengine is given byTTc14sqc4πππ ππΏπ TL(7)The efficiency of the endoreversible heat engine π0 (π 1)is a convenient choice to relate the ratio of the heat enginetemperature; thus (4) could be rearranged as given bySProcessesπππ 1 π0 π π .πβπβ1-2s: isentropic compression1-2: irreversible compression2s-3, 2-3: constant temperature heat addition(8)Equations (6)β(8) are used to derive the explicit expressionfor the rate of heat input to the engine, given by3-4s: isentropic expansion3-4: irreversible expansionπππβ πβ ππ»4s-1, 4-1: constant temperature heat rejectionFigure 2: Schematics of π-π diagram of the irreversible heat engine.cooling rate or the rate of heat rejection from the heat engine,ππ , is given byππ ππ (πππ ππΏπ ) .ππ.ππ(2)In this equation, ππ is the thermal conductance at the cold sideof the heat engine, ππ is the temperature of the working fluidat the cold side of the engine, and ππΏ is the temperature of thecold reservoir. The net power rate, π€, extracted by the heatengine, follows the first law of thermodynamics for a cyclicprocess and is given by((1 π0 ) ππ )π 1(1 π0 ) ((1 π0 ) (π /π )).(9)In this equation π is the ratio between the temperatures ofthe reservoirs, ππΏ /ππ», and π is the ratio between the thermalconductance ratio ππ /πβ .As stated earlier, there are different criteria that arecommonly used to describe the performance of heat engines.In this study the following criteria are considered:(i) the net power extracted from the heat engine (givenby (3));(ii) the net power density defined as the net powerextracted divided by the maximum volume (see [32]for more details);(3)(iii) the net efficient power defined as the efficiencymultiplied by the net power;The second law of thermodynamics, accounting for internalirreversibilities, is given by(iv) the net efficient power density defined as the netefficient power divided by the maximum volume ofthe working fluid;π€ πβ ππ .ππβ π π 0.πβππ(4)In (4) π is the irreversibility factor; its value falls in the 0 π 1 range.The efficiency of the heat engine is defined as the ratiobetween net power extracted from the heat engine and therate of heat absorbed by it. In mathematical symbols theefficiency is given byπ 1 1 π0ππ 1 ,πβπ (5)where π0 is the efficiency of the endoreversible heat engine(π 1).Equations (1) to (4) are manipulated and rearranged inthe following calculations in order to explicitly progress to the(v) Results being valid for ideal gas (10).The maximum volume of the working fluid, πmax (expressedin SI units), is adopted from [32] and is given byπmax ππ π πππmin.(10)In this equation π is the mass of the working fluid, π π is theideal gas coefficient (assuming ideal gas as the working fluid),and πmin is the minimum pressure in the cycle. It is interestingto note that the criteria summarized in the previous pageby (i)β(iv) could be addressed by the work criteria function,which is given byWCF ππΌ πβπ½πmax.(11)
Physics Research InternationalWCF π (πΌ, π½, π, π, π , π ; π0 ) .(12)To facilitate the generation of the plots given in the nextsection, the temperatures at both sides of the heat engine, asderived in explicit form, are shown for the convenience of thereader. The temperature of the working fluid at the hot side ofthe engine is given byπβ ππ»(1 1/ππβ) .πβ πππ»(13)The temperature of the working fluid at the cold side of theengine (using (8)-(9) and (13)) is given by1/ππππ ππ» (1 π0 ) (1 (1 π0 ) πππ 1(1 π0 ) ((1 π0 ) (π /π ))).(14)In the next section the WCF is considered numerically andsample plots are given for illustrating its usage in analyzingdifferent criteria for the irreversible heat engine.3. Numerical ConsiderationsIn this section the work criteria function is used to illustratethe different criteria aforementioned earlier. The Newtonianheat transfer law is used in the illustration, but other casescould be presented, following the same procedure.3.1. Newtonian Heat Transfer Law (π 1). The followingfigures (Figures 3β9) are given for the case of the Newtonianheat transfer law, with π 0.5 (a typical value of Rankinecycle or steam turbines, for which ππ» 600 K and ππΏ 300 K). Figure 3 shows the power output of the endoreversibleheat engine relative to its maximum value. The plot showscurves of the power for different values of the irreversibilityfactor, π , in the 0.85β1 range.By consulting Figure 3 two points should be explained insome detail. First, the maximum efficiency is reduced fromPower relative to its maximum value at R 1versus efficiency for different values of irreversibility parameter R10.90.80.70.60.5R cyFigure 3: Power relative to its maximum value at π 1 and π 1versus Newtonian heat transfer law efficiency (π 1). The plotsrepresent values of π in the 0.85β1 range. The ratio between thetemperatures of the reservoirs, ππΏ /ππ» is π 0.5, a typical valuefor a Rankine cycle (for the steam turbine working between 600 Kand 300 K).Power density relative to its maximum value at R 1versus efficiency for different values of irreversibility parameter RPower density relative to its maximumvalue at R 1In this equation πΌ and π½ are integers to represent the criteriaconsidered in the study. The study uses πΌ 0, π½ 0 to givethe expression of the heat input to the heat engine (see (1) and(9)). The power output extracted by the heat engine is givenwhen πΌ 1 and π½ 0. The power density criterion is definedwhen πΌ 1 and π½ 1. The efficient power criterion is definedusing πΌ 2 and π½ 0. From the suggested formalism (see(11)), one could consider the criterion defined by πΌ 2, π½ 1 to mean the efficient power density extracted by the heatengine. The WCF, therefore, provides a general statement todefine efficient power and efficient power density of any order.The WCF could be viewed as a one-dimensional functionfor a specific choice of variables or parameters involved. Inthis study, the parameter π0 was chosen to be variable; itsvalue spans the range from zero up to Carnot efficiency (0 π0 1 π). The other parameters are each chosen to define aspecific criterion. The general criteria suggested by the workfunction is given byPower relative to its maximumvalue at R 1410.90.80.70.60.50.40.30.20.10R 10.950.900.8500.10.20.3Efficiency0.40.5Figure 4: Power density relative to its maximum value at π 1 andπ 1 versus Newtonian heat transfer law efficiency (π 1). Theplots represent values of π in the 0.85β1 range. The ratio betweenthe temperatures of the reservoirs, ππΏ /ππ» is π 0.5, a typical valuefor a Rankine cycle (for the steam turbine working between 600 Kand 300 K).the Carnot efficiency to values given by (5). Second, themaximum power is reduced and its efficiency is shifted to theleft. If we consider the extreme case, or π 0.85, one couldobserve that the maximum efficiency and the efficiency atmaximum power were reduced by approximately 20%, whilethe maximum power was reduced by 42%. Similar behavioris observed when considering the other performance characteristic curves, such as power density (Figure 4), efficientpower (Figure 5), and efficient power density (Figure 6). It isimportant to note that the criteria considered in the plots, inascending order, show a shift to the right in the location ofthe maximum value of the criterion under consideration. Onecould conclude that, for a higher order of the efficient power
5Power density relative to its maximum value versusEfficient power relative to its maximum value atR 1 versus efficiency for different values ofirreversibility factor R10.8R 10.950.60.900.85efficiency for different k values10.40.2000.10.20.3Efficiency0.40.5Figure 5: Efficient power relative to its maximum value at π 1and π 1 versus Newtonian heat transfer law efficiency (π 1). Theplots represent values of π in the 0.85β1 range. The ratio between thetemperatures of the reservoirs, ππΏ /ππ» is π 0.5, a typical value fora Rankine cycle (or the steam turbine working between 600 K and300 K).Efficient power density relative to its maximum value atR 1 versus efficiency for different values ofPower density relative to its maximum valueEfficient power relative to its maximumvalue at R 1Physics Research Internationalk 0.50.90.8k 10.70.60.50.40.30.20.1000.10.40.5Efficient power density relative toits maximum value at R 1Figure 6: Efficient power density relative to its maximum value atπ 1 and π 1 versus Newtonian heat transfer law efficiency(π 1). The plots represent values of π in the 0.85β1 range. Theratio between the temperature of the reservoirs, ππΏ /ππ» is π 0.5,a typical value for a Rankine cycle (for the steam turbine workingbetween 600 K and 300 K).or the efficient power density, a higher shift to the right in theefficiency value is produced.The effect of thermal conductance is reported in Figures7 and 8, for which inclusive values of π in the 1β10 rangeare shown on the plots. The reasonable general conclusionis depicted, as the value of π changes drastically, of theefficiency at maximum criteria changed approximately by3%. For comparison purposes, power, power density, efficientpower, and efficient power density criteria for the π 1 caseare shown on the plots given by Figure 9. The conclusionsstated above are now shown explicitly, as could be noticed inthis figure.Efficient power density relative to itsmaximum nt power density relative to its maximum valueversus efficiency for different values of kR 100.20.3EfficiencyFigure 7: Power density relative to its maximum value at π 1 andπ 1 versus Newtonian heat transfer law efficiency (π 1). Theplots represent values of π in the 1β10 range. The ratio between thetemperatures of the reservoirs, ππΏ /ππ» is π 0.5, a typical value fora Rankine cycle (for the steam turbine working between 600 K and300 K).irreversibility parameter R10.90.80.70.60.50.40.30.20.10k 100.9k 0.50.80.7k 1 k ure 8: Efficient power density relative to its maximum value atπ 1 and π 1 versus Newtonian heat transfer law efficiency(π 1). The plots represent values of π in the 1β10 range. The ratiobetween the temperatures of the reservoirs, ππΏ /ππ» is π 0.5, atypical value for a Rankine cycle (for the steam turbine workingbetween 600 K and 300 K).3.2. Radiative Heat Transfer Law (π 4). The work criteriafunction could be used to easily analyze different valuesfor different values of π (the exponent power found in theheat transfer rate expressions). For demonstrating its use, theradiative heat transfer law is considered. Figure 10 shows thecriteria (similar to Figure 9) with one exceptionβthe typicalvalue of π is 0.3 (a typical value of the gas turbine dictatedby ππ» 1000 K and ππΏ 300 K). Although the location of
6Physics Research InternationalNormalized work criteria function relative toits maximum value versus efficiency for n 1Normalized work criteria function10.90.80.70.60.50.4Normalized powerNormalized power densityNormalized efficient powerNormalized efficient power zed work criteria function relativeto its maximum valueFigure 9: Power, power density, efficient power, and efficient powerdensity relative to its maximum value at π 1 and π 1 versusNewtonian heat transfer law efficiency (π 1). The ratio betweenthe temperatures of the reservoirs, ππΏ /ππ» is π 0.5, a typical valuefor a Rankine cycle (for the steam turbine working between 600 Kand 300 K).Normalized work criteria function relative to itsmaximum value versus efficiency for n 410.90.80.70.60.50.40.30.20.10Normalized powerNormalized power densityNormalized efficient powerNormalized efficient power density00.10.20.30.4Efficiency0.50.6density, a new criterion used for describing the performancecharacteristics of heat engines. The WCF suggests efficientpower and efficient power density of different orders, represented by the exponents πΌ and π½.Sample plots are given in Section 3 illustrating the simplicity of the WCF. The Newtonian heat transfer law (π 1)served as a working example to present and compare thecriteria mentioned above. The conclusions from these plotsfor the arbitrarily chosen value of π 0.85 (ideally the valueof π should approach 1) were as follows: (1) the maximalefficiency is reduced according to (5) by approximately 20%;(2) the maximum criteria were reduced by 42%; (3) thelocations of the efficiency at maximum criteria were shiftedto the left, similar to the shift in the maximal efficiency; (4)the smaller the value of the heat conductance ratio, the morepower that could be extracted from the engine; and (5) thelocation of the efficiency at maximum criteria is shifted to theright while comparing the criteria considered above in theirorder of presentation.The radiative heat transfer law was considered briefly inFigure 10. Similar conclusions are drawn as for the Newtonianheat transfer law with the following exceptions: the value ofthe efficiency at maximum criteria is different and the reservoirs temperature is 0.3, which is a typical value representinggas turbine.Comparing the plots given in Section 3 with what wasobserved by previous studies, the following conclusion couldbe derived regarding the practical efficiency of real heatengines.The efficiencies of any criteria of any order alwaysfall between two extremesβthe Carnot efficiency and theCurzon-Ahlborn efficiency.Conflict of Interests0.7Figure 10: Power, power density, efficient power, and efficient powerdensity relative to its maximum value at π 1 and π 1 versusradiative heat transfer law efficiency (π 4). The ratio between thetemperatures of the reservoirs, ππΏ /ππ» is π 0.3, a typical value fora Brayton cycle (for the gas turbine working between 1000 K and300 K).the efficiency at the maximum criteria considered is changed,conclusions similar to those drawn for the case of π 1 arenevertheless valid.4. Summary and ConclusionsIn the current study three criteria of the irreversible heatengine in finite time are reconsidered for a more generalheat transfer law (as given by (1) and (2)). Power, powerdensity, and efficient power were cast in a functional formcalled the work criteria function (WCFβsee (11)-(12)). Thisformulation enabled the introduction of the efficient powerThe author declares that there is no conflict of interestsregarding the publication of this paper.References[1] F. L. Curzon and B. Ahlborn, βEfficiency of a carnot engine atmaximum power output,β American Journal of Physics, vol. 43,pp. 22β24, 1975.[2] I. I. Novikov, βThe efficiency of atomic power stations,β Atommaya Energiya, vol. 3, p. 409, 1957, English translation in Journalof Nuclear Energy, vol. 7, p. 125 , 1958.[3] B. Andresen, βCurrent trends in finite-time thermodynamics,βAngewandte Chemie-International Edition, vol. 50, no. 12, pp.2690β2704, 2011.[4] L. Chen, C. Wu, and F. Sun, βFinite time thermodynamicoptimization or entropy generation minimization of energysystems,β Journal of Non-Equilibrium Thermodynamics, vol. 24,no. 4, pp. 327β359, 1999.[5] C. Wu, L. Chen, and J. Chen, Eds., Recent Advances in FiniteTime Thermodynamics, Nova Science Publishers, New York, NY,USA, 1999.[6] L. Chen and S. Fengrui, Advances in Finite Time Thermodynamics: Analysis and Optimization, Nova Science, New York, NY,USA, 2004.
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Irreversible heat engine qh qc T h T H Tc T L W F : Schematics of irreversible heat engines showing the components and variables involved. following the de nitions given in [ ], and its maximum value (MEPD). Details of the criteria function are given in the next section. 2. The Work Criteria Function e irreversible heat engine, as considered .
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