Entropy production and lost work for some irreversible processes Francesco Di Liberto To cite this version: Francesco Di Liberto. Entropy production and lost work for some irreversible processes. Philosophical Magazine, Taylor & Francis, 2007, 87 (3-5), pp.569-579. 10.1080/14786430600909006. hal-00513741 HAL Id: hal-00513741 https://hal.archives-ouvertes.fr/hal-00513741 Submitted on 1 Sep 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Philosophical Magazine & Philosophical Magazine Letters For Peer Review Only Entropy production and lost work for some irreversible processes Journal: Philosophical Magazine & Philosophical Magazine Letters Manuscript ID: TPHM-06-Apr-0109.R1 Journal Selection: Philosophical Magazine Date Submitted by the 09-Jun-2006 Author: Complete List of Authors: di Liberto, Francesco; Università di Napoli, INFN.CNR-CNISM, Dipartimento di Scienze fisiche Keywords: statistical physics, thermodynamics, transformations Keywords (user supplied): irreversibility, entropy production, Clausius inequality http://mc.manuscriptcentral.com/pm-pml Page 1 of 12 Philosophical Magazine & Philosophical Magazine Letters 1 1 2 3 Entropy production and lost work for some irreversible processes 4 5 6 Francesco di Liberto 7 8 Dipartimento di Scienze Fisiche 9 10 Università di Napoli “Federico II” 11 Complesso universitario Monte S. Angelo 12 Via Cintia - 80126 Napoli (Italy) 13 [email protected] 14 tel. + 39 081 676486 - fax + 39 081 676346 15 16 For Peer Review Only 17 In this paper we analyse in depth the Lost Work in an irreversible process (i.e. 18 WLost = WRe v −WIrrev ) . This quantity is also called ‘degraded energy’ or ‘Energy 19 unavailable to do work’. Usually in textbooks one can find the relation W ≡ T∆S , 20 Lost U 21 which, for many processes , is not suitable to evaluate the Lost Work. Here we find for 22 W a more general relation in terms of internal and external Entropy production, π 23 Lost int 24 and π ext , quantities which enable also to write down in a simple way the Clausius 25 inequality. Examples are given for elementary processes. 26 27 28 Keywords : irreversibility, entropy production, adiabatic process 29 30 1. Introduction 31 32 33 Entropy production, a fascinating subject, has attracted many physics researches even in cosmological physics 34 35 [1], moreover in the past ten years there has been renewed interest in thermodynamics of heat engines; many 36 papers address issues of maximum power, maximum efficiency and minimum Entropy production both from 37 38 practical and theoretical point of view [2-6]. 39 40 One of the main points in this field is the analysis of Available Energy and of the Lost Work. Here we 41 42 give a general relation between Lost Work and Entropy production, merging together the pioneering papers of 43 44 Sommerfeld (1964), Prigogine (1967), Leff (1975) and Marcella (1992), which contain many examples of 45 such relation, and the substance-like approach to the Entropy of the Karlsruhe Physics Course due mainly to 46 47 Job (1972), Falk, Hermann and Schmid (1983) and Fuchs (1987). 48 49 It is well known [7-13] that for some elementary irreversible process, like the irreversible isothermal 50 51 expansion of an ideal gas in contact with an heat source T , the work performed by the gas in such process W 52 53 is related to the reversible work WRe v (i.e. the work performed by the gas in the corresponding reversible 54 55 process) by means of the relation 56 (1) 57 W = WRe v − T ∆SU 58 where ∆S is the total entropy change of the universe (system + external heat sources). The degraded energy 59 U 60 T∆SU is usually called WLost ‘the Lost work’, i. e.: WLost = WRe v − W http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters Page 2 of 12 2 1 2 3 4 the work that could have been performed in the related reversible process (here the reversible expansion); it is 5 also called ‘energy unavailable to do work’. 6 7 By the energy balance, the same relation holds for the amount of heats extracted from the source T 8 9 = − ∆ 10 Q QRe v T SU 11 12 13 Therefore T∆SU is also called the ‘Lost heat QLost ’, i.e. the additional heat that could have been drawn from 14 ‡ 15 the source in the related reversible process . 16 For Peer Review Only 17 The total variation of Entropy , ∆SU , is usually called ‘Entropy production’. The second Law claims that 18 19 ∆SU ≥ 0 20 21 The relation between Entropy production and WLost (or QLost ) is the main subject of this paper. In Sec.3 22 23 we will find a relation more general than relation (1) . To introduce the subject let us remind the steps that 24 25 lead to the relation (1).[9] 26 27 For a process (A—>B) in which the system (for example, the ideal gas) absorbs a given amount of 28 heat Q from the heat source at temperature T and performs some work W , the entropy production of the 29 ext 30 31 Universe, i. e. the variation of Entropy of the system+variation of Entropy of the external source, , is 32 Q 33 ∆SU ≡ ∆S sys + ∆Sext = ∆S sys − (2) 34 Text 35 36 From the energy balance ∆U sys =Q −W it follows 37 38 Text ∆SU = Text ∆S sys − ∆U sys −W (3) 39 40 If the process is reversible then ∆S =0 and W ≡ W = T ∆S − ∆U , 41 U Re v ext sys sys 42 ∆ = − = 43 Therefore Text SU WRe v W WLost (4) 44 45 which defines the Lost Work and proves relation (1). There are however some irreversible processes for 46 which relation (4) is not suitable to evaluate the Lost Work, for example the irreversible adiabatic processes, 47 48 [11] in which there is some W , some Entropy production ∆S , but no external source T . 49 Lost U ext 50 In general for an irreversible process, ∆S >0 , . from relation (3), it follows 51 U 52 W < W (5) 53 Re v 54 55 i.e. the Reversible Work is the maximum amount of work that can be performed in the given process 56 57 58 59 ‡ For an irreversible compression T∆SU is sometime called WExtra or QExtra [10] i.e. the excess of work performed on the 60 system in the irreversible process with respect to the reversible one (or the excess of heat given to the source in the irreversible process). In a forthcoming paper we will show that WExtra is related to the environment temperature and to the entropy productions http://mc.manuscriptcentral.com/pm-pml Page 3 of 12 Philosophical Magazine & Philosophical Magazine Letters 3 1 2 In Sec. 3 we evaluate WLost for some simple irreversible processes, refine relation (4) taking account of 3 4 internal and external irreversibility and give a general procedure to evaluate the Lost Work. Such procedure 5 6 follows from the analysis of Sec.2 where it has been shown that often the total Entropy production is due to 7 8 the entropy production of the sub-systems. When the subsystems are the system and the external source, their 9 entropy productions has been called respectively internal and external, i.e. ∆S = π + π 10 U int ext 11 In Sec.2 the entropy balance and the entropy productions for irreversible processes are analyzed by 12 13 means of the substance-like approach. In the following the heat quantities Q ’s are positive unless explicitly 14 15 stated and the system is almost always the ideal gas. 16 For Peer Review Only 17 2. Entropy production for irreversible processes 18 19 In this Section are given some examples of Entropy production for elementary processes. First we analyse the 20 21 reversible isothermal expansion (A-->B) of one mole of monatomic ideal gas at temperature T which 22 23 receives the heat Q from a source at temperature T . For the ideal gas we have 24 25 ∆S gas = Sin − Sout (6) 26 27 QRe v VB 28 where Sout = 0 and the Entropy which comes into the system is S In = = R ln , since 29 T VA 30 B B 31 V Q = δQ = PdV = RT ln B . The heat which flows from the heat source into the gas is Q , which 32 Re v ∫ Re v ∫ V Re v 33 A A A 34 is also the work performed by the system in the reversible isothermal expansion. The increase in Entropy for 35 36 B δQ V 37 ∆ = = B the gas is S gas ∫ R ln ; R=8.314 J/mol. K° is the universal constant for the gases. 38 A T VA 39 40 For the heat Source it holds 41 ext ext ext 42 ∆S = Sin − Sout (7) 43 44 − Q Q where ∆S ext = rev , S ext = 0 and S ext = Re v . 45 T in out T 46 47 For the Universe ∆S = ∆S + ∆S ext = 0. In this example (a reversible process) the Entropy is conserved. 48 U gas 49 50 Let us turn to the irreversibility and take a look at the irreversible isothermal expansion at temperature T of 51 one mole of monatomic ideal gas from the state A to the state B (let, for example, P = 4P ).