Equivalence of 'Reversible' and 'Irreversible' Entropy Modeling
MMartti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 1 (19) Manuscript Espoo R&D Center v01 Kemira 21.06.2020
Equivalence of “Reversible” and “Irreversible” Entropy Modeling Martti Pekkanen Department of Chemical Engineering, Helsinki University of Technology, Espoo, Finland Present address: Espoo R&D Center, Kemira Oyj, Espoo, Finland E-mail: [email protected] Abstract There are currently two – main – continuum models of entropy: a “reversible” (“equilibrium”) entropy model and an “irreversible” (“non-equilibrium”) entropy model. It is shown that the “reversible” entropy model and the “irreversible” entropy model are equivalent with respect to entropy accumulation – which entails same values of entropy change and same values of entropy, given entropy value at reference state. The context of the analysis is continuum physics. The non-continuous physical reality poses restriction to domains of continuous field equations: lower limits for the sizes of the systems and upper limits for the time rates of change of the phenomena. The domain of the analysis is the “mechanical theory of heat” of Clausius, i.e. the heat and work phenomena of i) heat transfer, ii) heat generation, iii) heat absorption, and iv) work transfer. The equivalence within heat and work phenomena entails equivalence within all phenomena artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 2 (19) Manuscript Espoo R&D Center v01 Kemira 21.06.2020 INTRODUCTION 1.1
Two Models of Entropy There are currently two – main – continuum models of entropy: the “reversible” (“equilibrium”) model based on Clausius [1, 2] and the “irreversible” (“non-equilibrium”) model of Onsager [3], Prigogine [4] and others. The “reversible” entropy model is based on the “second fundamental equation” of the “mechanical theory of heat”, Clausius [1], p. 355, [2], p. 90, dQdS T , (1) in which dS is entropy accumulation, dQ is heat absorption, and T is temperature. The “irreversible” entropy model uses the concept of entropy flow by heat flow divided by temperature, Onsager [3], p. 421, Prigogine [4], p. 16, QS JJ T , (2) in which S J is entropy flow, Q J is heat flow, and T is temperature. 1.2 “Reversibility” According to Clausius [1], p. 366, [2], p. 110, equation (1) holds true – i.e. may be used – for “reversible” phenomena, only. According to current engineering practice and textbooks , equation (1) may be used – i.e. holds true – for not “reversible” phenomena, also. This contradiction is relevant for the practical engineer, at least. 1.3 Equivalence It will be shown, below, that the “reversible” entropy model and the “irreversible” entropy model are equivalent with respect to entropy accumulation – which entails same values of entropy change and same values of entropy, given entropy value at reference state. The equivalence shows that the contradiction has not practical, engineering consequences. The context of the analysis is continuum physics. The non-continuous physical reality poses restriction to domains of continuous field equations: lower limits for the sizes of the systems and upper limits for the time rates of change of the phenomena. The domain of the analysis is the “mechanical theory of heat” of Clausius [1, 2], i.e. the heat and work phenomena of i) heat transfer, ii) heat generation, iii) heat absorption, and iv) work transfer. See appendix A artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 3 (19) Manuscript Espoo R&D Center v01 Kemira 21.06.2020 “REVERSIBLE” ENTROPY MODELING The context of the “mechanical theory of heat” of Clausius [1, 2] are the heat and work phenomena of i) heat transfer, ii) heat generation, iii) heat absorption, and iv) work transfer. The “reversible” entropy model is based on the “second fundamental equation” of the “mechanical theory of heat”, Clausius [1], p. 355, [2], p. 90, dQdS T , (1) in which dS is entropy accumulation, dQ is heat absorption, and T is temperature. According to Clausius [1], p. 366, [2], p. 110, equation (1) holds true for “reversible” phenomena, only. According to Clausius [1], p. 133, [2], p. 212, phenomena with temperature difference and heat generation are not “reversible”. According to Clausius [1, 2], thus, equation (1) does not hold true real phenomena, because all conceivable real phenomena within the “mechanical theory of heat” of Clausius [1, 2] involve either temperature difference or heat generation. Time explicitly, equation (1) may expressed as dS Qdt T , (3) and using quantities per volume, in a “local” form, as S Qt T . (4) The theory is not self-evident, e.g. “Generations of mathematicians have tried to make sense of what Clausius wrote.”, Truesdell [5], p. 13. A tilde is used to denote a time-rate of a quantity and primes are used to signify a quantity per area or per volume. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 4 (19) Manuscript Espoo R&D Center v01 Kemira 21.06.2020 “IRREVERSIBLE” ENTROPY MODELING The “irreversible” (“non-equilibrium”) entropy model is based on Onsager [3] and Prigogine [4] and presented, e.g., in de Groot [6] and de Groot and Mazur [7]. The “irreversible” entropy model uses the concept of entropy flow by heat flow divided by temperature, Onsager [3], p. 421, Prigogine [4], p. 16, QS JJ T , (2) in which S J is entropy flow, Q J is heat flow, and T is temperature. The concept of entropy flow by heat flow divided by temperature follows from Onsager [3], p. 407: “[…] the rate of production of entropy per unit volume of the conductor equals [unnumbered] in which J are “flows” and X are “forces”. The dimensional homogeneity of this equation requires that, Onsager [3], p. 406: “[…] the ‘force’ that drives the heat flow J is , [unnumbered]” which, in a case with heat flow only, leads to the entropy generation S S Q Q Q
J X J TT T . (5) Now, entropy flow by heat flow divided by temperature of equation (2) is needed to conform the entropy generation of equation (5) with Onsager equation (5.6), Onsager [3], p. 421, “The rate of local accumulation of heat equals / div [...]Tds dt J (5.6)” i.e. to have div 1div grad div Q Q Q S S
J Jds J T Jdt T T T . (6) Equations (2) and (5) are the expressions of entropy flow and entropy generation, respectively, of the “irreversible” (“non-equilibrium”) entropy model – in a case with heat flow only. The extended expressions are presented, e.g., by de Groot and Mazur [7], p. 24. The theory is not self-evident, e.g. “[…] the entropy production (de Groot and Mazur, 1962) […] is both logically and structurally incoherent”, Lavenda [8], p. 150. The time rate of generation of an extensive quantity is not the time derivative of the quantity. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 5 (19) Manuscript Espoo R&D Center v01 Kemira 21.06.2020 HEAT Heat Q and work W are conceptual, high level bookkeeping quantities that – through heat transfer transfer Q and work transfer transfer W – incorporate all possible mechanisms of energy transfer. For simplicity , it is taken as axioms that Heat has zero value anywhere anytime. (AX-Q) Work has zero value anywhere anytime. (AX-W) Heat transfer is energy transfer due to temperature difference. (AX-QT) Work transfer is energy transfer not due to temperature difference. (AX-WT) Thus, heat transfer and work transfer form a dichotomy of energy transfer. This is used, e.g., in the energy balance of a closed system (the “first law of thermodynamics”), Et Q W , (7) which may be expresses as ( ) ( )IN OUT IN OUT dE Q Wdt , (8) and as ( ) ( )IN OUT IN OUT dE dQ dW . (9) Because heat has zero value anywhere anytime, the values of the three heat quantities – of the heat and work phenomena – must obey, anytime anywhere, that GEN ABS
Q Q Q . (10) which may be expresses as ( ) IN OUT GEN ABS
Q Q Q , (11) and as ( ) IN OUT GEN ABS dQ dQ dQ . (12) “Transfer” refers to flux with respect to bulk matter. See appendix B. The aim of the axioms is simplicity, i.e. the avoidance of justification (and long presentation), and not axiomatization. See appendix B. Because heat and work have no value anywhere anytime, there is no heat or work transport. Thus, to simplify the equations, the transfer of heat and work is not denoted, i.e. transfer Q Q and transfer W W . artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 6 (19) Manuscript Espoo R&D Center v01 Kemira 21.06.2020 EQUIVALENCE 5.1
Entropy Balance In the context of continuum physics, the generic entropy S balance for a stationary system, is
10, 11 bulk transfer GEN
S S St u S , (13) in which
S t is entropy accumulation, bulk S u is entropy “transport”, i.e. entropy flux due to flux of bulk matter, transfer S is entropy “transfer”, i.e. entropy flux with respect to bulk matter, and GEN S is entropy generation. For a closed system, which excludes fluxes of matter, equation (13) is transfer GEN S St S . (14) For simplicity, this entropy balance of a closed system is considered in the context the “mechanical theory of heat” of Clausius [1, 2], i.e. within the heat and work phenomena of i) heat transfer, ii) heat generation, iii) heat absorption, and iv) work transfer. 5.2 “Reversible” Entropy Model In the “reversible” entropy model, entropy transfer and entropy generation, respectively, for heat and work phenomena are modeled as transfer S , (15) ABSGEN
QS T . (16) The insertion of equations (15) and (16) into the entropy balance equation (14) gives ABS
QSt T , (17) which is a specific entropy balance of a closed system for heat and work phenomena, according to the “reversible” entropy model. Equation (17) may expressed, in a “global” form, as ABS
QdSdt T , (18) and, further, time implicitly, as ABS dQdS T . (19) The comparison with equation (1) shows that equations (17-19) are expressions of the “second fundamental equation” of the “mechanical theory of heat”, Clausius [1], p. 366, [2], p. 195 – with the subscript for absorption added, for clarity. For balance, see appendix B. The domain of validity of this continuous field equation is limited by the non-continuous physical reality, see 6 Continuum Physics. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 7 (19) Manuscript Espoo R&D Center v01 Kemira 21.06.2020 “Irreversible” Entropy Model In the “irreversible” entropy model, entropy transfer and entropy generation, respectively, for heat and work phenomena, e.g. based on equations (20, 21) of de Groot and Mazur [7], p. 24, “ nS q k kk T J J J , (20)” “ n rkq k k j jk j
T T J AT T T T T J J F v , (21)” are modeled as transfer T QS , (20) GENGEN
QS TT T Q . (21) The insertion of equations (20) and (21) into the entropy balance equation (14) gives GEN
QS Tt T T T
Q Q , (22) which is a specific entropy balance of a closed system for heat and work phenomena according to the “irreversible” entropy model. Using equation (10) – based on axiom (AX-Q) that heat has zero value anywhere anytime – GEN ABS
Q Q Q , (10) in equation (22), gives equation (17) ABS
QSt T , (17) which is another expression for the specific entropy balance of a closed system for heat and work phenomena according to the “irreversible” model. 5.4 Equivalence If heat has zero value anywhere anytime, equations (17) and (22) are the same, one equation and, thus, equivalent with respect to entropy accumulation, which may be expressed concisely as ABS GEN
Q QS Tt T T T T
Q Q . (23) Equivalence with respect to entropy accumulation entails same values of entropy change and same values of entropy, given entropy value at reference state. The equivalence in the context of the Onsager Reciprocal Relations, Onsager [3], is considered in appendix C. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 8 (19) Manuscript Espoo R&D Center v01 Kemira 21.06.2020 CONTINUUM PHYSICS The analysis demonstrates that, in the context of continuum physics, within heat and work phenomena, both the “reversible” and the “irreversible” entropy model lead to the continuous field equation (23) ABS GEN
Q QS Tt T T T T
Q Q . (23) For equation (23) to be useful, matter must be assumed continuous with continuous physical properties. With this assumption, equation (23) describes a continuous entropy density field. The non-continuous physical reality poses restriction to domains of continuous field equations: There are lower limits for the sizes of the systems and there are upper limits for the time rates of change of the phenomena, for which the continuum assumption is valid. For simplicity, two axioms for continuum physics are taken: Matter is continuous with continuous physical properties. (AX-CP1) Systems are not smaller and time rates of change of phenomena are not larger than the limits posed by the non-continuous physical reality. (AX-CP2) The locations of the limits vary case by case and are not considered, here. These axioms make possible, e.g., the following two differential equations for temperature T and specific entropy s , respectively – based on energy balance and entropy balance, respectively – for a stationary, closed system, for a case with no heat generation and constant physical properties ABSp p
QTt c c Q , (24) ABS
Qst T T Q . (25) The two equations specify well-defined continuous field equations of temperature and specific entropy, respectively, with minimum discretizations and maximum time rates of change of heat flow depending on the specifics of the case. The ubiquitous use of equations analogous to equation (24) – for not essentially zero and time independent temperature gradients – implies the adequacy of the two axioms. The analogous statistical nature of entropy and temperature, Landau and Lifshitz [10], p. 35, implies that equation (25) is analogous to equation (24) in validity. Truesdell and Toupin [9], p. 226-228 It seems that the purpose of “local equilibrium”, e.g. de Groot and Mazur [7], p. 23, is the same as the purpose of axiom (AX-CM2), while the assumption of “local equilibrium” seems more restrictive. “Local equilibrium” entails local “equilibrium”, i.e. local “high degree of” uniformity and time independence of values of observable quantities, Landau and Lifshitz [10], p. 6, e.g. essentially zero and time independent temperature gradients, while axiom (AX-CP2) entails not too large time rate of change of the temperature gradients. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 9 (19) Manuscript Espoo R&D Center v01 Kemira 21.06.2020 SUMMARY The analysis demonstrates that the “reversible” entropy model and the “irreversible” entropy model are equivalent with respect to entropy accumulation within heat and work phenomena – if heat has zero value anywhere anytime. Equivalence with respect to entropy accumulation entails same values of entropy change and same values of entropy, given entropy value at reference state. Both models lead to the same continuous field equation (23) within heat and work phenomena ABS GEN
Q QS Tt T T T T
Q Q . (23) The non-continuous physical reality poses restriction to the domain of this continuous field equation: lower limits for the sizes of the systems and upper limits for the time rates of change of the phenomena. Accordingly, if entropy accumulation is calculable using the “irreversible” entropy model, the entropy accumulation is calculable using the “reversible” entropy model with the same result. This despite the different (sub)models used for entropy transfer and entropy generation in the two entropy models as presented in Table 1. (sub)model for entropy transfer (sub)model for entropy generation sum of (sub)models “reversible” entropy model ABS QT ABS QT “irreversible” entropy model T Q GEN
QTT T Q GEN QT Q Table 1. (sub)models used in the entropy models This equivalence within heat and work phenomena entails equivalence within all phenomena , ,, ,2 ABS QW GEN QWGEN O GEN O
Q QS S T St T T T T
Q Q . (26) in which QW refers to heat and work phenomena and O refers to “other” (than heat and work) phenomena. This equivalence justifies the use of the “reversible” entropy model for practical, engineering calculations for real physical phenomena, none of which are “reversible”. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 10 (19) Manuscript Espoo R&D Center v01 Kemira 21.06.2020 References 1 R. Clausius, The Mechanical Theory Of Heat, With Its Applications To The Steam–Engine And To The Physical Properties Of Bodies (John Van Voorst, 1867) 2 R. Clausius, The Mechanical Theory of Heat (Macmillan and Co, 1879) 3 L. Onsager, Reciprocal Relations in Irreversible Processes, Phys. Rev., Vol 37, pp. 405-26 (1931) 4 I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, 3rd ed. (Interscience Publishers, 1967) 5 C. Truesdell, Rational Thermodynamics, 2nd ed. (Springer-Verlag, 1984) 6 S.R. de Groot, Thermodynamics of Irreversible Processes (North-Holland, 1951) 7 S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, 1962) 8 B.H. Lavenda, Thermodynamics of Irreversible Processes (Dover Publications, 1978) 9 C. Truesdell and R. Toupin, The Classical Field Theories in S. Flügge, ed., Encyclopedia of Physics, Vol III/1 (Springer-Verlag, 1960) 10 L.D. Landau and E.M. Lifshitz, Statistical Physics, Part 1, 3rd ed. (Elsevier; 1980) 11 M. Zemansky, Heat and Thermodynamics, 3rd. ed. (McGraw-Hill 1951) 12 D. Kondepudi and I. Progogine, Modern Thermodynamics: From Heat Engines to Dissipative Structures, (Wiley, 1998) 13 O. Reynolds, Papers On Mechanical And Physical Subjects, Vol III (Cambridge University Press, 1903) 14 D.G. Miller, Thermodynamics of Irreversible Processes. The Experimental Verification of the Onsager Reciprocal Relations, Chem. Rev., Vol 60, No 1, pp. 15–37 (1960) 15 C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, 1976) artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 11 (19) Espoo R&D Center Kemira Appendix A
Appendix A: Practical Calculations and Textbooks 1 “REVERSIBILITY” The “second fundamental equation” of the “mechanical theory of heat”, Clausius [1], p. 355, [2], p. 90, is dQdS T . (II) According to Clausius [1], p. 366, [2], p. 110, equation (II) holds true for “reversible” phenomena, only. According to Clausius [1], p. 133, [2], p. 212, phenomena with temperature difference and heat generation by are not “reversible”. 2 PRACTICAL CALCULATIONS 2.1
Temperature Difference Consider continuous, i.e. time independent, isobaric, lossless heating of pure nitrogen from p = 1.0 bar, T = 0.0 C to T = 100 C with high flow of 3 bar saturated steam (T = 134 C). The case calculated by Aspen Plus ® , of Aspen Technology, Inc., is presented in figure A.1. Figure A.1: Heat transfer with temperature difference Consider batch, i.e. time dependent, isobaric, lossless heating of an ideal gas with C p = 29.16 J/K/mol within a closed system from p = 1.0 bar, T = 0.0 C and S = -2.4 J/K/mol to T = 100 C. The entropy modeling, using Clausius equation (II), is ln 9,1 J/K/mol T T T T pIN ABS pT T T T
C dTdQ dQ TdQdS CT T T T T (A.1)
2, 4 9,1 J/K/mol 6, 7 J/K/molS S dS (A.2) It is seen that the result for S is the same as calculated by Aspen Plus ® , in figure A.1. Thus, the numerical results obtained by Clausius equation (II) conform with the results of entropy modeling in engineering practice for heat transfer with temperature difference – explicitly excluded from the domain of the Clausius equation (II), Clausius [1], p. 133, [2], p. 214. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 12 (19) Espoo R&D Center Kemira Appendix A Heat Generation Consider continuous, i.e. time independent, adiabatic, lossless compression of pure nitrogen from p = 1.0 bar, T = 0.0 C to p = 3.0 bar and the same with losses with efficiency of 0.5. The cases calculated by Aspen Plus ® , of Aspen Technology, Inc., are presented in figure A.2. Figure A.2: Heat generation Consider batch, i.e. time dependent, adiabatic, lossless compression of an ideal gas with C p = 29.16 J/K/mol within a closed system from p = 1.0 bar, T = 0.0 C and S = -2.4 J/K/mol to p = 3.0 bar and the same with losses with efficiency of 0.5. The heat generation, equal to the work input “lost”, is
11 2, , 1 kkGEN IN lost IN ideal
RT pdQ dW dW k p . (A.3) The entropy modeling, using Clausius equation (II), is ln lossless T pABS GEN p losslessT
C dTdQ dQ TdQdS CT T T T T . (A.4) The results are presented in Table A.1. Table A.1. It is seen that the results for T and S are the same as calculated by Aspen Plus ® , in figure A.2. Thus, the numerical results obtained by Clausius equation (II) conform with the results of entropy modeling in engineering practice for heat generation – explicitly excluded from the domain of the Clausius equation (II), Clausius [1], p. 133, [2], p. 214. efficiency 1,0 0,5 -dW,IN 2 094,6 4 189,2 J/moldQ,IN 0,0 0,0 J/moldW,lost 0,0 2 094,6 J/moldQ,GEN 0,0 2 094,6 J/moldQ,ABS 0,0 2 094,6 J/moldS 0,0 6,9 J/molT,2 100,5 201,0 CS,2 - 2,4 4,5 J/K/molInput specifications grayed. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 13 (19) Espoo R&D Center Kemira Appendix A TEXTBOOKS Zemansky [11], pp. 173-4, p. 179 (underline added): “If i S is the entropy at the initial state and f S that of the final state, then we have the result rR f ii dQ S ST […] The subscript R […] indicates that the preceding equation is true only if dQ is transferred reversibly.” “When a system undergoes an irreversible process between an initial equilibrium state and a final equilibrium state, the entropy change of the system is equal to rf i R i dQS S T where R indicates any reversible process arbitrarily chosen by which the system may be brought from the given initial state to the given final state.” Kondepudi and Prigogine [12], pp. 84-85 (underline added): “Using (3.3.3), if the entropy S of a reference or standard state is defined, then the entropy of an arbitrary state X S can be obtained through a reversible process that transforms the state 0 to the state X (Fig. 3.6). dQS S T (3.4.1) […] In a real system the transformation from state 0 to state X occurs in a finite time and involves irreversible processes along the path I. In classical thermodynamics it is assumed that every irreversible transformation that occurs in nature can also be achieved through a reversible process for which (3.4.1) is valid.” Thus, according to Zemansky [11] and Kondepudi and Prigogine [12], an equation that holds true for “reversible” phenomena, i.e. does not hold true for “irreversible” phenomena, may be used for “irreversible” phenomena. In both cases the equation is the “second fundamental equation” of the “mechanical theory of heat”, Clausius [1], p. 355, [2], p. 90, dQdS T . (II) Accordingly, the presentations are internally inconsistent and in contradiction with the explicit claim by Clausius [1], p. 133, [2], p. 214. If the claim by Zemansky [11] and Kondepudi and Prigogine [12] is true – as it seems to be – this makes the concept of “reversibility” redundant. Zemansky [11], p. 173, Kondepudi and Prigogine [12], p. 79 The claim seems justified by entropy history independence, Zemansky [11], p. 173, Kondepudi and Prigogine [12], p. 85. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 14 (19) Espoo R&D Center Kemira Appendix B
Appendix B: Balance The balance proposition may be taken as an axiom , as in Reynolds [13], p. 9: “AXIOM I: Any change whatsoever in the quantity of any entity within a closed surface can only be effected in one or other of the two distinct ways: (1) it may effected by the production or destruction of the entity within the surface, or (2) by the passage of the entity across the surface.” For simplicity, it is taken as an axiom that For any extensive quantity, for any system, and for any instant or interval of time, accumulation equals net input plus net generation. (AX-B) The balance proposition may be expressed as ACC IN OUT GEN , (B.1) where the terms have meaning with respect to a system C and its boundary B, as in figure B.1, only.
Figure. B.1 The system (i.e. control volume, balance volume) and the terms of the balance 3.1 “Transit”, “Transport” and “Transfer” The terms “transport” and “transfer” are used with varying meanings. Here, the terms refer to the following dichotomy of the flux of an extensive quantity transit transport transfer
X X X , (B.2) in which “transit” refers to the total flux, “transport” refers to the flux due to flux of bulk matter, and “transfer” refers to the flux with respect to bulk matter (possible at boundaries of closed systems). 3.2
Balance For a stationary system C with boundary B, the balance of the extensive quantity X may be expressed, using the dichotomy of “transport” and “transfer” for the term (IN-OUT), as B bulk B transfer GENC B B C d X dV X dA dA X dVdt n u n X , (B.3) from which, using the divergence theorem, bulk transfer GEN
X X Xt u X . (B.4) or just “set”, as in Truesdell and Toupin [9], p. 468 or, equivalently, as net net ACC IN GEN or as
ACC IN OUT GEN DES A tilde is used to denote a time-rate of a quantity and primes are used to signify a quantity per area or per volume.
ACCGEN OUTC BIN artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 15 (19) Espoo R&D Center Kemira Appendix B
Example: Energy The balance of the extensive quantity of energy, E, based on equation (B.4), is bulk transfer GEN
E E Et u E . (B.5) First, take energy conservation, i.e. energy no-generation anywhere anytime, and express this as GEN E . (B.6) Second, take energy transfer equal to heat transfer plus work transfer and express this as transfer E Q W . (B.7) Third, take a closed system, the definition of which excludes the flux of matter – both bulk matter and component matter – through the boundary and thus transport of any quantity, e.g. energy, with matter, i.e. transport matter E E u . (B.8) The insertion of equations (B.6-B.8) into equation (B.5) gives Et Q W , (B.9) which may be expresses as ( ) ( )IN OUT IN OUT dE Q Wdt , (B.10) and as ( ) ( )IN OUT IN OUT dE dQ dW , (B.11) and as dE dQ dW , (B.12) which equation almost equals the “first fundamental equation” of the “mechanical theory of heat”, Clausius [1], p. 366, [2], p. 195 dQ dU dW , (B.13) which equation is being called the “first law of thermodynamics”. . The derivation shows that equation (B.9) is a combination of the propositions of i) balance, ii) energy conservation, iii) energy transfer equal to heat transfer plus work transfer, and iv) closed system. The “first law of thermodynamics” according to Clausius equation (B.13), first, shows ignorance of the concept of balance and must use “sign convention” to conform to energy balance and, second, is commonly taken to express “conservation of energy” using an equation of “internal energy”. Clausius [1], pp. 251-252: “[…] to include under the common name energy, both heat and everything that heat can replace. I have no hesitation, therefore, in adopting, for the quantity U, the expression energy of the body”. Clausius [2], p. 31: “In what follows the quantity U will therefore be called the Energy of the body.” artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 16 (19) Espoo R&D Center Kemira Appendix C
Appendix C: Onsager Reciprocal Relations 1
ONSAGER RECIPROCAL RELATIONS According to Onsager [3], p. 407, [originally equation unnumbered]: “[…] the rate of production of entropy per unit volume of the conductor equals ” (C.1) This is the fundamental equation of the ORR with respect to entropy. Only electric current and heat flow will be considered, below, as these are the first two phenomena considered by Onsager [3]. For electric current, I J , and heat flow, Q J , the fundamental equation of the ORR with respect to entropy, equation (C.1), is rewritten as S I I Q Q
J X J XT . (C.2) in which S is entropy generation. ONSAGER ENTROPY BALANCE Onsager [3], p. 421: “The rate of local accumulation of heat equals / div [...]Tds dt J (5.6)” writing s for the local entropy density […] If we write * ( ) / n n S J J T d (5.7)” for the entropy given off to the surroundings, […]” Onsager [3], thus, in equation (5.7) defines the concept of entropy flow equal to heat flow per temperature QS JJ T . (C.3) Given the entropy flow of equation (5.7), the entropy balance equation (5.6) becomes, for heat flow only, div 1div div grad Q QS S Q
J Jds J J Tdt T T T , (C.4) and with entropy generation due to heat generation term added div 1div div grad Q Q GENS S Q
J J Qds J J Tdt T T TT . (C.5) The time rate of generation of an extensive quantity is not a time derivative of the quantity. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 17 (19) Espoo R&D Center Kemira Appendix C ELECTRIC CURRENT For electric current, I J , using J LX , take I I I
J L X UR , (C.6)
Q J R . (C.7) The insertion into equation (C.2), with electric current, only, gives grad GEN III I IS
QJ UJ X J RT T T T . (C.8) With electric current only and with entropy generation in equation (C.8), the entropy balance (C.5) becomes ,2 Q GEN IGENQ
J QQds J Tdt T T TT . (C.9) The entropy balance of a closed system for heat and work phenomena according to the “irreversible” model is equation (22), which, with no heat flow, Q , reduces to GEN
QdSdt T . (C.10) The entropy balance of a closed system for heat and work phenomena according to the “reversible” model is equation (17), which, with no heat flow, Q , and using ABS GEN
Q Q Q , gives GEN
QdSdt T . (C.11) Equations (C.10) and (C.11) are the same, one equation and correspond to equation (C.9). Thus, the “irreversible” model of entropy and the “reversible” model of entropy are equivalent with respect to entropy accumulation in this case of the ORR. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 18 (19) Espoo R&D Center Kemira Appendix C HEAT FLOW For heat flow Q J , using J LX , take grad Q Q Q
J L X T , (C.12) , GEN Q Q . (C.13) The insertion into equation (C.2), with heat flow, only, gives grad Q Q QS
J X J TT T . (C.14) This equation is of a form different than equation (C.8). An ad-hoc solution according to Onsager [3], p. 406, for this is: “In corresponding units the ‘force’ which drives the flow of heat will be ”. (C.15) This means that equation (C.12) is replaced by Q Q Q
J L X TT , (C.16) and equation (C.14) becomes ( grad ) 1 grad
Q Q QS Q
J X J T J TT T T . (C.17) With heat flow only and with entropy generation in equation (C.17), the entropy balance (C.5) becomes div1div grad Q QGENQ
J JQds J Tdt T T TT . (C.18) The entropy balance of a closed system for heat and work phenomena according to the “irreversible” model is equation (22), which, with no heat generation, GEN Q , reduces to St T Q . (C.19) The entropy balance of a closed system for heat and work phenomena according to the “reversible” model is equation (17), which, with no heat generation, GEN Q and using ABS GEN
Q Q Q , gives St T Q . (C.20) Equations (C.19) and (C.20) are the same, one equation and correspond to equation (C.18). Thus, the “irreversible” model of entropy and the “reversible” model of entropy are equivalent with respect to entropy accumulation in this case of the ORR. artti Pekkanen Equivalence of “Reversible” and “Irreversible” Entropy Modeling 19 (19) Espoo R&D Center Kemira Appendix C ONSAGER RECIPROCAL RELATIONS AND ENTROPY Onsager [3], p. 406: “In the following, a general class of reciprocal relations in irreversible processes will be derived from the assumption of microscopic reversibility. No further assumptions will be necessary, except certain theorems borrowed from the general theory of fluctuations.” Onsager [3], p. 413: “Now the collision is in effect a kind of transition leading from a state c haracterized by one pair of velocities ( , )v v to another state ( , )v v . The requirement of microscopic reversibility enters through the condition that the transitions: ( , ) ( , )v v v v and ( , ) ( , )v v v v must occur equally often when the system has reached thermodynamic equilibrium.” If it is, indeed, the case that no further assumptions are necessary for the derivation of the ORR, then the fundamental equation of the ORR with respect to entropy expressing “the production of entropy”, Onsager [3], p. 407, “ ” (C.1) is not necessary for the ORR. Further, this fundamental equation of the ORR with respect to entropy does not follow from the ORR, neither from Onsager [6], p. 406, “ X R J R JX R J R J (1.1)” “
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R R (1.2)” nor from Onsager [6], p. 408, “ J L X L XJ L X L X (2.1)” “
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L L (2.2)” in which J are “flows”, X are “forces”, R are “resistances”, and L are “conductances”. Accordingly, if Onsager equation (C.1) both is not necessary for the ORR and does not follow from the ORR, then the “reversible” entropy model – instead of the “irreversible” entropy model – may be used in the context of the ORR, also. The “theorems borrowed from the general theory of fluctuations” refer to “Einstein, Ann. d. Physik 33, 1275 (1910)” and “P. and T. Ehrenfest, Enz. d math. Wiss. IV. 32”.23