Thermoelectric Thomson's relations revisited for a linear energy converter
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Thermoelectric Thomson’s relations revisited for a linear energy converter
S. Gonzalez–Hernandez and L. A. Arias–Hernandez
Dpto. de Física, Escuela Superior de Física y Matemáticas,Instituto Politécnico Nacional, U. P. Adolfo López Mateos, Zacatenco,C.P. 07738, México D.F., México, Email: [email protected] (Dated: 19/dic/2017)In this paper we revisit the thermocouple model, as a linear irreversible thermodynamic energyconverter. As is well known, the linear model of the thermocuple is one of the classics in thisbranch. In this model we note two types of phenomenological coefficients: the first comes from somemicroscopic models, such as the coefficient associated with the electric conductivity, and the secondcomes from experimental facts such as the coefficient associated with the thermoelectric power. Weshow that in the last case, these coefficients can be related to the operation modes of the converter.These relationships allow us to propose a generalization of the first and second Thomson’s relations.For this purpose we develop the ideas of non-isothermal linear converters, operated directly (heatengine) and indirect (refrigerator). In addition to this development we analyze the energy describedby these converters.
I. INTRODUCTION
Thermoelectricity is a seminal phenomenon in Non-Equilibrium Thermodynamics; within the effects that constitutethis phenomenon, three are well known T. J. Seebeck discovered the electricity generated by the application of heatto the junction of two different materials (1821, Seebeck effect) [1, 2], Jean C. A. Peltier found a temperature gradientin the junction under isothermal conditions due an electrical current (1834, Peltier effect) [3], and W. Thomsonpredicted and observed the heating or cooling of a current-carrying conductor with a temperature gradient (1851,Thomson effect) [4, 5]. Thomson’s experiments allowed him to find two relations between these effects: one was asubtle connection between the Peltier effect and the Sebeeck effect, called Second Thomson’s Relation (STR). Theother was a relation between the three effects, called First Thomson’s Relation (FTR). It was not until the advent ofthe linear theory of non–equilibrium processes, established by L. Onsager [6, 7], that it was possible to satisfactorilydemonstrate both relations.L. Onsager first and later several authors [4, 6–10], derived the phenomenological equations of the thermocouple.Begining with the entropy production of thermoelectric phenomenon and considering the electrochemical potentialand the fluxes and forces on the system, we obtain the generalized equations [4], (cid:20) − J N J Q (cid:21) = (cid:20) L L L L (cid:21) (cid:20) T ∇ µ ∇ (cid:0) T (cid:1) (cid:21) , (1)with − J N the electrical current (the generalized flux J ), J Q the heat flux (the generalized flux J ), L ′ ij s the Onsagercoefficients. For the Seebeck effect, we can take as the generalized driven force the electric potential X = ∇ µ/eT ,and take as the driver generalized force the temperature gradient X = −∇ (1 /T ) . These gradients are between thewelding points of materials A and B (see Figure 1a). Then we get the phenomenological Onsager’s equations: (cid:20) J J (cid:21) = (cid:20) L L L L (cid:21) (cid:20) X X (cid:21) . (2)where L ij = (cid:16) ∂J i ∂X i (cid:17) eq . Now, from the entropy production of the thermocouple, σ = J X + J X > (3)we can establish the relation, | J X | > | J X | , with J X < and J X > , agree with the definition of thedriven and driver forces respectively. Then, we can associated the first term of the entropy production to a poweroutput (by temperature unit) and the second to a power input (by temperature unit), and build a steady state LinearEnergy Converter (LEC) [11, 12] (see Figure 1b). This array is a nonzero entropy production and a nonzero poweroutput converter, because of its interactions with the surroundings ( X i and J i ). Using the work of Caplan and Essig[11], it is possible to make a first step towards a linear description of a LEC. In general, the governing fluxes J i of a realsystem are usually very complicated and non—linear functions of the generalezd forces X i . However, the linear regimeallows us to give a fair enough description of the phenomenon. These authors, based on the analysis of equations (2)introduced the so called coupling coefficient q ( L ij ) , which comes directly from the second law of thermodynamics [12].This is a dimensionless parameter that measures the degree of coupling between the spontaneous and nonspontaneousfluxes, ≥ q ≡ L L L ≥ . (4)In addition, we can take into account a parameter introduced by Stucki [13] which measures the relation between thetwo forces X and X as follows: x ≡ r L L X X ←− driven f orce ←− Driver f orce , (5)where x ∈ [ − , is called the force ratio; also we can build J /J = p L /L the ratio between the fluxes.On the other hand, one of the most important features of an irreversible converter, is the amount of energyexchanged with the surroundings to do work or acomplish another type of objective. This feature is usually knownas the energetics of the LEC [12]. We can write some functions that characterize this energetics, in terms of theparameters q , x , L , and the force X .This paper is organized as follows: in Section II we present a non–isothermic LEC and different working regimensof this converter, transferred here from Finite–Time Thermodynamics (FTT) [15–22]. The converter can be operated a) b)Figure 1. Steady non-isothermic Linear Energy Converters. a) The heat engine as a direct linear energy converter (D–LEC).We can describe this engine with the scheme shown here, a system with two fluxes ( J D , J D ) and two forces ( X D , X D ),where J D is the input heat flux, then | J D X D | is the power input (by temperature unit) and | J D X D | is the power output(by temperature unit) of the converter. b) The refrigerator as an inverse linear energy converter (I–LEC). In this case we havetwo fluxes ( J I , J I ) and two forces ( X I , X I ), but now | J D X D | is the power output (by temperature unit) and | J D X D | is the power input (by temperature unit) of the refrigerator. as a heat engine (direct energy converter) or as a refrigerator (inverse energy converter). In Section III we presentthe deduction of the phemomenological coefficients of thermocouple, starting from the phenomenological equationsand the general form of its entropy production. Then we introduce the operation modes built in Section II with theobjective to rewrite the Thomson’s relations considering the thermocouple as a LEC. Finally, in IV we present someconclusions concerning our results. II. NON–ISOTHERMIC LEC OPTIMIZATION
Of all actual energy converters, a very large portion of them use gradients of temperature. A set of these thermalengines are converters as the thermocouple and other systems which contains pairs of fluxes that give us cross–effects,such as the Soret effect or Reynolds effect [14]. With the purpose of make a general study of the energetics of thesekind of phenomena, in the next paragraphs we will take the entropy production in two cases: when the heat flux isa spontaneous flux (see Fig. 1a) and when this flux is non-spontaneous (see Fig. 1b). We call the first case directconverter and the second case inverse converter. Later we will use some known objective functions of models ofirreversible energy converters studied in other contexts, and built the equivalent objective functions for these newmodels.
A. Heat engine (direct LEC)
One of the most common thermal engines, is that exchanging an amount of energy with the surroundings to do work,known as a heat engine. In this case a gradient of temperature promotes a flux against any other gradient (gravity,electric field, etc.). Some models of this kind of engines have been proposed in the context of Linear IrreversibleThermodynamics, Finite Time Thermodynamics and other constructions within Non–Equilibrium Thermodynamics[12, 22].Now we can use the entropy production of the LEC, given in general form by eq. (3), and take as the driver fluxthe heat flux, and as the driven flux any other flux against a generalized force. For this reason we will call this engine“direct linear energy converter” (D–LEC). In this case the force ratio will be x D = r L L X D X D . (6)Now, using the q and x D parameters we can write the flows J D and J D as follows, J D ( x D , q ) = (cid:16) qx D (cid:17) L X D a ) (cid:16) x D q (cid:17) L X D b ) , (7)and J D ( x D , q ) = ((cid:16) qx D (cid:17) L X D a )( qx D + 1) L X D b ) . (8)We note that both 7a and 7b as 8a and 8b are equivalent since it can be reached from one to another by performingthe proper substitution of the force ratio and the coupling parameter. Now, we make an additional hypothesis aboutthe driver force; we will suppose that the temperature gradient is constant and of the form, X D = 1 T c − T h > , (9)with T c the temperature of the “cold” reservoir and T h the temperature of the “hot” reservoir. Due to this hypothesisthe D–LEC is a steady state converter.
1. D–LEC Dissipation Φ ∗ D On the basis of the analysis of the entropy production (Eq. 3) it is possible to construct several objective functionsfor this D–LEC. The first function that we can construct is a function called dissipation ( Φ D ). At first approximationthis function can be considered as measuring the part of energy that is used only for the coupling between the driverand the driven flux. We define Φ D in terms of generalized forces and fluxes through the entropy production as follows[5, 23], Φ D ≡ T c ( J D X D + J D X D ) = T c X D (cid:18) J D X D X D + J D (cid:19) = η C (cid:0) x D + 2 x D q + 1 (cid:1) L X D , (10)here we use the explicit form of the temperature gradient to obtain η C = T c X D = (1 − T c /T h ) , substitute the Eqs.(7b) and (8b) in Eq. (3), and we get Eq. (10) in terms of η C , x , q , L and X D . Finally we normalize the dissipationfunction by the constat L X D : Φ ∗ D ( x D , q, η C ) = Φ D L X D = (cid:0) x D + 2 qx D + 1 (cid:1) η C . (11)This expression is analog to that published by Arias–Hernandez et al for a steady state isothermic–LEC (see ec. 8 of[12]). Hereinafter we will consider normalized functions such that F ∗ = F/L X D . The nomalized dissipation Φ ∗ D isplotted versus the force ratio x in Fig. 2a, in this graphic we observe that Φ ∗ D has a minimum. We can optimize theD–LEC, with the purpose that it operates in a working regime of minimum dissipation ( mdf ), by finding the value ofthe force ratio x mdf which satisfies the equation ∂ x D Φ ∗ D ( x D , q, η C ) | x mdf = 0 , x mdf ( q ) = − q. (12)
2. D–LEC Power output P ∗ D Another objective function that we could built is the power output of the D–LEC. From the dissipation function (Eq.3) we note that the first term T c J D X D < , which corresponds to the driven flux promoted against a generalizedgradient, has units of energy per second, which can be taken as the power output P ≡ − T c J D X D of the D–LEC.Now if we take 5 and 7b and replace them in P we obtain, P ∗ D ( x D , q, η C ) = − x D ( x D + q ) η C . (13)This function is plotted in Figure 2a and we can observe that it has a maximum, so there exists a x MP D solution of ∂ x D P ∗ D ( x D , q, η C ) | x MPD = 0 and a maximum power output working regime (
M P D ) is possible to operate the D–LEC,if x MP D ( q ) = − q . (14) a) - - - - - D N o r m a li z e dob j ec ti ve f un c ti on s F D * Η F D * P D * W DG * E DG * b) F D * H x i , q , Η c L F D MPD * F D M Η * F D M H E, W L DG * c) Η H x i , q , Η c L Η MP D Η M H E, W L DG Η M Η d) P D * H x i , q , Η c L P D MPD * P D M Η * P D M H E, W L DG * Figure 2. a) Different objective functions for the steady state non–isothermic D–LEC: Dissipation function Φ ∗ D , Efficiency η (notnormalized), Power output P ∗ D , Generalized ecological function E ∗ DG , Generalized omega function Ω ∗ DG . Here we take q = 0 . and η C = 0 . . b) Comparative plot of the dissipation function at different working regimes, note Φ ∗ D ( x mdf , q ) = Φ ∗ D mdf = q → −−−→ .c) Comparative plot of the efficiency at different working regimes, note η ( x mdf , q ) = η mdf = q → −−−−→ x →− q η C . d) Comparative plot ofthe power output at different working regimes, note P ∗ D ( x mdf , q ) = P ∗ D mdf = 0
3. D–LEC Efficiency η We can define the irreversible efficiency of the D–LEC, as the power output divided by the input heat flux η ≡ P/J ,and using Eqs. (13) and (8b) we get, η ( x D , q, η C ) = − η C x D ( x D + q )1 + qx D . (15)Note that the efficiency is not a function of L X D . We plot η versus x D and see in Figure 2 that it has a maximum.This maximum is given by, x Mη ( q ) = − q p − q , (16)obtained from the equation ∂ x D η ( x D , q, η C ) | x Mη = 0 . Therefore, the D–LEC can operate in an optimum efficiencyworking regime ( M η ).
4. D–LEC Generalized ecological function E ∗ DG Using the characteristic functions we can built functions that accomplish other objectives, for example a goodtrade–off between the dissipation and the power output. Within the context of Finite Time Thermodynamics (FTT),in 1991, F. Angulo–Brown [15] proposed the Ecological Function, E = P D − Φ D , as this good trade–off function. If weoperate the heat engine at maximum ecological working regime, the engine reaches around 80% of the power outputof the M P D –working regime and 30% of the dissipation of this regime [16]. Later the Generalized Ecological Functionwas proposed [17, 24], that guaranteed the best trade–off between the power output and dissipation, through thefunction g ED ( η ) = η/ ( η C − η ) [15–18] evaluated at the efficiency of the M P D –working regime, E DG = P D − g EMP D Φ D .Evaluating the efficiency (Eq. 15) at x MP D = − q/ we get: η ( x MP D , q, η C ) = η MP D ( η C , q ) = q − q ) η C , (17)and substituting in g , g ED [ η MP D ( η C , q )] = g EMP D ( q ) = q − q . (18)Note that in the limit of ideal coupling we have lim q → g EMP D ( q ) = 1 and E DG = E .Finally, using Eqs. (13), (11) and (18) we write the generalized ecological function for the D–LEC as, E ∗ DG ( x D , q, η C ) = x D (cid:2) x D − q (cid:0) q − (cid:1)(cid:3) + q (cid:0) − x D (cid:1) q − η C . (19)We show the plot of E DG versus x , for a given q and η C , in Figure 2a and observe that this function has a maximum.Then the generalized ecological function can be used to optimize the operation D–LEC at this point ( x ME DG ); wecall this the M E DG –working regime. Solving the equation ∂ x D E ∗ DG ( x D , q, η C ) | x MEDG = 0 we obtain x ME DG , x ME DG ( q ) = − q (cid:0) q − (cid:1) q − . (20)
5. D–LEC Generalized omega function Ω ∗ DG The last objective function that we built for the D–LEC, is the generalized omega function. In 2001 within thecontext of FTT, Calvo–Hernandez et al proposed a unified optimization criterion for energy converters, based on themaximum of the omega function
Ω = E eu − E lu [19]. This function mades a trade–off between the effective usefulenergy E eu ≡ E u − r min E i , and the lost useful energy E lu ≡ r max E i − E u , where E u is the useful energy of theheat engine, r min is its minimum performance, E i is the input energy and r max is its maximum performance. Theperformance of the engine is defined as r ≡ E u /E i . The generalized function Ω DG = E eu − g Ω MP D E lu , was introducedby Tornez in 2006 [24], where g Ω MP D is the function g Ω D ( η ) = η/ ( η Mη − η ) for the omega function evaluated in the M P D –efficiency (Eq. 17), g Ω D [ η MP D ( η C , q )] = g Ω MP D ( q ) = p − q + 1 p − q − ! . (21)Operating the engine in the M Ω DG –working regime achieves the best compromise between E eu and E lu . In the contextof this model we can use the dissipation to define the input energy of the D–LEC: E i ≡ T J X = η C ( qx + 1) L X .On the other hand its useful energy is the power output E u ≡ P D , so the performance for the D–LEC is, r D ( x D , q ) = − x D ( x D + q )( qx D + 1) , (22)which is related to the efficiency in the following manner r D = η/η C , from this relation we conclude that the minimumperformance of the D–LEC is r min = 0 , and the maximum is r max = η Mη /η C , where η Mη is the efficiency evaluted at x Mη (Eq. 16) and is given by, η ( x Mη , q, η C ) = η Mη ( η C , q ) = η C q p − q ! (23) D–LEC x mdf = − q x Mη = − q √ − q x MP D = − q x M ( E, Ω) DG = − q ( q − ) ( q − ) working regimes Φ ∗ D ( x D , q ) = q → −−−→ η C ( − q ) √ − q η C (cid:16) − q (cid:17) η C − ( q − q ) ( q − ) η ( x D , q ) = η mdf q → −−−−→ x →− q η C η C (cid:18) q √ − q (cid:19) − η C q ( q − ) − η C q ( q − )( q − ) ( q − )( − q + q ) P ∗ D ( x D , q ) = 0 η C p − q (cid:18) q √ − q (cid:19) η C (cid:0) q (cid:1) η C q ( q − )( q − ) ( q − ) Table I. The table shows the characteristic functions for different working regimes of the steady state non–isothermic D–LEC,which are: Dissipation function Φ ∗ D , efficiency η and Power output P ∗ D , evaluated at the optimal values of the force ratio of theseworking regimes, which are respectively, minimum function of dissipation x mdf , maximum efficiency x Mη , maximum outputpower x MP D . Added to these optimal points we include the force ratio of the maximum generalized ecological and maximumgeneralized omega regime x M ( E, Ω) DG . Substituting E u , E i , r min , r max and g Ω MP D in the definitions of E eu , E lu and Ω DG we obtain, Ω ∗ DG ( x D , q, η C ) = x D (cid:2) q (cid:0) q − (cid:1) − x D (cid:3) + q (cid:0) x D − (cid:1)(cid:16)p − q − (cid:17) η C . (24)In Figure 2 we show that Ω ∗ DG reaches it maximum at x M Ω DG , x M Ω DG = − q (cid:0) q − (cid:1) q − , (25)which is the solution of ∂ x D Ω ∗ DG ( x D , q, η C ) | x M Ω DG = 0 .
6. Energetics of the D–LEC
The energetics of the D–LEC is shown in Figure 2. We must note that x ME DG = x M Ω DG , therefore the characteristicfunctions in these working regimes are the same. We show the characteristic functions of the above working regimesin Table II A 6If we observe the curves of the characteristic functions from Figure 2, we will see that each of them represents amode of operation that fulfills some objective of the thermodynamic process, and that the condition to operate theD-LEC optimally, to meet this objective, is to achieve the corresponding force ratio x i , that is, the way in whichthe flow handled through its associated potential is generated, and the potential against which the handler flux doeswork, which is subject to a certain degree of fixed coupling given by the design of the converter.Based on the criteria analyzed here Φ D , P D , η and ( E, Ω) DG , we search for quotient ratios compatible with thedifferent optimization criteria (see Figs. 2). For the D-LEC we observe the criteria comparatively ( mdf , M η , M P D and M ( E, Ω) DG ). Note that the efficiency of a non-isothermal linear energy converter, working in the differentregimes saves the following hierarchy η C > η Mη > η M ( E, Ω) DG > η MP D (Fig. 2c). In the same way we can hierarchizethe output power of this converter operating in different modes (Fig. 2d), such that P ∗ DM ( E, Ω) DG > P ∗ DMP D >P ∗ DMη > P ∗ Dmdf . The dissipation of this converter Φ ∗ D evaluated in the different working regimes satisfies the followinghierarchy (Fig. 2b), Φ ∗ DMP D > Φ ∗ DM ( E, Ω) DG > Φ ∗ DMη > Φ ∗ Dmdf . Finally, note that in the limit of strong coupling,the force ratios corresponding to the operating modes of the D-LEC, comply with the following hierarchical order x mdf = x Mη < x M ( E, Ω) DG < x MP D (tab. II A 6). B. Refrigerator (inverse LEC)
About 30% of world’s energy is used to promote heat fluxes against temperature gradients; these processes can becalled inverse conversion of energy. In particular, when the objective of the engine is to extract a heat flux from abody, we could say that we have a refrigerator.Now if we want to use the force ratio (Eq. 5) introduced by Stucki [13] it is necessary to write it for the case whenthe system is operating in an inverse mode, since that in the refrigerators the driven flux is J I = ˙ Q cI and the driverflux will be J I which are associated with the driven and driver forces X I and X I respectively. Then the force ratiofor the I–LEC is, x I = r L L X I X I . (26)Now we can write the fluxs J I and J I in terms of the inverse force ratio and the coupling coefficient as, J I = ( (1 + qx I ) L X I a ) (cid:16) qx I (cid:17) L X I b ) , (27)and J I = (cid:16) x I q (cid:17) L X I a ) (cid:16) qx I (cid:17) L X I b ) . (28)We will then extend the proposal of Jiménez de Cisneros et al [25] for the refrigeration cycles. We take as the drivenforce the following force, X I = 1 T h − T c . (29)where T h and T c have the same meaning as in the D–LEC.
1. I–LEC Dissipation Φ I The dissipation function for the I–LEC that operates between these two reservoirs can be defined as Φ I = T h σ ,using the entropy production Eq. (3), substituting x I , q we get, Φ I = T h ( J I X I + J I X I ) = ( T h X I ) (cid:18) x I + 2 qx I + 1 x I (cid:19) L X I = − ǫ C (cid:18) x I + 2 qx I + 1 x I (cid:19) L X I , (30)and normalizing by T h L X I we obtain, Φ I ( x I , q ) = x I + 2 qx I + 1 x I . (31)The plot of this function versus x I for q and ǫ C fixed, shows a minimum in the Fig.3a, and this minimum is reachedat x I mdf given by ∂ x I Φ I ( x I , q ) (cid:12)(cid:12)(cid:12) x Imdf = 0 , x I mdf ( q ) = − q . (32)and the corresponding working regime of the I–LEC ( mdf –working regime), will be obtained when we evaluate itscharacteristic functions in x I mdf ( q ) = − /q . a) - - - - - I N o r m a li z e dob j ec ti ve f un c ti on s F I ð Ε F I ð Q cI ð W IG ð E IG ð b) F I ð H x i , q , Ε c L F I M H E, W L IG ð F I M Ε ð c) Ε H x i , q , Ε c L Ε M Εð Ε M H E, W L IG ð d) Q c ð H x i , q , Ε c L Q cI M H E, W L IG ð Q cI M Ε ð Figure 3. Different objective functions for the steady state non–isothermic I–LEC: Dissipation function Φ I , Coefficient ofperformance ǫ (not normalized), Cooling power ˙ Q cI , Generalized ecological function E IG , Generalized omega function Ω IG . Herewe take q = 0 . and ǫ C = 2 . b) Comparative plot of the dissipation function at different working regimes, note Φ I (cid:0) x I mdf , q (cid:1) =Φ I mdf = q → −−−→ . c) Comparative plot of the Coefficient of performance at different working regimes, note ǫ (cid:0) x I mdf , q (cid:1) = ǫ I mdf = q → −−−−→ x →− q ǫ C . d) Comparative plot of the Cooling power at different working regimes, note J I = ˙ Q cI ( x I mdf , q ) =˙ Q cI mdf = q → −−−→ .
2. I–LEC Coefficient of performance ǫ C As is well known the amount of heat flux that can be driven by a refrigerator depends on the temperature differencebetween the reservoirs. The greater the difference, the lower the engine performance. This performance is measuredby the Coefficient of Performance (COP) built with heat flux extracted to the cold reservoir divided by the powerinput to the I–LEC, ǫ = ˙ Q cI /P . In terms of generalized fluxs and forces we can write, ǫ ( x I , q, ǫ C ) = J I T h J I X I = J I X I ( T h X I ) J I X I = − ǫ C x I ( x I + q )1 + qx I , (33)using the Eqs. (26), (27a) and (28b). From Eq. (33) we see that ǫ has the same form of η , but the values interval for ǫ C is [0 , ∞ ) . COP is plotted in Fig.3a and shows a maximum given by the solution of ∂ x I ǫ ( ǫ C , q, x I ) | x Mǫ = 0 , x Mǫ = − q p − q , (34)we notice that x Mη = x Mǫ . At this point we get an operating regime for the I–LEC at maximum COP, the M ǫ –workingregime.0
I–LEC x I mdf = − q x Mǫ = x Mη x M ( E, Ω) IG = qq − (cid:16) √ − q (cid:17) working regimes Φ I ( x I , q ) = q → −−−→ ( − q ) (cid:16) √ − q (cid:17) q q + (cid:16) √ − q (cid:17) q − p − q − ǫ ( x I , q ) = ǫ mdf q → −−−−→ x →− q ǫ C ǫ C (cid:18) q √ − q (cid:19) − ǫ C q h q − (cid:16) − √ − q (cid:17)ih (cid:16) √ − q (cid:17) − q ih (cid:16) √ − q (cid:17) − q i J I = ˙ Q cI ( x I , q ) = q → −−−→ ǫ C p − q − ǫ C (cid:16) − q + 2 p − q (cid:17) Table II. The table shows the characteristic functions for different working regimes of the steady state non–isothermic I–LEC,which are: Dissipation function Φ ∗ I , Coefficient of performance ǫ and Cooling power ˙ Q cI , evaluated at the optimal values ofthe force ratio of minimum function of dissipation x I mdf , maximum coefficient of performance x Mǫ , and the force ratio of themaximum generalized ecological and maximum generalized omega regimes x M ( E, Ω) IG .
3. I–LEC Generalized ecological function E IG The generalized ecological function for an irreverible model of a FTT–refrigerator: E RG = P e − g REMǫ
MAX T h σ , wasintroduced by Tornez in 2006 [24], it was defined as a function whose objective is to obtain the best trade–off betweenthe cooling power P e and the entropy production T h σ of the refrigerator, and the parameter g RE ( ǫ ) = ǫ C ǫ/ ( ǫ C − ǫ ) evaluated at half of the maximum COP, g REMǫ
MAX , guarantees this best trade–off. We will define the generalizedecological function for the I–LEC as the difference between the heat flux J I = ˙ Q cI and the dissipation function Φ I ,in the following manner, E IG = . Q cI − g IE (cid:16) ǫ Mǫ (cid:17) Φ I , (35)where ǫ ( x Mǫ , q, ǫ C ) = ǫ Mǫ ( ǫ C , q ) = ǫ C q p − q ! , (36)then g IE ( ǫ Mǫ /
2) = ǫ C ( ǫ Mǫ / / [ ǫ C − ( ǫ Mǫ / will be, g IE (cid:16) ǫ Mǫ (cid:17) = ǫ c q (cid:16) p − q (cid:17) − q . (37)Substituting the generalized fluxes and forces (Eqs. 26, 27 and 28) and Eq. (37) in Eq. (35) and normalizing by T h L X I , we obtain the generalized ecological function E IG in terms of x I , q and ǫ C , E IG ( x I , q, ǫ C ) = − ǫ C qx I qx I x I + 2 qx I + 12 (cid:16) p − q (cid:17) − q . (38)This function has a maximum (see Fig. 3a) therefore we can operate the I–LEC at this point and obtain a maximum E IG working regime ( M E IG ). To this end we take ∂ x I E IG ( x I , q, ǫ C ) (cid:12)(cid:12)(cid:12) x MEIG = 0 and solve for x ME IG , x ME IG ( q ) = 2 qq − (cid:16) p − q (cid:17) , (39)then we can substitute this solution in the characteristic functions of the I–LEC to get the energetics of this workingregime.1
4. I–LEC Generalized omega function Ω IG The generalized omega function Ω RG = E eu − g R Ω Mǫ MAX E lu was proposed by Tornez for an irreversible FTT–refrigerator[24], the meaning of E eu and E lu are the same as in the case of Ω , but with the performance of the refrigerator givenby r ≡ ( P e /P ) = ǫ , with P e the cooling power (useful energy, E u ) and P the power supplied (input energy, E i ). Thisobjective function proposes a trade–off between the effective useful energy E eu and the lost useful energy E lu . Theparameter g R Ω Mǫ MAX corresponds to the function g R Ω ( ǫ ) = ǫ/ ( ǫ MAX − ǫ ) [24] evaluated at the half of maximum COP,therefore g R Ω ( ǫ MAX /
2) = 1 . Following these definitions we define the generalized omega function for the I–LEC as, Ω IG = E Iue − E Ilu , (40)where E Iue = . Q cI , because E u = J I = . Q cI and the minimum performance of the I–LEC is r min = 0 , and E lu = ǫ Mǫ T h J I X I − J I , with the maximum performance r max = ǫ Mǫ and E i = T h J I X I . Substituting the fluxes andforces (Eqs. 27 and 28) in E Iue and E Ilu and the inverse force ratio (Eq. 26), we obtain the generalized omegafunction for the I–LEC, Ω IG ( x I , q, ǫ C ) = − ǫ C (cid:18) q + x I x I (cid:19) + q p − q ! (cid:18) qx I x I (cid:19) , (41)here we used the factor of normalization T h L X I .In Fig. 3a we observe that this function could give us the M Ω IG –working regime by solving ∂ x I Ω IG ( x I , q, ǫ C ) (cid:12)(cid:12)(cid:12) x M Ω IG =0 , and obtain the inverse force ratio for this regime, x MΩ IG ( q ) = 2 qq − (cid:16) p − q (cid:17) , (42)to evaluate the characteristic functions and get the energetics of the I–LEC working in this regime shown in Fig. 3b.
5. Energetics of the I–LEC
The energetics of the I–LEC is shown in Figure 3. In this case we observe x ME IG = x M Ω IG as for the D–LECcase, therefore the characteristic functions in these working regimes are the same. The functions that describe theenergetics of the I–LEC are shown in Table II B 3.We can see from the Figure 3 the comparison between the optimization criteria ( Imdf , M ( E, Ω) IG , M ǫ , ˙ Q cI ).We note that the COP of this non-isothermal linear converter working in the various operating regimes, keeps thefollowing hierarchy ǫ C > ǫ Mǫ > ǫ M ( E, Ω) IG , in the same way as the same converter; we observe the hierarchy ofdissipation and cooling load (see Figures 3b and 3d) under different operating modes: Φ IM ( E, Ω) DG > Φ IMǫ > Φ
Imdf and ˙ Q cIM ( E, Ω) DG > ˙ Q cIMǫ , respectively. III. THERMOELECTRIC THOMSON’S RELATIONS FOR A NON–ISOTHERMIC LEC
In this section we will make a proposal to introduce several working regimes in the thermoelectric phenomena theory,constructed within the LIT. We use our previous models of a steady linear energy converter (D–LEC & I–LEC), forsmall △ T , to describe a thermocouple subject to a heat flux, given by J i = J Q (Eq. 8b for i = D or Eq. 28b for i = I )and a charge flux J i = − J N (Eq. 7b for i = D or Eq. 27b for i = I ). We will introduce these working regimes throughthe use of Eq.47 as the revisited Seebeck power ξ = ∂ T V | J i . Also, we will place a non-resistive load (the systemtransfer work to the surroundings) or a battery (the surroundings transfer work to the system) in the thermocoupleat temperature T ′ , between the points a and b ; with these elements we could tune–in the flux J i for each operationmode (see Fig.4). For example, in the case of the minimum dissipative mode the current J D ( q, x mdf ) = 0 (D–LEC),so we will have a load such that it allows no passage of electric current but offers no resistance to the heat flow [4].Now we will deduce the two phenomenological coefficients from the definitions of the electric ( c ) and heat ( κ ) conductivities. The conductivity c is defined as the electric flux per unit potential gradient in an isothermal system2 a)Figure 4. a) Outline of a thermocouple with a variable electric current J i = A i ( x ij , q ) L ∇ T , where A i = D,I ( x ij , q ) = n(cid:16) x Dj q (cid:17) , (cid:16) qx Ij (cid:17)o and j = Mη, M ( E, Ω) DG , MP D , M ( E, Ω) IG or Mǫ , respectively. These charge fluxes can betuned–in with a non–resistive load for a D–LEC or a battery for an I–LEC. ( ∇ T = 0 ). Additionally, if the system is homogeneous then ∇ µ = ∇ µ e substituting these conditions in the generalizedequations for the thermocouple (Eqs. 1a), we obtain, c = e L T ⇒ L = cTe , (43)where e is the electric charge. The heat conductivity κ is defined as the heat flux per unit temperature gradient forzero electric field in an homogeneous medium, introducing this definition in Eqs. (1) we get, κ = L T ⇒ L = κT . (44)These direct coefficients correspond to the well known phenomelogical laws, Ohm’s law and Fourier’s law. A. Second Thomson’s relation
On the other hand we will perform a procedure to obtain the cross coefficients. First, we consider a flux J i = − J N of the form J i ( x ij , q ) = A i ( x ij , q ) L ∇ T (see Eqs. 7b and 27b), where, A i ( x ij , q ) = ( x Dj q , f or i = D qx Ij , f or i = I , (45)and j = M η, M ( E, Ω) DG , M P D , M ( E, Ω) IG or M ǫ respectively. Replacing J i in (1a) we have ∇ µ = − [ A i ( x ij , q ) − L T L ∇ T (46)The Seebeck effect is the phenomenon that consists of the production of an electromotive force emf in a thermocoupleunder the condition of a null electric current J N = 0 . For our proposal to introduce different modes of operation wewill consider a flux J i compatible with each mode.Now integrating and rewriting Eq. (46) in terms of µ ´ a and µ ´ b see (Fig. 4) we obtain µ ´ b − µ ´ a = − [ A i ( x ij , q ) − ˆ (cid:18) L A T L A − L B T L B (cid:19) dT, (47)3rewrite 47 in terms of µ and µ we obtain the following, µ ´ b − µ ´ a = − [ A i ( x ij , q ) −
1] ( µ − µ ) . (48)But, because there is no temperature gradient across the voltmeter, the voltage is given as follows V = 1 e (cid:0) µ ´ b − µ ´ a (cid:1) = − e [ A i ( x ij , q ) −
1] ( µ − µ ) = − [ A i ( x ij , q ) − ˆ (cid:18) L A eT L A − L B eT L B (cid:19) dT, (49)the thermoelectric power (Seebeck power) under the condition J i ( x mdf , q ) = 0 is defined as follows ξ mdfAB = ∂V∂T (cid:12)(cid:12)(cid:12)(cid:12) J i =0 = − (cid:20)(cid:18) − L B eT L B (cid:19) − (cid:18) − L A eT L A (cid:19)(cid:21) , (50)in the same way we define the new Seebeck power, for the general case J i = J i ( x ij , q ) : ξ AB = ∂V∂T (cid:12)(cid:12)(cid:12)(cid:12) J i = − [ A i ( x ij , q ) − (cid:20)(cid:18) − L B eT L B (cid:19) − (cid:18) − L A eT L A (cid:19)(cid:21) . (51)The absolute Seebeck power is defined as ξ ≡ − [ A i ( x ij , q ) − L A eT L A . (52)Now we have been able to calculate the values of the phenomenological coefficients, which have remained in termsof the Seebeck power (52), the electric conductivity (43), the thermal conductivity (44) and the operating constant A i (Eq. 45), L = T ce , L = − T cξ [ A i ( x ij , q ) − e , L = T κ. (53)If we accept the electrical conductivity c , the thermal conductivity κ and the absolute thermoelectric power ξ asthe three physically significant dynamic properties of a medium in addition to force ratio and the coupling parameter,we can eliminate the three phenomenological coefficients and therefore rewrite the kinetic equations (1) as follows: − J N = (cid:18) T ce (cid:19) T ∇ µ + (cid:26) − T cξ [ A i ( x ij , q ) − e (cid:27) ∇ T , (54) J Q = (cid:26) − T cξ [ A i ( x ij , q ) − e (cid:27) T ∇ µ + T κ ∇ T , (55)As is well known, the Peltier effect describes the way in which the heat of an isothermal welding ( ∇ T = 0) producedby an electric current evolves. Under the condition that the process of heat evolution in the welding is isothermal,the dynamic equations (1) take the following form, J Q = − T ξ [ A i ( x i , q ) −
1] ( eJ N ) , (56)where J Q = J BQ − J AQ = − T ( ξ B − ξ A )[ A i ( x i ,q ) − ( eJ N ) . On the other hand the Peltier coefficient π AB is defined as π AB = J Q eJ N = − T ( ξ B − ξ A )[ A i ( x i , q ) −
1] = − T ξ AB [ A i ( x i , q ) − , (57)this last relation is called the second Thomson’s relation, which shows a subtle relation between the Seebeck powerand the Peltier coefficient.4 B. First Thomson’s relation
In this section we will make a deduction of the first Thomson’s relation, for which we will proceed in a habitualway [4, 8]. We will begin considering the Fig. 1a which is a synthesized description of a thermocouple. We must fixour attention in the soldering; suppose that there is a charge unit transfer along the thermocouple which inevitablycauses several transfers of energy in the system, the analysis of these energy transfers will help us to construct anenergy balance equation from which we obtain the first Thomson’s relation.Before starting with the analysis of the energy balance it is necessary to define the Thomson coefficient τ which isdefined as the Thomson heat absorbed per unit temperature gradient and per unit electric current τ ≡ T homson heat ( eJ N ) ∇ T = T dξdT (58)Consider that the load unit passes the welding to temperature T in the clockwise direction from B to A ; this causesa heat to be absorbed from the source due to the Peltier effect π BA , the load now on the material A absorbs a heatof Thomson τ A dT , the load follows its path through the circuit in such a way that, at the time of traversing thewelding that lies at ( T + dT ) in a clockwise manner from A to B in such a way that the system absorbs Peltierheat ( π AB + dπ AB ) the charge in its path traverses material B where it absorbs a heat due to the Thomson effect ( dV = − dV ) . Finally, when the charge crosses the battery performs a work equal to the emf that produces thebattery ( dV = − dV ). We must mention that we have not considered the contributions of heat in the balance due tothe heat of Joule since in our analysis this is small in comparison to contributions due to the heat of Thomson.Now if we equalize the total energy that is absorbed by the system along the path of the circuit with the work doneon the battery we get the following − π AB + ( π AB + dπ AB ) + ( τ A − τ B ) dT = dV, (59)which can be rewritten as follows τ A − τ B = − dπdT + dVdT (60)using (50) and (57) dVdT = ξ AB = − [ A i ( x i , q ) − π BA T , (61)therefore dπ AB dT + ( τ A − τ B ) = ξ AB , (62)or dπ AB dT + ( τ A − τ B ) = − [ A i ( x i , q ) − π AB T , (63)Which is the first Thomson’s relation for any mode of operation, when x mfd = − q (minimum dissipation function)reproduces the first Thomson’s relation (see Table .III B). IV. CONCLUDING REMARKS
For the deduction of two Thomson’s relations, usually two experiments are carried out which imply that thethermocouple transfers energy in the form of work to its surroundings (Seebeck effect) or receives energy in the formof work of these (Peltier effect). In order to build a model in the context of linear irreversible thermodynamics, whichtakes into account these exchanges of work, here we proposed a non–isothermal energy converter, since of the fluxs5
Relation D–LECworking regimes x mdf x Mη x MP D x M ( E, Ω) DG FTR dπ AB dT + ( τ A − τ B ) = π AB T √ − q π AB T π AB T q ( q − ) ( q − ) π AB T STR π AB = T ξ AB (cid:16) p − q (cid:17) T ξ AB T ξ AB ( q − ) q ( q − ) T ξ AB I–LECworking regimes x I mdf x Mǫ = x Mη x M ( E, Ω) IG FTR dπ AB dT + ( τ A − τ B ) = q π AB T √ − q π AB T − (cid:20) q − (cid:16) √ − q (cid:17) (cid:21) π AB T STR π AB = q T ξ AB (cid:16) p − q (cid:17) T ξ AB − (cid:20) q − (cid:16) √ − q (cid:17) (cid:21) T ξ AB Table III. The table shows the first and the second Thomson’s relations (FTR) and (STR) evaluated in optimal force ratio ofdifferent objective functions, in the case when the system operates as a D–LEC, the evaluation points are minimum dissipationfunction x mdf , maximum efficiency x Mη , maximum power output x MP D , and the point of maximum ecological function whichis equivalent to the maximum omega function x M ( E,Ω ) DG . Similary when the system operates as a I–LEC, the force ratio fromwhich the Thomson’s relations will be evaluated are the minimum dissipation function x I mdf , maximum COP x Mǫ and theforce ratio of maximum ecological function and maximum omega function x M ( E,Ω ) IG . involved in this phenomenon one of them is of heat and occurs between two heat reservoirs whose temperatures arefixed ( T and T + △ T ). This converter works in two modes, such as a heat engine where a spontaneous heat fluxpromotes a non–spontaneous (D-LEC), and as a refrigerator where a spontaneous flow of any nature promotes anon–spontaneous heat flow (I -LEC). For these two converters we find different working regimes (steady states) thatcorrespond to a specific relation between the force ratios (operation) and the coupling coefficient (design).This fact allows that for each of given regimes, an expression for J D and the force X D (Eq. 7b) or, whereappropriate, for J I and the force X I (Eq. 27b), so as to ensure that reciprocity relationships are satisfied in eachregime. Thus, we propose that the extra-thermodynamic information required to find the cross-coefficients in anysystem, given the linear relation between fluxs and forces and the bilinear form of entropy production [26], can beobtained from the relation between the force ratio x i and the coupling coefficient q that provides the thermodynamicoptimization. From the above we can say that the direct coefficients can be deduced from microscopic models or fromphenomenological laws such as Ohm’s law and Fourier’s law, while the cross–coefficients necessarily come from anexperiment that involves the interaction of the system with the surroundings .When applying these results to the thermoelectric phenomena, we find that for the working regime that correspondsto a minimum dissipation function of the thermocouple, operating as D-LEC, recovers the already known secondThomson’s relation (see Table III B). This is so because the dissipation in this regime is minimal and therefore theproduction of entropy also, in fact can be verified that in this regime J D ( x mdf , q ) = 0 , that it is precisely the conditionof open circuit that is used to deduce this relation. Similarly, for the regime of minimum dissipation function, we recoverthe well known first Thomson’s relation. Additionally, a new set of Thomson relations is obtained; this comes fromthe thermodynamic optimization, which provides information on how to transfer the system work to the surroundings,for example, in the case of the work regime of maximum output power of the D-LEC, the second relation given by π AB = 2 T ξ AB is obtained, while for the first relation we obtain dπ AB /dT + ( τ A − τ B ) = π AB / T . These two relationsare like this because a non–resistive load has been placed between points a and b which the thermocouple transfersa maximum amount of work consistent with the flux of electric charge given by J D ( x MP D , q ) = − ( T c/e ) ( ∇ µ/eT ) (see Eq. 27a). In the case of the Peltier effect (I-LEC) at maximum generalized ecological function, the current thatmust force the battery (external work) in the thermocouple must be, J I ( x MP D , q ) = (1 /
3) (
T c/e ) ( ∇ µ/eT ) (see Eq.27), at the strong coupling condition. Applying systematically the energetics of the linear stationary energy converterdeveloped in Section II, we obtain 6 new Thomson’s "second relations" and as many Thomson’s "first relations". Infact, from the results shown in Table III B, we see that in the condition of strong coupling, the Peltier heat in thedifferent working regimes satisfies π mdfAB = π I mdf AB = π MηAB = π MǫAB < π M ( E, Ω) DG AB < π M ( E, Ω) IG AB < π MP D AB .Then the concept of energy converter, isothermal [27] and non-isothermal, developed from the division of theproduction of entropy into two subsets of products, one made up of the products of fluxes and forces that contributepositively to this, and the other formed by those products that contribute to it negatively, it can be useful to explorethe cross-contributions to the flows that intervene in the system (Eqs. 7, 8, 27 and 28) as has been shown in thisarticle. Another example is found in the reference mentioned above, where it is shown that given the elements ofan electrical circuit and the different work regimes there is a specific relationship between them. This allows us to6 - - - - - D E ffi c i e n t po w e r P Η D * Figure 5. The figure shows the normalized Efficient Power P ∗ η for the steady state non–isothermic D–LEC; here we take q = 0 . and η C = 0 . . affirm that the use of optimization criteria developed for other converter models [28] in other contexts opens up thepossibility of designing new experiments within linear irreversible thermodynamics. Acknowledgement
We thank RMMA and FAB for stimulating discussions, suggestions and invaluable help in the preparation of themanuscript. This work was supported in part by EDI–SIP–COFAA–IPN, CONACYT, Mexico.
APPENDIX
The efficient power P η is defined as P η = P η , which can be written in terms of generalized fluxs and forces asfollows, P η = ( T c J D X D ) J D = η c [ x ( x + q )] ( qx + 1) L X , (64)or P ∗ η = η c [ x ( x + q )] ( qx + 1) . (65)Optimizing the Equation 64 with respect to x , it is possible to obtain the maximun efficient power force ratio x MP η (see Figure 5), x MP η ( q ) = 46 q − q + (cid:0) q (cid:1) − (cid:16) (cid:0) q (cid:1) (cid:17) q , (66)and we derive the energetics of the D–LEC from Eq. (66). [1] Seebeck, T. J., Ueber den Magnetismus der galvenischen Kette, Royal Prussian Academy of Science (1821).[2] Seebeck, T.J., Ueber die magnetische Polarisation der Metalle und Erze durch Temperatur-Differenz, Annalen der Physik82 (1826), pp. 1-20.[3] Peltier, J.C.A., "Nouvelles experiences sur la caloricite des courants électrique", Annales de Chimie et de Physique 56(1834), pp. 371--386.7