Irreversible thermodynamics, a.k.a. Non-equilibrium thermodynamics (an introduction)

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1 Process Engneernghermodynamcs course # v. 05 Irreversble thermodynamcs, a.k.a. Non-equlbrum thermodynamcs (an ntroducton) Ron Zevenhoven Åbo Akadem Unversty hermal and Flow Engneerng aboratory / Värme- och strömnngsteknk tel. 33 ; ron.zevenhoven@abo.f Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 ÅA Equlbrum, classcal thermodynamcs vs. rreversble (non-equlbrum) thermodynamcs Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3

2 Irreversble thermodynamcs / Classcal thermodynamcs deals wth drvng forces for chemcal reactons and wth equlbrum, wth equlbrum descrptons followng from the st and nd laws of thermodynamcs for closed systems. Open systems wth heat, mass and/or electrcty transported over system boundares are much more mportant n engneerng applcatons. Change and movement s more common than equlbrum state (also n nature as a result of the constant energy nflux from the sun). Pc: Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 Equlbrum (HK65, chapter 4) / Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 4/3

3 Equlbrum (HK65, chapter 4) / Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 5/3 Irreversble thermodynamcs / Irreversble thermodynamcs addresses nonequlbrum stuatons, assumng reversblty on a small scale (.e. local equlbrum) and lnear transport processes. A startng pont was the work of homson (later ord Kelvn) on thermo-electrcty,.e. nteractng transport of heat and electrc charge n the 850 s he man goal s to descrbe nteractng transport processes, takng nto account entropy producton and the nd aw of thermodynamcs An Estonan-German physcst homas Seebeck (770-83) twsted two wres of dfferent metals together and heated the pont at where they were joned. He produced a small current of electrcty. hs s called thermoelectrcty and s known n physcs as the "Seebeck Effect". Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 Pc:

4 ransport processes (lnear) / dx Q d n dc Fourer : Q Fck : M D A dx A dx dv dvy Ohm : I Newton : mm xy A dx dx n eral : X flux transport coeff. drvng force (voltage V, specf. conductance σ = /ρ, where spec. resstance ρ = R A/Δx) Entropy producton and Fourer s law: Heat transfer between a and b : ds a b ds dq dq S Q Q A a b dt ΔEx o d exergy loss : ΔEx o S o Q A volume dx A m R = electrcal resstance, Ω Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 7/3 ransport processes (lnear) / Q Smlar for electrc current (or mass dffuson or flud flow): V ΔEx o S ds dt * o ds V S exergy loss : dt V ΔEx o dv volume dx hs gves the eral descrpton = X, and S X Fourer :, Q ; where X and X Δ ; s the drvng force for Ohm :, ΔV Cross-effects and nteractons can be descrbed too: where X j j X ; and and (Onsager' s recprocal relatons) X X V = voltage * no mass or heat flows ds S dt X Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 8/3

5 ransport processes (lnear) /3 For example, smultaneous heat and mass transfer where Q QQ QQ X MQ Q QM QM X and ~ heat conductvty λ, s the thermodffuson coeffcent; and hs gves the eral descrpton M ; MM X X ~ mass dffuson coeffcent he recprocal relatons are also known as the 4 th aw of hermodynamcs. S S for entropy producton wth two coupled flows X X X X ; mples that and X ( M, MQ X, )X X Q and MM X M j j D X X j Note: may be < 0! Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 9/3 Irreversble thermodynamcs /3 Irreversble or non-equlbrum thermodynamcs descrbes transport processes n systems not n global equlbrum. he nd aw s reformulated n terms of entropy producton, Ṡ, assumng local equlbrum. he approach s very powerful for the analyss of smultaneous transfer of heat and mass, or mass and electrc charge, etc, as for example found n S membrane separatons or electrolyte systems, or systems where gravty o W S S X lost s mportant Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 0/3

6 8..05 Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 Drvng forces; entropy producton For heat exchange the entropy producton rate s the product of the thermodynamc drvng force X = Δ(/) and the resultng flow = Q. For more eral systems, for example an solated system separated nto sectons by a membrane permeable only to one speces (e.g. speces ): X µ n p V Q S µ µ dt dn p p dt dv dt dq dt ds V, n, p,, µ V, n, p,, µ Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 An example Source: K07 z y x and volume S Here / / / :

7 ÅA Maxmum entropy producton Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 Maxmum entropy producton For combned transport processes, the fnal outcome s not only determned by balances for conserved propertes such as mass, energy, charge and momentum For smulantaneous transport the ncreased degree of freedom results n nteractons he fnal outcome s then more strongly governed by the nd law of thermodynamcs and, unless the devatons from equlbrum are small, the process tends to entropy producton maxmse Maxmum entropy analyss s also used as a statstcs method Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 Pc:

8 Max. entropy producton example/ An electrc heatng element dstrbutes heat Q nto Q + Q whle heatng up two dfferent streams. Input energy Q results n temperature h for the heatng element. he system s well nsulated. For both streams the energy balance equaton gves Q = ṁ Δ c p = U A Δ lm,, and Q = Q + Q s fxed. (Assume A = A, or even U A = U A ). How wll the nput heat energy Q be dstrbuted? Flow ṁ,, c p Flow ṁ, 3, c p Heat Q Flow ṁ,, c p Q = h Q Flow ṁ, 4, c p Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 Max. entropy producton example/ For one of the streams the entropy eraton s S, ds m cp, d m d m ln( Q whle also ; smlar for the other stream m c p, he result follows from max {Ṡ (Q ) + Ṡ (Q )} c p, c p, ) Flow ṁ,, c p Flow ṁ, 3, c p Heat Q Flow ṁ,, c p Q = h Q Flow ṁ, 4, c p Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 6/3

9 Max. entropy producton example/3 hus, ds dq m, m c p, c ds dq p,, Q, m c c (Q Q ) whch can be solved for Q, gvng Q = Q - Q, fnally gvng temperatures 3 and 4. whch gves p, m p, Flow ṁ,, c p Flow ṁ, 3, c p Heat Q Flow ṁ,, c p Q = h Q Flow ṁ, 4, c p Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 7/3 ÅA hermo-electrcty Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3

10 Entropy eraton / Consder a slce of a materal that conduct heat and electrcty; the heat flux Ф (W/m ) and electrc current densty I (A/m ). V= electrc potental = temperature he entropy producton rate per unt volume (area dx) ds /dt for the heat flow s gven by Pc, source: B Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 9/3 Entropy eraton / hen for the total volume A dx: See also sldes 7,8 ds''' ds dt volume dt '' dv dx For the electrc current densty (Ohm s law): wth σ = /ρ, wth specfc electrcal resstance ρ he entropy producton as a result of Ohmc losses, per unt volume: see also next slde Pc, source: B Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 0/38

11 Entropy eraton /3 hen for the total volume: he processes can be coupled usng: wth Note : ds''' dq dt volume I dr area dx '' I area dr dx '' '' I ρ dx I dx σ σ dv dx wth resstance R( Ω) Pc, source: B Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 I dv dv R dv Φ I 0 d d d Entropy eraton /4 d 0 Accordng to Ohm s law, f d = 0: wth resstance R = ρ dx/a, for thckness dx, area A Fourer s law follows from I = 0: wth Φ = G d, heat conductance G = λ A/dx. A thrd materal property s needed to descrbe the cross phenomena: the Seebeck coeffcent, θ, defned as: Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku Pc, source: B0 /38

12 Entropy eraton /5 hs gves for current I: Elmnatng the voltage gradent dv/dx from the expressons gves whch wth conductance G = λ area/dx can also be wrtten as combnng Fourer s law and the Seebeck effect. thermo-electrcty Pc, source: B Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 3/38 he Seebeck effect For combned heat flow and electrc current, wth I = 0: hs means that the voltage dfference ΔV ( thermo-current ) for system n the Fg. can be calculated as: Δ Δ the can be hermometry,.e. a thermocouple Reference temperature Pc, source: B0 Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3

13 he Pelter effect / Consder agan combned heat flow and electrc current, but now wth d = 0. he Onsager expressons now gve: and Combnng d = 0 wth gves whch mples that an electrc current nvolves a heat flow as well, whch enters and leaves the materal wthout causng heatng or coolng. Pc, source: B0 Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 he Pelter effect / If, however, two dfferent materals a and b are used (as n a thermocouple), a heatng or coolng effect s obtaned at the contact pont of materals a and b. he netto heat flow, Ф ab s determned by the Seebeck coeffcents for the two materals, θ a and θ b : he property s referred to as the Pelter-coeffcent for the materal set a-b. Pc, source: B0 Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3

14 ÅA Power from osmoss Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3 Work from a salne power plant / he mxng of sea water wth fresh water gves a (mxng) exergy effect that can be exploted Installng a membrane system at 0-50 m below the fresh water ntake allows for a sgnfcant extra hydropower effect Pc, source: KB Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 8/3

15 Work from a salne power plant / Osmoss leads to transport of water across the membrane from the fresh water to the salt water sde soluton hs gves a hydrostatc pressure dfference Pc, source: KB Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 9/3 Work from a salne power plant /3 he fresh water wll go through the membrane, aganst a pressure dfference; the pressure on the sea water sde s not hgh enough to prevent fresh water movement. Pc, source: KB Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 30/3

16 Work from a salne power plant /4 In realty, only part of the potental energy s recovered; the unt won t be as low as 63 m below the fresh water ntake. Entropy producton n the membrane (combned heat and mass transfer) s mportant Pc, source: KB Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku 3/3 Sources B0: Bart, G.C.. Advanced thermodynamcs (n Dutch) course compendum Delft Unv. of echnol., Delft (00) Chapter B97: Bejan, A. Advanced engneerng thermodynamcs ohn Wley & Sons (997) Chapter FFK88: Förland, K.S., Förland,., Kjelstrup, S. Irreversble thermodynamcs. apr Akademsk Förlag, rondhem (988) HK65: Hatsopoulos, G.N., Keenan,.H. Prncples of eral thermodynamcs. R.E. Kreger Publ. Co. (965) Chapter 4 H84: Hoodoorn, C.. Advanced thermodynamcs (n Dutch) course compendum Delft Unv. of echnol., Delft (984) K07: Koper, G..M. An ntroducton to chemcal thermodynamcs, VSSD, Delft (007) Chapter 3 KB08: Kjelstrup, S., Bedeaux, D. Non-equlbrum thermo-dynamcs of heteroeous systems, World Scentfc (008) KBG0: Kjelstrup, S., Bedeaux, D., ohanessen, E., Gross,. Non-equlbrum thermodynamcs for engneers, World Scentfc (00) SAKS04: de Swaan Arons,., van der Koo, H., Sankaranarayanan, K. Effcency and sustanablty n the effcency and chemcal ndustres. Marcel Dekker, New York (NY) 004 Åbo Akadem Unv - hermal and Flow Engneerng Pspankatu 8, 0500 urku /3