Module 8 Non equilibrium Thermodynamics

Transcription

1 Modul 8 Non quilibrium hrmodynamics

2 ctur 8.1 Basic Postulats

3 NON-EQUIIRIBIUM HERMODYNAMICS Stady Stat procsss. (Stationary) Concpt of ocal thrmodynamic qlbm Extnsiv proprty Hat conducting bar dfin proprtis z Spcific proprty lim m 0 Z m

4 NON-EQBM HERMODYNAMICS Postulat I Although systm as a whol is not in qlbm., arbitrary small lmnts of it ar in local thrmodynamic qlbm & hav stat fns. which dpnd on stat paramtrs through th sam rlationships as in th cas of qlbm stats in classical qlbm thrmodynamics.

5 Postulat II NON-EQBM HERMODYNAMICS S & F κ κ Entropy gn rat affinitis fluxs

6 NON-EQBM HERMODYNAMICS Purly rsistiv systms Flux is dpndnt only on affinity at any instant t at that instant t Systm has no mmory -

7 NON-EQBM HERMODYNAMICS Coupld Phnomnon κ κ ( F F, F, ; xtnsiv.), prop Sinc k is 0 whn affinitis ar zro, k 1 + jk F j 2! j i j ijk F i F j +

8 NON-EQBM HERMODYNAMICS whr jκ κ F j ijκ 0 ; F 2 i κ F j 0 kintic Coff jk ( F ) 0 F, jk, 1 Postulat III Rlationship btwn affinity & flux from othr scincs

9 NON-EQBM HERMODYNAMICS Hat Flux : Q k ( ρ C ) α yy yy ρ Momntum : M u μ y ν ( ρ u ) y Mass : m D c y Elctricity : λ E y

10 NON-EQBM HERMODYNAMICS Postulat IV Onsagr thorm {in th absnc of magntic filds} jk kj

11 NON-EQBM HERMODYNAMICS Entropy production in systms involving hat Flow 1 2 A x

12 NON-EQBM HERMODYNAMICS Q k x Q A s Q k x Entropy gn. pr unit volum s x + s, x

13 NON-EQBM HERMODYNAMICS Q d 1 x+ 1 1 Q Q 2 x d S& S Q Q 2 d

14 NON-EQBM HERMODYNAMICS Entropy gnration du to currnt flow : I I A I A λ de Hat transfr in lmnt lngth δq de I

15 NON-EQBM HERMODYNAMICS Rsulting ntropy production pr unit volum Q S & δ A. de ( )

16 NON-EQBM HERMODYNAMICS otal ntropy prod / unit vol. with both lctric & thrmal gradints S& S& + S& Q Q 2 d de Q FQ +. F affinity affinity

17 NON-EQBM HERMODYNAMICS 1 F Q 2 1 F d de

18 Analysis of thrmo-lctric circuits Addl. Assumption : hrmo lctric phnomna can b takn as INEAR RESISIVE SYSEMS K jk F j {highr ordr trms ngligibl} Hr K 1,2 corrsp to hat flux Q, lc flux

19 Analysis of thrmo-lctric circuits Abov quations can b writtn as F + F Q QQ Q Q F + Q Q Substituting for affinitis, th xprssions drivd arlir, w gt QQ d 1 de Q Q 2 dx dx F Q d 1 dx 2 de dx

20 Analysis of thrmo-lctric circuits W nd to find valus of th kintic coffs. from xptly obtainabl data. Dfining lctrical conductivity λ as th lc. flux pr unit pot. gradint undr isothrmal conditions w gt from abov de de λ dx dx λλ

21 End of ctur

22 ctur 8.2 hrmolctric phnomna

23 Analysis of thrmo-lctric circuits h basic quations can b writtn as F + F Q QQ Q Q F + Q Q Substituting for affinitis, th xprssions drivd arlir, w gt QQ d 1 de Q Q 2 dx dx F Q d 1 dx 2 de dx

24 Analysis of thrmo-lctric circuits W nd to find valus of th kintic coffs. from xptly obtainabl data. Dfining lctrical conductivity λ as th lc. flux pr unit pot. gradint undr isothrmal conditions w gt from abov de de λ dx dx λλ

25 Analysis of thrmo-lctric circuits Considr th situation, undr coupld flow conditions, whn thr is no currnt in th matrial, i.. 0. Using th abov xprssion for w gt 0 2 de d dx dx Q d dx de dx Q Sbck 0 ffct

26 Analysis of thrmo-lctric circuits or de d Q 0 Sbck coff. de d 0 α > Q α α λ 2 Using Onsagr thorm Q Q α λ 2

27 Analysis of thrmo-lctric circuits Furthr from th basic qs for & Q,for 0 w gt QQ d Q Q d Q 2 dx dx + QQ 2 Q Q d dx

28 Analysis of thrmo-lctric circuits For coupld systms, w dfin thrmal conductivity as k his givs k Q ( d dx) 0 QQ + 2 Q Q

29 Analysis of thrmo-lctric circuits Substituting valus of coff., Q, Q calculatd abov, w gt QQ 2 2 k ( ) +α α λ 2 ( ) k + λα 2

30 Analysis of thrmo-lctric circuits Using ths xprssions for various kintic coff in th basic qs for fluxs w can writ ths as : Q d dx de dx ( ) k + α 2 λ α λ d α λ dx de λ dx

31 Analysis of thrmo-lctric circuits W can also rwrit ths with fluxs xprssd as fns of corrsponding affinitis alon : Q k + α dx λ k de α λ + 2 k + λα dx k + λα 2 Q Using ths qs. w can analyz th ffct of coupling on th primary flows

32 PEIER EFFEC Undr Isothrmal Conditions de λ dx a Q, ab b Hat flux Qa α ; α a Qb b

33 PEIER EFFEC Hat intraction with surroundings Qab Qa Qb ( α ) a αb Pltir ff. π ab Pltir coff. Π ( α ) ab a α b Klvin Rlation

34 PEIER REFRIGERAOR a : Cu b : F α α 13.7 μv C u F 0 K Q ab? 20 Amp. ~ 270K μv 13.7 K Smi ( 270K )( 20Amp).074W conductors α α 423μv a b K : 3 Bi N 2 P

35 HOMSON EFFEC otal nrgy flux thro conductor is + E Q, surr E Q Using th basic q. for coupld flows E k + α + x k + ( α E) x + ( ) E Q Q

36 HOMSON EFFEC h hat intraction with th surroundings du to gradint in E is d Q, surr Ex+ d k + α x E x d E ( + E)

37 HOMSON EFFEC Sinc is constant thro th conductor d 2 dk d Q, surr k x 2 + dα d + α + de

38 HOMSON EFFEC Using th basic q. for coupld flows, viz. d de α λ λ abov q. bcoms (for homognous d matrial, k const.; const. ) 2 d Q, surr dα λ homson hat oulan hat

39 dα HOMSON EFFEC Q, γ homson coff rvrsibl hating or cooling xprincd du to currnt flowing thro a tmp gradint d Comparing w gt γ d α d

40 HOMSON EFFEC W can also gt a rlationship btwn Pltir, Sbck & homson coff. by diffrntiating th xp. for π ab drivd arlir, viz. dπ d ab Π ab ( α α ) a ( + a b α α ) a b b dα d ( α α ) + ( γ γ ) a b a b dα d

41 End of ctur

42 Analysis of thrmo-lctric circuits Abov quations can b writtn as F + F Q QQ Q Q F + Q Q Substituting for affinitis, th xprssions drivd arlir, w gt QQ d 1 de Q Q 2 dx dx F Q d 1 dx 2 de dx

43 Analysis of thrmo-lctric circuits W nd to find valus of th kintic coffs. from xptly obtainabl data. Dfining lctrical conductivity λ as th lc. flux pr unit pot. gradint undr isothrmal conditions w gt from abov de de λ dx dx λλ