Irreversibility of Processes in Closed System

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Godwin Beasley
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1 Unversty of Segen Insttute of Flud- & hermodynamcs 5 2/1 Irreversblty of Processes n Closed System m G 2 m c 2 2, p, V m g h h 1 mc 1 1 p, p, V G J.P. Joule Strrng experment v J.B. Fourer Heat transfer m g h h = mc > < 1 2 Gay-Lussac Gas expanson = > < U(,V,m) = U(,V + V,m) V > < I.G. : =

2 Unversty of Segen Insttute of Flud- & hermodynamcs Gas Expanson Process Irreversblty VELO > 5 2/2 Σ REV EQ Σ F > VELO Σ REV EQ = Σ F = Reversblty

3 Unversty of Segen Insttute of Flud- & hermodynamcs 2 nd Law (1): Clausus Entropy (t) I e (t) B 1234 U e (t) PC dq rev S S ( )...d Clausus Equalty dq rev ( = ) 5 = + = 2/3 Q dq Entropymeter = 2 I R dt Carnot-Relaton Q Q W =, η = = 1 Q Q c W= Q Q Q

4 Unversty of Segen Insttute of Flud- & hermodynamcs Gbbs Equaton (Smple System) 5 2/4 st 1 Law : nd 2 Law: du = dq pdv rev ds = dq rev p,,v Σ dw = pdv Gbbs: 1 p ds = du + dv dq rev = m = const C S S d' dv' 2 2 V S S V p,v' = = + + V V ' (V ) V () Calorc EOS hermal EOS

5 Unversty of Segen Insttute of Flud- & hermodynamcs 5 2/5 Entropy of Ideal Substances Ideal Gas Ideal Lqud EOS: pv= CEOS: m M R H= H + mc p R p S = S + m cp ln ln M p R V S = S + m cv ln ln + M V EOS: ρ= const = 1 v CEOS: c= const H= H + mc + mv p p S= S + mcln

6 Unversty of Segen Insttute of Flud- & hermodynamcs 5 2/6 U U S(U,n...n ) S c ln 1 n 1 = + + = 1 cn 1 = = 1 1 = = 1 1 = = 1 = 1 = 1 = 1 S(U,n...n ) S (s Rln x )n G(,n...n ) G (c s Rln x )( ) c ln n G(,n...n ) G ( s Rln x ) n U pv U = U = U + c ( )n,v = v n H = H = H + c ( ) + pv n µ (,n...n ) = µ + ( )s + c ln R( )lnx 1 µ (,n...n ) µ = s R ln x 1 Incompressble, deal flud mxture (c = const,v = const, = 1...)

7 Unversty of Segen Entropy of Water (H 2 O) K. Stephan, W. Wagner IAPS (1985) Insttute of Flud- & hermodynamcs 5 2/7

8 Unversty of Segen Insttute of Flud- & hermodynamcs 2 nd Law (2): Clausus Inequalty 5 2/8 dm Σ Σ dw = pdv IRR : REV : Σ: dq S S + s dm ( ) ( ) dq Exchange of mass heat work Quasstatc changes of state ( +d) Closed Systems: IRR : REV : A dq ds s dn dq + ( ) =, dm = S S = 1 ( ) ( )... World Statement?

9 Unversty of Segen Insttute of Flud- & hermodynamcs 5 2/9 Interpretatons of the Clausus Entropy (S()) 1. M. Planck (1885) Measure for probablty, tendency, preference of a system to actually realze a certan equlbrum state (). ( 2 -gas n steel bottle) 2. L. Boltzmann (189) Measure for molecular dsorder n a system n state (). (Crystal, lqud, gas, plasma...) 3. C. Shannon (1948) Measure for lack of knowledge of the mcro-.e. molecular state of a system (Σ) n a gven macroscopc state ()

10 Unversty of Segen Insttute of Flud- & hermodynamcs Strrng Experment of J.P Joule 5 2/1 st 1 Law: U + m gh = U+ m gh w w m m w g h h CEOS: nd 2 Law (1), I.G.: U= U + mc v = + S S mcv ln :,h :, h nd 2 Law (2): S> S > h< h ( <,h> h ) Reversed process not possble!

11 Unversty of Segen Insttute of Flud- & hermodynamcs 5 2/11 Maxmum Work of a Mass Flow (H. von Helmholtz (189)) Smple open system m dm dw t Σ : h,s,,p, ρ dm ( ) Σ = dq : h,s, =,p, ρ m ( ) Statonary State = s t st 1 Law : U = Q + W + h h m = Q nd 2 Law: S = + ( s s ) m + Ps = P Wt h h s s m Ps W = e (, )m + P P ex = t x ex = Ps

12 Unversty of Segen Insttute of Flud- & hermodynamcs hermodynamcs of Processes (1) 5 2/12 Dscrete System, Exchange of Heat, Work, Mass Balance Equatons : Gbbs Equaton : n ( ) A U = Q p V + h n n 1 p ds = du + dv µ dn dq Clausus Inequalty : S S = + s dn Fundamental Inequalty: ( ) ( ) t,a 1 1 p p U µ µ + V + n dt... all t, hermal energy work mass exchange = dq dw U Σ V, n p ( ) ( ) = dn J dt

13 Unversty of Segen Insttute of Flud- & hermodynamcs 5 2/13 Process Equatons (Flux-Force-Relatons) (Eckart Onsager Mexner Prgogne) hermal energy Heat transfer Ut = F(%) u Mechancal work Vt = F(%) v Mass transfer n t = F (%) = 1..., = 1...A ( ) 1 1 p p µ k µ k (%) =,,,k = 1..., = 1...A F... Functons or Functonals s t of ther arguments

14 Unversty of Segen Insttute of Flud- & hermodynamcs Classfcaton of Process Equatons 5 2/14 F(%) t Functon (t) Functonal (- <s t) s ( ) ( ) ( ) nd Fundamental Inequalty 2 Law, () heorem JUK, 1968 xf x,x dt... all t F x,x = =, = 1... Lnearty IP LPS P = x F x,x on-lnearty IP PS (?) IP ( ) = F x,x L x x ( ) ( ) k k k Onsager-Casmr-Relatons L =εε L, ε, ε =± 1 IP k k k k F x,x = L x x + LPS k k k F x,x klm ( x) x x x ( 5) + M + klm k l m... Lnear Passve Functonal (J. Mexner, H. Köng, 1964)

15 Unversty of Segen Insttute of Flud- & hermodynamcs 5 2/15 Dmensonal Analyss Phenomenologcal Coeffcents and Functons: Buckngham s heorem (π heorem): Y = Y( 1... M) Dmensonal Matrx Basc Unts System (SI-System) [ Y] G 1, G 2, G g M = Π φ π π Y (... ) = 1 M = Π = 1 1 M r Rank: [ ] M βjk k j= 1 j k π =Π, π = 1 µ r ln φ =φ + φ ln π = 1

16 Unversty of Segen Dmensonal Analyss Insttute of Flud- & hermodynamcs 5 2/16 ln φ π... π =ψ ln π...ln π... π 1 M r 1 M r M r M r k k,k M r (... ) C ( ln ln ln...) 1 M r k k kl k l k k,l M r M r γ ln π k k β k C..., C e =ψ + β ln π + γ ln π ln π + O 3 φ π π = π β + γ π + δ π π + π π ψ (... π ) aylor seres expanson of the reduced functon φ π : Energy: Scale shft nvarance! 1 M r

17 Unversty of Segen Insttute of Flud- & hermodynamcs Example 1: Velocty of Molecules n an Ideal Gas 5 2/17 Lst of relevant varables, parameters, constants: w = w (, p, V m, M, R) M = 5 G = 5 r = 5 M r = w R = Const, Calbraton Experment: M Const = 3, w = 3R M

18 Unversty of Segen Insttute of Flud- & hermodynamcs 5 2/18 Process Calculaton (Intal value problem, ODE) A Σ : (t) = U(t), V(t), n (t) = n Accompanyng equlbrum ntensve parameters at tme (t): S = S(U,v,n...n ) 1 p µ ds = du + dv dn (t), p(t), µ (t) Process equatons for U, V,n, aylor-seres expanson: 1 2 Σ : (t + t) = U(t + t) = U(t) + F u(t) t + F u( t) ( ) ( ) ( ) 1 ( ) 2 n (t + t) = n (t) + F (t) t + F (t)( t) +... Iteraton procedure 2

19 Unversty of Segen Insttute of Flud- & hermodynamcs 5 2/19 Statonary Processes and States U =,V = = A = = 1 n n, 1... Fundamental Inequalty,A ( ) µ µ ( ) Ps = n, Process Equatons (Flux Force Relatons) Mass transfer: n = F (./.), = 1..., = 1...A ( ) ( ) µ µ ( ) (./.) =, = 1..., = 1...A

Introduction to Statistical Methods

Introduction to Statistical Methods Introducton to Statstcal Methods Physcs 4362, Lecture #3 hermodynamcs Classcal Statstcal Knetc heory Classcal hermodynamcs Macroscopc approach General propertes of the system Macroscopc varables 1 hermodynamc