GMS Equations From Irreversible Thermodynamics

Transcription

1 GMS Equatios From Irreversible hermodyamics ChE 6603 Refereces E. N. Lightfoot, rasport Pheomea ad Livig Systems, McGraw-Hill, New York R. B. Bird, W. E. Stewart ad E. N. Lightfoot, rasport Pheomea 2 d ed., Chapter 24 McGraw-Hill, New York D. Jou, J. Casas-Vazquez, Exteded Irreversible hermodyamics, Spriger-Verlag, Berli R. aylor, R. Krisha Multicompoet Mass rasfer, Joh Wiley & Sos, R. Haase, hermodyamics of Irreversible Processes, Addiso-Wesley, Lodo,

2 Outlie Etropy, Etropy trasport Etropy productio: forces & fluxes Species diffusive fluxes & the Geeralized Maxwell-Stefa Equatios Heat flux hermodyamic oidealities & the hermodyamic Factor Example: the ultracetrifuge Fick s law (the full versio) Review 2

3 A Perspective Referece velocities Allows us to separate a species flux ito covective ad diffusive compoets. Goverig equatios Describe coservatio of mass, mometum, eergy at the cotiuum scale. GMS equatios Provide a geeral relatioship betwee species diffusio fluxes ad diffusio drivig force(s). So far, we ve assumed: Ideal mixtures (ielastic collisios) small pressure gradiets Goal: obtai a more geeral form of the GMS equatios that represets more physics Body forces actig differetly o differet species (e.g. electromagetic fields) Noideal mixtures Large pressure gradiets (cetrifugal separatios) 3

4 Etropy Etropy differetial: ds =de + pdv otal (substatial/material) derivative: D Dt t + v µ i dω i µ i = µ i /M i e v Chemical potetial per uit mass Iteral eergy Specific volume ρ Ds Dt = ρde Dt + pρdv Dt ρ =1/v ρ Ds Dt = ρde Dt p ρ Dρ Dt µ i ρ Dω i Dt µ i ρ Dω i Dt ρ De Dt = q τ : v p v + f i j i Dρ Dt = ρ v ρ Dω i Dt = j i + σ i 4

5 Etropy rasport ρ Ds Dt = q τ : v p v + f i j i + p ρ ρ v = q τ : v + f i j i + µ i j i µ i σ i, µ i ( j i + σ i ), chai rule... (αβ) =α β + β α ρ Ds Dt = 1 1 µi q µ i j i + q j i 1 τ : v + 1 f i j i 1 µ i σ i rasport of s Productio of s 5

6 ρ Ds Dt = 1 q µ i j i rasport of s 1 µi + q j i 1 τ : v + 1 f i j i 1 µ i σ i Productio of s Now let s write this i the form: j s = 1 q µ i j i diffusive trasport of etropy 1 µi σ s = q j i 1 τ : v + 1 f i j i 1 µ i σ i, = q l µi j i 1 f i 1 τ : v 1 µ i σ i σ s = q l ρ Ds Dt = j s + σ s µi j i,p µ i + V i p f i τ : v M i Λ i Look at this term (etropy productio due to species diffusio) = µ i + 1 µ i p p + 1,p µ i, = 1 1 µ i M i p p +,p µ i, = 1 Vi p +,p µ i M i µ i σ i productio of etropy 6

7 Part of the Etropy Source erm j i,p µ i + V i p f i M i Λ i j i Λ i = cr d i = c i,p µ i +(φ i ω i ) p ω i ρ = j i Λ i 1 ρ p + ω k f k ρω i (u i v),p µ i + f i ω k f k k=1 k=1 Vi 1 p f i + M i ρ Why ca we add this arbitrary term? What does this term represet? ω k f k, = (u i v) c i,p µ i +(φ i ω i ) p ρω i f i ω k f k k=1, cr d i = cr d i (u i v), = cr 1 ρω i d i j i k=1 ω i M i = x i M φ i = c i Vi µ i = µ i M i From physical reasoig (recall d i represets force per uit volume drivig diffusio) or the Gibbs- Duhem equatio, j i = ρω i (u i v) V i Partial molar volume. d i =0 7

8 he Etropy Source erm - Summary ρ Ds Dt = j s + σ s j s = 1 q µ i j i σ s = q l = q l 1 From the previous slide: j i Λ i = cr d i j i cr d i = c i,p µ i +(φ i ω i ) p ω i ρ j i,p µ i + V i p f i τ : v M i Λ i cr d i j i τ : v µ i σ i f i ω k f k k=1 µ i σ i Iterpretatio of each term??? 8

9 σs Forces Fluxes σ s = q l cr d i j i τ : v µ i σ i Fudametal priciple of irreversible thermodyamics: σ s = α J α F α Flux, J α q j i τ Force, F α l cr v d i Fluxes are fuctios of: hermodyamic state variables:, p, ωi. Forces of same tesorial order (Curie s postulate) What does this mea? More soo J α = J α (F 1,F 2,...,F β ;, p, ω i ) J α = Jα F β + O (F β F γ ) F β β β L αβ F β L αβ J α F β L αβ = L βα Lαβ - Osager (pheomeological) coefficiets 9

10 Species Diffusive Fluxes esorial order of 1 ay vector force may cotribute. Idex form: Flux: J α q j i τ Force: F α l cr d i v -1 dimesioal matrix form From irreversible thermo: cr j i = L ij d j L iq l ρ j Fick s Law: j i = ρ D ij d j Di l Dij - Fickia diffusivity D i - hermal Diffusivity (j) = ρ [L](d)+ l (β q ) (j) = ρ[d](d) (D ) l Geeralized Maxwell-Stefa Equatios: x i x j ji d i = j j l ρð ij ω i ω j j=i α ij = 1 Ð ij x i x j αij j=i D i D i ρ j ρ(d) = [B o ](j) l [Υ](D ) 10

11 Costitutive Law: Heat Flux esorial order of 1 ay vector force may cotribute. q = L qq l L qi cr Flux: J α q j i τ Force: F α l cr d i v d i Choose L qq =λ to obtai Fourier s Law Dufuor effect - mass drivig force ca cause heat flux! Usually eglected. q = λ + Fourier h i j i Species + cr D x i x j ji j j ρ i Ð ij ρ j j=i Dufour here we have substituted the RHS of the GMS equatios for di. Note: the Dufour effect is usually eglected. he Species term is typically icluded here, eve though it does ot come from irreversible thermodyamics. Occasioally radiative terms are also icluded here... 11

12 Observatios o the GMS Equatios d i = cr d i = c i,p µ i +(φ i ω i ) p ω i ρ What have we gaied? hermal diffusio (Soret/Dufuor) & its origis. ypically eglected. Full diffusio drivig force Chemical potetial gradiet (rather tha mole fractio). More later. Pressure drivig force. Whe will φ i ω i? More later. Body force term. Does gravity eter here? x i J j x j J i cd ij l x i x j α ij f i k=1 ω k f k Osager coefficiets themselves ot too importat from a practical poit of view. Still do t kow how to get the biary diffusivities. 12

13 &K 2.2 he hermodyamic Factor, Γ µ i = µ i (, p, x j ) 1 µ i,p µ i = x j x P j,p, µ i (,p)=µ i + R l γ i x i Γ ij δ ij + x i l γ i x j d i = 1 d i = x i R,pµ i + 1 c t R (φ i ω i ) p,p,σ x i R,pµ i = x i R = x i R 1 = x i = = 1 1 Γ ij x j + 1 c t R (φ i ω i ) p 1 1 µ i x j x j,,p,σ R l γ ix i x j l x i x j + l γ i x j δ ij + x i l γ i x j Γ ij x j c t R c t R f i f i x j,,p,σ,p,σ ω k f k k=1,p,σ ω k f k k=1 x j, γ - Activity coefficiet May models available (see &K Appedix D) x j, Note: for ideal gas, p = c t R 13

14 &K Example: he Ultracetrifuge Used for separatig mixtures based o compoets molecular weight. Cosider a closed system... depleted i dese species Ω f = f i = Ω 2 r For a closed cetrifuge (o flow) with a kow iitial charge, what is the equilibrium species profile? 14

15 Species equatios: t = i + s i steady, 1D, o reactio i r =0 GMS Equatios: i = v r + j i,r =0 j i,r = J i,r =0 he geeralized diffusio drivig force: 1 d i = Γ ij x j + 1 c t R (φ i ω i ) p ω iρ c t R 1 d i = 0= 1 x i J j x j J i cd ij =0 dx j Γ ij dr = 1 c t R (ω i φ i ) dp dr Γ ij dx j dr + 1 c t R (φ i ω i ) dp dr ω iρ c t R f i Ω 2 r ω k f k k=1 ω k Ω 2 r k=1 For a ideal gas mixture, φi = xi, ad Γij = δij. dx i dr = 1 c t R (ω i x i ) dp dr We do t kow dp/dr or xi0 (compositio at r = 0). 15

16 Species mole balace: dx i dr = 1 c t R (ω i x i ) dp dr rl 0 cx i 2πr dr = rl 0 c x i 2πr dr * idicates the iitial coditio (pure stream). For species i, rl 0 px i r dr = p x i r 2 L 2 Must kow p(r) ad xi(r) to itegrate this. Species mole balace costrais the species profile solutio (dictates the species boudary coditio) Mometum: ρv t at steady state (o flow): = (ρvv) τ p + ρ dp dr = ρ dp dr = ρω2 r = pm R Ω2 r ω i f r,i = ρω 2 r ω i f i We do t kow p0 (pressure at r = 0). he mometum equatio gives the pressure profile, but is coupled to the species equatios through M. 16

17 otal mole balace (at equilibrium): rl 0 rl 0 V c dv = V cr dr = c r2 L 2 pr dr = p r2 L 2 c dv * idicates the iitial coditio (pure stream). dv = L2πrdr c = p R Substitute p(r) ad solve this for p0... otal mole balace costrais the pressure solutio (dictates the pressure boudary coditio) Solve these equatios: With these costraits: dx i dr = M R (ω i x i )Ω 2 r rl 0 px i r dr = p x i r 2 L 2 dp dr = ρω2 r = pm R Ω2 r rl 0 pr dr = p r2 L 2 Note: M couples all of the equatios together ad makes them oliear. Optio A: 1. Guess xi0, p0. 2. Numerically solve the ODEs for xi, p. 3. Are the costraits met? If ot, retur to step 1. Optio B: ry to simplify the problem by makig approximatios. Note: for tips o solvig ODEs umerically i Matlab, see my wiki page. 17

18 Example: separatio of Air ito N2, O2. Cetrifuge diameter: 20 cm Approximatio Level 1 Approximate M as costat, (MO2+MN2)/2, for the pressure equatio oly. his decouples the pressure solutio from the species ad gives a easy aalytic solutio for pressure profile. Solve species equatios umerically, give the aalytic pressure profile. Air iitially at SP Approximatio Level 2 Approximate M as costat, (MO2+MN2)/2, for the species ad pressure equatios. Obtai a fully aalytic solutio for both species ad pressure. p (atm) 1e1 1 1e 2 1e RPM 50,000 RPM 100,000 RPM 150,000 RPM fully umeric costat M r (m) O 2 Mole Fractio ,000 RPM 50,000 RPM 100,000 RPM 500,000 RPM umeric approximate 1 approximate r (m) 18

19 d i = d i = = 1 Fick s Law (revisited) x i x j ρd ij ji ω i j j ω j x j J i x i J j cd ij Igorig thermal diffusio, Notes: [D]=[B] -1 [Γ] l Γ ij x j + 1 c t R (φ i ω i ) p (J) = c[b] 1 [Γ]( x) 1 l c t R For ideal mixtures: [Γ]=[I] x i x j αij I the biary case: D11=Γ11Ð12 x i x j α ij J = c[b] 1 (d) l (D ) f i ω k f k k=1 p R [B] 1 ((φ) (ω)) 2 his is the same [B] matrix as before (&K eq ) ρ R [B] 1 [ω] ((f) [ω](f + f )) 3 How do we iterpret each term? Whe is each term importat? 19

20 Review: Where we are, where we re goig Accomplishmets Defied referece velocities ad diffusio fluxes Goverig equatios for multicompoet, reactig flow. mass-averaged velocity Established a rigorous way to compute the diffusive fluxes from first priciples. Ca hadle diffusio i systems of arbitrary complexity, icludig: oideal mixtures, EM fields, large pressure & temperature gradiets, multiple species, chemical reactio, etc. Simplificatios for ideal mixtures, egligible pressure gradiets, etc. Solutios for simple problems. Still Missig: Models for biary diffusivities. Give a model, we are good to go! Roadmap: Models for biary diffusivities. (&K Chapter 4) - we wo t cover this... Simplified models for multicompoet diffusio Iterphase mass trasfer (surface discotiuities) urbulece - models for diffusio i turbulet flow. Combied heat, mass, mometum trasfer. 20