GMS Equations From Irreversible Thermodynamics

Transcription

1 GMS Equatos From Irreversble hermodyamcs ChE 6603 Refereces E. N. Lghtfoot, rasport Pheomea ad Lvg Systems, McGraw-Hll, New York 978. R. B. Brd, W. E. Stewart ad E. N. Lghtfoot, rasport Pheomea 2 d ed., Chapter 24 McGraw-Hll, New York D. Jou, J. Casas-Vazquez, Exteded Irreversble hermodyamcs, Sprger-Verlag, Berl 996. R. aylor, R. Krsha Multcompoet Mass rasfer, Joh Wley & Sos, 993. R. Haase, hermodyamcs of Irreversble Processes, Addso-Wesley, Lodo, 969.

2 Outle Etropy, Etropy trasport Etropy producto: forces & fluxes Speces dffusve fluxes & the Geeralzed Maxwell-Stefa Equatos Heat flux hermodyamc odealtes & the hermodyamc Factor Example: the ultracetrfuge Fck s law (the full verso) Revew 2

3 A Perspectve Referece veloctes Allows us to separate a speces flux to covectve ad dffusve compoets. Goverg equatos Descrbe coservato of mass, mometum, eergy at the cotuum scale. GMS equatos Provde a geeral relatoshp betwee speces dffuso fluxes ad dffuso drvg force(s). So far, we ve assumed: Ideal mxtures (elastc collsos) small pressure gradets Goal: obta a more geeral form of the GMS equatos that represets more physcs Body forces actg dfferetly o dfferet speces (e.g. electromagetc felds) Nodeal mxtures Large pressure gradets (cetrfugal separatos) 3

4 Etropy Etropy dfferetal: ds =de + pdv otal (substatal/materal) dervatve: D Dt µ d! t + v µ = µ /M e v Chemcal potetal per ut mass Iteral eergy Specfc volume Ds Dt = De Dt + p Dv Dt =/v Ds Dt = De Dt p D Dt µ D! Dt µ D! Dt De Dt = r q : rv pr v + X f j D Dt = v D Dt = j + 4

5 Etropy rasport Ds Dt = q : v p v + = q : v + f j + f j + p v µ j µ, µ ( j + ), cha rule... ( ) = + Ds Dt = q µ j rasport of s + q j µ : v + f j µ Producto of s 5

6 Ds Dt = q µ j rasport of s + q j µ : v + f j µ Producto of s Ds Dt = Now let s wrte ths the form: j s + s j s = q µ j dffusve trasport of etropy s = q j µ : v + q = l j µ f s = q l j,p µ + V p f M µ Look at ths term (etropy producto due to speces dffuso) = = = µ M µ p f j µ, : v µ + p + V M p +,p µ : v µ p + p,p µ,,p µ, µ producto of etropy Note that we have t completed the cha rule here. We wll apply t to speces later... 6

7 Part of the Etropy Source erm j,p µ + V p M j = 0 f cr d = c,p µ +( )p f k f k (u v) 2 = " j,p µ + k= p + k f k k= V p f + M Why ca we add ths arbtrary term? What does ths term represet? #! k f k,! = (u v) 6 4 c,p µ +( )p f k f k 7C k= 5A, {z } cr d = cr d (u v), = cr d j k= 3 = x M M = c V µ = µ M From physcal reasog (recall d represets force per ut volume drvg dffuso) or the Gbbs- Duhem equato, j = (u v) V Partal molar volume. d =0 7

8 he Etropy Source erm - Summary Ds Dt = j From the prevous slde: s + s d j j = cr j s = q µ j cr d = c,p µ +( )p f k f k s = q l = q l j,p µ + V p f M cr d j 2 : v 3 µ 4 : v k= µ Iterpretato of each term??? 8

9 σs Forces Fluxes s = q l cr d j : v µ Fudametal prcple of rreversble thermodyamcs: s = J F Flux, J q j Force, F l cr v d Fluxes are fuctos of: hermodyamc state varables:, p, ω. Forces of same tesoral order (Cure s postulate) What does ths mea? More soo J = J (F,F 2,...,F ;, p, ) J = J F + O (F F ) F L F L J F L = L Lαβ - Osager (pheomeologcal) coeffcets 9

10 Speces Dffusve Fluxes esoral order of ay vector force may cotrbute. Idex form: Flux: J q j Force: F l cr d v - dmesoal matrx form From rreversble thermo: cr j = L j d j L qr l j Fck s Law: j = Geeralzed Maxwell-Stefa Equatos: x x j j j j d = rl Ð j j j6= D j d j D r l Dj - Fcka dffusvty D - hermal Dffusvty j = Ð j x x j j6= D D j j (j) = [L](d)+ l ( q ) (j) = [D ](d) D r l (d) = [B o ](j) rl [ ](D ) 0

11 Costtutve Law: Heat Flux esoral order of ay vector force may cotrbute. q = L qq l L q cr Flux: J q j Force: F l cr d v d Choose L qq =λ to obta Fourer s Law Dufuor effect - mass drvg force ca cause heat flux! Usually eglected. q = r + {z } Fourer h j {z } Speces + cr D x x j j j j Ð j j j6= {z } Dufour here we have substtuted the RHS of the GMS equatos for d. Note: the Dufour effect s usually eglected. he Speces term s typcally cluded here, eve though t does ot come from rreversble thermodyamcs. Occasoally radatve terms are also cluded here...

12 Observatos o the GMS Equatos d = What have we gaed? hermal dffuso (Soret/Dufuor) & ts orgs. ypcally eglected. Full dffuso drvg force Chemcal potetal gradet (rather tha mole fracto). More later. Pressure drvg force. cr d = c,p µ +( )p f Whe wll φ ω? More later. Body force term. Does gravty eter here? x J j cd j x j J l x x j j k= k f k Osager coeffcets themselves ot too mportat from a practcal pot of vew. Stll do t kow how to get the bary dffusvtes. 2

13 &K 2.2 he hermodyamc Factor, Γ µ = µ (, p, x j ),pµ = j µ (,p)=µ + R l x j + x l x j d = µ x j,p, P x j d = x R,pµ + c t R (,p, jx j + c t R ( x R,p µ = x R = x R = x = = )p )p µ x j,p, x j, R l x x j l x x j,p, + l x j j + x l x j j x j c t R c t R f f k=,p, k=,p, k f k k f k x j, x j, γ - Actvty coeffcet May models avalable (see &K Appedx D) x j, Note: for deal gas, p = c t R 3

14 &K Example: he Ultracetrfuge Used for separatg mxtures based o compoets molecular weght. Cosder a closed system... depleted dese speces f = f = 2 r For a closed cetrfuge (o flow) wth a kow tal charge, what s the equlbrum speces profle? 4

15 Speces equatos: t = steady, D, + s o reacto r =0 = v r + j,r =0 j,r = J,r =0 GMS Equatos: d = x J j x j J =0 cd j he geeralzed dffuso drvg force: d = jrx j + c t R ( 0= j dx j dr + c t R ( )rp ) dp dr c t R c t R f 2 r X k= k f k!! k 2 r k= j dx j dr = c t R ( ) dp dr For a deal gas mxture, φ = x, ad Γj = δj. dx dr = c t R ( x ) dp dr We do t kow dp/dr or x0 (composto at r = 0). 5

16 Speces mole balace: dx dr = c t R ( x ) dp dr Z rl 0 cx 2 r dr = Z rl 0 c x 2 r dr * dcates the tal codto (pure stream). For speces, Z rl 0 px r dr = p x r 2 L 2 Must kow p(r) ad x(r) to tegrate ths. Speces mole balace costras the speces profle soluto (dctates the speces boudary codto) Mometum: v t at steady state (o flow): dp dr = = ( vv) p + dp dr = 2 r = pm R X 2 r f r, = 2 r f We do t kow p0 (pressure at r = 0). he mometum equato gves the pressure profle, but s coupled to the speces equatos through M. 6

17 otal mole balace (at equlbrum): Z Z rl 0 Z rl 0 V c dv = Z V cr dr = c r2 L 2 pr dr = p r2 L 2 c dv * dcates the tal codto (pure stream). dv = L2 rdr c = p R Substtute p(r) ad solve ths for p0... otal mole balace costras the pressure soluto (dctates the pressure boudary codto) Solve these equatos: Wth these costrats: dx dr = M R ( Z rl 0 x ) 2 r px r dr = p x r 2 L 2 dp dr = Z rl 0 2 r = pm R pr dr = p r2 L 2 2 r Note: M couples all of the equatos together ad makes them olear. Opto A:. Guess x0, p0. 2. Numercally solve the ODEs for x, p. 3. Are the costrats met? If ot, retur to step. Opto B: ry to smplfy the problem by makg approxmatos. Note: for tps o solvg ODEs umercally Matlab, see my wk page. 7

18 Example: separato of Ar to N2, O2. Cetrfuge dameter: 20 cm Approxmato Level Approxmate M as costat, (MO2+MN2)/2, for the pressure equato oly. hs decouples the pressure soluto from the speces ad gves a easy aalytc soluto for pressure profle. Solve speces equatos umercally, gve the aalytc pressure profle. Ar tally at SP Approxmato Level 2 Approxmate M as costat, (MO2+MN2)/2, for the speces ad pressure equatos. Obta a fully aalytc soluto for both speces ad pressure. p (atm) e e 2 e 4 50,000 RPM 00,000 RPM 50,000 RPM 000 RPM fully umerc costat M r (m) O 2 Mole Fracto ,000 RPM 50,000 RPM 00,000 RPM 500,000 RPM umerc approxmate approxmate r (m) 8

19 Fck s Law (revsted) d = d = = x x j D j x j J cd j j x J j jx j + c t R ( j j j l )p l c t R x x j j x x j j J = c[b] f k= k f k (d) l (D ) hs s the same [B] matrx as before (&K eq ) Igorg thermal dffuso, (J) = c[b] [ ](x) p R [B] (() ( )) 2 R [B] [ ] ((f) [ ](f + f )) 3 Notes: [D]=[B] - [Γ] For deal mxtures: [Γ]=[I] I the bary case: D=ΓÐ2 How do we terpret each term? Whe s each term mportat? 9

20 Revew: Where we are, where we re gog Accomplshmets Defed referece veloctes ad dffuso fluxes Goverg equatos for multcompoet, reactg flow. mass-averaged velocty Establshed a rgorous way to compute the dffusve fluxes from frst prcples. Ca hadle dffuso systems of arbtrary complexty, cludg: odeal mxtures, EM felds, large pressure & temperature gradets, multple speces, chemcal reacto, etc. Smplfcatos for deal mxtures, eglgble pressure gradets, etc. Solutos for smple problems. Stll Mssg: Models for bary dffusvtes. Gve a model, we are good to go! Roadmap: Models for bary dffusvtes. (&K Chapter 4) - we wo t cover ths... Smplfed models for multcompoet dffuso Iterphase mass trasfer (surface dscotutes) urbulece - models for dffuso turbulet flow. Combed heat, mass, mometum trasfer. 20