GMS Equations From Irreversible Thermodynamics
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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
GMS Equations From Irreversible Thermodynamics
GMS Equatios From Irreversible hermodyamics ChE 6603 Refereces E. N. Lightfoot, rasport Pheomea ad Livig Systems, McGraw-Hill, New York 1978. R. B. Bird, W. E. Stewart ad E. N. Lightfoot, rasport Pheomea
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