CBE 548: dnced rnspr Phenen II Spring Finl E Prble. rbirry Frulin f he escripin f Mss rnsfer he ulicpnen Fick diffusiiies re defined ih he flling equins nd cnsrins (BSL p. 767) cr (.) (.) N c (.) be used in he generlied Fick equins cnsiuie equins fr he ss diffusie flu f cpnen relie he cener f ss elciy f he fr (BSL eq (4.-) p. 767) d N c (.4) here he diffusinl driing frces re gien by (BSL eq (4.-8) p. 766) crd c p g g R (.5) Fr binry isherl diffusin e frequenly rie Fick s l s (.6) here nd. Rigrusly derie he relinship beeen nd. Se ll ssupins de. he relinship shuld be epressed eclusiely in ers f le frcins ss frcins nd. Se f he flling relins y be useful yu. G G (Gibbs-uhe equin binry isherl isbric) p p G G R (relin beeen pril lr Gibbs free energy nd ciiy) (fr binry syses nly) N c
Sluin: Fr isherl binry diffusin in he bsence f eernl frces nd pressure diffusin he generlied Fick equins eq. (.4) bece d d (.7) Equin (.) beces fr binry syse (.8) hich cn be rerrnged s (.9) Subsiuin f equin (.9) in equin (.7) yields d d d d (.) he diffusinl driing frces fr his syse crd c R (.) Subsiuin f equin (.) in equin (.) yields (.) We cn eque eq. (.) nd (.6) yield Siplificin yields (.) (.4) Furher siplificin yields (.5)
Reeber h fr binry syse (.6) S h equin (.5) beces (.7) Fr equilibriu herdynics under cnsn eperure nd pressure cndiins he Gibbs uhe equin ses p p G G (.8) he pril lr Gibbs free energy cn be reled he ciiy i R G G (.9) G R G (.) here G is he lr enhlpy f cpnen in he pure se. he deriie f he pril lr Gibbs free energy ih respec cpsiin is p p p R G R G (.) Subsiuin f (.) in (.8) yields p p (.) Subsiuin equin (.) in (.7) is (.) Yu cn see h equin (.) is syeric ih respec inerchnge f nd subscrips s i us be.
Prble. Cheicl Penil Grdien rien iffusin Cnsider he sedy se behir f hree cpnen fluid lced in n isherl nd isbric syse beeen bundries. he herdynic se f he bundry = is defined by he le frcin f =.5 le frcin f =.84 eperure = K nd pressure p = br. he herdynic se f he bundry = is defined by he le frcin f =. le frcin f =.89 eperure = K nd pressure p = br. he cheicl penil f cpnen i in ulicpnen n der Wls gs is gien by N c bi Rb i i R i () b i i i here R is he gs cnsn is he lr lue i is he herl de Brglie elengh. Fr his eple e ill se ll f he n der Wl b preers (ll b i nd b i ) er. he lues f re s flls: = = = = = = = = = Jules- /le. Cnsider he lr lue be cnsn =.5 - /le. sks. () Using finie difference frul deerine he erge le frcin grdiens fr ech cpnen bsed n he bundry lues. (b) Bsed n he sign f he le frcin grdiens in hich direcin uld yu epec he diffusie flu f ech species be? (c) Using finie difference frul deerine he erge cheicl penil grdiens fr ech cpnen bsed n he bundry lues. (d) Bsed n he sign f he cheicl penil grdiens in hich direcin uld yu epec he diffusie flu f ech species be? (e) Bsed n yur cnclusins in prs (b) nd (d) hich flues ill ne cully bsere hse gien in pr (b) r pr (d)? Why? (f) Wh is he cn er gien he rnspr phenen ehibied by ne f he cpnens? (g) Ne cheicl engineering uni perin in hich his rnspr phenen is frequenly eplied. 4
Sluin: Cnsider he sedy se behir f hree cpnen fluid lced in n isherl nd isbric syse beeen bundries. he herdynic se f he bundry = is defined by he le frcin f =.5 le frcin f =.84 eperure = K nd pressure p = br. he herdynic se f he bundry = is defined by he le frcin f =. le frcin f =.89 eperure = K nd pressure p = br. he cheicl penil f cpnen i in ulicpnen n der Wls gs is gien by b i R i i i Rb i b i N c i () here R is he gs cnsn is he lr lue L i is he herl de Brglie elengh. Fr his eple e ill se ll f he n der Wl b preers (ll b i nd b i ) er. he lues f re s flls: = = = = = = = = = Jules- /le. Cnsider he lr lue be cnsn =.5 - /le. sks. () Using finie difference frul deerine he erge le frcin grdiens fr ech cpnen bsed n he bundry lues...5.5..84.8.89..88 (.) (.) (.) (b) Bsed n he sign f he le frcin grdiens in hich direcin uld yu epec he diffusie flu f ech species be? One uld epec h species diffuse fr high le frcin l le frcin. Cpnen uld e he bundry =. Cpnen uld e he bundry =. Cpnen uld e he bundry =. (c) Using finie difference frul deerine he erge cheicl penil grdiens fr ech cpnen bsed n he bundry lues. he cheicl penil epressins re 5
6 k B (.) k B (.) k B (.) he erge cheicl penil grdiens re R R R (4.) R (4.) R (4.) Nuericl eluin yields 4.6 kj/l/ (5.) -966 kj/l/ (5.) 96 kj/l/ (5.) (d) Bsed n he sign f he cheicl penil grdiens in hich direcin uld yu epec he diffusie flu f ech species be?
One uld epec h species diffuse fr high cheicl penil l cheicl penil. Cpnen uld e he bundry =. Cpnen uld e he bundry =. Cpnen uld e he bundry =. (e) Bsed n yur cnclusins in prs (b) nd (d) hich flues ill ne cully bsere hse gien in pr (b) r pr (d)? Why? One ill bsere he flues prediced in pr (d) becuse pr (d) is bsed n he herdynic driing frce fr diffusin. Flling he cheicl penil grdien ill led he syse ler free energy. Here becuse cpnen inercs re frbly ih cpnen hn i des ih cpnen he dnge f he energeic driing frce ueighs he disdnge f he enrpic driing frce sscied ih ging up cncenrin grdien. (f) Wh is he cn er gien he rnspr phenen ehibied by ne f he cpnens? Cpnen ne ill disply uphill diffusin here i diffuses up he cncenrin grdien. (g) Ne cheicl engineering uni perin in hich his rnspr phenen is frequenly eplied. In liquid-liquid ercin gd slen is used erc slue fr less gd slen. he gdness f slen is relly n indicr f he cheicl penil f he slue in h slen. hus ne cn erc slue higher cncenrin in he gd slen hn s riginlly presen in he less gd slen due he erll reducin in free energy. cde shing he clculins is gien bel. 7
clse ll; cler ll; fr lng e; preers R = 8.4; J/l/K = ; K p = 5; P =.; J-^/le^ = ; bi = ; bundry ne (=) =.5; =.84; = - - ; i = ** + **; = (R* + sqr( (R*)^ - 4*p*i) )/(*p) = (kb* - sqr( (kb*)^ - 4*p*i) )/(*p) bundry (=) =.; =.89; = - - ; i = ** + **; = (R* + sqr( (R*)^ - 4*p*i) )/(*p) = (kb* - sqr( (kb*)^ - 4*p*i) )/(*p) ssue lr lue is cnsn ( ke he prble esier fr he e) his nuber is gien. g = ( + )/ = g; = g; le frcin grdiens grd = - grd = - grd = - cheicl penil grdiens grdu = R**lg( (*)/(*) ) + **( / - / ) grdu = R**lg( (*)/(*) ) + **( / - / ) grdu = R**lg( (*)/(*) ) 8
Prble. ifferenil blnces () erie he cninuiy equin fr cnicl pipe here he crss secinl re is funcin f il psiin nd here is nly spil riin in prperies in he il diensin. (b) erie he ss blnce fr cpnen fr cnicl pipe here he crss-secinl re is funcin f il psiin nd here is nly spil riin in prperies in he il diensin. (c) Wh is he equin fr he ss frcin f fr n isherl binry syse in he b bsence f cnecin nd he presence f firs rder recin B here nd Ficks l is gien s nd he recin re is gien s r k. ssue he densiy diffusiiy nd recin re cnsn re ll cnsn. Pu his equin in he siples fr pssible. (d) Wh des he sedy se prfile f he ss frcin f lk like in pipe f cnsn crss-secinl re nd he bsence f recin? (e) Cpring he resuls fr he cnsn crss-secinl re pipe nd he rying crsssecinl re pipe fr he cse ihu recin hich ill yield higher cpsiin f he ule fr he se inle cndiins? Sluin () erie he cninuiy equin fr cnicl pipe here he crss-secinl re is funcin f il psiin nd here is nly spil riin in prperies in he il diensin. ccuulin = in u + generin cc (.) here he differenil lue is. he in nd u ers re in (.) u (.) Subsiue in he blnce equin (.4) iide by he lue (.5) ke he lii s he differenil lengh ges er. 9
(.6) Use he prduc rule spli ers. (.7) he secnd er disppers if he crss-secinl re is cnsn. (b) erie he ss blnce fr cpnen fr cnicl pipe here he crss-secinl re is funcin f il psiin nd here is nly spil riin in prperies in he il diensin. ccuulin = in u + generin cc (.) here he differenil lue is. he in nd u ers fr cnecin nd diffusin re in (.) u (.) he generin er is gen r (.4) Subsiue in he blnce equin r (.5) iide by he lue r (.6) ke he lii s he differenil lengh ges er.
r (.7) Use he prduc rule spli ers. r (.8) Yu cn siplify his by subrcing he cninuiy equin fr his equin. r (.9) (c) Wh is he equin fr he ss frcin f fr n isherl binry syse in he bsence f cnecin nd he presence f firs rder recin B here b nd Ficks l is gien s nd he recin re is gien s k r. ssue he densiy diffusiiy nd recin re cnsn re ll cnsn. Pu his equin in he siples fr pssible. r (.) (.) k r (.) Subsiuin f he diffusie flu recin re nd crss-secinl re in equin (.) yields k b b (.) his cn be furher siplified s k b b (.4) (d) Wh des he sedy se prfile f he ss frcin f lk like in pipe f cnsn crss-secinl re nd he bsence f recin?
(.5) his sluin f his OE is srigh line. (.6) (e) Wh des he sedy se prfile f he ss frcin f lk like in he pipe ih linerly rying crss-secinl re nd he bsence f recin? b (.7) b Le X X b X (.8) b X X b b (.9) Inegre. X b (.) X b X b X (.) b Subsiue in fr X. b b (.) b (.) b Inegre
b b (.4) b b (e) Cpring he resuls fr he cnsn crss-secinl re pipe nd he rying crsssecinl re pipe fr he cse ihu recin hich ill yield higher cpsiin f he ule fr he se inle cndiins? b b he quesin is heher he quniy hich is he fcr in frn f he b b iniil slpe in equin (.4) is greer hn r less hn. If b is psiie nd he pipe is dierging hen he cpsiin f he ule ill be ler hn he cnsn crss-secinl re pipe. If b is negie nd he pipe is cnerging hen he cpsiin f he ule ill be higher hn he cnsn crss-secinl re pipe. ypicl Pl (n required) shn bel
he flling equins re prided fr he e. he cninuiy equin is (.) here is he ss densiy is he cener-f-ss elciy nd is ie. In BSL his is equin (.-4) n pge 77 []. he ss blnce n cpnen is N R r (.) i i here is he ss frcin f cpnen is he diffusie ss flu f cpnen relie he cener-f-ss elciy N R is he nuber f independen cheicl recins in he syse nd r i is he re f prducin f cpnen in recin i in unis f ss/lue/ie. In BSL his is equin (9.-4) n pge 584. he enu blnce is p ˆ (.) here p is he pressure is he er sress ensr nd ˆ is he specific eernl field ipsed by fr eple griy. his equin is he difference f equin (.-9) n pge 8 f BSL nd he cninuiy equin equin(). he energy blnce is Uˆ ˆ Uˆ ˆ q p (.4) here Uˆ is he specific (per ss) inernl energy ˆ is he specific penil energy due n eernl field nd q is he he flu due cnducin. his is equin (.-9) n pge 6 in BSL. 4