CBE 548: Advanced Transport Phenomena II Spring, 2010 Final Exam

Transcription

1 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

2 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)

3 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.

4 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

5 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 (.) (.) (.) (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 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?

7 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

8 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

9 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

10 (.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.

11 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?

12 (.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

13 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

14 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