Evaluation the reflection coefficient of polymeric membrane in concentration polarization conditions

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1 PRACE ORYGINALNE Pom. Md. 0, 4,, 9 ISSN Copyrgt by Wrocaw Mdca Unvrsty Korna Bato, Izaba Śęza-Procaza, Andrzj Śęza Evauaton t rfcton coffcnt of poymrc mmbran n concntraton poarzaton condtons Ocna współczynna odbca mmbrany pomrowj w warunac poaryzacj stężnowj Dpartmnt of Informatcs for Economcs, Unvrsty of Economcs n Katowc Insttut of Martng, Częstocowa Unvrsty of Tcnoogy n Częstocowa Dpartmnt of Pubc Hat, Częstocowa Unvrsty of Tcnoogy n Częstocowa Summary Introducton. T rfcton coffcnt of t mmbran (σ s on of t basc paramtrs of t poymr mmbran transport. Cassca mtods usd to dtrmn ts paramtr rqur ntnsv mxng of two soutons sparatd by a mmbran to mnat t ffcts of concntraton poarzaton. In t ra condtons, spcay n boogca systms, ts rqurmnt s cangng. Tus, concntraton boundary ayrs, wc ar t ssnc of t pnomnon of concntraton poarzaton, form on bot sds of t mmbran. Purpos. T man am of ts papr s to dtrmn wtr t vau of rfcton coffcnt n a concntraton poarzaton condtons dpnd on t concntraton of soutons and ydrodynamc stat of concntraton boundary ayrs. Matras and mtods. In ts papr, w usd t modayss mmbran of cuos actat (Npropan and aquous gucos soutons as t rsarc matras. Formasm of nonqubrum trmodynamcs and Kdm-Katcasy quatons wr our rsarc toos. Rsuts. Drvd matmatca quatons dscrb t rato of rfcton coffcnts n a concntraton poarzaton condtons (σ S and n trms of omognty of t soutons (σ. Ts rato was cacuatd for t confguraton n wc t mmbran was orntd orzontay. It was sown tat ac of t curvs as a bffurcaton pont. Abov ts pont, t vau of t rfcton coffcnts dpndd on t concntraton of t souton, t confguraton of t mmbran systm and t ydrodynamc concntraton boundary ayrs. Bow ts pont, t systm dd not dstngus t gravtatona drctons. Concuson. T vau of rfcton coffcnt of t modayss mmbran n a concntraton poarzaton condton (σ S s dpndnt on bot t soutons concntraton and t ydrodynamc stat of t concntraton boundary ayrs. T vau of ts coffcnt s t argst n t stat of forcd convcton, owr n natura convcton stat and t owst n dffusv stat. Obtand quatons may b rvant to t ntrprtaton of mmbran transport procsss n condtons wr t assumpton of omognty of t souton s dffcut to mpmnt (Pom. Md. 0, 4,, 9. Ky words: osmoss, rfcton coffcnt, concntraton boundary ayrs, Kdm Katcasy quatons. Strszczn Wprowadzn. Współczynn odbca mmbrany (σ naży do grupy podstawowyc paramtrów transportowyc mmbrany pomrowj. Kasyczna mtodya orśana tgo paramtru wymaga ntnsywngo mszana roztworów rozdzanyc przz mmbranę, w cu mnacj ftów poaryzacj stężnowj. W warunac rzczywstyc, a szczgón w uładac boogcznyc, wymóg tn jst trudny do razacj. W zwązu z tym po obydwu stronac mmbrany tworzą sę stężnow warstwy granczn, stanowąc stotę zjawsa poaryzacj stężnowj. C. Cm pracy jst sprawdzn, czy wartość współczynna odbca wyznaczana w warunac poaryzacj stężnowj, zaży od stężna roztworów stanu ydrodynamczngo stężnowyc warstw grancznyc. Matrał mtody. Matrałm badawczym była mmbrana modazacyjna z octanu cuozy (Npropan wodn roztwory guozy. Narzędzm badawczym jst formazm trmodynam nrównowagowj oraz równana Kdm- Katcasy go.

2 Korna Bato t. a Wyn. Wyprowadzono równana matmatyczn opsując stosun współczynnów odbca w warunac poaryzacj stężnowj (σ S w warunac jdnorodnośc roztworów (σ. Wyonano obczna tgo stosunu da onfguracj, w tóryc mmbrana była zorntowana oryzontan wyazano, ż ażda z rzywyc posada punt bfuracyjny. Powyżj tgo puntu, wartość stosunu współczynnów odbca zaży zarówno od stężna roztworów, onfguracj uładu mmbranowgo oraz stanu ydrodynamczngo stężnowyc warstw grancznyc. Ponżj tgo puntu, uład n rozróżna runów grawtacyjnyc. Wnos. Wartość współczynna odbca modazacyjnj mmbrany pomrowj w warunac poaryzacj stężnowj (σ S, jst zażna od stężna roztworów od stanu ydrodynamczngo stężnowyc warstw grancznyc. Wartość tgo współczynna jst najwęsza w stan onwcj wymuszonj, mnjsza w stan onwcj swobodnj, a najmnjsza w stan dyfuzyjnym. Otrzyman równana mogą mć znaczn da ntrprtacj procsów transportu mmbranowgo w warunac, w tóryc założn o jdnorodnośc roztworów jst trudn do razacj (Pom. Md. 0, 4,, 9. Słowa uczow: osmoza, współczynn odbca, stężnow warstwy granczn, równana Kdm Katcasy go Introducton T rfcton (σ, ydrauc prmabty (L p and dffusv prmabty (ω coffcnts ar t trad of mmbran transport coffcnts rsutng from t A. Katcasy and O. Kdm trmodynamc formasm [ ]. Ts formasm s basd on t quatons dscrbng t voum (J v and sout (J s fuxs. For omognous soutons of nonctroyts, ts quatons can b wrttn as Jv = Lp ( ΔP σδπ ( J = ωδπ + C ( σ ( s J v wr L p ydrauc prmabty coffcnt, σ t coffcnt of rfcton, ΔP = P P ydrostatc prssur dffrnc, Δπ = RT(C C osmotc prssur dffrnc, RT product of gas constant and trmodynamc tmpratur, C and C soutons concntratons n t cambrs sparatd by a mmbran, ω sout prmabty coffcnt, C = (C C [n(c C ] 0.5(C + C t avrag concntraton of t souton n a mmbran. Homognty of soutons can b accompsd by tr vgorous strrng wt a mcanca strrrs pacd n t soutons on bot sds of t mmbran. Suc condtons can b nsurd ony n t macroscopc mmbran systms [4]. In boogca systms t omognty of t soutons s dffcut to acv [5 8]. Coffcnt σ n t condtons of soutons omognty can b dfnd by t foowng xprsson rsutng from Eq. ( ΔP σ = ( Δπ J v = 0 Vaus of ts coffcnt ar n t foowng rang 0 σ. If σ = 0, tn t mmbran s ndscrmnat. T fufmnt of condton σ = s rqurd for t sm-prmab mmbran []. Eqs. ( and ( can b appd for bot a mmbran consdrd as bac box and a porous mmbran. For porous mmbran t rfcton coffcnt t rfrrd to ndvdua por of mmbran and quas or 0 [9]. In cas of non-omognous soutons,.. wtout mcanca strrng, concntraton boundary ayr (, ar cratd on bot sds of t mmbran (M, and tratd as qud mmbrans [07]. T tcnss of ts ayrs n t stady-stat quas to δ and δ. Importanty, concntratons boundary ayrs (, ar componnts of t compx /M/ and trfor ty do not appar as sparat objcts. Craton of ts ayrs and tr tm-spac vouton s a manfstaton of concntraton poarzaton ffct. For ayrs and as w as for compx /M/, crtan transport proprts can b assgnd accordng to Kdm-Katcasy trmodynamc formasm. Ts mans tat t Kdm- Katcasy quatons can b usd for t anayss of transport n t concntraton poarzaton condtons. Smar to prvous papr [8] w consdr sngmmbran systm sown n Fg.. Ts fgur ustrats a mmbran systm wt concntraton boundary ayrs (, cratd on bot sds of t mmbran. T mmbran, wc s an ntgra part of ts systm, s ctronutra and sctv for dssovd substancs. Mmbran s mountd n t orzonta pan and sparats compartmnts ( and ( fd wt dutd and mcancay unstrrd soutons of t sam substancs wt concntratons of C and C (C > C at t nta momnt. Ony at t nta momnt (t = 0, ts soutons ar omognous trougout t soutons and at t mmbran ntrfac. Trfor, n stady-stat concntraton of soutons on t contacts /M and M/ cang accordng to t vaus of C and C (C > C > C, C > C > C. In stady-stat, t voum and sout fuxs troug t mmbran ar dnotd as t J vm and J sm, rspctvy. For t stuaton sown n Fg., t fuxs J vm (J vm = J vs < J v and J sm (J vm = J ss < J s can b dscrbd usng Eqs. ( and ( [,4]. Ts quatons can b wrttn as J vm = L ΔP L σ RT ( C C (4 p s p J = ω RT C C + J ( σ( C + C (5 sm m ( vm Transport proprts of ayrs and ar caractrzd by t coffcnts of: rfcton fufng t condton σ = σ = 0, t coffcnts of dffuson

3 Poymrc mmbran J ss P C J s C J sm J vm M C J s C P J vs Fg.. A sng-mmbran systm: M mmbran; and t concntraton boundary ayrs; P and P mcanca prssurs; C and C concntratons of soutons outsd t boundars; C and C t concntratons of soutons at boundars /M and M/ ; J vm t voum fuxs troug mmbran M; J vs t voum fux troug compx /M/ ; J s, J s and J sm t sout fuxs troug ayrs, and mmbran; J ss t sout fuxs troug a compx /M/ [9] Ryc.. Uład jdno-mmbranowy: M mmbrana; stężnow warstwy granczn; P and P cśnna mcanczn; C C stężna roztworów na zwnątrz warstw; C C stężna roztworów na grancac /M M/ ; J vm strumń objętoścowy przz mmbranę M; J vs strumń objętoścowy przz omps /M/ ; J s, J s J sm strumn substancj rozpuszczonj przz warstwy, mmbranę; J ss strumń substancj rozpuszczonj przz omps /M/ [9] D and D and coffcnts of sout prmabty ω and ω, rspctvy. Coffcnts ω and ω ar assocatd wt tcnsss δ and δ and dffuson coffcnts D and D, by xprsson ω = D (RTδ and ω = D (RTδ [0]. Sout fuxs troug t ayrs and ar ndcatd by J s and J s, rspctvy. It can b dscrbd by usng Eq. ( [,4] J = ω RT C C + J ( C + C (6 s ( v J = ω RT C C + J ( C + C (7 s ( v Voum and sout fuxs troug compx /M/, dnotd by J vs and J ss, rspctvy, can b rprsntd by usng t foowng xprssons [] J vs = L ΔP L σ RT( C C (8 p s p S J = ω RT C C + J ( σ ( C + C (9 ss S ( vs T rfcton (σ S and sout prmabty (ω S coffcnts dscrb t transport proprts of t compx /M/. In addton, coffcnts σ S and ω S, tat appar n Eqs. (8 and (9, can b dfnd by t foowng xprssons rsutng from Eqs. (8 and (9 [] ΔPS σ S = (0 Δπ Jvs = 0 J ω S = ( ss Δ π Jvs = 0 S In stady-stat, t foowng ratons J s = J sm = J s = J ss and J vm = J vs ar fufd. T coffcnts ω S, ω m, ω and ω for bnary soutons, n condton of dffuson (J vs = 0, ar ratd wt foowng xprsson ω S = ωω ω (ω ω + ωω + ωω []. Ts xprsson can aso b wrttn as ω S = ωd D [D D + RTω(D δ + D δ ] [,0]. T rato of coffcnts ω S and ω, dfns a dmnsonss dffuson coffcnt of concntraton poarzaton (ζ s [] D D ζs = D D + ωrt ( D δ + D δ ( T vaus of ts coffcnt fuf t raton: (ζ s mn ζ s. Ts mans tat t concntraton poarzaton s maxma wn ζ s (ζ s mn and mnma wn ζ s. By agbrac transformng of Eqs. (5 (7, n t stady stat, xprssons for t concntratons C and C can b drvd [5]. Consdrng ts xprssons n Eq. (5, w obtan J + λ J + λ J + λ = 0 ( vm vm vm wr λ = β L p [β ΔP S σrt(α χ ] λ = β 0 L p [β ΔP S σrt(α χ ] λ = L p [β 0 ΔP S σrt(α 0 χ 0 ] α 0 = C D δ ωrt + C D δ (ωrt + D δ α = 0,5[(ωRT + D δ (C C + σ(c D δ + + C D δ α = 0.5[C + σ(c C ] β 0 = D δ ωrt + D δ (ωrt + D δ β = 0.5σ(D δ D δ β = 0.5 ( σ χ 0 = C D δ ωrt + C D δ (ωrt + D δ χ = 0.5[(ωRT + D δ (C C σ(c D δ + + C D δ

4 4 Korna Bato t. a χ = 0.5[C σ(c C ] In prvous paprs [5] w sowd tat Eq. ( dscrbd t souton voum fux n condton of concntraton poarzaton. Ts quaton can b usd to dtrmn t ffct of concntraton poarzaton on t vau of rfcton coffcnt of t mmbran. In ts papr, sutab xprssons wr drvd for t rfcton coffcnt of t mmbran, undr dffusv and dffusv-convctv condtons for J vm = 0. Ts xprssons sowd tat t vau of rfcton coffcnt of t mmbran dtrmnd undr condtons of concntraton poarzaton s dpndnt, among otrs, on a tcnss of concntraton boundary ayrs (δ, concntraton of soutons (C, C and ydrodynamc stat of concntraton boundary ayrs controd by t concntraton Rayg numbr (R C. As an xamp, t obtand quatons wr appd for cuos mmbran and aquous gucos soutons. T study was carrd out for t mmbran transport procsss undr condtons n wc omognty of t soutons s dffcut or vn mpossb to acv. Exprssons for Rato of Rfcton Coffcnts For J vm = 0 n Eq. ( and upon smp agbrac transformatons, w obtan ΔPS Δπ D D = σ Jvm = 0 D D + ωrt ( D σ + Dσ = σ S (4 In ts quaton, σ S s t rfcton coffcnt of /M/ compx. Ts coffcnt was dtrmnd xprmntay n t macroscopc mmbran systms. It soud b notd tat vn an ntns strrng of soutons wt a mcanca strrr dd not fuy mnat t concntraton boundary ayrs. Trfor, a cacuatd vau ΔP S (for J vm = 0, undr condtons of concntraton poarzaton, was owr tan a vau of ΔP (for J vm = 0 dtrmnd n a omognous souton condtons. In mcroscopc systms, suc as boogca systms, wr t us of strrng of soutons sparatd by t mmbran s dffcut or vn mpossb, t coffcnt σ S, nstad of σ, s appontd. By transformng Eq. (5, w obtan an xprsson tat nabs t vauaton of t mpact of concntraton poarzaton on a vau of rfcton coffcnt of a mmbran n dffusv ( = d and dffusv-convctv ( = stats. = (5 σ σ σ + ωrt + D D Tcnsss δ and δ prsntd n t abov quaton can b dtrmnd by optca mtods [ 4] or by voum fux masurmnt mtod [4,7]. Wn t concntratons boundary ayrs ar symmtrca, and t dffuson coffcnts (D d, D d ar ndpndnt of t concntraton of soutons sparatd by t mmbran, tn condtons δ d = δ d = δ d and D d = D d = D d ar fufd. Trfor for t stat of dffuson (nonconvctv, Eq. (5 can b wrttn as = d d + ωrt σ (6 σ Dd In t dffusv-convctv stat accptanc of constant vau of ts coffcnts s arg approxmaton. Trfor, n ordr to dscrb t dffuson coffcnts undr dffusv-convctv condton (D, D, t xprssons prsntd n a prvous papr [4] can b usd gαc δ D = [( C C ( C C ] (7 ν R D gαc δ = ν R C C [( C C ( C C ] (8 T dffrnc C C can b cacuatd, tang nto account Eq. (5 and (9 n condton wn J vm = J vm = 0. As a rsut of ratvy smp transformatons, w can wrt C C = ζ ( C C (9 s wr ζ s = ω S /ω. T coffcnt ζ s s gvn by Eq. (5. Ts coffcnt can b dtrmnd xprmntay [5]. Insrtng Eq. (9 n Eqs. (7 and (8, w obtan D D gαcδ ( ζs( C C = (0 ν RC gαc δ ( ζs( C C = ( ν RC wr g accraton of gravty, α C = ( ρ/ C/ρ and α C = ( ρ/ C/ρ ratv cang n mass dnsty (ρ, ρ wt t concntraton, ν and ν nmatc vscosty, R C and R C concntraton Rayg numbr. By ntroducng Eqs. (0 and (, nto t Eq. (5, w obtan σs σ = ωrt + g( ζ ( C s ν R C αcδ C ν R + α δ C C ( Assumng tat n Eq. ( condtons δ = δ = δ, ν = ν = ν, α C = α C = α C and R C = D C = R C w gt = ( σ 4ωRTνR C + gα ( ζ ( C C δ C s

5 Poymrc mmbran 5,8,6,4 δ d = f( C δ = f( C δ [mm],,0 0,8 0,6 0, C [mo m ] Fg.. Dpndnc of tcnss of concntraton boundary ayrs (δ on t avrag concntraton of gucos (ΔC for t sng-mmbran systm basd on Śęza t a. papr [4]. Curv ustrats a dffusv part ( = d, wras curv a dffusvconvctv part ( = of t caractrstcs δ = f(δc Ryc.. Zażność grubośc stężnowyc warstw grancznyc (δ od różncy stężń guozy (ΔC da uładu jdno-mmbranowgo opracowan na podstaw pracy Śęza t a. [4]. Krzywa ustruj część dyfuzyjną ( = d natomast rzywa część dyfuzyjno-onwcyjną ( = caratrysty δ = f(δc Rsuts and Dscusson W cacuatd t rato (σ S /σ = d, for t Npropan mmbran and aquous gucos soutons undr sotrma condtons (T = 95 K, usng Eqs. (6 and (, for t dffusv and dffusv-convctv condtons. Dffusv condtons occur wn t mmbran systm s orntd n confguraton A for ΔC > 0 and n confguraton B for ΔC 5 mo m. Convctvdffusv condtons occur wn t mmbran systm s orntd n confguraton B and ΔC > 5 mo m. T mmbran transport paramtrs,.. ydrauc prmabty (L p, rfcton (σ and sout prmabty (ω coffcnts, prsntd n a prvous papr []. Tr vaus ar: L p = 5 0 m N s, σ = and ω = mo N s. W prvousy sowd dpndncs of δ = f(δc for confguratons A and B [4]. Hr w modfd ts dpndncs (s. Fg. assumng tat foowng condtons wr fufd: ΔC < 0 (confguraton A and ΔC > 0 (confguraton B. To cacuat (σ S /σ m = d (for dffuson condtons, w usd Eq. (7 and t constant vaus of gucos dffuson coffcnt n an aquous souton of gucos (D d and t unvrsa gas constant (R: D d = m s and R = 8. J mo K. To stmat (σ S /σ = (for t dffusv-convctv condtons, w usd Eq. ( and t foowng data ρ = ρ ( + α C C, ν = ν ( + γ C, wr α C = ρ ρ/ C = m mo and γ = ν ν/ C = m mo, (ρ = 998 g m, ν = m s. Vaus ζ s drvd from t dpndncs ζ s = f(δc wr prsntd n a prvous papr [5]. T crtca vau of concntraton Rayg numbr (R C cr = 709. was usd prvousy [4]. Ratons (σ S /σ = d, = f(δc cacuatd basd Eqs. (6 and ( wr sown by sod n n Fg. 4. To cacuat t rato σ S /σ, t us consdr Eqs. ( and (8, assumng t condton J v = J vs = 0. Dvdng bot sds of ts quatons, w can wrt σs σ Jv = Jvs = 0 ΔP = S ΔP (4 From t abov xprsson, t rsuts tat to dtrmn t rato σ S /σ for mmbrans orntd n a orzonta pan, t s suffcnt to dsgnat t foowng dpndncs ΔP S = f(δc for J vs = 0 and ΔP = f(δc for J v = 0, for tr ngatv or postv ΔC n a srs of ndpndnt xprmnts. To fuf t condton ΔC < 0, souton at a concntraton of C fd compartmnt abov t mmbran and souton wt a concntraton of C compartmnt bow t mmbran. Ts confguraton of t mmbran systm s dnotd by A. Wn ΔC > 0, a souton wt a concntraton of C fd a compartmnt bow t mmbran, a souton wt a concntraton of C fd a compartmnt abov t mmbran (confguraton B. Rato σ S /σ, cacuatd on t bass of t dpndncs ΔP S = f(δc and

6 6 Korna Bato t. a 0,5 0,4 ζ s =f( C curv ζ v =f( C curv ζ 0, 0, C [mo m ] Fg.. Dpndnc of a dffusv-convctv part ( of t concntraton poarzaton coffcnt (ζ on an avrag gucos souton concntraton (ΔC for t sng-mmbran systm. Curv ustrats t raton ζ s = f(δc for t dffuson concntraton poarzaton coffcnt (ζ s and curv ustrats t raton ζ v = f(δc for osmotc concntraton poarzaton coffcnt (ζ v Ryc.. Zażność częśc dyfuzyjno-onwcyjnj ( współczynna poaryzacj stężnowj (ζ od różncy stężń guozy (ΔC da uładu jdno-mmbranowgo. Krzywa ustruj zażność ζ s = f(δc da współczynna dyfuzyjngo współczynna poaryzacj stężnowj (ζ s. Krzywa ustruj zażność ζ v = f(δc da współczynna osmotyczngo współczynna poaryzacj stężnowj (ζ v. 0,5 0,9 xprmnta tortca σ s /σ 0,6 0, C [mo m ] Fg. 4. Dpndnc of a rato of rfcton coffcnts n condtons of concntraton poarzaton (σ S and souton omognty (σ on dffrnc n gucos concntraton (ΔC n t sng-mmbran systm, cacuatd from Eqs. (6 for ΔC 0.5 mo m and ( for ΔC > 0.5 mo m Ryc. 4. Zażność stosunu współczynna odbca w warunac poaryzacj stężnowj (σ S w warunac jdnorodnośc roztworów (σ od różncy stężń guozy (ΔC da uładu jdno-mmbranowgo, obczon na podstaw równana (6 da ΔC 0.5 mo m równana ( da ΔC > 0.5 mo m

7 Poymrc mmbran 7 0, 0,4 xprmnta tortca (σ S /σ con. 0,6 0,08 0, C [mo m ] Fg. 5. Dpndnc of a convcton part of rfcton coffcnts on t dffrnc n gucos concntraton (ΔC. In t sngmmbran systm, t rfcton coffcnt σ S s cacuatd n condtons of concntraton poarzaton and t rfcton coffcnt σ s cacuatd n condtons of omognty of soutons. Dpndncs wr cacuatd from Eqs. (5 ( and (6 (sod n for ΔC > 0.5 mo m Ryc. 5. Zażność onwcyjnj częśc stosunu współczynna odbca w warunac poaryzacj stężnowj (σ S w warunac jdnorodnośc roztworów (σ od różncy stężń guozy (ΔC da uładu jdno-mmbranowgo, obczon na podstaw równań (5 ( na podstaw równana (6 (na cągła da ΔC > 0,5 mo m ΔP = f(δc, was rprsntd ( n Fg. 4. Ts fgur sows good corraton (wtn 6% masurmnt rror rang btwn xprmnta ( and tortca (sod n rsuts. T grap aso ndcats tat for t ΔC fufng t condton 5 mo m ΔC 5 mo m, σ S /σ dos not dpnd on ΔC. For ΔC fufng t condton ΔC < 5 mo m, σ S /σ dcrass nary, and for ΔC > 5 mo m, σ S /σ ncrass nary wt ncras of ΔC t absout vau. Morovr, bot t xprmnta and t tortca dpndnc for 5 mo m > ΔC > 5 mo m s asymmtrc to t axs of ordnats. In prvous paprs [4] w namd t rato σ S /σ as t coffcnt of t osmotc concntraton poarzaton and w dnotd t rato by ζ v. For comparson, Fg. prsnts t rsuts of cacuatons ζ v = σ S /σ = f(δc. Ts fgur sows tat ζ v = ζ vs wtn 0% stmaton rror. Basd on t fndngs prsntd n Fg. 4, undr convctv condtons rato σ S /σ can b cacuatd as a dffrnc σ con = σ σ d (5 Insrtng Eqs ( and (7 n Eq. (5, t foowng xprsson can b wrttn as σs σ con ωrt [ grtαc ( ζs( C C δ δd νdd RC] = [ gα ( ζ ( C C δ + 4ωRTνR ][ D + ωrtδ ] C s C d d (6 T dpndncs (σ S /σ con. = f(δc, cacuatd on t bass of Eqs. (6 and (7 ar sown n Fg. 5. In Fg. 5, t sod n ustratng t dpndncs (σ S /σ con. = f(δc tat wr cacuatd from Eq. (6, and t symbos ( ustratd t rsuts obtand from Eq. (5. Ts fgur sows tat t rsuts obtand by bot mtods ar consstnt wtn 0% rror rang of rato (σ S /σ con stmaton. Fg. 4. sows tat a pont wt coordnats (σ S /σ con. = 0 and ΔC = 0.5 mo m srvs as a bfurcaton pont. Ts bfurcaton pont s a bordr btwn dffusv-convctv and dffusv stat. Ts s on of many xamps of t ro of structurtrmodynamc nvronmnt n trmodynamc systms [6]. Concussons Prsntd matmatca xprssons ustrat t dpndnc of t rato (σ S /σ on pyscocmca paramtrs of t soutons (ρ, ν, D, mmbran transport paramtrs (ω m and concntratons Rayg numbr

8 8 Korna Bato t. a (R C. Ts xprssons wr drvd from nonnar Eq. ( wr t fux rat n a concntraton poarzaton was dscrbd. Tang nto account t obtand xprssons, t cacuatons of rato σ S /σ wr prformd for Npropan mmbran and aquous soutons of gucos. T cacuatd curvs prsntd n Fg. 4 and 5 av a bfurcaton pont n wc R C = (R C crt. Abov ts pont,.. n convctv ara, σ S /σ dpnds bot on souton concntraton and t ydrodynamc stat of concntraton boundary ayrs. Bow ts pont, t systm s n t ara of dffuson (nonconvcton and σ S /σ dpnds soy on t concntraton of t soutons. In boogca systms, obtand rsuts may b appd for ntrprtaton of mmbran transport procsss undr condtons of concntraton poarzaton [5 8, 7]. Ltratur [] Kdm O., Katcasy A.: Trmodynamcs anayss of t prmabty of boogca mmbrans to non-ctroyts. Bocm. Bopys. Acta (958, 7, [] Katcasy, A., Curran, P. F.: Nonqubrum trmodynamcs n bopyscs. Harvard Unv. Prss, Cambrdg, (965. [] Kargo M., Kargo A.: Passv mass transport procsss n cuar mmbrans and tr bopysca mpcatons. In: K. Vafa (Ed. Porous mda: appcaton n boogca systms and botcnoogy. CRC Prss, Boca Raton, 0, [4] Śęza A., Grzgorczyn S., Jas-Śęza J., Mcasa-Małca K.: Natura convcton as an asymmtrca factor of t transport troug porous mmbran. Transp. Porous Md. (00, 84, [5] Myamoto Y., Yuosa H., Iga T., Hanano M.: Dtrmnaton of t mmbran prmabty coffcnt and t rfcton coffcnts by t two-dmnsona amnar fow mod for ntstna prfuson xprmnts. Bocm. Bopys. Acta (986, 854, 997. [6] Hamada Y., Ima M.: Effct of ntracuar unstrrd ayr on apparnt rfcton coffcnt for ura n nnr mduary coctng duct: a computr smuaton. Exp. Npro. (995,, 0 0. [7] Tyr M. T., Ko S., Sands P.: T dtrmnaton of mmbran transport paramtrs wt t c prssur prob: tory suggst tat unstrrd ayrs a sgnfcant mpact. Pant, C Envron. (005 8, [8] Km Y. Y Q., Rnardt H., Stud E.: Frutr quantftaton of t ro of ntrna unstrrd ayrs drvng t masurmnt of transport coffcnts n ggant ntrnods of Cara by nw stop-fow tcnqu. J. Exp. Bot. (006, 57, [9] Kargo M., Kargo A.: Mcanstc quatons for mmbran substanc transport and tr dntfy wt Kdm-Katcasy quatons. Bopys. Cm. (00, 0, 77. [0] Barry P. H., Damond J. M.: Effcts of unstrrd ayrs on mmbran pnomna. Pyso. Rv. (984, 64, [] Śęza A., Dworc K., Andrson J.E.: Gravtatona ffcts on transmmbran fux: t Rayg-Tayor convctv nstabty. J. Mmbran Sc. (985,, 7 8. [] Śęza A.: Irrvrsb trmodynamc mod quatons of t transport across a orzontay mountd mmbran. Bopys. Cm. (989, 4, 90. [] Śęza A.: Mmbran transport of t non-omognous non-ctroyt soutons: matmatca mod basd on t Kdm-Katcasy and Rayg quatons. Pom. Md. (007, 7, [4] Grzgorczyn S., Jas-Śęza J., Mcasa-Małca K., Śęza A.: Transport of non- ctroyt soutons troug mmbran wt concntraton poarzaton. Gn. Pyso. Bopys. (008, 7, 5. [5] Jas-Śęza J., Oszówa K. M., Śęza A.: Ocna wartośc współczynna osmotyczngo van t Hoffa wwarunac poaryzacj stężnowj uładu mmbranowgo. Pom. Md. (0, 4, [6] Rubnstn, I., Zatzman, B.: Ectro-osmotcay nducd convcton at a prmsctv mmbran. Pys. Rv. E (000, 6, 8 5. [7] Jas-Śęza J., Oszówa K.M., Śęza A.: Estmaton of tcnss of concntraton boundary ayrs by osmotc voum fux dtrmnaton. Gn. Pyso. Bopys. (0, 0, [8] Kargo A.: Effct of boundary ayrs on rvrs osmoss troug a orzonta mmbran. J. Mmbr. Sc. (999, 59, [9] Jas-Śęza J., Oszówa K. M., Śęza A.: Ocna wartośc różncy stężń dtrmnującj transport mmbranowy w warunac poaryzacj stężnowj. Pom. Md. (00, 40, [0] Gnzburg B. Z., Katcasy A.: T frctona coffcnts of t fows of non-ctroyts troug artfca mmbrans. J. Gn. Pyso. (96, 47, [] Śęza A., Dworc K., Śęza I. H., Wąs S.: Prmabty coffcnt mod quatons of t compx: mmbran-concntraton boundary ayrs for trnary nonctroyt soutons. J. Mmbr. Sc. (005, 67, [] Dworc K.: Intrfromtrc nvstgaton of t nar-mmbran dffuson ayrs. J. Bo. Pys. (995,, [] Frnándz-Smpr J., Ruz-Bvá F., Garca-Agado P., Sacdo-Díaz R.: Vsuazaton and modng of t poarzaton ayr and rvrsb adsorpton procss n PEG-0000 dad-nd utraftraton. J. Mmbr. Sc. (009, 4, [4] Putnvtt, B. A., Gunasgaran, G. S., Agrawa, Y. K., Arar, J. H.: Lngt of nar- wa pums n turbunt convcton, J. Fud Mc. (0, 685, [5] Śęza A., Grzgorczyn S., Bato K.: Rsstanc coffcnts of poymr mmbran wt concntraton poarzaton. Transp. Porous Md. (0, 95, 570. [6] Kondpud D., Prgogn I.: Modrn trmodynamcs. From at ngns to dsspatv structurs. Jon Wy & Sons, Ccstr, 006. [7] Pappnmr J. R.: Ro of pr-pta unstrrd ayrs n absorpton of nutrnts for t uman jjunum. J. Mmbran Bo. (00, 79,

9 Poymrc mmbran 9 Adrs do orspondncj: Dr Korna Bato Katdra Informaty Eonomcznj Unwrsytt Eonomczny u. Bogucca B, Katowc -ma: orna.bato@u.atowc.p