Effect of Wall Absorption on dispersion of a solute in a Herschel Bulkley Fluid through an annulus

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1 Available online a Advances in Applied Science Reseach,, 3 (6): ISSN: CODEN (USA): AASRFC Effec of Wall Absopion on dispesion of a solue in a Heschel Bulley Fluid hough an annulus B. Ramana and G. Saoamma Dep. of Applied Mahemaics, Si Padmavai Mahila Univesiy, Tiupai, Andha Padesh - 575, India ABSTRACT The combined effec of yield sess, ievesible bounday eacion on he dispesion of a ace in annula egion and powe index on dispesion pocess is sudied using genealized dispesion model. The sudy descibes he dispesive anspo following he inecion of a ace in ems of hee effecive anspo coefficiens, viz. he exchange, he convecion and he dispesive coefficiens. The effec of powe index, annula gap, yield sess and wall absopion paamee on he above hee effecive anspo coefficiens is discussed. The effec of flow paamees on mean concenaion is sudied. Key-Wods Genealized dispesion model; Heschel Bulley fluid; Seady flow INTRODUCTION In dispesion of a solue in an annula flow when i is eihe ievesibly absobed o undegoes an exchange pocess a he bounday have seveal applicaions. In mass-ansfe devices, he fomaion of concenaion gel and polaizaion layes a ansfe sufaces degades he pefomance of such devices. The efficiency of such devices, fouling of sufaces inhibis he efficiency. I is esablished ha he fouling in he sufaces can be educed when he mixing wih in he flow condui is inceased. The sudy of mass and hea ansfe in an annula flow is significan in view of is abundan applicaions in exohemic chemical eacions, geohemal enegy ecovey, nuclea eacos. The annula chomaogaphic mehod used fo sepaaion of meals, sugas and poeins ec. [, ] is anohe example. The indicao diluion echnique using cahees o inec dye and o wihdaw blood samples is an example in clinical medicine. Davidson and Schoe [3] sudied he paen of dispesion and upae an inhaled slug of issue soluble gas wih in a bonchial wall as an assembly of saigh igid ubes wih absobing wall of finie hicness. They obseved geae peneaion fo lowe solubiliy of he gas and he mixing coefficien was shown o decease wih disance ino he lung o a value, which may be much smalle han he molecula diffusiviy. Phillips e al. [4] invesigaed he anspo of a ace hough a wall laye consising of a ube conaining flowing fluid suounded by a wall laye in which he ace was soluble. They obseved ha he effecive convecion and dispesion coefficiens wee of lile use in pedicing he ime - vaying concenaion a a fixed posiion as he spaial concenaion pofile became Gaussian only ove he longe ime scale. Jayaaman e al. [5] exended he heoeical model of Davidson and Schoe [3] o sudy he dispesion of solue in a fluid flowing hough a cuved ube wih absobing walls by using a mahemaical model of an infiniely long condui defined by wo concenic cuved cicula pipes. Thei esuls based on peubaion and specal mehods, confimed he ealie expeimenal findings ha he influence of 3878

2 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): seconday flows on he dispesion was educed if he ace was vey soluble in he wall. Nagaani e al. [6] analyzed he effec of wall absopion on he dispesion of solue in a Casson fluid flowing in an annulus. Blai and Spanne [7] epoed ha blood obeys Casson equaion only in a limied ange, excep a vey high and vey low shea ae and ha hee is no diffeence beween he Casson plos and he Heschel Bulley plos of expeimenal daa ove he ange whee he Casson plo is valid. I is obseved ha he Casson fluid model can be used fo modeae shea aes γ < /s in smalle diamee ubes wheeas he Heschel Bulley fluid model can be used a sill lowe shea ae of flow in vey naow aeies whee he yield sess is high [8]. Fuhe, he mahemaical model of Heschel Bulley fluid also descibes he behaviou of Newonian fluid, Bingham fluid and powe law fluid by aing appopiae values of he paamees viz. yield sess and powe law index. In his pape an aemp has been made o sudy he effec of inephase mass ansfe a he oue bounday in he sudy of dispesion of solue in a Heschel-Bulley fluid flowing in an annulus. Secion pesens he mahemaical fomulaion of he poblem wih appopiae iniial and bounday condiions. Secion 3 gives he soluion of he mahemaical model by using he modified deivaive expansion mehod. In secion 4 he effec of wall absopion paamees, yield sess and annula gap and he powe law index on he hee dispesion coefficiens viz. absopion, convecion and dispesion coefficiens and mean concenaion is discussed. Secion 5 descibes he applicaion of he mahemaical model o he dispesion in a caheeized aey. The conclusions ae pesened in secion 6 Mahemaical Fomulaion Figue shows he schemaic diagam of he annula geomey. The adius of he oue ube is a and ha of he inne ube is a wih <. Fo a fully developed lamina flow of a Heschel Bulley fluid in an annulus, he convecive diffusion equaion, which descibes he local concenaion C of a solue as a funcion of axial disance z, adial disance and ime in non-dimensional fom is given by + w = ( ) C Pe z + () wih he non-dimensional vaiables C w Dm z Dm C =, w =, =, z =, =, Pe = C w a a w a aw Dm () whee w is non-dimensional axial velociy of he fluid, D m is coefficien of molecula diffusion, assumed o be consan. C is he efeence concenaion, Pe is Pecle numbe, w is he chaaceisic velociy. Fig Schemaic diagam of a caheeized aey Iniial and Bounday condiions A an insan of ime, he amoun of ace lef in he sysem, is convecive velociy and he exen of shea disibuion depend upon he iniial dischage. The iniial disibuion is aen a = as he case when he solue of mass m is inoduced insananeously a he plane z =, unifomly ove he coss-secion of a cicle of adius, da (whee < d < ) concenic, in he annula egion, wih he ube. Thus, he iniial disibuion of he solue assumed in a vaiable sepaable fom in ems of he non-dimensional quaniies is given by 3879

3 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): C(,z, ) = ψ (z)b() fo < < (3) wih ψ (z) =δ(z) / ((d )Pe) (4.) whee δ (z) denoes he Diac dela funcion, and () B = < d d < (4.) In he cuen model we conside he heeogeneous eacion mechanism occuing a he oue wall ( = ) of he ube so ha (, z,) = C (, z,) whee is he non-dimensional wall eacion paamee. In addiion we have he bounday condiion a = (5) (, z,) = (6) descibing he impemeable wall of he inne ube. and C (,, ) = (,, ) = (7) in non- dw The consiuive equaion fo a Heschel Bulley fluid elaing he sess (τ ) and shea ae d dimensional fom is given by dw τ = τ n y + ( ) if τy d τ (8.) and dw d =, if τ τy (8.) whee τ τ =, µ (w / a) τy τ y = (8.3) µ (w / a) τ and τy ae he dimensional shea sess and yield sess especively and τ y is dimensionless yield sess. The velociy equaions ae given by n n + λ λ w () = w () = Λ d n d if λ (9.) w = w + () = w p = w ( λ) = w + + ( λ ) = consan if λ λ (9.) n n ++ λ λ w () = w () = Λ d n d if λ (9.3) whee Λ (9.4) = λ λ = τy is he widh of he plug flow egion and λ = λ λ, (9.5) 388

4 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): The supescips + and + + epesens he shea flow egion λ and λ, especively, and he supescip epesens he plug flow egion λ λ. Fom eq (9.4) and fom (9.) and using (9.5), we obain λ λ n n n ( λ ( Λ+λ ) )/ ) d (( λ ( Λ+λ ))/ ) d nλ ( λ ( Λ+λ ) )/ ) d+ nλ (( λ ( Λ+λ ))/ ) ) Λ+λ Λ+λ n d = which is an inegal equaion o be solved fo λ numeically by using Regula falsi mehod, and λ can be obained fom equaion (9.4), once λ is nown Mehod of soluion In ode o solve he convecion-diffusion equaion () along wih he associaed ses of iniial and bounday condiions (3-7), we expess he concenaion following [9] as C (,, z) = Cm f (, ) = () whee he aveage concenaion, Cm Cm = + D (, z,) D C m = C m is expessed as D C d () Muliplying equaion () by Pe and inegaing wih espec o fom o yields w(, ) C(,z,) d () Inoducing () ino (), he following dispesion model fo C m is obained as Cm = K () = (3) whee K () δ, f = + D (,) D f (,) w(,) d =,, (4) Pe whee δ denoes he Konece dela. The exchange coefficien K () accouns fo he non zeo solue flux a he bounday and will be zeo if hee is no wall absopion a he wall. K and K coespond o he convecive and dispesion coefficiens especively. Sanaasubamanian and Gill [9] showed ha equaion (3) can be uncaed by neglecing highe ode ems involved K 3, K 4 ec and hen genealized dispesion model aes he fom m m Cm = K()Cm + K() + K () Using equaions (), (3) in equaion () and equaing coefficiens of diffeenial equaions fo f is obained as Cm (5), =,,,----- he following se of f = f ( ) w(, )f + f Kif i Pe i= =,, (6) whee f - = = f -. Fom eq (3), (4) and () we ge Cm (, z) = D C(,, z) d = D ψ(z) I (7.) 388

5 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): B() which gives f (, ) = (7.) D I and f (, ) = =,... ( 7.3) f (,) = f (,) f (, ) = fo = =,,...,,... (7.4) (7.5) f (, ) d = δ, / D (7.6) The coupled equaions (4) and (6) ae solved one afe he ohe o obain he soluion of (f, K ) =,,. Evaluaion of f (, ) and K () The funcions f and exchange coefficien K ae independen of velociy field, and hence can be solved diecly. Fom equaion (6) fo =, we obain f f = ( ) f K Inoducing he ansfomaion K ( η ) dη f (,, = e g, ) ( equaion (8, aes he fom g g = ( ) (8) (9) () The iniial and bounday condiions on g ae B () g (, ) = () D I g (,) = g (,) () g (, ) = (3) The soluion of g saisfying he iniial and bounday condiions () - (3) is given by A µ g(, ) E( µ ) e = J(u ) = (4) whee E ( µ ) = [ J ( µ ) Y ( µ ) Y ( µ ) J ( µ ) ] (5) and µ ae he oos of he anscendenal equaion ( Y ( µ ) J ( µ ) Y ( µ ) J ( µ ) ) + ( Y ( µ ) J ( µ ) Y ( µ ) J ( µ )) µ (6) = µ n ( ) J( µ ) B () E ( µ ) d and An = (7) ( µ + ) E ( µ ) µ E I ( µ ) and J, J, Y, Y ae he Bessel funcions of fis ind, second ind of ode zeo and one especively. Fom (7.6), we have f (,) d = / (8) D 388

6 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): Using he ansfomaion (9) and equaion (8), we obain e K ( η ) dη = / D I (9) Fom (9) and (9), he expession f (,) can be obained as f (, ) = g (, ) D (3) I which can be wien as f D (3) (, ) = D E ( µ ) D D3 µ Using equaions (4) and (3), he exchange coefficien K () is obained as D = µ K () D D3 D 3 (3) µ The expessions fo f (, ) and K () educe o hose deived by Sanaasubamanian and Gill (973) in he limiing case. Asympoic Expansions fo f s and K s fo =,,, fo seady flow The coupling beween f (, ) and K () and solving he ansfomed poblem fo, say f (, ) would esul in complicaed inegal equaion in K (,. The highe ode funcions ae even moe difficul o solve. Hence, we esic ouselves o he asympoic seady-sae epesenaions of f (, ) and K () fo he case of seady flow, we will obain soluions (f, K ), =,, fo lage imes, so ha he dispesion model defined in (5) is a epesenaion of he asympoic esuls unde seady sae condiions. As, he asympoic epesenaion fo f, K, K and K ae f (, ) = µ E( µ ) D D4 (33) K ( ) = µ (34) µ K = w( ) E ( µ ) d (35) ( µ + ) E( µ ) µ E ( µ ) Bn I3 f = E( µ n) E( µ n) (36) n= J( µ n) I4 whee B n s ae given by Bn = [w() + K] f() E( µ n) d ( µ µ n ) J( µ n) [ w( ) + K] f E( µ ) d K = (37) Pe f E( µ ) d A n µ whee D = ; D = e n ; ( ) J(u n ) D 3 = J ( µ )Y ( µ ) Y ( µ )J ( ) ; D 4 = ( µ ) Y ( µ ) Y ( µ ) J ( ) ; I = B ( d ; I = g n n n µ n (, ) n ) J µ d ; I 3 = E ( µ d ; I 4 = E ( µ d ) The mean concenaion C m is obained fom equaion () wih iniial and bounday condiions given by (3) and (7) is ) 3883

7 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): z C (,z) exp ( m = ζ ) (38) (Pe) πξ 4ξ whee ζ( ) = K ( η) η, z (, z) = z + K ( η) η, ξ( ) = K ( η) η d d d RESULTS AND DISCUSSION In he pesen analysis, he developmen of he dispesive anspo following he inecion of a chemically acive ace in a solven flowing hough an annulus wih eacive oue wall has been invesigaed. The fluid flowing hough he annulus is aen as Heschel- Bulley fluid. The analysis uses he deivaive expansion mehod poposed by [9]. The effec of modeae wall absopion, in he pesence of a coaxial inne ube, on he hee effecive anspo coefficiens, viz. he exchange, he convecion and he dispesion coefficiens is sudied. The value of he wall absopion paamee is aen in he ange of. o. The aio of he adius of inne cylinde o ha of he oue cylinde is vaied fom. o.4. The yield sess τ y is aen o ange fom.5 o.5. The esuls ae discussed fo diffeen fluids Newonian ( τ y =, n = ), Powe law ( τ y =, n = ), Bingham ( τ y, n = ) and Heschel-Bulley fluids ( τy, n = ). Asympoic absopion coefficien K Figue shows he vaiaion of he absopion coefficien (- K ) wih and fo lage imes. When he wall absopion paamee is, he value of - K inceases fom 5.65 in he case of a ube o 9.37 in an annula flow ( =.4). Thus, hee is moe absopion of solues a he wall in an annulus compaed o he ubula flow. 9 8 = =. =. =.3 = K Fig Vaiaion of negaive asympoic absopion coefficien K wih absopion paamee fo diffeen values of Asympoic convecion coefficien K Figue 3 (a) pesens he asympoic convecion coefficien K vesus he wall absopion paamee fo vaiaion of. In he absence of solue flux acoss he wall( = ), when he fluid flowing is Newonian, -K = / [] indicaes ha he solue is being conveced along he annulus wih an aveage velociy of he flow. In he case of pipe flow of a Newonian fluid flow [9], when wall eacion is pesen, he solue disibuion moves wih a velociy geae han he aveage velociy. Bu in an annula flow, he solue is obseved o be conveced along wih a velociy lowe han he aveage velociy of he flow when he yield sess is as small as.5. In he pesence of cahee ( =.) he - K educes o half of is value coesponding o ha in a ubula flow when =., τ y =

8 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): and n =. K is noiced o decease wih decease in annula gap. Fo =, convecion coefficien deceases fom.39 ( =.) o. 48 ( =.4) =. =. =. =.3 =.4. τ y =.5 τ y =. τ = K.3 - K (a). - - (b).4.35 Newonian Bingham Powe law HBF.3 - K (c) Fig 3 Vaiaion of negaive asympoic convecion coefficien K wih absopion paamee fo (a) diffeen values of fo τ y =.5, n = (b) fo diffeen values of τy when n = (c) fo diffeen fluids The asympoic convecion coefficien K educes Fig 3 (b) fom.393 ( τ y =.5) o.56 ( τ y =.5) wih incease of yield sess when =. Fig 3c gives he educion in he asympoic convecive coefficien ( K ) due o non-newonian heology. The dop in he K is much moe apid in case of Powe law and Heschel- Bulley fluids when compaed o Bingham fluids. When he fluid is Casson K assumes less value han he coesponding values in Heschel-Bulley fluid [6]. 3885

9 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): x = =. =. =.5 =..4 x -3. τ y =.5 τ y =. τ y =.5. K -/Pe.8 K -/Pe (a) - - (b) x -3 Newonian Bingham Powe law HBF 3.5 K -/Pe (c) Fig 4 Vaiaion of asympoic dispesion coefficien K /Pe wih absopion paamee (a) fo diffeen values of when τ y =.5, n = (b) fo diffeen values of τ y when =., n = (c) fo diffeen fluids when =. Asympoic dispesion coefficien K Fig. 4 shows he vaiaion of asympoic dispesion coefficien K (fom which he addiive conibuion of he axial diffusion /Pe has been deduced) agains he wall absopion paamee fo diffeen values of, τ y and n. The effec of on K is significan unlie in he case of K and -K. As he dispesion coefficien K which is a muli fold decease fom is value fo no wall eacion, which is.3-3 when =., n = and τ y =.5. (K / Pe ) deceases fom.653 (in he case of ubula flow) o.77 (in an annula flow) =., when =., τ y =.5 and n =. 3886

10 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): The absopion of solues is moe as he annula gap deceases, which is also obseved by [6]. Consequen o he bounday absopion, he axial dispesion inhibis by he ansvese diffusion and hus he axial dispesion coefficien deceases. I is seen ha fom fig 4 (a), K deceases fom.74-3 when is as small as. o.5-5 when =. fo =. and τ y =.5. This also noiced by [] in he case of solue dispesion in an annulus wih no wall absopion when he fluid flowing in Newonian. Fom fig 4(b), i is obseved ha he effec of yield sess is o educe he dispesion coefficien when τ y =.5 he value of K ( ) is educed 3 imes of he value (.6-3 ) coesponds o he value when τ y =.5 fo =..4 =. =.. =. = C m 6 C m (a) (b) Newonian Bingham Powe law HBF Newonian Bingham Powe law HBF C m C m (c) (d) (a) diffeen values of when =., Fig 5 Vaiaion of mean concenaion C m wih ime fo τ y =.5, z =.5, n = (b) diffeen values of when =, τ y =.5, z =.5, n = (c) diffeen fluids when =., =., z =.5 (d) diffeen fluids when =, =., z =.5 Fig 4 (c) pesens he dispesion coefficien fo vaiaion of n and τ y. I is obseved ha when =., hee is a heefold educion in he dispesion coefficien in he case of Heschel- Bulley fluid coesponding o ha of a Newonian case when =. 3887

11 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): Mean Concenaion The mean concenaion C m as a funcion of ime is ploed in fig.5 (a, b, c, d) fo vaiaion in and n. The pea mean concenaion inceases fom (4.497) o (.76) when changes fom. o. when =., τ y =.5 and n =, when =, hee is a seven fold educion in he mean concenaion when inceases fom. o. The vaiaion of non- Newonian heology on pea mean concenaion when =., =., z =.5 is descibed in fig 5a. The pea mean concenaion in he case of Heschel- Bulley fluid is highe compaed o is coesponding value in he cases of powe and Newonian fluids. The behavio is oally evesed when = 5 Applicaion o caheeized aey The invesigaions of he pesen mahemaical model ae elevan in undesanding he cahee inseion elaed aifacs in flow measuemens in he cadiovascula sysem. Cahees ae used o inec he dye and o ae blood samples fo he pupose of measuemens [, ].The inseion of a cahee ino he blood vessels can be one of he significan causes fo he fomaion eddies and mixing of blood. This mehod of aing blood samples invaiably inoduces some disoion in he ime-concenaion cuve and hus he ecoded cuve does no epesen he in siu concenaion ime elaion a he wihdawal sie [3]. The expeimenal sudies of [4, 5], epo he calibaions based on mahemaical models fo he emoval of he cahee disoion. The esuls obained fom he mahemaical model fo emoval of cahees inoduced disoion of indicao diluion cuves of [6] showed a shape and highe pea and less smooh han he measued cuves. The analysis of dispesion pocess wih bounday absopion is impoan o undesand he indicao diluion echnique as lung and blood vessels involve conducive walls. The obecive is o povide a coecion fo he cahee induced eos in he measued values based on he concep of longiudinal diffusion of subsances due o he combined acion of convecion and diffusion. In his mahemaical model, he oue ube epesens he aey wall wih conducance. The inne coaxial ube can coespond o a cahee. The annula gap, o equivalen o he aio of he adius of he cahee o he aey, is vaied in he ange. o.4. The wall absopion paamee quanifies he pemeabiliy of he aeial wall and is values ae vaied fom. o, so ha he esuls can be elaed o physiological siuaions. Table Vaiaion of K, K and K wih and fo τ y =.5, n = =. =. = K K K K / Pe K K K / Pe K K / Pe Table epesens he vaiaion of he asymmeic absopion coefficien K, convecion coefficien K and he dispesion coefficien K /Pe wih he wall absopion paamee and annula gap. I is obseved ha he pesence of he cahee inhibis he convecion coefficien [deceases fom.3565 (ube) o.879 ( =.)] and dispesion coefficien [deceases fom.5 ( =.) o. ( =.3)] when =., τ y =.5 and n = Table gives he values of he convecive and dispesion coefficiens fo diffeen values of yield sess and annula gap when n = and =. Thee is a double fold educion in he convecive coefficien and dispesion coefficien due o he pesence of a cahee ( =.). As he cahee size inceases ( =.4) hee is a fivefold diminuion in K when ( =.) when τ y =. while he educion faco in he dispesion coefficien is. Wih incease in he yield sess he asympoic convecion coefficien deceases fom.667 (τ y =.) o.3 (τ y =.5) and he coesponding educion in K /Pe is double fold. The combined impac of he pesence o incease in adius of he cahee and yield sess is o diminuive he convecion and dispesion coefficiens. 3888

12 B. Ramana e al Adv. Appl. Sci. Res.,, 3(6): Table Vaiaion of K and K wih and τ y fo =, n = τ y =. τ y =.5 K K / Pe K K / Pe CONCLUSION The dispesion of a solue in a Heschel- Bulley fluid in an annula egion is analyzed using he genealized dispesion model. The effec of wall conducance a he oue bounday is consideed. Following he inecion of a ace he enie dispesion pocess is descibed by he hee effecive anspo coefficiens, viz. he exchange coefficien, he convecion and he dispesion coefficiens. I is obseved ha he analyic absopion coefficien -K, inceases fom 5.67, in he case of a ubula flow ( = ) o 9.37 fo he annula flow ( =.4) fo lage values of wall absopion paamee ( = ). Thus he absopion of solues is favoued a he walls in an annulus. The absopion and convecion coefficiens ae found o incease wih wheeas he dispesion coefficien is seen o decease wih. The absopion coefficien is independen of yield sess and powe law heology. The yield sess and powe law heology educe he convecion and dispesion coefficiens. The esuls of he mahemaical model ae discussed o undesand he dispesion pocess in a caheeized aey. The pesence of a cahee and incease in he size of he cahee favous he dyes o ohe solues o ge ou of he blood vessels fo all values of he absopion paamee. The non- Newonian and powe law heology has a significan effec on he convecion and dispesion coefficiens. REFERENCES [] Caa G, Decali JP, Byes CH, Sisson, WG, Chem. Engg. Commun,989, 79, 7-7 [] Bloomingbug GF, Caa G, The chem. Engg. J., 994, 55, B9-B7. [3] Davidson MR, Schoe RC, J. Fluid Mech., 983, 9, [4] Phillips CG, Kaye SR, Robinson CD, J. Fluid Mech., 995, 97,373-4 [5] Jayaaman G, Pedley TJ, Goyal A, QJMAM, 998, 5, [6] Nagaani P, Saoamma, G, Jayaaman, G, Aca Mechanica, 8,, [7] Blai GWS, Spanne DC, An inoducion o Bioheology. Elsevie Scienific Publishing Company Amsedam, Oxfod, New Yo,974, [8] Tu C, Deville M, J. Biomech., 996, 9, [9] Sanaasubamanian R, Gill WN, Poc. Roy. Soc. London A, 973, 333, 5-3. [] Rao AR, Deshiacha KS, ZAMM, 987, 67, [] McDonald D. A, Blood flows in Aeies. Edwad Anold, London,974 [] Mahie R T, Blood Flow Measuemen in Man, Casle House Publicaions Ld, 98 [3] Milno WR, Jose AD, J. Appl. Physiol., 96, 5, [4] Goesy CA, Silveman M, Am. J. Physiol. 963, 7, [5] Paish D, Gibbons GE, Bell JW, J. Appl. Physiol.,96, 7, [6] Spaacino G, Vicini P, Bonadonna R, Maaccini, P, Lehovia, M, Feannini, E, Cobelli C, Med Biol Eng Compu,997, 35,

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion