Unsteady State Molecular Diffusion

Transcription

1 Chaper. Differeial Mass Balae Useady Sae Moleular Diffusio Whe he ieral oeraio gradie is o egligible or Bi <<, he mirosopi or differeial mass balae will yield a parial differeial equaio ha desribes he oeraio as a fuio of ime ad posiio. For a biary sysem wih o hemial reaio, he useady sae moleular diffusio is give by = (D ) (.-) For oe-dimesioal mass rasfer i a slab wih osa D ad oveive odiios of h m ad,, equaio (.-) is simplified o = D (.-) h m,,if h m,,if - = Figure.- Oe-dimesioal useady mass rasfer i a slab. Equaio (.-) a be solved wih he followig iiial ad boudary odiios I. C. =, (, ) = i B. C. =, = = ; =, D = = h m ( f, ) I geeral, he oeraio wihi he slab depeds o may parameers besides ime ad posiio. = (,,,i,,,, D, h m ) -

2 The differeial equaio ad is boudary odiios are usually haged o he dimesioless forms o simplify he soluios. We defie he followig dimesioless variables Dimesioless oeraio: θ = K ' K ',, i, =K, + θ (,i K, ) Dimesioless disae: = = Dimesioless ime or Fourier umber: D = F o = = D Fo K is he equilibrium disribuio oeffiie. Subsiuig T,, ad i erms of he dimesioless quaiies io equaio (.-) yields (,i, ) D D θ = (,i, ) Fo θ θ θ = Fo (.-3) Similarly, he iiial ad boudary odiios a be rasformed io dimesioless forms θ (, ) = θ = = ; θ = = Bi m θ (, ), where Bi m = hm K ' D Therefore θ = f(, F o, Bi m ) The dimesioless oeraio depeds θ oly o, F o, ad Bi m. The mass rasfer Bio umber, Bi m, deoes raio of he ieral resisae o mass rasfer by diffusio o he eeral resisae o mass rasfer by oveio. Equaio (.-3) a be solved by he mehod of separaio of variables o obai θ = = where he oeffiies C are give by C ep( ζ F o ) os(ζ ) (.-4) C = 4siζ ζ + si(ζ ) ad ζ are he roos of he equaio: ζ a(ζ ) = Bi m. -

3 Table.- liss he Malab program ha evaluaes he firs e roos of equaio ζ a(ζ ) = Bi m ad he dimesioless oeraios give i equaio (.-4). The program use Newo s mehod o fid he roos (see Review). Table.- Malab program o evaluae ad plo θ = = C ep( ζ Fo) os(ζ ) % Plo he dimesioless oeraio wihi a slab % % The guess for he firs roo of equaio za(z)=bi depeds o he Bio umber % Bio=[ if]'; alfa=[ ]; zea=zeros(,);=zea; Bi=; fprif('bi = %g, New ',Bi) Bi=ipu('Bi = '); if legh(bi)>;bi=bi;ed % Obai he guess for he firs roo if Bi> z=alfa(); else z=ierp(bio,alfa,bi); ed % Newo mehod o solve for he firs roos for i=: for k=: a=a(z);ez=(za-bi)/(a+z(+aa)); z=z-ez; if abs(ez)<., break, ed ed % Save he roo ad alulae he oeffiies zea(i)=z; (i)=4si(z)/(z+si(z)); fprif('roo # %g =%8.4f, C = %9.4e\',i,z,(i)) % Obai he guess for he e roo sep=.9+i/; if sep>pi; sep=pi;ed z=z+sep; ed % % Evaluae ad plo he oeraios hold o Fop=[..5 ]; s=-:.5:; osm=os('s); for i=:5-3

4 Fo=Fop(i); hea=.ep(-fozea.^)osm; plo(s,hea) ed grid label('');ylabel('thea') Bi =.5 Roo # =.6533, C =.7e+ Roo # = 3.93, C = e- Roo # 3 = 6.366, C =.4335e- Roo # 4 = , C = -.56e- Roo # 5 =.66, C = 6.68e-3 Roo # 6 = , C = -4.64e-3 Roo # 7 = 8.876, C =.87e-3 Roo # 8 =.39, C = -.69e-3 Roo # 9 = 5.56, C =.579e-3 Roo # = 8.9, C = -.483e-3 Figure.- shows a plo of dimesioless oeraio θ versus dimesioless disae a various Fourier umber for a Bio umber of.5. Temperaure disribuio i a slab for Bi =.5 Fo=..9.8 Fo=.5.7 Fo=.6 Thea.5.4 Fo=.3.. Fo= Figure.- Dimesioless oeraio disribuio a various Fourier umber. For he roos of equaio ζ a(ζ ) = Bi m, le f = ζ a(ζ) Bi m The f = a(ζ) +ζ( + a(ζ) ); -4

5 The differeial oduio equaio for mass rasfer i he radial direio of a ifiie ylider wih radius R is = D r r r r (.-5) The differeial oduio equaio for mass rasfer i he radial direio of a sphere wih radius R is = D r r r r (.-6) Equaios (.-5) ad (.-6) a be solved wih he followig iiial ad boudary odiios I. C. =, (r, ) = i B. C. r =, r r= = ; r = R, D r r= R = h m ( f, ) The soluio of equaio (.-5) for he ifiie ylider is give as θ = = C ep( ζ F o ) J (ζ ) (.-7) where J (ζ ) is Bessel fuio of he firs kid, order zero. The oeffiie C are o he same as hose i a slab. The soluio of equaio (.-6) for a sphere is give as θ = = si( ζ C ep( ζ r) F o ) ζ r (.-8) Sie lim si( ζ r) r ζ r = lim ζ os( ζ r) r ζ =, i should be oed ha a r = θ = = C ep( ζ F o ) For oe-dimesioal mass rasfer i a semi-ifiie solid as show i Figure.-3, he differeial equaio is he same as ha i oe-dimesioal mass rasfer i a slab = D -5

6 Semi-Ifiie Solid Figure.-3 Oe-dimesioal mass rasfer i a semi-ifiie solid. We osider hree ases wih he followig iiial ad boudary odiios Case : I. C.: (, ) = i B. C.: (, ) = s, (, ) = i Case : I. C.: (, ) = i B. C.: D = = N, (, ) = i Case 3: I. C.: (, ) = i B. C.: D = h m ( f, ), (, ) = i = ll hree ases have he same iiial odiio (, ) = i ad he boudary odiio a ifiiy (, ) = i. However he boudary odiio a = is differe for eah ase, herefore he soluio will be differe ad will be summarized i a able laer.. pproimae Soluios The summaio i he series soluio for rasie diffusio suh as equaio (.-4) a be ermiaed afer he firs erm for F o >.. The full series soluio is θ = = The firs erm approimaio is C ep( ζ F o ) os(ζ ) (.-4) θ = C ep(- ζ F o ) os(ζ ) (.-) where C ad ζ a be obaied from Table.- for various value of Bio umber. Table.- liss he firs erm approimaio for a slab, a ifiie ylider, ad a sphere. Table.-3 liss he soluio for oe-dimesioal hea rasfer i a semi-ifiie medium for hree differe boudary odiios a he surfae =. Table.-4 shows he ombiaio of oe-dimesioal soluios o obai he muli-dimesioal resuls. -6

7 Table.- Coeffiies used i he oe-erm approimaio o he series soluios for rasie oe-dimesioal oduio or diffusio Bi m PNE W INFINITE CYINDER SPHERE ζ (rad) C ζ (rad) C ζ (rad) C

8 Table.- pproimae soluios for diffusio ad oduio (valid for Fo>.) D Fo = = D r, θ = K ' K ',, i, Diffusio i a slab, θ = C ep(-ζ F o ) is defied as he disae from he eer of he slab o he surfae. If oe surfae is isulaed, is defied as he oal hikess of he slab. θ = θ os(ζ M ) ; = si( M ζ ζ ) θ Diffusio i a ifiie ylider θ = θ J (ζ r M θ ) ; = J (ζ ) M ζ Diffusio i a sphere θ = θ si(ζ r M 3θ ) ; = 3 [si(ζ ) ζ os(ζ )] ζ r M ζ If he oeraio a he surfae,s is kow K, will be replaed by,s ζ ad C will be obaied from able a Bi m = Noaio: = oeraio of speies i he solid a ay loaio a ay ime,s = oeraio of speies i he solid a he surfae for >,i = oeraio of speies i he solid a ay loaio ad a =, = bulk oeraio of speies i he fluid surroudig he solid K, = = oeraio of speies i he solid ha is i equilibrium wih, M = amou of rasferred io he solid a ay give ime M = amou of rasferred io he solid as (maimum amou rasferred) hm Bi m = = raio of ieral resisae o mass rasfer by diffusio o eeral mass K ' D rasfer by oveio h m = k = mass rasfer oeffiie = for a slab wih hikess or a slab wih hikess ad a impermeable surfae = r o for radial mass rasfer i a ylider or sphere wih radius r o K = equilibrium disribuio oeffiie D = diffusiviy of i he solid -8

9 Table.-3 Semi-ifiie medium Cosa Surfae Coeraio: (, ) =,s, s, i, s = erf D ; N = D = = D (, ), s i π D Cosa Surfae Flu: N (=) = N (, ),i = N π D ep 4 D N erf D D The omplemeary error fuio, erf(w), is defied as erf(w) = erf(w) K ', i,, i = erf Surfae Coveio: D = h m ( f, ) = D hm hm ep + K ' D K ' D erf hm + D K ' D Noaio: = oeraio of speies i he solid a ay loaio a ay ime,s = oeraio of speies i he solid a he surfae for >,i = oeraio of speies i he solid a ay loaio ad a = f = oeraio of speies i he liquid a he solid-liquid ierfae a ay ime, = bulk oeraio of speies i he fluid surroudig he solid K, = = oeraio of speies i he solid ha is i equilibrium wih, h m = k = mass rasfer oeffiie K = equilibrium disribuio oeffiie D = diffusiviy of i he solid -9

10 Table.-4 Mulidimesioal Effes r o r o (r,) r (r,,) The oeraio profiles for a fiie ylider ad a parallelpiped a be obaied from he oeraio profiles of ifiie ylider ad slabs. [ fiie ylider ] = [ ifiie ylider ] [ slab ] [ parallelpiped ] = [ slab ] [ slab ] [ slab 3 ] S(, ) (, ) K ' K ', i,, Semi-ifiie solid P(, ) (, ) K ' K ', i,, Plae wall C(r, ) ( r, ) K ' K ', i,, Ifiie ylider -