February 28, 2013 COMMENTS ON DIFFUSION, DIFFUSIVITY AND DERIVATION OF HYPERBOLIC EQUATIONS DESCRIBING THE DIFFUSION PHENOMENA Mental Experiment regarding 1D randm walk Cnsider a cntainer f gas in thermal equilibrium and intrduce a very small amunt f the labeled gas ( special gas) at a single spt in the cntainer. The mlecules are mving nly by randm mtin as per the kinetic thery f gases. The special r labeled gas mlecules are identical in prperties t the rest f the gas. T further simplify the treatment withut lss in generality let us fcus n a phenmenn ccurring in ne dimensin alng the x-axis. We assume that in the ther directins perpendicular t x- axis there are n cncentratin gradients f the special gas. Cnsider nw an imaginary plane f unit area A (Insert a diagram as sketched in class) perpendicular t the x- directin and cmpute the net flw f the mlecules f the special gas acrss that plane during time t. In rder t cunt the number f special mlecules that crss the plane in a time interval t, we first cunt the number f mlecules in a vlume t adjacent t the plane. Here is the average actual mlecular velcity in the x- directin. The number f special mlecules that passes frm left t right is n t and frm right t left n t. Here, n -, n + are the number density (number per unit vlume) f special mlecules t the left and t the right f ur dividing plane at X. The mlecular flux, als called net mlecular current (mlecule/cm 2 s) frm left t right, i.e. in the directin f the x-axis then is n t n t t (n n ) (1) 1
The questin arises where d we cunt ur mlecular densities, n-, n +? We must cunt at psitins where mlecules started their flight, i.e at a distance equal t the mean free path away frm the plane at X. Nw we wuld like t use a cntinuus functin, c, t represent the cncentratin f the special mlecules. Then n n dc dc x 2 dx dx Ignre the factr 2 and substitute eq. (2) int eq. (1). dc dc D dx dx Hence, mlecular diffusivity D is prprtinal t the mean free path and mean mlecular velcity as indicated by eq. (4) (2) (3) D (4) The abve develpment uses many apprximatins, instead f x, n + and n - measured at distance, while fr mlecules that d nt travel perpendicularly t ur plane it is a slant distance that cunts. A mre careful analysis reveals s that 1 3 dc dx (3a) D 1 3 (4a) The mean free path expresses hw far the mlecules travel between cllisins (n the average). We can als talk abut the average time between cllisins,. It is knwn that (5) By cllisin crss sectin we mean the area within which the center f ur particle (mlecule) must be lcated if it is t cllide with a particular ther mlecule. 2
c 2 (r 1 2 r 2 2 ). Then c 2 n 1 (6) There is ne cllisin n the average when the particle ges a distance in which the gathering mlecules culd just cver the ttal area. Sme additinal cmments: If we have special mlecules (S) and backgrund mlecules, then there is a specific frce F acting n S imparting an average velcity n S. drift F m Drift velcity is prprtinal t this frce! (n general name fr cnstant f prprtinality). In electrical prblems where F qe (charge x field) drift qe where is mbility). (7) Let us adpt that name drift F (7a) Then m (8). Inic cnductivity drift qe q V b ( ddrift x t ) ins arrive at a plate. There are c i ins per unit vlume. The number reaching the plate is (c i A drift t). Each in carries charge q. The charge cllected = qc i A drift t. Current I V I q 2 c i b V R b R q 2 c i charge cllcted unit time qc i A drift Relatinship between resistances and mlecular prperties m 3
S referring t the diffusin flux f (3a) we can represent it as: i kt dc dx D i kt The diffusin cefficient is kt times the mbility cefficient. This is called Ficks first law and the expressin fr the flux is repeated belw. D dc dx Ficks 2 nd law then states: (Acc) (in) (ut) x c t x xx (3b) (4b) Upn dividing with x and taking the limit ne gets: c t x D 2 c x 2 This is a 1D representatin f the Ficks 2 nd law f diffusin. As a cnsequence f the nature f the abve, the respnse t an impulse at x = 0 at t= 0 is felt instantaneusly at all x (althugh expnentially small). The infinite speed f signal prpagatin by diffusin equatin is a paradx due t the theretical nature f the diffusin equatin. Hwever, the mlecules d nt cver the free path instantaneusly. S the time when cncentratins are cunted at the left and right side f the plane at X differs frm the time when the mlecular current is measured by apprximately the mean time between cllisins r /. Mathematically, the flux is evaluated at t based n cncentratin difference that existed at t. (t ) D c(t) x This yields (by expansin in Taylr series) (9) (t ) (t) t Dc x This frmulatin avids the infinite speed f prpagatin. It has been derived by (10) 4
Maxwelll (1867) Fick (1926) Davidv (1935) Gldstein (1951) Davies (1954) Cattine (1948) Vermtte (1958) Alternative derivatin f Ficks law Derivatin f Ficks law is nly valid if c varies slwly s that ne term Taylr expansin is permissible in equatin (10). When that is nt the case and cncentratin changes nticeably during r n the length scale, a mre general apprach is needed. - Cnsider 1D general case when c can change cnsiderably during a time perid f r n the length f mean path. - N cnvective flw Cnsider particles in randm walk alng x axis, v +, v - are velcities in each directin. Particle interacts instantaneusly with surrundings and as a cnsequence changes directins. Each particle has the prbability p + that it will mve with v + and prbability p = 1 p + that it will mve with v -. Time spent mving in a certain directin is a randm variable characterized by prbability depending n this perid. We cunt again particles crssing frm left t right and thse frm right t left, at a plane at x. The particles reaching x frm the left at time t had their last cllisin at the time t, and at the pint x - v ( 0) and btained the velcity v + in the directin f the plane at x, mved during time withut cllisins and did nt disappear during time f travel due t reactin. The ttal number f cllisins at pint x v and at time t, in terms f cntinuus functin c(x,t) describing the distributin f particles alng the x-axis, is equal t c( x v, t ) / where is the mean perid f time between cllisins. The flux f particles at x and at t frm left t right is j (x,t) p v G()H() c(x v,t )d (11) 5
where G() = prbability that a particle survives a time withut cllisins H() = prbability that a particle survives a time withut reacting. Similarly p j ( x, t) G( ) H( ) c( x v ; t ) d (12) v If the prbability that a particle mves withut cllisin during time lnger than a certain value des nt depend n the histry f particle mvement, then fr arbitrary t 1 and t 2. Nw G(t 1 )G(t 2 ) G(t 1 t 2 ) s that (13) G(t) e t / If we cnsider a first rder reactin, then H(t) e kt Mass balance requires p v p v The cncentratin is defined by j (14) (15a) c (15b) j The net flux in the psitive x-directin is: j j j Prbabilities f all permissible events sum t ne: (15c) p + +p - = 1 (15d) Cnsider nw j + and fr ease f writing drp the subscript + fr nw. Then, upn substituting equatins (13) and (14) int equatin (12) we get: with j pv e a c(x v,t )d (16) a (k 1 ) (17) 6
Integratin by parts f eq. (16) can be perfrmed as indicated belw: dv e a d 1 a ea and yields: u c(x v,t ) du (v c x c t )d j pv 1 a ea c(x v,t ) 1 a term int tw, n multiplicatin by ur cnstant a ne gets: e a (v c x c t )d (18) Upn substituting the integratin limit in the first term abve and splitting the secnd pv pv a c pv a c aj c( x, t) v e x v t d e d (, ) x (19) t Differentiatin f eq. (16) with respect t psitin and time yields: j pv x j pv t e e a a c d x c d t (20) (21) Substitutin f eqs. (20) and (21) int eq. (19) yields: aj pv j c x j t Nw this has been derived and is valid fr j +. A similar expressin, nly with the sign fr j x changed, because nw ne deals with resulting equatins are shwn belw: (k 1 ) j p c j t j x (22) c(x v ;t ), can be derived fr j -. The (23) (k 1 ) j p v c j t j x (24) 7
Divide eq.(23) with and (24) with and add them up. (k 1 ) j j p p c 1 j t 1 j j t x j (25) x Recall eqs. (15b), (15c) and (15d) and use them in eq. (25) t btain: 1 1 c j ( k ) c c (26) t x Upn cancellatin f a cmmn term this becmes: c t j x k c 0 (27) Subtract nw eq. (24) frm eq. (23) t get: (k 1 )( j j ) p p v Upn substitutin f eqs. (15a) (15c) we get: c t ( j j ) j x j x (28) ( k 1 ) j j t j x j x (29) Nw we can add and subtract the fllwing terms, given by expressin (30), hence effectively adding zer) t eq. (29). j x j x v j x v j x Upn, use f (15c) and regruping we get equatin (31): j k 1 j t j v x j v x Hwever frm eq. (15b) it fllws that v v j v x c j j v v (32) x x x Substitutin int eq. (31) yields: j x (30) (31) (k 1 ) j j t (v v ) j x v c v x (33) 8
Upn multiplicatin by we get: (k 1) j j t (v v ) j x v v c x 0 (34) In summary, hyperblic equatins with finite speed f prpagatin describe the diffusin prcess mre accurately. These equatins are: c j kc 0 t x j (1 k ) j ( v t v j ) v x c v x 0 (35) (36) Thus, equatins (35) and (36) need t be slved simultaneusly fr cncentratin, c(x,t), and fr the flux, j (x,t). this frms a system f hyperblic equatins and the speed f transmissin f infrmatin is acknwledged as finite! Read treatment by Richard Feynman in The Feynman Lectures n Physics by Feynman, Leightn and sands. Read applicatins by Westerterp and clleagues t packed bed and ther chemical engineering equipment.. Wave mdel fr lngitudinal dispersin: analysis and applicatins Westerterp, K.R. (Twente Univ f Technlgy); Dil'man, V.V.; Krnberg, A.E.; Benneker, A.H. Surce: AIChE Jurnal, v 41, n 9, Sep, 1995, p 2029-2039 ISSN: 0001-1541 CODEN: AICEAC Publisher: AIChE Abstract: An analysis and applicatins f the wave mdel fr lngitudinal dispersin are presented. Asympttic frms f the wave mdel are cnsidered and analytical slutins f typical linear statinary and nnstatinary prblems f chemical reactr engineering interest are btained and cmpared t thse fr the Fickian dispersin mdel. The wave mdel leads t efficient analytical slutins fr linear prblems, which in principle differ frm the slutins f the Fickian dispersin mdel; nly fr slwly varying cncentratin fields d the slutins f bth mdels apprach each ther. Spatial and time mments f the cncentratin distributin are btained fr pulse-dispersin prblems; the first three spatial mments f the mean, variance, and skewness have exact, large-time asympttic frms in the case f Taylr dispersin. Old experiments that culd nt be explained with the standard dispersin mdel are recnsidered and explained: the change with time f the variance f a cncentratin 9
pulse when the flw directin is reversed and the difference in values f the apparent axial dispersin cefficient and the back-mixing cefficient in a rtating disk cntactr. The experimental determinatin f mdel parameters is discussed. (29 refs.) Wave mdel fr lngitudinal dispersin: Applicatin t the laminar-flw tubular reactr Krnberg, A.E. (Twente Univ f Technlgy); Benneker, A.H.; Westerterp, K.R. Surce: AIChE Jurnal, v 42, n 11, Nv, 1996, p 3133-3145 ISSN: 0001-1541 CODEN: AICEAC Publisher: AIChE Abstract: The wave mdel fr lngitudinal dispersin, published elsewhere as an alternative t the cmmnly used dispersed plug-flw mdel, is applied t the classic case f the laminar-flw tubular reactr. The results are cmpared in a wide range f situatins t predictins by the dispersed plug-flw mdel as well as t exact numerical calculatins with the 2-D mdel f the reactr and t ther available methds. In many practical cases, the slutins f the wave mdel agree clsely with the exact data. The wave mdel has a much wider regin f validity than the dispersed plug-flw mdel, has a distinct physical backgrund, and is easier t use fr reactr calculatins. This prvides additinal supprt t the thery develped elsewhere. The prperties and the applicability f the wave mdel t situatins with rapidly changing cncentratin fields are discussed. Cnstraints t be satisfied are established t use the new thery with cnfidence fr arbitrary initial and bundary cnditins. 10