Mathematical Models of Dusty Gas Flow through Porous Media

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1 Mathematical Mdels f Dusty Gas Flw thugh Pus Media M.H. HAMDAN Depatment f Mathematical ciences Univesity f New Bunswick P.O. Bx 5050, aint Jhn, New Bunswick, EL 4L5 CANADA hamdan@unb.ca Abstact:- This wk epts n the ecent advances in the cntinuum appach t dusty gas flw mdeling thugh istpic pus stuctues. This appach has eceived cnsideable attentin ve the last half centuy due t the need t develp dusty gas flw mdels capable f descibing natual and industial tanspt phenmena, including subsuface tanspt f disslved suspended paticulates, design f liquid-dust sepaats, and analysis and design f filtatin systems. A numbe f mdels have ecently been develped t descibe gas paticulate flw thugh pus media, and accunt f bth the macscpic flw behavi as well as the micscpic inteactins that aise due t the pus micstuctue. Detailed knwledge f pus micstuctues leads t a bette undestanding f the inteactins between the phases invlved, and f the fces exeted by the pus matix n the flwing phases. Mathematical idealizatin f pus micstuctues has been implemented in the mdels, which descibe vaius dusty gas flw situatins and paticle tanspt thugh pus stuctues. We discuss in this wk mdels that assume eithe a unifm vaiable distibutin f paticles in the flw field, and mdels that pvide f mdeling flexibility using phase patial pessues. Key-Wds:- Pus Media, Paticle-Laden, Intinsic lume Aveaging. Intductin Paticle-laden flw thugh pus media is encunteed in a numbe f industial applicatins and natual settings, including deep-bed filtatin pcesses, [8, 9]; the design f liquid-dust sepaats, [6, 3]; subsuface tanspt f cntaminants, [9]; mvement f nutients int plant ts; gund-wate ecvey; sil cntaminatin by heavy metals; tanspt f sluies thugh pus stuctues, [0,, ], in additin t the gethemal and gechemical applicatins. These and vaius the applicatins emphasize the fundamental imptance f mdeling paticle-laden flw thugh pus media, and the slutin t initial and bunday-value pblems in this field, [, 3, 6,, 7]. The pblem f paticle-fluid flw thugh pus media invlves dynamics f a thee-phase system that is cmpsed f a caie fluid (fluid-phase), a dilute f paticles a suspensin (paticle-phase), and a flw dmain (a pus medium). The pus matix may be cnsideed a statinay phase in this thee phase system (hence thee ae essentially tw flwing phases), the pus matix may undeg sme fagmentatin and sme f its cnstituents ae added t the flw field as a thid phase. In this latte case, the psity f the medium changes due t the blcking f cnnected pes (thus educing the effective psity f the medium), due t the ceatin f me cnnected pes (thus inceasing the effective psity). Tansient fagmentatin and the assciated changes in psity ende psity a functin f psitin and time. In il esevi analysis, if a pus laye is impaied by the settling f paticles n the slid pus matix cnstituents, thus cnsticting the pes, the esult is pemeability eductin with advese effects n esevi yield (cf. [0, 5, 33] and the efeences theein). Of paticula inteest t the cuent wk ae mathematical mdels that descibe paticle-laden flw thugh istpic pus media, unde the assumptin f n fagmentatin f the slid matix. These mdels have been typically based n the cntinuum appach and the aveaging f affman s dusty gas flw equatins, [8], ve a epesentative elementay pus vlume. A numbe f mdels ae available t descibe the flw f a fluid-paticle mixtue in eithe cnstant psity vaiable psity media, wheein the paticle distibutin is eithe unifm vaiable, [, 7, 6, 8, 9, 0,, 3, 4, 30]. IN: IBN:

2 If the numbe density is nt assumed t be cnstant (that is, the paticle distibutin is nn-unifm), the mdel equatins culd descibe a flw situatin with a pssibility f paticle settling n the slid matix. Analysis f the behaviu f the dust paticles has been pvided in the wk f iyyam and Hamdan [30], and was fcused n analyzing the fces exeted by the pus matix n the paticle-phase and t initially mdel the diffusin and dispesin pcesses that take place. Building n the analysis pvided by iyyam and Hamdan, [30], the cuent wk pvides analysis f the cmplete mdel descibing paticle-fluid flw thugh an istpic pus stuctue cnsisting f a igid, slid matix f vaiable psity. The effects f the pus micstuctue n the flwing phases ae analyzed when the pus mateial is f the ganula type. The geneal case f nn-unifm paticle distibutin is cnsideed. Gvening Equatins The time-independent flw f a viscus, incmpessible paticle-laden flw is gvened by the fllwing cupled set f field equatins, [5], witten hee in a fm suitable f the vlume aveaging pcess: Fluid-phase cntinuity equatin U = 0. () Fluid-phase mmentum equatin 9µ ρ f UU = P+ µ U + α( U ) ρ f G. () Dust-phase cntinuity equatin α = 0. (3) Dust-phase mmentum equatin 9µ ρ p α = α( U ) ρ pαg+ ρ fαg. (4) whee U and ae the fluid-phase and paticle-phase velcity fields, espectively, P is the fluid pessue, ρ is the fluid density, ρ p is the paticle density, a is the spheical paticle diamete,α is the paticle distibutin, G is the lcal gavitatinal acceleatin, and µ is the fluid viscsity cefficient. The abve equatins epesent a deteminate system f eight scala equatins in the eight unknwns, U,, α, and P. If the paticle distibutin α is unifm thughut the flw field, equatins () thugh (4) epesent an ve-detemined system f eight equatins in seven unknwns U,, P. In de t btain a deteminate system when α is unifm, it is pssible t intduce a paticle-phase patial pessue in equatin (4), intepeted as the pessue necessay t be applied t maintain a unifm distibutin paticles in the flw dmain. The intductin f the paticle-phase patial pessue allws f sme mdeling flexibility f the flw at hand, [3]. Ou inteest is t develp a cntinuum mdel t descibe the flw f a paticle-fluid mixtue thugh an istpic pus mateial, with nn-unifm paticle distibutin. The equatins gvening the flw in fee space, i.e. equatins (), (), (3), and (4), abve, will be aveaged ve a Repesentative Elementay lume. The effects f the pus micstuctue n the flwing mixtue will be accunted f thugh the cncept f a Repesentative Unit Cell, [,, 3, 4]. 3 The Aveaging Appach Intinsic vlume aveaging and its ules have been descibed elsewhee (cf. [4, 3]), hweve, we summaize the appach in what fllws f ease f efeence. Let be a Repesentative Elementay lume, RE, defined in [4, 3, 4] t be a cntl vlume that cntains fluid and pus matix in the same pptin as the whle pus medium. The psity,, f this cntl vlume is the same as that f the whle pus medium, and is defined as the ati f the pe vlume t the bulk vlume f the RE. ymblically, psity is defined as = f (5) IN: IBN:

3 whee is the effective pe vlume within the RE that cntains the fluid and paticle phases. The vlumetic phase aveage f a quantity ψ is defined as: <ψ > = ψd φ (6) and the cespnding intinsic phase aveage (that is, the vlumetic aveage f ψ ve the effective pe space, φ ) is defined as: <ψ > = ψd. (7) φ Relatinship between the vlumetic phase aveage and the intinsic phase aveage can be seen fm equatins (5), (6), and (7), as: < ψ >= < ψ >. (8) The fllwing aveaging theems ae then applied t equatins (), (), (3) and (4). Letting ψ be a scala quantity, ψ a vect quantity, and c a cnstant (whse aveage is itself), then: ( i)... < cψ >= c< ψ > = c < ψ >. (ii) < ψ >= < ψ > + ψ nd whee is the suface aea f the RE that is in cntact with the flwing phases, and n is the unit nmal vect pinting int the slid. The quantity ψ is the deviatin f the aveaged quantity fm its tue (micscpic) value. whee ψ,ψ ae tw vlumetically additive quantities. ( v)... < ψ ψ >=< ψ >< ψ >+< ψ ψ = < ψ > < ψ >. + < ψ ψ 4 The Aveaged Equatins The abve aveaging ules ae applied t equatins () thugh (4) t btain the aveaged equatins belw. F cnvenience f ntatin, we dente < U > = u, < > = v, < P > = p, < G> = g, and α > = Γ. < Fluid-phase cntinuity equatin: u+ U nd = 0. (9) Fluid-phase mmentum equatin: ρ f uu = p + µ u 9µ + [ v u] Γ 9µ + < α > [ ] < > < U > - ρ f < U > < U > ρ fg ρ f < G > + ( µ U np ) d + ( µ U ρ f UU n) d. (0) Paticle-phase cntinuity equatin: > > ( iii)... < ψ >= < ψ >+ ψ nd < ψ > + ψ nd. ( iv)... < ψ mψ >=< ψ > m< ψ >= < ψ > m < ψ > = Γv+ < α > + Paticle-phase mmentum equatin: α nd = 0. () IN: IBN:

4 9µ ρ p Γvv = [ u v] Γ + ( ρ f ρ p ) Γg+ ( ρ f ρ p ) < α > < G > 9µ + < α > [ ] < U > < > ρ p < α > < > < > α ( n) d. () Equatins (9) and (0) epesent the fluid-phase intinsic vlume aveaged cntinuity and mmentum equatins, espectively, while equatins () and () epesent the paticle-phase intinsic vlume aveaged cntinuity and mmentum equatins, espectively. The deviatin tems and the suface integals appeaing in these equatins cntain the necessay infmatin n the inteactins between the flwing phases and the pus medium. 5 Analysis 5. uface Integals Invlving the Fluid- phase elcity In the study f gas-paticulate flw thugh fee-space, that is in the absence f a pus matix, it is custmay t impse a n-slip cnditin n the fluid-phase velcity [5]. F the incmpessible fluid at hand, cnsevatin f mass tanslates int vanishing nmal cmpnent f velcity. This can be seen by invking the divegence theem, and making use f the fluid-phase cntinuity equatin (), as fllws: nd = Ud = U 0. (3) Equatin (3) indicates that if the fluid-phase velcity appeas explicitly in a suface integal then the suface integal vanishes. Accdingly, the tem U nd vanishes in the fluid-phase cntinuity equatin (9), and the tem ( µ U ρ f UU n) d vanishes in the fluid-phase mmentum equatin (0). The fluid-phase cntinuity and mmentum equatins thus take the fllwing fms, espectively: u = 0. (4) ρ f uu = p + µ u 9µ + [ v u] Γ 9µ + < α > [ ] < > < U > ρ f < U > < U > ρ fg ρ f < G > + ( µ U np ) d. (5) 5. uface Integals Invlving the Paticle-phase elcity F the paticle phase, using the divegence theem and making use f the paticle-phase cntinuity equatin (3), we btain α nd = αd = 0. (6) We pint ut that (6) is valid f bth unifm and nn-unifm paticle distibutin. While this equatin gives the impessin that a n-slip cnditin shuld be impsed n the paticle-phase n a macscpic slid bunday, this in fact is nt the case as can be explained in the fllwing, [3, 4, 30]: ) As ealized in [30], equatin (6) indicates that the nmal cmpnent f the pduct f the paticle-phase velcity and the paticle distibutin vanishes. This is due t the absence f paticle shea. This in tun implies that hyddynamic settling f paticles n the slid matix is pssible if the paticle distibutin is nn-unifm. Paticles may settle n the slid matix, eflect back int the flw field, set int mtin paticles aleady settled. In effect, the integal nd is nt necessaily ze. ) If the paticle distibutin is unifm, then (6) implies that α is cnstant n the paticle-phase steamlines f tw-dimensinal flw. This is an imptant ealizatin that adds a meaning t the cncept f unifm paticle distibutin, α = cnstant, and can be seen by fist expessing the paticle-phase cntinuity equatin (3) as: IN: IBN:

5 α + α = 0. (7) Nw, if the paticle distibutin is unifm, then = 0 and (7) educes t: α = 0. (8) Equatin (8) implies that α is cnstant alng the paticle-phase steamlines in the case f twdimensinal flw. In light f (6), the paticle-phase cntinuity equatin () educes t: Γv + < α > = 0. (9) Nw, the suface integal α( n) d appeaing in equatin () epesents a shea fce. Due t the absence f paticle shea, this integal vanishes and equatin () educes t: 9µ ρ p Γvv = [ u v] Γ + ( ρ f ρ p ) Γg+ ( ρ f ρ p ) < α > < G > 9µ + < α > [ ] < U > < > ρ p < α > < > < >. (0) 5.3 Analysis f the Deviatin Tems The deviatin tems appeaing in the fluid-phase mmentum equatins (5), in the paticle-phase cntinuity equatin (9), and in the paticle-phase mmentum equatins (0), ae elated t the hyddynamic dispesin f the aveage phase velcities in the pus medium. Hyddynamic dispesin thugh pus media is the sum f mechanical dispesin (due t ttusity f the flw path and the pus micstuctue) and mlecula diffusin f the fluid-phase vticity, [3, 4, 30]. The deviatin tems < U U > f equatin (5), and < α > f equatin (0), invlve pducts f deviatins f aveage phase velcities. These ae inetial tems epesentative f mechanical dispesin due t the pus micstuctue. In pus media whee velcity and psity gadients ae nt high, these tems ae small, and hence can be neglected. Hweve, they may be f significance in media with high psity gadients, and may thus be mdeled using dynamic diffusivity [30]. The tem < G > appeaing in equatins (5) and (0) epesents a negligible deviatin f the aveage lcal gavitatinal acceleatin, hence igned. The tem < α > < U > < > ] f [ equatin (0), and its negative in equatin (5), epesents dispesin f the paticles due t fluctuatins in the aveage elative velcity vect. These have been analyzed in [, 30], and summaized in what fllws f the sake f cmpletin. In case f a unifm distibutin f paticles, α is cnstant and α < > = 0. Thus, the dispesin vect vanishes. F a nn-unifm paticle distibutin, hyddynamic dispesin may eithe be mdeled as a Fuie diffusin pcess, as diffusin expessed as a pduct f a diffusin cefficient vect, δ, and a numbe density diving diffeential, < Γ> Γd, whee Γ d is an aveage efeence paticle distibutin. In this latte case we have, []: < α > < U > < > ] = δ Γ Γ ]. [ [ d () Equatins (5) and (0) thus take the fllwing fms, espectively: ρ f uu = p + µ u 9µ + [ v u] Γ 9µ δ [ Γ Γ ] d ρ fg + ( µ U np ) d. () 9µ ρ p Γvv = [ u v] Γ + (ρ f ρ p ) Γg 9µ + δ [ Γ Γ ] d. (3) The tem < α > appeaing in the paticlephase cntinuity equatin (9) invlves the pduct f deviatins f paticle-phase velcity and paticle IN: IBN:

6 distibutin. Fllwing [, 30], we wite the paticlephase cntinuity equatin (9) in the fllwing fm: Γv = < α > < >. (4) Assuming that the amunt f paticle tansfe t be a functin f the ttal suface aea f the slid matix,, and a tansfe cefficient, ε, equatin () may be expessed in the fllwing fm, [, 30]: Γv = ε. (5) 5.4 The Dacy Resistance Tem The effects f the pus matix n the flwing mixtue ccu thugh the ptin f the suface aea f the slid that is in cntact with the flwing phases. The suface integal in () that invlves the pessue deviatin and the velcity gadient cntains the necessay infmatin t quantify these effects. In case f single phase flw thugh cnstant psity pus media, this tem may µ be identified with the Dacy esistance: u, whee k k is the pemeability, [7]. In case f paticle-fluid flw, Dacy esistance may be expessed in tems f the µ elative velcity, as: ( u v). Hweve, accuate k evaluatin f the suface integals depends n the knwledge f the pus micstuctue and its gemetic desciptin. me imptant micstuctue desciptins have been epted in the wk f Du Plessis [], Du Plessis and Masliyah [3, 4], and Du Plessis and Diedichs []. F example, in thei desciptin f ganula micstuctue, Du Plessis and Masliyah [4] pvided the fllwing value f the suface integal that appeas in equatin (), abve: ( µ U np ) d = 3 [3 f ( τ )(3τ ) l / τ ] µ u (6) wheein f is the pduct f the Reynlds numbe and the fictin fact assciated with the flw f a dusty fluid thugh pus media; and τ is the ttusity f the medium. These have been based n the cncept f a Repesentative Unit Cell (RUC), [3,4], that is defined as the minimal RE in which the aveage ppeties f the pus medium ae embedded. Du Plessis and Masliyah [4] pvided the fllwing elatinship between psity and ttusity f ganula RUC: / 3 τ = [ ( ) ] /. (7) 6 Cnclusin In the cuent wk we emplyed the methd f intinsic vlume aveaging t develp equatins f mtin gvening the flw f a paticle-fluid mixtue thugh istpic pus media with vaiable psity. We have cnsideed the geneal case f nn-unifm paticle distibutin. The final fms f the aveaged equatins ae as fllws. Fluid-phase cntinuity equatin: u = 0. Fluid-phase mmentum equatins: ρ f uu = p + µ u 9µ + [ v u] Γ 9µ δ [ Γ Γ ] d ρ fg + F whee, f ganula media, F= ( µ U np ) d = 3 [3 f ( τ )(3τ ) l / τ ] µ u. Paticle-phase cntinuity equatin: Γv = ε. Paticle-phase mmentum equatins: 9µ ρ p Γvv = [ u v] Γ + (ρ f ρ p ) Γg 9µ + δ [ Γ Γ ] d. IN: IBN:

7 F cmputatinal pupses, the fllwing ntatin is used in the final fm f the equatins: q = u; q =v. Refeences: [] F.M. Allan and M.H. Hamdan, Fluid-paticle Mdel f Flw thugh Pus Media: The Case f Unifm Paticle Distibutin and Paallel elcity Fields, Applied Mathematics and Cmputatin, l.83, #, 006, pp [] M.M. Awatani and M.H. Hamdan, me Admissible Gemeties in the tudy f teady Plane Flw f a Dusty Fluid thugh Pus Media, Applied Mathematics and Cmputatin, l. 00#, 999, pp [3]. M.M. Awatani and M.H. Hamdan, Nn-eactive Gas-Paticulate Mdels f Flw thugh Pus Media, Applied Mathematics & Cmputatin, l. 00#, 999, pp [4] Y. Bachmat and J. Bea, Macscpic Mdeling f Tanspt Phenmena in Pus Media, I: The Cntinuum Appach, Tanspt in Pus Media, l., 986, pp [5]. R.M. Ban and M.H. Hamdan, The tuctue f epaated Dusty Gas Flw at Lw and Mdeate Re, Int. J. f Engineeing cience, l. 7#3, 989, pp [6]. R.M. Ban and M.H. Hamdan, The teady Mtin f an Incmpessible Dusty Gas in Pus Media, Applied Mathematics & Cmputatin, l. 37#3, 990, pp [7]. R.M. Ban and M.H. Hamdan, On the Dacy- Lapwd-Binkman-affman Dusty Fluid Flw Mdels in Pus Media. Pat II: Applicatins t Flw int a Tw-Dimensinal ink, Applied Mathematics & Cmputatin, l. 54#, 993, pp [8] C. Ch and C. Tien, Analysis f Tansient Behavi f Deep-bed Filtatin, Cllid Inteface cience, l. 69, 995, pp [9] C.. Chysikpuls, E.A. udias and M.M. Fyillas, Mdeling f Cntaminant Tanspt Resulting fm Disslutin f Nn-aqueus Phase Liquid Pls in atuated Pus Media, Tanspt in Pus Media, l. 6, 994, pp [0] F. Civan and M.L. Rasmussen, Analytical Mdels f Pus Media Impaiment by Paticles in Rectilinea and Radial Flws, In Handbk f Pus Media, secnd editin, K. afai Ed., Tayl and Fancis, New Yk, 005, pp [] J.P. Du Plessis, Analytical Quantificatin f Cefficients in the Egun Equatin f Fluid Fictin in a Packed Bed, Tanspt in Pus Media, l. 6, 994, pp [] J.P. Du Plessis and G.P.J. Diedeicks, Pe-scale Mdeling f Intestitial Phenmena. In Fluid Tanspt in Pus Media, J.P. Du Plessis, ed., 997, pp Cmputatinal Mechanics Publicatins. [3] J.P. Du Plessis and J.H. Masliyah, Mathematical Mdeling f Flw thugh Cnslidated Istpic Pus Media, Tanspt in Pus Media, l. 3, 988, pp [4] J.P. Du Plessis and J.H. Masliyah, Flw thugh Istpic Ganula Pus Media, Tanspt in Pus Media, l. 6, 99, pp [5] M.R. Fste, P.W. Duck, and R.E. Hewitt, The Unsteady Kaman Pblem f a Dilute Paticle uspensin, Fluid Mechanics, l. 474, 003, pp [6]. M.H. Hamdan, Gas-Paticulate Flw thugh Istpic Pus Media: Pat I: Intinsic lume Aveaging, Develpments in Theetical and Applied Mechanics, l. XI, 99, pp. II [7] M.H. Hamdan, M.H. ingle-phase Flw thugh Pus Channels: A eview. Flw Mdels and Channel Enty Cnditins, Applied Mathematics and Cmputatin, 6 (,3), 994, pp [8]. M.H. Hamdan and R.M. Ban, A Dusty Gas Flw Mdel in Pus Media, Cmputatinal & Applied Mathematics, l. 30, 990, pp [9]. M.H. Hamdan and R.M. Ban, On the Dacy-Lapwd-Binkman-affman Dusty Fluid Flw Mdels in Pus Media. Pat I: Mdels IN: IBN:

8 Develpment, Applied Mathematics and Cmputatin, l. 54#, 993, pp [0]. M.H. Hamdan and R.M. Ban, Gas-Paticulate Flw thugh Istpic Pus Media. Pat II: Bunday and Enty Cnditins, Applied Mechanics f the Ameicas, l., 993, pp []. M.H. Hamdan and R.M. Ban, Numeical imulatin f Inetial Dusty Gas Mdel f Flw thugh Natually Occuing Pus Media, Develpments in Theetical and Applied Mechanics, l. XII, 994, pp [] M.H. Hamdan and K.D. awalha, Dusty Gas Flw thugh Pus Media, Applied Mathematics and Cmputatin, l. 75#, 996, pp Applied Mathematics and Mechanics, l.9, N.4, 008, pp [3] D. Thmas, P. Penict, P. Cntal, D. Leclec, and J. endel J., Clgging f Fibus Filtes by lid Aesl Paticles: Expeimental and Mdeling tudy. Chemical Engineeing cience, l. 56, 00, pp [3]. Whitake, The Methd f lume Aveaging. Kluwe Academic Publishes, 999, Ddecht. [33] R.C.K. Wng and D.C. A. Mettananda, Pemeability Reductin in Qishn andstne pecimens due t Paticle uspensin Injectin, Tanspt in Pus Media, 8, (00), pp [3] M.H. Hamdan and H.I. iyyam, Tw-pessue Mdel f Dusty Gas Flw thugh Pus Media, Int. J. Applied Mathematics, l., N. 5, 009, pp [4] M. H. Hamdan and H. I. iyyam, On the Flw f a Dusty Gas with Cnstant Numbe Density thugh Ganula Pus Media, Applied Mathematics and Cmputatin, l. 09, 009, pp [5] T. Iwasaki, me Ntes n and Filtatin, Ameican Wate Wks Assciatin, 9(0), 937, pp [6] D.C. Mays and J.R. Hunt, 005, Hyddynamic Aspects f Paticle Clgging in Pus Media. Envinmental cience and Technlgy, l. 39, 005, pp [7] M. Pat, On the Bunday Cnditins at the Macscpic Level, Tanspt in Pus Media, 54, 989, pp [8] P.G. affman, On the tability f Lamina Flw f a Dusty Gas, Fluid Mechanics, l. 3 Pat, 96, pp [9] M.M. hama and Y.C. Ytss, A Netwk Mdel f Deep Bed Filtatin Pcesses, AIChE J., 33(0), 987, pp [30] H.I. iyyam and M.H. Hamdan, Analysis f Paticulate Behaviu in Pus Media, IN: IBN: