Application of the advection-dispersion equation to characterize the hydrodynamic regime in a submerged packed bed reactor

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1 548 opyrigh 24 Tech Science Press Applicaion of he advecion-dispersion equaion o characerize he hydrodynamic regime in a submerged packed bed reacor Anónio Albuquerque, 1, Adério Araújo 2, Ercília Sousa 2 Summary The hydraulic characerisics of a laboraory submerged packed bed, filled wih a volcanic sone, pozzuolana, have been experimenally invesigaed hrough racer ess. Ses of essays a flow raes from 1 o 2.5 l/h in clean condiions were performed. The resuls showed a considerable amoun of dispersion hrough he filer as he hydraulic loading was changed, indicaing a mulipliciy of hydrodynamic saes, approaching is behavior o plug flow. An analyical soluion for he advecion-dispersion equaion model have been developed for a semi-infinie sysem and we have considered an appropriae physical boundary condiion. A numerical simulaion using finie difference schemes is done aking ino accoun his paricular boundary condiion ha changes according o he flow raes. Proper formulaion of boundary condiions for analysis of column displacemens experimens in he laboraory is criically imporan o he inerpreaion of observed daa, as well as for subsequen exrapolaion of he experimenal resuls o ranspor problems in he field. Inroducion The experimens were carried ou on a pilo scale packed bed (Fig. 1) made of ubular acrylic glass wih 7 cm inernal diameer, 41 cm oal packing lengh, submerged wih 3 cm of waer level. The filer was filled wih a homogeneous pozzuolana maerial wih 4 mm of effecive diameer and porosiy of.52. Five pors have been used o collec samples. The flow raes were measured by a perisalic pump. Experimens have been performed a flow raes of 1., 2. and 2.5 l/h, a differen carbon concenraions for a 33 cm packing lengh. These experimens will allow he sudying of he hydrodynamic characerisics along he filer. We injeced 1 ml of a racer (Blue Dexran) impulse immediaely above he liquid level being he response evaluaed by measuring he absorbance a 61 nm of colleced samples a equal ime periods. In verical columns, especially if he raio lengh/diameer is oo large (Bedien e al [3]), he effecs of liquid flow in he horizonal direcion x is considered no imporan compared wih he flux in he verical direcion z. In hese condiions, he mechanism of advecion, dispersion and exchange reacion in an isoropic and homogeneous packed bed under seady-sae condiions, are generally described by he well-known adveciondispersion equaion, see for insance, Ogaa and Banks [7], van Genuchen and Alves [8], van Genuchen and Parker [9], Levespiel [5], Bedien e al [3], R +V z = D 2 z 2, (1) 1 Deparameno de Engenharia ivil e Arquiecura, Universidade da Beira Inerior, Porugal 2 Deparameno de Maemáica, Universidade de oimbra, Porugal Proceedings of he 24 Inernaional onference on July, 24, Madeira, Porugal

2 549 opyrigh 24 Tech Science Press Figure 1: Schemaic represenaion of he experimenal apparaus where is he solue concenraion, D is he dispersion coefficien, V is he average porewaer velociy, is he ime and z is he disance. The parameer R accouns from possible ineracions beween he chemical and he solid phase of he soil. Here, we consider here is no ineracions beween he chemical and he solid phase and herefore R = 1. We consider a dimensionless parameer, called Pécle number, Pe = V L, where L is D he packing lengh. The Pécle number describes he relaive influence of he effecs caracerised by advecion-dispersion problems which involve a non-dissipaive componen and a dissipaive componen. The Pécle number also deermines he naure of he problem, ha is, he Pécle number is low for dispersion-dominaed problems and is large for advecive dominaed problems. Our ineres is in he soluion of The model problem +V z = D 2 z 2 (2) for >, z wih an iniial condiion (z,) = f (z) (3) and subjec o he boundary condiions lim (z,) = and (,) = g(),. (4) x Proceedings of he 24 Inernaional onference on July, 24, Madeira, Porugal

3 55 opyrigh 24 Tech Science Press The exac soluion of he problem (2)-(4) can be found using Laplace Transforms in and we will ge he soluion (z, ) = Z 1 π 1 π Z + V+z 2 D g( ˆτ)G (z, ˆτ)dˆτ + 1 π Z + V z 2 D f (z V + 2 Dξ)e ξ2 dξ f ( z V + 2 Dξ)e V z/d e ξ2 dξ, (5) where he funcion G (z, ˆτ) is given by G z (z, ˆτ) = 2 ˆτ) e (z V Dˆτ 3/2 2 /4Dˆτ. For our paricular case we have ha he iniial condiion is given by f (z) =. We need o deermine he boundary condiion, g(), which represens he solue concenraion on he inflow boundary. We have he following physical parameers: V in j denoes he volume of injeced racer; V sl is he volume of he liquid on he op of he packed bed; M is he mass injeced; sl is he concenraion of he liquid level where he racer is absorbed before going ino he packed bed hrough he media op and Q denoes he flow rae. M s We have ha sl = and he physical boundary condiion is given by he V in j +V sl following exponenial decay g() = sl e Q/V sl. (6) This condiion is obained considering ha he inflow concenraion is governed by he differenial equaion, dg d = Q V sl g wih g() = sl (7) which describes he inflow decay by a rae of Q/V sl. Noe ha for our specific case where he iniial condiion is given by (z,) = and he inflow is governed by (6) we have he analyical soluion (z,) = 1 Z g( ˆτ)G (z, ˆτ)dˆτ. (8) π Numerical soluion using a finie difference scheme To derive finie differences we suppose here are approximaions U n := {U j n } o he values (x j, n ) a he mesh poins x j = j x, j =,1,2,... Proceedings of he 24 Inernaional onference on July, 24, Madeira, Porugal

4 551 opyrigh 24 Tech Science Press If we choose a uniform space sep x and ime sep, here are wo dimensionless quaniies very imporan in he properies of a numerical scheme µ = D ( x) 2, ν = V x. The quaniy ν is usually called he ouran (or FL) number. We use he usual cenral, backward and second difference operaors, U j := 1 2 (U j+1 U j 1 ), U j := U j U j 1, and δ 2 U j := U j+1 2U j +U j 1 o describe he finie difference scheme. onsider he approximaion formula U n+1 j = [1 ν + ( 1 2 ν2 + µ)δ 2 + ν( 1 6 ν2 6 µ)δ2 ]U n j. (9) This scheme was firs proposed by Leonard [4] using conrol volume argumens. However, i can also be obained using a cubic expansion by inerpolaing Uj 2 n as well as U j 1 n, Un j and Uj+1 n, as we can see in Moron and Sobey [6]. The model problem we are ineresed in is defined on he half-line wih an inflow boundary condiion (,) = g(), where g() is defined by (6). onsequenly we consider U n = g(n ). (1) The scheme (9) is a higher order scheme and i uses wo poins upsream. Therefore i can no be applied on he firs inerior poin of he mesh. A his paricular poin we need o apply a numerical boundary condiion. To deermine he numerical boundary condiion we use for inerpolaion he poins U n, U 1 n, U 2 n and U 3 n and we bring in a forward hird difference insead of a backward hird order difference o yield U n+1 1 = [1 ν + ( 1 2 ν2 + µ)δ 2 + ν( 1 6 ν2 6 µ)δ2 + ]U n 1, (11) where + is he forward operaor defined by + U j := U j+1 U j. For more informaion on his and oher numerical boundary condiions see for insance Sousa and Sobey [1]. The use of his downwind hird difference does no affec accuracy since sill based on a cubic local approximaion. However, i does have some penalies in erms of sabiliy. Some more ineresing discussions could be done on he righ choice of he numerical boundary condiion which is independen of he physical boundary condiion (1). Proceedings of he 24 Inernaional onference on July, 24, Madeira, Porugal

5 552 opyrigh 24 Tech Science Press Numerical resuls versus experimenal resuls In his secion we presen he numerical resuls ha adjus he essays for hree differen flow raes. Table 1 shows he values of differen parameers necessary o he evaluaion of he inflow boundary condiion defined by (6). We can observe ha we have a differen boundary condiion for each flow rae Q. We show, in Fig. 2 and Fig.3, he experimenal resuls and he numerical simulaions for differen flow raes. The numerical resuls allow us o deermine he Pécle number, ha is helpful in he characerizaion of he hydraulic condiions. Q (l/h) V in j (ml) V sl (ml) M s (mg) sl (mg/l) Table 1. Parameers relaed o he deerminaion of he physical inflow boundary condiion. 5 Q=2.5 Q=2 5 Q=2.5 Q=2 4 Q= Q= (a) (b) Figure 2: (a) Experimenal resuls for flow raes Q = 1, 2, 2.5; (b) Numerical simulaion for flow raes Q = 1, 2, 2.5. The resuls lead us o conclude ha, according o he range of hydraulic loading applied, a large amoun of diffusion occurs in he filer bed. This occurrence is associaed o he likely combinaion of facors such as dead zones, immobile zones, shor-circuiing and diffusion (boh mechanical dispersion and molecular diffusion). The analyical soluion represened by (5) for he semi-infiniive sysem can accuraely predic he experimenal curves and may be applied o resuls from finie experimens as he one here menioned. To he numerical simulaion we use a numerical scheme quie appropriaed since when we have significan values of diffusion we need a larger sabiliy region, ha is, we need he mehod o converge o he analyic soluion in a region where we can have grea accuracy and a he same ime we are allowed o have significan diffusion. More experimens are in progress considering differen organic loadings a differen hydraulic loadings. Proceedings of he 24 Inernaional onference on July, 24, Madeira, Porugal

6 553 opyrigh 24 Tech Science Press (a) (b) (c) Figure 3: The same as Fig. 2 bu wih he numerical resuls and experimenal resuls in he same figure: (a) Q = 1: V =.828, D/V L =.65, Pe = 15.3 (b) Q = 2: V =.144, D/V L =.56, Pe = 17.8 (c) Q = 2.5: V =.168, D/V L =.54, Pe = 18.5 Reference 1. Nishioka, T. and Aluri, S. N. (1982): Analysis of Surface Flaws in Pressure Vessels, Journal of Pressure Vessel Technology, Vol. 14, pp Aluri, S. N. (1997): Srucural Inegriy and Durabiliy, Tech Science Press. 3. Bedien, P. Rifai, H. and Newell,. (1999): Ground waer conaminaion ranspor and remediaion. 2nd ediion, Prenice Hall PTR, New Jersey, USA. 4. Leonard, B.P. (1979): A sable and accurae convecive modeling procedure, ompuer Mehods in Applied Mechanics and Engineering Vol. 19, Levenspiel, O. (1986): The hemical Reacor Omnibook, O.S.U., Book Sore Inc, New York, USA. 6. Moron, K.W. and Sobey, I.J. (1993): Discreisaion of a convecion-diffusion equaion, IMA Journal of Numerical Analysis Vol. 13, Ogaa, A. and Banks, R. (1961): A soluion of he Differenial equaion of Longiudinal Dispersion in Porous Media, U.S. Geol.Survey, Paper 411-A, 7pp. 8. van Genuchen, M. and Parker, J. (1984): Boundary condiions for displacemen experimens hrough shor laboraory soil columns. J. Soil Sci. Soc. Ame., Vol. 48, 4, van Genuchen, M. and Alves, W. (1982): Analyical soluions of he one-dimensional convecive-dispersive solue ranspor equaion. Technical Bullein N Agriculural Research Service. USDA Riverside, alifornia. USA, 149pp. 1. Sousa, E. and Sobey, I.J. (22): On he influence of numerical boundary condiions, Applied Numerical Mahemaics Vol. 41, Proceedings of he 24 Inernaional onference on July, 24, Madeira, Porugal