Mixing in an agitated tubular reactor. J.J. Derksen. School of Engineering, University of Aberdeen, Aberdeen, UK.

Transcription

1 Mxng n an agtated tubular reactor J.J. Derksen School of Engneerng, Unversty of Aberdeen, Aberdeen, UK jderksen@abdn.ac.uk Submtted to Specal Issue WCCE10/CFD of CJCE January 018 Revson submtted February 018 Accepted March 018 Abstract We analyze, through numercal smulatons, the sngle-phase lqud flow and assocated passve scalar mxng n a tubular reactor that s agtated by lateral shakng whch nduces the moton of a sold mxng element nsde the reactor. The Reynolds number assocated to the shakng moton s n the range 1,00 5,600. Dependent on ts specfc value we perform drect or large-eddy smulatons. A fxed-grd lattce- Boltzmann method s used for solvng the flud flow. The movng boundary condton at the surface of the mxng element s dealt wth by means of an mmersed boundary method. To quantfy mxng, a transport equaton for a passve scalar s solved n conjuncton wth the flow dynamcs. Keywords Tubular reactor, mxng, drect & large-eddy smulaton, mmersed boundary method, scalar transport 1

2 1 Introducton A novel type of contnuous mxng system s shown n Fgure 1. It conssts of an outer tube wth crcular cross secton that undergoes a snusodal moton n the lateral horzontal drecton. Ths n turn nduces the lateral moton and rotaton of a sold nternal mxng element (called the nternal n the remander of ths paper) that has the form of a round tube wth holes, as depcted n the fgure. Ths through-flow mxng system can be used as a reactor, a blender, as well as a system to promote sold partcle suspenson and agtaton. In ths paper we only consder ts sngle-phase operaton. The workng flud s a lqud of constant densty and vscosty. The research descrbed n ths paper s purely numercal and of an exploratory nature. The am s to show the feasblty, as well as the lmtatons, of our numercal approach to solve flud flow and scalar transport n ths geometrcally and knematcally complex flow system under transtonal and mldly turbulent condtons. Such smulatons then wll allow us n future work to make predctons of the performance of ths mxng devce under specfc process condtons that can be compared wth expermental data as well as wth the performance of other contnuous flow mxng devces at comparable condtons. As can be judged from Fgure 1, the dmensonalty of the parameter space of ths mxng devce s extensve and contans the frequency and ampltude of the shakng moton, lqud propertes, the throughflow rate, gravtatonal acceleraton and the densty of the nternal, as well as geometrcal (.e. aspect) ratos. Numercal smulatons are an effectve way to explore for process desgn purposes large parts of the parameter space and we plan to do so n future research. Future research also should nvolve the assessment of numercal effects most mportantly establshng grd ndependence as well as valdaton by means of expermental data. Ths bref paper s organzed n the followng manner: We wll start by defnng the flow system and characterzng t n terms of a set of dmensonless numbers. Then we make a number of smplfyng assumptons. Subsequently turbulence treatment and numercal methods are brefly dscussed, wth

3 referencng to the lterature for further detals. In the Results secton the frst part descrbes flow feld results, the second part passve scalar transport. The paper s closed wth a Summary & Conclusons secton. Flow system The geometrcal layout of the agtated tubular reactor s gven n Fgure 1. The man geometrcal parameters are the nner dameter of the outer tube (D) and ts length L, as well as the dameter of the nternal ( D ). Mxng s enhanced by holes n the cylndrcal nternal. The placement and dameters of the holes along wth further detals are provded n Fgure 1 that also defnes the coordnate system. Fgure 1. Flow geometry. An nternal mxng element (n green) can move nsde a tube of dameter D that s shaken n the x-drecton. The nternal feels a net gravty force that acts n the negatve z-drecton. The overall flow s n the y-drecton; the smulatons assume perodc condtons n y-drecton. The dmensons as gven n the drawng relatve to the tube dameter are: D= 0.6D, L= 1.8D, dh= 0.5D, t= 0.013D. The orgn of the coordnate system s at the center of the outer tube and at ts front. 3

4 The nternal has three degrees of freedom, as specfed n Fgure. These are the lateral horzontal locaton of ts center x, ts vertcal locaton z and ts rotaton angle α along the y-axs. The nternal does not touch the outer tube wth ts surface. It has spacers that keep a mnmum dstance equal to ε between the nner surfaces of the outer tube and the outer surface of nternal. The workng flud s Newtonan wth knematc vscosty ν and densty ρ. The sold materal the nternal s made of has densty ρ = 8.0ρ, except for one smulaton that has a densty rato of 4.0. The nternal feels a net gravty s force n the negatve z-drecton. Fgure. Schematc of the degrees of freedom of the nternal ( x, z and angle α ), the assocated veloctes ( v, v,ω ) and the mnmum dstance ε between nternal and tube; ε= 0.038D. x z The entre system s agtated by shakng t n a snusodal way n the x-drecton: the dsplacement has ampltude A and frequency f: Asn( π ) ft. Ths shakng nduces moton of the nternal whch n turn nduces mxng. When n operaton, the mxer has a contnuous through flow n the y-drecton wth volumetrc flow rate φ V gvng rse to a superfcal velocty of U 4φV πd. The above set of process parameters has been combned to a set of non-dmensonal numbers where the length scale s mostly taken as the outer tube dameter D. Other dmensons of the system can be derved from the aspect ratos as gven n Fgures 1 and. The through-flow Reynolds number s defned 4

5 as Re φ UD ; the shakng moton s characterzed by a Reynolds number ν Re s π fad, a Strouhal ν number Str fd U, and the ampltude over dameter rato A D. The rato of shakng nerta and net gravty of the nternal s expressed as ( f) ( ρ ρ) ρa π θ g s. Fnally we have the densty rato ρ ρ s. The smulatons are performed n a reference frame that moves wth the shakng moton. Ths mples that an nertal body force s appled to the lqud: f ρ ( π ) sn( π ) = A f ft ex. Furthermore we apply perodc condtons n the y-drecton. Ths means that the geometry as shown n Fgure 1 can be seen as a secton taken from a much longer mxng system. A major smplfcaton n the current study s that we set the volumetrc flow rate φ V = 0. In ths case, the mxng s only due to the shakng moton. It also mples that Reφ = 0 and that the Strouhal number becomes nfnte and, n ths stuaton, s not a meanngful dmensonless parameter. When the dstance between the outer surface of the nternal and the nner surface of the tube equals ε, the nternal colldes. In the smulatons we have appled two collson models. In the hard model a fully elastc collson s performed and the velocty of the nternal changes nstantaneously. In the soft model, once the dstance ε s reached, an elastc sprng s actvated that apples a normal repulsve force on the nternal proportonal to the overlap: = ( ε ) F k s, wth k the sprng constant and s < ε the s dstance between nternal and outer tube. The hard as well as the soft model assume frctonless collsons so that the angular velocty Ω of the nternal does not change upon collson. 3 Numercs The lattce-boltzmann method has been appled to solve the flud flow. The specfc numercal scheme s due to Somers [1]. The scheme apples a unform, cubc (edge length ) grd wth the flow varables defned n the cube centers. Off-grd boundary condtons, such as the no-slp condtons on the tube 5

6 surface and on the nternal, are enforced through an mmersed boundary method []. These surfaces are represented by a set of closely spaced ponts (nearest neghbor spacng typcally 0.7 ); for the nternal ths set of ponts s shown n Fgure 3. At these ponts we force the flud to have the same velocty as the local velocty of the sold surface. The mmersed boundary method used s based on nterpolaton of veloctes and extrapolaton of forces []. Veloctes at the mmersed boundary ponts are estmated by nterpolaton of the surroundng grd veloctes; the forces actng on the flud at the mmersed boundary ponts are extrapolated to the surroundng grd nodes. Integraton of the mmersed boundary forces over the surface of the nternal gves the total hydrodynamc force F and torque T on the nternal. The x and z component of F as well as the y component of T are taken nto account when solvng the equatons of moton of the nternal. Fgure 3. Representaton of the nternal mxng element by a closely spaced set of ponts on ts surface. The total number of ponts s approxmately 70,000. In ths study, the cubc grd has a resoluton such that the tube dameter D spans 156 lattce spacngs. Ths leads to an overall grd node count of approxmately 5 mllon. The number of mmersed boundary ponts on the nternal s 70,000; on the outer tube t s 140,000. At ths stage of the research we have not yet consdered grd effects and thus are not able to show grd convergence. Tme steppng n 6

7 lattce-boltzmann methods s largely dctated by constrants on the Mach number [3]. In the smulatons presented here, t takes 10,000 tme steps to complete one shakng cycle,.e. f t = The three equatons of moton of the nternal (translatonal n x and z drecton, rotatonal along the y-axs) are solved wth a splt-dervatve method [4]. The forces on the nternal are gravty and hydrodynamc force F, as well as the collson force f a soft collson model s appled. The hydrodynamc torque T on the nternal s ncluded n the rotatonal equaton of moton. In addton to solvng the flow and the dynamcs of the nternal, we have solved the transport of a passve scalar accordng to a convecton-dffuson equaton: c + =Γ u c c t (1) wth c the scalar concentraton, u the velocty feld that s the result of the lattce-boltzmann flow solver, and Γ the dffuson coeffcent. Equaton 1 s dscretzed accordng to a fnte volume method on the same (unform, cubc) grd as appled by the lattce-boltzmann scheme, and explctly updated n tme wth the same tme step as used n the lattce-boltzmann method. We apply flux lmters [5] to suppress false dffuson. In the present work we set Γ= 0 so that the (lmted) levels of dffuson observed n the smulatons are due to the fnte resoluton of the grd and stll some numercal dffuson. Procedures smlar to the ones proposed n [6] were used for settng non-penetraton boundary condtons at the movng sold surfaces of the nternal, as well as for assgnng scalar concentratons to grd nodes that are uncovered due to the moton of the nternal. In ths work, the Reynolds numbers assocated to the shakng moton ( Re s ) were n the range 1,00 5,600. By default t was attempted to solve the flow wthout the use of a turbulence model. Unphyscal nstabltes occurred when Re s exceeded approxmately 3,000. In such stuatons we swtch to largeeddy smulatons (LES) and apply a standard Smagornsky subgrd scale model [7] wth Smagornsky constant C = 0.1. In the LES cases, the scalar dffuson coeffcent Γ s kept at zero level,.e. we do not S apply a fnte level turbulent Schmdt number. 7

8 4 Results 4.1 Flud flow A base case has been defned wth the followng dmensonless parameters: Re s =5,600, A D =1.13, ρs ρ =8.0, and θ= 0.37 and the geometry as defned n Fgures 1 and. Ths case apples hard collsons between nternal and outer tube, and a large-eddy approach for turbulence modellng. In Fgure 4 mpressons of the flow feld n a xz-cross secton passng through four holes n the nternal ( y= L 4 ). The shakng moton s started wth the nternal n the centre of the outer tube ( x= z = 0 ) and flud at rest ( u= 0 everywhere). The velocty magntude contour plots n Fgure 4 llustrate how the flow develops from ths ntal condton. These and all subsequent mages are n the reference frame movng wth the shakng moton. Intally ( t= 0.5 f ) the nternal moves down due to net gravty and to the rght snce the shakng moton starts to the left. In the subsequent frames we see an overall ncrease n the velocty magntude. The velocty vector plot shows an nstantaneous flow feld after the system has fully developed. It also gves an mpresson of the level of resoluton of the smulatons. 8

9 Fgure 4. Left: nstantaneous velocty vectors n an xz cross secton of the flow at y= L 4 after the flow s fully developed. Rght: mpressons of how the flow starts n terms of velocty magntude contours n the same xz-plane. From top to bottom: tf = 0.5, 0.5,1.0,.0. Base-case condtons. Under base-case condtons, the moton of the nternal quckly settles n a perodc state, as can be judged from the tme seres of the vertcal (center) locaton of the nternal n Fgure 5. For an effectve operaton of ths mxng devce, nerta of the nternal appears mportant. If we change the densty rato from ρs ρ =8.0 (base-case) to 4.0 and keep all other parameters the same, the nternal gets hardly agtated (see the nd tme seres n Fgure 5) whch has detrmental effects on the mxng performance. 9

10 Fgure 5. Tme seres of the vertcal (z) coordnate of the center locaton of the nternal. Red curve: basecase (whch has a densty rato ρs ρ =8.0); blue curve: same as base-case except that ρs ρ =4.0. At Re s =,300 we are able to get stable results wthout the use of a subgrd-scale model. If we compare these drect smulaton results wth ones obtaned wth a Smagornsky subgrd-scale model at the same Reynolds number we see hardly any dfference n the flow feld as well as n the levels of agtaton of the nternal. As an llustraton we show, n Fgure 6, very close agreement of the tme seres of the vertcal poston of the nternal as obtaned wth LES and DNS. These results mply that our procedure s able to represent n a smooth way the transton from lamnar to turbulent flow. Modellng the collsons of the nternal s an mportant aspect of the smulatons. In the hard collson model the velocty of the nternal changes nstantaneously at the moment of the collson. Gven the slghtly compressble nature of the lattce-boltzmann scheme [3] ths generates waves n the flud. These waves are unphyscal f the ntenton s to smulate an ncompressble flow. The waves can be avoded by makng the collsons soft and usng an elastc sprng that generates a repulsve force on the nternal to push t away from the outer wall. The choce of collson model has some mpact on the moton of the nternal as can be seen n Fgure 7. The soft model leads to some vbratons of the nternal. On average, however, the soft and hard model result n very smlar moton of the nternal. Wthout any benchmark or expermental data t s hard to judge whch collson model gves more relable results. In 10

11 the remander of ths paper when dscussng scalar transport results have been generated by usng the hard model. Fgure 6. Tme seres of the vertcal (z) coordnate of the center locaton of the nternal. Comparson between LES and DNS at Re s =,300. The rest of the condtons are the same as for the base-case. Fgure 7. Tme seres reflectng the dfferences as a result of collson modellng; hard versus soft collsons. The parameter d s the dstance of the nternal from the wall so that a collson takes place f d ε=1. Base-case condtons except that Re s =,300. DNS flow smulatons. 4. Passve scalar transport In order to drectly test the mxng performance of the agtated tubular reactor, the transport equaton of a passve scalar concentraton c (Eq. 1) was solved n conjuncton wth the flow dynamcs. The ntal condton for the smulatons nvolvng scalar transport s a fully developed flow feld and a segregated scalar concentraton feld. The latter means that we set c = 1 for z > 0, and c = 0 for z < 0. 11

12 Fgure 8. Scalar mxng n the form of concentraton contours n the xz-plane at y= L 4. Top row: Re s =5,600 (LES); bottom row: Re s =1,00 (DNS). From left to rght tf = 0.0, 0., 0.4, 0.8,1.6 wth at t = 0 a fully developed flow and a segregated scalar feld. In Fgure 8 mpressons of the mxng process are shown for two dfferent Reynolds numbers Re s. We see the formaton of scalar concentraton stratons and n the course of tme a homogenzaton of the concentraton feld wth faster mxng for the hgher Reynolds number. To quantfy mxng we defne the scalar varance as σc c c c wth denotng spatal averagng over nstantaneous realzatons of the concentraton feld, so that σ c s a functon of tme. In Fgure 9 we show an example of a σ c tme seres where the spatal average s taken over the center vertcal (yz) plane. Gven that at tme zero the scalar s segregated, the startng value of σ c s approxmately 1. We see the decay of scalar varance and t beng faster for the hgher Reynolds number. If σ c = 0.0 s set as the level for a practcally homogeneous system, then for Re s =5,600 homogenzaton s acheved after less than 10 shakng perods; for Re s =1,00 t takes at least 15 perods. The perodc fluctuatons are due to the perodc moton of the nternal. 1

13 Fgure 9. Tme seres of scalar varance tme for Re s =5,600 (blue curve) and Re s =1,00 (red curve). σc c c c n the yz-plane wth x=0 as a functon of 5 Summary & conclusons In ths paper we have shown the feasblty of performng sngle-phase flow and mxng smulatons of a novel agtated tubular reactor. Smulatons so far have been on a systems operatng n the transtonal / early turbulent regme. It was shown that we can make a smooth transton from drect (.e. no turbulence model) to large-eddy smulatons. The smulaton procedure s able to apply a hard as well as a soft model for the collsons of the nternal mxng element wth the outer tube wall. It was hghlghted that the nternal mxng element needs suffcent nerta (.e. mass) to be suffcently agtated by the shakng moton of the outer tube. Includng passve scalar varance calculatons allow for drectly quantfyng the mxng process, for nstance n terms of the decay tme of scalar varance. Ths paper descrbes a feasblty study so that much more work needs to be done before applyng the smulatons for reactor desgn. The next steps are verfcaton (e.g. establshng grd convergence) and valdaton (e.g. comparson wth expermental data). Then we wll be ncludng through-flow so that we 13

14 wll see mxng not only due to the moton of the nternal but also due to an average axal velocty. It wll be nterestng to quantfy the relatve mportance of these two contrbutons to mxng as a functon of operatonal condtons. Acknowledgement Sncere thanks to Andrew Bayly and Y He (Unversty of Leeds, UK) for brngng ths flow system to my attenton. References [1] Somers, J.A., Drect smulaton of flud flow wth cellular automata and the lattce-boltzmann equaton, Appl. Sc. Res. 51, (1993). [] Derksen, J. and H. E. A. Van den Akker, Large-eddy smulatons on the flow drven by a Rushton Turbne, AIChE J. 45, 09 1 (1999). [3] Succ, S., The lattce Boltzmann equaton for flud dynamcs and beyond, Clarendon Press, Oxford (001). [4] Shardt, O. and J.J. Derksen, Drect smulatons of dense suspensons of non-sphercal partcles, Int. J. Multphase Flow 47, 5 36 (01). [5] Sweby, P.K., Hgh resoluton schemes usng flux lmters for hyperbolc conservaton laws, SIAM J. Numer. Anal. 1, (1984). [6] Derksen, J.J., Scalar mxng by granular partcles, AIChE J. 54, (008). [7] Smagornsky, J., General crculaton experments wth the prmtve equatons: 1. The basc experment, Mon. Weather Rev. 91, (1963). 14