Title: INfluence of drag and turbulence modelling on CFD predictions of solid liquid suspensions in stirred vessels

Transcription

1 Title: INfluence of dag and tubulence modelling on CFD pedictions of solid liquid suspensions in stied vessels Autho: A. Tambuini A. Cipollina G. Micale A. Bucato M. Ciofalo PII: S (13) DOI: Refeence: CHERD 1399 To appea in: Received date: Revised date: Accepted date: Please cite this aticle as: Tambuini, A., Cipollina, A., Micale, G., Bucato, A., Ciofalo, M.,INfluence of dag and tubulence modelling on CFD pedictions of solid liquid suspensions in stied vessels, Chemical Engineeing Reseach and Design (2013), This is a PDF file of an unedited manuscipt that has been accepted fo publication. As a sevice to ou customes we ae poviding this ealy vesion of the manuscipt. The manuscipt will undego copyediting, typesetting, and eview of the esulting poof befoe it is published in its final fom. Please note that duing the poduction pocess eos may be discoveed which could affect the content, and all legal disclaimes that apply to the jounal petain.

2 HIGHLIGHTS Solid-liquid suspensions in stied tanks ae investigated by CFD. Patial to complete suspension conditions wee studied. Altenative models fo inte-phase dag foce and tubulence closue ae tested. Results ae validated against a lage numbe of expeimental data. Asymmetic k- model plus Bucato et al coection is found to be a good compomise. 1 Page 1 of 51

3 INFLENCE OF DRAG AND TRBLENCE MODELLING ON CFD PREDICTIONS OF SOLID LIQID SSPENSIONS IN STIRRED VESSELS A. Tambuini a, A. Cipollina a, G. Micale a *, A. Bucato a, M. Ciofalo b a Dipatimento di Ingegneia Chimica, Gestionale, Infomatica, Meccanica b Dipatimento Enegia, Ingegneia dell'infomazione e Modelli Matematici nivesità di Palemo, Viale delle Scienze Ed. 6, Palemo (ITALY) * Coesponding autho: giogiod.maia.micale@unipa.it Abstact. Suspensions of solid paticles into liquids within industial stied tanks ae fequently caied out at an impelle speed lowe than the minimum equied fo complete suspension conditions. This choice allows powe savings which usually ovecome the dawback of a smalle paticle-liquid intefacial aea. Despite this attactive economical pespective, only limited attention has been paid so fa to the modelling of the patial suspension egime. In the pesent wok two diffeent baffled tanks stied by Rushton tubines wee simulated by employing the Euleian-Euleian Multi Fluid Model (MFM) along with eithe the Sliding Gid algoithm (tansient simulations) o the Multiple Refeence Fame technique (steady state simulations). In paticula, a compaison of altenative modelling appoaches fo inte-phase dag foce and tubulence closue is pesented. The esults ae evaluated against a numbe of expeimental data concening sediment featues (amount and shape) and local axial pofiles of solids concentation, with emphasis on the patial suspension egime. 2 Page 2 of 51

4 Results show that some of the appoaches commonly adopted to account fo dense paticle effects o tubulent fluctuations of the volumetic factions may actually lead to substantial discepancies fom the expeimental data. Convesely simple models which do not include such additional effects give the best oveall pedictions in the whole ange of patial to complete suspension conditions. Key wods: Computational Fluid Dynamics (CFD); stied tank; solid liquid suspension; dag foce; tubulence model; multiphase flow. 3 Page 3 of 51

5 1. INTRODCTION AND LITERATRE REVIEW 1.1 Patial and complete suspension conditions Many eseach effots have been devoted to the investigation of solid-liquid stied tanks in the last decades (Zwieteing, 1958; Nienow, 1968; Boune and Shama, 1974; Yamazaki et al, 1986; Baesi and Baldi, 1987; Oldshue and Shama, 1992; Amenante et al., 1998; Micale et al., 2000; Riege, 2000; Angst and Kaume, 2006; Sadeshpande et al., 2009; Tambuini et al., 2009b,2013a; Jiout and Riege, 2011; Montante et al., 2012). The minimum impelle speed fo complete suspension conditions, N js is the most investigated topic since it is known to epesent a good compomise between the eduction of agitation costs and the enhancement of mass tansfe pocesses. Many studies focus on the poposal of methods to measue N js (Zwieteing, 1958, Musil and Vlk, 1978; Bohnet and Niesmak, 1980; Rewatka et al., 1991; Micale et al., 2002; Zhu and Wu, 2002; Jiout et al., 2005; Ren et al., 2008; Bucato et al., 2010; Tambuini et al., 2011c,2012c; Selima et al., 2008) o CFD models fo its pediction (Kee and Tan, 2002; Wang et al., 2004; Muthy et al., 2007; Hosseini et al., 2010; Tambuini et al., 2012a). Despite this taditional inteest fo the assessment of N js, in many industial solid-liquid stied eactos the best compomise between cost eduction and pocess efficiency is achieved by opeating at an impelle speed lowe than N js (Oldshue, 1983; Riege et al., 1988; Van de Westhuizen and Deglon, 2007; Jafai et al., 2012; Wang et al., 2012). Van de Westhuizen and Deglon (2007) investigated a mechanical flotation cell and stated that paticles sedimentation and associated losses may become dastic only when the system is opeated at an N < 0.6 N js (unde gassed condition). Jafai et al. (2012) epoted data concening the opeation of an industial sluy eacto fo the gold cyanidation pocess at N = 0.8 N js ; a choice which (i) did not appeciably affect the eaction selectivity and pocess yield and (ii) allowed a substantial eduction of enegy consumption with espect to complete suspension conditions. Also, they epoted that a decease of 4 Page 4 of 51

6 ~0.55 M$/yea in poduct value due to opeating at N = 0.5 N js coesponded to an enegy saving of ~1M$/yea, with a ~0.45 M$/yea incease in net pofit. In this egad, the availability of data on the patial suspension conditions may be vey useful fo the industy. Such infomation could allow economical analyses to be pefomed leading to the optimal choice of N. Notwithstanding the inteest expessed so fa at the industial level fo patial suspension conditions, only a few data can be found in the liteatue fo this paticula egime and many of them wee collected with diffeent aims. Fo instance, except fo a few cases (Tambuini et al., 2011a,b, 2012b), all liteatue woks pesenting CFD data on systems unde patial suspension conditions ae devoted to the evaluation of N js (Kee and Tan, 2002; Oshinowo and Bakke, 2002; Wang et al., 2004; Ochieng and Lewis, 2006; Muthy et al., 2007; Kasat et al., 2008; Panneeselvam et al., 2008; Hosseini et al., 2010): most of the citeia adopted in the liteatue to estimate N js equie data at N < N js and N > N js thus leading to the necessity of pefoming simulations unde patial suspension conditions. Howeve, none of these woks fully investigated the patial suspension egime. A univesal CFD model capable to manage all the types of solid-liquid suspensions within stied tanks does not exist yet. Fo the case of complete suspension, it is possible to find in the liteatue diffeent fomulations fo the inte-phase dag foce teatment as well as fo the tubulence closue. 1.2 Inte-phase dag foce The inte-phase dag foce is one of the most cucial factos affecting both solids suspension and distibution. Gidaspow s dense paticle effect (Gidaspow, 1994) is the most accepted fomulation of the inte-phase dag foce which takes into account the paticle-paticle inteaction effect (Gidaspow, 1994; Ochieng and Onyango, 2008; Scully and Fawley, 2011): accoding to this fomulation, the highe the local solid volume faction, the highe the inte-phase dag foce as a consequence of moe intense paticle-paticle inteactions. When the solid loading is vey high, the 5 Page 5 of 51

7 adoption of Egun s equation (Egun, 1952) fo the inte-phase dag foce is commonly suggested (CFX-4 Documentation, 1994; Ochieng and Onyango, 2008) although it was oiginally fomulated fo the case of fixed beds of paticles. In some cases, the two pevious appoaches ae simultaneously employed fo the simulation of the same system, the fome being employed in the domain egions with low solid volume factions and the latte in the domain egions with highe solid volume factions (Gidaspow, 1994; CFX-4 Documentation, 1994; Ochieng and Onyango, 2008) thus leading to a discontinuity in the elation between inte-phase dag foce and solid volume faction. Recently, Tambuini et al. (2009a) poposed a piecewise coelation employing the two above fomulations along with a linea intepolation between them fo an intemediate ange of solid volume factions theeby avoiding any discontinuity and poviding a monotonic behaviou in the inte-phase dag foce vs solid volume faction elation. 1.3 Tubulence closue As fa as the tubulence closue is concened, the k- tubulence model (along with the Euleian- Euleian teatment of the two phase system) is the most widely employed fo solid-liquid systems. In paticula, fou main two-phase extensions of the standad k- tubulence model (homogeneous, pe phase, dispesed and asymmetic) can be encounteed in the liteatue. In the homogeneous appoach, only one k and one equations ae solved, whee the physical popeties of the mixtue ae adopted: the two phases shae the same k and ε value and the tanspot equations fo k and ε have no inte-phase tubulence tansfe tems. In the pe phase, o phase-specific, fomulation, the tubulence model equations ae solved fo each phase. Additional tems efeing to the modelling of inte-phase tanspot of k and have to be included in the equations elevant to each phase. Altenatively, in the dispesed appoach, suitable fo dilute suspensions, fluctuating quantities of the dispesed phase ae computed as functions of the mean chaacteistics of 6 Page 6 of 51

8 the continuous phase and the atio of the paticle elaxation time and eddy-paticle inteaction time (Gosman et al., 1992). Continuous phase tubulence is modelled using the standad k- model including exta tems which account fo the influence of the dispesed phase on the continuous one (Feng et al., 2012). Pedictions of tubulence quantities fo the dispesed phases ae obtained using the Tchen theoy of dispesion of discete paticles by homogeneous tubulence (Hinze, 1975). Recently Tambuini et al. (2011a) poposed the adoption of the asymmetic k- tubulence model and applied it to dense solid-liquid suspensions at N N js : accoding to this appoach, since many paticles may be unsuspended unde patial suspension conditions, only the tubulence of the liquid phase was accounted fo and no tubulent viscosity was calculated fo the solid phase. Montante and Magelli (2005) compaed the fist thee fomulations. They obseved that using moe computationally demanding appoaches like the pe phase fomulation does not lead to any significant impovement ove the homogeneous fomulation. Futhemoe, the homogeneous k-ε tubulence model povides a satisfactoy epesentation of the solids distibution thoughout the vessel fo a numbe of cases involving dense suspensions in stied tanks povided that N N js (Montante et al., 2001; Micale et al., 2004; Montante and Magelli, 2005; Khopka et al., 2006; Kasat et al., 2008; Tambuini et al., 2009a). On the basis of the above consideations, the pesent wok aims at compaing diffeent CFD models including altenative fomulations of eithe the inte-phase dag foce o the tubulence closue in ode to identify the best modelling pocedue to deal with the incomplete suspension egime. 7 Page 7 of 51

9 2 EXPERIMENTAL DATA The data employed fo the validation of the CFD simulations deive fom expeiments made by the authos (Tambuini et al., 2011a) and fom the liteatue (Micheletti et al., 2003). Such data ae elevant to two vey simila systems both consisting of solid-liquid suspensions in a cylindical flatbottomed baffled tank stied by a standad six-bladed Rushton tubine. A) The fist tank, sketched in Fig. 1, has intenal diamete T equal to 0.19 m, impelle diamete D equal to T/2 and impelle cleaance C equal to T/3, and is filled up to a height H=T. H=T C=T/3 W=T/10 D=T/2 T T = 0.19 m B=D/5 A=D/4 Fig. 1: System A The expeimental data (Tambuini et al., 2011a) egading this system ae elevant to thee diffeent suspensions of glass ballottini ( =2500 kg/m 3 ) in deionized wate with diamete ange and mass factions of: 8 Page 8 of 51

10 m, 33.8% w solid /w liquid (efeed as w/w in the following); m, 16.9% w/w; m, 33.8% w/w. These data povide the suspension cuve (i.e. the mass faction of suspended paticles x susp against the impelle speed N) and the height of the sediment h sed visible on the lateal wall midway between two subsequent baffles. B) The second tank, investigated by Micheletti et al. (2003), is a flat bottomed tank with T=H=0.29m, impelle diamete D equal to T/3 and impelle cleaance C equal to T/3. The expeimental data egad a suspension in deionized wate of m glass paticles ( =2470 kg/m 3 ) with 25% w/w. These data povide axial pofiles of solid concentation. Moe pecisely, local steady state solid concentations measuements wee pefomed by a conductivity pobe at a adial position R/T=0.35, midway between subsequent baffles and at diffeent heights of the tank. The above expeimental data ae the only ones available in the liteatue concening paticle distibution unde patial suspension conditions. This is the main eason why all the pesently epoted simulations wee limited to a Rushton tubine. The void faction of the paticle bed lying on the bottom unde no agitation conditions was estimated to be about 40% fo both systems on the basis of the liteatue (Tambuini et al., 2011a) and specific measuements. 3 MODELLING RANS simulations of the systems descibed in the pevious section wee pefomed by adopting the Euleian-Euleian Multi Fluid Model (MFM), available as a standad option in the commecial finite volume CFD code Ansys (R) CFX4.4. With espect to moe ecent eleases, this code vesion 9 Page 9 of 51

11 offes a geate flexibility in designing use-defined outines fo non standad poblems, thus, it has been extensively applied by the pesent authos to a vaiety of non standad CFD poblems (Micale et al., 1999; Di Piazza and Ciofalo, 2002). In the pesent wok, the choice of this code allowed the implementation of the Excess Solid Volume Coection solids edistibution algoithm descibed in section 4. Accoding to the MFM fundamentals, the two phases ae teated as two intepenetating continua: the continuity and momentum equations ae solved fo each phase, thus obtaining sepaate flow field solutions fo the liquid and the solid phase simultaneously. The two phases shae the same pessue field. Diffeent modelling appoaches wee implemented and tested in the pesent wok: all the combinations simulated ae summaized in Tab.1, while the main featues of each combination (i.e. tested modelling appoach) ae classified and descibed in the following paagaphs. Modelling appoach Refeence Model Asymmetic_1 tem Additional tems aising fom tubulence closue in continuity equation Intephase Dag Foce in momentum equations Tubulence Model Gidaspow-Egun Liquid phase Gidaspow's dense Piecewise No tems only Both phases Standad (eq.6) paticle effect coelation Homogeneous k- Asymmetic k- X DPE X X PwC X Homogeneous_no tems X X X Homogeneous_2 tems X X Table 1: Summay of the modelling appoaches featues 3.1 Refeence Model X X X X X X X X X A vey simple model was employed to evaluate its capability of pedicting the whole set of the expeimental data. As it will be bette descibed in the following, this model accounts only fo the tubulence of the liquid phase and adopts only a vey standad fomulation of the dag foce. Moeove, also the tubulence fluctuations of the volumetic factions ae not taken into account. As a second step, moe efined modelling appoaches will be also tested. This simple model epesents 10 Page 10 of 51

12 Page 11 of the efeence modelling appoach with espect to which, all the othe appoaches can be egaded as modifications. It was aleady employed by the pesent authos (Tambuini et al., 2011a) and will be only biefly descibed in the following. Continuity equations: Assuming both phases to be incompessible, fo each phase, the continuity equation is witten as a function of the elevant volume faction: 0 t (1) 0 t (2) whee the subscipts and efe to the continuous and dispesed phases espectively, is volumetic faction, ρ is density and is mean velocity. Clealy, 1 (3) Momentum equations: Assuming both phases to behave as Newtonian fluids, one has M g P t T t (4) M g P t T (5) whee g is gavity acceleation, is viscosity, t is tubulent viscosity, P is pessue and M is the inte-phase momentum tansfe tem. Notably, M was consideed to be equal to the dag foce, while contibutions due to othe foces wee neglected as suggested by the liteatue fo simila

13 systems (Ljungqvist and Rasmuson, 2001; Fan et al., 2005; Fletche and Bown, 2009; Sadeshpande et al., 2011). In paticula, accoding to Fletche and Bown (2009) the vitual mass and the lift foces have a negligible effect on the system dynamics; similaly, the Basset foce in most cases is found to be much smalle than the dag foce (Khopka et al., 2006; Kasat et al., 2008). Also, Tatteson (1991) estimated vitual mass and lift foces to be much smalle than the dag foce and, thus, to be negligible when / > 2 (as in the pesent wok). Intephase Dag Foce: M C 3 C D, tub 4 d p The paticle dag coefficient based on two phases slip velocity was estimated by means of the Clift et al. coelation (1978): p CD, slip Re (fo Re p p 1000) (7a) Re C D, slip 0.4 whee Re is the slip velocity Reynolds numbe: p (fo Re p >1000) d p Re p (8) The influence of the fee steam tubulence on paticle dag was accounted fo by employing the coelation by Bucato et al. (1998), Bucato CD, tub CD, slip d p (6) (7b) (9) whee is the Kolmogoov length scale, calculated by (10) whee the dissipation of tubulent kinetic enegy is povided by the tubulence model. 12 Page 12 of 51

14 Table 2 shows the coection of C D,slip povided by eq. 9, fo the case of the Refeence model. The values povided in the table ae elevant to the aveage value of in the computational domain. Clealy, the coection is cucial fo lage paticles and high impelle speeds. 231 m Bucato et al. Coelation N [RPM] d p / C D,tub /C D,slip SYSTEM A N [RPM] SYSTEM B 550 m 655 m Bucato et al. coelation Bucato et al. coelation d p / C D,tub /C D,slip N [RPM] d p / C D,tub /C D,slip Tab. 2: Influence of liquid fee steam tubulence on dag coefficient. k- tanspot equations: t t t t k k k k t T t C C 2 k 1 k t (11) 2 T C2 k (12) (13) The pesent vesion of the k- model was named Asymmetic k- tubulence model by Tambuini et al. (2011a). 13 Page 13 of 51

15 Altenatives to specific aspects of the efeence modelling appoach will be pesented late on. In paticula altenative inte-phase dag foce fomulations and diffeent tubulence closues will be tested. 3.2 Modifications to the Inte-phase Dag Foce Dense Paticle Effect (DPE) The Dense Paticle Effect (DPE) modification, following Gidaspow (1994), intoduces an additional facto E in the inte-phase dag foce equation to account fo paticle-paticle inteactions in dense suspensions (i.e. high solids loading) so that eq.(6) is ewitten as: M Bucato 3 C D, tub E C 4 d p whee E (14a) (14b) As shown by eq.(14b), lage solid volume factions lead to an enhancement of the dag foce as a consequence of moe fequent paticle-paticle collisions Piecewise Coelation (PwC ) This inte-phase dag foce modelling was poposed by Tambuini et al. (2009a) and successfully applied fo the case of the stat-up dynamics of a dense solid-liquid suspension. Thee diffeent equations ae used fo the computation of the inte-phase dag foce, each one fo a specific solids volumetic faction ange. low volumetic factions (0 < β < β_min ): equations 14 ae applied, yielding: 14 Page 14 of 51

16 M Bucato 3 C D, tub * 1 C 4 d p (15a) high volumetic factions ( β_max < β < β_packed ): the model adopts the inte-phase foce initially poposed by Egun (1952) to deal with closely packed fixed-bed systems: M 2 C d p d p (15b) intemediate volume factions ( β_min < β < β_max ): fo this ange a linea intepolation of the two pevious equations is used in ode to avoid any discontinuity in the C αβ - β elation: this unphysical behaviou would occu if only equations 15a and 15b wee employed. Adopting this intepolation with β_min and β_max set to 0.35 and 0.45, along with a C D computed via slip velocity (equation 7), povides in most cases a monotonic dependence of C vs (Tambuini et al., 2009a) M C C _ min C C _ max _ min _ max _ min _ min 3.3 Modification to the tubulence closue This sub-section is devoted to the desciption of the modelling appoaches which pesent some modification in the tubulence closue with espect to the Refeence Model. The tubulence closue concens both the continuity equations (because of the poducts between the fluctuating volumetic factions and the fluctuating velocity components) and the momentum equation (because of the (15c) poducts between the fluctuating velocity components). In the Refeence Model no tubulence closue is adopted fo the continuity equation and, as fa as the momentum equation is concened, only the tubulence closue of the liquid phase is addessed (asymmetic k- tubulence model). 15 Page 15 of 51

17 Page 16 of The phase specific teatment, in which k and fields ae sepaately computed fo each phase, was not taken into consideation in the pesent study because, as anticipated in the intoduction, it involves a geate complexity in the modelling of inte-phase momentum tansfe tems and does not offe significant advantages with espect to the homogeneous and the asymmetic models tested hee (Montante and Magelli, 2005) Homogeneous k- tubulence model Momentum equations: As concens the modification of the efeence momentum equation, both phases ae consideed hee to be tubulent and a tubulent viscosity appeas in the momentum equations of both phases. As aleady mentioned in the intoduction the homogeneous k-ε tubulence model was found to povide a fai epesentation of the solid distibution thoughout the vessel fo a numbe of cases dealing with dense suspensions in stied tank (Montante et al., 2001; Micale et al., 2004; Montante and Magelli, 2005; Khopka et al., 2006; Kasat et al., 2008). Theefoe, a homogeneous k-ε tubulence model was employed hee fo compaison puposes with the Refeence Model. k and tanspot equations emain pactically the same with espect to the single-phase case, but no volume factions ae pesent and all physical popeties appeaing in these equations ae the mixtue aveaged popeties. The intephase dag foce fomulation is equal to that of the Refeence Model. M g P t T t (16) M g P t T t (17) 2 k C t (18a)

18 Page 17 of k C t (18b) k- tanspot equations: T t k t k k k t (19) k C k C k t T t t (20) Velocities and physical popeties appeaing in equations ae the mixtue aveaged popeties. (21) 1 (22) (23) t t t (24) Continuity equations Tubulence closue may give ise to altenative fomulations of the continuity equation (Table 1). In the fist and simple case, the continuity equations ae eqs. 1 and 2: the tubulent dispesion of volumetic factions is neglected (Muthy et al., 2007; Hosseini et al., 2010; Wang et al., 2010; Liu and Baigou, 2013), i.e. the continuity equation does not include the tems aising fom Reynolds aveaging. Summaizing, this fome appoach includes tubulence closue tems only in the momentum equations. The Refeence model case utilizes this appoach. When a homogeneous tubulence model is used, the neglect of tubulence dispesion tems in the continuity equations gives ise to the teatment called Homogeneous_no tems in Table 1.

19 In a second case, named hee Homogeneous_2 tems, the homogeneous model is used fo tubulence and the two continuity equations include additional tems (Zhao et al, 2013): t t t 2 0 t 2 0 t t whee t is a tubulent Schimdt numbe which is assumed to be equal to 0.8 as suggested by the liteatue (Montante and Magelli, 2007). Accoding to Montante et al. (2001), contay to what happens with dynamic phenomena, the sensitivity of the esults to this paamete is negligible in steady-state solid-liquid systems. Summaizing, the Homogeneous_2 tems appoach includes tubulence closue tems in both the continuity and in the momentum equations by means of the homogeneous k- model. Finally a thid vaiant consists of adopting the asymmetic tubulence model and, coheently, in including the dispesion tem only in the liquid phase continuity equation (eq. 25), while neglecting it in equation 26 fo the solids phase continuity. This appoach is dubbed Asymmetic_1 tem in Table 1. The coesponding esults obtained with this appoach tuned out to be simila to (and actually slightly wose than) those elevant to the Refeence Model and thus will not be epoted in the following fo the sake of bevity. All these vaiants ae elevant to the Reynolds aveaging of the govening (continuity and momentum) equation. The Fave aveaging is a well-known altenative: in this case, no additional (25) (26) tems aise in the momentum equation while an additional tem named tubulent dispesion foce has to be included in the momentum equation (Sadeshpande et al., 2011; Gohel et al., 2012). The Reynolds aveaging along with the employment of diffusion tems in the continuity equation (to model the poduct of fluctuating volumetic faction and the fluctuating velocity) on the one hand 18 Page 18 of 51

20 and the Fave aveaging along with the inclusion of a suitable tubulence dispesion foce in the momentum equation on the othe hand, can be consideed as just diffeent ways to get consistent esults. Fo this eason, only Reynolds aveaged cases ae investigated in the pesent pape. 4 NMERICAL DETAILS Cental diffeences wee employed fo all diffusive tems and the QICK thid-ode upwind scheme fo the advective ones. The segegated SIMPLEC algoithm was adopted to couple pessue and velocity. Baffles and impelle blades wee consideed as thin sufaces. No-slip bounday conditions wee assumed fo all the tank boundaies with the exception of the top suface whee fee-slip conditions wee employed to simulate a fee suface. As fa as the teatment of the impelle-baffle elative otation is concened, both the steady state Multiple Refeence Fame (MRF) by Luo et al. (1994) and the time dependent Sliding Gid (SG) algoithm by Muthy et al. (1994) wee adopted in the pesent wok. Tambuini et al. (2011a) caied out both MRF and SG simulations on system A and found vey simila esults as egads suspension cuves and sediment height. As a consequence, the MRF technique was adopted to simulate the system A in view of its lage computational savings. Convesely, Tambuini et al. (2013b) have shown that diffeent pedictions can be found by employing the SG o the MRF appoach fo the case of system B whee local quantities ae pedicted: as a diffeence fom the suspension cuves (integal data) of system A case, the CFD pediction of local solids concentation pofiles (data elevant to system B) equies a moe accuate calculation, especially if diffeent modelling appoaches have to be caefully compaed. In accodance with the liteatue (Ochieng and Lewis, 2006; Panneeselvam et al., 2008) moe accuate pedictions of the solid-liquid flow field can be povided by the tansient SG simulation. Theefoe, the tansient SG appoach was adopted to simulate the system B investigated by 19 Page 19 of 51

21 Micheletti et al. (2003). A time step equal to the time the impelle needs to sweep an azimuthal angle equal to a cell (one cell time step) was adopted fo all the SG simulations pefomed. The position of both the suface sepaating the two domains both in the SG and in the MRF famewok was chosen in accodance with the assumptions of Luo et al. (1994) at the dimensionless adial position R = 0.62 (T/2). Accoding to the systems axial symmety, only one half of the tank was included in the computational domain and two peiodic boundaies wee imposed along the azimuthal diection. The stuctued gid chosen fo the discetization of system A encompasses cells distibuted as along the azimuthal, axial and adial diection, espectively. The computational gid is fine in the poximity of the impelle whee the lagest gadients of the flow vaiables ae expected. A simila stuctued gid with computational cells (distibuted as along the axial, adial and azimuthal diection espectively) was employed fo system B of Micheletti et al. (1994). As well-known in the liteatue, the choice of adopting a completely hexahedal gid may allow, fo any given equied accuacy, a discetization degee much lowe than that necessay fo a tetahedal discetization of the computational domain (Scully and Fawley, 2011; Tambuini et al., 2012d). In ode to assess the gid dependence of the CFD esults, Tambuini et al. (2011a) and Tambuini (2011) employed computational gids up to 8 times fine fo the two systems and found no significant gid dependence. In paticula the mean value of the discepancy between the finest gid and the gid adopted hee was found to be ~1.0% fo which is the most significant vaiable in solid-liquid systems. Theefoe, only coase gid simulations wee pefomed fo the pupose of the pesent wok in view of its compaative natue and of the lage numbe of test cases investigated. The Excess Solid Volume Faction Coection (ESVC) algoithm fistly poposed by Tambuini et al. (2009a) and successively optimized by Tambuini et al. (2011a) was adopted since patial suspension conditions wee investigated in the pesent wok. This algoithm guaantees that the maximum packing volumetic faction _packed is not exceeded in the computational domain even at 20 Page 20 of 51

22 vey low impelle speeds, when a sediment is pesent. It opeates iteatively at the end of each SIMPLEC iteation focing the volumetic factions exceeding _packed (solid-excess) to be distibuted among the neighbouing computational cells and eventually moved towads egions whee no solid-excess is pesent. The nsuspended Solid Citeion (SC) poposed by Tambuini et al. (2011a) was employed to compute the mass faction of suspended paticles. In accodance with this citeion solid paticles ae judged as being unsuspended if and only if they belong to cells whee _packed. Both these algoithms wee found suitable to deal with solid-liquid suspensions in baffled stied tank unde patial to complete suspension egimes (Tambuini et al., 2011a, 2012a,b and 2013b). Typically in MRF steady state simulations SIMPLEC iteations wee found to be sufficient fo vaiable esiduals to settle to vey low values in all the investigated cases and to allow the ESVC algoithm to efficiently opeate. As concens the time dependent SG simulations of system B, 100 full evolutions wee found sufficient to each steady state conditions, in ageement with the liteatue (Micale et al., 2004; Tambuini et al., 2009a, 2011a, 2013b,c). The numbe of SIMPLEC iteations pe time step was set to 30 to allow esiduals to settle befoe moving to the next time step. As fa as the initial guess o initial condition is concened, the fluid was assumed to be still and the paticles wee consideed to be motionless on the vessel bottom and at thei maximum packing value _packed. 5 RESLTS AND DISCSSION 5.1 Results of the Refeence Model The compaison between the expeimental values elevant to system A and system B and the coesponding CFD pedictions povided by the Refeence Model ae shown in the following 21 Page 21 of 51

23 Figues 2 and 3. Expeimental data include the suspension cuves and the nomalized sediment height as visually obseved on the lateal wall midway two subsequent baffles fo the case of system A and local axial pofiles of solid concentation fo the case of system B. As fa as the suspension phenomenon is concened, the discepancy between the expeimental and the CFD data elevant to system A is acceptable. The suspension cuves ae shown in Fig.2a whee some oveestimations of the expeimental data can be obseved, especially fo the case of the lagest paticles. Only fo the case of the low-concentation/small-paticles case a slight undeestimation is found at about pm. The sediment height is anothe featue of the suspension phenomenon and is epoted in Fig.2b. In this case a vey good ageement is found: only slight diffeences can be obseved. As it concens the pediction of the paticle distibution phenomenon, the pediction of the expeimental local axial pofiles of solid concentation via the Refeence Model fo system B is epoted in Fig.3: each pofile is elevant to a specific impelle speed anging fom highly incomplete (i.e. 400pm) to complete (i.e. 1100pm > N js ) suspension conditions. All the expeimental pofiles ae satisfactoily followed by the CFD data, although the paticle distibution degee is slightly undeestimated. Moe pecisely, at 400pm and 500pm the expeimental data at the vey top of the vessel ae not pefectly matched by the CFD pedictions; while a somewhat lage discepancy can be obseved below the impelle at impelle speeds of 600pm and 700pm. 22 Page 22 of 51

24 hsed/h [-] xsusp [-] N [pm] Exp_231micon_33.8% Exp_231micon_16.9% Exp_550 micon_33.8% CFD_231micon_33.8% CFD_231micon_16.9% CFD_550 micon_33.8% (a) (b) Exp_231micon_33.8% Exp_231micon_16.9% Exp_550micon_33.8% CFD_231micon_33.8% CFD_231micon_16.9% CFD_550micon_33.8% N [pm] Fig. 2: Refeence Model MRF simulations vesus expeimental data (Tambuini et al., 2011a) fo system A. (a) suspension cuves, (b) nomalized sediment heights. 23 Page 23 of 51

25 z/h [-] Expeimental_400pm 0.8 Expeimental_600pm 0.7 Refeence Model_400pm 0.6 Refeence Model_600pm / _av [-] z/h [-] Expeimental_500pm 0.8 Expeimental_700pm 0.7 Refeence Model_500pm 0.6 Refeence Model_700pm / _av [-] Fig. 3: Local axial pofiles of solid concentation (system B): Refeence Model SG simulations vesus expeimental data (Micheletti et al., 2003). Summaizing, all the expeimental data ae faily well pedicted by the vey simple Refeence Model, although thee seems to be still oom fo futhe impovements. In the following, some modifications to the pesent Refeence Model will be tested with the aim at impoving the pedictions of the expeimental data. Theefoe, in the following the discussion will be focused on the influence of the modelling modifications descibed in section 3 (concening inte-phase dag foce and tubulence closue) on the pediction of the expeimental data. 5.2 Influence of model changes System A Intephase Dag Foce modelling Suspension Cuves As it can be seen in Fig.4, neithe of the poposed modifications to the intephase dag foce povides satisfactoy esults: both of them significantly oveestimate the expeimental data as well as the pedictions of the Refeence Model, especially at the intemediate impelle speeds. The pesent modifications appea to lead to a pematue suspension of a lage amount of solid paticles. Howeve, both the Dense Paticle Effect appoach and the Piecewise Coelation appoach ae able to pedict the typical sigmoidal tend of the suspension cuve. 24 Page 24 of 51

26 As egads the DPE appoach, the dag foce enhancement due to the dense paticle effect may cause an oveestimation of the momentum exchange between phases thus poviding a suspended solid faction highe than the expeimental one. The highe the local the highe the influence of the dense paticle effect tem, so one may expect a lage influence at low N when the paticle distibution degee is vey small. Actually, when paticles ae lying packed on the bottom as sediment they show the same = -packed, so that the dag foce enhancement is the same in all cells whee sediment is pesent, independently of the impelle speed. When the impelle speed is vey low, the dag foce is not sufficient to suspend many paticles (even if the DPE enhancement is accounted fo) because the slip velocity between the liquid flow and the sediment is still too low. As the agitation speed inceases, the slip velocity inceases (the liquid velocity inceases, while the solids velocity emains still low) and the dag foce eaches its citical value fo the suspension of a lage faction of paticles (i.e. when the cuve stats to ise with a lage slope). In geneal, when a DPE appoach is employed the dag foce eaches its citical value at impelle speeds lowe than those of the Refeence Model, thus yielding an oveestimation of the amount of suspended paticles. As fa as the Piecewise Coelation (PwC) appoach is concened, Fig.4 shows that at vey low impelle speeds the mass factions of suspended solids pedicted by the PwC and DPE methods ae vey simila thus suggesting that even the incease of the dag foce induced by the adoption of the Egun equation is not sufficient to allow paticles to suspend when the slip velocity is low. At highe impelle speeds the ove-pediction povided by the PwC appoach appeas to be simila and slightly lage than that of the DPE model. This behaviou is confimed by Fig.5 whee the compaison of the same appoaches is pesented fo the case of a solid-liquid suspension with the same geomety and paticle size, but a lowe solids loading. The Egun equation povides a lage inceasing of the intephase dag foce thus leading to x susp values highe than those pedicted by the DPE appoach. 25 Page 25 of 51

27 xsusp [-] N [pm] Expeimental Njs_Zwieteing MRF_Refeence Model MRF_DPE MRF_PwC Fig. 4: MRF simulations vesus suspension cuve expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 33.8% w/w : compaison of diffeent dag foce fomulations. xsusp [-] N [pm] Expeimental Njs_Zwieteing MRF_Refeence Model MRF_DPE MRF_PwC Fig. 5: MRF simulations vesus suspension cuve expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 16.9% w/w : compaison of diffeent dag foce fomulations. As a diffeence with the pevious case, hee a lage oveestimate of the expeimental data is obseved fo the two altenative appoaches even at the lowest impelle speeds. The pedictions of Hosseini et al. (2010) concening cloud height and homogenization degee show a simila oveestimate at the lowe impelle speeds fo a simila solids loading (10% w/w ). By a close 26 Page 26 of 51

28 inspection of Figs 4 and 5 it is possible to note that at, fo example, 150pm the x susp values pedicted by the DPE and the PwC appoaches fo the case of the moe diluted suspension (Fig.5) ae about twice those pedicted fo the case with the lage solids loading (Fig.4). Independently of the solids loading, when the impelle speed is vey low, paticles ae packed ove the bottom exhibiting the maximum allowed volume faction as well as the same paticle-paticle inteactions. As a consequence the DPE appoach on the one hand, and the PwC appoach on the othe hand, poduce dag foce enhancements which emain constant with the solids loading thus pedicting a simila quantity of paticles to be suspended. Fo the moe dilute suspension this quantity coesponds to a lage faction of suspended paticles. At highe impelle speeds, no diffeences between the x susp values pedicted by the DPE and the PwC appoaches ae appeciable. This occuence is not supising as fo modeate solids loadings, when most paticles ae suspended thoughout the vessel, is elatively low. As a consequence, the PwC appoach switches to equation 15a, thus yielding dag foce pedictions identical to the DPE appoach. As it is shown in Fig.6, the two appoaches povide ove-pedictions lage than those of the Refeence Model also fo the case of the solid-liquid suspension with lage paticle size thus confiming that taking into account the paticle-paticle inteactions via the poposed dag foce modifications is not suitable fo patial suspension conditions. As concens the compaison between the DPE and the PwC appoaches, Fig.6 shows that they povide pactically identical esults thus suggesting that fo the pesent case the Egun equation and the Gidaspow coection poduce the same incement of the inte-phase dag foce. This diffeence with espect to the pevious cases (whee diffeent incements wee found) may be linked to the pesence of the Bucato et al dag-tubulence coelation (1998) whose effect is negligible fo the lowe paticle diamete case as it can be seen in Tab.2. In accodance with this coelation, the highe the paticle diamete, the highe the enhancement of the dag coefficient. This coection does not appea in the Egun equation, thus suggesting that the gap in dag foce pedictions 27 Page 27 of 51

29 between the two appoaches fo the lowe paticle size cases may be bidged by the action of the dag coefficient enhancement due to the liquid fee steam tubulence. This hypothesis might be confimed by obseving in Fig.6 that at 368pm and 413pm the DPE appoach yields x susp slightly lage than those by PwC appoach since the highe the impelle speed, the lowe the Kolmogoov length scale, the highe the dag coefficient enhancement xsusp [-] N [pm] Expeimental Njs_Zwieteing MRF_Refeence Model MRF_DPE MRF_PwC Fig. 6: MRF simulations vesus suspension cuve expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 33.8% w/w : compaison of diffeent dag foce fomulations. Sediment Height The following Figs 7,8 and 9 show the nomalized sediment height fo the same configuations fo which Figs 4,5 and 6 epot suspension cuves. Notably, the sediment height data epesent only a local piece of infomation and should be egaded as less decisive fo compaison puposes than the moe global infomation povided by the suspension cuves. By obseving Fig.7 (small paticles, highe solids loading) consideations simila to those pesented in the fome sub-section fo Fig.4 can be made. Both the DPE and the PwC appoaches yield an undeestimation of both the expeimental data and the CFD pedictions obtained by the Refeence Model. Also, PwC yields lage undeestimations than DPE. 28 Page 28 of 51

30 0.25 Expeimental hsed/h [-] MRF_Refeence Model MRF_DPE MRF_PwC N [pm] Fig. 7: MRF simulations vesus sediment height expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 33.8% w/w : compaison of diffeent dag foce fomulations. In Fig.8 (small paticles, lowe solids loading) a supisingly diffeent behaviou can be obseved: as a diffeence with the coesponding suspension cuve esults in Fig.5, hee all the appoaches yield vey simila esults and pedict values of h sed in good ageement with the expeiments. This occuence can be explained by consideing the 3D visualization of the sediment estimated by the simulations, shown in Fig.9. Hee the computational cells whee _packed (sediment) ae indicated in gey. Contous of the solids volumetic faction on a vetical diametical plane ae also epoted. The DPE and the PwC esults exhibit a sediment distibution along the peipheal wall (and thus a h sed value) simila to the efeence case, but diffe in the way sediment is suspended fom the cental egion of the tank bottom (and thus in the x susp value). 29 Page 29 of 51

31 hsed/h [-] Expeimental MRF_Refeence Model MRF_DPE MRF_PwC N [pm] Fig. 8: MRF simulations vesus sediment height expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 16.9% w/w : compaison of diffeent dag foce fomulations. The impelle speed of 171pm showed in Fig.9 is just an example: the same behaviou is also obsevable at 125pm and 147pm. Such behaviou is not in accodance with the extensively validated Refeence Model esults and with expeimental visualization (Tambuini, 2011) whee paticles seem to suspend pefeentially fom the peipheal egion of the sediment. Convesely, the analysis of the 3-D sediment visualization fo the othe two suspension cases showed that the thee model appoaches pedict simila sediment shape evolution as a function of the impelle speed. Clealy, a featue like the shape of the sediment is the esult of a delicate balance among diffeent foces including pessue gadients, dag and tubulent stesses so that it is not supising that even slight model changes may esults in significant diffeences in the pedictions. Fig.10 (lage paticles, high solids loading) is pefectly in accodance with the coesponding Fig.6: the DPE and the PwC appoaches povide the same unde-pedictions of the expeiments and of the data obtained by applying the Refeence Model. 30 Page 30 of 51

32 Refeence Model DPE PwC Fig. 9: 3-D sediment visualization plot upon contou plots of solid volumetic factions on a vetical diametical plane at 171pm fo the case of 231 m ballottini paticles at 16.9% w/w : compaison of diffeent dag foce fomulations hsed/h [-] N [pm] Expeimental MRF_Refeence Model MRF_DPE MRF_PwC Fig. 10: MRF simulations vesus sediment height expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 33.8% w/w : compaison of diffeent dag foce fomulations Tubulence closue Suspension Cuves Fig.11 epots expeimental and computational esults concening the suspension cuves fo system A with small paticles and high solids loading. The adoption of the homogeneous k- tubulence 31 Page 31 of 51

33 model along with no additional tems in the continuity equation (Homogeneous_no tems) povides esults which ae simila to those obtained by the Refeence Model, even close to the expeimental data at the intemediate impelle speeds. The eason of this finding is closely elated to the homogeneous k- tubulence model fundamentals. The two phases shae the same tubulence kinetic enegy k and the same dissipation of tubulence kinetic enegy which ae calculated by employing the mixtue quantities. At low to intemediate impelle speeds, when a sediment is pesent, the mixtue velocity used by the tubulence model nea the sediment-fluid inteface esults to be quite lowe than that of the liquid phase, theeby leading to compute small velocity gadients, smalle tubulence poductions and thus a less intensive tubulence. This fact causes the suspension of a quantity of paticles lowe than that of the Refeence Model. As a diffeence, when the additional tems ae included in the continuity equation (Homogeneous_2 tems), vey lage oveestimation of x susp ae obtained, even at vey low impelle speeds (Fig.11). This occuence is linked to the natue of the additional tems including the Laplacian of the volumetic faction. The action of such tem is paticulaly stong at the sediments inteface and esults in making this inteface a diffuse one fom which solid paticles ae moe easily dagged by the liquid. In accodance with Mesmann et al. (1998) the suspension of solid paticles is stictly linked to two diffeent phenomena: the bottom lifting and the avoidance of settling. Which of them may be the one contolling the paticle suspension depends on paticle and liquid popeties, on solids hold up, on impelle to tank diamete atio. By applying the decisive citeion fo suspension by Mesmann et al. (1998) (see following eq.27) it esults that bottom lifting is the phenomenon contolling the solid suspension fo all the cases investigated in the pesent wok, i.e. the amount of enegy necessay to allow the paticles to lift fom the tank bottom BL is highe than that necessay to avoid the suspended paticles to settling AS (the atio of eq.27 esults highe than 1 fo all the cases studied hee): 32 Page 32 of 51

34 BL AS T 527 D 5/ 2 d p H A 1/8 n 1 1/ 4 _ av 1 _ av (27) whee A is Achimedes numbe and n is an exponent depending on the value of A. Full details can be found in the wok by Mesmann et al. (1998). The inclusion of the additional tems in the continuity equation lagely aids the bottom lifting phenomenon thus allowing a lage amount of paticles to be pematuely suspended. xsusp [-] Expeimental Njs_Zwieteing Refeence Model MRF_Homogeneous_no tems MRF_Homogeneous_2 tems N [pm] Fig. 11: MRF simulations vesus suspension cuve expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 33.8% w/w : compaison of diffeent tubulence closues. The elative effectiveness of altenative appoaches appeas to be quite independent of paticle concentation and mean diamete. By obseving Figs. 12 (small paticles, low solids loading) and 11 (lage paticles, high solids loading) consideations simila to those elevant to Fig.11 can be made: the x susp values pedicted by the Homogeneous_no tems appoach ae lowe than those obtained by the Refeence Model and geneally close to the expeimental data. Convesely the adoption of the Homogeneous_2 tems appoach gives ise fo both suspensions to ove-pedictions at all the impelle speeds and paticulaly at low speeds. This esults is in ageement with the findings by Fletche and Bown (2009) who found that the homogeneous and the asymmetic k- models 33 Page 33 of 51

35 povide simila esults, but the homogeneous model leads to a lowe paticle dispesion. They stated also that the inclusion of tubulent dispesion (as done in the case of Homogeneous_2 tems appoach) leads to a geat enhancement of paticle dispesion and that the soundness of its adoption at low impelle speeds has to be caefully validated. xsusp [-] N [pm] Expeimental Njs_Zwieteing Refeence Model MRF_Homogeneous_no tems MRF_Homogeneous_2 tems Fig. 12: MRF simulations vesus suspension cuve expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 16.9% w/w : compaison of diffeent tubulence closues. xsusp [-] Expeimental Njs_Zwieteing Refeence Model MRF_Homogeneous_no tems MRF_Homogeneous_2 tems N [pm] Fig. 13: MRF simulations vesus suspension cuve expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 33.8% w/w : compaison of diffeent tubulence closues. 34 Page 34 of 51

36 Sediment Height In Fig.14 the compaison of the sediment heights povided by the diffeent tubulence closues fo the case of m glass ballottini paticles with a solid loading of 33.8% w/w is shown. The Refeence Model and the Homogeneous_no tems pedictions appea to be quite simila and in a good ageement with the expeimental data. The Homogeneous_2 tems appoach povides vey low h sed values. This esult is in accodance with the lage undeestimation of the amount of unsuspended paticles pesented in Fig.11. hsed/h [-] Expeimental MRF_Refeence Model MRF_Homogeneous_no tems MRF_Homogeneous_2 tems N [pm] Fig. 14: MRF simulations vesus sediment height expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 33.8% w/w : compaison of diffeent tubulence closues. As concens the small paticle, low solids loading case, Fig.15, by compaing the pedictions of the Refeence Model and the Homogeneous_no tems, it appeas that these ae vey simila only fo impelle speeds lowe than 180pm. This is in accodance with the esults shown in Fig.12, whee simila x susp values can be obseved fo the two appoaches in the same impelle speed ange. As a 35 Page 35 of 51

37 diffeence with Fig.12, at impelle speeds highe than 180pm, the Refeence Model pedictions ae close to the expeimental data than the Homogeneous_no tems ones. hsed/h [-] Expeimental MRF_Refeence Model MRF_Homogeneous_no tems MRF_Homogeneous_2 tems N [pm] Fig. 15: MRF simulations vesus sediment height expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 16.9% w/w : compaison of diffeent tubulence closues. The Homogeneous_2 tems appoach leads to h sed values lowe than the expeiments and the othe pedictions. At 147pm and 171pm the discepancies between the Homogeneous_2 tems and the elevant expeimental h sed ae lowe than the coesponding discepancies between the x susp values obsevable in Fig.12. The use of Homogeneous_2 tems appoach esults into the suspension of most paticles fom the bottom cente (Fig.16), in a simila way as peviously shown in Fig.9 fo the influence of the dag model in Section 5.2.1, thus poviding lagely diffeent x susp at simila h sed. As aleady stated fo the cases of the DPE and PwC appoaches, the vigoous suspension fom the bottom cente is not consistent with the expeimental evidence fo a T/2 Rushton tubine thus suggesting that the Homogeneous_2 tems is not suitable to deal with the case unde study. 36 Page 36 of 51

38 Refeence Model Homogeneous_no tems Homogeneous_2 tems Fig. 16: 3-D sediment visualization plot upon contou plots of solid volumetic factions on a vetical diametical plane at 171pm fo the case of 231 m ballottini paticles at 16.9% w/w : compaison of diffeent tubulence closues As fa as the case of a suspension with lage paticle size is concened, the compaison among the diffeent modelling appoaches is shown in Fig.17. hsed/h [-] Expeimental MRF_Refeence Model MRF_Homogeneous_no tems MRF_Homogeneous_2 tems N [pm] Fig. 17: MRF simulations vesus sediment height expeimental data (Tambuini et al., 2011a) fo the case of m glass ballottini paticles with a solid loading of 33.8% w/w : compaison of diffeent tubulence closues. 37 Page 37 of 51