Modelling vortex formation in an unbaffled stirred tank reactors

Transcription

1 Modelling votex fomation in an unbaffled stied tank eactos G. M. Catland Glove and J. J. Fitzpatick * Cuent Addess: Foschungzentum Rossendof, Institut fü Sicheheitsfoschung, P. O. Box , D Desden, Gemany Tel: ; g.glove@fz-ossendof.de; * Depatment of Pocess & Chemical Engineeing, Univesity College, Cok, Ieland Tel: ; j.fitzpatick@ucc.ie;

2 Abstact Agitating liquids in unbaffled stied tank leads to the fomation of a votex in the egion of the impelle shaft when opeating in the tubulent flow egime. A numeical model is pesented hee that captues such a votex. The volume of fluid model, a multiphase flow model was employed in conjunction with a multiple efeence fame model and the Shea Stess Tubulence model. The dimensions of the tank consideed hee, wee m fo the liquid depth and tank diamete with a m diamete impelle at a height of m. The impelle consideed was an eight-bladed paddle type agitato that was otating with an angula velocity of 7.54 ad s -1 (72 pm) giving a Reynolds numbe of 10 5 and Foude numbe of Peliminay esults of a second investigation into the effect of liquid phase popeties on the votex fomed ae also pesented. Keywods: Computation Fluid Dynamics, Stied Tanks, Fee Suface Votices, Volume of Fluid Intoduction Stied tanks ae used acoss a wide spectum of the pocess industy to blend fluids, dispesing two immiscible fluids (i.e. gas-liquid o liquid-liquid), to suspend solids and to pomote the tanspot of heat and mass paticulaly when pefoming eactions. A lage numbe of stied tanks ae designed accoding to standad o commonly used specifications, whee the hydodynamic phenomena and mixing chaacteistics have been widely investigated [1]. Using non-standad tank design will esult in

3 diffeent fluid flow phenomena that influences the opeational chaacteistics of the pocesses consideed. Fo example emoving the baffles will change the flow chaacteistics and theefoe the mixing ate, thus alteing the effectiveness of the tank design fo eaction and phase contacting pocesses. Suspending o dissolving solids into a liquid medium is an impotant pocess in the food industy and a significant facto is the ate at which paticles sink, theefoe influencing the design and opeation of such stied tanks [2]. Fo floating o slowly sinking paticles, unbaffled stied tanks ae geneally used, as the spial flow stuctues that ae fomed, tend to daw such paticles into the liquid medium [2]. These flow stuctues esult fom the otation of the impelle that can simply be descibed as a pimay flow otating about the agitato shaft and a seconday flow field of two lage eddies that fom above and below the stie. The seconday votices cause the liquid suface to be pushed up at the tank wall and sucked down nea the impelle shaft. This esultant suction foms a whilpool like votex that entains o daws the floating o slowly sinking solid paticles into the liquid medium. Note that baffles ae not geneally used fo the sinking of paticles as the flow stuctues fomed by the baffles suppess the development of the votex at the impelle shaft. To chaacteise the conditions equied to cause the floating paticle to sink, votex fomation is an impotant facto that is dependent on the suction ate of liquid though the impelle [2]. Votex shape and depth is dependent on the agitation ate, the liquid medium viscosity and tank geomety. Fo example the depth of the impelle below

4 the liquid suface will cause ai entainment if the impelle is close to the suface and this educes the futhe away the bottom of the votex gets fom the agitato. An investigation into the influence that the concentation of died milk solids had on the fluid flow phenomena obseved in an unbaffled stied tank was pefomed [3]. The died milk solids wee added to a known volume of wate that was agitated by a paddle-bladed type impelle. The died milk solids became dissolved in the liquid phase, thus changing the physical chaacteistics of the liquid phase. As the died milk solid content was inceased up to a concentation of 55%, the solid-liquid mixtue viscosity inceased exponentially, paticulaly fo concentations geate than 40%. At the same time the liquid mixtue density deceased fom aound 1000 kg m -3 to nealy 700 kg m -3 (Fig. 1A). In this state, the liquid containing the dissolved milk solids stated to behave like a gel. Theefoe with the change in volume and the shea egime, adjustments wee made to the impelle location and agitation ate to maintain a constant votex depth. The effect of these changes to the opeation of the tank caused the paticle sink time to decease to negligible levels (Fig 1B) [3]. This aised questions about what caused the paticle sink ate to decease so makedly i.e. could such effects be the esult of changes to the fluid popeties, the incease in the suction ate of liquid though the impelle, the incease in the powe consumption, changes to the flow stuctue (typified by the alteed shape of the votex) o as a combination of these effects. It is poposed that a numeical model of the unbaffled stied tank can be used to explain the change in the phenomena obseved in the expeimental investigation [3]. This numeical model can be implemented though the use of computational fluid

5 dynamics (CFD) solves, fo example ANSYS CFX [4] o FLUENT [5] with the application of flow models that include volume of fluid, otational flow and tubulence. The modelling of stied tanks has been long established to chaacteise otational flow [4-21]. Fo the majoity of numeical studies of stied tanks the liquid suface is not consideed to defom with agitation (i.e. though the use of a fee-slip suface whee the velocity consideed is that of the fluid and not influenced by the bounday laye). The use of the fee slip suface is validated by the conditions in the tank that educe the likelihood of votex fomation (i.e. baffles [8-9, 13, 15, 18], location, size and shape of the impelle and the tank flow egime [6-9, 11-21]). The votex fomed in the unbaffled stied tank was numeically investigated [10, 14] to good effect by achieving a close compaison with the expeimental esults [22]. The technique used in [10] is compaatively difficult to implement as the fee suface is consideed as pat of the mesh. Theefoe, the mesh defoms to the shape of the votex o fluid suface as a conveged solution of the flow field is calculated [10]. The numeical investigation focussed on an expeimental investigation of flow stuctues in a m diamete stied tank that was agitated by an eight-bladed paddle type impelle [22]. A second numeical investigation was ecently pefomed that utilised the volume of fluid method that applied seveal tubulence models (SST and vaious Reynolds stesses models) that epoduced the flow fields to a easonable accuacy [14]. Axial, adial and tangential velocity pofiles with seconday flow pattens wee obtained fom measuements made on the stied tanks with pitot tubes

6 [22]. The effect that these paametes had on the fee liquid suface fo baffled and unbaffled tanks opeating unde the tubulent flow egime was also discussed. These investigations [10, 14, 22] povide the physical basis to validate an altenative appoach to modelling the fee suface of the liquid. This appoach, descibed in the following sections, utilises both the volume of fluid and multiple efeence fame techniques to assess the effect that agitation has on the fomation of a votex in an unbaffled stied tank. The validated model is intended fo use in an investigation of the effect of the change in the physical chaacteistics of the liquid in the flow phenomena obseved when died milk solids wee added to wate as descibed above. Initial esults of this investigation ae pesented ae also pesented hee. Mathematical Models Fluid Flow To model the shape of the liquid suface in an unbaffled stied tank a volume of fluid model and a multiple efeence fame model ae used to modify the Navie-Stokes equations (1) and (2) [4]. t t = S = ( ρ ) + ( ρ U) SM T ( ( ) ( ρu) + ρu U μ U + ( U) M p (1) (2)

7 Rotational motion is intoduced to the Navie-Stokes equations (1) and (2) using a multiple efeence fame model. The model used hee divides the tank into two sepaate domains, one that is otating (in the egion of the impelle) and a second that is stationay (encompassing the est of the tank). An inteface between both domains is constucted such that stict consevation of the fluxes in all equations occus, especially with any pitch changes that occu as the otating conditions ae applied to impelle egion. Additional otational foces (Coiolis, (3), and centifugal, (4), foces) that influence the flow in stied tanks ae then intoduced as souce tems, (5), in the momentum tanspot equation, (2) [4]. S Co = 2ρω U (3) S cfg = ρω ( ω ) (4) S M,ot = S Co + S cfg (5) The volume of fluid method is used to assess the motion of the liquid being agitated in the stied tank. This method (also known as a homogeneous multiphase flow) teats both the velocity and pessue fields as the same fo all the phases consideed though conditions in (6) and (7). Theefoe, only one set of the Navie-Stokes equations wee used to assess the flow phenomena using (1) and (2). The effect of each phase is intoduced though a volume faction equation, (8), and elations that calculate the effect that the volume faction of each phase has on the local fluid density, (9), and

8 viscosity, (10). A souce tem, (11), is then included in (2) to account fo the effect of buoyancy foces on eithe fluid, whee ρ ef, the efeence density, is geneally the lighte phase [4]. U = U, 1 α N α P (6) p α = p α =1...N fo all P (7) N P α α= 1 = 1 (8) ρ = N P α= 1 αρ α (9) μ = N P α= 1 αμ α (10) S M,buoy = ( ρ ρ )g ef (11) To model the inteface between the two fluids consideed (ai and wate in this case), a continuum suface foce model was employed. This equies the definition of a pimay fluid (wate) and a seconday fluid (ai). Essentially, the model is a foce balance that is applied to egion of the inteface between the phases by using equations (12) to (15). Whee δ αβ is the inteface delta function (i.e. is zeo away

9 fom the inteface) and κ αβ detemines the cuvatue of the suface though the inteface nomal vecto [4]. F αβ = f αβ δ αβ (12) f αβ = σαβκαβn αβ + σ s (13) δ = αβ αβ (14) καβ = n αβ (15) Tubulence Tanspot As the motion of the liquid phase in the stied tank is a fully developed tubulent flow (Reynolds numbe geate than 10 3 ), an appopiate tubulence model must be consideed. Examining the flow domain, a flow sepaation occus ove the paddle blade impelle, theefoe to model this sepaation accuately the Shea Stess Tanspot (SST) model can be used to account fo these effects. The SST model is a two-equation eddy viscosity model, (16) and (17). This model diffes fom the k-ε tubulence and the k-ω tubulence model by employing blending functions and an altenative eddy viscosity to assess the diffeent scales of tubulence that can be chaacteised by two equation models. The SST model woks by estimating the egion nea to wall whee the ω equation (17) woks well in appoximating the smallscale fluctuations. The model then uses elationships (18) to estimate the smalle

10 scale fluctuations in tems of the eddy dissipation ate in the flow field away fom the walls whee the ε equation woks well. This means that the wall conditions can be assessed automatically though the functions (19)-(26) defined below [4]. t t ω = μ ϕ ( ) ( ) t ρk + ρuk = μ + k + P β ρkω ( ρω) + ( ρuω) ε β k φ = B ( 1 1 ) φ 2 1 φ 1 + B 4 ( ) B 1 = tanh ag 1 k3 μ t = μ + ω + α ϕω ϕ ω 2 ( 1 B ) 2ρ k ω β ρω ω2 k 3 ω P k 3 k (16) (17) (18) (19) (20) ag 1 k 500ν 4ρk = min max,, 2 β ωy y ω CD kωϕ ω2 y 2 (21) 1 CD kω = max 2ρ k ω,1.0* 10 ϕω2ω 2 ( ) B 2 = tanh ag 2 10 (22) (23) 2 k 500ν ag 2 = max, 2 (24) β ωy y ω a k 1 ν t = (25) max( a1ω, λb2 ) ν t = μt ρ (26)

11 Case Specifications Tank Dimensions The expeimental study of Nagata [22] included the effect of baffled and unbaffled configuations of the stied tank in the completely tubulent flow egime using an eight-bladed paddle type impelle with a diamete of m, whee the blade dimensions ae m by 0.06 m. The impelle (Fig. 2) was located on the centeline of a m diamete vessel at a height of m fom the base of the unbaffled vessel that was filled to a depth of m with wate [22]. The impelle was mounted on a shaft m in diamete that otated about the vessel centeline at 72 pm [22]. A second smalle stied tank of diamete m was used to investigate the effect of died milk solids concentation on the votex fomed (Fig. 3). In this case, the impelle was mounted on a m diamete shaft that was moved to etain a constant depth of m between the liquid suface and the impelle as the volume of the liquid in the stied tank inceased with the died milk solids added [3]. The impelle was a six-bladed paddle, whee the blades ( m high) wee attached to a m hub to give impelle diamete of m [3]. Fo the two cases consideed at 5 and 20 % died milk solids concentation the agitato was otated at 280 and 288 pm [3]. Note that the shaft was only located above the impelle, theefoe the mesh was esolved to give to a point below the impelle as in Fig. 3 by following ideal mesh geneation techniques fo tiangula segments [23].

12 Impelle Region To model the otation of the impelle, the tank consideed is split into two egions, one about the impelle and the othe egion is the emainde of the tank. Theefoe, fo the study based on the investigation of Nagata [22], a egion fom m (shaft) to m (oute egion limit) by 0.16 m high was otated though 45º (fo easons of symmety and computational efficiency an eighth segment of the whole tank was modelled). At the cente of the egion, an intenal thin wall epesenting the impelle was pojected m into the domain, fom the shaft (see Fig 2A). Note that the blade is 0.05 m fom the edge of the impelle egion in the vetical and hoizontal diections. The emaining conditions include an extenal wall fo the impelle shaft, a peiodic pai inteface fo flow into and out of the segment and thee fozen oto intefaces on the sufaces lying next to the oute egion (i.e. top and bottom sufaces plus the suface at m). The fozen oto inteface condition was used to assess the flux of the otating and stationay fluids acoss the bounday between the impelle and oute flow egions. Two meshes wee defined fo the domain, a hexahedal mesh of and a tetahedal mesh of elements. A otational velocity of 7.54 ad s -1 o 72 pm was applied to the fluid in this egion to epesent the motion of the impelle. The impelle egion fo the 5 and 20 % died milk solids case was constucted in a simila fashion (Fig. 3A). Howeve, the gap between the edge of the impelle and the blade was set to 0.01 m athe than 0.05 m and defined as a segment with an angle of

13 60 due to the fewe numbe of blades used hexahedal elements wee used to mesh the impelle egion that was otated at 280 and 288 pm espectively with the inteface conditions defined as above. Oute Region The oute egion of the tank based on the studies of Nagata [22] is an eighth segment of the whole tank is 0.75 m high by m wide fom the shaft (at = m) to the oute wall (at = m). This mesh includes a zone above the liquid suface, so that the changes in liquid height ae not impeded by the top suface of the domain). Between m and m in the vetical plane and up to a adius of m, a section of the domain is emoved to allow the Impelle Region to be inseted (see Fig 2B). The oute egion is consideed as stationay egion, without application of the otational flow equations. Conditions applied to the domain include a otating wall condition fo the shaft (otating at 7.54 ad s -1 ) and a stationay oute wall fo the base and the suface at m, as the Oute Region is in the stationay fame of efeence. A peiodic pai condition is applied to the sufaces that will inteact with the othe segments in vessel, and thee fozen oto intefaces to allow fluxes fom the Impelle Region to ente the domain (i.e. the sufaces at the heights m and m and at the adius of m). An opening condition is applied to the top suface with an ai volume faction of 1 and a static pessue of 0 kg m -1 s -2 to minimise influences of the ai pessue and velocity on the liquid velocity field. Note that the tubulence condition applied to the top was the default intensity and length scale. Two meshes wee

14 defined fo the domain that coesponded to the meshes fo the impelle egion. Thus, a hexahedal mesh of and a tetahedal mesh of elements wee constucted. Simila conditions wee applied to oute egions (Fig. 3B-C) fo the cases whee the viscosity and density was modified by the addition of died milk solid (5 and 20 dissolved milk solids %). In the expeiments, a constant depth between the impelle and the liquid suface was maintained [3]. This was because the density of the liquid and the mass dissolved changed with the incease in the died milk solid content. To accommodate this change in the liquid height the height of the domain was defined as m with a adius of m and an angle of 60. The section of the tank that was emoved fo the impelle egion was m in adius and m high and the height of the lowe fozen oto inteface was changed fom m fo the 5 % died milk solids content case to m fo the 20 % case. The meshes fo the 5 and 20 % cases wee defined with and hexahedal elements espectively. Fluid Conditions To model the fee suface of the liquid phase fo the tank based on the expeimental investigation of Nagata [22], the fluid phases wee defined as ai and wate that both expeience an opeating pessue of kg m -1 s -2 and a tempeatue of 298 K (Table 1). The definition of the liquid phase popeties fo the effect of the dissolution of the died milk solids ae also found in Table 1. Note that the suface tension coefficient fo the inteface between the ai and wate phases was specified as

15 0.072 N m -1 and was assumed to emain constant fo both the solids concentations consideed. This was used in the continuum-suface foce model whee the wate phase was consideed as the pimay fluid. The expessions (26) to (28) ae defined to initialise the volume faction and pessue fields pio to stating the simulation. Note that z i is the initial liquid depth and z n is used as a unit length to non-dimensionalise the elation fo a step function to detemine the local ai faction as eithe zeo o one. A = step ( z z ) z n i (27) = 1 W A (28) p = ρ W *g * W ( z z) i (29) Solve Conditions The specifications equied to obtain a solution include the use of the steady state fom of the tanspot equations. Second-ode discetisation was specified fo all the tanspot equations consideed [4]. The unde-elaxation applied to each equation was 0.1 s except fo the volume faction equation, which had an unde-elaxation facto of 0.01 s fo expeimental study of Nagata [22] and the espective unde-elaxation fo the dissolved milks solids cases wee 0.01 s and s. Note that the solve that was employed utilised the time-dependent tem to unde-elax each equation to achieve a

16 pseudo-steady state when convegence is achieved [4]. The body foce contol method was selected with the aveaged oot mean squaed type of convegence fo the mean esiduals set to an eo of 10-5 to give stable values of velocity and volume faction. This was achieved afte about 650 and 1800 iteations fo the hexahedal and tetahedal Nagata cases and 6000 and 7500 iteations fo the dissolved milk solids cases. The incease in the solution time was the esult of the lowe undeelaxation facto that was equied to obtain a stable solution. Results The esults of the simulations descibed above ae pesented below fo the validation against the expeimental investigation of Nagata [22] and the modification of the fluid popeties by the dissolution of died milk solids [3]. Nagata Test Case [22] Fou plots wee used to depict the expeimental vaiation of the velocity pofiles in the top half of the stied tank and show the influence that the flow stuctues have on the liquid suface shape [22]. The flow paametes include the tangential, axial and adial velocities with seconday flow pattens in the fom of steamfunction contous. To compae and validate the model, field plots of the liquid suface and the seconday flow pattens (in the fom of a vecto field) ae pesented in Fig. 4 and Fig. 5. The tangential velocities fo both the expeimental and numeical pofiles (Fig. 6) ae pesented next and ae followed by the espective axial (Fig. 7) and adial velocities (Fig. 8).

17 Fig. 4 depicts the change in the liquid depth fom the impelle shaft to the tank wall, as the liquid is agitated by the stie. The vetical and hoizontal lines define the expeimental limits of the liquid suface shape [22], with the adial distance of 0.12 m indicating the location at which the defomed suface cosses the initial liquid depth. The thee hoizontal limits indicate the depth of the bottom of the votex (0.521 m), the initial liquid depth (0.585 m) and the depth of the liquid at the vessel wall (0.603 m). The fee-suface pofiles fo both the hexahedal and tetahedal cases show easonable ageement with the expeimental suface. At the vessel wall, the inteface cosses the height of m at a volume faction of 0.8 and 0.5 fo the espective meshes and at a faction of 0.8 at m at the impelle shaft fo both cases. At 0.12 m fom the impelle, whee in the expeimental study the gas-liquid inteface cosses the initial liquid height, the volume faction is low (less than 0.1) as the bulk of the liquid cosses ove this height at a adius of up to 0.2 m fo both cases. Note that the pofiles found in the hexahedal case ae much close togethe and ae not as cuved as the tetahedal case o the expeimental esults suggest. Fig. 5 illustates the vecto field epesentation of the seconday flow pofiles and on compaison with the seconday flow steamlines, it is appaent that the fom of the flows ae diffeent fom one anothe. This is appaent fom the asymmety in the tetahedal case, as in the top half of the tank, the jet fom the impelle is pointed upwad whilst the hexahedal case has a jet that points hoizontally to the wall with two symmetic votices above and below the impelle. Both cases diffe fom the expeimental steamlines [22], in that the jet appeas to die out ove a distance of 10 cm fo the simulations, whilst the votex above the impelle is obseved to each the

18 wall fo the expeimental study. This featue of the flow is consistent with late investigations [6-21]. Howeve, it must be noted that the tank configuation modelled in [21] is makedly diffeent with thee impelles agitating the liquid. Othe votices ae fomed between the whee the jet dies out and the tank wall and both effects suggest that thee is an imbalance in the enegy impated into the flow by the impelles. The effect of this is to limit the fomation of the lage scale votices that influence the flow stuctue both above and below the impelle. The effect that this has on the velocity pofiles in the uppe half of the tank can be obseved Fig. 6 to 8. Fig. 6 depicts the tangential velocity pofiles fo both the expeimental and numeical investigations, espectively. A single peak is obseved fo the expeimental pofiles above the impelle at a adius of 0.12 m, whee maximum velocities of 0.8 to 0.9 m s -1 wee obseved. A singula peak is also obseved fo the pofiles with the tetahedal meshed case, but the location of the peaks ae found between 0.14 with maximum velocities of up to 0.76 m s -1, theefoe unde-pedicting the velocities by up to 0.2 m s -1. Howeve, the hexahedal case is diffeent in that thee is no significant peak, as the velocities ae of the same ode of magnitude nea to the wall. Bette ageement between the expeimental and numeical flow pofiles is appaent in the adial velocities in the egion close to the impelle (Fig. 7). Howeve, peak velocities in the steam away fom the impelle ae 0.09 and 0.17 m s -1 fo the hexahedal and tetahedal cases. The unde-pediction in the velocities fo the tetahedal case can be explained by the angle of the flow steam away fom the blade tip (with espect to the hoizontal axis) as between (see Fig. 5) so that a

19 significant popotion of these vectos is also found in the axial diection. This diffes fom the expeimental data and hexahedal case whee hoizontal jets ae depicted. The axial pofiles fo the simulations pesented in Fig. 8 do not coespond to the expeimental velocities. The eason why can be explained by the diffeences obseved in the jet fom the impelle dying out in the middle of the tank as depicted in Fig. 5 and 7, the effect of this influences the flow stuctues in the est of the tank. It must be noted that even though the stuctual diffeences ae appaent, velocities of a simila magnitude ae obseved fo both cases and this has a small influence on the shape of the fee suface. It is possible to compae the pofiles in Fig. 8 with the axial pofiles pesented in [11-12] fo both expeimental and numeical systems unde conditions of lamina flow. Similaities wee obseved in the shape of the pofiles, though no quantitative compaisons can be made as the cases consideed ae vey diffeent due to the tank geometies consideed. Died Milk Solids Cases [3] The flow fields fom the initial simulations of the expeimental study descibed above, whee died milk solids wee dissolved in a lite of wate up to concentation of 55% dissolved milk solids ae pesented in Fig. 9. The meshes fo both these cases ae hexahedal in fom, as the flow in both the uppe and lowe halves of the tank was found to be moe symmetical fo the hexahedal case study based on the expeiments of the Nagata [22]. Anothe eason was that the fee-suface of the hexahedal case was also moe compact than the tetahedal case. Nevetheless, the flow stuctues in the liquid wee not epoduced as accuately as is possible, the phenomena in the egion of the fee suface is still good enough fo futhe application paticulaly as

20 the tank consideed hee is consideably smalle in scale (diamete of m compaed to m). Thus, it is possible to detemine the flow conditions that influence the sinking of died milk paticles befoe they can be dissolved. Note that in these simulations it was assumed that the died milk solids wee completely dissolved. Theefoe, the solid paticles had no influence on the flow stuctues in the tank. The fee suface pofiles fo both the 5 and 20 % dissolved milk solids cases depicted in Fig. 9 display a shap gadient fo the factions of 0.6 and 0.8 between the initial liquid height (0.087 and m) and the base of the votex ( and m). The lowe factions of 0.2 and 0.4 have a shallowe gadient that is compaable to the liquid that is found above the initial liquid height. Howeve, the gadient in the feesuface elevation is lowe fo the 20% dissolved solids cases. The majo diffeence between both cases is that the jet of liquid fom the impelle blade is angled upwads at aound 10 and quickly dies out fo the 5 % case. The 20 % case is angled downwads at aound 10 and the jet eaches the tank wall. Fo the 5 % case thee ae two seconday flow votices above the impelle, with the stonge velocities obseved in the eddy next to the defomed suface geneated by the liquid flow, whilst only a single votex appeas above the impelle in the 20 % case. The diffeences in the obseved flow stuctues cause the diffeences in the fom of the pedicted fee suface pofiles. Compaison of flow vaiables between numeical cases The stuctue of the fee sufaces fo all the cases shows the captues quantitatively the physicality of the phenomena obseved in expeiments if not qualitatively.

21 Theefoe, insight into the sinking chaacteistics obseved in milks solid study can be obtained fom examining the liquid velocities and the pessues (Fig. 10, 11 and 12). Fig. 10 depicts the adial velocities at the fee suface. Fo most of the tank adius the obseved velocities ae negative (i.e. the liquid velocity component is pointing towads the impelle). This suggests that paticles would be moved in the diection of the impelle. Howeve, fo both the hexahedal and tetahedal cases of the Nagata [22] study and the 5 % dissolved milk solids these velocities ae small (>-0.04), suggesting that this effect is not stong. Yet fo the 20 % solids case the obseved velocity component has a stonge velocity (nealy -0.10) close to the tank wall. Refeing back to Fig. 1, it was obseved that thee was a stong eduction in the sink time between the 5 and 20 % died milk solids cases, fo an appaently small change in physical popeties, geometical configuation and agitation ate. Theefoe, this diffeence between the two pofiles could help explain the change in the sink times obseved. Examining the axial velocity pofiles in Fig. 11 at the fee suface of both the hexahedal and tetahedal meshed cases of the expeimental study Nagata [22], the pofiles ae simila to measued pofiles. Theefoe, the ealie supposition that the flow stuctues ae boadly coect is backed up by these pofiles. Howeve, caution must be used as these pofiles ae not found in the est of the tank and that the velocities obseved hee ae faily small at < In the case of the 5 and 20 % solids cases, the velocities ae much highe in magnitude, with a stong peak at a thid of the adius fo both dissolved milk solids cases that coespond to the stong change the liquid suface. A second peak in the axial velocity is also obseved fo the 20 %

22 case at appoximately two thids of the adius. The velocities show that thee is a geate suction foce that would act on died milk paticles when they ae added to the liquid. Howeve, it is impotant to note that the velocities at a dimensionless adius of less than 0.25 coespond to the egion whee the 80% faction contou coesponds to the location of the impelle as seen in Fig. 9. Fig. 12 depicts the dynamic pessues obseved at the fee suface. The pofiles have two foms, with a peak at 0.3 to 0.4 of the tank diamete fo all cases. Howeve, in two cases the pessue educes as the adius is inceased (tetahedal mesh case and the 20 % solids case) and fo the othe two cases thee is a second peak close to the wall with a plateau in between. Theefoe, little can be deived fom the impact of the dynamic pessue on the flow phenomena in the died milk solids cases as a esult of the pofiles obseved hee. Futhe effects to elucidate influences on the suface fomation and the paticle sink ate may be obtained fom flow ates into and out of the impelle egion and the pessues acting on the liquid at locations between the impelle and the liquid suface (Table 2). Between the two cases based on the studies of Nagata [22], diffeences aise fom the aea integals of the meshes, as the hexahedal mesh has lage cosssectional aeas. Yet the velocities, the volume and mass flow ates ae not significantly diffeent. Fo the dissolved milk solids cases, the only noticeable diffeence is in the mass flow ates, which is a esult of the change in the physical popeties of the mixtues. Note that the negative value fo the wall value coesponds to mass leaving the impelle zone.

23 Conclusions A easonable epesentation of the flow phenomena in an unbaffled stied tank opeating in the tubulent flow egime was achieved using a volume of fluid multiphase model in conjunction with a multiple efeence fame model. Theefoe, the technique discussed was used to assess the impact that fluid popeties have on the fluid flow phenomena obseved when died milk solids ae added to a liquid at concentations of 5 and 20 % in a much smalle tank. Whee changes to the local pessues and flow ates can indicate the causes behind the eduction of paticle sink times that wee obseved expeimentally [3]. The esults indicate that even with a small change in the physical popeties obseved between the 5 and 20 % cases a significant change in the adial and axial velocities at the fee suface ae obseved. Some questions emain ove the accuacy of the technique used due to the significant diffeences obseved in the axial velocity pofiles fo the expeimental and numeical cases. To impove the accuacy of the modelling techniques, it would be pudent to investigate the phenomena in a simila vessel opeating unde the tubulent flow egime by exploiting the advances in expeimental techniques since the oiginal investigation was pefomed (i.e. Lase Dopple Anemomety and Paticle Image Velocimety). Enhancements that could be made to the numeical model would include consideing the whole domain (all 360 of the tank, athe than a 45 o 60 segment [14, 19]) and the use of an altenative tubulence modelling techniques. Such techniques consideed include Lage Eddy, Detached Eddy o Scale Adaptive Simulations o the Reynolds Stess model that would impove calculation of tubulence in the flow field. This is especially impotant in the estimating the

24 tubulent enegies in flow-steam moving away fom the impelle and paticulaly fo cases consideing lage tanks [6-10, 14, 16-17]. Futhe validation would include the modelling an unbaffled stied tank opeating unde a egime without significant suface defomation [11-12]. This would be a good test to obseve whethe o not the volume of fluid model is influenced by the flow stuctues that wee depicted hee in such system whee the liquid suface is not affected by the hydodynamics. Acknowledgements We would like to acknowledge the suppot of the EU though Maie Cuie sponsoship of the eseach (Contact numbe HPMD-CT ) and Ansys CFX Limited fo enabling this wok to be pesented. Nomenclatue Latin Symbols a = speed of sound (m s -1 ) ag B = agument tem in the SST model = blending function CD = function tem assessing the effect of the wall in the SST model F = continuum suface foce f = continuum suface foce coefficient g = gavitational acceleation vecto [ ] (m s -2 ) k = kinetic enegy (m -2 s -2 )

25 N n = numbe of phases = inteface nomal vecto p = pessue (kg m -1 s -2 ) S t = volume faction of a phase = location vecto = souce tem = time (s) U = velocity (m s -1 ) y z z i z n = distance to the neaest wall (m) = location in z-diection (m) = initial liquid height of the liquid phase (m) = unit length to non-dimensionalise the step function (m) Geek Symbols α 1 = model constant fo the eddy fequency equation = 5/9 α 2 = model constant fo the eddy fequency equation = 0.44 α 3 = model constant fo the eddy fequency equation β 1 = model constant fo the eddy fequency equation = β 2 = model constant fo the eddy fequency equation = β 3 = model constant fo the eddy fequency equation β = model constant fo the eddy fequency equation = c μ = 0.09 δ = inteface delta function ε = dissipation ate (m -2 s -3 )

26 κ λ = suface cuvatue = stain ate μ = dynamic viscosity (kg m -1 s -1 ) ν = kinematic viscosity (kg m -1 s -1 ) ρ = density (kg m -3 ) σ = suface tension coefficient (N m -1 ) φ k1 = tubulent Pandtl numbe fo the kinetic enegy equation = 2 φ k2 = tubulent Pandtl numbe fo the kinetic enegy equation = 1 φ k3 = tubulent Pandtl numbe fo the kinetic enegy equation φ ω1 = tubulent Pandtl numbe fo the eddy fequency equation = 2 φ ω2 = tubulent Pandtl numbe fo the eddy fequency equation = 1/0.856 φ ω3 = tubulent Pandtl numbe fo the eddy fequency equation ω = tubulent eddy fequency (s -1 ) ω = angula velocity (ad s -1 ) φ = tubulence model constant Mathematical opeatos Σ s = sum opeato = patial diffeential opeato = Gad = Gad ove the suface. = dot poduct = tenso poduct

27 Subscipts A = ai phase index buoy = buoyancy foce tem Co = Coiolis foce tem cfg M P ot ef T t W α αβ β = centifugal foce tem = mass = phase index (fo total numbe of phases) = otational foces fo the otating efeence fame model = efeence value = tanspose of the matix = tubulent = wate phase index = pimay phase index = tem indicating inteaction between two phases = seconday phase index Refeences [1] Coulson J. M., Richadson, J. F., Backhust, J. R. & Hake, J. H. (1990). Chemical Engineeing Volume 1 (Fluid flow, Heat Tansfe & Mass Tansfe). Oxfod: Pegamon Pess. [2] Feudig, B., Hogekamp, S., & Schubet, H. (1999) Dispesion of powdes in liquids in a stied vessel. Chemical Engineeing and Pocessing 38, 525. [3] Fitzpatick, J. J., Webe, N., Schobe, C., Weidendofe, K. & Teunou, E. (2002). Effect of viscosity and votex fomation on the mixing of powdes in

28 wate to fom high concentation solutions in stied-tanks. Wold Congess on Paticle Technology 4, Sydney, Austailia. [4] ANSYS Incopoated (2005). ANSYS CFX Release 10.0 Documentation. Cannonsbug PA, USA, Ansys Incopoated. [5] Fluent Incopoated, Use Guide, Fluent Incopoated, Lebanon, New Hampshie, USA, [6] Alexopoulos, A. H., Maggiois, D. & Kiapaissides, C. (2002). CFD analysis of tubulence non-homogeneity in mixing vessels: A two compatment model. Chemical Engineeing Science 57, [7] Amenante, P., Chou C. C., & Hemajani, R. R. (1994). Compaison of expeimental and numeical fluid viscosity distibution pofiles in an unbaffled mixing vessel povided with a pitched-blade tubine. Institution of Chemical Enginees Symposium Seies 136, 349. [8] Bucato, A., Ciofalo, M., Gisafi, F., & Micale, G. (1998). Numeical pediction of flow fields in baffled stied vessels: A compaison of altenative modelling appoaches. Chemical Engineeing Science 53, [9] Bucato, A., Ciofalo, M., Gisafi, F., & Tocco, R. (2000). On the simulation of stied tank eactos via computational fluid dynamics. Chemical Engineeing Science 55, 291. [10] Ciofalo, M, Bucato, A., Gisafi, F. & Toaca, N. (1996). Tubulent flow in closed and fee-suface unbaffled tanks stied by adial impelles. Chemical Engineeing Science 51, [11] Dong, L., Johansen, S. T., & Engh, T. A. (1994a). Flow induced by an impelle in an unbaffled tank - I. Expeimental. Chemical Engineeing Science 49, 549.

29 [12] Dong, L., Johansen, S. T., & Engh, T. A. (1994b). Flow induced by an impelle in an unbaffled tank - II. Numeical modelling. Chemical Engineeing Science 49, [13] Gobby, D. (2002). Mixing of Newtionian and non-newtonian fluids in an agitated baffled tank. CFX Validation Repot CFX-VAL12/1202. Oxfod: CFX Limited. [14] Haque, J. N., Mahmud, T., & Robets, K.J. (2006). Modeling tubulent flows with fee-suface in unbaffled agitated vessels. Industial Engineeing and Chemisty Reseach 45, [15] Jones, R. M., Havey, A. D. & Achaya, S. (2001). Two-equation tubulence modelling fo impelle stied tanks. Jounal of Fluids Engineeing Tansactions of the Ameican Society of Mechanical Enginees 123, 640. [16] Montante, G., Lee, K. C., Bucato, A. & Yianneskis, M. (2001). Numeical simulations of the dependency of flow patten on impelle cleaance in stied vessels. Chemical Engineeing Science 56, [17] Mununga, L., Houigan, K., & Thompson, M. (2003). Numeical study of the effect of blade sze on pumping effectiveness of a paddle blade impelle in an unbaffled mixing vessel. Thid Intenational Confeence on CFD in the Mineals and Pocess Industies CSIRO, Melboune, Austalia. [18] Ranade, V. V. (1997). An efficient computational model fo simulating flow in stied vessels: a case of Rushton tubine. Chemical Engineeing Science 52, [19] Vezicco, R., Iaccaino, G., Fatica M., & Olandi, P. (2004). Flow in an impelle stied tank using an immesed bounday method. AIChE Jounal 50, 1109.

30 [20] Wang, F., Wang, W., Wang, Y., & Mao Z. -S. (2003). CFD Simulation of solid liquid two-phase flow in baffled stied vessels with Rushton tubines. Thid Intenational Confeence on CFD in the Mineals and Pocess Industies CSIRO, Melboune, Austalia. [21] Zalc, J. M., Szalai, E. S., Alvaez, M. M. & Muzzio, F. F. (2002). Using CFD to undestand chaotic mixing in lamina stied tanks. AICHE Jounal 48, [22] Nagata, S. (1975). Mixing: Pinciples and Applications. New Yok: Wiley. [23] ANSYS Incopoated (2005). ICEM CFD Release 10.0 Documentation. Cannonsbug PA, USA, Ansys Incopoated. List of Figues Fig. 1. Change of physical popeties, paticle and agitation ates with solids content in an unbaffled stied tank eacto A: Viscosity,, (kg m -1 s -1 ) and density,, (kg m -3 ); B: Sink time,, (min) and agitation ate, Δ, (pm); Fig. 2. Tank geomety fo the Nagata case [21]: A: Inne egion; B: Oute egion; C: Stied tank with inne and oute egions in the coect locations; Fig. 3. Tank geomety fo the dissolved died milk solids cases [3]: A: Inne egion; B: Oute egion fo the 5 % died milk solids content; C: Oute egion fo the 20 % died milk solids content;

31 Fig 4. Lines at = 0.12 m, z = , and m indicate the limits of the liquid suface epoted by Nagata [21]. The cuved pofiles ae the pedicted suface shape at liquid volume factions of 0.2, 0.4, 0.6, and 0.8, between light gey and black espectively fo the hexahedal (A) and tetahedal (B) cases. Fig 5. Seconday flow velocity vectos in the plane of the impelle blade (i.e. at a tangent to the plane) fo the hexahedal (A) and tetahedal (B) meshes; highest velocity in the blade jetsteam wee m s -1 and m s -1 ; note that fo easons of claity only 2000 vectos ae depicted. Fig. 6. Pofiles of the tangential velocity (m s -1 ) at thee heights: shot dashed ( m) long dashed (0.3325) and solid ( m) lines fo the expeimental pofiles epoted by Nagata [21] (thick gey lines), hexahedal meshed element domain (thick black lines) and the tetahedal meshed domain (thin black lines). Fig. 7. Pofiles of the adial velocity (m s -1 ) at thee adial positions: shot dashed (0.04 m) long dashed (0.16) and solid (0.26 m) lines fo the expeimental pofiles epoted by Nagata [21] (thick gey lines), hexahedal meshed element domain (thick black lines) and the tetahedal meshed domain (thin black lines). Fig. 8. Pofiles of the axial velocity (m s -1 ) at thee heights: shot dashed ( m) long dashed (0.3925) and solid ( m) lines fo the expeimental pofiles epoted by Nagata [21] (thick gey lines), hexahedal meshed element domain (thick black lines) and the tetahedal meshed domain (thin black lines).

32 Fig. 9. Velocity vecto field and volume faction contous of the 5% (A) and 20 % (B) dissolved milk solids mixtue, whee the limits indicate the initial liquid height and the measued depth of the votex. Highest velocities in the blade jetsteam wee m s -1 and m s -1 espectively. Suface pofiles at volume factions of 0.2, 0.4, 0.6, and 0.8, between light gey and black espectively. Fig. 10. Pofiles of the adial velocities at the fee suface location coesponding to the liquid volume faction of 0.8 fo the both hexahedal (thick black line) and tetahedal (thin black line) meshes with Nagata and the 5 (thick gey line) and 20 % (thin gey line) dissolved milk solids cases. Note that thee is non-dimensionalisation with espect to the tank adii and the impelle tip speeds. Fig. 11. Pofiles of the axial velocities at the fee suface location coesponding to the liquid volume faction of 0.8 fo the both hexahedal (thick black line) and tetahedal (thin black line) meshes with Nagata and the 5 (thick gey line) and 20 % (thin gey line) dissolved milk solids cases. Note that thee is non-dimensionalisation with espect to the tank adii and the impelle tip speeds. Fig. 12. Pofiles of the dynamic pessues at the fee suface location coesponding to the liquid volume faction of 0.8 fo the both hexahedal (thick black line) and tetahedal (thin black line) meshes with Nagata and the 5 (thick gey line) and 20 % (thin gey line) dissolved milk solids cases. Note that thee is non-dimensionalisation with espect to the tank adii and the impelle tip speeds. Table 1: Fluid definitions and bounday conditions

33 Table 2: Flow paametes extacted fom the suface zones on the impelle egion.