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Reaction-precipitation by a spinning disc reactor: Influence of hydrodynamics on nanoparticles production
Chemical Engineering Science 76 (2012) 73–80
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Reaction-precipitation by a spinning disc reactor: Influence of hydrodynamics on nanoparticles production B. de Caprariis n, M. Di Rita, M. Stoller, N. Verdone, A. Chianese Department of Ingegneria Chimica Materiali Ambiente, Sapienza University of Rome, via Eudossiana 18, 00184 Rome, Italy
a r t i c l e i n f o
abstract
Article history: Received 20 September 2011 Accepted 26 March 2012 Available online 6 April 2012
The main aim of this work is to show the usefulness of a Computational Fluid Dynamics (CFD) model to interpret the results obtained from a reaction-precipitation process by using a spinning disc reactor (SDR). First, an experimental investigation was performed to produce nanoparticles of hydroxyapatite by a reaction at ambient temperature between aqueous solutions of calcium chloride and ammonium phosphate in presence of ammonium hydroxide to achieve an operating pH of 10. An SDR was adopted. It consists of a PVC disc rotating at a speed in the range 58–147 rad/s. Nanoparticles in the range 78– 350 nm were obtained by varying the rotational speed and the reagent solutions injection points. A CFD model was then developed to predict the hydrodynamics and the volume fraction of the reagent solutions in the liquid layer. The numerical simulation was performed using Ansys Fluent 12 and the volume of fluid model (VOF) approach was adopted. After the validation of the model on literature data, the CFD model was used to interpret the experimental results. It was shown that the size of the produced nanoparticles are inversely proportional to the specific dispersed energy at a reagent feed location and that at the highest adopted rotational speed an almost complete mixing of the reagent solutions is attained in the liquid layer. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Precipitation Hydrodynamics Mixing Hydroxyapatite Spinning disc reactor CFD
1. Introduction Spinning disc reactor (SDR) is an effective apparatus for the production of nanoparticles by wet chemical synthesis. Rotation of disc surface at high speed creates high centrifugal fields which promote thin film flow with a thickness in the range 50–500 mm. Films are highly sheared and have numerous unstable surface ripples, giving rise to intense mixing. SDR has been successfully applied to a range of processes which are micromixing dependent: polymerisation, competitive organic reactions and reactionprecipitation. The benefit of mixing intensification to precipitation process is to reduce the mixing time down to 1 ms or less, so that the sum of reaction time and the mixing time is shorter than the induction nucleation time and the maximum product supersaturation is locally achieved where reaction takes place. This condition leads to the maximum allowable value of nucleation rate and of generated nuclei as well. Moreover, due to the very high nucleation rate a low value of supersaturation residues for the subsequent processes of particles growth and aggregation and, as a consequence, nanoparticles may be produced.
n
Corresponding author. Tel.: þ39 064 4585 600; fax: þ 39 06 482 7453. E-mail address: [email protected] (B. de Caprariis).
0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.03.043
The performances of an SDR as a device for a reactionprecipitation process are investigated in many research works. Baffi et al. (2002) studied the precipitation of barium sulphate by an SDR, 0.5 m in diameter, operating at rotational speed between 200 and 1000 rpm. They showed that at a rotational speed greater than 800 rpm the performances were similar to that of a T-mixer, in terms of number of generated particles per unit volume and micromixing time, which was of the order of 1 ms. In both the cases a complete micromixing was considered to be attained, but according to these Authors the energy consumption is much less when the rotating device is used. SDR is also used to perform gas– liquid reaction. Trippa et al. (2002) demonstrated the use of an SDR for the production of calcium carbonate particles from dissolved CO2. The capability of this intensified system to generate a wide range of particle sizes and morphologies through control of the operating conditions, such as rotational speed, was clearly shown. Raston et al. (2006) produced b-carotene nanoparticles with spinning disc reactor and they obtained good results on controlling size and shape of the nanoparticles. All these works emphasised the great importance of hydrodynamics on the produced particles size. Therefore, the investigation on hydrodynamics in the liquid layer over the disc surface, particularly in correspondence of the reagents feed points, is of major importance to interpret the precipitation process over an SDR. Many works have been devoted to predict hydrodynamics of
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a liquid film over a spinning disc by means of a simulation model. Hydrodynamics of the liquid film was first studied in 1916 when Nusselt (1916) modelled the downward flow of a condensing liquid film. He considered that the liquid film was stable with no shear at the gas–liquid interface and that the acceleration was due exclusively to the gravitational force. The Nusselt analysis constituted a basis for more sophisticated models. In case of a liquid flow over a rotating disc higher accelerations have to be considered since they laid to thinner film and instabilities over the surface. Espig and Hoyle (1965) found that the predicted film thickness is 40–50% greater than that one predicted by the Nusselt’s theory. They attributed this difference to the formation of waves on the film surface. These Authors experimentally observed three different flow regimes over the rotating disc: a waveless laminar flow near the inlet region, a zone characterised by an asymmetric wave formation and a turbulent region at the disc periphery produced by a combination of asymmetric and helical waves. Charwat et al. (1972) observed a decay of the waves with increasing radius and so the formation of a new laminar zone after the turbulent one. Those results were confirmed by the work of Butuzov and Puhovoi (1976). Woods and Watts (1973) studied the heat and mass transfer on rotating discs. They identified three different wavy regimes. To describe the hydrodynamic over the rotating disc, they proposed a two-dimensional model, known as Pigford model. This model took into account inertial and viscous influences and was given by the following two equations: vr
@vr v2t 12p2 r 2 K 1 vt 3 ¼ vr @t r Q2
ð1Þ
vr
@vt vr vt 12p2 r 2 K 2 vt 2 þ ¼ vr ðr ovt Þ @t r Q2
ð2Þ
where K1 and K2 were empirical correction factors to adjust the viscous terms in the equation. Burns et al. (2003) performed experimental tests to investigate the hydrodynamics of the flow over a spinning disc. They compared the Pigford model with the Nusselt theory and demonstrated that the Nusselt model is not able to predict hydrodynamic behaviour of liquid film on a rotating disc when the inertial forces are higher than the viscous one, that is for low values of the Ekman number (E¼ n/h2o). In the Burns’ work the radial velocity profile was analysed and three zones were identified (Fig. 1): the injection zone where the liquid flow is slowed down by viscous drag, the acceleration zone where the high centrifugal acceleration increases the radial flow
velocity and the synchronized zone where the fluid radial velocity is close to the disc rotational speed. In the synchronized zone the film flow behaviour becomes similar to that calculated by Nusselt model. The hydrodynamics of the liquid film over a spinning disc was recently investigated by Bhatelia et al. (2009) using the computational fluid dynamics. They performed 3D simulations of the liquid layer and successfully compared the simulation results with the Burns experimental data. The only research work dealing with the simulation of a precipitation process performed over a rotating disc (Plasari et al., 2001), concerns a two phase solid-liquid system. In this work CFD technique aimed at the prediction of the reactor hydrodynamics and simulation of the precipitation process. In this latter respect, a two-step process of nucleation and growth was considered, whereas aggregation was neglected. According to the Authors, some discrepancies still remained between experiments and calculation. When precipitation takes place in presence of intensified micromixing conditions the size distribution of nanoparticles are mainly determined by a two step process nucleation-aggregation. In fact, in presence of complete micromixing between the reagents, almost all the supersaturation is consumed by nucleation, just a small residual local supersaturation is available and it is realistic to think that it is consumed during the aggregation process. According to Hounslow and Mumtaz (2000) aggregation may be described as a two step process consisting of shear stress ortokinetic collisions between two particles, and the subsequent cementation of the two particles at their point of contact due to the internal diffusion of solute ions. Fluid shear plays a double opposite role: it increases the number and intensity of particles collisions and reduces the timescale for bridge formation, thus lowering the effect of the collisions effect. The above mentioned authors showed that the aggregation of nanoparticles of calcium oxalate, in a Poiseuille flow crystallizer, is reduced in presence of more intensified conditions since the disruptive action of the fluid shear stress on aggregation was predominant. The results of this work showed that the mixing process intensification enhanced nucleation and reduced aggregation leading to reduction of nanoparticles size. Process precipitation over the surface of a horizontal rotating disc gives rise to a three phase gas–liquid–solid systems with a three dimensional structure. In presence of multiple reagent streams we may have a very complex map of the reagent concentrations with variations over each plane, positioned at different layer height. The process performances are dramatically dependent on the mixing phenomena between the reagents. The rotational speed is the most important operating parameter of an SDR, since it plays the main influence on micromixing, and as a consequence on nucleation and aggregation rate. In the present work the effect of the rotational speed of an SDR on the precipitation of hydroxyapatite was investigated. Then, an attempt for the interpretation of the experimental results was made on the basis of the prediction of the reagent concentrations in the fluid layer, by using a CFD simulation model.
2. Experimental set-up
Fig. 1. Radial velocity profile, three flow regimes identification. Picture taken from Burns et al. (2003).
An SDR was used to produce nanoparticles of hydroxyapatite (HAP) QUOTE by chemical precipitation. The experimental device, schematized in Fig. 2 consists of a cylindrical case with an inner disc made by PVC and 8.5 cm in diameter. Rotational velocity of the disc may be increased up to 147 rad/s. The reagent solutions are fed over the disc at a distance of 5 mm from the disc surface through tubes, 1 mm in diameter. The produced suspension draws off in continuous mode from the periphery of disc and then suddenly from the cylindrical case. The reaction takes place
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Fig. 2. Schematisation of the spinning disc reactor.
between calcium chloride and ammonium phosphate, in presence of ammonium hydroxide: 10CaCl2 þ 6ðNH4 Þ2 HPO4 þ 8NH4 OH-Ca10 ðPO4 Þ6 ðOHÞ2 þ 20NH4 Cl þ6H2 O Ammonium hydroxide is used to attain a pH value equal to 10, and as a consequence a high yield of the reaction to HAP (Pang and Bao, 2003). Reaction was conducted at room temperature. Three solutions were fed over the disc surface: the two reagent solutions at a distance from the disc centre of 2 or 3 cm and an aqueous solution of ammonium hydroxide at the disc centre In particular, a 10% aqueous solution of NH4OH at a flow rate of 80 ml/min was fed at disc centre, whereas the two reagent aqueous streams had both a flow rate of 100 ml/min, and a mass fraction of 5,6% of CaCl2 and 3,5% of (NH4)2HPO4, respectively. The calcium/phosphate (Ca/P) ratio of 1.67, corresponding to stechiometric HPA, was respected: this condition is considered by many researchers very important in order to obtain nanopowder of hydroxyapatite with high purity Martins et al., 2008). Particles in the produced suspension were measured by a DLS instrument (Brookhaven Plus 91). To contrast the formation of aggregates during the measurement a particular procedure was adopted. A 25 ml solution of NaOH 0.1 M at pH 10 with 0.2 g of the surfactant Twin 60 was first prepared, then five drops of the produced suspension were injected into this solution and submitted to ultrasonication for 3 min, before the size measurement by the DLS instrument. A typical size distribution measurement of the HAP nanoparticles is reported in Fig. 3. It has to be noticed that the produced nanoparticles, even in the nanometres range, are aggregation of single particles of 10 nm diameter. This is, in fact, the dimension of a single crystallite estimated from a X-ray diffractometer using the Sherrer formula (Burgue´s and Clemente, 2001) for particles produced at the maximum rotational speed. Several runs were made at different rotational speed in the range 58–147 rad/s by feeding the reagents at a radial distance of 2 cm from the centre. A few runs were operated at a constant speed of 147 rad/s by feeding the reagent at a radial distance of 3 cm to evaluate the effect of the feeding point. The obtained results reported in Fig. 4 show that the mean size of the produced particles is inversely proportional to the disc rotational velocity and decreases for a feed point of 3 cm from the centre. The minimum size, equal to 78 mm was obtained using a rotational velocity of 147 rad/s and feeding points of reagents 3 cm from the disc centre.
Fig. 3. Size distribution measurement of HAP nanoparticles obtained with a rotational speed of 147 rad/s.
3. Computational model 3.1. The governing equations The numerical simulations were performed using Ansys Fluent 12, a finite volume based commercial CFD package. The volume of fluid (VOF) model (Hirt and Nichols, 1981) was applied, because it is known as one of the most suitable multiphase models to track the air–liquid interface (see for example the work by Bhatelia et al. (2009)). Four phases were considered: the gas phase, composed by air, and three liquid phases for the three injections. In the VOF model the momentum equation depends on the volume fraction of all the phases trough the fluid density (r) and viscosity (m): @ !! ! !T ! ! ðrvÞ þ r:ðr v v Þ ¼ rp þ rm r v þ r v þ r g þ F ð3Þ @t where QUOTE is the external body force, in this case composed by centrifugal and Coriolis forces. The fluid density and viscosity are calculated as volume-mean value, and span over all phases: P P r ¼ ni¼ 1 aqi rqi , m ¼ ni¼ 1 aqi mqi . In each cell the properties concern only one phase or a mixture of phases if the interphase is present. The volume fraction of each phase is governed by the continuity equation:
daq ! þ v :raq ¼ 0: dt
ð4Þ
One phase can be chosen as the one satisfying the conservation of mass, n X
aqi ¼ 0
ð5Þ
i¼1
and is referred as ‘‘primary phase’’. In this work the air is chosen as primary phase. To model the mixing of the liquid components on the disc the species transport model was used, considering that in this case each species corresponds to one liquid phase. The mass conservation equation was solved for each considered species: ! @ ! ðrY i Þ þ r:ðr v Y i Þ ¼ r: J i @t
ð6Þ
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Fig. 4. Average size of the produced particles, varying the rotational speed and the injections position. Blue points and red point refer to an injection point at 2 cm and 3 cm from the disc centre, respectively. (For interpretation of the references to colour in this figure legend, the reader is reffered to the web version of this article.)
! where J i ¼ rDim rY i is the diffusive contribute. Since each species was considered, in first approximation, as composed by only water, the self-diffusivity coefficient of water in water (Wang, 1951) was considered (Dim ¼ 2:12:105 cm2 =s) for all the species. The above mentioned approximation can be justified by the relatively small concentrations of the reagents in water (less than 6% b.w.), so that the solutions can be considered diluted. The turbulence plays an important role in this process as it increases the mixing and mass transfer coefficients of the reactants. There are three possible approaches to solve problems that involve turbulence: the Reynolds Mean Navier Stokes (RANS), the Large Eddy Simulation (LES) and the Direct Numerical Simulation (DNS), listed in order of increasing accuracy in flow modelling and of computational cost. RANS resolves the transport equations considering mean flow quantities and only the whole turbulence scales are modelled. DNS directly solves the Navier Stokes equations without using any modelling. The LES model is a compromise: it is able to directly compute the large energy-carrying structures while the small scales are modelled. Choosing a turbulence model is not trivial, since both laminar and turbulent flow are present (near the centre of the disc and at the disc periphery, respectively), however, the LES model seemed to be appropriate. In it the large eddies, that are dictated by the problem geometry, are directly solved using the Navier Stokes equation. Thus the LES model was used in the simulations. The small eddies that are considered to be homogeneous and isotropic, so independent of the flow regime, are modelled by using one sub-grid models that describes interactions between the resolved and the unresolved scales. The governing equations for LES are obtained by spatially filtering over small scales. Filtering eliminates the eddies whose scales are smaller than the filter width or grid spacing used in the computations. The filtered Navier Stokes equations are as follow: @s @ @ @ @p @t ðrvi Þ þ ðrvi vj Þ ¼ m ij ij @t @xj @xj @xi @xj @xj
ð7Þ
where sij is the stress tensor due to molecular viscosity defined by @vi @vj 2 @v þ m l dij 3 @xl @xj @xi
sij ¼ m
ð8Þ
and tij is the subgrid-scale stress defined by:
tij ¼ Rvi vj Rvi vj
ð9Þ
The use of the VOF model for multiphase and the LES formulation to model turbulence make the simulation computational expensive. When the VOF model is used, transport equation is solved for each secondary phase. LES requires finer meshes than those used for RANS calculation and has to be run for a sufficiently long time to obtain stable statistic of the modelled flow. Therefore the computational cost of this process is very high. 3.2. Model geometry The setup of the disc to be modelled consists of an 85 mm diameter disc and three feed injectors, 1 mm in diameter, placed 5 mm above the disc surface. One injector is at the disc centre, the other two are opposite and located at 2 or 3 cm from the centre. The hydrodynamics of the flow over a rotating disc is hard to be modelled. The film thickness varies from 300 mm near the feed injection to 50 mm at the disc extremity. The disc diameter is about three orders of magnitude greater than the maximum film thickness. Therefore, to accurately analyse the system hydrodynamics and at the same time to save computational time, a fine grid on the vertical direction is needed while a large one can be used in the radial direction. Moreover a 3D model has to be adopted since asymmetric wave structures are present on the film surface (Bhatelia et al., 2009). A structured grid of cubic cells was used. The shape of the cell has been chosen to contain the numerical diffusion, since in this way the flux results always perpendicular to a cube face. Since we use the LES model, the grid is a part of the model itself. The cell width was set to 500 mm, that is 12 times the Kolmogorov scale of the system. The adopted Kolmogorov scale corresponds to a micromixing time, calculated by Bourne equation (Eq. 14), equal to 1 ms, usually considered a suitable value to obtain complete micromixing. To demonstrate that the solution is grid independent, we constructed a finer grid with a cell width of 8 times the Kolmogorov scale and we successfully compared the results concerning the film thickness, considering only the central injection. To minimise the number of cells and save computational time, the total volume was reduced moving the injection points from
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Fig. 5. Schematic representation of the computational domain and boundary conditions used in the simulations.
Table 1 Simulation settings and numerical details. Settings
Choice
Temporal discretization Pressure-velocity coupling Pressure Momentum Volume fraction
II order SIMPLE PRESTO! QUICK QUICK
Numerical details Multiphase model Turbulence model Grid size
VOF LES 540,000cells
the real height of 5 mm to 1 mm, and computing the new velocity of the feeding jet (Fig. 5). It has been verified that the choice of a reduced height of the volume has no influence on the hydrodynamic behaviour. To optimise the dimension of each cell the following approach was adopted. The total volume was divided into horizontal cylinders of different heights. The height of each cylinder increases with the distance from the disc surface of the air–water interface. Each cylinder is divided into cells along the radial and azimuthal directions. The cells have same height of the cylinder and their volume increases with the distance from the disc centre. The minimum cell height is 5 mm, on the disc surface, and the maximum is 150 mm, in the air region. To match the film shape the grid was refined along multiple iterations using the simulation results. The final grid consisted of 540,000 cells and 17 cylinders.
Fig. 6. Comparison between CFD and experimental data (Burns et al., 2003), radial velocity profile.
Fig. 7. Comparison between the CFD data and the Nusselt model results. Radial velocity profile are obtained with a rotational speed of 146.5 rad/s.
3.3. Numerical approach The disc geometry was created by using Gambit 2.4 software, and then it was imported into the processor Ansys Fluent 12. A segregated time dependent solver in Fluent was used. In Table 1 simulation settings and numerical details have been reported. The implicit body force treatment and the Courant number 0.25 for the volume fraction calculation were used. To ensure the convergence, the time step has to be very small (1e-05) and at the early stage of the calculation the maximum number of iterations per time step very high. The criteria of convergence for all the equations were set at 1e-04. The boundary conditions were: mass flow inlet for the three liquid feeds and pressure outlet for the outflow since the flow is not completely developed at the exit.
4. Model validation The CFD model was validated on the experimental data reported by Burns et al. (2003). The data describes the radial velocity of a water stream over a rotating disc. The radius of the disc was 15 cm and the water stream was injected at the disc centre through a 5 mm tube. Since in this case the area to be simulated is too large, to save computational time the grid was restricted to simulate only the disc from the centre to half radius. The grid was built following the same criteria of the previous section and using the same cell size, but, since the disc radius was
Fig. 8. Ekman number plot. Ratio of film thickness calculated with the CFD model and with the Nusselt model for varying Ekman number.
increased, the cell number of the validation grid results to be 1,628,804. The rotational speed of the disc was 21 rad/s and the water flow rate 0.01 kg/s. In Fig. 6, the experimental and computed radial velocities are reported as a function of the radial position. The comparison shows a good agreement between experimental and simulated data. It is interesting to compare for the case examined in this work the radial velocity and the film thickness predicted by means of the simplified Nusselt model described by Wood and Watts (1973) and by the CFD model here presented. The Nusselt model assumes that the flow is fully laminar over the disc and that the film thickness is: hN ¼
3Q n 2pr 2 o2
1=3 ð10Þ
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and the radial velocity: Q v¼ 2prhN
ð11Þ
In Fig. 7 the radial velocity profiles are compared. The three different zones mentioned in the Burns’ work injection, acceleration and synchronization (see Fig. 1), can be clearly identified in the profile predicted by the CFD model. The two models agree just in the synchronized zone, when the distance from the centre of the disc is greater than 1 cm and the Ekman number greater than 1.62 (Fig. 8).
by using the CFD model to evaluate the specific energy dispersion at the reagents feed points. According to Moore (1996), the specific energy power dispersed on a rotating disc can be calculated as:
e¼
1 ððr 2 Uo2 þv2re Þðr 2i Uo2 þ v2ri ÞÞ 2Ut res e
where t res is the residence time of the liquid solution on the rotating disc between an external radius,r e , where the fluid velocity is vre and an internal radius,r i where the fluid velocity is vri . The residence time was calculated with CFD simulations as: t res ¼
5. Results and discussion First of all, the thickness of the liquid layer over the SDR surface at the different applied rotational speeds was predicted by the developed model. The obtained results are reported in Fig. 9. The layer thickness, around 300 mm near the disc centre, progressively decreases along the disc radius and at the disc periphery almost asymptotic values of less than 100 mm are reached. The thickness is inversely proportional to the rotational speed as expected. The minimum value of layer thickness, equal to 50 mm is attained at disc periphery for a rotational speed of 146.5 rad/s. SDR is an apparatus that utilises the centrifugal forces to produce thin liquid film. The intensification of the mixing process of the reagent solutions in the liquid film is essential to maximise the concentration of the reaction product, its local supersaturation and the nucleation rate as well. The most important parameter affecting the mixing intensity is the local specific dispersed power, e, which represents the energy transferred to the fluid by the disc. Its relationship with the mixing time at macro, meso and micro scale is reported in Table 2. For all scales mixing time is inversely proportional to the specific dispersed energy. Since shorter is the micromixing time more effective is the local mixing between the reagent solutions, at the higher energy dispersion higher values of reaction product concentration and nucleation rate occur and smaller values of nanoparticles size, as a consequence, are expected. We made an attempt to check the inverse relationship between the energy dispersion and the size of the nanoparticles obtained in this work,
ð12Þ
2Uðr e r i Þ vre þ vri
ð13Þ
The values of the specific dispersion energy determined for a feeding point 2 cm far from the disc at the adopted values of rotational speed are reported in Fig. 10. In this figure the value of e calculated for a feeding point 3 cm far from the disc centre and at 146.5 rad/s is also reported. Finally, in Fig. 11 the obtained mean size of the produced HAP nanoparticles vs. the specific dispersed energy at the reagents injection point are reported in a log–log scale. The data are well fitted by the equation and the confidence interval of the slope value at a probability of 95% is 70.095. It is interesting to observe that the slope of the log–log linear correlation between mean diameter and e is close to 0.5, which is the dependence both of micromixing and mesomixing times with
Fig. 10. Specific dispersed power as a function of the rotational speed and reactants injection position. Blue points and red point refer to a feed points at 2 cm and 3 cm from the disc centre, respectively. (For interpretation of the references to colour in this figure legend, the reader is reffered to the web version of this article.)
Fig. 9. Film thickness profile varying rotational speed obtained with the three injections.
Table 2 Dependence of the mixing time on the specific dispersed power. (Johnson and Prud’homme, 2003). Type of mixing
Macromixing
Mesomixing
Micromixing
Relationship
tpe 1/3
tp(n/e)1/2
tp(n/e)1/2Sc
Fig. 11. Correlation between HAP mean particle size and specific dispersed power (Blue points and red point are referred to reagent injection point 2 cm and 3 cm far from the disc centre, respectively). (For interpretation of the references to colour in this figure legend, the reader is reffered to the web version of this article.)
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respect to the specific energy dispersion (see Table 2). It confirms the major role played by micromixing and mesomixing on the determination of the produced nanoparticles size. In conclusion, increasing the speed rotation and the distance of the reactant feed from the disc centre, the particle diameter decreases, due to higher efficiency of the mixing processes at micro and meso scales. For each experimental run, the mixing time at the feeding point was calculated by means of the semi-empirical expression suggested by Bourne et al. (1981): n0:5
t ¼ 12
e
ð14Þ
Increasing the rotational speed from 58 to 146.7 rad/s and the distance from the centre of the disc, the mixing time decreases from 3.9 to 0.9 ms. For rotational speed higher than 120 rad/s the mixing time approaches values around 1 ms; which is the typical value for the attainment of micromixing (Baffi et al., 2002). The implemented CFD model can also predict at any point of the liquid layer the fraction volume of each solution adopted in the experimental work, i.e., ammonia, calcium chloride and ammonia phosphate aqueous solution. This kind of information can be very useful for qualitatively estimating the effect of the rotational speed on the mixing degree of each solution over the disc. In Fig. 12 the map of the volume fraction of each aqueous solution in the layer section 5 mm from the disc surface is reported for two values of speed rotation: 58 and 147 rad/s. First of all, ammonia solution, fed at the disc centre, results to be absent in the examined layer section for the smaller rotation speed. As a matter of fact from the CFD model results this solution is present only in upper sections of the layer, that is because the ammonia aqueous stream is pushed upward by the reagent
79
streams. It is not the case when a rotational speed of 147 rad/s is applied. In fact, the ammonia solution volume fraction at large rotational speed is present wherever in the considered layer section. The presence of ammonia solution all around the disc is important to assure the local attainment of a pH value of 10 and a high selectivity of the reaction to hydroxyapatite, as well. The volume fraction distribution of the two reagents over the considered layer section at a constancy of rotational speed is similar. At 58 rad/s the mixing is very pour at the injection point. The volume fraction progressively decreases along the azimuthal abscissa due to the mesomixing, but still at some radial distance from the injection point volume fractions are far to be constant. On the contrary at 147 rad/s a very different picture of each reagent solution volume fraction appears. To match the stoichiometric ratio between the reagents the volume fraction for the two reagents and the ammonia solutions has to be 0.35 and 0.3, respectively. From the figure it can be noticed that the solutions are in the correct stoichiometric ratio. At each reagent injection point the other solutions are already present and very early along the disc an almost homogeneous fraction volume of each reagent solution is attained. The pictures in Fig. 12 gives an impressive confirmation of the effect of the disc rotational speed on the mixing effectiveness and support the hypothesis of an almost complete mixing at micro scale by operating at 147 rad/s.
6. Conclusions SDR is an effective apparatus to produce nanoparticles by reaction precipitation process in continuous mode.
Fig. 12. Representation of volume fraction of each aqueous solution at different disc rotational speed. (A) 58 rad/s, (B)147 rad/s.
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This apparatus in fact progressively intensifies the mixing phenomena in the liquid layer, as much as the rotational speed is increased, until micromixing conditions are established. In this work a CFD model of the liquid layer over the rotating disc surface have been implemented and used to interpret the results of a reaction precipitation process for the production of nanoparticles of hydroxyapatite. By increasing the speed of rotation between 58 and 147 rad/s nanoparticles from 350 to around 110 nm in size were obtained for an injection point 2 cm far from the disc centre. A further size reduction was obtained by locating the feed point at 3 cm from the disc centre because of a higher local specific dispersed energy: in this case by operating at 147 rad/s nanoparticles smaller than 90 nm in size were produced. The simulation of the liquid layer by the developed CFD model allowed to show the inverse proportionality between the produced nanoparticles size and the specific dispersed energy at the injection point. Moreover the prediction of the distribution of the volume fraction of the three adopted aqueous streams in a section of the liquid layer gave an impressive picture of the change of the mixing effectiveness when the rotational speed is increased.
Nomenclature D E F g h hN p
m Q r t tres v v Y
mass diffusion coefficient, m2/s Ekman number external body forces, N gravity, kg/m2 film thickness, m Nusselt film thickness, m pressure, Pa filtered pressure, Pa volumetric flow rate, m3/s radius, m time, s residence time of the fluid on the disc, s fluid velocity, m/s fluid filtered velocity, m/s mass fraction
Greek letters
a e m n r t o
phase volume fraction dispersed specific power, W/kg liquid viscosity, kg/m.s liquid kinematic viscosity, m2/s liquid density, kg/m3 mixing time, s rotational speed, rad/s
Abbreviations PRESTO Pressure staggering option QUICK Quadratic Upstream Interpolation for Convective Kinematics SIMPLE Semi-Implicit Method for Pressure Linked Equations
References Baffi, G., Cafiero, M.L., Chianese, A., Jachuck, R.J., 2002. Process intensification: precipitation of barium sulphate using a Spinning disc reactor (SDR). Ind. Eng. Chem. Res. 41, 5240–5246. Bhatelia, T. J., Utikar, R.P., Pareek, V.K., Tade, M.O., 2009. Characterizing liquid film thickness in spinning disc reactors. 7th International Conference on CFD in the Minerals and Process Industries. Melbourne, Australia, December 9–11. Bourne, J.R., Kozicki, F., Rys, P., 1981. Mixing and fast chemical reaction – I; test reactions to determine segregation. Chem. Eng. Sci. 36, 1643. Burgue´s, T.J., Clemente, R.R., 2001. Hydroxyapatite precipitation in a semibatch process. Cryst. Res. Technol. 36 (8–10), 1075–1082. Burns, J.R., Ramshav, C., Jachuck, R.J, 2003. Measurement of liquid film thickness and the determination of spin-up radius on a rotating disc using an electrical resistance technique. Chem. Eng. Sci. 58, 2245–2253. Butuzov, A.I., Puhovoi, I.I., 1976. On regimes of liquid film flows over a rotating surface. J. Eng. Phys. 31, 217. Charwat, A.F., Kelly, R.E., Gazley, C., 1972. The flow and stability of thin liquid films on a rotating disc. J. Fluid Mech. 53, 227–255. Espig, H., Hoyle, R., 1965. Waves in a thin layer on a rotating disk. J. Fluid Mech. 22, 671–677. Hirt, C.W., Nichols, B.D., 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225. Hounslow, M.J., Mumtaz, H.S., 2000. Aggregation during precipitation from solution: an experimental investigations using a poiseuille flow. Chem. Eng. Sci. 55, 5671–5681. Johnson, B.K., Prud’homme, R.K., 2003. Chemical processing and micromixing in confined impinging jets. AlChE J. 49 (9), 2264–2282. Martins, M.A., Santos, C., Almeida, M.M., Costa, M.E.V., 2008. Hydroxyapatite micro- and nanoparticles: nucleation and growth mechanisms in the presence of citrate species. J. Colloid Interface Sci. 318, 210–216. ¨ Nusselt, W., 1916. Die Oberflachenkondensation des Wasserdampfes. Zoologische Verhandelingen Deutsch Ingenieurwessen 60 (27), 541–546. Pang, Y.X., Bao, X., 2003. Influence of temperature, ripening time and calcinations on the morphology and crystallinity of hydroxyapatite nanoparticles. J. Eur. Ceram. Soc. 23, 1697–1704. Plasari, E., Muhr, H., Rousseaux, J.-M., Vial, C., 2001. CFD simulation of precipitation in the sliding-surface mixing device. Chem. Eng. Sci. 56 (4), 1677–1685. Raston, C.L., Anantachoke, N., Makha, M., Reutrakul, V., Smith, N.C., Saunders, M., 2006. Fine tuning the production of nanosized b-carotene particles using spinning disk processing. J. Am. Chem. Soc. 128 (42), 13847–13853. Trippa, G., Hetherington, P., Jachuck, R., 2002. Process intensification: precipitation of calcium carbonate from the carbonation reaction of lime water using a spinning disc reactor, Proceedings of 15th International Symposium on Industrial Crystallization, Sorrento, Italy, September 15–18. Wang, J.H., 1951. Self-diffusion and structure of liquid water. measurement of selfdiffusion of liquid water with deuterium as trace. J. Am. Chem. Soc. 73, 510–513. Wood, R.M., Watts, B.E., 1973. The flow, heat and mass transfer characteristics of liquid films on rotating discs. Trans. Inst. Chem. Eng. 51, 315–322.
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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Reaction-precipitation by a spinning disc reactor: Influence of hydrodynamics on nanoparticles production B. de Caprariis n, M. Di Rita, M. Stoller, N. Verdone, A. Chianese Department of Ingegneria Chimica Materiali Ambiente, Sapienza University of Rome, via Eudossiana 18, 00184 Rome, Italy
a r t i c l e i n f o
abstract
Article history: Received 20 September 2011 Accepted 26 March 2012 Available online 6 April 2012
The main aim of this work is to show the usefulness of a Computational Fluid Dynamics (CFD) model to interpret the results obtained from a reaction-precipitation process by using a spinning disc reactor (SDR). First, an experimental investigation was performed to produce nanoparticles of hydroxyapatite by a reaction at ambient temperature between aqueous solutions of calcium chloride and ammonium phosphate in presence of ammonium hydroxide to achieve an operating pH of 10. An SDR was adopted. It consists of a PVC disc rotating at a speed in the range 58–147 rad/s. Nanoparticles in the range 78– 350 nm were obtained by varying the rotational speed and the reagent solutions injection points. A CFD model was then developed to predict the hydrodynamics and the volume fraction of the reagent solutions in the liquid layer. The numerical simulation was performed using Ansys Fluent 12 and the volume of fluid model (VOF) approach was adopted. After the validation of the model on literature data, the CFD model was used to interpret the experimental results. It was shown that the size of the produced nanoparticles are inversely proportional to the specific dispersed energy at a reagent feed location and that at the highest adopted rotational speed an almost complete mixing of the reagent solutions is attained in the liquid layer. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Precipitation Hydrodynamics Mixing Hydroxyapatite Spinning disc reactor CFD
1. Introduction Spinning disc reactor (SDR) is an effective apparatus for the production of nanoparticles by wet chemical synthesis. Rotation of disc surface at high speed creates high centrifugal fields which promote thin film flow with a thickness in the range 50–500 mm. Films are highly sheared and have numerous unstable surface ripples, giving rise to intense mixing. SDR has been successfully applied to a range of processes which are micromixing dependent: polymerisation, competitive organic reactions and reactionprecipitation. The benefit of mixing intensification to precipitation process is to reduce the mixing time down to 1 ms or less, so that the sum of reaction time and the mixing time is shorter than the induction nucleation time and the maximum product supersaturation is locally achieved where reaction takes place. This condition leads to the maximum allowable value of nucleation rate and of generated nuclei as well. Moreover, due to the very high nucleation rate a low value of supersaturation residues for the subsequent processes of particles growth and aggregation and, as a consequence, nanoparticles may be produced.
n
Corresponding author. Tel.: þ39 064 4585 600; fax: þ 39 06 482 7453. E-mail address: [email protected] (B. de Caprariis).
0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.03.043
The performances of an SDR as a device for a reactionprecipitation process are investigated in many research works. Baffi et al. (2002) studied the precipitation of barium sulphate by an SDR, 0.5 m in diameter, operating at rotational speed between 200 and 1000 rpm. They showed that at a rotational speed greater than 800 rpm the performances were similar to that of a T-mixer, in terms of number of generated particles per unit volume and micromixing time, which was of the order of 1 ms. In both the cases a complete micromixing was considered to be attained, but according to these Authors the energy consumption is much less when the rotating device is used. SDR is also used to perform gas– liquid reaction. Trippa et al. (2002) demonstrated the use of an SDR for the production of calcium carbonate particles from dissolved CO2. The capability of this intensified system to generate a wide range of particle sizes and morphologies through control of the operating conditions, such as rotational speed, was clearly shown. Raston et al. (2006) produced b-carotene nanoparticles with spinning disc reactor and they obtained good results on controlling size and shape of the nanoparticles. All these works emphasised the great importance of hydrodynamics on the produced particles size. Therefore, the investigation on hydrodynamics in the liquid layer over the disc surface, particularly in correspondence of the reagents feed points, is of major importance to interpret the precipitation process over an SDR. Many works have been devoted to predict hydrodynamics of
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a liquid film over a spinning disc by means of a simulation model. Hydrodynamics of the liquid film was first studied in 1916 when Nusselt (1916) modelled the downward flow of a condensing liquid film. He considered that the liquid film was stable with no shear at the gas–liquid interface and that the acceleration was due exclusively to the gravitational force. The Nusselt analysis constituted a basis for more sophisticated models. In case of a liquid flow over a rotating disc higher accelerations have to be considered since they laid to thinner film and instabilities over the surface. Espig and Hoyle (1965) found that the predicted film thickness is 40–50% greater than that one predicted by the Nusselt’s theory. They attributed this difference to the formation of waves on the film surface. These Authors experimentally observed three different flow regimes over the rotating disc: a waveless laminar flow near the inlet region, a zone characterised by an asymmetric wave formation and a turbulent region at the disc periphery produced by a combination of asymmetric and helical waves. Charwat et al. (1972) observed a decay of the waves with increasing radius and so the formation of a new laminar zone after the turbulent one. Those results were confirmed by the work of Butuzov and Puhovoi (1976). Woods and Watts (1973) studied the heat and mass transfer on rotating discs. They identified three different wavy regimes. To describe the hydrodynamic over the rotating disc, they proposed a two-dimensional model, known as Pigford model. This model took into account inertial and viscous influences and was given by the following two equations: vr
@vr v2t 12p2 r 2 K 1 vt 3 ¼ vr @t r Q2
ð1Þ
vr
@vt vr vt 12p2 r 2 K 2 vt 2 þ ¼ vr ðr ovt Þ @t r Q2
ð2Þ
where K1 and K2 were empirical correction factors to adjust the viscous terms in the equation. Burns et al. (2003) performed experimental tests to investigate the hydrodynamics of the flow over a spinning disc. They compared the Pigford model with the Nusselt theory and demonstrated that the Nusselt model is not able to predict hydrodynamic behaviour of liquid film on a rotating disc when the inertial forces are higher than the viscous one, that is for low values of the Ekman number (E¼ n/h2o). In the Burns’ work the radial velocity profile was analysed and three zones were identified (Fig. 1): the injection zone where the liquid flow is slowed down by viscous drag, the acceleration zone where the high centrifugal acceleration increases the radial flow
velocity and the synchronized zone where the fluid radial velocity is close to the disc rotational speed. In the synchronized zone the film flow behaviour becomes similar to that calculated by Nusselt model. The hydrodynamics of the liquid film over a spinning disc was recently investigated by Bhatelia et al. (2009) using the computational fluid dynamics. They performed 3D simulations of the liquid layer and successfully compared the simulation results with the Burns experimental data. The only research work dealing with the simulation of a precipitation process performed over a rotating disc (Plasari et al., 2001), concerns a two phase solid-liquid system. In this work CFD technique aimed at the prediction of the reactor hydrodynamics and simulation of the precipitation process. In this latter respect, a two-step process of nucleation and growth was considered, whereas aggregation was neglected. According to the Authors, some discrepancies still remained between experiments and calculation. When precipitation takes place in presence of intensified micromixing conditions the size distribution of nanoparticles are mainly determined by a two step process nucleation-aggregation. In fact, in presence of complete micromixing between the reagents, almost all the supersaturation is consumed by nucleation, just a small residual local supersaturation is available and it is realistic to think that it is consumed during the aggregation process. According to Hounslow and Mumtaz (2000) aggregation may be described as a two step process consisting of shear stress ortokinetic collisions between two particles, and the subsequent cementation of the two particles at their point of contact due to the internal diffusion of solute ions. Fluid shear plays a double opposite role: it increases the number and intensity of particles collisions and reduces the timescale for bridge formation, thus lowering the effect of the collisions effect. The above mentioned authors showed that the aggregation of nanoparticles of calcium oxalate, in a Poiseuille flow crystallizer, is reduced in presence of more intensified conditions since the disruptive action of the fluid shear stress on aggregation was predominant. The results of this work showed that the mixing process intensification enhanced nucleation and reduced aggregation leading to reduction of nanoparticles size. Process precipitation over the surface of a horizontal rotating disc gives rise to a three phase gas–liquid–solid systems with a three dimensional structure. In presence of multiple reagent streams we may have a very complex map of the reagent concentrations with variations over each plane, positioned at different layer height. The process performances are dramatically dependent on the mixing phenomena between the reagents. The rotational speed is the most important operating parameter of an SDR, since it plays the main influence on micromixing, and as a consequence on nucleation and aggregation rate. In the present work the effect of the rotational speed of an SDR on the precipitation of hydroxyapatite was investigated. Then, an attempt for the interpretation of the experimental results was made on the basis of the prediction of the reagent concentrations in the fluid layer, by using a CFD simulation model.
2. Experimental set-up
Fig. 1. Radial velocity profile, three flow regimes identification. Picture taken from Burns et al. (2003).
An SDR was used to produce nanoparticles of hydroxyapatite (HAP) QUOTE by chemical precipitation. The experimental device, schematized in Fig. 2 consists of a cylindrical case with an inner disc made by PVC and 8.5 cm in diameter. Rotational velocity of the disc may be increased up to 147 rad/s. The reagent solutions are fed over the disc at a distance of 5 mm from the disc surface through tubes, 1 mm in diameter. The produced suspension draws off in continuous mode from the periphery of disc and then suddenly from the cylindrical case. The reaction takes place
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75
Fig. 2. Schematisation of the spinning disc reactor.
between calcium chloride and ammonium phosphate, in presence of ammonium hydroxide: 10CaCl2 þ 6ðNH4 Þ2 HPO4 þ 8NH4 OH-Ca10 ðPO4 Þ6 ðOHÞ2 þ 20NH4 Cl þ6H2 O Ammonium hydroxide is used to attain a pH value equal to 10, and as a consequence a high yield of the reaction to HAP (Pang and Bao, 2003). Reaction was conducted at room temperature. Three solutions were fed over the disc surface: the two reagent solutions at a distance from the disc centre of 2 or 3 cm and an aqueous solution of ammonium hydroxide at the disc centre In particular, a 10% aqueous solution of NH4OH at a flow rate of 80 ml/min was fed at disc centre, whereas the two reagent aqueous streams had both a flow rate of 100 ml/min, and a mass fraction of 5,6% of CaCl2 and 3,5% of (NH4)2HPO4, respectively. The calcium/phosphate (Ca/P) ratio of 1.67, corresponding to stechiometric HPA, was respected: this condition is considered by many researchers very important in order to obtain nanopowder of hydroxyapatite with high purity Martins et al., 2008). Particles in the produced suspension were measured by a DLS instrument (Brookhaven Plus 91). To contrast the formation of aggregates during the measurement a particular procedure was adopted. A 25 ml solution of NaOH 0.1 M at pH 10 with 0.2 g of the surfactant Twin 60 was first prepared, then five drops of the produced suspension were injected into this solution and submitted to ultrasonication for 3 min, before the size measurement by the DLS instrument. A typical size distribution measurement of the HAP nanoparticles is reported in Fig. 3. It has to be noticed that the produced nanoparticles, even in the nanometres range, are aggregation of single particles of 10 nm diameter. This is, in fact, the dimension of a single crystallite estimated from a X-ray diffractometer using the Sherrer formula (Burgue´s and Clemente, 2001) for particles produced at the maximum rotational speed. Several runs were made at different rotational speed in the range 58–147 rad/s by feeding the reagents at a radial distance of 2 cm from the centre. A few runs were operated at a constant speed of 147 rad/s by feeding the reagent at a radial distance of 3 cm to evaluate the effect of the feeding point. The obtained results reported in Fig. 4 show that the mean size of the produced particles is inversely proportional to the disc rotational velocity and decreases for a feed point of 3 cm from the centre. The minimum size, equal to 78 mm was obtained using a rotational velocity of 147 rad/s and feeding points of reagents 3 cm from the disc centre.
Fig. 3. Size distribution measurement of HAP nanoparticles obtained with a rotational speed of 147 rad/s.
3. Computational model 3.1. The governing equations The numerical simulations were performed using Ansys Fluent 12, a finite volume based commercial CFD package. The volume of fluid (VOF) model (Hirt and Nichols, 1981) was applied, because it is known as one of the most suitable multiphase models to track the air–liquid interface (see for example the work by Bhatelia et al. (2009)). Four phases were considered: the gas phase, composed by air, and three liquid phases for the three injections. In the VOF model the momentum equation depends on the volume fraction of all the phases trough the fluid density (r) and viscosity (m): @ !! ! !T ! ! ðrvÞ þ r:ðr v v Þ ¼ rp þ rm r v þ r v þ r g þ F ð3Þ @t where QUOTE is the external body force, in this case composed by centrifugal and Coriolis forces. The fluid density and viscosity are calculated as volume-mean value, and span over all phases: P P r ¼ ni¼ 1 aqi rqi , m ¼ ni¼ 1 aqi mqi . In each cell the properties concern only one phase or a mixture of phases if the interphase is present. The volume fraction of each phase is governed by the continuity equation:
daq ! þ v :raq ¼ 0: dt
ð4Þ
One phase can be chosen as the one satisfying the conservation of mass, n X
aqi ¼ 0
ð5Þ
i¼1
and is referred as ‘‘primary phase’’. In this work the air is chosen as primary phase. To model the mixing of the liquid components on the disc the species transport model was used, considering that in this case each species corresponds to one liquid phase. The mass conservation equation was solved for each considered species: ! @ ! ðrY i Þ þ r:ðr v Y i Þ ¼ r: J i @t
ð6Þ
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Fig. 4. Average size of the produced particles, varying the rotational speed and the injections position. Blue points and red point refer to an injection point at 2 cm and 3 cm from the disc centre, respectively. (For interpretation of the references to colour in this figure legend, the reader is reffered to the web version of this article.)
! where J i ¼ rDim rY i is the diffusive contribute. Since each species was considered, in first approximation, as composed by only water, the self-diffusivity coefficient of water in water (Wang, 1951) was considered (Dim ¼ 2:12:105 cm2 =s) for all the species. The above mentioned approximation can be justified by the relatively small concentrations of the reagents in water (less than 6% b.w.), so that the solutions can be considered diluted. The turbulence plays an important role in this process as it increases the mixing and mass transfer coefficients of the reactants. There are three possible approaches to solve problems that involve turbulence: the Reynolds Mean Navier Stokes (RANS), the Large Eddy Simulation (LES) and the Direct Numerical Simulation (DNS), listed in order of increasing accuracy in flow modelling and of computational cost. RANS resolves the transport equations considering mean flow quantities and only the whole turbulence scales are modelled. DNS directly solves the Navier Stokes equations without using any modelling. The LES model is a compromise: it is able to directly compute the large energy-carrying structures while the small scales are modelled. Choosing a turbulence model is not trivial, since both laminar and turbulent flow are present (near the centre of the disc and at the disc periphery, respectively), however, the LES model seemed to be appropriate. In it the large eddies, that are dictated by the problem geometry, are directly solved using the Navier Stokes equation. Thus the LES model was used in the simulations. The small eddies that are considered to be homogeneous and isotropic, so independent of the flow regime, are modelled by using one sub-grid models that describes interactions between the resolved and the unresolved scales. The governing equations for LES are obtained by spatially filtering over small scales. Filtering eliminates the eddies whose scales are smaller than the filter width or grid spacing used in the computations. The filtered Navier Stokes equations are as follow: @s @ @ @ @p @t ðrvi Þ þ ðrvi vj Þ ¼ m ij ij @t @xj @xj @xi @xj @xj
ð7Þ
where sij is the stress tensor due to molecular viscosity defined by @vi @vj 2 @v þ m l dij 3 @xl @xj @xi
sij ¼ m
ð8Þ
and tij is the subgrid-scale stress defined by:
tij ¼ Rvi vj Rvi vj
ð9Þ
The use of the VOF model for multiphase and the LES formulation to model turbulence make the simulation computational expensive. When the VOF model is used, transport equation is solved for each secondary phase. LES requires finer meshes than those used for RANS calculation and has to be run for a sufficiently long time to obtain stable statistic of the modelled flow. Therefore the computational cost of this process is very high. 3.2. Model geometry The setup of the disc to be modelled consists of an 85 mm diameter disc and three feed injectors, 1 mm in diameter, placed 5 mm above the disc surface. One injector is at the disc centre, the other two are opposite and located at 2 or 3 cm from the centre. The hydrodynamics of the flow over a rotating disc is hard to be modelled. The film thickness varies from 300 mm near the feed injection to 50 mm at the disc extremity. The disc diameter is about three orders of magnitude greater than the maximum film thickness. Therefore, to accurately analyse the system hydrodynamics and at the same time to save computational time, a fine grid on the vertical direction is needed while a large one can be used in the radial direction. Moreover a 3D model has to be adopted since asymmetric wave structures are present on the film surface (Bhatelia et al., 2009). A structured grid of cubic cells was used. The shape of the cell has been chosen to contain the numerical diffusion, since in this way the flux results always perpendicular to a cube face. Since we use the LES model, the grid is a part of the model itself. The cell width was set to 500 mm, that is 12 times the Kolmogorov scale of the system. The adopted Kolmogorov scale corresponds to a micromixing time, calculated by Bourne equation (Eq. 14), equal to 1 ms, usually considered a suitable value to obtain complete micromixing. To demonstrate that the solution is grid independent, we constructed a finer grid with a cell width of 8 times the Kolmogorov scale and we successfully compared the results concerning the film thickness, considering only the central injection. To minimise the number of cells and save computational time, the total volume was reduced moving the injection points from
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Fig. 5. Schematic representation of the computational domain and boundary conditions used in the simulations.
Table 1 Simulation settings and numerical details. Settings
Choice
Temporal discretization Pressure-velocity coupling Pressure Momentum Volume fraction
II order SIMPLE PRESTO! QUICK QUICK
Numerical details Multiphase model Turbulence model Grid size
VOF LES 540,000cells
the real height of 5 mm to 1 mm, and computing the new velocity of the feeding jet (Fig. 5). It has been verified that the choice of a reduced height of the volume has no influence on the hydrodynamic behaviour. To optimise the dimension of each cell the following approach was adopted. The total volume was divided into horizontal cylinders of different heights. The height of each cylinder increases with the distance from the disc surface of the air–water interface. Each cylinder is divided into cells along the radial and azimuthal directions. The cells have same height of the cylinder and their volume increases with the distance from the disc centre. The minimum cell height is 5 mm, on the disc surface, and the maximum is 150 mm, in the air region. To match the film shape the grid was refined along multiple iterations using the simulation results. The final grid consisted of 540,000 cells and 17 cylinders.
Fig. 6. Comparison between CFD and experimental data (Burns et al., 2003), radial velocity profile.
Fig. 7. Comparison between the CFD data and the Nusselt model results. Radial velocity profile are obtained with a rotational speed of 146.5 rad/s.
3.3. Numerical approach The disc geometry was created by using Gambit 2.4 software, and then it was imported into the processor Ansys Fluent 12. A segregated time dependent solver in Fluent was used. In Table 1 simulation settings and numerical details have been reported. The implicit body force treatment and the Courant number 0.25 for the volume fraction calculation were used. To ensure the convergence, the time step has to be very small (1e-05) and at the early stage of the calculation the maximum number of iterations per time step very high. The criteria of convergence for all the equations were set at 1e-04. The boundary conditions were: mass flow inlet for the three liquid feeds and pressure outlet for the outflow since the flow is not completely developed at the exit.
4. Model validation The CFD model was validated on the experimental data reported by Burns et al. (2003). The data describes the radial velocity of a water stream over a rotating disc. The radius of the disc was 15 cm and the water stream was injected at the disc centre through a 5 mm tube. Since in this case the area to be simulated is too large, to save computational time the grid was restricted to simulate only the disc from the centre to half radius. The grid was built following the same criteria of the previous section and using the same cell size, but, since the disc radius was
Fig. 8. Ekman number plot. Ratio of film thickness calculated with the CFD model and with the Nusselt model for varying Ekman number.
increased, the cell number of the validation grid results to be 1,628,804. The rotational speed of the disc was 21 rad/s and the water flow rate 0.01 kg/s. In Fig. 6, the experimental and computed radial velocities are reported as a function of the radial position. The comparison shows a good agreement between experimental and simulated data. It is interesting to compare for the case examined in this work the radial velocity and the film thickness predicted by means of the simplified Nusselt model described by Wood and Watts (1973) and by the CFD model here presented. The Nusselt model assumes that the flow is fully laminar over the disc and that the film thickness is: hN ¼
3Q n 2pr 2 o2
1=3 ð10Þ
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and the radial velocity: Q v¼ 2prhN
ð11Þ
In Fig. 7 the radial velocity profiles are compared. The three different zones mentioned in the Burns’ work injection, acceleration and synchronization (see Fig. 1), can be clearly identified in the profile predicted by the CFD model. The two models agree just in the synchronized zone, when the distance from the centre of the disc is greater than 1 cm and the Ekman number greater than 1.62 (Fig. 8).
by using the CFD model to evaluate the specific energy dispersion at the reagents feed points. According to Moore (1996), the specific energy power dispersed on a rotating disc can be calculated as:
e¼
1 ððr 2 Uo2 þv2re Þðr 2i Uo2 þ v2ri ÞÞ 2Ut res e
where t res is the residence time of the liquid solution on the rotating disc between an external radius,r e , where the fluid velocity is vre and an internal radius,r i where the fluid velocity is vri . The residence time was calculated with CFD simulations as: t res ¼
5. Results and discussion First of all, the thickness of the liquid layer over the SDR surface at the different applied rotational speeds was predicted by the developed model. The obtained results are reported in Fig. 9. The layer thickness, around 300 mm near the disc centre, progressively decreases along the disc radius and at the disc periphery almost asymptotic values of less than 100 mm are reached. The thickness is inversely proportional to the rotational speed as expected. The minimum value of layer thickness, equal to 50 mm is attained at disc periphery for a rotational speed of 146.5 rad/s. SDR is an apparatus that utilises the centrifugal forces to produce thin liquid film. The intensification of the mixing process of the reagent solutions in the liquid film is essential to maximise the concentration of the reaction product, its local supersaturation and the nucleation rate as well. The most important parameter affecting the mixing intensity is the local specific dispersed power, e, which represents the energy transferred to the fluid by the disc. Its relationship with the mixing time at macro, meso and micro scale is reported in Table 2. For all scales mixing time is inversely proportional to the specific dispersed energy. Since shorter is the micromixing time more effective is the local mixing between the reagent solutions, at the higher energy dispersion higher values of reaction product concentration and nucleation rate occur and smaller values of nanoparticles size, as a consequence, are expected. We made an attempt to check the inverse relationship between the energy dispersion and the size of the nanoparticles obtained in this work,
ð12Þ
2Uðr e r i Þ vre þ vri
ð13Þ
The values of the specific dispersion energy determined for a feeding point 2 cm far from the disc at the adopted values of rotational speed are reported in Fig. 10. In this figure the value of e calculated for a feeding point 3 cm far from the disc centre and at 146.5 rad/s is also reported. Finally, in Fig. 11 the obtained mean size of the produced HAP nanoparticles vs. the specific dispersed energy at the reagents injection point are reported in a log–log scale. The data are well fitted by the equation and the confidence interval of the slope value at a probability of 95% is 70.095. It is interesting to observe that the slope of the log–log linear correlation between mean diameter and e is close to 0.5, which is the dependence both of micromixing and mesomixing times with
Fig. 10. Specific dispersed power as a function of the rotational speed and reactants injection position. Blue points and red point refer to a feed points at 2 cm and 3 cm from the disc centre, respectively. (For interpretation of the references to colour in this figure legend, the reader is reffered to the web version of this article.)
Fig. 9. Film thickness profile varying rotational speed obtained with the three injections.
Table 2 Dependence of the mixing time on the specific dispersed power. (Johnson and Prud’homme, 2003). Type of mixing
Macromixing
Mesomixing
Micromixing
Relationship
tpe 1/3
tp(n/e)1/2
tp(n/e)1/2Sc
Fig. 11. Correlation between HAP mean particle size and specific dispersed power (Blue points and red point are referred to reagent injection point 2 cm and 3 cm far from the disc centre, respectively). (For interpretation of the references to colour in this figure legend, the reader is reffered to the web version of this article.)
B. de Caprariis et al. / Chemical Engineering Science 76 (2012) 73–80
respect to the specific energy dispersion (see Table 2). It confirms the major role played by micromixing and mesomixing on the determination of the produced nanoparticles size. In conclusion, increasing the speed rotation and the distance of the reactant feed from the disc centre, the particle diameter decreases, due to higher efficiency of the mixing processes at micro and meso scales. For each experimental run, the mixing time at the feeding point was calculated by means of the semi-empirical expression suggested by Bourne et al. (1981): n0:5
t ¼ 12
e
ð14Þ
Increasing the rotational speed from 58 to 146.7 rad/s and the distance from the centre of the disc, the mixing time decreases from 3.9 to 0.9 ms. For rotational speed higher than 120 rad/s the mixing time approaches values around 1 ms; which is the typical value for the attainment of micromixing (Baffi et al., 2002). The implemented CFD model can also predict at any point of the liquid layer the fraction volume of each solution adopted in the experimental work, i.e., ammonia, calcium chloride and ammonia phosphate aqueous solution. This kind of information can be very useful for qualitatively estimating the effect of the rotational speed on the mixing degree of each solution over the disc. In Fig. 12 the map of the volume fraction of each aqueous solution in the layer section 5 mm from the disc surface is reported for two values of speed rotation: 58 and 147 rad/s. First of all, ammonia solution, fed at the disc centre, results to be absent in the examined layer section for the smaller rotation speed. As a matter of fact from the CFD model results this solution is present only in upper sections of the layer, that is because the ammonia aqueous stream is pushed upward by the reagent
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streams. It is not the case when a rotational speed of 147 rad/s is applied. In fact, the ammonia solution volume fraction at large rotational speed is present wherever in the considered layer section. The presence of ammonia solution all around the disc is important to assure the local attainment of a pH value of 10 and a high selectivity of the reaction to hydroxyapatite, as well. The volume fraction distribution of the two reagents over the considered layer section at a constancy of rotational speed is similar. At 58 rad/s the mixing is very pour at the injection point. The volume fraction progressively decreases along the azimuthal abscissa due to the mesomixing, but still at some radial distance from the injection point volume fractions are far to be constant. On the contrary at 147 rad/s a very different picture of each reagent solution volume fraction appears. To match the stoichiometric ratio between the reagents the volume fraction for the two reagents and the ammonia solutions has to be 0.35 and 0.3, respectively. From the figure it can be noticed that the solutions are in the correct stoichiometric ratio. At each reagent injection point the other solutions are already present and very early along the disc an almost homogeneous fraction volume of each reagent solution is attained. The pictures in Fig. 12 gives an impressive confirmation of the effect of the disc rotational speed on the mixing effectiveness and support the hypothesis of an almost complete mixing at micro scale by operating at 147 rad/s.
6. Conclusions SDR is an effective apparatus to produce nanoparticles by reaction precipitation process in continuous mode.
Fig. 12. Representation of volume fraction of each aqueous solution at different disc rotational speed. (A) 58 rad/s, (B)147 rad/s.
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This apparatus in fact progressively intensifies the mixing phenomena in the liquid layer, as much as the rotational speed is increased, until micromixing conditions are established. In this work a CFD model of the liquid layer over the rotating disc surface have been implemented and used to interpret the results of a reaction precipitation process for the production of nanoparticles of hydroxyapatite. By increasing the speed of rotation between 58 and 147 rad/s nanoparticles from 350 to around 110 nm in size were obtained for an injection point 2 cm far from the disc centre. A further size reduction was obtained by locating the feed point at 3 cm from the disc centre because of a higher local specific dispersed energy: in this case by operating at 147 rad/s nanoparticles smaller than 90 nm in size were produced. The simulation of the liquid layer by the developed CFD model allowed to show the inverse proportionality between the produced nanoparticles size and the specific dispersed energy at the injection point. Moreover the prediction of the distribution of the volume fraction of the three adopted aqueous streams in a section of the liquid layer gave an impressive picture of the change of the mixing effectiveness when the rotational speed is increased.
Nomenclature D E F g h hN p
m Q r t tres v v Y
mass diffusion coefficient, m2/s Ekman number external body forces, N gravity, kg/m2 film thickness, m Nusselt film thickness, m pressure, Pa filtered pressure, Pa volumetric flow rate, m3/s radius, m time, s residence time of the fluid on the disc, s fluid velocity, m/s fluid filtered velocity, m/s mass fraction
Greek letters
a e m n r t o
phase volume fraction dispersed specific power, W/kg liquid viscosity, kg/m.s liquid kinematic viscosity, m2/s liquid density, kg/m3 mixing time, s rotational speed, rad/s
Abbreviations PRESTO Pressure staggering option QUICK Quadratic Upstream Interpolation for Convective Kinematics SIMPLE Semi-Implicit Method for Pressure Linked Equations
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