Experimental investigation of a two-zone model for semi-batch precipitation in stirred-tank reactors

Experimental investigation of a two-zone model for semi-batch precipitation in stirred-tank reactors

Chemical Engineering Science 207 (2019) 258–270 Contents lists available at ScienceDirect Chemical Engineering Science journal homepage: www.elsevie...

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Chemical Engineering Science 207 (2019) 258–270

Contents lists available at ScienceDirect

Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces

Experimental investigation of a two-zone model for semi-batch precipitation in stirred-tank reactors H. Rehage, F. Nikq, M. Kind ⇑ Karlsruhe Institute of Technology (KIT), Institute of Thermal Process Engineering, Germany

h i g h l i g h t s

g r a p h i c a l a b s t r a c t

 LME model for semi-batch

precipitation in stirred-tanks is presented.  Model considers stirrer type, size and rotational speed.  Model accuracy is validated with experimental comparison setup.  Mixing zone parameterized by pipein-pipe geometries.

a r t i c l e

i n f o

Article history: Received 12 November 2018 Received in revised form 21 May 2019 Accepted 17 June 2019 Available online 18 June 2019 Keywords: Precipitation Zone model Barium sulfate Semi-batch

a b s t r a c t A zone model for the semi-batch precipitation process of sparingly soluble salts in stirred-tank reactors is presented. The low product solubility of these substances leads to high levels of supersaturation during the process. Consequently, nucleation and growth rates are fast and solids formation only takes place in a part of the reactor close to the feed pipe. The presented local mixing environment (LME) model extends an existing model for semi-batch processes, which consists of two zones. In the mixing zone, a steady-state plug-flow reactor (PFR) is used to imitate the local flow environment of the feed. The tank is approximated as a well-mixed storage tank outside the mixing zone. Exchange streams between the two zones are estimated by dimensional analysis considering stirrer type, size and rotational speed. The correctness of the two-zone hypothesis and the accuracy of the LME model is validated by barium sulfate precipitation in an experimental comparison setup using different stirrer types, rotational speeds and feed rates. The PFR is represented in the experiments as a pipe-in-pipe reactor in jet-in-cross-flow (JICF) or coaxial-flow (COAX) arrangement. The experimental results show that the semi-batch precipitation of sparingly soluble salts in stirred-tanks can be successfully simplified by the assumption of a PFR mixing zone. The LME model is simple to implement, scalable and reaches acceptable results in the experimental validation. It is therefore a promising model for future application in process simulation of industrial precipitation processes. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Precipitation is an important industrial solids formation process and a standard operation to produce catalysts (Kawabata et al. ⇑ Corresponding author. E-mail address: [email protected] (M. Kind). https://doi.org/10.1016/j.ces.2019.06.024 0009-2509/Ó 2019 Elsevier Ltd. All rights reserved.

2006), color pigments or pharmaceutical products (Rogers et al. 2004). Stirred-tank reactors in batch, semi-batch or continuous operation mode are the principle precipitation apparatus in industry. Precipitation of sparingly soluble substances leads to high levels of supersaturations during the mixing step. Nucleation and growth rate depend strongly on the supersaturation. Therefore, the characteristic precipitate of a sparingly soluble salt comprises


H. Rehage et al. / Chemical Engineering Science 207 (2019) 258–270

Nomenclature Aplane Ashell B ci C dp D hblade K sp L L50;3 N Nq q3 Q circ Qf Q jicf=coax Q prim r pos R Re Sa

axial primary flow surface (m2 ) radial primary flow surface (m2 ) tank baffle width (m) molar concentration of component i (mol=LÞ impeller off-bottom clearance (m) diameter of circulation flow pipe (m) stirrer diameter (m) stirrer blade high (m) 2 solubility product (mol =L2 Þ particle diameter (lm) volume-based median of PSD (lmÞ stirrer rotational speed (rpm) impeller flow number (-) volume-based particle size distribution i (1=lm) circulation volume flow (L/min) feed volume flow (L=min) cross-flow or coaxial volume flow (L=min) impeller primary flow (m3 /s) radial position of feed pipe (m) free lattice ion ratio (-) Reynolds number (-) activity-based supersaturation (-)

of a high number of small-scale particles, which are generated during a short reaction time. If the reaction time scale is smaller than the relevant mixing time scale of the process, the mixing process can strongly influence solid product attributes, such as the particle size distribution (PSD) (Metzger and Kind 2016b; Gradl et al. 2006; Öncül et al. 2009). Process simulation is a powerful tool which supports engineers during process design, optimization and operation (Ingham et al. 2008). Although the possibilities to calculate dynamic precipitation processes in commercial software have increased during the last decade (Jones et al. 2005), precipitation process simulations are still limited to material systems with a high solubility. For these substances, reaction time scales are long compared to the mixing time scale. Thus, the whole tank reactor can be simplified to a well-mixed reaction environment. Precipitation of sparingly soluble substances in stirred-tank reactors, however, still poses a serious problem to process simulation. The solid formation is localized in a small volume near the feed pipe due to the short reaction times. The attributes of the solid product are, in this case, only dependent on the local mixing environment close to the feed point. The fluid outside of this reaction zone only influences the process by defining the composition of the fluid in which the feed is mixed in the reaction zone. Obviously, a zone model is necessary for process simulation. Several research groups have investigated zone models for mixing processes in stirred-tank reactors in the past, which could be used to simulate semi-batch precipitation of sparingly soluble salts. A timeline of important publications is given in Fig. 1. A general issue of these models is the unknown reaction volume and the local flow behavior in the stirred-tank. One approach is the subdivision of the reactor volume in different ideally mixed zones

t pro T  uf  ujicf=coax V0 V1

ci e


l q smacro


process time (min) inner diameter of tank reactor (m) average feed velocity (m/s) average environment velocity (JICF/COAX) (m/s) educt volume (LÞ tank volume (LÞ activity coefficient (-) energy dissipation (m2 /s3 ) velocity ratio (-) viscosity (mPas) density (kg/m3 Þ macro mixing time constant (s) computational fluid dynamic confined imping jet mixer coaxial-flow jet-in-cross-flow local mixing environment plug-flow reactor particle size distribution storage tank

(Fox and Varma 2003). For these zones, simplified and averaged flow properties are assumed. The convection field, produced by the stirrer, is approximated by exchange streams between the zones. The unknown flow properties, such as velocity or average 

energy dissipation rate e, may be accessed by numerical simulations. Alexopoulos et al. (2002) divided the continuous stirredtank reactor into impeller zone and circulation zone. The average energy dissipation ratios and the volumes of both zones were extracted from computational fluid dynamic (CFD) simulation results. Application of the model to a liquid-liquid dispersion showed satisfactory agreement between model and experimental data. Vicum et al. (2004) found satisfactory agreement between the experimental results of a reactive mixing process in a semibatch stirred-tank reactor and an engulfment model as well as a CFD-based approach. The CFD-based model was found to be computationally expensive but more accurate than the engulfment model. The CFD-based model was also applied to precipitation in stirred-tank rectors by Bałdyga et al. (2005). Alopaeus et al. (1999) separated the reactor into eleven zones of different mixing intensities to investigate breakage and coalescence in liquid-liquid systems. Although various phenomena in liquid-liquid systems could be described correctly with this approach, the high number of zones is accompanied by an uneconomical computational effort for process simulation purposes. Bourne and Yu (1994) developed a multi-zone model with six zones, which was validated by parallel chemical reactions. They introduced the experimental flow model to approximate the complex convection and energy dissipation field in the reactor without CFD simulations. Zauner and Jones (2000) further developed the segregated feed model of Villermaux, (1989) and successfully applied it to semi-batch precipitation. The segregated feed model divides the reactor into

Fig. 1. Timeline for important publications on zone models for stirred-tank reactors.


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two feed zones and one bulk zone with separated mass and population balances. A similar approach can be found in van Leeuwen et al. (1996). In this paper, the three-zone approach (two feed and one bulk zone) presented by Gösele and Kind (1991) was used to derive a precipitation model for stirred-tank reactors which includes micro and meso mixing effects. Although most of the approaches described achieved acceptable results, two problems arise from the assumption of an ideal mixed reaction zone. Firstly, it is difficult to define the volume of the reaction zone correctly, since the implicit assumption of an ideal mixture in this zone is only a hypothetical concept. The complex mixing interaction of the feed with the tank fluid is strongly simplified. Secondly, application and scale-up of these models is difficult, since time-intensive CFD calculations are necessary for each new reactor configuration (Alexopoulos et al. 2002). Furthermore, the models become quite complex if numerous zones are used to describe the flow field in the tank. However, for process simulation, the number of zones and equations must be small to allow for low simulation effort. A different type of published model (e.g. Bałdyga and Bourne 1999; Kim and Tarbell 1996) does not separate the reactor into only well-mixed environments. Instead, a plug-flow reactor is used to describe the local mixing process in the feed region. The semibatch model presented by Bałdyga and Bourne (1999) belongs to this model type. As shown in Fig. 2, the stirred-tank reactor in semi-batch operation mode (a) is represented in the Bałdyga model (b) by an equivalent circuit consisting of a well-mixed storage tank (ST) connected to a PFR. In the PFR, the feed volume flow Q f is mixed with the volume flow Q circ;1 from the ST. The volume flow Q circ;2 designates the backflow to ST. With the implementation of a mixing model for the PFR (Bałdyga and Bourne (1999) used an Engulfment mixing model) a process simulation of the complete semi-batch process is possible. However, the assumption of an equivalent mixing process in the PFR only makes sense if relevant processes (e.g. solid formation or chemical reaction) are localized close to the feed pipe. We identify two major drawbacks which are hindering a direct use of the model by Bałdyga and Bourne (1999) for process simulation of precipitation. Firstly, it is still necessary to find a correct 

value for the average turbulent dissipation rate e in the PFR (either by CFD simulation or literature research). Secondly, Bałdyga and Bourne (1999) did not explain how to estimate Q circ;1 for a given stirrer type, stirrer size and rotational velocity. Again, a CFD simulation is necessary, which is a huge effort and therefore hinders direct application of the model. In this paper, we further develop the semi-batch model of Bałdyga and Bourne (1999) to a new model to increase its applicability. We extend the model of Bałdyga and Bourne (1999) with a

new method for calculating the exchange volume flow Q circ;1 based on dimensional analysis. Type, size and rotational speed of the stirrer are considered. Comparable to the idea of the experimental flow model by Bourne and Yu (1994), this method allows the estimation of the central parameter Q circ;1 without additional CFD simulations. We do further assume, that the e value of the PFR and the modelled semi-batch reactor is on a comparable scale if similar boundary conditions are applied in both flow situations. 

Thus, compared to most models presented above, the e value is not directly matched from the CSTR to the PFR. Instead, with application of the correct boundary conditions for the PFR, the ‘‘local mixing environment” for the feed fluid is assumed to be comparable to the feed fluid in the stirred-tank reactor. This leads to a similar energy dissipation and thus to a similar timescale of micro mixing. Consequently, the resulting model is referred as the local mixing environment (LME) model. Aim of this paper is an experimental validation to verify, if the LME model is a suitable candidate for process simulation of semibatch precipitations. The underlying assumption of simplifying the semi-batch process by an equivalent circuit is only a hypothesis by Bałdyga and Bourne (1999), since it has not yet been selectively validated. Furthermore, the concept of a similar ‘‘mixing environment” between fluid flow in stirred-tanks and the PFR needs validation. For this reason, we do not compare process simulation results with experimental data in this work. Instead, we used a dedicated experimental setup to validate the two following hypotheses: (1) A PFR circuit can be applied for high supersaturated systems to generate an analogous mixing environment compared to a standard semi-batch precipitation in stirred-tank reactors. (2) The LME model correctly estimates the exchange volume flow between the two zones.

2. Local mixing environment model This Section provides an overview about the LME model. In the first part, the zone model concept of Bałdyga and Bourne (1999) is explained and adapted to the precipitation process of sparingly soluble salts. Furthermore, the extension by an exchange volume flow calculation method for different stirrer types, sizes and rotational velocities is presented. The last part provides a classification of the model in the background of classical mixing theory.

2.1. General model setup The LME model extends the semi-batch model of Bałdyga and Bourne (1999) to a more applicable process model and adapts it to the precipitation of sparingly soluble salts. In Fig. 3, the general concept of the LME model is demonstrated. The semi-batch stirred-tank precipitation reactor (a) is represented by an equivalent circuit of two interconnected zones (b), plug-flow reactor (PFR) and well-mixed storage tank (ST). In the PFR, the feed stream Q f is mixed with Q circ;1 . Consequently, Q circ;2 can be calculated without consideration of density changes by Eq. (1).

Q circ;2 ¼ Q circ;1 þ Q f

Fig. 2. Schematic representation of semi-batch precipitation in a stirred-tank (a) and two-zone model proposed by Bałdyga and Bourne (1999) (b).


In the PFR, the coupled processes of mixing, supersaturation buildup and depletion by nucleation and growth are taking place. Due to the high nucleation and growth rates of sparingly soluble substances, the solid formation is already finished when the fluid leaves the PFR. The ST serves as a residence time element and represents the reactor fluid volume V 1 which is outside of the reaction


H. Rehage et al. / Chemical Engineering Science 207 (2019) 258–270

Fig. 3. Schematic representation of semi-batch precipitation process (a) and LME model process abstraction (b).

zone (Fig. 3). Both zones (j) are connected by volume flows Q circ;j with the component (index i) concentrations ci;j and the solid q3;j ðLÞ PSDs. L designates the particle diameter. Compared to the fast primary processes, the time scales of secondary processes, such as aggregation, are large, since they are not directly affected by the high level of supersaturation (Marchisio et al. 2006). Therefore, the secondary processes (e.g. aggregation) are assumed to take place only in the ST. Advantages of the proposed zone model are the steady-state reaction zone and the separation of fast primary and slow secondary processes. Mixing relevant information is reduced to the PFR which simplifies further modelling (e.g. CFD simulation). With this equivalent circuit of PFR and ST, the process dynamics of the semi-batch precipitation process can be depicted. Besides mixing relevant data for the mixing process inside the 

PFR (e.g. average energy dissipation e) only Q circ;1 remains as an undefined variable since Q f is generally a given input parameter. Instead of using CFD to extract this value, a simplified method to correlate stirrer type and size, feed point position and the volume flow rate Q circ;1 based on dimensional analytic considerations was developed. 2.2. Exchange volume flow calculation The LME model follows the idea that a comparable local mixing environment is generated between the stirred-tank and the PFR. At this point it must be differentiated which type of pipe-in-pipe mixer is used to realize the PFR behavior. For stirrers with a mainly radial flow direction, the jet-in-cross flow (JICF) arrangement would be the closed approximation. For stirrers with mainly axial flow direction, however, the coaxial-flow (COAX) arrangement is more suitable. The idea of the LME model is, that if similar values 

for the average feed velocity uf and the average environment veloc

ity ujicf=coax are present for PFR and semi-batch reactor, a similar mixing situation can be reached. This similarity includes meso and micro mixing effects and therefore a similar energy dissipation. Of course, the dissipation field in CSTRs is different from the one obtained in a pipe flow. However, probably that does not affect the solid formation in the case of relevant feed rates since the feed entering the environmental fluid also strongly influences the energy dissipation field. The variables considered to derive the average environment velocity for radial and axial pumping stirrers are depicted in Fig. 4. The impeller primary flow Q prim can be calculated with the impeller flow number N q , the rotational velocity N and the stirrer diameter D by Eq. (2).

Fig. 4. Relevant parameters for simplified calculation of the average environment  velocity ujicf=coax for the reaction zone based on stirrer type, size and radial position of the feed pipe. Uniform flow over the cylinder surface Ashell at the feed position rpos is assumed for stirrers with mainly radial main flow direction (e.g. a Rushton turbine). Uniform flow over the covered surface Aplane is assumed for stirrers with a mainly axial flow direction (e.g. a propeller stirrer).

Q prim ¼ Nq  N  D3


The flow number N q depends on several parameters (e.g. tank size, stirrer type and fluid properties) and is accessible for various impeller types (Zlokarnik 2001). A cross-sectional area must be 

defined to calculate the average environment velocity ujicf=coax from the impeller primary flow. The calculation procedure for the average environment velocity depends on the stirrer type. If the main delivery stream of the stirrer is orientated radial to the stirrer axis, the mixing zone is parameterized as JICF. As shown 

in Fig. 4 (left), the average environment velocity ujicf can then be calculated through division of Q prim by the cylinder crosssectional area Ashell , which is defined by the radial position of the feed pipe r pos and the height of the stirrer blade hblade (see Eq. (3)). 

ujicf ¼

Q prim Nq ND3 ¼ Ashell 2pr pos hblade


As shown in Fig. 4 (right), a different approach is used to calcu

late the average environment velocity ucoax for stirrers with a mainly axial pumping direction. Following Eq. (4), the impeller primary flow is divided by the surface Aplane which is covered by the stirrer blades during rotation to account for the coaxial arrangement. 

ucoax ¼

Q prim Nq ND3 4 ¼ p 2 ¼ Nq ND Aplane p D 4


The radial position of the feed pipe is not considered for axial pumping directions. However, the feed pipe must be positioned close to the stirrer for correctly estimating the average environ

ment velocity ucoax . Finally, the flowrate for the pump, which realizes the flow in the circuit for the experimental validation, can be calculated considering JICF or COAX arrangement (Fig. 5). Eq. (5) is used to calculate 

Q circ;1 with the average environment velocity ujicf=coax and the circuit pipe diameter dp .

Q circ;1 ¼

p 4


dp ujicf=coax


It is obvious that the assumption of a uniform velocity distribution over the cylinder surface is a quite coarse simplification of the real velocity distribution in a stirred-tank. However, as it will be shown in Section 4, a satisfactory model accuracy can be achieved even with these simple derivations. The model described is simple to implement and allows the estimation of Q circ;1 without timeintensive CFD calculations.


H. Rehage et al. / Chemical Engineering Science 207 (2019) 258–270

Fig. 5. Relevant pipe-in-pipe mixer parameters for JICF (a) and COAX (b) arrangement.

The Reynolds numbers of the stirred-tank reactor (ReT ) and the pipe-in-pipe mixers (ReP ) are not relevant as input variables for the LME model. However, both are calculated for comparison to mixing experiments in literature by Eqs. (6) and (7) using average values 

for fluid density q and dynamic viscosity


ReT ¼

ND2 q


ReP ¼


ujicf=coax dp q 



2.3. Mixing time scale analysis

V Q prim

3. Materials and methods This Section introduces the material barium sulfate, the experimental setup for the dynamic and steady-state precipitation experiments, the experimental procedures and the measurement methods. 3.1. Material The precipitation of barium sulfate from aqueous Na2 SO4 and BaCl2 solution was used for the experimental validation. Mixing of these educts leads to barium sulfate precipitation according to Eq. (9):

BaCl2 ðaqÞ þ Na2 SO4 ðaqÞ ! BaSO4 ðsÞ þ 2NaClðaqÞ

The classical mixing theory distinguishes three different mixing scales, namely macro, meso and micro mixing. Macro mixing describes mixing on the reactor scale. For stirred-tank reactors, a common expression for the macro mixing time smacro is given by Eq. (8) with V designating the total fluid volume in the stirredtank.

smacro ¼

The process of macro mixing is therefore depicted in the LME model by the equivalent circuit itself. Macro mixing time constants will be given for informational purposes, although the model does not directly use it as input parameters. In contrast to macro mixing, the processes of meso and micro mixing are on the scale of the pipe-in-pipe mixer. For a process simulation with the LME model, the relevant mechanisms for meso and micro mixing must be implemented in the reaction zone to account for the local mixing processes. However, as this paper does only intend to verify the general idea of the LME model and the equivalent circuit assumption, it is not necessary to discuss micro and meso processes of mixing at this point. It is part of the model assumptions that micro and meso mixing are imitated correctly in the pipe-in-pipe mixer by application of the correct boundary conditions.


Barium sulfate precipitation is well-researched in literature and often used to study the influence of mixing on fast solids formation (Metzger and Kind 2016a; Judat and Kind 2004; Gradl et al. 2006; Schwarzer et al. 2006; Bałdyga et al. 1995; Marchisio et al. 2002). The investigation of precipitation at high supersaturation is possible due to the low solubility of barium sulfate with the solubility product K sp ð25 CÞ ¼ 9:82  1011 mol L2 (Monnin 1999). The saturation Sa;0 was calculated from Eq. (10) with cBa2þ ;free and cSO4 2 ;free designating the free ion concentrations of the product 2



Sa;0 ¼ c

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cBa2þ ;free  cSO4 2 ;free K sp


The model equation by Pitzer (1973) was used to calculate the activity coefficients c for a target supersaturation of Sa;0 ¼ 1000 and a free lattice ion ratio of R0 ¼ 5. Eq. (11) was used to calculate the value for R0 .

R0 ¼

Fig. 6. Schematic depiction of experimental simulation for validation of the LME model assumptions. Product particle attributes of semi-batch precipitation with radial (a) or axial main flow direction (c) of the stirrer are compared to final particle attributes results from an experimental setup with an external reaction zone. The mixing zone is parameterized as a JICF (b) or COAX reactor (d).

cBa2þ ;free cSO4 2 ;free



The high level of supersaturation guarantees the model requirement of short precipitation time scales (ms) and, therefore, a localized reaction zone. The index 0 indicates that the values for Sa and R are referred to a complete and immediate mixture of the educt volumes. The real values for Sa and R depend on the local mixing situation and show a strong variation during the process. The free lattice ion ratio of R0 ¼ 5 was chosen to stabilize the product suspension. Direct dilution with the fluid outside of the reaction zone takes place in the semi-batch process. In the pipe circuit, however, the surrounding fluid amount is limited. This effect could lead to a higher particle number density and, therefore, a promoted aggregation rate in the circuit for non-stabilized suspensions. Suppression of aggregation effects is, therefore, necessary to be able to compare product volume-based PSDs of both

H. Rehage et al. / Chemical Engineering Science 207 (2019) 258–270

experiments. Kucher et al. (2006) and Kügler et al. (2015) showed that a value of R0 ¼ 5 is sufficient for the suppression of aggregation in confined imping jet mixers (CIJM), which is, compared to JICF or COAX a much smaller mixing reactor geometry. Since JICF and COAX are furthermore asymmetrical mixing reactors, the required excess of barium ions for colloidal stabilization was investigated in preliminary steady-state JICF experiments. Results of this study are given in Section 4. Barium chloride dihydrate ( 99% p.a., Carl Roth) and sodium sulfate ( 99% p.a., Carl Roth) solved in desalinated water were used as educts for all experiments. Concentrations are specified in Table 1. Calculations of the ReT and ReP numbers require average values for fluid density and viscosity. Density was averaged out of the  ¼ 1064; 2 kg1 m3 . An average value educt densities to a value of q  ¼ 9:4  107 m2 =s, was calculated from values  =q for the viscosity, l published by Metzger and Kind (2016c):

3.2. Dynamic experiments The LME model is experimentally simulated to validate the assumption of the local mixing zone abstraction by an external pipe-in-pipe mixer. Therefore, an experimental setup is used which enables precipitation experiments in a pipe-in-pipe mixer connected to a well-mixed storage tank and allows comparison of the volume-based PSD to the one obtained by a standard semi-batch precipitation (bulk). This procedure allows the selective assessment of the LME model accuracy in estimating the exchange volumetric flow based on dimensional analytic data. Furthermore, the general validity of a mixing zone approximation by a pipe-in-pipe mixer is investigated.

Table 1 Educt concentrations for S 0 ¼ 1000; R0 ¼ 5, values measured at 20 °C.

BaCl2 solution Na2 SO4 solution

  1 c mol L1

V 0 ðLÞ

qðkg1 m3 Þ

0.580 0.144

5.37 5.37

1099.6 1031.5


The experimental process plant is depicted in Fig. 7. It consists of a temperature-controlled 11-L tank reactor (A), a 6-L feed container (B) and an external pipe circuit (C) with a mixing reactor (D) for the zone model experiments. The plant can be equipped with a mixing reactor in either JICF or COAX configuration. The diameter of all feed piping is 1/800 ; the circuit piping diameter is 3/400 . The tank reactor with four baffles can be equipped with two different stirrer types, a Rushton turbine with six blades and a pitched blade (45° angle) turbine with four blades. A technical drawing of the reactor, the two stirrers and the sampling position is given in Fig. 8. Relevant dimensions are listed in Table 2. T designates the tank diameter, C the off-bottom clearance and B the baffle width. P indicates the sampling position. The Rushton turbine (diameter DB ) was used exemplarily for stirrers with a mainly radial flow direction. The pitched blade turbine (diameter DA ) was chosen for the validation of stirrer types with axial main flow direction. The feed pipe position for the bulk experiments was orientated on the LME model (see Fig. 4). For radial stirrers, the feed pipe was positioned at r pos ¼ 0:044 m. The horizontal position of the feed inlet was adjusted to the upper stirrer blade edge. The radial position of rpos ¼ 0:052 m was chosen for the pitched blade turbine to account for the larger stirrer diameter. The feed inlet for the pitched blade stirrer was positioned 10 mm below the stirrer to ensure a fully axial flow environment for the feed. The JICF and COAX geometries were constructed with an inner diameter of dp ¼ 0:019m. A feed pipe with an internal diameter of 4 mm was used in bulk and pipe-in-pipe experiments. Check valves upstream of the feed pipe were found to be necessary to prevent backflow into the feed pipe due to pressure fluctuations. The mixing reactor was made of polymethylmethacrylate to allow visual observation of the precipitation process. Stainless steel was used for the feed pipe and hose connections. The reactors in both configurations are depicted in Fig. 9. The feed pipe outlet for JICF configuration was positioned 3 mm inside the cross-flow to prevent wall effects. The circulation volume flow Q circ;1 was calculated according to the LME model (Eq. (2)–(5)), as described in Section 2.2. The input parameters are the rotational speed N of the reference bulk experiments and constant parameters. The software CheCalc (2017) was

Fig. 7. Simplified process scheme of the experimental setup for bulk and two-zone precipitation experiments.


H. Rehage et al. / Chemical Engineering Science 207 (2019) 258–270

Fig. 8. Sectional drawing of the 11-L tank reactor with six-blade Rushton turbine and 45° pitched blade turbine with four blades. P indicates the sampling position.

Table 2 Relevant dimensions for stirred tank reactor.

Size (m) Relative size







0.1 0.42 T

0.084 0.35 T

0.084 0.35 T

0.024 0.1 T

used to calculate the pumping factor N q , dependent on stirrer type and scale, tank size, material attributes and process parameters. The resulting LME model parameters for both stirrers are given in Table 3. Using the value for Ashell or Aplane and the reactor pipe diameter of dp ¼ 0:019m, Q circ was calculated with Eqs. (3)–(5) for different stirrer rotational velocities. The resulting circulation flow rates for the investigated process points are given in Table 4 and Table 5. Parameters for the feed are given in Table 6. All experiments were

carried at a tank fluid temperature of 20 °C. For the calculation of smacro the fixed reference volume V ¼ 10.74 L (final filling) was assumed. The total process time tpro varied, dependent on the applied feed rate, between 14  55min. 3.2.1. Bulk mode experiments A standard semi-batch precipitation was performed in bulk mode. The valves for the external circuit were therefore closed and 5.37 L BaCl2 solution was poured into the tank reactor (50% of final filling). The rotational velocity of the stirrer was adjusted to the value desired. The Na2 SO4 solution was added with the predefined feed rate Q f (controlled by Ismatec IP65 pump with Krohne Optiflux 5300) from the feed container directly into the tank reactor, until a total addition volume of 5.37 L was reached (scales controlled). The total process time of the precipitation was therefore

Fig. 9. Schematic drawing of JICF (left) reactor and COAX reactor (right).

Table 3 Input parameters for the LME model. Stirrer type


N q ðÞ

hblade ðmÞ

r pos ðmÞ

Ashell ðm2 Þ

Aplane ðm2 Þ

Rushton turbine (six blades)






Pitch blade turbine (45°, four blades)





H. Rehage et al. / Chemical Engineering Science 207 (2019) 258–270 Table 4 LME model environment data calculated for the comparison of bulk with Rushton turbine and JICF two-zone experiment. NðrpmÞ







ReT ðÞ smacro ðsÞ DB Nðm1 s1 Þ ujicf ðm1 s1 Þ

6255 30.2 0.07 0.075

12,510 15.1 0.14 0.15

18,765 11.3 0.21 0.23

25,021 7.5 0.28 0.3

31,275 6.0 0.35 0.38

37,530 5.0 0.45 0.46

Q circ;1 ðL1 min1 Þ ReP ðÞ













Table 5 LME model environment data calculated for comparison of bulk with pitched blade turbine and COAX two-zone experiment. NðrpmÞ






ReT ðÞ smacro ðsÞ DA Nðm1 s1 Þ

17,730 8.1 s 0.17 0.17

26,595 5.4 0.26 0.26

35,461 4.1 0.34 0.34

44,325 3.2 0.43 0.43

53,191 2.7 0.51 0.51











ucoax ðm1 s1 Þ Q circ;1 ðL1 min1 Þ ReP ðÞ

Table 6 Total process time and feed volume flow for different feed velocities with a constant feed pipe diameter of 4mm. uf ðm1 s1 Þ





tpro ðminÞ

55 0.1

27 0.2

18 0.3

14.5 0.4

Q f ðL1 min1 Þ

dependent on the chosen feed rate (see Table 6). Sampling was carried out directly after the end of feed addition close to the stirrer (see Fig. 6 for sampling position). The stirrer was set to a rotational velocity of 300 rpm during sampling. 3.2.2. Two-zone experiments The 5.37 L BaCl2 solution was poured into the tank reactor while in the two-zone mode. The rotational speed of the stirrer was adjusted to the fixed value of 300 rpm to simulate a well-mixed storage tank. The valves for the external circuit were opened and the peristaltic pump (Pump 2, Apex 20 by Bredel) was adjusted to the calculated value of Q circ (see Table 4 and Table 5), controlled by a magnetic flow meter (Promag 33A), and run for several minutes to remove the air in the circuit. A pulsation dampener (PD;

SentryTM by BlacohTM) was activated in some experiments to suppress pulsation (disabled by default). This PD consists of a gas volume which is separated from the fluid volume by a flexible membrane. The pipe connection to the feed pipe in the tank reactor was closed. Instead, the Na2 SO4 solution was added through the external pipe-in-pipe reactor with the predefined feed rate Q feed until a total addition volume of 5.37 L was reached. Sampling was carried out close to the stirrer directly after the end of feed addition (see Fig. 6 for sampling position).

3.3. Steady-state experiments An additional steady-state precipitation setup was used to investigate precipitation in the mixing reactor without process dynamic influence (see Fig. 10). No circulation loop was implemented in this setup. The mixing reactor was directly connected to an additional product container. Similar pumps were used for pump 1 and 2, compared to the dynamic setup, to generate the volume flow Q jicf=coax and the feed volume flow Q f . For some experiments, a gear pump (Verder H5F) was used for pump 2 instead of the peristaltic pump (Apex20 by Bredel). The pumps 1 and 2 were adjusted to the desired flowrate during the experimental

Fig. 10. Simplified process scheme of the experimental setup for steady-state precipitation experiments. A gear pump was used for pump 1. Pump 2 was equipped with either a gear pump or a peristaltic pump, depending on the experiment.


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procedure. After a few seconds of precipitation to reach steadystate, sampling was carried out downstream of the mixing reactor. 3.4. Analytics The volume-based PSDs for the dynamic experiments were measured with the static light scattering measurement system Mastersizer 3000 (dynamic experiments) and 3000E (steadystate experiments) by Malvern Panalytical. Measurements were carried out in deionized water using a Hydro EV wet dispersion system. Samples were measured with a laser occlusion of 10%. A refractive index of 1.643 and absorption coefficient of 0.1 were used to measure the volume-based PSD. Furthermore, scanning electron microscope pictures were taken for visual inspection of the particle morphology. For sample preparation, the process suspension was centrifuged twice for 10 min at 2000 rpm in a SIGMA 2-16KC centrifuge. A washing step with deionized water was performed after the first centrifugation. The product was mortared after it had been dried for 24 h at 50 °C in a drying cabinet. Finally, the solid particles were sputtered with 3.5 nm platin and investigated in a LEO 1530 scanning electron microscope. 4. Results In the first part of this Section, the results of preliminary particle stabilization and parameter range experiments are presented. In the second part, the results from comparison experiments between bulk and two-zone process are given. Finally, the possible influence of the circuit pump type is further investigated. 4.1. Parameter range for feed and cross-flow velocity

Fig. 11. Exemplarily steady-state JICF precipitations with different velocity ratios f.   Velocity parameters: uf ¼ 0:13 m1 s1 and ujicf ¼ 0:23 m1 s1 (top),    uf ¼ 0:52 m1 s1 and ujicf ¼ 0:23 m1 s1 (center), uf ¼ 0:52 m1 s1 and  ujicf ¼ 0:15 m1 s1 (bottom).

wall influence is negligible for all process points. The mixing reactors of three steady-state precipitations with different velocity ratios are shown in Fig. 11. It was noticeable for velocity ratios abovef ¼ 3.5 that the feed stream interacts with the opposite wall. Consequently, all experiments were carried out at f 3:5. A mini

mal value of uf ¼ 0:13m1 s1 was used for all experiments to prevent the influence of backmixing. Backmixing becomes relevant for slow feed velocities (Vicum et al., 2004). 4.2. Particle stabilization

The correct process parameter range must be selected to validate the LME model as well as the two-zone hypothesis. The main condition for choosing the right processes parameters in the described experimental setup is a noticeable influence of the mixing process on the volume-based PSD of the precipitate. If the rotational speed of the stirrer or the feed velocity is too high, the resulting particle attributes are no longer influenced by mixing. In this case, the bulk and the corresponding two-zone experiment would show similar volume-based PSDs even if the equations of the LME model to calculate the circulation volume flow are not correct. Consequently, preliminary tests were carried out to investigate the maximum stirrer velocity and feed velocity for mixing limited precipitation. As a result, the volume-based PSD measured in the reactor with a Rushton turbine stirrer became independent of the stirrer speed for a rotational velocity above 300 rpm. Since the dis-

The influence of the free lattice ion ratio R0 on the colloidal stability of the product particles was investigated by steady-state precipitations in the JICF reactor. As already mentioned, colloidal stability was reached for R0  5 in CIJMs according to the literature. The excess of barium ions in JICF precipitation is, due to the asymmetric mixing process, higher in JICF mixers as in CIJMs. It therefore could be expected, that sufficient stabilization may even be reached for R0 < 5. However, as it can be seen in Fig. 12, solutions with R0 < 5 still showed significant aggregation of particles. Since a value of R0 ¼ 6 did not further improves colloidal stability, all further experiments were carried out with R0 ¼ 5.

sipated energy e in the COAX reactor is known to be lower than in the JICF reactor, this parameter limit could also be used for the pitched blade turbine experiments. Furthermore, the feed velocity should not exceed a maximum value to ensure a free stream mixing environment in the pipe-inpipe mixer. The influence of the pipe wall in the JICF mixer is no longer negligible for high feed velocities. The momentum ratio of is a main criterion for the penetration depth of the feed stream. Since both educts have almost equal densities, the momentum ratio can be simplified to the velocity ratio f, which is defined in Eq. (12). 




Precipitations with different velocity ratios were inspected visually in the steady-state experimental setup to ensure that the

Fig. 12. Influence of Q jicf on the average particle diameter in steady-state experiments with JICF reactor. Particle size L50;3 was measured with static light scattering.


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4.3. Experimental simulation Experimental static light scattering results from bulk and twozone experiments are shown in Fig. 13 (left). Bulk experiments  f ¼ 0:39 m1 s1 with the Rushton turbine at N ¼ 150 rpm and u (red line) are compared to the corresponding JICF two-zone exper1

iment with Q circ;1 ¼ 4:4 L1 min (dashed red line). Both result in a single peak particle distribution with L50;3 300 nm. The particles generated in the two-zone experiment are slightly larger compared to the bulk experiment. Larger particles (L50;3 1 lm) with a broader distribution are formed in the bulk experiment  f ¼ 0:13 m1 s1 . The corresponding twofor N ¼ 50 rpm and u 1

shows a similar zone experiment with Q circ;1 ¼ 1:3 L1 min volume-based PSD with slightly larger particles. In Fig. 13 (right), bulk experiments with the pitched blade turbine are compared to the corresponding two-zone experiment with the COAX mixing reactor. A high similarity can be found between the N ¼ 100 rpm and N ¼ 200 rpm bulk experiments and the corresponding COAX two-zone experiments with 1

Fig. 14. The SEM pictures of the bulk (left) and corresponding two-zone experiment (right). Bulk experiments were performed with the pitched blade turbine at  N ¼ 100 rpm and uf ¼ 0:13 m1 s1 . The COAX two-zone experiments were per 1 formed with Q circ ¼ 2:9 L1 min and uf ¼ 0:13 m1 s1 .


Q circ ¼ 2:9 L1 min and 5:8 L1 min . These results strongly support the hypothesis of a correct abstraction of the local mixing environment by the LME model. Besides the systematic generation of slightly larger particles in the two-zone circuit, the JICF and COAX reactors seem to be able to generate a similar mixing environment compared to their bulk reference. SEM pictures were taken to investigate the particle morphology for products of bulk and two-zone experiments. Exemplary SEM results of the corresponding bulk and two-zone experiment are shown in Fig. 14. As it can be seen, using the two-zone process instead of the bulk process does not influence the particle morphology. The particle morphology (plate like) comes close to the one obtained by Kucher et al. (2006) in a CIJM for Sa;0 ¼ 500 and R0 ¼ 10. This indicates that the effective supersaturation is, due to mixing effects, below the value of Sa;0 ¼ 1000. However, the missing difference of the particle morphology between bulk and two-zone precipitation further supports the validity of the LME model hypotheses. A substantial change in the mixing process and consequently a different level of supersaturation would have been observable by a change in particle morphology. It can be assumed from the presented results that the assumptions of the LME model (e.g. assuming a simplified velocity field around the stirrer) can approximate the circuit stream correctly to a certain extent. However, the model should not only be validated for process points chosen exemplarily. The influence of different stirrer speeds and feed velocities were investigated

separately over the full range of process parameters (see Table 4–6) to further investigate the two-zone idea and the LME model validity. The volume-weighted average particle diameters L50;3 of the bulk process with the pitched blade turbine and the COAX twozone experiments are compared in Fig. 15 (left). 1

A constant feed rate of Q f ¼ 0:1 L1 min was applied for different rotational speeds. The values of the volume-based particle diameter for bulk and two-zone show a logarithmic dependency on the stirring rate, indicating operation in the mixing limited process region. A close agreement between zone model and bulk experiments for all the investigated rotational velocities can be 

found. Different feed velocities from uf ¼ 0:13  0:52 m1 s1 were used at a constant stirrer rotational velocity of N ¼100 rpm. Results are shown in Fig. 15 (right). The two-zone experiments were performed with the same feed velocities and circulation vol1

ume flow Q circ ¼ 2:9 L1 min . A high agreement can be found, which is independent of the stirrer rotational speed or feed velocity. Particles resulting from the two-zone process are slightly larger than the particles formed during the bulk process throughout the full process range. The bulk process with the Rushton turbine is compared with the two-zone process for different stirrer rotational speeds and a con f = 0.26 m1 s1 . Results are shown in Fig. 16 stant feed velocity of u (left). In Fig. 16 (right), results are shown with feed velocities from

Fig. 13. The volume-based PSD of bulk (Rushton turbine) and corresponding JICF two-zone experiments for two different rotational speeds and feed velocities (left). The volume-based PSD of bulk (pitched blade turbine) and corresponding COAX two-zone experiment for two different rotational speeds and feed velocities (right). The volumebased PSDs were measured with static light scattering.


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Fig. 15. Influence of the stirring rate on the average particle size in bulk (pitched blade turbine) and corresponding COAX two-zone experiments with Q f ¼ 0:1 L1 min (left). Influence of different feed velocities on the average particle size in bulk (pitched blade turbine) and corresponding COAX two-zone experiments at N ¼ 100 rpm (right). The volume-weighted PSD was measured with static light scattering.

 f = 0.13 to 0.52 m1 s1 and a constant stirrer rotational speed of u N ¼150 rpm for the bulk and the JICF two-zone setup. A close agreement between both processes can be found in both parameter studies. Similar to the results with the COAX mixing zone, the JICF results further support the LME model assumptions and the idea of a process abstraction by the equivalent circuit. The particles originating in the two-zone process are slightly larger compared to bulk values using the same process conditions, independent of the mixing reactor or process conditions. Several influence factors can cause this effect (see Section 5). The influence of pulsation, as one possible explanation for this effect, was further experimentally investigated.

Although visual inspection showed a lower pulsation, we found that the application of a PD for the validation of the two-zone approach does not improve the setup. Large particle agglomerates were formed inside the PD despite the stabilization by R0 ¼ 5. A comparison of particle attributes between bulk and two-zone approach, therefore, no longer made sense. Different pump setups were integrated in the steady-state setup to investigate the circuit pump influence on the zone model validation further (see Fig. 10). Three different setups were chosen for pump 1: The peristaltic pump, the peristaltic pump with PD directly behind the pump outlet and a gear pump (Verder H5F). The volume-weighted average 

particle diameter for different environment velocities ujicf with 

4.4. Pulsation influence A peristaltic pump was used in the experimental two-zone setup for generating the circuit stream, since one requirement for the circuit pump was the gentle delivery of particles. Peristaltic pumps produce a pulsating volume flow which influences the mixing process in the pipe-in-pipe mixer and, therefore, influences the product attributes. To overcome this issue, the PD (further description in Section 3.2) was integrated behind the pump to lower the pulsation in the mixing zone.

constant feed velocity uf = 0.13m1 s1 are compared in Fig. 17. As it can be seen, the choice of pump influences the average size of the particles precipitated, even in a steady-state setup. Interestingly, the peristaltic pump with PD generates the largest particles, although the PD was installed upstream compared to the mixing reactor. An influence of the PD on the solid product, besides the influence on the mixing process, was therefore impossible. One explanation for the deviation between two-zone and bulk process could be the usage of a peristaltic pump in the circuit, as the near pulsation free gear pump generated systematically smaller

Fig. 16. Influence of the stirring rate on the average particle size in the bulk (Rushton turbine) and corresponding JICF two-zone experiments with Q f ¼ 0:2 L=min (left). Influence of different feed velocities on the average particle size in bulk (Rushton turbine) and corresponding JICF two-zone experiments at N ¼ 150 rpm (right). The volumebased PSD was measured with static light scattering.

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Fig. 17. Average particle diameter L50;3 of steady-state precipitation product (COAX) for different cross-flow velocities and pump setups: peristaltic pump without pulsation dampener (PD), peristaltic pump with PD and a gear pump only.  A constant feed rate of uf = 0.13 m1 s1 was applied. The volume-based PSD was measured with static light scattering.

particles. However, further experiments should be carried out with experimental setups allowing the investigation of different pulsation more selectively.


between bulk and two-zone experiments showed strong similarity in volume-based PSD and morphology over a range of feed velocities and stirrer rotational velocities. However, a constant deviation between equivalent circuit and bulk process remained. The particles generated in the two-zone process were found to be slightly larger than the particles from the reference bulk process. The influence of the circuit pump pulsation was investigated as one possible reason for this deviation. A PD could not be integrated to reduce pulsation, since a direct influence on particle attributes was observed. However, a comparison of the steady-state COAX precipitation showed that the pump type influences the volume-based PSD. Further research with an experimental setup which allows the setting of specific pulsation parameters, such as amplitude or frequency, is necessary to clarify the influence of pulsation. Additionally, future work should verify the LME model assumptions regarding different reactor scales, which were not covered in this research. Overall, it can be stated that the semi-batch precipitation process can be simplified by using a two-zone model for sparingly soluble substances with high accuracy when it comes to prediction of particle size and distribution. Integration of the LME model as a shortcut model for semi-batch reactors in process simulation systems is therefore promising and can reduce the simulation effort by several scales.

7. Outlook 5. Discussion Several influence factors could have caused the smaller size of the particles precipitated in the two-zone equivalent circuit experiments. As described in Section 2, an error is introduced by the simplified calculation of the exchange volume flows. Furthermore, the micro and meso mixing times in the PFR could deviate from the stirred tank reactor although similar boundary conditions are applied. Replacing the CSTR with the JICF reactor probably does influence the integral scale of turbulence and thus affects the timescale for meso mixing. As shown in Section 4.4, also experiential decisions (e.g. the selected pump type) can influence the validation results. Additionally, the pitched blade turbine does not provide a fully axial flow in the stirred-tank. This could lead to an overestimation of the circulation volume flow by the LME model for this stirrer type. However, besides the observed deviation in particle size, the LME model seems to be a suitable candidate for process simulation of semi-batch precipitation. The assumption of a PFR as reaction zone simplifies the determination of important mixing

An outlook regarding the multiple ways in which the LME model can be used besides process simulation is shown in Fig. 18. Steady-state results of a coaxial mixing reactor with 

uf ¼ 0:13 m1 s1 and Q circ;1 ¼ 5:8 L1 min1 are compared to results of the corresponding dynamic processes with Q circ;1 ¼ 5:8 L1 min1 for the two-zone and the corresponding N ¼ 200 rpm for the bulk process. As it can be seen, the zone model approach can be used to estimate the size of the first particles which are precipitated in the dynamic process. Since the steady-state experiment requires a smaller amount of educt solutions and an immediate result is gained, use of the LME model can reduce the number of necessary scale-up experiments in fine chemistry process development. An aspect which should be clarified in future research is the supersaturation range, for which the LME model can be applicated. The assumption of a localized reaction zone was validated for high values of supersaturation only. Therefore, additional experimental studies should verify the supersaturation limit, for which the

parameters (e.g. e). Computational effort for a pipe-in-pipe mixer CFD simulation is reduced by several scales compared to a CSTR as reactor geometry, since the pipe-in-pipe mixer is a simple and steady-state mixing geometry. 6. Conclusion In this work, the LME model for process simulation of semibatch precipitation of sparingly soluble salts in stirred-tank reactors was presented. The LME model extends the existing model of Bałdyga and Bourne (1999) for a calculation method for the internal exchange volume flows. The model assumptions were validated by barium sulfate precipitation in dedicated experimental setups. It was shown in the experimental validation that an equivalent circuit of PFR and well-mixed storage tank can be used successfully to approximate the semi-batch process for highly supersaturated precipitations. The simple and scalable LME model performed well in estimating the necessary circuit volume flows. The comparison

Fig. 18. Comparison of steady-state, bulk and two-zone volume-based PSD for similar process conditions. The COAX reactor was used for the steady-state and the two-zone experiment. The reactor was equipped with the pitched blade turbine for the bulk experiment.


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two-zone assumption holds. This is especially interesting, as for very low values of supersaturation the stirred tank reactor can, due to the slow solid formation, be approximated as well-mixed system. Therefore, hybrid models can possibly be applied for moderate supersaturations which can neither be treated as fully wellmixed reactors nor be fully covered by the two-zone model only from a simulative point of view. The second important aspect that will need further investigation is the reaction volume for secondary processes. Although secondary processes (e.g. aggregation, breakage) were not relevant for this work, they play a major role in industrial precipitation processes. It should therefore be selectively investigated, if the assumption of the LME model (well-mixed tank for secondary processes) process is appropriate for relevant aggregation and which mixing volume has to be assumed for the well-mixed tank reacor. Declaration of Competing Interest None Acknowledgements The authors would like to thank the German Research Foundation (DFG) for financial support under the Priority Program SPP1679 ‘‘Dynamic simulation of interconnected solid processes” (KI 709/26-3). Furthermore, we thank R. Lavrenev for experimental support. We thank the laboratory for Electron Microscopy (LEM) at Karlsruhe Institute of Technology for providing the particle SEM pictures. References Alexopoulos, A.H., Maggioris, D., Kiparissides, C., 2002. CFD analysis of turbulence non-homogeneity in mixing vessels: a two-compartment model. Chem. Eng. Sci. 57 (10), 1735–1752. Alopaeus, V., Koskinen, J., Keskinen, K.I., 1999. Simulation of the population balances for liquid–liquid systems in a nonideal stirred-tank. Part 1 description and qualitative validation of the model. Chem. Eng. Sci. 54 (24), 5887–5899. Bałdyga, J., Podgórska, W., Pohorecki, R., 1995. Mixing-precipitation model with application to double feed semibatch precipitation. Chem. Eng. Sci. 50 (8), 1281–1300. Bałdyga, J., Bourne, J.R., 1999. Turbulent Mixing and Chemical Reactions. Wiley. Bałdyga, J., Makowski, Ł., Wojciech, O., 2005. Interaction between Mixing, chemical reactions, and precipitation. Ind. Eng. Chem. Res. Bourne, J.R., Yu, S., 1994. Investigation of micromixing in stirred-tank reactors using parallel reactions. Ind. Eng. Chem. Res. 33 (1), 41–55. CheCalc, 2017. CheCalc. Accessed 2 November 2018. . Fox, R.O., Varma, A., 2003. Computational Models for Turbulent Reacting Flows. Cambridge Univ Press. Gösele, W., Kind, M., 1991. Versuche zum Einfluß der Vermischung auf die Qualität eines kontinuierlich gefällten Produktes. Chem. Ing. Tech. 63 (1), 59–62. https:// doi.org/10.1002/cite.330630113. Gradl, J., Schwarzer, H.-C., Schwertfirm, F., Manhart, M., Peukert, W., 2006. Precipitation of nanoparticles in a T-mixer: coupling the particle population dynamics with hydrodynamics through direct numerical simulation. Particul. Process. A Special Issue of Chem. Eng. Process. 45 (10), 908–916. https://doi.org/ 10.1016/j.cep.2005.11.012.

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