Dynamic Simulation of Technical Precipitation Processes

Abstract

Precipitation of sparingly soluble salts is a widely applied industrial unit operation to produce color pigments or nutritional additives. Simulation of this unit operation on a flowsheet level would be a useful tool to simplify process development and optimization. However, the numerical effort of simulating the industrial standard apparatus for precipitation, the stirred-tank reactor (STR), is generally too high for process flowsheet simulation. This high computational cost is due mostly to the complex coupling of mixing and solids formation and the inhomogeneous reaction environment in STRs. Handling of this multiscale challenge in a short time scale, thus, requires the development of numerically efficient short-cut surrogate models. In this chapter, we provide an overview of the results from of our project aiming to develop a dynamic precipitation model for flowsheet simulation using the example of semi-batch precipitation of barium sulfate. This chapter covers the full development progress with step-by-step increasing complexity from steady-state to semi-batch process scale. Multiple experimental setups are used to proof the model hypotheses. The steady-state and dynamic semi-batch precipitation model are exemplarily implemented in the flowsheet framework Dyssol. By using these flowsheet units, the specific process dynamics of semi-batch precipitation processes is investigated. It is, furthermore, demonstrated that using dynamic process parameters (e.g. increasing impeller rotational speed) might be a suitable method to optimize the product particle size distribution (PSD) for semi-batch precipitations in the future.

1 Introduction

Precipitation is an important solids formation process which is extensively applied in the chemical and pharmaceutical industry. Among the products typically produced by precipitation are active pharmaceutical ingredients, color pigments and catalyst materials. Sparingly soluble salts are a specific class of material which pose a serious challenge for industrial process development. They are often precipitated by mixing two aqueous salt solutions, each of them carrying one of the reactive ion types. Due to the low product solubility in the mixture, solid formation is triggered after mixing both educts. As the supersaturation reached for sparingly soluble salt precipitation is generally on a high level compared to other crystallization processes, the increased nucleation and growth rates cause a small time scale of solids formation $\tau_{\text{sf}}$ [s]. If $\tau_{\text{mix}} /\tau_{\text{sf}} \ge 1$ with $\tau_{\text{mix}}$ [s] designating the time scale of mixing, the latter can crucially influence the PSD of the solid product. As consecutive process steps, such as centrifugation, rely heavily on the product PSD, knowledge about the process functionality between process parameters and the PSD is required for process development.

The standard apparatus for technical precipitation is the STR in batch, semi-batch or continuous operation mode. The semi-batch operation is often the method of choice, especially for fine chemical products, as flexible production is possible here. A central drawback of the semi-batch operation, besides the process dynamics, is the localized reaction zone. The solids formation does not take place in the full STR domain due to the low $\tau_{\text{sf}}$. It is, instead, limited to a small volume around the feed pipe. This volume is also designated as the reaction zone. The local mixing conditions in this reaction zone have a significant impact on the final product PSD and, therefore, must be considered for simulation. It is often required to perform numerically expensive computational fluid dynamics (CFD) simulations for information about the local mixing process. This aspect is aggravated by the fact, that even while using CFD, the definition of the exact reaction volume is unclear and, thus, can often only be roughly estimated.

Flowsheet simulation simplifies process development and optimization and is widely applied in process engineering, but it requires short-cut modeling of the different unit operations involved. To date, commercial flowsheet frameworks, such as Aspen Plus or gProms, enable the steady-state and dynamic simulation of precipitation, but the underlying models are restricted to processes with only low levels of supersaturation. As they rely on the mixed-suspension, mixed-product-removal (MSMPR) concept ($\tau_{\text{mix}} /\tau_{\text{sf}} < 1$), neither different zones in the reactor nor the influence of mixing on the PSD are considered. Therefore, it has not been possible to simulate sparingly soluble salts on a process flowsheet level yet. Although numerically reduced models for semi-batch precipitation exist in literature, these models do not reach the numerical efficiency required for process flowsheet simulation. Consequently, new short-cut models must be developed to allow the process flowsheet simulation of semi-batch precipitation of sparingly soluble salts.

The aim of this project is the development of a dynamic model for precipitation which operates on time scales suitable for process flowsheet simulation. We use the knowledge gained by investigating the influence of mixing on precipitation by CFD methods [1,2,3,4] to develop a numerically efficient model for steady-state precipitation in confined impinging jet mixers (CIJMs). The latter are simple static mixing geometries in comparison to the complex dynamic STRs. The steady-state model is selectively validated by simple and complex process flowsheet simulations [5] and, furthermore, applied to investigate different mixing models from literature. The dynamic semi-batch model is developed according to the current state of the art in literature, as a considerable amount of literature exists on the modeling of mixing influenced precipitation in STRs (Fig. 1).

Fig. 1

Literature approaches for modeling mixing influenced precipitation in STRs. The CFD simulations resolve the full flow field a, zone models use discrete zones with exchange streams b and mechanistic models use a plug flow reactor (PFR) mixing zone c. To date, no models with a computational efficiency suitable for process flowsheet simulation exist d

The models with highest computational cost fully resolve the dynamic fluid flow with CFD (a), coupled with population balance equation (PBE) approaches, to track the solids formation [6,7,8,9,10,11,12,13,14]. These studies, supported by experimental work [15,16,17], offered further insights into the complex dynamics of precipitation processes in stirred tanks. Of course, these models are numerically intense and, therefore, not suitable for process flowsheet simulation.

Zone models (b) use distinct reactor zones instead of CFD simulations to depict the fluid dynamics. Consequently, these models are much faster to calculate [8, 18,19,20,21,22,23].

Mechanistic models (c) are even further reduced. They combine a well-mixed bulk fluid (BF) with a plug flow reactor (PFR) which represents the mixing and reaction zone in the stirred tank. One of the most significant mechanistic models for semi-batch precipitation was presented by Bałdyga and Bourne [24], who used the engulfment model (E-model) to simulate the influence of mixing on the product PSD. This type of model was also used by [22, 25, 26].

The mechanistic models are not as accurate as the CFD models, but they offer fast computational speed. However, as illustrated in Fig. 1, the mechanistic models in literature do not reach the time scales required for process flowsheet simulation. Nevertheless, it has been shown within this project that it is possible to further improve their numerical efficiency. We modify the mechanistic model of [24] with an additional approximation method, leading to a hybrid model design. This new approximation method increases the calculation speed by several orders of magnitude. Furthermore, we validated the equivalent concept of BF and PRF proposed by Bałdyga and Bourne [24]. Subsequently, the final semi-batch model was implemented into Dyssol and validated by experiments. Furthermore, we applied the model exemplarily to improve the final product PSD by using dynamic process parameters.

2 Materials and Methods

Section 2.1 introduces the model material barium sulfate, which was used for all simulations and experiments within this project. Section 2.2 deals with the different types of experiments which were conducted to validate our model assumptions. Section 2.3 presents and summarizes the most relevant model equations and the Simulation Setups investigated.

2.1 Materials

We used barium sulfate precipitation from aqueous sodium sulfate and barium chloride solution for our studies. Barium sulfate precipitation is well-investigated in literature and a typical model material for research on mixing influenced precipitation. During contact of both educt solutions, immediate reaction to solid barium sulfate according to Eq. (1) takes place.

$$ {\text{BaCl}}_{2} \left( {\text{aq}} \right) + {\text{Na}}_{2} {\text{SO}}_{4} \left( {\text{aq}} \right) \to {\text{BaSO}}_{4} \left( {\text{s}} \right) + 2{\text{NaCl }}\left( {\text{aq}} \right) $$

(1)

The most significant process parameter for precipitation is the activity-based saturation $S_{a}$ [–], as it impacts the nucleation and growth rate directly. The functionality for $S_{a}$ is given in Eq. (2).

$$ S_{a} = \gamma_{ \pm } \sqrt {\frac{{\tilde{c}_{{{\text{Ba}}^{2 + } }} \cdot \tilde{c}_{{{\text{SO}}_{4}^{2 - } }} }}{K}} $$

(2)

$K\,[{\text{mol}}^{2} {\text{m}}^{ - 6} ]$ designates the solubility product, $\gamma_{ \pm }$ [–] the average activity coefficient and $\tilde{c}\,[{\text{mol}}^{1} {\text{m}}^{ - 3} ]$ the molar concentrations of the reactive ions. The real values for saturation vary greatly during the process, as the saturation buildup depends on local mixing attributes and changes due to the process dynamics. We, therefore, use the index $1{\text{:}}1$ as a reference to a well-mixed, 1:1 volumetric mixture of both educt solutions. Therefore, $S_{a}^{1:1}$[–] provides a coarse estimation of the general level of saturation within the process.

In this work, we used colloidal stabilization to prevent particle aggregation, as aggregation is not part of our model yet. Colloidal stabilization was reached by an excess of barium ions using a lattice ion ratio of $R^{1:1} = \tilde{c}_{{{\text{Ba}}^{2 + } }} /\tilde{c}_{{{\text{SO}}_{4}^{2 - } }} = 5$ for all experiments and simulations. Several literature studies have confirmed that this excess of barium ions is sufficient to prevent aggregation in CIJMs [27, 28]. We, furthermore, showed in [29] that $R^{1:1} = 5$ is also a suitable value for colloidal stabilization of barium sulfate in other mixing geometries.

Our experiments and simulations were performed at a supersaturation level of $S_{a}^{1:1} = 1000$. By using this high level of supersaturation, the consequently low time scale of solids formation guarantees an influence of mixing on the PSD for standard process parameters for semi-batch STR and CIJM precipitation. The concentrations required to achieve $S_{a}^{1:1} = 1000$ and $R^{1:1} = 5$ were calculated with a Pitzer model approach. The resulting educt concentrations of $\tilde{c}_{{{\text{BaCl}}_{2} ,0}} = 0.58\,{\text{mol}}/{\text{L }}$ and $\tilde{c}_{{{\text{Na}}_{2} {\text{SO}}_{4} ,0}} = 0.144\,{\text{mol}}/{\text{L}}$ were used for all experiments and simulations presented in this work. The index 0 indicates $t = 0$, with $t\,[{\text{s}}]$ as the process time. The educt solutions were prepared by solving ${\text{Na}}_{2} {\text{SO}}_{4}$ and ${\text{BaCl}}_{2} \cdot 2{\text{H}}_{2} {\text{O}}$ (>99.99% w/w by Carl Roth) in deionized water.

2.2 Experiments

We conducted different types of experiments (Experimental Setups A–D) to provide a step-by-step validation of our model. Section 2.2.1 presents the experimental setups for validation of the steady-state model. Section 2.2.2 presents the setup for the dynamic semi-batch experiments and the “experimental simulation” setup to allow a specific validation of the equivalent circuit concept for semi-batch precipitation.

2.2.1 Steady State Experiments

The experimental Setup A for steady-state precipitation in CIJMs is illustrated in Fig. 2. Two flow-controlled gear pumps P1 and P2 (MCP-Z by Ismatec) provided the educt solution volume flows of $Q_{{{\text{Na}}_{2} {\text{SO}}_{4} /{\text{BaCl}}_{2} }} = 24 - 150 \,{\text{ml}}/{ \hbox{min} }$ (1:1 volumetric mixture) for the CIJM mixer unit. The educts were prepared as described in Sect. 2.1 and filled into the educt tanks, which are temperature-controlled at $T = 20 \,^\circ {\text{C}}$. Subsequently, the pumps were adjusted to the target volume flows. Samples were taken at the CIJM outlet after several minutes and analyzed with methods described in Sect. 2.2.3, to check for a constant PSD indicating a steady-state operation. A technical drawing of the CIJM with indications is given in Fig. 3. More details on this experiment are provided in [5].

Fig. 2

Reprinted with permission from [5]

Experimental Setup A with pumps (P), educt storage tanks and CIJM reactor (indications in mm). Flow meters for P1 and P2 are neglected due to visibility reasons.

Fig. 3

Reprinted with permission from [5]

Experimental Setup B to investigate the influence of recycle streams with pumps (P), mixer (M), splitter (S), and CIJM precipitation reactor. Flow meters for all pumps are neglected in the figure due to reasons of visibility.

The experimental Setup B extends Setup A by a recycle stream to investigate its influence on the product PSD experimentally (Fig. 3). We used the peristaltic pump DULCO flex by Prominent for P3, which was flow-controlled by a magnetic inductive flow meter (Optiflux by Krohne). The mixer (M) and splitter (S) in Fig. 3 were realized as three-way T ball valves (KHTC 3/18 T by Landefeld), since classifying effects were not expected for the small particle size of the precipitate. The CIJM was constructed according to the indications in Fig. 3. Samples were taken after the splitter to check for a constant PSD. Setup B experiments generally reached steady-state after twelve minutes or less.

We adjusted the educt concentrations and educt volume flows according to Table 1 to investigate the impact of the recycle ratio $\beta = \dot{M}_{\text{circ}} /\dot{M}_{{{\text{mix}},2}}$ independently of changes for $S_{a}^{1:1}$, $R^{1:1}$ or the energy dissipation in the CIJM. $\dot{M}\,[{\text{kg}}^{1} {\text{s}}^{ - 1} ]$ designates the mass flow, including liquid and solid phase, and $x_{j}^{\text{L}} = M_{j}^{\text{L}} /M^{\text{L}}$ are the component mass fractions in the liquid phase (L). Neither educt solution contains a solid phase. Further details on this experiment are provided in [5].

Table 1 Educt solution composition for recycle stream simulations. Reprinted with permission from [5]

2.2.2 Dynamic Experiments

We used the plant in Fig. 4 for bulk semi-batch experiments (Type C) and two-zone experiments (Type D) to validate the equivalent circuit hypothesis for the semi-batch model. Further information on the underlying idea of this experiment is provided in Sect. 2.3.2.1. The plant consists of a 11-L tank reactor (1), a 6-L feed container (2), and an external pipe-circuit (3) with a mixing reactor (4). Only a short overview on the experimental procedure is given in this article. Further details are provided in [29].

Fig. 4

Reprinted with permission from [29]

Simplified process scheme of experimental Setup C and D for bulk and two-zone precipitation experiments.

A technical drawing of the tank reactor is given in Fig. 5. It was constructed according to DIN 28131 [30] with an inner tank diameter of $T_{\text{T}} = 240\,{\text{mm}}$, four baffles of size $B_{\text{T}} = 0.1\,T_{\text{T}}$, a six-blade Rushton Turbine of diameter $D_{\text{T}} = 0.35\,T_{\text{T}}$ and an off-bottom clearance of $C_{\text{T}} = 0.35\,T_{\text{T}}$. The feed pipe with inner diameter $d_{\text{prim}} = T_{\text{T}} /60$ was positioned at a radial distance to the stirrer axis of $R_{\text{T}} = 0.183\,T_{\text{T}}$ with a feed off-bottom clearance of $H_{\text{T}} = 0.421\,T_{\text{T}}$. The product sampling position is indicated by P in Fig. 5.

Fig. 5

Schematic (left) and technical drawing (right) of STR with four baffles and six-blade Rushton Turbine stirrer

The process plant (Figs. 4 and 5) enables two different types of experiments: Bulk and two-zone experiments.

2.2.2.1 Bulk Experiments (Experimental Setup C)

A standard semi-batch procedure is performed in bulk experiments. Therefore, the valves to the external loop (3, 4) are closed and the feed pipe is positioned in the tank reactor (1). Initially, $5.37\,{\text{L}}$ ${\text{BaCl}}_{2}$ solution was poured into the tank reactor, which corresponds to 50% of final filling. The rotational velocity was adjusted to the target value. After the process started, the ${\text{Na}}_{2} {\text{SO}}_{4}$ solution was added through the feed pipe by using a defined feed volume flow $Q_{\text{prim}}$ until a total addition volume of $5.37\,{\text{L}}$ ${\text{Na}}_{2} {\text{SO}}_{4}$ solution was reached. Subsequently, probes of solid product were taken at position P in the STR.

2.2.2.2 Two-Zone Experiments (Experimental Setup D)

The feed pipe in two-zone experiments is not positioned inside the tank reactor (A). Instead, it is positioned in the external mixing reactor (C). The valves to the external loop were opened and Pump 2 was adjusted to the calculated value for the circulation flow. The stirring rotational speed was always adjusted to $N = 300\,{\text{rpm}}$, as the stirred tank represented the well-mixed BF. At the start of the process, the ${\text{Na}}_{2} {\text{SO}}_{4}$ solution was added through the feed pipe into D, until a total addition volume of 5.37 L ${\text{Na}}_{2} {\text{SO}}_{4}$ solution was reached. Sampling was carried out at position P in the STR.

2.2.3 Analytics

The volume-based PSD $q_{3} (L ) ( {\text{m}}^{ - 1} )$ for particles from steady-state experiments was measured by dynamic light scattering (DLS) using a Zetasizer Nano ZS by Malvern. Additionally, scanning electron microscopy (SEM) was used as a second measurement technique to calculate the number-based PSD $q_{0} \left( L \right)({\text{m}}^{ - 1} )$. Probes for SEM analysis were centrifuged three times at 11,000 rpm, including two intermediate washing steps with deionized water. Subsequently, the product was dried for 24 h at 50 ℃ in a drying cabinet and sputtered with 3.5 nm platin for the investigation in a LEO 1530 SEM. Diameters of 500 particles at different locations were determined by graphical evaluation to calculate the $q_{0} \left( L \right)$ distribution.

The particle sizes for dynamic semi-batch experiments exceeded the measurement range of DLS, as the mixing intensity in the STR is several orders lower than the one reached in CIJMs. Therefore, we measured the particle size for these experiments by static light scattering (SLS) using Mastersizer 3000 or 3000E by Malvern. Measurements were carried out in deionized water using a Hydro EV wet dispersion system by Malvern. Samples were measured with a laser occlusion of 10%, a refractive index of $n_{\text{r}} = 1.643$ and an absorption coefficient of $\alpha_{\text{abs}} = 0.1$.

2.3 Simulation

Section 2.2.1 introduces the model for steady-state precipitation. Section 2.2.2 presents the dynamic semi-batch model.

2.3.1 Steady-State Precipitation Model

The model equations are given in Sect. 2.3.1.1. The mixing model is presented in Sect. 2.3.1.2. Section 2.3.1.3 provides information about the Simulation Setups. All sections only represent the most important equations and information. Consultation of [5] is recommended for a more detailed view of the steady-state precipitation model.

2.3.1.1 Model Equations

During steady-state precipitation, both educt solution A and B with educt volume flows $Q_{\text{A}}$ and $Q_{\text{B}}$ are mixed along the mixer length coordinate z [m]. An exemplarily illustration for the balance volume in CIJMs and the spatial discretization is given in Fig. 6.

Fig. 6

Reprinted with permission from [5]

Spatial discretization of the CIJM geometry with typical supersaturation $S_{a}$ and total particle density $n_{\text{t}}$ (left). Balance volume for steady-state precipitation, shown exemplarily with two educt environments (A, B) and one mixed environment (M) (right).

The process of turbulent mixing is complex, as eddies of multiple size scales are involved in the mixing process. However, several mechanistic models have been developed to account for the mixing process in a simplified way. In our project, we used the micro mixing model proposed by Metzger and Kind [4]. Mechanistic mixing models divide the liquid phase into different zones (index k) with volume $V_{k}^{\text{L}} \,[{\text{m}}^{3} ]$ to track the status of mixing $(\alpha_{k} = V_{k}^{\text{L}} /V^{\text{L}} )$. $V^{\text{L}} \,[{\text{m}}^{3} ]$ designates the total volume of liquid phase including all zones. Due to the mixing process, these volume fractions $\alpha_{k}$ change along the mixer length coordinate z. The balance volume is the well-mixed reaction volume fraction (M in this example), which will grow along the mixer length coordinate z by engulfing the unmixed educt fluid A and B.

The PBE is based on the particle density $n = dn_{\text{t}} /dL\,[{\text{m}}^{ - 4} ]$ in the reaction zone M, with $n_{\text{t}} \,[{\text{m}}^{ - 3} ]$ as the total number of particles per volume suspension and L [m] as the particle diameter. As the rising saturation level S_a inside the reaction zone will trigger solids formation, a rising total particle number density can be observed along z (Fig. 6).

The PBE in Eq. (3) is used to calculate n along the z coordinate of the mixer. $B\,[{\text{m}}^{ - 4} {\text{s}}^{ - 1} ]$ designates the nucleation rate and $G = dL/dt[{\text{m}}^{1} {\text{s}}^{ - 1} ]$ the particle growth rate, $\bar{u}_{\text{out}} [{\text{m}}^{1} {\text{s}}^{ - 1} ]$ the average velocity at the mixer outlet. $n_{{{\text{A}}/{\text{B}}}} [{\text{m}}^{ - 4} ]$ are the particle densities in the educt mixing zones A and B, which are only relevant if the educt solutions already contain particles. The last two terms in Eq. (3) are exemplarily adapted to the mixing model by Metzger and Kind [4]. These terms must be changed if other mixing models are investigated.

$$ \frac{dn}{dz} + n \cdot \frac{{d{ \ln }\left( {\alpha_{\text{M}} } \right)}}{dz} = \frac{1}{{\bar{u}_{\text{out}} }}\left( {B - \frac{{d\left( {G n} \right)}}{dL}} \right) - \frac{{n_{\text{A}} }}{{\alpha_{\text{M}} }}\frac{{d\alpha_{\text{A}} }}{dz} - \frac{{n_{\text{B}} }}{{\alpha_{\text{M}} }}\frac{{d\alpha_{\text{B}} }}{dz} $$

(3)

The semi-empirical Eq. (4), considering homogenous and heterogenous nucleation, is used in the model to calculate the nucleation rate. $J_{{{ \hbox{max} }, {\text{hom}}/{\text{het}}}}$ $[ {\text{m}}^{ - 3} {\text{s}}^{ - 1} ]$ and ${\text{C}}_{{{ \hom }/{\text{het}}}}$ [–] are material specific constants. The dirac-delta function $\delta (L_{\text{crit}} )$ $[ {\text{m}}^{ - 1} ]$ is used to include the nuclei at the critical nucleation radius $L_{\text{crit}}$ $[ {\text{m]}}$.

$$ B = \delta \left( {L_{\text{crit}} } \right) \cdot \left( {J_{{{ \hbox{max} }, {\text{hom}}}} \cdot e^{{ - C_{ \hom } \cdot { \ln }(S_{\text{a}} )^{ - 2} }} + J_{{{ \hbox{max} }, {\text{het}}}} \cdot e^{{ - C_{\text{het}} \cdot { \ln }(S_{\text{a}} )^{ - 2} }} } \right) $$

(4)

The size of the thermodynamically stable nuclei $L_{\text{crit}}$ depends on the supersaturation and is calculated by Eq. (5) following the classical nucleation theory. The solid-liquid interface tension $\gamma_{\text{sl}}$ $[N^{1} {\text{m}}^{ - 1} ]$, the molecular volume of the solid $V_{{{\text{mol}},{\text{s}}}}$ $[ {\text{m}}^{3} ]$, the Boltzmann constant $k_{\text{B}}$ $[ {\text{m}}^{2} {\text{kg}}^{1} {\text{s}}^{ - 2} {\text{K}}^{ - 1} ]$, the number of ions $\nu_{\text{s}}$ [–] and the system temperature T [K] are the relevant variables.

$$ L_{\text{crit}} = \frac{{4 \cdot \gamma_{\text{sl}} \cdot V_{{{\text{mol}},{\text{s}}}} }}{{\nu_{\text{s}} \cdot k_{\text{B}} \cdot T \cdot { \ln }\left( {S_{\text{a}} } \right)}} $$

(5)

A size-dependent growth rate was implemented by Eq. (6), proposing a diffusion-limited growth mechanism. $\bar{D}_{{{\text{ri}},{\text{sol}}}} [{\text{m}}^{2} {\text{s}}^{ - 1} ]$ designates the average diffusion coefficient of the reactive ions (ri) in the solvent (sol), $\tilde{\rho }_{\text{s}}$ $[ {\text{mol}}^{1} {\text{m}}^{ - 3} ]$ designates the molar density of the solid and Sh the Sherwood number. Due to the small particle size, $Sh = 2.0$ was assumed for all simulations.

$$ G = Sh \cdot \frac{{2\bar{D}_{{{\text{ri}},{\text{sol}}}} }}{{L \cdot \tilde{\rho }_{\text{s}} }} \cdot \sqrt K \cdot \left( {S_{a} - 1} \right) $$

(6)

$\bar{D}_{{{\text{ri}},{\text{sol}}}}$ is calculated by Stokes-Einstein Eq. (7). $\mu$ $[ {\text{kg}}^{1} {\text{m}}^{ - 1} {\text{s}}^{ - 1} ]$ designates the dynamic viscosity of the solvent and $\bar{L}_{{{\text{mol}},{\text{ri}}}}$ $[ {\text{m]}}$ the average molecular diameter of the reactive ions.

$$ \bar{D}_{{{\text{ri}},{\text{sol}}}} = \frac{{k_{\text{B}} T}}{{3 \pi \mu \bar{L}_{{{\text{mol}},{\text{ri}}}} }} $$

(7)

The solution composition changes along z, as ions are mixed into the reaction zone and depleted by solids formation. Consequently, the concentration balance for all ionic components in the liquid phase (index m) is given in Eq. (8). The last two terms in Eq. (8) are exemplarily adapted to the mixing model by [4]. These terms must be changed if other mixing models are investigated. Differences between the densities of the mixing environments are neglected for Eq. (8).

$$ \frac{{d\tilde{c}_{m} }}{dz} + \tilde{c}_{m} \cdot \frac{{d{ \ln }\left( {\alpha_{\text{M}} } \right)}}{dz} = \frac{{d\tilde{c}_{{m,{\text{sf}}}} }}{dz} - \frac{{\tilde{c}_{{m,{\text{A}}}} }}{{\alpha_{\text{M}} }}\frac{{d\alpha_{\text{A}} }}{dz} - \frac{{\tilde{c}_{{m,{\text{B}}}} }}{{\alpha_{\text{M}} }}\frac{{d\alpha_{\text{B}} }}{dz} $$

(8)

The solid formation reduces the ion concentration according to Eq. (9), with $\vartheta_{{m,{\text{sf}}}}$ [–] as stochiometric coefficient of ion type m in the solids formation reaction. Spherical particles are assumed with $dV_{p} /dL = \pi L^{2} /2$. $V_{p}$ designates the volume of a single particle. $\vartheta_{{m,{\text{sf}}}}$ obtains a negative value for educts of the solid formation reaction. If ions are not part of the solids formation reaction, $\vartheta_{{m,{\text{sf}}}} = 0$.

$$ \frac{{dc_{{m,{\text{sf}}}} }}{dz} = \frac{\pi }{2}\frac{{\vartheta_{{m,{\text{sf}}}} \cdot \tilde{\rho }_{\text{s}} }}{{ \bar{u}_{\text{out}} }} \cdot \mathop \int \limits_{L}^{{}} n\left( L \right) G\left( L \right) L^{2} dL $$

(9)

The saturation $S_{\text{a}}$ is not directly calculated by the model. Instead, the model is connected to the software PhreeqC to calculate the activity coefficients. Further details on this software connection or additional equations for the steady-state model (e.g. for $\mu$) can be found in [5].

We used a high-resolution finite-volume scheme with a van Leer flux limiter to solve the PBE. More information on the solver and its control is provided in [5]. The material constants for barium sulfate can be found in [5].

2.3.1.2 Mixing Model

We investigated different mixing models for CIJMs to find the most promising candidate for process flowsheet simulation. We applied the micro-mixing model by Metzger and Kind [4] for most of the steady-state simulations conducted within this project. The model consists of three mixing zones, two educt zones (A, B) and one well-mixed reaction zone (Fig. 7). The model by Metzger and Kind [4] is predictive for the given setup but does not follow the physical concept of the engulfment theory directly. Consequently, it is more an empirically based than a physically based model.

Fig. 7

Temporal mixing volume fractions evolution for the model by [4]

The temporal evolution of the volume fractions is given by Eqs. (10–12). $E = 0.058\,\bar{\varepsilon }^{0.5} \nu^{ - 0.5}$ $[ {\text{s}}^{ - 1} ]$ designates the engulfment constant. $\bar{\varepsilon }$$[ {\text{m}}^{2} {\text{s}}^{ - 3} ]$ is the average energy dissipation and $\nu$ $[ {\text{m}}^{2} {\text{s}}^{ - 1} ]$ the kinematic viscosity.

$$ \frac{{d\alpha_{\text{A}} }}{dz} = - \frac{E}{{\bar{u}_{\text{out}} }} \cdot \alpha_{\text{A}} \cdot \left( {1 - \alpha_{\text{A}} } \right) $$

(10)

$$ \frac{{d{\alpha }_{\text{B}} }}{dz} = - \frac{E}{{\bar{u}_{\text{out}} }} \cdot \alpha_{\text{B}} \cdot \left( {1 - \alpha_{\text{B}} } \right) $$

(11)

$$ \frac{{d\alpha_{\text{M}} }}{dz} = \frac{E}{{\bar{u}_{\text{out}} }} \left( {\alpha_{\text{A}} \cdot \left( {1 - \alpha_{\text{A}} } \right) + \alpha_{\text{B}} \cdot \left( {1 - \alpha_{\text{B}} } \right)} \right) $$

(12)

2.3.1.3 Simulation Setups

This section introduces the steady-state Simulation Setups. Flowsheet Simulation Setup A (Fig. 8) was designed according to Experimental Setup A and represents a stand-alone simulation of CIJM precipitation. The input concentrations of the educt solutions were defined according to the educt concentrations presented in Sect. 2.1 and the input volume flows were varied according to the experiments described in Sect. 2.2.1. Further details of Setup A simulations are given in [5].

Fig. 8

Reprinted with permission from [5]

Simulation flowsheet for steady-state model validation (Setup A).

The recirculation flowsheet (Simulation Setup B, Fig. 9) was constructed according to the Experimental Setup B (Fig. 3, Sect. 2.2.1). We used the units for ideal mixing/ideal splitting implemented in Dyssol [31] for Splitter and Mixer. Further details regarding the simulations and the Dyssol solver configurations can be found in [5]. The process conditions for the simulations with different recycle ratios $\beta = \dot{M}_{\text{circ}} /\dot{M}_{{{\text{mix}},2}}$ were chosen according to Table 1.

Fig. 9

Simulation flowsheet for investigating the influence of the recycle stream on the precipitation process. Reprinted with permission from [5]

2.3.2 Dynamic Semi-Batch Precipitation Model

Section 2.3.2.1 provides an overview of the model equations. The mixing model is explained in Sect. 2.3.2.2. Section 2.3.2.3 presents the newly developed approximation method, which is the reason for the outstanding numerical performance of the model. Section 2.3.2.4 provides information regarding the Simulation Setups.

2.3.2.1 Model Equations

The semi-batch model is based on the mechanistic model proposed by [24], who divided the semi-batch STR in a PFR connected to well-mixed stirred bulk fluid (BF). We could prove experimentally in [29] that this assumption can be used to model the semi-batch process. The relevant process parameters are the feed mass flow $\dot{M}_{\text{prim}}$, the outlet mass flow $\dot{M}_{\text{out}}$, the secondary inlet mass flow $\dot{M}_{ \sec }$ and the impeller rotational speed $N$. All of these process parameters can be either steady-state or dynamic. The semi-batch process itself will be dynamic in any case.

Figure 10 illustrates the main variables of the model. The BF’s mass $M_{\text{BF}}$ consists of a liquid (L) and a solid phase (S) with the mass fractions $\xi_{\text{BF}}^{\text{S}} = M_{\text{BF}}^{\text{S}} /M_{\text{BF}}$ and $\xi_{\text{BF}}^{\text{L}} = M_{\text{BF}}^{\text{L}} /M_{\text{BF}}$. Without the presence of other phases, $\xi_{\text{BF}}^{\text{L}}$ can be calculated by the closure $\xi_{\text{BF}}^{\text{L}} = 1 - \xi_{\text{BF}}^{\text{S}}$. The mass fractions of the components in the liquid phase are defined by $x_{{j,{\text{BF}}}}^{\text{L}} = M_{{j,{\text{BF}}}}^{\text{L}} /M_{\text{BF}}^{\text{L}}$ with j as the index for all components in the liquid phase. The relative mass of the particles in each size class is defined by $w_{i}^{\text{S}} = m_{i}^{\text{S}} /M_{\text{BF}}^{\text{S}}$.

Fig. 10

Process variables for the semi-batch process a and the equivalent circuit of PFR and BF b

The system state vector of the BF is given by $\vec{B}\left( t \right) = \left( {M_{\text{BF}} \,\xi_{\text{BF}}^{\text{S}} \,x_{{j,{\text{BF}}}}^{\text{L}} \,w_{{i,{\text{BF}}}}^{\text{S}} } \right)^{\text{T}}$. With five components ($j = 5$) in the liquid phase (e.g. ${\text{H}}_{2} {\text{O}},{\text{Ba}}, {\text{Na}}, {\text{Cl}}, {\text{SO}}_{4}$) and 150 particle size classes $(i = 150)$, $\vec{B}\left( t \right)$ consists of 157 entries. No particles are considered in the feed stream $\dot{M}_{\text{prim}}$ $(\xi_{\text{prim}}^{\text{S}} = 0 w_{{i,{\text{prim}}}}^{\text{S}} = 0)$. Furthermore, we assume the presence of only one solvent. The BF volume $V_{\text{BF}}$ is assumed to correspond to the volume of pure solvent $(V_{\text{BF}} = x_{{{\text{H}}_{2} {\text{O}},{\text{BF}}}} M_{\text{BF}}^{\text{L}} /\rho_{{{\text{H}}_{2} {\text{O}}}} )$.

The streams, which are interconnecting BF and PFR are $\vec{S}_{{{\text{circ}},1}}$ (entering the PFR) and $\vec{S}_{{{\text{circ}},2}}$ (leaving the PFR). The stream vector $\vec{S}$ is defined by $\vec{S} = (\dot{M} \,\xi^{\text{S}}\, x_{j}^{\text{L}}\, w_{i}^{\text{S}} )^{\text{T}}$. Corresponding to the BF, the variables for the stream vector are defined by $\xi_{{{\text{circ}},1}}^{\text{S}} = \dot{M}_{{{\text{circ}},1}}^{\text{S}} /\dot{M}_{{{\text{circ}},1}}$, $\xi_{{{\text{circ}},1}}^{\text{L}} = \dot{M}_{{{\text{circ}},1}}^{\text{L}} /\dot{M}_{{{\text{circ}},1}}$, $x_{{j, {\text{circ}},1}}^{\text{L}} = \dot{M}_{{j,{\text{circ}},1}}^{\text{L}} /\dot{M}_{{{\text{circ}},1}}^{\text{L}}$ and $w_{i}^{\text{S}} = \dot{m}_{i}^{\text{S}} /\dot{M}_{{{\text{circ}},1}}^{\text{S}}$.

The balance equations for the BF can be derived with the variables given. $M_{\text{BF}}$ will change due to the incoming and outgoing streams, according to Eq. (13).

$$ \frac{{dM_{\text{BF}} }}{dt} = \dot{M}_{ \sec } + \dot{M}_{{{\text{circ}},2}} - \dot{M}_{\text{out}} - \dot{M}_{{{\text{circ}},1}} $$

(13)

The temporal evolution of the solids phase fraction is described by Eq. (14). No source term for solids formation must be considered for the BF balance equation, as solids formation only takes place in the PFR.

$$ \begin{aligned} & \frac{{d\xi_{\text{BF}}^{\text{S}} }}{dt} + \xi_{\text{BF}}^{\text{S}} \frac{{d{ \ln }\left( {M_{\text{BF}} } \right)}}{dt} \\ & \quad = \frac{1}{{M_{\text{BF}} }} \cdot \left( {\dot{M}_{ \sec } \,\xi_{ \sec }^{\text{S}} + \dot{M}_{{{\text{circ}},2}}\, \xi_{{{\text{circ}},2}}^{\text{S}} - \left( {\dot{M}_{\text{out}} + \dot{M}_{{{\text{circ}},1}} } \right) \,\xi_{\text{BF}}^{\text{S}} } \right) \\ \end{aligned} $$

(14)

The component balances are given by Eq. (15).

$$ \begin{aligned} & \frac{{dx_{{j, {\text{BF}}}}^{\text{L}} }}{dt} + x_{{j, {\text{BF}}}}^{\text{L}} \frac{{d{ \ln }\left( {\xi_{\text{BF}}^{\text{L}} } \right)}}{dt} + x_{{j, {\text{BF}}}}^{\text{L}} \frac{{d{ \ln }\left( {M_{\text{BF}} } \right)}}{dt} \\ & \quad = \frac{1}{{M_{\text{BF}}\, \xi_{\text{BF}}^{\text{L}} }} \cdot \left( {\dot{M}_{ \sec}\, \xi_{ \sec }^{\text{L}}\, x_{{j, {\text{sec}}}}^{\text{L}} + \dot{M}_{{{\text{circ}},2}} \,\xi_{{{\text{circ}},2}}^{\text{L}}\, x_{{j, {\text{circ}},2}}^{\text{L}} } \right. \\ & \quad \quad \left. { - \left( {\dot{M}_{\text{out}} + \dot{M}_{{{\text{circ}},1}} } \right)\,\xi_{\text{BF}}^{\text{L}} x_{{j, {\text{BF}}}}^{\text{L}} } \right) \\ \end{aligned} $$

(15)

Nucleation or growth are not relevant for the BF, as both take place only in the PFR. Consequently, the particle mass fractions for the BF can be calculated by Eq. (16).

$$ \begin{aligned} & \frac{{dw_{{i,{\text{BF}}}}^{\text{S}} }}{dt} + w_{{i,{\text{BF}}}}^{\text{S}} \cdot \frac{{d{ \ln }\left( {x_{\text{BF}}^{\text{S}} } \right)}}{dt} + w_{{i,{\text{BF}}}}^{\text{S}} \cdot \frac{{d{ \ln }\left( {M_{\text{BF}} } \right)}}{dt} \\ & \quad = \frac{1}{{M_{\text{BF}} \,x_{\text{BF}}^{\text{S}} }} \cdot \left( {\dot{M}_{ \sec } \,\xi_{ \sec }^{\text{S}} \,w_{{i,{ \sec }}}^{\text{S}} + \dot{M}_{{{\text{circ}},2}}\, \xi_{{{\text{circ}},2}}^{\text{S}} \,w_{{i,{\text{circ}},2}}^{\text{S}} } \right. \\ & \quad \quad \left. { - \left( {\dot{M}_{\text{out}} + \dot{M}_{{{\text{circ}},1}} } \right) \,\xi_{\text{BF}}^{\text{S}} \,w_{{i,{\text{BF}}}}^{\text{S}} } \right) \\ \end{aligned} $$

(16)

The unknown variables of this model are the circulation mass flow rate $\dot{M}_{{{\text{circ}},1}}$ and the vector $\vec{S}_{{{\text{circ}},2}}$ representing the fluid after precipitation in the PFR. A closure for $\dot{M}_{{{\text{circ}},1}}$ based on the stirrer type and size was developed by using the conceptual idea of the similarity between the local reaction zone in a stirred tank with Rushton Turbine and a JICF precipitation. Calculation of $\vec{S}_{{{\text{circ}},2}}$ is explained in Sect. 2.3.2.3. Explicit Euler’s method is used to solve Eqs. (13–16) with a static time discretization $\Delta t = 0.5$. This value for $\Delta t$ is appropriate for the Simulation Setups C and E.

2.3.2.2 Mixing Models

The semi-batch model uses the steady-state model presented in Sect. 2.3.1 to calculate the PFR. The mixing model for CIJMs was replaced by a mixing model for the jet in cross flow (JICF) mixing of the feed volume flow and the circulation flow. As shown in [29], a JICF mixer can be used to imitate the local flow environment around the feed pipe in STRs. The reaction zone is, therefore, defined as $Q_{\text{prim}}$ (P), which engulfs $Q_{{{\text{circ}},1}}$ (C) over the mixer length coordinate $z$. A possible influence of meso mixing must be additionally considered, as the fluid must be meso mixed first to start micro mixing. Consequently, zone ${\text{C}}_{\text{meso}}$ is introduced for the fluid of $Q_{{{\text{circ}},1}}$ which is already meso mixed and therefore, can act as an engulfment environment for P (Fig. 11).

Fig. 11

Fluid prim (P) engulfing fluid circ, 1 (C) in the E-model by [24]. ${\text{C}}_{\text{meso}}$ designates the fluid which is already meso mixed and, thus, provides the environment for the engulfment process of P

We used the E-model by [24] considering micro and meso mixing to depict the evolution of P over z. The timescale for meso mixing was calculated with $\tau_{\text{meso}} = 1.2\, d_{\text{prim}}^{2/3}\, \bar{\varepsilon }^{ - 1/3}$ according to [32].iterations on a flowsheet

$$ \frac{{d\alpha_{\text{P}} }}{dz} = \frac{E}{{\bar{u}_{{{\text{circ}},2}} }}\alpha_{\text{P}} \left( {1 - \frac{{\alpha_{\text{P}} }}{{\alpha_{\text{u}} }}} \right) $$

(17)

$$ \alpha_{\text{u}} = \frac{{\alpha_{{{\text{P}}0}} }}{{\alpha_{{{\text{P}}0}} + \left( {1 - \alpha_{{{\text{P}}0}} } \right) \cdot { \exp }\left( {z/\left( {\tau_{\text{meso}} \bar{u}_{\text{mix}} } \right)} \right)}} $$

(18)

The mixing model requires an average energy dissipation $\bar{\varepsilon }$, which was correlated by steady-state experiments.

Replacing the mixing model in the steady-state model also required adaptation of the PBE and the component balances. For precipitation in the PFR, ${\text{P}}$ is the balance volume instead of M. Equation (3) and Eq. (8) were, therefore, replaced by Eq. (19) and Eq. (20), respectively.

$$ \frac{dn}{dz} + n \cdot \frac{{d{ \ln }\left( {\alpha_{\text{P}} } \right)}}{dz} = \frac{1}{{\bar{u}_{{{\text{circ}},2}} }}\left( {B - \frac{{d\left( {G n} \right)}}{dL}} \right) - \frac{1}{{\alpha_{\text{P}} }}\frac{{d\left( {n_{{{\text{C}}_{\text{meso}} }} \alpha_{{{\text{C}}_{\text{meso}} }} } \right)}}{dz} - \frac{{n_{\text{C}} }}{{\alpha_{\text{P}} }}\frac{{d\alpha_{\text{C}} }}{dz} $$

(19)

$$ \frac{{d\tilde{c}_{m} }}{dz} + c_{m} \cdot \frac{{d{ \ln }\left( {\alpha_{\text{P}} } \right)}}{dz} = \frac{{d\tilde{c}_{{m,{\text{sf}}}} }}{dz} - \frac{1}{{\alpha_{\text{P}} }}\frac{{d\left( {\tilde{c}_{{m,{\text{C}}_{\text{meso}} }} \alpha_{{{\text{C}}_{\text{meso}} }} } \right)}}{dz} - \frac{{\tilde{c}_{{m,{\text{C}}}} }}{{\alpha_{\text{P}} }}\frac{{d\alpha_{\text{C}} }}{dz} $$

(20)

2.3.2.3 Approximation Method

The reason for the outstanding numerical performance in our semi-batch model in comparison to mechanistic models in literature is the approximation method which we developed within this project. This method takes advantage of the fact that the PFR is a steady-state system and, therefore, will only show a different output signal if its input signals, $\vec{S}_{{{\text{circ}},1}}$ and $\vec{S}_{\text{prim}}$, deviate significantly compared to the prior iteration. In a typical mechanistic model from literature, the PFR consumes 99.7% of computational time, whereas the BF model only requires 0.3% (measured in our example case). The BF model can be calculated much faster than the PFR, since the coupling between mixing, nucleation and growth does not have to be solved in the BF. Consequently, the PFR calculation is the bottleneck of mechanistic models. Improvements which simplify or skip the PFR calculation can, thus, increase numerical efficiency by several orders of magnitude.

The approximation method is illustrated in Fig. 12. n designates the iteration index. The vector $\vec{S}_{{{\text{circ}},2}}$ can either be gained by solving the PFR (high numerical effort) or approximating its result by a component balance. We assume for this additional balance that the saturation in $\vec{S}_{{{\text{circ}},2}}$ is fully depleted to thermodynamic equilibrium. For the example of barium sulfate (two reactive ions, $\vartheta_{\text{Ba}} = \vartheta_{{{\text{SO}}_{4} }} = 1$), the resulting ion concentrations in $\vec{S}_{{{\text{circ}},2}}$ can be calculated by Eq. (21).

Fig. 12

Approximation method for the short-cut modeling of the semi-batch process dynamics

$$ \tilde{c}_{{{\text{Ba}},{\text{circ}},2}} = 0.5 \cdot \left( {\tilde{c}_{{{\text{Ba}},{\text{circ}},1}} - \tilde{c}_{{{\text{SO}}_{4} ,{\text{circ}},1}} } \right) + \sqrt {0.25\left( {\tilde{c}_{{{\text{Ba}},{\text{circ}},1}} - \tilde{c}_{{{\text{SO}}_{4} ,{\text{circ}},1}} } \right)^{2} + K} $$

(21)

$\dot{M}_{{n,{\text{circ}},2}}$, $\xi_{{{\text{n}},{\text{circ}},2}}^{\text{S}}$ $x_{{n,{\text{j}},{\text{circ}},2}}^{\text{L}}$ can, thus, be calculated by simple balance equations without solving the PFR. Only the PSD is not calculated. It is, instead, approximated by using the PSD from the last iteration ($w_{{i,{\text{circ}},2}}^{\text{S}} = w_{{i - 1,{\text{circ}},2}}^{\text{S}}$). The approximation method, thus, introduces an error on the PSD but increases computational speed by several orders of magnitude.

The recalculation frequency $f_{\text{rec}} = n_{\text{rec}} /n_{\text{it}}$ is the number of recalculations of the PFR divided by the number of iterations. The approximation method is not active for $f_{\text{rec}} = 1$ as the PFR is calculated on each iteration. The value $f_{\text{rec}} = 1/10$ means that after one calculation, the approximation method is applied for the next nine iteration steps.

2.3.2.4 Simulation Setups

Two different Simulation Setups were investigated for the semi-batch precipitation model (Fig. 13). Simulation Setup C was designed according to Experimental Setup C. All simulation parameters were adapted to the experimental STR with the six-blade Rushton turbine stirrer. We conducted simulations for $N = 50 - 200\,{\text{rpm}}$ and $Q_{\text{prim}} = 0.1 - 0.4 \,{\text{L}}^{1} { \hbox{min} }^{ - 1}$ according to the experiments. The results from Setup C simulations, therefore, allow one to validate the semi-batch model.

Fig. 13

Simulation Setups: constant feed rate and constant stirring rate for validation (C) and dynamic increase of the stirring rate (E)

Furthermore, Simulation Setup E was designed to demonstrate on the example of a dynamic stirring rate that there is currently unused potential to influence the process dynamics. We use a linearly increasing stirring rate to influence the process dynamics of the semi-batch process (Fig. 14). The feed volume flow is kept constant at $Q_{\text{prim}} = 0.2 \,{\text{L}}^{1} { \hbox{min} }^{ - 1}$.

Fig. 14

Linear increase of the rotational speed versus time (Setup E)

3 Results and Discussion

This section presents selected results from steady-state and dynamic experiments and simulations. The results of the steady-state simulation are compared to the corresponding validation experiments in Sect. 3.1. Furthermore, the model is applied to investigate the influence of recycle streams on the product PSD. Section 3.2 deals with the results from “experimental simulation” to verify whether an equivalent circuit of PFR and well-mixed BF can be used to simulate a semi-batch STR. Section 3.3 compares the dynamic semi-batch simulation to experimental data and, furthermore, investigates the influence of dynamic process parameters on the product PSD.

3.1 Steady-State Precipitation

The experiments and simulations of Setup A aim at validating our steady-state model for CIJM precipitation and investigating its predictiveness towards the PSD. In addition to the micro mixing model by Metzger and Kind [4], two other models by [24, 33] are compared, which are not explicitly discussed in Sect. 2.3.1.2.

As shown in Fig. 15, the median of the volume-based PSD $(L_{50,3} )$ for precipitated particles in Setup A experiments can be well-predicted with a steady-state precipitation model using the micro mixing model by Metzger and Kind [4]. Interestingly, the model by Bałdyga and Bourne [24] predicts larger particles compared to the other mixing models. We could furthermore show by mathematical analysis that the models by Metzger and Kind [4] and Schwarzer [33] do not implement the engulfment theory correctly. Both models proved to be suitable for our simulations but should be designated as empirical models.

Fig. 15

Median of volume-based PSD $( L_{50,3} )$ versus jet velocity with for simulations with different micro mixing models by [24, 33, 4] and experiments. Particle size measured by DLS

We, furthermore, investigated the influence of recycle streams on steady-state precipitation by Setup B experiments and simulations. As observable in Fig. 16, the PSD becomes increasingly multimodal with increasing splitting factor $\beta$. This multimodality is caused mostly by recycled particles passing the CIJM more often and, therefore, growing more than particles passing the CIJM only once. Furthermore, the size of the smallest particles in the first peak, which are the particles after a single precipitation in the CIJM, is also increased for $\beta \ge 0.1$ in a nonlinear way (Fig. 16).

Fig. 16

Reprinted with permission from [5]

Simulated influence of the splitting factor $\beta$ on the number-based PSD $\left( {q_{0} } \right)$ of the product stream for ideal mixing in the CIJM.

The nonlinear system behavior for increased values of the splitting factor $\beta$ is also observable for the saturation curve along the z-axis (Fig. 17). Whereas the saturation does not significantly differentiate for $\beta \le 0.2$, major changes can be observed for $\beta = 0.3$. As we could show in [5], the reason for this faster supersaturation depletion is the faster depletion of reactive ions, which becomes relevant when larger particles are present.

Fig. 17

Reprinted with permission from [5]

Saturation $S_{\text{a}}$ over the CIJM z-axis in flowsheet B for different splitting factors $\beta$ for ideal mixing in the CIJM.

The particles show, independently of the value chosen for the splitting factor $\beta$, no differences in morphology, see Fig. 18.

Fig. 18

Reprinted with permission from [5]

REM pictures of barium sulfate precipitated for splitting factors $\beta = 0$ (left) and $\beta = 0.3$ (right).

Nevertheless, the number distribution based on the particles counted (see Sect. 2.2.3) reveals a similar size shift effect comparable to the simulation. The differences between the PSD ($q_{0}$) for a splitting factor of $\beta = 0$ and of $\beta = 0.2$ are shown exemplarily in Fig. 19.

Fig. 19

Reprinted with permission from [5]

Experimental SEM results for splitting factors $\beta = 0.0$ and $\beta = 0.2$.

The experimental results analyzed with SLS in comparison to simulation results are given in Fig. 20. As observable, the model can predict the average particle size correctly even for a complex flowsheet including a recycle stream. The simulation with a splitting factor of $\beta = 0.3$ ($\bar{u}_{\text{jet}} = 12.7 {\text{m}}^{1} {\text{s}}^{ - 1}$) required 26 iterations on a flowsheet level with a total computational time requirement of $\tau_{\text{ct}} = 5.6 \,{\text{s}}$ (ct—computational time) in Dyssol on one core of an Intel Core i7-7700 3.60 GHz processor. Consequently, even for complex flowsheets, the steady-state model combines reasonable accuracy with a high numerical efficiency.

Fig. 20

Reprinted with permission from [5]

Median of volume-based PSD ($L_{50,3}$) versus the splitting factor $\beta$ and two different mixing assumptions (ideal mixed and micro mixing model by [4]).

3.2 Validation of the Equivalent Circuit Model Concept

The two-zone experiments (Setup D, Sect. 2.2.2.2) were performed to verify whether the assumption of a well-mixed stirred tank connected to a PFR is, in principle, a suitable analogon for depicting the semi-batch process. As Bałdyga and Bourne [24] did not selectively investigate this aspect before using this model idea for simulation, we performed an “experimental simulation” by comparing particle sizes from the bulk semi-batch and corresponding two-zone equivalent circuit experiments. This method allows us to approximate the error which originates from this model concept. The corresponding value of $Q_{{{\text{circ}},1}}$ was calculated for a given stirring rate N of the bulk process by using the method published in [29]. The two-zone experiment was then performed according to the procedure described in Sect. 2.2.2.2 over a wide range of process conditions.

Figure 21 illustrates an exemplary result using the example of two process conditions, firstly, with $150\,{\text{rpm}}$ and $u_{\text{prim}} = 0.39\,{\text{m}}/{\text{s}}$ and, secondly, with $50\,{\text{rpm}}$ and $u_{\text{prim}} = 0.13\,{\text{m}}/{\text{s}}$. The resulting PSDs are quite similar with slightly larger particles generated in the two-zone process. This effect of slightly larger particles was observed over a wide range of process conditions and two different stirrer types in [29]. Therefore, it can be concluded that the equivalent circuit assumption and the correlation for $Q_{{{\text{circ}},1}}$ introduces a small error on the final PSD. However, this error is not significant in the context of process flowsheet simulations, where the simplified model assumptions only allow for a coarse estimation of the target variables. Consequently, replacing the semi-batch process by the equivalent circuit of PFR and well-mixed BF is a suitable modeling strategy for process flowsheet simulation.

Fig. 21

Reprinted with permission from [29]

The volume-based PSD $(q_{3} )$ of bulk (Rushton turbine stirrer) and corresponding JICF two-zone experiments for two different rotational speeds and feed velocities.

3.3 Dynamic Simulation

Results from semi-batch process simulation are compared to experimental data in Sect. 3.3.1. Furthermore, the numerical efficiency of the model of the newly developed approximation method used is demonstrated. Section 3.3.2 provides results regarding the influence of the semi-batch process dynamics for steady-state boundary conditions. Section 3.3.3 illustrates how dynamic boundary conditions might be used to optimize the product PSD for semi-batch processes in the future.

3.3.1 Validation and Numerical Efficiency

An exemplary result from validation of the semi-batch model for different stirrer rotational speeds is given in Fig. 22. As observable, the simulation predicts the experimental results well, with an error for $L_{50,3}$ below $100\,{\text{nm}}$. We also tested different feed volume flow ratios, with the result that higher deviations between model and experiments occurred for $Q_{\text{prim}} \ge 0.3\,{\text{L}}/{ \hbox{min} }$. Consequently, we did not investigate these process conditions further, as a model refinement is required to depict high feed volume flows correctly.

Fig. 22

Median of volume-based PSD $(L_{50,3} )$ versus impeller speed for experiments and Setup C simulation

The computational time of the model depends mostly on $f_{\text{rec}}$, the PSD discretization (${\Delta }L$, $L_{ \text{max} } - L_{ \text{min} }$) and time discretization $\Delta t$. All influencing factors were investigated separately to ensure that none of them influences the results significantly. Reliable results can be gained with 100 equally distributed particle size classes (${\Delta }L = 22\,{\text{nm}}$) from $L_{ \text{min} } = 22\,{\text{nm}}$ to $L_{ \text{max} } = 2.2\,{\mu m}$ and $\Delta t = 0.5$. These values are, therefore, used as default values for all simulations.

The recalculation frequency $f_{\text{rec}}$, which is part of the approximation method described in Sect. 2.3.2.3, is the main reason for the outstanding numerical performance of our surrogate model compared to mechanistic models from literature. An exemplarily chosen case with Simulation Setup C, $Q_{\text{prim}} = 0.2\,{\text{L}}/{ \text{min} }$, $N = 100\,{\text{rpm}}$ illustrates the massive improvement of numerical efficiency if the PFR is not recalculated in every timestep. Without the approximation method $(f_{\text{rec}} = 1)$, this simulation case requires approximately $\tau_{\text{ct}} \approx 58\,{ \hbox{min} }$ on a single core of an Intel Core i-7 7700 CPU with 3.60 GHz. As shown in Fig. 23, the calculation speed can be improved by several orders of magnitude by using a lower value for $f_{\text{rec}}$. Interestingly, even for $f_{\text{rec}} = 1/30,$ which requires $\tau_{\text{ct}} \approx 2.1\,{ \hbox{min} }$ instead of $\tau_{\text{ct}} \approx 58\,{ \hbox{min} }$, the accuracy of the model is only insignificantly diminished.

Fig. 23

Median of volume-based PSD ($L_{50,3}$) for recalculation frequencies of $f_{\text{rec}} = \frac{1}{2}$ to $f_{\text{rec}} = \frac{1}{80}$. Simulations performed at $Q_{\text{prim}} = 0.2\,{\text{L}}/{ \hbox{min} }$, $N = 100\,{\text{rpm}}$ and, $\Delta t = 0.5\,{\text{s}}$ with Setup C simulation. Computational time $\tau_{\text{ct}}$ referred to a single core of an Intel Core i-7 7700 CPU with 3.60 GHz

3.3.2 Process Dynamics for Steady-State Boundaries

A standard semi-batch operation with static feed rate and impeller rotational speed shows a dynamic PSD evolution, for which several mechanisms are responsible: Firstly, the reduction of reactive ion concentration by dilution and solids formation reaction decreases the supersaturation level in the PFR over time. Secondly, particles already present in the reactor can pass the PFR additional times and grow more. At a later stage of the process, these recycled particles can massively decrease the supersaturation in the PFR if a specific amount and size of particles is reached to offer the particle surface for a fast depletion of reactive ions in the PFR.

Figure 24 illustrates a typical PSD evolution using the example of a Setup C simulation with $Q_{\text{prim}} = 0.2\,{\text{L}}^{1} { \hbox{min} }^{ - 1}$ and $N = 100\,{\text{rpm}}$. The total process time of this simulation is $\tau_{\text{pro}} = 27\,{ \hbox{min} }$. A primary peak can be observed at the start of the process. These are the first particles precipitated in the PFR. During the process, the PSD shifts to larger particle sizes with a secondary peak observable.

Fig. 24

Temporal evolution of PSD for Setup C simulation with $Q_{\text{prim}} = 0.2\,{\text{L}}/{ \hbox{min} }$ and $N = 100\,{\text{rpm}}$

3.3.3 Dynamic Optimization by Dynamic Boundary Conditions

Further analysis of $S_{a} \left( {z,t} \right)$ and of $n \left( {z,t} \right)$ confirm that supersaturation decreases over the process time. As this is assumed to be the main reason for the widening of the PSD over the process time, Setup E was used to investigate whether this specific process dynamics can be counteracted by increasing the stirring rate over time. As higher mixing intensities lead to the generation of smaller particles in the PFR, this effect could be of possible use to counteract the increase of the particle size due to lower supersaturation. We, therefore, used a dynamic stirring rate for Simulation Setup E, which was increased from 100 to 300 rpm, as shown in Fig. 14.

A narrower PSD can be reached by dynamically increasing the stirring rate, as observable in Fig. 25. The overall effect is not significant compared to the experimental reproducibility due to the process condition chosen. However, this simulation illustrates how dynamic optimization might be performed by using dynamic process parameters for model-based process control.

Fig. 25

Variation of static stirring rate (100, 300 rpm) and linear increase (dynamic) from 100 to 300 rpm impacting the final PSD (Simulation Setup E)

4 Conclusion

In this contribution, a steady-state and a dynamic semi-batch surrogate model for precipitation of sparingly soluble salts were presented. Both models reach the numerical efficiency required for application in flowsheet process simulations.

The steady-state model solves the coupled mixing and solids formation process along a z coordinate of the mixer (plug flow assumption). The mixing process is modeled by the interaction of different fluid environments according to the E-model by Bałdyga and Bourne [24]. The PSD is calculated by solving a PBE under consideration of nucleation and diffusion-limited particle growth. The steady-state model was implemented in Dyssol and validated with two Experimental Setups: A simple stand-alone precipitation experiment and a complex flowsheet with a recycle stream connecting inlet and outlet to investigate the influence of recycle streams on the PSD. Experimental validation proved that the model predicts the experimental outcomes well, also for the complex flowsheet with recycle streams involved. Furthermore, we were able to investigate the influence of the recycle stream ratio on the PSD both numerically and experimentally.

The dynamic semi-batch model is based on a semi-batch model concept by Bałdyga and Bourne [24], who divided the semi-batch stirred tank in a well-mixed BF and a PFR reactor as the mixing and reaction zone. As Bałdyga and Bourne [24] did not experimentally proof the idea of a PFR-BF equivalent circuit, we developed the concept of an “experimental simulation,” which compares experimental PSDs of the equivalent circuit to the PSD resulting from semi-batch bulk experiments. Although there is a small error introduced by the equivalent circuit concept, this concept proved to be a suitable simulation strategy for process flowsheet simulation. To implement the model, the steady-state model (with minor adaptations) was used to solve the PFR with high computational speed. It was, furthermore, possible to increase the computational speed of the model significantly by developing a hybrid modeling technique. Within this hybrid model design, which can be applied for every mechanistic model using a BF-PFR equivalent circuit in literature, the PFR is not recalculated on each iteration but, instead, most of the timesteps are approximated by simpler equations. Simulation time scales suitable for dynamic process flowsheet simulation can be reached by utilizing this new approximation method. The final dynamic semi-batch model was implemented in Dyssol. Except for high feed volume flows, the model predicted the experimental data well. The dynamic process simulations show that the wide PSD obtained by semi-batch precipitation originates from the semi-batch process dynamics. As an outlook to future work, it is, furthermore, demonstrated that dynamic process parameters might be used to optimize semi-batch precipitation processes.