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.
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- \( B \) :
-
Nucleation rate \( [ {\text{m}}^{ - 4} {\text{s}}^{ - 1} ] \)
- \( B_{\text{T}} \) :
-
Baffles size [m]
- \( C_{\text{T}} \) :
-
Stirrer off-bottom clearance [m]
- \( \tilde{c} \) :
-
Molar concentration \( [ {\text{mol}}^{1} {\text{m}}^{ - 3} ] \)
- \( C \) :
-
Nucleation kinetics constant [–]
- \( \overline{D}_{{{\text{ri}},{\text{sol}}}} \) :
-
Average diffusion coefficient of reactive ions (ri) in solvent (sol) \( [ {\text{m}}^{2} {\text{s}}^{ - 1} ] \)
- \( D_{\text{T}} \) :
-
Impeller diameter [m]
- \( d_{\text{prim}} \) :
-
Inner feed pipe diameter [m]
- \( E \) :
-
Engulfment constant \( [ {\text{s}}^{ - 1} ] \)
- \( f_{\text{rec}} \) :
-
Recalculation frequency [–]
- \( G \) :
-
Growth rate \( [ {\text{m}}^{1} {\text{s}}^{ - 1} ] \)
- \( H_{\text{T}} \) :
-
Feed pipe off-bottom clearance [m]
- \( J \) :
-
Nucleation kinetics factor \( [ {\text{m}}^{ - 3} {\text{s}}^{ - 1} ] \)
- \( K \) :
-
Solubility product \( [ {\text{mol}}^{2} {\text{m}}^{ - 6} ] \)
- \( k_{\text{B}} \) :
-
Boltzmann constant \( [ {\text{m}}^{2} {\text{kg}}^{1} {\text{s}}^{ - 1} {\text{K}}^{ - 1} ] \)
- \( L \) :
-
Particle diameter [m]
- \( \bar{L}_{{{\text{mol}},{\text{ri}}}} \) :
-
Average molecular diameter of reactive ions [m]
- \( L_{\text{crit}} \) :
-
Critical nucleation radius [m]
- \( L_{ \text{min} } \) :
-
Minimal particle size of PSD grid [m]
- \( L_{ \text{max} } \) :
-
Maximal particle size of PSD grid [m]
- \( L_{50,3} \) :
-
Median of volume-based PSD [m]
- \( {\Delta }L \) :
-
PSD grid spacing [m]
- \( M \) :
-
Mass [kg]
- \( \dot{M} \) :
-
Mass flow \( [ {\text{kg}}^{1} {\text{s}}^{ - 1} ] \)
- \( \tilde{M} \) :
-
Molar mass \( [ {\text{kg}}^{1} {\text{mol}}^{ - 1} ] \)
- \( m_{i} \) :
-
Mass of particles in class \( i \) [kg]
- \( N \) :
-
Stirrer rotational speed \( [ {\text{s}}^{ - 1} ] \)
- n :
-
Particle number density \( [ {\text{m}}^{ - 4} ] \)
- \( n_{{{\text{A}}/{\text{B}}}} \) :
-
Particle number density in zone A or B \( [ {\text{m}}^{ - 4} ] \)
- \( n_{\text{t}} \) :
-
Total particle density \( [ {\text{m}}^{ - 3} ] \)
- \( n_{\text{r}} \) :
-
Refractive index [–]
- \( Q \) :
-
Volume flow \( [ {\text{m}}^{3} {\text{s}}^{ - 1} ] \)
- \( q_{0} \) :
-
Number-based PSD \( [ {\text{m}}^{ - 1} ] \)
- \( q_{3} \) :
-
Volume-based PSD \( [ {\text{m}}^{ - 1} ] \)
- \( R \) :
-
Free lattice ion ratio [–]
- \( R_{\text{T}} \) :
-
Radial position of feed pipe [–]
- \( S_{a} \) :
-
Activity-based saturation [–]
- \( T_{\text{T}} \) :
-
Tank diameter [m]
- \( T \) :
-
Temperature [K]
- \( t \) :
-
Process time [s]
- \( \Delta t \) :
-
Semi-batch model timestep [s]
- \( \bar{u} \) :
-
Average velocity \( [ {\text{m}}^{1} {\text{s}}^{ - 1} ] \)
- \( V_{{{\text{mol}},{\text{s}}}} \) :
-
Molecular volume of solid [m3]
- \( V^{L} \) :
-
Liquid phase volume [m3]
- \( V_{\text{BF}} \) :
-
Bulk fluid volume [m3]
- \( w_{i} \) :
-
Particle mass fraction in particle size class i [–]
- x :
-
Mass fraction [–]
- \( x_{j}^{\text{L}} \) :
-
Component mass fractions in liquid phase [–]
- z :
-
Mixer length coordinate [m]
- \( \alpha_{\text{abs}} \) :
-
Absorption coefficient [–]
- \( \alpha_{k} \) :
-
Volume fraction of zone k [–]
- \( \beta \) :
-
Splitting factor/recycle ratio [–]
- \( \gamma_{ \pm } \) :
-
Activity coefficient [–]
- \( \gamma_{\text{sl}} \) :
-
Solid-liquid interface tension \( [ {\text{N}}^{1} {\text{m}}^{ - 1} ] \)
- \( \delta \) :
-
Dirac-delta function \( [ {\text{m}}^{ - 1} ] \)
- \( \bar{\varepsilon } \) :
-
Average energy dissipation \( [ {\text{m}}^{2} {\text{s}}^{ - 3} ] \)
- \( \mu \) :
-
Dynamic viscosity \( [ {\text{Pa}}^{1} {\text{s}}^{1} ] \)
- \( \nu \) :
-
Kinematic viscosity \( [ {\text{m}}^{2} {\text{s}}^{ - 1} ] \)
- \( \xi^{\text{S}} \) :
-
Mass fraction of solid phase [–]
- \( \xi^{\text{L}} \) :
-
Mass fraction of liquid phase [–]
- \( \upsilon_{\text{s}} \) :
-
Number of ions for solids formation [–]
- \( \vartheta_{{m,{\text{sf}}}} \) :
-
Stochiometric coefficients of solids formation reaction [–]
- \( \tilde{\rho }_{\text{s}} \) :
-
Molar density of solid [\( {\text{kg}}^{1} {\text{m}}^{ - 3} \)]
- \( \tau_{\text{mix}} \) :
-
Time scale of mixing [s]
- \( \tau_{\text{sf}} \) :
-
Time scale of solids formation [s]
- \( \tau_{\text{ct}} \) :
-
Computational time (real time) [s]
- i :
-
Particle size class index
- j :
-
Compound index in liquid phase (solved ions and solvent)
- k :
-
Mixing fraction index
- m :
-
Index for solved ions in liquid phase
- \( w \) :
-
Index for solvents in liquid phase
- 0:
-
Index for initial value \( (t = 0) \)
- sf:
-
Solids formation
- mix:
-
Mixing
- ri:
-
Reactive ions
- sol:
-
Solvent
- out:
-
Outlet
- mol:
-
Molecular
- L:
-
Liquid phase
- S:
-
Solid phase
- 1:1:
-
1:1 mixture of educt fluids for well-mixed condition
- BF:
-
Bulk fluid
- CFD:
-
Computational fluid dynamics
- CIJM:
-
Confined impinging jet mixer
- STR:
-
Stirred-tank reactor
- DLS:
-
Dynamic light scattering
- E-model:
-
Engulfment model
- MSMPR:
-
Mixed-suspension, mixed product-removal
- JICF:
-
Jet in cross flow
- PBE:
-
Population balance equation
- PFR:
-
Plug flow reactor
- PSD:
-
Particle size distribution
- SEM:
-
Scanning electron microscopy
- SLS:
-
Static light scattering
- Sh :
-
Sherwood number [–]
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Rehage, H., Kind, M. (2020). Dynamic Simulation of Technical Precipitation Processes. In: Heinrich, S. (eds) Dynamic Flowsheet Simulation of Solids Processes. Springer, Cham. https://doi.org/10.1007/978-3-030-45168-4_4
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DOI: https://doi.org/10.1007/978-3-030-45168-4_4
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