ABSTRACT
This paper reviews the current literature on extraction column simulation when using droplet population modelling. In that regard, numerical methods will be briefly discussed and background information to necessary kernels and correlations as well as parameter evaluation will be presented. This comprises physical and chemically enhanced mass transfer, droplet movement and interactions in stirred columns. Furthermore, the benefits of CFD calculations to resolve local hydrodynamic details or ultra-fast simulation concepts for model-based forward control are covered. Finally, two case studies for an RDC and Kühni extraction column provide in detail the methodology on how to handle a design case which in principle can be extended to other extraction apparatuses.
Nomenclature
Roman symbols
Ac | = | Column cross-sectional area (m2) |
bn | = | Adjustable parameters for breakage models n =1, 2, 3 (-) |
B, D | = | Birth and death source terms in Equation (2) (m−3 s−1) |
c | = | Solute concentration (kg m−3) |
c | = | Continuous phase (-) |
cn | = | Adjustable parameters for coalescence models n =1, 2, 3, … (-) |
CD | = | Drag coefficient (-) |
CKH | = | Constant in Equation (19)&(20) (-) (116) |
Cσn | = | Interfacial tension constants in Equation (114) (-) |
d | = | Dispersed phase (-) |
d | = | Droplet diameter (m) |
d30 | = | Volume mean drop diameter (m) |
d32 | = | Sauter mean drop diameter (m) |
dM | = | Mother droplet diameter (m) |
dN | = | Inner orifice (nozzle) diameter (m) |
dsw | = | Henschkes’ velocity model parameter (rigid-internal circulation) (m) |
dη | = | Droplet diameter with dominating viscous forces (m) |
dσ | = | Droplet diameter with dominating surface tension (m) |
dΤ | = | Droplet diameter with dominating inertia forces (m) |
D | = | Diffusion coefficient (m2 s−1) |
Dax | = | Axial dispersion coefficient (m2 s−1) |
DCol | = | Column diameter (m) |
DR | = | Rotor diameter (m) |
DS | = | Stator diameter (m) |
DSh | = | Rotating shaft diameter (m) |
f | = | Number density function (m−3) |
F | = | Function/objective function (-) |
FA | = | Buoyancy force (N) |
Fη | = | Viscous force (N) |
Fσ | = | Force by surface tension (N) |
FΤ | = | Force by inertia (N) |
g | = | Gravitational acceleration equal to 9.8 m s−2 (m s−2) |
h | = | Collision rate frequency (m3 s−1) |
Hc | = | Compartment height (m) |
Hcd | = | Hamaker coefficient (N m) |
HCol | = | Column height (m) |
k | = | Individual mass transfer coefficient (m s−1) |
kv | = | Slowing factor (-) |
K | = | Equilibrium constant (-) |
Kb | = | Learning parameters for the breakage term in Equation (13) (-) |
Kc | = | Learning parameters for the coalescence term in Equation (13) (-) |
Kdf | = | Consistency factor (-) |
Kε | = | Constant in Equation (62)&(63) (-) |
Koy | = | Overall mass transfer coefficient in the dispersed phase (m s−1) |
L | = | Length of column section (m) |
Ṁ | = | Mass flow rate (kg s−1) |
n | = | Stoichiometric constant (-) |
n | = | Number of droplet classes (-) |
nblade | = | Number of rotor blades (-) |
ndf | = | Flow index (-) |
ni | = | Number density of droplets in class i with diameter di (m−3) |
nu | = | Variable exponent in Equation (44).(-) |
N | = | Rotor speed, number of droplets (s−1) |
Nc | = | Number of stirred compartments (-) |
Np | = | Number of pivots/particles (-) |
Ny | = | Number concentration (m−3) |
o | = | Number of data (-) |
p(d) | = | Breakage probability (-) |
pc | = | Coalescence probability (-) |
P | = | Vector of chemical properties in (-) |
P | = | Power (kg m2 s−3) |
Pd | = | Drop size density distribution function m−1) |
q1, q2 | = | Constants in Equation (99) (-) |
qn | = | Adjustable parameters for daughter droplets number n=1, 2, 3, (-) |
Q | = | Volumetric flow rate (m3 s−1) |
r, r’ | = | Radius of droplet (m) |
Ṙ(z,t) | = | Velocities for internal coordinate (m−3 s−1) |
sn | = | Adjustable parameters for slowing factor models n=1, 2, 3. (-) |
S | = | Source term (m−3 s−1) |
t | = | Time (s) |
u | = | Droplet velocity (m s−1) |
uN | = | Velocity in the orifice (m s−1) |
us | = | Superficial velocity (m s−1) |
V,v | = | Volume (m3) |
VT | = | Compartment volume (m3) |
x, y | = | Weight fraction of solute in the referred phase (kg kg−1) |
Ẋ(z,t) | = | Velocities for external coordinate (m−3 s−1) |
z | = | Space coordinate (m) |
Greek lettters
αsw | = | Steepness of crossover parameter of Henschkes’ velocity model (-) |
αdef | = | Parameter of Henschkes’ velocity model of deformed droplets (-) |
α15 | = | Parameter of Henschkes’ velocity model of oscillating droplets (-) |
α16 | = | Characterize the sharpness of the transaction between velocities (-) |
βn | = | Daughter droplet distribution based on droplet number (m−1) |
ϑ | = | Mean number of daughter droplets (-) |
Γ | = | Breakage frequency (s−1) |
δ(z-zy) | = | Dirac delta function (m−1) |
Δ | = | Increment (-) |
ε | = | Energy dissipation (mechanical power dissipation per unit mass) (m2 s−3) |
κf | = | Forward reaction constant (-) |
κr | = | Reward reaction constant (-) |
η | = | Dynamic viscosity kg m−1 s−1) |
λ | = | Coalescence efficiency (-) |
λη | = | Viscosity ratio of dispersed to continuous phase (-) |
ν | = | Kinematic viscosity (ν= η/ρ) (m2 s−1) |
ρ | = | Density (kg m−3) |
σ | = | Surface tension (N m−1) |
τm | = | Residence time (s) |
φs | = | Relative free cross-sectional stator area (m2 m−2) |
ϕ | = | Holdup/volume fraction (volume concentration) (-) |
ψ | = | Internal and external coordinates vector [d cy z t] (-) |
ω | = | Coalescence rate (m3 s−1) |
ωu | = | Angular velocity (ωu= 2 π N) (s−1) |
Subscripts
Aq | = | Aqueous |
b | = | Breakage |
c | = | Coalescence, concentration |
cal | = | Calculated |
cir | = | With internal circulation |
crit | = | Critical |
d | = | Droplet |
def | = | Deformed |
exp | = | Experimental |
I | = | Species index |
in | = | Inlet (feed) |
k | = | Model number: 1, 2, 3, … |
max | = | Maximum |
min | = | Minimum |
n | = | Number of parameters: 1, 2, 3, … |
opti | = | Optimum |
org | = | Organic |
os | = | Oscillating |
r | = | Relative |
rig | = | Rigid drop condition |
s | = | Superficial |
sph | = | Spherical |
stab | = | Stable |
t | = | Terminal |
tot | = | Total |
x | = | Aqueous or heavy (continuous) phase |
y | = | Organic or light (dispersed) phase |
Superscript
. | = | Derivative with respect to time |
* | = | Equilibrium |
b | = | Breakage |
c | = | Coalescence |
d | = | Droplet |
in | = | Inlet |
List of dimensionless
= | Archimedes number | |
= | Bodenstein number of the continuous phase | |
= | Bodenstein number of the dispersed phase | |
= | Eötvös number | |
= | modified Froude number | |
= | Morton number | |
= | Power number (Newton number) | |
= | Ohnesorge number | |
= | Péclet number of the continuous phase | |
= | Péclet number of the dispersed phase | |
= | Reynolds number in | |
= | Reynolds number of rotor/agitator | |
= | Reynolds number of the continuous phase | |
= | Reynolds number of the dispersed phase | |
= | Schmidt number of the continuous phase | |
= | Schmidt number of the dispersed phase | |
= | Sherwood number of the continuous phase | |
= | Sherwood number of the dispersed phase | |
= | Weber number | |
= | Droplet Weber number | |
= | Droplet Weber number[Citation171] | |
= | Modified Weber number[Citation173] | |
= | Modified Weber number[Citation118] | |
= | Weber number for droplet formation[Citation156] | |
= | Modified Schlichting laminar Weber number[Citation169] |
List of abbreviations
ADQMOM | = | Adaptive Direct QMOM |
ba-w | = | butyl acetate (d)-water |
ba-a-w | = | butyl acetate (d)-acetone-water |
BVSQMOM | = | bivariate sectional QMOM |
CFD | = | Computational Fluid Dynamics |
CM | = | Classes Method |
Corr. | = | Correlation |
DN | = | Diameter Nominal |
DPBM | = | Droplet Population Balance Model |
DSD | = | Droplet Size Distribution |
EFCE | = | European Federation of Chemical Engineering |
EFPT | = | Extended Fixed Pivot Technique |
Equation | = | Equation |
Exp. | = | Experimental |
FPM | = | Finite Pointset Method |
FPQMOM | = | Fixed Pivot QMOM |
iso-w | = | isotridecanol (d)-water |
LFPQMOM | = | Local Fixed Pivot QMOM |
LLEC | = | Liquid-Liquid Extraction Column |
LLECMOD | = | Liquid-Liquid Extraction Column MODule |
MOM | = | Method Of Moments |
MOPOSPM | = | modified OPOSPM |
M-QMOM | = | Modified QMOM |
MPEM | = | Moving Particle Ensemble Method |
NQMOM | = | Normalized QMOM |
OMST | = | Online Monitoring and Simulation Tool |
OPOSPM | = | One Primary One Secondary Particle Method |
Opti. | = | Optimized |
PBE | = | Population Balance Equation |
PBM | = | Population Balance Model |
POPMOD | = | Population Balance Module |
PPBLab | = | Particulate Population Balance Laboratory |
RDC | = | Rotating Disc Contactor |
ReDrop | = | Representative DROPs |
Ref. | = | Reference |
rpm | = | revolution per minute |
SDPBE | = | Spatially Distributed Population Balance Equation |
Sim. | = | Simulated |
SQMOM | = | Sectional Quadrature Method Of Moments |
t-w | = | toluene (d)-water |
t-a-w | = | toluene (d)-acetone-water |
Acknowledgments
The authors wish to thank the German Science Foundation (DFG, Bonn) for financial support and PD Dr.-Ing. M. W. Hlawitschka for his fruitful discussions.