Population Balances for Extraction Column Simulations—An Overview

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.

KEYWORDS:

• Liquid extraction
• column simulation
• population models
• OPOSPM
• parameter estimation
• correlations

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 Table 1 (-)
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

$\mathrm{A}\mathrm{r}=\frac{{\rho }_c\mathrm{\Delta }\rho g{d}^3}{{\eta }_c^2}$	=	Archimedes number
$B{o}_x=\frac{Lu}{D_{ax,x}}$	=	Bodenstein number of the continuous phase
$B{o}_y=\frac{Lu}{D_{ax,y}}$	=	Bodenstein number of the dispersed phase
$E\text{"}o=\frac{g\mathrm{\Delta }\rho d^2}{\sigma }$	=	Eötvös number
$F{r}_{N,\mathrm{\Delta }\rho }=\frac{u_N^2}{d_{N}g}{\left(\frac{{\rho }_y}{\mathrm{\Delta }\rho }\right)}^{5/4}$	=	modified Froude number
$Mo=\frac{g\mathrm{\Delta }\rho {\eta }_c^4}{{\rho }_c^2{\sigma }^3}$	=	Morton number
$N_P=\frac{P}{N^3{D_R}^5{\rho }_x}$	=	Power number (Newton number)
$Oh=\frac{{\eta }_y}{\sqrt{\sigma d{\rho }_y}}$	=	Ohnesorge number
$P{e}_x=\frac{du}{D_x}$	=	Péclet number of the continuous phase
$P{e}_y=\frac{du}{D_y}$	=	Péclet number of the dispersed phase
$\mathrm{R}{\mathrm{e}}_D=\frac{{\rho }_x{D_R}^2{\omega }_u}{{\eta }_x}$	=	Reynolds number in Table 8
$\mathrm{R}{\mathrm{e}}_{\mathrm{R}}=\frac{{\rho }_x{D_R}^{2}N}{{\eta }_x}$	=	Reynolds number of rotor/agitator
$R{e}_x=\frac{{\rho }_{x}du}{{\eta }_x}$	=	Reynolds number of the continuous phase
$R{e}_y=\frac{{\rho }_{y}du}{{\eta }_y}$	=	Reynolds number of the dispersed phase
$\mathrm{S}{\mathrm{c}}_{\mathrm{x}}=\frac{{\eta }_x}{{\rho }_x{D}_x}$	=	Schmidt number of the continuous phase
$\mathrm{S}{\mathrm{c}}_{\mathrm{y}}=\frac{{\eta }_y}{{\rho }_y{D}_y}$	=	Schmidt number of the dispersed phase
$\mathrm{S}{\mathrm{h}}_{\mathrm{x}}=\frac{k_{x}d}{D_x}$	=	Sherwood number of the continuous phase
$\mathrm{S}{\mathrm{h}}_{\mathrm{y}}=\frac{k_{y}d}{D_y}$	=	Sherwood number of the dispersed phase
$\mathrm{W}\mathrm{e}=\frac{{\rho }_x{D_R}^3{N}^2}{\sigma }$	=	Weber number
$\mathrm{W}{\mathrm{e}}_{\mathrm{d}}=\frac{{\rho }_{x}d{D_R}^2{N}^2}{\sigma }$	=	Droplet Weber number
$\mathrm{W}{\mathrm{e}}_{\mathrm{d},\mathrm{G}\mathrm{o}}=\frac{{\rho }_x{\epsilon }^{2/3}{d}^{5/3}}{\sigma }$	=	Droplet Weber number[171]
$W{e}_{\mathrm{m},\mathrm{B}\mathrm{a}}=\frac{{\rho }_{x}d{D_R}^2\left(N^2-{N_{crit}}^2\right)}{\sigma }$	=	Modified Weber number[173]
$\mathrm{W}{\mathrm{e}}_{\mathrm{m}\mathrm{o}\mathrm{d}}=\frac{{\rho }_x^{0.8}{\eta }_x^{0.2}d{D}_R^{1.6}{(2\pi )}^{1.8}(N^{1.8}-N_{crit}^{1.8})}{\sigma }$	=	Modified Weber number[118]
$\mathrm{W}{\mathrm{e}}_{\mathrm{N}}=\frac{u_N^2{d}_N{\rho }_y}{\sigma }$	=	Weber number for droplet formation[156]
$\mathrm{W}{\mathrm{e}}_{\mathrm{S}\mathrm{L}\mathrm{M}}=\frac{{{\rho }_x}^{0.5}{{\eta }_x}^{0.5}d{D}_R\left(N^{1.5}-{N_{crit}}^{1.5}\right)}{\sigma }$	=	Modified Schlichting laminar Weber number[169]

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.