Mathematical modelling and control of agitated extraction columns
Liquid-liquid extraction has long been known as a unit operation that plays an important
role in industry. This process is well known for its complexity and sensitivity to operation
conditions. This thesis presents an attempt to explore the dynamics and control of this process
using a systematic approach and state of the art control system design techniques.
The process was studied first experimentally under carefully selected. operation
conditions, which resembles the ranges employed practically under stable and efficient
conditions. Data were collected at steady state conditions using adequate sampling techniques'
for the dispersed and continuous phases as well as during the transients of the column with the
aid of a computer-based online data logging system and online concentration analysis.
A stagewise single stage backflow model was improved to mimic the dynamic op.eration
of the column. The developed model accounts for the variation in hydrodynamics, mass
transfer, and physical properties throughout the length of the column. End effects were treated
by addition of stages at the column entrances. Two parameters were incorporated in the model
namely; mass transfer weight factor to correct for the assumption of no mass transfer in the.
settling zones at each stage and the backmixing coefficients to handle the axial dispersion
phenomena encountered in the course of column operation. The parameters were estimate.d by ..
minimizing the differences between the experimental and the model predicted concentration
profiles at steady state conditions using non-linear optimisation technique. The estimated
values were then correlated as functions of operating parameters and were incorporated in·the ..
model equations. The model equations comprise a stiff differential~algebraic system. This
system was solved using the GEAR ODE solver. The calculated concentration profiles were
compared to those experimentally measured. A very good agreement of the two profi1es was'
achieved within a percent relative error of ±2.S%.The developed rigorous dynamic model of the extraction column was. \ls~d to derive
linear time-invariant reduced-order models that relate the input variables (agitatorsjJeed, .
solvent feed flowrate and concentration, feed concentration andflowrate) to the output
variables (raffinate concentration and extract concentration) using the asymptotic method of
The reduced-order models were shown to be accurate in capturing the dynamic
behaviour of the process with a maximum modelling prediction error of I %. The simplicity
and accuracy of the derived reduced-order models allow for control system design and
analysis of such complicated processes.
The extraction column is a typical multi variable process with agitator speed and
solvent feed flowrate considered as manipulative variables; raffinate concentration and extract
concentration as controlled variables and the feeds concentration and feed flowrate as
disturbance variables. The control system design of the extraction process was tackled as
multi-loop decentralised S1S0 (Single Input Single Output) as well as centralised MIMO
(Multi-Input Multi-Output) system using both conventional and model-based control
techniques such as IMC (Internal Model Control) and MPC (Model Predictive Control).
Control performance of each control scheme was. studied in terms oJ stahilit~, speed of
response, sensitivity to modelling errors (robustness), setpoint tracking capabilities ahd load
For decentralised control, multiple loops were assigned to pair .each manipulated
variable with each controlled variable according to the interaction analysis and other pairing
criteria such as relative gain array (RGA), singular value analysis (SVD) and Jacobi
eigenvalue criterion. Loops namely Rotor speed-Raffinate concentration and Solvent flowrateExtract
concentration showed weak interaction. Multivariable MPC has shown more effective
performance compared to other conventional techniques since it accounts for loops
interaction, time delays, and input-output variables constraints