CGP006- Advanced Process Design and Optimisation Loughborough University
B Benyahia

Advanced Process Design and Optimisation

To be submitted electronically via Learn by Friday 26th March 2021
before 16:00
Crystallisation is a key separation and purification technology in the pharmaceutical industry.
The driving force for crystallisation is supersaturation which is commonly generated by
cooling, addition of antisolvent or solvent evaporation. A batch cooling crystallisation process
is available to recover paracetamol from water. The initial concentration of paracetamol is
0.0256 g/g solvent and the initial temperature is 315 K. Supersaturation is generated by
cooling. Figure 1 illustrates the principle of cooling crystallisation. The system, which is initially
undersaturated at point A, is cooled down and crosses the solubility curve at point B. In fact,
nucleation does not occur at this stage and further cooling is required to reach the limit of the
metastable zone (point C), where the nucleation occurs spontaneously. The nuclei undergo
growth and as a result, supersaturation drops gradually bringing the system closer to the
solubility curve. Further cooling is then required to maintain higher supersaturation to help
the crystals grow larger while maximising solute (paracetamol) recovery. At the end, the batch
process is terminated at point D close to the solubility curve.

Figure 1. Phase diagram and principle of cooling crystallisation

The mathematical model of Paracetamol batch crystallisation is described below. The
standard method of moment is used for the population balance which captures all variations
in the population of crystals du to nucleation and growth. This population balance is critical
to predict the crystal size distribution over time.

= , (1)

= 0, (2)
CGP006- Advanced Process Design and Optimisation Loughborough University
B Benyahia

= 21, (3)

= 32 (4)

= − 32 (5)
Where is the ith moment and C the concentration of Paracetamol in water (g/g water).
is the shape factor ( = 0.24 ) and the density of the crystals ( = 1.296e6 g −3).
and are respectively growth and nucleation kinetics given by

= ( − ) (6)
= ( − ) (7)

Where , , , are the kinetic parameters and ( − ) is the supersaturation.
The concentration at saturation, (solubility curve) is given by the polynomial function
= 1.5846e − 5 × 2 − 9.0567e − 3 × + 1.3066 (8)

where T is the process (solution) temperature.
The mathematical model above allows the prediction of the key features of the products
obtained. For example, the number based mean crystal size (d) and the coefficient of
variation (CV) can be obtained by
= 1

= �2012 − 1 (10)

Several experimental measurements of the mean crystal size and concertation are available
in table 1 below.

Table1. Experimental measurements of the mean crystal size and concentration
Time (min) 0 20 32 45 60 80
Mean size (m) 0 1.91E-05 2.71E-05 3.02E-05 3.15E-05 3.30E-05
(g/g solvent) 0.0256 0.0221 0.0175 0.0149 0.0132 0.0125

CGP006- Advanced Process Design and Optimisation Loughborough University
B Benyahia
1. Simulate the process using random values of ( ∈ [2.018 , 5.020]) and
( ∈ [0.001 , 0.5]). Show and comment on your results.
[10 marks]

2. Estimate the kinetic parameters and using the experimental data available in
Table 1. Show that the prediction capability of the model is acceptable and use a
quantitative criterion to prove that the model fits well with the data.
We assume that the exponent parameters are given by = . and = .

[20 marks]

3. Develop a dynamic optimisation strategy to maximize the mean crystal size. Justify the
algorithm/method used and show and comment on your results.
Tips: to address the optimisation problem,
• Use at least 5 linear intervals for the dynamic cooling profile and use either the
temperature or the cooling rate as the vector of decision variables
• The final temperature of the batch crystallisation process must be 283 K
• Only cooling is allowed (no heating)
• The cooling rate bounds are: – 2 ℃/min and 0 ℃/min
[20 marks]

4. Using a similar approach as question 3, develop a dynamic optimisation strategy to
minimise the batch time.
[20 marks]

5. Develop a dynamic multiobjective optimisation (MOO) strategy by considering two
cost functions: maximising the mean crystal size and maximising yield. Formulate and
solve the MOO problem. Justify the algorithm/method used and show and comment
on the Pareto front. Suggest an operating point from the Pareto front and justify your
choice. Is it possible to consider an additional objective function? Justify your answer.

[30 marks]

Requirements: A written report with the worked solutions must be submitted electronically via
Learn by the submission deadline stated on CASPA. The report must include ORIGINAL figures
properly labelled and numbered (and cited in the text) with the mathematical formulation of all
optimisation problems. The minimum admissible font size is Times New Roman 11 or Arial 10,
single paragraph, minimum margins of 20 mm all round. There is a strict page limit of nine (8) A4
pages for the full report – any content beyond 8 pages will NOT be marked. Submit the report with
a front cover page including your student ID (which does not count to page limit). Please save your
file as a PDF file as “StudentID-CGP0062021” – this will help us saving time during marking. You
should submit all MATLAB codes as a zipped file under the same name “StudentID-CGP0062021”