Question 6 of 6: Control of Copolymer Composition in a Semi-Batch Reactor

Question 6: Control of Copolymer Composition in a Semi-Batch Reactor (20 points)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

This is a composition-uniformity problem: in a batch copolymerization the two monomers react at different rates, so the instantaneous copolymer composition drifts as the faster monomer is depleted, producing a compositionally heterogeneous (blocky, broad) product. The task is to hold the instantaneous copolymer composition F₁ constant by continuously replacing the more reactive monomer as it is consumed — a semi-batch “starved / power-feed” policy — and to do so under a controller that can infer, in real time, how fast the reaction is running.

Information required. (i) The monomer reactivity ratios r₁ and r₂, which set the instantaneous composition through the Mayo–Lewis equation$$F_1=\frac{r_1f_1^2+f_1f_2}{r_1f_1^2+2f_1f_2+r_2f_2^2}$$where f₁, f₂ are the mole fractions of the two monomers in the reactor. (ii) The target copolymer composition F₁^* and the corresponding reactor feed composition f₁^* obtained by inverting Mayo–Lewis. (iii) The polymerization rate law and initiator decomposition kinetics (so the consumption rate of each monomer can be predicted). (iv) The heat of copolymerization ΔH_r and the jacket/reactor heat-transfer parameters (UA, flow, temperatures), so reaction rate can be inferred calorimetrically. (v) An online or inferential composition/conversion measurement (reaction calorimetry, densitometry, on-line GC or in-line spectroscopy).

Design of the control system. The scheme is a feedforward–feedback (cascade) structure:

1. Set the target and invert Mayo–Lewis. Choose the desired uniform composition F₁^*, then solve the Mayo–Lewis equation for the reactor mole fraction f₁^* that yields it. Holding f₁ at f₁^* throughout the run guarantees a constant instantaneous composition.
2. Feedforward feed rate. To keep f₁ constant, the more reactive monomer must be added at exactly its consumption rate. That rate equals the overall polymerization rate times the instantaneous mole fraction of monomer 1 entering the chain, both computable from the kinetics once the rate R_p is known — this is the model-based feedforward signal.
3. Calorimetric soft sensor. With conversion negligible during heat-up, the running reaction rate is obtained from the jacket energy balance: the heat removed by the cooling water, $Q_{rem}=\dot m_w c_p\Delta T$, equals the reaction heat $(-\Delta H_r)R_pV$ (allowing for accumulation), so R_p — and hence conversion and composition — is estimated in real time from the same cooling-water measurements the temperature loop already uses.
4. Feedback trim. Compare the estimated (or measured) composition with F₁^* and use a PI controller to trim the feedforward feed rate, correcting for model error, impurities and drift. The existing jacket temperature loop (steam→cooling-water throttling) runs underneath as the inner cascade, keeping temperature — and therefore r₁, r₂ and the kinetics — constant so the composition model stays valid.

In short: hold temperature constant with the jacket cascade so the reactivity ratios are fixed; compute the required reactive-monomer feed from Mayo–Lewis plus the calorimetrically-estimated rate (feedforward); and close a slower composition feedback loop to reject model error. The result is a copolymer of uniform composition despite the intrinsic reactivity difference.