## 2 Description of the Integrated ABE Fermentation-Membrane Pervaporation Process  
The process considered in this investigation is the biochemical production of butanol via acetone-butanol-ethanol (ABE) fermentation. ABE fermentation is a biphasic biological production involving acidogenesis and solventogenesis. Acidic metabolites accumulate during the former phase and are subsequently assimilated into industrial relevant solvents such as butanol. Butanol is an important chemical precursor and is currently being produced commercially via the hydroformylation of propylene. ABE fermentation can potentially be used for commercial production of butanol instead of the aforementioned petrochemical-based process. This interest in the development of bio-based processing alternatives is motivated by the enviable characteristic of biobutanol as a renewable liquid fuel. The ABE fermentation process suffers from low butanol yield and productivity because of the co-production of acetone, ethanol, and organic acids. Moreover, the concentration of butanol in the final fermentation broth is limited to very low concentrations (typically \(<\)13 g/L) due to butanol toxicity to the producing bacteria [4]. Consequently, the recovery of ABE solvents is energy-intensive and costly. At such low concentrations, the energy requirement for product recovery is greater than the energy content of the product itself [5]. Significant experimental efforts have been made to optimize ABE fermentation for improved butanol yields. One such method is extractive fermentation,a continuous process in which fermentation and solvent separation are integrated, thereby partly alleviating product inhibition, reducing downtime associated with batch processing, and increasing overall butanol concentration. Researchers have proposed various mathematical models describing cell growth and variations in the concentration of metabolites over time, by accounting for their rates of production and consumption in the biochemical reaction pathway to adequately represent the experimental observations [6; 7]. These mathematical models of ABE fermentation provide cost- and time-effective simulations for fermentation design and optimization studies. Therefore, extractive fermentation can be adequately modelled, allowing researchers to further generate new testable hypotheses and proposals, which can significantly improve the overall process and identify optimal operating conditions. For the continuous integrated fermentation process, pervaporation is an attractive separation method that can be used to continuously recover butanol from the ABE fermentation broth [8]. Pervaporation is a membrane-based separation process of binary or multicomponent liquid mixtures, and, as its name implies, it involves permeation through a dense hydrophobic membrane and evaporation on the permeate side. The permeating vapor is then recovered by condensation. Polydimethylsiloxane (PDMS) membrane is regarded as a promising candidate for butanol separation, as it exhibits high and stable performance in butanol recovery from ABE solvents via pervaporation [9]. The continuous integrated fermentation-membrane pervaporation process that was optimized in this work is presented schematically in Figure 1. This process mainly consists of a continuous fermenter, a microfiltration unit, and a membrane pervaporation system. For a set of input operating conditions of the overall process, the dynamic mass balance of each individual component of the process is solved with time until a steady state is achieved. The main input variables of the process, i.e., the decision variables in the optimization problem, are the fermenter dilution rate, the inlet sugar concentration, the cell retention factor, and the membrane area of the pervaporation unit as dictated by the number of membrane modules and the number of membrane stacks per module.