Dynamic modeling of syngas fermentation in a continuous stirred‐tank reactor: Multi‐response parameter estimation and process optimization

Abstract Syngas fermentation is one of the bets for the future sustainable biobased economies due to its potential as an intermediate step in the conversion of waste carbon to ethanol fuel and other chemicals. Integrated with gasification and suitable downstream processing, it may constitute an efficient and competitive route for the valorization of various waste materials, especially if systems engineering principles are employed targeting process optimization. In this study, a dynamic multi‐response model is presented for syngas fermentation with acetogenic bacteria in a continuous stirred‐tank reactor, accounting for gas–liquid mass transfer, substrate (CO, H2) uptake, biomass growth and death, acetic acid reassimilation, and product selectivity. The unknown parameters were estimated from literature data using the maximum likelihood principle with a multi‐response nonlinear modeling framework and metaheuristic optimization, and model adequacy was verified with statistical analysis via generation of confidence intervals as well as parameter significance tests. The model was then used to study the effects of process conditions (gas composition, dilution rate, gas flow rates, and cell recycle) as well as the sensitivity of kinetic parameters, and multiobjective genetic algorithm was used to maximize ethanol productivity and CO conversion. It was observed that these two objectives were clearly conflicting when CO‐rich gas was used, but increasing the content of H2 favored higher productivities while maintaining 100% CO conversion. The maximum productivity predicted with full conversion was 2 g·L−1·hr−1 with a feed gas composition of 54% CO and 46% H2 and a dilution rate of 0.06 hr−1 with roughly 90% of cell recycle.

origins; it may be, for example, (a) syngas produced via gasification of a wide range of feedstocks, including municipal solid waste and lignocellulosic biomass; (b) off-gas from steel production and cement industries; (c) CO 2 captured from power plants blended with H 2 from renewable electricity, generated via electrolysis; and (d) reformed biogas (Liew et al., 2016). There has been a great expansion of gas fermentation technology over the last few years; at least three commercial-scale ethanol plants are currently under construction (LanzaTech, 2018) or have started operation (China News Service, 2018), and many pilot plants have already been in operation for long periods of time (Liew et al., 2016). Different studies indicate that the process can play an important role in the development of a sustainable bioeconomy, being comparable to other lignocellulosic processes in terms of cost, energy efficiency, and environmental impact, while also permitting feedstock flexibility (Liew et al., 2016;de Medeiros, Posada, Noorman, Osseweijer, & Filho, 2017;Pardo-Planas, Atiyeh, Phillips, Aichele, & Mohammad, 2017;Roy, Dutta, & Deen, 2015). From the point of view of process systems engineering, however, there is still vast room for improvement, from strain enhancement and efficient product separation, to the integrated optimization of process parameters. With that in mind, in this article, we address specifically the syngas fermentation bioreactor, coveting the presentation and analysis of a model that can be useful in optimization frameworks.
Models to describe syngas fermentation are still scarce in the literature, and only a few authors have attempted to adjust kinetic expressions to experimental data. Younesi, Najafpour, and Mohamed (2005) and Mohammadi, Mohamed, Najafpour, Younesi, and Uzir (2014) adjusted logistic curves to the growth of Clostridium ljungdahlii on artificial syngas using experimental data from batch fermentation essays in serum bottles. Mohammadi et al. (2014) were also able to fit Gompertz equations to their experimental profiles of product formation, and uptake rate equations for CO, presenting estimations of kinetic parameters that were later adopted by Chen, Gomez, Höffner, Barton, and Henson (2015) in their dynamic Flux Balance Analysis (FBA) model of a syngas fermentation bubble column. The latter was the first application of FBA in a dynamic model for syngas fermentation and the first spatiotemporal model of this process, but it was not compared with experimental data. The same group also published an improved version of their model, applied for CO fermentation with Clostridium autoethanogenum and considering uptake parameters obtained and protected by LanzaTech (Chen, Daniell, Griffin, Li, & Henson, 2018). Furthermore, Jang, Yasin, Park, Lovitt, and Chang (2017) simulated CO fermentation in a batch culture of Eubacterium limosum KIST612 using a dynamic model with kinetic parameters previously estimated by Chang, Kim, Lovitt, and Bang (2001), but this process results in the formation of acetic acid as the only product, which has lower a value than ethanol.
In the present study, a dynamic model was constructed for syngas fermentation with ethanol production in a continuous stirred tank reactor (CSTR). The unknown model parameters were estimated with a multi-response minimization framework using experimental culture data from the literature and the significance of parameters was assessed with statistical analysis and generation of confidence intervals. The model was then used to study the effects of different process conditions (i.e., gas composition, dilution rate, gas residence time [GRT], and cell recycle), as well as the sensitivity of the kinetic parameters, and a multiobjective optimization was conducted for maximization of productivity and conversion. Although similar studies exist for other process, such as acetone-butanol-ethanol (ABE) fermentation (see e.g., Buehler & Mesbah, 2016), to our knowledge, there are no previous studies contemplating upon parameter estimation, statistical treatment, sensitivity analysis, and multiobjective optimization of syngas fermentation; therefore, this work was devised to fill this lacuna.

| MODEL DESCRIPTION
The dynamic model developed in this study describes a stirred tank with continuous supply of syngas and batch or continuous flow of liquid, with or without cell recycle. It accounts for two phases (G/L) and seven species-CO, H 2 , CO 2 , ethanol (C 2 H 6 O or EtOH), acetic acid (C 2 H 4 O 2 or HAc), water, and biomass-comprising 13 state variables which are the concentrations of the six chemical compounds in the gas C G,j (mmol/L) and in the liquid C L,j (mmol/L)-where j = CO, H 2 , CO 2 , EtOH, HAc, H 2 O-as well as the concentration of biomass in the liquid C X (g/L). Two types of input are provided to the modeling framework (a) kinetic parameters, which define the relations between biochemical reaction rates and concentrations of chemical species and cells-these parameters are estimated in this study; and (b) operating conditions, such as gas flow rate, dilution rate, agitation rate, and syngas composition-these are specified for each of the cases analyzed in this work and their effects are further evaluated.
Fitting the model parameters with literature data turned out to be a challenge due to several reasons. First, the number of experimental papers on syngas fermentation is relatively small compared with other types of fermentation; and an even smaller number provides data without coproduction of other chemicals, such as butanol and butanediol. Among these, some provide exploratory data of very long cultures in which several accidents or interventions occur, and others fail to provide clear information about the process conditions (e.g., often the gas flow rates are omitted from the text, probably because they were not fixed during the experiment). In the present work, the model parameters were estimated for five different case studies from three different papers (C1; Phillips, Klasson, Ackerson, Clausen, & Gaddy;Phillips, Klasson, Clausen, & Gaddy, 1993); (C2; Gaddy et al., 2007);(C3A,B,C;Maddipati, Atiyeh, Bellmer, & Huhnke, 2011). These case studies have in common the use of continuous supply of syngas mixtures in stirred tanks and the formation of acetic acid and ethanol as the only products. Table 1 presents the main differences between the five scenarios, apart from the liquid medium composition, which is omitted due to space limitations. It is worth noting that C2 actually consists of 35 steadystate points obtained under different conditions of gas composition, flow rates and agitation, while C1 and C3A,B,C comprise dynamic data. C3 is one case study subdivided in three, that is, all of the process conditions are the same, except for the concentration of yeast extract or corn steep liquor.
The next three subsections present the modeling approach for the specific production/consumption rates of species due to cell fermentation (Reaction rates); the mass balance equations considering in/out flows, gas-liquid mass transfer, fermentation, and cell recycle (Mass balance equations); and the calculation of special terms that appear in the mass balance equations (Calculation of special terms).

| Reaction rates
C. ljungdahlii and other acetogens assimilate CO, H 2 , and CO 2 through the Wood-Ljungdahl pathway to produce acetyl-CoA, which is then used to produce cell biomass and products, as schematized in In theory, acetyl-CoA reduction towards ethanol is possible with aldehyde dehydrogenase, but this route is always thermodynamically less favorable and actually infeasible if H 2 is the electron donor (Bertsch & Müller, 2015). Indeed, Richter et al. (2016) found with proteome analysis of C. ljungdahlii that ethanol was produced exclusively through the AOR route. Ethanol production is favored when acetate accumulates inside the cell due to growth limiting conditions (i.e., biomass cannot be produced) or due to low extracellular pH (Richter et al., 2016). In the latter case, undissociated acetic acid, which is prevalent under pH lower than 4.76 (acetic acid pKa), diffuses freely through the cell membrane due to its neutral charge; however it dissociates again in the cytosol where the pH is close to neutrality and it cannot be exported through the cell membrane without active transport processes (i.e., using cellular energy), thus leading to the accumulation of acetate and protons inside the cell. In C. ljungdahlii, Richter et al. (2016) reported that the enzymes needed for the synthesis of ethanol were always available in excess and, as reducing equivalents are constantly being provided by the oxidation of CO and H 2 (see Equations (1) and (2) catalyzed by carbon monoxide dehydrogenase and hydrogenase, respectively), the authors suggest that ethanol is formed as soon as undissociated acetic acid and reducing equivalents reach a threshold concentration required to make the reduction of acetic acid thermodynamically feasible.
With that in mind, we propose a kinetic model following the stoichiometry of the reactions presented in Equations (3)-(6), which intend to generally represent the chemical reactions catalyzed by the cell. The model accounts for the following assumptions: (a) The uptake rates of CO and H 2 follow Monod kinetics with inhibition by substrate and product (Equation (7)); (b) acetic acid and ethanol inhibit substrate uptake with standard inhibition kinetics (Equation (7b)), but ethanol inhibition is only activated after a threshold concentration is achieved; (c) CO inhibits the uptake of H 2 but not CO (Equation (7c))this was decided after preliminary estimation routines showed that a CO inhibition constant for CO uptake could not be estimated with the experimental data used here; (d) biomass growth is a function of the uptake rates of CO and H 2 (Equation (8)) and cell death (Equation (9)), and its composition is assumed constant; (e) acetic acid is produced from CO (Equation (3)) and H 2 /CO 2 (Equation (4)); (f) ethanol is produced exclusively through reduction of acetic acid (Equations (5) and (6)), with reaction rates that are hyperbolic functions of the acetic acid concentration, also mimicking Michaelis-Menten kinetics (Equation (10)); (g) the effects of pH are not directly included in the model, but it is assumed that the estimated values of the kinetic parameters associated with acetic acid uptake and reduction will reflect the pH conditions adopted in the experiments used for the parameter estimation. With these assumptions, we may calculate the specific reaction rates k R ν (mmol·g −1 ·hr −1 )-where k indicates the reaction's equation number, that is, Equations (3-6)-and the specific consumption/production rates of species j, j ν (mmol.g −1 .hr −1 ), where a negative sign in the value of j ν indicates that the species is consumed, otherwise it is produced.
The specific biomass growth rate μ (h −1 ) is then calculated from these uptake rates via yield coefficients Y X,j (g/mol) for both substrates as shown in Equation (8). Although H 2 is not a source of carbon, it is coupled with the consumption of CO 2 and it has also been shown to be associated with the growth rate (Mohammadi et al., 2014). The death rate r d is a function of cell concentration as shown in Equation (9), where k d is the death constant estimated in this study. It is worth noting that, with this equation, the growth rate is also affected by the concentration of inhibitors (ethanol, acetic acid, and CO), and the effects of other nutrients and maintenance issues are expressed in the yield coefficients and the death constant.
The reaction rates of acetic acid reduction (AcR), that is, k R ν for k = 5 and 6, are calculated with Equations (10a) and (10b), where the parameters j max, AcR ν and K s j , AcR (j = CO, H 2 ) are estimated in this study.
The condition in Equation (10a) should be read as "for j = CO and k = 5, or for j = H 2 and k = 6." The expressions F AcR,j are only used to make the equations clearer; they are not model parameters. The idea behind this set of equations is that acetic acid is reduced with hyperbolic kinetics limited by its concentration (Equation (10b)), and the consumption rate of CO or H 2 necessary to provide reducing equivalents to these reactions is bounded by the total uptake rates previously calculated from Equation (7); thus it can be easily verified tends to the expression F AcR,j when the uptake of CO or H 2 is significantly larger than F AcR,j , whereas it tends to −ν j /2 when |ν j /2| is smaller than F AcR,j (the division by 2 is due to the stoichiometric coefficient of CO and H 2 in Equations (5) and (6)).
The remaining substrate that is consumed can then be assumed to be used in Equations (3) and (4), and the corresponding reaction rates are calculated from Equation (11), where j AcR, ν is the reaction rate of AcR (Equations (5) or (6)) using substrate j (i.e., CO or H 2 ), for example, , AcR CO ν in Equation (11) corresponds to R 5 ν as calculated from Equation (10). The total consumption/production rates of other components then follow the stoichiometry of Equations (3-6) as calculated with Equations (12-15).
F I G U R E 1 Schematic representation of syngas fermentation metabolism in Clostridium ljungdahlii under acidic pH, including acetic acid diffusion through the cell membrane. In this study, ethanol formation is considered possible only via the AOR pathway. ALDH, aldehyde dehydrogenase; AOR, aldehyde ferredoxin oxidoreductase For NC species j in the liquid phase, For the biomass concentration (in the liquid phase): The gas-liquid equilibrium factors ∈ m j NC , ∈ m j C in Equations (16-20) are described in Equations (21) and (22), where R = 8.314 Pa·m 3 /mol·K is the ideal gas constant; MM L and ρ L refer to liquid phase molar mass (kg/mol) and density (kg/m 3 ) assumed pure water at 36°C; and the respective physical parameters-Henry's law constants H j (Pa), saturation pressures P sat,j (Pa) and infinite-dilution activity coefficients ∞ j γcan be found in the Table S1. V L and V G are the volumes (L) of liquid and gas inside the reactor; Q G,in and Q G,out are the gas volumetric flow rates (L/hr) in/out the vessel, with the latter calculated as described in Calculation of special terms; k L a j are mass transfer coefficients calculated as described in Calculation of special terms; Q L is the liquid volumetric flow rate (L/hr). The specific rates ν j , μ, and r d were presented in Reaction rates and are calculated accordingly at each time point; subscript in refers to inlet gas and liquid concentrations; and XP is the cell purge fraction, that is, the fraction of cells that are not recycled to the vessel.

| Calculation of special terms
Certain terms that appear in the right-hand side of the ordinary differential equations (ODEs), but which are not state variables, are calculated as explained in the following.
2.3.1 | Outlet volumetric gas flow rate Q G,out Q G,out is calculated from a mole balance in the gas phase considering isobaric conditions inside the vessel. Taking into account the mass transfer of NC species (j ∈ NC) from gas to liquid and the mass transfer of C species (j ∈ C) from liquid to gas, the total gas mole flow rate leaving the reactor is calculated at each time with Equation (23). Q G,out is then calculated with the assumption of ideal gas in Equation (24).

| Mass transfer coefficients
The mass transfer coefficient k L a for air in water at T = 36°C is calculated via Equations (25) (Equations (25c) and (25d)), where P g /V L is the impeller power per unit volume, which is estimated from the impeller ungassed power P ug (Equation (26)) and the correlation for the ratio P g /P ug in Equations (27) (Cui, Van der Lans, & Luyben, 1996). The weighting factor f 0 is an unknown parameter which is estimated in this study. In Equations (25) In all cases, the reactor is assumed to have a height/diameter ratio of 2 and an impeller diameter of 40% the reactor diameter, as standard in New Brunswick Bioflo bioreactors.
The individual k L a j for each species is then obtained from the reference air-water k L a by applying the penetration theory as in Equation (28) (Talbot, Gortares, Lencki, & de la Nouë, 1991), where Df j is the mass diffusivity of species j in water (Table S1).

| NUMERICAL METHODS
The dynamic fermentation model described by the ODEs, Equations (16)-(20), and its supplemental algebraic equations in Model description represent a nonlinear algebraic-differential system which demands specialized numerical solvers for stiff problems. In the present case, the ode15s variable-order method from MATLAB was used for time integration from a feasible initial condition, given the appropriate value of the vector of model parameters in Equation (29).
The β vector of parameters (N P x 1,N P = 15) comprises the 14 kinetic parameters explained in Reaction rates, as well as the k L a weighting factor f 0

| Estimation of model parameters
The unknown model parameters β were estimated asβ using the maximum likelihood principle (MLP; Himmelblau, 1970), with the experimental data from the case studies presented in Table 1 where r j,i are known response-experiment factors and 2 σ ε is the unknown fundamental variance (Himmelblau, 1970).
With Equation (30) It can also be shown (Himmelblau, 1970) with assumptions (A1), initial point was known. In both cases, sensible lower and upper bounds were stipulated forβ. These bounds are displayed in The factors r j i , (j = 1…N R , i = 1…N E ) of the variance model of experimental responses in Equations (30) and (31) The experimental response values were read from the dynamic profiles (C1 and C3) or steady-state outcomes (C2) reported in the case studies considered here (see Table 1). For C1 and C3, the

| Significance of parameters
The confidence intervals of the estimated parametersˆk β were calculated with Equation (33), where t 1 -α/2 is the abscissa at with this work. The F test to reject the null hypothesis (i.e., parameter β k is significant) with 95% probability is given (Himmelblau, 1970) in is the abscissa at ( − )⋅ 1 100% α probability (α = .05) of the Fisher PDF with degrees of

| Steady-state sensitivity and multiobjective optimization
After the estimation of kinetic parameters, the model was used to study the effects of several process conditions on the steady-state productivity of ethanol (i.e., C L,EtOH • D rate ). The steady states were obtained by integrating the ODE system until all the state variables showed absolute gradients smaller than 10 −6 . This procedure was found to be faster than solving the system of nonlinear algebraic equations, as this required the initial guesses to be very close to the actual solutions. It can also be shown that, for a wide range of initial conditions, the steady state was stable and independent of such specifications (phase-portraits depicting the dynamic trajectories are presented in Figure S2), therefore an arbitrary set of initial conditions equal to those of case study C1 was used. With this framework, the sensitivity was analyzed with respect to the gas composition (varying the molar fractions of CO and H 2 ), the GRT, the dilution rate (D rate ), and also to the kinetic parameters under different conditions of GRT and D rate . On the basis of these results, the process was optimized using multiobjective genetic algorithm for the maximization of two conflicting objectives: Ethanol productivity and CO conversion. The decision variables were three operating conditions (GRT, D rate , and XP-cell purge fraction) and nine kinetic parameters, which could possibly be tuned with the design of the nutrient medium, the choice of strain and/or genetic engineering. In a last study, the H 2 :CO ratio was also included as a decision variable.
For this optimization routine, the bounds were specified based on the ranges of kinetic parameters estimated for the five case studies (see Table S3).

| Parameter estimation and confidence intervals
The full parameter vector ̲ β in Equation (29) was first estimated with the data from C1. Since C2 employed the same strain, the maximum uptake rates (ν max ), saturation constants (K s ), and inhibition constants (K I ) were fixed and the remaining eight parameters were reestimated with the steady-state data from C2. It should also be said that the ethanol inhibition term was excluded from this case study since the reported data did not achieve the threshold concentration Case C2 (steady state) showed the highest deviations from the experimental data as well as parameter uncertainties, which is probably due to the large range of process conditions encompassed by the data. It should also be noted that it is unclear whether the medium composition was kept fixed or not during these experiments. Nonetheless, the model was still good at capturing the tendency of the data, especially the concentrations of products and cells. In comparison with C1, which used the same strain, the AcR parameters were more favorable to ethanol production, that is, with higher 2 were similar if we consider the confidence intervals.
The last parameter, f 0 , indicates the level of coalescence in the liquid, with higher f 0 (as in cases C1 and C2) meaning the liquid is highly noncoalescing and thus enables higher gas-liquid mass transfer coefficients. It is worth noting that f 0 increased from Case C3A to C3C and specially from C3A to C3B (when 1 g/L yeast extract was replaced with 10 g/L corn steep liquor).

| Sensitivity of process conditions and kinetic parameters
With the fitted models, the performance of the bioreactor was evaluated for different conditions of gas composition, dilution rate and gas flow rates. For these sensitivity analyses, the parameter vector estimated in C2 was used as basis. The effects of syngas composition are depicted in Figure 5 for ethanol productivity and CO conversion. It can be seen that both responses are enhanced with the CO content, but there is a maximum outcome at H 2 :CO close to 1 and the peak is slightly dislocated to the left (higher H 2 :CO) for CO conversion. This result suggests that the syngas composition can be tuned to improve the performance of the bioreactor, but the optimal composition would, of course, depend on the balance between extra productivity/conversion in the bioreactor and extra energy costs in upstream operations (gasification and gas conditioning). It was also observed that cell mass concentration always increased with the fraction of CO, going from near 1 g/L at low values up to 11 g/L with pure CO (figure shown in Figure S5).
Assuming fixed gas composition of a CO-rich gas, the response surfaces shown in Figure 6 were generated to illustrate the effects of dilution rate (D rate ) and GRT with cell recycle (10% purge) and without. Both cases demonstrate how lower values of GRT (i.e., higher gas flow rates) enhance the productivity due to higher supply of substrate as well as higher gas-liquid mass transfer coefficient, although at the expense of the CO conversion. Moreover, as typically observed in chemostat cultures, the productivity is a concave function of the dilution rate with a clear maximum-in this case, also dependent on the gas flow rate.
From Figure 6, it is also clear that cell recycle enhances the ethanol productivity (the maximum increases from around 1.13-1.44 g·L −1 ·hr −1 ) and broadens the region of operation without cell wash-out. Moreover, the maximum cell mass concentration increases from 4.5 to around 16 g/L when cell recycle is used (response surfaces shown in Figure S6). The effects of agitation are not shown here, but response surfaces with this variable can be found in Figure S3. Evidently, increasing the agitation rate also enhances the mass transfer of CO and H 2 between the gas and the liquid, which allows for higher conversions and ethanol productivity; however, at the price of higher energy consumption. The same response surfaces were constructed for H 2 -rich gas (see Figure S4), which had overall the same shape and tendencies as Figure 6.
In accordance with Figure 5, it was also observed that increasing the

| Optimization of ethanol productivity and CO conversion
The solutions to three optimization runs are shown in Figure 8.
The multiobjective optimization was first solved for three operating conditions (GRT, D rate , and XP) and nine kinetic parameters (all excepting the saturation and inhibition constants, which were fixed at the values obtained for C2). The lower and upper bounds were chosen based on the intervals of the parameters estimated in Table 2 (see Table S3); for the operating conditions, the GRT was free to vary in the range 5-50 min, D rate in the range 0.005-0.2 hr −1 and XP in the range 0.1-1 (this meaning no cell recycle). In the first run, the gas composition was fixed for a CO-rich gas, that is,  Finally, a third optimization was conducted adopting the H 2 :CO ratio in the feed gas as a new decision variable, which was allowed to vary between 0 (pure CO) and 3 (25% CO and 75% H 2 ). Also for this case, shown in Figure 8c, the solutions did not form a Pareto front, as 100% CO conversion was attainable for a wide range of productivities. Figure   8c is hence analogous to Figure 8b  AcR 2 = 396 mmol/L, and k d = 0.00546 hr −1 . It is noteworthy that all of these parameters, with the exception of the yield coefficients, remain relatively close to the nominal parameters estimated for C2, in fact inside their confidence intervals, which suggests that efforts should be concentrated on enhancing the cell yields and not, for example, the maximum uptake rates (at least for the conditions of gas-liquid mass transfer encompassed by this study). Even though ν max is directly associated with the cell's capacity to take up substrate, the uptake rate might just be limited by the mass transfer, such that after a certain point there would be no actual gain with increasing ν max .
Another result from the optimization studies is that high productivities would be attained with very large cell concentrations reaching up to 30 g/L (results shown in Figure S8), although this depends on the gas composition. For example, 1 g·L −1 ·hr −1 of ethanol productivity would be attainable with H 2 -rich gas with operating conditions and kinetic parameters that result in 10 g/L of cell mass, while for CO-rich gas the cell concentration would be a little over 20 g/L for the same ethanol productivity.
Agitation rate and gas recycle rate are also important operating variables which were not included in this study, but should be evaluated in the future with the inclusion of power consumption as a third objective function. It is possible, for example, that under certain conditions the gains in productivity and conversion might compensate for any extra spending with electricity. Other reactor designs should also be evaluated, such as bubble column, gas-lift, and membrane reactors. Ultimately, however, the bioreactor should be optimized simultaneously with other unit operations, such as gasification and distillation, since optimal conditions in one unit might lead to worse outcomes in other units with respect to economic and/or environmental issues.

| CONCLUSIONS
A dynamic model was presented for the production of ethanol via syngas fermentation in a CSTR, and unknown kinetic parameters were estimated with literature data employing different conditions of gas flow rate, dilution rate, syngas composition, and medium composition. The modeling framework was then used to evaluate the effects of different input variables on the outcomes of ethanol productivity and gas conversion, and it was observed that cell recycle rate, gas flow rate, and H 2 content had clear positive effects on the productivity, while the dilution rate gives a different maximum depending on the other variables. Moreover, the kinetic parameters were found to have different sensitivity patterns depending on the process conditions, for example some of them having larger effects on the productivity when higher gas flow rates are used.
Since these parameters are specific to the type of strain and composition of the liquid medium, we conducted an optimization of productivity and conversion using operating conditions and kinetic parameters as decision variables, thereby showing the possibility of attaining higher values of both responses at the same time. Implementation of the results predicted in this work would require further studies connecting the kinetic parameters to the exact aspects of the liquid medium and strain capabilities, as well as more experiments investigating the inhibitory effects of products and CO. Therefore, as more experimental data become available, the modeling framework presented here can be used to re-estimate parameters, generate more accurate results and provide new insights for integrated process optimization.