Modeling, simulation and optimal control strategy for batch fermentation processes

International Journal of Industrial Chemistry (2019) 10:67–76 https://doi.org/10.1007/s40090-019-0172-9 RESEARCH Modeling, simulation and optimal control strategy for batch fermentation processes Neba Fabrice Abunde1,3 · Nana Yaw Asiedu2 · Ahmad Addo1 Received: 4 December 2017 / Accepted: 3 February 2019 / Published online: 18 February 2019 © The Author(s) 2019 Abstract The use of fermenters at large scale is usually hampered by sub-optimal conditions in terms of yield and productivity, along with the low tolerance of strains to process stresses, such as substrate and product toxicity, and other fermentation inhibitors. Attempts to improve the industrial efficacy of fermenters have been in the areas of genetic engineering to improve strain tolerance, but this usually involves detailed and unfeasible mechanistic studies. Statistical designs of experiments have also been used to optimize industrial fermenters but this again often results in local optima due to the relatively small-dimensional space covered by the experiments. Mathematical techniques have recorded great successes and regarding ethanol fermentation with sorghum extracts, previous work has modeled and established the presence of product inhibition, however, did not consider other degrees of freedom (temperature and pH) that minimize the effect of such inhibitions. This paper includes the description of a batch alcohol fermentation process that has been optimized using a technique based on the application of mathematical modeling and optimal control. Calculus of variation is introduced as a valuable tool to derive and solve the necessary conditions for optimality, and the obtained results show the optimal temperature and pH profiles for the fermentation of sorghum extracts. A Simulink model of the fermentation process shows that using the proposed control strategy increases ethanol yield by 14.18%, cell growth by 71.96% decreases the residual substrate by 84.77%. Keywords Alcoholic fermentation · Mathematical modeling · Ethanol inhibition · Optimal control simulation · Sorghum extracts List of symbols Eg Activation energy for cell growth (cal∕mol) Gs Yield coefficient of cell based on substrate utilization (g∕g h) Kip Product inhibition coefficient on product formation (100 g∕g) Ksp Substrate saturation (Monod) constant for product formation (g∕100 g) Ksx Substrate saturation (Monod) constant for cell growth (g∕100 g) * Nana Yaw Asiedu 1 Department of Agricultural and Biosystems Engineering, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana 2 Department of Chemical Engineering, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana 3 Department of Civil and Environmental Engineering, Norwegian University of Science and Technology, Trondheim, Norway Mp Specific rate of ethanol production by a maintenance metabolism (g∕g h) Ms Specific rate of substrate consumption for cell maintenance (g∕g h) Tmax Maximum fermentation temperature (°C) Tmin Minimum fermentation temperature (°C) Yp Yield coefficient of cell based on substrate utilization (g∕g) Yx Yield coefficient of cell based on substrate utilization (g∕g) k1 Empirical constant in pH model (mol∕l) k2 Empirical constant in pH model (mol∕l) kd Cell death rate (h−1 ) kg Pre-exponential Arrhenius constant for growth pHmax Maximum pH in the fermenter pHmin Minimum pH in fermenter qmax Maximum specific rate of product formation ( h−1 ) qp Specific rate of product formation (h−1 ) 𝜇max Maximum specific growth rate ( h−1 ) A Weight coefficient for product formation in optimization problem (dimensionless) 13 Vol.:(0123456789) 68 International Journal of Industrial Chemistry (2019) 10:67–76 B Weight coefficient for temperature control in optimization problem (dimensionless) C Weight coefficient for pH control in optimization problem (dimensionless) J Performance index for optimal control problem P Concentration of product (g∕100 g) R Universal gas constant (cal∕K mol) S Concentration of substrate (g∕100 g) X Concentration of biomass (Mcells∕0.1 ml) t Batch fermentation time ( h) T Fermentation temperature (°C) 𝜇 Specific rate of cell growth (h−1 ) Introduction Sorghum, a cereal which belongs to the family Gramineae is now used in most breweries as locally available alternative to imported barley malt. In a generalized view of processing and brewing sorghum, though involves several unit operations, the fermentation step is the crux of the process, regarded as the heart of the entire production where a near optimal environment is desired for microorganisms to grow, multiply and produce the desired product [3]. However, the use of fermenters at a large scale is usually hampered by sub-optimal conditions in terms of yield and productivity, resulting from low tolerance of strains to process stresses, such as substrate and product inhibition, and other fermentation inhibitors [10, 14, 17]. In several attempts to improve the industrial efficacy of fermenters, a variety of approaches have been proposed; genetic techniques involving detailed, mechanistic studies of metabolic pathways, inherently involving inverse problem that cannot be understood with certainty [7]; statistical design of experiments [3], which again requires the construction of expensive prototype systems and most often leads to local optima due to the relatively small dimensional space covered by the experiments [3, 23]. Alternatively, design and optimization of bioreactors can be enhanced via validated mathematical models developed from mechanistic studies that lead to a more in depth understanding of process stresses such as ethanol inhibition [23]. In this regard, optimal temperature profiles have been determined to maximize beer flavor [19], maximize ethanol formation from sugarcane molasses (Marcus and Normey-Rico [13], minimize acetyl acetate production [9], maintain cell viability and reduce glycerol production [5]. However, the aforementioned as well as other studies have focused on temperature and rarely pH for optimization of batch fermentation processes. Fermentation principles consist of exploiting the metabolic reactions that take place in the cell of a microorganism for the production of valuable products [16]. To activate the metabolic pathways of interest 13 within the cell, specific environmental conditions (temperature, pH, nutrient concentration) are applied to enable the yeast cell grow and produce the required ethanol. In addition, due to the dynamic nature of the culture medium, yeast cells often suffer from various stresses resulting from both the environmental conditions, and from both product and or substrate imbibition [2]. To maximized ethanol yield, all the main aspect (ethanol inhibition kinetics, temperature and pH) should be considered simultaneously [8]. This paper presents the optimal pH and temperature profiles in the alcoholic fermentation of sorghum extracts using a linear product inhibition mo (...truncated)