## Mathematical modelling of bioenergy systems for stability analysis and parametric sensitivity

##### Permanent link

https://hdl.handle.net/10037/20637##### Date

2021-03-19##### Type

Doctoral thesisDoktorgradsavhandling

##### Author

Das, Subhashis##### Abstract

This thesis presents a research study on bioreactor design and biochemical processes for conversion of biomass into bioenergy and develops mathematical models that can be used for designing better reactors and control systems. Good design of the bioreactors and understanding of the biochemical process is essential to control the process for maximizing the yield. Improving the performance efficiency of bioreactors is necessary for large-scale biomass deployment to energy conversion systems and their economic viability.

Large scale production mostly deploys continuous stirred tank bioreactors (CSTBR) involving single, or polycultures. Many micro-environmental parameters in the reactor such as system pH, dilution rate, inlet substrate concentration etc. simultaneously affect the process and also the production of the desired output. Due to concurrent influence on many variable-limits, the process has multiple steady states, and the slight variation in one parameter leads to deviation from a steady-state operation and microbial growth is affected reactor may stall.

Bioprocesses within these reactors can be expresses as a set of nonlinear equations. The output variables (e.g. bioenergy products; gaseous or liquids) depend simultaneously upon a parametric range of a number of input variables such as pH, dilution rate, temperature, substrate concentration etc. It may be inferred that the biosystems are parametrically sensitive with respect to specific parameters. Thus, the sensitivity of a biosystem is studied for understanding the operation of a reactor. From the process engineering perspective, it is a challenge to determine a priori region of parametric sensitivity, i.e., to determine the set of values of system parameters beyond which the biosystem becomes highly sensitive.

Kinetics constants for the mathematical model need to be determined experimentally. Thus the experiments were performed using two types of lactic acid bacteria, namely, *Pediococcus acidilactici* and *Lactobacillus casei* using the batch reactor to find the optimum pH for microbial growth and other kinetic variables that define microbial growth characteristics. From the experiments, the kinetic constants for both bacteria strains were found. The maximum specific growth rate (μ_{max}) and substrate saturation constant (K_{s}) provide the guideline for a working range of feed stream flow rate and its concentration for designing continuous processes. For *Pediococcus acidilactici*, the optimum pH value, maximum specific growth rate (μmax) and substrate saturation constant (k_{s}) to be 6.7, 1.0775 h^{−1} and 4.5017gL^{-1}, respectively. The kinetic constants for *Lactobacillus casei*, the optimum value of pH, maximum specific growth rate (μ_{max}) and substrate saturation constant (K_{S}) were found to be 6.75, 0.6 h^{−1} and 0.814gL^{-1}, respectively. Although both the strains are LAB and their optimum pH are quite similar, other kinetic parameters are different.

When the product is gaseous such as methane or hydrogen, that itself can retard the growth of bacteria and its own production if allowed to accumulate in the reactor headspace. In order to study the effect of accumulated hydrogen on the process in a reactor, the third series of batch experiments were conducted by using a hydrogen-producing bacteria, *Clostridium acetobutylicum*. From these experiments, the maximum specific microbe growth rate (μ_{max}), substrate saturation constant (k_{s}), a critical hydrogen concentration at which growth ceased (H_{2}^{*}) were determined, 0.976 h^{−1}, 0.63 ± 0.01 g/L, and 24.74 mM, respectively. The degree of inhibition was 0.4786.

Two models based on a set of ordinary differential equations were developed to derive a dimensionless multiplicity criterion, ω, that indicates a set of values of input parameters corresponding to multiple steady states in a reactor. Using the kinetic variables obtained experimentally model can quantitatively predicting the parametric range of operating variables for steady operation of the process and optimal yield for a reactor

Parametric sensitivity of pH with respect to input variables specifically dilution rates and concentrations of nutrient and alkali stream for pH control in the regions of multiple and unique steady states in a CSTBR was determined using the mathematical model. The parametric sensitivity of pH was observed over the entire region of operation under study, and the model estimation was in agreement with those of the experimental observations.

The first model studied the nonlinear behaviour of a CSTBR using *Lactobacillus casei*. Parameter space was determined where the system exhibits sensitive behaviour through normalized objective sensitivity. Parametric range of inputs for controlling optimal pH range conducive for the microbial growth was determined. The influence of input parameters, which are directly intricate pH of the system, is observed by determining normalized objective sensitivity of pH. A generalized criterion, i.e., a specific range of certain input parameters, e.g., θ, R and pH_{0} corresponding to the system's maximum sensitivity was determined.

The second mathematical model studied the reactor process for producing biohydrogen. The multiplicity analysis of steady states was determined using the classical theory of bifurcation analysis with the help of local stability analysis. It was found that in a particular range of d_{1} (a dimensionless form of dilution rate of feed stream), from 0.44 to 0.4453, the CSTBR operation becomes unstable as it comes across multiple steady states condition. On the other hand, CSTBR enters to instability due to another essential operating parameter X_{20} (a dimensionless form of feed substrate concentration) when the operating region of X_{20} within 9.5717 to 13.658 where steady-states of CSTBR system bifurcated to multiple steady states.

The present study also endorses some scope for further research works, such as more output variables namely, temperature, the concentration of byproducts needing to be analysed for designing and safe operation of a continuous stirred tank bioreactors. In the case of biohydrogen production, there are other metabolites such as volatile fatty acids (VFA) produce along with the hydrogen can alter the system pH, which can be unfavourable for microorganisms and shift the metabolic pathway of microbial reactions. So, the influence of the concentration of VFA and adopt a kinetic model is recommended to study further. The validation of the model-predicted data with the experimental study shows the critical value of input parameters for which the CSTBR becomes sensitive. The interaction between input parameters or the points corresponding to the limits of the region of instability is recommended to be investigated. Moreover, the impact of model kinetic parameters on system stability is the scope where further work is suggested.

The mathematical models presented in this thesis can be used to investigate the operation of similar processes in CSTBRs. Using the kinetic parameters, relevant to different microbial growth, the developed model can be used to perform a stability analysis of a CSBTR and obtain parametric sensitivity regions of the process. Such quantitative analysis of CSTBRs will benefit in selecting design and adopting strategies for safe, controlled and economical utilization of CSTBRs. The information obtained from the present research study contributes to the general field of bioenergy conversion processes and the design and optimization of bioreactors, such as the production of biomethanol ethanol and biohydrogen.

##### Has part(s)

Paper 1: Das, S., Calay, R.K., Banerjee, A., Chowdhury, R. & Bhattacharya, P. (2016). Parametric sensitivity of pH and steady-state multiplicity in a continuous stirred tank bioreactor (CSTBR) using a lactic acid bacterium (LAB), *Pediococcus acidilactici*. *Journal of Chemical Technology and Biotechnology, 91*(5), 1431. Also available at https://doi.org/10.1002/jctb.4740.

Paper 2: Das, S., Calay, R.K. & Chowdhury, R. (2020). Parametric Sensitivity of CSTBRs for *Lactobacillus casei*: Normalized Sensitivity Analysis. *ChemEngineering, 4*(2), 41. Also available in Munin at https://hdl.handle.net/10037/18734.

Paper 3: Das, S., Calay, R.K., Chowdhury, R. & Nath, K. (2020). Product Inhibition of Biological Hydrogen Production in Batch Reactors. *Energies, 13*(6), 1318. Also available in Munin at https://hdl.handle.net/10037/18614.

##### Publisher

UiT Norges arktiske universitetUiT The Arctic University of Norway

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