Mathematical modelling and analysis of deformable perturbed continuously stirred tank reactors (cstrs)

Abstract

In this dissertation, deterministic and stochastic mathematical models for a deformable per turbed continuously stirred tank reactor (CSTR) with exothermic and endothermic reactions have been developed and analysed. The Ordinary Differential Equations (ODEs) were obtained by using the Reynold transport theorem and Stochastic Differential Equations (SDEs) were de rived in the Ito sense from the developed classical deterministic models. There were four types ˆ of SDEs formulation, namely, additive SDE, multiplicative SDE, parameter perturbation SDE and transition probabilities SDE. The numerical results of the developed models were obtained and analysed through statistical and Bayesian methods. These methods were Classical Least Squares (LSQ) and Markov chain Monte Carlo (MCMC) for ODES while the Euler-Maruyama technique was used to simulate the SDEs. The LSQ numerical findings showed that the mea surements fit theoretical models well provided that the noise intensity ranges between 0 and 0.5. The MCMC results identified the parameters posterior means and the credible intervals in which models parameters must be oscillating. The PRCCs with Latin Hypercube Sampling technique were applied to check the sensitivity and uncertainty quantification of estimated parameters against the models’ response. Some of the parameters of models were found to be highly and positively correlated with models’ states and others were highly and negatively correlated with models’ state variables. For example, seven parameters were found to be highly correlated with exothermic CSTR model whilst six parameters were identified to be highly correlated with en dothermic CSTR model. This implies that those parameters have to be controlled and treated carefully as the increase or decrease in their values significantly impact the models’ outcomes. For the case of stochastic part, simulations of SDEs revealed that high fluctuations notably af fect trajectories of the variables. The overall numerical results obtained seem to be reliable and have shown an insight in describing the dynamics of the CSTR deterministic and stochastic models with detailed mathematical and statistical information. So, the formulated models were analysed, validated and can be used to model and describe various mechanical, biological and chemical processes such as filtration, anaerobic respiration and combustion among others.