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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. |
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