This research explores the recent advancements in mathematical modelling of heat transfer and fluid flows, emphasizing fractional calculus, non-Newtonian fluid dynamics, and nanofluid models. We discuss the use of fractional derivatives in modelling complex thermal and flow behaviours in various engineering contexts, including permeable surfaces and porous media. Applications range from industrial heat exchangers to biomedical devices. Numerical methods such as the Iterative Power Series (IPS) technique and Finite Element Methods (FEM) are employed to solve the nonlinear equations governing these phenomena. A comprehensive comparison of these methods highlights their strengths and limitations in terms of accuracy, convergence, and computational efficiency. The results reveal significant insights into the optimization of thermal systems, with fractional models demonstrating superior adaptability to anomalous flow and heat transfer conditions. The findings contribute to the development of more efficient and effective engineering designs, and the study suggests directions for future research in this field.

In this work, various algorithms of PID (Proportional-Integral-Derivative) controllers such as series, parallel, series together with derivative filter and parallel together with derivative filter form are implemented for the comparative analysis of the closed loop responses of an important parameter in head box of a paper machine in a paper making process industry called consistency, which has FOPDT (First Order Plus Dead Time) dynamics. Also the comparison is made between the closed loop responses achieved from different values of derivative filter factor for non-ideal situations. The steady state and dynamic characteristics are analyzed and compared and important inferences are derived.

In present analysis, a trickle bed reactor process is selected for control system analysis. This system is modelled as a third order plus delay time (TOPDT) model. Since, very few controller design methods are available for TOPDT model, this model is approximated to integrator plus date time (IPDT) model. Suitable controllers based on Ziegler and Nichols, Hay, Alfaro Rulz, Gorecki and Leonard tuning techniques are designed. Important steady state and transient time response characteristics are derived and compared for these models with designed PID controllers using various controller tuning methods in closed loop feedback system.

The idea of fractional order controller in combination with Nelder Mead algorithm is proposed in this paper for controlling a multivariable industrial scale polymerization reactor. The concept of decentralization is also applied to identify the pairing and decentralization of loops for better performance of the controller. PID controller is tuned using Cohen-Coon (CC) tuning technique. CC tuning technique is more flexible and suitable in comparison to conventional Ziegler-Nichols tuning technique. Using Nelder-Mead optimization technique, the Fractional Order-PID (FO-PID) controller parameters including the special parametric values are calculated which plays a very important role in tuning of FO-PID controllers finely and for optimal performance of these controllers. Simulation results are obtained and dynamic characteristics and performance indices are calculated and analyzed.