Kinetics of in situ epoxidation of hemp oil under heterogeneous reaction conditions: an overview with preliminary results

Epoxidised hemp oil (EHO) was synthesised in the laboratory by reacting hemp oil (HO) with peroxyacetic acid (PA) in a batch reactor. The peroxyacetic acid was formed in situ from acetic acid (AA) and hydrogen peroxide (H2O2) in the presence on an acidic ion exchange resin (Amberlite IR-120) as catalyst. The overall reaction can be thought of as having two components. The first being epoxidation, a homogenous reaction which occurs at the interface of the aqueous phase and the HO phase while the second is the formation of PA, a heterogeneous reaction at the interface of the aqueous phase and the solid catalyst phase. The overall reaction kinetics were modelled by applying the Langmuir-Hinshelwood-Hougen-Watson (LHHW) model to heterogeneous reactions. Of the steps in the reaction it is postulated that the formation of PA is rate limiting, while the epoxidation occurs comparatively fast negating the requirement for an additional homogenous model. The diffusion steps in the reaction are also ignored in the kinetic model as it is believed that their effects are negligible due to intensive mixing in the batch reactor. Experiments were used to determine the optimal molar ratios of reactants and it was found that at these conditions 88% conversion of double bonds to epoxy groups occurred. The kinetic model was found to be in good agreement with the experimental results

A stable and accurate control-volume technique based on integrated radial basis function networks for fluid-flow problems

Radial basis function networks (RBFNs) have been widely used in solving partial differential equations as they are able to provide fast convergence. Integrated RBFNs have the ability to avoid the problem of reduced convergence-rate caused by differentiation. This paper is concerned with the use of integrated RBFNs in the context of control-volume discretisations for the simulation of fluid-flow problems. Special attention is given to (i) the development of a stable high-order upwind scheme for the convection term and (ii) the development of a local high-order approximation scheme for the diffusion term. Benchmark problems including the lid-driven triangular-cavity flow are employed to validate the present technique. Accurate results at high values of the Reynolds number are obtained using relatively-coarse grids

Linearized Asymptotic Stability for Fractional Differential Equations

We prove the theorem of linearized asymptotic stability for fractional differential equations. More precisely, we show that an equilibrium of a nonlinear Caputo fractional differential equation is asymptotically stable if its linearization at the equilibrium is asymptotically stable. As a consequence we extend Lyapunov's first method to fractional differential equations by proving that if the spectrum of the linearization is contained in the sector \{\lambda \in \C : |\arg \lambda| > \frac{\alpha \pi}{2}\} where $\alpha > 0$ denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable