Fractional Diffusion in Gaussian Noisy Environment

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: D(α)tu(t,x)=Bu+u⋅W˙H, where D(α)t is the Caputo fractional derivative of order α∈(0,1) with respect to the time variable t, B is a second order elliptic operator with respect to the space variable x∈Rd and W˙H a time homogeneous fractional Gaussian noise of Hurst parameter H=(H1,⋯,Hd). We obtain conditions satisfied by α and H, so that the square integrable solution u exists uniquely

Development of a methanol to hydrocarbons process over zeolite coatings in a microstructured reactor

In this work, the hydrothermal synthesis of ZSM-5 and its coating with controllable crystal size and Si/Al ratio has been performed. The obtained catalysts have been studied in the methanol to hydrocarbon (MTH) reaction. This reaction is the last step in an integrated fuel processor for the conversion of bio resources to liquid fuels. The development of ZSM-5 coatings has been supported by advanced characterization and testing of catalysts for the determination of property/performance relationships. An optimal synthesis time of 72 h was found to provide the highest crystallinity of ZSM-5 coatings. The larger crystal size of ZSM-5 coatings leads to a higher selectivity towards gasoline (C8-11) hydrocarbons. The selectivity towards the gasoline fraction over ZSM-5 coatings with a thickness of 14 μm was similar to that of an industrial ZSM-5 catalyst, however the yield of the undesirable aromatics by-products was reduced by half due to shorter diffusion pathways in thin catalyst layers. In an attempt to improve the yield of the C8-11 hydrocarbons, two post-synthesis modifications have been performed: Ca ion-exchange and desilication by alkaline treatment. The maximum gasoline selectivity over Ca-H-ZSM-5 was observed at a Ca/H ratio of 0.1 while the longest lifetime in the reaction was observed at the ratio of 0.2. Mesoporosity has been introduced into microporous ZSM-5 catalysts. The obtained meso-microporous ZSM-5 coating show 3 times lifetime and 2.7 times selectivity towards C8-11 hydrocarbon fraction than microporous coating in the MTH reaction. Lumped kinetics of MTH reaction over H-ZSM-5 were used to design a microstructured reactor/heat-exchanger (MRHE) with reaction channels coated with the ZSM-5 catalyst. 2D and 3D convection and conduction heat transfer models coupled with the MTH reaction kinetics were employed to investigate temperature distribution in the MRHE. The effect of the dimension of the microreactor/ heat-exchanger and flow condition on the temperature field has been studied. The 2D model under-predicts the magnitude of temperature gradient. The optimised reactor configuration shows a temperature gradient of 21 K in the reaction channels

Fractional Diffusion in Gaussian Noisy Environment

Three types of stochastic partial differential equations are studied in this dissertation. We prove the existence and uniqueness of the solutions and obtain some properties of the solutions. Chapter 3 studies the linear stochastic partial differential equation of fractional orders both in time and space variables. We prove the existence and uniqueness of the solution and calculate the moment bounds of the solution when the noise has Reisz kernel as space covariance. Along the way, we obtain some new properties of the fundamental solutions. Chapter 4 studies the time-fractional diffusion with fractional Gaussian noise. We obtain conditions so that the square integrable solution exists uniquely. Chapter 5 studies the time-fractional diffusion where the Gaussian noise is general in time with space covariance given by fractional, Riesz and Bessel kernel