In the following:
The continuous-flow stirred tank reactor (CSTR) is a standard tool used to investigate the behaviour of chemical processes subject to nonlinear kinetics. A recently proposed variation of the CSTR is the extended continuous-flow stirred tank reactor (ECSTR). This consists of a standard CSTR attached to an environmental tank reactor with mass transfer occurring between them through a membrane. The attraction of studying a reaction scheme in an ECSTR, rather than a CSTR, is that this offers the possibility of modifying the behaviour with a larger parameter dimension. We investigate how the behaviour of a standard non-linear chemical mechanism, quadratic autocatalysis subject to linear decay, changes when it is studied in an ECSTR rather than a CSTR.
M.I. Nelson and E. Balakrishnan. Quadratic autocatalysis in an extended continuous-flow stirred tank reactor (ECSTR). Applied Mathematical Modelling, 40(1), 363-372, 2016. http://dx.doi.org/10.1016/j.apm.2015.05.009.
Semi-analytical solutions for cubic autocatalytic reactions are considered in a circularly symmetric reaction-diffusion annulus. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for this novel geometry. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found using a degenerate Hopf bifurcation analysis. A novel feature of this geometry is the effect, of varying the width of the annulus, on the static and dynamic multiplicity. The results show that for a thicker annulus, Hopf bifurcations and multiple steady-state solutions occur in a larger portion of parameter space. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations.
M.R. Alharthi^{p}, T.R. Marchant and M.I. Nelson. Cubic autocatalysis in a reaction-diffusion annulus: semi-analytical solutions. Zeitschrift für angewandte Mathematik und Physik, 67, article 65. 2016. http://dx.doi.org/10.1007/s00033-016-0660-0.
In this paper, seaweed cultivation is mathematically modelled. The potential use of the crop to consume by-products from ethanol production is considered with a feasibility study and simple financial model. The growth of seaweed is described using differential equations and considering various factors such as solar radiation.
W.L. Sweatman, G. Mercer, J. Boland, N. Cusimano, A. Greenwood, K. Harley, P. van Heijster, P. Kim, J. Maisano, M.I. Nelson, and G. Pettet. Seaweed cultivation and the remediation of by-products from ethanol production: a glorious green growth. In T. Farrell and A.J. Roberts, editors, Proceedings of the 2014 Mathematics and Statistics in Industry Study Group, MISG-2014, volume 56 of ANZIAM Journal, pages M1--M29, May 2016. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/9402 .
The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reaction-diffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full non-smooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail.
H.Y. Alfifi^{p}, T.R. Marchant, and M.I. Nelson. Non-smooth feedback control for belousov-zhabotinskii reaction-diffusion equations: semi-analytical solutions. Journal of Mathematical Chemistry, 54:1632--1657, 2016. http://dx.doi.org/10.1007/s10910-016-0641-8.
We develop a mathematical model describing the operation of autothermal processes. Autothermal reactors provide considerable thermal efficiency over conventional reactors. The reaction mechanism investigated is A-> B -> C, where the reactions occur in a two reactor cascade. Specific features of coupled endothermic and exothermic reactions are taken into account. Particular considerations are presented and discussed for different catalysts to obtain 90% conversion into product.
M.M. Saleh^{p} and M.I. Nelson. Maximizing product concentration in a diabatic multistage reactor. In Mark Nelson, Dann Mallet, Brandon Pincombe, and Judith Bunder, editors, Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J., pages C101--C124, 2016. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/10387.
The activated sludge process is the most widely used biological wastewater treatment method for domestic and industrial wastewaters. One drawback of this process is the production of `sludge.' The expense for treating excess sludge accounts for 50-60% of the operating costs in a wastewater treatment plant. Traditional methods for disposing of excess sludge are increasingly regulated due to environmental concerns about the presence of potentially toxic elements in the sewage sludge. A promising method to reduce excess sludge production is to increase its biodegradability by disintegrating it within the bioreactor. We extend an earlier model by allowing the sludge disintegration reactions to occur at a finite rate. Furthermore, we include a more realistic model for the hydrolysis of particulate matter by allowing the rate of hydrolysis to saturate at high particulate concentrations. We analyze the steady-state operation of an activated sludge system incorporating a sludge disintegration unit to prevent excess sludge production.
F.S. Saadi, M.I. Nelson, and A.L Worthy. Sludge disintegration model with finite disintegration rate. In Mark Nelson, Dann Mallet, Brandon Pincombe, and Judith Bunder, editors, Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J., pages C346--C363, 2016. http://dx.doi.org/10.21914/anziamj.v57i0.10385.
In this study, we investigate a model of an activated sludge process connected to a sludge disintegration unit. We formulate a model of a sludge disintegration unit where disintegration processes occur at a finite rate. We then take an appropriate asymptotic limit to obtain a model where these processes occur infinitely rapidly. We use our limiting model to show that a previously proposed infinite-rate model is formulated incorrectly. Our principle aim is to investigate how the disintegration rate in the sludge disintegration unit affects the formation of sludge in the reactor. In the limiting case of an infinite disintegration rate, we show that there is a critical value for the sludge disintegration factor, above which the reactor system is guaranteed to be in a state of negative excess sludge production. For the case of finite rate processes, we show that if the disintegration rate is sufficiently high, then the error when assuming an infinite rate is less than 10% of the exact value using a finite rate. In these cases, the behavior of the reactor can be estimated within experimental error by assuming an infinite rate. We also show that if the reaction rate in the sludge disintegration unit is sufficiently small, then there is no longer a critical value for the sludge disintegration factor above which the reactor operates in a state of negative excess sludge production for all residence times. Instead, negative excess sludge production can only be achieved when the residence time is sufficiently large.
R.T. Alqahtani, M.I. Nelson, Annette L. Worthy. Sludge disintegration. Applied Mathematical Modelling, 40: 7830--7843, 2016. doi:10.1016/j.apm.2016.03.040.