Many problems in chemical-reactor engineering are modelled by a system of coupled partial differential equations (PDEs). Knowledge about regions in parameter space in which hysteresis and oscillatory behaviour occur is important in the design, start-up and control of chemical reactors as these phenomena can adversely affect reactor performance. Thus it is important to be able to accurately determine the location of these regions, without resorting to tedious and time-consuming direct numerical integration of the governing equations.
Typically, the bifurcation points of a system of coupled PDEs are found using numerical methods which requires the solution of a large set of linear equations at each iteration step. These methods are computationally expensive and normally applied to one-dimensional problems only. A naive approach to dealing with a spatially distributed model is to discretise it to produce a set of ordinary differential equations and to then use standard numerical bifurcation algorithms. For 1-d models good results can be obtained for discretisations using of the order of 10^{2} points. However, for 2-d and 3-d models the growth in nodal points makes it impractical to compute the Jacobian matrix and eigenvectors using such a direct approach (Balakotaiah & Khinast, 2000)
For problems in one spatial dimension an efficient, and powerful technique, is to combine Liapunov-Schmidt reduction with shooting methods (Balakotaiah & Khinast, 2000; Subramanian & Balakotaiah, 1996). This method leads to a systematic procedure to obtain bifurcation diagrams for distributed reactor models in which the steady-state models are characterised by a single intrinsic state variable. The drawback of this method is that it is strictly limited to one-dimensional problems. For problems in higher dimensions Subramanian & Balakotaiah (1996) suggest that the model equations be averaged in the spatial directions in which the variables do not change rapidly.
An alternative averaging method uses the Liapunov-Schmidt technique to perform spatial homogenization over small scales. Averaging is done over the local (transverse) dimensions. This leads to a series solution in a small parameter which is the ratio of local diffusion time to convection time of the system. This method has been applied to convention-diffusion-reaction (CDR) equations to obtain low-dimensional models. The first-order term in the series expansion parameter is sufficient to retain all the qualitative features of the CDR model (Chakraborty & Balakotaiah, 2002a). In the low-dimensional models the concentration of a chemical species is not given by a single equation but by a system of differential-algebraic equations. The differential equations describes the variation of the `cup-mixing' concentration with the residence time whilst the algebraic equations capture mixing on local scales. Typically, one algebraic equation is used to model micromixing. (For non-isothermal systems there are multiple temperature equations).
In a bifurcation context this model has been applied to an isothermal autocatalytic reaction in a laminar flow tubular reactor (Chakraborty & Balakotaiah, 2002a), to investigate the effects of mixing on an autocatalytic reaction occurring a CSTR (Chakraborty & Balakotaiah, 2002b), for a first-order non-isothermal reaction in an adiabatic tubular reaction and a cooled CSTR (Chakraborty & Balakotaiah, 2004), to investigate the effects of mixing times on competitive-consecutive reactions in an adiabatic CSTR (Chakraborty & Balakotaiah, 2004). The methodology developed in these papers is limited to CDR models with Neumann (zero flux), Robin or periodic boundary conditions. Furthermore, the averaged models exist only when the local diffusion time is much smaller than the convective and characteristic reaction times.
My colleague Dr T.R. Marchant has developed a semi-analytical method that can be used to study partial differential equations. In this a Galerkin method is used to obtain a semi-analytical approximation to the governing partial differential equations. This involves approximating the spatial structure of the solutions using a series of orthogonal basis functions that satisfy the boundary conditions. The semi-analytical method is then obtained by averaging the governing partial differential equations. This procedure gives a system of ordinary differential equations (ODEs), which can be readily analysed using the standard techniques of bifurcation/singularity theory. In cases where the averaging can not be integrated explicitly, the semi-analytical method is given by a system of integrodifferential equations. This method is not restricted to problems in one spatial dimension. The semi-analytical method is an efficient and simple way of generating information about the static and dynamic multiplicity of a system of PDEs, particularly for systems in higher dimensions.
Marchant (2002) used a semi-analytical method to examine the Gray-Scott cubic autocatalytic scheme in a reaction-diffusion cell. This involved approximating the governing PDEs by ODEs. The semi-analytical model was then analysed using a local stability analysis and singularity theory to determine the regions of parameter space in which the various bifurcation patterns and Hopf bifurcations occurred. An excellent comparison was obtained between the semi-analytical results and numerical solutions of the governing PDEs.
Marchant & Nelson (2004) used the semi-analytical method to examine the problem of heat-mass transfer within a porous catalytic pellet. (This work is described below). As the Arrhenius law cannot be integrated explicitly, the semi-analytical model is now given by a system of integrodifferential equations. Nelson et al (2007) used the semi-analytical method to examine self-heating in compost piles due to biological effects. (This work is described below). As the Arrhenius law cannot be integrated explicitly, the semi-analytical model is now given by a system of integrodifferential equations.
References
Applied the semi-analytical method to the problem of heat and mass transfer within a porous catalytic pellet in which an irreversible first-order exothermic reaction occurs. This is a classic problem in chemical reactor engineering that has been much studied as it exhibits both static and dynamic multiplicity. It is described by a system of two coupled reaction-diffusion equations. The pellet problem was investigated in both one- and two-dimensional slab geometries. Comparison with numerical solutions of the governing reaction-diffusion equations and with other results in the literature shows that the semi-analytical solutions are extremely accurate.
T.R. Marchant and M.I. Nelson. Semi-analytical solutions for one and two-dimensional pellet problems. Proceedings of the Royal Society of London A, 460, 2381-2394, 2004.
For more details on the spontaneous combustion of compost heaps please read this page.
M.I. Nelson, T.R. Marchant, G.C. Wake, E. Balakrishnan, and X.D. Chen. Self-heating in compost piles due to biological effects. Chemical Engineering Science 62(17), 4612-4618, 2007.
The DOI (Digital Object Identifier) link for this article is http://dx.doi.org/10.1016/j.ces.2007.05.018.
Dr T.R. Marchant | 2003-Present |