Modelling Flow Distributed Oscillations In The CDIMA Reaction

1. Modelling Flow Distributed Oscillations In The CDIMA Reaction Jonathan R Bamforth, Serafim Kalliadasis, John H Merkin, Stephen K Scott School of Chemistry, Department of Chemical Engineering and Department of Applied Mathematics. University Of Leeds, Leeds, LS2 9JT.

2. Introduction • Kuznetsov et al.1 recently proposed a pattern-formation mechanism termed ‘flow distributed oscillations’ • Flow distributed oscillations do not rely on any difference in diffusion coefficients (Turing patterns) or flow velocities (DIFICI patterns) of the different reacting species. • Kuznetsov et al. and Andrésen et al.2 determined the conditions for such pattern formation through analysis and computation based on the Brusselator model. • Kærn and Menzinger3 demonstrated this phenomena experimentally exploiting the Belousov-Zhabotinsky reaction. • In this present work the conditions for flow distributed oscillations are studied numerically using a model of the CDIMA reaction.

3. Inflow Outflow Plug Flow Reactor CSTR Experimental Configuration • The experimental set-up imagined consists of a plug flow reactor maintained at its inflow by a CSTR. • The dynamical behaviour is closely related to the reactant concentrations for which the system would show temporal patterns in a well stirred batch reactor.

4. Model And Governing Equations • Following the Lengyel-Epstein model4, the important intermediate species concentration are iodide I and chlorite ClO2  and these participate in the following three component stoichiometric processes: • MA + I2 IMA + I + H+ • ClO2 + I  ClO2  + ½I2 • ClO2  + 4I + 4 H+ Cl  + 2I2 + 2H2O

5. Model And Governing Equations • The governing mass balance equations for the plug flow reactor allowing for diffusion and advection can be written in the following dimensionless form: • Where u and v are the dimensionless intermediate species I and ClO2 .  and  are dimensionless kinetic parameters and P is the dimensionless flow velocity in the plug flow reactor.  is the ratio of dimensionless diffusion coefficients which is fixed at 1 to prevent any Turing type instabilities.

6. Boundary and Initial Conditions • The inflow boundary condition of the plug flow reactor is maintained by a CSTR. • Parameter values are chosen so that the operating conditions of the CSTR are stable. • The outflow adopts zero-flux boundary conditions, allowing no mass transfer back into the plug flow reactor. • The plug flow reactor is initially at steady state with parameter values chosen so that the system is unstable.

7. Stability Analysis • Using the dispersion relation it is possible to determine regions of different behaviour. • Types of behaviour include: • Absolutely unstable region • Convectively unstable region • Region of stationary patterns • These different types of dynamical behaviour can be achieved by manipulating the flow velocity through the plug flow reactor for a given set of parameter values.

8. Stability Loci In The -P Parameter Plane • The change from absolute to convective instability is given by: • The change to stationary patterns is given by: • The change from unstable to stable behaviour is given by:

9. Absolute Instability • A spatially distributed system is absolutely unstable if a wave packet propagates with a front but no back. • Perturbations initially increase exponentially in time at any fixed point in the laboratory frame and the system does not return to the uniform steady state after it has been disturbed.

10. Convective Instability • A spatially distributed system is convectively unstable if a small perturbation induces a local growth away from the spatially uniform steady state , but that disturbance then propagates forward as a wave packet growing in size and with a definite front and back. • At any given location the system experiences a finite disturbance from the steady state as the wave front arrives but later returns to this state as the wave back passes.

11. Stationary Patterns • A stable stationary pattern is produced when the system is in a convectively unstable state and the flow velocity is above P,cr. • The wavelength of the stationary pattern varies linearly with the flow velocity.

12. Experimental Operating Conditions • All the dynamical behaviour seen arises at operating conditions that could easily be arranged in experimental reactors used for other studies. • Typical values for initial concentrations of reactants are: • [I2]0 = 103 mol dm3 • [ClO2]0 = 5  103 mol dm3 • [MA]0 = 103 mol dm3 • Taking the diffusion coefficients to be 2  105 cm2 s-1 and the values for the rate constants suggested by Lengyel and Epsteinthe flow velocity at which stationary patterns occur is of the order 10 mm min1and the wavelength of the patternsapproximately 0.5 mm.

13. References 1. S.P. Kuznetsov, E.Mosekilde, G.Dewel and P.Borckmans, J. Chem. Phys., 1997, 106, 7609. 2. P.Andrésen, M.Bache, E.Mosekilde, G.Dewel and P.Borckmans, Phys. Rev. E, 1999, 60, 297. 3. M.Kærn, M.Menzinger, Phys. Rev. E, 1999, 60, R3471. 4. I.Lengyel and I.R. Epstein, Science, 1991, 251, 650. 5. J.R.Bamforth, S.Kalliadasis, J.H.Merkin and S.K.Scott, Phys. Chem. Chem. Phys., in press.

14. Subcritical Nature of P,cr • The onset of stationary patterns was found to be subcritical in nature. • This gives a region of bistability over a range of P. • A bifurcation diagram illustrates this behaviour. • Further confirmation is achieved by identifying a critical threshold at which a perturbation to the initial steady state will cause the system to move to the stationary pattern state.