Harvinder Sidhu

Combustion waves for gases (Le = 1) and solids (Le→∞)

The traditional combustion problems of calculating flame speeds for a premixed gaseous fuel and for a premixed solid fuel are revisited using a simpler (than previously) non–dimensional temperature. It turns out to be possible to carry out asymptotic calculations for flame speed and the agreement with corresponding numerical calculations is remarkably good. In each case the uniqueness of the speed is considered using phase plane methods, with a little effort to determine the nature of the ‘cold’ critical point. Consideration of the stability of the travelling combustion wave fronts suggests a period doubling route to chaos for the premixed solid fuel (as the exothermicity is decreased) and corresponds with previous work using different non–dimensional temperature and parameters.

Evans Function Stability of Combustion Waves

In this paper we investigate the linear stability and properties, such as speed, of the planar travelling combustion front. The speed of the front is estimated both analytically, using the matched asymptotic expansion, and numerically, by means of the shooting and relaxation methods. The Evans function approach extended by the compound matrix method is employed to numerically solve the linear stability problem for the travelling wave solution.

An oscillatory route to extinction for solid fuel combustion waves due to heat losses

A numerical method is used to show that heat loss increases can lead to a period–doubling route to the cessation of propagation of solid fuel combustion. Oscillatory combustion waves are found in certain regions of the parameter space. The behaviour of these oscillatory waves becomes more complex as the heat loss is increased until extinction of the combustion reaction occurs. Large excursions in temperature, above the adiabatic temperature, are possible in the non–adiabatic case close to this extinction point.

Finite difference schemes for multilayer diffusion

Although numerical methods have been developed for diffusion through single layer materials, few have been developed for multiple layers. Diffusion processes through a multilayered material are of interest for a wide range of applications, including industrial, biological, electrical, and environmental areas. We present finite difference schemes for multilayered materials with a range of matching conditions between the layers, in particular for a jump matching condition. We show the finite difference methods are flexible, simple to implement, and help illustrate interesting behaviour in multilayered diffusion.

Evans function stability of non-adiabatic combustion waves

In this paper we investigate the linear stability and properties of the planar travelling non–adiabatic combustion front for the cases of zero and non–zero ambient temperature. The speed of the front is estimated numerically using the shooting and relaxation methods. It is shown that for given parameter values the solution either does not exist or there are two solutions with different values of the front speed, which are referred to as ‘fast’ and ‘slow’. The Evans function approach extended by the compound–matrix method is employed to numerically solve the linear–stability problem for the travelling–wave solution. We demonstrate that the ‘slow’ branch of the solutions is unstable, whereas the ‘fast’ branch can be stable or exhibits Hopf or Bogdanov–Takens instability, depending on the parameter values.

Period doubling and chaotic transient in a model of chain-branching combustion wave propagation

The propagation of planar combustion waves in an adiabatic model with two-step chain-branching reaction mechanism is investigated. The travelling combustion wave becomes unstable with respect to pulsating perturbations as the critical parameter values for the Hopf bifurcation are crossed in the parameter space. The Hopf bifurcation is demonstrated to be of a supercritical nature and it gives rise to periodic pulsating combustion waves as the neutral stability boundary is crossed. The increase of the ambient temperature is found to have a stabilizing effect on the propagation of the combustion waves. However, it does not qualitatively change the behaviour of the travelling combustion waves. Further increase of the bifurcation parameter leads to the period-doubling bifurcation cascade and a chaotic regime of combustion wave propagation. The chaotic regime has a transient nature and the combustion wave extinguishes when the bifurcation parameter becomes sufficiently large. For Lewis numbers of fuel close to unity, the parameter regions where pulsating solutions exist become very close to each other and this makes it difficult to experimentally observe the period-doubling. It is shown that the average velocity of pulsating waves is less than the speed of the travelling wave for the same parameter values.

Analysis of the activated sludge model (number 1)

Abstract We analyse a model for the activated sludge process occurring in a biological reactor without recycle. The biochemical processes occurring within the reactor are represented by the activated sludge model number 1 (ASM1). In the past the ASM1 model has been investigated via direct integration of the governing equations. This approach is time-consuming as parameter regions of interest (in terms of the effluent quality leaving the plant) can only be determined through laborious and repetitive calculations. In this work we use continuation methods to determine the steady-state behaviour of the system. In particular, we determine bifurcation values of the residence time, corresponding to branch points, that are crucial in determining the performance of the plant.

Combustion waves in a model with chain branching reaction and their stability

In this paper the travelling wave solutions in the adiabatic model with the two-step chain branching reaction mechanism are investigated both numerically and analytically in the limit of equal diffusivity of reactant, radicals and heat. The properties of these solutions and their stability are investigated in detail. The behaviour of combustion waves are demonstrated to have similarities with the properties of non-adiabatic one-step combustion waves in that there is a residual amount of fuel left behind the travelling waves and the solutions can exhibit extinction. The difference between the non-adiabatic one-step and adiabatic two-step models is found in the behaviour of the combustion waves near the extinction condition. It is shown that the flame velocity drops down to zero as the extinction condition is reached. Prospects of further work are also discussed.

A dynamical systems model of the limiting oxygen index test

(Received Novembe 29r , 1998; revised February 2, 1999)AbstractOxygen index methods have been widely used to measure the flammability of polymericmaterials and to investigate the effectiveness of fire-retardants. Using a dynamical systemsframework we show how a limiting oxygen index can be identified with an appropriatebifurcation.The effectiveness of fire-retardants in changing the limiting oxygen index is calculated byunfolding the bifurcation point with a suitable non-dimensionalised variable whic, h dependsupon the mode of action of the additive. In order to use this procedure it is essential themodel is non-dimensionalised so as to retain the variables of interest as distinct continuationparameters.

Oscillatory thermal-diffusive instability of combustion waves in a model with chain-branching reaction and heat loss

In this paper we investigate the properties and the linear stability of premixed combustion waves in a non-adiabatic thermal-diffusive model with a two-step chain-branching reaction mechanism. Here we focus only on the emergence of the pulsating instabilities, and the stability analysis is carried out for Lewis numbers for fuel greater than one, and various values of Lewis number for radicals. We consider the problem in two spatial dimensions to allow perturbations of a multidimensional nature. It is demonstrated that the flame speed as a function of the parameters is a double-valued C-shaped function, i.e. for a given set of parameter values there are either two solutions, fast and slow solution branches, propagating with different speed, or the combustion wave does not exist. The extinction of combustion waves occurs at finite values of the parameters and non-zero flame speed. The flame structure demonstrates a slow recombination regime behaviour with negligible fuel leakage for the fast solution branch away from the extinction condition. For parameter values close to the extinction condition and on the slow solution branch, the fuel leakage is significant and a fast recombination regime is observed. It is demonstrated that two types of instabilities emerge in the model: the uniform planar and the travelling instability. The slow solution branch is always unstable due to the uniform perturbations. The fast solution branch is either stable or loses stability due to the travelling or uniform perturbations. The switching between the onset of various regimes of instability is due to the bifurcation of co-dimension two. In the adiabatic limit this bifurcation is found for Lewis number for fuel equal to one, whereas in the non-adiabatic case it moves towards values above unity. The properties of the travelling instability are studied in detail.