Multi-channel monolith reactors as dynamical systems

Combustion and Flame 134 (2003) 193 205 Multi-channel monolith reactors as dynamical systems A. James a, *, J. Brindley b, A.C. McIntosh b a Department of Applied Maths, University of Leeds, Woodhouse Rd, Leeds, LS2 9JT, UK b Department of Fuel and Energy, University of Leeds, Woodhouse Rd, Leeds, LS2 9JT, UK Received 17 October 2002; received in revised form 19 March 2003; accepted 4 April 2003 Abstract A model for an array of channels in a catalytic monolith reactor is presented and its qualitative behavior is examined, using a dynamical systems approach. The model uses a coupled map lattice system to represent a row of channels across the monolith. The aim is to provide a simple model that can give qualitative solutions for a wide range of parameter values. The model exhibits multiple stable solutions and periodic solutions which have been seen experimentally in monoliths [1,2] and single-channel models [3] but have not been explored theoretically in multi-channel models. This multi-channel model displays richer behavior than is possible in a single channel model; path-following techniques are used to explore the distribution in parameter space of the varying phenomenology. 2003 The Combustion Institute. All rights reserved. Keywords: Catalytic combustion; Multi-channel model; Coupled map lattice 1. Introduction Catalytic monoliths have a wide range of uses including pollution control in vehicle exhausts. A typical monolith consists of a large number of parallel passageways often in a honeycomb type arrangement. The surface of each passageway is covered with a highly porous washcoat containing the catalyst. Monoliths are well suited to pollution control as they have a high surface area to volume ratio, and successful operation requires a low pressure drop in comparison to other types of reactor e.g. fixed-bed. Most mathematical models of monoliths concentrate on modelling a single channel of the monolith in isolation. However; many questions which arise naturally in the use of monoliths cannot be answered by models focusing on a single channel i.e., what are the effects of the catalyst deteriorating in some channels? * Corresponding author. Tel.: 44-0-114-225-2835; fax: 44-0-114-225-3066. E-mail address: a.james@shu.ac.uk (A. James). What if some channels become blocked? If a single channel stops working will this affect surrounding channels? What are the effects of different external conditions, e.g. cold/hot climates? How do these different scenarios affect the overall performance of the monolith? To answer such questions one must model the entire monolith. Models which do this are usually referred to as multi-channel models. Compared with the large body of work pertaining to single channel models there has been relatively little study of multi-channel models. Early multichannel models focused on describing the heat transfer within a monolith and ignored both homogeneous and heterogeneous reactions in the flow through the channels. Flytzani-Stephanopoulos et al. [7] developed a model for heat transfer in a honeycomb monolith with no reactions and achieved good comparisons with experimental results. Cybulski and Moulijn [8] also examined the case for no reaction and formulated a model for heat transfer between channels in a sinusoidal monolith. Both models use boundaryvalue ODEs to describe the steady state temperatures 0010-2180/03/$ see front matter 2003 The Combustion Institute. All rights reserved. doi:10.1016/s0010-2180(03)00094-4

194 A. James et al. / Combustion and Flame 134 (2003) 193 205 Nomenclature Variables (a * denotes the corresponding dimensional quantity) T g (i) Gas temperature in the ith channel T L (i) Left surface temperature in the ith channel T R (i) Right surface temperature in the ith channel y g (i) Gas mass fraction in the ith channel y L (i) Left surface mass fraction in the ith channel y R (i) Right surface mass fraction in the ith channel t Time Parameters A c Pre-exponential factor for catalysed reaction 2.3 10 5 ms 1 A g Pre-exponential factor for gas-phase reaction 2 10 9 m 3 kg 1 s 1 C ox Initial [O 2 ] 0.12 kg m 3 * C pr Initial concentration of propane 4 10 3 kg m 3 c p Specific heat capacity (gas) 10 3 J kg 1 K 1 c pc Specific heat capacity (ceramic) 10 3 Jkg 1 K 1 * E c Activation energy for catalysed reaction 9 10 4 J mol 1 E g Activation energy for gas-phase reaction 1.1 10 5 J mol 1 ( H) Heat release 4.5 10 7 Jkg 1 h Heat transfer coefficient 120 W m 2 K 1 h D Mass transfer coefficient 0.1 m s 1 L Length of reactor 0.1 m N Number of channels P Power input R Universal Gas Constant 8.315 J mol 1 K 1 S Cross-sectional area of channel 1.3 10 6 m 2 S c Cross-sectional area of solid 1.3 10 6 m 2 T 0 Inlet temperature of gas 550 K T amb Ambient temperature 300 K u 0 Inlet axial velocity 10 m s 1 y 0 Inlet mass fraction 2%* Wall width 0.3 10 3 Ceramic thermal conductivity 0.5 W m 1 K 1 Gas density 0.6 kg m 3 c Ceramic density 2 10 3 kg m 3 * Wetted perimeter 4.5 10 3 m Non-dimensional groups Bi Biot number D C Catalytic Damkohler number D H Homogeneous Damkohler number J H Heat transfer number of the gas J W Heat transfer number of the wall J D Mass transport number P Non-dimensionalised power input Gas-ceramic conversion ratio Adiabatic flame temperature c Non-dimensional catalytic activation energy g Non-dimensional homogeneous activation energy For a fuller explanation of the non-dimensionalisation see the Appendix. Numerical values marked are taken from Spence et al. [4] from Bennett et al. [5],* from Wanker et al. [6]. of the gas phase and wall of each channel. The channels are then coupled using heat transfer terms. Significant contributions have been made by Kolaczkowski et al. [9] who used a differential algebraic system of ODEs to describe the heat transfer in a monolith operating at steady state with a non-reacting gas. They then tested this model against the experimental data given by Flytzani-Stephanopoulos et al. [7] and showed that it gave good quantitative results. The work was then continued in Worth et al. [10] where a two step reaction was included. Later Kolaczkowski and Worth [11] described in more detail the numerical scheme necessary to solve this model, and several sets of parameter values were chosen to compare the cases for metal and ceramic monoliths. Finally, Worth et al. [12] included radiative exchange between channels, introducing an approximation for the integral equations which arise in the model. This model is a highly complex creature; thus, whilst computations for particular parameter values appear to give excellent comparisons with experimental data, its complexity makes it difficult to explore extended regions of parameter space.

A. James et al. / Combustion and Flame 134 (2003) 193 205 195 Fig. 1. A schematic diagram showing the temperature variables within the array of channels. Each channel has three corresponding fuel concentration variables. Transient models have also been proposed. Recently, Jahn et al. [13] have developed a PDE model of a monolith with 24 channels. It contains three spatial dimensions and time. The model is discretized to several tens of thousands of ODEs and integrated to give transient solutions. They found that on a fast workstation one minute of real time in the model required about one hour to simulate. Tischer et al. [14] have also performed numerical simulations using a highly detailed 2D/3D transient model. Both these studies gave results for several sets of parameter values but made no comparisons with experimental results. While these complex models are useful for generating simulations of specific operating conditions, they are not well suited to carry out an exploration and classification of qualitative behavior over a range of parameter space, and the possibility of unexpected or undesirable behavior may be missed. For example, none of them mention the possibility of multiple steady states or of instabilities. In contrast, both multiple stable steady-states and periodic behavior have been shown to exist in a single channel model [3] and there is evidence of these phenomena in experimental data. Cimino et al. [1] report experimental results where, as the inlet gas temperature is varied, the monolith exhibits the classic ignition and extinction hysteresis behaviour. Euzen et al. [2] showed that a monolith can display oscillatory as the catalyst deactivates with age. The aim of this paper is to extend the approach used in James et al. [3] to develop a simple model for a monolith which can be used to explore the qualitative behavior patterns, both steady state and transient, available throughout a range of parameter space. With the essence of these behavior patterns captured in a simple model, the more detailed approaches of [9 14] will be necessary to simulate the phenomenology in detail and under-pin the design of monoliths for practical use. 2. Model and methodology To focus on the effect of spatial coupling of a number of channels in as simple a configuration as possible, a one dimensional array of channels is considered. This may be thought of as a single row of channels in a rectangular monolith (see Fig. 1), or, perhaps as a single radial row in a circular monolith. Specifically, consider a row of N channels as shown in Fig. 1; the leftmost channel is at the center of the monolith, the rightmost at the outer edge. Three temperature variables are assigned to each channel; T g (i) the gas temperature of the ith channel, and two surface temperatures T L (i) and T R (i). The surface temperatures can be taken to depict the left and right surfaces of the channel, with respect to the array shown in Fig. 1. Each channel has three corresponding fuel concentration variables, y g (i), y L (i) and y R (i). The choice of array configuration and variable positioning is very similar to that of Worth et al. [12] although they use a different notation and refer to inside and outside ring surface as oppose to the left and right notation used here. The main modelling assumptions are: Pressure drop in the channels is negligible. There is a constant temperature and mass fraction at the channel inlet and the radial flow distribution at the inlet is flat.

196 A. James et al. / Combustion and Flame 134 (2003) 193 205 Diffusion is negligible. All physical properties of the gas and the solid are independent of temperature. Both the gas-phase and catalyst reactions have simple one-step chemistry. Convection of both mass and fuel occurs in the gas phase and this term is linearised. Heat transfer occurs between the gas and solid phase and also between channels. Mass transfer occurs between the gas and solid phase. With the exception of the linearization of the convection terms these are relatively standard modelling assumptions for a single channel plug-flow model of a catalytic monolith [3,5,15 17]. The homogeneous reaction term is not a common feature in models of catalytic combustion; however, it is included here to allow the model to be as general possible. The extension to a multi-channel model by including heat transfer between channels has also been seen previously [8, 12]. The variables have no space dependence and represent the value near the outlet of the channel. Approximating the convection term by linearization allows the model to retain the simplicity of a single-channel model and remain open to powerful numerical techniques for ODEs (see Appendix for derivation and description of parameters). Using these assumptions the mass and energy balances for each internal channel yield the following non-dimensionalised equations dt L i dt H T g i T L i J W T R i 1 T L i D C y L i exp c T L i dt g i dt T g i 1 J L H T L i 2T g i T R i D H y g i exp g T g i dt R i dt H T g i T R i J W T L i 1 (1) (2) T R i D C y R i exp c (3) T R i dy L i dt D y g i y L i D C y L i exp c T L i (4) dy g i dt y g i 1 J L D y L i 2y g i y R i dy R i dt D y g i y R i D H y g i exp g T g i (5) D C y R i exp c (6) T R i Finally the inner- and outermost channels must be considered. Symmetry across the center of the monolith is assumed, this removes the solid-phase heat transfer term (i.e. J W 0) in the T L (1) equation. This assumption excludes any asymmetric solutions. While asymmetrical behavior has been seen in monolith reactors it is usually attributed to inhomogeneities in the monoliths construction rather than intrinsic asymmetric behaviour. It would be a trivial problem to alter this model either to include the possibility of such asymmetric solutions or to include inhomogeneities in the monolith s structure. For the outermost channel a Biot number edge condition with an external heat source is assumed, viz dt R N dt H T g N T R N Bi T amb T R N D C y R N exp c T R N P (7) From a mathematical viewpoint, the set of 6N equations are identical to those for a coupled map lattice (CML) [18]. Such maps have been widely used to describe spatially distributed systems in which local time evolution occurs at each of a lattice of points between which some coupling or information flow takes place. Here each point is a channel, whose state is defined by T g,t L,T R,y g,y L, and y g, and whose local evolution is described by equations (1 6). The ith point is coupled to its neighbours i 1 and i 1 through the wall heat transfer term. The methodology to be used is then that of dynamical systems as developed for CMLs, to determine the qualitative behaviour of the system. The use of CML models is not new in combustion problems, e.g. Epstein and Showalter [19] reviewed the use of CML models to study pattern formation in chemical oscillators. A monolith modelled in this way is similar to an array of coupled tank reactors connected in parallel, although the coupling is via heat transfer between channels rather than fuel transfer. However, the novelty in this study is that the approach of the

A. James et al. / Combustion and Flame 134 (2003) 193 205 197 Table 1 Standard parameter values J H J D J W D C D H g s Bi 7 4 0.02 5.3 10 4 1.6 10 4 23 20 1.5 3 10 4 3 10 4 1 See the nomenclature and appendix for details. These are not intended as exact values but rather as orders of magnitude and as a starting point for further exploration. simple CML model is applied to a very specific engineering system, i.e. a monolith reactor, rather than more generic chemical reactions. 3. Results The model is simply a set of time dependent ODEs, and so is amenable to solution by a variety of methods. Since the system is stiff, all integration is done using Richardson extrapolation and the Bulirsch-Stoer method [see 20, chap. 16.4]. Standard pathfollowing techniques [21] were used with the addition of a globally convergent Newton s method for the root-finding [see 20, chap. 9.7]. The standard values for the non-dimensional parameters, taken from other authors experimental work, are given in Table 1. These values are to be seen as a starting point for an exploration of the surrounding parameter space rather than a fixed set of values. Figure 2a,b shows the results of integrating the system with respect to time from two sets of initial values with a power source of P 0.7 applied to the outer channel. The size of the power source is directly related to temperature, i.e. in a single tube monolith with no reaction the steady state temperature will equal P 1. In Fig. 2a the initial temperature of the monolith is T(i) 1.6, (n.b. the temperatures are non-dimensionalised with respect to the inlet temperature). It is clear that there is no ignition event and the array quickly reaches a constant temperature. Figure 2b shows the same set of parameter values but with the initial temperature of the monolith as T(i) 1.65, here there is a clear ignition event. Ignition initially occurs over almost the entire crosssection of the reactor and the only radial temperature gradient is at the outer edge and due to the nonadiabatic edge condition. The heat quickly spreads to these outer channels giving an almost uniform radial temperature. The temperature rise for these parameter values is quite high and is representative of monolith being used for catalytic combustion. Monoliths used for pollution control usually have a lower temperature rise. The consequences of varying the adiabatic flame rise, are discussed briefly in the next section. This type of ignition event where the whole monolith lights simultaneously is seen experimentally in Jahn et al. [22]. 3.1. Structural stability A structural stability analysis is essential to establish the credibility of any model. It is rare for any model of a physical situation to be exact; usually approximations have been made and often, especially in the case of chemical reactions, the values of the physical parameters are not exactly known. A structural stability analysis can throw light on these issues Fig. 2. Two sets of initial conditions leading to different solutions. All parameters set to standard values (Table 1) and P 0.7 for an array of 12 channels. The z-axis shows the temperature of the gas within the channels (T g (i)). The ignited solution starts at T(i) 1.65 and the unignited solution has initial condition T(i) 1.6. All units are nondimensionalised as given in the text.

198 A. James et al. / Combustion and Flame 134 (2003) 193 205 Fig. 3. The solution curve for the standard set of parameters (Table 1). The units are non-dimensionalised as given in the text. The temperature is the averaged value across all channels of the monolith. The points A, B, C refer to the profiles in Fig 4. by examining the effects of small changes in parameter values, establishing the generic behavior predicted by the model and, in particular, looking for any bifurcational changes that may occur, implying qualitative changes in the behavior of the real system. The analysis is carried out using path-following techniques that can reveal both the stable and unstable solutions as a parameter is varied. The power input is chosen as the primary continuation parameter to allow for easier comparison with previous models [3] and experimental work [23]. However, other continuation parameters could be chosen for example the inlet fuel concentration or temperature. Figure 3 shows the entire solution curve as the external power source is varied. The y-axis shows an averaged value of the temperature across the monolith. The upper and lower solution branches are stable and the whole of the intermediate branch is unstable. The ignition fold is located at P 2.5 and the extinction fold at P 0.2. As the power is increased from P 2toP 2.5 the system goes through a number of small ignition like events as the outer channels progressively approach ignition but the inner channels remain unignited. Temperature and fuel concentration profiles across the monolith for the three co-existing solutions A,BandCatP 1.96 are shown in Fig. 4. It is clear that the only fully ignited solution is C, on the upper branch. Figure 5 although 1 shows a brief structural stability analysis of the model. The standard solution curve of Fig. 3 is shown in each diagram. In addition to this each diagram contains two more solution curves each generated by increasing or decreasing a single parameter. Throughout extensive numerical experimentation the intermediate tangle of solution curves which lies approximately between P 0.3 and P 1.8 was always unstable. These branches Fig. 4. The three solutions corresponding to the marked points in Fig 3. a) the temperature profiles across the array; b) the fuel profiles. correspond to solutions which would never be seen in a physical situation; they are ignored in all further discussions. 3.1.1. The ignition event As mentioned previously, at the standard set of parameter values there are two ignition folds. The first is an ignition-like event where the outer channels of the array ignite; the second is the full ignition event where all the channels reach their ignited state. It is clear from the structural stability analysis that the first ignition-like event is very stable to parameter changes. Its position moves only as a result of changes in the Biot number or the ambient temperature. Conversely the full ignition event is very sensitive to changes in parameters. It moves to higher power values if either of the Damkohler numbers, the adiabatic flame rise or the mass transfer is decreased, or either of the activation energies, the heat transfer in the gas or the number of channels is increased. Physically, this could correspond to a decrease in the reactor length, or an increase in the velocity of the gas passing through the monolith. The full ignition event can also move to lower power values, however, and it is worth noting that if the position of the full

A. James et al. / Combustion and Flame 134 (2003) 193 205 199 Fig. 5. Structural stability. Each figure shows the standard solution curve as in Fig 3, in addition each parameter is perturbed upwards and downwards to illustrate the effects of varying the parameters. Stability of the solutions is not shown. ignition fold moves to a lower power than that of the smaller ignition-like fold, then the latter event becomes the full ignition event. A good example of this is the solution curves for changing catalytic activation energy; at c 19.8 the full ignition event clearly occurs at P 2. Figure 7 shows examples of the gas temperature solutions around these ignition events. The standard solution curve is labelled to show the illustrated solutions and all the integrations have the same initial conditions. In Fig. 7a, P 1.8, the system is wholly unignited. In Fig. 7b, P 2.3, this is above the ignition-like fold but below the full ignition fold, in this case the system very quickly reaches a steady state where the outer channels are partially ignited, the inner channels remain unignited. Finally, Fig. 7c shows P 2.5 the system quickly reaches the partially ignited solution and finally after a significant lapse of time the inner channels ignite to give the fully ignited solution. 3.1.2. The extinction event The structural stability of the extinction fold mirrors that of the ignition folds. Again, there are often two extinction events. In this case the first is often an extinction-like event, very stable at P 0.2. At this point the temperature in the outer channels starts to fall significantly but the inner channels remain fully

200 A. James et al. / Combustion and Flame 134 (2003) 193 205 ignited. The full extinction event is less stable to parameter changes. For some ranges of parameter space the full extinction fold moves to negative values of P and the system is autothermal. These ranges include increases in either of the Damkohler numbers or the adiabatic flame rise or decreases in the catalytic activation energy relative to the standard values. It must be noted that, even when the system is autothermal, the solution at the autothermal point is rarely the fully ignited one. At P 0.2 the extinction-like event occurs and unburnt fuel is able to leave the monolith through the outer channels. Figure 8 shows examples of the temperature profiles around these extinction events. The solution curve at c 19.8 is used as a good example of the Fig. 6. Comparison of the simplified CML model (solid line) and the standard plug-flow model used by James et al. [3] (dotted line) at the parameter values given in Table 2.

A. James et al. / Combustion and Flame 134 (2003) 193 205 201 Fig. 7. The evolution of the gas temperature at the three points, A, B, C on the standard solution curve (Fig 3) demonstrating the spatio-temporal structure of the three types of solution near the point of ignition. phenomena. A labelled version of this curve is shown. All the integrations have the same initial conditions. In Fig. 8a P 0.15 and the solution remains ignited. In Fig. 8b the power input is lowered to P 0.03 and the solution quickly becomes partially extinguished, the temperature drops significantly in the outer channels and the inner channels remain fully lit. Finally, in Fig. 8c there is no power input, P 0, and all the channels are fully extinguished though only after a significant amount of time. There are other interesting features of the structural stability analysis. When is decreased, Fig. 5xi, e.g. by lowering the fuel concentration, the power input needed to ignite the monolith is greatly increased. If the homogeneous reaction is removed by setting D H 0 the resulting solution curve is virtually indistinguishable from the curve for D H 1.2 10 3 shown in Fig. 5ii. Altering the number of channels, Fig. 5x, has a particularly dramatic effect on the ignition point of the curve, this is to be expected for the type of edge condition used here, clearly a monolith with more channels would need a higher power input on the edge channel to achieve ignition. However if the numbers of channels was increased whilst simultaneously altering the heat transfer co-efficients to represent narrower tubes (i.e., keeping the overall monolith diameter constant) the effect is substantially lessened. Alternatively, using a different continuation parameter, e.g., inlet temperature, also lessens the effects of changing the number of channels. 3.2. Comparison with similar models It is useful for any simplified model to be compared to more sophisticated models to establish that any essential features have not been lost. Figure 6 takes the results from James et al. [3, Fig. 3] which show the solution curve for a standard single channel plug-flow model at the parameter values given in Table 2 (dotted line). These are compared to the the coupled map lattice model at the same parameter values for a single, symmetric, adiabatic channel (solid line). However, the plug-flow results give the wall temperature at the channel inlet, whilst the CML model gives wall temperature corresponding at the outlet. Overall the comparison is favourable, the CML model now has a clear ignition and extinction fold like the plug-flow model. The extinction fold has moved to a negative power value making the system autothermal. The main difference between the two

202 A. James et al. / Combustion and Flame 134 (2003) 193 205 Fig. 8. The evolution of the gas temperature at the three points A, B, C on the solution curve for c 19.8, demonstrating the spatio-temporal structure of the three types of solution near the point of extinction. models is that for the CML model the two folds are at less extreme power values i.e. the hysteresis fold covers a smaller range of power values. 3.3. Other types of behavior Figure 9 shows a solution curve with all parameter values as in Table 1, with the exception of, which is increased to 2.22 10 3. This solution curve has two Hopf bifurcations, the first on the upper solution branch at P 2.5 with associated stable periodic solutions of significant amplitude between P 1.5 and P 2.0. The maximum and minimum values of these periodic solutions are shown. The upper stable solution between P 2.0 and P 2.5 is also periodic but with very small amplitude and is invisible on the scale of the figure. This Hopf bifurcation can be seen for a range of values. In general, as increases the bifurcation travels along the upper solution branch to higher power values. The second Hopf bifurcation is on the lower stable solution branch at P 1.7. This also has associated stable periodic orbits but again the amplitudes are very small. Figure 10 shows an example of the periodic solutions in Fig. 9 with P 1.8. The amplitude of the oscillations is largest in the centre of the monolith and is centred on the upper solution branch. In each oscillation the entire array heats and then cools. 3.4. Fuel maldistribution A common use of multi-channel models is to investigate the phenomena of fuel maldistribution. Table 2 Parameter values for comparison with a plug-flow model J H J D J W D C D H g s Bi 1 1 1 1.2 10 6 1 10 6 23 20 1 3 10 4 3 10 4 0 These parameters are taken from James et al. [3].

A. James et al. / Combustion and Flame 134 (2003) 193 205 203 Fig. 9. Solution curve with Hopf bifurcations and associated periodic orbits. All parameters are as in Table 1 except 2.22 10 3. Fig. 11. The two co-existing solutions for the fuel maldistribution example. This occurs when the fuel entering the monolith is not fully premixed and some channels have a higher fuel concentration than others. In this model this is easily incorporated by modifying equation 5 to give dy g i dt y g i J L D y L i 2y g i y R i D H y g i exp g (8) T g i In the channels where the usual amount of fuel enters the monolith, 1 as previously. Figure 11 shows an example where the four central channels have 25% extra fuel, this is similar to the example in [10]. To aid the comparison the external edge condition has been changed to adiabatic, i.e., B i 0. The model shows results which are qualitatively similar, except for a significant detail, this model shows there are two co-existing solutions. The first is unignited, the second is similar to that found in other models, i.e., the central channels, where the fuel input is higher, have a higher temperature. Fig. 10. An example of the periodic solutions shown in Fig. 9atP 1.8. 4. Summary and main conclusions A simple mathematical model for a multi-channel catalytic monolith has been presented. The model uses a linearised convection term which enables it to be expressed in terms of ODEs rather than PDEs. This simplification permits a dynamical systems approach and the numerical methods to be used include path-following techniques, which can find both stable and unstable solutions. The model is designed to give qualitative results over wide parameter ranges and it can be easily adapted to include turbulent or laminar flow profiles, fuel maldistribution, 2-D channel arrays etc. An overview reveals the types of behavior the model can display. These include co-existing solutions, hysteresis, Hopf bifurcations and periodic orbits. These phenomena have all been seen experimentally in monoliths [1,2] but have not been reproduced in multi-channel numerical models. This illustrates the strength of this simple dynamical system modelling. It reveals in an economical way the range of behaviour, much of it unwelcome in an engineering sense, which could occur if design or operating conditions were badly chosen. When a comprehensive map of behaviour in parameter space has been established, large scale numerics on much more realistic models (eg Kolaczkowski and Worth [11], Tischer et al. [14] can be used to establish quantitative results. The two approaches compliment each other in a mutually advantageous way. In this particular configuration the occurrence in a simple model of interesting behavior as seen in experiments, for wholly realistic parameter values, adds urgency to the need for further development and exploration of more sophisticated and complex numerical models.

204 A. James et al. / Combustion and Flame 134 (2003) 193 205 Acknowledgments A.J. was supported in this work by an EPSRC grant. Appendix A1 The homogeneous (gas-phase) reaction term in the ith channel is R g i A g C ox C pr y g i exp E g (A2) RT* g i, thus the gas phase energy balance in the ith channel of the array yields dt * g i S c p dt* S c T * g i T 0 pu 0 h T * L L i T * g i h T * R i T * g i S H R g i. The heterogeneous (surface) reaction term is (A3) R L,R i A c C pr y L,R i exp E c (A4) RT* L,R i, Both the heterogeneous and homogeneous reaction terms have the same heat of reaction but they have different activation energies and pre-exponential factors to represent the gaseous fuel being burnt with and without a catalyst respectively. The catalyst energy balance in the left side of the channel surface is dt * L i S c c c pc h T * dt* g i T * L i T * R i 1 T * L i H R L i, (A5) which contains heat transfer from the neighbouring channel. The equation for the right side is identical except for heat transfer with the (i 1)th channel rather than the (i 1)th. The mass balance in the gas phase is S dy* g i dt* S u y* g i y 0 0 h L D y* L i y* g i h D y* R i y* g i SR g i. (A6) The solid-phase mass balance for the left side of the channel surface is dy* L i S c c dt* h D y* g i y* L i R L i. (A7) An interesting question that arises at this point is the use of c. It is not immediately clear whether the gas density or the ceramic/washcoat density should be used at this point. The mass transfer occurs between the washcoat and the gas-phase so here it is more appropriate to use the gas density. However, the correct density for the transient term is less clear. To address this problem extensive parameter space explorations have been made to uncover the consequences of an incorrect choice of parameter value at this stage. A similar question arises over the choice of cross-sectional area to use, however this choice is of less significance because S c S, whereas the two densities differ by approximately 10 3. Temperatures are non-dimensionalised with respect to the inlet gas temperature, T 0, the mass fractions with respect to the inlet mass fraction, y 0, and time with respect to u 0 /L viz T g,l,r T * g,l,r y T g,l,r y* g,l,r t t*u 0 0 y 0 L. (A8) Thus rewriting the differential equations (A3,A5,A6,A7) yields dt g i dt T g i 1 J L H T L i 2T g i T R i D H y g i exp g T g i dt L i dt H T g i T L i J W T R i 1 T L i D C y L i exp c T L i dy g i dt (A9) (A10) y g i 1 J L D y L i 2y g i y R i dy L i dt D y g i y L i where D H y g i exp g T g i D C y L i exp c T L i (A11) (A12)

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