A Singular Perturbation Problem for Steady State Conversion of Methane Oxidation in a Reverse Flow Reactor

The performance of a reverse flow reactor (RFR) is strongly influenced by the switching time used to alternate the flow direction. This research aimed to study the effect of the switching time on reactor dynamics including the heat propagation along the bed and reaction rate in the oxidation methane for low concentration using catalytic reverse flow reactor. The experimental results show that the reverse flow operating mode can influence heat propagation along the reactor and reaction conversion. Based on the three switching times was tested, the temperature dynamics formed were in the sliding regime. The effect of switching time on RFR on conversion is very significant. When compared to steady operation, RFR operation provides the highest conversion at smaller switching times. At large switching times, the effect of reversal of flow direction becomes less dominant and reactor behavior approaches steady state.

Process intensification in reverse flow reactors to boost various industrial applications: A review

Methane as a potential green house gases contributor which gives 21 GWP has to be mitigated to diminish the global warming effect. High concentration methane can be easily converted into CO 2 by mean oxidation. However lean methane can only be oxidized in catalytic reaction system as the catalyst lowers the reaction temperature up to 400 o C. Nevertheless, this is still high temperature to achieve by low concentration and low temperature feed. It still needs preheating the feed until its reaction condition reached which can be supplied outside or within the system called auto thermal. One of promising auto thermal reactor is reverse flow reactor which is the reactor that its feed flow periodically switches to make the heat trapped inside the reactor. In this work we have designed reverse flow reactor by one-dimensional model, pseudohomogeneous for mass and heterogeneous for energy to burn lean methane from a station compressor. The critical parameter of switching time on the system of periodical reversal is also presented

This paper suggests some estimators for population mean of the study variable in simple random sampling and two-phase sampling using information on an auxiliary variable under second order approximation. Bahl and Tuteja (1991) and Singh et al. (2008) proposed some efficient estimators and studied the properties of the estimators to the first order of approximation. In this paper, we have tried to find out the second order biases and mean square errors of these estimators using information on auxiliary variable based on simple random sampling and two-phase sampling. Finally, an empirical study is carried out to judge the merits of the estimators over others under first and second order of approximation.

The catalytic oxidation of methane for abating the emission vented from coal mine or natural gas transportation has been known as most reliable method. A reverse flow reactor operation has been widely used to oxidize this methane emission due to its capability for autothermal operation and heat production. The design of the reverse flow reactor requires a proper kinetic rate expression, which should be developed based on the operating condition. The kinetic rate obtained in the steady state condition cannot be applied for designing the reactor operated under unsteady state condition. Therefore, new approach to develop the dynamic kinetic rate expression becomes indispensable, particularly for periodic operation such as reverse flow reactor. This paper presents a novel method to develop the kinetic rate expression applied for unsteady state operation. The model reaction of the catalytic methane oxidation over Pt/-Al2O3 catalyst was used with kinetic parameter determined from laboratory experiments. The reactor used was a fixed bed, once-through operation, with a composition modulation in the feed gas. The switching time was set at 3 min by varying the feed concentration, feed flow rate, and reaction temperature. The concentrations of methane in the feed and product were measured and analysed using gas chromatography. The steady state condition for obtaining the kinetic rate expression was taken as a base case and as a way to judge its appropriateness to be applied for dynamic system. A Langmuir-Hinshelwood reaction rate model was developed. The time period during one cycle was divided into some segments, depending on the ratio of CH4/O2. The experimental result shows that there were kinetic regimes occur during one cycle: kinetic regime controlled by intrinsic surface reaction and kinetic regime controlled by external diffusion. The kinetic rate obtained in the steady state operation was not appropriate when applied for unsteady state operation. On the other hand, the kinetic rate expression obtained in the unsteady state operation fitted quite well. It was proven that in one cycle period the kinetic rate would always shift according to the ratio of CH4/O2.

The effect of water onto the rate and selectivity of methane and ethane oxidation in the conditions of oxidative coupling of methane (OCM) is studied. The effect of water strongly depends on the OCM catalyst composition: whereas PbOX/Al2O3 undergoes an irreversible deactivation, no effect of water is observed on both reaction rate and OCM selectivity over La/MgO. Over NaWMn/SiO2 catalyst both rate of reaction and selectivity are enhanced by water addition to the feed at low conversions during methane oxidation. In the case of ethane oxidation, the rate of reaction is strongly affected by water addition, whereas selectivity to ethylene does not change at equal conversions. At increasing concentration of water in the feed gas its relative effect onto the methane oxidation substantially decreases, so the increasing concentration of water above ~8 vol% does not further enhance the rate of methane oxidation. The observed effects are explained by the participation of water in the active site turnover, namely in its re-oxidation by shifting the prevailing re-oxidation route from filling surface oxygen vacancies to oxidative dehydrogenation of surface hydroxy-groups.

Reverse flow reactors are regarded as an appropriate tool for removing gaseous pollutants from industrial effluents. This paper presents the results of an experimental study on a particular reverse flow reactor with periodic variations in inlet concentration; these experimental results are compared with those obtained by numerical simulation studies. Catalytic oxidation of methane is chosen as model reaction. When the inlet concentration is low (i.e., close to extinction limits) and periodically varied, the interaction between the feeding cycle period and the flow-reversal cycle period may lead to reactor instability; adjusting the period of flow reversal is one way to maintain thermal stability in such situations. Results of numerical investigations reveal that if the flow-reversal cycle time is an odd multiple of the inlet variation cycle time, a harmonic response will occur, with sharply increased variation of the maximum temperature. Basically, it is the inequality of total methane input during the two half-cycles during flow-reversal operations that cause reactor instability. Adjusting the phase between the two cycle periods may ease the inequality of the inlet concentration values, which procures a second strategy to enhance reactor stability. The temperature maximum is very sensitive to the heat-transfer parameters, when periodic inlet variations are introduced.

Manipulation of the catalyst surface coverage for improved conversion and selectivity substantially requires more frequent flow reversals than the classical reverse flow operation for energy saving does (Chapter 1). 2.A reverse flow reactor should be operated at comparable switching time and residence time in order to manipulate the reactor's selectivity (Chapter 3).3. The concept of reverse flow operation with concentration programming provides an engineering tool to control conversion and selectivity, with three benefits: ammonia feedstock savings, increased ammonia conversion, and increased selectivity to NO; a drawback is the decrease of production rate (Chapter 4).4. The concept of reverse flow operation with reactor side feeding shows a fourfold superior performance: higher conversion of ammonia, higher selectivity to NO, no dead gas volume, and high production rate (Chapter 5).5. Intensification of a university study program must assure that the students' residence time distribution for completing their study follows a plug flow model.6.The concept of periodic reactor operation is natural.7. So called relaxation oscillation occurs when the cycling time period is much less than the system's time constant.However, this is not the case during reverse flow operation when operating the reactor at extremely short cycling time.8. The porosity of a fixed bed can be determined elegantly by a combination of reverse flow operation and cycling of the feed concentration.9. Joint bachelor-master-doctor programs improve the selectivity to master and doctor degrees as well as the productivity of a university.10. Assalamu'alaikum is a time-independent greeting.

Singular perturbation problems (SPP) have several origins in applied mathematics. One comes from fluid dynamics and results in linear boundary value problems containing a small parameter e (the coefficient of viscosity) such that for e → 0 the differential equation loses the highest derivative (see Exercise 1 below). Others originate in the study of nonlinear oscillations with large parameters (van der Pol 1926, Dorodnicyn 1947) or in the study of chemical kinetics with slow and fast reactions (see e.g., Example (IV.1.4)).

A new method is proposed for constructing a combined shock-capturing scheme that monotonically localizes shock wave fronts and, at the same time, has increased accuracy in smoothness regions of calculated generalized solutions. In this method, the solution of the combined scheme is constructed using monotonic solutions of a bicompact scheme of the first order of approximation in time and the fourth order of approximation in space obtained for different time steps in the entire computational domain. This construction method is much simpler than a previously proposed method. Test calculations are presented that demonstrate the advantages of the new scheme compared to the WENO5 scheme of the fifth order of approximation in space and the third order of approximation in time.

The International Workshop on Multi-Rate Processes and Hysteresis conference series focuses on singular perturbation problems and hysteresis as common strongly nonlinear phenomena occurring in mathematical, physical, economical, engineering and information systems. The term ‘strongly nonlinear’ means, in particular, that linearization will not encapsulate the observed phenomena. Singular perturbation problems and hysteresis can be manifested at different stages of the same or similar processes. Furthermore, a number of fundamental hysteresis models can be considered as a limit of time relaxation processes, or admit an approximation by a differential equation, which is singular with respect to a particular parameter. However, interaction between researchers in the areas of systems with time relaxation and systems with hysteresis (and between the ‘multi-rate’ and ‘hysteresis’ research communities) has so far been limited, and there is little cross-fertilization of ideas. It is the aim of the conference series to fill this gap.The 5th International Workshop on Multi-Rate Processes and Hysteresis (MURPHYS 2010) was hosted by the Pollack Mihály Faculty of Engineering, University of Pécs, Hungary, from 31 May to 3 June 2010, on the occasion of Pécs being the Cultural Capital of Europe in 2010. The workshop was organized in cooperation with University College Cork, Ireland, which hosted all of the previous Workshops: International Workshop on Multi-rate Processes and Hysteresis (University College, Cork, Ireland, 31 March–5 April 2008). Proceedings are published in Journal of Physics: Conference Series volume 138. See http://euclid.ucc.ie/appliedmath/murphys2008/murphys2008.htm; International Workshop on Multi-rate Processes and Hysteresis (University College, Cork, Ireland, 3–8 April 2006). Proceedings are published in Journal of Physics: Conference Series volume 55. Further information is available at http://Euclid.ucc.ie/murphys2006.htm; International Workshop on Hysteresis and Multi-scale Asymptotic (University College, Cork, Ireland, 17–21 March 2004). Proceedings are published in Journal of Physics: Conference Series volume 22. Further details are available at http://Euclid.ucc.ie/hamsa2004.htm; International Workshop on Relaxation Oscillations and Hysteresis (University College, Cork, Ireland, 1–6 April 2002). The related collection of invited lectures was published as a volume Singular Perturbations and Hysteresis, SIAM, Philadelphia, 2005. International Workshop on Geometrical Methods of Nonlinear Analysis and Semiconductor Laser Dynamics (University College Cork, Ireland, 5–6 April 2001). A collection of invited papers has been published as a special issue of Proceedings of the Russian Academy of Natural Sciences: Nonlinear dynamics of laser and reacting system, available at http://euclid.ucc.ie/appliedmath/gmna2001/ProcGMNA2001p1.pdf. Among the aims of this and previous workshops were: to bring together the leading experts in singular perturbation and hysteresis phenomena in applied problems; to discuss important problems in the areas of reacting systems, semiconductor lasers, shock phenomena, economic modelling, fluid mechanics, electrical engineering and modelling biological systems with emphasises on hysteresis and singular perturbations; to learn and share modern techniques in areas of common interest. The International Workshop on Multi-rate Processes and Hysteresis (Pollack Mihály Faculty of Engineering, University of Pécs, Hungary, 31 May–3 June 2010) brought together about 50 scientists who are actively researching the areas of dynamical systems with hysteresis and singular perturbations with applications to physical, engineering and economic systems. The countries represented at the Workshop included the Czech Republic, Germany, Hungary, Ireland, Israel, Italy, Poland, Romania, Russia, the United Kingdom and USA. Workshop photo 31 May 2010 Sponsorship of the Workshop by the Pollack Mihály Faculty of Engineering, University of Pécs (Hungary), University College Cork (Ireland), University of Pécs (Hungary), The University of Texas at Dallas (USA), and the Cultural Capital of Europe 2010, Pécs (Hungary), is gratefully acknowledged.The Editors and Organizers of the Workshop are sincerely grateful to Dr Géza Várady, Ms Andrea Zseni and Mr Ádám Schiffer of the Pollack Mihály Faculty of Engineering, University of Pécs, and Dr Alexander Pimenov of University College Cork for managing the organization of the conference and for the assistance in formatting of all the manuscripts.More information about the workshop can be found at http://murphys5.pmmk.pte.hu/Amalia Ivanyi, Péter Iványi, Dmitrii Rachinskii and Vladimir A Sobolev EditorsMURPHYS 2010, PMMK PTE, 31 May – 3 June 2010Sponsored by POLLACK MIHÁLY FACULTY OF ENGINEERING, UNIVERSITY OF PÉCS PÉCSI TUDOMÁNYEGYETEM UNIVERSITY OF PÉCS Cultural Capital of Europe 2010, Pécs, Hungary International Steering Committee Z I BalanovIsrael M BrokateGermany R CrossUK K DahmenUSA M DimianRomania G FriedmanUSA A Ivanyi (Co-Chairman)Hungary P Iványi (Co-Chairman)Hungary L KalachevUSA P KrejčíCzech Republic R O'Malley (Co-Chairman)USA A Pokrovskii (Co-Chairman)Ireland N PopovicUK D Rachinskii (Co-Chairman)Ireland S S SazhinUK V Sobolev (Co-Chairman)Russia S SzabóHungary C VisoneItalyInternational Program Committee G AlmásiHungary Z BalanovIsrael M BrokateGermany R CrossUK K DahmenUSA M DimianRomania G FriedmanUSA A Ivanyi (Co-Chairman)Hungary P Iványi (Co-Chairman)Hungary S JeneiHungary G KádárHungary L KalachevUSA R KersnerHungary G KovácsHungary P KrejčíCzech Republic P M KuczmannHungary P P O'KaneIreland R O'Malley (Co-Chairman)USA A Pokrovskii (Co-Chairman)Ireland N PopovicUK D Rachinskii (Co-Chairman)Ireland B V H ToppingUK V C VisoneItaly

A brief survey on numerical methods for solving singularly perturbed problems

Combined oxidation and reforming of methane to produce pure H 2 in a membrane reactor

Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.

In §6.2 in the last chapter we briefly discussed, by way of example, some singular perturbation problems in linear differential equations which could be solved using the exponential method developed there. In this chapter we discuss some more generally applicable singular perturbation techniques. These are extremely powerful and let us consider nonlinear problems, involving a small or large parameter, with a view to extracting the principle features of the solutions which cannot be found by classical methods. A fundamental property of nonlinear problems that we shall be interested in is that the main features of the solutions are not contained in the linearized problem as we see in §7.2 below.