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There currently exist two methods for analysing an explosive mode introduced by chemical kinetics in a reacting process: the Computational Singular Perturbation (CSP) algorithm and the Chemical Explosive Mode Analysis (CEMA). CSP was introduced in 1989 and addressed both dissipative and explosive modes encountered in the multi-scale dynamics that characterize the process, while CEMA was introduced in 2009 and addressed only the explosive modes. It is shown that (i) the algorithmic tools incorporated in CEMA were developed previously on the basis of CSP and (ii) the examination of explosive modes has been the subject of CSP-based works, reported before the introduction of CEMA.

In order to obtain physical insights on ammonia combustion, which is characterised by exceptionally long ignition delays and increased NOx emissions, the autoignition dynamics of an ammonia/air mix...

A demonstration is provided on how algorithmic asymptotic analysis of multi-scale pharmacokinetics (PK) systems can provide (1) system level understanding and (2) predictions on the response of the model when parameters vary. Being algorithmic, this type of analysis is not hindered by the size or complexity of the model and requires no input from the investigator. The algorithm identifies the constraints that are generated by the fast part of the model and the components of the slow part of the model that drive the system within these constraints. The demonstration is based on a typical monoclonal antibody PK model. It is shown that the findings produced by the traditional methodologies, which require significant input by the investigator, can be produced algorithmically and more accurately. Moreover, additional insights are provided by the algorithm, which cannot be obtained by the traditional methodologies; notably, the dual influence of certain reactions depending on whether their fast or slow component dominates. The analysis reveals that the importance of physiological processes in determining the systemic exposure of monoclonal antibodies (mAb) varies with time. The analysis also confirms that the rate of mAb uptake by the cells, the binding affinity of mAb to neonatal Fc receptor (FcRn), and the intracellular degradation rate of mAb are the most sensitive parameters in determining systemic exposure of mAbs. The algorithmic framework for analysis introduced and the resulting novel insights can be used to engineer antibodies with desired PK properties.

Synergistic octane blending behavior of ethanol with gasoline and its surrogates has been observed by many researchers. The nonlinear octane boosting tendency is observed at mid and high molar blen...

Reactions contributing to the generation of the explosive time scale that characterise autoignition of homogeneous stoichiometric CH4/air mixture are identified using two different chemical kinetics models; the well known GRI-3.0 mechanism (53/325 species/reactions with N-chemistry) and the AramcoMech mechanism from NUI Galway (113/710 species/reactions without N-chemistry; Combustion and Flame 162:315-330, 2015). Although the two mechanisms provide qualitatively similar results (regarding ignition delay and profiles of temperature, of mass fractions and of explosive time scale), the 113/710 mechanism was shown to reproduce the experimental data with higher accuracy than the 53/325 mechanism. The present analysis explores the origin of the improved accuracy provided by the more complex kinetics mechanism. It is shown that the reactions responsible for the generation of the explosive time scale differ significantly. This is reflected to differences in the length of the chemical and thermal runaways and in the set of the most influential species.

A model reduction methodology, based on the quasi steady-state approximation (QSSA), is employed for the construction of reduced mechanisms in the case of an n-heptane/air premixed flame. Several issues related to the construction of these reduced mechanisms are discussed; such as the influence of the size of the starting skeletal mechanism, the stiffness reduction, and the truncation/simplification of (i) the expressions of the global rates and (ii) the steady-state relations. The starting point for the reduction is two skeletal mechanisms that involve 177/768 and 66/326 species/reactions, respectively [J. Prager, H.N. Najm, M. Valorani, and D.A. Goussis, Skeletal mechanism generation with CSP and validation for premixed n-heptane flames, Proc. Combust. Inst. 32 (2009), pp. 509–517] and which were derived from the detailed mechanism of Curran et al. [H.J. Curran, P. Gaffuri, W.J. Pitz, and C.K. Westbrook, A comprehensive modeling study of iso-octane oxidation, Combust. Flame 129 (2002), pp. 253–280], whi...

The multi-scale dynamics of a two-compartment with first order absorption Target-Mediated Drug Disposition (TMDD) pharmacokinetics model is analysed, using the Computational Singular Perturbation (CSP) algorithm. It is shown that the process evolves along two Slow Invariant Manifolds (SIMs), on which the most intense components of the model are equilibrated, so that the less intensive are the driving ones. The CSP tools allow for the identification of the components of the TMDD model that (i) constrain the evolution of the process on the SIMs, (ii) drive the system along the SIMs and (iii) generate the fast time scales. Among others, such diagnostics identify (i) the factors that determine the start and the duration of the period in which the ligand-receptor complex acts and (ii) the processes that determine its degradation rate. The counterintuitive influence of the process that transfers the ligand from the tissue to the main compartment, as it is manifested during the final stage of the process, is studied in detail.

Normal alkanes constitute a significant fraction of transportation fuels, and are the primary drivers of ignition processes in gasoline and diesel fuels. Low temperature ignition of n-alkanes is driven by a complex sequence of oxidation reactions, for which detailed mechanisms are still being developed. The current study explores the dynamics of low-temperature ignition of n-hexane/air mixtures, and identifies chemical pathways that characterize the combustion process. Two chemical kinetic mechanisms were selected as a comparative study in order to better understand the role of specific reaction sequences in ignition dynamics: one mechanism including a new third sequential O 2 addition reaction pathways (recently proposed by Wang et al. 2017), while the other without (Zhang et al. 2015). The analysis is conducted by applying tools generated from the computational singular perturbation (CSP) approach to two distinct ignition phenomena: constant volume and compression ignition. In both cases, the role of the third sequential O 2 addition reactions proves to be significant, although it is found to be much more pronounced in the constant volume cases compared to the HCCI. In particular, in the constant volume ignition case, reactions present in the third sequential O 2 addition reaction pathways (e.g., KDHP → products + OH) contribute significantly to the explosivity of the mixture; when accounted for along with reactions P(OOH) 2 + O 2 → OOP(OOH) 2 and OOP(OOH) 2 → KDHP + OH, they decrease ignition delay time of the mixture by up to 40%. Under HCCI conditions, in the first-stage ignition, the third-O 2 addition reactions contribute to the process, although their role decays with time and becomes negligible at the end of the first stage. The second ignition stage is dominated almost exclusively by hydrogen-related chemistry.

The effect of CO 2 dilution on the adiabatic and isochoric autoignition of CH 4 /air mixtures is analyzed with Computational Singular Perturbation (CSP) algorithmic tools, with a particular emphasis on the determination of the features of the chemical dynamics that control ignition delay and emission formation. Increasing CO 2 dilution causes longer ignition delays, lower final temperatures and decreased formation of NO and CO. These effects of CO 2 dilution are shown to be entirely thermal, contrary to what happens with dilution with H 2 O, which also has chemical activity and can reduce ignition delay. For the same initial mole fraction of the diluent, the decrease in final temperature and in NO concentration is larger in the CO 2 case whereas the decrease in CO is larger in the H 2 O case. The thermal effect of CO 2 is entirely analogous with those of dilution with the chemically inert Ar, only stronger for the same percentage of initial dilution, because of the larger specific heat of CO 2 . The reactions that have the largest contribution to the characteristic explosive time scale of the system during ignition delay (H 2 O 2 (+M) → OH + OH(+M), CH 3 O 2 + CH 2 O → CH 3 O 2 H + HCO, CH 4 + CH 3 O 2 → CH 3 + CH 3 O 2 H, H + O 2 → O + OH, etc.) are not substantially affected by CO 2 dilution, neither are the species that are pointed by CSP (CH 3 O 2 , H 2 O 2 , CH 2 O, etc.) as having the largest impact on the this timescale. The same holds for the modes that control CO and NO formation. The results point to the possibility of cold exhaust gas recirculation being used in order to produce mixtures with longer ignition delays and therefore substantial resistance to uncontrolled ignition.

The dynamics of a homogeneous adiabatic autoignition of an ammonia/air mixture at constant volume was studied, using the algorithmic tools of Computational Singular Perturbation. Since ammonia combustion is characterized by both unrealistically long ignition delays and elevated NO x emissions, the time frame of action of the modes that are responsible for ignition was analyzed by calculating the developing time scales throughout the process and by studying their possible relation to NO x emissions. The reactions that support or oppose the explosive time scale were identified, along with the variables that are related the most to the dynamics that drive the system to an explosion. It is shown that reaction H 2 O 2 (+M) → OH + OH (+M) is the one contributing the most to the time scale that characterizes ignition and that its reactant H 2 O 2 is the species related the most to this time scale. These findings suggested that addition of H 2 O 2 in the initial mixture will influence strongly the evolution of the process. It was shown that ignition of pure ammonia advanced as a slow thermal explosion with very limited chemical runaway. The ignition delay could be reduced by more than two orders of magnitude through H 2 O 2 addition, which causes only a minor increase in NO x emissions.

This paper examines the influence of wall reactions on the generation of the explosive time scale that characterizes ignition delay around the third explosion limit of a stoichiometric H2/a...

The Computational Singular Perturbation (CSP) algorithm is employed in order to determine how H 2 O -dilution influences ignition delay and chemical paths that generate NO during isochoric homogenous lean CH 4 /air autoignition. Regarding the ignition delay, it is shown that H 2 O -dilution enhances reactivity, mainly due to the increased OH production throughout the explosive stage via reaction H 2 O 2 ( + H 2 O ) → OH + OH ( + H 2 O ) . With regard to NO generation, the relative importance of thermal and chemical effects are examined and it is concluded that both are important. The thermal effects result in a lower temperature at the end of the explosive stage, while the most notable chemical effect is the lower level of O after this stage, mainly due to the effect of H 2 O -dilution on the equilibrium of the reaction O + H 2 O ↔ OH + OH . The depletion of O, together with the thermal effect, causes a substantial decrease in final NO generation.

The dynamics of homogeneous isochoric explosions of dimethylether (DME)/air and ethanol (EtOH)/air mixtures were studied and compared at relatively low initial temperatures (≤ 700 K) using algorithmic tools derived from the methodology of computational singular perturbation. In the DME case, it is shown that the low-temperature oxidation is dominated by reactions involving heavy carbonaceous species, in contrast to the high-temperature case where oxidation is dominated by reactions involving light carbonaceous species. Moreover, it is demonstrated that the outcome of the competition between two specific reactions is the cause of the exhibited negative temperature coefficient (NTC). In the EtOH case, the analysis points to the importance of carbonaceous species and in particular acetaldehyde, which is different from what happens at elevated initial temperatures (above 1000 K), where hydrogen chemistry dominates the entirety of the oxidation process. The autoignition dynamics of both mixtures are shown to b...

A detailed analysis is reported on a multiscale pharmacokinetic model, simulating the interaction of a drug with its target, the binding of the compounds and the outcome of their interaction. The analysis is based on the algorithmic computational singular perturbation (CSP) methodology. Among others, the analysis concludes that the partial equilibrium approximation and the quasi-steady-state approximation (PEA and QSSA) are valid in two distinct stages in the evolution of the process. Similar conclusions are reached from the algorithmic criteria for the validity of the QSSA and PEA. The reactions in the pharmacokinetic model that (i) generate the fast time scales, (ii) generate the constraints in which the system evolves and (iii) drive the system at various phases are identified, with the use of algorithmic CSP tools. These identifications are very important for the improvement of the model and for the identification of ways to control the evolution of the process. Regarding the qualitative understanding of the process, the present analysis systematises the findings in the literature and provides some new insights.

The effect of selected additives on the ignition delay of ethanol (EtOH)/air and dimethylether (DME)/air mixture is investigated. Computational Singular Perturbation (CSP) tools are utilized in an effort to determine algorithmically which species to select as additives and it is established that CSP can identify species whose addition to the mixture can affect ignition delay. However, this is not a necessary condition for additives to be effective. Additives that are not identified by CSP can have a substantial effect on ignition delay, provided that they drastically alter the prevailing chemistry, by altering the instant in time when the thermal runaway regime develops. Some of the additives that were studied computationally are unstable radicals whose injection in practical mixtures is unrealistic. However, chemically stable, relatively light species were also determined that can drastically affect ignition delay, such as hydrogen peroxide, formaldehyde and acetaldehyde.

When the fast dissipative time scales become exhausted, the evolution of reacting processes is characterized by slower time scales. Here the case where these slower time scales are of explosive character is considered. This feature allows for the acquisition of significant physical understanding; among others, the identification of intermediates in the reacting process that can be used as additives for the control of the ignition delay. The case of the homogeneous autoignition of CH 4 /air mixtures is analyzed here and the effects of adding the stable intermediates CH 2 O and H 2 O 2 to the fuel. These two species are identified as those relating the most to the explosive mode that causes autoignition, throughout the largest part of the ignition delay. Small quantities of these species in the initial mixture decrease considerably the ignition delay, by expediting the development of the thermal runaway.

The autoignition kinetics of DME/air and EtOH/air stoichiometric mixtures are compared with the use of algorithmic tools from the CSP method at a range of initial conditions that refers to the operation of reciprocating engines. DME and EtOH are two isomer fuels, with the potential for production from renewable sources, that have virtually identical thermochemistry; i.e. very closely equal heat of combustion and adiabatic flame temperature. These isomer fuels have drastically different ignition delays because of their different kinetics. In particular, the first and largest part of the ignition delay in the DME and EtOH cases is dominated by two different sets of components of carbon chemistry, while the last and shortest part is dominated by the same hydrogen chemistry. Considering sufficiently large initial temperatures, in the DME case the time scale that characterizes autoignition in the first part is promoted by single-carbon chemistry and is opposed mainly by recombination of CH 3 radicals. On the contrary, in the EtOH case the two-carbon chain retains its bond in that part. Therefore, the hydrogen chemistry plays an important role in promoting the generation of the time scale that characterizes autoignition from the start of the process, while the reactions that oppose the generation of this time scale involve HO 2 and H 2 O 2 and they are not as effective as the reactions opposing ignition for DME. These features generate a substantially shorter ignition delay for EtOH. This situation is reversed for sufficiently low initial temperatures due to the shift in relative importance between internal and external H-abstraction that occurs as temperature increases.

Appropriate algorithmic tools are employed for the analysis of the explosive modes developing during the autoignition of homogeneous mixtures. The ability of these tools to provide significant physical understanding is demonstrated in the case of the homogeneous ignition of a stoichiometric H2/air mixture, modelled by two different chemical kinetics mechanisms. It is shown that the ignition process evolves in two stages. The first stage is characterised by the development of two explosive timescales (one fast and one slow), that lead the system away from equilibrium. As the end of the first stage is approached, the two explosive timescales converge, they merge and then they disappear. In the second stage only dissipative timescales develop, which drive the system all the way to equilibrium. It is shown that throughout the first stage the fast explosive timescale is generated by chain reactions. The slow explosive timescale is initially generated by an initiation reaction that produces the radicals require...

A geometrical interpretation is provided for the validity of the model reduction methodology based on Partial Equilibrium Approximation , by comparing its algorithm to that of the methodology based on Singular Perturbation Analysis . The cases where the former methodology fails to provide a valid reduced model, while the latter one succeeds, are explored. It is shown that the failure of the Partial Equilibrium Approximation is due to the inability of (i) the stoichiometric vectors of the reactions considered in partial equilibrium to provide a leading order approximation of the fast directions in phase space or/and (ii) the equilibration of the related forward and backward rates to provide a leading order approximation of the equilibrations that develop under the action of the fast time scales. These issues are illustrated geometrically, by analyzing a number of simple examples.

The fast/slow dynamics of the Rossler model in the chaotic regime are compared to the dynamics of the reduced (slow) model governing the flow along the slow invariant manifold, on which the trajectory is restrained to evolve during the slow part of the attractor. It is shown that the dynamics of the reduced model incorporate the slow time scales of the full model. However, instead of the fast dissipative time scale of the full model that restrains the trajectory on the slow invariant manifold, the reduced model generates a new time scale that (i) relates to the curvature of the manifold and (ii) becomes faster as higher order corrections are incorporated in the approximation of the manifold and the reduced model. It is shown that this new time scale does not characterize the motion of the solution on the manifold but it can be employed as a signal for the strengthening or weakening of the manifold, depending on whether it relates to components of the reduced model that tend to lead its solution towards equilibrium or away from it. Finally, it is demonstrated that the new time scale introduced by the reduced model has a significant role in determining the maximum accuracy that can be delivered by the singular perturbation methodology.

The development of time scales that characterize the initiation of an auto-ignition processes is supported by specific reactions and is opposed by some others. An algorithmic tool which identifies these two sets of reactions is validated on the basis of the auto-ignition of a homogeneous stoichiometric methane/air mixture at 50 bar and 1100 K. It is shown that this process is characterized by two time scales, the fastest of which relates to the generation of the required for ignition radical pool and the temperature increase. This time scale associates mainly to carbon chemistry, except the very last period of its presence where it relates to hydrogen/oxygen-chemistry. The analysis allows for the quantitative assessment of several chemical phenomena that have been associated with the abnormal long ignition delay of methane, such as the formation of C2H6 and the relative importance of the formation of CH2O and CH3O as means of consumption of CH3 radicals. The identification of the reactions supporting or opposing the initiation of the auto-ignition is most useful when it is desired to control the ignition process.

Lactate in the brain is considered an important fuel and signalling molecule for neuronal activity, especially during neuronal activation. Whether lactate is shuttled from astrocytes to neurons or from neurons to astrocytes leads to the contradictory Astrocyte to Neuron Lactate Shuttle (ANLS) or Neuron to Astrocyte Lactate Shuttle (NALS) hypotheses, both of which are supported by extensive, but indirect, experimental evidence. This work explores the conditions favouring development of ANLS or NALS phenomenon on the basis of a model that can simulate both by employing the two parameter sets proposed by Simpson et al. (J Cereb. Blood Flow Metab., 27:1766, 2007) and Mangia et al. (J of Neurochemistry, 109:55, 2009). As most mathematical models governing brain metabolism processes, this model is multi-scale in character due to the wide range of time scales characterizing its dynamics. Therefore, we utilize the Computational Singular Perturbation (CSP) algorithm, which has been used extensively in multi-scale systems of reactive flows and biological systems, to identify components of the system that (i) generate the characteristic time scale and the fast/slow dynamics, (ii) participate to the expressions that approximate the surfaces of equilibria that develop in phase space and (iii) control the evolution of the process within the established surfaces of equilibria. It is shown that a decisive factor on whether the ANLS or NALS configuration will develop during neuronal activation is whether the lactate transport between astrocytes and interstitium contributes to the fast dynamics or not. When it does, lactate is mainly generated in astrocytes and the ANLS hypothesis is realised, while when it doesn't, lactate is mainly generated in neurons and the NALS hypothesis is realised. This scenario was tested in exercise conditions.

The Michaelis-Menten reaction scheme is among the most influential models in the field of biochemistry, since it led to a very popular expression for the rate of an enzyme-catalysed reaction. After the realisation that this expression is valid in a limited region of the parameter space, two additional expressions were later introduced. The range of validity of these three expressions has been studied thoroughly, since the significance of a reliable rate is not based only on the accuracy of its predictive abilities but also on the physical insight that is acquired in the process of its construction. Here a new expression for the rate is introduced that is valid in practically the full parameter space and reduces to the expressions in the literature, when considering appropriate limits. The new expression is produced by employing algorithmic tools for asymptotic analysis, so that its construction is not hindered by the dimensional form and the complexity of the full model or by a wide parameter range of interest. These tools can be employed for the derivation of enzyme rate expressions of much more complex kinetics mechanisms.

The results of a detailed, step-by-step, asymptotic analysis of the NF-κB signaling system are reported, in the case where the system exhibits limit cycle behavior. The analysis is based on the dimensional form of the model and exploits the various fast/slow time scale gaps that develop as the solution evolves along the limit cycle. It is shown that under the action of fast time scales of dissipative character, the limit cycle is confined on a low-dimensional surface in the phase space. The cycle can be divided in three parts, each one related to a different characteristic time scale. The first part refers to the slow rate of NF-κB release in the cytoplasm. The second part refers to the even slower rate by which the free NF-κB enters into the nucleus. The last part refers to the fast rate by which the nuclear NF-κB is first produced and then depleted. It is demonstrated that these and many other findings (regarding the fast variables, the equilibrated fast processes, the rate limiting and/or driving steps, etc.) can be acquired by simple algorithmic tools.

The development of dissipative time scales, which are much faster than the rest, force the solution of a system of 1st order ODEs to land and then evolve on a slow invariant manifold. The slow evolution on this manifold can be approximated by a reduced system, which is constructed with asymptotic analysis and is free of the fast scales. In this work, the dynamics of the reduced system are compared to the fast and slow dynamics of the original system. The comparison is based on the van der Pol system, which is typical of those exhibiting relaxation–oscillation behavior and is characterized by successive dissipative and explosive phases. Special attention is given to the dissipative phase, during which the solution evolves on a slow invariant manifold and the fast/slow time scale gap tends to contract. It is shown that one cannot predict from the dynamics of the van der Pol system the approaching explosive phase and the associated degeneracy of the manifold. Such a prediction is possible only through the dynamics of the reduced system, which involves an explosive component that is absent in the original system. This explosive component accelerates as higher order effects are incorporated in the reduced system, resulting in a smaller gap between the fast time scale of the original system and the fast time scale of the reduced one. This feature sets a limit on the accuracy singular perturbation techniques can deliver, even in the presence of a reasonably large fast/slow time scale gap in the dynamics of the original system. Given a fast/slow system, predicting forthcoming changes in the dimensions of the manifold is very important when it is desired to identify the underlying major physical mechanisms or to control the process.

The quasi steady state and partial equilibrium approximations are analysed in the context of a system of nonlinear differential equations exhibiting multiscale behaviour. Considering systems in the most general and dimensional form , it is shown that both approximations are limiting cases of leading-order asymptotics. Algorithmic conditions are established which guarantee that the accuracy and stability delivered by the two approximations are equivalent to those obtained with leading-order asymptotics. It is shown that the quasi steady state approximation is a limiting case of the partial equilibrium approximation. Algorithms are reported for the identification of the variables in quasi steady state and/or of the processes in partial equilibrium.

Large-scale models of cellular reaction networks are usually highly complex and characterized by a wide spectrum of time scales, making a direct interpretation and understanding of the relevant mechanisms almost impossible. We address this issue by demonstrating the benefits provided by model reduction techniques. We employ the Computational Singular Perturbation (CSP) algorithm to analyze the glycolytic pathway of intact yeast cells in the oscillatory regime. As a primary object of research for many decades, glycolytic oscillations represent a paradigmatic candidate for studying biochemical function and mechanisms. Using a previously published full-scale model of glycolysis, we show that, due to fast dissipative time scales, the solution is asymptotically attracted on a low dimensional manifold. Without any further input from the investigator, CSP clarifies several long-standing questions in the analysis of glycolytic oscillations, such as the origin of the oscillations in the upper part of glycolysis, the importance of energy and redox status, as well as the fact that neither the oscillations nor cell–cell synchronization can be understood in terms of glycolysis as a simple linear chain of sequentially coupled reactions.

We demonstrate the feasibility of a new strategy for the construction of an adaptive chemistry model that is based on an explicit integrator stabilized by an approximation of the Computational Singular Perturbation (CSP)-slow-manifold projector. We examine the effectiveness and accuracy of this technique first using a model problem with variable stiffness. We assess the effect of using an approximation of the CSP-slow-manifold by either reusing the CSP vectors calculated in previous steps or from a pre-built tabulation. We find that while accuracy is preserved, the associated CPU cost was reduced substantially by this method. We used two ignition simulations – hydrogen–air and heptane–air mixtures – to demonstrate the feasibility of using the new method to handle realistic kinetic mechanisms. We test the effect of utilizing an approximation of the CSP-slow-manifold and find that its use preserves the order of the explicit integrator, produces no degradation in accuracy, and results in a scheme that is com...

We use a procedure based on the decomposition into fast and slow dynamical components offered by the Computational Singular Perturbation (CSP) method to generate automatically skeletal kinetic mechanisms for the simplification of the kinetics of n-heptane oxidation. The detailed mechanism of the n-heptane oxidation here considered has been proposed by Curran et al. and involves 561 species and 2538 reactions. After carrying out a critical assessment of important aspects of this procedure, we show that the comprehensive skeletal kinetic mechanisms so generated are able to reproduce the main features of n-heptane ignition at various initial pressures and temperatures and equivalence ratios. A by-product of the algorithm that generates the skeletal mechanisms is the identification of the network of important species and reactions at a given state of the kinetic system. The analysis of this network is carried out by resorting to a visual representation of the pathways at selected time instants of the ignition process. Visual inspection of the pathways enables the identification and comparison of the relevant kinetic processes as obtained at different ignition regimes. The graphs are generated by interfacing the model reduction procedure with the open-source package graphviz.

The reactions responsible for the initial development of the time scale that characterizes the autoignition of a homogeneous stoichiometric CH4/air mixture are identified algorithmically using the computational singular perturbation method. The analysis considers a wide range of initial temperatures and pressures, located below and above the explosion limit. The major role of reactions CH3+O2→CH3O+O (at high temperatures), CH3+O2→CH2O+OH (at low temperatures), and CH2O+O2→HCO+HO2 (at intermediate temperatures) is identified. The results provide some additional insights regarding the significance to the initiation of the process of some reactions that are generally considered to become active at later times.

The relation between the iterative algorithms based on the computational singular perturbation (CSP) and the invariance equation (IE) methods is examined. The success of the two methods is based on the appearance of fast and slow time scales in the dynamics of stiff systems. Both methods can identify the low-dimensional surface in the phase space (slow invariant manifold, SIM), where the state vector is attracted under the action of fast dynamics. It is shown that this equivalence of the two methods can be expressed by simple algebraic relations. CSP can also construct the simplified non-stiff system that models the slow dynamics of the state vector on the SIM. An extended version of IE is presented which can also perform this task. This new IE version is shown to be exactly similar to a modified version of CSP, which results in a very efficient algorithm, especially in cases where the SIM dimension is small, so that significant model simplifications are possible.

A differential system that models the circadian rhythm in Drosophila is analyzed with the computational singular perturbation (CSP) algorithm. Reduced nonstiff models of prespecified accuracy are constructed, the form and size of which are time-dependent. When compared with conventional asymptotic analysis, CSP exhibits superior performance in constructing reduced models, since it can algorithmically identify and apply all the required order of magnitude estimates and algebraic manipulations. A similar performance is demonstrated by CSP in generating data that allow for the acquisition of physical understanding. It is shown that the processes driving the circadian cycle are (i) mRNA translation into monomer protein, and monomer protein destruction by phosphorylation and degradation (along the largest portion of the cycle); and (ii) mRNA synthesis (along a short portion of the cycle). These are slow processes. Their action in driving the cycle is allowed by the equilibration of the fastest processes; (1) the monomer dimerization with the dimer dissociation (along the largest portion of the cycle); and (2) the net production of monomer+dimmer proteins with that of mRNA (along the short portion of the cycle). Additional results (regarding the time scales of the established equilibria, their origin, the rate limiting steps, the couplings among themore » variables, etc.) highlight the utility of CSP for automated identification of the important underlying dynamical features, otherwise accessible only for simple systems whose various suitable simplifications can easily be recognized.« less

In systems of stiff Ordinary Differential Equations (ODEs) both fast and slow time scales are encountered. The fast time scales are responsible for the development of low-dimensional manifolds on which the solution moves according to the slow time scales. In this paper, methodologies for constructing highly accurate (i) expressions describing the manifold, and (ii) simplified non-stiff equations governing the slow evolution of the solution on the manifold are developed, according to an iterative procedure proposed in the Computational Singular Perturbation (CSP) method. It is shown that the increasing accuracy achieved with each iteration is directly related to the time rates of change of the CSP vectors spanning the manifold along the solution trajectory. Here, an algorithm is presented which implements these calculations and is validated on the basis of two simple examples.

The interaction of a two-dimensional counter-rotating vortex-pair with a premixed methane-air flame is analyzed with the Computational Singular Perturbation (CSP) method. It is shown that, as the fastest chemical time scales become exhausted, the solution is attracted towards a manifold, whose dimension decreases as the number of exhausted time scales increases. A necessary condition for a chemical time scale to become exhausted is that it must be much faster than the locally prevailing diffusion and convection time scales. Downstream of the flame, the hot products are in a regime of near-equilibrium, characterized by a large number of exhausted fast chemical time scales and the development of a low dimensional manifold, where the dynamics are locally controlled by slow transport processes and slow kinetics. In the flame region, where intense chemical and transport activity takes place, the number of exhausted chemical time scales is relatively small. The manifold has a large dimension and the driving time scale is set by chemical kinetics. In the cold flow region, where mostly reactants are present, the flow regime can be described as frozen, as the active chemical time scales are much slower than the diffusion and convection time scales; the driving scale set by diffusion. The algebraic relations among the elementary rates, which describe the manifold, are discussed along with a classification of the unknowns in three classes: i) CSP radicals; ii) trace; and, iii) major species. It is established that the optimal CSP radicals must be: i) strongly affected by the exhausted fast chemical time scales; and, ii) significant participants in the algebraic relations describing the manifold. The identification of CSP radicals, trace and major species, is a prerequisite for simplification or reduction of chemical kinetic mechanisms.

A numerical model for planar premixed flames of methane in ceramic porous media has been developed to improve the understanding of the structure of such flames. The model successfully reproduces experimental data for both single- and two-layer surface flames. The success is attributed to the detail given to the boundary conditions and the radiation modelling, which was done by solving the radiation transfer equation inside the porous medium without any simplifying models. Surface-stabilized flames yielded S L/S L0 1 and their energy balance was similar to that of a free flame, which implies that the burning velocity acceleration is due to the...

An algorithm is presented for the construction of global reduced mechanisms, based on concepts from the Computational Singular Perturbation method. Input to the algorithm are (i) the detailed mechanism, (ii) a representative numerical solution of the problem under investigation, and (iii) the desired number of steps in the reduced mechanism. The algorithm numerically identifies the “steady-state” species and fast reactions and constructs the reduced mechanism. The stoichiometric coefficients are constant and are connected to the non “steady-state” species, while the related rates involve the slow elementary rates only. The proposed method is applied to a laminar premixed CH4/Air flame and a complex detailed chemical kinetics mechanism, consisting of 279 reactions and 49 species and accounting for both thermal and prompt NOx production. A seven-step mechanism is constructed which is shown to reproduce the species profiles and the laminar burning velocity very accurately over a wide range of values for the initial mixture composition and temperature. In addition, it is shown that the seven-step mechanism introduces much lower time scales than the detailed mechanism does. Since the proposed procedure for constructing reduced mechanisms is fully algorithmic and requires minor computations, it is very much suited for the simplification of large detailed mechanisms.