Quasi-steady State Assumption Research Articles

A New Approach to the Analytical Treatment of Steam-Assisted Gravity Drainage: A Prescribed Interface Model

Summary The majority of the models in the literature for the steam-assisted-gravity-drainage (SAGD) process solve the problem of conductive heat transfer ahead of a moving hot interface using a quasisteady-state assumption and extend the solution to the base of the steam chamber where the interface is not moving. This approach, as discussed by Butler (1985) and Reis (1992), results in inaccurate or sometimes infeasible estimations of the oil-production rate, steam/oil ratio (SOR), and steam-chamber shape. In this work, a new approach for the analytical treatment of SAGD is proposed in which the problem of heat transfer is directly solved for a stationary source of heat at the base of the steam chamber, where the oil production occurs. The distribution of heat along the interface is then estimated depending on the geometry of the steam chamber. This methodology is more representative of the heat-transfer characteristics of SAGD and resolves the challenges of those earlier models. In addition, it allows for the extension of the formulations to the early stages of the process when the side interfaces of the chamber are almost stationary, without loss of the solution continuity. The model requires the overall shape of the steam chamber as an input. It then estimates the movement of chamber interfaces using the movement of the uppermost interface point and by satisfying the global material-balance requirements. Oil-production rate and steam demand are estimated by Darcy's law and energy-balance calculations, respectively. The result is a model that is applicable to the entire lifetime of a typical SAGD project and provides more-representative estimations of in-situ heat distribution, bitumen-production rate, and SOR. With the improved knowledge obtained on the fundamentals of heat transfer in SAGD, the reason for the discrepancies between the various earlier models will be clarified. Results of the analytical models developed in this work show reasonable agreement with fine-scale numerical simulation, which indicates that the primary physics are properly captured. In the final section of the paper, the application of the developed models to two field case studies will be demonstrated.

On the role of vector modeling in a minimalistic epidemic model.

The motivation for the research reported in this paper comes from modeling the spread of vector-borne virus diseases. To study the role of the host versus vector dynamics and their interaction we use the susceptible-infected-removed (SIR) host model and the susceptible-infected (SI) vector model. When the vector dynamical processes occur at a faster scale than those in the host-epidemics dynamics, we can use a time-scale argument to reduce the dimension of the model. This is often implemented as a quasi steady-state assumption (qssa) where the slow varying variable is set at equilibrium and an ode equation is replaced by an algebraic equation. Singular perturbation theory will appear to be a useful tool to perform this derivation. An asymptotic expansion in the small parameter that represents the ratio of the two time scales for the dynamics of the host and vector is obtained using an invariant manifold equation. In the case of a susceptible-infected-susceptible (SIS) host model this algebraic equation is a hyperbolic relationship modeling a saturated incidence rate. This is similar to the Holling type II functional response (Ecology) and the Michaelis-Menten kinetics (Biochemistry). We calculate the value for the force of infection leading to an endemic situation by performing a bifurcation analysis. The effect of seasonality is studied where the force of infection changes sinusoidally to model the annual fluctuations of the vector population. The resulting non-autonomous system is studied in the same way as the autonomous system using bifurcation analysis.

A much better replacement of the Michaelis–Menten equation and its application

Michaelis–Menten equation is a basic equation of enzyme kinetics and gives acceptable approximations of real chemical reaction processes. Analyzing the derivation of this equation yields the fact that its good performance of approximating real reaction processes is due to Michaelis–Menten curve (8). This curve is derived from Quasi-Steady-State Assumption (QSSA), which has been proved always true and called Quasi-Steady-State Law by Banghe Li et al. [Quasi-steady state laws in enzyme kinetics, J. Phys. Chem. A 112(11) (2008) 2311–2321].Here, we found a polynomial equation with total degree of four [Formula: see text] (14), which gives more accurate approximation of the reaction process in two aspects: during the quasi-steady-state of the reaction, Michaelis–Menten curve approximates the reaction well, while our equation [Formula: see text] gives better approximation; near the end of the reaction, our equation approaches the end of the reaction with a tangent line the same to that of the reaction process trajectory simulated by mass action, while Michaelis–Menten curve does not. In addition, our equation [Formula: see text] differs to Michaelis–Menten curve less than the order of [Formula: see text] as [Formula: see text] approaches [Formula: see text].By considering the above merits of [Formula: see text], we suggest it as a replacement of Michaelis–Menten curve. Intuitively, this new equation is more complex and harder to understand. But, just because of its complexity, it provides more information about the rate constants than Michaelis–Menten curve does.Finally, we get a better replacement of the Michaelis–Menten equation by combing [Formula: see text] and the equation [Formula: see text].

Maini's many contributions to mathematical enzyme kinetics: A review

Modelling enzyme kinetics has a long history, as one of the earliest areas where mathematics was utilized to understand biological (in this case, biochemical) phenomena. By using differential equations and the law of mass action, many researchers have discovered information about the mechanisms of enzyme workings in the more than 100 years since Leonor Michaelis and Maud Menten published their classic paper in 1913. This work uses he laws obeyed by molecules during a reaction, translates them into mathematical equations, and then compares the results predicted by the equations to data obtained in laboratory conditions. This can inform whether the mechanism used to create the mathematical model is consistent with experimental results.Philip K. Maini has contributed to several significant results in the field of mathematical modelling in enzyme kinetics. Along with his collaborators, he has published results analyzing and extending the Quasi-Steady State Assumption, applying it to special enzyme reactions, and considering special conditions. This review paper will provide an overview of the contributions made by PK Maini to the field of modelling enzyme kinetics, summarize the main results and discuss their significance.

Point island dynamics under fixed rate deposition

In this paper we consider the dynamics of point islands during submonolayer deposition, in which the fragmentation of subcritical size islands is allowed. To understand asymptotics of solutions, we use methods of centre manifold theory, and for globalisation, we employ results from the theories of compartmental systems and of asymptotically autonomous dynamical systems. We also compare our results with those obtained by making the quasi-steady state assumption.

Effects of cascading failure intervals on synchronous stability

Cascading failure analyses are mainly based on simultaneous or quasi-steady-state (QSS) assumptions, which are inappropriate in cases of rapid cascading failure analysis and control. Fault interval is considered in the stability analysis of cascading failure for the first time in this article. Based on the extended equal area criterion (EEAC), a transient stability analysis method is proposed for cascading failure in consideration of intervals. The results show the periodicity of stable margin variations with intervals. The conclusions are validated in a Hamiltonian system, and the effects of time-varying factors are analyzed. The OMIB case shows that it may be over-optimistic to assess stability levels under simultaneous or QSS assumptions. The validity of all conclusions in this paper is verified with the New England system and with an actual power system.

Numerical Investigation of Combustion Characteristics of a Wave Rotor Combustor Based on a Reduced Reaction Mechanism of Ethylene

To improve simulations of the flame and pressure wave propagation process and investigate the combustion characteristics of a wave rotor combustor (WRC), direct relation graphs with error propagation (DRGEP), quasi-steady-state assumption (QSSA), and sensitivity analysis were used to establish a reduced reaction mechanism comprised of 23 species and 55 elementary reactions, based on the LLNL N-Butane mechanism. The reduced reaction mechanism of ethylene was combined with an eddy dissipation concept (EDC) model to simulate the flame propagation characteristics in a simplified WRC channel. The effects of spoilers with different blockage ratios and hot-jets of different species on combustion characteristics of flame propagation and pressure rise in the WRC channel were investigated. When the heated inert air was used as hot-jet, the ignition delay time of WRC would increase, which indicated that the activity of the burned gas from the hot-jet igniter would affect the ignition delay time. The spoiler facilitates the coupling of flame and shock waves to reduce the coupling time and distance. With the blockage ratio of the spoiler increasing, the coupling time and distance would be reduced.

Dynamical reduction of linearized metabolic networks through quasi steady state approximation

Metabolic modeling has gained accuracy in the last decades, but the resulting models are of high dimension and difficult to use for control purpose. Here we propose a mathematical approach to reduce high dimensional linearized metabolic models, which relies on time scale separation and the quasi steady state assumption. Contrary to the flux balance analysis assumption that the whole system reaches an equilibrium, our reduced model depends on a small system of differential equations which represents the slow variables dynamics. Moreover, we prove that the concentration of metabolites in quasi steady state is one order of magnitude lower than the concentration of metabolites with slow dynamics (under some flux conditions). Also, we propose a minimization strategy to estimate the reduced system parameters. The reduction of a toy network with the method presented here is compared with other approaches. Finally, our reduction technique is applied to an autotrophic microalgae metabolic network. © 2018 American Institute of Chemical Engineers AIChE J, 65: 18–31, 2019

The Rosenzweig-MacArthur system via reduction of an individual based model.

The Rosenzweig-MacArthur system is a particular case of the Gause model, which is widely used to describe predator-prey systems. In the classical derivation, the interaction terms in the differential equation are essentially derived from considering handling time vs. search time, and moreover there exist derivations in the literature which are based on quasi-steady state assumptions. In the present paper we introduce a derivation of this model from first principles and singular perturbation reductions. We first establish a simple stochastic mass action model which leads to a three-dimensional ordinary differential equation, and systematically determine all possible singular perturbation reductions (in the sense of Tikhonov and Fenichel) to two-dimensional systems. Among the reductions obtained we find the Rosenzweig-MacArthur system for a certain choice of small parameters as well as an alternative to the Rosenzweig-MacArthur model, with density dependent death rates for predators. The arguments to obtain the reductions are intrinsically mathematical; no heuristics are employed.

Steady thermocapillary migration of a droplet in a uniform temperature gradient combined with a radiation energy source at large Marangoni numbers.

Steady thermocapillary droplet migration in a uniform temperature gradient combined with a radiation energy source at large Reynolds and Marangoni numbers is studied. To reach a terminal quasisteady process, the magnitude of the radiation energy source is required to preserve the conservative integral thermal flux across the surface. Under a quasisteady state assumption, an analytical result for the steady thermocapillary migration of a droplet at large Reynolds and Marangoni numbers is derived by using the method of matched asymptotic expansions. It is shown that the thermocapillary droplet migration speed increases as the Marangoni number increases, while a radiation energy source with a sine square dependence is provided.

A comprehensively validated compact mechanism for dimethyl ether oxidation: an experimental and computational study

Dimethyl ether (DME) is regarded as one of the most promising alternatives to fossil fuels used in compression ignition engines. In order to critically evaluate its overall combustion behaviour via numerical simulations, an accurate as well as compact kinetic mechanism to describe its oxidation is most essential. In the present study, a short kinetic mechanism consisting of 23 species and 89 reactions is proposed to describe the oxidation of DME. This is based on the detailed San Diego mechanism. The short mechanism accurately reproduces the available experimental data for ignition delays, laminar flame speeds, and species profiles in flow reactors as well as jet-stirred reactors. To assess the validity of this reaction mechanism in non-premixed systems, extinction strain rates of DME–air mixtures, which are not available in the literature, are measured in a counter-flow diffusion flame burner as a part of the present work. The 23 species reaction mechanism is also able to predict the experimental data for extinction within the uncertainty limits. This mechanism is further reduced by introducing quasi-steady state assumptions for six intermediate species to finally obtain a 14-step global kinetic scheme. A code is developed in MATLAB to obtain these 14 global steps and their corresponding rate expressions in terms of the individual reaction rates. The 14-step mechanism performs as good as the 23 species mechanism for all the experimental data sets tested for.

Reaction Mechanism Reduction for Ozone-Enhanced CH4/Air Combustion by a Combination of Directed Relation Graph with Error Propagation, Sensitivity Analysis and Quasi-Steady State Assumption

In this study, an 18-steps, 22-species reduced global mechanism for ozone-enhanced CH4/air combustion processes was derived by coupling GRI-Mech 3.0 and a sub-mechanism for ozone decomposition. Three methods, namely, direct relation graphics with error propagation, (DRGRP), sensitivity analysis (SA), and quasi-steady-state assumption (QSSA), were used to downsize the detailed mechanism to the global mechanism. The verification of the accuracy of the skeletal mechanism in predicting the laminar flame speeds and distribution of the critical components showed that that the major species and the laminar flame speeds are well predicted by the skeletal mechanism. However, the pollutant NO was predicated inaccurately due to the precursors for generating NO were removed as redundant components. The laminar flame speeds calculated by the global mechanism fit the experimental data well. The comparisons of simulated results between the detailed mechanism and global mechanism were investigated and showed that the global mechanism could accurately predict the major and intermediate species and significantly reduced the time cost by 72%.

Efficient Solution of Washcoat Diffusion-Reaction Problem for Real-Time Simulations

As countries around the globe adapt more stringent emissions standards set by Real Driving Emissions (RDE) legislation, mathematical models are becoming ever more widely used as plant models for devising vehicle control strategies. It is important for the model to run on Hardware-in-Loop (HIL) and engine control unit (ECU) systems which have significantly less computational power and memory than modern personal computers. Washcoat diffusion limitations play a very important role in the efficient design of a catalytic converter. Numerical solution of aftertreatment models that include diffusion-reaction equations in the washcoat are computationally demanding. There are several simplified approaches proposed in the literature for the solution of diffusion-reaction equations in the washcoat to avoid the computational demand of the full numerical solution. In this paper, we use the recently proposed asymptotic solution and compare the results with that of the full numerical solution for the following aftertreatment reactor models with both single- and dual-layer washcoat configurations for the practical range of operating conditions; three-way catalyst (TWC), diesel oxidation catalyst (DOC), Selective Catalytic Reduction (SCR), and ammonia slip catalyst (ASC). These reactor models are constructed using published kinetic mechanisms and represent the global kinetics mechanisms (including non-linear reaction orders and inhibition functions) commonly used in the aftertreatment modeling community. We also discuss the importance of adaptive mesh, quasi-steady state assumption, and occurrence of concentration jumps in the simulation of aftertreatment reactors.

Effect of explicit dynamics of free virus and intracellular delay

In this paper, based on quasi steady-state assumption, a phenomenological model of viral infections with intracellular delay is proposed to explore viral infection process. The existence as well as stability of equilibria and bifurcation are studied. The results indicate that the basic reproduction number is a sharp threshold parameter to determine the outcomes of infection when the intracellular delay is omitted. Otherwise, Hopf bifurcation may occur under some conditions. Combining with numerical simulations, we observe that the explicit dynamics of free virus can affect the dynamics behavior, the fluctuation scenario may appears in the case of full logistic proliferation term even if there is no intracellular delay. In addition, there are different patterns of bifurcation under different drug effectiveness and patient’s age interacting with intracellular delay, which indicates that intracellular delay may induce complicated kinetics of virus infection.

Improved skeletal reduction on multiple gasoline-ethanol surrogates using a Jacobian-aided DRGEP approach under gasoline compression ignition (GCI) engine conditions

As an octane enhancer in commercial gasoline, ethanol is widely used by its oxygenated and renewable natures. In this study, five gasoline-ethanol blended fuels are studied with anti-knock index of 87–92: (1) Haltermann CARB LEV III E10 gasoline, (2) FACE C gasoline with 7% ethanol (by vol), (3) FACE C with 14% ethanol (by vol), (4) FACE J with 23% ethanol (by vol), (5) FACE J with 36% ethanol (by vol). First, surrogate models are formulated for the five fuels. By conducting a Jacobian-aided directed relation graph with error propagation (DRGEP) approach, a much improved performance is shown over the original counterpart for the multiple heavy fuel surrogate mechanism reduction. Based on the analysis, skeletal mechanisms generated after a hump region with an acceptable error in graph-based approaches is shown to deteriorate the further reduction potential. Starting from a detailed 1134-species mechanism, a 130-species skeletal mechanism and an 80-species reduced mechanism are finally developed with high fidelity under gasoline compression ignition (GCI) engine conditions, combining the DRGEP, target search algorithm (TSA) and quasi-steady state assumption (QSSA) approaches. Additionally, the low temperature heat releases in homogeneous charge compression ignition (HCCI) engine are validated under wide ranges of intake temperatures and pressures.

Development of a numerical model for fluid-structure interaction analysis of flow through and around an aquaculture net cage

In the present work, we developed a numerical model for fluid-structure interaction analysis of flow through and around an aquaculture net cage. The numerical model is based on the coupling between the porous media model and the lumped mass structural model. A novel interface was implemented to ensure efficient data exchange and element mapping between the fluid and structural solver via random-access memory. The main idea is to apply a static mesh in the fluid model, in case that large deformation of the net structure reduces the quality of the mesh. Then the geometry of the net cage was approximated by a set of dynamic porous zones, where the grid cells were updated at every iteration based on the transferred nodal positions from the structural model. A time stepping procedure was introduced, so the solver is applicable in both steady and unsteady conditions. In order to reduce the computational effort, sub-cycling was applied for the structural solver within each time step, based on the quasi-steady state assumption. The numerical model was validated against experiments in both steady and unsteady conditions. In general, the agreement is satisfactory.

Terminal thermocapillary migration of a droplet at small Reynolds numbers and large Marangoni numbers

In this paper, the overall steady-state momentum and energy balances in the thermocapillary migration of a droplet at small Reynolds numbers and large Marangoni numbers are investigated to confirm the quasi-steady state assumption of the system. The droplet is assumed to have a slight axisymmetric deformation from a sphere shape. It is shown that under the quasi-steady state assumption, the total momentum of the thermocapillary droplet migration system at small Reynolds numbers is conservative. The general solution of the steady momentum equations can be determined with its parameters depending on the temperature fields. However, a nonconservative integral thermal flux across the interface for the steady thermocapillary migration of the droplet at small Reynolds numbers and large Marangoni numbers is identified. The nonconservative integral thermal flux indicates that no solutions of the temperature fields exist for the steady energy equations. The terminal thermocapillary migration of the droplet at small Reynolds numbers and large Marangoni numbers cannot reach a steady state and is thus in an unsteady process.

Linear and nonlinear analyses of the effect of chemical reaction on the onset of buoyancy‐driven instability in a CO2 absorption process in a porous medium or Hele‐Shaw cell

AbstractTo understand CO2 absorption into a basic solution saturated porous medium or a Hele‐Shaw cell, the effect of the chemical reaction between dissolved CO2 and the basic reactant in the solution on the onset of a buoyancy‐driven instability in a gas absorption process is analyzed theoretically. For the infinitely fast reaction, new linear and non‐linear stability equations are derived without the quasi‐steady state assumption (QSSA) and solved analytically. The present stability analysis without the QSSA shows that the important parameters are the dimensionless densification numbers and the reactant concentration ratio. For the physically unstable system, the chemical reaction can induce a non‐monotonic density profile and has an important effect on the onset of instability motion. The effect of the reactant concentration ratio on the stability of the system is strongly dependent on the physical situation. Here, we call the system which is physically stable when there is no chemical reaction as the physically stable system. Even for this physically stable system, the chemical reaction makes the system unstable and induces the gravitational fingering instability motion. Also, it is found that the stability of the physically stable system is relatively insensitive to the non‐monotonicity of the density profile.

Automatic reduction and optimisation of chemistry for turbulent combustion modelling: Impact of the canonical problem

In a preliminary part, reduced chemical schemes from the literature on methane/air turbulent combustion are tested in one-dimensional premixed flames at various equivalence ratios. As discussed by Peters (1985), for the very same number of chemical species transported with the flow, a scheme introducing essential intermediate species through analytical relations derived from quasi-steady state and equilibrium assumptions, provides much better predictions than the global schemes limited to the transported species. Along these lines, a fully automated reduction procedure is discussed. The calibration of reduced chemistry usually relies on canonical problems, also computed with reference detailed chemical schemes. To cover at once a given range of chemical compositions, equivalence ratios and temperatures, the chemical properties of stochastic particles, submitted to micro-mixing and chemical reactions, are combined with the computation of deterministic one-dimensional composition-space trajectories, issued from an arbitrary number of inlet conditions. Along these trajectories, well-established methods to reduce the numbers of species and reactions are applied, cast in a fully automated manner. The rates of the resulting reduced scheme are then optimised, following a genetic algorithm, to match the detailed chemistry response. The discussed strategy is applied to methane/vitiated-air combustion. The accuracy of the resulting reduced scheme is evaluated in the simulation of freely propagating one-dimensional premixed flames, at various equivalence ratios, and also in strained diffusion flames in a one-dimensional counter-flowing jet configuration, up to the quenching point. The accurate reproduction of the flame speed response versus equivalence ratio requires adding premixed flames as target in the optimisation loop. Once the flame speed is captured, the quenching scalar dissipation rate of the diffusion flames is also well reproduced, as anticipated by Peters in a relation between the flame speed and the quenching scalar dissipation rate (Peters, 1991, 2000). Finally, the optimisation of the chemical rates is demonstrated under specific operating conditions, featuring four inlets with a large variety of flow compositions.

Fully three dimensional numerical analysis of industrial scale silicon Czochralski growth with a transverse magnetic field

Fully three dimensional (3D) numerical simulation model of the heat and mass transfer in the silicon (Si) Czochralski growth with a transverse magnetic field (MCZ) was developed. Time transient calculation of heat and mass transfer in the industrial scale crystal growth of 300mm diameter was performed under the quasi steady state assumption. The model predicts the completely asymmetrically 3D melt flow and off-centered temperature distributions, different from the previous works. The temperature distribution measured experimentally by thermocouples, beneath the melt surface along the azimuthal direction, was almost the same as the model prediction, providing a strong support for the 3D simulation model.