Analytical Derivation of Moment Equations in Stochastic Chemical Kinetics

The master probability equation captures the dynamic behavior of a variety of stochastic phenomena that can be modeled as Markov processes. Analytical solutions to the master equation are hard to come by though because they require the enumeration of all possible states and the determination of the transition probabilities between any two states. These two tasks quickly become intractable for all but the simplest of systems. Instead of determining how the probability distribution changes in time, we can express the master probability distribution as a function of its moments, and, we can then write transient equations for the probability distribution moments. In 1949, Moyal defined the derivative, or jump, moments of the master probability distribution. These are measures of the rate of change in the probability distribution moment values, i.e. what the impact is of any given transition between states on the moment values. In this paper we present a general scheme for deriving analytical moment equations for any N-dimensional Markov process as a function of the jump moments. Importantly, we propose a scheme to derive analytical expressions for the jump moments for any N-dimensional Markov process. To better illustrate the concepts, we focus on stochastic chemical kinetics models for which we derive analytical relations for jump moments of arbitrary order. Chemical kinetics models are widely used to capture the dynamic behavior of biological systems. The elements in the jump moment expressions are a function of the stoichiometric matrix and the reaction propensities, i.e. the probabilistic reaction rates. We use two toy examples, a linear and a non-linear set of reactions, to demonstrate the applicability and limitations of the scheme. Finally, we provide an estimate on the minimum number of moments necessary to obtain statistical significant data that would uniquely determine the dynamics of the underlying stochastic chemical kinetic system. The first two moments only provide limited information, especially when complex, non-linear dynamics are involved.     

具有随机时延和丢包的网络化离散系统的动态输出反馈H_∞控制

针对前向通道和反馈通道均存在随机时延和丢包现象的网络控制系统(NCSs),研究了关于离散时域下鲁棒H∞控制器的设计问题。采用伯努利分布描述随机时延和丢包现象,将闭环NCSs建模为随机参数系统。根据Lyapunov稳定性理论和增广状态空间法,得到闭环NCSs均方指数稳定的H∞性能判据;利用LMI技术和锥补线性化算法,给出动态输出反馈控制器的设计方法。最后,把这种方法应用于搅拌斧反应器中,仿真验证了所提控制器设计方法的有效性。     

Probabilistic model of dispersed turbulent flow in channels with rough walls

Abstract

An equation for the probability density function (PDF) for particle velocity and coordinates in a gas turbulent flow is derived. The system of equations for the first and second moments of particle velocity fluctuations is obtained. Using a method similar to Grad’s method, an approximate solution of the PDF equation was found. Based on this approximate solution, the system of equations for the averaged concentration, velocity, and second moments of particle velocity fluctuations was closed. Also, using an approximate solution, the boundary conditions on the rough wall of the channel were obtained. The boundary conditions self-consistently take into account the direction of the velocity vector of particles colliding with the surface, as well as the direction of the normal to a random plane that simulates the roughness. The fundamental difference between the mechanisms of generation of random motion of particles in channels with smooth and rough walls is shown.

Copyright © 2020 American Association for Aerosol Research     

Eulerian Monte Carlo method for the joint velocity and mass-fraction probability density function in turbulent reactive gas flows

A new Eulerian (field) Monte Carlo method for solving an equation that describes the one-time one-point probability density of species mass fractions in turbulent reactive gas flows has been proposed in a previous article. In the present paper, this method is extended to an equation for the joint velocity and mass-fraction probability density function. The method is based on passing from Lagrangian variables used in Lagrangian Monte Carlo methods to Eulerian variables. In this manner, stochastic ordinary differential equations for the Lagrangian trajectories of fluid particles are transformed to partial stochastic equations. As compared to the classical hydrodynamics, the stochastic velocity field satisfies only the mean continuity constraint and not the instantaneous one. As a consequence, one has to introduce a stochastic density, which differs from the physical density but has the same mean value. The case of the mass-fraction probability density is revised. The equations differ from those derived previously: they can be written in divergent form. Both formulations, however, are statistically equivalent. __________ Translated from Fizika Goreniya i Vzryva, Vol. 42, No. 6, pp. 144–155, November–December, 2006.     

萃取柱内液-液两相流CFD-PBM模拟研究进展

对萃取柱内CFD-PBM模拟研究进行了较详细的综述,包括其基本理论、不同的求解方法及模拟研究现状等. CFD-PBM模拟的基本方程包括流动方程和群体平衡方程,其相互耦合,群体平衡方程涉及破碎与聚并2个关键模型. 群体平衡模型的求解方法包括直接离散化方法、矩量法、正交矩量法、直接正交矩量法、分段正交矩量法等,对这些方法的原理、优点和缺点进行了综述. 目前国际上关于萃取柱内CFD模拟采用较多的是简单的欧拉-欧拉两相流模拟,考虑液滴尺寸分布和进一步的浓度分布的群体平衡模型应用较少. 完善伴随传质的液-液分散体系的群体平衡模型,并将其应用于不同类型的萃取柱中,是萃取分离学科的重要任务.     

Optimal discrete stochastic control theory for process application

The design of discrete feedback controllers which minimize some linear function of the variances of the output deviations from target subject to possible constraints on the variances of the inputs, for linear systems subject to stochastic disturbances, is treated from two points of view: (1) using transfer function models to characterizing the process dynamics and autoregres-sive-moving-average models to characterize the stochastic disturbances, and then solving the optimal control problem using an approach due to Box and Jenkins and a discrete version of the Wiener-Newton theory; and (2) using state variable models to characterize both the dynamic and stochastic parts of the system, and then solving the optimal control problem using the results of dynamic programming and Kalman filtering. Practical considerations such as model forms, their identification and estimation, and the development of variance relationships that are necessary for the application of these two approaches in the process industries are discussed. The relationship between and a comparison of these two approaches is made.     

Difference Equations for the Higher-Order Moments and Cumulants of the INAR(1) Model

Recently, as a result of the growing interest in modelling stationary processes with discrete marginal distributions, several models for integer value time series have been proposed in the literature. One of these models is the INteger-AutoRegressive (INAR) model. Here we consider the higher-order moments and cumulants of the INAR(1) process and show that they satisfy a set of Yule–Walker type difference equations. We also obtain the spectral and bispectral density functions, thus characterizing the INAR(1) process in the frequency domain. We use a frequency domain approach, namely the Whittle criterion, to estimate the parameters of the model. The estimation theory and associated asymptotic theory of this estimation method are illustrated numerically.     

Solution of the population balance equation using the sectional quadrature method of moments (SQMOM)

A discrete framework is introduced for simulating the particulate physical systems governed by population balance equations (PBE) with particle splitting (breakage) and aggregation based on accurately conserving (from theoretical point of view) an unlimited number of moments associated with the particle size distribution. The basic idea is based on the concept of primary and secondary particles, where the former is responsible for distribution reconstruction while the latter is responsible for different particle interactions such as splitting and aggregation. The method is found to track accurately any set of low-order moments with the ability to reconstruct the shape of the distribution. The method is given the name: the sectional quadrature method of moments (SQMOM) and has the advantage of being not tied to the inversion of large sized moment problems as required by the classical quadrature method of moments (QMOM). These methods become ill conditioned when a large number of moments are needed to increase their accuracy. On the contrary, the accuracy of the SQMOM increases by increasing the number of primary particles while using fixed number of secondary particles. Since the positions and local distributions for two secondary particles are found to have an analytical solution, no large moment inversion problems are anymore encountered. The generality of the SQMOM is proved by showing that all the related sectional and quadrature methods appearing in the literature for solving the PBE are merely special cases. The method has already been extended to bivariate PBEs.     

Probabilistic reactor design in the framework of elementary process functions

Computational process models in combination with innovative design methodologies provide a powerful reactor design platform. Yet, model-based design is mostly done in a pure deterministic way. Possible uncertainties of the underlying model parameters, prediction errors due to simplifying assumptions regarding the reactor behavior and suboptimal realizations of the design along the reaction coordinate are in general not considered. Here we propose a systematic design approach to directly account for the impact of such variabilities during the design procedure. The three level design approach of Peschel et al. (2010) based on the concept of elementary process functions (EPF) serves as basis. The dynamic optimizations on each level are extended within a probabilistic framework to account for different sources of randomness. The impact of these sources on the performance prediction of a design is quantified and used to robustify the reactor design aiming at a more reliable performance and thus design prediction. The uncertainties of model parameters, non-idealities of the reactor behavior and inaccuracies in the design are included via statistical moments. By means of the sigma point method (Julier and Uhlmann, 1996) random variables are mapped to the design objective space via the nonlinear process model. Importantly, this work introduces a full probabilistic orthogonal collocation approach, i.e. random and stochastic variables can be described. Whereas the former one relates to randomness independent on the reaction time (e.g. kinetic model parameters or initial conditions), the latter one describes stochasticity along the reaction time (e.g. fluctuating pressure or temperature control). As an example process the hydroformylation of 1-dodecene in a thermomorphic solvent system consisting of n-decane and N,N-dimethylformamide is considered.Our probabilistic EPF approach allows designing robust optimal reactors, which operate within an estimated confidence at their expected optimum considering almost any kind of randomness arising in the design procedure. An additional value is that with increased predictive power of the reactor performance its embedding in an overall process is strongly simplified.     

Comparison of parameter estimation methods in stochastic chemical kinetic models: Examples in systems biology

Stochastic chemical kinetics has become a staple for mechanistically modeling various phenomena in systems biology. These models, even more so than their deterministic counterparts, pose a challenging problem in the estimation of kinetic parameters from experimental data. As a result of the inherent randomness involved in stochastic chemical kinetic models, the estimation methods tend to be statistical in nature. Three classes of estimation methods are implemented and compared in this paper. The first is the exact method, which uses the continuous‐time Markov chain representation of stochastic chemical kinetics and is tractable only for a very restricted class of problems. The next class of methods is based on Markov chain Monte Carlo (MCMC) techniques. The third method, termed conditional density importance sampling (CDIS), is a new method introduced in this paper. The use of these methods is demonstrated on two examples taken from systems biology, one of which is a new model of single‐cell viral infection. The applicability, strengths and weaknesses of the three classes of estimation methods are discussed. Using simulated data for the two examples, some guidelines are provided on experimental design to obtain more information from a limited number of measurements. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1253–1268, 2014     

复合材料层合板非线性系统的随机响应分析

工程结构中,复合材料的几何参数往往具有随机性质,如何研究随机参数非线性系统的随机响应及统计特性,这对结构的可靠性设计和优化设计有着非常重要的意义。本文应用摄动法和随机中心差分法,建立了复合材料非线性系统的振动方程和计算模型,研究了复合材料层合板具有随机参数的非线性系统在确定性荷载下的随机响应,数值算例说明了本算法的正确性。     

Stochastic simulation of population balance models with disparate time scales: Hybrid strategies

The Monte Carlo methods have been an effective tool for the numerical solution of population balance models (PBMs). They are particularly useful for complex multidimensional problems. Less attention has been paid to solving population balance models where some species are away from the thermodynamic limit (very dilute or finite) and other species can be considered deterministic (high concentration). These types of problem often result in a stochastic system with rates spanning orders of magnitude for different mechanisms. Using the exact Monte Carlo solution to solve these types of problem is very inefficient because of the simulation time spent sampling fast events. These fast events are associated with species with large populations for which a single event does not change the population appreciably. This frequent sampling of fast events becomes a bottleneck during a simulation in which many single MC steps are required to make an appreciable change in the population.In this work, a hybrid solution strategy is developed to effectively solve this type of problem. The method implements the self-consistent fast/slow partitions used to solve stochastic equations in chemical kinetics. One strategy is found on the capacity of a coarse-grained Markov model called particle ensemble random product (PERP) to accelerate the simulation of fast events of PBMs (Chem. Eng. Sci. 63, 7649–7664; Chem. Eng. Sci. 63, 7665–7675). A second strategy approximates the fast events using mass conservation equations. These models are coupled with the exact MC simulation of slow events. Two extreme cases of heterocoagulation are studied to demonstrate these hybrid strategies.     

MODELLING AND SCALEUP OF DISPERSED PHASE LIQUID-LIQUID REACTORS

The problem of modelling dispersed phase liquid-liquid reactors is discussed from a global view. The two major areas of microscopic and macroscopic problems are addressed. The microscopic problem is concerned with the determination of the local rate of transfer of reactants and/or products between the two phases. Attention is focussed on approaches to obtain kinetic data; and results on recent important chemical systems such as nitrations, metal chelation reactions, and phase transfer catalytic reactions are discussed. The liquid jet recycle reactor is pointed out as a useful tool for obtaining laboratory data. Recent works employing the classical film and penetration theories to obtain flux expressions for complex reactions are described. The macroscopic problem deals with the reactor design question. Various models proposed to account for macromixing and/or micromixing effects are categorized into noninteraction and coalescence- dispersion or interaction models. The former approach includes the axial dispersion and CSTR models and can predict conversions at the extremes of micromixing. (See the previous paper in this series by Nauman for complete discussion of these models.) The basic formulations of these models and results are discussed in this paper, The latter approach discussed here includes population balance equations and Monte Carlo simulation methods. The ability of Monte Carlo simulation techniques to predict the effect of intermediate degrees of micromixing on conversion is demonstrated. The potential of the Monte Carlo simulation technique to account for local variations in dispersion properties, model droplet rate processes, and model complex reaction systems is also shown. (See the paper by Patterson in this series for more discussion of the application of Monte Carlo stimulation to complex reactions.)     

Separation and energy efficiency of packed apparatuses for purifying gases from aerosols

A mathematical model of the turbulent transport of liquid aerosol particles has been considered using an approach in which the transport of a finely dispersed phase is conditionally described as a diffusion process with mass-transfer theory equations. The main parameter in this formulation is the coefficient of the velocity of particle transport (turbulent migration), which is an analogue of the mass transfer coefficient. The use of flow pattern models with a bulk source of the mass of sedimenting particles in a random packed bed and wire-mesh demisters is demonstrated. The equations for calculating the separation zone length and the process efficiency have been derived with allowance for the inlet section and the back mixing of the gas flow. Some calculation results are given for the aerosol separation efficiency and the energy coefficient compared with experimental data.     

Modeling an annular flow tubular reactor

A stochastic model representing annular flow in a tubular reactor is proposed. Numerical simulation was utilized to generate sample paths fitting the residence time distributions (RTD) of the system. The model was constructed from basic diffusion equations with the additional consideration of random effects disturbing the system, thus yielding a stochastic partial differential equation. The stochastic model is simulated using the Euler–Maruyama procedure. Experimental data from three tubular polymerization reactors were well fitted by the model. The model encompassed the two deterministic parameters, mean residence time and Peclet number, as well as three stochastic parameters: stochastic relevance (b), updated time (ΔT) and the seed that begins the Wiener process. The satisfactory results indicate that the model constitutes an important step toward comprehending the complex fluid dynamics of tubular flow systems.     

Big data for microstructure‐property relationships: A case study of predicting effective conductivities

The analysis of big data is changing industries, businesses and research as large amounts of data are available nowadays. In the area of microstructures, acquisition of (3‐D tomographic image) data is difficult and time‐consuming. It is shown that large amounts of data representing the geometry of virtual, but realistic 3‐D microstructures can be generated using stochastic microstructure modeling. Combining the model output with physical simulations and data mining techniques, microstructure‐property relationships can be quantitatively characterized. Exemplarily, we aim to predict effective conductivities given the microstructure characteristics volume fraction, mean geodesic tortuosity, and constrictivity. Therefore, we analyze 8119 microstructures generated by two different stochastic 3‐D microstructure models. This is—to the best of our knowledge—by far the largest set of microstructures that has ever been analyzed. Fitting artificial neural networks, random forests and classical equations, the prediction of effective conductivities based on geometric microstructure characteristics is possible. © 2017 American Institute of Chemical Engineers AIChE J, 63: 4224–4232, 2017     

Spatial distributions of mean age and higher moments in steady continuous flows

Transport equations and boundary conditions for spatial distribution of age moments in steady continuous flows are derived. Mean age is the first moment. The coefficient of variation is obtained from the second moment. Mixing‐cup averaged mean age and higher moments across the exit plane are identical to the corresponding moments of the residence‐time distribution. Numerical solutions for a 2‐D (two‐dimensional) reactor are studied and compared with those from a transient tracer equation. Agreement is excellent. Local tracer distribution function curves reveal that mean age is located on the long tail for both convection dominated short circuiting paths and diffusion dominated dead zones. Computing cost for the mean age and higher moment equations is orders of magnitude lower than that for the transient tracer concentration equation, making this mean age method an efficient tool to study mixing in steady continuous flow systems. © 2010 American Institute of Chemical Engineers AIChE J, 2010