Математическая модель катализатора синтеза с локальными центрами реакции

The article considers a catalyst granule with a porous ceramic passive substrate and point active centers on which an exothermic synthesis reaction occurs. A rate of the chemical reaction depends on the temperature according to the Arrhenius law. Heat is removed from the pellet surface in products of synthesis due to heat transfer. In our work we first proposed a model for calculating the steady-state temperature of a catalyst pellet with local reaction centers. Calculation of active centers temperature is based on the idea of self-consistent field (mean-field theory). At first, it is considered that powers of the reaction heat release at the centers are known. On the basis of the found analytical solution, which describes temperature distribution inside the granule, the average temperature of the reaction centers is calculated, which then is inserted in the formula for heat release. The resulting system of transcendental algebraic equations is transformed into a system of ordinary differential equations of relaxation type and solved numerically to achieve a steady-state value. As a practical application, the article considers a Fischer-Tropsch synthesis catalyst granule with active cobalt metallic micro-particles. Cobalt micro-particles are the centers of the exothermic reaction of hydrocarbons macromolecular synthesis. Synthesis occurs as a result of absorption of the components of the synthesis gas on metallic cobalt. The temperature distribution inside the granule for a single local center and reaction centers located on the same granule diameter is found. It was found that there is a critical temperature of reactor exceeding of which leads to significant local overheating of the centers - thermal explosion. The temperature distribution with the local reaction centers is qualitatively different from the granule temperature, calculated in the homogeneous approximation. It is shown that, in contrast to the homogeneous approximation, the temperature of the granule surface with local centers cannot serve as a reliable criterion for the thermal state of the synthesis centers inside the granule.This work was supported by RFBR grant № 17-08-00376.Рассматривается гранула катализатора с пористой керамической пассивной основой и с точечными активными центрами, на которых происходит экзотермическая реакция синтеза. Скорость химической реакции зависит от температуры по закону Аррениуса. С поверхности гранулы тепло отводится в продукты синтеза за счет теплоотдачи. В нашей работе впервые предложена модель для расчета стационарной температуры гранулы катализатора с локальными центрами реакции. Расчет температуры активных центров основан на идее самосогласованного поля. Вначале считается, что мощности тепловыделения центров реакции известны. На основе найденного аналитического решения, описывающего распределение температуры внутри гранулы, рассчитывается средняя температура центров реакции, которая затем подставляется в формулу для тепловыделений. Полученная система трансцендентных алгебраических уравнений преобразуется к системе обыкновенных дифференциальных уравнений релаксационного типа и решается численно до достижения стационарного значения. В качестве примера рассмотрена гранула катализатора синтеза Фишера–Тропша с активными микрочастицами металлического кобальта. Микрочастицы кобальта являются центрами экзотермической реакции синтеза высокомолекулярных углеводородов. Синтез происходит в результате абсорбции компонентов синтез-газа на металлическом кобальте. Найдено распределение температуры внутри гранулы для одиночного локального центра и центров реакции, расположенных на одном диаметре гранулы. Установлено существование критической температуры реактора, превышение которой приводит к существенному перегреву локальных центров – тепловому взрыву. Распределение температуры с локальными центрами реакции качественно отличается от температуры гранулы, рассчитанной в гомогенном приближении. Показано, что в отличие от гомогенного приближения температура поверхности гранулы с локальными центрами не может служить однозначным критерием теплового состояния центров синтеза внутри гранулы.    Работа поддержана фондом РФФИ грант № 17-08-00376

Флуктуации скорости частицы в вязком газе со случайной скоростью в виде суммы двух коррелированных цветных шумов

The article focuses on methods for studying the phenomenon of two-phase turbulent flows. The turbulence effect on the movement of solid particles in a viscous gas is under study. Dynamics of particles movement in a gas is written in the Stokes approximation, which allows us to suppose the dynamic relaxation time to be a constant value.The random gas velocity is modeled by the sum of two correlated random noises. It is shown that this approach makes it possible to model noise of any structural complexity. The paper describes two research methods based on fundamentally different Euler and Lagrange approaches to the description of a continuous medium. The first approach uses a well-known generalization of the spectral analysis technique for random processes, a popular method for studying turbulence. The second approach implementation is based on the modern generalizations of the theory of numerical algorithms for solving stochastic ordinary differential equations. The spectral method is used to obtain analytical expressions of correlation functions and variance of random processes describing the velocity of gas and solid particles. The qualitative difference between the correlation of fluctuations of modulated random velocities and the behavior of correlations in the case of a single-component gas velocity composition is analyzed. A method of direct numerical simulation for studied processes based on the numerical solution of a stochastic ordinary differential equations system is proposed and analyzed in detail. An array of statistical data obtained as a result of direct numerical modeling is collected and processed. Analytical results are compared qualitatively with numerical results. The influence of input parameters on the character of turbulent flow is studied. The dynamic relaxation time has a significant effect on the complexity of the autocorrelation function of the particle velocity and the response function of particles to gas velocity fluctuations. It is shown that the obtained functions tend to the known results of the standard theory. The considered methods for describing two-phase turbulent flows hold promise for further research.В статье основное внимание уделено методам исследования явления двухфазных турбулентных течений. Изучается влияние турбулентности на характер движения твёрдых частиц в вязком газе. Динамика движение частиц в газе записывается в приближении Стокса, что позволяет считать время динамической релаксации постоянной величиной. Случайная скорость газа моделируется суммой двух коррелированных случайных шумов. Показано, что такой подход позволяет моделировать шумы любой структурной сложности. Изложены два метода исследования, основанные на принципиально разных подходах Эйлера и Лагранжа к описанию сплошной среды. Первый подход использует известное обобщение техники спектрального анализа для случайных процессов – популярного метода при изучении турбулентности. Второй подход реализуется на основе современных обобщений теории численных алгоритмов для решения стохастических обыкновенных дифференциальных уравнений. Спектральным методом получены аналитические выражения корреляционных функций и дисперсий случайных процессов, описывающих скорость газа и твердых частиц. Проанализировано качественное отличие корреляции флуктуаций модулированной случайных скоростей от поведения корреляций в случае однокомпонентного состава скорости газа. Предложен и детально разобран метод прямого численного моделирования изучаемых процессов, на основе численного решения системы стохастических обыкновенных дифференциальных уравнений. Собран и обработан массив статистических данных, полученных в результате прямого численного моделирования. Аналитические результаты качественно сопоставляются с численными.  Исследовано влияние входных параметров на характер турбулентного течения. Время динамической релаксации оказывает существенное влияет на сложность автокорреляционной функции скорости частиц и функцию отклика частиц на флуктуации скорости газа. Показано, что полученные функции стремятся к известным результатам стандартной теории. Рассмотренные методы описания двухфазных турбулентных течений являются перспективными для будущих исследований

Влияние микроструктуры турбулентности на диффузию тяжелых инерционных частиц

Based on the spectral expansion of Euler correlation of the carrier medium the a closed system of functional equations for the Lagrange spectra of heavy inertial particles and the velocity fluctuations of the carrier medium on the particle trajectory have been obtained. To split the fourth moments the approximation of quasinormality and velocity fluctuations of particles is performed by a random Gaussian process. The approximate self-consistent method is proposed for solving the resulting system of functional equations. The influence of the particle inertia, the velocity of the averaged slip and microstructure of velocity fluctuations of the medium on the parameters of the chaotic motion of an impurity has been studied. It is shown that the difference in integral time scales of Eulerian and Lagrangian correlations is associated with the spatial microstructure of velocity fluctuations of the medium. It is established that in the absence of mass forces, the coefficient of the stationary diffusion of inertial particles is always greater than the diffusion coefficient of inertialess impurity. The dependence of the turbulent diffusion coefficient of particles impurity on the structural parameter of turbulence has been illustrated. The spectrum of Euler correlations of medium velocity fluctuations is modeled by Karman distributions. The influence of the particle inertia, the velocity of the averaged slip and microstructure of velocity fluctuations of the medium on the parameters of the chaotic motion of an impurity has been studied. It is shown that the difference in integral time scales of Eulerian and Lagrangian correlations is associated with the spatial microstructure of velocity fluctuations of the medium. It is established that in the absence of mass forces, the coefficient of the stationary diffusion of inertial particles is always larger than the diffusion coefficient of inertialess impurity. The dependence of the turbulent diffusion coefficient of particles impurity on the structural parameter of turbulence has been illustrated.DOI: 10.7463/mathm.0215.0776054На основе спектрального разложения корреляции Эйлера несущей среды получена система замкнутых функциональных уравнений для спектров Лагранжа тяжелой инерционной частицы и флуктуаций скорости несущей среды на траектории частицы. При расщеплении четвертых моментов используется приближение квазинормальности и аппроксимация флуктуаций скорости частиц случайным процессом Гаусса. Предложен приближенный самосогласованный метод решения полученной системы функциональных уравнений. Спектр корреляций Эйлера флуктуаций скорости среды моделируется распределениями Кармана. Исследовано влияние инерции частиц, скорости осредненного скольжения и микроструктуры флуктуаций скорости среды на параметры хаотического движения примеси. Показано, что отличие во временных интегральных масштабах корреляции Эйлера и Лагранжа связано с пространственной микроструктурой флуктуаций скорости среды. Установлено, что в отсутствии массовых сил коэффициент стационарной диффузии инерционных частиц всегда выше, чем коэффициент диффузии безынерционной примеси. Проиллюстрирована зависимость коэффициента турбулентной диффузии примеси от структурного параметра турбулентности.DOI: 10.7463/mathm.0215.077605

Нестационарные эффекты в реакторе Фишера-Тропша с неподвижным слоем частиц катализатора

Based on analysis of small temperature disturbances in the Fischer-Tropsch reactor with a fixed bed of catalyst particles various scenarios of thermal instability were investigated. There are two possible scenarios of thermal instability of the reactor. First, thermal explosion may occur due to growth of temperature disturbances inside a catalytic granule. Second scenario connected with loss of thermal stability as a result of an initial increase in temperature in the reactor volume. The boundaries of thermal stability of the reactor were estimated by solving the eigenvalue problems for spherical catalyst particles and cylindrical reactor. Processes of diffusional resistance inside the catalytic granule and heat transfer from wall of the reactor tube are taken into account. Estimation of thermal stability area is compared with the results of numerical simulation of behavior of temperature and concentration of synthesis gas.DOI: 10.7463/mathm.0115.0777093На основе анализа малых возмущений температуры в реакторе Фишера-Тропша с неподвижным слоем гранул катализатора исследуются сценарии потери тепловой устойчивости. Установлены два сценария потери тепловой устойчивости реактора. Во-первых, возможна потеря тепловой стабильности внутри каталитических гранул, приводящая к росту их температуры. Во-вторых, неконтролируемое увеличение температуры в объеме реактора. Границы, термической стабильность получены на основе решения задач на собственные значения для сферических гранул катализатора и цилиндрического реактора. Моделируются процессы диффузионного торможения в порах гранул, тепловыделение в объеме гранул и охлаждение стенок реактора. Оценки границ термической стабильности сопоставляются с результатами численного моделирования нестационарного поведения температур и концентрации синтез-газа.DOI: 10.7463/mathm.0115.077709

SIMULATION OF HYDRODYNAMICS AND OF HEAT-MASS TRANSFER IN DISPERSED TURBULENT STREAMS

There the hydrodynamical and heat-mass transfer processes in two-phase turbulent streams with the impurities of particles or drops have been studied. The description manner of dispersed turbulent streams, which is new in essence, has been developed on basis of use of equation for the function of distribution probability density of characteristics of dispersed phase. The new results about the formation mechanisms of averaged and pulsation characteristics of dispersed impurity, about the inverse influence of particles on the flow characteristics of carrier phase have been obtained. The new computation procedure of rate of deposition of particle impurity in the tubes has been created. The new results about the collision process of particles in the non-uniform turbulent stream have been obtained. The gasification simulator of cokes of high-ash coals has been constructed. The new physicochemical simulator of break-down of organic heat carriers under the conditions of liquid-phase oxidation has been developed. The physicochemical simulator of processes in the absorption zone of sulphur-purifying plant to catch the sulphur dioxide from the smoke gases of high-capacity steam electric stations has been createdAvailable from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio

Estimation of Covid-19 Infection Growth Rate Based on the Imbedding Method

An analysis is presented of universal stages in the development of the COVID-19 virus infection epidemic. It is assumed that during a pandemic the growth rate in the number of infections in a country occurs similarly to the process of virion replication in an infected organism. The exponent of the growth rate of the number of infection cases reflects not only biomedical parameters of a virus infection but also distinctive features of the social behavior of the population of a country. The dynamics of the change in the growth rate exponents is simulated by a system of relaxation-type ordinary differential equations. In applied mathematics, based on the imbedding method, limit values are forecast for exponents of the growth rate of the number of cases using the currently available experimental data. These values “forecast ahead” are subsequently approached by the actual exponents of growth in the number of infectees. The search for unknown parameters is carried out by minimizing a specially constructed quadratic functional accounting for all confirmed cases of COVID-19 infection. The functional minimum is found via iterations by solving a system of ordinary differential equations

Effect of Turbulence Internal Structure on Diffusion of Heavy Inertial Particles

Based on the spectral expansion of Euler correlation of the carrier medium the a closed system of functional equations for the Lagrange spectra of heavy inertial particles and the velocity fluctuations of the carrier medium on the particle trajectory have been obtained. To split the fourth moments the approximation of quasinormality and velocity fluctuations of particles is performed by a random Gaussian process. The approximate self-consistent method is proposed for solving the resulting system of functional equations. The influence of the particle inertia, the velocity of the averaged slip and microstructure of velocity fluctuations of the medium on the parameters of the chaotic motion of an impurity has been studied. It is shown that the difference in integral time scales of Eulerian and Lagrangian correlations is associated with the spatial microstructure of velocity fluctuations of the medium. It is established that in the absence of mass forces, the coefficient of the stationary diffusion of inertial particles is always greater than the diffusion coefficient of inertialess impurity. The dependence of the turbulent diffusion coefficient of particles impurity on the structural parameter of turbulence has been illustrated. The spectrum of Euler correlations of medium velocity fluctuations is modeled by Karman distributions. The influence of the particle inertia, the velocity of the averaged slip and microstructure of velocity fluctuations of the medium on the parameters of the chaotic motion of an impurity has been studied. It is shown that the difference in integral time scales of Eulerian and Lagrangian correlations is associated with the spatial microstructure of velocity fluctuations of the medium. It is established that in the absence of mass forces, the coefficient of the stationary diffusion of inertial particles is always larger than the diffusion coefficient of inertialess impurity. The dependence of the turbulent diffusion coefficient of particles impurity on the structural parameter of turbulence has been illustrated.DOI: 10.7463/mathm.0215.077605

Particle Velocity Fluctuations in Viscous Gas with Random Velocity as the Sum of Two Correlated Color Noises

The article focuses on methods for studying the phenomenon of two-phase turbulent flows. The turbulence effect on the movement of solid particles in a viscous gas is under study. Dynamics of particles movement in a gas is written in the Stokes approximation, which allows us to suppose the dynamic relaxation time to be a constant value.The random gas velocity is modeled by the sum of two correlated random noises. It is shown that this approach makes it possible to model noise of any structural complexity. The paper describes two research methods based on fundamentally different Euler and Lagrange approaches to the description of a continuous medium. The first approach uses a well-known generalization of the spectral analysis technique for random processes, a popular method for studying turbulence. The second approach implementation is based on the modern generalizations of the theory of numerical algorithms for solving stochastic ordinary differential equations. The spectral method is used to obtain analytical expressions of correlation functions and variance of random processes describing the velocity of gas and solid particles. The qualitative difference between the correlation of fluctuations of modulated random velocities and the behavior of correlations in the case of a single-component gas velocity composition is analyzed. A method of direct numerical simulation for studied processes based on the numerical solution of a stochastic ordinary differential equations system is proposed and analyzed in detail. An array of statistical data obtained as a result of direct numerical modeling is collected and processed. Analytical results are compared qualitatively with numerical results. The influence of input parameters on the character of turbulent flow is studied. The dynamic relaxation time has a significant effect on the complexity of the autocorrelation function of the particle velocity and the response function of particles to gas velocity fluctuations. It is shown that the obtained functions tend to the known results of the standard theory. The considered methods for describing two-phase turbulent flows hold promise for further research