Modelling and numerical study of the physicochemical laws of 1,4-cis-polyisoprene obtained in the presence of modified catalytic systems

2020 ◽  
pp. 7-17
Author(s):  
Эльдар Наилевич Мифтахов ◽  
Светлана Анатольевна Мустафина ◽  
Семен Израилевич Спивак
Keyword(s):  
Mathematical Model ◽  
Numerical Study ◽  
Continuous Process ◽  
Kinetic Equations ◽  
Molecular Characteristics ◽  
Batch Reactors ◽  
Active Centers ◽  
Catalytic Systems ◽  
Continuous Mixing ◽  
The Mathematical Model

Построена математическая модель, описывающая процесс получения полиизопрена в присутствии микрогетерогенных титансодержащих каталитических систем с учетом динамики активных центров. Построение модели носило поэтапный характер, при котором модель, описывающая периодический процесс, была дополнена рекуррентными соотношениями с целью описания процесса в каскаде реакторов идеального перемешивания. Численное решение прямой задачи позволило провести анализ молекулярно-массового распределения получаемого продукта в зависимости от исходной загрузки. The aim of this work is building a mathematical model that describes the process of producing polyisoprene in the presence of microheterogeneous titanium-containing catalytic systems, taking into account the dynamics of active centers and numerical calculation of the main physicochemical properties of the resulting product. The main methodology for compiling the mathematical model is the application of the kinetic approach, which consists of compiling and numerically solving the kinetic equations for all types of particles involved in the process. Previously, the distribution of active centers of the applied catalytic system was studied. The confirmed polycentricity of the used catalyst, represented by two types of active centers, is reflected in the nature of the description of the mathematical model. To reduce the system of differential equations to the final form, the method of moments was applied, which allows evaluating the complex properties of the obtained product by its averaged molecular characteristics. A model that permits describing the scale of batch reactors was modified by a macrokinetic module that takes into account possible energy and hydrodynamic laws of the process. For this, recurrence relations were introduced that are valid for the description of continuous mixing reactors. The software package developed by the authors of the work allows carring out a computational experiment for various technological conditions for conducting a continuous process. Based on the results of its launch, the dependences of conversion and averaged molecular characteristics on time in the context of each polymerizer were obtained. Thus, the numerical solution of the direct problem is capable analyzing the molecular weight distribution of the obtained product depending on various parameters of the initial load, for both the batch and continuous modes of the process. The dependences obtained by calculation showed satisfactory agreement with experimental data

Calculations of three-dimensional viscous flow in a multiblade centrifugal fan by modelling blade forces

2003 ◽  
Vol 217 (3) ◽  
pp. 287-297 ◽  
Author(s):  
S-J Seo ◽  
K-Y Kim ◽  
S-H Kang
Keyword(s):  
Mathematical Model ◽  
Turbulent Flows ◽  
Numerical Study ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Centrifugal Fan ◽  
Complex Flows ◽  
Flow Through ◽  
Volume Method ◽  
The Mathematical Model

A numerical study is presented for Reynolds-averaged Navier-Stokes analysis of three-dimensional turbulent flows in a multiblade centrifugal fan. Present work aims at development of a relatively simple analysis method for these complex flows. A mathematical model of impeller forces is obtained from the integral analysis of the flow through the impeller. A finite volume method for discretization of governing equations and a standard k-ɛ model as turbulence closure are employed. For the validation of the mathematical model, the computational results for velocity components, static pressure, and flow angles at the exit of the impeller were compared with experimental data. The comparisons show generally good agreement, especially at higher flow coefficients.

scholarly journals Numerical study of the effect of droplet coagulation intensity on polydisperse aerosol fraction distribution

2021 ◽  
Vol 1 (1) ◽  
pp. 73-80
Author(s):  
D.A. Tukmakov ◽  
Keyword(s):  
Mathematical Model ◽  
Boundary Conditions ◽  
Volume Content ◽  
Fine Particles ◽  
Stokes Equations ◽  
Numerical Study ◽  
Dispersed Phase ◽  
Mass Force ◽  
Carrier Medium ◽  
The Mathematical Model

The paper is devoted to the study of the effect of the intensity of aerosol fluctuations on the dis-tribution of fractions of the dispersed component of the coagulating aerosol. Oscillations of aerosol in closed channel are numerically modeled in operation. To describe the dynamics of the carrier medium, a two-dimensional non-stationary system of Navier-Stokes equations for compressed gas is used. They are written taking into account interfacial power interaction and interfacial heat ex-change. To describe the dynamics of the dispersed phase, a system of equations is solved for each of its fractions. It includes an equation of continuity for the “average density” of the fraction, equa-tions of preservation of spatial components of the pulse and an equation of preservation of thermal energy of the fraction of the dispersed phase of the gas suspension. Phase-to-phase power interac-tion included Archimedes force, attached mass force, and aerodynamic drag force. Heat exchange between the carrier medium-gas and each of the fractions of the dispersed phase was also taken into account. The mathematical model of dynamics of polydisperse aerosol was supplemented by the mathematical model of collision coagulation of aerosol. For the velocity components of the mixture, uniform Dirichlet boundary conditions were set. For the remaining functions of the dynamics of the multiphase mixture, uniform Neumann boundary conditions were set. The equations were solved by the explicit McCormack method with a nonlinear correction scheme that allows to obtain a mono-tone solution. As a result of numerical calculations, it was determined that in the vicinity of the os-cillating piston, an area with an increased content of coarse particles is formed. The coagulation process results in a monotonous increase in volume content of the coarse particle fraction and a mo-notonous decrease in volume content of fine particles. Increasing the intensity of gas fluctuations leads to intensification of the process of coagulation of aerosol droplets.

scholarly journals MODELO MATEMÁTICO PARA LA TRANSFRUCTOSILACIÓN DE ESTEVIÓSIDO CON - FRUCTOFURANOSIDASA INMOVILIZADA

2019 ◽  
pp. 13-18
Keyword(s):  
Experimental Data ◽  
Mathematical Model ◽  
Immobilized Enzyme ◽  
Kinetic Equations ◽  
Particle Diffusion ◽  
Palabras Clave ◽  
El Sistema ◽  
Partition Equilibrium ◽  
The Mathematical Model ◽  
Total Enzyme

MODELO MATEMÁTICO PARA LA TRANSFRUCTOSILACIÓN DE ESTEVIÓSIDO CON - FRUCTOFURANOSIDASA INMOVILIZADA MATHEMATICAL MODEL FOR TRANSFRUCTOSYLATION OF STEVIOSIDE USING ImmOBILIZED - FRUCTOFURANOSIDASE Fiorella Cárdenas T. Departamento de Ingeniería Química, Universidad de Tohoku, Japón DOI: https://doi.org/10.33017/RevECIPeru2010.0002/ RESUMEN En el presente articulo, un modelo matemático fue diseñado para la transfructosilación de esteviósido con b-fructofuranosidasa inmovilizada considerando el equilibrio de partición entre la fase liquida y la fase de partícula, difusión y reacción dentro de la partícula. Las constantes cinéticas fueron estimadas mediante el ajuste de las ecuaciones cinéticas con los resultados experimentales de la síntesis de fructosil-esteviósido con enzima libre. Las otras constantes fueron estimadas mediante el ajuste del modelo matemático con resultados experimentales de coeficiente de equilibrio de partición, y síntesis de fructosil-esteviósido con enzima inmovilizada. El modelo describe la formación de fructosil-esteviósido en el tiempo para varias condiciones de concentración inicial de sustrato y enzima total en el sistema. Palabras clave: transfructosilación, fructosil-esteviósido, análisis cinético, modelo matemático. ABSTRACT In the present article, a mathematical model for the b- immobilized fructofuranosidase was constructed considering the partition equilibrium between the liquid phase and the particle, diffusion and reaction within the particle. The kinetic constants were estimated by fitting the kinetic equations with the experimental data of the fructosyl-stevioside experiments using suspended enzyme. The other constants were estimated by fitting the mathematical model with experimental data of partition equilibrium coefficient, and fructosyl-stevioside synthesis using immobilized enzyme. The model could describe the fructosyl-stevioside formation under several conditions of initial concentration of substrate and total enzyme in the system. Keywords: Transfructosilation, fructosyl – stevioside, kinetic analysis, mathematical model.

Study of Macrosegregation Defects Formation Caused by Double Diffusive Convective Flow during Solidification of a Binary Alloy

2009 ◽  
Vol 283-286 ◽  
pp. 340-345 ◽  
Author(s):  
Farid Mechighel ◽  
Mahfoud Kadja ◽  
Mohammed El Ganaoui ◽  
Bernard Pateyron
Keyword(s):  
Mathematical Model ◽  
Finite Volume ◽  
Binary Alloy ◽  
Convective Flow ◽  
Continuum Model ◽  
Numerical Study ◽  
Finite Volume Approach ◽  
The Mathematical Model ◽  
Double Diffusive

Numerical study of both the solidification of the binary alloy Al-4.1%wtCu and macrosegregation defects formation have been carried out. The mixture continuum model was used in the development of the mathematical model representing the solidification phenomena. This model included the conservative equations (mass, momentum, energy and species); these equations were numerically solved by using a finite volume approach.

A Model for Drilled Cooling in the Valve Bridge of Cylinder Head

2011 ◽  
Vol 52-54 ◽  
pp. 338-342
Author(s):  
Xiao Liu ◽  
Wei Zheng Zhang ◽  
Chang Hu Xiang
Keyword(s):  
Heat Transfer ◽  
Mathematical Model ◽  
Heat Transfer Coefficient ◽  
Diesel Engine ◽  
Theoretical Analysis ◽  
Transfer Coefficient ◽  
Cylinder Head ◽  
3D Model ◽  
Numerical Study ◽  
The Mathematical Model

To evaluate the efficiency of drilled cooling in the valve bridge of cylinder head, theoretical analysis for the drilled cooling is carried out, and a mathematical model for the enhanced cooling is presented based on a simplified 3D model. The mathematical model is validated by numerical study on the heat transfer with and without drilled cooling, which is carried out through fluid-solid coupling. The correlation between the velocity in the drilled passage and heat transfer coefficient was also analyzed. The results can be used to solve the heat transfer in enhanced diesel engine.

scholarly journals Numerical Study of Being Forced Leather Cap Type Pig in Straight Gas Pipeline

2016 ◽  
Vol 10 (1) ◽  
pp. 141-148
Author(s):  
Liqiong Chen ◽  
Yunyun Li ◽  
Xiaoxiao Chen ◽  
Yilan Zhan ◽  
Meijuan Dang
Keyword(s):  
Mathematical Model ◽  
Finite Element ◽  
Numerical Study ◽  
Geometric Model ◽  
Gas Pipeline ◽  
Average Force ◽  
Oil Pipeline ◽  
Finite Element Software ◽  
Operation Rules ◽  
The Mathematical Model

The research on pipeline pigging technology is significant for the operation and management of pipeline. Domestic and foreign scholars usually research the operation rules of pigging in oil pipeline. There are few studies about gas pipeline pigging running because of running rate. The author established the force calculating model and corresponding numerical methods of leather cap type pig in gas pipeline. The model is based on geometric model of oil pipeline pigging. Combining pigging operation parameters with records in September 2013 and February 2014 at Bei Neihuan, the thesis used mathematical method and finite element software respectively to verify the mathematical model. The mathematical results described the average force of cup. The results indicated that the reason of the breaking of the cup is the force, instead of the cup material, temperament extrinsic reasons, etc. The force is larger than the tensile strength of the cup. The results of ANSYS finite element software simulation described the force of different parts of the cup. It is found that the force exceeding of the cup anti-pull force strength in the upper and lower sides of the cup is larger. Both results showed that using the mathematical model can quickly calculate cup pigging force conditions and determine the cause of damage to the cup. It can improve the efficiency of pigging.

scholarly journals Mathematical model of the elastoplastic shell collapse, taking into account the possible development of the process instability

2019 ◽  
Author(s):  
А.S. Novoseltsev ◽  
A.V. Babkin
Keyword(s):  
Mathematical Model ◽  
External Surface ◽  
Numerical Study ◽  
Nonstationary Problem ◽  
Tangential Direction ◽  
Initial Time ◽  
Explosive Load ◽  
Adjustment Procedure ◽  
Elastoplastic Shell ◽  
The Mathematical Model

The mathematical model for the subsequent numerical study of the shaped charge liner collapse affected by external surface forces simulating an explosive load is presented. The basic liner was considered as an originally cylindrical compressible elastoplastic shell within the framework of a two-dimensional flat nonstationary problem of continuum mechanics. To ensure the rationality of the modeling and numerical calculation at the initial time the design fragment was discriminated in the liner by central beams. Deformation of the fragment being a part of the shell was taken into account by the boundary conditions of cyclic repeatability in the tangential direction. For numerical solving the well-known Wilkins Lagrangian method was used, which was refined in terms of the relations describing the mechanical behavior of an elastoplastic medium. Additionally, a self-developed grid adjustment procedure was used, excluding the appearance of highly elongated cells in the calculation. The instability of the shell deformation was initiated by harmonic surface perturbations, initially assigned on the outer or inner surfaces. The degree of instability was assessed by the deviation of the disturbed surface (or the boundary of the so-called stream-forming layer) from the cylindrical one. The used finite-difference algorithms are implemented in the form of appropriate calculation programs. A number of computational verification measures was performed proving the viability of the developed mathematical model and the possibility of its further use

scholarly journals KINETIC HETEROGENEITY OF POLYMER PRODUCTS OBTAINED IN THE PRESENCE OF MICROHETEROGENIC CATALYTIC SYSTEMS BASED ON GEL CHROMATOGRAMS

2021 ◽  
Vol 18 (38) ◽  
pp. 27-37
Author(s):  
Eldar N. MIFTAKHOV ◽  
Svetlana A. MUSTAFINA ◽  
Ildus Sh NASYROV ◽  
Azat Kh DAMINOV
Keyword(s):  
Molecular Weight ◽  
Average Molecular Weight ◽  
Polymerization Process ◽  
Fredholm Integral Equations ◽  
Molecular Characteristics ◽  
Active Centers ◽  
Catalytic Systems ◽  
Polymer Product ◽  
Software Algorithms ◽  
Kinetic Heterogeneity

Background: the polymer product obtained in the presence of microheterogeneous catalytic systems is characterized by fairly molecular weight distribution (MWD), resulted from kinetically nonequivalent active centers (ACs) in the system that initiate the polymerization process. The nature and composition of ACs are determined by setting and solving an inverse problem on the formation of MWD. This problem is acute because revealing the nature of the kinetic heterogeneity explains changes in the molecular and consumer parameters of the product for different catalyst compositions and propagation modes in polymerizations. Aim: This study aimed to develop methods and algorithms for interpreting gel chromatograms to analyze the kinetic heterogeneity of a polymer product obtained industrially in microheterogeneous catalytic systems. Methods: the solution method is based on the assumption that the formed MWD is a superposition of distributions inherent in each type of ACs. Since the problem in the final formulation refers to the Fredholm integral equations of the first kind, the regularization method of A. N. Tikhonov is used for its numerical solution, with the original problem being preliminary discretized. This methodology and the developed software algorithms were used to determine the kinetic heterogeneity of titanium- and neodymium-containing catalytic systems. Results and discussion: The MWD analysis revealed two types of ACs with an average molecular weight of ATi-lnM = 11.3 and BTi-lnM = 13.2 in the titanium catalyst and three types of ACs ANd-lnM = 11.1, BNd-lnM = 12.7 and CNd-lnM = 14 for the neodymium catalyst, respectively. Conclusions: repeated computational experiments under different polymerization conditions and requirements for the preparation of a catalytic system make it possible to reveal a relationship with the resulting heterogeneity of ACs. It allows us to set and solve problems of controlling the molecular characteristics of the resulting polymer product.

scholarly journals Numerical study of a modified Trombe wall solar collector system

2009 ◽  
Vol 13 (1) ◽  
pp. 195-204 ◽  
Author(s):  
Snezana Dragicevic ◽  
Miroslav Lambic
Keyword(s):  
Mathematical Model ◽  
Numerical Study ◽  
Operational Parameters ◽  
Collector System ◽  
Fast Running ◽  
Simulation Techniques ◽  
Solar Radiation Intensity ◽  
Trombe Wall ◽  
Solar Wall ◽  
The Mathematical Model

The paper presents numerical analysis of efficiency of the modified Trombe wall with forced convection. The analyzed system comprises a double glass glazing, and a massive wall with opening and central channel in it. In order to increase the efficiency, a fan is provided at the bottom vent of the wall. It is more advanced as compared with simple Trombe solar wall with a relatively low thermal resistance, which is taken as a reference in experimental analysis. The mathematical model, composed for the massive solar wall efficiency, is usually very complicated and assessment of the thermal behavior requires the use of thermal simulation techniques. This paper presents steady-state and one-dimensional mathematical model for simplified analysis of thermal efficiency of modified Trombe solar wall. The results from presented model were analyzed to predict the effects of variations in the operational parameters on the solar wall efficiency: solar radiation intensity, air velocity in the entrance duct, and room air temperature. The results have been compared with the available experimental study, and the comparison has shown satisfactory agreement. The obtained results have be used for simple and fast running design tools that designers can use in the early phases of the design process for approximate calculations of efficiency of the passive solar heating systems.

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