scholarly journals Mathematical model of coal self-heating in a stack

2021 ◽  
Vol 318 (3) ◽  
pp. 90-96
Author(s):  
K.M. Akhmetov ◽  
◽  
G.S. Shaikhova ◽  
V.V. Zhurov ◽  
E.N. Khmyrova ◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Diffusion Coefficient ◽  
Differential Equations ◽  
Specific Heat ◽  
Heat Release ◽  
Temperature Profile ◽  
Exponential Growth ◽  
Self Heating ◽  
The Mathematical Model

The article presents a mathematical model of coal self-heating in the stack in which the heat exchange and gas exchange processes are described by a system of two non-linear differential equations of the second order with respect to the temperature t of coal self-heating and the volume fraction C of oxygen in the voids of the stack with boundary and initial conditions. The differential equations took into account that self-heating of coal in the stack and appearance of spontaneous combustion are observed in a relatively small layer adjacent to the surface of its contact with the air and called the zone of oxygen influence. In the mathematical model, the influence on the process of coal self-heating of parameter F- specific heat release power was taken into account, which in addition characterises the stability of coal during storage. When compiling the differential equations, such physical parameters as thermal conductivity, diffusion coefficient, specific heat capacity of coal in the stack, bulk density, thermal effect of oxidation, stack voidness, temperature coefficient of exponential growth of heat release power were also used. For numerical implementation of the mathematical model, dimensionless variables and criteria were introduced, which allowed us to apply the net method. Analysis of the obtained results allowed to get: change in the stack temperature profiles with time; change in the stack oxygen concentration profiles with time; influence on the stack temperature profile of the specific heat release power; influence on the stack temperature profile of the parameter characterizing exponential growth of heat release intensity with temperature increase. It has been determined that the dynamics of coal self-heating in the stack is mostly influenced by the Lykov criterion, proportional to the diffusion coefficient, and the Nusselt criterion related to the effective thermal conductivity and to the effective thermal diffusivity of coal. The obtained results suggest that self-heating in the stack is due on the one hand to intensive penetration of air oxygen and on the other hand to a weakened heat transfer. Self-heating and the transition of self-heating into ignition are associated with the occurrence of turbulent diffusion in the stack, arising from increased thermal blowing, whose impact can be enhanced by directing it perpendicular to the surface of the stack.

Glycemia monitoring

1998 ◽  
Vol 2 ◽  
pp. 23-30
Author(s):  
Igor Basov ◽  
Donatas Švitra
Keyword(s):  
Diabetes Mellitus ◽  
Mathematical Model ◽  
Numerical Methods ◽  
Differential Equations ◽  
Blood Sugar ◽  
Self Regulation ◽  
Physiological System ◽  
Non Linear ◽  
The Mathematical Model

Here a system of two non-linear difference-differential equations, which is mathematical model of self-regulation of the sugar level in blood, is investigated. The analysis carried out by qualitative and numerical methods allows us to conclude that the mathematical model explains the functioning of the physiological system "insulin-blood sugar" in both normal and pathological cases, i.e. diabetes mellitus and hyperinsulinism.

Mathematical model implementation in conditions of uncertainty and possible risks

2021 ◽  
pp. 137-145
Author(s):  
A. Kravtsov ◽  
◽  
D. Levkin ◽  
O. Makarov ◽  
◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Layer Structure ◽  
Boundary Value ◽  
Goal Function ◽  
Cauchy Problems ◽  
System Of Differential Equations

The article presents the theoretical and methodological principles for forecasting and mathematical modeling of possible risks in technological and biotechnological systems. The authors investigated in details the possible approach to the calculation of the goal function and its parameters. Considerable attention is paid to substantiating the correctness of boundary value problems and Cauchy problems. In mechanics, engineering, and biology, Cauchy problems and boundary value problems of differential equations are used to model physical processes. It is important that differential equations have a single physically sound solution. The authors of this article investigate the specific features of boundary value problems and Cauchy problems with boundary conditions in a two-point medium, and determine the conditions for the correctness of such problems in the spaces of power growth functions. The theory of pseudo-differential operators in the space of generalized functions was used to prove the correctness of boundary value problems. The application of the obtained results will make it possible to guarantee the correctness of mathematical models built in conditions of uncertainty and possible risks. As an example of a computational mathematical model that describes the state of the studied object of non-standard shape, the authors considered the boundary value problem of the system of differential equations of thermal conductivity for the embryo under the action of a laser beam. For such a boundary value problem, it is impossible to guarantee the existence and uniqueness of the solution of the system of differential equations. To be sure of the existence of a single solution, it is necessary either not to take into account the three-layer structure of the microbiological object, or to determine the conditions for the correctness of the boundary value problem. Applying the results obtained by the authors, the correctness of the boundary value problem of systems of differential equations of thermal conductivity for the embryo is proved taking into account the three-layer structure of the microbiological object. This makes it possible to increase the accuracy and speed of its implementation on the computer. Key words: forecasting, risk, correctness, boundary value problems, conditions of uncertainty

scholarly journals Computational Study of the Coupled Mechanism of Thermophoretic Transportation and Mixed Convection Flow around the Surface of a Sphere

Molecules ◽  
2020 ◽  
Vol 25 (11) ◽  
pp. 2694
Author(s):  
Amir Abbas ◽  
Muhammad Ashraf ◽  
Yu-Ming Chu ◽  
Saqib Zia ◽  
Ilyas Khan ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Mixed Convection ◽  
Computational Study ◽  
Convection Flow ◽  
Mixed Convection Flow ◽  
Partial Differential ◽  
Primitive Form ◽  
The Mathematical Model

The main goal of the current work was to study the coupled mechanism of thermophoretic transportation and mixed convection flow around the surface of the sphere. To analyze the characteristics of heat and fluid flow in the presence of thermophoretic transportation, a mathematical model in terms of non-linear coupled partial differential equations obeying the laws of conservation was formulated. Moreover, the mathematical model of the proposed phenomena was approximated by implementing the finite difference scheme and boundary value problem of fourth order code BVP4C built-in scheme. The novelty point of this paper is that the primitive variable formulation is introduced to transform the system of partial differential equations into a primitive form to make the line of the algorithm smooth. Secondly, the term thermophoretic transportation in the mass equation is introduced in the mass equation and thus the effect of thermophoretic transportation can be calculated at different positions of the sphere. Basically, in this study, some favorite positions around the sphere were located, where the velocity field, temperature distribution, mass concentration, skin friction, and rate of heat transfer can be calculated simultaneously without any separation in flow around the surface of the sphere.

Low-cost flow-rate estimate in separate layers along gas wells from temperature and pressure profiles

2013 ◽  
Vol 53 (1) ◽  
pp. 285
Author(s):  
Emile Barrett ◽  
Imran Abbasy ◽  
Chii-Rong Wu ◽  
Zhenjiang You ◽  
Pavel Bedrikovetsky
Keyword(s):  
Thermal Conductivity ◽  
Mathematical Model ◽  
Flow Rate ◽  
Low Cost ◽  
Pressure Sensors ◽  
Gas Wells ◽  
Production Rates ◽  
Pressure Profiles ◽  
Temperature And Pressure ◽  
The Mathematical Model

Estimation of rate profile along the well is important information for reservoir characterisation since it allows distinction of the production rates from different layers. The temperature and pressure sensors in a well are small and inexpensive; while flow meters are cumbersome and expensive, and affect the flow in the well. The method presented in this peer-reviewed paper shows its significance in predicting the gas rate from temperature and pressure data. A mathematical model for pressure and temperature distributions along a gas well has been developed. Temperature and pressure profiles from nine well intervals in field A (Cooper Basin, Australia) have been matched with the mathematical model to determine the flow rates from different layers in the well. The presented model considers the variables as functions of thermal properties at each location, which is more accurate and robust than previous methods. The results of tuning the mathematical model to the field data show good agreement with the model prediction. Simple and robust explicit formulae are derived for the effective estimation of flow rate and thermal conductivity in gas wells. The proposed approach has been applied to determine the well gas rate and formation thermal conductivity from the acquired well pressure and temperature data in field A. It allows for recommending well stimulation of layers with low production rates.

scholarly journals Mathematical Model of Spatial Motion of the Controlled Parachute-Tether System of the Wind Kite Type

2021 ◽  
Vol 24 (4) ◽  
pp. 17-24
Author(s):  
V.M. Churkin ◽  
T.Yu. Churkina ◽  
A.M. Girin
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Solid Body ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Spatial Motion ◽  
Mathematical Task ◽  
Spatial Movement ◽  
Elastic Thread ◽  
The Mathematical Model

Mathematical modeling is created for the mathematical task of spatial motion of the controlled parachute-tether system of the “wind kite” type. The mathematical model parachute-tether system consists of a model of the main parachute and a model of the braking parachute. The parachutes are connected by the tether. The model of the main parachute is supposed to be the solid body. This solid body has two planes of symmetry. The braking parachute is the solid body with axial symmetry. The tether model is an absolutely flexible elastic thread. The tether is connected by ideal hinges with the main parachute and braking parachute. The control of the main parachute is carried out by changing the length of the control slings. Changing the length causes deformation of the dome. This is the reason for the change in its aerodynamics. Maneuvering of the main parachute occurs in the vertical plane, when the length of the control slings changes simultaneously. Maneuvering of the main parachute in space is carried out when the length of the control slings changes, when the slings are given a travel difference. The system of dynamic and kinematic equations is designed for calculating the controlled spatial movement of the main parachute, braking parachute and tether. The option exists when the mass of the tether and the forces applied to the tether cannot be neglected. The motion of the tether is represented by the equations of motion of an absolutely flexible elastic thread in projections on the axis of a natural trihedron. The mathematical model is represented by a system of ordinary differential equations and partial differential equations. The problem is solved using various numerical methods. The solution is possible with the help of an integrated numerical and analytical approach as well.

scholarly journals Sensitivity analysis and practical identifiability of the mathematical model for partial differential equations

2021 ◽  
Vol 2092 (1) ◽  
pp. 012012
Author(s):  
O Krivorotko ◽  
D Andornaya
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Sensitivity Matrix ◽  
Partial Differential ◽  
Identifiability Analysis ◽  
Practical Identifiability ◽  
Eigenvalue Method ◽  
Simulation Results ◽  
The Mathematical Model

Abstract A sensitivity-based identifiability analysis of mathematical model for partial differential equations is carried out using an orthogonal method and an eigenvalue method. These methods are used to study the properties of the sensitivity matrix and the effects of changes in the model coefficients on the simulation results. Practical identifiability is investigated to determine whether the coefficients can be reconstructed with noisy experimental data. The analysis is performed using correlation matrix method with allowance for Gaussian noise in the measurements. The results of numerical calculations to obtain identifiable sets of parameters for the mathematical model arising in social networks are presented and discussed.

scholarly journals Mathematical Model with Feedback Loop in Company Control

2021 ◽  
Vol 114 ◽  
pp. 01020
Author(s):  
Arkadiy Plotnikov ◽  
Tatyana Goryacheva ◽  
Flyura Kazakova ◽  
Ekaterina Zakharchenko
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Feedback Loop ◽  
Structural Model ◽  
Production Costs ◽  
Operator Form ◽  
Single Industry ◽  
Dynamic Relationships ◽  
The Mathematical Model ◽  
Further Development

This article is devoted to the further development of feedback loop models. They are used in the management of a single-industry firm. The development of the studying consists in the mathematical modeling of transients and assessing their impact on economic indicators. To solve this problem, the following was done in the article. A structural model of the functioning of a single-industry firm (SIF) is presented. It is built in accordance with the theory of automatic control (TAC) and includes elements: a management unit, a production unit, a sales unit, information nodes, as well as a feedback loop (FL). Equations and relationships describing the logic of the functioning of the company as a production system are presented. They allowed us to derive dynamic relationships and differential equations that reflect feedback loops on revenue and production costs. The system of expressions in operator form is presented, which describes the contour of the FL SIF. It has the form of a system of differential equations. It forms the basis of the mathematical model of SIF in the control system. This model made it possible to obtain a graphical interpretation of transients with closed and open FL based on the use of the Mathcad editor. Transients are fluctuations in sales volumes and production costs in the presence of disturbing influences. This is the scientific result and determines the novelty of the article.

ALGORITHM OF USING HURWITZ CONDITIONS TO STUDY STABILITY OF STEADY STATES OF CONTINUOUS FERMENTATION PROCESSES WITH NONLINEAR KINETICS

2017 ◽  
pp. 125-135
Author(s):  
Aleksey Georgievich Borodkin ◽  
Elena Lvovna Gordeeva ◽  
Yuliya Lvovna Gordeeva ◽  
Yuriy Alekseevich Komissarov
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Continuous Fermentation ◽  
Raw Materials ◽  
Sufficient Conditions ◽  
Steady States ◽  
Microbiological Synthesis ◽  
Hurwitz Matrix ◽  
First Approximation ◽  
The Mathematical Model

The authors of the article produce the algorithm to evaluate the stability of the steady states for microbiological synthesis using mathematical models for kinetics based on the unstructured approach. The object under study is a continuous biotechnological process aiming to obtain a target product, in addition to biomass. The estimation of stability is performed using Hurwitz matrix. In this process the use of raw materials containing components for additional producing of the primary substrate are possible. The algorithm includes five steps: formulation of the mathematical model that takes into account biological and technological limits; definition of steady states (one or several) for which it is necessary to evaluate conditions of stability and to calculate process parameters for the adopted steady states resulted from solving a system of nonlinear algebraic equations of the mathematical model; formulation of the equations of the first approximation (the system of first-order linear differential equations with constant coefficients) and evaluation of the coefficients corresponding to the adopted steady states; formation of Hurwitz matrix and calculation of its items; calculation of necessary and sufficient conditions for stability using Hurwitz matrix. Two methods of calculating Hurwitz matrix have been shown. The first method is implemented by transforming the system of four differential equations of the first approximation into one fourth-order ordinary differential equation. In the course of transformation there were determined calculated ratios for coefficients of the matrix. In the second method the coefficients are calculated by solving vector equation for the system of first approximation with the introduction of eigen values. The algorithm has been used for evaluating steady states of microbiological synthesis for lactic acid production from raw materials containing the component reproducing the main substrate. The estimation has been carried out for two steady states, one of which is optimal. For numerical values of the constants (from the literature) both steady states were stable.

SELF-SIMILAR PROBLEMS FOR MODELING THE SURFACE CHEMICAL REACTIONS WITH THE GRAVITATION

1998 ◽  
Vol 3 (1) ◽  
pp. 25-32
Author(s):  
Jânis Cepîtis ◽  
Harijs Kalis
Keyword(s):  
Boundary Layer ◽  
Mathematical Model ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Chemical Reactions ◽  
Glass Fibre ◽  
Surface Chemical ◽  
Self Similar ◽  
Surface Chemical Reactions ◽  
The Mathematical Model

The mathematical model of a chemical reaction which takes place on the surface of the uniformly moving vertically imbedded glass fibre material is considered. The effect of gravitation is taken into account. Boussinesq's and boundary layer fittings allow to derive boundary value problems for self‐similar systems of ordinary differential equations.

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