scholarly journals Analysis of the non-stationary mathematical model of a simple hydraulic network with a centrifugal pump

Vestnik IGEU ◽  
2020 ◽  
pp. 64-70
Author(s):  
V.A. Naumov
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Centrifugal Pump ◽  
Similarity Criteria ◽  
Dimensionless Form ◽  
Maximum Volume ◽  
Stationary Flows ◽  
Variable Level ◽  
Hydraulic Network ◽  
The Mathematical Model

Simple hydraulic networks with a centrifugal pump are not only part of complex networks, but are also widely used in Autonomous water supply and Sewerage systems. The mathematical model of simple networks taking into account the variable level of liquid in reservoirs includes the well-known Bernoulli equation for non-stationary flows. Published works on this problem do not take into account the non-stationary nature of the flow due to the variable liquid level. The conditions for using the quasi-stationary model are not discussed. Similarity criteria for the issue were not found. The purpose of the study is to analyze the non-stationary mathematical model of the object, including the definition of criteria for similarity of the problem and their impact on the solution. The well-known equations of fluid quantity balance and Bernoulli for non-stationary flows with smoothly changing characteristics were used as a mathematical model of a simple hydraulic network. The pressure characteristic of a centrifugal pump is approximated by a well-established dependence in the form of a square three-member. The system of differential equations was reduced to a dimensionless form. Analytical and numerical methods were used to solve the problem. The analysis of the mathematical model of pumping liquid by a centrifugal pump in a hydraulic network with a variable level was carried out. The dimensionless form of the system of equations allowed us to determine three similarity criteria for the problem, including the analog of the Struhal number Str. The analytical solution to the Cauchy problem is found in the quasi-stationary formulation (Str = 0). The solution of the problem in the full statement is obtained by the numerical method. The results of the study of the influence of similarity criteria on the solution are presented. The dimensionless flow rate of the liquid decreases with increasing Str values. In this case, the maximum volume of liquid and the time to reach it increases. Increasing the values of the other two criteria leads to an increase in both the flow rate and the maximum volume of the liquid. The analytical solution in the quasi-rational formulation can be used only for Str < 0,1. The results obtained can be used in the design of Autonomous Water supply and Sewerage systems. Further research for the non-self-similar area of hydraulic resistance and for variable fluid viscosity is promising.

scholarly journals A Mathematical Model of Epidemics—A Tutorial for Students

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1174 ◽  
Author(s):  
Yutaka Okabe ◽  
Akira Shudo
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Analytical Solutions ◽  
Sir Model ◽  
Exact Analytical Solutions ◽  
The Mathematical Model ◽  
Epidemic Diseases

This is a tutorial for the mathematical model of the spread of epidemic diseases. Beginning with the basic mathematics, we introduce the susceptible-infected-recovered (SIR) model. Subsequently, we present the numerical and exact analytical solutions of the SIR model. The analytical solution is emphasized. Additionally, we treat the generalization of the SIR model including births and natural deaths.

Research on Tool-Path Generating Methods for NC Machining of Centrifugal Pump Vanes

2010 ◽  
Vol 26-28 ◽  
pp. 982-987
Author(s):  
Xiu Ting Wei ◽  
Qiang Du ◽  
Jing Cheng Liu
Keyword(s):  
Mathematical Model ◽  
Centrifugal Pump ◽  
Iterative Algorithm ◽  
Tool Path ◽  
Tool Path Generation ◽  
Terminal Condition ◽  
Path Generation ◽  
Contact Parameter ◽  
Nc Program ◽  
The Mathematical Model

The universal NC program was inefficient for machining centrifugal pump vanes and a new tool-path generation method along section lines was proposed in this paper. After analyzing curve characteristics of centrifugal pump vanes, the mathematical model for calculating cutting step-lengths and determining the next tool contact parameter ui+1 through step estimating method was put forward. Furthermore, a new iterative algorithm was implemented, along with its terminal condition simplified. The proposed algorithm provided a new idea for developing special NC program of impellers.

Torus-Like Balloons

2020 ◽  
Vol 56 ◽  
pp. 59-65
Author(s):  
Ivïalo M. Mladenov ◽  
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Elliptic Integrals ◽  
Deployable Structures ◽  
The Future ◽  
The Mathematical Model

The concepts for inflatable deployable structures have been under development and evaluation for many years. Strangely enough only the Mylar balloon up to now has been described adequately. Here we provide the mathematical model and its analytical solution for the torus-like balloons. Their characteristics and shapes are described explicitly in terms of elliptic integrals. The obtained results are commented shortly and the possible directions for the related studies in the future are outlined.

scholarly journals Physical modeling of a skip pneumatic winder

2021 ◽  
pp. 64-73
Author(s):  
Aleksandr Leontiev ◽  
Keyword(s):  
Mathematical Model ◽  
Physical Modeling ◽  
Aerodynamic Characteristics ◽  
Volumetric Efficiency ◽  
Similarity Criteria ◽  
Experimental Sample ◽  
Point Analysis ◽  
Analytical Phase ◽  
The Mathematical Model ◽  
Parameters Calculation

Introduction. The analytical phase of research on mine skip pneumatic winders has been passed, so the theoretical provisions have to be tested by the methods of physical modeling which is aimed at confirming the mathematical model adequacy and assessing the effectiveness of different types of sealing devices. Research methods. Physical modeling phases have been formulated, including modeling by geometric and aerodynamic similarity criteria, constructing aerodynamic characteristics of the installation, carrying out experiments with non-contacting and combined seals, and calculating the values of the installation volumetric efficiency based on the experimental data obtained. Research results. The lifting time of the skip model with different masses of material and seal types has been determined. The installation working points in the “flow rate–pressure” coordinate system have been identified, and the values of the volumetric efficiency have been calculated for each working point. Analysis of the results. A satisfactory convergence of calculated and experimental parameters of the physical model has been established. The model's volumetric efficiency has reached a technically acceptable level. The expected value of the experimental model’s volumetric efficiency has been calculated according to the similarity constants. Conclusions. The model's study revealed the convergence of the experimentally obtained volumetric efficiency of the model with its calculated values and proved the applicability of the mathematical model to experimental sample parameters calculation. The volumetric efficiency of the installation with both non-contacting and combined seals is quite high allowing to recommend the studied sealing devices for mine pneumatic winders.

scholarly journals On the Sodium Concentration Diffusion with Three-Dimensional Extracellular Stimulation

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Luisa Consiglieri ◽  
Ana Rute Domingos
Keyword(s):  
Mathematical Model ◽  
Boundary Condition ◽  
Analytical Solution ◽  
Parabolic Equations ◽  
Three Dimensional ◽  
Sodium Concentration ◽  
Dynamical Boundary Condition ◽  
Sodium Diffusion ◽  
The Mathematical Model ◽  
Concentration Diffusion

We deal with the transmembrane sodium diffusion in a nerve. We study a mathematical model of a nerve fibre in response to an imposed extracellular stimulus. The presented model is constituted by a diffusion-drift vectorial equation in a bidomain, that is, two parabolic equations defined in each of the intra- and extra-regions. This system of partial differential equations can be understood as a reduced three-dimensional Poisson-Nernst-Planck model of the sodium concentration. The representation of the membrane includes a jump boundary condition describing the mechanisms involved in the excitation-contraction couple. Our first novelty comes from this general dynamical boundary condition. The second one is the three-dimensional behaviour of the extracellular stimulus. An analytical solution to the mathematical model is proposed depending on the morphology of the excitation.

scholarly journals Growth of N-dimensional spherical bubble within viscous, superheated liquid: Analytical solution

2019 ◽  
pp. 380-380
Author(s):  
Gamiel Shalaby ◽  
Ali Abu-Bakr
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Variable Viscosity ◽  
Superheated Liquid ◽  
Spherical Bubble ◽  
The Mathematical Model

In this paper, we present the study of the bevaiour of spherical bubble in N-dimensions fluid. The fluid is a mixture of vapour and superheated liquid. The mathematical model is formulated in N-dimensions fluid on the basis of continuity and momentum equations, and solved its analytically. The variable viscosity is taken in an account problem. The obtained results show that the radius of bubble increases with the decreasing of the value of N-dimensions.

AI OPTIMIZATION OF THE EXOTHERMIC REACTION OF ETHYLENE OXIDE WITH WATER

2021 ◽  
pp. 2150033
Author(s):  
Khaled A. Al-Utaibi ◽  
Ayesha sohail ◽  
Andleeb Zafar ◽  
Rana Talha ◽  
Sadiq M. Sait
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Ethylene Oxide ◽  
Ethylene Glycol ◽  
Exothermic Reaction ◽  
Plug Flow ◽  
Material Balance ◽  
Chemical Species ◽  
Future Research ◽  
The Mathematical Model

A computational framework, for the numerical approximation of the exothermic reaction of ethylene oxide (EO) with water, to form ethylene glycol is presented in this paper. Ethylene Glycol also known as Mono-ethylene Glycol (MEG), is a diol with a boiling of 198[Formula: see text]C and conventionally produced through hydrolysis of ethylene oxide which is obtained through the oxidation of ethylene. It is used as an excellent automobile coolant as the 1:1 ratio mixture of MEG with Water boils at 129[Formula: see text]C and freezes at [Formula: see text]C. Other than its use as an antifreeze, it is also used as a reagent during the production of polyester fibers, pharmaceutics, cosmetics, hydraulic fluids, printing inks, explosives, polyesters and paint solvents. The mathematical model presented here, consists of an energy balance and a material balance system, described in an axisymmetric coordinate system. The optimized resulting values using the artificial intelligence approach are summarized in this paper. We derive an analytical solution. The analytical solution for the mathematical model equations is in general not possible for this model but it may be possible to derive an analytical solution to this mathematical model if we consider the equation for the conservation of material (chemical species) as a formulation for plug flow and isothermal conditions. Noteworthy findings are reported in this paper for future research.

scholarly journals Mathematical model of centrifugal compressor vaneless diffuser based on CFD calculations

2020 ◽  
Vol 178 ◽  
pp. 01014
Author(s):  
Olga Solovyeva ◽  
Aleksandr Drozdov
Keyword(s):  
Mathematical Model ◽  
Centrifugal Compressor ◽  
Similarity Criteria ◽  
Relative Width ◽  
Inlet Flow ◽  
Flow Angle ◽  
Gas Dynamic ◽  
Modelling Method ◽  
Vaneless Diffusers ◽  
The Mathematical Model

The approximate engineering techniques based on mathematical modelling are used in centrifugal compressor design. One of such methods is the well-proven Universal Modelling Method, developed in the scientific and research laboratory “Gas dynamics of turbo machines”, SPbPU. In the modern version of the compressor model, vaneless diffusers mathematical model was applied based on a generalization of the CFD calculations. The mathematical model can be used for vaneless diffusers with a relative width in the range of 1.4 – 10.0%, with a radial length up to 2.0, in the range of inlet flow angles 10 to 90 degrees, the inlet velocity coefficient in the range of 0.39 – 0.82, Reynolds number varying from 87 500 to 1 030 000. The model was also used for calculating low-flow-rate model stages with narrow diffusers with diffusers’ relative width in the range of 0.5 – 2.0%. The mathematical model showed lesser accuracy. To widen the model applicability, new series of CFD-calculations were executed. A series of vaneless diffusers was designed with relative width in the range of 0.6 – 1.2%, The gas-dynamic characteristics of loss coefficients and outlet flow angle versus inlet flow angle of diffuser were calculated. Regression analysis was used to process the calculated data. System of algebraic equations linking geometric, gas-dynamic parameters and similarity criteria was developed. The obtained equations are included in a new mathematical model of the Universal Modelling Method.

scholarly journals COMPUTER SIMULATION AND ANALYTICAL SOLUTION OF FOUR BAR MECHANISM

2020 ◽  
pp. 120-128
Author(s):  
Darina Hroncová
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Mathematical Models ◽  
Computer Model ◽  
Analytical Methods ◽  
Kinematic Analysis ◽  
Linkage Mechanism ◽  
The Creation ◽  
Four Bar Linkage ◽  
The Mathematical Model

Urgency of the research. The use of computers in technical practice leads to the extension of the possibility of solving mathematical models. This makes it possible to gradually automate complex calculations of equations of mathematical models. It is necessary to input the relevant inputs of the mathematical model, to build a simulation computer model and to monitor and evaluate the output results using a computer's output device. Target setting. The possibilities of modeling a four-bar linkage mechanism by classical analytical methods and methodsusing computer modeling are presented in this paper.The problem is to describe the creation of a computer model and to show the mathematical model and its solution in the classical ways. Actual scientific researches and issues analysis. The inspiration for the creation of the article was the study of the mechanisms in the work [1-3] and the study of other resources available in library and journal materials, as well as prepared study materials for students of Technical university Kosice. Uninvestigated parts of general matters defining. The question of building a real mechanism model. The possibilities to building a real model, based on the result of simulation. The research objective. The aim of this paper is to develop a functional model of the mechanism in ADAMS/View and Matlab and its complete kinematic analysis.The statement of basic materials.The task was to create a computer model in MSC Adams and Matlab and to perform a four-bar linkage mechanism kinematic analysis. At the same time the classical procedure of analytical methods of kinematic analysis was described. Kinematic сharacteristics of driven members and their selected points were determined. The movement of the parts of the mechanism in its significant points was analyzed. The results of the solution were shown in both programs in graphical form. Kinematic analysis was performed by both vector and graphical methods. Finally, the results with a graphical representation of parameters such as angular displacement, angular velocity and angular acceleration of mechanism members are presented in this work. The results of these solutions are created in the form of graphs. To ensure that the results do not differ from the model real, a good computer model gradually was created by its verification and modification, which is one of the advantages of MSC Adams. The practical applicability of the mathematical model was limited by the existence of an analytical solution. Conclusions. The development of computer technology has expanded the limit of solvability of mathematical models and made it possible to gradually automate the calculation of equations of mathematical models. In a computer model the auto-mated calculation can be treated as a real object sample. In various variations of calculation, we can monitor and measure the behavior of an object under different conditions, under the influence of different inputs. Graphical and vector methods were used for classical analytical methods. MSC Adams and Matlab were used for the automated calculations.

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