Progress in Analytical Modeling of Water Hammer

2021 ◽  
Author(s):  
Kamil Urbanowicz ◽  
Haixiao Jing ◽  
Anton Bergant ◽  
Michał Stosiak ◽  
Marek Lubecki
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Dynamic Pressure ◽  
Complete Solution ◽  
Water Hammer ◽  
Inverse Transformation ◽  
Mathematical Form ◽  
Dimensionless Time ◽  
Wide Range ◽  
Analytical Formulas

Abstract In this paper analytical formulas of water hammer known from the literature are simplified to the shortest possible mathematical form based on dimensionless parameters: dimensionless time, water hammer number, etc. Novel formulas are determined, for example for the flow velocity and wall shear stress in the Muto and Takahashi solution. A complete solution in the Laplace domain is presented and the problem of its inverse transformation is discussed. A series of comparative studies of analytical solutions with numerical solutions and the results of experimental research were carried out. The compared analytical solutions, taking into account the frequency-dependent nature of the hydraulic resistances, show very good agreement with the experimental results in a wide range of water hammer numbers, in particular when Wh ≤ 0.1. On the other hand, it turned out that the analytical model based on the quasi-steady friction in great detail simulates dynamic pressure response in systems characterized by a high value of the water hammer number Wh ≥ 0.5.

An Approximate Method for Transient Radial Flow

1962 ◽  
Vol 2 (03) ◽  
pp. 225-256 ◽  
Author(s):  
G. Rowan ◽  
M.W. Clegg
Keyword(s):  
Infinite Series ◽  
Analytical Solutions ◽  
Radial Flow ◽  
Numerical Solutions ◽  
Basic Theory ◽  
Variable Pressure ◽  
Flow Problems ◽  
Disturbed Zone ◽  
Approximate Analytical Solutions ◽  
Wide Range

Abstract The basic equations for the flow of gases, compressible liquids and incompressible liquids are derived and the full implications of linearising then discussed. Approximate solutions of these equations are obtained by introducing the concept of a disturbed zone around the well, which expands outwards into the reservoir as fluid is produced. Many important and well-established results are deduced in terms of simple functions rather than the infinite series, or numerical solutions normally associated with these problems. The wide range of application of this approach to transient radial flow problems is illustrated with many examples including; gravity drainage of depletion-type reservoirs; multiple well systems; well interference. Introduction A large number of problems concerning the flow of fluids in oil reservoirs have been solved by both analytical and numerical methods but in almost all cases these solutions have some disadvantages - the analytical ones usually involve rather complex functions (infinite series or infinite integrals) which are difficult to handle, and the numerical ones tend to mask the physical principles underlying the problem. It would seem appropriate, therefore, to try to find approximate analytical solutions to these problems without introducing any further appreciable errors, so that the physical nature of the problem is retained and solutions of comparable accuracy are obtained. One class of problems will be considered in this paper, namely, transient radial flow problems, and it will be shown that approximate analytical solutions of the equations governing radial flow can be obtained, and that these solutions yield comparable results to those calculated numerically and those obtained from "exact" solutions. It will also be shown that the restrictions imposed upon the dependent variable (pressure) are just those which have to be assumed in deriving the usual diffusion-type equations. The method was originally suggested by Guseinov, whopostulated a disturbed zone in the reservoir, the radius of which increases with time, andreplaced the time derivatives in the basic differential equation by its mean value in the disturbed zone. In this paper it is proposed to review the basic theory leading to the equations governing the flow of homogeneous fluids in porous media and to consider the full implications of the approximation introduced in linearising them. The Guseinov-type approximation will then be applied to these equations and the solutions for the flow of compressible and incompressible fluids, and gases in bounded and infinite reservoirs obtained. As an example of the application of this type of approximation, solutions to such problems as production from stratified reservoirs, radial permeability discontinuities; multiple-well systems, and well interference will be given. These solutions agree with many other published results, and in some cases they may be extended to more complex problems without the computational difficulties experienced by other authors. THEORY In order to review the basic theory from a fairly general standpoint it is proposed to limit the idealising assumptions to the minimum necessary for analytical convenience. The assumptions to be made are the following:That the flow is irrotational.That the formation is of constant thickness.Darcy's Law is valid.The formation is saturated with a single homogeneous fluid. SPEJ P. 225^

A Mathematical Model Development for Simulating Nitrate Pollutant Transport along a River

2021 ◽  
Vol 57 ◽  
pp. 149-168
Author(s):  
Kayode O. Olowe ◽  
Muthukrishnavellaisamy Kumarasamy
Keyword(s):  
Mathematical Model ◽  
Water Quality ◽  
Surface Water ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Nitrate Concentration ◽  
Pollutant Transport ◽  
Spatial And Temporal Variation ◽  
Wide Range ◽  
Natural Rivers

Contamination of surface water bodies by a wide range of organic and inorganic pollutants has been a serious problem in the recent time, these have an effect on human and aquatic animals. The water quality deterioration calls for regular monitoring of the water quality in order to maintain the health and sustainability of the aquatic ecosystems. Accurate monitoring of discharged pollutants into the rivers may be time taking and labour intensive. Water quality models are significant tools for simulating water quality and controlling the surface water pollution. The purpose of this study is to develop a simplified mathematical model which is hybrid cells in series model (HCIS) to simulate the spatial and temporal variation of nitrate concentration in natural rivers. The HCIS model was formulated to serve as an alternative method to the Fickian based models. Analytical solutions for the first order reaction kinetics of nitrate with the advection and dispersion process were derived using Laplace transformation technique. The model considered the effect of nitrate concentration at several points along the river downstream by considering the transformation of nitrite to nitrate through nitrification process. In addition, the uptake of nitrate by algae for its growth and conversion of nitrate to nitrogen gas due to denitrification process were considered. The HCIS-NO3 model was applied to uMgeni River, South Africa to investigate the nitrate concentration along the river. Furthermore, the quantitative measures based on the coefficient of determination (R2) and standard errors (SE) were used to evaluate the performance of the model. The result shows that the simulated values agreed with the measured values of nitrate concentration in the river which resulted in a R2 value of 0.72 and a low standard error. Analytical solutions of HCIS - NO3 model were compared with the numerical solutions of the Fickian based ADE model for hypothetical problems. Comparison of the responses indicates that the HCIS - NO3 and ADE- NO3 models were in good agreement. The study shows that the hybrid model is a simple and effective tool for simulating pollutant transport in natural rivers.

Analytical Solutions of Bimodal Gaussian Processes’ Fatigue Damages

2017 ◽  
Author(s):  
Wenbo Huang
Keyword(s):  
Analytical Solutions ◽  
Gaussian Processes ◽  
High Frequency ◽  
Numerical Solutions ◽  
Low Frequency ◽  
Frequency Domain Method ◽  
Load Combination ◽  
Analytical Formulas ◽  
Reduction Effect ◽  
Damage Estimates

Based on the rain-flow counting technique, a frequency domain method is developed for calculating the fatigue damages caused by the bimodal Gaussian loads. Firstly, by considering the reduction effect of low frequency loads on high frequency ones, the amplitudes of the small rain-flow cycles are obtained. Secondly, the amplitudes of the large rain-flow cycles are determined by means of Turkstra’s rule of load combination. Moreover, based on the numerical solutions developed, the two analytical formulas for the damage estimates of small and large cycles are developed. Both numerical and analytical solutions are benchmarked against the rain-flow damage estimates and compared with the existing ones. The numerical analyses show that the damages estimated by the new method are close to the rain-flow damages.

On the Frequency Response of Viscous Compressible Fluids as a Function of the Stokes Number

1970 ◽  
Vol 92 (2) ◽  
pp. 333-347 ◽  
Author(s):  
F. R. Goldschmied
Keyword(s):  
Numerical Solutions ◽  
Dynamic Pressure ◽  
Stokes Number ◽  
Pressure Response ◽  
Compressible Fluids ◽  
Tube Length ◽  
Dimensionless Frequency ◽  
Amplitude Distortion ◽  
Chamber Volume ◽  
Wide Range

Air, carbon dioxide, and helium test data are presented for the experimental verification of Iberall’s analysis of the dynamic pressure response of viscous compressible fluids in rigid tubes with deadened volume termination against oscillatory frequency. A graphical display is given of numerical solutions of Iberall’s theory over a wide range as a function of the Stokes number and of a dimensionless frequency. Rapid engineering solutions are presented for the following problems: Given a tube, a chamber volume, and a fluid, determine the maximum frequency to be transmitted at ±10 percent amplitude distortion. Given a tube, a chamber volume, and a fluid, sketch the dynamic pressure response curve. Given a chamber volume, a fluid, and a specified frequency to be transmitted within ±10 percent amplitude distortion, plot allowable tube length against tube diameter.

Verification of ANSYS and Matlab Conduction Results Using Analytical Solutions

2020 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
Giampaolo D'Alessandro ◽  
James V. Beck
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Early Time ◽  
The Body ◽  
Numerical Code ◽  
Transient Temperature ◽  
Dimensionless Time ◽  
Time Step ◽  
First Case

Abstract A two-dimensional transient thermal conduction problem is examined and numerical solutions to the problem generated by ANSYS and Matlab, employing the finite element method, are compared against an analytical solution. Various different grid densities and time-step combinations are used in the numerical solutions, including some as recommended by default in the ANSYS software, including coarse, medium and fine spatial grids. The transient temperature solutions from the analytical and numerical schemes are compared at four specific locations on the body and time-dependent error curves are generated for each point. Additionally, tabular values of each solution are presented for a more detailed comparison. The errors found in the numerical solutions by comparing them directly with the analytical solution vary depending primarily on the time step size used. The errors are much larger if calculated using the analytical solution at a given time as a basis of the comparison between the two solutions as opposed to using the steady-state temperature as a basis. The largest errors appear in the early time steps of the problem, which is typically the regime wherein the largest errors occur in mathematical solutions to transient conduction problems. Conversely, errors at larger values of dimensionless time are extremely small and the numerical solutions agree within one tenth of one percent of the analytical solutions at even the worst locations. In addition to difficulties during the early time values of the problem, temperatures calculated on convective boundaries or prescribed-heat-flux boundaries are locations generating larger-magnitude errors. Corners are particularly difficult locations and require finer gridding and finer time steps in order to generate a very precise solution from a numerical code. These regions are compared, using several grid densities, against the analytical solutions. The analytical solutions are, in turn, intrinsically verified to eight significant digits by comparing similar analytical solutions against one another at very small values of dimensionless time. The solution developed using the Matlab differential equation solver was found to have errors of a similar magnitude to those generated using ANSYS. Two different test cases are examined for the various numerical solutions using the selected grid densities. The first case involves steady heating on a portion of one surface for a long duration, up to a dimensionless time of 30. The second test case involves constant heating for a dimensionless time of one, immediately followed by an insulated condition on that same surface for another duration of one dimensionless time unit. Although the errors at large times were extremely small, the errors found within the short duration test were more significant.

Analytical Solution of Temperature Field in Micro-Poiseuille Flow With Constant Wall Temperature

2008 ◽  
Author(s):  
Masoud Darbandi ◽  
Salman SafariMohsenabad ◽  
Shidvash Vakilipour
Keyword(s):  
Temperature Field ◽  
Wall Temperature ◽  
Analytical Solutions ◽  
Analytical Study ◽  
Numerical Solutions ◽  
Experimental Studies ◽  
Constant Wall Temperature ◽  
Power Series Method ◽  
The Past ◽  
Wide Range

The analytical study of microchannels has been considered as a preliminary approach to alleviate the difficulties which are normally encountered in numerical and experimental studies. Among the analytical solutions, those with high robustness and low complexities are certainly more attractive. In this work, we present a theoretical approach to predict the temperature field in micro-Poiseuille channel flow with constant wall temperature. The use of power series method simplifies the solution in the current analytical approach. The current analytical derivations are examined for channels with both hot-wall and cold-wall conditions. The current solutions agree well with the numerical solutions for a wide range of Knudsen numbers. Contrary to the past analytical solutions and in spite of using a simple and robust approach, the current formulations predict the temperature field in the channel readily.

Magnetoencephalography inverse problem in the spheroid geometry

2019 ◽  
Vol 27 (2) ◽  
pp. 159-169 ◽  
Author(s):  
Petr I. Karpov ◽  
Tatyana Zakharova
Keyword(s):  
Inverse Problem ◽  
Exact Solution ◽  
Magnetic Fields ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Approximate Analytical Solutions ◽  
Wide Range ◽  
Ill Posed ◽  
The Brain

AbstractThe inverse problem of magnetoencephalography is ill-posed and difficult for both analytical and numerical solutions. Additional complications arise from the volume (passive) currents and the associated magnetic fields, which strongly depend on the brain geometry. In this paper, we find approximate analytical solutions for the forward and the inverse problems in the spheroid geometry. We compare the obtained results with the exact solution of the forward problem and deduce that for a wide range of parameters our approximation is valid. The analysis sheds new light on the role of the volume magnetic fields for solving the inverse problem of magnetoencephalography.

Calculation of plasma centrifugal spraying parameters of fine granules of heat-resistant nickel alloys

2020 ◽  
pp. 471-474
Author(s):  
M.G. Yagodin ◽  
E.I. Starovoytenko
Keyword(s):  
Powder Metallurgy ◽  
Nickel Alloys ◽  
Dynamic Pressure ◽  
Calculated Data ◽  
Powder Size ◽  
Metal Powders ◽  
Heat Resistant ◽  
Gas Dynamic ◽  
Spraying Parameters ◽  
Wide Range

The equipment for the production of wide range of metal powders purposed for powder metallurgy is described. The possibility for producing of powders by the plasma centrifugal spraying is considered taking into account the gas dynamic pressure. The calculated data on the powder size for different materials are given.

Electromechanical fields in a hollow piezoelectric cylinder under non-uniform load: flexoelectric effect

2021 ◽  
pp. 108128652110207
Author(s):  
Olha Hrytsyna
Keyword(s):  
Analytical Solutions ◽  
Point Group ◽  
The Body ◽  
Small Scale ◽  
Gradient Theory ◽  
Local Mass ◽  
Wide Range ◽  
Mass Displacement ◽  
Local Mass Displacement ◽  
Local Gradient

The relations of a local gradient non-ferromagnetic electroelastic continuum are used to solve the problem of an axisymmetrical loaded hollow cylinder. Analytical solutions are obtained for tetragonal piezoelectric materials of point group 4 mm for two cases of external loads applied to the body surfaces. Namely, the hollow pressurized cylinder and a cylinder subjected to an electrical voltage V across its thickness are considered. The derived solutions demonstrate that the non-uniform electric load causes a mechanical deformation of piezoelectric body, and vice versa, the inhomogeneous radial pressure of the cylinder induces its polarization. Such a result is obtained due to coupling between the electromechanical fields and a local mass displacement being considered. In the local gradient theory, the local mass displacement is associated with the changes to a material’s microstructure. The classical theory does not consider the effect of material microstructure on the behavior of solid bodies and is incapable of explaining the mentioned phenomena. It is also shown that the local gradient theory describes the size-dependent properties of piezoelectric nanocylinders. Analytical solutions to the formulated boundary-value problems can be used in conjunction with experimental data to estimate some higher-order material constants of the local gradient piezoelectricity. The obtained results may be useful for a wide range of appliances that utilize small-scale piezoelectric elements as constituting blocks.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Sign in / Sign up

Export Citation Format

Share Document