Mechanical Energy Equation Research Articles

The Relationships in the Law of Conservation of Mechanical Energy and the Ideal Gas Equation in Argon Molecules based on Moldy and Gnuplot

Science and technology have been progressing by leaps and bounds over the past few decades all over the world, resulting in the rapid development of advanced computer programs. Under such circumstances, we are able to prove plenty of theories in a new way with the help of different kinds of modern computer programs. For example, we are capable of investigating the relationship between kinetic energy and potential energy, as well as the connection between temperature and pressure, by applying computer programs to draw graphs. In this paper, moldy and gnuplot which are progressive computer programs are used to paint the graphs about the two relationships in the conservation law of mechanical energy and ideal gas equation respectively in argon gas with the help of the information obtained from the Internet. These two computer programs not only remove the burden of calculation, but also make the relationship clearer. In the end, this paper proves that the sum of kinetic energy and potential energy in argon molecules is invariable in a closed system under different temperatures. Pressure is also tested to have a positive linear association with temperature although the density changes when the argon gas is an ideal gas.

Total mechanical energy budget in open-channel transitions with turbulent flows

Abstract The issue of mechanical energy budget has received a lot of attention in open-channel flows which are frequently turbulent. Turbulent motions play a major role in the total mechanical energy of the flow field. It is difficult to evaluate total mechanical energy balance and energy loss for 3-D turbulent flows in irregular open channels due to the formation of vortices and the presence of secondary currents. In 3-D turbulent flows, the effect of these phenomena has often been considered as the empirical coefficient of energy loss. In this study, these details in the total mechanical energy budget are investigated through theoretical analysis and numerical simulation, a new form of the total mechanical energy equation and a relationship for the energy loss in 3-D turbulent flows are derived, and their applications are examined in open-channel transitions. For this purpose, the Navier-Stokes equations are solved numerically, and the results are used to obtain the various terms in the total mechanical energy balance equation and analyze the details. The relationships derived in this study generally support the concept of the total mechanical energy budget to comprehend the turbulence parameters affecting the mechanical energy balance in various open-channel flows.

Standardization of Mechanical Specific Energy Equations and Nomenclature

SummaryThis paper recommends standardized names and equations for the two most common uses of mechanical specific energy (MSE) concepts: “total MSE” and “downhole MSE.” These names and their equations should be used uniformly in all applications, including electronic drilling recorder (EDR) picklists, rigsite surveillance, engineering surveillance, data analytics, research, and technical publications.MSE, used as a metric for drilling efficiency, is a mathematical calculation of the energy used per volume of rock drilled. The downhole MSE equation calculates the efficiency of the bit alone, while the total MSE equation includes both the bit and drillstring. Those who use MSE in surveillance or analytics know the negative effects created by the lack of standardization over the years; it is certainly not a new problem.The lack of standardized nomenclature has resulted in the use of the same name for different equations, or different names are given for two equations that are identical. This affects the ability of drill teams to engage vendors in the redesigning of performance limitations or to communicate new operational practices between teams and rigs. In addition to standardizing nomenclature, this document corrects a mathematical error that is common in calculating the total MSE.The concern with the inconsistencies has increased as MSE has become a key element in many automated optimization schemes. Inconsistencies or uncertainties in the basis of MSE values calculated in real time or shared in large data sets will affect the industry’s ability to develop useful analytics or to automate rig control platforms and data-driven decisions.This paper also includes a discussion of the MSE measurement errors and their effect on calculated values, which is of particular interest to controls engineers and those involved in data analytics. Examples are provided to illustrate how the two different MSE values are used in field operations. Also, a substantial list of current and potential future uses of MSE is included to encourage better MSE-based practices to potentially lead to the development of new uses in the future, including automation.This ad hoc MSE Standardization Committee is a volunteer group with representation from operators, rig contractors, service companies, and data acquisition vendors. The guidance given reflects their shared experience in utilizing MSE in surveillance and analytics, and the recommended equations are technically correct.

An analytical initial orbit determination method using two observations from a bistatic radar

An analytical initial orbit determination (IOD) method using two observations from a bistatic radar is proposed. Each observation contains bistatic range and the doppler frequency shift, azimuth angle and elevation angle. This problem arises from the challenging catalogue process for small space debris, which features sparse radar observations. By incorporating doppler measurements into the original Lambert’s IOD problem which uses two position vectors and can only be solved iteratively, we show that an analytical solution can be obtained. The specific angular momentum conservation equation and the specific mechanical energy equation are used as kinematic constraints. An ad hoc coordinate system is proposed to derive the solution. The geometric dilution of precision (GDOP) metric is analysed to express the IOD accuracy using a linearization method. Simulations are carried out to demonstrate the performance of the proposed method.

Analytical and numerical investigation of mechanical energy balance and energy loss of three-dimensional steady turbulent flows in open-channels

Abstract Study about the mechanical energy balance and the energy loss of 3-D turbulent flows in open-channels has its own complexities. The governing equation of the mechanical energy in turbulent flows has been previously known and includes turbulence parameters that their calculations or measurements are not easy. In this study, a form of the total mechanical energy equation that leads to a number of significant physical insights is analytically investigated, from which analytical relationships for the energy loss estimation in 3-D turbulent flows are defined. The effect of different turbulence parameters is reflected on the new relationships and analyzed by equalizations replacing unknown correlations with closure approximations using the numerical turbulence simulation. In order to investigate the application of the analytical relationships, numerical simulations are performed by using OpenFOAM software to solve the Navier-Stokes equations with the RSM turbulence model in open-channels with different geometries. Then, the contribution of the turbulence parameters to the total mechanical energy balance is evaluated in uniform and nonuniform turbulent flows and their difference is analyzed, that leads to identify the parameters affecting the friction and local losses. The results demonstrate that the magnitudes of the turbulent diffusion, the work done by the viscous stresses pertaining to the mean motion and the viscous diffusion of the turbulence energy are substantially smaller than the other terms of the total energy equation for turbulent flows in open-channels with different geometries, while the effect of the variations of the turbulence kinetic energy and the work done by the turbulence stresses, that has not been considered in the previous mechanical energy equations, is more important in complex flows. From a practical viewpoint, in order to study the details of the total mechanical energy balance and the energy loss in 3-D turbulent flows with the presence of the secondary currents, the proposed method can be useful.

A numerical derivation of internal energy equation under a staggered mesh system for 1-D thermal hydraulic system analysis codes

This paper introduces a modified internal energy equation derived for multiphase flow in various flow conditions for a staggered mesh system. The pressure drop and heat dissipation terms in the internal energy conservation equation currently being developed based on the assumption that the scalar and momentum variables, which originate from the total energy and mechanical energy equations, respectively, are all located in the cell center of the control volume were redefined such that two different pressures and velocities stored in both cell centers and faces were imposed for the internal energy conservation. To achieve this, first, a modified internal energy conservation equation for a staggered mesh was derived by subtracting the mechanical energy equation from the total energy equation. The equation was then discretized classifying each term by its origin; variables that originated from the total energy equation were defined in the cell center, whereas terms that came from the mechanical energy equation were identified at faces. Since the discretized form of the proposed equation contained face velocity and cell pressure for the heat dissipation and pressure drop terms, respectively, these two terms were calculated implicitly leading to enhanced numerical stability. The accuracy of the modified internal energy equation in predicting the system pressure, fluid temperature, and heat balance for various flow channels was assessed. The verification of the proposed equation was completed through simulations of multiple theoretical problems including saturated liquid depressurization, adiabatic expansion of hydrogen, heat transfer in a helical steam generator, and energy transfer in a converging pipe. An improved result was obtained with the modified equation as the numerical calculation results agreed well with the analytic solutions with relative deviations less than 0.1% for most cases, while the solution obtained by the conventional internal energy equation showed a significant amount of deviation from the analytic solution.

Numerical simulation of separation flows induced by shockwave using an anisotropic turbulence model

This paper studies the performance of a nonlinear anisotropic turbulence model developed in the basis of a statistical partial average scheme. The first order velocity of turbulent fluctuations, retained by a novel average scheme, and turbulent length scale can be determined from the momentum equations and mechanical energy equation of the fluctuation flow, respectively. The two physical quantities are readily to construct the nonlinear anisotropic eddy viscosity tensor and to significantly improve computational results. Two typical shock-wave-boundary-layer-interaction (SWBLI) separation flows are deeply studied using CFD method. One is an incident oblique shock wave impinging on a flat plate with a turbulent boundary layer, and the other one is a transonic turbulent separation flow in a converging-diverging transonic diffuser. Comparisons between the computational results and experimental data are carried out for velocity profiles, density contours, pressure distribution, skin friction coefficient, Reynolds stress as well as streamline vectors distribution. Without using any empirical coefficients and wall functions, the numerical results are in good agreement with the available experimental data, which further confirms that the nonlinear anisotropic eddy viscosity tensor is of the decisive factor for the success of the computational results.

Mathematical Modeling of Spray Impingement and Film Formation on Pharmaceutical Tablets during Coating

The application of coating films is an important step in the manufacture of pharmaceutical tablets. Understanding the phenomena taking place during coating spray application provides important information that can be used to reduce the number of defective tablets and select the optimal conditions for the coating process. In this work, we investigate spray impact and film spreading on a tablet while this passes through the spray-zone in a rotating coating drum. To simulate spray impingement, we developed an one-dimensional (1D) spreading model that is based on the mechanical energy equation. We assumed the spray to be uniform and we divided it into arrays of droplets that impinge successively on the substrate orthogonally to its surface. In the mechanical energy equation that describes the coating spreading, we accounted for the rate of work done on the surface of the liquid coating film by the impinging droplets that leads to volume change (film spreading and thickness increase). The novel model we propose in this work can calculate the coating spreading rate and thickness. We implemented the mathematical model employing the gPROMS Modelbuilder platform. To study the effect of coating properties and process parameters on the film spreading rate and on the final liquid film thickness, we performed variance-based sensitivity analysis. The model predictions are in good agreement with experimental data found in the literature.

Modelagem de um Tanque de Escorva para Bomba em Sistema de Descarte de Óleo

This study aimed to evaluate and create a mathematical model for a priming system of a centrifugal pump, used to transport oil from a storage tank to the production line. Bettini (2007) proposes an equation for validation of a priming tank using water in laboratory tests. The present work is a similar system, but using oil, where the common obstructions of a foot valve, located at the old alignment of pump suction, led to the replacement of that valve by a priming tank system, located upstream the pump. A mathematical model based on the mechanical energy equation and Boyle's law to model the priming tank was created. The results obtained with the model have been validated by comparing with the real system operation, showing excellent agreement. The useful volume ranges of the priming tank, as well as its range of variation during system operation, were very close to those calculated. The efficiency of the system over time was also proven, considering one year after the modification maintenance interventions were not necessary in the system versus the five maintenances required in the year prior to the modification. Keywords: Centrifugal pump, priming tank, Boyle's Law.

UN ENFOQUE DE ESCALAMIENTO PARA LA TEORÍA DE BETZ-JOUKOWSKY

The Betz-Joukowsky (BJ) theory is the starting point to study wind turbines because it provides the upper bound of their power at the megascale. However, not all the assumptions that support this theory have been clearly identified in the literature. In this work, an upscaling approach is applied to the mass, linear momentum and mechanical energy equations at the microscopic scale in order to obtain their megascale counterparts. The analysis allows a systematic identification of the assumptions and restrictions involved in the BJ theory, which were found to serve as a guideline for the shape and size of the averaging domain. The derivations show that the actuator disc concept is indispensable in order to recover Froude's theorem. In addition, numerical experiments evidence that the restrictions related to neglecting mass transport and air power at the lateral surfaces of the averaging domain are the hardest ones to satisfy. The results from this work expand the current understanding of the BJ theory and open new research lines to improve it by identifying its ancillary assumptions and restrictions.

Dimensionless analytical solutions for steady-state fire smoke spread through high-rise shaft

Analytical solutions in terms of dimensionless numbers for the smoke spread through high-rise shafts during fires are essential to provide a fundamental understanding of smoke transport physics, which is a complex coupled heat and mass transfer problem. Existing solutions are often dimensional based on simplification of the problem such as assuming adiabatic conditions. In order to obtain the dimensionless analytical solutions, energy balance equation, mechanical energy equation and mass balance equation were established for smoke spread in high-rise buildings under both mechanical and natural venting conditions. Experiments were designed and conducted on two scaled shafts with different sizes and materials, and the measured results were compared to the dimensionless analytical solutions. It was found that the dimensionless analytical solutions could predict temperature profiles, mass flow rate and neutral plane level accurately. The effect of the adiabatic assumption on the accuracy was also discussed. For example, due to the adiabatic assumption, the error of the calculated mass flow rate required during mechanical venting to maintain a high-rise shaft smoke free was found to increase with a dimensionless number, ω, defined by the geometrical and thermal properties of the shaft.

Mechanical energy and equivalent differential equations of motion for single-degree-of-freedom fractional oscillators

This paper addresses the total mechanical energy and equivalent differential equation of motion for single degree of freedom fractional oscillators. Based on the energy storage and dissipation properties of the Caputo fractional derivatives, the expression for total mechanical energy in the single degree of freedom fractional oscillators is firstly presented. The energy regeneration due to the external exciting force and the energy loss due to the fractional damping force during the vibratory motion are analyzed. Furthermore, based on the mean energy dissipation and storage in the fractional damping element in steady-state vibration, two new concepts, namely mean equivalent viscous damping and mean equivalent stiffness are suggested and the above coefficient values are evaluated. By this way, the fractional differential equations of motion for single-degree-of-freedom fractional oscillators are equivalently transformed into integer-order ordinary differential equations.

Macroscopic momentum and mechanical energy equations for incompressible single-phase flow in porous media.

Modeling flow in porous media is usually focused on the governing equations for mass and momentum transport, which yield the velocity and pressure at the pore or Darcy scales. However, in many applications, it is important to determine the work (or power) needed to induce flow in porous media, and this can be achieved when the mechanical energy equation is taken into account. At the macroscopic scale, this equation may be postulated to be the result of the inner product of Darcy's law and the seepage velocity. However, near the porous medium boundaries, this postulate seems questionable due to the spatial variations of the effective properties (velocity, permeability, porosity, etc.). In this work we derive the macroscopic mechanical energy equation using the method of volume averaging for the simple case of incompressible single-phase flow in porous media. Our analysis shows that the result of averaging the pore-scale version of the mechanical energy equation at the Darcy scale is not, in general, the expected product of Darcy's law and the seepage velocity. As a matter of fact, this result is only applicable in the bulk region of the porous medium and, in the derivation of this result, the properties of the permeability tensor are determinant. Furthermore, near the porous medium boundaries, a more novel version of the mechanical energy equation is obtained, which incorporates additional terms that take into account the rapid variations of structural properties taking place in this particular portion of the system. This analysis can be applied to multiphase and compressible flows in porous media and in many other multiscale systems.

Inkjet Printing and the Steady State Macroscopic Mechanical Energy Balance (SSMMEB) Equation

The author is a co-author on a Wiley book under preparation entitled the “Handbook of Industrial Inkjet” (chief editor Werner Zapka). One area of relevant new content in this book is in surface manufacturing, which refers to the customization of a finished product — usually a manufactured good — by using printing and/or printing-like processes. Surface manufacturing is a form of additive manufacturing in which there is a template in the form of a partially finished object upon which to use as a surface for the (usually product-finishing) custom manufacturing. There are four major types of inkjetting that can be considered for a role in surface manufacturing. After describing these, we then consider their use in surface manufacturing in light of the Steady State Balance Macroscopic Mechanical Energy (SSMMEB) Equation. Some approaches to designing for variability are then discussed.

Theoretical analysis and numerical simulation of mechanical energy loss and wall resistance of steady open channel flow

The mechanical energy loss and the wall resistance are very important in practical engineering. These problems are investigated through theoretical analysis and numerical simulation in this paper. The results are as follows. (1) A new mechanical energy equation for the total flow is obtained, and a general formula for the calculation of the mechanical energy loss is proposed. (2) The general relationship between the wall resistance and the mechanical energy loss for the steady channel flow is obtained, the simplified form of which for the steady uniform channel flow is in consistent with the formula used in Hydraulics deduced by π theorem and dimensional analysis. (3) The steady channel flow over a backward facing step with a small expansion ratio is numerically simulated, and the mechanical energy loss, the wall resistance as well as the relationship between the wall resistance and the mechanical energy loss are calculated and analyzed.

Simulations of three-dimensional, self-oscillating vocal fold replicas with liquid-filled cavities

Synthetic, self-oscillating vocal fold replicas are used in bioreactors to study the response of injected cells to phono-mimetic vibrations. In this type of replica, cells are contained within a cylindrically shaped cavity that runs anterioposteriorly and that is located near the replica surface. During vocal fold replica flow-induced vibration, the cavity deforms, thereby subjecting the cells to a periodically varying stress field. The characteristics of this stress field are unknown, and experimental determination is not presently feasible. Therefore, in this research, numerical simulations of these self-oscillating, cell-containing synthetic replicas are studied. A three-dimensional vocal fold model containing a liquid-filled cavity, developed using the commercial software package ADINA, is described. The effects of glottal airflow are modeled using the mechanical energy equation to simulate the pressure distribution along the replica airway surface. The liquid-filled cavity is modeled through a fully ...

Confined compressive strength model of rock for drilling optimization

Abstract The confined compressive strength (CCS) plays a vital role in drilling optimization. On the basis of Jizba's experimental results, a new CCS model considering the effects of the porosity and nonlinear characteristics with increasing confining pressure has been developed. Because the confining pressure plays a fundamental role in determining the CCS of bottom-hole rock and because the theory of Terzaghi's effective stress principle is founded upon soil mechanics, which is not suitable for calculating the confining pressure in rock mechanics, the double effective stress theory, which treats the porosity as a weighting factor of the formation pore pressure, is adopted in this study. The new CCS model combined with the mechanical specific energy equation is employed to optimize the drilling parameters in two practical wells located in Sichuan basin, China, and the calculated results show that they can be used to identify the inefficient drilling situations of underbalanced drilling (UBD) and overbalanced drilling (OBD).

Energy balance in the δ-SPH scheme

Abstract An in-depth analysis of the energy balance in the δ -SPH model has been carried on. In comparison to the standard SPH scheme, the mechanical energy equation of the δ -SPH variant is characterized by a further term that is generally dissipative and is related to the diffusive operator inside the continuity equation. The behaviour and the structure of such a term have been studied in detail and a number of specifically conceived test cases have been considered, highlighting that the dissipative term is generally small and it mainly acts when spurious high-frequency acoustic components are excited. In spite of such a dissipation mechanism, the δ -SPH appears more accurate than the standard SPH scheme even in simulating inviscid fluids.

Anisotropy of the Reynolds stress tensor in the wakes of wind turbine arrays in Cartesian arrangements with counter-rotating rotors

A 4 × 3 wind turbine array in a Cartesian arrangement was constructed in a wind tunnel setting with four configurations based on the rotational sense of the rotor blades. The fourth row of devices is considered to be in the fully developed turbine canopy for a Cartesian arrangement. Measurements of the flow field were made with stereo particle-image velocimetry immediately upstream and downstream of the selected model turbines. Rotational sense of the turbine blades is evident in the mean spanwise velocity W and the Reynolds shear stress −vw¯. The flux of kinetic energy is shown to be of greater magnitude following turbines in arrays where direction of rotation of the blades varies. Invariants of the normalized Reynolds stress anisotropy tensor (η and ξ) are plotted in the Lumley triangle and indicate that distinct characters of turbulence exist in regions of the wake following the nacelle and the rotor blade tips. Eigendecomposition of the tensor yields principle components and corresponding coordinate system transformations. Characteristic spheroids representing the balance of components in the normalized anisotropy tensor are composed with the eigenvalues yielding shapes predicted by the Lumley triangle. Rotation of the coordinate system defined by the eigenvectors demonstrates trends in the streamwise coordinate following the rotors, especially trailing the top-tip of the rotor and below the hub. Direction of rotation of rotor blades is shown by the orientation of characteristic spheroids according to principle axes. In the inflows of exit row turbines, the normalized Reynolds stress anisotropy tensor shows cumulative effects of the upstream turbines, tending toward prolate shapes for uniform rotational sense, oblate spheroids for streamwise organization of rotational senses, and a mixture of characteristic shapes when the rotation varies by row. Comparison between the invariants of the Reynolds stress anisotropy tensor and terms from the mean mechanical energy equation indicate correlation between the degree of anisotropy and the regions of the wind turbine wakes where turbulence kinetic energy is produced. The flux of kinetic energy into the momentum-deficit area of the wake from above the canopy is associated with prolate characteristic spheroids. Flux upward into the wake from below the rotor area is associated with oblate characteristic spheroids. Turbulence in the region of the flow directly following the nacelle of the wind turbines demonstrates greater isotropy than regions following the rotor blades. The power and power coefficients for wind turbines indicate that flow structures on the order of magnitude of the spanwise turbine spacing that increase turbine efficiency depending on particular array configuration.

The mechanical energy equation for total flow in open channels

The mechanical energy equation is a fundamental equation of a 1-D mathematical model in Hydraulics and Engineering Fluid Mechanics. This equation for the total flow used to be deduced by extending the Bernoulli's equation for the ideal fluid in the streamline to a stream tube, and then revised by considering the viscous effect and integrated on the cross section. This derivation is not rigorous and the effect of turbulence is not considered. In this paper, the energy equation for the total flow is derived by using the Navier-Stokes equations in Fluid Mechanics, the results are as follows: (1) A new energy equation for steady channel flows of incompressible homogeneous liquid is obtained, which includes the variation of the turbulent kinetic energy along the channel, the formula for the mechanical energy loss of the total flow can be determined directly in the deduction process. (2) The theoretical solution of the velocity field for laminar flows in a rectangular open channel is obtained and the mechanical energy loss in the energy equation is calculated. The variations of the coefficient of the mechanical energy loss against the Reynolds number and the width-depth ratio are obtained. (3) The turbulent flow in a rectangular open channel is simulated using 3-D Reynolds averaged equations closed by the Reynolds stress model (RSM), and the variations of the coefficient of the mechanical energy loss against the Reynolds number and the width-depth ratio are discussed.