Calculation Of Kinetic Energy Research Articles

A spectroscopically determined potential energy surface for the ground state of H216O: A new level of accuracy

The potential energy function for the electronic ground state of the water molecule has been obtained by fitting rotation-vibration term values involving J≤14 for 24 vibrational states of H216O together with 25 additional vibrational term values belonging to higher excited states. The fitting was carried out by means of an exact kinetic energy Hamiltonian. It was found that the differences between the exact kinetic energy calculations and calculations with the morbid program (i.e., calculations with an approximate kinetic energy operator) depend only very slightly on the parameters of the potential. This fact allowed us to make an inexpensive fitting using the morbid approach and still get the accuracy obtainable with the exact kinetic energy Hamiltonian. The standard deviation for 1600 term values was 0.36 cm−1. For 220 ground state energy levels the standard deviation was 0.03 cm−1. With the fitted potential, calculations of term values with J≤35 were carried out. This showed the excellent predictive power of the new potential. For the J=20 term values in the vibrational ground state, the deviations from experiment are typically below 0.2 cm−1. The discrepancy for the observed level with the highest Ka value, JKaKc = 20200, is only 0.008 cm−1. The calculated term value for the observed level with the highest J, 35035, deviates 0.1 cm−1 from experiment. Because of the level of accuracy achieved in these calculations, we can for the first time demonstrate the breakdown of the Born–Oppenheimer approximation for the water molecule. The high Ka level calculations allow us to show that the rotational energy level structure in water is at least of a very different nature than the fourfold cluster structures observed for H2Se and calculated for H2S, H2Se, and H2Te.

Fluorescence spectra and torsional potential functions for trans-stilbene in its S and S1(π,π*) electronic states

The fluorescence excitation spectra and dispersed fluorescence spectra of trans-stilbene have been recorded and analyzed. Vibrational assignments for the eight low-frequency modes have been made for both the S0 and S1(π,π*) electronic states, and these differ substantially from those of previous workers. Two-dimensional kinetic and potential energy calculations were carried out in order to determine the potential energy surfaces for the two phenyl internal rotations ν37 and ν48. The function V(φ1,φ2)= 1/2V2(2+cos 2φ1+cos 2φ2)+V12 cos 2φ1 cos 2φ2 +V12′ sin 2φ1 sin 2φ2, with V2=1550 cm−1, V12=337.5 cm−1, and V12′ = 402.5 cm−1 for the S0 state and with V2=1500 cm−1, V12=−85 cm−1, and V12′ = −55 cm−1 for the S1(π,π*) state fits the observed data (nine frequencies for S0 and six for S1) extremely well. The barriers to simultaneous internal rotation of both phenyl groups are given by twice the V2 values. The fundamental frequencies for these torsions are ν37=9 cm−1 and ν48=118 cm−1 for the S0 state and ν37=35 cm−1 and ν48=110 cm−1 for the S1 excited state. The third torsion ν35, which is the internal rotation about the C=C bond, was assigned at 101 cm−1 for the S0 state based on a series of overtone frequencies (202, 404 cm−1, etc.). For S1, ν35=99 cm−1 based on observed frequencies at 198, 396 cm−1, etc. Kinetic energy calculations were also carried out for this mode, and a one-dimensional potential energy function of the form V(θ)=1/2V1(1−cos θ)+1/2V2(1−cos 2θ)+1/2V4(1−cos 4θ) was utilized to reproduce the frequencies for the ground state. For the excited state, an additional V8 term was added in order to fit the data for the trans potential energy well. The data indicate that the trans→twist barrier for the S1 state is higher than 1400 cm−1. However, a somewhat revised frequency assignment would be compatible with a barrier of 1250 cm−1, which is close to the value of 1200 cm−1 determined from dynamics studies.

Heat Transfer Predictions With Extended k–ε Turbulence Model in Radial Cooling Ducts Rotating in Orthogonal Mode

Standard and extended k–ε turbulence closure models have been employed for three-dimensional heat transfer calculations for radially outward flow in rectangular and square cooling passages rotating in orthogonal mode. The objective of this modeling effort is to validate the numerical model in an attempt to fill the gap between model predictions and the experimental data for heat transfer in rotating systems. While the trend of heat transfer predictions by the standard k–ε turbulence model is satisfactory, the differences between the data and the predictions are approximately 30 percent or so in the case of high rotation number flow. The extended k–ε turbulence model takes an approach where an extra “source” term based on a second time scale of the turbulent kinetic energy production rate is added to the equation for the dissipation rate of turbulent kinetic energy. This yields a more effective calculation of turbulent kinetic energy as compared to the standard k–ε turbulence model in the case of high rotation number and high density ratio flow. As a result, comparison with the experimental data available in the literature shows that an improvement of up to a significant 15 percent (with respect to data) in the heat transfer coefficient predictions is achieved over the standard k–ε model in the case of high rotation number flow. Comparisons between the results of the standard k–ε model and the extended formulation are made at different rotation numbers, different Reynolds numbers, and varying temperature ratio. The results of the extended k–ε turbulence model are either as good or better than those of the standard k–ε model in all these cases of parametric study. Thus, the extended k–ε turbulence model proves to be more general and reduces the discrepancy between the model predictions and the experimental data for heat transfer in rotating systems.

Kinetic Energies and Exchange Model Potentials Calculated with a Simple Analytical Representation of Atomic Densities

A simple analytical representation of Hartree-Fock atomic densities recently devised defines ρ(r) as a sum of exponentially decaying functions. This basis set-independent representation is here applied to the calculation of kinetic energies, making use of various well-known kinetic energy functionals. Despite its extreme simplicity, these functions provide highly reliable representations of ρ(r), allowing the calculation of kinetic energies to within 1% or less. This representation of the density is also analyzed in exchange-only functional calculations. A semiempirical model exchange potential developed before (Pacios, L. F. J. Phys. Chem. 1992, 96, 7294) is here modified and analyzed

Equivalent continuum models of large platelike lattice structures

A new continuum method based on the concept of energy equivalence is introduced to represent a large platelike periodic lattice structure as an equivalent continuum plate. The procedure for developing an equivalent continuum plate involves the use of existing well-defined finite-element matrices for easy calculation of strain and kinetic energies contained in a representative lattice cell of the lattice plate. The strain and kinetic energies contained in the representative lattice cell of lattice plate and the finite-element of equivalent continuum plate are expressed in terms of continuum degrees-of freedom introduced in this paper with corresponding reduced stiffness and mass matrices. The structural properties of the equivalent continuum plate are then obtained by simply equating the reduced stiffness and mass matrices for the equivalent continuum plate to those for the lattice plate. Free vibration analyses for the continuum plates are conducted to evaluate the continuum method proposed in this paper. Numerical tests show that the present continuum method gives very reliable structural and dynamic properties compared with other well-recognized continuum methods.

Graph theoretical approach—II. determination of generalized forces for a class of systems consisting of particles and springs

The process of determining the generalized forces (as a function of generalized coordinates and system parameters) of a system consisting of mass particles and springs is often extremely laborious. In this paper, a method is presented where the derived formulae have a form of evident functions not only of generalized coordinates and system parameters but also of the topological structure of a system. This paper contains the derivation of general formulae for potential energies of gravity and elastic forces as well as for the generalized forces related to both potential and nonpotential forces acting on a system of particles. It is assumed that the system of particles has a topological tree structure, and that linear springs are situated arbitrarily between the particles. The method is based on a graph-theoretical approach and is consistent with the kinetic energy calculation presented in Arczewski (J. Franklin Inst., Vol. 329, pp. 469–481, 1992).

Application of graph theory to the mathematical modelling of a class of rigid body systems

The process of determining equations of motion of a system consisting of rigid bodies and springs is often extremely laborious. In this paper, a method is presented whereby all terms of a system of equations for planar rigid body systems are calculated systematically. It is assumed that the system of rigid bodies has a topological tree structure. The method is based on a graph-theoretical approach and is consistent with that for the calculation of kinetic and potential energies presented by Arczewski ( J. Franklin Inst., Vol. 324, pp. 351–367 and pp. 369–386, 1987). The proof of the method and an example are provided.

Kinetic energy functions for the ring-bending and ring-twisting vibrations of asymmetrically substituted six-membered rings

Vector models were developed for calculating the kinetic energy (reciprocal reduced mass) expressions for the ring-bending and single bond ring-twisting vibrations of six-membered rings with one double bond. These models are valid for asymmetrically substituted rings. A two-dimensional kinetic energy calculation was carried out and the twist-bend cross terms were also calculated. The reduced masses calculated for several oxygen-containing analogs of cyclohexene increase as the ring bends or twists. Kinetic energy expressions as functions of the vibrational coordinates were derived.

Iso-erodent map of Zambia: Part I: The calculation of erosivity indices from a rainfall data bank

The first stage in the establishment of an iso-erodent map of Zambia is the development of a Zambian rainfall data bank, the calculation and discussion of three erosivity indices and correlating the erosivity indices with rainfall amount. The data bank is made up of 10 years of data from eight different meteorological stations representing the different rainfall zones in Zambia. The data bank includes 1. 1. total rainfall amounts split into component rainfall amounts with constant intensity 2. 2. kinetic energy calculations 3. 3. the erosivity indices. Three erosivity indices have been calculated: E130 (WISCHMEIER 1959), KE (KINNELL 1981) and ΣN p I p (LAL 1976). Daily amount of rainfall explained 96, 95, and 89% of the variation in those indices respectively. Although the three indices are of comparable value, advantages and disadvantages of each of them are discussed. This work will be the basis for modelling the monthly erosivity and finally for making the iso-erodent map of Zambia (see part II, this volume).

Influence of initial and boundary conditions on vortex ring development

THIS paper presents the results of an experimental investigation comparing the mass and energy content of fully formed laminar vortex rings in air with that of the original pulse that generated them for a variety of initial and boundary conditions. In particular, the fractional entrainment of mass and the partition of initial energy between kinetic energy of translation and kinetic energy of rotation are studied. It is found that these characteristics depend strongly on the boundary conditions. A technique is presented which enables calculation of kinetic energy of rotation from motion picture sequences. The ratio of characteristi c translational speed to characteristic rotational speed is shown to be a useful parameter for correlation of data. Data on vortex size and speed are presented using this correlation, and it is seen that all data, regardless of initial and boundary conditions, fall on a single curve. A theoretical curve is derived, and it is seen that the data compare well with it. Contents In many applications utilizing nonsteady flow, it is of importance to understand the mechanisms of mass, momentum, and energy exchange between a primary flow and a secondary flow. The present work attempts to shed light on such mechanisms by ultilizing a simple example: the formation of vortex rings. A sketch of the system under study is shown in Fig. 1. We consider that, at time £ = 0, a uniform pulse is initiated with velocity Uj(t) for a duration Tp through a chimney of height H and diameter D0. Note that, when //=0, we have a simple orifice. After a certain period of time, the pulse becomes a fully formed vortex ring of diameter D which propagates at a speed U. The nature of the vortex ring will depend on the boundary conditions (H, DQ) and on the initial conditions luj(t), T p }. Since all of the vorticity essential to the existence of a vortex is generated in boundary layers prior to formation, different geometries will produce different amounts of vorticity. Furthermore, in the case of the simple orifice, vorticity of opposite sense is generated on the outside of the orifice by the boundary layer formed from the entrained flow, passing over the solid surface, thereby impeding the influx of entrained flow and diminishing the net available vorticity. These effects will result in the generation of vortices with different sizes and different propagation speeds, as well as differences in internal flowfields. In the experimental investigation, vortex rings were generated in still air for various sizes of simple orifices and chimneys by means of a large loudspeaker energized by a step in voltage, producing a controllable jet-pulse intensity. The pulse intensity and duration were monitored by means of a hot-film anemometer. Oscillographic records of the

Application of graph theory to the determination of potential energy of systems consisting of rigid bodies and springs

The process of determining the potential energy, as a function of generalized coordinates, of a system consisting of rigid bodies and springs is often extremely laborious. In this paper, a method is presented by means of which all the potential energy terms of a system are calculated in a systematic manner. It is assumed that the system of rigid bodies has a topological tree structure, and that linear springs are situated arbitrarily between the rigid bodies. The method is based on a graph-theoretical approach and is consistent with that of the calculation of kinetic energy presented in ( 2 , J. Franklin Inst., Vol. 324, No. 3, pp. 351–367, 1987). The paper contains the derivation of general formulae for potential energies of gravity and elastic forces. These formulae are implemented for the case of a planar system; an example is also provided.

Long‐term in situ calculations of kinetic energy and Reynolds stress in a deep sea boundary layer

Long‐term measurements of Reynolds stress and kinetic energy are necessary to characterize the bottom boundary layer during storms which transport sediment in the High Energy Benthic Boundary Layer Experiment area on the Nova Scotia Rise. The benthic acoustic stress sensor current meter system employed an on‐board microcomputer to process 2‐Hz velocity vector data into 20‐min averages of velocity vectors and the turbulent Reynolds stress tensor. The data indicate a bottom boundary layer, which is usually transitional, with bottom roughness scaling between viscous (z0 = v/9u*) and rough (z0 = ks/30). During the periods of transitional flow the logarithmic layer was not a useful indicator of boundary stress. Direct stress and energy measurements showed that the stress was much less than that indicated by the slope of the transitional mean logarithmic velocity profile. Several westward high‐velocity events, which were modulated by a significant tidal component, were recorded. It appears from this data set that the mean westward flow must be accompanied by a train of internal tidal waves to achieve the velocities necessary to be the sediment transport events identified as “benthic storms”.

Kinetic Energy Calculations for Non‐Newtonian Fluids in Circular Tubes

ABSTRACTExpressions to calculate the kinetic energy of Herschel‐Bulkley fluids flowing in circular tubes are presented. The influence of yield stress, consistency coefficient and flow behavior index are investigated. An equation ‐ based on theoretical considerations ‐ for determining kinetic energy from a kinetic energy correction factor is developed. Solutions are given in a graphical format to facilitate kinetic energy calculations.

Dynamic analysis of practical blades with shear center effect

Shear center effects on the natural frequencies and mode shapes of rotating and non-rotating practical blades are considered. An 18 degrees of freedom thick beam finite element is developed. Bending and shear force displacements and slopes, and torsional displacements are taken as degrees of freedom at both ends of the element. Total blade deflection slopes are considered as composed of bending and shear force deflection slopes in calculations of blade strain and kinetic energy. This element is compared with the existing thin and thick beam finite elements, and theoretical models. Results obtained for the vibration characteristics of rotating and non-rotating non-uniform aerofoil cross-sectioned blades are compared with the available calculated and experimental values. In all cases considered the element exhibits good convergence characteristics and produces accurate results.

High-energy, Low-velocity Close-range Shotgun Wounds

A review is made of 42 close-range shotgun wounds. The massive injuries had many of the characteristics attributed to high-velocity missile wounding. Calculations of kinetic energy show magnitudes at muzzle velocity in the range of those generated by high-velocity rifle rounds. This is due to the very great mass. While emphasis has been placed in the literature on the factor of velocity, it is stressed that the mass is also very important since the equation for kinetic energy is the product of the square of velocity and mass. It is kinetic energy which relates to tissue injury. Medical writings should return velocity to its proper place as a factor and not an end-point of energy calculations.

Truncation error terms in the kinetic energy calculation in the HELP algorithm and their consequences

When the HELP computer code is applied to conical-shaped charge warhead problems, the computed internal energy predicts a thermal state completely different than that indicated by experiments. The cause of this phenomenon is the numerical interchange of the kinetic energy to internal energy which is generated by terms of the order of the truncation error in the kinetic energy calculation. A correction is given and qualitative thermal agreement is achieved for the first time between a HELP calculation and experimental evidence. The effects of the original kinetic energy calculation on the accuracy of the internal energy is problem dependent and criteria for such a determination are given in terms of the truncation error terms for the internal energy. A mesh refinement study further illustrates the consequences of the original and modified kinetic energy calculations. The same phenomenon can occur in other computer codes with a Particle-In-Cell based algorithm and the given correction is applicable.

The Physical Oceanography of Two Rings Observed by the Cyclonic Ring Experiment. Part II: Dynamics

Abstract Data from the 1977 Cyclonic Ring Experiment is used to examine the, density, velocity, vorticity and energy distributions in a Gulf Stream cyclonic ring. A time series on ring BOB provides information on the temporal changes in these parameters. The rings are shown to have an interior in near-solid-body rotation surrounded by a frontal jet exhibiting a maximum in potential vorticity. There is evidence that the baroclinic structure of the rings extends to the bottom and that there is significant energy in vertical modes higher than the first baroclinic mode. Calculations of available potential energy (APE) and kinetic energy are presented along with the time rates of change in the APE. The APE is partitioned into the energy in the mean ring and a set of perturbations about the mean. The temporal variations in the perturbation APE are large enough to be important to the adjustment and decay of mean APE in the ring. Ring spindown is discussed in terms of the observations and several ring models prop...

Kinetic energy and temperature profiles of shockwaves in solids

Tsai and MacDonald (1973, 1978) published a critical analysis of the authors' molecular dynamic simulation of shockwave propagation. Most of their criticism was directed to the authors' definition of temperature and kinetic energy. It is shown that their analysis is incorrect and that the authors' calculation of kinetic energy is correct and the definition of temperature is reasonable. Most of the misunderstanding arose from the other not understanding that all the authors' calculations were performed in a stationary frame of reference. It is further shown that, when there is a variation in the planar velocity, the Tsai-MacDonald local kinetic temperature is neither equal to the local kinetic energy (expressed as temperature) nor does it have the characteristics of temperature.

A couple-stress theory for gridwork-reinforced media

A two-dimensional continuum model with couple stress for a gridwork-reinforced composite is developed. The derivation is based on the calculation of equivalent potential and kinetic energies stored in the representative medium. Hamilton's principle is used to derive the equations of motion and the boundary conditions. Deformation variables are defined and the constitutive relations are subsequently derived. The in-plane transverse vibration problem is investigated as an evaluation example for which both the continuum approach and the discrete element method are employed to compute the frequencies. Numerical results show that solutions according to both methods agree reasonably well.

Electron spectroscopy of transition metal oxide surfaces

A review of XPS and AES studies performed on clean transition metal oxide (TMO) surfaces is presented. A discussion of the main aspects of these spectra and the difficulties encountered in their interpretation is given. Recent developments in the calculation of binding energies and Auger kinetic energies are described and used to discuss interpretations of chemical shifts in XPS and AES. A possible correlation between extra-atomic relaxation and the ionicity of the compound, for the zinc chalcogenides and zinc halides, is put into perspective. Multiplet splitting is briefly discussed. The influence of the number of unpaired spins and of the nature of the chemical bond, is emphasized. Spectra of the oxides MnO, CoO and ZnO are shown and discussed together with the data of the metals Co, Ni, Cu and compounds of Cr, Mn, Fe. Multiple excitations are shown to be linked to the paramagnetic character of the compound. This is illustrated by the CuO, Cu 2O, NiO, CoO and MnO spectra. Multiple effects in the corresponding Auger spectra are also discussed. It is shown that in some compounds the doubly excited state is sufficiently long lived to give rise to shake-up satellites in Auger spectra (NiO), but not in others (CuO and CoO). Valence band studies using XPS and AES are described. XPS-valence band spectra have been interpreted on the basis of band structure calculations (ZnO and ReO 3) or crystal field theory (NiO and Cr 2O 3). For ReO 3 cross-section modulation effects are emphasized. The valence band structures of V 2O 5 and of lower oxides (V 4O 9 and V 6O 13), present on V 2O 5 after different reducing treatments, are discussed. A discussion of the influence of valence band structure in Auger spectra, based on the comparison of the O (KLL) spectra of different oxides, is presented. Finally, broadening effects, mainly lifetime broadening, of both photolines and Auger transitions are discussed. For the Zn chalcogenides the width of certain Zn lines in different chemical environments, is apparently linked to other properties, such as the ionicity of the bond.