Mechanical Energy Balance Research Articles

鋼ガス圧接部ふくらみの熱間せん断による非破壊検査（第３報） 有限要素法による鋼ガス及び熱間せん断時の変形と応力の解析

In the previous report, an approximate model which describes the mechanism of bulge removal by hot shearing was presented. But the details of stress distribution and plastic deformation mechanism in this process which have influence on the occurrence of a linear defect could not be known experimentally because of its high temperature. In this report, the problem of determining temperature and stress histories, stress distributions during gas pressure welding of steel and subsequent hot shearing is solved on the basis of coupled thermal and mechanical energy balances using the finite element method. A nonlinear heat transfer analysis and a thermo-elastic-plastic stress analysis are performed simultaneously taking into account the temperature dependence of the material properties, heat losses due to convection and radiation, and phase transformation.Tensile stress is confirmed to be generated just under the edge in hot shearing which opens the joint when the welding condition is not suitable. The computational results are compared with experimental ones, showing good correlation between them.

Nonlinear Hertzian Indentation Fracture Mechanics

Indentation cracking under blunt indenters is analyzed using nonlinear fracture mechanics. The usual assumptions of linear elastic fracture mechanics have been replaced with a nonlinear load vs load‐point displacement curve while assuming the material is linear elastic. The load, the load‐point displacement, and a function of the crack area have been related to the crack driving force, J, while assuming a cone fracture under the Hertzian sphere. Experimentally, it was found that the load‐displacement curve during loading, cracking, and unloading is nonlinear. The crack length is empirically shown to be proportional to the load‐point displacement for several indenters. The experimentally measured relations between indenter load, load‐point displacement, and crack geometries are then analyzed with mechanical energy balances based on the similitude of crack lengths with load‐point displacements. The Hertz hardness that describes the nonlinear load vs load‐point displacement relation during cracking is derived on the constant J line in load‐displacement space. Finally, well‐known experimental expressions that relate load to crack length are shown to be indistinguishable from the load‐point displacement analysis reported.

Application of the meta‐channel concept: Construction of the meta‐channel hydraulic geometry for a natural catchment

AbstractThis is the second in a series of three papers about the meta‐channel concept which illustrates the derivation of the principles behind the concept, the construction of the hydraulic geometry and the application of the concept to flood routing, respectively. It was shown in the first of these that a channel network in a catchment can be conceptualized into a single ‘effective channel’ representation: a meta‐channel. This study uses this conceptualization to show how such a meta‐channel can be constructed. The techniques derived are applied to one catchment in New Zealand. We derive hydraulic geometries expressed as functions of flow distance throughout this catchment based on the Leopold and Maddock power laws. This derivation uses classical published values for hydraulic geometry coefficients and exponents, regional parameterization of the index flood relationship for New Zealand as a whole, together with local knowledge regarding the order of magnitude of the channel roughness. Conservation principles derived from the continuity and mechanical energy balance equations are used to construct the hydraulic geometry of the meta‐channel of this catchment. A meta‐channel long profile is established and compared against the mainstream long profile. The effectiveness of the Leopold and Maddock power law assumptions is tested by comparing the derived hydraulic geometry against available field cross‐sectional data for the gauging site at the outlet of the catchment.

Spiral instabilities in the flow of highly elastic fluids between rotating parallel disks

Experimental observations and linear stability calculations are presented for the stability of torsional flows of viscoelastic fluids between two parallel coaxial disks, one of which is held stationary while the other is rotated at a constant angular velocity. Beyond a critical value of the dimensionless rotation rate, or Deborah number, the purely circumferential, viscometric base flow becomes unstable with respect to a nonaxisymmetric, time-dependent motion consisting of spiral vortices which travel radially outwards across the disks. Video-imaging measurements in two highly elastic polyisobutylene solutions are used to determine the radial wavelength, wavespeed and azimuthal structure of the spiral disturbance. The spatial characteristics of this purely elastic instability scale with the rotation rate and axial separation between the disks; however, the observed spiral structure of the secondary motion is a sensitive function of the fluid rheology and the aspect ratio of the finite disks.Very near the centre of the disk the flow remains stable at all rotation rates, and the unsteady secondary motion is only observed in an annular region beyond a critical radius, denoted R*1. The spiral vortices initially increase in intensity as they propagate radially outwards across the disk; however, at larger radii they are damped and the spiral structure disappears beyond a second critical radius, R*2. This restabilization of the base viscometric flow is described quantitatively by considering a viscoelastic constitutive equation that captures the nonlinear rheology of the polymeric test fluids in steady shearing flows. A radially localized, linear stability analysis of torsional motions between infinite parallel coaxial disks for this model predicts an instability to non-axisymmetric disturbances for a finite range of radii, which depends on the Deborah number and on the rheological parameters in the model. The most dangerous instability mode varies with the Deborah number; however, at low rotation rates the steady viscometric flow is stable to all localized disturbances, at any radial position.Experimental values for the wavespeed, wavelength and azimuthal structure of this flow instability are described well by the analysis; however, the critical radii calculated for growth of infinitesimal disturbances are smaller than the values obtained from experimental observations of secondary motions. Calculation of the time rate of change in the additional viscous energy created or dissipated by the disturbance shows that the mechanism of instability for both axisymmetric and non-axisymmetric perturbations is the same, and arises from a coupling between the kinematics of the steady curvilinear base flow and the polymeric stresses in the disturbance flow. For finitely extensible dumb-bells, the magnitude of this coupling is reduced and an additional dissipative contribution to the mechanical energy balance arises, so that the disturbance is damped at large radial positions where the mean shear rate is large.Hysteresis experiments demonstrate that the instability is subcritical in the rotation rate, and, at long times, the initially well-defined spiral flow develops into a more complex three-dimensional aperiodic motion. Experimental observations indicate that this nonlinear evolution proceeds via a rapid splitting of the spiral vortices into vortices of approximately half the initial radial wavelength, and ultimately results in a state consisting of both inwardly and outwardly travelling spiral vortices with a range of radial wavenumbers.

Mixed‐Layer Hydrothermal Reservoir Model

This paper describes a new epilimnetic mixing formulation for two‐dimensional reservoir models. This formulation incorporates a wind‐mixing algorithm based on a mixed‐layer theory developed previously for the upper ocean. Given an input of relevant meteorological parameters, the algorithm computes a heat‐flux source term and a mixed‐layer thickness using a mechanical energy balance approach. The computed heat‐flux profile is used as a source term to the two‐dimensional (longitudinal‐vertical) transport equation for temperature. This transport equation is coupled with a corresponding laterally averaged hydrodynamic model developed previously. The combined formulation is solved using a semi‐implicit, central‐differencing scheme. Overall, this modeling approach requires minimal parameterization of wind‐mixing processes. The combined model was adapted to an elongated (10.6‐km‐long, 0.8‐km‐wide) reservoir system, the Wanaque Reservoir, New Jersey. Wanaque Reservoir data were collected semimonthly during a cali...

Environmental correlate to underwater acoustic noise in the Arctic Ocean

In an effort to find a physical parameter to use as a predictor of acoustic noise in the Arctic Ocean, a modified version of the AIDJEX plastic ice model was used to study the mechanical energy balance of the sea ice cover in the vicinity of long-term acoustic measurements taken in the Lincoln Sea in 1989–90. It was assumed that pa−pw (the rate at which the air works on the ice minus the rate at which the ice works on the water) was the power available for generating noise and that a fixed fraction of this power was converted locally into acoustic noise intensity in the water. The dominant environmental variable in this term was the geostrophic wind velocity, derived from barometric pressure measurements from The Coordinated Arctic Buoy Program [Colony and Rigor, APL/Washington]. A relative noise intensity was determined by calculating the quantity pa−pw on a grid over the Arctic Ocean, weighting by transmission loss from the grid points to the measurement site, and summing. Monthly time series correlation coefficients between pa−pw and 15-Hz noise intensity were generally near 0.6. [Sponsored by the AEAS Program of the Office of Naval Research.]

Analysis of a membrane enclosed catalytic reactor for butane dehydrogenation

A mathematical model consisting of component mass balances and a mechanical energy balance was formulated and simulated for butane dehydrogenation in a cocurrent, packed bed, isothermal membrane enclosed catalytic reactor at 723 K and 152 kPa (1.5 atm ). Properties of a y-alumina membrane with 4-nm diameter pores were used in the simulations. The mechanical energy balance accounted for the axial pressure change and affected the flows in the reactor, but due to the low reaction rate, it did not significantly alter the conversion calculated. Butane conversions higher than the equilibrium conversion were calculated. A single reactor system that consists of a packed catalyst bed enclosed in a tube with an impermeable section at the reactor entrance, followed by a permeable section was proposed. For maximum butane conversion, the length of the impermeable section was determined to be about 2.4 cm, and the amount of butane converted in this section was 2.2% (equilibrium conversion at these conditions was 10.3% ). For reactors with butane space times less than 0.1 sec, permeation reduced the butane conversion. Gas that entered the reactor on the low pressure side displayed a reversal in the permeation direction. This phenomenon is observed for the gases that permeate through the membrane by Knudsen diffusion, and have sufficient reactor space time. This sufficient space time limit was 0.5 sec for the investigated system.

Analysis of a membrane enclosed catalytic reactor for butane dehydrogenation

A mathematical model consisting of component mass balances and a mechanical energy balance was formulated and simulated for butane dehydrogenation in a cocurrent, packed bed, isothermal membrane enclosed catalytic reactor at 723 K and 152 kPa (1.5 atm). Properties of a γ-alumina membrane with 4-nm diameter pores were use din the simulations. The mechanical energy balance accounted for the axial pressure change and affected the flows in the reactor, but due to the low reaction rate, it did not significantly alter the conversion calculated. Butane conversion higher than the equilibrium conversion were calculated. A single reactor system that consists of a packed catalyst bed enclosed in a tube with an impermeable section at the reactor entrance, followed by a permeable section was proposed. For maximum butane conversion, the length of the impermeable section was determined to be about 2.4 cm, and the amount of butane converted in this section was 2.2% (equilibrium conversion at these conditions was 10.3%). For reactors with butane space times less than 0.1 sec, permeation reduced the butane conversion. Gas that entered the reactor on the low pressure side displayed a reversal in the permeation direction. This phenomenon is observed for the gases that permeable through the membrane by Knudsen diffusion, and have sufficient reactor space time. This sufficient space time limit was 0.5 sec for the investigated system.

Can steady-state momentum equations be used in modelling pressurization of adsorption beds?

Modelling of pressurization and blowdown steps of a pressure swing adsorption (PSA) cycle has been carried out using a steady-state momentum equation (Darcy or Ergun) locally in the bed. There has been some controversy about the validity of such an approach since Darcy's or Ergun's equations are strictly valid only for quasi-stable flow, and during pressurization at short times, the flow is unstable. In order to check the assumptions usually made in modelling isothermal pressurization, three models are developed. Model I includes mass balance and unsteady-state mechanical energy balance for a packed bed with non-porous, inert particles; Model II neglects the kinetic energy change in the mechanical energy balance and Model III considers a steady-state mechanical energy balance. The latter model is the one used in previous work. The numerical tool used to solve all the model equations is the moving finite element method (MFEM) with polynomial approximation of any degree in each element developed by C. Sereno 1. Comparison of simulation results for the various models showed that steady-state momentum equations can be safely used in modelling pressurization of adsorbers.

CONTINUUM MECHANICS APPLIED TO A VERY THIN LIQUID FILM: ESTIMATION OF SURFACE TENSION FOR THE n-ALKANES

Abstract This analysis demonstrates that, at least in some cases, continuum mechanics can be successfully applied in problems involving very thin films, so long as proper recognition is given to long-range intermolecular forces and to the finite size of molecules. In this particular case, the integral mechanical energy balance is used to analyze a simple thought experiment in which a thin liquid film is slowly stretched. The result is an expression for surface tension as a function of the Hamaker constant describing liquid-liquid, long-range intermolecular forces, the molecular weight, and the liquid density, which is in good agreement with previous experimental measurements for n-alkanes.

The Mean Power of Forces and Moments in Unsteady Aerodynamics

AbstractThe balance of mechanical energy serves as the starting point for a unified and consistent approach to the two basic and inherent mechanisms in unsteady aerodynamics, propulsion and flutter. Both phenomena are mutually dependent and interchangeable with each other by a mere variation of the reduced frequency as the governing magnitude with respect to unsteady effects.

Phase distribution during gas—liquid flow through horizontal dividing junctions

A model has been developed for the description and prediction of the phase distribution during gas—liquid flow with small liquid holdup values ( ε L < 0.06 ) through horizontal tubes with a horizontal branch. This so-called Double Stream Model has been derived from the steady-state macroscopic mechanical energy balance (extended Bernoulli equation), applied to the “inlet-to-run” stream and the “inlet-to-branch” stream of both the gas phase and the liquid phase. From the formulation of the Double Stream Model it follows that the branch liquid mass intake fraction λ L , which can be calculated according to the procedure given in appendix A, is a function of: • — the branch gas mass intake fraction λ G , • — the geometry of the junction, and • — the ratio κ of kinetic energies per unit of volume of gas and liquid respectively in the inlet. The value of κ, can be calculated from the densities and mass flow rates of gas and liquid, the liquid holdup and the velocity profiles of both gas and liquid in the inlet of the junction. The Double Stream Model has been verified with data obtained from experiments carried out at the University of Amsterdam and at institutes in Tulsa and Manitoba.

Sea-ice mechanical energy balance: nearshore Chukchi Sea, 1982

The mechanical energy balance of sea ice provides information about ice dynamic behavior, driving forces and the constitutive law. The energy balance equation, formed as the product of ice velocity with the ice momentum balance equation, describes changes to the kinetic and potential energy densities as power is input to the ice by wind and current. The momentum balance equation may also be used to describe the ice-stress divergence, air stress, and water stress, but the scalar form of the energy balance is simpler to understand. This paper provides new interpretations of several terms in the energy balance equation, in particular power input by air and water stress and by sea-surface tilt. Barometric pressure fields and drifting buoys deployed on the Chukchi Sea ice cover during 1982 provide wind, ice motion and current measurements that allow each term in the energy balance equation to be evaluated as a function of time. Magnitudes of power input by wind and current show how the energy balance is decomposed and help describe the relative importance of these driving forces. In the nearshore Chukchi Sea during February, March and April 1982, both wind and current provided significant forcing of the ice. Ice stress was also important and, at times, dominated other terms in the mechanical energy balance.

Sea-ice mechanical energy balance: nearshore Chukchi Sea, 1982

The mechanical energy balance of sea ice provides information about ice dynamic behavior, driving forces and the constitutive law. The energy balance equation, formed as the product of ice velocity with the ice momentum balance equation, describes changes to the kinetic and potential energy densities as power is input to the ice by wind and current. The momentum balance equation may also be used to describe the ice-stress divergence, air stress, and water stress, but the scalar form of the energy balance is simpler to understand. This paper provides new interpretations of several terms in the energy balance equation, in particular power input by air and water stress and by sea-surface tilt. Barometric pressure fields and drifting buoys deployed on the Chukchi Sea ice cover during 1982 provide wind, ice motion and current measurements that allow each term in the energy balance equation to be evaluated as a function of time. Magnitudes of power input by wind and current show how the energy balance is decomposed and help describe the relative importance of these driving forces. In the nearshore Chukchi Sea during February, March and April 1982, both wind and current provided significant forcing of the ice. Ice stress was also important and, at times, dominated other terms in the mechanical energy balance.

A model for predicting liquid route preference during gas—liquid flow through horizontal branched pipelines

A model has been developed for the calculation of liquid route preference during separated gas—liquid flow with small liquid holdup values of ε L < 0.06, through horizontal tubes with a horizontal branch. This so-called Double Stream Model (DSM) has been derived from the steady-state macroscopic mechanical energy balance (extended Bernoulli equation), applied to the “inlet-to-run” stream and the “inlet-to-branch” stream of both the gas phase and the liquid phase. According to the DSM, whose application is given in Appendix A, the branch liquid mass intake fraction λ L is a function of: (1) the branch gas mass intake fraction λ G , (2) the geometry of the junction, and (3) the ratio κ of the kinetic energy per unit volume of the inlet gas flow and that of the inlet liquid flow. The value of κ depends on the densities, mass flow rates, holdup values and velocity profiles of gas and liquid in the inlet. The DSM has been verified with experimental data from three institutes obtained with gas-liquid systems with different values of transport properties and three types of dividing junctions, namely sharp-edged regular, radiused regular and radiused reduced tees.

THE INVERSE FLUIDIZATION AIRLIFT BIOREACTOR, PART I: HYDRODYNAMIC STUDIES

The inverse fiuidization airlift bioreactor offers a simple solution to the problem of handling shear sensitive cultures and/or systems requiring biofilm growth control. This unit combines in a single vessel the hydrodynamic behavior of concentric tube airlift aerators or contactors and liquid-solid fluidized beds. The effects of the diameter of the concentric tubes, liquid level, bed properties and gas flow rate on gas holdup, bed expansion and liquid circulation velocity were investigated in this study. The Zuber and Findlay relation (1965) gives satisfactory results for gas holdup. The inverse bed expansion can be predicted by the Richardson and Zaki correlation (1953). The liquid circulation velocity can be calculated using a mechanical energy balance

On the development of a model for predicting phase separation phenomena in dividing two-phase flow

A model to predict the dividing flow characteristics for annular flow in a T-junction is proposed consisting of mixture and vapour phase continuity equations, two pressure change correlations and a closure relationship. The pressure change from the inlet through the run of the T is modelled by way of a balance of axial momentum at the junction based on a separated flow assumption. The branch pressure change is modelled using a balance of mechanical energy for the branching flow consisting of reversible and irreversible components. The closure relationship links the phase separation characteristics with the junction pressure changes. It involves a balance between pressure and inertia forces within the junction volume defining a dividing surface for each phase between the run and branch flows. The branch quality is then determined using a well-defined inlet flow distribution. The model is capable of predicting the experimentally observed phase separation characteristics from three independent studies of annular/steam—water and air-water flow in dividing T-junctions.

Energy Budget in Atomization

An equation of mechanical energy balance in a liquid jet atomizing in an ambient gas is derived. The time rate of change of kinetic energy of the fluctuating disturbance in a given volume of the jet is shown to be equal to the sum of four types of works done per unit time on the jet and the energy dissipation rate through the agent of viscosity. The four types of works involved are the work by the surface tension, the work by the pressure fluctuation in the ambient gas, the work done by the fluctuating pressure in the jet, and the work by the viscous stress. Numerical results obtained for a wide range of relevant parameters are used to show that the surface tension work is negative in jet atomization. This is contrary to the situation in the breakup of an ink jet for which the surface tension work is positive, and thus the breakup is due to capillary pinching. It is shown that the work by fluctuating gas pressure is responsible for the atomization process, since the pressure work term is the dominant positive term in the energy budget of the jet atomization.

Brittle frictional mountain building: 1. Deformation and mechanical energy budget

An active fold‐and‐thrust belt is analogous to a wedge of soil or snow that forms in front of a moving bulldozer; such wedges exhibit a critical taper and a regional state of stress that is everywhere on the verge of Coulomb failure. The width of such a critically tapered fold‐and‐thrust belt does not depend on its brittle strength or frictional properties but rather on the accretionary influx rate of fresh material at its toe and on the rate of erosion; a steady state fold‐and‐thrust belt is one in which the accretionary influx is balanced by the erosive efflux. Rocks are accreted at the toe and then horizontally shortened as they are transported toward the rear; those that enter lower in the accreted section are more deeply buried before being uplifted by erosion. Mass balance and isotropy constrain the kinematics of this large‐scale deformation, enabling us to infer the trajectories, residence times, and stress‐strain histories of rocks incorporated into eroding fold‐and‐thrust belts. A typical rock resides in the steady state Taiwan wedge for 2–3 m.y. before it is uplifted and eroded; during its motion through the wedge, it experiences strain rates in the range 10−13 to 10−14 s−1. The mechanical energy budget of brittle frictional mountain building is described by the equation , where is the rate of work performed on the base and front of the fold‐and‐thrust belt by the subducting plate, is the rate at which energy is dissipated against friction on the decollement fault, is the rate at which energy is dissipated by internal frictional processes within the deforming brittle wedge, and is the rate of work performed against gravitational body forces in a reference frame attached to the overriding plate. The total mechanical power being supplied by the subducting Eurasian plate to the active fold‐and‐thrust belt in Taiwan is slightly over 3 GW. Approximately 60% of this work of steady state mountain building is being dissipated against friction on the decollement fault, and about another 25% is being dissipated against internal friction; this leaves only 15% or roughly half a gigawatt available to do useful work against gravity. In general, fold‐and‐thrust belts with moderate pore fluid pressures are dominated by work done against friction on the decollement fault; however, those with nearly lithostatic pore fluid pressures may be dominated by work done against gravity. Internal frictional dissipation is always less than basal frictional dissipation, as it is in Taiwan. An alternative and equivalent description of the mechanical energy balance of a steady state fold‐and‐thrust belt is provided by the equation . In this version the quantity on the left, , is the rate at which work is performed on the back of the wedge by the overriding plate and is the rate of work performed against gravity in a reference frame attached to the subducting plate. The latter quantity is always positive for any critically tapered fold‐and‐thrust belt whose decollement fault dips toward its rear, in contradiction to the central premise of the gravity spreading theory of fold‐and‐thrust tectonics.

Pressure drop and flow rate measurements in a human aortic bifurcation cast for steady and pulsatile flow

Pressure drop and flow rate measurements in a rigid cast of a human aortic bifurcation under both steady and physiological pulsatile flow conditions are reported. Integral momentum and mechanical energy balances are used to calculate impedance, spatially averaged wall shear stress and viscous dissipation rate from the data. In the daughter branches, steady flow impedance is within 30% of the Poiseuille flow prediction, while pulsatile flow impedance is within a factor of 2 of fully developed, oscillatory, straight tube flow theory (Womersley theory). Estimates of wall shear stress are in accord with measurements obtained from velocity profiles. Mean pressure drop and viscous dissipation rate are elevated in pulsatile flow relative to steady flow at the mean flow rate, and the exponents of their Reynolds number dependence are in accord with available theory.