Semi-theoretical Equations Research Articles

Jet impingement and the circular hydraulic jump on adverse-sloped beds

Hydraulic jump type stilling basins are the most common hydraulic structures for decreasing the excess kinetic energy of flow and are divided into three types: classical, radial and circular basins. Previous studies have reported that the hydraulic characteristics of flow are more favourable in circular basins than in the other types. Therefore, theoretical/experimental modelling was applied to examine the characteristics of circular hydraulic jumps on adverse-sloped beds in circular stilling basins. The present theoretical model applies the momentum and energy equations to derive equations for the sequent depth ratio and relative energy loss of circular hydraulic jumps and classical hydraulic jumps (CHJs). Analysing the effectiveness of the dominant independent parameters showed that increasing the sequent radius ratio up to 2 and the bed slope up to −5% decrease the sequent depth ratio and length of the circular hydraulic jumps by about 30% and 60%, respectively, whereas the relative energy loss increases by about 30%. In addition to the correlation coefficient, different error functions were determined to analyse the precision of the derived semi-theoretical equations. The present circular stilling basin acts like a plunge pool, having a small water depth for low discharges to have a submerged jet condition, downstream of large dams.

Insights Into Estimation of Sand Permeability: From Empirical Relations to Microstructure-based Methods

Abstract This study evaluates various methods for estimating soil permeability using microtomography-derived data and compares them to the conventional laboratory approaches. Different methods, including measurement in custom-designed permeameter at micro-CT-compatible scale, empirical equations, simulated sifting, semi-theoretical equations, pore-network modeling, and lattice-Boltzmann simulations, were applied to samples of sandy soils with distinct microstructural properties. The empirical equations showed varied results, highly dependent on the method chosen. The simulated sifting method was able to adequately estimate the granulometric properties of the soil, allowing the use of empirical permeability formulations for substantially small samples. Semi-theoretical equations based on the microstructural properties presented reasonable agreement for some samples. The pore-network modeling approach demonstrated computational efficiency but lacked accuracy. The lattice-Boltzmann method required significant computational resources but did not provide substantially closer alignment with the measured hydraulic properties of some samples. None of the simulations was able to properly determine the permeability of silty and organically contaminated sand. The study highlights the complexity of permeability estimation, emphasizing the need for choosing volumes of interest, resolution of micro-CT scans, and methods that match specific soil characteristics and available computational resources.

Comparative studies on isothermal attachment ventilation based on vertical walls, square and circular columns

Ventilation and air distribution methods play an important role in indoor air quality and thermal environments. Effective airflow distribution within indoor environments has always been a great cause of concern. Attachment ventilation has drawn considerable interest since it is more energy efficient and space saving than traditional mixing or displacement ventilation methods. As a newly developed and applied attachment ventilation method, vertical wall-based attachment ventilation (WAV) was first proposed in the 1990’s by the authors. Later, the concepts of square column-based attachment ventilation (SAV) and circular column-based attachment ventilation (CAV) were also presented to meet the needs of more industrial applications. In this paper, three attachment ventilation methods— i.e., WAV, SAV and CAV are compared based on experimental studies and numerical simulations. For the vertical attachment region, semi-theoretical equations characterizing the dimensionless centerline velocity of WAV, SAV and CAV are proposed. The velocity profiles of all three types of attachment ventilation methods are similar. Of the three attachment ventilation methods, for the same distance y* along the attached wall measured from the slot inlet, the dimensionless centerline velocity of WAV is the largest, while those of SAV and CAV remain almost identical. In the horizontal air reservoir region (air lake zone), the semi-theoretical equations to predict dimensionless centerline velocity also are presented. The results show that for the same distance x normal to the wall/column, the dimensionless centerline velocity of WAV is the largest, followed by SAV and CAV. Additionally, unified semi-theoretical equations are presented for predicting the jet centerline velocity decay of attachment ventilation, which is helpful for the design of air distribution systems in heating, ventilation and air conditioning (HVAC) engineering applications.

Impact of process parameters and sludge properties on electro-dewatering performance

In this paper, the semi-theoretical kinetics equations were proposed and the electro-dewatering experiments of sludge were conducted to analyze the kinetics laws of electro-dewatering under constant voltage and constant current. The results showed that the decline rate of moisture content remained stable under constant current, while the decline rate of moisture content kept decaying in the operating mode of constant voltage. No matter in what kind of operating mode, the change in decline rate of moisture content was consistent with that of filtrate flow rate, both of which were essentially dependent on the current. The follow-up electro-dewatering tests under constant voltage were carried out by using three different kinds of sludge to understand the mechanism of electro-dewatering. The results showed that the main parameters that controlled the electro-dewatering behavior were measured current, conductivity and zeta potential of sludge. When treated at 30 V, sludge C reached the lowest moisture content, owing to its higher conductivity and zeta potential. However, the poorer electro-osmotic effect of sludge C led to its higher average unit energy consumption. By comparison, an increase of applied voltage was more effective for electro-dewatering process of sludge C but at the expense of higher energy consumption, while sludge A and B were more inclined to improve the electro-dewatering effect by extending the dewatering time. Finally, the derived equations were demonstrated by the experiments, which could provide guidance for the practical applications.

Numerical modelling of landslide-tsunami propagation in a wide range of idealised water body geometries

Large landslide-tsunamis are caused by mass movements such as landslides or rock falls impacting into a water body. Research of these phenomena is essentially based on the two idealised water body geometries (i) wave flume (2D, laterally confined wave propagation) and (ii) wave basin (3D, unconfined wave propagation). The wave height in 2D and 3D differs by over one order of magnitude in the far field. Further, the wave characteristics in intermediate geometries are currently not well understood. This article focuses on numerical landslide-tsunami propagation in the far field to quantify the effect of the water body geometry. The hydrodynamic numerical model SWASH, based on the non-hydrostatic non-linear shallow water equations, was used to simulate approximate linear, Stokes, cnoidal and solitary waves in 6 different idealised water body geometries. This includes 2D, 3D as well as intermediate geometries consisting of “channels” with diverging side walls. The wavefront length was found to be an excellent parameter to correlate the wave decay along the slide axis in all these geometries in agreement with Green's law and with diffraction theory in 3D. Semi-theoretical equations to predict the wave magnitude of the idealised waves at any desired point of the water bodies are also presented. Further, simulations of experimental landslide-tsunami time series were performed in 2D to quantify the effect of frequency dispersion. This process may be negligible for solitary- and cnoidal-like waves for initial landslide-tsunami hazard assessment but becomes more important for Stokes-like waves in deeper water. The findings herein significantly improve the reliability of initial landslide-tsunami hazard assessment in water body geometries between 2D and 3D, as demonstrated with the 2014 landslide-tsunami event in Lake Askja.

Predicting macroturbulence energy and timescales for flow over a gravel bed: Experimental results and scaling laws

The fluvial science community lacks predictive equations for macroturbulence energy and timescales, which may serve useful for predicting sediment transport in rivers. Therefore, we investigated the systematic variation of macroturbulence scales by varying hydraulic parameters in an experimental facility, collecting velocimetry data and measuring macroturbulence scales with spectral analyses, and testing scaling laws for macroturbulence prediction that consider inner variables, outer variables, and turbulence kinetic energy. Experiments spanned across 11 energy gradients and aspect ratio combinations including 116 testing conditions each with 30 minute instantaneous turbulence time-series collected. Results showed velocimetry data and time-average turbulence parameters supported the presence of an equilibrium outer region and coherent double-layer assumed in the experimental design. Macroturbulence and bursting wavenumbers were constant in the outer region when scaled with the flow depth, and were independent of distance above the boundary, aspect ratio, energy gradient, Froude number, or Shields parameter. Data results agreed with the derivation herein and showed macroturbulence energy scales with the streamwise turbulence kinetic energy. Semi-theoretical equations for turbulence kinetic energy scaled macroturbulence energy reasonably well. A novel equation is presented for the macroturbulence period, and the equation's transferability is supported by consistency with past macroturbulence studies, bursting formula, and flow visualization studies. Novel equations are presented for macroturbulence energy, and transferability is supported on theoretical and empirical grounds; however open questions remain regarding a universal function for streamwise turbulence kinetic energy and the net contribution of macroturbulence to total turbulence energy. Results showed the macroturbulence wavenumber required a relatively long duration of data (i.e., 17 min) to stabilize. Results were used to formulate a time scale for minimum measurement length of turbulence data when information about macroturbulence is sought after. We discuss macroturbulence linkages with sediment transport, including, prediction of the sediment carrying capacity via macroturbulence formula as one application to help close the gap between turbulence and fluvial sediment transport.

Mixing properties of Tris(2-ethylhexyl) phosphate with alkanes at different temperatures and data treatment using several correlation equations based on Eyring's absolute reaction theory

Densities and viscosities of binary mixtures of Tris(2-ethylhexyl) phosphate with n-heptane, or n-octane have been determined over the entire range of composition at T = (298.15, 303.15, and 308.15) K and atmospheric pressure. The excess molar volumes are calculated and fitted to a Redlich–Kister type equation. The excess molar volumes are negative over the whole composition range. Several semi-theoretical equations based on Eyring's absolute rate theory and the excess Gibbs free energy function have been applied to correlate the experimental viscosity data and compared. The evaluated models include the Eyring–Van Laar, Eyring–Margules, Eyring–NRTL, Eyring–UNIQUAC and the Grunberg–Nissan equations.

Drop characteristics of free-falling nappe for aerated straight-drop spillway

A parallel study of regression and semi-theoretical equations is presented to forecast the drop force and the drop length of a free-falling nappe for an aerated straight-drop spillway. The ratio between the critical velocity and the exit velocity at the drop wall was introduced to establish semi-theoretical equations. Regression techniques were applied by using the dimensionless channel bed slope and the drop number to describe the drop characteristics. There were 64 groups of test data to calibrate the parameters of equations. Another 36 groups of test data were used to verify the developed model. The simulation results indicate that the semi-theoretical method provides reliable estimates for the drop force, and that the regression method can offer accurate predictions for the drop length.

Fluids of pseudo-hard bodies: From simulations to equations of state

Thermodynamic properties of a number of fluids of pseudo-hard bodies (PHB), representing non-additive excluded volume effects in selected polar and associating real fluids, have been studied by Monte Carlo (MC) simulations. Virial coefficients up to the fourth one have been determined and used in various semi-theoretical equations of state. It is shown that the perturbed virial expansion (PVE) about an appropriately chosen effective hard sphere (HS) reference is able to describe the thermodynamic behavior of the considered fluids quite accurately.

Physical explanation of the empirical coefficients of gas–liquid mass transfer equations

Equipment design and scale up is one of the biggest problems that chemical engineers face. A lot of research has been done at laboratory and pilot plant scale. However, experimental data are, many times, useless in equipment scale up because they overlook the processes involving bubbles. Therefore, a new approach considering the hydrodynamics of the bubbles is proposed to explain and understand the mass transfer rates. Semi-theoretical equations have been developed for bubble columns and stirred tanks based on hydrodynamic considerations of the flow, the bubbles and dispersions in the tank. These equations are able to explain the physical meaning, identify the effect of the scale on mass transfer rates and even predict the coefficients of the empirical correlations for k L a based on the hydrodynamic processes that bubbles experience. It can be concluded that the proportional constant of the correlations for k L a depends on the bubble size, on the physical and transport properties and on the size of the tank because it affects the mixing. Meanwhile, the exponents related to the power input and the superficial gas velocity depend on bubble break up and coalescence, and the dispersions generated. In the case of stirred tanks, the physical effect of the impeller on the bubbles also plays an important role on the exponents.

Characterization of the microtube emitters used in a novel micro-sprinkler

The microtube is a simple and cheap emitter that was widely used throughout the world in the early days of drip irrigation. Its length can be adjusted according to the pressure distribution along the lateral line and the discharge from the microtube can be adjusted by its length. This not only counters the pressure loss due to pipe friction but also makes it suitable for undulating and hilly conditions, where pressure in the lateral line varies considerably according to the differences in elevation. This is the major problem facing the designer, i.e., emitter flow changes as the acting pressure head changes. In this study, a novel micro-sprinkler system is proposed that uses microtube as the emitter and where the length of the microtube can be varied in response to pressure changes along the lateral to give uniformity of emitter discharges. The objective of this work is to develop and validate empirical and semi-theoretical equations for the emitter hydraulics. Laboratory testing of two microtube emitters of different diameter over a range of pressures and discharges was used in the development of the equations relating pressure and discharge, and pressure and length for these emitters. The equations proposed will be used in the design of the micro-sprinkler system, to determine the length of microtube required to give the nominal discharge for any given pressure. The semi-theoretical approach underlined the importance of accurate measurements of the microtube diameter and the uncertainty in the estimation of the friction factor for these tubes.

Psychrometric charts for water vapour in natural gas

Psychrometric charts present in a graphical form physical property data for gas systems involving one or more non-condensing gas components and a single condensing vapour component. Originally limited to studying the humid air system of water vapour in dry air, the theory behind preparing psychrometric charts for water vapour in natural gas is presented. To illustrate the technique two charts are prepared for a system in which the natural gas is 80.0 mol% methane, 15.0 mol% ethane and 5.0 mol% propane. One chart is prepared for the temperature range of 0 to 60 °C at a system pressure of 101.3 kPa and the other chart for the range of 40 to 110 °C at a system pressure of 400.0 kPa. The charts are based upon semi-theoretical equations and make use of published physical property data and correlations. The behaviour of the gas mixture is characterized by the virial equation of state truncated at the third term. The charts are constructed with the dry bulb temperature and absolute humidity scales as the orthogonal axes. Curves of constant adiabatic saturation temperature, constant relative humidity, constant gas specific volume and constant enthalpy deviation are plotted on the charts. Using the theory presented psychrometric charts for any temperature range, system pressure and natural gas composition may be prepared.

Prediction of Microwave Drying Behavior of Pharmaceutical Powders Using Thin-Layer Models

Common pharmaceutical excipients and active ingredients, wetted with specific solvents, were dried under selected continuous power microwave and pulsed power microwave-vacuum conditions in an experimental system. Irrespective of the drying technique, a typical drying profile, with a constant drying rate stage followed by two falling rate periods, was exhibited. The experimental moisture loss data were fitted to semi-theoretical and empirical thin-layer drying equations and the models compared on the basis of three statistical parameters. The drying characteristics were satisfactorily described by the Lewis, Page, Logarithmic, Chavez-Mendez et al., and Midilli et al. models, with the latter providing the best representation of the data.

Thin-Layer Modeling of Microwave, Microwave-Convective, and Microwave-Vacuum Drying of Pharmaceutical Powders

Common pharmaceutical excipients and active ingredients, wetted with specific solvents, were dried under selected microwave, microwave-convective, and microwave-vacuum conditions in an experimental drying system (2.45 GHz, 90 W). The generalized drying profile was independent of drying technique and material characteristics, with a constant rate stage and two falling rate periods being observed. Moisture removal rate was, however, dependent on the drying technique (pressure and presence/absence of external heating source), and powder and solvent properties (dielectric, physical and thermal). The experimental moisture loss data were fitted to selected semi-theoretical and empirical thin-layer drying equations. The correlations were compared according to two statistical tests, namely root mean square error and reduced chi-square. The microwave, microwave-convective, and microwave-vacuum drying behaviors were adequately represented by the Page, Logarithmic, Chavez-Mendez et al. and Midilli et al. equations, with the latter providing the best description of the experimental results.

Preparation of psychrometric charts for water vapour in Martian atmosphere

Psychrometric charts for condensing water vapour in Martian atmosphere at four different pressures are presented. The charts are based upon semi-theoretical equations and make use of published physical property data and correlations. The behaviour of the vapour phase is characterised by the Virial Equation of State truncated at the third term. The solubility of gas in water is also considered. The charts are constructed with the dry bulb temperature and absolute humidity scales as the orthogonal axes. Curves of constant adiabatic saturation temperature, constant relative humidity, constant gas specific volume and constant enthalpy deviation are plotted on the charts.

Evaporative Heat Transfer of Annular Two-phase Flow of Air-water in a Small Vertical Tube

The evaporative heat transfer of the non-boiling annular two-phase flow of air-water in a small vertical tube with uniform wall heat flux was studied both theoretically and experimentally. A simplified two-phase flow boundary layer model was used to calculate the thickness of the water film attached to the wall, and from the liquid film thickness the evaporative heat transfer coefficients of the annular two-phase flow were obtained. Theoretical equations and semi-theoretical equations were proposed for predicting the evaporative heat transfer of the annular two-phase flow of air-water in a small vertical tube. The semi-theoretical prediction agrees well with the experimental data. The mechanism of the heat transfer enhancement is the evaporation of the thin liquid film attached on the wall.

Psychrometric charts for organic vapours 1. Ketones in air

Psychrometric charts for five different condensing ketone vapours in air at 100 kPa are pressures are presented. The charts are based upon semi-theoretical equations and make use of published physical property data and correlations. The behaviour of the vapour phase is characterised by the virial equation of state truncated at the third term. The solubility of gas in the liquid ketones is also considered. The charts are constructed with the dry bulb temperature and absolute humidity scales as the orthogonal axes. Curves of constant adiabatic saturation temperature, constant relative humidity, constant gas specific volume and constant enthalpy deviation are plotted on the charts.

Study of the kinetics of the cementation of silver ions onto copper in a rotating cylinder system from acidic sulphate solutions

The kinetics of the cementation of silver ions onto copper from acidic sulphate solutions were investigated in a rotating cylinder system. The influence of several parameters on the course of the reaction, such as cylinder rotation speed, initial concentration of silver ions, temperature, concentration of sulphuric acid and the presence or absence of oxygen in the system, was investigated, and the reaction was found to follow first-order kinetics. The initial rate of the reaction is limited by diffusion through the mass transfer boundary layer. The experimental values of the initial rate constant calculated from both methods used for recording the changes of the silver ions concentration in the solutions corresponded ideally to the values predicted by semi-theoretical equations describing the turbulent flow in the rotating cylinder system. After the initial period of cementation, an enhancement of the rate was observed. In oxygenated solutions, rate enhancement is associated mainly with changes in the structure of the deposit involving an increase in the effective surface area during the process. However, the rate enhancement phenomenon in the absence of oxygen is attributed not only to an increase in the effective surface area but also to the chemical reaction between the Cu + and Ag + ions.

Preparation of psychrometric charts for alcohol vapours in nitrogen

Psychrometric charts for three different systems involving condensing alcohol vapour in nitrogen are presented. The alcohols considered are methanol, ethanol and n-propanol. The charts are based upon semi-theoretical equations and make use of published physical property data and correlations. The behaviour of the vapour phase is characterised by the Virial Equation of State truncated at the third term. The solubility of nitrogen in the liquid alcohols is also considered. The charts are constructed with the dry bulb temperature and absolute humidity scales as the orthogonal axes. Curves of constant adiabatic saturation temperature, constant relative humidity, constant gas specific volume and constant enthalpy deviation are plotted on the charts.

Thermal conductivity, elastic modulus, and coefficient of thermal expansion of polymer composites filled with ceramic particles for electronic packaging

The effective thermal conductivity, elastic modulus, and coefficient of thermal expansion of epoxy resins filled with ceramic fillers like silica, alumina, and aluminum nitride were determined. The data obtained was compared with theoretical and semitheoretical equations in the literature that are used to predict the properties of two phase mixtures. It was found that Agari's model provided a good estimate of the composite thermal conductivity. The Hashin-Shtrikman lower bound for composite modulus fits the modulus data fairly well at low concentrations of the filler. Also, it was found that the coefficients of thermal expansion of the filled composites lie in between Schapery's upper and lower bounds. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 74: 3396–3403, 1999