Use Of Shape Factors Research Articles

Improvement and extension of Cordier diagram for hydraulic turbines

The Cordier diagram for hydraulic turbines has been accurately improved and extended. The first part of this work completes the Cordier diagram by adding the effect of the number of injector nozzles in Pelton turbines. The Cordier lines (or bands) for Francis and Kaplan turbines have been recalculated based on the use of most available empirical relations without artificial smoothing of the boundaries between the different types of hydraulic turbines. Both turbine type and impeller diameter can be more accurately determined. The part of extending the Cordier diagram is based on the application of design equations for hydraulic turbines and the introduction of the turbine shape factor. This shape factor has a clear physical meaning in contrast to the diameter number in the conventional Cordier diagram. For Pelton turbines, it is representative of the ratio between runner diameter and jet diameter; for Francis and Kaplan turbines, it is representative of the ratio between the height of the wicket gate and the diameter of the impeller. The Cordier diagram for the diameter number can be replaced by the new diagram for the turbine shape factors. Exclusively for Francis turbines, the impeller shape factor also combines the blade profile (angle) at the impeller inlet with design parameters. The method presented in this paper can be well applied to create the extended Cordier diagram for the use of shape factors for pumps and compressors.

Cleat-scale modelling of the coal permeability evolution due to sorption-induced strain

Coal permeability is known to be affected by the sorption-induced strain. Indeed, coal swells with gas sorption and shrinks with desorption, which is likely to modify the cleat aperture and thus the permeability. Coal permeability evolution is crucially important for either Coalbed Methane production (CBM) or Carbon dioxide Capture and Storage (CCS).A model is developed at the scale of the fractures to take into account the hydro-mechanical couplings observed experimentally. It is implemented in the finite element code Lagamine. This numerical model is developed at the scale of the fractures and the matrix blocks. Fractures are modelled with interface elements specially adapted to manage sorption/desorption by taking into account the Langmuir's isotherm (Langmuir, 1918). In the matrix blocks, the sorption strain is assumed proportional to the adsorbed gas content. Depending on the boundary conditions, sorption strain affects the stress state and thus the fracture aperture. Contrary to macroscale models (Bertrand et al. 2017), this model does not require the use of shape factors since the geometry can be explicitly represented. The implementation of the model is validated by comparison with the analytical solution on a simple geometry. The model is finally used to simulate gas injection on a representative elementary volume (REV) made up few matrix blocks. This modelling highlights the interests of a numerical approach to compute the permeability evolution compared to a porosity-based model.

Shape of re-entrant particles — characterization regarding particle–fluid interaction

Particle–fluid interaction plays an important role in the design, rating and operational tuning of many industrial equipments (e.g., elutriators, settling chambers, cyclones, hydrocyclones, centrifuges etc.). The core of this interaction is the drag force exerted by the fluid on suspended particles, which, for a given fluid, depends mainly on the fluid-particle relative velocity, as well as on the particle size and shape. Traditionally, particle shape is quantified by the use of shape-factors, the most commonly used being the sphericity. However, sphericity does not discriminate well the effects of particle shape, when the particle surface is markedly re-entrant. In this work the terminal velocities of re-entrant particles, however geometrically simple, were accurately measured while falling under gravity in Newtonian liquids. A total of 77 carefully crafted particles of various materials were tested, at least in triplicate, giving rise to 2798 values of terminal velocities. The strategy was to test particles of the same material, volume and sphericity, but with different re-entrant shapes. Two shape factors were combined with sphericity (ϕ), to account for the effects of particle shape on particle–fluid interaction. A correlation for the terminal velocity using the three shape factors was developed in terms of drag coefficient (CD) and particle Reynolds number (Rep).

후판 압연공정에서 선단부 굽힘 예측을 위한 롤 바이트 형상맵 기법에 관한 연구

The front end bending(FEB) behavior of material that usually occurs in plate rolling is investigated. In this paper, a rollbite profile map approach that systematically predicts the FEB slope is presented. It is based on the concurrent use of shape factors and reduction ratios to ensure an accurate value of the FEB and its slope. In order to obtain the unit roll-bite profile map, the FEB slope model was decomposed into a temperature deviation component and a roll-velocity deviation component. By mapping the results of a series of finite element analyses to the unit functions of the roll-bite profile map, it was possible to obtain a realistic prediction of the FEB slope applicable to an actual plate rolling process. Thereby, the usefulness of the present approach is clearly demonstrated.

Longitudinal Dispersion Characteristics of Rivers and Natural Streams in Greece

An optimized parameterization of the non-dimensional longitudinal dispersion coefficients in Greek rivers, $$K_{x_a } $$ , is described in terms of parameters derived from cross-section river geometry and longitudinal velocity data. Optimization is accomplished, using various combinations of the selected parameters, either for subsets of $$K_{x_a } $$ values, according to the cross-section aspect ratio range, or for the entire $$K_{x_a } $$ population. The use of shape factors associated with the river cross-section transverse profiles of depth-mean longitudinal velocities and the normalized intensities of longitudinal velocity variations from the overall mean cross-sectional velocity improves the parameterization. The statistical behavior of $$K_{x_a } $$ in terms of skewness, flatness and other shape coefficients is also examined. $$K_{x_a } $$ histograms show that these coefficients have a log-normal distribution. An empirical expression for $$K_{x_a } $$ is also proposed that takes into account both contributions from the transverse velocity profile and from the vertical profile of longitudinal velocities, at each cross-section, in shaping the value of dispersion coefficients when the aspect ratio of the cross-section is of O(1) and/or when $$K_{x_a } $$ is of O(1).

An Evaluation of Geometric Factors Used in the Two-Stage Borehole Test (ASTM D6391-99) Using the Finite Element Method

Abstract Geometric or shape factors are constants used in the interpretation of in-situ hydraulic conductivity tests, such as the Two-Stage Borehole (TSB) test in ASTM D6391-99. The finite boundary condition shape factors used with the TSB test are an extension of those derived by Hvorslev (1951) for infinite boundary conditions. The accuracy of this extrapolation is investigated here, using the finite element method (FEM) as a tool for deriving shape factors for finite boundary conditions. The utilized FEM has been validated against closed-form solutions for other test geometries. The resulting errors from the incorrect use of shape factors with the TSB test are discussed, and new shape factors derived from the FEM are presented in the form of a chart.

Experimental investigation of models of sound wave propagation in air-saturated porous media

Several theoretical models of sound wave propagation in air-saturated porous material are studied making use of precise measurements of both physical (porosity, flow resistivity, tortuosity) and acoustical (characteristic impedance, propagation constant) parameters. The aspects covered include the use of shape factors and the adequacy of frequency-dependent dynamic viscosity functions. Experimental results on both model materials and real material are used, with special attention to the low-frequency/high-flow resistivity regime for solid frame porous materials.

Surface area estimation of potato tubers

Surface areas of tubers of the potato cultivars Cara, Desiree, King Edward and Maris Piper were estimated empirically by measuring the area of adhesive tape required to cover each tuber surface. Measuring the area of thin peelings of all four cultivars underestimated tuber surface area, the degree of underestimation increasing with decreasing tuber size. ‘Shape factors’ (D), calculated as:D = surface area/volume2/3, varied with tuber size and variety (overall mean = 5.070±0.0245;n=56), but would be suitable for estimating the surface areas of similarly-sized tubers of a single cultivar. A mathematical model which used a surface integral to estimate a tuber's surface area from its axial dimensions and volume closely approximated to the empirical estimates of surface area for all four cultivars, and would be preferable to the use of shape factors when estimating surface areas of potatoes with variable shapes or unknownD values.

Approximate Calculation of Multidimensional Solidification by Using Conduction Shape Factors

When solidification occurs with conduction controlling and small temperature differences present, quasi-steady models become advantageous. In this paper, the use of conduction shape factors to predict freezing rate and heat transfer in various two-dimensional objects is investigated. Simple equations for calculating these quantities are obtained, and compared to existing results. The use of shape factors to model solidification with time-dependent boundary conditions is illustrated.

A Streamline Model for Secondary Recovery

Abstract A mathematical model particularly suitable for secondary recovery predictions is described. The model is based upon the flow lines generated by the superposition of line sources and sink solutions and is easily adaptable to arbitrary well patterns and fluid displacement mechanisms. Nonunity mobility ratios and reservoir stratification may be modeled. The model may be used with relatively small digital computing equipment. Previous models of this type have used the generated flow lines to outline "flow bundles", thus requiring a prior knowledge of the geometric shape of each of these bundles so that the flow through each could be computed. The modeling method described does not require any such prior knowledge of these flow lines since the volumetric flow in a stream bundle is computed as each chosen streamline is generated. This feature makes the model particularly adaptable to arbitrary well patterns. patterns. Examples are given to show the application of this model to both single and multiphase flow. Introduction Streamline models of secondary recovery projects result from the use of line source and sink solutions to the diffusivity equation to represent injection and production wells. Muskat described these solutions and their application to simplified problems. He used semi-analytic techniques to problems. He used semi-analytic techniques to obtain breakthrough sweep efficiencies for regular, dispersed injection patterns at a mobility ratio of 1. Collins described a finite-difference approximation for determining streamlines for unit mobility ratios in arbitrary well patterns. Both Muskat and Collins described image-well techniques for mathematically bounding the area being studied. Collins' finite-difference method is particularly adaptable to the use of a high-speed computer for obtaining streamlines and travel times along streamlines. Higgins and Leightons have described a technique for approximating the waterflood recovery of oil using streamlines generated by single-fluid how models (such as the method described by Collins). Their technique uses these streamlines to divide the total flow area into "stream channels", which flow in parallel between injection and production wells. Each stream channel is divided into a number of recovery rectilinear flow cells, in series, which closely approximate the shape of the stream channel. Through the use of shape factors determined for each flow cell, a Buckley and Leverett type of frontal displacement in each stream channel is computed. Combining the results from all stream channels gives the waterflood production history by this method. Once the stream channels, the flow cells, and the shape factors have been determined, a single computer program is used to obtain the production history. production history. Hauber has also described a method using stream channels formed by single-fluid streamlines. In this method, the cross-sectional area of each stream channel, as a function of length along a center streamline, is determined mathematically from the stream function, and an integration technique is used to determine the flow through each stream channel. Hauber applied this method to piston-like displacement in a five-spot injection piston-like displacement in a five-spot injection pattern. pattern. The stream-channel concept has been used and elaborated upon in subsequent publications dealing with the displacement of oil by fluids of unequal mobilities. This use of stream channels requires a prior knowledge of the single-fluid streamlines for the well spacing to be studied, so that flow cells and their shape factors, or the area functions of the channels, may be determined. The purpose of this paper is to describe a method which produces equivalent results but requires no prior knowledge of the streamlines or stream channels to be used. It is not necessary to know the eventual destinations of the "stream channels" in order to obtain production histories for arbitrary well patterns and either of the types of displacement patterns and either of the types of displacement mechanism mentioned above. SPEJ P. 7

A Method for the Routine Measurement of Particle Shape Factors in the Sieve Range

SIZE distribution, cannot alone describe a particle population, and for some purposes it becomes necessary to measure the particle shape within closely sized ranges. Some instances of the use of shape factors are in estimating approximate surface areas, calculating the number of particles in a size range and investigating the variation of particle shape with size. Since these applications are not confined to one field or industry, interest in shape factor is widespread.

Experimental Evaluation of Human Shape Factors With Respect to Floor Areas

Abstract Results are presented for the shape factor of an average standing person with respect to energy in the form of heat received from a floor area. The use of shape factors is essential in determining conditions of comfort in a heated or cooled enclosure.