Head Coefficient Research Articles

A new direction for shallow refraction seismology: integrating amplitudes and traveltimes with the refraction convolution section

The refraction convolution section (RCS) is a new method for imaging shallow seismic refraction data. It is a simple and efficient approach to full‐trace processing which generates a time cross‐section similar to the familiar reflection cross‐section. The RCS advances the interpretation of shallow seismic refraction data through the inclusion of time structure and amplitudes within a single presentation. The RCS is generated by the convolution of forward and reverse shot records. The convolution operation effectively adds the first‐arrival traveltimes of each pair of forward and reverse traces and produces a measure of the depth to the refracting interface in units of time which is equivalent to the time‐depth function of the generalized reciprocal method (GRM). Convolution also multiplies the amplitudes of first‐arrival signals. To a good approximation, this operation compensates for the large effects of geometrical spreading, with the result that the convolved amplitude is essentially proportional to the square of the head coefficient. The signal‐to‐noise (S/N) ratios of the RCS show much less variation than those on the original shot records. The head coefficient is approximately proportional to the ratio of the specific acoustic impedances in the upper layer and in the refractor. The convolved amplitudes or the equivalent shot amplitude products can be useful in resolving ambiguities in the determination of wave speeds. The RCS can also include a separation between each pair of forward and reverse traces in order to accommodate the offset distance in a manner similar to the XY spacing of the GRM. The use of finite XY values improves the resolution of lateral variations in both amplitudes and time‐depths. The use of amplitudes with 3D data effectively improves the spatial resolution of wave speeds by almost an order of magnitude. Amplitudes provide a measure of refractor wave speeds at each detector, whereas the analysis of traveltimes provides a measure over several detectors, commonly a minimum of six. The ratio of amplitudes obtained with different shot azimuths provides a detailed qualitative measure of azimuthal anisotropy and, in turn, of rock fabric. The RCS facilitates the stacking of refraction data in a manner similar to the common‐midpoint methods of reflection seismology. It can significantly improve S/N ratios.Most of the data processing with the RCS, as with the GRM, is carried out in the time domain, rather than in the depth domain. This is a significant advantage because the realities of undetected layers, incomplete sampling of the detected layers and inappropriate sampling in the horizontal rather than the vertical direction result in traveltime data that are neither a complete, an accurate nor a representative portrayal of the wave‐speed stratification. The RCS facilitates the advancement of shallow refraction seismology through the application of current seismic reflection acquisition, processing and interpretation technology.

Resolving refractor ambiguities with amplitudes

Amplitudes are used to constrain refraction models. This study demonstrates that the refraction time section generated through the convolution of forward and reverse refraction traces together with a static shift facilitates the convenient recognition of amplitude variations related to changes in refractor wavespeed. For large contrasts in wavespeeds between the upper layer and the refractor, the head coefficient is approximately proportional to the ratio of the specific acoustic impedances. Since the convolution operation effectively multiplies the amplitudes of the forward and reverse arrivals, the convolved amplitudes are proportional to the square of this ratio. In general, the higher the contrast in the refractor wavespeed and/or density, the lower the amplitude. The regions recognized in the wavespeed analysis function correlate with those determined with amplitudes, thereby providing an additional constraint on inversion with model‐based approaches.

Can amplitudes resolve ambiguities in refraction inversion?

The inversion of seismic refraction data with model-based methods is inherently ambiguous, and artefacts, which are geologically plausible and significant, can be introduced by the algorithms used to generate the starting model. In many cases, the minimum variance criterion of the generalized reciprocal method (GRM) can resolve whether lateral variations in the refractor wavespeeds are genuine, or whether they are artefacts of the inversion algorithm. As an additional constraint, this paper demonstrates that any genuine lateral changes in refractor wavespeed should also have an associated amplitude expression.Amplitudes are not commonly used in most seismic refraction studies, mainly because the very large geometric spreading component masks any variations related to geology. The amplitude decay is usually much more rapid than the commonly assumed inverse distance squared function, which only applies after the signal has travelled several wavelengths in the refractor.This study demonstrates that the refraction time section generated through the convolution of forward and reverse refraction traces, shows the same structure on the refracting interface, in units of time, as would be produced by the conventional reciprocal method (CRM) or the GRM. The traveltimes, which are contained within the phase spectra, are added with convolution.The amplitude spectra are multiplied, which is sufficient both to compensate for the large geometric effects and to facilitate convenient recognition of amplitude variations related to changes in refractor wavespeed. Convolution emphasises these amplitude variations through the squaring of the head coefficients and the convenient inclusion of transmission losses. In general, the higher the refractor wavespeed and/or density, the lower the amplitude.The results are applicable to the search for massive sulphide orebodies under electrically conductive regoliths, where other traditional exploration techniques, such as electrical and electromagnetic methods, may not be fully effective. These targets would be characterised by an increase in the thickness of the regolith and a decrease in amplitude caused by an increase in the density.

Evaluation of Blade Passage Analysis Using Coarse Grids

This paper presents the results of a study using coarse grids to analyze the flow in the impellers of an axial flow pump and a mixed flow pump. A commercial CFD code (FLOTRAN) is used to solve the 3-D Reynolds Averaged Navier Stokes equations in a rotating cylindrical coordinate system. The standard k−ε turbulence model is used. The meshes for this study use 22,000 nodes and 40,000 nodes for the axial flow impeller, and 26,000 nodes for the mixed flow impeller. Both models are run on a SPARCstation 20. This is in contrast to typical analyses using in excess of 100,000 nodes. The smaller mesh size has advantages in the design environment. Stage design parameters for the axial flow impeller are, rotational speed 870 rpm, flow coefficient ϕ=0.13, head coefficient ψ=0.06, and specific speed 2.97 (8101 US). For the mixed flow impeller the parameters are, rotational speed 890 rpm, flow coefficient ϕ=0.116, head coefficient ψ=0.094, and specific speed 2.01 (5475 US). Evaluation of the models is based on a comparison of circumferentially averaged results to measured data for the same impeller. Comparisons to measured data include axial and tangential velocities, static pressure, and total pressure. A comparison between the coarse and fine meshes for the axial flow impeller is included. Results of this study show that the computational results closely match the shapes and magnitudes of the measured profiles, indicating that coarse CFD models can be used to accurately predict performance. [S0098-2202(00)02202-1]

3‐D Viscous Flow Analysis of a Mixed Flow Pump Impeller

This paper presents the results of a study using a coarse grid to analyze the flow in the impeller of a mixed flow pump. A commercial computational fluid dynamics code (FLOTRAN) is used to solve the 3‐D Reynolds Averaged Navier Stokes equations in a rotating cylindrical coordinate system. The standard k − ε turbulence model is used. The mesh for this study uses 26,000 nodes and the model is run on a SPARCstation 20. This is in contrast to typical analyses using in excess of 100,000 nodes that are run on a super computer platform. The smaller mesh size has advantages in the design environment. Stage design parameters are, rotational speed 1185 rpm, flow coefficient φ = 0.116, head coefficient ψ = 0.094, and specific speed 2.01 (5475 US). Results for the model include circumferentially averaged results at the leading and trailing edges of the impeller, and analysis of the flow field within the impeller passage. Circumferentially averaged results include axial and tangential velocities, static pressure, and total pressure. Within the impeller passage the static pressure and velocity results are presented on surfaces from the leading edge to the trailing edge, the hub to the shroud, and the pressure surface to the suction surface. Results of this study are consistent with the expected flow characteristics of mixed flow impellers, indicating that small CFD models can be used to evaluate impeller performance in the design environment.

Numerical Simulation of the Two-Dimensional Unsteady Interaction of a Centrifugal Impeller with its Volute Casing by Vortex Method.

An accurate numerical analysis of the flows associated with impeller-volute configurations in turbomachinery can be extremely helpful in optimizing the performance of centrifugal turbomachinery. In the present study, without introducing the periodicity of the blade-to-blade stream, the two dimensional unsteady features of the flows through a centrifugal impeller rotating in a volute casing of a pump are directly simulated by using an advanced vortex method. Results are presented in the form of the flow pattern of discrete vortices, volocity vector, theoretical head coefficient of impeller and the unsteady hydraulic radial force on impeller. Some of the results are compared with experimental data and the agreement is found to be good.

Transient Ground‐Water Flow Simulation Using a Fuzzy Set Approacha

AbstractThe expense of characterizing aquifer spatial variability often results in a lack of available or realistically obtainable direct measurement data for ground‐water system simulation. As a consequence ground‐water models that are able to utilize imprecise or “soft” information need to be developed. In this paper, a methodology based on fuzzy set theory is developed to incorporate imprecise data into transient ground‐water flow simulation. The imprecise model parameters may come from indirect measurements, expert judgment, and subjective interpretation of available information. Fuzzy numbers are used to represent imprecise parameters. They are also used as a measure of the uncertainty associated with the hydraulic head due to the imprecision of input data. A fuzzy ground‐water flow model is developed by linking the finite‐difference method with fuzzy number representations. Fuzzy number operations (α‐level cuts) are used to solve the resulting fuzzy ground‐water flow model and are extended to consider the dependencies among hydraulic head coefficients. With the fuzzy number inputs, the transient fuzzy ground‐water flow model provides a direct measure of hydraulic head uncertainties in the time domain. The model outputs can be used as the inputs for subsequent risk analysis and decision‐making processes. The fuzzy modeling technique can handle imprecise information directly without generating a large number of realizations. It is also flexible as it can handle different types of membership functions describing fuzzy input parameters. The methodology can be used to combine data with different levels of quality into ground‐water flow models and provides a realistic method to handle parameter imprecision, especially expert judgment and subjective information. A numerical model based on the methodology was tested against the Theis analytical solution for a homogeneous aquifer with pumping. The tested model was also applied to two different heterogeneous flow fields to demonstrate the methodology.

Numerical Analysis of Separated Flow through a Centrifugal Impeller in Volute Casing by the Discrete Vortex Method.

Separated flow through a centrifugal impeller in a volute casing has been analyzed by the discrete vortex method. The results on velocity distribution at 40% of the design flow rate were in good agreement with those of Miner's experiment, which was performed under the same conditions with the same impeller and volute geometry. The flow pattern change due to interference effects between the blades and the volute tongue causes periodical change in the flow rate through one blade passage and in the torque of one blade during rotation. Head coefficients obtained from the head difference between the impeller center and the exit of the volute became smaller than the theoretical head coefficients calculated from the impeller torque because of the mixing loss.

Numerical analysis of turbulent flow around a small propeller fan operating at the inlet of open chamber

Performance characteristics of a small propeller fan are numerically investigated solving the continuity and Reynolds-averaged Navier-Stokes equations. The Reynolds stresses for turbulent transport are modelled using a k-.epsilon. turbulence model. The present numerical procedure is constructed using the Finite Volume Method with the SIMPLE algorithms. The performance parameters obtained from the calculations are compared with the measured values for the various flow rates. A performance test of the fan shows different characteristics between a radial type at small flow rates and an axial type at large flow rates. Comparisons between the predictions and the measurements show that the predicted results are in good agreement with the measured values and reasonably reproduce the sharp variations of the power and head coefficient around a flow coefficient .PHI.=0.3. These comparisons indicate that the present numerical method is capable of resolving the performance characteristics with reasonable accuracy. At low flow rates, it is found that the flow enters the fan in an axial direction and is discharged radially outward at the tip which happens in the centrifugal fan. The centrifugal effect makes a significant difference in the characteristics of a fan at the low and high values of flow coefficient.

3‐D Viscous Flow Analysis of an Axial Flow PumpImpeller

A commercial CFD code is used to compute the flow field within the first stage impeller of a two stage axial flow pump. The code solves the 3‐D Reynolds Averaged Navier Stokes equations in a rotating cylindrical coordinate system using a standard k − ε turbulence model. Stage design parameters are, rotational speed 870 rpm, flow coefficient φ = 0.12, head coefficient ψ = 0.06, and specific speed 2.86 (8070 US). Results from the study include relative and absolute velocities, flow angles, and static and total pressures. Comparison is made to measured data available for the same impeller at two planes, one upstream of the impeller and the other downstream. The comparisons are for circumferentially averaged results and include axial and tangential velocities, impeller exit flow angle, static pressure, and total pressure. Results of this study show that the computational results closely match the shapes and magnitudes of the measured profiles, indicating that CFD can be used to accurately predict performance.

Steady State Groundwater Flow Simulation With Imprecise Parameters

A methodology based on fuzzy set theory is developed to incorporate imprecise parameters into steady state groundwater flow models. In this case, fuzzy numbers are used to represent parameter imprecision. As such, they are also used as a measure for the uncertainty associated with the hydraulic heads due to the imprecision in the input parameters. The imprecise input parameters may come from indirect measurements, subjective interpretation, and expert judgment of available information. In the methodology, a finite difference method is combined with level set operations to formulate the fuzzy groundwater flow model. This fuzzy modeling technique can handle imprecise parameters in a direct way without generating a large number of realizations. Two numerical solution methods are used to solve the fuzzy groundwater flow model: the groundwater model operator method proposed in this methodology and the iterative algorithm based on conventional interval arithmetics. The iterative method is simple but may overestimate the uncertainty of hydraulic heads. The groundwater model operator method not only provides the hull of the solution set for the hydraulic heads but also considers the dependence of hydraulic head coefficients which are functions of imprecise parameters. Sensitivity analysis shows that the dependence of hydraulic head coefficients has a critical impact on the model results, and neglecting this dependence may result in significant overestimation of the uncertainty of hydraulic heads. A numerical model based on the methodology is tested by comparing it with the analytical solution for a homogeneous radial flow problem. It is also applied to a simplified two‐dimensional heterogeneous flow case to demonstrate the methodology.

Determination of the heat transfer capability of laser mirrors with cooled cells

A mathematical model of steady-state heat transfer in a laser mirror involving cooled prismatically shaped cells has been developed. Using cooling systems with hexahedral and tetrahedral cells (by the number of side walls) as examples, the influence of the mirror illumination nonuniformity, reflector thickness, and other parameters on the effective heat-transfer coefficient and thermal head coefficient is investigated; the physical limits for heat-transfer characteristics in the case of an unlimited increase in heat transfer from the surfaces of the cell walls have been determined.

Shut-off Head of Centrifugal Pumps and Fans

A simple model is proposed for estimating the contribution of the volute to the shut-off head of a centrifugal pump or fan. The model is based on an assumed linear distribution of tangential velocity in the plane of the cut water which satisfies approximately the continuity condition of zero net flow into the outlet duct. The contribution of the impeller is assumed to be that given by a solid body rotation at the angular velocity of the pump from the bore of the inlet duct to the impeller tip. The simple radial equilibrium equation is then used to calculate the static head rise in both the impeller and volute. The resultant prediction of shut-off head, which only requires as input the angular velocity and four radii, has been compared to test data on various pump series made available by courtesy of two European manufacturers. In all of the series the impeller diameter has been varied between 100 and 90–80 per cent of its design value and has been tested in the design volute. Over the series the predicted shut-off head coefficient (gH30/U22) was between 1 and 6 per cent lower than the measured values, which themselves varied over the range 0.5–0.6. The predicted contribution from the volute varied typically between 20 and 30 per cent of the total head rise over the range of impeller size, emphasizing the significant contribution from the volute and its dependence on the impeller/volute geometry. Since a review of the available literature did not show any previous work of a fully consistent nature on this topic the proposed model as described in detail below is offered as a fairly accurate prediction technique for design purposes.

Experiments With a Low-Specific-Speed Partial Emission Centrifugal Compressor

This paper describes experiments that were conducted several years ago on a single-stage open-face centrifugal partial emission compressor with a nondimensional specific speed of 0.15 (19.3 dimensional). The purpose of the experiments was experimentally to evaluate the partial emission compressor as a possible candidate for a high-speed low-flow natural gas fuel pump, the alternative to which was a larger heavier reciprocating compressor. A review and re-analysis of the test data were conducted. They used updated compressor data reduction performance procedures and flow models to redefine the experimental compressor as a reference case in the low-specific-speed range of single-stage centrifugal compressors. Normalized performance characteristics are presented for the experimental compressor in terms of efficiencies, diffuser recovery, and head coefficients versus diffuser relative flow parameter, or choke margin. Peak efficiency, obtained at a De Laval number of 1.16, and nondimensional specific speed of 0.15, was 34.5 percent. Stage performance is found to be dominated by diffuser geometry and choke margin, plus windage losses.

Performance Characteristics of Two- and Three-Dimensional Impellers in Centrifugal Compressors

The overall performance of two- and three-dimensional impellers of a centrifugal compressor were tested and compared. A closed-loop test stand with Freon gas as the working fluid was employed for the experiments. The inlet and outlet velocity distributions of all impellers were measured using three-hole cobra probes. As a result, it has been revealed that three-dimensional impeller in terms of efficiency, head coefficient, and operating range. Further, it has also been clarified that the impeller slip factor is affected by blade angle distribution.

Reynolds Number Effects on Regenerative Pump Performance

Reynolds number effects on the performance of a conventional design regenerative pump were investigated, using glycerine-water mixtures, between an impeller tip speed Reynolds number, RT, of 5.0×103 (all glycerine) and 1.6×106 (all water). Results show that the maximum efficiency, nm, can be expressed in terms of an output to loss ratio, nm/1−nm, which varies as RT0.203 for 2.0×104 < RT < 1.6×106 and as RT1.156 for RT < 2.0×104. These results are consistent with efficiency behavior reported in similar investigations on other types of turbomachines. Further, the design point flow coefficient increased over the range of Reynolds number investigated, while the design point head coefficient exhibited a maximum within this range. In addition, marked departure from scaling behavior occurred in the lower Reynolds number range. Finally, the correlation among torque coefficient, head coefficient, and flow coefficient previously established by the author was further verified and followed scaling behavior for the higher Reynolds number range.

A Study on Unstable S-Shape Characteristic Curves of Pump Turbines at No-Flow

A small model of a Francis-type pump turbine was tested without the rotating shroud and the radial distribution of pressure in the rotor was measured using pressure taps on the stationary shroud. The mean tangential velocity component in the rotor was measured with a laser velocimeter, and the flow pattern was visualized using a camera which rotated with the rotor. Comparing data at the no-flow condition in the turbine mode with those in the pump mode, the effect of semi-open spaces at the inlet and at the exit of blade rows on the difference of pressure rise between the two modes was recognized, and the flow patterns and the distributions of tangential velocity component in these zones were utilized to understand the phenomena. Furthermore, some discussion is included on the relationship between the design head coefficient in the turbine mode and the S-shape characteristics.

Comparison of ring head and SPT head write fields

The coefficients of Fan ring head and single-pole-type (SPT) head of various gap widths/pole thicknesses were evaluated and the head field distribution for various configurations and flying heights were calculated. For easy visualization in comparison with Karlquist field the head fields are expressed in reduced units as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H\min{xy}\max{F} = H\min{xy}\max{K}+ H\min{xy}\max{C}</tex> where F, K, and C stands for Fan, Karlquist, and correction terms, respectively. The results show the following. 1) The ring head coefficients are independent of gap width while the SPT head coefficients vary with the ratios of pole thickness to flying heights. The latter coefficients approach that of the ring head in the limit if it is properly redefined, indicating similarity and dissimilarity between the two head fields. 2) In the gap region the write field of a ring <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">head F\min{Rx}\max{F}</tex> bends downward in comparison with Karlquist field, reaching approximately 0.82 at very low flying heights. 3) The x-component ring head field <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H\min{Rx}\max{F}</tex> differs considerably from the y-component SPT head field <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H\min{py}\max{F}</tex> when measured along the direction of media motion. The two fields behave similarly to each other only if each is measured along the direction of its own image plane.

Analyses of the characteristics of a centrifugal impeller with leading edge cavitation by mapping methods.

The characteristics of a centrifugal impeller under a condition with leading edge cavitation are analyzed by using conformal mapping methods. It is assumed that the thickness of the cavity is small, and linear cavity models are used. Concerning the treatment of the Bernoulli equation, two different models are considered. The first one is based on a full Bernoulli equation in a rotating frame. In the second model, the Bernoulli equation is linearized on the assumption that the disturbance due to cavity is small. The second model predicts shorter cavity, but the differences in the pressure distribution and in the head coefficient are small for the conditions with the same cavity length. The results of the first model are in general agreement with those by a singularity method and experiments.

On the Evaluation of Reynolds Number and Relative Surface Roughness Effects on Centrifugal Compressor Performance Based on Systematic Experimental Investigations

This paper summarizes the results of systematic investigations into the Reynolds number effects. It is based on performance map measurements carried out on a compressor test rig which was constructed primarily for this purpose. The measurements were performed for stages with different flow coefficients (0.004 ≦ φ1 ≦ 0.05), with different gases (air, nitrogen, helium, freon) and in the inlet pressure range 0.2 bar ≦ p1 ≦ 40 bar. By analogy with the turbulent flow in technically rough pipes, semi-empirical correlations are derived concerning the effects of the Reynolds number and the relative surface roughness on the characteristic performance parameters (efficiency, flow coefficient, head coefficient, work coefficient). For the detailed design calculation of individual stages, provision is made for the different effects on the hydraulic flow losses and the disk friction losses. Simplified correlations are given for the conversion of characteristics measured during thermodynamic performance tests. The correlations are applied to various single and multistage compressors, and the results compared with measured performance characteristics in the Reynolds number range 6 × 103 ≦ Ret ≦ 1.1 × 107. The good correspondence obtained forms the basis for recommending the application of these simplified relationships for the improvement of centrifugal compressor performance test codes (e.g. ASME PTC-10 and ISO TC 118).