Minimum Froude Number Research Articles

Modeling minimum transport boundary for pneumatic conveying of powders

Accurate prediction of blockage conditions or the minimum transport boundary is essential for the reliable design and operation of a pneumatic powder conveying system. Many existing empirical models for minimum transport boundary do not consider essential powder properties and operating conditions, such as loose poured bulk density, particle size, and air density. Based on the conveying results of 13 different powders, this paper has developed a new empirical model for the minimum transport boundary. The model includes a Froude number based on particle size and bulk density and a dimensionless gas density term, which makes the model inherently adaptable to variations in powder properties and operating conditions. Results of validation show that the new model provides a significantly improved prediction of minimum Froude Number (in the range of 7 to 13% relative error only) compared to the existing models, which provided relative errors in the range of 19 to 67%.

Water Flow Simulation of Tigris River Between Samara and Baghdad Based on HEC-RAS Model

This work aims to use the digital elevation model (DEM) or 3-Dimensional surface to model and analyze the water flow in the Tigris River. The first stage of this study is based on filed data acquisition survey at 38 stations. These stations were adopted to implement the one-dimensional steady and unsteady numerical flow models. The HEC-RAS software was used to implement these models. The most important results of the steady-state model were that the maximum discharge and velocity and minimum flow area were 638.17 m3/sec, 2.12 m/sec and 301.42 m2 respectively. These values occurred at station 66+700. While, the minimum discharge and velocity and maximum flow area were 504.5 m3/sec, 0.13m/sec and 4758.13 m2 respectively. The results of unsteady state were analyzed based on the recorded discharge in the year 2018. The maximum and minimum discharge (flow velocity) were 638.12 m3/sec (2.42 m/sec) at the stations 94+666+7 and 635.57 m3/sec(0.14 m/sec) at station 00+00, respectively. The maximum and minimum Froude number were 0.52 and 0.02 at stations 94+666+7 and 04+800, respectively. These results help to identify the flow choking areas that cause defects. In addition, it can be the base for determining the critical sedimentation stations in which the bed level rises and aggravate the flow choking problem within the considered reach of the Tigris River.

Roll waves and their generation criteria

ABSTRACT Pulsating waves (also known as roll waves) might occur on the free surface of extreme events like mud and debris flows, among others, usually intensifying the caused damage. This technical note aims to inform about the roll wave phenomenon developing in a free-surface laminar flow, and analyze its generation criteria, centered on the concepts of Froude number and disturbance frequency. The complete linear stability analysis of the new depth-averaged model was proven a useful theoretical tool in determining new generation criteria for roll waves developing in non-Newtonian fluids. The results showed that the roll wave generation depends on two criteria: the first is associated to the minimum Froude number, and the second is related to the cut-off frequency. In addition, we have confirmed that the new generation criteria can be verified via numerical simulation based on a second model with full equations (Fluent software). Globally, the emergence of roll waves is favored by the non-Newtonian properties of the flowing fluid and the fact that the cut-off frequency decreases along with the minimum Froude number. Lastly, both generation criteria were tested in order to examine a case study involving the occurrence of roll waves in a watershed.

Effects of an Upstream Sluice Gate and Holes in Pooled Step Cascade Weirs on Energy Dissipation

Pooled step cascade weirs are hydraulic structures that are preferred over ordinary weirs due to their application on steep slopes and to their effectiveness in obtaining water aeration. However, these structures are subject to failure due to scour being initiated downstream, or to breakdown due to the high energy of impinging water. In this paper, an experiment was presented to study the effect of the number and ratio of holes in weirs to improve energy dissipation. To improve energy dissipation, cascade weirs with different configurations of openings were proposed. Furthermore, a sluice gate at the upstream of a cascade weir was proposed as a new technique to increase energy dissipation and reduce scour at downstream. Ten models of pooled step cascade weirs (nine models with different hole configurations and one with a sluice gate upstream), in addition to the ordinary weir, were constructed to investigate the effect of the sluice gate and holes on energy dissipation; the range of each type of flow (nappe, transition, and skimming); the relevant Froude number; and the ratio of water depth to critical depth over pooled step cascade weirs. To implement this, the ratio of holes to the area of the weir were between 0 and 21.22% and the number of holes used were one hole, two holes, and three holes. The results concluded that the best method to obtain maximum energy dissipation was putting a sluice gate at the upstream of a solid pooled step cascade up to a discharge of 2.0 l/s. The most effective ratio of holes to weir area was 7.15% at the minimum Froude number when the discharge exceeded 2.0 l/s. Moreover, the ratio of upstream water depth to critical depth decreased with increasing discharge. The nappe, transition, and skimming flow types were increased more on a pooled stepped cascade weir of 7 mm opening diameter with three holes (ratio of 7.15%) than the other models. The Froude number decreased as the hole numbers increased at high discharges for the pooled stepped cascade weir, especially with the hole diameter of 12 mm. Some empirical equations were established to represent the relation between the upstream head and discharge for each case.

Floodplain persistence and dynamic-equilibrium conditions in a canyon environment

Canyon river systems are laterally constrained by steep walls, strath terraces, and bedrock intrusions; however, semialluvial reaches are nested within these environments as discontinuous floodplains along the river margins. These semialluvial floodplains provide an example of dynamic-equilibrium set within high-energy fluvial systems, marking areas where the river is free to alter its boundary conditions. Most research has focused on hydraulic conditions necessary for floodplain formation and persistence in unconfined systems, whereas little is known about canyon streams. This paper focuses on (1) characterizing dynamic-equilibrium, (2) describing the controls on floodplain formation and distribution, and (3) evaluating the performance of extremal hypotheses to identify dynamic-equilibrium and floodplain persistence in high-energy, semiconfined canyon environments.These objectives were addressed with field and numerical data derived from a one-dimensional hydraulic model for bankfull and 100-year return interval flood events, supported by closely spaced cross sections for the lower 38-km canyon reach of the Deadwood River (Idaho). Under bankfull conditions, critical energy thresholds for equilibrium floodplain persistence at this study site present the upper limits of: slope=0.018, shear stress=175N/m2, and specific stream power=400W/m2. Channel and floodplains near equilibrium, quantified with a near-zero sediment transport divergence (Exner equation), were successfully identified by the minimum unit stream power extremal hypothesis and to a lesser degree by the other extremal hypotheses that minimize energy expenditure (minimum specific stream power, minimum total stream power, and minimum Froude number), provided backwater environments and major tributaries could be identified. Extremal results were compared to hydraulic geometry relations to evaluate how closely equilibrium floodplains approached values for unconfined alluvial river systems.

Control of instabilities in non-Newtonian free surface fluid flows

Free surface flows in inclined channels can develop periodic instabilities that are propagated downstream as shock waves with well-defined wavelengths and amplitudes. Such disturbances are called “roll waves” and are common in channels, torrential lava, landslides, and avalanches. The prediction and detection of such waves over certain types of structures and environments are useful for the prevention of natural risks. In this work, a mathematical model is established using a theoretical approach based on Cauchy’s equations with the Herschel–Bulkley rheological model inserted into the viscous part of the stress tensor. This arrangement can adequately represent the behavior of muddy fluids, such as water–clay mixture. Then, taking into account the shallow water and the Rankine–Hugoniot’s (shock wave) conditions, the equation of the roll wave and its properties, profile, and propagation velocity are determined. A linear stability analysis is performed with an emphasis on determining the condition that allows the generation of such instabilities, which depends on the minimum Froude number. A sensitivity analysis on the numerical parameters is performed, and numerical results including the influence of the Froude number, the index flow and dimensionless yield stress on the amplitude, the wavelength of roll waves and the propagation velocity of roll waves are shown. We show that our numerical results were in agreement with Coussot’s experimental results (1994).

Experimental studies of air pocket movement in a pressurized spillway conduit

Undesired air entrainment in a bottom outlet conduit causes pressure transients, leading to conduit vibrations, blowbacks and discharge pulsations and thus endangers operational safety. In this study, the propagation velocity of a solitary air pocket and the characteristics of its critical velocity were examined in experiments conducted using a 240-mm-diameter pipe. Air pocket movement depends on the pipe diameter, slope, roughness and air pocket size. The critical pipe Froude number for initiating downstream movement of an air pocket is smaller in a larger pipe, most likely due to the scale effect and/or to a smaller reduction in the effective cross-sectional area. The critical velocity in rough pipes was found to be independent of the air pocket size. A minimum Froude number was suggested for a rough pipe instead of a critical pipe Froude number because the air removal process was found to involve successive air losses from the air pocket caused by turbulence.

Hydraulic Jumps in Sloping Channels: Sequent Depth Ratio

A large variety of hydraulic jumps on horizontal and sloping inverts at the end of an ogee standard weir is investigated. An ogee standard weir was used to create supercritical flow and slopes of 0.0, −0.025 , −0.05 , −0.075 , and −0.10 were built downstream of the weir. Based on the momentum equation in the horizontal direction, a method to predict the sequent depth ratio is presented. The theory agrees well with the results of the writers and previous investigators. A correlation was developed to predict the minimum Froude number needed to establish jumps on negative slopes. Observations showed that in those cases where the gravity force component in the jump was opposite to the flow direction, the water surface of the surface roller became undular and unstable. The hydraulic jump on an entirely adverse slope was almost impossible to control. The analysis of experimental data showed that the negative slope of the basin reduces the sequent depth ratio, while a positive slope increases the sequent depth r...

Advantages of the hollow (concave) turbine for multi-phase agitation under intense operating conditions

The mixing literature on hollow blade turbines (HBTS), for operation in fully turbulent flow, is reviewed and compared with the results of our own studies. The SCABA 6SRGT is shown to have an almost identical pumping rate to a disc turbine, when compared at the same diameter and specific power. An equation is proposed for the effect of scale and blade geometry on the power number of a range of concave hollow blade agitators. The “flooding–loading” condition is revisited. It is found that, when compared at conditions above the minimum Froude number required to disperse gas, the HBT designs are as energetically efficient as Rushton turbines for dispersing gas. If we compare them on an “ungassed” power basis, as is the usual literature case, then the HBT is more efficient because of their ability to disperse gas without significant loss of power. The much lower power number resulting from the streamlined blade design of the HBTs also ensures that they achieve the minimum Froude number required to disperse gas at a much lower power than a RT. A simple method to avoid “flooding” for radial turbines, based on this work, is proposed. Under fully loaded conditions the hollow blade turbines will handle high gas rates without significant loss of power and this ability is a function of the degree of streamlining. For the suspension of high levels of solids the D = T / 2 hollow blade turbines, at a clearance of T / 4 , are found to be amongst the most efficient agitators especially under gassed conditions, where almost no effect of gassing on the just suspension speeds were noted.

Minimum froude number and the equilibrium of alluvial sand rivers

AbstractRivers adjust towards an equilibrium condition, the stability of which depends upon a set of controlling factors expressed by the Froude number. As alluvial river channels approach stable conditions, the Froude number of the channel flow will tend to attain a minimum value which reflects minimum bed material motion and maximum channel stability, under the constraints imposed by water discharge, sediment load, and particle size. Computer simulations for sand bed rivers show that the Froude number of the flow tends to a minimum value when the equilibrium river tends to a certain hydraulic geometry. Evidence from 57 alluvial sand material rivers and stable canals shows that this simulated hydraulic geometry with minimum Froude number corresponds to the natural equilibrium state.

Calculation of primary bubble volume in gravitational and centrifugal fields

AbstractA one‐stage model of the formation of primary bubbles is presented in which the bubble volume is deduced from an equilibrium of buoyancy, viscosity, inertia and surface tension forces. In contrast to the two‐stage model, presented by Kumar and Kuloor, it was not assumed that the drag coefficient in bubble expansion can be described by the same constants as in the steady‐state bubble ascent. The constants were adapted in such a way that the introduction of an additional bubble volume was not necessary. It was demonstrated that this model describes the bubble formation in gravitational and centrifugal fields equally well and, furthermore, is also applicable to structurally viscous liquids, provided that the effective shear gradient is calculated from the equilibrium of shearing and buoyancy forces. The model is based on the assumption of a constant volumetric flow rate during bubble formation and, for this reason, a minimum Froude number is necessary in analogy to the weeping limit for sieve plates. The normalized presentation permits simple operation. The possibility of applying the model to drop formation was confirmed by comparison of experimental values with those, predicted by the model.