Modified Froude Number Research Articles

Effects of particle density and fluid properties on mono-dispersed granular flows in a rotating drum

Due to their simple geometric configuration and involved rich physics, rotating drums have been widely used to elaborate granular flow dynamics, which is of significant importance in many scientific and engineering applications. This study both numerically and experimentally investigates dry and wet mono-dispersed granular flows in a rotating drum, concentrating on the effects of relative densities, ρs−ρf, and rotating speeds, ω. In our numerical model, a continuum approach based on the two-phase flow and μI theory is adopted, with all material parameters calibrated from experimental measurements. It is found that, in the rolling and cascading regimes, the dynamic angle of repose and the flow region depth are linearly correlated with the modified Froude number, Fr*, introducing the relative density. At the pore scale, flow mobility can be characterized by the excess pore pressure, pf. To quantify the variance of the local pf, it is specifically nondimensionalized as a pore pressure number, K, and then manifested as a function of porosity, 1−ϕs. We find K(ϕs) approximately follow the same manner as the Kozeny–Carman equation, K∝ ϕs2/1−ϕs3. Furthermore, we present the applicability of the length-scale-based rheology model developed by Ge et al. [“Unifying length-scale-based rheology of dense suspensions,” Phys. Rev. Fluids 9, L012302 (2024)], which combines all the related time scales in one dimensionless number G, and a power law between G and 1−ϕs/ϕc is confirmed. This work sheds new lights not only on the rigidity of implementing continuum simulations for two-phase granular flows, but also on optimizing rotating drums related engineering applications and understanding their underlying mechanisms.

Experimental and theoretical study on liquid film thickness of upward annular flow in helically coiled tubes

The liquid film thickness is crucial for studying the thermal hydraulic mechanism of the annular flow region in helically coiled tubes (HCTs). This paper introduces a refined experimental study on the liquid film thickness of annular flow in HCTs, utilizing a newly developed liquid film sensor. The experimental results indicate that the smaller coil diameters and pitches result in thinner average liquid film thicknesses. The average liquid film thickness decreases with increasing superficial gas velocity and decreasing superficial liquid velocity. However, the liquid film thickness at different circumferential positions on the cross-section of the tube exhibits varying sensitivities to superficial gas and liquid velocities. The experimental data reveal that the existing typical correlation formula for the average liquid film thickness of annular flow in straight tubes does not apply to HCTs. Consequently, a new prediction model is proposed for the average liquid film thickness of the annular flow region in HCTs, based on the modified Froude number, modified Dean number, Ekman number, and Reynolds number. This model comprehensively incorporates the structural characteristics of HCTs and fluid properties, and its validity is verified through the utilization of available and current data in the literature.

A water model study on mixing behaviour in a Kaldo furnace using acid–base neutralization decolourization method combined with RGB-based image analysis

Mixing time is usually used as an important parameter for quantifying the mixing performance of multiphase flow in a metallurgical bath. However, the topic of the mixing time of a Kaldo furnace for copper anode slime treatment has received little attention from scholars to date; this concept is important for guiding the optimization of high-efficiency smelting in the Kaldo furnace. In the present work, the mixing behaviour of gas-liquid two-phase flow in a Kaldo furnace was simulated using a scaled-down 1:4.5 water model fabricated from transparent acrylic plastic. A visualization technique and a quantitative characterization method for mixing performance in the Kaldo furnace—acid-base neutralization decolourization combined with RGB-based image analysis—were innovatively proposed. This approach can not only determine the global mixing level η¯(t) and track the frequency distribution of the local mixing performance ξk(t) of the fluid region during the mixing process but also quantify the mixing time. In addition, based on the proposed method and by using dimensional analysis, the effects of operating parameters (such as the gas flow rate Q, injection lance height h, rotation speed ω, liquid volume fraction φ, inclination angle θ, and liquid viscosity μl) on the mixing time of gas-liquid two-phase flow in the molten bath were discussed. On this basis, the empirical equation of the mixing time as a function of h, ω, φ, the modified Froude number (Fr'), μl and θ was established as the first quantitative relationship of the mixing time for the Kaldo furnace.

Flow instabilities of second-grade fluid over a stretching sheet

The work at hand studies thin-film flow instabilities in a non-Newtonian second-grade fluid moving over a stretching sheet. The analysis considers the free surface evolution equation of the thin second-grade viscoelastic fluid model based on the long-wave approximation method. We undertook a linear stability analysis using a normal mode approach and obtained expressions for the linear time growth rate, linear phase speed, and critical wavenumber, which illustrates the linear stability structure of the flow system. Furthermore, the multi-scale analysis of the fluid’s weakly non-linear stability characteristics reveals the presence of subcritical instability in the linear stability zone. The study shows that the varying flow parameters affect the threshold amplitude, which influences the stable and unstable zones in the subcritical region. The outcomes show a stabilizing effect spanning the range of non-dimensional parameters, such as the second-grade parameter ranging from 0 to 100 in values, the surface tension parameter ranging from 0 to 100 in values, and the modified Froude number from 0 to 100 in values.

Flow pattern criteria for vapor condensation inside a downward inclined tube

Vapor condensation in inclined tubes is a widely used process in various industries, but currently there is no verified map available for predicting flow patterns during condensation. This study aimed to develop a method to predict flow patterns during condensation in downward inclined tubes. The literature on condensation flow patterns in downward inclined tubes was thoroughly investigated, and this paper presented a summary of experimental data on condensation in downward inclined tubes, covering inclination angles from 7° to 75° (downward flow), tube diameters from 5.79 to 25.0 mm, pressures from 0.1 to 1.0166 MPa, mass fluxes from 8.1 to 705 kg/m2s, and three fluids including water and refrigerants. By comparing the existing flow pattern maps with the data, it was found that the existing flow pattern maps were deficient and did not predict the condensation flow pattern well. Based on extensive condensation flow pattern data, a set of criteria for predicting the flow pattern during condensation, using the modified Froude number F and the Lockhart-Martinelli parameter X as map coordinates, was proposed. The results showed good prediction accuracy for inclined in-tube condensation flow patterns.

Critical balance and scaling of strongly stratified turbulence at low Prandtl number

We extend the scaling relations of strongly (stably) stratified turbulence from the geophysical regime of unity Prandtl number to the astrophysical regime of extremely small Prandtl number applicable to stably stratified regions of stars and gas giants. A transition to a new turbulent regime is found to occur when the Prandtl number drops below the inverse of the buoyancy Reynolds number, i.e. $Pr\,Rb<1$ , which signals a shift of the dominant balance in the buoyancy equation. Application of critical balance arguments then derives new predictions for the anisotropic energy spectrum and dominant balance of the Boussinesq equations in the $Pr\,Rb\ll 1$ regime. We find that all the standard scaling relations from the unity $Pr$ limit of strongly stratified turbulence simply carry over if the Froude number, $Fr$ , is replaced by a modified Froude number, $Fr_M\equiv Fr/(Pr\,Rb)^{1/4}$ . The geophysical and astrophysical regimes are thus smoothly connected across the $Pr\,Rb=1$ transition. Applications to vertical transport in stellar radiative zones and modification to the instability criterion for the small-scale dynamo are discussed.

Temperature Dependence on Transient Hydrodynamic Gravity Driven Flow Down an Inclined Plane

Transient hydrodynamic gravity driven flow of a viscous incompressible fluid down an inclined plane occupying a semi-infinite region of space bounded by an infinite hot plate is studied. The effect of modified Froude number plays an important role in determining the flow situation. Exact solution is obtained by employing Laplace transform method. This problem is concerned with a car – following theory subjected to traffic flow with a decisive importance to a variation of depth with a slope leading to a flood wave. A reducible influence of a frictional drag has an active influence to the flow reversal characteristics with time variation satisfying the separation due to the existance of a modified Froude number. The stabitity of a roll wave is determined by the critical Froude number in a time varying flow. Numerical results of velocity distributions and frictional shear stress are depicted graphically for different values of flow parameters. Numerical results of a critical Froude number are presented in tables. Temperature analysis has been presented analytically with a view to point out the expression of Nusselt number at the plate.

Investigation of Optimum Scale-Up of Media Stirred Mill Using the Discrete Element Method

Media stirred mills have been used in various fields to efficiently produce fine products. The scale-up of such mills is required to increase efficiencies and applications for industrial processes. Despite such circumstances, established scale-up laws remain unavailable, and scale-up has been conducted empirically. This study aims to propose essential requirements for the scale-up of a media stirred mill according to experiments and discrete element method (DEM) simulations. The experimental results were evaluated qualitatively by using particle size distributions of ground products and fitted using grinding kinetic theory as a quantitative evaluation. The grinding performance in the laboratory model was equivalent to the results using the scaled-up model (SU-model) with the modified Froude number, regardless of the rotation speed. DEM simulations identified the main factor that contributed to the agreement. The average collision energy of the media particles was almost identical. To set optimal conditions to fulfill this core requirement, the SU-model operation conditions need to follow the modified Froude number, and the intervals of each arm should be fixed to the same distance. These insights encourage the application of scaled-up media stirred mills.

Experimental study on the effects of exhaust vent layout on plug-holing for ceiling central smoke extraction in road tunnel fires

In this study, a series of reduced-scale experiments are performed to study the phenomenon of plug-holing induced by ceiling central smoke extraction in road tunnel fires. By changing the side length of the vent perpendicular to the longitudinal direction of a tunnel (vent width), the heat release rate and the area of the ceiling extraction vent, the influencing factors for the occurrence of plug-holing are analysed. The experimental results show that increasing any of the three aforementioned factors can increase the smoke exhaust flow rate corresponding to the occurrence of plug-holing. The experimental results are then compared with the results obtained by previous research, and it is found that the previous research is no longer applicable with respect to the change in the vent width. On this basis, we propose a modified Froude number to predict the occurrence of plug-holing, considering the influence of vent width. The critical value corresponding to the plug-holing for the newly proposed Froude number is a constant having a value of 4.71 (Frc = 4.71), regardless of the variation in the abovementioned three influencing factors. The results can be used as a reference for the design of ceiling central smoke extraction in road tunnels.

Dynamic response of debris flows impacting curved joint check dams

The dynamic impact force of debris flow on dams with a curved upstream face curved was investigated using laboratory experiments and a theoretical approach. Equations describing the impact force and maximum run-up height were derived. The experiments and theoretical considerations reveal that the impact force and maximum run-up height are mainly controlled by the Froude number (Fr) reduced by the cosine of the channel slope angle (cosα). Both the impact force and the maximum run-up height have a quadratic relation with the modified Froude number (Fr/(cosα)0.5). The experimental data and the results of the theoretical approach are in good agreement, indicating that the theoretical approach can be used in practical applications. It is concluded that the comparison between the curved-joint dam and the more conventional sharp-joint dam shows no differences in the maximum impact force and run-up height for the same modified Froude number. With the sharp-joint dam, the peak values of the impact force are reached more quickly than with the curved-joint dam.

Scale model experiments of fire whirls over the non-fuel zone around an L-shaped fire source

The Hifukusho-ato fire whirl (HAFW) is a catastrophic fire accident in Tokyo in 1923 and a representative fire whirl phenomenon that occurs over the non-fuel zone around an L-shaped fire source. Previous studies have mainly focused on fire whirls above fire sources, while very limited work reported the HAFW-type fire whirls, which were found to be scale-dependent (more easily to occur in a large-scale wind tunnel test facility than small-scale ones due to the strong viscous effects in the latter) and thus not precisely discussed. In this study, a scale-independent method is proposed by fitting an additional fire source in the inside bend of an L-shaped fire source to mimic the flame-overhang region observed in the large-scale tests; the HAFW-type fire whirl is then successfully reconstructed in a small-scale test model (1/10,000th of the HAFW prototype). Meanwhile, the results show that the average fire whirl occurrence position in the flow direction in a large-scale test model (1/2,000th of the HAFW prototype) does not collapse with those in a small-scale test model (1/10,000th of the HAFW prototype without additional fire sources installed), but with those in the aforesaid scale-independent test model; such collapse can be summarized by a linear fitting correlation between a normalized fire whirl occurrence position and a modified Froude number under different fire source scales and lateral wind velocities. Based on the above experimental facts, updated scaling law to reproduce HAFW-type fire whirls and similar fire whirl occurrence positions is provided. This work advances a fundamental understanding of fire whirls generated by an L-shaped fire source; thus, helping guide fire-rescue operations, and benefiting future fire whirl model development.

Experimental study on burning rates of large-scale hydrocarbon pool fires under controlled wind conditions

Open square fuel pools with areas of 50 m2, 100 m2, 200 m2, 380 m2, and 400 m2 were used to burn RP-5 aviation fuel and diesel oil, under a well-controlled and regulated horizontal wind field with speeds ranging from 0 to 17 m/s by a large-scale open jet wind tunnel. The mean mass burning rate of the fuel in the quasi-steady period was taken as the representative value. Under the calm condition, the burning rates of the two fuels were proportional to the ratios of the ideal heat of combustion to the effective heat of gasification. The burning rates of different size pool fires were little affected by various wind conditions. With the increase of wind speed, all pool fires exhibited a common pattern in the burning rate, a steady growth followed by a level-off constant value. The traditional Froude number (Fr) was unable to predict the behavior of large-scale pool fires under various wind speeds. A modified Froude number (Fr* = Fr·Re0.2) was developed based on the heat transfer correlation for turbulent mixed convection on a horizontal flat plate and successfully correlated a total of nine different large-scale pool fire experimental data under wind conditions.

Experimental investigation on the fire and thermal smoke spread in tunnels: Effects of fire elevation

Previously, studies related to tunnel fires predominately concentrated on the fire at ground level while the risk of elevated fires caused by different vehicle heights was mostly ignored. In the current research, pool fires elevated at seven levels were experimentally studied in two reduced-scale model tunnels with both natural and longitudinal ventilation. Fuel burning rate, temperature profile, and thermal smoke stratification affected by the fire elevation were successively explored. Experimental results declared the major findings as 1) Fuel burning rate of the 8 cm × 8 cm pool fire was indistinctly affected by the fire elevation while it increased rapidly with the elevated height as pool size increased; 2) In the natural-ventilated tunnel, the maximum temperature significantly increased with the fire elevation and was found to be well predicted by the modified models with a linear correlation versus Q2/5Hd¯ or QD∗2/5·DfHd¯ and a constant value equal to 2.9. When longitudinal ventilation was activated, flame deflection, shortened back-layering length, and decreased maximum temperature were respectively observed. 3) Interface of the smoke-air layer was lifted, and thermal smoke stratification was promoted by the fire elevation. Good agreement was achieved between the present measurements and temperature quotient proposed by Newman. Correlation by using modified Froude number denoted different characteristics of smoke stratification of ground fire and elevated fire.

Inclination Effect on Mixing Time in a Gas–Stirred Side–Blown Converter

Small‐scale physical models are commonly used to investigate gas‐stirred processes in steelmaking practice. The argon oxygen decarburization (AOD) converter is among various processes widely used in the metallurgy field and utilizes side blowing of oxygen and inert gas for mixing in the bath. Herein, the effect of the converter inclination on mixing time and jet‐penetration length with a side‐blown physical model is investigated. Scaling with the modified Froude number is applied on data from a real industrial AOD converter to achieve a system with reasonable gas flow rates. During the experiments, water is used to simulate liquid steel and air is blown through side‐mounted nozzles for stirring. A NaCl tracer is added and subsequent conductivity measurements are used to measure mixing time. Overall, the penetration length is shown to be independent of inclination angle. The mixing time is found to be influenced by the change of bath height to diameter ratio, change of geometry in the bath volume, gas flow rate, and the intensified wave motion at the interface caused by the inclination of the vessel. The mixing time increase with 14% when 14° angle is applied.

Bubble behavior in cylindrical and square vessels under centric mechanical stirring

Water model experiments were carried out to investigate the bubble behavior in cylindrical and square vessels under centric mechanical stirring. The bubble behavior in the square vessel was investigated in detail by using a high-speed camera to record the transient images of the bubbles. An image analysis software was used to obtain the bubble diameter. The results showed that the centric mechanical stirring in the square vessel was suitable for breakage and dispersion of bubbles, but not suitable for that in the cylindrical vessel. Increasing the impeller blade length and impeller rotation speed was beneficial to disintegrate and disperse bubbles widely. The bubble diameter decreased with the increase in the Weber number and increased slightly with the increase in the modified Froude number. The dimensionless correlation equation of bubble diameter was obtained by the dimensional analysis method.

Study of the critical velocity of the tunnels using an analytical approach

Fire safety is one of the major design issues of tunnel engineers. Longitudinal ventilation is a common method for exhausting smoke and hot gases in tunnel fires. The minimum ventilation air velocity along the tunnel to prevent smoke back layering is called the critical velocity. This critical velocity is the key element in designing the longitudinal exhaust system. Several researchers have studied the impact of different parameters on critical velocity, including tunnel geometry, tunnel height, and fire magnitude, numerical and experimentally. In this study, an analytical solution was used to solve a third-order non-linear differential equation to determine the critical velocity of the tilted tunnel.Dimensional analysis can significantly reduce the costs associated with the experimental study in the full-scale tunnels. The Froude number representing the power of the buoyancy force against the inertial force is widely used in the critical velocity studies. This study validated our analytical results with critical values obtained from an experimental and numerical simulation in a scaled model with a ratio of 1:8.The results showed that the values calculated for the critical velocity employing an analytical solution were lower than the numerical and experimental studies' values.Data from the latest international standard was used to enhance the precision of the critical velocity calculation. We showed that using a modified Froude number can significantly increase the accuracy of the analytical solution. The critical velocity values obtained using the modified Froude number were then compared with experimental results from full-scale tests.This study emphasized that the analytical solution for the critical velocity saves a significant amount of time compared to the iterative solutions while keeping the accuracy in a reasonable range.

General equations for landslide-debris impact and their application to debris-flow flexible barrier

The runup height and impact load of landslide debris are the key parameters for design of debris hazard mitigation structures. However, the movement of obstacle (e.g., deformation of flexible barrier) still cannot be quantitatively considered in the existing analytical models. This study proposes general equations for landslide-debris impact under vertical jet and momentum jump modes. Especially, the influence of flow regime and obstacle movement (in the form of a modified Froude number), state of granular material, compressibility (density change), and flow resistance within the jump volume are explicitly considered in the momentum jump model. Through comparison between model prediction and experimental results, it is found that the vertical jet model prediction agrees well with the physical measurements. While the discrepancy between the momentum jump model and experimental results mainly comes from the inappropriate characterization of the unsteady flow properties (velocity and flow depth). Systematic parametric study demonstrates that the functionality of debris-flow flexible barrier primarily relies on the deflection of barrier, and the state of granular material also affects the impact load of dry granular flows.

Interface-Resolved Simulation of Bubbles–Metal–Slag Multiphase System in a Gas-Stirred Ladle

A grid-filtered scale interface-resolved volume of fluid (VoF) model coupling a sub-grid scale large eddy simulation (LES) was used to directly simulate bubble behavior and the evolution of multiphase interfaces and free surfaces in a bubble–metal–slag multiphase system. The model has been applied to an industrial 150-ton gas-stirred ladle. The results show transient behavior of the slag eye, with large variations over time of the eye size. Bubble detachment frequency and rise characteristics (including coalescence and breakup) directly affect the size, position and variation of the slag eye. The effects of the argon flow rate, slag thickness, slag viscosity, and slag–steel interfacial energy were investigated; the latter two have little effect. The calculated average slag eye size agrees with previously reported results for industrial ladles. A new correlation for the time-averaged slag eye area, based on the modified Froude number, is proposed for industrial ladles.

Effect of Froude Number on Submerged Gas Blowing Characteristics.

The flow behavior of gas in compressible and incompressible systems was investigated at an ambient temperature in an air–water system and at an operating process temperature in the IronArc system, using computational fluid dynamics. The simulation results were verified by experiments in the air–water system and established empirical equations to enable reliable predictions of the penetration length. The simulations in the air–water system were found to replicate the experimental behavior using both the incompressible and compressible models, with only small deviations of 7–8%. A lower requirement for the modified Froude number of the gas blowing to produce a jetting behavior was also found. For gas blowing below the required modified Froude number, the results illustrate that the gas will form large pulsating bubbles instead of a steady jet, which causes the empirical equation calculations to severely underpredict the penetration length. The lower modified Froude number limit was also found to be system dependent and to have an approximate value of 300 for the studied IronArc system. For submerged blowing applications, it was found that it is important to ensure sufficiently high modified Froude numbers of the gas blowing. Then, the gas penetration length will remain stable as a jet and it will be possible to predict the values using empirical equations.

Accelerating Taylor bubbles within circular capillary channels: Break-up mechanisms and regimes

In the present paper, an enhanced Volume of Fluid model is applied for the conduction of parametric numerical simulations, to investigate break-up phenomena of accelerating, elongated, vapour bubbles, within circular mini-channels. The effect of fundamental controlling parameters in the resulting break-up characteristics is investigated. Four different series of parametric numerical simulations of isolated vapour bubbles within mini-channels are performed, examining the effects of the imposed pressure difference between the inlet and the outlet of the channel, the surface tension, the effect of the applied heat flux as well as the initial liquid film thickness between the bubbles and the channel, on the developed vapour/liquid interface dynamics. The overall dimensionless number ranges examined are 6.76 < We < 1474.7, 0.007 < Ca < 0.14, 694 < Re < 12541 and by introducing a modified Froude number in order to account for the flow acceleration, 1 < Fr* < 21.86. These dimensionless number ranges are selected in order to overlap with experimental observations in zero-gravity Pulsating Heat Pipe experiments that constitute the motivation for the present numerical investigation. The proposed simulation results identify three prevailing regimes. A “full break-up” regime, a “partial break-up” regime and a “no break-up” regime. The entrainment of liquid droplets at the trailing edge of the vapour slugs is in most cases responsible for their subsequent “full break-up”, into a leading and a trailing bubble, as it is identified from the numerical simulations. Moreover, the applied heat flux does not influence the resulting break-up regimes. Finally, these identified break-up regimes, are grouped together into a well-defined flow map with respect to the We and Fr* numbers.