Understanding the heat and mass transfer phenomena occurring during critical flow of two-phase mixtures is of primary importance in the safety analyses of pressurized water, boiling water and liquid-metal-cooled nuclear reactors. It has been shown that during a blowdown incident, the critical flow rate of the two-phase mixture may be affected by a variety of parameters such as the fluid stagnation conditions, the configuration of the blowdown vessel, the length and diameter of the exhaust duct, the purity of the liquid and the local and frictional pressure losses in the flow channel. The complexity of the thermodynamic phenomena taking place during the blowdown process resulted in many studies which compare a particular theory with selected sets of experimental data. However, in the absence of an adequate theory which is applicable over the entire range of the parameters encountered in the nuclear industry, there is a tendency to rely on semi-empirical correlation of the existing data.The main objective of this paper is to provide a general review of two-phase critical flow from the viewpoint of the needs of thermal-hydraulic systems codes and to conduct a systematic evaluation of the existing data and theoretical models in order to quantify the validity, under various conditions, of several of the more widely used critical flow models. Ten different critical flow models have been formulated and tested in this study against an extensive set of data from critical flow experiments with water as the test fluid. Results of the present study are expected to enhance the understanding of the predictive capabilities and limitations of the critical flow models currently used in the power industry.
The development of practical and accurate methods to measure two-phase mass flow rates is of prime interest to applied nuclear reactor safety research. This article summarizes a comparison and evaluation of four commonly used mass flow rate devices. The particular systems investigated include (a) the true mass flow meter (TMFM), (b) the radionuclide technique, (c) the combination of a free field drag disk-turbine meter-transducer (DTT) and a gamma densitometer, and (d) the combination of a venturi nozzle and a full flow turbine meter. The experiments were performed under similar conditions in steady-state steam-water flow. The flow direction upstream of the instruments was horizontal except for the last method. The pressures varied between 3 and 9 MPa, and the highest values of the mass flow rate, the quality were 5 kg/s and 90 per cent respectively. The test matrix included wave-, slug- and annular flow. The measuring techniques are described briefly and a classification is proposed, which is based on the different ways of mass flow rate evaluation. The experimental results show that the accuracy of some methods is distinctively dependent on phase distribution (flow regime). Simple calibration correlations were developed to account for these effects.
An optical measurement method for two-phase flow pattern characterization in microtubes has been utilized to determine the frequency of bubbles generated in a microevaporator, the coalescence rates of these bubbles and their length distribution as well as their mean velocity. The tests were run in a 0.5 mm glass channel using saturated R-134a at 30 °C (7.7 bar). The optical technique uses two laser diodes and photodiodes to measure these parameters and to also identify the flow regimes and their transitions. Four flow patterns (bubbly flow, slug flow, semi-annular flow and annular flow) with their transitions were detected and observed also by high speed video. It was also possible to characterize bubble coalescence rates, which were observed here to be an important phenomena controlling the flow pattern transition in microchannels. Two types of coalescence occurred depending on the presence of small bubbles or not. The two-phase flow pattern transitions observed did not compare well to a leading macroscale flow map for refrigerants nor to a microscale map for air–water flows. Time averaged cross-sectional void fractions were also calculated indirectly from the mean two-phase vapor velocities and compared reasonably well to homogeneous values.
This paper reports the results of an experimental investigation on direct contact boiling of immiscible liquids. The continuous phase, water, is under stagnant conditions, while the dispersed one, Freon 114, is injected in the test section with different velocity and thermodynamic conditions through a nozzle. The injection system has been designed to vary the quantity of injected refrigerant and/or the liquid injection velocity. The test section is a Plexiglas vertical cylinder 72 mm i.d., 2.0 m long. Experimental data are obtained from high-speed movies of the continuous phase level during and after the Freon 114 injection, as well as from the movies of the rising boiling dispersed phase (injected under nearly saturation conditions). Vaporization rate has been characterized as a function of thermal hydraulic conditions (i.e. water temperature, system pressure and Freon mass flow rate). Direct contact boiling efficiency was derived by the evaluation of the fraction of Freon that did not undergo the boiling process during the transit in the test section.
Subcooled flow boiling heat transfer for refrigerant R-134a in vertical cylindrical tubes with 0.83, 1.22 and 1.70 mm internal diameter was experimentally investigated. The effects of the heat flux, q″ = 1–26 kW/m2, mass flux, G = 300–700 kg/m2 s, inlet subcooling, ΔTsub,i = 5–15 °C, system pressure, P = 7.70–10.17 bar, and channel diameter, D, on the subcooled boiling heat transfer were explored in detail. The results are presented in the form of boiling curves and heat transfer coefficients. The boiling curves evidenced the existence of hysteresis when increasing the heat flux until the onset of nucleate boiling, ONB. The wall superheat at ONB was found to be essentially higher than that predicted with correlations for larger tubes. An increase of the mass flux leads, for early subcooled boiling, to an increase in the heat transfer coefficient. However, for fully developed subcooled boiling, increases of the mass flux only result in a slight improvement of the heat transfer. Higher inlet subcooling, higher system pressure and smaller channel diameter lead to better boiling heat transfer. Experimental heat transfer coefficients are compared to predictions from classical correlations available in the literature. None of them predicts the experimental data for all tested conditions.
Experimental investigation of two-phase flow patterns for refrigerant R-134a and air–water in horizontal tubes with inside diameter from 1.0 to 3.0 mm was performed. The air–water test results agree very well with previous work. However, R-134a flow leads to a shift in the slug to annular transition to lower value of gas velocity. The locations of bubble to plug and slug flow transition are also significantly affected by the working fluids properties. We concluded that, in addition to buoyant force and turbulent fluctuations, surface tension force is also an important parameter for flow pattern determination in small tubes. Surface tension force causes the system to act to minimize its interfacial area. It tends to keep bubbles retaining its circular shapes and also to keep the liquid holdup between the tube walls to retard the transition from slug to annular. Since the surface tension of air–water is much larger than that of R-134a, it makes the intermittent to bubble flow transition occurs earlier for air–water than for R-134a. And also leads to a shift in the slug to annular transition to lower value of gas velocity for R-134a.
The equations describing the radial encroachment of a viscous liquid into a homogeneous, anisotropic porous medium are formulated and solved by two approximate methods. An analytical approximation is in good agreement with a finite element numerical solution, provided the angular component of the pressure gradient in an elliptical coordinate system is small. In the specific case where one of the principal flow directions is perpendicular to the flow plane, treatment of experimental flow data in accord with the analytical approximation determines the principal in-plane permeabilities and the degree of in-plane anisotropy. In the general case, the analysis yields effective permeabilities that are functions of the principal permeabilities and the orientation of the principal coordinate system.
In the present work, heat transfer augmentation by coiled wire inserts during forced convection condensation of R-22 inside horizontal tubes has been studied. The test-condenser consisted of four separate coaxial double pipe test sections assembled in series. It was a counterflow heat exchanger where R-22 condensed inside the inner tube by rejecting heat to the coolant water flowing in the annulus. Coiled wires with three different wire diameters of 0.65, 1.0 and 1.5 mm and three different coil pitches of 6.5, 10.0 and 13 mm were made and used in full length of test-condenser. The use of helically coiled wires were found to increase the condensing heat transfer coefficients by as much as 100% above the plain tube values on a nominal area basis.
This paper proposes an extension scheme for the application of the single phase multi-block lattice Boltzmann method (LBM) to the multiphase Gunstensen model, in which the grid is refined in a specific part of the domain where a fluid–fluid interface evolves, and the refined grid is free to migrate with the suspended phase in the flow direction. The method is applicable to single and multiphase flows, and it was demonstrated by simulating a benchmark single phase flow around a 2D asymmetrically placed cylinder in a channel and for investigating the shear lift of 2D neutrally buoyant drop in a parabolic flow.
Discrete element Newtonian dynamics simulations have been carried out of filling and discharge under gravity of non-cohesive discs (in two dimensions) and spheres (in three dimensions) from model hoppers. The current model improves that developed previously by us (Langston et al., 1994) in several respects. We introduce a continuous and gradual hopper filling method, a more realistic normal-tangential interaction between the particles, particle size polydispersity, and the model is extended from two to three dimensions (3D). The hopper discharge rate has been computed as a function of material head height, outlet size and the hopper half-angle. The model results are, in general, in very good agreement with established literature empirical predictions. The hopper wall stresses have been compared in the static state after filling and in the dynamic state during discharge. Generally there is encouraging agreement with predictions from the continuum differential slice force balance method, with significant improvements over our previous work. We have also observed, for the first time in a discrete element simulation of hoppers, the appearance of rupture zones within the material and associated wall stress peaks where the rupture zones intersect with the hopper wall. We consider that the current model is more successful than the previous one because the particle interactions include a much greater level of frictional “engagement” at low loads, with less variation at high loads.
The influence of particle rotation on the stability of the grain wake is investigated experimentally for particle Reynolds numbers less than 300. Particle rotation may be present in most industrial and geophysical two-phase flows and imparts significant differences to the structure of the wake when compared to cases where no rotation is present. An oil-filled flume has been used to investigate the dimensions of the wake region and frequency of wake eddy shedding for isolated, spherical and spheroidal particles at rotation rates up to 10 revolutions/second. A parameter, β, is defined which expresses the ratio of the peripheral surface velocity of the particle to the relative velocity between the two phases. For both shaped particles, the wake is destroyed and absent at β>0.5, whilst the size of the wake region is significantly smaller for 0<β<0.5 when compared to the wake dimensions at zero rotation. At 0<β<0.5 for the spherical grain, the frequency of eddy shedding is modulated and equal to the rotation rate, w, whilst for spheroidal grains this frequency is 2w as alternate wakes are shed into the flow as the particle adopts two oblate and prolate orientations in each revolution. At greater rotation rates, an increasingly dominant shear layer is formed on the underside of the grain and generated by fluid being dragged over the rotating particle and sheared against the freestream flow beneath the particle. Where the wake region is destroyed through particle rotation (β>0.5), turbulence enhancement in the freestream may still be present due to the generation of this lower shear layer. This work suggests that a principal mechanism of turbulence enhancement by large grains in two-phase flows involves the influence of particle rotation. If particle rotation is present, turbulence enhancement may occur at much lower particle Reynolds numbers than previously assumed, and at higher rotation rates turbulence enhancement of the freestream flow may occur in the absence of the wake region. Previous numerical modelling of two-phase flows which invokes wake instability to explain turbulence enhancement in the absence of rotation, may therefore require modification.
The design of tight-lattice pressurized water reactors requires the knowledge of CHF in tight rod bundles. Experimental investigations on CHF behavior in tight hexagonal 37-rod bundles were performed by using the model fluid Freon-12. About 400 CHF data points have been obtained in a range of parameters: pressure 1.0–2.7 MPa, mass flux 1.4–4.5 Mg/m2 s and bundle exit steam quality −0.4 to +0.2. It is found that the effect of different parameters on CHF in the tight rod bundle is similar to that in tube geometries. The present test results agree also quantitatively well with the CHF data obtained in tubes of comparable hydraulic diameters. Some in the open literature well known CHF correlations proposed for rod bundles under-predict the test results significantly.
An extensive study of the most important hydrodynamic characteristics of fairly large-scale bubble plumes was conducted using several measurement techniques and a variety of tools to analyze the data. Particle image velocimetry (PIV), double-tip optical probes (OP) and photographic techniques were extensively applied to measure bubble and liquid velocities, void-fraction and bubble sizes. PIV measurements in a vertical plane crossing the centre of the injector provided the instantaneous velocity fields for both phases, as well as hydrodynamic parameters, such as the movement of the axis of the plume and its instantaneous shape. Statistical studies were performed using image processing to determine the distribution of the apparent instantaneous plume diameter and centreline position. An important finding was that there is little instantaneous spreading of the bubble plume core; the spreading of the time-averaged plume width (as measured from the time-averaged void-fraction and time-averaged liquid velocity fields) is largely due to plume meandering and oscillations. The liquid-phase stress tensor distributions obtained from the instantaneous velocity data indicate that, for the continuous phase, these stresses scale linearly with the local void-fraction in the range of 0.5% < α < 2.5%. The bubbles were found to be ellipsoidal, with shape factor e ≈ 0.5.
Numerical simulations are carried out to describe the dense zone of a spray where very little information is available, either from experimental or theoretical approaches. Interface tracking is ensured by the level set method and the ghost fluid method (GFM) is used to capture accurately sharp discontinuities for pressure, density and viscosity. The level set method is coupled with the VOF method for mass conservation.The level set–VOF coupling is validated on 2D and 3D test cases. The level set–ghost fluid method is applied to the Rayleigh instability of a liquid jet. Preliminary results are then presented for 3D simulation of the primary break-up of a turbulent liquid jet with the level set–VOF–ghost fluid method.
Experiments on flow boiling heat transfer and critical heat flux (CHF) of the dielectric coolant FC-72 are carried out for four in-line simulated electronic chips of 10 mm × 10 mm, for both flush-mounted and protruded chips on one wall of a vertical rectangular channel. The fluid velocity and subcooling are varied from 4.2 to 78 cm/s (ReL=1.0×103 to 3.0 × 104), and from 15 to 33°C, respectively. The fully-developed nucleate boiling regime is not affected by changes in flow velocity and subcooling, whereas the surface temperatures decrease with increasing flow velocity and subcooling in the partial boiling regime. CHF generally increases with the degree of subcooling and with velocity at higher inlet velocities, but velocity has little effect at values less than 20 cm/s. The surface temperatures for the flush-mounted chips are lower than those for protruding chips, and the CHF data for the flush-mounted chips are higher than those for protruding chips; the differences between these two cases increase with increasing velocity.
Forced flow and natural circulation void fraction data are presented for steam-water upflow in a 73.9 mm pipe. Nondevelopment apparent in some of the data is due to bubble retardation at a nearby upstream bend. Through the use of suitable approximations, a previous analysis of distributed and annular flow voidage has provided a simple but accurate predictive method for the “developed flow” component of the data.
In order to investigate the effects of electric fields on the behavior of a bubble attached to a wall, basic experiments on the deformation and departure of a bubble are carried out under d.c./a.c. applied voltages. In the present study, three types of electrodes are used, to examine the bubble behavior by the nonuniformity of the electric field. For a d.c. electric field, as the applied voltage increases the bubble attached to a wall is more extended in the direction parallel to the imposed electric field, thus the aspect ratio and the contact angle also increase. The bubble departure volume in a nonuniform electric field decreases continuously, while that in a uniform electric field is nearly constant. The present results by the d.c. electric field show that the bubble behavior is significantly affected by the degree of the in omogeneity of the electrode configuration. For an a.c. electric field, as a preliminary analysis, a general theory on an equivalent dynamic system is extended to derive the bubble oscillation frequency. The analysis predicts that the bubble departure occurs near a resonant frequency. It is observed that the departure process of a bubble in the a.c. electric field is associated with the bubble oscillation which is composed of three different regions. Near the critical voltage, the bubble departure volume drops suddenly. It is also found that the reduction of the bubble departure volume by the a.c. electric field is more effective than in the d.c. electric field case.
A series of experiments on two-phase gas–liquid flow patterns in a test tube with a length of 356 mm and an inside diameter of 10 mm were performed aboard the Mir Space Station in August 1999. Carbogal and air were used as the liquid and the gas phase, respectively. In the present paper, the experimental results at the background microgravity environment of the Mir Space Station (no more than 10−5g) were reported. Five kinds of flow patterns, namely dispersed bubble flow, bubble flow, slug flow, slug–annular transitional flow, and annular flow, were observed in the space experiment. Due to the small length-to-diameter ratio of the test tube used in the present study, the observed flow patterns should be considered to be developing ones. The experimental results were compared with the model proposed previously which accounts for the entrance effects on the flow pattern transitions. A good agreement between the predictions and the experimental data was obtained. Some widely used models developed based on the analysis of fully developed two-phase flow at microgravity were also compared with the present data in order to make evaluations of these models and to have some insights on the flow evolution.
The steady-state two-dimensional flow which takes place in a half space of two-phase suspension bounded by an infinite stretching sheet is investigated. This problem is used as a vehicle for the study of certain aspects of two-phase stagnation-point flow. Attention is focused on the singularity in the particle-phase density distribution in the vicinity of the stagnation point predicted by certain existing theories. It is found that the inclusion of a particle-phase pressure gradient in the governing equations eliminates singular behavior in the particle-phase density distribution and makes possible the calculation of stagnation-point solutions throughout the entire range of inverse Stokes numbers.
A new formula for the pressure recovery in an abrupt expansion based on the superficial velocities of the two phases is proposed. Its predictive accuracy and that of other models from the literature are compared with new experimental data from stationary two-phase flow experiments in a diffusor with a very steep opening contour. Only the new model exhibits good agreement between predicted and measured experimental steam-water and air-water data. Furthermore, the new formula is verified by experimental data of other authors obtained with different expansion geometries and flow conditions.
A stochastic model based on the Boltzmann kinetic equation and employing the comprehensive treatment of the dynamics of binary droplet collisions is suggested to describe the droplet size and spatial distribution in dense spray. The model is valid for highly non-equilibrium impinging sprays in which the inertia of the droplets is very high and dynamic coupling with the gas is low. A Monte Carlo simulation procedure is developed for the solution of the kinetic equation. A model is used to analyse the absorption of a gas in a liquid spray in an impinging streams absorber. It is demonstrated that droplet collisions result mainly in coalescence, and reduce the overall droplet concentration and the interphase area in the reactor. The results of the analysis of the vaporization of a pentane spray in an impinging streams combustor are presented. It is shown that while droplet collisions reduce the vaporization rate by deflecting droplets out of the reactor and by coalescence, collision-induced fragmentation strongly affects the droplet size distribution and increases the fuel vaporization rate. The obtained results indicate that in the high velocity combustion of light fuels the collision-induced fragmentation of fuel droplets has a profound effect on the droplet size and spatial distribution.
The enhancement of the gas-liquid mass transfer rates in aqueous slurries containing small activated carbon particles was studied in a semi-batchwise operated stirred cell absorber with a plane interface. The maximum observed enhancement factors for absorption of propane, ethene and hydrogen in the aqueous slurries were 3.6, 3.3 and 2.0, respectively. It was shown that for our results and those reported in the literature the maximum enhancement factor, Emax, decreases with increasing liquid side gas-liquid mass transfer coefficient, kl. From all the experimental data the following relationship is found: although different types of activated carbon particles and differences in sizes were used by the various research groups. To describe these results a simple theory is presented.
Air entrainment by plunging jets and the consequent production of bubbles play an increasing role in industrial and environmental applications. Generally, air entrainment can be seen as the consequence of two complementary mechanisms: (1) the interfacial shear along the liquid jet interface which drags down an air boundary layer and (2) the air entrapment process at the point of impact of the plunging jet with the receiving pool. The latter process is usually dominated by growing interfacial disturbances travelling on the jet. The great variety of these disturbances is represented by the concept of dynamic interfacial roughness of which a definition is provided and justified. To measure this quantity, an easy-to-use optical technique is designed. The dynamic interfacial roughness and the ability to entrap air by a plunging jet are found to depend on the hydrodynamical noise applied to the jet. The optical technique is presented as a very simple tool to optimally devise the internal design of nozzles and then, to control air entrainment.
Trajectories are measured and compared with computed trajectories of solid particles with a diameter of 1–2 mm in downward gas flow near a solid cylinder with a diameter, dc, of 25 mm. The Reynolds number based on dc has been varied from 3000–13,000. The particle Reynolds numbers, based on the relative velocity |UG − Up|, ranged from 0 to 2000. Of all forces other than gravity, drag is dominant, although the pressure gradient and added mass forces for Rec > 10,000 have the same order of magnitude. The Basset force can be neglected. The correlation , originally derived by Sridhar and Katz (1995) for the lift force coefficient of bubbles, has been found to be appropriate for freely moving solid particles with shear number less than 0.04.
The force and torque acting on an accelerating rigid body of arbitrary shape, moving at low Reynolds number through a fluid at rest in infinity, are considered. The expressions found for the force due to pure translation and the torque due to pure rotation of the body each include three tensors which relate the acceleration and velocity to the force and the torque. In the case of combined translation and rotation, three “coupling tensors” are added to each of the above expressions. These expressions are extended for the case of a particle, immersed in a quiet fluid and acted upon by an impulse. Generalized Faxen's theorems are derived for non-steady flows which do not vanish in infinity. Finally, the effect of non-zero initial velocity of the fluid and the body is considered. The stop distance is shown to depend linearly on the initial velocity of the body through a displacement tensor which consists of the traditional quasi-stationary term and an additional tensor. This additional tensor depends on the geometry of the body and on the initial velocity field of the fluid. It is infinite if the kinetic energy of the initial field is infinite. Likewise, the expression for the force acting on the body contains an additional term which depends on time, on the geometry of the problem and on the initial velocity field.
We have performed a direct numerical simulation of dilute turbulent
particulate flow in a vertical plane channel, fully resolving the phase
interfaces. The flow conditions are the same as those in the main case of
"Uhlmann, M., Phys. Fluids, vol. 20, 2008, 053305", with the exception of the
computational domain length which has been doubled in the present study. The
statistics of flow and particle motion are not significantly altered by the
elongation of the domain. The large-scale columnar-like structures which had
previously been identified do persist and they are still only marginally
decorrelated in the prolonged domain. Voronoi analysis of the spatial particle
distribution shows that the state of the dispersed phase can be characterized
as slightly more ordered than random tending towards a homogeneous spatial
distribution. It is also found that the p.d.f.'s of Lagrangian particle
accelerations for wall-normal and spanwise directions follow a lognormal
distribution as observed in previous experiments of homogeneous flows. The
streamwise component deviates from this law presenting significant skewness.
Finally, a statistical analysis of the flow in the near field around the
particles reveals that particle wakes present two regions, a near wake where
the velocity deficit decays as 1/x and a far wake with a decay of approximately
The instability-governed entrainment rate of the lower surface of a subcooled water column accelerated upwards by an expanding steam mass is measured. It is found that the entrainment rate is approximately proportional to the fourth root of the acceleration. This would be the case if the characteristic length scale in the late stages of Taylor instability were governed by linear instability theory. In addition to the linear displacement measurements, the steam pressure in the lower driver section was monitored as a function of time. Estimates of the concentration, radius and age distribution of the entrained droplet population were made by modeling the bubble-and-spike breakup into discrete droplets. This allows the steam condensation rate, and hence the steam pressure, at each instant of time to be computed. This is compared with the observed steam pressure history. Reasonable agreement is found. One can thus estimate the reduction in work potential in the case of a steam explosion in the lower plenum of a pressurized-water nuclear reactor.
The mechanisms of acceleration-induced breakup of liquid drops are reviewed briefly. Data on acceleration-induced fragmentation of liquid drops have been collected from the literature and are presented on a consistent basis. Included are critical Weber number data, breakup time data, velocity history data and fragment size data. A triangular relationship based on the concept of a critical Weber number, breakup time data and velocity history data is presented which permits prediction of the maximum size of stable fragments.
The paper concerns the effect of particle inertia on acceleration statistics. A simple analytical model for predicting the acceleration of heavy particles suspended in an isotropic homogeneous turbulent flow field is developed. This model is capable of describing the influence of both Stokes and Reynolds numbers on the particle acceleration variance. Comparisons of model predictions with numerical simulations are presented.
The conditions allowing particle suspension in turbulent flow are of interest
in many applications, but understanding them is complicated both by the nature
of turbulence and by the interaction of flow with particles. Observations on
small particles indicate an invariance of acceleration PDFs of small particles
independent of size. We show to be true the postulated role of particle/fluid
interaction forces in maintaining suspension. The 3D-PTV method, applied for
two particle phases (tracers and inertial particles) simultaneously, was used
to obtain velocity and acceleration data, and through the use of the particle's
equation of motion the magnitude of forces representing either the flow or the
particle interaction were derived and compared. The invariance of PDFs is shown
to extend to the component forces, and lift forces are shown to be significant.
In this communication we propose a correlation for the drag coefficient on a liquid drop suddenly exposed to a high-speed airstream. The correlation is a bi-power law in the Ohnesorge and Weber numbers. The correlation predicts the acceleration of the drop in terms of known quantities.
A numerical formulation for Eulerian–Lagrangian simulations of particle-laden flows in complex geometries is developed. The formulation accounts for the finite-size of the dispersed phase. Similar to the commonly used point-particle formulation, the dispersed particles are treated as point-sources, and the forces acting on the particles are modeled through drag and lift correlations. In addition to the inter-phase momentum exchange, the presence of particles affects the fluid phase continuity and momentum equations through the displaced fluid volume. Three flow configurations are considered in order to study the effect of finite particle size on the overall flowfield: (a) gravitational settling, (b) fluidization by a gaseous jet, and (c) fluidization by lift in a channel. The finite-size formulation is compared to point-particle representations, which do not account for the effect of finite-size. It is shown that the fluid displaced by the particles plays an important role in predicting the correct behavior of particle motion. The results suggest that the standard point-particle approach should be modified to account for finite particle size, in simulations of particle-laden flows.
A mathematical model for dilute bubble plumes is derived from the two-fluid model equations. This is coupled to a mass transfer model to get a closed CFD formulation. The mass transfer equations used are the same as those implemented in the 1D model proposed, so as to get a CFD formulation and a 1D integral formulation that are fully consistent. In fact, the 1D model can be rigorously derived from the CFD one. The mathematical derivation is detailed pointing out the approximations involved.Results of both models for typical conditions of isolated aeration plumes in deep wastewater reservoirs are presented. Good agreement is reported between them, emphasizing on the most relevant variables such as gas dissolution rates, gas holdup, liquid’s velocity and bubbles’ radius. Furthermore, entrainment rates evaluated from the CFD results are shown to lie within the experimental range. Finally, CFD-based assessment of the approximations involved in the 1D model proves them to hold within a few percents of relative accuracy. A solid basis for applying CFD models to aeration plumes, as natural extensions of the popular integral models, emerges from the investigation.
Several rheological constitutive equations for the modeling of dense suspensions in nonlinear shear flows have been developed over the last three decades. Although these models have been able to predict the correct steady-state solid-phase concentration profile, none have been able to follow the transient experimentally measured concentration profile over a range of suspended particle radii with a consistent set of diffusion coefficients. In this research, two improvements are made to the diffusive-flux model, namely, modeling the diffusion coefficients as linear functions of the so-called nonlinearity parameter and adding slip boundary conditions at the wall. A particle-level explanation for the linear dependence of the diffusion coefficients on the nonlinearity parameter is provided. With these two improvements, it is shown that the modified diffusive flux model can accurately predict the transient solid-phase concentration profile in a Couette device over a wide range of particle radii.
The measurement accuracy of a mono-fiber optical probe is studied experimentally using isolated bubbles rising freely in a still liquid. The dwell time of the probe tip within the gas phase, which is obtained from both the optical probe signal and high-speed visualization, is compared with the value expected for a non-perturbed bubble. The difference originates mainly from the intrusive nature of the optical probe, which modifies the bubble behavior when it comes into contact with the probe tip. This interaction increases the dwell time if the bubble is pierced by the probe near its pole, and shortens it for piercing near the equator. The mean dwell time, obtained by averaging for various piercing locations, is shortened and the local void fraction indicated by the probe is thus underestimated. It is shown that the void fraction error can be correlated with a modified Weber number, and this correlation is helpful for sensor selection and for uncertainty estimate. In addition, the distribution of gas dwell time usually differs from the response expected for an ideal probe. This deviation results from the dependence of the dwell time error on the piercing location. The dwell time distribution can be used to infer the dependence of the dwell time on the piercing location. Finally, the deformation of long fibers during the bubble-probe interaction significantly increases the measurement error. Observed results are consistent with data of Andreotti (2009), which were measured in an airlift flow, suggesting that present results are applicable also to the case of moving liquid. Conclusions of this study could be applied also to conductivity probes or more generally to the interaction of a bubble with any kind of thin, intrusive sensor or fiber.
The Lagrangian–Eulerian (LE) approach is used in many computational methods to simulate two-way coupled dispersed two-phase flows. These include averaged equation solvers, as well as direct numerical simulations (DNS) and large-eddy simulations (LES) that approximate the dispersed-phase particles (or droplets or bubbles) as point sources. Accurate calculation of the interphase momentum transfer term in LE simulations is crucial for predicting qualitatively correct physical behavior, as well as for quantitative comparison with experiments. Numerical error in the interphase momentum transfer calculation arises from both forward interpolation/approximation of fluid velocity at grid nodes to particle locations, and from backward estimation of the interphase momentum transfer term at particle locations to grid nodes. A novel test that admits an analytical form for the interphase momentum transfer term is devised to test the accuracy of the following numerical schemes: (1) fourth-order Lagrange Polynomial Interpolation (LPI-4), (3) Piecewise Cubic Approximation (PCA), (3) second-order Lagrange Polynomial Interpolation (LPI-2) which is basically linear interpolation, and (4) a Two-Stage Estimation algorithm (TSE). A number of tests are performed to systematically characterize the effects of varying the particle velocity variance, the distribution of particle positions, and fluid velocity field spectrum on estimation of the mean interphase momentum transfer term. Numerical error resulting from backward estimation is decomposed into statistical and deterministic (bias and discretization) components, and their convergence with number of particles and grid resolution is characterized. It is found that when the interphase momentum transfer is computed using values for these numerical parameters typically encountered in the literature, it can incur errors as high as 80% for the LPI-4 scheme, whereas TSE incurs a maximum error of 20%. The tests reveal that using multiple independent simulations and higher number of particles per cell are required for accurate estimation using current algorithms. The study motivates further testing of LE numerical methods, and the development of better algorithms for computing interphase transfer terms.
This paper aims at investigating the detailed structure of turbulent non-reacting dilute spray flows using advanced laser diagnostics. A simple spray jet nozzle is designed to produce a two-phase slender shear flow in a co-flowing air stream with well-defined boundary conditions. The carrier flow is made intentionally simple and easy to model so that the focus can be placed on the important aspects of droplet dispersion and evaporation, as well as turbulence–droplet interactions. Phase Doppler interferometry is employed to record droplet quantities, while planar laser-induced fluorescence imaging is applied separately to obtain acetone vapour data. Measurements are conducted for four acetone spray jets in air at several axial stations starting from the nozzle exit. The combined liquid and vapour mass fluxes of acetone integrated across the jet at downstream locations agree satisfactorily with the total mass flow rate of acetone injected.
Acoustic wave propagation in a monodisperse suspension of varying solids concentration was modeled exactly for wavelengths much larger than the particle size, including unsteady viscous effects, for the situation where particle interactions are predominantly inviscid. Inviscid particle interactions are addressed in terms of the added mass coefficient, which is sensitive to the solids concentration, the direction of insonification and the anisotropy of the particle arrangement. The sound speed and attenuation were calculated and compared to experimental results for a wide range of ka, where k is the wavenumber (=2Π/λ) and a is the particle radius. The attenuation, which is a strong function of ka, is seen to have non-monotonic behavior with respect to the solids fraction at low frequencies and it becomes monotonic at high frequencies. In general, the effect of ka on sound speed is seen to be small in comparison. The comparison with experiments shows that at values of ka near 1, effects associated with multiple scattering begin to affect acoustic propagation sufficiently to cause marked deviation between the present theory and measurements.
Sound-assisted fluidization of nonfluent 0.5–45 μm catalyst particles has been studied with a 145 mm i.d. column. Different amounts of solids of weight W ranging from 1 to 3 kg have been charged in the column. A loudspeaker generated an acoustic field, above the bed, with a sound pressure level SPL (referred to 20 μPa) varying from 110 to 140 dB and a frequency f varying from 30 to 1000 Hz. The improvement of the quality of fluidization obtained with certain combinations of W, SPL and f has been attributed to the breakup of clusters originally forming the bed into subclusters. For given W and SPL, the ranges of frequency within which channel-free homogeneous fluidization could be obtained have been determined, and within these ranges the kinds of curves for sizes of subclusters ds as a function of the frequency have been outlined. The nonmonotonic form of these curves could not be explained by means of the original sound-assisted fluidization model, which assumes a rigid cluster-subcluster structure. The existence of elastic forces between clusters and subclusters, assumed by the cluster/subcluster oscillators model, yields theoretical ds versus f curves with the same trend as those from experiments.
An equation set for multidimensional, time variant, inviscid flow of a condensing vapour is presented. The equations include the effects of relative motion between the primary gas phase and the suspended liquid droplets. They have been formulated with steam turbine applications in mind but are also relevant to problems of gas-particle and liquid bubble flow.It is shown that the critical velocity in one dimensional choking of low pressure wet steam is identical with the “frozen” speed of acoustic propagation, and the variation of choking mass flow with respect to equilibrium based calculations is described. Results obtained with two different models of droplet growth are compared, and simple formulae for calculating limiting values of choking flow are given. A generalised loss coefficient including the effects of thermodynamic and kinematic non-equilibrium is introduced.
The behaviour of drops in an acoustic levitator is simulated numerically. The ultrasound field is directed along the axis of gravity, the motion of the drop is supposed to be axisymmetric.The flow inside the drop is assumed inviscid (since the time intervals considered are short) and incompressible.First, as a test case, we consider a stationary ultrasound wave. We observe, as in previous experimental and theoretical works, that stable drop equilibrium shapes exist for acoustic Bond numbers up to a critical value. The critical value depends on the dimensionless wave number of the ultrasound. Beyond the critical value, we still observe equilibrium drop shapes, but they are not purely convex (i.e. “dog-bone” shaped) and found to be unstable.Next we modulate the ultrasound pressure level (SPL) with a frequency ω2, which is comparable to the first few drop resonance frequencies, and a small modulation amplitude. Simulations and experiments are performed and compared; the agreement is very good. We further on investigate numerically the more general case of an arbitrary ω2 (still comparable to the first few drop resonance frequencies, yet). A very rich drop dynamics is obtained. We observe that a resonant drop break-up can be triggered by an appropriate choice of the modulation frequency. The drop then disintegrates although the acoustic Bond number remains below its critical value.Finally we change the modulation frequency linearly with time, sweeping over a window containing the drop's first eigenfrequency ω2(res). After ω2 has crossed ω2(res), in the range of validity of the inviscid approximation, the drop equatorial radius oscillates between well-defined saturation values. For small modulations the range of oscillation grows linearly with the modulation amplitude. For larger modulations, however, a substantial increase in the oscillation range of the drop equatorial radius is observed in the case of down-sweep; the increase does not occur in up-sweeps of the modulation frequency. We compare our results with experimental findings and in particular the so-called jump phenomenon, as well as with experimental and numerical results from the literature.
This work centers on the transport processes in turbulent liquid films undergoing heating or surface evaporation. A new semi-empirical turbulence model is presented. It consists of a single continuous eddy-viscosity profile which spans the entire liquid layer. Numerical calculations reveal good agreement with experimental data for freely-falling films. It was found that it is impossible to obtain universal correlations for different fluids in terms of Reynolds and Prandtl numbers alone since the heat transfer data display strong dependence on the Kapitza number below Re = 10,000. The Kapitza number accounts for the effects of surface tension and viscosity on the turbulence structure near the free surface of the film. Turbulent-film correlations similar to those used in conventional internal or external flows are recommended for higher Reynolds numbers.
A new model to calculate the two-phase pressure drop across a sudden contraction in a duct area was developed and checked against data recently obtained with mixtures of air and water, aqueous glycerol, watery calcium nitrate and with the freon 12. In addition, data available from the open literature are considered. In the model all relevant physical parameters are included and, in contrast to correlations published so far, the entrainment of liquid in the gas stream is taken into consideration. The predictions are validated for a wide range of conditions, pipe diameters and physical properties typically encountered in industrial piping systems. Calculations based on the new approach are sufficiently accurate for engineering purposes.
Flow patterns, void fraction and friction pressure drop measurements were made for an adiabatic, vertical up-and-down, two-phase flow of air–water mixtures across a horizontal in-line, 5×20 tube bundle with a pitch-to-diameter ratio of 1.28. The flow patterns in the cross-flow zones were obtained and flow pattern maps were constructed. The data of average void fraction were less than the values predicted by a homogenous flow model and showed a strong mass velocity effect, but were well-correlated in terms of the Martinelli parameter Xtt and liquid-only Froude number FrLO. The two-phase friction multiplier data could be well-correlated with the Martinelli parameter.
The vertical two-phase cross-flow was studied on the shell side of a horizontal tube bundle. The section used consists of 5 tubes with 20 mm dia. The tubes are arranged in 10 rows on a square pitch with a pitch ratio of 1.5 in a rectangular shell. Adiabatic flows of air water mixtures were tested in a large superficial velocity range, for liquid (0.001–0.65 m/s) and for gas (0.047–9.3 m/s), respectively. Flow patterns were established and visual observation together with photographic data and a video-film were used. These results, existing flow pattern data and flow pattern maps for the cross-flow in tube bundles were employed to work out a general flow pattern map.
The effect of surface tension on adiabatic two-phase flow across a bank of 100 μm diameter staggered circular micro pillars, 100 μm long with pitch-to-diameter ratio of 1.5, for Reynolds number between 5 and 50, was investigated. Experiments with ethanol were performed and compared to results with water. Flow maps revealed similar flow patterns, but the transition lines were different for the two liquids. Void fraction measurements of the two fluids were also compared, and no significant deviations were observed. The two-phase pressure drop characteristics were significantly affected by the reduction in surface tension. Interfacial friction was attributed to this deviation, and a two-fluid model was developed to account for surface tension force. In addition, a modified form of Chisholm correlation was developed that accounts for surface tension.
This paper presents an analytical and numerical approach to describe the capillary infusion of a liquid across an array of parallel micro-cylinders. Based on a series of simplifying assumptions, the model proposes a method to average the varying capillary pressure and introduces a technique to assess the inhibiting effects of the gas entrapped between the micro-cylinders as the liquid radially ingresses into the micro-cylinder array. The proposed averaging scheme of the capillary pressure is an improvement over previous analytical approaches, as it accounts for the physics of wetting at the micro-scale. The equations are non-dimensionalized and the role of various parameters such as gas entrapment and average capillary pressure is explored. The inclusion of the gas displacement phenomenon requires only one parameter to be determined empirically, in order to describe the impregnation behavior of the liquid. Two limiting cases when all gas is allowed to escape through the impregnating liquid and when all the gas is entrapped within are identified. This study should prove useful in understanding the role of entrained gas and capillarity on impregnation mechanics in micro-scale porous media.
This paper presents an experimental approach to study the capillary impregnation of a liquid across an array of parallel micro-cylinders. The technique presented successfully validates the theoretical findings of part I, and provides a methodology to calculate the fill time and the overall capillary impregnation dynamics for any arbitrary cylindrical sample made of aligned micro-cylinders, by taking into account the role of the trapped gas, which opposes and slows down the inward capillary flow.
We present different simulations of primary atomization using an adaptive Volume-of-Fluid method based on octree meshes. The use of accurate numerical schemes for mesh adaptation, Volume-of-Fluid advection and balanced force surface tension calculation implemented in Gerris, the code used to perform the simulations included in this work, has made possible to carry out accurate simulations with characteristic scales spreading over several orders of magnitude. The code is validated by comparisons with the temporal linear theory for moderate density and viscosity ratios, which basically corresponds to atomization processes in high pressure chambers. In order to show the potential of the code in different scenarios related to atomization, preliminary results are shown in relation with the study of the two-dimensional and 3D temporal and spatial problem, the influence of the injector and the vortex generated inside the chamber, and the effect of swirling at high Reynolds numbers.
Improved numerical methods and physical models have been applied to droplet collision modeling. Numerically, an adaptive collision mesh method is developed to calculate collision rate. This method produces a collision mesh that is independent of the gas phase mesh and adaptively refined according to local parcel number density. An existing model describing the satellite droplet formation during the collision process is improved to reflect the experimental findings that the satellite droplets are much smaller than the parent droplets. The adaptive collision mesh and the improved satellite model have been used to simulate three impinging spray experiments. The model was able to qualitatively predict the occurrence of small satellite drops and bi-modal post-collision drop size distributions. The effect of the collision mesh and the satellite droplet model on a high-speed non-evaporating diesel spray is also assessed.