Maximum Stable Size Research Articles

Structure and solvation of confined water and water-ethanol clusters within microporous Brønsted acids and their effects on ethanol dehydration catalysis.

Aqueous-phase reactions within microporous Brønsted acids occur at active centers comprised of water-reactant-clustered hydronium ions, solvated within extended hydrogen-bonded water networks that tend to stabilize reactive intermediates and transition states differently. The effects of these diverse clustered and networked structures were disentangled here by measuring turnover rates of gas-phase ethanol dehydration to diethyl ether (DEE) on H-form zeolites as water pressure was increased to the point of intrapore condensation, causing protons to become solvated in larger clusters that subsequently become solvated by extended hydrogen-bonded water networks, according to in situ IR spectra. Measured first-order rate constants in ethanol quantify the stability of SN2 transition states that eliminate DEE relative to (C2H5OH)(H+)(H2O) n clusters of increasing molecularity, whose structures were respectively determined using metadynamics and ab initio molecular dynamics simulations. At low water pressures (2-10 kPa H2O), rate inhibition by water (-1 reaction order) reflects the need to displace one water by ethanol in the cluster en route to the DEE-formation transition state, which resides at the periphery of water-ethanol clusters. At higher water pressures (10-75 kPa H2O), water-ethanol clusters reach their maximum stable size ((C2H5OH)(H+)(H2O)4-5), and water begins to form extended hydrogen-bonded networks; concomitantly, rate inhibition by water (up to -3 reaction order) becomes stronger than expected from the molecularity of the reaction, reflecting the more extensive disruption of hydrogen bonds at DEE-formation transition states that contain an additional solvated non-polar ethyl group compared to the relevant reactant cluster, as described by non-ideal thermodynamic formalisms of reaction rates. Microporous voids of different hydrophilic binding site density (Beta; varying H+ and Si-OH density) and different size and shape (Beta, MFI, TON, CHA, AEI, FAU), influence the relative extents to which intermediates and transition states disrupt their confined water networks, which manifest as different kinetic orders of inhibition at high water pressures. The confinement of water within sub-nanometer spaces influences the structures and dynamics of the complexes and extended networks formed, and in turn their ability to accommodate the evolution in polarity and hydrogen-bonding capacity as reactive intermediates become transition states in Brønsted acid-catalyzed reactions.

Droplet size prediction model based on the upper limit log-normal distribution function in venturi scrubber

Abstract Droplet size and distribution are important parameters determining venturi scrubber performance. In this paper, we proposed physical models for a maximum stable droplet size prediction and upper limit log-normal (ULLN) distribution parameters. For the proposed maximum stable droplet size prediction model, a Eulerian-Lagrangian framework and a Reitz-Diwakar breakup model are solved simultaneously using CFD calculations to reflect the effect of multistage breakup and droplet acceleration. Then, two ULLN distribution parameters are suggested through best fitting the previously published experimental data. Results show that the proposed approach provides better predictions of maximum stable droplet diameter and Sauter mean diameter compared to existing simple empirical correlations including Boll, Nukiyama and Tanasawa. For more practical purpose, we developed the simple, one dimensional (1-D) calculation of Sauter mean diameter.

Droplets behavior of subcooled dispersed phase under nucleate boiling of continuous phase of liquid emulsion

The paper considers the influence of the dispersed phase of the emulsion upon the nature of heat and mass transfer in systems of immiscible liquids. In particular, is studied the interfacial interaction and breakage of subcooled drops of the dispersed phase during bubble boiling of the continuous phase of the emulsion. They predetermine the complex nature of the joint behavior of hydrodynamic and thermal processes. It is known that the controlled conditions for dispersion of emulsions in MQCL method under the minimum amount of cooling lubrication be able to improve the quality of metal processing by 10–18% (Maruda et al., 2017). Phase changes liquid generate in it a special kind of pulsation motion: «hot» turbulence of the superheated continuous phase. She is generated by process formation of nuclei and growth of steam bubbles accompanying by the pressure pulsations. Thus heat energy is converted into mechanical energy of the perturbed by the movement of vapor bubbles in evaporating liquid. The formula for the evaluating perturbations of the continuous phase of the emulsion under the influence of vapor bubbles is proposed. The behavior of these fluctuations depends on the frequency of the viable vapor phase nuclei formation related to the heat flow density. The dependence, obtained by Labuntsov (2000) for boiling homogeneous liquid, is used as a characteristic of the thermal velocity.Taking into account the character of the boiling of the continuous phase of emulsion, the generalization of the Kolmogorov-Hinze model has been performed for breakage drops in isothermal flow of emulsion under the influence of turbulent fluctuations of velocity. A correlation model of “thermal” breakup is proposed, which is complementary to the physical representations of the breakup for subcooled drops during the liquid emulsion boiling. Using dimensionless criteria was justified the form of the correlation for the maximum stable droplet size and thermal velocity fluctuations. This makes it possible to relate mechanisms hydrodynamic and thermal processes at use of usual computer tools used in practice for engineering calculations. The correlation for the maximum stable droplet size and thermal velocity fluctuations was developed in a dimensionless form. The results of the model studies were evaluated using the experimental data on the breakup of drops of oil in coolant “oil-water” (Januszkiewicz et al., 2004). They were obtained under conditions of creeping flow in a narrow channel, eliminating the creation of ordinary hydrodynamic turbulence. A good agreement with experimental data indicates that the present model would be promising for the extension of model representations about the breakup of droplets in complex processes of heat and mass transfer in liquid emulsions.

Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models

Abstract. The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work, we investigate various implicit–explicit (IMEX) additive Runge–Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. The IMEX formulations considered include horizontally explicit – vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case, the impact of solving nonlinear systems in each implicit ARK stage in a linearly implicit fashion is also explored.The accuracy and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. HEVI splittings that treat some vertical dynamics explicitly do not show a benefit in solution quality or run time over the most implicit HEVI formulation. While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a globally coupled system. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy for some methods. Overall, the third-order ARS343 and ARK324 methods performed the best, followed by the second-order ARS232 and ARK232 methods.

Structural instability of large-scale functional networks.

We study how large functional networks can grow stably under possible cascading overload failures and evaluated the maximum stable network size above which even a small-scale failure would cause a fatal breakdown of the network. Employing a model of cascading failures induced by temporally fluctuating loads, the maximum stable size nmax has been calculated as a function of the load reduction parameter r that characterizes how quickly the total load is reduced during the cascade. If we reduce the total load sufficiently fast (r ≥ rc), the network can grow infinitely. Otherwise, nmax is finite and increases with r. For a fixed r(< rc), nmax for a scale-free network is larger than that for an exponential network with the same average degree. We also discuss how one detects and avoids the crisis of a fatal breakdown of the network from the relation between the sizes of the initial network and the largest component after an ordinarily occurring cascading failure.

Development of a unified oil droplet size distribution model with application to surface breaking waves and subsea blowout releases considering dispersant effects

An oil droplet size model was developed for a variety of turbulent conditions based on non-dimensional analysis of disruptive and restorative forces, which is applicable to oil droplet formation under both surface breaking-wave and subsurface-blowout conditions, with or without dispersant application. This new model was calibrated and successfully validated with droplet size data obtained from controlled laboratory studies of dispersant-treated and non-treated oil in subsea dispersant tank tests and field surveys, including the Deep Spill experimental release and the Deepwater Horizon blowout oil spill. This model is an advancement over prior models, as it explicitly addresses the effects of the dispersed phase viscosity, resulting from dispersant application and constrains the maximum stable droplet size based on Rayleigh-Taylor instability that is invoked for a release from a large aperture.

High‐pressure homogenization induced degradation of amylopectin in a gelatinized state

An investigation of the degradation mechanism of a highly branched polymer during a high‐pressure homogenization (HPH) process was performed using gelatinized waxy maize starch as a substrate. The degradation mechanism was studied using high‐performance size‐exclusion chromatography equipped with multi‐angle laser‐light scattering and refractive index detectors. The HPH induced an exponential molecular degradation trend that was a function of HPH pressure/cycles. The results revealed that HPH preferentially disrupted larger molecules and caused their size distribution (or molecular weight) to a narrow range, resulting in a maximum stable size (or molecular weight). A final z‐average radius of gyration of about 20 nm for amylopectin was observed after HPH for eight cycles at 100 MPa or two cycles at 150 MPa, which was due to the midpoint scission mechanism during the HPH treatment. The increase in molecular density after HPH verified the breakdown of the α‐1,4‐glycosidic bonds among the clusters of (midpoint scission) amylopectin.

Experimental Visualization of Cryogenic Backflow Vortex Cavitation with Thermodynamic Effects

The world’s first test facility that allows the visualization of cavitation on a rotating inducer in both cryogen and water was used for comparing the cavitation features in liquid nitrogen at 77.9 K and water at 292.5 and 333.5 K. The test inducer was a triple-threaded helical one with a diameter of 65.3 mm and a rotational speed range of 3500–6000 rpm. The backflow vortex cavitation on the rotating inducer was quantitatively measured in the considered fluids. From the results, it was inferred that the backflow vortex cavitation orbital rate and its orbital diameter around the inducer axis depend on the head coefficient but are almost independent of the cavitation number and kind of fluid. Furthermore, the diameter of each backflow vortex cavitation column depends on the head coefficient, cavitation number, and kind of fluid. At the same head coefficient and cavitation number, the diameter in liquid nitrogen is 2.4 times smaller than that in water because of the thermodynamic effects. Moreover, the appearance of the cavitation is “foggy” in nitrogen and “foamy” in water. This difference can be explained by the maximum stable size of cavitation bubbles derived from the critical Weber number theory. Based on this theory, each cavitation bubble at the tip vortex regions in nitrogen is four times smaller than that in water.

Non-linear plasma wake growth of electron holes

An object's wake in a plasma with small Debye length that drifts across the magnetic field is subject to electrostatic electron instabilities. Such situations include, for example, the moon in the solar wind and probes in magnetized laboratory plasmas. The instability drive mechanism can equivalently be considered drift down the potential-energy gradient or drift up the density-gradient. The gradients arise because the plasma wake has a region of depressed density and electrostatic potential into which ions are attracted along the field. The non-linear consequences of the instability are analysed in this paper. At physical ratios of electron to ion mass, neither linear nor quasilinear treatment can explain the observation of large-amplitude perturbations that disrupt the ion streams well before they become ion-ion unstable. We show here, however, that electron holes, once formed, continue to grow, driven by the drift mechanism, and if they remain in the wake may reach a maximum non-linearly stable size, beyond which their uncontrolled growth disrupts the ions. The hole growth calculations provide a quantitative prediction of hole profile and size evolution. Hole growth appears to explain the observations of recent particle-in-cell simulations.

Experimental determination of maximum effective hydrodynamic stress in multiphase flow using shear sensitive aggregates

Maximum effective hydrodynamic stress, , responsible for the breakup of aggregates with size comparable to Kolmogorov eddies, was experimentally determined in an aerated stirred tank. The proposed method is based on the measurement of the maximum stable aggregates size consisting of poly(methyl methacrylate) nanoparticles. The fractal aggregates were broken under various operating conditions in an aerated stirred tank and calibrated with known flow conditions using contracting nozzles to convert the measured aggregate sizes into hydrodynamic stress. It was found that can vary substantially among studied conditions and its magnitude depends on the controlling mechanism including gas jet during bubble formation, bubble rise, bubble burst at the gas–liquid interface or the turbulence generated by the impeller. The measured values are in good agreement with literature data which supports the applicability of this method to characterize the maximum effective hydrodynamic stress in complicated multiphase flow. © 2015 American Institute of Chemical Engineers AIChE J, 61: 1735–1744, 2015

Unified maximum particle size prediction for turbulent dilute dispersions

Abstract A new maximum stable particle size model for turbulent dilute dispersion flow is proposed, which includes both droplets and bubbles. The proposed model incorporates, for the first time, the combined effects of the dispersed-phase density and viscosity. The model has been tested against a database consisting of 169 sets corresponding to turbulent dilute dispersions showing a good agreement.

Determination of maximum turbulent energy dissipation rate generated by a rushton impeller through large eddy simulation

Large eddy simulation (LES) of a stirred tank equipped with a Rushton impeller and four cylindrical baffles was used to characterize the flow pattern and to assess the maximum turbulent kinetic energy dissipation rate εmax. While the shorter baffle‐impeller distance significantly affects the radial velocity profile and the trailing vortices expansion, the flow field in the impeller vicinity is comparable to that of a standard setup with rectangular baffles connected to the wall. The phase‐resolved profile of εmax indicates its very strong variation from 10 · N3 D2 to 130 ± 13 · N3 D2. When using peak values of the corresponding hydrodynamic stress , the maximum stable aggregate size measured in the same stirred tank closely correlates with breakage data obtained under laminar conditions using the same initial aggregates. This indicates that the same mechanism was involved in the aggregate breakup under both conditions, allowing us to predict aggregates breakup under various conditions. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3642–3658, 2013

Formation of oil droplets in plasticized starch matrix in simple shear flow

This paper describes the effect of simple shear flow on the formation of triglyceride oil droplets in a plasticized starch matrix. An in-house developed shearing device was used that enabled the application of controlled shear flow and rheological characterization of the native maize starch–triglyceride blends at shear stresses of up to 37kPa. Due to the high viscosity of starch matrix, the viscosity ratio of the continuous starch phase and the dispersed triglyceride phase varied between 10−7 and 10−5. It was possible to create small droplets with a droplet diameter of 2.1μm using simple shear flow only. An increase in shear rate had no influence on droplet diameter. However, an increase in oil content led to a vast increase in droplet diameter indicating the occurrence of coalescence. The results further show that the maximum stable droplet size in plasticized starch is significantly smaller (up to 100 times) than the predicted values for a Newtonian matrix. The differences of plasticized starch to Newtonian matrices are discussed in detail.

Reply to the comment by Mike R. James et al. on “It takes three to tango: 2. Bubble dynamics in basaltic volcanoes and ramifications for modeling normal Strombolian activity”

[1] The current paradigm for normal eruptions at Stromboli volcano and Strombolian‐type activity more generally posits that each eruption represents the burst of a large pocket of gas, commonly referred to as slug, at the free surface of themagma column. This slug model has been investigated and refined primarily through analog fluid dynamical experiments at the laboratory scale. There is no doubt that these studies have advanced our understanding of Strombolian eruptions considerably. However, given the very fundamental status of the slug model for our current thinking about Strombolian‐type eruptions, it is paramount to carefully assess all underlying assumptions of the model. One important uncertainty lies in the scaling behavior of the observed slug dynamics in the laboratory. Scale invariance requires all nondimensional numbers to be identical, but it is generally not feasible to match all of the nondimensional numbers characterizing the volcanic conduit exactly in a laboratory experiment. Numerical computations offer a means of directly comparing slug dynamics at drastically different scales [Suckale et al., 2010b]. We find that the very large slugs in volcanic conduits are more prone to dynamic instabilities and breakup than the comparatively small slugs in laboratory settings. This finding in itself does of course not ‘disprove’ the slug model: it merely points to potentially important differences in slug stability at volcanic scales as opposed to laboratory scales. [2] Being careful and critical of the errors and interpretations of numerical simulations is important; similarly, the appropriateness of laboratory analogs must also be examined. James et al. [2011] raise a valid point asking whether our simulations are capable of reproducing stable slug rise in water. Figure 1 shows a computation performed for the experimental parameters provided by James et al. [2011]. In agreement with the analog experiments, we do indeed find that this slug rises stably in water, i.e., the slug ascends buoyantly within the conduit without experiencing ‘catastrophic’ breakup as defined by Suckale et al. [2010b]. Thus the experiments by James et al. [2011] do not appear to contradict our computations. We also reproduce computationally stable slug flow at finite Reynolds number (Re) shown in the experiment specified by Jaupart and Vergniolle [1989] [Suckale et al., 2010b]. It is worth noting that in both cases, the set of nondimensional numbers characterizing the experiment are not identical to those thought to be representative of the volcanic conduit and thus potential differences in the fluid dynamical behavior are not unexpected. [3] That we observe stable slugs in water but not in magma supports an argument raised by James et al. [2011], namely that Re is an insufficient parameter for comparing slug dynamics at different spatial scales. We welcome their discussion of the drawbacks of using a single Re to describe slugs. Focusing on Re to characterize our computations was a purely pragmatic decision: First, Re is commonly used for the purpose of comparing fluid dynamical computations at different scales. Second, we were able to use estimates for Re characterizing Strombolian slugs based on observational data measured at Stromboli [Vergniolle and Brandeis, 1996]. Third, we observed a correlation between Re and breakup within the regime we investigated, which is not unexpected since Re is an indirect measure of the size of a gas bubble or slug. That being said, a more comprehensive set of nondimensional numbers is undoubtedly needed to evaluate scaling behavior of slugs for different fluid dynamical regimes, including nondimensional criteria that capture the wavelength of potential interface instabilities in relation to the curvature of the interface and the time scale associated with instability growth to the time scale associated with advection of the instability along the interface. Our study is a first step toward gaining a better understanding of slug stability. [4] From a theoretical point of view, very large and dynamic gas volumes are not expected to be indefinitely stable. The light gas slug moves underneath a column of very heavy magma and this unstable density stratification is prone to the formation of Rayleigh‐Taylor instabilities at the essentially flat upper surface of the slug. Grace et al. [1978] developed a semiempirical rationalization of this process that affords rough estimates of maximum stable sizes as listed in Table 1 of Suckale et al. [2010b]. The appeal of this model lies primarily in its simplicity, but this very simplicity also Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada.

Mechanism of Degradation of Starch, a Highly Branched Polymer, during Extrusion

An investigation of the mechanisms of degradation of a branched polymer in extrusion was performed using starch as substrate. Starch has the advantage that the distribution of degree of polymerization of individual branches can be readily obtained using a debranching enzyme and also that it does not undergo any reaction except scission during extrusion, thereby aiding mechanistic interpretation. Various starches, containing a range of the highly branched amylopectin component and the much less branched amylose component, were extruded in the presence of water and glycerol as plasticizers with extruder barrel temperatures ranging from 50 °C at the hopper zone through to 140 °C near the die exit. Analysis by size-exclusion chromatography of both whole and debranched samples subject to various levels of extrusion showed that the extrusion degradation process involved preferential cleaving of larger molecules, while causing the size distribution to narrow and converge toward a maximum stable size. This is ana...

What determines size distributions of heavy drops in a synthetic turbulent flow?

Abstract. We present results from an individual particle based model for the collision, coagulation and fragmentation of heavy drops moving in a turbulent flow. Such a model framework can help to bridge the gap between the full hydrodynamic simulation of two phase flows, which can usually only study few particles and mean field based approaches for coagulation and fragmentation relying heavily on parameterization and are for example unable to fully capture particle inertia. We study the steady state that results from a balance between coagulation and fragmentation and the impact of particle properties and flow properties on this steady state. We compare two different fragmentation mechanisms, size-limiting fragmentation where particles fragment when exceeding a maximum size and shear fragmentation, where particles break up when local shear forces in the flow exceed the binding force of the particle. For size-limiting fragmentation the steady state is mainly influenced by the maximum stable particle size, while particle and flow properties only influence the approach to the steady state. For shear fragmentation both the approach to the steady state and the steady state itself depend on the particle and flow parameters. There we find scaling relationships between the steady state and the particle and flow parameters that are determined by the stability condition for fragmentation.

When Ancient Lithuanian State Became an Empire and Ceased to Be It? An Answer to Lithuanian Question Using an Estonian Method

The article introduces a Lithuanian reader into the ideas of distinguished American-Estonian political scientist Rein Taagepera on comparative cliometrics of empires and uses them to detect main thresholds in the imperial history of ancient Lithuania. For this goal, its size-time integral (called in the paper Taagepera integral) was measured. It is used to calculate the maximum stable size (M), emergence (20% of M), adulthood (80% of M), failure (50% of M) dates, rise and duration times of the ancient Lithuanian empire. According to author's calculations, ancient Lithuania emerged as an empire already under Mindaugas rule (before 1263), and its rise time was about 100 years. Its adulthood came in the early sixties of the XIV century, and it failed in 1569, when just before the Lublin Union, Poland annexed the southern provinces of the Grand Duchy of Lithuania. So its duration time was slightly more above 200 years. These chronological conclusions are made under the assumption that the old Lithuanian state survived until 1795. If this assumption is exchanged for another that says that 1569 was also the endpoint of the ancient Lithuanian state, only slight changes are to be made in these conclusions.

On facets of stable set polytopes of claw-free graphs with stability number three

Abstract Providing a complete description of the stable set polytopes of claw-free graphs is a long-standing open problem since almost twenty years. Eisenbrandt et al. recently achieved a breakthrough for the subclass of quasi-line graphs. As a consequence, every non-trivial facet of their stable set polytope is of the form k ∑ v ∈ V 1 x v + ( k + 1 ) ∑ v ∈ V 2 x v ⩽ b for some positive integers k and b, and non-empty sets of vertices V 1 and V 2 . Roughly speaking, this states that the facets of the stable set polytope of quasi-line graphs have at most two left coefficients. For stable set polytopes of claw-free graphs with maximum stable set size at least four, Stauffer conjectured in 2005 that this still holds. It is already known that some stable set polytopes of claw-free graphs with maximum stable set size three may have facets with up to 5 left coefficients. We prove that the situation is even worse: for every positive integer b, there is a claw-free graph with stability number three whose stable set polytope has a facet with b left coefficients.

Density-contrast instabilities in finite element modelling of slow viscous flow

Finite element methods are often used to model Earth processes involving slow viscous or viscoelastic flow. Inertial terms of the Navier-Stokes equations are neglected in very slow flows, so timestep size is not limited by the Courant instability. However, where there is advection of density contrasts in a gravitational field, over-advection can lead to numerically induced flow oscillations. We derive analytic results for the maximum stable timestep size in two cases: a free surface over a fluid of uniform density, and a free surface kept level by sedimentation/erosion, but with a density gradient in the underlying medium. Using parameters appropriate to the Earth's crust we show that the density-contrast instability occurs for timesteps larger than 3000 years for the constant-density case. For a fluid with a density gradient of 10 kg/m3 per km the solution is stable for timesteps up to about 200,000 years if full erosion/sedimentation is implemented.

Modeling of Drop Coalescence in Isotropic Turbulent Flow

Coalescence of drops in fully developed turbulent flow depends on the size of drops and the properties of the flow. By comparing the size of drops with the Kolmogoroff length scale, collision frequencies of drops have been determined. If λ>λ0, then the collision frequency of drops in a gaseous medium is given as ωc∼(ϵR/a2)1/3 and for λ<λ0, it is given as ωc∼(ϵR/νc)1/2 in a liquid medium. New expressions for the fluctuation, thinning, and breaking of the intervening film between drops that is formed due to collisions were also suggested. At various Re and Mo numbers, for the calculation of maximum stable sizes of drops, some equations are suggested. In order to evaluate coalescence and break up rates a new dimensionless number, , is introduced. This number is defined as the ratio of the energy of the turbulent flow to surface energy. As a result of the coalescence of drops, evolution of the distribution function with time is determined from the solution of the Focker‐Planck equation. Comparisons of the calculated drop sizes with the experimental data reported in literature showed good agreement.