Modeling drop breakage using the full energy spectrum and a specific realization of turbulence anisotropy

The assumption of homogeneous isotropic turbulence when modeling drop breakage in industrially relevant geometries is questionable. We describe the development of an anisotropic breakage model, where the anisotropy is introduced via the inclusion of a perturbed turbulence spectrum. The selection of the perturbed spectrum is itself motivated by our previous large-eddy simulations of high-pressure homogenizers. The model redistributes energy from small to large scales, and assumes that the anisotropic part of the Reynolds stresses is confined to the energy-containing range. The second-order structure function arising from the perturbed spectrum is used in the standard framework of Coulaloglou and Tavlarides to calculate breakage frequency. While the base model exhibits non-monotonic behavior (by predicting a maximum value for a certain drop size), the effect of anisotropy is shown to increase breakage frequency in length scales larger than this peak, thereby reducing non-monotonicity. This effect is more pronounced for small turbulence Reynolds numbers.     

Droplet breakage and coalescence models for the flow of water-in-oil emulsions through a valve-like element

An experimental and theoretical study was carried out regarding the evolution of droplet size distributions for the flow of water in oil emulsions through a valve-like element that simulates a mixing valve. The water droplets had diameters in the 0.1–100 μm range, which includes droplets that are either smaller or larger than the Kolmogorov length scale for the experimental conditions. Droplet breakage and coalescence models that can be used for this size range were proposed. A simplified population balance model was developed to interpret the experimental data and solved by the method of classes. The model parameters were estimated by the orthogonal distance regression method. The simplifying assumption of the model was verified by global optimization using a parallel implementation of the particle swarm optimization method. The agreement between simulated and experimental droplet volume size distributions was good. The predictions of the DeBroukere mean diameter, d₄₃, were unbiased with mean errors of 8%.     

Development of a unified framework for calculating molecular weight distribution in diffusion controlled free radical bulk homo-polymerization

In the present work, two different approaches to model diffusion controlled free radical polymerization, namely the free volume model and the entanglement theory are compared. These approaches are applied to methyl methacrylate bulk polymerization in a batch reactor to calculate the conversion, total radical concentration, the number and weight average molecular weights as well as the entire molecular weight distribution as a function of the polymerization time and the process conditions. All the diffusion-controlled phenomena were taken into account, including gel, glass and cage effects as well as residual termination. The molecular weight distribution is calculated by direct numerical integration of a large system of non-linear ordinary differential equations describing the conservation of the mass of macromolecular species in the batch reactor. Model predictions are in good agreement with available experimental data for conversion, number and weight average molecular weights as well as the entire molecular weight distribution, thus justifying the ability of these models to describe the main issues of the diffusion-controlled free radical polymerization.     

A review of the statistical turbulence theory required extending the population balance closure models to the entire spectrum of turbulence

The models for fluid particle breakage and coalescence due to turbulence are generally limited to the inertial subrange of isotropic turbulence and infinite Reynolds numbers. A rigorous procedure for extending the fluid particle breakage and coalescence closures to the entire spectrum of isotropic turbulence and for a wider range of Reynolds numbers can be established based on statistical turbulence theory. The modeling procedure consists of a three‐dimensional (3‐D) literature model energy spectrum for the dissipation, inertial, and energy containing subranges of isotropic turbulence, and an exact literature integral relation for determining the second‐order longitudinal structure function from the 3‐D energy spectrum. A review of the requisite statistical turbulence theory and the use of the model energy spectrum is provided, because the necessary details are not easily accessible in the chemical engineering literature and misconceptions are found in the publications of the previous modeling attempts. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1795–1820, 2016     

A note deriving the analytical solution for the wake theory of coalescence

The recent unsolved wake theory to describe bubble coalescence by Colella et al. [1999. A study on coalescence and breakage mechanisms in three different bubble columns. Chem. Eng. Sci. 54, 4767-4777] is solved analytically to give exact values of coalescence time in terms of bubble size and relative rise velocity. The solution is expressed using the tabulated exponential integral function.     

Model based on population balance for the simulation of bubble columns using methods of the least-square type

A novel model is presented which uses mass and momentum equations based on population balances to describe the dispersed phase of bubble columns. A set of integro-differential non-linear equations constitutes the model, which can be solved directly using the least-squares methods. The implementation for these is presented.The model presents a degree of flexibility, as different density functions (number, mass and volume) and different internal coordinates (bubble diameter, volume and mass) can be used.A simplified 1D model of a bubble column is solved using the least-squares spectral element method. The results obtained for two different gas volume fluxes at the inlet are compared with experiments by several authors. The simulations show good agreement with the measured data.     

Influence of ultrasonic conditioning of flocculant on the aggregation process in a tank with turbine mixer

Aggregation is a very widely encountered phenomenon in nature as well as in industrial processes. It plays an important role in processing of air and water, e.g. wastewater treatment plants, power plants, mining facilities. The new, cost-effective solutions are developed. Among the others modification of the flocculants by means of ultrasounds, mixtures with surfactants or polydispersive flocculant blends can be mentioned. Flocculants used in the aggregation process are long-chain polymers that can be of negative, positive or neutral charge. The aggregation caused by usage of such flocculants is the result of three different mechanisms: bridging, charge neutralization or electrostatic patch. The influence of ultrasonic field on the efficiency of different flocculants was investigated. Suspension of chalk in water was used and the measurements were taken in the laser particle sizer. The influence of the sonication time on flow curves is also presented.     

Influence of Distributed Particle Size on the Determination of the Parabolic Rate Constant for Oxidation by the Powder Method

The established analysis for the study of oxidation using powder specimens is based on the assumption of monosized particles. The experiments, however, are conducted on powders with a distributed particle size. Here we present a statistical approach for the calculation of the rate constant for oxidation. The results of the analysis are applied to new data on oxidation studies of dense powders of silicon carbonitride amorphous ceramics. The monosized model requires a wide range of values for the rate constant to fit the short term and the long-term data, leading to considerable ambiguity in the estimate of the parabolic rate constant, k _p, for oxidation. In contrast the statistical model fits over the entire range of data, yielding a much more reliable value for k _p. For example, the monosized approach gave a value in the range 19.7 × 10⁻¹⁸ < k _p < 2.7 × 10⁻¹⁸ m²/s. In contrast, the statistical model yields a specific value of 4.5 × 10⁻¹⁸ m²/s.     

Predicting the sizes of toluene-diluted heavy oil emulsions in turbulent flow Part 2: Hinze-Kolmogorov based model adapted for increased oil fractions and energy dissipation in a stirred tank

The applicability of several classical size prediction models to predict the Sauter mean diameters d₃₂ of four volume fractions of toluene-diluted heavy oil in water emulsions was studied. Energy dissipation in a fully baffled cylindrical tank stirred at varied speeds by a Rushton turbine was related to the d₃₂ measured with a Mastersizer 2000 laser light scattering instrument after 75 min of agitation. At low oil fractions of 0.01 droplet breakage behaviour was in accordance with the breakage dominance assumption of Hinze-Kolmogorov (H-K) theory. At intermediate oil fractions of 0.05 and 0.1, the system behaved similarly with lower rates of breakage than at 0.01. At high oil fractions of 0.3, the droplet sizes were the largest. Here, coalescence may have played a larger role due to increased collision frequency, while turbulence dampening caused larger eddies and reduced breakage. The resilience to breakage due to the surface elasticity was assumed to be constant for all oil fractions. The experimental diameters compared with the diameters calculated from several size-predictive models showed unsatisfactory predictions at higher oil fractions of 0.05-0.3. By modification of existing constants and coefficients of a H-K-based model that accounts for breakage and coalescence and by using iterative techniques, specific predictive equations for each mixing speed and a new averaged equation were developed. Good agreement between measured and predicted sizes was achieved.     

基于群体平衡理论的水合物聚集动力学模型

水合物颗粒间的聚集是导致油气输送管道堵塞的重要原因,建立聚集模型能更好地进行水合物聚集的防治。首先,假定管道中水合物颗粒粒径连续分布并忽略对流扩散的影响,建立了以群体平衡模型为基本框架的水合物聚集动力学模型。重点考虑水合物的聚集和破碎,该模型的核心主要包括聚集核和破碎核两部分。其中,聚集核包括水合物颗粒间的碰撞频率和碰撞后的聚集效率,破碎核包括水合物颗粒的破碎频率和破碎后子颗粒的粒径分布函数。接着,以前人研究为基础并结合水合物颗粒的自身特性,对模型聚集核和破碎核的计算方法进行了选取和改进。最后,采用计算流体力学方法对模型进行了求解并将求解结果与相关实验数据进行了对比分析。该模型可为管道流动安全保障提供技术支持。     

Control strategies for intermittently mixed, forcefully aerated solid-state fermentation bioreactors based on the analysis of a distributed parameter model

This paper tests different control strategies based on classic proportional integral derivative (PID) and advanced dynamic matrix control (DMC) algorithms for an intermittently stirred, forcefully aerated solid-state fermentation bioreactor. The study was done using a distributed parameter model to reproduce the main operating features of this type of bioreactor. There is predicted to be a remarkable improvement in the bioreactor productivity when control strategies are implemented. For this type of bioreactor, the temperature and water content of the substrate bed can be controlled by saturating the air at the air inlet but manipulating its temperature, coupled with a strategy of water replenishment when the water content of the bed falls below a threshold. Dynamic matrix control is superior to PID control; however, a specific convolution matrix for different stages of the fermentation is necessary due to the changing behavior of the system. This work shows the benefit of mathematical modeling, since the many different operating conditions investigated via simulations would not have been economically feasible to undertake experimentally with a large-scale bioreactor. The results obtained provide an excellent starting point for such large-scale experimental work.     

Combination of a Monte Carlo approach with the contact time distribution concept for the steady-state modeling of an isothermal heterogeneous coordinated anionic ring opening polymerization reactor

In this paper, the possibility of combining a molecular level Monte Carlo simulation with a chemical engineering model of an isothermal fixed-bed reactor is demonstrated. This approach is applied to the isothermal heterogeneous coordinated anionic ring opening polymerization of ε-caprolactone. The contact time distribution (CTD) concept as defined by Orcutt et al. (Chem. Eng. Prog. Symp. Ser. 38 (58) (1962) 1), Nauman and Collinge (Chem. Eng. Sci. 23 (1968a) 1309) and Shinnar et al. (Chem. Eng. Sci. 27 (1972) 1627) is used. The contact time is the time spent by a reactant within the pores of the pellets constituting the fixed-bed. The classical monodisperse model represents the hydrodynamic and mass transfer in the reactor. The CTD is calculated from this model according to Shinnar et al. theory. From that point, the reactor outlet monomer conversion can be calculated analytically, the ε-caprolactone consumption being a first-order process. Furthermore, the molecular size distribution (MSD) of the oligomers at the reactor outlet is computed on the basis of Monte Carlo simulations. Comparisons with experimental data are given.     

Semi-empirical model of toluene transport in polyethylene membranes based on the data using a new type of apparatus for determining gas permeability, diffusivity and solubility

Solubility, diffusivity and permeability of toluene vapors in low-density polyethylene (LDPE) membranes of various thicknesses (approximately 48, 93, 138 and 187 μm) at different temperatures 30, 40 and 50 °C in the range of relative vapor pressure p/p⁰=(0.05; 0.95) were measured using new type of permeation apparatus. Moreover, special construction of the new cell enables determination of the permeant amount sorbed in the membrane in the steady state operation of vapor permeation.The simple semi-empirical model of toluene transport in a polyethylene membrane based on relation between experimentally obtained effective diffusion coefficients and concentration dependent diffusion coefficients evaluated from experiments on a new permeation apparatus was proposed. The model enables estimation of toluene fluxes, sorption in the steady state of vapor permeation and concentration profiles in a polyethylene membrane from equilibrium sorption isotherms and effective diffusion coefficients. Very good agreement between experimental and calculated values from the proposed model was obtained.     

In situ syntheses of a suspension agent based on a styrene–acrylic acid copolymer for the suspension polymerization of styrene

We carried out the suspension polymerization of styrene, initiated with benzoyl peroxide at 80°C, in the presence of the simultaneous polymerization of acrylic acid in the water phase, initiated by potassium peroxidisulfate (KPS) at the same temperature. The polymerization in the water phase was started at certain times after the beginning of the polymerization of styrene. Then, a continuous addition of KPS was carried out at a given landing rate and during variable feeding times. The water‐phase polymerization actually produced a copolymer of styrene and acrylic acid, which displayed surface‐active properties. The particle size distribution depended on the variables mentioned earlier (starting time, KPS feeding rate, and addition time), being controlled by the molecular weight, and on the composition of the copolymer produced and its availability at the increasing conversion of styrene. A second distribution of submicronic particles was produced. Both families of particles had about the same molecular weight. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 86: 3271–3285, 2002     

Morphological study of two‐phase polymer blends during compounding in a novel compounder on the basis of elongational flows

A new laboratory‐scale mixing device based on an original concept was built and tested. This device has important technical features such as tightness to liquids and gases, the possibility of direct specimen molding after mixing, and easy handling of reactive systems. In comparison with existing laboratory mixers, the flow in this mixer is characterized by a high contribution from elongational flow. Morphological data on model polystyrene/poly(methyl methacrylate) blend systems have proved the high distributive and dispersive mixing efficiency in comparison with a classical rotational batch mixer. The influence of different experimental parameters such as the flow rate, mixing time, mixing element geometry, and viscosity ratio of blends is characterized and discussed. Much finer dispersions have been obtained with this new device versus those obtained with a conventional mixer with equivalent specific energy input. © 2010 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

Prediction of fire classification for wood based products. A multivariate statistical approach based on the cone calorimeter

Wood has long traditions as a building material, and is often used in construction elements, and as interior and exterior surfaces in the Nordic countries. In most applications, there are reaction to fire requirements to products used as surfaces, e.g. in escape routes and larger public spaces. Most wood products will therefore have to be treated with fire retardant (FR) agents to fulfil the strict requirements to properties connected to heat release and flame spread. Unfortunately, FR agents usually also increase the smoke production, as they cause a more incomplete combustion of the wood. The wood product manufacturers seek to find the optimal amount of FR additives where both heat release and smoke production in the classifying test are within the requirements given in the building regulations. This paper describes models for prediction of the European reaction to fire classes of wood products. The models are based on multivariate statistical analysis, and use test results from the cone calorimeter test as input. The presented models are, with very good precision, able to predict which Euroclass and additional smoke class a wood based product would obtain if it were to be tested in the single burning item test. Copyright © 2006 John Wiley & Sons, Ltd.     

Unified equation of state based on the lattice fluid theory for phase equilibria of complex mixtures part I. Molecular thermodynamic framework

Consistent calculation of fugacities of fluid mixtures remains as one of the most important subjects in contemporary molecular thermodynamics. In practice, equations of state (EOSs) and g^E-models have been used. However, most EOSs are erroneous for condensed phases at high densities and g^E-models are inapplicable for pressuresensitive systems. Recently to remedy the shortcomings in both approaches, there has been a surge of new g^E-EOS mixing rules. By equating any set of EOS and g^E-models, the limitations in both approaches could be resolved significantly. However, the self-consistency in the underlying concept of those mixing rules remains controversial. During the last several years, the present authors proposed a new lattice-fluid EOS and its simplification relevant to phase equilibrium calculations. Without employing any g^E-EOS mixing rule and with only two parameters for a pure component and one adjustable interaction energy parameter for a binary mixture, results obtained to date demonstrated that the EOSs are quantitatively applicable to a great variety of phase equilibrium properties of mixtures, especially, for complex and/or macromolecular systems. In the present article we summarize the EOSs and extended the applications to liquid-liquid Equilibria. In part I, we discussed briefly the molecular thermodynamic aspects of general derivation of the EOS and a brief discussion of applying the EOSs to pure fluids while the illustrative application to various real mixture systems is discussed in part II.     

Numerical simulation of the dynamic flow behavior in a bubble column: A study of closures for turbulence and interface forces

Numerical simulations of the bubbly flow in two square cross-sectioned bubble columns were conducted with the commercial CFD package CFX-4.4. The effect of the model constant used in the sub-grid scale (SGS) model, C_S, as well as the interfacial closures for the drag, lift and virtual mass forces were investigated. Furthermore, the performance of three models [Pfleger, D., Becker, S., 2001. Modeling and simulation of the dynamic flow behavior in a bubble column. Chemical Engineering Science, 56, 1737-1747; Sato, Y., Sekoguchi, K.,1975. Liquid velocity distribution in two-phase bubble flow. International Journal of Multiphase Flow 2, 79-95; Troshko, A.A., Hassan, Y.A., 2001. A two-equation turbulence model of turbulent bubbly flows. International Journal of Multiphase Flow 27, 1965-2000] to account for the bubble-induced turbulence in the k-ε model was assessed. All simulation results were compared with experimental data for the mean and fluctuating liquid and gas velocities. It is shown that the simulation results with C_S=0.08 and 0.10 agree well with the measurements. When C_S is increased, the effective viscosity increases and subsequently the bubble plume becomes less dynamic. All three bubble-induced turbulence models could produce good solutions for the time-averaged velocity. The models of Troshko and Hassan and Pfleger and Becker reproduce the dynamics of the bubbly flow in a more accurate way than the model of Sato and Sekoguchi. Based on the comparison of the results obtained for two columns with different aspect ratio (H/D=3 and H/D=6), it was found that the model of Pfleger and Becker performs better than the model of Troshko and Hassan, while the model of Sato and Sekoguchi performs the worst. It was observed that the interfacial closure model proposed by Tomiyama [2004. Drag, lift and virtual mass forces acting on a single bubble. Third International Symposium on Two-Phase Flow Modeling and Experimentation, Pisa, Italy, 22-24 September] performs better for the taller column. With the drag coefficient proposed by Tomiyama, the predicted slip velocity agrees well with the experimental data in both columns. The virtual mass force has a small influence on the investigated bubbly flow characteristics. However, the lift force strongly influences the bubble plume dynamics and consequently determines the shape of the vertical velocity profile. In a taller column, the lift coefficient following from the model of Tomiyama produces the best results.