A stochastic model for turbulent diffusion

An improved stochastic model for turbulent diffusion is developed. The model is free of arbitrary adjustable constants. The central objective, as in Taylor's approach, is to predict the turbulent diffusion (mean square displacement and/or concentration distribution) given certain statistical information about the turbulent velocity field.The applicability of the model to problems of turbulent diffusion in shear flows is explored. When the cross correlation between the fluctuating velocity and the stochastic forcing function is zero and the forcing function is white noise in character, the conditions necessary to describe Brownian motion are satisfied. However, when the cross correlation is other than zero and when, at the same time, the energy of the velocity fluctuation is concentrated at the lower frequencies, the equation can be used to describe turbulent diffusion. The deterministic coefficient β in the model and the nature of the stochastic forcing function can be calculated from known Eulerian statistics of the flow field.The method is applicable to non-homogeneous flows in three dimensions where correlation exists between fluctuating velocity components. Implementation of the model is most readily accomplished on a modern hybrid computer. Good agreement is obtained in comparisons of predicted results from the model with (a) theoretical solutions for homogeneous turbulent diffusion (b) experimental measurements of diffusion in a turbulent boundary layer and (c) measurements of diffusion in the atmospheric boundary layer.     

A diffusion coefficient model for polymer devolatilization

Polymer devolatilizers are in widespread use in the polymer industry for removing solvents and monomers from polymer melts prior to product fabrication. Design equations for describing the solvent flux usually include both the diffusion coefficient of the solvent in the polymer melt and the equilibrium concentration of the solvent at the polymer-vapor interface. Several models make the as sumption that the solvent diffusivity is constant over the ranges of solvent concentrations and temperatures in the devolatilizer. This is a critical assumption that may be difficult to check without obtaining diffusivity data at the operating temperatures and concentrations of the process equipment. There are three models that can be used for diffusion coefficients in devolatilizer design: the free volume model developed by Duda, Vrentas, and coworkers; a new linear model proposed in this study; and a constant diffusivity model, The linear model is obtained by combining a new correlation for solvent activity coefficients in molten polymers with free volume theory and linearizing the resulting equation. The error between using the complete free volume theory and using the linear model, or alternatively, using a constant diffusion coefficient, is calculated for several solvent-polymer systems. The linear model is convenient to use for determining the effects of the solvent activity coefficient on the diffusion coefficient. A method is presented for determining whether the complete model, the linear model, or the constant diffusivity model is appropriate for a given devolatilizer design.     

A theory of case II diffusion

A theory is proposed to explain the transport behaviour of organic penetrants in glassy polymers in terms of two basic parameters: the diffusivity of the penetrant, D, and the viscous flow rate of the glassy polymer, $\text{1}\text{η}{}_0$. The rate controlling process for transport in these systems is considered to be diffusion of solvent down an activity gradient coupled with time-dependent mechanical deformation of the polymer glass in response to the swelling stress. The theory combines these two factors and is able to predict a wide range of observed transport phenomena from Fickian diffusion kinetics at one extreme to so-called Case II and Super-Case II behaviour at the other. The existence of a sharp front separating swollen and unpenetrated polymer is shown to result from the concentration dependence of the viscous flow rate.     

A multiprocess eyring model for large strain plastic deformation

A multiprocess Eyring model is developed with a particular aim of predicting the localized instability occurring in “necking” polymers when cold‐drawn. Differences from using single and multiple Eyring processes are examined using a published data‐set for polypropylene test pieces; it is shown that a four Eyring process model can simultaneously fit both necking stretch ratio and draw force data for uniaxial stretching, whereas with a single process only one measurement could be fitted accurately. The multi process Eyring model is shown to give significantly more accurate predictions than a necking hyperelastic model. The multiprocess model is assessed against the same material undergoing a complex constant‐width elongation. It is shown that agreement is quantitatively good for both drawing force and surface deformation, with some minor differences in transverse force and surface stretch. A pronounced intermittent stretching pattern that is seen on the experimental test piece is replicated by the multiprocess Eyring simulation, but is absent using the hyperelastic model. A method is described to deform a photograph of the original specimen according to a finite element solution. The method is shown to give a clear indication of the accuracy of the model in predicting final form. © 2010 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

A mathematical model for diffusion with induced crystallization: 1

During the sorption of swelling agents in amorphous, but crystallizable, glassy polymers local crystallization takes place. Previous work has shown that the weight gain and overall crystallization kinetics depend on the polymer/penetrant interactions and the dimensions of the sample. In this paper we propose a mathematical model for the process which includes four important effects: time dependent polymer swelling, blocking of diffusion by crystallites, occlusion of penetrant by developing crystallites, and macrovoid formation. The model predicts four limiting regimes of behaviour according to the values of the dimensionless crystallization rate, Ω, and the dimensionless sample half thickness, λ_p. Analytical solutions are developed for each limiting case, with the corresponding criteria for their application to real systems. The model successfully predicts the variety of behaviours observed experimentally.     

A mathematical model for diffusion with induced crystallization: 2

This article extends the analysis of a mathematical model for solvent induced crystallization. By evaluating the model's parameters for polymer/penetrant systems studied previously we find the predicted behaviours to be consistent with those determined experimentally. Fitting a limiting solution to solvent transport data in poly(ethylene terephthalate) films gives the threshold concentrations for crystallization and the penetrant diffusivity in the amorphous component of the highly swollen polymer. Numerical solutions determine the model's behaviour in intermediate regions where analytical solutions do not apply. These predict negative curvature in plots of weight gain versus √t and a distinct crystallization front behind a swelling boundary; the former results from the rapid crystallization of swollen surface layers, while the latter indicates partial decoupling between solvent transport and polymer crystallization. We also analyse briefly the desorption process following solvent induced crystallization; the relative magnitude of initial sorption and desorption rates depends on the induced crystallinity and the crystallization rate.     

A diffusivity model for gas diffusion through fractal porous media

A fractal model for gas diffusivity in porous media is derived by using fractal theory and by considering rarefied gas effect in micro-/nano-channels/capillaries. The proposed gas diffusivity model is expressed as a function of micro-structural parameters (the fractal dimensions for pore area and for tortuosity of tortuous capillaries, porosity and pore sizes) of porous media. The effects of parameters such as porosity, microstructures of porous media and fractal dimensions on gas diffusivity are analyzed. The effective diffusivities predicted by the present fractal model are compared with the available experimental data, and a fair agreement between them is found when porosity is less than about 0.70.     

A thermodynamic base for the stress activated phase transition deformation model

The three essential factors of the Juska–Harrison stress activated phase transition (SAPT) deformation model are analyzed. Based on the concept of “mechanical melting”, a formula for the theoretical melting stress is derived from thermodynamics. A comparison of experimental and calculated results for a number of polymers shows that the strain energy is high enough to induce isothermal “mechanical melting” during extension, and the observed yield stress has the same order of magnitude as the theoretical melting stress.     

The droplet diffusion model for micromixing

The concept of segregation number leads to a simple but plausible model of micromixing. Initially, bulk mixing of fluid occurs from hydrodynamic mechanisms and it is this type of mixing that determines the residence time distribution. Co-mingling of material on a molecular scale requires diffusion, and the droplet model provides a convenient representation of such micromixing. The model can be used to relate the degree of segregation to a single, physically meaningful parameter. The model is easily applied to the prediction of reaction yields including complex kinetic schemes where diffusivities and hence levels of micromixing vary between the different molecular species.     

基于Simha-Somcynsky状态方程的高分子-溶剂体系扩散系数模型

将Simha-Somcynsky状态方程引入到Vrentas-Duda扩散系数模型，建立基于状态方程的溶剂在高分子中扩散系数模型。与原模型相比，改进的模型避免进行高分子的黏弹性实验测定，模型中高分子的相关参数仅由状态方程的特征温度和特征体积确定，提高了模型的预测能力。对苯、甲苯、乙苯、氯仿在聚苯乙烯、聚异丁烯和聚醋酸乙烯酯中的扩散系数计算结果表明，改进模型的预测值与实验值吻合较好。     

A multiscale model for quantifying helium diffusion in porous unsintered glass

Helium‐aided sintering of porous unsintered glass is a complex multiscale process, characterised by three different timescales, namely, that of helium diffusion, heat conduction, and radial shrinkage of the glass core. This work presents a multiscale model for quantifying heat and helium diffusion in a shrinking core system by decoupling the timescales based on their orders of magnitude. We obtain analytical solutions of our model, which allow us to quantify the spatio‐temporal profiles of temperature and helium concentration in the glass during the sintering process. Our results show that the introduction of helium increases the sintering rate of glass, and we conclude that pre‐sintering heating followed by helium‐aided sintering is better than simultaneous heating and helium diffusion. We also show that the pre‐sintering heating process for a standard glass sample should not be longer than an hour for the sake of heat economy, following which we may switch to the helium‐aided sintering process, where the sintering should occur under isothermal conditions for approximately 6 h. We perform dynamic simulations using glass porosity as a parameter, and find the sintering rate to be directly proportional to the initial porosity of the glass sample.     

A protein diffusion model of the sealing effect

Water transport across the arterial endothelium is believed primarily to occur through breaks in the tight junction strands at the cell periphery between neighboring cells. Additional proteins arriving at the tight junction can close these breaks, thereby attenuating this water flux. Motivated by evidence that the diffusion of presynthesized protein from the interior of the cell to and incorporation into the cell border is the mechanism of endothelial tight junctional sealing, we develop a diffusion-limited mathematical model of intercellular gap sealing. A single endothelial cell is represented as a thin, axisymmetric disk, initially containing a uniform distribution of junctional protein that does not interact with the apical or basal cell surfaces. Upon application of a transmural pressure gradient, water flows through the junctional cleft, and tight junction remodeling begins. We assume that proteins at the junction are instantaneously incorporated into its strand, dropping the free protein concentration at the cell periphery to zero. This sets the diffusion of intracellular proteins toward the junction in motion. The solution of this one-dimensional initial value problem provides excellent fits to current and previously published experimental data over a wide variety of conditions. It yields three physically meaningful parameters for each fit, including a protein diffusivity in the cytoplasm that varies little within experimental treatments. Statistical variation of these parameters allows rational comparison of experimental runs and identification of outlier runs.     

A model for calculation of undulatory deformation of sheet glass in lateral hardening

A method for calculating undulatory deformation of sheet glass moving in a heating furnace is refined. It is established that the standard calculation method applied to a multi-support scheme produces an approximately double error in estimating the deformation.     

A dispersion model of enhanced mass diffusion in nanofluids

An explanation for the enhancement of mass diffusion in nanofluids is presented using arguments based on dispersion in diluted fixed beds. Starting from the generalized Langevin equation, it is shown that the velocity field established around a Brownian nanoparticle is similar to the velocity field predicted by Brinkman equations leading to the analogy between dispersion in diluted fixed beds and dispersion in nanofluids. The proposed model predicts the order of magnitude of mass diffusion enhancement we observed recently (10-fold enhancement of rhodamine 6G mass diffusivity under the optimum conditions for a suspension of 10-nm alumina nanoparticles in deionized water). Contrarily to other Brownian motion-based models of diffusion in nanofluids, the proposed model samples the whole Maxwell–Boltzmann distribution of particle velocities rather than taking the non-representative root mean square velocity. The model also shows a strong dependence on the mass transfer Péclet number and, consequently, justifies the order of magnitude differences between the mass diffusivity and thermal conductivity enhancements reported in the literature.     

A diffusion model for particle mixing in a packed bed of burning solids

Particle mixing caused by grate movement in a packed bed of solids is an important process for biomass combustion and waste incineration. In this paper, a diffusion model for particle mixing in a burning bed is proposed and the related diffusion coefficient is measured. The diffusion model was incorporated into a combustion model for waste incineration in an actual full-scale bed and numerical calculations were carried to assess the effect of different mixing levels on the burning characteristics of the furnace. In-bed measurement of temperature, oxygen concentration and particle movement was also made using a special electronic device. It is found that the modelled flame front reaches the bed bottom at an earlier stage for a higher level of particle mixing; the average burning rate ranges from 0.05 to 0.13 kg/m² s and the mass loss rate for a higher mixing level can be twice of that for a lower mixing level. However, excessive mixing can cause significant delay in ignition or even extinction of the bed combustion; the obtained local air to fuel stoichiometric ratio covers a range from sub-stoichiometric (0.6 for the highest mixing level) to super-stoichiometric (1.6 for the lowest mixing level); the carbon in ash ranges from 3.5 to 10.5%; the most reasonable range of the particle-mixing (diffusion) coefficient is from 1.8 to 6.0 cm²/min for a full-scale bed, according to the calculation.     

A mesoscale model for diffusion and permeation of shale gas at geological depth

The demand on energy is rising and shale gas as an important unconventional energy resource has received worldwide attention. It has shown a significant effect on the world's energy structure after the commercial exploitation of shale gas in the United States. Understanding diffusion and permeation of shale gas at geological depths is quite essential, but it cannot be described by traditional Fick or Knudsen diffusion models. In this work, we use dual control volume–grand canonical molecular dynamics method to systematically investigate the permeation process of shale gas in montmorillonite (i.e., a clay mineral of shale) at different geological depths. Results indicate that temperature, pressure, and pore size have an important effect on the permeability, and Knudsen equation cannot describe the permeability of shale gas. Accordingly, on the basis of these simulated data, we propose a new mesoscale model to describe the permeability of shale gas at geological depths. The new mesoscale model shows extensive applicability and can excellently reproduce the extrapolation testing data, and it satisfactorily bridges the gap between Knudsen diffusion and Fick diffusion, which provides important fundamentals for exploitation of shale gas. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1059–1066, 2018     

Novel multi-scale diffusion model for catalytic methane combustion

A multi-scale model of methane catalytic combustion was built by a series of balance equations and diffusion equations, and these equations were solved through the computational fluid dynamics (CFD) software. The difference between this work and previous model is the diffusion process in catalyst coating was considered. By analyzing the methane conversion, temperature distribution and mass fraction contours of every component, the performance of multi-scale model was compared with that of the pure CFD model without diffusion. The effects of diffusion, methane concentration, flow rate on the methane conversion and temperature distribution of monolithic reactor were also evaluated and discussed by the multi-scale model. The multi-scale model showed better accuracy than the pure CFD model without diffusion process. Different methane concentrations and gas flow rates had enormous effects on the methane conversion and temperature. Therefore, it was beneficial to the reaction process to adjust the methane concentration and gas flow rate appropriately.     

Thermomechanical studies of polymer deformation II. A thermodynamic analysis of a viscoelastic material in sinusoidal deformation

In this study, a linearly viscoelastic polyurethane film was subjected to continuous, sinusoidal deformation in a new isothermal deformation calorimeter, whose design details were recently reported (1). Internal energy and entropy of the polymer at each state in the deformation cycle were computed from heat rate and work rate data. This was made possible by using linear viscoelasticity theory to predict the irreversible entropy production. Thermal data were corrected for instrument time lag.