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Transport and particulate processes are ubiquitous in environmental, industrial and biological applications, often involving complex geometries and porous media. In this work we present a general population balance model for particle transport at the pore-scale, including aggregation, breakage and surface deposition. The various terms in the equations are analysed with a dimensional analysis, including a novel collision-induced breakage mechanism, and split into one- and two-particles processes. While the first are linear processes, they might both depend on local flow properties (e.g. shear). This means that the upscaling (via volume averaging and homogenisation) to a macroscopic (Darcy-scale) description requires closures assumptions. We discuss this problem and derive an effective macroscopic term for the shear-induced events, such as breakage caused by shear forces on the transported particles. We focus on breakage events as prototype for linear shear-induced events and derive upscaled breakage frequencies in periodic geometries, starting from nonlinear power-law dependence on the local fluid shear rate. Results are presented for a two-dimensional channel flow and a three dimensional regular arrangement of spheres, for arbitrarily fast (mixing-limited) events. Implications for linearised shear-induced collisions are also discussed. This work lays the foundations of a new general framework for multiscale modelling of particulate flows.

We deal with multidimensional regularized systems of equations for the one-velocity and one-temperature inert gas mixture dynamics consisting of the balance equations for the mass of components and the momentum and total energy of the mixture, with diffusion fluxes between the components as well as the viscosity and heat conductivity terms. The regularizations are kinetically motivated and aimed at constructing conditionally stable symmetric in space discretizations without limiters. We consider a new combined form of regularizing velocities containing the total pressure of the mixture. To confirm the physical correctness of the regularized systems, we derive the balance equation for the mixture entropy with the non-negative entropy production, under generalized assumptions on the diffusion fluxes. To confirm nice regularizing properties, we derive the systems of equations linearized at constant solutions and provide the existence, uniqueness and L2-dissipativity of weak solutions to an initial-boundary problem for them. For the original systems, we also discuss the related Petrovskii parabolicity property and its important corollaries. In addition, in the one-dimensional case, we also present the special three-point and symmetric finite-difference discretization in space of the regularized systems and prove that it inherits the entropy correctness property. We also give results of numerical experiments confirming that the discretization is able to simulate well various dynamic problems of contact between two different gases.

In this contribution, crystallization was performed to assess the kinetics of nucleation and crystal growth of L-lactide. In most common solvents, this compound shows very high solubility even at low temperatures, which could be challenging for crystallization process design. In the first part of this paper, the anti-solvent effects of n-hexane on solutions of L-lactide in tetrahydrofuran (THF) were investigated through studying the influence of solvent compositions on the solubility. Thanks to these effects, the solubility of the interested compound can be adjusted to desired degrees of supersaturation by adding suitable amounts of the anti-solvent. In the second part, a solvent composition at a mass ratio of 45/55 (n-hexane/THF) was chosen, and an isothermal seeded crystallization process was implemented. The evolution of the particle sizes and changes in the solute concentration profile of this process were monitored. Based on the obtained data, a widely used model, i.e., the population balance equation (PBE), was then utilized to model the crystal size distribution (CSD). Reasonable assumptions were made to reduce the mathematical complexity of the PBE. In the simplified model, only crystal growth and secondary nucleation were considered for model formulation, with assumptions of the size-independent growth rate and negligible size of nuclei. The kinetic parameters were estimated by using the seed and final-time crystal density functions in combination with variations in the concentration of the mother liquor. Indeed, the numerical solution for the one-dimensional problem of the L-lactide crystallization based on the estimated parameters gained a relatively good agreement with the determined CSD. Furthermore, the obtained model also correlated well with the variations in the solute concentration of the mother liquor. In short, this simple approach can be used for predicting the productivity and CSD of the L-lactide crystallization.

To improve the efficiency of existing networks, special mathematical models for assessing the losses and temperature of conductors in real time can be used, with the climatic factors being taken into account. The approximate analytical solution of the nonlinear differential equation of heating and cooling of the insulated conductor with numerical method simulation of heat transfer is proposed comparison in this work. The solution is based on lowering the degree of temperature of the conductor using the least squares method in the integral form. A positive feature of the proposed solution is its universality. It allows the analysis of overhead conductors both with and without insulation. The developed method is almost as accurate as the calculation of the conductor temperature by numerical methods. The reliability of the heat balance equation of overhead power lines at non-stationary thermal mode developed by this method is confirmed by comparison with the results obtained by the finite elements method.

Methods for reconstruction of river runoff have been an important component of paleo-climatology and paleo-hydrology. The lower Heihe River was studied from 2002 to 2008 to reconstruct the annual average runoff at the Zhengyi Gorges hydrological station using measured data and improved water and energy balance equations. The results indicate that the reconstructed annual runoff and the measured runoff values are not equivalent. However, the difference between reconstructed and measured runoff values is within 10 %, which suggests that water and energy balance models can successfully reconstruct runoff from rivers in arid regions. Regional differences imply that parameters must be corrected when applying water and energy balance equations to prevent large errors. In addition, the latent heat of evaporation in arid lakes is greater than their net radiation, according to latent and sensible heat calculations, which demonstrates the existence of the “cold island effect”.

The moving track of journal bearing changes with the time in the condition of dynamic loading. The force balance equation of journal bearing is established, and the generalized Reynolds equation, the oil film thickness equation of spiral oil wedge journal bearing under dynamic loading are gained, which is based on axial inertia force, bearing capacity and dynamically loading. By finite difference method, Euler method and Reynolds boundary condition, the generalized Reynolds equation and force balance equations are solved simultaneously, the periodic moving track of journal bearing at different times is solved. The results show that the circumferential pressure, axis displacement, axis velocity, axis acceleration velocity of journal bearing change periodically as time goes. The influence of dynamical loading on pressure distribution of oil film and axis locus is analyzed.

The present study investigates the kinetics of lead and copper removal from oil-field brine by potential sorption. A population balance equation, coupled with a mass balance equation, was used in the estimation of kinetic parameters. Metal removal was performed by potential sorption of lead and copper through CaCO3 precipitates induced by the reaction of Na2CO3 and CaCl2. The oil-field brine was selected from an oil well in Gachsaran, Iran. The crystal size distribution of the solid phase was measured by dynamic laser scattering analyzer, and the liquor phase was analyzed using atomic adsorption. The morphology of calcium carbonate particles was illustrated using scanning electron microscopy and X-ray diffraction. The results showed that the presence of copper and lead decreases the average size distribution of calcium carbonate particles by influencing the kinetic parameters. Lead and copper concentrations were reduced from 2.911 to 0.127 ppm (95.63% removal) and 0.476 to 0.025 ppm (94.74% removal), respectively, in exchange for 12 g CaCO3 consumption per 100 ml oil-field brine.

It is understood that ensuring equation balance is a necessary condition for a valid model of times series data. Yet, the definition of balance provided so far has been incomplete and there has not been a consistent understanding of exactly why balance is important or how it can be applied. The discussion to date has focused on the estimates produced by the general error correction model (GECM). In this paper, we go beyond the GECM and beyond model estimates. We treat equation balance as a theoretical matter, not merely an empirical one, and describe how to use the concept of balance to test theoretical propositions before longitudinal data have been gathered. We explain how equation balance can be used to check if your theoretical or empirical model is either wrong or incomplete in a way that will prevent a meaningful interpretation of the model. We also raise the issue of “ $I(0)$ balance” and its importance.

Experimental and theoretical techniques have been developed to quantify foamy oil behaviour of solvent-heavy oil systems at bubble level during a gas exsolution process. During constant composition expansion (CCE) tests, we artificially induced foamy oil dynamics for solvent-heavy oil systems by gradually reducing pressure and recorded the changed pressures and volumes in an isolated PVT setup at a given temperature. By discretizing gas bubbles on the basis of the classical nucleation theory, we theoretically integrated the population balance equation (PBE), Fick’s law, and the Peng–Robinson equation of state (PR EOS) to reproduce the experimental measurements. Pseudo-bubblepoint pressure for a given solvent-heavy oil system can be increased with either a lower pressure depletion rate or a higher temperature, during which gas bubble growth is facilitated with a reduction in viscosity and/or an increase in solvent concentration, but gas bubble nucleation and mitigation is hindered with an increase in solvent concentration. Compared to CO2, CH4 is found to yield stronger and more stable foamy oil, indicating that foamy oil is more stable with a larger amount of dispersed gas bubbles at lower temperatures. Using the PR EOS together with the modified alpha functions at Tr = 0.7 and Tr = 0.6, the absolute average relative deviation (AARD) is reduced from 4.58% to 2.24% with respect to the predicted pseudo-bubblepoint pressures.