Clet Numbers Research Articles

Highly accurate and numerical tractable coupling of nanoparticle nucleation, growth and fluid flow

Supersaturation as the thermodynamic driving force of particulate precipitation couples the mixing of solute compounds with the solid formation consisting of nucleation and molecular growth. Because the process spans many orders of magnitude in space and time, gathering spatio-temporal information by experiments is hardly achievable. Numerical simulation of the nonlinear, non-local balance equations for mass, momentum and dispersed phase provide detailed solution information in space and time. We here introduce a novel, highly accurate and very efficient numerical formulation of the population balance equations for the dispersed phase coupled to the concentration and velocity field. We reformulate the problem in terms of a fixed-point equation solving the population balance equation along Lagrangian paths with the recently developed exact Method of Moments (eMoM, Pflug et al., 2020) coupled to the concentration field (Eulerian frame). The analytical reformulation provides the information about the full particle size distribution while the computational effort remains in the order of classical moment methods and is much less than numerical schemes approximating the full particle size distribution such as finite volume methods. Furthermore, we explore the difference between an Euler-Euler and Euler-Lagrange coupling considering moderate to large P é clet numbers for the solutes and particles. It turns out that the mean particle size as a solution of the population balance is much more sensitive to the numerical discretization of the advection term than the approximation of the number density function. Based on this finding, we eventually explore the impact of particle diffusion on the solid formation process and give guidelines, when particle diffusion should be taken into account.

Structure-Preserving Numerical Method for Ampere-Nernst-Planck Model

Charge dynamics play essential role in many practical applications such as semiconductors, electrochemical devices and transmembrane ion channels. An Amp\'{e}re-Nernst-Planck (ANP) model that describes charge dynamics via concentrations and the electric displacement is able to take effects beyond mean-field approximations into account. To obtain physically faithful numerical solutions, we develop a structure-preserving numerical method for the ANP model whose solution has several physical properties of importance. By the Slotboom transform with entropy-mean approximations, a positivity preserving scheme with Scharfetter-Gummel fluxes is derived for the generalized Nernst-Planck equations. To deal with the curl-free constraint, the dielectric displacement from the Amp\'{e}re equation is further updated with a local relaxation algorithm of linear computational complexity. We prove that the proposed numerical method unconditionally preserves the mass conservation and the solution positivity at the discrete level, and satisfies the discrete energy dissipation law with a mild time-step restriction. Numerical experiments verify that our numerical method has expected accuracy and structure-preserving properties. Applications to ion transport with large convection, arising from boundary-layer electric field and Born solvation interactions, further demonstrate that the ANP formulation with the proposed numerical scheme has attractive performance and can effectively describe charge dynamics with large convection of high numerical cell P\'{e}clet numbers.

Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution

We study the interactions between classical elementary waves and delta shock wave in quasilinear hyperbolic system of conservation laws. This governing system describes a thin film of a perfectly soluble anti-surfactant solution in the limit of large capillary and P$ \acute{e} $clet numbers. This system is one of the example of non-strictly hyperbolic system whose Riemann solution consists of delta shock wave as well as classical elementary waves such as shock waves, rarefaction waves and contact discontinuities. The global structure of the perturbed Riemann solutions are constructed and analyzed case by case when delta shock wave is involved.

Operating regimes of a chemical reactor with longitudinal and transverse mixing in the case of large P�clet numbers

A two-dimensional model of a chemical reactor with longitudinal and transverse mixing is investigated in the case of large Peclet numbers calculated from the effective thermal conductivity in the transverse direction. For this model the existence of at least one steady-state regime has been demonstrated [1], sufficient criteria of its uniqueness have been determined, an asymptotic expansion of the solution has been constructed in the case of small Peclet numbers, and the critical ignition and quenching parameters have been found. In this paper the other limiting case of the model, in which heat is propagated in the transverse direction much more slowly than it is transported by the flow along the reactor (large Peclet numbers), is analyzed in detail. An asymptotic expansion of the solution which closely coincides with the data of numerical calculations is constructed. The critical quenching and ignition conditions of the process are determined.

Some general invariance relations in problems of convective heat and mass transfer at large p�clet numbers

Some general invariance relations are obtained for the integral diffusion fluxes of the reactant on the surface of one or several reacting particles of arbitrary shape in Stokes flow of a viscous incompressible fluid around the particles at large Peclet numbers. The case of irrotational flow is also considered.

Convective mass transfer between a solid spherical particle and a fluid at large p�clet numbers

Convective mass transfer between a solid spherical particle and a fluid at large p�clet numbers

Diffusion to a particle at large p�clet numbers and small Reynolds and Hartmann numbers

A study is made of the influence of a homogeneous magnetic field on the mass transfer for a spherical solid particle and a liquid drop in a flow of a viscous electrically conducting fluid. The previously obtained [1] velocity field of the fluid is used to calculate the concentration distribution in the diffusion boundary layer, the density of the diffusion flux, and the Nusselt number, which characterizes the mass transfer between the particle and the surrounding medium.

Intensification of mass transfer in a current-carrying fluid at high P�clet numbers

Two axisymmetric problems relating to mass transfer in a system consisting of a solid body and an electrically conducting fluid are examined.

Asymptotic of solution of problem of convective diffusion to a drop with large p�clet numbers and finite reynolds numbers

Asymptotic of solution of problem of convective diffusion to a drop with large p�clet numbers and finite reynolds numbers

Diffusion to a chain of drops (bubbles) at large p�clet numbers

The problem of diffusion of a substance, dissolved in a flow, to absorbing drops (bubbles) moving one after another in a viscous incompressible fluid is investigated. An approximate analytic expression is obtained for the differential and integral flows of the substance to the surface of each drop with consideration of the changes of the concentration and velocity fields due to the presence of other drops. A chain of spherical drops of equal radius arranged on the axis of a uniform forward flow is examined. It is shown that if the distance between drops, referred to the radius of the drops, satisfies the inequality 1≪l≪P1/2 (P is the Peclet number), then the integral inflow of the substance to the surface of the second drop of the chain is 2.41 times less than the integral inflow to the first (the drops are enumerated along the flow); the total diffusion flow to the surface of an arbitrary drop with number k is determined by the expression Ik=I1[k1/2 − (k−1)1/2], where Ik is the total flow to the first drop of the chain. The case of diffusion interaction of a solid particle and drop is examined. It is shown that for particles moving one after another with the same velocity in a fluid at rest the presence of a drop before the solid particle leads to a marked decrease of the total diffusion flow of the solid particle [by O(P1/6) times], whereas the presence of a solid particle before a drop does not affect (in the main approximation with respect to the characteristic diffusion parameter) the total flow of the latter.I k=I i[k 1/2−(k−1)1/2]

Diffusion to a drop at high P�clet numbers and finite reynolds numbers

Diffusion to a drop at high P�clet numbers and finite reynolds numbers