Convective Diffusion Process Research Articles

Mathematical Modeling of the Dynamics of Nonequilibrium in Time Convection–Diffusion Processes in Domains with Free Boundaries

A mathematical model is constructed that describes the dynamics of fractional-differential locally nonequilibrium in time convection---diffusion process of soluble substances in plain-vertical established filtration with free boundary. The respective boundary-value problem is formulated and the technique is outlined to derive its approximate solution. Parallel algorithms for calculation of cluster systems are developed, the results of testing the response of parallel algorithms for GPU and the results of numerical experiments on simulation of the dynamics of the migration process under study are presented.

Mechanistic understanding of temperature-driven water and bacterial infiltration during hydrocooling of fresh produce

Freshly harvested fruits and vegetables (produce) are chilled to extend their shelf life but the chilling process increases opportunities for contamination by pathogenic bacteria carried by water through openings such as stem scars. Using tomato as a representative system, a 3D porous medium transport model is developed. The model simulates the transport of water vapor, liquid water, bacteria, and energy in light of convection–diffusion processes driven by pressure gradients from condensation inside as well as by water concentration gradients between the cooling water and that in the tomato. Results show that increasing the rate of cooling (i.e., creating a higher temperature difference) increases the rate of infiltration due to higher-pressure gradients. A higher temperature differential drives bacteria further into the tomato. A less hydrated tomato will incur deeper and more extensive infiltration. Size, permeability, and diffusivity play a less significant role. The novel mechanistic understanding our study provides should aid in designing safer hydrocooling processes.

Liquid Gradient Refractive Index Microlens for Dynamically Adjusting the Beam Focusing

An in-plane liquid gradient index (L-GRIN) microlens is designed for dynamically adjusting the beam focusing. The ethylene glycol solution (core liquid) withde-ionized (DI) water (cladding liquid) is co-injected into the lens chamber to form a gradient refractive index profile. The influences of the diffusion coefficient, mass fraction of ethylene glycol and flow rate of liquids on the refractive index profile of L-GRIN microlens are analyzed, and the finite element method and ray tracing method are used to simulate the convection-diffusion process and beam focusing process, which is helpful for the prediction of focusing effects and manipulation of the device. It is found that not only the focal length but the focal spot of the output beam can be adjusted by the diffusion coefficient, mass fraction and flow rate of liquids. The focal length of the microlens varies from 942 to 11 μm when the mass fraction of the ethylene glycol solution varies from 0.05 to 0.4, and the focal length changes from 127.1 to 8 μm by varying the flow rate of the core liquid from 0.5 × 103 to 5 × 103 pL/s when there is no slip between the core and cladding inlet. The multiple adjustable microlens with a simple planar microfluidic structure can be used in integrated optics and lab-on-chip systems.

Mathematical Modeling of the Fractional Differential Dynamics of the Relaxation Process of Convective Diffusion Under Conditions of Planned Filtration

To describe the dynamics of the relaxation process of convective diffusion under conditions of planned filtration, the authors propose a mathematical model based on the fractional derivative equation with respect to time. The boundary-value problem corresponding to this model is formulated. A parallel computing algorithm based on the locally one-dimensional scheme is developed. The results of numerical implementation of the solution are presented.

An efficient approach for calculating default probabilities for cash-flow based project finance with reserve account

The quantitative assessment of risks associated with several types, eg, rating methods for cash-flow driven projects, can be reduced to determining the probability that a random variable, for instance representing a cash-flow, drops below a given threshold. That probability can be derived in an analytic closed form, if the underlying distribution is not too complex. However, in practice there is often a reserve account in place, which saves excess cash to reduce the volatility of the cash-flow available for debt service. Due to the reserve account, the derivation of a solution in an analytic closed form is even in the case of rather simple underlying distributions, eg, independent Gaussian distribution, not feasible. In this paper, we present two very efficient approximation methodologies for calculating the probability that a random variable falls under a threshold allowing the presence of a reserve account. The first proposed approach is derived using transition probabilities. The resulting recursive scheme can be implemented easily and yields fast and stable results even in the case of dependent cash-flows. The second methodology uses the similarity of the considered stochastic processes with convection-diffusion processes and combines the stochastic transition probabilities with the finite volume method, which is well known for solving partial differential equations. We present numerical results for some realistic test problems demonstrating convergence of order h for the transition probability based approach and $$h^2$$h2 for the combination with the finite volume method for sufficiently smooth probability distributions.

Counter-extrapolation method for conjugate interfaces in computational heat and mass transfer

In this paper a conjugate interface method is developed by performing extrapolations along the normal direction. Compared to other existing conjugate models, our method has several technical advantages, including the simple and straightforward algorithm, accurate representation of the interface geometry, applicability to any interface-lattice relative orientation, and availability of the normal gradient. The model is validated by simulating the steady and unsteady convection-diffusion system with a flat interface and the steady diffusion system with a circular interface, and good agreement is observed when comparing the lattice Boltzmann results with respective analytical solutions. A more general system with unsteady convection-diffusion process and a curved interface, i.e., the cooling process of a hot cylinder in a cold flow, is also simulated as an example to illustrate the practical usefulness of our model, and the effects of the cylinder heat capacity and thermal diffusivity on the cooling process are examined. Results show that the cylinder with a larger heat capacity can release more heat energy into the fluid and the cylinder temperature cools down slower, while the enhanced heat conduction inside the cylinder can facilitate the cooling process of the system. Although these findings appear obvious from physical principles, the confirming results demonstrates the application potential of our method in more complex systems. In addition, the basic idea and algorithm of the counter-extrapolation procedure presented here can be readily extended to other lattice Boltzmann models and even other computational technologies for heat and mass transfer systems.

Phase noise, and phase models: Recent developments, new insights and open problems

This paper presents new insights and a review of recent results on the application of phase models to phase noise analysis in nonlinear oscillators. It is shown that an approach based on stochastic processes and the theory of stochastic differential equations offers several advantages. It is also shown that white noise is responsible for phase diffusion and frequency shift in oscillators, that is, the phase noise problem is best described as a convection diffusion process. An approach for the derivation of a reduced phase model and the associated Fokker-Planck equation is presented. These equations allow an easy determination of fundamental properties as the expectation value for the oscillation frequency, the probability distribution of the phase, the correlation function and the power spectral density.

Analysis of the tunable liquid gradient index based on optofluidics

A new type of tunable gradient-index (GRIN) microfluidic lens is proposed, which is based on the convection-diffusion process. By using the finite element method, the spatial distribution of the refractive-index (RI) determined by the concentration distribution in the microchannel is analyzed. Results show that the distribution of RI in the microchannel can be tuned by different flow rates of the core and cladding liquid streams. Furthermore, although taking into consideration the existence of the viscous resistance between the channel wall and the flowing stream, the RI distribution is still dependent on the combination of flow rates of the core and cladding streams. This microfluidic lens can be applied to dynamic adjustment of focusing, splitting or bending light beams. It therefore may have extensive applications in the optofluidic optical detecting system and microscale imaging system.

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A theoretical description of the convection-diffusion process in a homogeneous system enabling estimation of diffusion coefficients employing commercially available Y-junction microchannel is presented. A detailed numerical analysis based on finite volumes and finite differences, namely the explicit, implicit and Crank-Nicolson method, was performed and analyzed on the same domain in order to verify the proposed models. All numerical approaches provided stable solutions with certain numerical variations depending on the number of iterations defined by the mesh density. In addition, the method was validated with measurements of diffusion coefficients of some selected components in the short Y-junction microchannel. Benefits and possible pitfalls of this estimation method are discussed.

Phase Noise and Noise Induced Frequency Shift in Stochastic Nonlinear Oscillators

Phase noise plays an important role in the performances of electronic oscillators. Traditional approaches describe the phase noise problem as a purely diffusive process. In this paper we develop a novel phase model reduction technique for phase noise analysis of nonlinear oscillators subject to stochastic inputs. We obtain analytical equations for both the phase deviation and the probability density function of the phase deviation. We show that, in general, the phase reduced models include non Markovian terms. Under the Markovian assumption, we demonstrate that the effect of white noise is to generate both phase diffusion and a frequency shift, i.e. phase noise is best described as a convection-diffusion process. The analysis of a solvable model shows the accuracy of our theory, and that it gives better predictions than traditional phase models.

Artemether-Soluplus Hot-Melt Extrudate Solid Dispersion Systems for Solubility and Dissolution Rate Enhancement with Amorphous State Characteristics.

This work studied artemether (ARTM) solid dispersion (SD) formulation using mixture of polymer excipient Soluplus, PEG 400, Lutrol F127, and Lutrol F68 melts at temperatures lower than the melting point of ARTM using a laboratory-size, single-screw rotating batch extruder. The effects of three surfactants PEG 400, Lutrol F127, and Lutrol F68 and parameters like mixing temperature, screw rotating speed, and residence time were systematically studied. SEM, XRD, and FT-IR were employed to investigate the evolution of ARTM's dissolution into the molten excipient. Differential scanning calorimetry (DSC) was used to quantitatively study the melting enthalpy evolution of the drug. The results showed that the dissolution rate increased with increasing the ratio of polymer and surfactant to that of drug. It was concluded that the dissolution of the drug in the polymer melt is a convective diffusion process and that laminar distributive mixing can significantly enhance the dissolution rate. The aqueous solubility and dissolution rate of prepared solid dispersion were significantly enhanced. In vitro antimalarial studies revealed marked improvement in IC50 values. Thus hot-melt extrusion (HME) is a promising technology for improving solubility and dissolution profile of ARTM.

The use of incomplete LU decomposition in modeling convection-diffusion processes in an anisotropic medium

The three-dimensional stationary convection-diffusion problem with mixed derivatives, describing the convection-diffusion processes in an anisotropic medium, is considered. The solution of this problem in an anisotropic medium is of significant interest for various models. A finite-difference upwind approximation of this problem is performed on a thirteen-point stencil. Sufficient conditions for the matrix of the system of linear algebraic equations (SLAE) obtained after natural ordering of the grid domain to be an M-matrix are proved. The SLAE is numerically solved by the method of incomplete LU decomposition, which converges for M-matrices. Numerical experiments were performed, which showed the efficiency of this method for solving the convection-diffusion problem both in isotropic and anisotropic media. A drawback of the method is the restriction on the coefficients multiplying mixed derivatives, which enters into the set of sufficient conditions for the matrix of the operator to be an M-matrix.

Approximated model of primary cosmic radiation

The galactic cosmic rays (CR) are the main ionization source at altitude of ~ 3 to 35 km in the atmosphere. An analytical model is proposed, which describes the intensity of galactic cosmic rays in the inner heliosphere (where energy losses have important role) and the outer heliosphere (where convection-diffusion processes are dominant) [Buchvarova et al, 2011]. On the base of computed model parameters for 1 AU integral CR spectra D(>E) for the Earth and outer planet are obtained at solar minimum and maximum. The obtained results are compared with experimental data and theoretical results.

The asymptotic solutions for boundary value problem to a convective diffusion equation with chemical reaction near a cylinder

The work deals with a boundary value problem for a quasilinear partial elliptical equation. The equation describes a stationary process of convective diffusion near a cylinder and takes into account the value of a chemical reaction for large Peclet numbers and for large constant of chemical reaction. The quantity the rate constant of the chemical reaction and Peclet number is assumed to have a constant value. The leading term of the asymptotics of the solution is constructed in the boundary layer as the solution for the quasilinear ordinary differential equation. In this paper, we construct asymptotic expansion of solutions for a quasilinear partial elliptical equation in the boundary layer near the cylinder.

Controllable formation of aromatic nanoparticles in a three-dimensional hydrodynamic flow focusing microfluidic device

Herein we investigate the formation of aromatic organic nanoparticles in a three-dimensional (3D) hydrodynamic flow focusing microfluidic device. We demonstrate a microfluidic based solvent/non-solvent exchange technique that enables controllable formation of aromatic nanoparticles with tunable size and size distribution. The 3D focusing is achieved by hydrodynamically focusing the sample stream with sheathed streams in both horizontal and vertical directions, and solvent/non-solvent exchange happens between the solvent contained sample stream and non-solvent contained horizontally sheathed streams. The 3D focusing effect was visualized using florescence confocal microscopy and the dynamics of solvent (DMF) depletion in the focused stream was calculated using a finite element computation software package COMSOL Multiphysics. By analyzing the results of self-assembled aromatic nanoparticles in the microfluidic device, we find that the speed of DMF depletion strongly influences the size and size distribution of self-assembled aromatic nanoparticles, and a rapid depletion of DMF is critical for achieving small aromatic nanoparticles with narrow size distribution. This work suggests that our 3D hydrodynamic flow focusing microfluidic device with the ability to precisely control the convective-diffusion process, and continuously vary the fluid flow conditions is promising for studying the formation of nanomaterials.

Robust state estimation for a class of convection-diffusion-reaction systems

This paper presents some state reconstruction scheme for a class of infinite dimensional systems, from the only knowledge of boundary conditions. The considered model is typical of convection-diffusion processes with possible additional reaction phenomenon. The state reconstruction is based on a simple boundary output injection and a full convergence analysis. The robustness of the estimation with respect to modeling errors is also discussed. It is shown that if one employs more sensors to get more measurements, the robustness can be reinforced. An illustrative example with pollutants transport problem is presented accordingly. The proposed approach is finally validated in simulation by using the newly developed Lattice Boltzmann Method.

Numerical modeling of the dynamics of a convection diffusion process locally non-equilibrium in time

The dynamics of a locally non-equilibrium (in time) process of convection diffusion of pollutions during 2D stationary scheduled filtration of underground water to a water intake is mathematically modeled.

Classical implicit travelling wave solutions for a quasilinear convection–diffusion equation

We discuss classical implicit solutions to the partial differential equationut=(H(u))xx+(G(u))x,a general convection–diffusion PDE with particular subcases appearing in many areas of fluids and astrophysics. As an illustrative example, and to compare our results with those present in the literature, we frequently consider travelling wave solutions for the quasilinear PDEut=(um)xx+(un)x,which has been used to describe the flow of viscous fluids on an inclined bed and as a model of convection–diffusion processes. When n⩾m>1, this equation can be used to model the flow of a fluid under gravity through a homogeneous and isotropic porous medium. The travelling wave ODE for both the general and more specific cases have a first integral which is used to obtain an implicit solution for the travelling wave profiles. We should mention that, for some values of m, the implicit relation can be solved in closed form for explicit exact solutions. In the case of n=2m−1, solving the implicit relation gives a general way of obtaining the solutions found in Vanaja [Vanaja, V., 2009. Physica Scripta 80, p. 045402] where the travelling wave solutions for the cases (m,n)=(2,3) and (m,n)=(3,5) were explicitly constructed using a more complicated ansatz method. For other more complicated cases where inversion cannot be performed, we apply the method of series reversion to construct series solutions from the implicit relations. Furthermore, we deduce the dependence of travelling wave solutions on the wave speed, even in cases where the explicit exact solution cannot be found.

Direct two-dimensional electrochemical impedance spectra simulation for solid oxide fuel cell

A two-dimensional (2D) EIS simulation approach is developed by solving a SOFC unit cell model with imposed sinusoidal voltage perturbations at different frequencies. The transient SOFC unit cell model describes the intricate interdependency among the ionic/electronic conduction, multi-component species transport, electrochemical reaction processes and electrode microstructure as well as the coupling processes of mass, energy, momentum transport within flow channels. The model calculates the local transient response and impedance spectra as a function of channel position. The effects of the reaction depletion, product accumulation as well as the temperature variation along the flow channels on the EIS spectra are numerically simulated with a counter-flow mode. The results show that the convection-diffusion process along the flow channel has significant effects on the low frequency half circle of the impedance spectra. The temperature oscillations accumulate along the flow channels, and then affect the current responses which probably lead to an electro-thermal impedance effects.

Mathematical Modeling of Dynamicsof the Process of Filtration Convective Diffusion under the Condition of Time Nonlocality

Mathematical modeling of dynamics of locally unbalanced in time process of convective diffusion of soluble substances for two-dimensional steady plane-vertical filtration with a free surface is performed.