6.2 Solutions of diffusion equations for constant ternary interdiffusion coefficients

2005 ◽  
pp. 372-375 ◽  
Author(s):  
M. A. Dayananda
Keyword(s):  
Diffusion Equations ◽  
Interdiffusion Coefficients

scholarly journals InfPolyn, a Nonparametric Bayesian Characterization for Composition-Dependent Interdiffusion Coefficients

Materials ◽  
2021 ◽  
Vol 14 (13) ◽  
pp. 3635
Author(s):  
Wei-W. Xing ◽  
Ming Cheng ◽  
Kaiming Cheng ◽  
Wei Zhang ◽  
Peng Wang
Keyword(s):  
Interdiffusion Coefficient ◽  
Diffusion Equations ◽  
Numerical Inversion ◽  
Extended Model ◽  
Parametric Form ◽  
Physical Processes ◽  
Large Margin ◽  
Statistical Framework ◽  
Fitting Method ◽  
Interdiffusion Coefficients

Composition-dependent interdiffusion coefficients are key parameters in many physical processes. However, finding such coefficients for a system with few components is challenging due to the underdetermination of the governing diffusion equations, the lack of data in practice, and the unknown parametric form of the interdiffusion coefficients. In this work, we propose InfPolyn, Infinite Polynomial, a novel statistical framework to characterize the component-dependent interdiffusion coefficients. Our model is a generalization of the commonly used polynomial fitting method with extended model capacity and flexibility and it is combined with the numerical inversion-based Boltzmann–Matano method for the interdiffusion coefficient estimations. We assess InfPolyn on ternary and quaternary systems with predefined polynomial, exponential, and sinusoidal interdiffusion coefficients. The experiments show that InfPolyn outperforms the competitors, the SOTA numerical inversion-based Boltzmann–Matano methods, with a large margin in terms of relative error (10x more accurate). Its performance is also consistent and stable, whereas the number of samples required remains small.

The Growth Rates of Intermediate Phases in Co/Si Diffusion Couples: Bulk Versus Thin-Film Studies

1989 ◽  
Vol 148 ◽  
Author(s):  
C. H. Jan ◽  
J. C. Lin ◽  
Y. A. Chang
Keyword(s):  
Experimental Data ◽  
Growth Rates ◽  
Film Studies ◽  
Bulk Diffusion ◽  
Diffusion Equations ◽  
Diffusion Couples ◽  
Intermediate Phases ◽  
Diffusion Controlled ◽  
Composition Profiles ◽  
Interdiffusion Coefficients

ABSTRACTBulk diffusion couples of Co/Si were annealed at 800, 900, 1000, 1050 and 1100°C for periods ranging from 24 hours to one month. Growth rates of the intermediate phases, Co2Si, CoSi and CoSi2, as well as the composition profiles across the couples were determined by optical microscopy and electron probe microanalysis (EPMA). Using the solution to the multiphase binary diffusion equations and the experimental data, the interdiffusion coefficients for Co2Si, CoSi and CoSi2 are obtained as a function of temperature. The activation energies obtained are 140, 160 and 190 KJ/mole for Co2Si, CoSi and CoSi2, respectively. The generally small interdiffusion coefficient of CoSi2 and its high activation energy cause the growth rate of CoSi2 to be extremely small at low temperatures.The interdiffusion coefficients for Co2Si, CoSi and CoSi2 at 545°C are obtained by extrapolation of the high-temperatures data. Using these data and solving numerically the diffusion equations with the appropriate boundary conditions, the growth of Co2Si, CoSi and CoSi2 is calculated as a function of time. The calculated results are in good agreement with the experimental data reported in the literature. This study demonstrates clearly that the initial absence of the CoSi2 phase is due to diffusion-controlled rather than nucleation-controlled kinetics. This phenomenon may be quite common in many thin-fiflm metal/Si couples.

Computer Simulation of Amperometric Biosensor Response to Mixtures of Compounds

2002 ◽  
Vol 7 (2) ◽  
pp. 3-14 ◽  
Author(s):  
R. Baronas ◽  
J. Christensen ◽  
F. Ivanauskas ◽  
J. Kulys
Keyword(s):  
Computer Simulation ◽  
Enzymatic Reaction ◽  
Diffusion Equations ◽  
Response Data ◽  
Finite Difference Technique ◽  
Stationary Diffusion ◽  
Biosensor Array ◽  
Menten Kinetic ◽  
Biosensor Response

A mathematical model of amperometric biosensors has been developed. The model bases on non-stationary diffusion equations containing a non-linear term related to Michaelis-Menten kinetic of the enzymatic reaction. The model describes the biosensor response to mixtures of multiple compounds in two regimes of analysis: batch and flow injection. Using computer simulation, large amount of biosensor response data were synthesised for calibration of a biosensor array to be used for characterization of wastewater. The computer simulation was carried out using the finite difference technique.

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Kolmogorov and classic diffusion equations

1988 ◽  
Vol 53 (6) ◽  
pp. 1181-1197
Author(s):  
Vladimír Kudrna
Keyword(s):  
Mass Transfer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Parabolic Type ◽  
Diffusion Equations ◽  
Stochastic Motion ◽  
Energy Component ◽  
Classic Form

The paper presents alternative forms of partial differential equations of the parabolic type used in chemical engineering for description of heat and mass transfer. It points at the substantial difference between the classic form of the equations, following from the differential balances of mass and enthalpy, and the form following from the concept of stochastic motion of particles of mass or energy component. Examples are presented of the processes that may be described by the latter method. The paper also reviews the cases when the two approaches become identical.

Certain problems with the application of stochastic diffusion processes for description of chemical engineering phenomena; Diffusion equations in curvilinear coordinates

1991 ◽  
Vol 56 (3) ◽  
pp. 602-618
Author(s):  
Vladimír Kudrna
Keyword(s):  
Heat Transfer ◽  
Mass Transport ◽  
Probability Density ◽  
Chemical Engineering ◽  
Diffusion Processes ◽  
Curvilinear Coordinates ◽  
Diffusion Equations ◽  
Parabolic Partial Differential Equations ◽  
Spherical Coordinates ◽  
Transformation Procedure

Parabolic partial differential equations used in chemical engineering for the description of mass transport and heat transfer and analogous relationship derived in stochastic processes theory are given. A standard transformation procedure is applied, allowing these relations to be generally written in curvilinear coordinates and particular expressions for cylindrical and spherical coordinates to be derived. The relation between the probability density for the position of a discernible particle and the concentration of a set of such particles is discussed.

scholarly journals Discrete effect on single-node boundary schemes of lattice Bhatnagar–Gross–Krook model for convection-diffusion equations

2019 ◽  
Vol 31 (01) ◽  
pp. 2050017
Author(s):  
Liang Wang ◽  
Xuhui Meng ◽  
Hao-Chi Wu ◽  
Tian-Hu Wang ◽  
Gui Lu
Keyword(s):  
Boundary Condition ◽  
Relaxation Time ◽  
Lattice Boltzmann ◽  
Diffusion Equations ◽  
Bgk Model ◽  
Distance Ratio ◽  
Single Node ◽  
Convection Diffusion ◽  
Velocity Models ◽  
Discrete Effect

The discrete effect on the boundary condition has been a fundamental topic for the lattice Boltzmann method (LBM) in simulating heat and mass transfer problems. In previous works based on the anti-bounce-back (ABB) boundary condition for convection-diffusion equations (CDEs), it is indicated that the discrete effect cannot be commonly removed in the Bhatnagar–Gross–Krook (BGK) model except for a special value of relaxation time. Targeting this point in this paper, we still proceed within the framework of BGK model for two-dimensional CDEs, and analyze the discrete effect on a non-halfway single-node boundary condition which incorporates the effect of the distance ratio. By analyzing an unidirectional diffusion problem with a parabolic distribution, the theoretical derivations with three different discrete velocity models show that the numerical slip is a combined function of the relaxation time and the distance ratio. Different from previous works, we definitely find that the relaxation time can be freely adjusted by the distance ratio in a proper range to eliminate the numerical slip. Some numerical simulations are carried out to validate the theoretical derivations, and the numerical results for the cases of straight and curved boundaries confirm our theoretical analysis. Finally, it should be noted that the present analysis can be extended from the BGK model to other lattice Boltzmann (LB) collision models for CDEs, which can broaden the parameter range of the relaxation time to approach 0.5.

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