Technology & Engineering
Diffusion Equation
The diffusion equation is a partial differential equation that describes how a quantity such as heat, mass, or momentum diffuses through a medium over time. It is widely used in engineering and technology to model various physical processes, such as heat conduction in materials, chemical diffusion in solutions, and the spread of pollutants in the environment.
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6 Key excerpts on "Diffusion Equation"
eBook - PDFPartial Differential Equations
A First Course
- Rustum Choksi(Author)
- 2022(Publication Date)
- American Mathematical Society(Publisher)
As a simple illustration 1 , consider a swarm of millions of bees. Each bee has a particular distinct motion, but from the perspective of swarm tracking it would be overwhelming, and moreover unnecessary, to track each trajectory. Rather one focuses on the overall motion of the swarm and its density variation. 1 The swarm of bees is to illustrate our focus on collective behavior, not as an example of a system which would be modeled by the Diffusion Equation. 269 270 7. The Diffusion Equation It is when describing a certain ubiquitous collective behavior of many sys-tems that the Diffusion Equation proves invaluable. To quote the excellent text on sta-tistical mechanics by Sethna 2 : You may think that Newton’s law of gravitation, or Einstein’s refinement to it, is more fundamental than the Diffusion Equation. You would be correct; gravitation applies to everything. But the simple macroscopic law of gravitation emerges, presumably, from a quantum exchange of immense numbers of virtual gravitons just as the Diffusion Equation emerges from large numbers of long random walks. The Diffusion Equation and other con-tinuum statistical mechanics laws are special to particular systems, but they emerge from the microscopic theory in much the same way as gravitation and the other fundamental laws of nature do. This is the source of many of the surprisingly simple mathematical laws describing nature. Roughly speaking, the Diffusion Equation models the concentration, or ensemble average, of some quantity which is diffusing through a medium. The meaning of diffusing will be made precise shortly. As Albert Einstein noted in one of his seminal papers of 1905 (his Annus Mirabilis 3 ), a natural way to interpret the Diffusion Equation is in terms of the probabilities for the position of a particle undergoing a certain random motion called Brownian motion . We will address this interpretation in Sections 7.3 and 7.4.
eBook - PDF- David C. Venerus, Hans Christian Öttinger(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
2 The Diffusion Equation One of the most famous equations in the field of transport phenomena is the Diffusion Equation. Its wide-ranging importance is underlined by the fact that, depending on the context, it is known by various names. In a proba-bilistic interpretation, it is usually referred to as a Fokker–Planck equation, which is a special type of Kolmogorov’s forward equation for memoryless stochastic processes. 1 In the context of Brownian motion, the name Smolu-chowski equation is most appropriate. The variety of names nicely indicates that this equation is not only useful for describing the transport phenomenon of mass diffusion; we will actually encounter it many times, in particular, also in the description of momentum and heat transport and in polymer kinetic theory. In the present chapter, we introduce it to describe the flow of probability. In doing so, we present the basic theme of transport phenomena, including some important concepts, tools, and results. 2.1 A Partial Differential Equation By a first glance at the Fokker–Planck or Diffusion Equation in one space dimension, ∂p ( t, x ) ∂t = − ∂ ∂x A ( t, x ) p ( t, x ) + 1 2 ∂ 2 ∂x 2 D ( t, x ) p ( t, x ) , (2.1) one recognizes a second-order partial differential equation for the evolution of some function p ( t, x ) of two real arguments involving coefficient functions A ( t, x ) and D ( t, x ). Here we simply assume D ( t, x ) ≥ 0, but later will show that this follows from the second law of thermodynamics (see Exercise 2.4 ). In many cases, the given coefficient functions A and D are independent of their first argument, t . The goal of this chapter is to bring the reader 1 See, for example, Chapter 3 of Gardiner, Handbook of Stochastic Methods (Springer, 1990). 8 The Diffusion Equation from this superficial perspective to a deep understanding of the physical meaning and implications of the Diffusion Equation ( 2.1 ).
eBook - ePubModels and Modeling
An Introduction for Earth and Environmental Scientists
- Jerry P. Fairley(Author)
- 2017(Publication Date)
- Wiley-Blackwell(Publisher)
CHAPTER 8 The transport equationChapter summary
Up to this point, we have concentrated on the problem of understanding potential flow—nominally, the flow of water in a porous medium, although the equations we have developed apply equally to species diffusion or the diffusion of heat. In this chapter, however, we will go beyond diffusion in a stationary medium and examine the movement of an intensive quantity that is being transported by a combination of diffusion and advection in a moving fluid. This conceptual model applies to the transport of heat by groundwater (e.g., in a hydrothermal system or in the vicinity of a buried package of high‐level nuclear waste), a conservative species, such as lithium bromide, added to groundwater as a tracer, or any contaminant that moves in solution as part of an advecting fluid. The transport equation is even general enough to cover situations such as modeling the temperature distribution in a tectonic plate at a spreading center or a subduction zone; as such, it is the most general equation we will examine in this text, and will require that we bring together our understanding of many of the most important aspects of the previous chapters into a coherent whole.8.1 The advection–dispersion equation
Our discussions of the Laplace equation, the Poisson equation, and the transient Diffusion Equation all share a common thread: in each of those three equations, the ultimate distribution of the dependent variable is driven by a process of diffusion. In essence, diffusion is a statistical‐mechanical process in which random motions of molecules drive the value of the dependent variable from regions of high concentration to regions of lower concentration; this is also called potential flow
- John S. Gulliver(Author)
- 2007(Publication Date)
- Cambridge University Press(Publisher)
2 The Diffusion Equation In this chapter, we will review various solution techniques for the Diffusion Equation, which is generally defined as the mass transport equation with diffusive terms. These techniques will be applied to chemical transport solutions in sediments. There are also a number of applications to chemical transport in biofilms. There are many other applications of the Diffusion Equation, including most of the topics of this text, but they require more background with regard to the physics of mixing processes, which will be addressed in later chapters. What is mass (or chemical ) transport ? It is the transport of a solute (the dissolved chemical) in a solvent (everything else). The solute is the dissolvee and the solvent is the dissolver. There are liquids that are generally classified as solvents because they typically play that role in industry. Some examples would be degreasing and dry cleaning solvents, such as trichloroethylene. In environmental applications, these “solvents” are the solutes, and water or air is usually the solvent. In fact, when neither water nor air is the solvent, the general term nonaqueous phase liquid is applied. A nonaqueous phase liquid is defined as a liquid that is not water, which could be composed of any number of compounds. The substance being transported can be either dissolved (part of the same phase as the water) or particulate substances. We will develop the Diffusion Equation by considering mass conservation in a fixed control volume. The mass conservation equation can be written as Flux rate − Flux rate + Source − Sink = Accumulation in out rate rate (2.1) Now that we have our mass conservation equation, we must decide which control volume would be the most convenient for our applications. The control volumes used most for this type of mass balance are given in Figure 2.1 . The general control volume, given in Figure 2.1 a, is used for descriptive purposes, to maintain generality.
eBook - PDFBrownian Motion
A Guide to Random Processes and Stochastic Calculus
- René L. Schilling(Author)
- 2021(Publication Date)
- De Gruyter(Publisher)
23 On diffusions Diffusion is a physical phenomenon which describes the tendency of two (or more) substances, e.g. gases or liquids, to reach an equilibrium: particles of one type, say B, move or “diffuse” in(to) another substance, say S, leading to a more uniform distribution of the type B particles within S. Since the particles are physical objects, it is reasonable to assume that their trajectories are continuous (in fact, differentiable) and that the movement is influenced by various temporal and spatial (in-)homogeneities which depend on the nature of the substance S. Diffusion phenomena are governed by Fick’s law. Denote by p = p(t , x) the concentration of the B particles at a space-time point (t , x), and by J = J(t , x) the flow of particles. Then J = −D ∂p ∂x and ∂p ∂t = − ∂J ∂x where D is called the diffusion constant which takes into account the geometry and the properties of the substances B and S. Einstein explained in 1905 the particle movement observed by Brown – under the assumption that the movement is temporally and spatially homogeneous – as a diffusion phenomenon: ∂p(t , x) ∂t = D ∂ 2 p(t , x) ∂x 2 hence p(t , x) = 1 √ 4πD e − x 2 4Dt . The diffusion coefficient D = RT N 1 6πkP depends on the absolute temperature T, the universal gas constant R, Avogadro’s number N, the friction coefficient k and the radius of the particles (i.e. atoms) P. If the diffusion coefficient depends on time and space, D = D(t , x), Fick’s law leads to a differential equation of the type ∂p(t , x) ∂t = D(t , x) ∂ 2 p(t , x) ∂x 2 + ∂D(t , x) ∂x ∂p(t , x) ∂x . In a mathematical model of a diffusion we could either use a macroscopic approach and model the particle density p(t , x), or a microscopic point of view modelling the movement of the particles themselves and determine the particle density. Using prob- ability theory we can describe the (random) position and trajectories of the particles by a stochastic process.
eBook - PDF- A. Eduardo Saez, James C. Baygents(Authors)
- 2014(Publication Date)
- CRC Press(Publisher)
As developed, the equation is valid for binary systems; that is, mixtures of two species A and B. However, the equation is also applicable to the transport of A in a multicomponent mixture in which A is dilute. In that case, the binary diffusivity, D AB , is replaced by the diffusivity of A in the mixture, D A , ∂ ∂ + ⋅ ∇ = ∇ + c t c D c R A A A A A v 2 (4.16) This equation, along with expressions of the species molar flux, is presented in expanded form for various coordinate systems in Appendix C. 59 Diffusive Transport For the particular case in which the material is not moving: v ≡ 0, Equation 4.16 is simplified by eliminating the convective term. This is the Diffusion Equation . In this chapter, we consider applications of the Diffusion Equation. 4.2 ONE-DIMENSIONAL, STEADY DIFFUSION The problems to be solved in this section correspond to binary systems or multicom-ponent systems in which the species whose transport is being studied (A) is present at dilute conditions (x A « 1), which makes Fick’s law applicable. For binary systems, the approach to a specific problem will depend on whether the mixture is dilute in A or not: 1. For dilute systems, Equation 4.16 with v = 0 applies. Note that taking v = 0 is really an approximation, since A will be diffusing and hence there will be motion in the material. However, bulk motion will be negligible due to the low concentrations of A. In terms of the representation of the molar flux of A, we can see that, since x A «1, we can state that N J A A ≈ * (4.17) and the components of the molar flux of A can be directly found from Fick’s law. 2. For concentrated binary systems, the continuity equation in terms of the molar flux (Equation 4.7) must be used, along with the general form of the flux equation (Equation 4.8). For these problems, information will be needed on both components (i.e., if solving for concentration profiles and flux of A, we will need to have information on the flux of B).
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