Diffusion equations in three dimensions

Exercise. A particle obeys the ordinary diffusion equation in three dimensions. It starts at a given point inside a given sphere. Find the probability distribution of its exit points on the sphere. 

The atmospheric diffusion equation in three dimensions requires horizontal boundary conditions, two for each of the x. y, and z directions. The only exception are global-scale models simulating the whole Earth s atmosphere. One usually specifies the concentrations at the horizontal boundaries of the modeling domain as a function of time ... 

The right-hand side of Eq. (12.89f) contains a factor 4 which does not appear in the definition of the dimensionless time for the UMDE, Eq. (12.16). This is because space is normalized by the half width of the electtode, w/2, while for the UMDE the radius a is used. The diffusion equation in three dimensions is then... 

Let us now consider a random walker in a three-dimensional cubic lattice. The atom will jump between sites of the normal lattice for a substitutional diffuser, and from interstitial to interstitial site for an interstitial diffuser. In the present case, the Einstein-Smoluchovskii equation for the diffusion coefficient in three dimensions which is a generalization of Equation 5.36, that is,... 

We discuss the diffusion process in three dimensions in the context of an anisotropic convection-diffusion equation for the density of particles. Our goal is to obtain the probabilistic solution to the initial value problem... 

According to classical equations of transla-tional Brownian motion in three dimensions, the mean diffusion distance of a particle during time t is (6 Dt)1 2. 

Self-diffusion and tracer diffusion are described by Equation 3-10 in one dimension, and Equation 3-8 in three dimensions. For interdiffusion, because D may vary along a diffusion profile, the applicable diffusion equation is Equation 3-9 in one dimension, or Equation 3-7 in three dimensions. The descriptions of multispecies diffusion, multicomponent diffusion, and diffusion in anisotropic systems are briefly outlined below and are discussed in more detail later. 

Steady-state diffusion in three dimensions with spherical symmetry (i.e., the concentration is a function of r only) is described by an ordinary differential equation (which is Equation 3-28 simplified for spherical symmetry, cf. Equation 3-66b later) ... 

Equation 7.52 is of central importance for atomistic models for the macroscopic diffusivity in three dimensions (see Chapter 8). For isotropic diffusion in a system of dimensionality, d, the generalized form of Eq. 7.52 is... 

Following the above equation it is obvious that if a plot of 1 /T versus 1 /if is linear, the system is categorized as ID. At low frequency, the ID diffusion breaks down because of inter-chain hopping and 2D or 3D behaviour is expected. In two dimensions, /(co) displays a logarithmic divergence, while in three dimensions, it is nearly constant. The crossover between ID and 2D or 3D regimes occurs at cu sD , which is the inter-chain diffusion rate.106... 

If all advection, dispersion, diffusion, and adsorption processes in three dimensions are considered, the three-dimensional MRTM governing equation can be expressed in the following form ... 

Fluid Flow in Two and Three Dimensions 11 Convective Diffusion Equation in... 

The factor of appears in equation (21-19) because molecules confined to narrow channels probably collide with the walls of a tube, for example, that are separated by 2(raverage), and the dimensionality of the system is 3 for random Brownian motion in three dimensions. In many cases, the factor of /3 in (21-19) is replaced by the kinetic theory prediction of y/S/jt when Knudsen is based on the average speed of the gas molecules (i.e., (u, ) = SRT/ttMW,). Now the Knudsen diffusion coefficient is given by 92% of (21-19) (see Moore, 1972, p. 124 Bird et al., 2002, pp. 23, 525 Dullien, 1992, p. 293 and Smith, 1970, p. 405). If the average pore size is expressed in angstroms and the temperature... 

Except for this section and Section 18.7. the solutions of the unsteady diffusion equation in one to three dimensions are beyond the scope of this book. Solutions to Eqs. (15-12c. d, e), the corresponding two-and three-dimension equations, and the equivalent heat conduction equations have been extensively studied for a variety of boundary conditions (e.g., Crank. 197S Cussler. 2009 Incropera et al 2011). Readers interested in unsteady-state diffusion problems should refer to these or other sources on diffusion. 

Diffusion results from Brownian motion, the random battering of a molecule by the solvent. Let s apply the one-dimensional random walk model of Chapter 4 (called random flight, in three dimensions) to see how far a peirticle is moved by Brownian motion in a time t. A molecule starts at position x = 0 at time t = 0. At each time step, assume that the particle randomly steps either one unit in the +x direction or one unit in the -x direction. Equation (4.34) gives the distribution of probabilities (which we interchangeably express as a concentration) c(x, N) that the particle will be at position x after N steps,... 

TWINKLE is a multidimensional spatial neutron kinetics code, whieh is patterned after steady-state codes currently used for reactor core design. The code uses an implicit finite-difference method to solve the two-group transient neutron diffusion equations in one, two, and three dimensions. The code uses six delayed neutron groups and contains a detailed multi-region fuel-clad-coolant heat transfer model for calculating point-wise Doppler and moderator feedback effects. The code handles up to 2000 spatial points and performs its own steady-state initialisation. Aside from basic cross-section data and thermal-hydraulic parameters, the code accepts as input basic driving functions, such as inlet temperature, pressure, flow, boron concentration, control rod motion, and others. Various edits are provided (for example, channel-wise power, axial offset, enthalpy, volumetric surge, point-wise power, and fuel temperatures). 

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