Diffusion number

The dimensionless parameter, DAtjAt is known as the diffusion number, Di. For stability of an explicit computational solution (i.e., for the solution to make any sense at all), this number must be less than 1/2. The logic behind this stability criterion can be seen in equation (E7.1.4). If Di is greater than 1 /2, then the relation between Q, +i and is negative, creating a computational solution that can have serious errors. 

We will select At = 3,600 s, which results in a diffusion number of 0.36. Our initial concentration in the surface control volumes is... 

The problem of Example 7.3 will again be solved with explicit and implicit exponential differences, and compared with the analytical solution, equation (E7.4.7). This solution is given in Figure E7.5.1. Note that the explicit solution is close to the analytical solution, but at a Courant number of 0.5, whereas the implicit solution could solve the problem with less accuracy at a Courant number of 5. In addition, the diffusion number of the explicit solution was 0.4, below the limit of Di < 0.5. The implicit solution does not need to meet this criteria and had Di = 4. 

Bidiagonal systems result from Eqs. (10.22) and (10.23). These can be solved efficiently using various routines. The algorithm is unconditionally stable and second-order accurate if the diffusion number At... 

The last three pi groups are well known in chemical engineering (113 is recognized as the Fourier reaction number (Fo ), 114 is the famous Biot diffusion number (Bi(j) and Hs is the Sherwood number (Sh)). 

The diffusion moisture-transfer number, with respect to the heat diffusion or the moisture- flow diffusion number (drying Luikov number [6.23, 6.28]) ... 

Figure 5 presents the data obtained with a diffuser-called diffuser number 1 - for particles generated at 1.09. 0.460 and 0.176 micron. It can be seen that to exclude the smaller particles, pressures greater than 85 psig were required while 60 psig were needed to exclude the 1.09 micron particles. A new diffuser - number 2 - was tested and It was found to require pressures of approximately 20 psig to exclude all ambient particles as shown In Figure 6. 

Consider convection diffusion toward a spherical particle of radius R, which undergoes translational motion with constant velocity U in a binary infinite diluted solution [3], Assume the particle is small enough so that the Reynolds number is Re = UR/v 1. Then the flow in the vicinity of the particle will be Stoke-sean and there will be no viscous boundary layer at the particle surface. The Peclet diffusion number is equal to Peo = Re Sc. Since for infinite diluted solutions, Sc 10 and the flow can be described as Stokesian for the Re up to Re 0.5, it is perfectly safe to assume Pec 1. Thus, a thin diffusion boundary layer exists at the surface. Assume that a fast heterogeneous reaction happens at the particle surface, i.e. the particle is dissolving in the liquid. The equation of convective diffusion in the boundary diffusion layer, in a spherical system of coordinates r, 6, (p, subject to the condition that concentration does not depend on the azimuthal angle [Pg.128]

Here the Peclet diffusion number is introduced swhl Uyr,n... 

We assume that deposition on the sphere is ideal, that is, each collision of a particle with the sphere results in the particle being captured. The factor of Brownian diffusion Dj,r = kTwhere Oj, is the particle s radius, is much smaller than the factor of molecular diffusion, therefore the Peclet diffusion number is Peo = Ua/Dhr 1- By virtue of this inequality (see Section 6.5), the diffusion flux of particles toward the sphere can be found by solving the stationary equation of convective diffusion with a condition corresponding to a thick or thin diffusion-boundary layer. Particles may then be considered as point-like, and the diffusion equation will become ... 

The advection of coolant in a time step is limited below one-third of the mesh size, which means that the Courant number is below one-third. The diffusion in a time step is limited below one-third of the mesh size, which means that the diffusion number is below one-third. 

Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

The oil droplets in a certain benzene-water emulsion are nearly uniform in size and show a diffusion coefficient of 3.75 x 10 cm /sec at 25°C. Estimate the number of benzene molecules in each droplet. 

LID) see Ref. 139. In this last method, a small area, about 0.03 cm radius, is depleted by a laser beam, and the number of adatoms, N(t), that have diffused back is found as a function of time. From Pick s second law of diffusion ... 

Micellization is a second-order or continuous type phase transition. Therefore, one observes continuous changes over the course of micelle fonnation. Many experimental teclmiques are particularly well suited for examining properties of micelles and micellar solutions. Important micellar properties include micelle size and aggregation number, self-diffusion coefficient, molecular packing of surfactant in the micelle, extent of surfactant ionization and counterion binding affinity, micelle collision rates, and many others. 

Model colloids have a number of properties that make them experimentally convenient and interesting systems to study. For instance, the timescale for stmctural relaxation of a colloidal fluid can be estimated as the time for a particle to diffuse a distance equal to its radius,... 

The tendency for particles to settle is opposed by tlieir Brownian diffusion. The number density distribution of particles as a function of height z will tend to an equilibrium distribution. At low concentration, where van T Ftoff s law applies, tire barometric height distribution is given by... 

Figure C2.7.13. Schematic representation of diffusion and reaction in pores of HZSM-5 zeolite-catalysed toluene disproportionation the numbers are approximate relative diffusion coefficients in the pores 1131.

These number fluctuations, i.e. the A fy ) tenn, will constantly tend to be eliminated by diffusion. On the other hand, because of the correlation between s and i, initial inliomogeneities in their spatial densities lead to the... 

Tlere, y Is the friction coefficien t of the solven t. In units of ps, and Rj is th e random force im parted to th e solute atom s by the solvent. The friction coefficien t is related to the diffusion constant D oflh e solven l by Em stem T relation y = k jT/m D. Th e ran doin force is calculated as a ratulom number, taken from a Gaussian distribn-... 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

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