Diffusion when negligible

Since the void fraction distribution is independently measurable, the only remaining adjustable parameters are the A, so when surface diffusion is negligible equations (8.23) provide a completely predictive flux model. Unfortunately the assumption that (a) is independent of a is unlikely to be realistic, since the proportion of dead end pores will usually increase rapidly with decreasing pore radius. 

When diffusion is assumed to be controlled by the boundary him, by implication, all other resistances to diffusion are negligible. Therefore, concentrations are uniform through the solid and local equilibrium exists between huid and solid. The whole of the concentration difference between bulk liquid and solid is conhned to the him. The rate of transfer into a spherical pellet may then be expressed as ... 

Fig. A2. Illustration of the pressure profiles through a rod when substantial bulk diffusion and negligible Poiseuille flow are occurring.

Effect of Chlorine on Oxidation-Enhanced Diffusion. If chlorine is added to oxygen in the furnace in sufficient concentrations such that stacking-fault retrogrowth occurs (46), then oxidation-enhanced diffusion becomes negligible (47). This result is believed to be due to the generation of vacancies at this Si-Si02 interface when Cl reacts with Si atoms on lattice sites to produce SiCl by the following reaction... 

We now consider some important limiting cases of this model. First, we note that when axial molecular diffusion is negligible, Eqs. (130)—(131) reduce to hyperbolic model ... 

The definitions of effective diffusivity tensors are key parameters in the solution of the transport equations above. For an isotropic medium, the effective diffusivity is insensitive to the detailed geometric structure, and the volume fraction of the phases A and B influences the effective diffusivity. When the resistance to mass transfer across the cell membrane is negligible, the isotropic effective diffusivity, Ds e = Dg eI may be obtained from Maxwell s equation... 

It must be kept in mind that these measurements were made with water, in which the molecular diffusivity is negligibly small. When the fluid is a gas, with a Schmidt number having the order of magnitude of unity, the Peclet number for molecular diffusion, being the product of Schmidt number and Reynolds number, has the order of magnitude of the Reynolds number. Even if the effect of molecular diffusion is multiplied by the fraction void and by another factor of %, to take care of... 

The equations governing the steady state, quasi-one-dimensional flow of a reacting gas with negligible transport properties can easily be obtained from equations (l-19)-(l-22). When transport by diffusion is negligible 0 and Dtj 0 for ij = 1,..., N the diffusion velocities, of course, vanish [FJ 0 for / = 1,..., N, see equation (1-14)]. If, in addition, transport by heat conduction is negligible (A 0) and = 0, then the heat flux q vanishes [see equation (1-15)]. Finally, in inviscid flow 0 and K 0), equations (1-16)-(1-18) show that all diagonal elements of the pressure tensor reduce to the hydrostatic pressure, pu = pjj — P33 = P-The steady-state forms of equations (1-20), (l-21a), and (1-22) then become... 

We now make two further assumptions, that molecular diffusion is negligible when compared with turbulent diffusion and that the atmosphere is incompressible. With reference to (3) this means that the first term on the right hand side of the equation is negligibly small compared... 

Knudsen diffusion becomes predominant when the mean-free path of the molecular species considered is larger than the pore diameter and hence, when the effects of the molecule-wall collisions become important. In liquid chromatography, Knudsen diffusion is negligible. 

The equations governing the steady-state, quasi-one-dimensional flow of a reacting gas with negligible transport properties can easily be obtained from equations (l-19)-(l-22). When transport by diffusion is negligible (D,j 0 and Dr,i 0 for i,j = 1,..., N), the diffusion velocities, of course, vanish - 0 for j = see equation (1-14)]. If, in addi-... 

The reactant molecules transfer from the entrance of the reactor to the neighbourhood of the catalyst pellets. This transfer takes place by convection and/or diffusion. When axial diffusion is negligible and radial diffusion is instantaneous, we get the simplest description for the bulk phase, that is one-dimensional piug/flbw. 

As shown in the previous section, scaling with geometric similarity, Sr = Sl = gives constant pressure drop when the flow is laminar and remains laminar upon scaleup. This is true for both liquids and gases. The Reynolds number and the external area increase as. Piston flow is a poor assumption for laminar flow in anyfhing but small tubes. Conversion and selectivity of the reaction is likely to worsen upon scaleup unless the pilot reactor is already so large that molecular and thermal diffusion are negligible on the pilot scale. Ways to avoid unpleasant surprises are discussed in Chapter 8... 

Inpolydisperse systems the blurriness of sedimentation boundary is related to both the diffusion and the differences in the sedimentation rates of particles having different sizes. In cases when diffusion is negligible, the c(R) dependence represents the shape of integral particle size distribution curve at any moment of time. 

In standard RDE theory, one also assumes a negligible radial diffusion contribution to the current at the edges of the disk. This is true if the disk radius is sufficiently large that diffusion to it can be treated as simple linear diffusion. When the disk electrode radius falls in the ultramicroelectrode regime, this is no longer the case, so different equations will apply to the rotating microdisk, as discussed in Section 9.7. 

When we operate a batch reactor, we inherently assume that molecular diffusion is negligible compared to the physically induced agitation of the reacting mixture. Figure 3.12 displays individual polynomial coefficients as a function of RPM (revolutions per minute), where RPM represents quality of agitation. This figure shows that the... 

When a tracer is injected via a drillhole into groundwater, it is subject to diffusion, dispersion, dilution and adsorption. Dispersion is the result of very small variations in the velocity of laminar flow through porous media. Molecular diffusion is negligible, unless the velocity of flow is unusually low. Even if these processes are not significant, flow through an aquifer may be stratified or concentrated along discontinuities. Therefore, a tracer may remain undetected unless the observation drillholes intersect these discontinuities. 

Other analogies. The Reynolds analogy assumes that the turbulent diffusivitiess, a, and are all equal and that the molecular difTusivities p/p, a, and are negligible compared to the turbulent diffusivities. When the Prandtl number (p/p)/a is 1.0, then pip = a also, for = 1.0,/i/p = D g. Then,(p/p + e,) = (a + a,) = (D g + e, ) and the Reynolds analogy can be obtained with the molecular terms present. However, the analogy breaks down when the viscous sublayer becomes important since the eddy difTusivities diminish to zero and the molecular diffusivities become important. 

According to the peak current, the concentrations of the reactant can be quantified when the diffusivity is negligible. The potential at which the peak current occurs can be used to identify the reaction. Based on the half-cell potential of the electrochemical reactions, the identification for reactions or reactants is listed extensively in handbooks and references. Amperometric sensors can be used very effectively to carry out qualitative and quantitative analyses of chemical and biochemical species. 

Self-Diffusion and Mutual Diffusion When each suspension or solute molecule is moving independently, the diffusion is a single-particle phenomenon. The latter is observed in the dilute solution limit where there are no other solute molecules in the neighborhood. When other solute molecules are nearby, the diffusion is strongly affected by the other solute molecules. The second terms in Eqs. 3.17 and 3.45 are not negligible any more. What DLS measures is 5(k, t), not 5i(k, r). Only when c c, 5(k, t) is equal to 5i(k, t). Otherwise, the apparent diffusion coefficient D estimated from the slope of gi(r) depends on c. We will learn how the apparent D depends on c. 

When surface diffusion is negligible, Pe, the adsorption and desorption rates are slow (the insoluble surfactant limit) and the bubble motion is steady, one obtains stagnant cap behavior for the steady motion. In this regime, Eq. (39) reduces to... 

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