Diffusion Zero penetrant concentration

Pi)si, and D all converge to the same value at the limit of zero penetrant concentration. This limiting value shall be denoted by a symbol D0. It is important to observe that Eqs. (2) and (3) are valid for one-dimensional diffusion. It appears that no corresponding equation is as yet known for diffusion in higher dimensions. 

According to eq. V- 123, the diffusion coefBcientat zero penetrant concentration Dc->o oisgivenby... 

Figure VI - 14 demonstrates that the diffusion coefficient can change by up to 10 orders of magnitude. Thus the diffusion coefficient of benzene in poly(vinyl alcohol) at zero penetrant concentration is less than 10- 9 nP/s [18], whereas the diffusion coefficient of water in hydrogels is greater than 10-9 nP/s, which is virmally equal to value of the self-diffusion coefficient of water.

Figure 5 shows the diffusion of a solute into such an impermeable membrane. The membrane initially contains no solute. At time zero, the concentration of the solute at z = 0 is suddenly increased to c, and maintained at this level. Equilibrium is assumed at the interface of the solution and the membrane. Therefore, the corresponding membrane concentration at z = 0 is Kc1. Since the membrane is impermeable, the concentration on the other side will not be affected by the change at z = 0 and will still be free of solute. This abrupt increase produces a time-dependent concentration profile as the solute penetrates into the membrane. If the solution is assumed to be dilute, Fick s second law Eq. (9) is applicable ... 

Figure 5 Diffusion into a semi-infinite membrane. The membrane initially contains no solute. At time zero, the concentration of the solution at z = 0 is suddenly increased to and maintained at cx. This abrupt increase produces time-dependent concentration profiles as the solute penetrates into the membrane.

Figure 6 Unsteady diffusion across a membrane. The membrane is initially free of solute. At time zero, the concentrations on the two sides of the membrane are increased to and maintained at c, and c2. The solute penetrates into the membrane from both sides, resulting in time-dependent concentration profiles within the membrane.

The terms 3Ji are here added to the right-hand sides of Eqs. (4.62) in order to allow for possible catalytic formation of the species at a wall boundary. In flame systems, a free radical arriving at a solid surface is assumed to be immediately destroyed there that is, we assume that all radical concentrations are zero at such surfaces. Computationally, radicals which diffuse to the surface are assumed to react at infinite rate with one of the major components of the mixture in such a way as to return a stable species to the gas phase. The terms SJ are necessary to maintain conservation at the boundary while allowing for this. The diffusive fluxes of the radicals are evaluated from Eq. (4.60), and the effect on other fluxes of their conversion to stable species at the boundary is calculated to give the dJ. The assumption of zero radical concentrations at the boundary (or at least zero penetration into the porous wall) is essential if reaction is to be confined to the gas phase. 

The diffusion in a semi-infinite slab is schematically sketched in Fig. 2.3-2. The slab initially contains a uniform concentration of solute ci oo. At some time, chosen as time zero, the concentration at the interface is suddenly and abruptly increased, although the solute is always present at high dilution. The increase produces the time-dependent concentration profile that develops as solute penetrates into the slab. 

Eor a food container, the amount of sorption could be estimated in the following way. Eor simple diffusion the concentration in the polymer at the food surface could be estimated with equation 3. This would require a knowledge of the partial pressure of the flavor in the food. This is not always available, but methods exist for estimating this when the food matrix is water-dorninated. The concentration in the polymer at the depth of penetration is zero. Hence the average concentration C is as from equation 9. 

The permeation technique is another commonly employed method for determining the mutual diffusion coefficient of a polymer-penetrant system. This technique involves a diffusion apparatus with the polymer membrane placed between two chambers. At time zero, the reservoir chamber is filled with the penetrant at a constant activity while the receptor chamber is maintained at zero activity. Therefore, the upstream surface of the polymer membrane is maintained at a concentration of c f. It is noted that c f is the concentration within the polymer surface layer, and this concentration can be related to the bulk concentration or vapor pressure through a partition coefficient or solubility constant. The amount... 

We note, finally, that with the help of equations (12) with a concrete form of the function F [e.g., (26)], it is also possible to solve the very interesting problem of the diffusion jump of fuel across the flame zone as is shown in Fig. 3, the concentration of the mutually penetrating substances in the transition across the reaction zone falls sharply, but does not become zero. Since the temperature and reaction rate also fall on both sides of the reaction zone, the concentration of fuel which has already reached a certain distance from the flame in the oxidation zone no longer changes. 

What penetration depth of oxygen would you expect in a sediment, if the removal of oxygen can be described in terms of a half-life of 5 days Sediments have a temperature of 10°C. Their porosity is at 70 percent. For the diffusion coefficients use Tables 3.1 and 3.2. The bottom water shall have an oxygen concentration of 200 imol T. Use an oxygen concentration less than 1 imol T as zero . 

In the case of short diffusion times (i.e., only near surface penetration), it can be useful to approximate a mineral with a planar boundary as a semi-infinite medium. For the case of diffusion from a well-stirred semi-infinite reservoir at concentration Co into a half space initially at zero concentration, the concentration distribution is given by... 

Figure 5.75 shows the bulk reaction rate of the biofilm k in dimensionless form as a function of the bulk concentration A with different biofilm characteristic values of A/B (Harremoes, 1978). Even though the depth of penetration may be small compared with the entire film thickness, the effect of internal diffusion resistances is to mask the true zero-order kinetics (see Equ. 5.111), yielding an apparent reaction order of one-half. 

In the case when an inversion is established from the ground up as in the late afternoon, the plume may be able to penetrate the inversion. The plume can now diffuse upward, but it is prevented from diffusing into the stable air below. This could be a favorable condition since the plume would tend to diffuse away above the stable layer resulting in zero ground-level concentration. However, eventually the material in the plume above the inversion comes to the ground and could cause a problem further downwind. 

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