Diffusion without convection

When the simplified mass transfer Peclet number is very small (i.e., <1), T 0.67 instead of unity because the numerator of T (i.e., lSA.eff. axial disp.) is based on unsteady-state pore diffusion without convection, whereas the denominator of T (i.e., a, ordinary) is measured in an unrestricted bulk fluid phase. In other words, the diffusivity in the numerator of T is reduced by porosity and tortuosity factors. 

Example 15-1. Steady-state diffusion without convection Low-tenqierature evaporation... 

Inserting these relations into Equation (2.97), one obtains a relation for the time response of the overvoltage at constant current when the rate is controlled by charge transfer and diffusion without convection. This relation can be written as... 

It is a typical feature of the diffusion processes at electrodes of small size, which are reached by converging diffusion fluxes, that a steady state can be attained even without convection (e.g., in gelled solutions). Such electrodes, which have dimensions comparable to typical values of 8, are called microelectrodes. 

NDIR-type C02 monitors are used in HVAC systems and for industrial applications. The portable units are usually battery operated, and their ambient sample is received by a combination of diffusion and convection effects, without using any pumps or filters. These units are provided with digital displays, one or two alarm settings, and digital output signals. 

Fig. 4. The poisoning factor iPdc its a function of a with 4 as a parameter. The diffusion regime without convection >s recovered for high values of 4. Uniform model.

Above the catalyst surface, substance A will be transported by diffusion, mainly in the x-direction. In a thin layer close to the wall, the diffusion boundary layer, mass transport by convection is negligible, and from (2.338) we obtain the diffusion equation valid for steady one-dimensional diffusion without chemical reaction... 

Diffusive renewal. Even without convection or electrolytic reversal, it is possible to obtain cyclic renewal of the diffusion layer simply by waiting long enough at the base potential for diffusion to replace the consumed electroreactant (46). 

Obviously, the Damkohler numbers are important when chemical reaction occurs, as illustrated by these two examples, which include diffusion and pseudo-homogeneous chemical reaction in porous catalytic pellets. Details of the diffusion equation without convection in nonreactive systems are summarized below for transient and steady-state analyses ... 

In a motionless electrolyte (namely without convection), such as a gel for example, one can estimate the time at which an interaction occurs between the electrodes in terms of mass transport. This is done by calculating the moment when the diffusion layer in a semi-infinite diffusion experiment (thickness of about -JW, see section 4.3.1.3) becomes approximately as thick as the inter-electrode distance. Therefore, in a system containing an electroactive species with a diffusion coefficient of about 1 (T crff s" and an inter-electrode distance of 3 mm, it would take an experiment lasting around 3 h for the diffusion layer to reach a thickness matching the interelectrode distance. 

Here again, in this system without convection, the composition throughout the electrolyte Is non-homogeneous, with the diffusion phenomena occupying the entire volume after a transient period. 

Steady-State Binaiy Fickian Diffusion and Mass Balances without Convection... 

Describe Fick s model of diffusion in words and equations, and use the model to solve steady-state binary diffusion problems without convection... 

For species in solution undergoing diffusion (without forced convection) the lowest frequency that can be used depends also on the hydrodynamic conditions. It is well known that the chronoamperometric curves in solutions might be measured up to 60 s, after which natural convection affects the linearity of diffusion. This means that measurements of the mass transport impedance are limited to approximately 0.1 Hz (or slightly lower). However, measuring diffusion in solid materials (e.g., hydrogen absorption in metals and alloys, lithium intercalation) is not influenced by convection, and the measurements might be carried out to the... 

The flux of species to an electrode is described by Pick s laws, which varies according to the electrode geometry flat electrode, cylindrical electrode, disk electrode, spherical electrode, etc. Applying electrochemical boundary conditions, the solution of the diffusion equation is obtained. Considering a chronoamperom-etry experiment at a disk microelectrode where a potential step is applied to the electrode and the reaction proceeds under diffusion control without convection or migration, the following equations are obtained ... 

MULLER - Do you think that this photochemically bistable system is a candidate for spatial pattern formation (without convective effects) under externally homogeneous illumination with the appropriate wavelength The chemical "feed-back loop" combined with diffusive transport might constitute a source of spatial symmetry-breaking, e.g. in an extended solution layer. 

At constant temperature, the gradient of concentration inside a solution (without convection or migration) produces a flow of matter in the opposite direction, which arises from random fluctuations in the positions of molecules in space. This phenomenon, denominated by isothermal diffusion, is an irreversible process. The gradient of chemical potential in the real solution is treated as the true virtual force producing diffusion. 

The statement by Maxwell quoted earlier suggests that diffusion and convection always occur together, that one cannot occur without the other. This fact sets diffusion apart from many other phenomena. For example, thermal conduction can certainly occur without convection. In contrast, diffusion generates its own convection, so that understanding the process can be much more complicated, especially in concentrated solutions. 

The finite element results obtained for various values of (3 are compared with the analytical solution in Figure 2.27. As can be seen using a value of /3 = 0.5 a stable numerical solution is obtained. However, this solution is over-damped and inaccurate. Therefore the main problem is to find a value of upwinding parameter that eliminates oscillations without generating over-damped results. To illustrate this concept let us consider the following convection-diffusion equation... 

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