Diffusive flux rate

Pick s Law. Pick s law is a physically meaningful mathematical description of diffusion that is based on the analogy to heat conduction (Pick, 1855). Let us consider one side of our control volume, normal to the x-axis, with an area 4r, shown in Figure 2.3. Pick s law describes the diffusive flux rate as... 

Convective Flux Rates. We will deal with the convective and diffusive flux rates separately. They will eventually be separated in the final diffusion equation, and it is convenient to make that break now. The x-component of the convective flux rate is equal to the x-component of velocity, u, times the concentration, C, times the area of our box normal to the x-axis. Therefore, in terms of convective flux rates, equation (2.9a) becomes... 

Diffusive Flux Rates. For net diffusive flux rate in the x-direction, equation (2.9a) becomes... 

A self-test of understanding flux rates would be to look at Figure 2.3 and write out the diffusive flux rates in the y- and z-directions on a separate sheet of paper. The result is similar to equation (2.12a) ... 

The diffusive flux rates would be treated similarly. The area of the control volume changing with radius is the reason the mass transport equation in cylindrical coordinates, given below - with similar assumptions as equation (2.18) - looks somewhat different than in Cartesian coordinates. 

Lake Sulfate Reductiona Diffusive Flux Rate of S Accumulation References... 

It is assumed that irreversible aggregation occurs on contact. The rate of coagulation is expressed as the aggregation flux J of particles towards a central particle. Using a steady-state approximation, the diffusive flux is derived to be... 

Diffusion in the bulk crystals may sometimes be short circuited by diffusion down grain boundaries or dislocation cores. The boundary acts as a planar channel, about two atoms wide, with a local diffusion rate which can be as much as 10 times greater than in the bulk (Figs. 18.8 and 10.4). The dislocation core, too, can act as a high conductivity wire of cross-section about (2b), where b is the atom size (Fig. 18.9). Of course, their contribution to the total diffusive flux depends also on how many grain boundaries or dislocations there are when grains are small or dislocations numerous, their contribution becomes important. 

The relationship between the diffusional flux, i.e., the molar flow rate per unit area, and concentration gradient was first postulated by Pick [116], based upon analogy to heat conduction Fourier [121] and electrical conduction (Ohm), and later extended using a number of different approaches, including irreversible thermodynamics [92] and kinetic theory [162], Pick s law states that the diffusion flux is proportional to the concentration gradient through... 

Under realistic conditions a balance is secured during current flow because of additional mechanisms of mass transport in the electrolyte diffusion and convection. The initial inbalance between the rates of migration and reaction brings about a change in component concentrations next to the electrode surfaces, and thus gives rise to concentration gradients. As a result, a diffusion flux develops for each component. Moreover, in liquid electrolytes, hydrodynamic flows bringing about convective fluxes Ji j of the dissolved reaction components will almost always arise. 

In an irreversible reaction that occurs under kinetic or mixed control, the boundary condition can be found from the requirement that the reactant diffusion flux to the electrode be equal to the rate at which the reactants are consumed in the electrochemical reaction ... 

It is a special feature of this diffusion situation that substance Red is produced by the chemical reaction, all along the diffusion path (i.e., sources of the substance are spatially distributed). For this reason the diffusion flux and the concentration gradient are not constant but increase (in absolute values) in the direction toward the surface. The incremental diffusion flux in a layer of thickness dx [ dJJdx)dx or -D (f-cldx ) dx] should be equal to the rate, v dx, of the chemical reaction in this layer. Hence, we have... 

Having defined the balance regions, the next task is to identify all the relevant inputs and outputs to the system (Fig. 1.10). These may be well-defined physical flow rates (convective streams), diffusive fluxes, but may also include interphase transfer rates. 

Change the program to calculate the effectiveness factor. Hint this can be done two ways, using diffusion fluxes and reaction rates. 

Mass conservation at the rhizoplane means that the diffusive flux towards the root, Eq. (4), must equal the rate of extraction by the root, Eq. (9), leading to the boundary condition... 

For these investigations, the UME was positioned in the aqueous subphase containing 0.1 M KNO3 and held at a potential to reduce oxygen at a diffusion-controlled rate, in order to promote the transfer of O2 from air (phase 2) to the aqueous solution (phase 1), with subsequent collection at the tip (Fig. 27). Under SECMIT conditions, the flux of oxygen from air to water is given by ... 

Equation (144) is the concentration profile as a function of the distance from the disk surface. The diffusion flux (intrinsic dissolution rate) is... 

The numerator of the right side of this equation is equal to the chemical reaction rate that would prevail if there were no diffusional limitations on the reaction rate. In this situation, the reactant concentration is uniform throughout the pore and equal to its value at the pore mouth. The denominator may be regarded as the product of a hypothetical diffusive flux and a cross-sectional area for flow. The hypothetical flux corresponds to the case where there is a linear concentration gradient over the pore length equal to C0/L. The Thiele modulus is thus characteristic of the ratio of an intrinsic reaction rate in the absence of mass transfer limitations to the rate of diffusion into the pore under specified conditions. 

In case of Fischer-Tropsch synthesis, we have to consider that the first-order reaction rate constant is related to the concentration in the gas phase (e.g., ce2), and that the diffusive flux in the liquid-filled pores is related to the concentration in the liquid (ce21). Thus, instead of Equation 12.10, we have to use... 

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