Diffusion equation cylindrical, steady-state

The experiment is carried out under diffusion control. Theoretical concentration profiles are calculated by solving Fick s second law of diffusion in the steady state with boundary conditions appropriate to the solution domain and to the substrate, taking into account its geometry and the type of reaction occurring on it. Assumption is made that the redox species are stable and not involved in a homogeneous reaction in solution. Two geometries known to produce steady-state concentration profiles have been considered (72,77) the hemisphere and the microdisk. The former only requires a radial dimension, and the diffusion equation can be solved analytically. The latter, on the other hand, necessitates cylindrical coordinates and the solution becomes much more complex. With the latter a closed form analytical ex-... 

In electrochemical systems with flat electrodes, all fluxes within the diffusion layers are always linear (one-dimensional) and the concentration gradient grad Cj can be written as dCfldx. For electrodes of different shape (e.g., cylindrical), linearity will be retained when thickness 5 is markedly smaller than the radius of surface curvature. When the flux is linear, the flux density under steady-state conditions must be constant along the entire path (throughout the layer of thickness 8). In this the concentration gradient is also constant within the limits of the layer diffusion layer 5 and can be described in terms of finite differences as dcjidx = Ac /8, where for reactants, Acj = Cyj - c j (diffusion from the bulk of the solution toward the electrode s surface), and for reaction products, Acj = Cg j— Cyj (diffusion in the opposite direction). Thus, the equation for the diffusion flux becomes... 

We consider steady-state, one-dimensional laminar flow (q ) through a cylindrical vessel of constant cross-section, with no axial or radial diffusion, and no entry-length effect, as illustrated in the central portion of Figure 2.5. The length of the vessel is L and its radius is R. The parabolic velocity profile u(r) is given by equation 2.5-1, and the mean velocity u by equation 2.5-2 ... 

The solution of the diffusion equation for the quasi-steady state in cylindrical coordinates shows that each dislocation line source will have a vacancy concentration diffusion field around it of the form... 

Consider steady state diffusion in a long a cylindrical annulus.[17] The governing equation and boundary conditions are ... 

Theoretical characteristics for the EQ mechanism, in the feedback mode under chronoamperometric and steady-state conditions, were obtained initially by solving numerically the diffusion equations in the axisymmetric cylindrical geometry defining the tip and substrate, in the SECM configuration (Fig. 1). The results were later generalized and simplified equations presented for the collection efficiency and the relationship between the substrate and tip responses (TG/SC mode) under steady-state conditions (5). 

In the absence of convective mass transfer and chemical reaction, calculate the steady-state liquid-phase mass transfer coefficient that accounts for curvature in the interfacial region for cylindrical liquid-solid interfaces. An example is cylindrical pellets that dissolve and diffuse into a quiescent liquid that surrounds each solid pellet. The appropriate starting point is provided by equation (B) in Table 18.2-2 on page 559 in Bird et al. (1960). For one-dimensional diffusion radially outward, the mass transfer equation in cylindrical coordinates reduces to... 

The equations for simultaneous pore diffusion and reaction were solved independently by Thiele and by Zeldovitch [16,17]. They assumed a straight cylindrical pore with a first-order reaction on the surface, and they showed how pore length, diffusivity, and rate constant influenced the overall reaction rate. Their solution cannot be directly adapted to a catalyst pellet, since the number of pores decreases going toward the center and assuming an average pore length would introduce some error. The approach used here is that of Wheeler [18] and Weisz [19], who considered reactions in a porous sphere and related the diffusion flux to the effective diffusivity, Z). The basic equation is a material balance on a thin shell within the sphere. The difference between the steady-state flux of reactant into and out of the shell is the amount consumed by reaction. 

In these equations, is the microparticle space coordinate and its half-dimension, is the non-dimensional concentration of the adsorbate in the micropores, the micropore diffusion coefficient and the microparticle shape factor (cr = 0 for plane, = 1 for cylindrical, and o- j, = 2 for spherical microparticle geometry). is the adsorption isotherm relation (generally nonlinear), which is again replaced by its Taylor series expansion (the coefficients of which, Op, b, ... depend on the steady-state pressure and concentration). The meaning of the boundary condition (11.33) is that the concentration profile in the microparticle is symmetrical, and of the boundary condition (11.34) that adsorption equilibrium is established at the micropore mouth. 

Impedance may also be studied in the case of forced diffusion. The most important example of such a technique is a rotating disk electrode (RDE). In a RDE conditions a steady state is obtained and the observed current is time independent, leading to the Levich equation [17]. The general diffusion-convection equation written in cylindrical coordinates y, r, and q> is [17]... 

The condition that the electrochemical processes occur largely in the region of the meniscus is only met if the thin film of electrolyte is absent or if the thickness is very small. The simple-pore model [64] is an example of the first case. The meniscus is assumed to form at the intersection of micropores with macropores. While the micropores are filled with electrolyte up to the intersection, the macropores are filled with gas. The meniscus may be treated as flat in a first approximation. The walls of micropores are the seat of the electrode reaction. The simple-pore model was suggested [64] as applying to non-wetted systems like the Teflon-bonded platinum black electrodes. The limiting current due to the diffusion of species into a micropore was derived [64] as the steady-state solution of the two-dimensional diffusion equation in cylindrical coordinates. The summation of the currents of the individual pores leads to ... 

The convective-diffusion Equation 7.8 in the case of two-dimensional system in a steady state [(dddt) = 0] in cylindrical coordinates is as follows ... 

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