Convective diffusion equation for

Under conditions of limiting current, the system can be analyzed using the traditional convective-diffusion equations. For example, the correlation for flow between two flat plates is... 

The convective diffusion equations for mass and energy are given detailed treatments in most texts on transport phenomena. The classic reference is... 

Solution The problem requires solution of the convective diffusion equation for mass but not for energy. Rewriting Equation (8.71) in dimensionless form gives... 

Since turbulent fluctuations not only occur in the velocity (and pressure) field but also in species concentrations and temperature, the convection diffusion equations for heat and species transport under turbulent-flow conditions also comprise cross-correlation terms, obtained by properly averaging products of... 

For problems involving gradients in chemical species, the convection-diffusion equations for the species are also solved, usually for N— 1 species with the Nth species obtained by forcing the mass fractions to sum to unity. Turbulence can be described by a turbulent diffusivity and a turbulent Schmidt number, Sct, analogous to the heat transfer case. 

Table 1 shows the particular forms of the convective diffusion equation for different geometries. It is fortunate that, due to the symmetrical nature of hydrodynamic electrodes, some of these terms may be neglected. Also, the major part of investigations conducted are under conditions of steady-state flow where dc/dt = 0. The exception to this is, of course, the cyclic operation of the DME. 

Since the form of the dimensionless convective-diffusion equation for tube and channel electrodes is exactly the same as for rotating electrodes, we can immediately conclude that the steady-state collection efficiency, N0, under conditions of uniform surface concentration at the generator electrode (which corresponds to the limiting current at the generator or to any point on a reversible wave) is, once again... 

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

Solving equation (7.55) for , the convection-diffusion equation for VP becomes... 

The convection-diffusion equation for y (u, f) will be of the same form as the rigid dumbbell model of section 7.1.6.2 except that the diffusivity must be replaced by Dr(u, i) to give... 

For the solution of a salt composed of two ionizable species (binary electrolyte), the four basic equations can be combined to yield the convective diffusion equation for steady-state systems ... 

The hydrodynamic conditions influence the concentration distribution explicity through the velocity term present in the convective diffusion equation. For certain well-defined systems the fluid flow equations have been solved, but for many systems, especially those with turbulent flow, explicit solutions have not been obtained. Consequently, approximate techniques must frequently be used in treating mass transfer. 

Ignoring edge effects, Levich 3 writes the convective diffusion equation for a rotating disc as ... 

When van der Waals and double-layer forces are effective over a distance which is short compared to the diffusion boundary-layer thickness, the rate of deposition may be calculated by lumping the effect of the particle-collector interactions into a boundary condition on the usual convective-diffusion equation. This condition takes the form of a first-order irreversible reaction (10, 11). Using this boundary condition to eliminate the solute concentration next to the disk from Levich s (12) boundaiy-kyersolution of the convective-diffusion equation for a rotating disk, one obtains... 

In Section 5.9, we show how to solve the convective-diffusion equation for the rotating disc electrode in order to calculate the diffusion-limited current. When the forced convection is constant, then dc/dt = 0, which simplifies the mathematical solution. 

To fully utilise the result expressed by equation (9.70), we must now consider the solution of the convective diffusion equation for a system of /i-type semiconductor particles as they diffuse towards an ORDE. Whilst... 

The convective diffusion equation for a particle of concentration cm and which contains m electrons in the conduction band and surface states is given by ... 

Laminar pulsatile flow in a tube Flow in a tube is in one direction, parallel to the electrode surface, (taken as the jr-direction). The time-dependent convective-diffusion equation for this geometry is given by equation (10.12). Mass transport to the surface of the electrode is thus determined both by the gradient perpendicular to the surface of the tangential flow, dujdy and the concentration gradient perpendicular to the surface ... 

The first approach is the discretization of the convection and the diffusion operators of the PDEs, which gives rise to a large (or very large) system of effective low-dimensional models. The order of these low-dimensional models depend on the minimum mesh size (or discretization interval) required to avoid spurious solutions. For example, the minimum number of mesh points (Nxyz) necessary to perform a direct numerical simulation (DNS) of convective-diffusion equation for non-reacting turbulent flow is given by (Baldyga and Bourne, 1999)... 

We can extend the hyperbolic model to cases in which the solute diffuses in more than one phase. A common case is that of a monolith channel in which the flow is laminar and the walls are coated with a washcoat layer into which the solute can diffuse (Fig. 4). The complete model for a non-reacting solute here is described by the convection-diffusion equation for the fluid phase coupled with the unsteady-state diffusion equation in the solid phase with continuity of concentration and flux at the fluid-solid interface. Transverse averaging of such a model gives the following hyperbolic model for the cup-mixing concentration in the fluid phase ... 

This is the general convective diffusion equation for particles in an isothermal gas when the particles are not subjected to any forces other than the convective motion of the gas and the molecular motion of the gas molecules. 

The convective-diffusion equation for solute (e.g., tracer) transport in both the axial and radial direction is... 

A uniformly accessible electrode is an electrode where, at the interface, the flux and the concentration of a species produced or consumed on the electrode are independent of the coordinates that define the electrode surface. The mass flux at the interface is obtained by solving the material balance equation. If migration can be neglected, the material balance equation for dilute electrol5 c solutions is reduced to the convective-diffusion equation. For an axisymmetric electrode, the concentration derivatives with respect to the angular coordinate 9 are equal to zero, and the convective-diffusion equation can be expressed in cylindrical coordinates as... 

The three-term convective-diffusion model provides the most accurate solution to the one-dimensional convective-diffusion equation for a rotating disk electrode. The one-dimensional convective-diffusion equation applies strictly, however, to the mass-transfer-limited plateau where the concentration of the mass-transfer-limiting species at the surface can be assumed to be both uniform and equal to zero. As described elsewhere, the concentration of reacting species is not uniform along the disk surface for currents below the mass-transfer-limited current, and the resulting nonuniform convective transport to the disk influences the impedance response. ... 

C. Deslouis, C. Gabrielli, and B. Tribollet, "An Analytical Solution of the Nonsteady Convective Diffusion Equation for Rotating Electrodes," Journal of The Electrochemical Society, 130 (1983) 2044-2046. 

We have previously written an expression for j n in Eq. (2-150), but this expression is in terms of the local bulk concentration evaluated at the interface, c, and thus to determine c we would need to solve bulk-phase transport equations. We will not pursue that subject here. However, when we use this material to solve flow problems, we will consider several cases for which it is not necessary to solve the full convection-diffusion equation for c. We will see that the concentration of surfactant tends to become nonuniform in the presence of flow -i.e., when u n and u v are nonzero at the interface. This tendency is counteracted by surface diffusion. When mass transfer of surfactant to and from the bulk fluids is added, this will often tend to act as an additional mechanism for maintenance of a uniform concentration T. This is because the rate of desorption from the interface will tend to be largest where T is largest, and the rate of adsorption largest where T is smallest. 

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