Convective diffusion equation species

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

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... 

Eggels and Somers (1995) used an LB scheme for simulating species transport in a cavity flow. Such an LB scheme, however, is more memory intensive than a FV formulation of the convective-diffusion equation, as in the LB discretization typically 18 single-precision concentrations (associated with the 18 velocity directions in the usual lattice) need to be stored, while in the FV just 2 or 3 (double-precision) variables are needed. Scalar species transport therefore can better be simulated with an FV solver. 

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. 

P 21] The mixing of gaseous methanol and oxygen was simulated. The equations applied for the calculation were based on the Navier-Stokes (pressure and velocity) and the species convection-diffusion equation [57]. As the diffusivity value for the binary gas mixture 2.8 x 10 m2 s 1 was taken. The flow was laminar in all cases adiabatic conditions were applied at the domain boundaries. Compressibility and slip effects were taken into account The inlet temperature was set to 400 K. The total number of cells was —17 000 in all cases. 

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). 

The convective diffusion equation is analogous to equations commonly used in dealing with heat and mass transfer. Similarly, if migration can be neglected in a multicomponent solution, then the convective diffusion equation can be applied to each species,... 

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 ... 

Hydrodynamic electrodes1 are electrodes which function in a regime of forced convection. The advantage of these electrodes is increased transport of electroactive species to the electrode, leading to higher currents and thence a greater sensitivity and reproducibility. Most of the applications of these electrodes are in steady-state conditions, i.e. constant forced convection and constant applied potential or current. In this case dc/dt = 0, which simplifies the solution of the convective diffusion equation (Section 5.6)... 

The concentration distribution of species in the diffusion layer, SD, obeys the convective-diffusion equation ... 

The permeate is continuously withdrawn through the membrane from the feed sueam. The fluid velocity, pressure and species concentrations on both sides of the membrane and permeate flux are made complex by the reaction and the suction of the permeate stream and all of them depend on the position, design configurations and operating conditions in the membrane reactor. In other words, the Navier-Stokes equations, the convective diffusion equations of species and the reaction kinetics equations are coupled. The transport equations are usually coupled through the concentration-dependent membrane flux and species concentration gradients at the membrane wall. As shown in Chapter 10, for all the available membrane reactor models, the hydrodynamics is assumed to follow prescribed velocity and sometimes pressure drop equations. This makes the species transport and kinetics equations decoupled and renders the solution of... 

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. ... 

K, dimensionless frequency associated, for example, with convective diffusion of species i, see, e.g., equations (11.66) and (13.34)... 

To increase the electrolyte conductivity, an additional ionic component that does not participate in the electrochemical reactions is often added to a solution. This nom-eactive component is called a supporting electrolyte or indifferent electrolyte. In the presence of a supporting electrolyte, there is a lowering of the electric field in solution, due to the electrolyte s high conductivity. Transport of the minor ionic species in solution is due primarily to diffusion and convection, in accordance with Equation (26.54) with VO = 0. Also, in the presence of a supporting electrolyte, the convective diffusion equation for a minor component in solution is written as... 

The dimensionless species concentration distribution (C) is described by the time-dependent convection-diffusion equation ... 

The chemical field is described by the convection-diffusion equation for all the species or. 

The simplest treatments of convective systems are based on a diffusion layer approach. In this model, it is assumed that convection maintains the concentrations of all species uniform and equal to the bulk values beyond a certain distance from the electrode, 8. Within the layer 0 x < 5, no solution movement occurs, and mass transfer takes place by diffusion. Thus, the convection problem is converted to a diffusional one, in which the adjustable parameter 8 is introduced. This is basically the approach that was used in Chapter 1 to deal with the steady-state mass transport problem. However, it does not yield equations that show how currents are related to flow rates, rotation rates, solution viscosity, and electrode dimensions. Nor can it be employed for dual-electrode techniques or for predicting relative mass-transfer rates of different substances. A more rigorous approach begins with the convective-diffusion equation and the velocity profiles in the solution. They are solved either analytically or, more frequently, numerically. In most cases, only the steady-state solution is desired. 

The interesting result shown by Eq. (3.4.14) is that the concentration distribution in a dilute binary electrolyte is governed by the same convective diffusion equation as for a neutral species even though there is a current flow. The potential distribution is given from Eq. (3.4.13), which is simply an expression of current continuity. We can see this from Eq. (3.4.4) for the current, which we may write, using... 

We shall not examine the equations of change in their most general form but will instead limit our attention to dilute or binary solutions in incompressible flow, without species production. For electrolyte solutions we will restrict ourselves to electrically neutral binary solutions or highly conducting solutions in which the electric field is small so that the convective diffusion equation for neutral species is applicable and the momentum and energy equations ate unaltered. For simplicity the transport and physical properties are also taken to be constant. 

These equations repeat those previously set down. Flete, u is the kinematic viscosity, and a is the thermal diffusivity. The subscripts have been dropped in the convective diffusion equation, and D can be the binary diffusion coefficient, the effective electrolytic diffusion coefficient, or the diffusion coefficient of the fth species. The molar concentration is to be interpreted in the same context. In the energy equation, sometimes referred to as the heat conduction equation in the form written, heat flux due to interdiffusion and due to viscous dissipation have been neglected as small. Heat sources are also absent. 

Since D plays the same role as the kinematic viscosity v, we may expect for large Schmidt numbers (v>D) that the viscous boundary layer thickness should be considerably larger than the diffusion boundary layer thickness. A consequence of this is that the velocity seen by the concentration layer at its edge is not the free stream velocity U but something much less, which is more characteristic of the velocity close to the wall (Fig. 4.2.1). We note also that since c is understood to be c, then in a multicomponent solution there may be as many distinct boundary layers as there are species, with the thickness of each defined by the appropriate diffusion coefficient. With this caveat in mind, we may write the convective diffusion equation for a two-dimensional diffusion boundary layer and estimate the diffusion layer thickness. 

It was shown in Section 3.4 that if the bulk of a dilute binary electrolyte solution may be assumed electrically neutral, then the distribution of reduced ion concentration is governed by the same convective diffusion equation as for a neutral species with an effective diffusion coefficient related to the difference in charge and diffusion coefficients of the positive and negative ions. Once the concentration distribution has been found, the potential distribution in the solution can be obtained by integrating the equation for current continuity (Eq. 3.4.16) to give... 

The phenomena and processes described can be modeled by convective diffusion equations with chemical reactions. In the simplest model, we may apply these equations in a cylindrical capillary and by means of a capillary model to a porous medium. Assuming dilute solutions, rapid chemical reactions, the double-layer thickness to the soil pore radius and the Peclet number based on the pore radius both small, the overall transport rate for the ith species in a straight cylindrical capillary is... 

For the electrochemical oxidation of a species A to in a microchannel, the convective-diffusive equation for mass transport under steady-state conditions is given by Eq. 2 below ... 

Many of the mixing simulations described in the previous section deal with the modeling of mass transfer between miscible fluids [33, 70-77]. These are the simulations which require a solution of the convection-difliision equation for the concentration fields. For the most part, the transport of a dilute species with a typical diSusion coeflEcient 10 m s between two miscible fluids with equal physical properties is simulated. It has already been mentioned that due to the discretization of the convection-diffusion equation and the typically small diffusion coefficients for liquids, these simulations are prone to numerical diffiision, which may result in an over-prediction of mass transfer efficiency. Using a lattice Boltzmann method, however, Sullivan et al. [77] successfully simulated not only the diffusion of a passive tracer but also that of an active tracer, whereby two miscible fluids of different viscosities are mixed. In particular, they used a coupled hydrodynamic/mass transfer model, which enabled the effects of the tracer concentration on the local viscosity to be taken into account. 

Cell Design Albery and coworkers [9-14] used tubular electrodes for ex situ electrochemical EPR experiments. The tubular electrode is equivalent to the channel electrode in all respects, except that the cross section is circular rather than rectangular [82, 137]. Like the later-developed channel flow cell, this setup (shown in Fig. 23) permits the interrogation of electrode reaction mechanisms of relatively long-lived radical species, [9-14] since the convective-diffusion equations are mathematically well defined, which at steady state are given by Eq. (37)... 

The liquid phase eonsists of pure water, while the gas phase has multi components. The transport of each species in the gas phase is governed by a general convection-diffusion equation in conjunction which the Stefan-Maxwell equations to account for multi species diffusion, as described in section 3, with the addition of a source term accounting for phase... 

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