Channel convective-diffusion equations

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 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... 

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 ... 

For hydrodynamic electrodes, in order to solve the convective-diffusion equation analytically for the steady-state limiting current, it is necessary to use a first-order approximation of the convection function(s) (such as the Leveque approximation for the channel). These approximate expressions for the steady-state mass transport limited currents were introduced in Section 4 (see Table 5). 

Next consider flow in what is called a T-sensor. Two flows come together, join, and traverse down one channel, as illustrated in Figure 10,9, This device is used in microfluidic medical devices, which are discussed further in Chapter 11, Here you will consider only the flow (which has no special utility until the convective diffusion equation is added in Chapter 11),... 

These interesting devices consist of a tube or duct within which static elements are installed to promote cross-channel flow. See Figure 8.5 and Section 8.7.2. Static mixers are quite effective in promoting radial mixing in laminar flow, but their geometry is too complex to allow solution of the convective diffusion equation on a routine basis. A review article by Thakur et al. (2003) provides some empirical correlations. The lack of published data prevents a priori designs that utilize static mixers, but the axial dispersion model is a reasonable way to correlate pilot plant data. Chapter 15 shows how Pe can be measured using inert tracers. 

The numerics in Table 16.2 make two points. One is that turbulence is difficult to achieve at the mesoscale and nearly impossible to achieve in micro- and nanoscale devices. The other point is that diffusion becomes so fast at the microscale that cross-channel (e.g., radial) mixing is essentially instantaneous for all but the very fastest reactions. Thus composition and temperature will be approximately uniform in the cross-channel direction. The solutions to the convective diffusion equations in... 

We now turn to the second criterion, in particular bearing in mind the criticism, alluded to above, about the difficulty associated with the theoretical description of processes at non-uniformly accessible electrodes. Again, we will compare and contrast the channel electrode and the RDE. Now the theoretical description of electrode reactions involves, typically, the solution of perhaps several coupled steady-state convective-diffusion equations of the form... 

The problem of edge effects at channel electrodes has also been considered by Cope and Tallman [51], but under the condition of inviscid flow that is, the solution was assumed to have a constant linear velocity, V. Thus, the convective-diffusion equation [as opposed to eqn. (11)]... 

We next consider the behaviour for DISP1 reactions at channel electrodes. The normalised steady-state convective-diffusion equations for this case, under the Levich approximation, are... 

At the channel electrode, this problem has been treated by Matsuda, initially employing the reaction-layer approximation [101], but subsequently the full coupled convective-diffusion equations were solved [102] (at the level of the Leveque approximation). It was shown that, to within 1%, the equation... 

The high sensitivity of the Allendoerfer cell makes it of great value in the detection of unstable radicals but, for the study of the kinetics and mechanism of radical decay, the use of a hydrodynamic flow is required. The use of a controlled, defined, and laminar flow of solution past the electrode allows the criteria of mechanism to be established from the solution of the appropriate convective diffusion equation. The uncertain hydrodynamics of earlier in-situ cells employing flow, e.g. Dohrmann [42-45] and Kastening [40, 41], makes such a computational process uncertain and difficult. Similarly, the complex flow between helical electrode surface and internal wall of the quartz cell in the Allendoerfer cell [54, 55] means that the nature of the flow cannot be predicted and so the convective diffusion equation cannot be readily written down, let alone solved Such problems are not experienced by the channel electrode [59], which has well-defined hydrodynamic properties. Compton and Coles [60] adopted the channel electrode as an in-situ ESR cell. 

The solution flow is normally maintained under laminar conditions and the velocity profile across the channel is therefore parabolic with a maximum velocity occurring at the channel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately determinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffusion equation for mass transport within the rectangular duct may be described by... 

For the fully developed velocity profile the mathematical problem again reduces to the solution of the steady form of the convective diffusion equation for the solute concentration. In contrast to the diffusion equation treated in the channel flow problem with soluble or rapidly reacting walls, it is necessary to include here the lateral convection term to account for the product removal through the membrane walls, putting... 

Reducing the solution of the convection-diffusion equation for the concentration field to a ID problem in the comoving frame-of-reference clearly reaches its limits if the fluid lamellae get deformed and are no longer arranged in parallel. Such a situation may occur if the channel depth is not small compared to its width and the design comprises a sudden expansion or contraction of the flow... 

In cases where hydrodynamic dispersion and the corresponding broadening of residence-time distributions deteriorate the performance of a process, the question arises as to which channel design minimizes dispersion. Already from the analysis of Taylor and Aris it becomes clear that an enhanced mass transfer perpendicular to the main flow direction reduces the broadening of concentration tracers. Such a mass-transfer enhancement can be achieved by the secondary fiow occurring in a curved channel. This aspect was investigated by Daskopoulos and Lenhoff [78] for ducts of circular cross section. They assumed the diameter of the duct to be small compared to the radius of curvature and solved the convection-diffusion equation for the concentration field numerically. More specifically, a two-dimensional problem defined on the cross-sectional plane of the duct was solved based on a combination of a Fourier series expansion and an expansion in Chebyshev polynomials. The solution is of the general form... 

The Reynolds number in microreaction systems usually ranges from 0.2 to 10. In contrast to the turbulent flow patterns that occur on the macroscale, viscous effects govern the behavior of fluids on the microscale and the flow is always laminar, resulting in a parabolic flow profile. In microfluidic reaction systems, where the characteristic length is usually greater than 10 pm, a continuum description can be used to predict the flow characteristics. This allows commercially written Navier-Stokes solvers such as FEMLAB and FLUENT to model liquid flows in microreaction channels. However, modeling gas flows may require one to take account of boundary sUp conditions (if 10 < Kn < 10 , where Kn is the Knudsen number) and compressibility (if the Mach number Ma is greater than 0.3). Microfluidic reaction systems can be modeled on the basis of the Navier-Stokes equation, in conjunction with convection-diffusion equations for heat and mass transfer, and reaction-kinetic equations. 

Application Channel electrodes have well-defined hydrodynamic properties [57], and the laminar flow convective-diffusion equation describing the mass... 

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)... 

A completely general solution of the governing convective diffusion equation (7.2.110), and equation (7.2.111), subject to the boundary conditions (7.2.114), (7.2.115) and (7.2.116) is not available. There are two types of solutions, similarity solutions and integral boundary layer solutions (apart from complete numerical solutions). Common to both of these solutions is the assumption that the particle concentration boundary layer is very thin compared to the membrane channel dimension normal to the axial flow further, the shear stress due to the axial velocity gradient in the particle concentration boundary layer is equal to that at the wall, namely... 

The distribution of concentration c(x, t) of a species flowing in the channel can be described by the following (time-dependent) convective-diffusion equation ... 

The calculation of rejected species concentration was performed by solving, in the module channel only, the convection-diffusion equation ... 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

Figure 9. Model of convective diffusion inside a tube channel (radius R) towards the walls of the tube considered as perfect active surfaces. A Poiseuille profile for the velocity (equation (35)) is also schematically shown with arrows on the right-hand side. The maximum velocity v is reached at the centre of the tube (r = 0)...

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

The diffusion battery consists of banks of tubes, channels, or screens through which a submicron aerosol passes at a constant flow rale. Particles deposit on the surface of the battery elements, and the decay in total number concentration along the flow path i measured, usually with a condensation particle counter. The equations of convective diffusion (Chapter 3) can be solved for the rate of deposition as a function of the particle diffusion coefficient. Because the diffusion coefficient is a monotonic function of particle size (Chapter 2), the measured and theoretical deposition curves can be compared to detennine the size for a monodisperse aerosol. 

Consider the governing equations that describe convection, diffusion, and chemical reaction in tube-wall duct reactors where expensive metal catalyst is coated on the inner walls of the flow channel. 

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