Channel flow boundary-layer approximation

There are numerous applications that depend on chemically reacting flow in a channel, many of which can be represented accurately using boundary-layer approximations. One important set of applications is chemical vapor deposition in a channel reactor (e.g., Figs. 1.5, 5.1, or 5.6), where both gas-phase and surface chemistry are usually important. Fuel cells often have channels that distribute the fuel and air to the electrochemically active surfaces (e.g., Fig. 1.6). While the flow rates and channel dimensions may be sufficiently small to justify plug-flow models, large systems may require boundary-layer models to represent spatial variations across the channel width. A great variety of catalyst systems use... 

The flow conditions are chosen to represent a range of gas-turbine-combustor conditions, covering a range of physical parameters that include inlet velocities from 0.5 to 5 m/s and pressures from 1 to 10 bar. These conditions can be characterized in terms of a Reynolds number based on channel diameter and inlet flow conditions, which is varied over the range 20 < Rej = V nd/v < 2000. The upper limit of Rej = 2000 is chosen to ensure laminar flow, hence removing the need to model turbulence. It should be noted that the validity of the boundary-layer approximations improve as the Reynolds number increases. 

FORTRAN computer program that predicts the species, temperature, and velocity profiles in two-dimensional (planar or axisymmetric) channels. The model uses the boundary layer approximations for the fluid flow equations, coupled to gas-phase and surface species continuity equations. The program runs in conjunction with CHEMKIN preprocessors (CHEMKIN, SURFACE CHEMKIN, and TRAN-FIT) for the gas-phase and surface chemical reaction mechanisms and transport properties. The finite difference representation of the defining equations forms a set of differential algebraic equations which are solved using the computer program DASSL (dassal.f, L. R. Petzold, Sandia National Laboratories Report, SAND 82-8637, 1982). 

By choosing a time step for the transient solid temperature equation long enough for gas-phase equilibration, the discretized, time-independent set of the Navier-Stokes equations under the boundary layer approximation can be solved for the flow field inside the catalytic channels [12] (quasisteady assumption for the gas-phase) using the CRESLAF package [13]. The applicability of the boundary layer approach in catalytic combustion at sufficiently large Reynolds numbers (Re > 20) has already been demonstrated [14]. The simplified equations thus become ... 

Equation (4) states that the linear deposition rate vj is a diffusion controlled boundary layer effect. The quantity Ac is the difference in foulant concentration between the film and that in the bulk flow and c is an appropriate average concentration across the diffusion layer. The last term approximately characterizes the "concentration polarization" effect for a developing concentration boundary layer in either a laminar or turbulent pipe or channel flow. Here, Vq is the permeate flux through the unfouled membrane, 6 the foulant concentration boundary layer thickness and D the diffusion coefficient. 

Catalytic combustion in a monolith channel provides an illustration of boundary-layer flow in a channel [322], Figure 17.18 shows a typical monolith structure and the particular single-channel geometry used in this example. Since every channel within the monolith structure behaves essentially alike, only one channel needs to be analyzed. Also a cylindrical channel is used to approximate the actual shape of the channels. 

When the inlet length is expressed in terms of number of gap widths , the difference between the flow in a tube and the flow in an annulus of narrow gap differs only by 25% [(0.05 - 0.04)/0.05]. This situation is an indication that the growth of the laminar boundary layers from the wall to the center of the channel is similar in both cases. Because duct friction coefficients, a measure of momentum transfer, do not vary by more than a factor of 2 for ducts of regular cross sections when expressed in terms of hydraulic diameters, the use of the inlet length for tubes or parallel plates can be expected to be a reasonable approximation for the inlet lengths of other cross sections under laminar flow conditions. In the annular denuder, the dimensionless inlet length for laminar flow development, L, can be expressed as... 

The concentration polarization occurring in electrodialysis, that is, the concentration profiles at the membrane surface can be calculated by a mass balance taking into account all fluxes in the boundary layer and the hydrodynamic conditions in the flow channel between the membranes. To a first approximation the salt concentration at the membrane surface can be calculated and related to the current density by applying the so-called Nernst film model, which assumes that the bulk solution between the laminar boundary layers has a uniform concentration, whereas the concentration in the boundary layers changes over the thickness of the boundary layer. However, the concentration at the membrane surface and the boundary layer thickness are constant along the flow channel from the cell entrance to the exit. In a practical electrodialysis stack there will be entrance and exit effects and concentration... 

Equations for concentration polarization have been derived for simple cases such as laminar flow of feed solution between parallel plates or inside hollow fibers. " Numerical solutions were required because of the developing concentration boundary layer and the gradual decrease in solution flow rate as permeation occurs. Exact solutions arc not available for the more important cases of flow outside hollow fibers or in the channels of a spiral-wound module, but an approximate analysis may still be helpful. 

The approximations given by Equations 8.35 are the solution to Leveque s problem given in Equation 8.30 with a linear wall reaction. Since the formulation of the problem leads to a linearized velocity profile in a planar boundary layer, laminar flows (parabolic velocity profiles) in curved channels are more susceptible to present higher deviations from these results. For a fully developed flow in a round tube, the error associated with Equation 8.35b is 1.4 and 0.13% for aPe ,lz equal to 100 and 1000, respectively. Lopes et al. [40] observed that these differences are visible mainly for Da — 00 and calculated corrections to account for these effects. It was shown that in the mass transfer-controlled limit. 

The Hagen-Poiseuille equation reported above assumes a no-slip condition at the capillary wall, that is, the molecules in the fluid layer in contact with the capillary wall have zero velocity. It should be noted that this is an approximation and there is no reason why particles at the wall should not have a finite velocity. The nature and very existence of slip is an intensely debated topic and is covered in detail elsewhere in this book. Several fluid flow experiments in nanometer channels showing huge slip lengths will be described later in this chapter. In these experiments, the slip length X has been calculated via the Hagen-Poiseuille equation with a slip wall boundary condition ... 

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