Channel flow boundary-layer equations

Given the scaling arguments in the previous sections, the axisymmetric channel-flow boundary-layer equations can be summarized as... 

The Von Mises transformation offers certain advantages in solving the channel-flow boundary-layer problems. However, it is certainly possible to solve the equations in their primative form. For a cylindrical channel, using constant fluid properties, consider the differences between the primative equations (Eqs. 7.44—7.47) and the Von Mises form (Eqs. 7.59-7.62). 

Given the particular circumstances of the flow in a long, narrow channel, explain the reduction of the governing equations to a boundary-layer form that accommodates the momentum and species development length. Discuss the essential characteristics of the boundary-layer equations, including implications for computational solution. 

Boundary-layer behavior is one of several potential simplifications that facilitate channel-flow modeling. Others include plug flow or one-dimensional axial flow. The boundary-layer equations, however, are the ones that require the most insight and effort to derive and to establish the ranges of validity. The boundary-layer equations retain a full two-dimensional representation of all the field variables as well as all the nonlinear behavior of Navier-Stokes equations. Nevertheless, when applicable, they provide a very significant simplification that can be used to great benefit in modeling. 

Figure 7.9 shows a cylindrical duct whose radius varies as a function of position, R(z). As long as the radius varies smoothly and relatively smoothly, the channel flow may be treated as a boundary-layer problem. Discuss what, if any, changes must be made to the boundary-layer equations and the boundary-condition specifications to solve the variable-area boundary-layer problem. 

The conservation equations for momentum, assuming horizontally homogeneous flow, need to be solved for each sub-basin. In straits or channels the boundary layer considered could often be represented by a channel flow model with wind stress, while in larger sub-basins an Ekman flow model can be applied. 

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. 

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations ... 

Following the very brief introduction to the method of lines and differential-algebraic equations, we return to solving the boundary-layer problem for nonreacting flow in a channel (Section 7.4). From the DAE-form discretization illustrated in Fig. 7.4, there are several important things to note. The residual vector F is structured as a two-dimensional matrix (e.g., Fuj represents the residual of the momentum equation at mesh point j). This organizational structure helps with the eventual software implementation. In the Fuj residual note that there are two timelike derivatives, u and p (the prime indicates the timelike z derivative). As anticipated from the earlier discussion, all the boundary conditions are handled as constraints and one is implicit. That is, the Fpj residual does not involve p itself. 

The boundary layer thickness 8 in Equation (4.11) is a function of the feed solution velocity u in the module feed flow channel thus, the term 8/D, can be expressed as... 

Use the linear Burgers equation for heat convection in a channel where the water is flowing with uniform velocity of 0.1 m/s across the cross section of the channel (boundary layers are neglected). The water is initially at 25°C throughout. At time t = 0 sec, waste heat is continuously rejected at x = 0 m, and the channel is long such that dT/dx = 0 for x > 1 m. The amount of heat rejected is 6.23 W/m2 for t > 0. Using the MacCormack explicit scheme, calculate the first 9 time steps to show the transient temperature distributions. 

Some of the more commonly used methods of obtaining solutions to problems involving natural convective flow have been discussed in this chapter. Attention has been given to laminar natural convective flows over the outside of bodies, to laminar natural convection through vertical open-ended channels, to laminar natural convection in a rectangular enclosure, and to turbulent natural convective boundary layer flow. Solutions to the boundary layer forms of the governing equations and to the full governing equations have been discussed. 

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). 

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. 

Cooling channels of the type depicted in Fig. 4.206 and c will first be discussed. For channels that are very long relative to the spacing of the vertical surfaces, the flow and heat transfer become fully developed (i.e., velocity and temperature profiles become invariant with distance along the channel) and are described by simple equations. For short channels or widely spaced vertical surfaces, a boundary layer regime is observed in which the boundary layers on the vertical walls remain well separated. In the latter case the heat transfer relation is similar... 

The differential equation may be seen to be exactly the same as Eq. (4.3.7) governing the developing diffusion layer in channel flow, with the boundary conditions the same as those appropriate to the case of a rapidly reacting wall, for which the solution is given by Eq. (4.3.19). 

This comparison focuses on the comer regions in square ducts that are nonexistent in tubes. In both configurations, the momentum boundary layer thickness is substantial (i.e., Effective/2) for fully developed laminar flow. The no-slip boundary condition for viscous flow near the walls increases the mass transfer boundary layer thickness and reduces the flux of reactants toward the catalytic surface relative to plug flow. This effect is significant in the comer regions of the channel with square cross section. Since the entire active surface in heterogeneous tubular reactors is equally accessible to reactants, one predicts larger conversion in tubes via equation (23-71) ... 

Gas flow through the small channels of a honeycomb matrix is nearly always laminar, and analytical solutions are available for heat and mass transfer for fully developed laminar flow in smooth tubes. In the inlet region, where the boundary layers are developing, the coefficients are higher, and numerical solutions were combined with the analytical solution for fully developed flow and fitted to a semitheoretical equation [14] ... 

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. 

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