Wall Boundaries

On no-slip walls zero velocity components can be readily imposed as the required boundary conditions (v = v, = 0 on F3 in the domain shown in Figure 3.3). Details of the imposition of slip-wall boundary conditions are explained later in Section 4.2. 

Figure 5.14 (a) The predicted velocity field corresponding to no-slip wall boundary conditions, (b) Tlie predicted velocity field corresponding to partial slip boundary conditions... 

After the imposition of no-slip wall boundary conditions the last term in Equation (5.64) vanishes. Therefore... 

Subject to the boundary conditions that must be imposed at the axis, at the inlet, and on the wall boundaries of the cyclone, Bloor and Ingham found that the solution for 4 may be approximated by the expression... 

The analysis of the behavior of the fluid temperature and the Nusselt number performed for a circular tube at the thermal wall boundary condition 7(v = const, also reflects general features of heat transfer in micro-channels of other geometries. 

One particular characteristic of conduction heat transfer in micro-channel heat sinks is the strong three-dimensional character of the phenomenon. The smaller the hydraulic diameter, the more important the coupling between wall and bulk fluid temperatures, because the heat transfer coefficient becomes high. Even though the thermal wall boundary conditions at the inlet and outlet of the solid wall are adiabatic, for small Reynolds numbers the heat flux can become strongly non-uniform most of the flux is transferred to the fluid at the entrance of the micro-channel. Maranzana et al. (2004) analyzed this type of problem and proposed the model of channel flow heat transfer between parallel plates. The geometry shown in Fig. 4.15 corresponds to a flow between parallel plates, the uniform heat flux is imposed on the upper face of block 1 the lower face of block 0 and the side faces of both blocks... 

The wall boundary condition applies to a solid tube without transpiration. The centerline boundary condition assumes S5anmetry in the radial direction. It is consistent with the assumption of an axis5Tnmetric velocity profile without concentration or temperature gradients in the 0-direction. This boundary condition is by no means inevitable since gradients in the 0-direction can arise from natural convection. However, it is desirable to avoid 0-dependency since appropriate design methods are generally lacking. 

Note that Equation (8.49) applies for every point except for y=Y where the wall boundary condition is used, e.g.. Equation (8.27). When i = 0, Ogidi— 1) = ow(+ ) ... 

The numerical techniques of Chapter 8 can be used for the simultaneous solution of Equation (9.3) and as many versions of Equation (9.1) as are necessary. The methods are unchanged except for the discretization stability criterion and the wall boundary condition. When the velocity profile is flat, the stability criterion is most demanding when at the centerline ... 

Using a first-order approximation for the derivative in Equation (9.4), the wall boundary condition becomes... 

Reactor wall thermal boundary conditions can have a strong effect on the gas flow and thus the deposition. Here, for example, we indicate how cooling the reactor walls can enhance deposition uniformity. We consider the results of three simulations comparing the effects of two different wall boundary conditions. Figure 4 shows how the ratio of the computed susceptor heat flux to the onedimensional heat flux varies with the disk radius for the different conditions (the Nusselt number Nu is a dimensionless surface heat flux). In two cases the reactor walls are held at 300 K (0 = 0), and in one case the walls are insulated ( 0/ r —... 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

A number of authors have considered channel cross-sections other than rectangular [102-104]. Figure 2.17 shows some examples of cross-sections for which friction factors and Nusselt numbers were computed. In general, an analytical solution of the Navier-Stokes and the enthalpy equations in such channel geometries would be involved owing to the implementation of the wall boundary condition. For this reason, usually numerical methods are employed to study laminar flow and heat transfer in channels with arbitrary cross-sectional geometry. 

In their studies of frichon factors in channels with a number of different cross-sechonal geometries, Shah [103] and Shah and London [102] also computed heat transfer properties. A few characterishc cross-sechons for which Nusselt numbers were obtained are displayed on the left side of Figure 2.17. Their results include both the Nusselt numbers for fixed temperature and fixed heat flux wall boundary condihons and are given as tabulated values for different geometric parameters. 

The mean velocity and turbulent diffusivity should approach zero at solid walls. In theory, this should be enough to keep particles from crossing wall boundaries. In practice, due to the finite time step, some particles will eventually cross wall boundaries and must be accounted for. 

Wall-boundary conditions in probability density function methods and application to a turbulent channel flow. Physics of Fluids 11, 2632-2644. 

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We defer the discussion of the effects of (r) until Section VII.C and begin with the special case, referred as a force-free diffusion, with a uniform distribution of electron spins outside the distance of closest approach with respect to the nuclear spin. Under the assumption of the reflecting-wall boundary condition at rjs = d, Hwang and Freed found the closed analytical form of the correlation function for translation diffusion (138) ... 

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