Infinite parallel plate model

Flow in a thin rectangular channel (Figure 4.2), such as that used in field-flow fractionation, can be treated in a manner similar to that used for cylindrical capillary tubes. If the drag at the edges of the channel is neglected (infinite parallel plate model), then the force balance expression (corresponding to Eq. 4.5 for capillary tubes) becomes... 

A field-flow fractionation (FFF) channel is normally ribbonlike. The ratio of its breadth b to width w is usually larger than 40. This was the reason to consider the 2D models adequate for the description of hydrodynamic and mass-transfer processes in FFF channels. The longitudinal flow was approximated by the equation for the flow between infinite parallel plates, and the influence of the side walls on mass-transfer of solute was neglected in the most of FFF models, starting with standard theory of Giddings and more complicated models based on the generalized dispersion theory [1]. The authors of Ref. 1 were probably the first to assume that the difference in the experimental peak widths and predictions of the theory may be due to the influence of the side walls. 

Heat transfer in microsystems under pressure-driven flow conditions is considered here. Figure 9.8 shows two large infinite parallel plates, separated by distance, H, with uniform surface heat flux q. This configuration can be used to model flow and heat transfer in rectangular channels with large aspect ratio. Velocity and temperature are assumed to be fully developed. Inlet and outlet pressures are P,- and P , respectively. 

The theory has been developed for two special cases, the interaction between parallel plates of infinite area and thickness, and the interaction between two spheres. The original calculations of dispersion forces employed a model due to Hamaker although more precise treatments now exist [194],... 

To solve Eqs. 12 and 13 with oscUlating velocity boundary conditions, simple models such as the Couette flow model and ID Stokes model have been used. These models ignore the finite size and edge effects, as both of them model the device as two infinitely large parallel plates with one (the proof mass) oscillating on the top of the other (substrate). The Couette model further assumes a steady flow, resulting in a linear velocity profile between the plates. As shown later, the quality factor obtained by these two models is overpredicted by a factor of two, indicating the importance of 3D effects. 

Up to this point, we have discussed an unrealistic semi-infinite solid with a laterally infinitely extended surface, so the actual dimension of the surface did not enter into our considerations. However, the finiteness of any real surface has important consequences for the surface dipole barrier. Consider a finite circular surface of radius R. Owing to the dipolar charge distribution, the surface can be modeled by a circular parallel-plate capacitor with charge density dipole moment S = [Pg.110]

For many applications, like chemical-vapor-deposition reactors, the semi-infinite outer flow is not an appropriate model. Reactors are often designed so that the incoming flow issues through a physical manifold that is parallel to the stagnation surface and separated by a fixed distance. Typically the manifolds (also called showerheads) are designed so that the axial velocity u is uniform, that is, independent of the radial position. Moreover, since the manifold is a solid material, the radial velocity at the manifold face is zero, due to the no-slip condition. One way to fabricate a showerhead manifold is to drill many small holes in a plate, thus causing a large pressure drop across the manifold relative to the pressure variations in the plenum upstream of the manifold and the reactor downstream of the manifold. A porous metal or ceramic plate would provide another way to fabricate the manifold. 

Very concise expressions can he developed for FFF if the following assumptions are made (1) the FFF channel can be modeled as the space between two parallel infinite plates, (2) the flow profile is parabolic, (3) a steady state concentration profile is established in the channel, and (4) the external field is imiform. With these assumptions, a simple mass balance across the cross section of the channel leads to the expression for the transversal flux Jx,... 

---