Parallel disks, radial flow

Radial Pressure Flow between Parallel Disks Solve the problem of radial pressure flow between two parallel disks. The flow is created by a pressure drop (P r 0 P r R). Disregard the entrance region, where the fluid enters from a small hole at the center of the top disk. 

Problem 4-8. Pressure-Driven Radial Flow Between Parallel Disks. The flow of a viscous fluid radially outward between two circular disks is a useful model problem for certain types of polymer mold-filling operations, as well as lubrication systems. We consider such a system, as sketched here ... 

Consider the steady-state, fully developed, incompressible flow between parallel disks, such as illustrated in Fig. 5.9. In concert with the Jeffery-Hamel assumptions that were made in the previous configurations, one can assume that only the radial velocity is nonzero. As a consequence the continuity and momentum equations reduce to the following ... 

Fig. 5.9 Illustrative geometry for the radial flow between parallel disks.

Fig. 5.11 The T( 2) function for radially inward flow between parallel disks.

In Section 5.5 it was shown that for radial flow between parallel disks that the convective term could not be retained. It appears here that for 0 0 the convective term can be retained, since the cross-stream momentum equation provides a relationship between / and the pressure gradient. In the limiting case of 0 = n/2 the convective terms vanishes in any case. 

As a partial check on the derivations in the conical coordinates, it should be possible to recover two, easily identified, special cases—the radial flow between parallel disks and the axial Poiseuille flow in an cylindrical annular gap. The parallel-disk flow (Section 5.5) is the case where 0 = 0, with x taking the role of r and y taking the role of z. In this case, h = De/2 + x = r. The momentum equations become... 

Problem 3-37. Taylor Dispersion with Streamwise Variations of Mean Velocity. We consider steady, pressure-driven axisymmetric flow in the radial direction between two parallel disks that are separated by a distance 2h. We assume that the volumetric flow rate in the radial direction is fixed at a value Q and that the Reynolds number is small enough that the Navier-Stokes equations are dominated by the viscous and pressure-gradient terms. Finally, the flow is ID in the sense that u = [nr(r, z), 0, 0]. In this problem, we consider flow-induced dispersion of a dilute solute. We follow the precedent set by the classical analysis of Taylor for axial dispersion of a solute in flow through a tube by considering only the concentration profile averaged across the gap, ( ) = f h dz. 

Example 2.4. Radial Flow of a Newtonian Fluid Between Two Parallel Disks... 

A possible problem of sealing the electrolyte path is found in the Foreman and Veatch cell. This can be avoided by placing the cells in a vessel. The best known example of this is the Beck and Guthke cell shown in Figure 8 (74). The cell consists of a stack of circular bipolar electrodes in which the electrolyte is fed to the center and flows radially out. Synthesis experience using this cell at BASF has been described (76). This cell exhibits problems of current by-pass at the inner and outer edge of the disk cells. Where this has become a serious problem, insulator edges have been fitted. The cell stack has parallel electrolyte flow however, it is not readily adaptable to divided cell operation. 

Consider one side of a disk whose diameter is large compared to 6, rotating in a large vessel of liquid. Since liquid rotates with the disk and acquires a radial as well as an angular motion, it must also flow toward the face of the disk. The velocity of the perpendicular flow can be considered independent of distance except very near the disk. The distance at which flow becomes essentially parallel to the disk can be called the hydrodynamic boundary layer thickness. The distance at which a concentration gradient begins was calculated by Levich (7) to be... 

A lubricant flows radially between two parallel circular disks from a radius ri to another f2 because of AP. 

Figure 7.3.14 illustrates the trajectory of a particle in between two consecutive disks (dashed lines). Define two coordinate axes the positive z-axis parallel to the disks, and in the direction of main liquid flow toward the main vertical device axis, and the positive r-axis radially... 

The overall sample geometry is governed by unidirectional heat flow. The only two practical geometries are either a slab-shaped solid with two parallel faces and heat flow perpendicular to these faces (Fig. 2a), most commonly used for polymers, or a right circular cylinder with heat flow in the radial direction and perpendicular to the axis (Fig. 2b), which is little used for pol3rmers. Because of the low thermal conductivity of polymers, the slab (or radius of the cylinder) is usually thin, so that heat losses in directions perpendicular to the desired heat-flow direction are minimized and the temperature drop AT is not excessive. Therefore, the preferred specimen shape is usually a thin disk with parallel faces and less commonly a long, thin rod or coaxial cylinder. 

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