Annular gap

Fig. 18. Stirred wet ball mills (a) vertical and (b) horizontal shaft, and (c) annular gap.

The annular gap mill shown in Fig. 20-36 is avariation of the bead mill. It has a high-energy input as shown in Fig. 20-37. It may be lined with polyurethane and operated in multipass mode to narrow the residence-time distribution and to aid cooling. 

FIG. 20-36 Annular gap-head mill (Welte Mahltechnik, Gmbh.) [Kolb, Ceramic Forum International, 70(5), 212-6 (1993)]. 

Among new mill developments, annular-gap bead mills and stirred bead mills are being used. These have a high cost, but result in a steep particle-size distribution when used in multipass mode [Kolb, Ceramic Forum International, 70(5), 212-216 (1993)]. Costs for fine grinding typically exceed the cost of raw materials. Produces are used for high-performance ceramics. 

The second half of the cooling air flow is directed to the front end of the rotor and from there through an aerodynamically shaped annular gap to the rotor surface, forming a film of cooling air down to the second stage. A small portion of this latter air flow is diverted to the labyrinths for sealing and cooling purposes. 

There is an analytical solution of the Navier-Stokes equations for the flow between two rotating cylinders with laminar flow (see e.g. [37]). The following equation applies for the velocity gradient in the annular gap in the general case of rotation of the outer cylinder (index 2) and the inner cylinder (index 1) ... 

MHz, from 20% w/w CTAB-D20 (41 °C) at observed, consistent with an ordered phase, different positions across the annular gap of a while near the outer wall the single peak of an cylindrical Couette cell and at an apparent isotropic phase is seen. In between, a mixed shear rate of 20 s 1. Near the inner wall, where phase region exists (adapted from Ref. [38]). 

In the Couette type, the material is contained in an annular gap between the inner cylindrical bob or spindle... 

As the name implies, the cup-and-bob viscometer consists of two concentric cylinders, the outer cup and the inner bob, with the test fluid in the annular gap (see Fig. 3-2). One cylinder (preferably the cup) is rotated at a fixed angular velocity ( 2). The force is transmitted to the sample, causing it to deform, and is then transferred by the fluid to the other cylinder (i.e., the bob). This force results in a torque (I) that can be measured by a torsion spring, for example. Thus, the known quantities are the radii of the inner bob (R ) and the outer cup (Ra), the length of surface in contact with the sample (L), and the measured angular velocity ( 2) and torque (I). From these quantities, we must determine the corresponding shear stress and shear rate to find the fluid viscosity. The shear stress is determined by a balance of moments on a cylindrical surface within the sample (at a distance r from the center), and the torsion spring ... 

Various arrangements at the bottom of the inner cylinder are available in Figure 3.2 an indentation is provided so that an air gap is formed and shearing in the sample below the inner cylinder is negligible. Another arrangement is to make the bottom of the inner cylinder a cone. When one of the cylinders is rotated, a Couette flow is generated with fluid particles describing circular paths. The only non-zero velocity component is ve and it varies in the r-direction. In order to minimize secondary flow (Taylor vortices) it is preferable that the outer cylinder be rotated however, in most commercial instruments it is the inner cylinder that rotates. In this case, the fluid s velocity is equal to IXR, at the surface of the inner cylinder and falls to zero at the surface of the outer cylinder. The shear stress is uniform over the curved surface of the inner cylinder and over the outer cylinder (to the bottom of the annular gap). 

For this purpose, the flow tube emptied directly into a high-pressure ion source. This source was essentially a sealed box with a gas inlet for the Cl reagent gas, a 0.58 mm hole to allow injection of a magnetically collimated electron beam, and a 0.99 mm hole to allow ions to exit into the mass spectrometer. The flow tube was coupled to the source using a 0.1 mm annular gap that thermally isolates the source from the flow tube, but allows little of the gas flow to escape. Even at a flow tube temperature of 1000 K, the source temperature increased no more than 50 K. To avoid any variations in source conditions with flow tube temperature, the source was thermostated to a constant temperature of 100 K. 

VFF devices consist of a stationary cyhnder, inside which a concentric cyhn-der rotates. The rotating movement of the inner surface of the annular gap creates a Taylor-Couette flow [16], generating Taylor vortices. The filter medium can... 

The following rod-and-guide example illustrates the behavior of these solutions, Fig. 4.4. Here a rod moves to the right with a constant velocity and the outer guide is held fixed. A pressure gradient is imposed. Two cases are considered one with a thin annular gap compared to the radius and the other a wide gap compared to the rod dimension. 

Fig. 4.4 Velocity profiles in the annular gap between a rod moving with velocity U to the right and a stationary guide. The figure on the left is for a relatively thin gap, rj/Ar = 10, and the solution on the right is for a relatively wide gap, rj/Ar = 0.05. The solutions are both parameterized by the nondimensional group, P = These solutions were...

As formulated in Section 4.2, the length scale for the nondimensionalization of the flow in an annular gap is the gap radius Ar. A more conventional length scale would be the hydraulic diameter,... 

Solve the Couette-flow problem to determine the velocity profile in the large annular gap that carries the primary flow. 

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... 

Fig. D.l Spreadsheet to solve the axial Couette-Poiseuille flow in a long annular gap. The problem is described and discussed in Section 4.2.2.

Flow birefringence of polymer solutions is, in general, measured with the aid of an apparatus of the Couette type, containing two coaxial cylinders. One of these cylinders is rotated at constant speed, the other is kept in a fixed position. The light beam for the birefringence measurement is directed through the annular gap between these cylinders, in a direction parallel with the axis of the apparatus. In this way, the difference of principal refractive indices An is measured just in the above defined plane of flow (1—2 plane). 

This type of apparatus has almost exclusively been used for investigations on dilute solutions. For a Newtonian fluid the velocity V at any point P in the annular gap between infinitely long cylinders reads 194) ... 

At the entrance to the annular gap the light-beam is confined in the radial direction by the edges of the cylinders (gap width 0.25 mms in the described apparatus) and by the rim of the circular hole D (0 = 2 mms) in the tangential direction. Two millimeters along the circumference of... 

Fig. 6.2. Photographs of the diffraction pattern which are observed in the plane of entrance to the annular gap. Left No extinction. Right Extinction. The dark zone in the middle (i.e. the isocline) is clearly noticeable...

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