Eccentric rotating cylinders

Cubers and pelletizers consist of a cylindrical die from hard steel, perforated with holes. Inside the die there are two rollers mounted on the periphery of the die. There are two designs of the machine as regards the rotating part. In one design, the die rotates while the rollers do not, and vice versa in the second. In another design of the machine, rollers have been replaced by an eccentrically rotating cylinder. 

The method of domain perturbations was used for many years before its formal rationalization by D. D. Joseph D. D. Joseph, Parameter and domain dependence of eigenvalues of elliptic partial differential equations, Arch. Ration. Mech. Anal. 24, 325-351 (1967). See also Ref. 3f. The method has been used for analysis of a number of different problems in fluid mechanics A. Beris, R. C. Armstrong and R. A. Brown, Perturbation theory for viscoelastic fluids between eccentric rotating cylinders, J. Non-Newtonian Fluid Mech. 13, 109-48 (1983) R. G. Cox, The deformation of a drop in a general time-dependent fluid flow, J. Fluid Mech. 37, 601-623 (1969) ... 

JACKSON, P.A., HDBATI, B. and MOBBS, F.R. "Secondary flows between eccentric rotating cylinders at sub-critical Taylor nunbers."... 

MOBBS, F.R. and OZOGAN, M.S. "Study of sub-critical Taylor votex flow between eccentric rotating cylinders torque measurements and visual observations." International journal of Heat and Fluid Flow, 1984, Vol. 5, no. 4, 251-253. 

CASTLE, P. and MC S, F.R. "I rodynamic stability of the flow between eccentric rotating cylinders." Proc. Instn. Mech Engrs, 1968, 182, 41-52. 

JCNES, C.D. "A study of secondary flow and turbulence between eccentric rotating cylinders." PhD. Thesis Mech. Eng. Dept., university of Leeds. 1973. 

O BRIEN, K.T. and MOBBS, F.R."The evolution of turbulence between eccentric rotating cylinders." 2nd Leeds-Lyon Symposium on Tribology. 1975, 51-56. 

Other eccentric rotating geometries (a) tilted rotating hemispheres (also called balance rheometer), (b) eccentric rotating cylinders, and (c) tilted rotating cone and plate. 

Eccentric rotating cylinders can give accurate and G data but only at rather small strains (Broadbent and Walters, 1971). At larger deformation, cavitation or extrusion of the sample from the gap can occur. This also seems to be the problem for such other eccentric geometries as the tilted rotating disks or cone and plate (Davis and Macosko, 1973 Walters, 1975). 

Fig. 7.9 Eccentric coaxial cylinders or journal bearing configuration. In the mixing protocol, the cylinders rotate alternately in the clockwise and counterclockwise directions.

As stated, we begin with the special problem of flow between two rotating cylinders whose axes are parallel but offset to produce the eccentric cylinder geometry shown in Fig. 5 1. In the concentric limit, this is the famous Couette flow problem, which was analyzed in Chap. 3. 

The sliding-vane compressor uses a single rotating element (see Figure 1-8). The rotor is mounted eccentric to the center of the cylinder portion of the casing and is slotted and fitted with vanes. The vanes are free to... 

Rotation of the pump rotor, which Is mounted eccentrically In the pump cylinder, traps entering vapor between rotor vane segments. As rotation continues, vapor is compressed then discharged into the exhaust box. Vapors then pass through four stages of Internal oil and smoke eliminators to remove 99.9% of lubricating oil from the exhaust. Oil is then returned to the recirculating oil system. 

There are several mechanisms by which wow can occur. One cause is a variation of rotational speed of the recording medium during either recording or playback. A further cause is eccentricity in the recording or playback process for disc and cylinder recordings, for example a hole which is not punched perfectly at the centre of a gramophone disc. Lastly it is possible for magnetic tape and optical film to become... 

To obtain governing equations and boundary conditions, we adopt the circular cylindrical coordinate system that is shown, along with a top view of the eccentric cylinder system, in Fig. 5-1. In this system, the origin of the coordinate system is chosen to be coincident with the central axis of the inner cylinder, which is assumed to have a radius a and an angular velocity 2 in the direction shown. The radius of the outer cylinder is assumed to be a( 1 + e), and its center is offset along the 0 = 0 axis relative to the center of the inner cylinder by an amount ask. In the journal-bearing problem, the outer cylinder does not rotate ( 2 = 0). The surface of the inner cylinder is thus... 

It is perhaps worthwhile to dwell briefly on the use of the journal-bearing configuration in practical lubrication applications. In such circumstances, the rotating inner cylinder is normally allowed to float to seek a position in which the hydrodynamic force precisely balances the load. We have analyzed the eccentric cylinder problem based on the picture in Fig. 5-1, in which the line of centers between the two cylinders is in the horizontal direction. In that configuration, there is a net vertical force but no horizontal force. Moreover, examination of (5-48) shows that the magnitude of the vertical force for a given pair of cylinders (so that a and e are fixed) is determined by 9 and A. The A dependence is contained in the factor... 

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