Concentric cylinders eccentricity

This result holds in particular for concentric spheres and very long concentric cylinders as here the assumption of isothermal surfaces applies more easily. If, however, body 1 lies eccentric in the enclosure surrounded by body 2, Fig. 5.64, then the two surfaces will generally not be isothermal, as the radiation flow is much higher in the regions where the two surfaces are close to each other than where a large distance exists between them. 

Enclosure problems (Fig. 4.1c) arise when a solid surface completely envelops a cavity containing a fluid and, possibly, interior solids. This section is concerned with heat transfer by natural convection within such enclosures. Problems without interior solids include the heat transfer between the various surfaces of a rectangular cavity or a cylindrical cavity. These problems, along with problems with interior solids including heat transfer between concentric or eccentric cylinders and spheres and enclosures with partitions, are discussed in the following sections. Property values (including P) in this section are to be taken at Tm = (Th+ TC)I2. 

Region between Concentric or Eccentric Cylinders. The geometry and dimensions are as shown in cross section in Fig. 4.36a, the two cylinders being assumed to have parallel axes. The dimension E represents the perpendicular distance from the axis of the inner cylinder to the axis of the outer cylinder. Thus, for concentric cylinders, E = 0. Each cylinder is taken to be isothermal but at a different temperature. 

FIGURE 436 Sketch of concentric and eccentric cylinder and sphere... 

The main sources of error in the concentric cylinder type measuring geometry arise from end effects (see above), wall shp, inertia and secondary flows, viscous heating effects and eccentricities due to misahgnment of the geometry [Macosko, 1994],... 

The case of eccentric cylinders, with its application to journal bearings, has not been as extensively investigated, but recent results for concentric cylinders have important implications for the eccentric cylinder case. 

The presence of weak vortex motions at Taylor ntmbers less than T have been reported by many investigators and have been described in detail by Jackson, Robati and Mobbs (3), for both concentric and eccentric cylinders. That these motions are due to end effects has been shown by numerical solutions of the Navier-Stokes equations for the flow in a finite length annulus. Solutions have been obtained by Alziary de Roquefort and Grillaud (4) for concentric cylinders with end plates rotating with the inner cylinder, Preston (5) for concentric cylinders with fixed end plates and El-Dujaily (6) for concentric and eccentric cylinders with fixed end plates. 

For eccentric cylinders, no systematic wave frequency measurements have been made, although O Brien and Mobbs (19) have ccnmented on the azimuthal dependence of the flow. Since x/d varies around the azimuth the situation is far more ccmplex than in the concentric cylinder case. 

EL-DUJAILK, M.J. "End effects on sub-critical and Taylor vortex flow between concentric and eccentric cylinders." KiD. Ttiesis. Department of Mechanical Engineering, university of Leeds, 1983. 

The bearing runout problems discussed above highlight the importance of mechanical construction in rheometers. Clearly a small departure from 90° between the axis of a small angle cone and the surface of its plate will cause the shear rate to be higher on one side than on the other. The same is true for eccentricity in the axes of concentric cylinders (eq. 5.3.43) but with typical gap sizes (> 0.5 mm), alignment is less critical than with the cone and plate. 

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 parameter k determines the degree of eccentricity. If k = 1, the two cylinders touch at 0 = 7r whereas they are concentric if k = 0. The range of allowable values for k is thus... 

This equation was shown to closely fit the E = 0 data of Scanlan et al. [241], which covered the ranges 1.3 x 103 < Ra < 6 x 108, 5 < Pr < 4000, and 1.25 < DJD, < 2.5. The measurements of Weber et al. [281] for eccentric spheres, where the displacement from the concentric position is vertically up or down, showed that for downward displacement E had little effect on Nu for 0 < EIL < 0.75, but for upward displacement with 0.25 < EIL < 0.75, the Nusselt number was observed to be about 10 percent higher than that given by Eq. 4.128. The same reservations as discussed for the cylinder when NuCond = Nu, apply here. 

E. M. Sparrow and M. Charmichi, Natural Convection Experiments in an Enclosure Between Eccentric or Concentric Vertical Cylinders of Different Height and Diameter, Int. J. Heat Mass Transfer (26/1) 133-143,1983. 

There have recently been rapid advances in our understanding of Taylor-Couette flow between a rotating inner cylinder and a concentric stationary outer cylinder. Hiis progress is reviewed and its implications in the case of eccentric cylinders or journal bearings considered. 

Much further work is required to extend the present knowledge of superlaminar flow between concentric cyliniders to the eccentric cylinder/journal bearing case. indications are that within the range of Taylor nonbers encountered in practice the flow consists primarily of Taylor vortices with superimposed travelling waves and any possible turbulence is weak. 

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