Concentric cylindrical surfaces

Spiral vortex. If a radial flow is superimposed upon the concentric flow previously described, the path lines will then be spirals. If the flow goes out through a circular hole in the bottom of a shallow vessel, the surface of liquid takes the form of an empty hole, with an air core sucked down the hole. If an outlet symmetrical with the axis is provided, as in a pump impeller, we might have a flow either radially inward or radially outward. If the two plates are a constant distance B apart, the radial flow with a velocity Vr is then across a series of concentric cylindrical surfaces whose area is 0.2nrB. Thus Q = 2nrBVr is a constant, from which it is seen that rVr is a constant. Thus the radial velocity varies in the same way with r that the circumferential velocity did in the preceding discussion. Hence the pressure variation with the radial velocity is just the same as for pure rotation. Therefore the pressure gradient of flow applies exactly to the case of spiral flow, as well as to pure rotation. 

Develop an expression similar to Eq. (12-53) for low-density conduction between concentric cylindrical surfaces. 

At each level, i, the horizontal stage is bounded by a plane of length idp and i — )dp from the entrance to the bed. Radial stages are bounded at j by concentric cylindrical surfaces, radius K j — )dp and Kjdp. The index i is an integer, but j assumes noninteger values (multiples of 1/2) for odd values of i. The radius, in terms of particle diameters, M, is also taken to be an integer... 

Fischer et al, [122] proposed a model to predict the adsorption isotherm of krypton in porous material at supercritical temperature. In their study, a model pore of infinite length is formed by concentric cylindrical surfaces on which the centers of solid atoms are located. The interaction between an adsorbate and an individual center on the pore wall is described by the LJ 12-6 theory, and the overall potential is the integral of this interaction over the entire pore surface. With thermodynamic relations, Fischer et al. obtained the functional dependence of the saturation adsorption excess and the Henry s law constant on the pore structure. The isotherm was then produced by the interpolation between Henry s law range and saturation range. They tested their theory with the adsorption of krypton on activated carbon. It was shown that, with information on the surface area of the adsorbent and thermodynamic properties of the adso bate, their model gives more than quantitative agreement with experimental data. If a few experimental data such as the Henry s law constant at one temperature are available, the isotherms for all temperatures and pressures can be predicted with good quality. 

Concentric spherical or infinite cylindrical surface zones, Ai inside = 1 f ai = Ai/Aa)... 

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 ... 

The parabolic diffusion model is used to indicate that diffusion controlled phenomena are rate limiting. It was originally derived based on radial diffusion in a cylinder where the chemical compound concentration on the cylindrical surface was constant, and initially the chemical compound concentration throughout the cylinder was uniform. It was also assumed that the diffusion of the compound of interest through the upper and lower faces of the cylinder was negligible. Following Crank [119], the parabolic diffusion model can be expressed as ... 

According to the MC simulation results mentioned above, if the diblock copolymer melts are confined in cylindrical pore or between two concentric curved surfaces with strong preference to one of the blocks, the... 

The mass conservation equation in rectangular coordinates x, y, and z (Equation 3.31) is suitable for catalyst pellets with a parallelepiped shape, in particular when they can be considered as a plate. For particles with cylindrical surfaces and for spheres, the cylindrical and spherical coordinate systems are more natural. For ax-isymmetrical concentration distributions in cylindrical coordinates Equation 3.30 becomes... 

There is thus in a small space a cylindrical porcelain surface very strongly heated and a concentric brass surface very cold. 

Continuously operated stirred tanks can only attain homogenization if q > q (where q is the liquid throughput through the vessel). The mixing time is then 0 oc V/q (where V is the liquid volume of the vessel). Knowledge of q, in the case of propeller stirrers, enables the flow velocity along the heat exchanging surfaces of one or more concentric cylindrical coils round the stirrer (and consequently the heat transfer rates) to be calculated. Such an installation may be indispensable in case of extremely exothermic reactions. 

Surface wave formation and stationary vortex formation are particularly effective demonstrations of convection enhancement via magnetic fields. A theoretical prediction for aqueous electrolytes was corroborated by experimental evidence produced in a concentric cylindrical cell using copper electrodes and aqueous cupric sulfate electrolytes. Similar observations were reported by Gak and co-workers. Self-sustained oscillations induced by nonhomogeneous periodic and aperiodic magnetic fields have been demonstrated in photographs showing contiguous six- and seven-vortex cluster domains in the experimental apparatus. 

An annular channel is the space between two concentric cylindrical tubes, shown in Figure 6.16. This geometry is used in double-pipe heat exchangers, as described later. The primary interest is on heat transfer on the outside surface of the inner tube. In evaluating the Reynolds, Nusselt, and... 

The primary current distribution equations were first derived by Kasper [11-13] for simple geometries of uniform distributions, such as infinite parallel plane plates, infinite height cylindrical surfaces, and concentric spheres (Figure 13.1). Further, some of these parameters were approached using the secondary current distribution. 

C, respectively, and the emissivities of A and B are 0,90 and 0.25, respectively. Both surfaces are gray, (a) Surfaces A and B are infinite parallel planes 3 m apart, (b) Surface is a spherical shell 3 m in diameter, and surface. S is a similar shell concentric with A and 0,3 m in diameter, (c) Surfaces A and B are fiat parallel squares 2 by 2 m, one exactly above the other, 2 m apart, (d) Surfaces A and B are concentric cylindrical tubes with diameters of 300 and 275 mm, respectively, (e) Surface A is an infinite plane, and surface B is an infinite row of lOO-mm-OD tubes set on 200-mm centers. (/) Same as (e) except that 200 mm above the centerlines of the tubes is another infinite plane having an emissivity of 0,90, which does not transmit any of the energy incident upon it. (g) Same as /) except that surface. 6 is a double row of lOO-mm-OD tubes set on equilateral 200-mm centers. 

Fig. 33. a Concentric cylindrical twist walls separating different chiral smectic regions. Inside each region the layer surface adopts a helicoidal shape, b Each cylindrical twist wall is comparable to that found in a conventional TGB phase, a helical shape being given to the screw dislocations constituting them (after Gilli and Kamaye [43])... 

Planar with circumferential flow, in which streamlines form concentric circles in the impingement plane Tubular, in which impinging streams meet at the cylindrical surface Annular, in which impinging streams meet in a sector of annulus... 

---