Concentric cylinders

If a fluid is placed between two concentric cylinders, and the inner cylinder rotated, a complex fluid dynamical motion known as Taylor-Couette flow is established. Mass transport is then by exchange between eddy vortices which can, under some conditions, be imagmed as a substantially enlranced diflfiisivity (typically with effective diflfiision coefficients several orders of magnitude above molecular difhision coefficients) that can be altered by varying the rotation rate, and with all species having the same diffusivity. Studies of the BZ and CIMA/CDIMA systems in such a Couette reactor [45] have revealed bifiircation tlirough a complex sequence of front patterns, see figure A3.14.16. 

A formal mathematical analysis of the flow in the concentric cylinder viscometer yields the following relationship between the experimental variables and the viscosity ... 

Figure 2.3 Definition of variables for concentric cylinder viscometers (a) the rotating cylinder and (b) the coaxial cylinders.

Figure 2.4 A commercial instrument, the Brookfield Digital Viscometer, based on the geometry of the concentric cylinder viscometer. (Photo courtesy of Brookfield Engineering Laboratories, Inc., Stoughton, Mass. 02072.)...

A fluid of viscosity 17 is confined within the gap between two concentric cylinders as shown in Fig. 2.3b. Consider a cylindrical shell of radius r, length 1, and thickness dr located within that gap. 

The concentric cylinder viscometer described in Sec. 2.3, as well as numerous other possible instruments, can also be used to measure solution viscosity. The apparatus shown in Fig. 9.6 and its variations are the most widely used for this purpose, however. One limitation of this method is the fact that the velocity gradient is not constant, but varies with r in this type of instrument, as noted in connection with Eq. (9.26). Since we are not considering shear-dependent viscosity in this chapter, we shall ignore this limitation. 

Normal Stress (Weissenberg Effect). Many viscoelastic fluids flow in a direction normal (perpendicular) to the direction of shear stress in steady-state shear (21,90). Examples of the effect include flour dough climbing up a beater, polymer solutions climbing up the inner cylinder in a concentric cylinder viscometer, and paints forcing apart the cone and plate of a cone—plate viscometer. The normal stress effect has been put to practical use in certain screwless extmders designed in a cone—plate or plate—plate configuration, where the polymer enters at the periphery and exits at the axis. 

Fig. 27. Concentric cylinder viscometer. R and R are the radii of the inner and outer cylinder, respectively, and Q is the relative angular velocity.

Coaxial (Concentric Cylinder) Viscometer, The eadiest and most common type of rotational viscometer is the coaxial or concentric cylinder instmment. It consists of two cylinders, one within the other (cup and bob), keeping the specimen between them, as shown in Figure 27. The first practical rotational viscometer consisted of a rotating cup with an inner cylinder supported by a torsion wire. In variations of this design the inner cylinder rotates. Instmments of both types ate useful for a variety of apphcations. 

The relationship between viscosity, angular velocity, and torque for a Newtonian fluid in a concentric cylinder viscometer is given by the Margules equation (eq. 26) (21,146), where M is the torque on the inner cylinder, h the length of the inner cylinder, Q the relative angular velocity of the cylinder in radians per second, T the radius of the inner cylinder wall, the radius of the outer cylinder wall, and an instmment constant. 

In addition to non-Newtonian flow, the main correction necessary for concentric cylinder measurements is that on account of end effects. Because the inner cylinder is not infinitely long, there is drag on the ends as well as on the face of the cylinder. The correction appears as an addition, to the length, b. The correction is best deterrnined by measuring the angular velocity and torque at several values of b, that is, at various depths of immersion. The data are plotted as M/Q vs b, and extrapolation is made to a value of at M/H = 0. The quantity (/i + h ) is substituted for b in the various equations. 

Dyna.mic Viscometer. A dynamic viscometer is a special type of rotational viscometer used for characterising viscoelastic fluids. It measures elastic as weU as viscous behavior by determining the response to both steady-state and oscillatory shear. The geometry may be cone—plate, parallel plates, or concentric cylinders parallel plates have several advantages, as noted above. 

With several springs, which function as torque gauges, and a number of spindles, viscosities can be measured up to 10 mPa-s with the Brookfield viscometer. The shear rates depend on the model and the sensor system they are ca 0.1 100 for the disk spindles, <132 for concentric cylinders, and <1500 for the cone—plate forlow viscosity samples. Viscosities at very low (ca 10 — 1 )) shear rates can be measured with the concentric... 

The Ravenfield model BS viscometer is a wide shear rate range iastmment with several possible measurement systems cone—plate, parallel plates, concentric cylinders, and taper plug. The last gives shear rates of up to 10 , and the cone—plate of up to 8 x lO". The viscosity range is... 

D, Outside diameter of wire or discharge electrode of concentric-cylinder type of electrical precipitator m ft ... 

The field strength is uniform between parallel plates, whereas it varies in the space between concentric cylinders, being highest at the surface of the central cylinder. After corona sets in, the current flow will become appreciable. The field strength near the center electrode will be less than given by Eq. (17-18) and that in the major portion of the clearance space will be greater and more uniform [see Eqs. (17-23) and (17-24)]. 

FIG. 24-4 Air-lift fermenters (a) Concentric cylinder (h) external recycle. 

In this apparatus the polymer melt is sheared between concentric cylinders. The torque required to rotate the inner cylinder over a range of speeds is recorded so that viscosity and strain rates may be calculated. 

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