Concentric cylinders viscometer

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.)...

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. 

Figure 3.37. Partial section of a concentric-cylinder viscometer...

Fig. 15. Effects of turbulent shear stress level and exposure time on cell viability measured by trypan blue staining. Cells were sheared in a concentric cylinder viscometer [1]...

Fig. 17. Response of CRL-8018 hybridoma cells to increasing levels of well-defined laminar shear in the concentric cylinder viscometer for 10 min. Spinner flask cultures were seeded with cells from routine T-flask cultures that were 3 days old. Cell samples were taken from the spinner flask cultures during late-exponential growth and sheared in the viscometer [17]...

Concentration standards, 75 750-751 Concentrators, sulfuric acid, 23 787 Concentric annular reactors, 23 544 Concentric cylinder viscometer, 27 733 Concentric hemispherical analyzer (CHA), 24 103-104, 105 energy resolution of, 24 106 Conching, milk chocolate, 6 363-364 Concomitant polymorphism, 8 69 CONCORD program, 6 10 76 752 Concrete(s)... 

Properties of EthanolIc Fructose Solutions. Published information on the properties of aqueous ethanolic fructose solution is very limited. As a result, solubility data from 25 to 60 °C was measured (Figure 1) and will be published separately. The equilibrium fructose/water mass ratio for zero alcohol, ranges from over 4 at 25 C to over 8 at 60 C (7,8). It can be seen that reasonable yields will only result with high alcohol additions (E/W at least 2). This is the range used in this study. Aqueous ethanolic solutions have a wide range of viscosities. These were measured at operating conditions using a Rheomat concentric cylinder viscometer. 

The same apparatus was used, but quantities of paste were removed to give an air space in the vessel. On rapid agitation the volume increased, dependent on the air content required. Paste viscosities were measured, using a Stormer viscometer, which is a type of concentric cylinder viscometer. Although it is possible to obtain results in absolute terms, for comparative purposes the times for 100 revolutions of the rotor under a fixed applied torque were recorded. 

In the next section we consider an experimental approach to viscosity. We generate the apparatus of interest by wrapping —in our imagination —the fluid in Figure 4.1 into a closed ring around the z axis. The two rigid surfaces then describe concentric cylinders, and the instrument is called a concentric-cylinder viscometer. 

CONCENTRIC-CYLINDER AND CONE-AND-PLATE VISCOMETERS 4.3a Concentric-Cylinder Viscometers... 

FIG. 4.3 Schematic representation of a concentric-cylinder viscometer (a) geometry of a cup and bob and (b) volume element within a liquid gap. 

EXAMPLE 4.1 Stress-Strain Relationship for a Concentric-Cylinder Viscometer with Small Gaps. Examine Equation (8) in the limit f -> 1 to show that the relationship reduces to Equation (1) under these conditions. 

EXAMPLE 4.2 Comparison Between Capillary Viscometers and Concentric-Cylinder Viscometers. Criticize or defend the following proposition A set of capillary viscometers with different radii can be used in much the same way as a concentric-cylinder viscometer with variable speed or gap width to conduct studies in which the rate of shear is an independent variable. 

Solution Equation (16) shows that the velocity gradient is not uniform in a capillary viscometer any more than it is in a concentric-cylinder instrument. The rate of shear dvldr is directly proportional to the radial distance from the axis of the cylinder. At the wall it has its maximum value, which is proportional to Rc] at the center of the tube it equals zero. Some intermediate value, say, the average, might be used to characterize the gradient in a given instrument. This quantity will be different for capillaries of different radii. All of this is similar to the situation in concentric-cylinder viscometers. 

FIG. 4.9 Experimental verification of Einstein s law of viscosity for spherical particles of several different sizes (Squares are yeast particles, Rs = 2.5 71m circles are fungus spores, Rs = 4.0 /xm triangles are glass spheres, R, = 80 /xm). Open symbols represent measurements in concentric-cylinder viscometers, and closed symbols represent measurements in capillary viscometers. (Data from F. Eirich, M. Bunzl, and H. Margaretha, Kolloid Z., 74, 276 (1936).)... 

Next let us consider how surface viscosities can be measured. A variety of methods for measuring rjs exist, including a method based on concentric rings, the two-dimensional equivalent of the concentric cylinder viscometer. We limit our discussion to the analog of the... 

Stress-strain relationship from a concentric-cylinder viscometer Capillary viscometers versus concentric-cylinder viscometers Inherent viscosity at low volume fractions Extent of hydration from intrinsic viscosity measurements Empirical determination of the Mark-Houwink coefficients Variation of viscosity with polymer configuration... 

C.G. Qiu and M.A. Rao, Effect of Dispersed Phase on the Slip Coefficient of Apple Sauce in a Concentric Cylinder Viscometer, J. Texture Studies, 16 179-192 (1989). 

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