Concentric cylinder viscometer fluids

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. 

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 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. 

Plant food dispersions such as tomato concentrates and concentrated orange juice are important items of commerce. The viscosity function and the yield stress are two important rheological properties that have received considerable attention. Corrections for slip, due to the formation of a thin layer of fluid next to solid surfaces, in a concentric cylinder viscometer depended on the magnitudes of applied torque and on the shear-thinning characteristics of the dispersion. Mixer viscometers were used for obtaining shear rate-shear stress and yield stress data, but the latter were higher in magnitude than those obtained by extrapolation of flow data. 

If the concentric cylinder viscometer has a narrow gap (i.e., 0.97 < R,IR < 1), with the inner cylinder rotating and the outer cylinder stationary, the shearing stress t, may be given for any fluid, newtonian or nonnewtonian, by the following expression ... 

In the analysis of raw data obtained with any type of rotational viscometer, it is assumed that the fiow field is known and simple. For example, in the conventional concentric-cylinder viscometer, it is assumed that the fluid moves in... 

The flow-property or rheological constants of non-Newtonian fluids can be measured using pipe flow as discussed in Section 3.5E. Another, more important method for measuring flow properties is by use of a rotating concentric-cylinder viscometer. This was first described by Couette in 1890. In this device a concentric rotating cylinder (spindle) spins at a constant rotational speed inside another cylinder. Generally, there is a very small gap between the walls. This annulus is filled with the fluid. The torque needed to maintain this constant rotation rate of the inner spindle is measured by a torsion wire from which the spindle is suspended. A typical commercial instrument of this type is the Brookfield viscometer. Some types rotate the outer cylinder. 

There are many different kinds of viscometers in use for measuring these properties. Perhaps the best for power law fluids would be either the concentric cylinder viscometer or the tube viscometer please refer to references 4,5, and 6 for details of these. 

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. 

Historically, viscosity measurements have been the single most important method to characterize fluids in petroleum-producing applications. Whereas the ability to measure a fluid s resistance to flow has been available in the laboratory for a long time, a need to measure the fluid properties at the well site has prompted the development of more portable and less sophisticated viscosity-measuring devices [1395]. These instruments must be durable and simple enough to be used by persons with a wide range of technical skills. As a result, the Marsh funnel and the Fann concentric cylinder, both variable-speed viscometers, have found wide use. In some instances, the Brookfield viscometer has also been used. 

A computer-controlled rheology laboratory has been constructed to study and optimize fluids used in hydraulic fracturing applications. Instruments consist of both pressurized capillary viscometers and concentric cylinder rotational viscometers. Computer control, data acquisition and analysis are accomplished by two Hewlett Packard 1000 computers. Custom software provides menu-driven programs for Instrument control, data retrieval and data analysis. 

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 ... 

We begin with a brief discussion of Newton s law of viscosity and follow this with a discussion of Newtonian flow (i.e., the flow of liquids that follow Newton s law) in a few standard configurations (e.g., cone-and-plate geometry, concentric cylinders, and capillaries) under certain specific boundary conditions. These configurations are commonly used in viscometers designed to measure viscosity of fluids. 

It is impossible to read much of the literature on viscosity without coming across some reference to the equation of motion. In the area of fluid mechanics, this equation occupies a place like that of the Schrodinger equation in quantum mechanics. Like its counterpart, the equation of motion is a complicated partial differential equation, the analysis of which is a matter for fluid dynamicists. Our purpose in this section is not to solve the equation of motion for any problem, but merely to introduce the physics of the relationship. Actually, both the concentric-cylinder and the capillary viscometers that we have already discussed are analyzed by the equation of motion, so we have already worked with this result without explicitly recognizing it. The equation of motion does in a general way what we did in a concrete way in the discussions above, namely, describe the velocity of a fluid element within a flowing fluid as a function of location in the fluid. The equation of motion allows this to be considered as a function of both location and time and is thus useful in nonstationary-state problems as well. 

To successfully measure non-Newtonian fluids, a known shear field (preferably constant) must be generated in the instrument. Generally, this situation is known as steady simple shear. This precludes the use of most single-point viscometers and leaves only rotational and capillary devices. Of these, rotational devices are most commonly used. To meet the criterion of steady simple shear, cone and plate, parallel plates, or concentric cylinders are used (Figure HI. 1.1). 

Figure 10.7 Two concentric cylinders with a viscous fluid between Margules viscometer.

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