42 Non-Newtonian Flow in the Process Industries Rotational viscometers Due to their relative importance as tools for the rheological characterisation of non-newtonian fluid behaviour, we concentrate on this class of rheometers by considering the two main types, namely; the controlled shear rate instruments (also known as controlled rate devices) and controlled stress instruments. Both types are usually supplied with the same range of measuring geometries, principally the concentric cylinder, cone-and-plate and parallel plate systems. The relative merits, potential drawbacks, working equations and other formulae associated with these designs have been described in great detail elsewhere (e.g., see Walters, 1975; Whorlow, 1992; Macosko, 1994) and so only their. most basic aspects are covered here The concentric cylinder geometry It is appropriate to begin by considering this geometry as it was the basis of the first practical rotational rheometer. Ideally the sample is contained in a narrow gap between two concentric cylinders (as shown in Figure 2.3). Typically the outer cup rotates and the torque T on the inner cylinder, which is usually. suspended from a torsion wire or bar, is measured Working equations relating the measured torque to the requisite shear stress, and angular velocity (of the cup) to the requisite shear rate, are widely available
Rheometry for non-newtonian fluids 43 Pointer Retaining spring R 1 h Rotating outer cylinder Stationary inner cylinder Figure 2.3 Partial section of a concentric-cylinder viscometer along with their derivations (see the list of references above). It is noteworthy that the working formulae quoted in many instances ignore the curvature of the surfaces of the measuring geometry. The determination of the shear stress and shear rate within the shearing gap is thus valid only for very narrow gaps wherein, the ratio of inner to outer cylinder radii, is >0.99. Several designs have been described which overcome end-effects due to the shear flow at the bottom of the concentric cylinder geometry. These include the recessed bottom system which usually entails trapping a bubble of air (or a low viscosity liquid such as mercury) beneath the inner cylinder of the geometry. Alternatively the Mooney-Ewart design, which features a conical bottom may, with suitable choice of cone angle, cause the shear rate in the bottom to match that in the narrow gap between the sides of the cylinders, see Figure 2.4. In this example the sample temperature is controlled by circulation of liquid through the outer cylinder housing (flow marked in and out in Figure 2.4) and h denotes the sample height within the shearing gap. The shear rate may be calculated from P D 2.7a R 1 where and R 1 are the outer and inner cylinder radii respectively, and is the angular velocity. For k > 0.99, the shear stress is given by: D T/2 1 h 2.7b
44 Non-Newtonian Flow in the Process Industries h In Out R 1 Figure 2.4 The Mooney-Ewart geometry To minimise end-effects the lower end of the inner cylinder is a truncated cone. The shear rate in this region is equal to that between the cylinders if the cone angle,, is related to the cylinder radii by: D tan 1 R 1 2.8 The main sources of error in the concentric cylinder type measuring geometry arise from end effects (see above), wall slip, inertia and secondary flows, viscous heating effects and eccentricities due to misalignment of the geometry [Macosko, 1994]. Secondaryflowsareof particularconcernin the controlled stress instruments which usually employ a rotating inner cylinder, in which case inertial forces cause a small axisymmetric cellular secondary motion ( Taylor vortices). The dissipation of energy by these vortices leads to overestimation of the torque. The stability criterion for a Newtonian fluid in a narrow gap is Ta D 2 2 R 1 3 R 1 / 2 < 3400 2.9 where Ta is the Taylor number. In the case of non-newtonian polymer solutions (and narrow gaps) the stability limit increases. In situations where the outer cylinder is rotating, stable Couette flow may be maintained until the onset of turbulence at a Reynolds number, Re, of ca. 50 000 where Re D R 1 / [Van Wazer et al., 1963]. 2.3.2 The wide-gap rotational viscometer: determination of the flow curve for a non-newtonian fluid An important restriction on the use of the concentric cylinder measuring geometry for the determination of the shear rate versus shear stress relationship for a non-newtonian fluid is the requirement, noted above, for a narrow shearing
Rheometry for non-newtonian fluids 45 gap between the cylinders. As indicated in the introduction to this chapter, direct measurements of shear rates can only be made if the shear rate is constant (or very nearly so) throughout the shearing gap but many coaxial measuring systems do not fulfill this requirement. In addition, many (if not most) non-newtonian fluid systems, particularly those of industrial or commercial interest such as pastes, suspensions or foods, may contain relatively large particles, or aggregates of particles. Thus the requisite shearing gap size to ensure that adequate bulk measurements are made, i.e., a gap size approximately 10 100 times the size of the largest particle size [Van Wazer et al., 1963], may conflict with the gap size required to ensure near constant shear rate, within the gap. Procedures for extracting valid shear stress versus shear rate data from measurements involving wide gap coaxial cylinder systems (the Brookfield viscometer being an extreme example of wide gap devices) are therefore of considerable interest in making quantitative measurements of the flow properties of non-newtonian process products. Most of these data-treatment procedures necessarily involve some assumption regarding the functional form of the flow curve of the material. One example is that made in the derivation of data from the Brookfield-type instrument, which assumes that the speed of rotation of the cylinder or spindle is proportional to the shear rate experienced by the fluid. This assumption implies that the flow curve is adequately described by a simple power-law (which for many shear-thinning non-newtonian fluids may be acceptable), but this assumption is widely taken to exclude all fluids which display an apparent yield stress and/or non-power law type behaviour. The starting point lies in considering the basic equation for the coaxial rotational viscometer, which has been solved by Krieger and co-workers for various sets of boundary conditions [Krieger and Maron, 1952]: D 1 2 R b R c f d 2.10 where is the angular velocity of the spindle with respect to the cup, is the shear stress in the fluid at any point in the system, f DPis the rate of shear at the same point and the subscripts b and c refer to the bob and the cup, respectively. The particular solution to equation (2.10) for a finite cylindrical bob rotating in an infinite cup can provide valuable quantitative rheological data for systems whose particulate constituents, and practical limitations on the size of the measuring geometry in terms of cylinder radius, preclude the use of conventional narrow gap geometries. The infinite cup boundary condition may be closely approximated by using a narrow cylindrical spindle (such as are supplied with instruments of the Brookfield type) in place of the more commonly used bob.
46 Non-Newtonian Flow in the Process Industries Assuming the infinite-cup boundary condition, c (shear stress on the cup) in equation (2.10) becomes equal to zero and the expression may be differentiated with respect to b giving: P b D f b D 2d/dln b D 2 b d/d b 2.11 and thus the rate of shear may be obtained by evaluating (graphically) either of the derivatives on the right-hand side of equation (2.11). The derivation of equation (2.11) assumes that a cup of infinite radius is filled with fluid. Implicitly this would exclude all systems which display a yield stress, as such systems would not behave as a fluid for values of stress below the yield value. As many non-newtonian systems are sufficiently structured to display an apparent yield stress, this requirement would appear to severely restrict the application of what would otherwise appear to be a very useful technique. However, on closer inspection, it has been shown that for a fluid which displays a yield stress, a more general derivation than that reported by Krieger and Maron [1952] may be obtained, and that the restriction of infinite outer boundary (i.e. cup) radius may in fact be eliminated [Jacobsen, 1974]. In a system which displays yield stress behaviour, the integral in the general expression for the rate of shear need not be evaluated from the bob all the way to the cup. This is due to the fact that, for such a system, no shearing takes place where is less than the yield value, 0. Thus the integral need only be evaluated from the bob to the critical radius, R crit, the radius at which D 0.Thisgives R crit D1/2 f d/ 2.12 R b where the critical radius, R crit, is given as: R crit D R b b / 0 1/2 2.13 This derivation relies on the fact that the condition of differentiability is not that one limit of the integral be zero (as is the case in the infinite cup solution) but that one limit be constant. Thus, for systems which may be described in terms of a constant value of yield stress, equation (2.12) may be differentiated, giving: P b D f b D 2d/dln b D 2 b d/d b 2.14 i.e. exactly the same result is obtained as that derived for the case of the infinite cup, equation (2.11). In practice, shear stress data are plotted against and the slopes d/d b are taken at each point. Given that the graphical solution may be somewhat tedious, and that a rapid evaluation of the general form of the flow
Rheometry for non-newtonian fluids 47 curve is often all that is required (e.g. in a product quality control context), the form of the versus b plots is sometimes taken as giving the general form of the correspondingp versus b curve (although, of course, the curves will differ quantitatively). In the absence of an apparent yield stress (over the experimental time-scale) the general character of thep versus b curves may sometimes be correctly inferred by this procedure: the situation is quite different when the system exhibits an apparent yield stress and this situation poses a trap for the unwary. An examination of equation (2.14) shows that for any fluid with a finite yield point, the versus b curve approaches the b axis at zero slope, due to the requirement for such a system that the shear rate must become zero at finite b. This may lead to apparent shear-thinning characteristics being ascribed to systems, irrespective of the actual form of their flow curves above the yield point, i.e., whether Bingham plastic, shear-thickening (with a yield stress), or shear-thinning (with a yield stress). An instrument called the rotating disk indexer (sic.) is also widely used in quality control applications and involves a rotating disc in a sea of fluid. Williams [1979] has described a numerical method for obtaining true P data with this device..