Rheometry for large-particulated fluids: analysis of the ball measuring system and comparison to debris flow rheometry

Abstract

For large-particulated fluids encountered in natural debris flow, building materials, and sewage treatment, only a few rheometers exist that allow the determination of yield stress and viscosity. In the present investigation, we focus on the rheometrical analysis of the ball measuring system as a suitable tool to measure the rheology of particulated fluids up to grain sizes of 10 mm. The ball measuring system consists of a sphere that is dragged through a sample volume of approximately 0.5 l. Implemented in a standard rheometer, torques exerted on the sphere and the corresponding rotational speeds are recorded within a wide measuring range. In the second part of this investigation, six rheometric devices to determine flow curve and yield stress of fluids containing large particles with maximum grain sizes of 1 to 25 mm are compared, considering both rheological data and application in practical use. The large-scale rheometer of Coussot and Piau, the building material learning viscometer of Wallevik and Gjorv, and the ball measuring system were used for the flow curve determination and a capillary rheometer, the inclined plane test, and the slump test were used for the yield stress determination. For different coarse and concentrated sediment–water mixtures, the flow curves and the yield stresses agree well, except for the capillary rheometer, which exhibits much larger yield stress values. Differences are also noted in the measuring range of the different devices, as well as for the required sample volume that is crucial for application.

Appendix: Problems encountered during BMS experiments

The basic experiment conducted with the BMS consists in measuring the torque T at a given rotational speed Ω while the sphere makes one full rotation. Problems encountered during the measurement of particulated fluids with the BMS are tool acceleration, wake formation and viscous overstream, sphere interaction with suspended particles, data scattering due to grain size and temporary jamming, and the effect of prestructured samples. These problems are typical for many rheometrical measuring techniques and devices, but are considered below.

Tool acceleration

In order to attain a specified rotational speed, the sphere must be accelerated, which requires an additional torque above that required to drag the sphere across a given fluid. Figure 15 shows the start-up flow for samples containing different maximum grain sizes and different sediment concentrations C _v. After acceleration, a steady drag flow around the sphere and sphere holder is achieved. In this regime, the measured torque is only dependent on the rheological properties of the fluid. Meaningful rheological data require steady-state flows, which must be established by experiments. As a consequence, only speed and torque data for the steady drag flow regime are used for any further rheological interpretation and for the conversion of measured into rheological data, respectively. Duration or number of data affected by the sphere acceleration regime primarily depends on the specified speed and, to a lesser extent, on the fluid characteristics. The influence of sphere acceleration is almost negligible for very small speeds but becomes very significant for high speeds. For example, for speeds equal to or larger than Ω = 2.25 rps, the sphere acceleration regime covers one third up to half of the sphere path of a BMS experiment.

Fig. 15

Basic BMS experiments using the sphere with the diameter D = 12 mm: development of the rotational speed Ω and the torque T measured along the path of one sphere rotation for a specified speed of Ω = 2.25 rps. a Sediment water mixture made of debris flow material with a maximum grain size d _max = 1 mm and a sediment concentration C _v = 0.502. b Maximum grain size d _max = 10 mm and a sediment concentration C _v = 0.595 [C _v = V _s/(V _s + V _w) with V _s = volume of sediment and V _w = volume of the water]

Wake formation and viscous overstream

In low viscous fluids (Newtonian or non-Newtonian, particulated or nonparticulated), it was observed that, for very high speeds of Ω = 2.25–4.5 rps, the accelerating sphere generated a wake in front of the sphere and the sphere holder. For higher viscous fluids, such a wake was not observed. For silicon oils with viscosities larger than 2 Pa·s, the oil was lifted up along the sphere holder for higher rotational speeds. This was interpreted as viscous overstream (Ω ≥ 1.35 rps for the medium viscous oil, η = 2 Pa·s, Ω = 0.135 rps in the case of the highly viscous oil η = 60 Pa·s). The measured torque might be biased due to this effect so that corresponding data should be interpreted with caution.

Sphere–particle interaction, data scattering, and temporary jamming

The flow around the measuring sphere is ideally the flow of a homogeneous one-phase system. However, interactions of larger particles with the measuring sphere create a complex flow situation, in particular when both the suspended particles and the measuring sphere are of the same size (Chhabra 2007). Data fluctuation as seen for the different sphere diameters (see next chapter) are, however, related to the different geometries rather than to sphere–sphere interaction. Therefore, we have not considered the interaction of the moving sphere in the suspension.

Data scattering and temporary jamming

Data fluctuation of both the rotational speed and the torque increases with an increasing maximum grain size d _max of the particles for the steady drag flow regime (as shown in Fig. 15). For relative maximum grain sizes d _max/D ≤ 0.125, the standard deviation of the speed σΩ was typically less than 1% and, including fluctuation, remained <4%. For the same range of the relative maximum grain size, the standard deviation of the torque σT was typically ~3% and, including fluctuation, remained <10%. A strong increase of the standard deviations, σT and σΩ, is observed for relative maximum grain sizes d _max/D > 0.125 and for fluids having d _max ≥ 5 mm grain size. The standard deviation of the rotational speed, σΩ, not only depends on the relative maximum grain size d _max/D but also strongly depends on the sediment concentration C _v of the fluid (Fig. 16). For highly concentrated, large-particulated fluids, the standard deviation of the rotational speed σΩ can thus attain values up to 100%. This behavior is mainly due to increasing friction and local or temporary jamming between the sphere, sphere holder, particles, and the container boundary along the sphere path, causing an increasing variation of the speed (Schatzmann et al. 2003b).

Fig. 16

Standard deviation of the rotational speed, σΩ, and the torque, σT, vs sediment concentration C _v for sediment–water mixtures with variable maximum grain size d _max. Average standard deviation values obtained for Ω = 0.0045, 0.0135, 0.045, 0.135, 0.45, 1.35, 2.25/2.7, and 4.5 rps

Influence of prestructured sample

During the first full rotation, the sphere is dragged through an undisturbed sample fluid, whereas, for the following rotations, the sphere is dragged through a prestructured sample due to the influence of the first rotation, namely, along the sphere path. Depending on the value of the measured torque for the first and the following rotations at a given speed, it is, in principle, possible to determine whether or not the fluid is completely relaxed between the first and the following rotations and whether or not the fluid endured fatigue (Schatzmann 2005).

Considerable side effects

The measuring sphere moves approximately at medium container depth through a sample fluid, which is at rest. Even though the sample fluid is stirred before every experiment, settling of particles may take place within measuring time. Although it is not expected that the settling particles influence the torque measurements, one must be aware of an inverse gradient of fluid concentration over the entire container depth. In fluids affected by particle settling, it is, thus, not clear whether at the sphere depth the rheological properties corresponding to the mean concentration of particles are measured.

The relatively small distances between the sphere and the container boundary (side and bottom), as well as the sphere holder, determine the drag flow and contribute to the total measured torque. The sphere holder is 0.6 mm thick and 3 mm long in the direction of the sphere path, and the immerged length varies between s _i = 11 mm for a D = 15-mm sphere and s _i = 18 mm for a D = 8-mm sphere. The distance between the container side and the sphere is s _w = 17 mm, and the distance between the container bottom and the sphere is s _b = 22 mm. While the influence of sphere holder and container boundary can be quantified for each sphere dragging across well-defined nonparticulated Newtonian fluids (Schatzmann 2005), some unpredictable effects remain when measuring in particulated Newtonian or non-Newtonian fluids.

One effect is, namely, the temporary transport of larger particles on top of the sphere while being supported backwards by the holder. In this case, the dragging body is not only the sphere and the holder but the sphere, the holder, and the large particle. As a consequence, increased torque values are measured in this case. A second effect is the temporary jamming of particles between the sphere, the container boundary, and/or the holder in high to highly concentrated mixtures producing relatively larger torque values. Both effects are difficult to assess based on the datasets of the torque and the speed in the steady drag flow regime because the effects are usually hidden behind the large data fluctuation in large-particulated fluids. One possibility to detect such an effect is the comparison of the torque data of two or several independent experiments performed at a specified speed. If the torque data of both experiments cover the same range, no such boundary effect is assumed to occur.

Another uncertainty is the influence of the container boundary on the yield stress fluids investigated in this study. The flow field of a sphere dragged across a yield stress fluid is characterized by a sheared zone around the dragged sphere and a nonsheared zone beyond the sheared zone (Beris et al. 1985; Chhabra and Uhlherr 1988). Generally, size and shape of the sheared zone depends on the sphere velocity, the parameters of the rheological model function of the fluid, and the confining boundary. Atapattu et al. (1995) measured size and shape of the sheared zone for spheres dragging across different yield stress fluids in tubes. Beaulne and Mitsoulis (1997) made numerical simulations and found a good agreement between the experimental data of Atapattu et al. (1995) and the numerical results. Based on the results of Beaulne and Mitsoulis (1997), the size of the sheared zone was estimated in the case of the present BMS. It was estimated that, for the different fluids and speeds investigated in the present study, the sheared zone must usually have spread to the container wall and bottom. This might not be the case for the small and medium spheres (D = 8, 12 mm) dragged at low velocities across the rather concentrated yield stress fluids. Because of the lack of appropriate technical equipment to measure the boundary of sheared and nonsheared zones in the present BMS device, this aspect was not further analyzed.