Limits for safe viscosity measurement of non-colloidal suspensions in rotational rheometry—a numerical simulation-based approach

Abstract

This work investigates transient non-colloidal suspension flows in cone-and-plate, plate-plate, and cylindrical geometries to assess particle motion’s impact on viscosity measurement. Mass and momentum conservation equations model the two-phase liquid–solid flow, with both phases treated as continuous in an Euler-Euler approach. Findings demonstrate rheometric flow induces particle motion, affecting suspension homogeneity and viscosity measurement over time. Both buoyancy and inertia effects drive particle motion, with buoyancy dominating at low shear rates and inertia at high shear rates. Particle volume fractions, shear rates, and liquid viscosity notably influence viscosity measurements. Measurements with concentric cylinders are the least affected by particle motion. Additionally, we propose a time limit and a critical Reynolds number in which particle motion does not affect the measurement of the suspension viscosity.

Appendix

A mesh sensitivity test was first carried out for a single-phase flow case. The outer radii of cone-plate and parallel-plate geometries are 17.5 mm, while the concentric cylinders have external and internal radii of R$_o$ (13.6 mm) and R$_i$ (12.54 mm), respectively. A 2$^{\circ }$-truncated cone is set for the cone-and-plate geometry, whereas the gap and the height in PP and CC geometries are 1 mm, H$_p$, and 37.6 mm, H$_c$, respectively. The liquid phase is a Newtonian fluid with a density of 840 kg m$^{-3}$ and a viscosity of 0.022 Pa s. A rotational speed of 28.6 rad s$^{-1}$ was imposed for all geometries. The computed torque on the CP, PP, and CC geometries was calculated on the cone surface, on the upper plate, and the internal cylinder by the Eq. 23:

$$\begin{aligned} T_{cal}= \int \tau _w rdA \end{aligned}$$

(23)

where $\tau _{\omega }$ is the shear stress on the wall.

Table 2 Comparison of the computed torque obtained for CP geometry, Newtonian fluid, and the five different meshes with the analytical solution

Table 3 Comparison of the computed torque obtained for PP geometry, Newtonian fluid, and the five different meshes with the analytical solution

Table 4 Comparison of the computed torque obtained for CC geometry, Newtonian fluid, and the five different meshes with the analytical solution

Analytical solutions for the torque on each geometry are as follows:

$$\begin{aligned} T_{CP} = \frac{2\mu \Omega \pi R^3}{3 \beta } \end{aligned}$$

(24)

$$\begin{aligned} T_{PP} = \frac{\mu \Omega \pi R^4}{2 H_p} \end{aligned}$$

(25)

$$\begin{aligned} T_{CC} = \frac{4 \mu \pi H_c R_i^2 R_o^2}{R_o^2 - R_i^2}\Omega _{R_i} \end{aligned}$$

(26)

where CP, PP, and CC represent cone-plate, plate-plate, and concentric cylinder geometries, respectively. $\Omega$ is the rotation speed, $\beta$ is the cone-and-plate angle, and R is the cone-and-plate and plate-plate radius.

Fig. 12

The 1$^{\circ }$ numerical grid for all three geometries. a Cone-plate, b parallel-plates, and c concentric cylinders

Fig. 13

Tangential velocity profile for all three geometries. a Cone-plate, b parallel-plates, and c concentric cylinders

Table 5 Torque obtained for CP and PP geometries, PVF of 0.2, and five different meshes

Tables 2, 3, and 4 compare computed torque for different meshes with the analytical torque. As noted, the calculated torque is quite insensitive to the grid refinement, with a maximum deviation of 0.11% between the least and the most refined meshes (Fig. 12). Comparison with the analytical solutions given by Eqs. 24 to 26 show a maximum difference of 0.31%. Although small, we attribute this discrepancy to the swirl flow in the r-z plane caused by centrifugal force, as already discussed by de Rosso and Negrao (2022).

The tangential velocity profile was monitored along the R direction (r* = 0, r* = 1) for all geometries. As shown in Fig. 13, the velocity profiles for all geometries are not only mesh-size independent but also agree with the theoretical velocity profiles given by Eqs. 24, 25, and 26, respectively.

Table 6 Torque obtained for CC geometry, PVF of 0.2, and five different meshes

Fig. 14

Particle volume fraction profiles at $z^*$ = 0.5 for all three geometries and five meshes as a function of the radial position. a Cone-plate, b parallel-plates, and c concentric cylinders

A mesh sensitivity test was also conducted for two-phase flow by considering the liquid phase as a Newtonian fluid with a density of 840 kg m$^{-3}$ and a viscosity of 0.022 Pa s. The dispersed phase comprises 50-$\mu$m diameter particles, a density of 1020 kg m$^{-3}$, and a particle volume fraction (PVF) of $\alpha _s$ = 0.2. The rotation speed considered for the CP and PP geometries is $\Omega$ = 28.6 rad s$^{-1}$, corresponding to shear rates of 818 s$^{-1}$ and 500 s$^{-1}$, respectively. For CC geometry, a rotation speed of $\Omega$ = 2.8 rad s$^{-1}$ is used, corresponding to a shear rate of 33.8 s$^{-1}$.

The computed torque and the PVF distribution in the radial direction are monitored to check for grid size sensitivity. Tables 5 and 6 present the torque for each mesh and the respective percentage deviation to the most refined grid torque. Figures 14 a, b, and c depict the PVF profile in the radial direction at $z^* = 0.5$ (the mid-height position which corresponds to the height of the cone-plate, plate-plate geometry and between the gap of concentric cylinders) for all three geometries. Despite the low torque variation with the grid size for all three geometries, the PVF profiles of CP and PP are sensitive to mesh refinement. Nevertheless, the higher the number of nodes, the less sensitive the PVF distribution to the mesh size, with mesh #4 being grid-independent for all three geometries. Mesh #4 is thus adopted to conduct the numerical simulations discussed in the next section.