Rheology of powders and nanopowders through the use of a Couette four-bladed vane rheometer: flowability, cohesion energy, agglomerates and dustiness

Abstract

A new four-bladed vane powder rheometer is proposed to study powder and nanopowder rheology at low and high shear rates. As shear rates increase, the powder flow will evolve from a Newtonian regime (low shear rates) to a Coulombic regime (moderate shear rates) and then to a kinetic regime (high shear rates) at which the powder may become self-fluidized because of intense particles collisions. Viscosity measurements of noncohesive glass beads at low shear rates (Newtonian and Coulombic regimes) do not strongly depend upon the particle size and can reach values much larger than commonly used viscosities in some CFDs models. Since, in the kinetic regime, the shear stress is a strong function of the particle size, agglomerate particle size of cohesive powders can be directly inferred from rheograms. Cohesive carbon black and silica nanopowders have been tested and compared to noncohesive glass beads microparticles, chosen as reference. For noncohesive glass bead micropowders, the “agglomerate” diameter average values found from rheological measurements are of the same order of magnitude as the primary diameters of the powder. This indicates that the kinetic theory of granular flows describes well these high shear rate regimes and that such theory can be used to estimate agglomerate diameters. It is also found that estimated agglomerate sizes of nanometric cohesive materials can reach sizes of hundred micrometers depending upon their cohesion strength. Such agglomerate size measurements have been corroborated with those made by microscopy.

Appendices

Appendix: Formulation of stress and strain tensors in two dimensions

In two dimensions, we can represent a stress tensor $\tilde{\sigma }$ as

$$\bar{\sigma } = \left[ {\begin{array}{*{20}c} {\sigma_{11} } & {\sigma_{12} } \\ {\sigma_{21} } & {\sigma_{22} } \\ \end{array} } \right]$$

with two vectors designing the normal (n ₁) and the tangent (n ₂) to a surface, defined in the (O ,i, j) coordinates as (Fig. 13)

$$n_{1} = \left( {\begin{array}{*{20}c} {\cos (\varphi )} \\ {\sin (\varphi )} \\ \end{array} } \right)$$

and $n_{2} = \left( {\begin{array}{*{20}c} { - \sin (\varphi )} \\ {\cos (\varphi )} \\ \end{array} } \right)$

Fig. 13

Construction of Mohr’s circle: a physical space, b stress space: At rest, τ _A = −τ _B . The Mohr’s circle representation provides an easy means to relate the Cartesian stresses to an equivalent stress in a given direction

So that the traction force T (attractive or repulsive) per unit surface (defined by the n _i vectors) is given by:

$$T_{n} = \left[ {\begin{array}{*{20}c} {\sigma_{11} } & {\sigma_{12} } \\ {\sigma_{21} } & {\sigma_{22} } \\ \end{array} } \right].\left( {\begin{array}{*{20}c} {\cos \left( \varphi \right)} \\ {\sin \left( \varphi \right)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\sigma_{11} \cos \left( \varphi \right) + \sigma_{12} \sin \left( \varphi \right)} \\ {\sigma_{21} \cos \left( \varphi \right) + \sigma_{22} \sin \left( \varphi \right)} \\ \end{array} } \right)$$

with

$$T_{n1} = \sigma_{1j} n_{j} = \sigma_{11} n_{1} + \sigma_{12} n_{2}$$

and

$$T_{n2} = \sigma_{2j} n_{j} = \sigma_{21} n_{1} + \sigma_{22} n_{2}$$

Or more generally (usually known as Cauchy’s formula):

$$T_{ni} = \sigma_{ij} n_{j}$$

The traction components are the normal and the tangential components, scalars $T_{n} \cdot n_{i}$ that can be expressed as

$$T_{n} \cdot n_{1} = \left( {\begin{array}{*{20}c} {\sigma_{11} \cos \left( \varphi \right)\, + \,\sigma_{12} \sin \left( \varphi \right)} \\ {\sigma_{21} \cos \left( \varphi \right) \,+\, \sigma_{22} \sin \left( \varphi \right)} \\ \end{array} } \right) \cdot \left( {\begin{array}{*{20}c} {\cos \left( \varphi \right)} \\ {\sin \left( \varphi \right)} \\ \end{array} } \right) = \sigma_{11} \cos \left( \varphi \right)^{2} \,+ \,\sigma_{12} \sin \left( \varphi \right)\cos \left( \varphi \right) \,+\, \sigma_{21} \cos \left( \varphi \right)^{2} \,+\, \sigma_{22} \sin \left( \varphi \right)^{2}$$

$$T_{n} \cdot n_{2} = \left( {\begin{array}{*{20}c} {\sigma_{11} \cos \left( \varphi \right)\,\,+\,} & {\sigma_{12} \sin (\varphi )} \\ {\sigma_{21} \cos \left( \varphi \right)\,\,+\,} & {\sigma_{22} { \sin }(\varphi )} \\ \end{array} } \right) \cdot \left( {\begin{array}{*{20}c} { - { \sin }(\varphi )} \\ {\cos (\varphi )} \\ \end{array} } \right)$$

$$T_{n2} \cdot n_{2} = - \sigma_{11} \cos \left( \varphi \right)\sin \left( \varphi \right) - \sigma_{12} \sin (\varphi )^{2} + \sigma_{21} { \cos }(\varphi )^{2} + \sigma_{22} { \sin }(\varphi )^{2}$$

These expressions can be simplified using the relations ${ \cos }(\varphi )^{2} = \frac{1}{2}(1 + \cos (2\varphi ))$; ${ \sin }(\varphi )^{2} = \frac{1}{2}(1 - \cos \left( {2\varphi } \right))$; and $\sin \left( \varphi \right)\cos \left( \varphi \right) = \frac{1}{2}{ \sin }(2\varphi )$

So the normal (σ) and tangent (τ) stresses in principal directions can be expressed as

$$\sigma = \frac{1}{2}\left( {\sigma_{11} + \sigma_{22} } \right) + \frac{1}{2}\left( {\sigma_{11} - \sigma_{22} } \right)\cos \left( {2\varphi } \right) + \sigma_{12} { \sin }(2\varphi )$$

and

$$\varvec{\tau}= \frac{1}{2}\left( {\varvec{\sigma}_{22} -\varvec{\sigma}_{11} } \right)\sin \left( {2\varvec{\varphi }} \right) +\varvec{\sigma}_{12} \varvec{cos}(2\varvec{\varphi })$$

(6)

Mathematical definition of angle of repose and internal friction angle of a powder

As can be seen from Eq. 6, there is an angle φ, for which the shear stress is nil. This angle is called the repose angle. For this angle, a particle is not subject to a shear force (τ = 0). For example, the particle could flow freely on the surface of the slopes of a heap since there is no shear forces. To help the representation of the normal and the tangential stress components, the Mohr’s circle (see below) is a convenient method. Measurements of these angles will be discussed later. At that given angle, there is a relationship between the normal stress difference σ ₂₂ − σ ₁₁ and the shear stress σ ₁₂. A representation of the physical plane (x ₁, x ₂), in which the angle between a surface normal and the horizontal is defined as φ, can also be represented in the stress plane (σ, τ) for which the angle is defined as 2φ, as shown in Fig. 13. In this figure, the vertical and horizontal stresses on the volume element are denoted as σ _a and σ _b while the shear stresses are denoted as τ _a and τ _b that are represented in the physical plane Fig. 13a, and in the stress plane in Fig. 13b. At equilibrium, τ _A  = −τ _B , as represented in Fig. 13b.

As will be seen later, at φ = θ (angle of repose), there exists the smallest Morh’s circle, for which the one of the normal stresses is zero and the other equals the diameter of the circle, and for which the shear stress is nil.

Definition of Mohr’s circles

Using the two preceding expressions for σ and τ, one can write

$$\left(\sigma - \frac{{\sigma_{11} + \sigma_{22} }}{2}\right)^{2} + \tau^{2} = \frac{1}{4}\left(\sigma_{11} - \sigma_{22}\right)^{2} +\, \sigma_{12}^{2}$$

which represents, in the σ; τ plane, a circle of origin $\left( {\frac{{\sigma_{11} + \sigma_{22} }}{2}, 0} \right)$ and of radius R whose value is, $R = \sqrt {\frac{1}{4}(\sigma_{11} - \sigma_{22} )^{2} + \sigma_{12}^{2} }$, which is called the Mohr’s circle.

Applications of Mohr’s circles: ratios of normal stresses in the case of noncohesive powders

The use of Mohr's circle representation allows us to quickly identify the normal and shear stresses on a powder volume element. Let us superpose on Fig. 13 b, in a stress plane, a Coulomb failure criterion illustrated by the straight line of slope θ, as shown in Fig. 14. This figure represents the normal stresses in abscissa and shear stresses in ordinates, on which the Coulomb failure locus straight line, of slope angle θ, cuts the ordinate axis at a the Cohesive stress C.

Fig. 14

Mohr’s circle, the radius of the circle is $\frac{{\varvec{\sigma}_{11} -\varvec{\sigma}_{22} }}{2}$ and the center is $\left( {\frac{{\varvec{\sigma}_{11} +\varvec{\sigma}_{22} }}{2},0} \right)$

From the Coulomb failure condition at point M;

$$\tau_{\text{ss}} = C + \sigma_{\text{ss}} { \tan }g (\theta )$$

where C is the cohesion shear stress. Physically, this cohesion stress represents the necessary stress constraint to make the powder flow at zero normal stresses.

One can write

$$\tau_{\text{m}} = C + \sigma_{\text{m }} {\rm \tan}g\left( \theta \right)$$

Or

$$\tau_{\text{m}} { \cos }(\theta ) = C { \cos }(\theta ) + \sigma_{\text{m }} sin\left( \theta \right)$$

$$\frac{{(\varvec{\sigma}_{11} -\varvec{\sigma}_{22} )}}{2} = \varvec{C}\cos (\varvec{\theta}) + \left( {\frac{{(\varvec{\sigma}_{11} +\varvec{\sigma}_{22} )}}{2}} \right)\sin (\varvec{\theta}),\,for\,\left( {\varvec{\sigma}_{11} >\varvec{\sigma}_{22} \varvec{ }} \right)$$

(7)

For noncohesive powders, (C = 0), the ratio of normal stresses can be expressed as:

$$\frac{{\sigma_{11} }}{{\sigma_{22} }} = \frac{{1 + { \sin }(\theta )}}{1 - \sin (\theta )}$$

This equation shows that as long as there is a nonzero angle of friction, θ, or angle of repose, α, the two normal stresses are not equal as it would be in the case in a fluid. One is larger than the other one by a ratio of $\frac{{1 + { \sin }(\theta )}}{1 - \sin (\theta )}$. This is schematically represented in the volume element below (Fig. 15): It is seen that if one wants to store a large quantity of granular material in a silo, only a small lateral confinement stress is needed. This feature is routinely used for the storage of large quantities of powders in silos of various geometries, as discussed by Jenike (1987), Johanson (1964). Note also that for a zero angle of friction powder (which, in this case, can be assimilated to a liquid), we obtain σ ₁₁ = σ ₂₂ indicating an isostatic stress which is basically the hydrostatic pressure for liquids.

Fig. 15

Illustration of the two normal stresses acting on a volume element and their ratio for a noncohesive powder