Understanding empirical powder flowability criteria scaled by Hausner ratio or Carr index with the analogous viscosity concept

Abstract

The viscosity concept is introduced to granular powders after the analogous temperature is defined consistently with thermodynamics, and the corresponding viscosity equation is obtained with the aid of Eyring's rate process theory and free volume concept. The popular empirical powder flowability criteria scaled with the Hausner ratio or Carr index are theoretically found only to work at special conditions and more universal criteria correlated with shearing conditions and particle physical properties are presented. Our results resolve a long time controversial mystery related to those two empirical indexes on powder flowability, and has a broad impact in many industrial areas. The work presented in this article may lay a foundation to scale powder flowability in a more fundamental and consistent manner.

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Flowability of granular powders is important in powder handling processes in many industrial areas like food, pharmaceuticals, minerals, and civil engineering, etc.^1–3 Objectively determining the flowability of granular powders is relatively difficult due to the complexity of particle materials: granular powders are usually non-continuous and compressible, and the regular fluid mechanics that works for continuum and incompressible thermal fluids may not be applicable to granular powders.^4,5 The Hausner ratio⁶ expressed as the tap density divided by the bulk density, H_r = ρ_tap/ρ_b, and related Carr index,⁷ CI = 1 − 1/H_r, are used to indicate the flowability of granular powders in a wide variety of industries.^8,9 Empirically, a Hausner ratio greater than 1.22 or Carr index greater than 0.18 is an indication of poor flowability, and a Hausner ratio below 1.18 or Carr index below 0.15 is regarded as good flowability.^10–12 However, much experimental evidence suggests that both the Hausner ratio and Carr index only work for some powder systems, and are incapable of acting as consistent and predictive flowability tools.¹³

For granular powders of relatively fast moving particles, granular temperatures are usually defined with the kinetic energy connection between the temperature and the velocity, 3/2k_BT = 1/2mv², where k_B is the Boltzmann constant, m is the mass of the particle, and v is the velocity of particles.^14–18 In this manner, the granular temperature remains the same original meaning as that in thermal systems and thus the thermodynamic principles could be applied to granular powders. For addressing flowability issues of granular powders, the viscosity is naturally thought to be the best parameter if this concept could analogously be introduced to the granular powders.

The analogous viscosity of granular powders could be potentially obtained with the Eyring rate process theory¹⁹ and the free volume concept.^20–22 The viscosities of liquids including both pure and mixtures, colloidal suspensions, and polymeric materials with and without an external electric field have been extensively addressed,^23,24 with the aid of Eyring's rate process theory and the free volume concept. The Eyring's rate process theory has been proved to be very powerful in revealing physical mechanisms of chemical reactions, dielectric relaxations, resonance energy transfer, and many other thermally activated motions.²⁵ The free volume concept has been frequently used to determine various equilibrium properties of both solid and liquids.^26,27 The popularly used empirical tap density equations of granular powders, the logarithmic and stretched exponential laws, are successfully derived with the Eyring's rate process theory and free volume concept.^28,29 All those evidences suggest that the Eyring's rate process theory and free volume concept are very useful tools in dealing with both thermal and athermal systems. Therefore, the analogous viscosity of granular powders could be potentially obtained with the Eyring rate process theory and the free volume concept. In such a methodology, viscosity equations across all systems from liquids, colloidal suspensions, polymeric materials, to granular matter, are obtained with same unified approaches.

Flow as a rate process was proposed by Eyring.^19,30 Eyring's theory is originally for molecular thermal systems, and the assumptions are pretty intuitive: a molecule moving from one equilibrium position to another may need to overcome an energy barrier, the activation energy, E₀; the applied shear field may lower the energy barrier in the flowing direction by ΔE, while it will raise the height of the energy barrier in the opposite direction by the same amount. Therefore, there are more molecules crossing over the energy barrier in the flowing direction than that in backward direction. The net rate expressed as the difference between the numbers of molecules crossing over the energy barriers in the flowing and opposite directions per unit time will determine the viscosity of the whole system. Consider that two layers of particles in a powder bed have a distance d apart and the relative velocity between those two layers is v, see Fig. 1. Under a shear stress σ, the viscosity, η, according to the definition, can be expressed as

Utilizing the Eyring's concept of the free volume that is unoccupied by molecules, Eyring et al. obtains the viscosity of liquids as shown below:¹⁹where V_m is the molar volume of a liquid, c is a constant dependent on how molecules pack in the system and equals to 2 for the cubic packing structure, R is gas constant, N_A is the Avogadro number, m is the mass of an individual molecule, k_B is Boltzmann constant, ΔE_vap is the molar vaporization energy required for a liquid transferred into a gas, T is the temperature, and E₀ is the activation energy. In analogy with Eyring's rate process theory, one may assume that a particle moving from one equilibrium position to another needs to overcome the energy activation barrier. The applied shearing force will reduce the height of the energy barrier in the flowing direction, while it will raise the height of the energy barrier in the opposite direction by the same amount. However, for applying eqn (2) to granular powders, the temperature must be analogously defined for attaining the same functionality as in thermal systems. Take a granular system under a simple shear shown in Fig. 1 as an example, if the shear stress is σ and the shear rate is [small gamma, Greek, dot above], then the force F and the velocity v may be expressed as:

F = σA, v = [small gamma, Greek, dot above]H (3)

where A is the area of the sample and H is the thickness of the sample. The injected energy rate from a simple shear field into the powder bed may be expressed as:

Ė = Fv = σ[small gamma, Greek, dot above]AH = σ[small gamma, Greek, dot above]V (4)

where V is the volume of the granular system, V = M/ρ_b = 4/3πr³Nρ_t/ρ_b, and ρ_t and ρ_b are the true and bulk density of the granular powder, r is the radius of particles, and N is the number of particles in the system. Using the kinetic energy connection for temperature below:the granular temperature of a sheared powder after a time period of t may be expressed as:where T_gp is the analogous temperature of a granular powder. The viscosity of a granular powder may thus be obtained by simply substituting T with T_gp expressed with eqn (6) and m with 4/3πr³ρ_t by the definition:since V_m is the molar volume of particles and thus may be represented as V_m = 4/3πr³N_Aρ_t/ρ_b, eqn (7) may be re-written as:ΔE_vap is the molar vaporization energy required for a liquid transferred into a gas, and may be considered as the energy that overcomes the binding interactions between molecules in liquid states. In a granular system, ΔE_vap may be considered as the energy required for separating particles to such a large distance that there is no interaction force between particles, i.e., ΔE_vap equals to the cohesive energy between particles, ΔE_coh. The activation energy E₀ may be expected to be some fraction of ΔE_vap:¹⁹with n is a number factor of the values from 2 to 5.¹⁹ For the simplicity reason, one may take n = 3 for granular powders. Under those assumptions and renaming ΔE_vap as ΔE_coh, eqn (8) may be rewritten as:

Eqn (10) clearly shows that viscosity of a granular powder is a function of particle size, true and bulk densities, packing structure, cohesive forces scaled with the separation energy, the applied shear stress, shear rate, and amazingly the time. Qualitatively, this equation makes sense, as powder flowability is empirically found to be complicated and is dependent on those parameters.

Fig. 1 Schematic diagram of a granular powder under a simple shear.

The cohesive energy between particles, ΔE_coh/N_A, is about 1.2 × 10⁻¹³ to 50 × 10⁻¹³ J, based on the direct Atomic Force Microscopy (AFM) measurements of interparticle forces.⁸ Let's first see how the viscosity is going to change with the true density of granular powders, as we empirically know that metallic particles usually flow much better than polymeric particles due to the true density differences. The predicted viscosity with eqn (10) vs. the true density of granular powders calculated under conditions r = 100 μm, ρ_b = 500 kg m⁻³, σ[small gamma, Greek, dot above]t = 0.1 for two cohesive energies ΔE_vap/N_A = 7 × 10⁻¹² J and ΔE_vap/N_A = 10⁻¹¹ J is shown in Fig. 2. The viscosity decreases sharply when the true density of materials changes from about one to three, which is consistent with empirical observations: organic or polymeric materials usually have a true density about 1 g cm⁻³ and those powders tend to have a poor flowability in comparison with metallic particles of a true density above 6 g cm⁻³. Even they have a same cohesive energy between particles and a same particle size, metallic particles should have a low viscosity, as predicted with eqn (10) and thus flow much better than low density particles. Another interesting thing shown in Fig. 2 is that particles of higher cohesive energy give larger viscosity initially at low density regions, but show lower viscosity at high density regions. This seems to be intuitive and reasonable, as particle with stronger interaction forces are hard to move at the beginning; once they start to flow, particles may flow together collectively due to strong cohesive forces, showing lower viscosities.

Fig. 2 The predicted viscosity with eqn (10) vs. the true density of granular powders calculated under conditions r = 100 μm, ρ_b = 500 kg m⁻³, σ[small gamma, Greek, dot above]t = 0.1 for two cohesive energies ΔE_coh/N_A = 7 × 10⁻¹² J and ΔE_coh/N_A = 10⁻¹¹ J.

The Hausner ratio⁶ and Carr index⁷ are frequently used to empirically scale the flowability of powders^8,9 and they are defined as:

where CI is Carr index and H_r is Hausner ratio, ρ_tap is the ultimate (equilibrium) tap density that will not change with the further increase of tap numbers. Obviously, CI = 1 − 1/H_r, and ρ_tap should be smaller than but quite close to the true density with the relationship, ρ_tap = Φ_mρ_t, where Φ_m is the maximum packing fraction at the equilibrium tapping state. One thus may reach:

Empirically, a Carr index greater than 0.18 (H_r is about 1.22) is considered to be an indication of poor flowability, and below 0.15 (H_r is about 1.18) is considered of good flowability,^10,11,19 though many experimental evidences suggest that both Carr index and Hausner ratio are quite scarce and inaccurate flowability indexes.¹³ Most food and pharmaceutical active ingredients and excipients are either small molecular organic materials or polymers, and the true densities of those materials are quite similar. It would be reasonable to assume that the true density is a constant. Differentiating the viscosity expressed in eqn (10) against (ρ_t/ρ_b) under an assumption that ρ_t is a constant and assuming dη/d(ρ_t/ρ_b) = 0 yields:

orwhen i.e. ΔE_vap/N_A = 8.37 × 10⁻¹³ J under conditions r = 100 μm and σ[small gamma, Greek, dot above]t = 0.072 Pa, Φ_m = 0.72, then ρ_tap/ρ_b = 1.2, quite close to the empirical criterion value, 1.22, as mentioned earlier. This point should be a critical turning point as it is obtained by assuming dη/d(ρ_t/ρ_b) = 0, implying that the viscosity would most likely go through a minimum. Fig. 3 shows the predicted viscosity obtained with eqn (10) vs. Hausner ratio expressed as ρ_tap/ρ_b under conditions ΔE_vap/N_A = 8.37 × 10⁻¹³ J, r = 100 μm, σ[small gamma, Greek, dot above]t = 0.072 Pa, ρ_t = 1000 kg m⁻³, Φ_m = 0.72. There is a minimum viscosity at ρ_tap/ρ_b = 1.2, consistent with the empirical criterion. Note that the minimum viscosity point should occur at the point where Hausner ratio is expressed with eqn (13), and may vary with systems. Eqn (10) together with eqn (13) solves the controversial mystery in literature why Hausner ratio could be used as a flowability index for some materials but fails to work for others. Note that Hausner ratio cannot be less than 1 by definition and the viscosity data obtained at ρ_tap/ρ_b < 1 and showed in Fig. 3 is only for the demonstration purpose of clearly showing the minimum viscosity point at ρ_tap/ρ_b = 1.2.

Fig. 3 The predicted viscosity with eqn (10) vs. Hausner ratio ρ_t/ρ_b, calculated under conditions ΔE_coh/N_A = 8.37 × 10⁻¹³ J, r = 100 μm and σ[small gamma, Greek, dot above]t = 0.072 Pa, ρ_t = 1000, Φ_m = 0.72.

In summary, by defining granular temperature analogously, the viscosity equation of granular powders is obtained with Eyring's rate process theory and Eyring's free volume method. It predicts that high density particles would likely to have low viscosity and the viscosity may go through a minimum against Hausner ratio. It clearly tells that the empirical flowability criterion scaled with the Hausner ratio and the Carr index can only work for special conditions. It also provide a theoretical basis for accurately estimating the turning point of Hausner ratio as shown in eqn (13), which could be useful in many industrial areas. The current work may provide an approach for quantitatively and predicatively scaling the flowability of granular powders based on the particle physical properties like density, particle size, and particle cohesive energy. It may have a significant impact on how industries efficiently and economically handle and deal with powder flowability issues. Academically, the work presented in this article at least provides an approach to define the viscosity of granular powders. More importantly, the viscosity equations of granular powders are derived with the same approaches as employed for that of liquids, colloidal suspensions, and polymeric materials; both granular athermal and conventional thermal systems are thus unified with a single methodology.

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