Development of a virtual Couette rheometer for aerated granular material

A novel rheometer to study the behavior of granular materials in an aerated bed of particles has been developed. The device, called the aerated bed virtual Couette rheometer, is shown to be able to measure the shear stresses in the quasi-static and in the intermediate flow regimes. To validate the instrument, a Newtonian fluid of known viscosity was first sheared. The device was then used for measuring the shear stress of nonaerated glass beads with three -3D printed cells of different sizes to validate the optimal radial position, r *, where both shear rate and shear stress are independent of the cell radius. The results for the nonaerated glass beads displayed Coulomb behavior. The same behavior was observed when the bed was aerated. These experiments also showed that for a fixed shear rate the shear stress decreases as the aeration velocity, U . increases.


| INTRODUCTION
Granular materials are widely produced and handled in the chemical and process industries. 1 Indeed, such materials can be found in powder-based unit operations such as granulation and drying; fluid catalytic cracking processes; coal gasification; fluidized beds. Consequently, understanding their flow behavior is of critical importance to help industrial practitioners handle and produce particulates in an efficient and less costly way.
Three regimes are generally observed in granular flow: the quasistatic (slow) regime, the inertial (rapid) regime, and the intermediate regime. The quasi-static regime, which can be described by plasticity models, 2,3 is characterized by the formation, rotation, and breakage of force chains. This regime has been extensively studied experimentally through a number of experimental techniques such as shear testers. [4][5][6][7][8][9] The inertial regime, where the kinetic theory 10,11 is usually adopted, is characterized by instantaneous and binary collisions; such behavior has been widely investigated in various experimental configurations such as fluidized beds. [12][13][14][15][16] The intermediate regime, which develops between the slow and rapid regimes, has become the subject of numerical and experimental studies. Jop et al. 17 proposed the following constitutive law that predicts the behavior of dense granular flows over the three regimes: where μ is the friction coefficient, I is the inertial number, I 0 is a model constant, μ s and μ 2 are the friction coefficients at low and high I, respectively. I depends on the shear rate, _ γ , on the particle diameter, d p , and on the applied pressure, P. Equation (1) states that the friction coefficient of the granular material has a critical value, μ s , at low shear rates and converges to a limiting value, μ 2 , at high inertial numbers.
Such a flow behavior in the quasi-static regime was successfully modeled by Equation (1) in different flow configurations, such as inclined planes, rotating drum and plane and annular shear flows, 18 where the friction coefficient, μ I ð Þ, is evaluated as the ratio of the shear stress, τ, to the applied pressure, P. Similarly, a static solid-like state in fluidized bed, where particles are aerated by gas velocity lower than the fluidization velocity, U mf , falls into the quasi-static regime. 19,20 In the static aerated bed, the fluidization index, (FI), which is defined as the ratio of the pressure drop to the total weight of the system, is less than 0.9. 21,22 The behavior of granular materials in the static aerated bed has been investigated in several studies. Klein et al. 19  showed a steady and almost linear decrease of the torque through the whole fluidization process suggesting that the internal structure of the bed, even when fluidized, does not change significantly. 29 Tardos et al. 9  The studies discussed show that aerated shearing devices, such as the aerated coaxial cylinder viscometer, 4,9,16,[26][27][28][29]31  In the present study, we verify first the applicability of the concept of the optimal radius r*, when shearing granular materials, by testing five different 3D printed geometries (two heights and three radiuses). The cells were tested with a Newtonian fluid (glycerol 99%) of known viscosity, then nonaerated glass beads (of three different sizes) were used; those powders are known to exhibit Coulomb flow behavior in the quasi-static regime. 6 The powders were then aerated with an aeration ratio U/U mf ranging from 0 to 0.

| Evaluation of the shear rate and shear stress
The shear rate, _ γ, and shear stress, τ, are computed from the rotational speed, ω, and the recorded torque, T, as follows:  The solution of the previous equation leads to the following relation 32 : To cover a large range of shear-thinning behaviors, two different values of n and n 0 need to be selected, for instance, n = 1 and n 0 = 0.15. It has been shown that closer n and n 0 values are more accurate is the determination of r*. 32 Consequently, as long as K τ and K _ γ are evaluated at the optimal position, r*, they can be considered as geometrical constants independent of n. The shear rate and stress, at r*, can be easily evaluated by rewriting Equations (2) and (3): The existence of r* was confirmed experimentally by Ait-Kadi et al. 32 while performing several experiments using standard and nonconventional Couette geometries to measure the viscosity of known Newtonian and non-Newtonian fluids. The results showed that, when evaluated at the optimal radial position, r*, the viscosities measured were in good agreement with their known values, independently of the device geometry. For the present study, r* was evaluated using Equation (4) Table 1.

| Experimental validation of the AB-VCR
Prior to performing shear tests on an aerated granular material bed, the AB-VCR was tested with a Newtonian liquid of known viscosity (glycerol). It was also tested with three different glass bead powders, which are known to behave as a Coulomb fluid in the quasi-static regime, 6 where the viscosity is inversely proportional to the shear rate. For these measurements, various cells geometries were tested to verify the applicability of Equations (5) and (6) to granular materials.

| Viscosity measurement of a Newtonian liquid
Glycerol (99% purity and viscosity 1.4 Pa/s) was initially tested. Four   Table 1  and Cell-2 (L = 70 mm and R i = 15 mm); hence, those two cells will be used for the glass beads measurements presented in the following.

| Glass beads powder measurements
The AB-VCR was tested with dry, noncohesive spherical glass beads with three different size distributions: a monodispersed sample with particle diameter d p-1 = 0.5 mm and two polydispersed samples with diameters d p-2 ranging from 0.05 to 0.08 mm and d p-3 ranging from 0.16 to 0.21 mm. The particles size ranges were obtained by mass distribution and will be referred to as d p-2 = 0.05-0.08 mm and d p-  Table 2.
The three glass bead samples were then tested using Cell-1, Cell-     For each studied glass beads, the shear stresses obtained with the different cells are averaged, and the corresponding results are presented in Figure 7. The error bars correspond to the SD which is consequent of the cell geometry.
The results presented in Figure 7a show that for all three studied powders the shear stress is independent of the shear rate. These  Figure 7b also show that the viscosity is independent of the particle size. There is a slight variation between the different measurements, which indicates that the evaluated viscosity depends on the material. It has been shown previously that the shear stress measurements are independent of the radius ( Figure 6). Figure 7b shows differences between the present study and the work by Marchal et al., 6 ; given that the particles used in the two studies are made of the same material, it is speculated that the variation observed  (5) is applied while assuming a uniform stress distribution along the height L, which might not be the case in granular materials.

| Aerated bed
Prior to perform shear tests with aeration, the AB-VCR was tested to determine to what extent, if any, the rotational speed of the inner cell altered the basic fluidized bed characteristics. To do so, the FT4 aeration vessel was used as a fluidized bed without cell where particles d p-2 and d p-3 were tested. The same fluidized bed configuration with Cell-2 was used to perform measurements at two rotational speeds (4 and 20 rpm). In Figure 8, the FI = ΔPA/mg is shown as a function of the normalized air velocity, U/U mf , where A is the aerated surface area, m is the total mass of particles in the bed (200 g), U is the air velocity, and U mf is the minimum fluidization velocity. The latter is evaluated experimentally from the fluidized bed tests performed with d p-2 and d p-3 particles and the corresponding values are presented in Table 2. A theoretical line, representing the perfect equilibrium where FI is equal to the ratio U/U mf FI = U/U mf , is also plotted in Figure 8 as reference (solid red line). The good agreement found between the F I G U R E 6 Evolution of shear stress as a function of shear rate obtained with various cells for particles with diameters: (a) d p-1 = 0.5 mm and (b) d p-2 = 0.05-0.08 mm. See Table 1  results with and without cell for both particle sizes and both rotational speeds ( Figure 8) indicates that having the cell rotating inside the aerated bed does not have a strong effect on its fluidization behavior.
However, it should be noted that a deviation from the linear behavior of the fixed bed is observed with the sample d p-2 , obtained without the use of the cell. We attribute this deviation to the small size of the particles, which are cohesive and therefore more difficult to fluidize.  For d p-1 particles, U mf was estimated theoretically 35 given that its value was higher than the maximum air velocity supported by the FT4 (40 mm/s). Therefore, the shear tests were only performed with two values of FI (0.1 and 0.2). For the three studied cells and for each shear rate, particle size, and FI, the torque was recorded for 60 s and it was found to be steady in time for all cases. The shear stress was then computed using Equation (5) and averaged over the last 40 s.
Good agreement was found between the results with different cells for one size of particle and one FI. For each size of particle and FI, the shear stresses obtained with three different cells, are averaged and the corresponding results are presented as a function of the shear rate in Figure 9 (only d p-2 and d p-3 are shown), where the error bars correspond to the SD of the cell geometry dependency.
The results presented in Figure 9 show that when the particles are aerated, good agreement is found between the shear stresses obtained using different cells, and negligible SD between the cells measurements is recorded (highest value around ±20 Pa which represents ±8% of the corresponding shear stress observed with d p-3 particles with FI = 0.2). In Figure 9 it can be seen that the shear stress decreases as FI increases. This might be explained by the fact that when aerated, the pressure acting on the particles decreases, which leads to a decreasing in viscosity. However, for a fixed shear rate, the shear stress decreases in a nonlinear way with increasing FI and an extrapolation of the results will lead to a shear stress equal to 0 for values of FI below 1, which is unlikely to happen. This observation might indicate that the studied powder will no longer be in the frictional quasi-static regime even if the aeration velocity is below U mf and one can assume that the transition to an intermediate regime will occur for FI < 1. The graphs presented Figure 9 also show that for all the studied FI and particle sizes, the shear stress in the aerated bed is independent of the studied shear rates, describing the Coulomb behavior observed without aeration (Figure 7). It should be noted that the same behaviors were observed with d p-1 particles (not shown in Figure 9). The average shear stress over the studied shear rates is evaluated for each FI and each particle size, and the corresponding values are reported in Table 2. For all studied FI > 0, good agreement is found between the average shear stresses, τ av , obtained for parti-  (1)). The experiments are assumed to be performed in the quasi-static regime (I < 0.01) where μ s is independent of the shear rate. Therefore, the pressure P, which is assumed to be constant for each FI, is estimated as the ratio of the average shear stress, τ av , to the steadystate friction coefficient, μ s . The values of the estimated pressures are reported in Table 2 together with the normalized pressure P* = PA/mg. For all studied values of FI, the estimated pressure is always lower than the sample weight, this with P* being systematically lower than 1.
The graph plotted in Figure 10 shows the comparison between the measured μ(I), which were evaluated using the estimated P, and the one evaluated with the constitutive law (Equation (1) The results presented in Figure 10 show that for an inertial number lower than 0.003, good agreement is found between the mea- as the inertial number increases. One should bear in mind that the experimental results presented in Figure 10 are obtained with an estimated value of the pressure, which might not be the real one. Nevertheless, these results validate qualitatively the methodology presented in this paper for the evaluation of the measurements of particle stresses in an aerated granular bed.

| CONCLUSIONS
In this study, a 3D printed cylindrical cell composed of six blades was adapted to the aeration kit of the Freeman FT4 rheometer in order to perform shear test experiments on aerated powders in a Couette flow configuration. This new rheometer prototype, named AB-VCR, was developed for the measurements of bulk solid stresses of granular material in an aerated bed. It was first validated with a Newtonian fluid (glycerol 99% of known viscosity) without aeration. The evaluated shear stress was found to be proportional to the applied shear rate describing the expected Newtonian behavior. The viscosity was found to be in good agreement with the theoretical value and independent of the geometry of the 3D printed cells. This prototype rheometer was also tested and validated for nonaerated glass bead powders (particle size ranging from 0.05 to 0.5 mm), where the shear stress was found to be independent of the applied shear rate (ranging from 3 to 22 s −1 ), confirming the Coulomb behavior of the granular materials in the quasistatic regime. 6 Similar behavior was obtained for the three studied samples and showed to be independent of the cell radius. The shear stress was also found to increase when the friction coefficient decreases. The device was finally tested as an aerated bed rheometer by varying the velocity of the air injected through the three glass bead samples studied. The aeration velocities were chosen according to the minimum fluidization velocity of each sample, in order to ensure that the shear tests were performed on a fixed aerated bed (U/U mf < 1). The main findings of the aerated bed measurements are summarized as follows: For a fixed aeration velocity, U, the evaluated shear stress was found to be independent of the imposed shear rate (ranging from 3 to 22 s −1 ) describing the Coulomb behavior observed without aeration for the three glass bead samples. The shear stress was also found to be decreasing with increasing the aeration velocity.
For a fixed FI, good agreement was found between the results obtained with samples having steady-state friction coefficients of similar value (i.e. μ s = 0.33 and 0.36), while the shear stress obtained with the sample having a higher steady-state friction coefficient (μ s = 0.46) was found to be systematically lower than the shear stress of the other two samples with lower friction coefficient.
Good agreement was found between the experimental μ(I), which was evaluated as the ratio of the shear stress, τ, to an estimated pressure P, and the one predicted by the model proposed by Jop et al.,17 for all studied FI and particle sizes.

ACKNOWLEDGMENT
Financial support from the EPSRC (grant no. EP/N034066/1) is kindly acknowledged.