# A study on the impact of shifted feeding of a segregating binary mixture in continuous powder mixing

Marie-Anne Batel, Carla Barcelo, Cendrine Gatumel, Henri Berthiaux

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Marie-Anne Batel, Carla Barcelo, Cendrine Gatumel, Henri Berthiaux. A study on the impact of shifted feeding of a segregating binary mixture in continuous powder mixing. Powder Technology, 2025, 452, pp.120564. 10.1016/j.powtec.2024.120564. hal-04869738

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# A study on the impact of shifted feeding of a segregating binary mixture in continuous powder mixing

Marie-Anne Batel, Carla M. Barcelo Chong, Cendrine Gatumel, Henri Berthiaux*

RAPSODEE Research Center, UMR CNRS 5302, IMT Mines Albi, Campus Jarlard, Cedex 09, 81013 Albi, France

## HIGHLIGHTS

* Feeding segregating powders close to the mixer outlet reduces the risk of de-mixing.

* Grain size differences cause radial and axial percolation, as well as trajectory segregation.

* Powder re-circulation within the mixer depends on the feed configuration.

* Steady-state is reached faster if the powders are fed at the same point.

## GRAPHICAL ABSTRACT

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Particle segregation

Powder mixing

Mixture homogeneity

Process start-up

Particulate solids

## ABSTRACT

Continuous mixing reduces the need to handle, transport or store mixtures. It may help reducing segregation issues and appear as a valuable alternative to the batch operations commonly employed. In the present work, we investigate the impact of the feed configuration on the continuous mixing process of two segregating products in an industrial mixer and a lab-scale mixer equipped with feeders that can be positioned at different points. To compare the different configurations, we are interested in three indicators: the amount of mixture produced during the transient phase at start-up, the internal composition profile of the powder mixture, as well as the homogeneity of the mixture at the mixer outlet once steady-state has been reached. The study of the mixture state within the mixers at steady-state shows that the composition profile does not depend on the equipment size, on the stirring device or on the particle size ratio. When powders are fed into different positions along the mixer, the powder fed the closest to the outlet hardly re-circulates forward because of the other powder already present. To limit transient regime at start-up, it is important that the first grains of each powder arrive at the equipment's outlet at the same time. We show that it is preferable to feed powders at the same time towards the mixer outlet to reach steady-state faster. We also find that the feed configuration has very little impact on the homogeneity of the mixture at the outlet once steady-state has been reached.

<footnote>* Corresponding author.
E-mail address: henri.berthiaux@mines-albi.fr (H. Berthiaux).</footnote>
# 1. Introduction

Powder mixing is an essential operation in many industries, for the production of pharmaceutical tablets, cosmetic powders, pigments, metal powders, cement, food mixtures, etc. The challenge when mixing powders is to avoid segregation, which has been studied extensively [1–3]. It is mainly caused by differences in particle size, density or shape. Segregation is favored when the mixture is made up of free-flowing powders, as the particles of these powders are highly mobile and rearrange themselves very readily. This type of mixture will be easy to mix, but also to de-mix if handled without care.

During batch mixing, segregation takes place in the mixer [4,5], but also afterwards when the mixture is handled, poured into a container, unloaded from a silo and transported [6]. To minimize these situations, powder mixing can be carried out in a continuous process. In this way, the mixture obtained at the mixer outlet can be directly granulated, compressed or packaged. Moreover, the mixture produced in a continuous mixer is often more homogeneous than that produced in a batch mixer [7–9]. Oka et al. [7] studied the performance of a batch mixer and a continuous mixer for different mixtures. These binary mixtures had different segregability due to differences in size and/or density between the powders. The authors showed that mixing performance depends on mixture segregability, and therefore on powder properties, particularly for batch mixing.

However, this does not prevent segregation to appear within the continuous mixer. Bekaert et al. [10] studied the impact of mixture properties and operating conditions on the performance of a continuous mixer, quantified by the variability of the active ingredient content in the mixture. According to the results of this study, long/large feeder deviations reduce mixing performance unless the mixer design is modified via the impeller configuration. The authors also highlighted the correlation between mixture properties and mixing performance using partial least squares regression. In particular, this study shows that high mixture flowability leads to poor mixing performance.

The main segregation mechanism observed in a convective continuous mixer operating in the dense flow regime is the percolation of fine particles through coarser ones, while the contrary is not possible because of a steric effect that is of the same nature as the Brazil nut effect [11]. This mechanism can take place in the radial direction, as well as in the axial direction. Another observed phenomenon is trajectory segregation due to the difference in air resistance and gravitational force as a function of particle size. Thus, when grains are thrown by blades, the finest grains will be thrown less far than the coarsest. In their work with a GCM500® continuous mixer, Ammarcha et al. [12] noted that the first particles to leave the mixer were predominantly coarse particles, as shown in Fig. 1 for two agitation speeds. They measured the accumulation of fine particles at the bottom of the equipment during its start-up, therefore pointing out this percolation phenomena. They concluded that segregation within the mixer may have an impact on mixture homogeneity at the outlet, not only during start-up, but also during any transient phase that is likely to happen during processing: feeders re-filling,

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Mixture composition at mixer outlet, initially richer in coarse particles during start-up, for two agitation speeds (Froude number equal to 0.5 and 1.9) [13].</figcaption></figure>

change in operating conditions, mixer discharge, etc. As the quality of the mixture cannot be guaranteed during these transient phases, the corresponding mass of product must be discarded. The associated loss, that may roughly be represented by 5 times the mean residence time or 5 times the hold-up weight (typically 10 kg in an industrial equipment), is far from being negligible. In the pharmaceutical industry, this discarded mass may therefore represent tenths of thousands of unit doses or tablets.

A possible way of reducing segregation during mixing is to modify the feed configuration by shifting the position of the feeders. With conventional feeding, all feeders are positioned at the extremity of the mixer which is opposite to the system's outlet. But because such horizontal mixers must create an important back-mixing to differ from plug flow, at least one feeder could be advantageously shifted along the mixer to feed one of the powders in the middle or even at the end of the mixer. Mizonov et al. [14] have carried out a theoretical study on the influence of feed position on the quality of the mixture leaving a continuous mixer. To this end, they have modelled a mixture of two powders subjected to segregation, for example a mixture of two powders of different particle sizes, with the aim of improving mixing, which means limiting segregation, knowing that the same mechanisms are responsible for mixing and segregation. At start-up, the modelled mixer is filled with one of the two powders. The authors have shown that the position of the second powder feeding influences the quality of the mixture at the mixer outlet, and that there is an optimum position depending on the difference in particle size. Indeed, if the finest powder feeding is shifted, this powder must spend enough time in the mixer to penetrate the powder bed and thus be mixed with the coarse particles, as illustrated in Fig. 2. But if it spends too much time in the mixer, it runs the risk of falling entirely to the bottom of the mixer. Shifting the feeding of one of the powders along a real equipment may therefore help in reducing segregation within the mixer, as well as quantities of out-of-specification product at the outlet. However, experimental evidence was lacking.

In the present study, we will first focus on the impact of feeders shifting on the composition profile of a segregating mixture inside two continuous mixers of different scales, namely a GCM 500 by Gericke and a home-made lab mixer. For this, our work will be supported by a binary mixture of free-flowing powders of different grain sizes. In a second part, we will pay attention to the impact of feeder shifting on the duration of the transient phase during start-up and mixture homogeneity obtained at steady-state for the lab-scale mixer only.

# 2. Material and methods

## 2.1. Powders studied

Free-flowing powders of different grain sizes, namely semolina and couscous, have been chosen for this study. Two types of semolina and couscous are considered in this work, depending on the mixer scale. Table 1 shows the main physical characteristics of the powders. The Carr indices are below 10, demonstrating the excellent flow properties of all four powders. The mixture of medium semolina and medium couscous has a $d_{50}$ ratio of 2.0 and the one made of fine semolina and fine

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Illustration of the mixer with shifted feeding.</figcaption></figure>
<caption>**Table 1**</caption>

Main physical characteristics of the powders. Particle size has been determined by Malvern Mastersizer® 3000 with dry dispersion, true density by the pycnometer and apparent bulk densities by Erweka “taptap” volumenometer. Carr index is deduced from apparent densities.

<table><thead><tr><th rowspan="2">Property</th><th colspan="3">Powders used in the industrial mixer</th><th colspan="3">Powders used in the lab-scale mixer</th></tr><tr><th>Medium semolina</th><th>Medium couscous</th><th>Ratio</th><th>Fine semolina</th><th>Fine couscous</th><th>Ratio</th></tr></thead><tbody><tr><td>d<sub>10</sub> (µm)</td><td>680</td><td>1375</td><td>2.02</td><td>197</td><td>830</td><td>4.21</td></tr><tr><td>d<sub>50</sub> (µm)</td><td>860</td><td>1700</td><td>1.98</td><td>307</td><td>1430</td><td>4.66</td></tr><tr><td>d<sub>90</sub> (µm)</td><td>980</td><td>1970</td><td>2.01</td><td>473</td><td>2470</td><td>5.22</td></tr><tr><td>Bulk density (kg.m<sup>-3</sup>)</td><td>759</td><td>762</td><td>1.00</td><td>709</td><td>765</td><td>1.08</td></tr><tr><td>Tapped apparent density (kg.m<sup>-3</sup>)</td><td>788</td><td>779</td><td>0.99</td><td>765</td><td>803</td><td>1.05</td></tr><tr><td>True density (kg.m<sup>-3</sup>)</td><td>1442</td><td>1452</td><td>1.01</td><td>1458</td><td>1426</td><td>0.98</td></tr><tr><td>Carr index (-)</td><td>3.7</td><td>2.2</td><td>0.59</td><td>7.3</td><td>4.7</td><td>0.65</td></tr></tbody></table>

couscous is 4.7. The segregability of the two mixtures is therefore probably different. As it can be seen, the densities of the powders are similar, as well as the Carr indices ratios which indicates that the powder flowabilities are close. The two powders can be separated by sieving, enabling them to be reused from one experiment to the next one.

Couscous was colored in black to differentiate the powders by their color, enabling image analysis of the mixture at the mixer outlet. Coloration was achieved by adsorption of an iodine solution onto the starch in the couscous. The solution was sprayed on while the grains were continuously agitated with a spatula. The grains were then air-dried. Several cycles of spraying and drying were necessary to obtain a uniform black color.

The segregability of the mixture is assessed by a segregation test developed by Massol-Chaudeur et al. [15], which consists in evaluating the homogeneity of the mixture after it has been poured into a heap in a thin cell. The powders are mixed in a Turbula® T2F batch mixer and the mixture, assumed to be homogeneous, is then poured into the cell. During pouring, the heap is observed to form in a succession of avalanches alternating from one side to the other. The layer of powder conveyed by an avalanche is not homogeneous, as couscous grains roll off the ends of the heap and semolina grains percolate through the couscous layers, as shown in Fig. 3. Binary mixtures of semolina and couscous will therefore be prone to segregation due to the free-flowing nature of the powders and the difference in grain size. In addition, composition has a major impact on mixture segregability. 13 samples were taken through small holes placed in the wall. The standard deviation between the compositions of the 13 samples reflects the heterogeneity of the mixture after pouring into a heap. Fig. 4 shows the

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Segregation test performed on a mixture of 30 % fine semolina by mass and 70 % fine couscous by mass.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Segregability of fine semolina and fine couscous mixtures according to their compositions.</figcaption></figure>

standard deviation of sample compositions as a function of the semolina fraction in the mixture of fine semolina and fine couscous. The results confirm that the mixture's segregability depends on its composition, a maximum being observed for the mixture containing nearly 30 % in fine semolina. This case demonstrates the highest standard deviation, indicating that this composition has a strong segregation potential.

## 2.2. Mixers and feeders

Two convective continuous mixers are studied in the present work. One is the Gericke GCM500® industrial mixer, whose semi-cylindrical tank measures 20 cm in diameter, 50 cm in length and 26.8 cm in height. The stirring device consists of a central screw combined with a series of blades mounted on a frame, which has been described by Marikh et al. [16]. The second mixer is a lab-scale mixer designed at the RAPSODEE laboratory. It has a volume of 2.9 L and is geometrically similar to the Gericke's GCM500®. The ratio between tank diameter and length is 0.4 for both mixers. Table 2 lists mixer, outlet dimensions and stirring devices characteristics.

Blade rotation speed can be expressed in terms of the Froude number $Fr$, which corresponds to the ratio of centrifugal force to gravitational force. This dimensionless number is defined by Eq. 1, where $R$ is the radius of the stirring device in m, $g$ is the acceleration of gravity in $m.s^{-2}$

<caption>**Table 2**</caption>

Geometric characteristics of the mixers studied.

<table><thead><tr><th></th><th>Industrial mixer</th><th>Lab-scale mixer</th></tr></thead><tbody><tr><th>Tank dimensions</th><td>Semi-cylinder<br>Diameter: 20 cm<br>Length: 50 cm<br>Height: 26.8 cm</td><td>Semi-cylinder<br>Diameter: 10 cm<br>Length: 26.4 cm<br>Height: 17.5 cm</td></tr><tr><th>Stirring device</th><td>Screw +14 blades inclined<br>at 45° mounted on a frame</td><td>14 blades inclined at a 5° angle<br>forward and the last 2 blades (before<br>the outlet) inclined at a 70° angle<br>forward et positioned opposite each<br>other</td></tr><tr><th>Outlet</th><td>Rectangle 4 cm wide and<br>7.5 cm high<br>Horizontal position:<br>centered on the axis of the<br>agitator shaft<br>Vertical position: at the<br>bottom of the tank</td><td>Rectangle 2.0 cm wide and 1.8 cm<br>high<br>Horizontal position: on one side<br>against the wall<br>Vertical position: centered on the<br>axis of the agitator shaft</td></tr></tbody></table>
while $\omega$ is the angular rotational speed in rad.s$^{-1}$.

$$Fr = \frac{R \times \omega^2}{g} \qquad (1)$$

Being both mixers nearly geometrically similar, and as we want to keep the same phenomenon at play in the two configurations, a dynamic similarity must also be ensured. For this, a same Froude number will be fixed in either equipment. However, filling ratios are different: 14 % in the industrial mixer and 58 % in the lab-scale mixer, considering that a filling degree of 100 % corresponds to filling up to the agitator shaft.

A look at Mellmann's bed behavior Diagram in Fig. 6 that establish the different flow regimes according to Froude number and filling degree, shows that a value of $Fr = 0,3$ would ensure the same flow regime, namely a cascading regime close to the limit of cataracting, in both cases. This corresponds to an angular rotational speed of 5.4 rad.s$^{-1}$ in the industrial mixer and 8.1 rad.s$^{-1}$ in the lab-scale mixer.

The same feeders are employed for both mixers, as shown in Fig. 5. These are Gericke's loss-in-weight GLD87®, feeding the mixer by regulating powder flow rates. Each feeder consists of a 15 L hopper surmounting a dosing system for which the screw speed is controlled by a regulation loop.

Medium semolina and medium couscous are processed in the industrial mixer, while fine semolina and fine couscous are used in the lab-scale mixer. We always consider a mixture of 50 % semolina and 50 % couscous, so that we can study the influence of the feed configuration under the same conditions. For mixing in the industrial mixer, we utilize a medium semolina feeder set to a flow rate of 5.6 g.s$^{-1}$ and a medium couscous feeder also set to 5.6 g.s$^{-1}$. For mixing in the lab-scale mixer, slightly lower flow rates are considered: 5 g.s$^{-1}$ for the fine semolina feeder and 5 g.s$^{-1}$ also for the fine couscous feeder. Averaged over all feed configurations, the hold-up mass at steady-state is 392 g in the industrial mixer and 422 g in the lab-scale mixer. The geometrical residence times $\tau$ in s are calculated according to Eq. 2, $M_{total}$ being the hold-up mass in g of semolina and couscous at steady-state and $Q_{total}$ the total flow rate in g.s$^{-1}$.

<figure><img src="image_8.png" /><figcaption>Fig. 6. Flow regime in mixers from Mellmann's Bed Behavior Diagram [18].</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Illustration of conventional feeding (a) and shifted feeding (b).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 5. Experimental set-up for continuous mixing (a) in the industrial mixer and (b) in the lab-scale mixer (1-feeders, 2-industrial mixer, 3-industrial mixer outlet, 4-lab-scale mixer, 5-lab-scale mixer and slope, 6- conveyor belt for image analysis).</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 8. Division of (a) the industrial mixer [19] and (b) the lab-scale mixer into five cells.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Illustration of the steady-state determination test on a fictive data set [21].</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. Device for separating the five cells of the lab-scale mixer.</figcaption></figure>

$$ \tau = \frac{Mtotal}{Qtotal} \qquad (2) $$

A dimensionless number N can be calculated by Eq. 3:

$$ N = \tau \times \frac{\omega}{2\pi} \qquad (3) $$

N is equal to 31 for the industrial mixer and 54 for the lab-scale mixer. This is a closed system with no dispersion across boundaries,

meaning that the geometrical residence time is equal to the mean residence time, so this dimensionless number would correspond to the number of blade passes. It may represent the amount of shear applied to the powder for dispersion while the powder is within the mixer. Consequently, a larger number of blade passes would contribute to a higher degree of mixture homogeneity at the mixer outlet [17].

In the conventional feed configuration, the feeders are positioned at the mixer inlet. For the purpose of this study, the feeders can also be shifted along the mixer, as illustrated in Fig. 7. Before the start of each experiment, the feeders are placed at a certain point between the mixer inlet and its outlet. They remain in this position for the duration of the experiment.

To study the composition profile inside the mixers, these are virtually divided into five cells. The five cells of the industrial mixer are of equal dimensions and are therefore 10 cm long (Fig. 8 (a)). The closest cell to the inlet is called cell 1 and the closest one to the outlet is called cell 5. Feeders can be positioned above any of the five cells of the industrial mixer, the conventional feeding being into cell 1. For this mixer, we will study configurations with at least one of the feeding placed in cell 1. To study the shifted feeding with both powders fed into the same cell, we will complete the study with both feedings in cell 2. For the lab-scale mixer, the division of the cells is made according to the blades, and therefore are of different lengths, as shown in Fig. 8 (b). Feeders cannot be positioned above cells 1 and 5. The conventional feeding position is therefore in cell 2, but the feeders can also be shifted above cell 3 or 4. The studied feed configurations for both mixers are listed in Table 3.

### 2.3. On-line mixture analysis

The lower part of the lab-scale mixer outlet is a 14.5 cm metal slope, inclined at 34° to the horizontal. This allows the powder to be discharged onto a moving conveyor belt. The mixture on the conveyor belt is analyzed by optical image processing using a grayscale linear CCD camera positioned above the conveyor belt. As couscous is colored in black and semolina is a yellow powder, the mixture composition can be determined by the images taken. The resulting video is a sequence of 3000*200 pixels images at a speed of 14 frames per second. The images are analyzed by double thresholding using the method developed by Demeyre [20] and Ammarcha et al. [12]. This enables us to identify the white pixels in each image corresponding to the semolina surface and the black pixels corresponding to the colored couscous surface. For each image, we calculate the fraction of white pixels. In this way, we can monitor fluctuations in the composition of the mixture via the evolution of the fraction of white pixels over time, which we will refer to as $x_k$. The mixture takes 10 s to reach the camera from the mixer outlet, without any particle motion. We remove this time from the camera's clock to consider the mixture directly at the mixer outlet. Although the fraction of white pixels is not directly linked to the fraction of semolina in the mixture by calibration, we consider that fluctuations in the fraction of white pixels reflect fluctuations in the fraction of semolina in the mixture.

The duration of the transient phase at start-up, beginning with the
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Sliding window for calculating the relative standard deviation with a sample size of (a) one image and (b) two images.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Composition profile at steady-state when both powders are fed into the same cell. The legend shows which of the 5 cells of the mixer the semolina and couscous are fed into. This shows a re-circulation of powders in the previous cells when both powders are fed into the same cell, and a wall effect at the industrial mixer inlet with an accumulation of semolina.</figcaption></figure>

mixer being empty, is determined from the evolution of the fraction of white pixels $x_k$. To do this, we need to detect the transition from transient state to steady-state. For powders, the attainment of steady-state is

difficult to detect, as $x_k$ still fluctuates significantly even when steady-state is reached, for a variety of reasons: the discrete nature of powders, the occurrence of “events”, such as the formation of heaps and
<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Composition profile at steady-state when powders are fed into adjacent cells. The legend shows which of the 5 cells of the mixer the semolina and couscous are fed into. This shows that the re-circulation of one powder is slowed down by the presence of the other powder closer to the mixer inlet.</figcaption></figure>

avalanches, the frequency of the blades and the heterogeneity of the mixture at the scale of scrutiny. Since we are not focusing on a precise application for the mixture produced, we do not set an acceptability level for fluctuations in $x_k$. We consider that steady-state is reached when $x_k$ fluctuations are not due to the transient state, but only to the nature of the powders and the mixing operation. A steady-state detection method for powder mixing was previously detailed in Batel et al. [21] and is shown in Fig. 9. First, $x_k$ is filtered to reduce the amplitude of fluctuations via a Savitzky-Golay filter. Next, the sliding standard deviation is calculated and compared with two thresholds. Steady-state begins when one point is below the lower threshold and none of the

following points are above the upper threshold.

From an industrial point of view, it would be more interesting to determine the mass of product to be discarded rather than the time during which the mixture must be discarded, especially if it is a high value-added product. A weighing scale connected to an acquisition system is placed under the container at the end of the conveyor belt. This enables us to determine the mass of the mixture leaving the mixer during the transient regime at start-up, bearing in mind that it takes 2.8 s for the mixture to go from the camera to the scale. Results will be expressed as the mass of mixture produced during the transient regime at start-up, which is the mass of “lost” mixture. For each feed configuration, we
<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 14.** Composition profile at steady-state when powders are fed into distant cells. The legend shows which of the 5 cells of the mixer the semolina and couscous are fed into. This shows that the re-circulation of one powder is slowed down by the presence of the other powder closer to the mixer inlet.</figcaption></figure>

perform 5 identical trials. Since steady-state detection is a delicate task, it can happen that one of the 5 trials gives a very different result. We therefore choose to keep the 4 trials giving the closest results for each feed configuration.

## 2.4. Homogeneity calculation within the mixer and at mixer outlet

The composition profile in the mixer is measured after the device has been switched off, once steady-state has been reached. The stirring device is carefully removed from the mixer, taking care not to destruct the mixture's structure. The five cells are then delimited using metal plates.

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

**Fig. 10** shows the device used in the lab-scale mixer to insert four metal plates vertically with the aid of a guide. The powder in each cell is then collected, either by spatula and brush in the industrial mixer, or by cyclone extraction in the lab-scale mixer. The mixture recovered from each cell is sieved to separate semolina and couscous, in order to determine the hold-up mass of each powder and the composition of the mixture.

The mixture quality within the mixers at steady-state will be quantified by the standard deviation of the semolina fraction between the five cells, the scale of scrutiny being therefore that of the cell. Depending on feed configurations, the total hold-up mass may be very rich in semolina
<figure><img src="image_21.png" /><figcaption>Fig. 15. Powder mass produced during the transient regime when powders are fed into the same cell (results of 4 trials), showing that the more the powders are fed towards the outlet, the lower is the mass of mixture produced during start-up.</figcaption></figure>

or very rich in couscous. The standard deviation allows us to account for deviations from the mean without depending on whether we use the semolina fraction or the couscous fraction, unlike the relative standard deviation.

Homogeneity of the mixtures at the mixer outlet at steady-state is determined from the fraction of white pixels in the images. We are interested here in fluctuations in the fraction of white pixels, and, as stated previously, do not directly relate the fraction of white pixels to the mass or number fraction of semolina in the mixture.

Steady-state is reached between 100 s and 200 s, depending on the feed configuration. We choose to use the data between 250 s and 350 s for all feed configurations to ensure steady-state. The relative standard deviation (RSD) is calculated from the mean $\bar{x}_i$ and the standard deviation $\sigma_i$ over a sliding window (Fig. 11 (a)) using Eq. 4 with the window

size equal to n = 20 samples, a sample being an image that corresponds to 0.7 g of mixture. Homogeneity at steady-state is quantified by the averaged RSD with $n_{SS}$ the number of points between startSS = 250 s and endSS = 350 s. The averaged RSD has been calculated over window size values ranging from 14 to 75 and has been found insensitive to n. As a result, n was chosen arbitrarily equal to 20.

$$ 
\begin{align*}
\bar{x}_i &= \frac{1}{n} \sum_{k=i-\frac{n-1}{2}}^{i+\frac{n-1}{2}} x_k \\
\sigma_i &= \sqrt{\frac{1}{n-1} \sum_{k=i-\frac{n-1}{2}}^{i+\frac{n-1}{2}} (x_k - \bar{x}_i)^2} \\
\text{Average RSD} &= \frac{1}{n_{SS}} \sum_{i startSS}^{endSS} \frac{\sigma_i}{\bar{x}_i} \tag{4}
\end{align*}
$$

We can vary the sample size to study the impact of the scale of scrutiny on the average RSD at steady-state. The sample size can be 1, 2, 3, 4, 5, 7, 10 or 15 images. When the sample size is two images, for example, we group the images two by two as shown in Fig. 11 (b).

## 3. Results and discussion

In this section, we present the composition profiles within the mixers, followed by the outlet mixture characteristics through the mass of mixture produced during start-up as well as homogeneity at steady-state.

### 3.1. Composition profiles within mixers

Our analysis will be based on whether the feeding locations of each powders are close to each other or not. As a consequence, we will first discuss the configurations for which both powders are fed into the same cell, then those with feeding into adjacent cells, and finally those with feeding into distant cells.

#### 3.1.1. Powders fed into the same cell

Fig. 12 shows the composition profile obtained at steady-state when both powders are fed into the same cells for each given equipment.

In the industrial mixer (Fig. 12 (a)), the fraction of semolina in the

<figure><img src="image_22.png" /><figcaption>Fig. 16. Powder mass produced during the transient regime when powders are fed into (a) adjacent cells or (b) distant cells (results of 4 trials). Feed configurations are arranged in ascending order of mass of mixture produced during start-up, showing that feeding powders into different cells increases the mass of mixture produced during start-up, particularly when couscous is fed closer to the outlet than semolina.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. 17. Steady-state mixture homogeneity for different feeding configurations. Error bars indicate the standard deviation over 5 trials.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 18. Influence of sample size on the steady-state mixture homogeneity. Each point represents the average of all configurations.</figcaption></figure>

hold-up mass of cell 1 is 0.83 when both powders are fed into cell 1, as compared with 0.54 when both powders are fed into cell 2. This is probably due to a wall effect, indicating that the powders should not be fed too close to the edge of the mixer. Small semolina particles must have accumulated during start-up in a stagnant zone at the bottom of the mixer near the wall. For this reason, the powders will never be fed into the first cell of the lab-scale mixer. Even when both powders are fed into cell 2 of the industrial mixer, the average fraction of semolina is 0.58 (Table 4), which is higher than the expected semolina fraction of 0.5. This is probably due to the accumulation of semolina at the bottom of the mixer during start-up thanks to a percolation phenomenon of the fine particles through the bed. When both powders are fed into cell 1 of the industrial mixer, the standard deviation is 0.11, which is twice the standard deviation when both powders are fed into cell 2 (Table 4). This is due to the very high semolina fraction in cell 1.

In the lab-scale mixer, the composition profile is similar, regardless of which cell the two powders are fed, as shown in Fig. 12 (b). The average semolina fraction is 0.48 when the powders are fed into cell 3, which is slightly lower than when they are fed into cell 2 or 4 (Table 4). The semolina fraction is slightly higher in the cell where the powders are fed (between 0.52 and 0.55), which may be again due to a slight accumulation of semolina at the bottom of the mixer. The close hold-up masses despite the difference in mixer scale implies a lower filling height in the industrial mixer. One hypothesis is that the height of semolina accumulated in the bottom of the mixer is similar for both

<caption>**Table 3**</caption>

Studied feed configurations. “S” and “C” represent the feeding of semolina and couscous respectively into one of the five mixer cells.

<table><thead><tr><th></th><th colspan="3">Industrial mixer</th><th colspan="3">Lab-scale mixer</th></tr></thead><tbody><tr><th>Powders fed into the same cell</th><td>S C<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td></tr><tr><th>Powders fed into adjacent cells</th><td>S C<br>1 2 3 4 5</td><td>C S<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td><td>C S<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td></tr><tr><th>Powders fed into more distant cells</th><td>S C<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td><td>C S<br>1 2 3 4 5</td><td>S C<br>1 2 3 4 5</td></tr></tbody></table>
<caption>**Table 4**</caption>

Average mass fraction content in semolina over the five cells, as well as associated standard deviation when both powders are fed into the same cell. Above-the-mean values of the average content are highlighted in pink, under-the-mean values in light blue.

<table><thead><tr><th rowspan="3"></th><th colspan="4">Industrial mixer</th><th colspan="4">Lab-scale mixer</th></tr><tr><th colspan="4">S C</th><th colspan="4">S C</th></tr><tr><th>1</th><th>2</th><th>3</th><th>4</th><th>1</th><th>2</th><th>3</th><th>4</th></tr></thead><tbody><tr><th>Average</th><td></td><td></td><td></td><td>0.66</td><td></td><td></td><td>0.50</td><td></td></tr><tr><th>Standard deviation</th><td></td><td>0.11</td><td></td><td>0.05</td><td></td><td>0.03</td><td></td><td>0.02</td></tr></tbody></table>

mixers, but that this represents a higher proportion of the hold-up mass in the industrial mixer due to the lower filling height. In the cells located between the inlet and the feeders, the semolina fraction is lower with, for example, a semolina fraction of 0.46 in cell 1 when feeding in cell 2. This means that coarse grains of couscous re-circulates more easily to the cells preceding the feeding when both powders are fed into the same cell. The standard deviation of the semolina fraction between the five cells, an indicator of mixture state, is at most 0.05 for the three feed configurations (Table 4). This means that there is good back-mixing, particularly when both powders are fed into cell 4 where the standard deviation is only 0.02.

### 3.1.2. Powders fed into adjacent cells

Fig. 13 shows the composition profiles when powders are fed into adjacent cells. In general, it can be seen that shifted feeding has little impact on the hold-up mass in the cell where the closest feeding to the outlet is located, and in subsequent cells up to the mixer outlet. In the industrial mixer (Fig. 13 (a)), when semolina is fed into cell 1 and couscous into cell 2, the fraction of semolina in cell 1 is 0.88. Because of the presence of the fine semolina grains, the coarse particles of couscous hardly re-circulates to cell 1. Conversely, when couscous is fed into cell 1 and semolina into cell 2, the semolina fraction in cell 1 is 0.24, which is closer to 0.5 than 0.88. This means that couscous re-circulates less than semolina, probably because the couscous grains cannot percolate between the semolina grains, creating a stagnant zone in cell 1. This phenomenon could be described as a relatively greater axial percolation of fine particles than for coarse particles due to a major steric hindrance of these. Similarly, radial percolation takes place when the semolina grains fall to the bottom of the mixer, while couscous grains remain at the surface, although the division into axial cells cannot reveal this phenomenon. In both configurations, cells 2 to 4 are very rich in semolina (semolina fraction between 0.58 and 0.69), as when both powders are fed into cell 2. The composition of cell 5 is 0.48 or 0.54, and is therefore close to the composition of the outlet mixture, namely 0.5 at steady-state.

Fig. 13 (b) and (c) shows the results for the lab-scale mixer when semolina and couscous are fed into cells 2 and 3, or 3 and 4. The overall composition of the hold-up mass is slightly further from 0.5 when couscous is fed closer to the outlet than semolina, than when semolina is fed closer to the outlet than couscous (Table 5). However, this phenomenon is much less pronounced in the lab-scale mixer than in the

industrial mixer, meaning that couscous re-circulates more in the lab-scale mixer than in the industrial mixer. This may be due to the convection provided by the blades, which is greater in the lab-scale mixer thanks to the higher filling ratio. When the couscous feeding is closer to the outlet than the semolina feeding, the semolina fraction in the cell just before the couscous feeding is 0.87 or 0.83. The semolina fraction in the preceding cells (closer to the inlet) is between 0.93 and 0.95. Conversely, when the semolina feeding is closer to the outlet than the couscous feeding, the semolina fraction in the cell just before the semolina feeding is 0.22 or 0.23. In contrast, the semolina fraction values in the preceding cells (closer to the inlet) are far apart: between 0.04 and 0.13. Indeed, when couscous is fed into cell 3 and semolina into cell 4, the semolina manages to re-circulate to cell 2, as the semolina grains can percolate axially between the couscous grains.

If couscous and semolina are fed together in cell 4, there is good back-mixing (Fig. 12 (b)), whereas if the powders are fed into different cells, the powder fed the closest to the inlet remains in the majority throughout the first part of the mixer (Figs. 13 and 14). This is the reason why the standard deviations are higher when powders are fed into different cells (Table 5). The four configurations where powders are fed into adjacent cells of the lab-scale mixer shows very close standard deviations: between 0.20 and 0.22. However, the standard deviation is always slightly higher when couscous is fed closer to the outlet than semolina. Once again, this may be due to the fact that the couscous cannot percolate axially through the semolina to move backwards and mix with semolina.

### 3.1.3. Powders fed into more distant cells

Fig. 14 shows the composition profile when powders are fed into more distant cells. As before, shifted feeding has little impact on the hold-up mass in the cell where the closest feeding to the outlet is located, and in subsequent cells up to the mixer outlet. In the industrial mixer, the composition profiles show that there are different phenomena depending on which powder is fed closer to the outlet, as shown in Fig. 14 (a), (b) and (c). When couscous is fed closer to the outlet than semolina, the semolina fraction in cells 1 and 2 lies between 0.90 and 1, showing once again that couscous hardly re-circulates to the industrial mixer inlet due to the presence of semolina. When the semolina is fed closer to the outlet than the couscous, the composition profile can be divided into 3 parts. The cells closest to the inlet are a stagnant zone in which the semolina fraction is between 0.08 and 0.14. The cell just

<caption>**Table 5**</caption>

Average mass fraction content in semolina over the five cells, as well as associated standard deviation when both powders are fed into adjacent cells. Above-the-mean values of the average content are highlighted in pink, under-the-mean values in light blue.

<table><thead><tr><th rowspan="3"></th><th colspan="4">Industrial mixer</th><th colspan="4">Lab-scale mixer</th></tr><tr><th colspan="4">S C</th><th colspan="4">S C</th></tr><tr><th>1</th><th>2</th><th>3</th><th>4</th><th>1</th><th>2</th><th>3</th><th>4</th></tr></thead><tbody><tr><th>Average</th><td></td><td></td><td></td><td>0.68</td><td></td><td></td><td>0.67</td><td></td></tr><tr><th>Standard deviation</th><td></td><td>0.13</td><td></td><td>0.16</td><td></td><td>0.22</td><td></td><td>0.20</td></tr></tbody></table>
before the couscous feeding has a semolina fraction between 0.38 and 0.52. The following cells have a semolina fraction close to that obtained with the other feed configurations. Thus, semolina re-circulates mainly in the cell preceding its feeding and a little in the preceding cells. The further apart the two powders are fed, the further the overall average semolina fraction is from 0.5. In the industrial mixer, the standard deviation is between 0.17 and 0.24 when powders are fed into distant cells (Table 6), whereas it is between 0.13 and 0.16 when powders are fed into adjacent cells (Table 5), and between 0.05 and 0.11 when powders are fed into the same cell (Table 4). Thus, the farther apart are the feeds, the higher is the standard deviation, and the poorer is the mixture quality in the mixer.

In the lab-scale mixer, as in the industrial-scale mixer, powders fed close to the outlet re-circulate mainly to the cell just before their feeding and slightly in the preceding cells. When semolina is fed into cell 2 and couscous into cell 4, the semolina fraction is 0.99 in cells 1 and 2, and 0.83 in cell 3, as shown in Fig. 14 (d). Conversely, when couscous is fed into cell 2 and semolina into cell 4, the semolina fraction is between 0.01 and 0.05 in cells 1 and 2, and 0.20 in cell 3. The standard deviation is slightly higher when couscous is fed closer to the outlet than semolina. This phenomenon is not found in the industrial mixer, which may be due to the fact that the grains do not move in the particle bed but above it, as the filling height is very low.

## 3.2. Analyzing outlet mixture characteristics

After analyzing the mixture quality within mixers at the scale of the cells, we will now focus on both the mixture quality at the outlet of the lab-scale mixer and the mass of powder “produced” during the transient start-up phase. Most of the time, this mass is regarded as an out-of-specification product and is discarded.

Fig. 15 shows the results when both powders are fed into the same cell. The more the powders are fed towards the outlet, the lower is the mass of mixture produced during start-up, which means that steady-state is reached quicker. The closer the feeders are located to the outlet, the shorter the geometrical residence time. It is 44.1 s, 43.3 s or 41.4 s when feeding into cell 2, 3 or 4 respectively. This means that, when fed close to the outlet, the powders spend less time in the mixer. It is possible that there is an optimum geometrical residence time for the powders to mix, without allowing sufficient time for them to de-mix afterwards. When the powders are fed into the same cell, there is apparently no link between the mixture state in the mixer described in the previous section and the mass of mixture produced during the transient start-up phase.

When the powders are fed into different cells (Fig. 16), the mass of mixture produced during start-up is higher than 900 g. This mass is therefore higher than when both semolina and couscous are fed into cell

3 or cell 4 (Fig. 15). When powders are fed into adjacent or more distant cells, the mass of mixture produced during start-up is higher when couscous is fed closer to the outlet than semolina. This phenomenon is more pronounced when powders are fed into distant cells. When there is little or no semolina in front of couscous particles, the couscous grains move very quickly towards the outlet as they are thrown by the blades, showing trajectory segregation. In addition, the couscous grains do not accumulate at the bottom of the mixer but roll over the powder bed. As a result, the mixture at the outlet is initially very rich in couscous and much time is needed before reaching a composition of 50 %, again resulting in a higher discarded mass. If the semolina feeding is located between the couscous feeding and the outlet, or at the same level as the couscous feed, it slows down the advance of the couscous grains, enabling the semolina and couscous grains to exit at similar times and thus to reach a composition of 50 % at the outlet more quickly. When the powders are fed into different cells, in order to limit the mass of mixture produced during start-up, it is preferable for the semolina feeding to be placed between the couscous feeding and the outlet, and for mixing to take place close to the outlet to reduce the risk of de-mixing, i.e. for the semolina to be fed into cell 4. Finally, the mass of mixture produced during start-up is very large if couscous is fed into cell 4, i.e. very close to the outlet.

The mixture at the mixer outlet must be at 50 % to reach steady-state. To reduce the mass of mixture produced during start-up, the first grains of semolina and the first grains of couscous must arrive at the mixer outlet nearly at the same time. Shifting the feeding of one of the powders alters the time at which the first grains arrive at the mixer outlet, specifically if fine particles are placed behind coarser ones.

Comparing the mass “produced” during start-up (Fig. 16) and the mixture state within the mixer at steady-state (Tables 5 and 6), we can see that the feed configurations with the highest RSDs on internal composition profile at steady-state are also the configurations with the lowest masses of mixture produced during start-up. Configurations in which semolina is fed into cell 4 show an RSD on internal composition profile of 74 % or 94 % and a mass of mixture produced during start-up of 940 g or 960 g on average. Conversely, configurations in which couscous is fed into cell 4 show an RSD on internal composition profile of 30 % and 33 % and a mass of mixture produced during start-up of 1130 g or 1140 g on average. It can be assumed that a high RSD on internal composition profile at steady-state is also high during transient state at start-up. This would mean that, despite very different compositions between the mixer cells, it is possible to reach steady-state quickly.

Fig. 17 shows homogeneity of the mixture at the outlet once steady-state is reached. The average RSD is between 11,1 % and 11,3 % when at least one of the powders is fed into cell 2, and between 11,6 % and 12,2 % when at least one of the powders is fed into cell 4. It seems that the

<caption>**Table 6**</caption>

Average mass fraction content in semolina over the five cells, as well as associated standard deviation when both powders are fed into distant cells. Above-the-mean values of the average content are highlighted in pink, under-the-mean values in light blue.

<table><thead><tr><th rowspan="3"></th><th colspan="8">Industrial mixer</th></tr><tr><th colspan="2">S</th><th colspan="2">C</th><th colspan="2">S</th><th colspan="2">C</th></tr><tr><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><th>Average</th><td colspan="2">0.76</td><td colspan="2">0.47</td><td colspan="2">0.78</td><td colspan="2">0.39</td><td colspan="2">0.86</td></tr><tr><th>Standard deviation</th><td colspan="2">0.17</td><td colspan="2">0.21</td><td colspan="2">0.18</td><td colspan="2">0.24</td><td colspan="2">0.18</td><td colspan="2">0.19</td></tr></tbody></table>

<table><thead><tr><th rowspan="3"></th><th colspan="8">Lab-scale mixer</th></tr><tr><th colspan="2">S</th><th colspan="2">C</th><th colspan="2">S</th><th colspan="2">C</th></tr><tr><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><th>Average</th><td colspan="2">0.76</td><td colspan="2">0.25</td><td colspan="2">0.78</td><td colspan="2">0.39</td><td colspan="2">0.86</td></tr><tr><th>Standard deviation</th><td colspan="2">0.25</td><td colspan="2">0.23</td><td colspan="2">0.18</td><td colspan="2">0.24</td><td colspan="2">0.18</td><td colspan="2">0.19</td></tr></tbody></table>
closer the powders are fed to the mixer outlet, the higher is the average RSD, and therefore the less homogeneous is the mixture obtained at steady-state. The average RSD at steady-state is around 11%–12% for all feed configurations. In order to determine whether the differences between the configurations are statistically significant, we consider that the 5 trials with conventional feeding represent a population following a normal distribution. The mean of the *n* = 5 trials is *μ* = 11.1% and the standard deviation is *σ* = 0.9%. Each configuration with shifted feeding represents one sample. A sample can be considered to belong to a population if the sample mean lies within the interval $[μ - z\frac{σ}{\sqrt{n}}; μ + z\frac{σ}{\sqrt{n}}]$ with *z* = 1.96 for a confidence level of 95%. Only the average of the points corresponding to feeding couscous into cell 2 and semolina into cell 4 does not belong to this interval. This means that only this configuration gives a mixture homogeneity at steady-state significantly different from that obtained with conventional feeding.

The RSD was determined for a sample size of 0.7 g, as this corresponds to the mass of powder present in an image. Since homogeneity is almost the same for all experiments, we wonder whether the scale of scrutiny is too large to discriminate between the different cases studied. Fig. 18 shows that increasing the scale of scrutiny makes the mixture appear better and better, especially around 0.7 g which is the scale of scrutiny previously used. This means that the resolution was sufficient to distinguish between different cases. This also emphasizes that defining the scale of scrutiny, which is related to the product's use, is the first step in any mixing study.

## 4. Concluding remarks

The composition profile in the mixer at steady-state does not seem to depend strongly on mixer size, stirring device and even particle size ratio, at least in the range reported here. When the powder feeders are placed in different cells, the internal composition profile depends mainly on the position of the feeder closest to the mixer outlet. The powders hardly re-circulate forward if another powder is already present, particularly when couscous is fed closer to the industrial mixer outlet than semolina, showing axial percolation difference of the particles, due to steric hindrance of couscous. This effect can sometimes be counterbalanced by trajectory segregation, which causes the couscous grains to be thrown further by the blades. Increasing the flow rate in the industrial mixer would increase the filling height and thus reduce the effect of the semolina layer accumulating at the bottom of the mixer because of radial percolation. This would bring the overall composition of the hold-up mass closer to 50%. The consideration of cells of large volume inside the mixers may hide some of the phenomena at play. It would therefore be interesting to evaluate homogeneity at a smaller scale of scrutiny, and more finely assess any differences between configurations. Segregation phenomena could also be studied in the case of a mixture of powders of different densities or shapes. For this, it would be necessary to consider adequate flow regimes, either cataracting to measure the influence of different densities or rolling to better measure a shape influence.

To ensure a low mass of mixture produced during start-up, it is preferable to feed the powders towards the mixer outlet to reduce the risk of de-mixing. It might be worthwhile increasing the agitation speed, as this would reduce the geometric residence time and therefore reduce the risk of de-mixing. It is also preferable to feed the powders into the same cell, or alternatively to feed the semolina (fine) closer to the outlet than the couscous (coarse) because of percolation issues, so that the first grains of semolina and the first grains of couscous arrive at the mixer outlet nearly at the same time. It would be interesting to combine a time shift with a space shift, by starting the feeding closest to the outlet a few seconds after the other feed. This would probably result in an internal composition profile far from 50% in all mixer cells, but our results show that despite the presence of very different compositions between the mixer cells, it is possible to reach steady-state quickly.

Shifted feeding could be modelled by taking axial and radial

percolation phenomena into account, as well as trajectory segregation, in order to highlight the different behaviors of powders in the absence or in the presence of another powder in the mixer. A DEM model integrated over a Markovian model would be useful to simulate grain displacements between the five cells of the mixer, considering axial segregation and segregation by trajectory thanks to the transition probabilities between the five cells. To account for radial segregation, a two-dimensional model would have to be developed, with, for example, a line of five cells representing the bottom of the mixer, another line corresponding to the inside of the powder bed and a third line for the surface flows. Simulations on a preliminary one-dimensional Markovian model show that it is possible to predict the composition of the mixture at the mixer outlet. This could make it possible to determine the time required to reach steady-state, and hence the mass of mixture produced during start-up. This will be part of future work from our team.

## CRediT authorship contribution statement

**Marie-Anne Batel:** Writing – original draft, Validation, Methodology, Investigation, Conceptualization. **Carla M. Barcelo Chong:** Methodology, Formal analysis, Data curation. **Cendrine Gatumel:** Writing – review & editing, Supervision. **Henri Berthiaux:** Writing – review & editing, Supervision.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

Data will be made available on request.

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