## 2 Computational Model  
The mixing process of powder detergent is carried out in a 3D mixer, shown in Figure 1, that replicates the Turbula(r) movement with a constant speed of 45 rpm. The components utilized to obtain the formulation are surfactant particles, sodium sulphate, sodium carbonate, coloured speckle, and liquid nonionic surfactant. Initially, every component is poured into the cup individually, except surfactant particles and liquid nonionic that are premixed before being poured. The cup has a capacity of 565 mL, and it was defined as having a filling grade of 70%. The properties of formulation components are shown in Table 1.  
The mixing process has been modelled by the DEM method, which is typically used to study the behaviour of individual particles when interacting with each other and boundaries. The particle's motion is governed by the equation of motion and the contact forces that act on them due to the interactions that occur with other particles and boundaries [6; 16]. These contact forces consist of normal force and tangential force, and they are calculated according to the contact model used in the DEM simulation. In this work, the hysteretic linear spring model was selected to calculate the normal force, the linear spring coulomb limit to calculate the tangential force, and a constant adhesive force model to take into account the adhesion due to liquid bridge forces. The gravity action was also taken into account. Rocky DEM 4.4.3 software was used to perform the computational simulation. From the geometry of the 3D mixer, only the parts that affect the mixing process were included in the DEM model. To model the powder components, spherical particles were used, and the liquid component was modelled as a liquid film added to the particles, and the liquid amount added to each particle can change when particles interact between them. Due to the particle size of the different components, the modelling of the mixture was expected to generate a high number of particles with a high computational cost. Thus, the Same Statistic Weight (SSW) method has been considered, according to [17], by which a scale factor is applied to the particle size of all the components, achieving a reduction in the particle number to be modelled but keeping the initial shape of the particle size distribution. Table 2 indicates the particle size of every component used in the DEM model, as well as the percentage by mass according to a standard formulation. The mechanical properties regarding interactions between particles, and between particles and boundaries, are presented in Table 3. The DEM model at the initial stage of the mixing process is shown in Figure 2, where a certain colour is assigned to every component. Initially, the powder components were totally segregated, and it was considered that the liquid component mass was equally distributed between particles of surfactant component. \begin{table}
\begin{tabular}{c c c} \hline
**Component** & **Particle Size Median (\(\mu\)m)** & **Bulk Density (kg/m\({}^{3}\))** \\ \hline Surfactant Particle & 443 & 830 \\ Sodium Sulphate & 207 & 1550 \\ Sodium Carbonate & 655 & 1150 \\ Coloured Speckle & 1000 & 800 \\ Liquid Nonionic Surfactant & - & 800 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Properties of formulation components. Figure 1: Geometry of the mixer. The Turbula(r) movement is a three-dimensional movement based on the Schatz six-revolute mechanism [19], which has been employed to mixing powders. To replicate this movement, which was analysed in [20; 21], an FEM model from the mixer geometry was created, and the parameterized position of every part during a cycle of movement was obtained by a rigid dynamics analysis. These position data were imported to the DEM software and applied to the DEM model. The movement described by the mixer model in each cycle is shown in Figure 3, representing the position of it every 45\({}^{\circ}\).