Modeling and Numerical Study of Ceramic Paste Extrusion

The extrusion processes of ceramic pastes, including 3D printing, are used for the production of high-value products. Ceramic paste extrusion is a complex process which depends on the paste rheological properties, die and extruder geometries, and operational parameters. Modeling and quantitative analysis of paste molding are important to design proper extrusion process for the production of high-value extrudates of desired strength, shape, and morphology. In this paper, the mathematical model of ram extrusion of ceramic materials is established, and the paste continuity and momentum equations for non-Newtonian fluid based on the modified Herschel-Bulkley viscous model were solved numerically. The effects of die geometry and paste feed rate on the distributions of paste velocity and pressure in the extruder and die were investigated numerically. As a result, the steeper radial profile of longitudinal velocity and higher value of longitudinal velocity were obtained in the narrow die. The pressure significantly increases in the die at a high feed rate, and the pressure profile is almost flat in the barrel. The rate of increase of the maximum pressure decreases with an increase of paste feed rate. The pressure steeply increases in the die of small diameter. The maximum pressure linearly increases with the ratio of die length to diameter.


Introduction
The extrusion process of ceramic pastes is commonly used for the production of high-value products, e.g., catalyst pellets for the chemical reactor (Devyatkov et al., 2016), honeycomb catalyst for purifying gas exhausted from an automobile (Govender and Friedrich, 2017), and ceramic packings for adsorption and direct heat transfer (Darakchiev et al., 2016). Recently, extrusion-based 3D printing was successfully applied for manufacturing of ceramic superconductors, zeolite monoliths, and porous scaffolds (Chen et al., 2019;Coucke et al., 2018;Vaezi et al., 2018). Ceramic paste extrusion is a complex process which depends on the paste rheological properties, die and extruder geometries, and operational parameters (Benbow and Bridgwater, 1993). The paste rheological properties are controlled by several factors including volume fraction of particles and their size distribution and shape, packing density and surface characteristics, as well as amount and properties of binder and other additives (Powell et al., 2013). Modeling and quantitative analysis of paste extrusion are important to design properly the extrusion process for the production of high-value extrudates of desired strength, shape, and morphology.
The objective of this paper is to establish appropriate mathematical model of ram extrusion of ceramic materials and use it for the optimization of the extrusion process. The paste extrusion will be modeled as a flow of non-Newtonian fluid. The paste continuity and momentum equations will be solved numerically using the modified Herschel-Bulkley viscous model. The effects of die geometry and paste feed rate on the distributions of paste velocity and pressure in the extruder and die will be investigated numerically.

Incompressible Navier-Stokes equations for non-Newtonian fluid
The modified Navier-Stokes equations for incompressible non-Newtonian fluid are formulated in the material domain d Ω ⊂ R , d=2,3, and in the time interval as (Roubicek, 2009) where v and p denote the unknown velocity and pressure, respectively, ρ stands for the paste density, f is the bulk force, and d is the spatial dimension. Here, τ is the viscous part of the stress tensor, depending on the symmetrized velocity gradient ( ) The model Eq.
(1) is complemented by the initial condition and the no-slip boundary condition 0, Other types of boundary conditions can also be prescribed, e.g., the Neumann boundary condition on the outflow part of the domain boundary. In the isothermal case, the momentum equation reads as follows The stress tensor can be represented as a combination of the volumetric and deviatoric stresses as whereas the deviatoric stress is given by Here, µ is the dynamic viscosity, which depends in general on the equivalent strain rate γ , and ε stands for the strain tensor defined as where s ∇ v is the symmetric part of the velocity gradient.
The equivalent strain rate γ and the equivalent deviatoric stress τ are defined as ( ) In the case of non-Newtonian fluid, the viscosity depends on the deformation process. To this end, various models for viscoplastic fluids have been established. In the present paper, we consider the Herschel-Bulkley model, which combines the existence of the yield stress, y τ , with the power law model for the viscosity (Herschel and Bulkley, 1926).
where k is the consistency parameter and n is the flow index. Thus, the deviatoric stress tensor is given by In the case when the rate of deformation tends to zero, there is a singularity in Eq. (6). In order to solve the model equations numerically, the regularized version of the Herschel-Bulkley model should be considered (Papanastasiou, 1987;dos Santos et al., 2011). Here, the modified version of the Herschel-Bulkley model is used as given by Li et al. (2013b):

Steady-state axisymmetric Extrusion Model Equations
In the present study, the ram extruder is considered as an axisymmetric domain. In this case, the steady-state extrusion model equations for momentum transfer in the radial and longitudinal directions, and mass conservation are formulated as follows: where r u and z u stand for the radial and longitudinal components of the velocity field v , respectively. Notice that the momentum equation for the angular direction is neglected due to the assumed axisymmetry. In this case, the relations ( )

Results and Discussion
In the present paper, the numerical solutions to the boundary value problem for Eq. (8) were obtained using the Finite Element package Comsol Multiphysics ® (COMSOL, 2018). The extruder geometry is shown in Figure 1. To study the effect of the ratio of die length to die diameter ( ) / L φ on extrusion characteristics, the die diameter was fixed at 3.18 mm and the die length was varied as 6.36 mm ( / 2 L φ = ), 12.72 mm ( / 4 L φ = ), 19.08 mm ( / 6 L φ = ), and 25.44 mm ( / 8 L φ = ). To investigate the effect of die diameter, the dies of a fixed length of 12.7 mm and varied diameters of 1.59 mm, 2.19 mm, 3.18 mm, and 6.35 mm were used in simulations. The viscosity experimental data by Thomas-Vielma et al. (2008) for the paste containing 50 vol. % of alumina (Alcoa, CT 3000, d50=0.8 µm), 25 vol. % of high-density polyethylene, 23 vol. % of paraffin wax, and 2 vol. % of stearic acid were fitted with the modified Herschel-Bulkley model by Eq. (7). The fitted parameters are summarized in Table 1, and the fitting results are illustrated in Figure 2.  The radial distributions of the longitudinal component of the velocity, z u , at the die end ( 0 z = ) are shown in Figure 3 for dies of various diameters. The extruder feed rate was kept constant at 0.002 kg/s. The radial profile of the longitudinal velocity is steeper in a narrow die.
The axial distributions of the longitudinal velocity at the die centreline are shown in Figure 4 for dies of various diameters. The longitudinal velocity is higher in the die region of the ram extruder with narrow die due to the larger average velocity at the constant feed rate.
The axial distributions of pressure at the die centreline are illustrated in Figure 5 for various feed rates. The pressure significantly increases in the die at a high feed rate and the pressure profile is almost flat in the barrel. The variation of the maximum pressure at the upper end of a barrel with feed rate is demonstrated in Figure 6. The rate of increase in maximum pressure decreases as the feed rate increase.  The effect of die length to diameter ratio on the axial pressure distribution is shown in Fig. 7. The pressure steeply increases in the die of small diameter. The maximum pressure linearly increases with the ratio of die length to diameter, as illustrated in Fig. 8.

Conclusions
The extrusion of ceramic materials for the production of high-value materials was studied in the present research. The mathematical model of ram extrusion was established, and the paste continuity and momentum equations were solved numerically for non-Newtonian fluid based on the modified Herschel-Bulkley viscous model. The effects of die geometry and paste feed rate on the distributions of paste velocity and pressure in the extruder and die were investigated numerically. As a result, the steeper radial profile of longitudinal velocity and higher value of longitudinal velocity were obtained in the narrow die. The pressure significantly increases in the die at a high feed rate, and the pressure profile is almost flat in the barrel. The rate of increase in maximum pressure decreases as the paste feed rate increases. The pressure steeply increases in the die of small diameter. The maximum pressure linearly increases with the ratio of die length to its diameter.