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The simulations were performed with the commercial software ANSYS FLUENT with the versions 6.3.26, 12.0.16, 12.1.2 and 13.0, which are comparable. An overview of the type of solver used by ANSYS can be found in (Kelecy, 2008). The solver used is an unsteady, segregated pressure-based solver. The segregated pressure-based algorithm solves the conservation equations sequentially. Because these equations are coupled, the solution loop must be carried out iteratively in order to obtain a converged numerical solution. Pressure and velocity were coupled either with the SIMPLE algorithm, Semi-Implicit Method for Pressure Linked Equations, or with the PISO schema, which stands for Pressure Implicit with Splitting of Operators and is available for unsteady calculations. The PISO pressure-velocity coupling algorithm has one predictor step and two corrector steps, extending the pressure correction procedure of the commonly used SIMPLE algorithm with a further corrector step (Versteeg & Malalasekera, 2007). The SIMPLE algorithm is a guess-and-correct procedure which uses a Poisson’s equation, developed from the continuity condition, for the pressure correction.

4. Simulated flow in solid bowl centrifuges

The previously published simulations of a tubular bowl centrifuge were carried out two-dimensionally (Spelter & Nirschl, 2010); hence it was not possible to include the real acceleration geometry in the model. For the work presented, all simulations were run three-dimensionally, including the acceleration and discharge geometry, and were compared with experimental data. The flow pattern was predicted for a tubular bowl centrifuge with a feeding system similar to the one used in a recent published work (Spelter, et al. 2011). The experimental data has been obtained by Laser Doppler Anemometry, a non-invasive technique for the determination of flow velocities.

4.1 Prediction of the flow pattern (k-e RNG model, transient solution) - tandem setup Figure 5 shows the tangential velocity profiles for four different rotational speeds at a constant throughput of 1 l/min. There is no significant deviation between the predicted tangential velocity and the rigid body rotation (RBR). The measurement of the tangential velocity for various rotational speeds and throughputs at different heights was in good agreement with the rigid body rotation as well (Spelter, et al. 2011). Thus it can be stated that the simulation is in good agreement with the experimentally determined tangential velocity profiles. Therefore it is possible to optimise existing acceleration geometries by using a feeding system similar to the one proposed. By calculating the present state and modifying the geometry, the changes in the acceleration efficiency can be evaluated by

means of CFD. Furthermore the calculation will help in the case of a scale-up of the centrifuge. The acceleration efficiency and the shear in the feed zone can be calculated before a pilot machine or a full scale centrifuge is built. With the knowledge of the flow behaviour, the geometry can be optimised, leading to an improved apparatus with less cost in development and testing.

Fig. 5. Comparison of calculated tangential velocity and rigid body rotation, 1 l/min

The tangential velocity of the fluid creates a pressure in the cylinder in addition to the stress in the rotor created by the rotating cylinder itself. For the case that the rotor is designed as an ideal cylinder, the pressure distribution and thus the tensile stress can be calculated analytically (Sokolow, 1971; Stahl, 2004). However in many industrial centrifuges, the rotor has a non-cylindrical shape. For these designs, the stress in the rotor is determined via finite element methods (FEM), but the arbitrary load by the rotating liquid has to be calculated separately. The geometry of the rotor can be imported from various CAD programs such as ProEngineer, Autodesk Inventor, Catia, etc.. The pressure on the different planes in the bowl is then accessible by calculating the pressure distribution for the highest rotational speed that occurs in the centrifuge. The combination of FEM and CFD leads to a complete virtual development and saves time and costs. The simulations will never substitute completely experimental evaluation of the separation efficiency and mechanical integrity of the centrifuge, but the scale-up factor may be significantly increased. The predicted pressure distributions in the cylinder are shown in Figure 6 for 9000 rpm (a) and for a range of rotational speeds in (b). The outlet is located at the top end of the cylinder. The predicted pressures in Figure 6 b) are compared with the analytically calculated pressure distribution for the different rotational speeds. All values are given in bar.

The gauge pressure must be taken into account for dimensioning the rotor. Furthermore it is possible to increase the density of the liquid to model the arbitrary load of a sediment on the bowl wall.

Fig. 6. Pressure distribution, all values in bar: a) 9000 rpm, pressure b) 1000-14000 rpm and comparison to analytical values

The axial velocity profiles are displayed as contour plots in Figure 7 for 1000 rpm (a) and 3000 rpm (b) at a constant throughput of 1 l/min. The inlet is located at the bottom while the liquid leaves the centrifuge at the top. The incoming liquid enters the centrifuge as a jet that does not widen up significantly towards the outlet. At 1000 rpm, the main axial flow takes place at a small circular cross-section that is not displaced in angular direction on the path towards the outlet bores. With increasing rotational speed, a small angular displacement of the axial flow path is predicted, as shown in Figure 7 b). This effect has been observed by Laser Doppler Anemometry as well.

Fig. 7. Axial velocity in m/s, throughput 1 l/min: a) 1000 rpm b) 3000 rpm

However, the predicted axial flow pattern does not depict the experimentally determined flow behaviour accurately. It has been observed that the inlet jet widens within the first 30 % of the rotor (Spelter, et al. 2011) until a parabolic profile is developed that is distributed over the whole circumference. The predicted axial flow profile close to the inlet is in good agreement with the experimental data but the change of flow pattern in direction of the angle of rotation is not correct. The axial flow profiles averaged over the length of the centrifuge between 1000 and 14 000 rpm for two angles of rotation and a constant throughput of 1 l/min are shown in Figure 8. The standard deviation is plotted in both diagrams. The scatter of the data is very high for the angle where the inlet is located, which is 0°. The diagram on the right side of Figure 8 shows the axial flow profiles for an angle of rotation of ±45° displacement to the inlet. There is no significant flow towards the exit of the rotor, only recirculating currents are predicted. The standard deviation of the data is small when compared to the angle of 0°. The origin of the high standard deviation of the data displayed in the diagram on the left in Figure 8 is caused by the change in flow pattern from the feed towards the liquid discharge. The radius of the inlet is higher than the radius of the outlet. Hence the main flow is shifted towards the rotational axis. These effects are displayed in Figure 9 for 1000 rpm (left) and 3000 rpm (right).

Fig. 8. Axial velocity, mean values from throughout the rotor: left: radius of inlet; right:

angular position ± 45°

The maximum of the axial velocity is close behind the inlet (H=0mm) at a radius of 42 mm, which corresponds to the inlet radius. Towards the liquid discharge, the maximum shifts to a radius of 37 mm, where the outlet pipes are located. The change in flow pattern is indicated by the arrows in Figure 9. This behaviour was detected for all rotational speeds and different throughputs. The changing flow profile explains the high standard deviation of the averaged profiles shown in the diagram on the left in Figure 8.

The deviation between simulation and experimental data with respect to the flow profile over the full angle of rotation may be explained by the turbulent model used. The modelling does not calculate the turbulences in detail, but estimates the degree of turbulence and thus extenuates the fluctuations at the inlet and outlet. Furthermore the flow between inlet and outlet may not be entirely turbulent, as mentioned in the introduction.

Fig. 9. Axial velocity, radius of inlet for different distances from inlet: left: 1000 rpm; right:

3000 rpm