Convergence to pseudo-steady state

Attainment of pseudo-steady state condition must be assessed to calculate power number and flow number when the flow is fully developed. Since the point comparison of velocity values of subsequent time iterations is sensible to velocity fluctuations due to blade frequency, criteria based on comparison of volume averaged quantities have been developed. Three different criteria are used in this work to assess convergence: (i) upflow profile stability, (ii) azimuthal momentum stability and (iii) power consumption stability.

Upflow is defined as the integral of upward directed fluxes across a section normal to the rotation axis. This parameter is used to measure the stirring capability of the reactor in the vertical direction (secondary circulation flow number). High values for secondary circulation number are necessary for reactors used to produce solid suspensions. Upward and downward fluxes balance at each time step for a given section as a consequence of mass conservation. If upflow is considered for a number of sections at different distance from the impeller, as shown is Figure A.1, an upflow profile is obtained along the rotation axis (see Figure A.2).

Figure A.1: Upflow regions for sections normal to the rotation axis: $v_z$ is upward directed in red regions. Section heights are $0.5$, $1.$, $1.5$ and $2.~m$ from cylindrical bottom.

Figure A.2: Upflow profile along the rotation axis: evolution in time.

At starting time, the main part of the fluid in the vessel is at rest and only the portion nearer to the impeller moves. As the flow develops, reaching the wall of the tank and the baffles, fluid streamlines are deflected upward and motion is progressively transferred to regions far from the impeller. The gradual movimentation of the fluid within the vessel can be observed from the evolution in time of upflow profiles. The upflow profiles gradually grows up far away from the impeller, until a steady state is reached in which no more perceivable variations are observed. This indicates that a steady state is achieved. Information obtained with this criterion were compared with the other two used in the work, and we found that convergence on upflow profile is a first but not ultimate measure of convergence to fully developed flow field. More detailed information about steady state achievement are obtained by criteria (ii) and (iii).

Variation in time of azimuthal momentum is used in conjunction with variation in power consumption to determine convergence to pseudo-steady state. Azimuthal momentum is used to assess convergence on primary azimuthal flow. Azimuthal momentum is defined as the volume integral of azimuthal velocity. Variation in time of this variable account for modification of the azimuthal structure of the flow. In the region near to the impeller rotation of the fluid is rapidly established. In the region far from the impeller (near the free surface) development of the radial motion is delayed until the flow field completely develops in the upward direction. Thus, azimuthal momentum variations are expected to be rather discontinuous in time. For each simulation azimuthal momentum is computed through time as reported in Figure A.3.

Figure A.3: Azimuthal momentum variation through time.

Figure A.4: Power consumption variation through time.

Variations in azimuthal momentum are found to correspond to variation in power consumption, confirming that azimuthal velocity distribution can deeply affect power consumption. Power consumption is calculated as explained in Section A.2.1 for different times. A curve representing the power variation in time is shown in Figure A.4. A larger value corresponding to static friction can be observed at starting time. Subsequent gradual modification in power consumption corresponds to extension of the rotational flow out of the impeller region (first steep increase) and to the upward development of the fluid flow in the vessel (first constant value region). The steep change in the final part of the graph ($80\%$ of the previous plateau) is due to the variation of azimuthal flow, as confirmed by direct comparison of Figure A.3 and  A.4.