As shown by Bridgeman [21
], model validation can be carried out by comparing the real power consumption (
) of the mixer with that obtained from the numerical simulation. Empirical power input measurements are determined from applied torque measured by a torque transducer. Thus, the power consumption is:
represents the rotating speed and T
is the applied torque.
In the numerical simulation, the torque imposed on the impellers is calculated through the post-processing tools of the software, by summing the cross products of the pressure and viscous force vectors with the moment vector (the vector from the specified moment center to the force origin) for each face. It is also noteworthy to mention that we neglect additional (and unknown) power losses induced by gear boxes and the like. The on-site power consumption of the mixer is 18.1
while the obtained power consumption from the simulation is equal to 24.5
, which differs by about 26% from the real value for power consumption. This difference can, e.g., be due to uncertainties in the estimation of rheological characteristics. However, the difference is acceptable, based on the difference ranges calculated in [21
]. Another method also exists for calculating the power consumption, which is based on the energy dissipation rate. In this method, the overall power consumption is estimated by integrating the local power consumption numerically over the entire volume of the tank, so the power consumption of the mixer is calculated via:
is the density,
denotes energy dissipation rate, and
indicates the volume. The obtained numerical power consumption based on this method is equal to 22.3
(around 9% lower than the one obtained via surface integrals and differing by about 23.4% from the real value for power consumption). Although the calculated power consumption via the energy dissipation rate is closer to the on-site data, this method is not seen as accurate as calculating the numerical torque [39
]. Thus, we apply the numerical torque methods for the remainder of the simulations.
3.2. Grid Independence
To evaluate the mesh independence, three different sizes for the elements are analyzed based on [18
]. Consequently, a series of simulations are conducted, in which the numbers of elements are chosen as 101,894, 222,058, and 462,912. For each simulation, the velocity profile at the vertical line, crossing a specific point (located with a 90-degree angular distance from the mixer, and at the same radial distance as the mixer), was computed for each mesh structure, and then the results were compared to each other, according to [26
]. Figure 2
depicts the velocity profile along the mentioned vertical line for the three mesh networks.
It was observed that the value for the velocity magnitude in the model with 101,894 elements differed more than 5.3% from that obtained by the model containing 222,058 elements, while the data of the model with 222,058 elements differed only about 0.5% compared to the model with 462,912 elements. Therefore, the velocity profiles for the models with 222,058 and 462,912 elements were in a very good agreement. In addition, based on the method in [40
], the amount of the grid convergence index (GCI) was calculated for the central point of the obtained velocity profile. The variables φ1
, and φ3
denote the velocity magnitude at the mentioned point for the cases with 101,894, 222,058, and 462,912 elements, respectively (see Table 3
). The variable φij
, and the ratios of the grid refinement are shown by r21
. Then the order of convergence (p) is determined by iteratively solving the 11th equation in [26
]. By calculating the relative error between the two finest meshes (e32
), the GCI value, with a safety factor of 1.25, is calculated as almost equal to 0.02%. It was, thus, concluded that the model with 222,058 elements is suitable for the simulation.
3.3. Contours and Vectors
Velocity data of the model are captured from the simulation results. By Figure 3
, the variation of the velocity magnitude in the range of 0 to 0.5 m/s is illustrated along two central vertical planes and three horizontal planes located at the heights of 0.5 m, 1 m, and 1.5 m, respectively. Indeed, the flow within the digester tank is mixed by a rotating impeller, which is employed to prepare a uniform distribution of the organic material in all parts of the tank. The streamlines of the fluid are obtained and illustrated in Figure 4
. The streamlines are captured from 50 points and started from three planes located horizontally at the height of 0.5 m, 1 m, and 1.5 m.
Additionally, for better understanding of the fluid motion, the absolute velocity vectors are presented in Figure 5
in the three horizontal and two vertical planes. The vector map in the horizontal planes, as well as the streamlines in Figure 4
, show that the fluid rotates in circular shape and is also agitated well. Likewise, the vector map of the vertical planes illustrates some moving directions which are shaped diagonally. In fact, from these diagonal rotations it is inferred that the material within the AD tank is mixed in an appropriate manner.
In the current study, the definition of dead zones is chosen according to the definition in [41
], where the regions with the sludge velocities less than 5% of the maximum velocity are denoted as dead volume. As in our case, the maximum velocity is between 0.45 and 0.50 m/s (close to the mixer). Therefore, the 0.02 m/s velocity is selected as the threshold for identifying dead volume.
reveals that the velocity magnitude in almost all elements of the tank is above the threshold value of 0.02 m/s. We, hence, conclude that the feedstock within the digester has been agitated properly and the type and the situation of the mixer is appropriate to the evaluated model. As depicted by the velocity contours, the minimum magnitude for velocity is reached even in the vicinity of the central square column. This is due to the no-slip boundary condition of the wall surfaces.
3.4. Data Analysis
For assessing a time-dependent model, it is necessary to carry out a transient simulation and compare its results with the steady-state simulation. After simulation in transient conditions for about 46 min, the corresponding velocity fields are investigated for specified time steps. The velocity magnitude at a specific point (1 m above the bottom and 4.8 m far from the central column) is plotted as a timeline in Figure 6
The velocity magnitude at the probed point increases approx. linear from 0.04 m/s (after 24 sec) to 0.31 m/s (after 7 min). At this point in time, the slope of the timeline declines, reaching 0.33 m/s after 11 min. This is already close to the maximum velocity of 0.36 m/s at the probed point—as derived from the steady-state simulation. As the difference to the steady state value is less than 10%, it can be estimated that the time to reach the steady state at the probe point is 11 min.
In order to investigate the velocity field along with the height of the tank, the average velocities of seven horizontal planes located at heights of 0.25, 0.5, 0.75, 1, 1.25, 1.5, and 1.75 m are computed, and shown by Figure 7
It is observed that the values for averaged velocities stay within the range between 0.35 m/s and 0.43 m/s. The velocities at the central parts of the digester are higher than the ones closer to the bottom and the top surface. This is because of the location of the mixer, which is also at the central height (1 m above the bottom of the tank).
Employing user-defined memory (as declared in [16
]), the total volume of the cells with a velocity magnitude below 0.02 m/s (which are considered as stagnant zone) is calculated. For the TS concentration of 12.1%, the stagnant zone value is computed as 1.66 m3
or 0.47% of the volume, respectively.
In order to estimate the effect of the TS concentration on the dead zone volume, the simulation is performed for TS concentrations of 2.5%, 5.4%, 7.5%, and 9.1%, considering individually their specific rheological properties according to [38
]. Figure 8
demonstrates that as the TS concentration increases from 2.5 to 12.1 the total value for dead volume also increases from 0.20% (0.71 m3
) to 0.47% (1.66 m3
), respectively. This shows an increase of about 133%, which can be considered as a significant variation in the total volume of dead zones, although all of the obtained values for the dead volume are negligible as compared to the total volume of the fluid. It is concluded that by increasing the TS concentration, the volume of dead zones increases.
In addition, employing Equation 4, the consumed power by the mixer is estimated for each model with specific TS concentration. The achieved values for the power consumption in the models with different TS concentrations are depicted by Figure 8
It is indicated that by enhancing the TS concentration from 2.5% to 12.1%, the power consumed by the mixer, which is rotating at the constant angular velocity of 300 rpm, increases from 17.7 kW to 24.5 kW. This shows an enhancement of about 27.7% in power consumption. Note that the increase in power consumption is not linear to the increase of TS but rather exponential. This is because of the nonlinear effect to viscous forces for higher TS concentrations.
In many cases, the appropriate agitation has another advantage, i.e., avoiding sedimentation. In order to determine whether the mixing is sufficient to prevent sedimentation of heavy solids in the digester, the velocity gradient is calculated. Calculating the velocity gradient (G
) has become a fundamental approach within the water and wastewater industry to classify mixing tanks. According to [42
can be estimated with local energy dissipation rate:
is the turbulent energy dissipation rate in mass unit and
is the kinematic viscosity.
Sindall et al. [13
] have suggested that the velocity gradient should lie between 7.2 and 14.5 s−1
. In this research, G
is determined by post-processing of the achieved data (utilizing volume integrals in the software) as 26 s−1
. Thus, mixing does not avoid formation of the sedimentation layer and an additional sewage pump regime is necessary to improve the performance of the AD. A solution could be reducing the amount of the mixer rotation speed; however, due to structural limitations, reducing the mixing speed is not possible in our case.
In order to investigate a possible prevention of sedimentation, an alternative mixing scenario is tested by situating the mixer 0.25 m lower than its previous location, thus aiming for increased agitation close to the bottom. Subsequently, the mixing quality is analyzed by calculating the dead volume and the velocity gradient. However, for the similar mixing rotation speed of 300 rpm, the dead volume does not decrease but instead increases to 2.60 m3. While this is still only about 0.73% of the digester volume, we denote that the effect is obviously contrary to the intended one (a 36% increase in the amount of dead volume as compared to the case where the mixer is at the central height). For both the velocity gradient and the power consumption, no significant difference is found for this scenario.
Evaluating the effect of mixing speed on the velocity gradient allows one to determine whether it is viable to increase the mixer rotation speed instead of using a sewage pump. The fluid with total solids of 12.1% is analyzed while imposing rotation speed values greater than 300 rpm. Based on the infrastructures, it is considered that the mixer rotates at the angular velocities of 300, 350, 400, 450, and 500 rpm. Similarly, the effect of mixing speed on the dead volume is analyzed. Figure 9
depicts the effects of angular velocity on the dead volume and velocity gradient, respectively.
As to the effect of the mixer rotation speed on the dead volume (Figure 9
), it is shown that the values for the dead volume decrease from 1.66 m3
for a rotation speed of 300 rpm to 0.12 m3
for a rotation speed of 500 rpm. In general, the dead volume decreases as the mixer rotation speed increases, but the relation is non-linear. This is different for the velocity gradient as this value increases from 26 s−1
for the mixer with the rotation speed of 300 rpm to 56.5 s−1
for the mixer with the rotation speed of 500 rpm. As the minimum mixer rotation speed (300 rpm) already leads to higher values in velocity gradient, increasing the speed does not help in keeping the G
value in its optimum range. Moreover, when the mixer rotates with the speed of 500 rpm, the power consumption increases to about 159.9 kW (according to Equation 5), which is quite high for such a plant size. Indeed, the amount of power consumption at 500 rpm is more than six times the one required for 300 rpm. For different mixer rotation speeds, the power consumption is summarized in Table 4