## 2 Materials and Methods
### Experimental Procedure  
To evaluate the effect of increasing TS concentration on rheological behaviour, TBS was conditioned with calcium carbonate, sand, and sodium bentonite that were separately added. Calcium carbonate (particle size 1-3 \(\upmu\)m) and sodium bentonite (particle size 35-105 \(\upmu\)m) in powdered form were purchased (Heiltropfen, London, UK) while sand (particle size 0.16-0.24 mm) was sampled as a grit removal residue in a full-scale wastewater treatment plant (WWTP). Sand was washed with distilled water to avoid the presence of organic matter; it was subsequently dried at 105 \({}^{\circ}\)C in an oven (Mermert, Schwabach, Germany) to avoid the presence of water which could alter the test.  
TBS was conditioned by adding NVS from 90 kg m\({}^{-3}\) up to 190 kg m\({}^{-3}\) of concentration. After that, the conditioned TBS was mixed for 4 h at 50-80 rpm to completely homogenize the sample before performing the rheological measurement. The value of VS, on the other hand, remained unchanged compared with the unconditioned sample.  
Rheological tests were performed using RC20 coaxial cylinders rheometer (RheoTec, Dresden, Germany) with a CC25 DIN configuration (i.e., spindle radius 12.5 mm; stator internal radius 13.56 mm). The following shear rate values were tested: 25 s\({}^{-1}\), 50 s\({}^{-1}\), 100 s\({}^{-1}\), 200 s\({}^{-1}\), 400 s\({}^{-1}\), 600 s\({}^{-1}\), 800 s\({}^{-1}\), and 1000 s\({}^{-1}\). The experimental procedure to acquire data for evaluating the flow curve of the testing sludge sample was carried out by maintaining each shear rate value for 150 s before increasing it at the next measurement step. Shear stress and apparent viscosity were acquired every 15 s (i.e., ten datasets for each share rate step). The shear stress values measured at each time and plotted in the following figures should not be intended as instantaneous values as they represent the average of the measures taken within the corresponding sampling time interval of 15 s. The mean and confidence interval of the shear stress measurements were calculated for each applied shear rate. Moreover, before carrying out the shear stress measurement at different shear rate steps, each sludge sample was preliminary subjected to a fixed shear rate (i.e., 400 s\({}^{-1}\)) for a constant time (i.e., 300 s) to cause structure breakdown and reduce the possible thixotropic influence on the rheological measurements.  
In a subsequent step, to evaluate a possible thixotropic property of TBS, the shear stress and the apparent viscosity were recorded at three diverse shear rates (100 s\({}^{-1}\), 400 s\({}^{-1}\), and 1000 s\({}^{-1}\)), each of which has been maintained for a suitably long-time interval (i.e., 1200 s). Shear rate values have been selected to provide an overview of the sludge behaviour within the investigated range. These values have been considered sufficient to obtain suitable information at this early stage of investigation, while the number of investigated shear rate values will be enhanced in subsequent studies.  
It must be pointed out that other experimental techniques can be applied for the thixotropic analysis of a non-Newtonian fluid, such as the 3-intervals thixotropy test (3 ITT) from the multiple interval thixotropic test protocol (miTT) [37]. The 3 ITT test can be conveniently used for applying an instant shear stress/shear rate deformation in order to simulate effects of pumping and stirring during food processing [38]. In any case, common full-scale applications of TBS in wastewater treatment plants involve overall steady state operating conditions with negligible shear stress/shear rate mean local variations.  
To reproduce full-scale operating conditions, the temperature during all tests was kept at 48 \(\pm\) 2 \({}^{\circ}\)C using a thermostatic bath (VELP Scientifica, Usmate Velate, Italy).