## 5 Conclusions  
This work aimed to quantify the influence of non-volatile media on rheological properties of TBS and to better explore the response of thermophilic sludge in case of prolonged shear rate. The results show a significant dependence of TBS from sodium bentonite concentration, while calcium carbonate and sand produced relatively lower influence. TBS response to diverse shear rates for longer period highlighted a thixotropic behaviour which is much pronounced at higher shear rate (1000 s\({}^{-1}\)). Double exponential fitting model was the best choice in terms of interpolation accuracy of the results and mathematical complexity for describing thixotropic behaviour in case of low (100 s\({}^{-1}\); AIC: \(-\)39.826) and high shear rate (1000 s\({}^{-1}\); AIC: 89.300), while single-exponential model represented the best option in case of medium shear rate (400 s\({}^{-1}\)) but with limited difference with double-exponential model (\(-\)55.641 vs. \(-\)41.035).  
This study represented a crucial step in gaining insight into the rheological properties of TBS, with special reference to thixotropic behaviour. Future research is needed to clarify a few aspects that have been pointed out in this work. The acquired experimental findings will be useful for managers of high-strength wastewater treatment plants, providing important information for improving the process efficiency.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app12105198/s1](https://www.mdpi.com/article/10.3390/app12105198/s1)]([https://www.mdpi.com/article/10.3390/app12105198/s1](https://www.mdpi.com/article/10.3390/app12105198/s1)), Section: Dependence of rheological parameters from the thermophilic sewage sludge composition; Figure S1: Shear stress of TBS sampled from (a) plant 1 and (b) plant 2 as a function of imposed shear rate. Section: Dependence of rheological parameters from the type of NVS; Figure S2: Viscosity as a function of shear rate during TBS conditioning with (a) calcium carbonate, (b) sand, and (c) sodium bentonite. Unconditioned situation is represented by 180 g L\({}^{-1}\) of TS concentration.  
Conceptualization, M.C.C., M.C.M., M.B., S.T. and S.M.; methodology, M.C.C., S.B., M.C.M., M.B., S.T. and S.M.; formal analysis, S.B.; investigation, S.B. and S.C.; resources, M.C.C., S.T. and S.M.; data curation, S.B.; writing-original draft preparation, S.B., F.M.C. and S.C.; writing-review & editing, S.T., M.C.M. and S.M.; visualization, F.M.C. and S.C.; supervision, M.C.C., M.B., S.T. and S.M.; project administration, M.C.C.; funding acquisition, M.C.C. All authors have read and agreed to the published version of the manuscript.  
The authors wish to thank ASMia S.r.l. (ASMortara S.p.a. group) for providing financial support for this research.  
Not applicable.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**AIC Index**} \\ \hline
**Shear Rate (s\({}^{-1}\))** & **Single Exponential** & **Double Exponential** & **Triple Exponential** \\ \hline
100 & \(-\)19.859 & \(-\)39.862 * & 14.883 \\
400 & \(-\)55.641 * & \(-\)41.035 & \(-\)26.070 \\
1000 & 102.200 & 89.300 * & 132.827 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Values of AIC index for different models in diverse shear rate conditions. (*): best conditions.  
Data Availability Statement: All data generated or analysed during this study are included in this published article.  
The authors wish to thank ASMia S.r.l. (ASMortara S.p.a. group) and Idroclean S.r.l. (Itelyum S.p.a. group) for their support during the research activities.  
The authors declare no conflict of interest.