# Thermal Hydrolysis of Sewage Sludge: A Case Study of a WWTP in Burgos, Spain
## Appendix A Mass and Energy Balance of the Thermal Hydrolysis Unit  
The mass and energy balances are represented in Equations (A1) and (A2) for the THP. These equations correspond to the \"_external equations_\" of the THP, associated with the control volume denoted by dotted lines in Figure 2. These equations were used to calculate values from the average cycle. THP was considered a single entity and only those streams interacting with the control surface were indicated in these equations. \[\text{M}_{\text{TS-pt}}+\text{M}_{\text{water-pt}}+\text{M}_{\text{Is}}=\text{ M}_{\text{TS-flash-tank}}+\text{M}_{\text{water-flash-tank}}+\text{M}_{\text{ pg}}\] (A1)\[(\text{M}_{\text{TS-pt*}}\text{C}_{\text{P-TS}}+\text{M}_{\text{water-pt*}}\text{C}_{ \text{P-water}})\;*\;\text{T}_{0}+\text{M}_{\text{Is}}*\text{h}_{\text{Is}}=( \text{M}_{\text{TS-flash-tank}}*\text{C}_{\text{P-TS}}\] \[+\text{M}_{\text{water-flash-tank}}*\text{C}_{\text{P-water}})\;*\; \text{T}_{\text{flash-tank}}+\text{M}_{\text{PS}}*\text{h}_{\text{PS}} \tag{1}\]  
The inlet stream to the THP corresponds to thickened sludge coming from the post-thickening section. Streams were assumed to be composed of total solids (TSs) and water, using this nomenclature as a suffix. For the mass balance M represents mass flow, \(\text{M}_{\text{TS-pt}}\) is, therefore, TS mass flow of sludge stream entering the pulper (which comes from the post-thickening section (suffix pt used)) and \(\text{M}_{\text{TS-flash-tank}}\) is the outlet sludge stream from the flash-tank (kg/h). Similarly, for the water contents of these streams (\(\text{M}_{\text{water-pt}}\) and \(\text{M}_{\text{water-flash-tank}}\)), \(\text{M}_{\text{Is}}\) is the mass flow of the live steam injected into the reactor (kg/h), and \(\text{M}_{\text{PS}}\) is the mass flow of process gases (kg/h). T\({}_{0}\) and \(\text{T}_{\text{flash-tank}}\) are the temperatures of the sludge stream entering the pulper (ambient temperature) and that of the flash-tank outlet stream (\({}^{\circ}\)C). \(\text{h}_{\text{Is}}\) and \(\text{h}_{\text{PS}}\) are the specific enthalpies of live steam and process gases, respectively (kJ/kg). These equations allow the calculation of \(\text{M}_{\text{Is}}\) and \(\text{M}_{\text{water-flash-tank}}\) given that the other variables are known parameters associated with the hydrolysis process. Specific heat of water and biosolids are represented by C\({}_{\text{P-water}}\) and C\({}_{\text{P-TS}}\) (kJ/kg\({}^{\circ}\)C). Equations (A3) and (A4) are the mass and energy balances for the flash-tank. \[\text{M}_{\text{TS-reactor}}+\text{M}_{\text{water-reactor}}=\text{M}_{\text{TS- flash-tank}}+\text{M}_{\text{water-flash-tank}}+\text{M}_{\text{fs-flash-tank}} \tag{2}\]  
\[(\text{M}_{\text{TS-reactor}}*\text{C}_{\text{P-TS}}+\text{M}_{\text{water- reactor}}*\text{C}_{\text{P-water}})\;*\;\text{T}_{\text{reactor}}=(\text{M}_{\text{TS-flash-tank}}*\text{C}_{\text{P-TS}} \tag{3}\]  
\[+\text{M}_{\text{water-flash-tank}}*\text{C}_{\text{P-water}})\;*\;\text{T}_{ \text{flash-tank}}+\text{M}_{\text{fs-flash-tank}}*\text{h}_{\text{fs-flash-tank}} \tag{4}\]  
Nomenclature is similar to that of the previous balance, using \(\text{M}_{\text{TS}}\) and \(\text{M}_{\text{water}}\) to represent TS and water mass flows. Suffix represents either the equipment evaluated or the origin of the incoming stream. Suffix fs denotes flash steam. The unknown variables to be calculated are, in this case, \(\text{M}_{\text{water-reactor}}\) and \(\text{M}_{\text{fs-flash-tank}}\). Mass and energy balances for the reactor are presented in Equations (A5) and (A6). Variables are denoted using a similar nomenclature to that of previous balances. The unknown variables are \(\text{M}_{\text{water-pulper}}\) and \(\text{T}_{\text{pulper}}\); thus, it is necessary to consider the effect of process gases based on Equations (A10) and (A11) to estimate values for \(\text{M}_{\text{fs-reactor}}\). This last stream contains VS and water as main components.