Tannins from Norway spruce (Picea abies [L.] Karst.) and Scots pine (Pinus sylvestris L.) bark were extracted with water at different temperatures (60–140 °C) in an ASE-350 system in order to optimize yield. In addition, the effect of chemicals such as urea, sodium bisulfite (NaHSO3), sodium carbonate (Na2CO3), and sodium benzoate on the yield was also investigated. Bark from debarking processes at both a sawmill and a pulp mill were included. The highest overall yield expressed as total dissolved solids (TDS) was obtained with hot water extraction of spruce bark at 140 °C. The TDS was 117 mg/g and it contained 47 mg/g tannins. With an increase in extraction temperature over 100 °C, the proportion of tannins decreased, whereas the proportion of carbohydrates increased. The addition of sodium carbonate improved yield within a 60–90 °C temperature range compared with pure water. Other chemicals did not improve the yield. Pine bark showed similar extraction yields to spruce bark but the proportion of tannins was lower in spruce than in pine. Pure water at 110 °C was chosen to be used for piloting in larger scale 300-liter extraction vessel. Based on the results, a machine-learning approach was applied using seemingly unrelated regression models (SUR). The models were able to predict the extracted tannin yields of spruce and pine bark when extractions were scaled up to 2 liters and then to 300 liters.