1. Conservation of Mass Problem

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3. Mass balance equation:
Class Problem: A large fishtank system (similar to yours, but much bigger) is used to store 773kg of water that is 0.15% wt NaCl. We wish to increase the salinity to 0.25% by adding salt water from a “salty tank” with a 5% wt NaCl solution. When water from the “salty tank” is added, an equal volume of water must leave the chamber through the overflow. Assume that the same mass of water that enters from the “salty tank” leaves through the overflow. If 20% of the water leaving through the overflow comes directly from the “salty tank” and the remainder that leaves is 0.15% wt NaCl, then what is the mass of “salty tank” water that should be added?
Mass balance equation:
Initial:
773𝑘𝑔
0.15% NaCl
99.85% water X
5% NaCl
95% Water
20% 𝑋
5% NaCl
95% water
𝑚 𝑖𝑛 − 𝑚 𝑜𝑢𝑡 + 𝑚 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 − 𝑚 𝑐𝑜𝑛𝑠𝑢𝑚𝑒𝑑 = 𝑚 𝑓𝑖𝑛𝑎𝑙 − 𝑚 𝑖𝑛𝑖𝑡𝑖𝑎𝑙
80% 𝑋
0.15% NaCl
99.85% water
Final:
773𝑘𝑔
0.25% NaCl
99.75% water
𝑚 𝑖𝑛 − 𝑚 𝑜𝑢𝑡 = 𝑚 𝑓𝑖𝑛𝑎𝑙 − 𝑚 𝑖𝑛𝑖𝑡𝑖𝑎𝑙
For simplicity, you could bring the “in” and “initial” to one side of the equal sign and “out” and “final” to the other side. This allows for all terms to remain positive.
𝑚 𝑖𝑛 + 𝑚 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝑚 𝑓𝑖𝑛𝑎𝑙 + 𝑚 𝑜𝑢𝑡
Overall equation:
Salt equation:
𝑚 𝑖𝑛 + 𝑚 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝑚 𝑓𝑖𝑛𝑎𝑙 + 𝑚 𝑜𝑢𝑡
𝑚 𝑖𝑛 + 𝑚 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝑚 𝑓𝑖𝑛𝑎𝑙 + 𝑚 𝑜𝑢𝑡
𝑋+773𝑘𝑔=773𝑘𝑔+0.2X+0.8X
0.05𝑋 (773𝑘𝑔)=0.0025(773𝑘𝑔)+0.05(0.2X) (0.8X)
𝑋+773𝑘𝑔=773𝑘𝑔+X
0.05𝑋 𝑘𝑔=1.9325𝑘𝑔+0.01X X
0.0388𝑋=0.773
𝑋=19.923𝑘𝑔