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Tutorials: Mathematical model
1. Emihle is an engineer at Cultimex water treatement plant. In this plant, there is a chemical
storage tank that is fed by chemical A and chemical B at a flow rate w 1 and w2 respectively.
The mixture exits the tank at a flow rate of w 3 and the density of the liquid is 800kg/m 3. This
is an open cylindrical tank 2.5m tall and 2m in diameter. Bormal procedure is to fill the tank
until it reaches a nominal value of 1.75m using a contanst flow of w 1 = 120kg/min, w2 =
100kg/min and w3 = 200kg/min. At that point the inlet flow w1 is adjusted so that the level
remains constant. However on this particular day corrosion of the tank has opened up a hole
in the wall at a height of 1m, producing a leak whose volumetric flowrate q4 (m 3/min) can be
approximated by q 4=0.025 √ ( h−1 ), Where h (m)
1.1 If the tank was initially empty, how long did it take for the liquid level to reach the
corrosion point.
1.2 If mass flowrate w1, w2, w3 are kept constant indefinitely, will the tank eventually
overflow.
2. Sihle has been tasked to prepare a sugar solution by mixing 100kg of sugar into 1000L of
water into a 2000L tank. Upon finishing dissolving the sugar, he realises he has actually put
in too much sugar 150kg. He decides to dilute the solution by feeding pure water at a rate of
15kg/min, while draining the tank at the same rate. How long will it take to reach the
desired sugar concentration
3. Consider a salt solution into a mixing tank at a rate of F 0 = 15m3/hr with 30% salt. This is fed
into a mixing tank initially containing 60% salt. The exiting stream is at F = 10m 3/hr and the
density of the solution is 2000kg/m3. What will be concentration in the exiting stream after
30min when the initial salt concentration in the tank was 60%.
4. Thilo has been tasked to prepare a salt solution by mixing 250kg of salt into 1000L of water
into a 2000L tank. Upon finishing dissolving the salt, he realises he has actually put in too
much salt 150kg. He decides to dilute the solution by feeding water at a rate of 25kg/min.
what will be mass of salt in the solution after 15min.
5. Two feed streams are introduced into a well stirred tank at a flowrate of W 1 (kg/hr) and
density ρ1 (kg/m3) and W2 (kg/hr). The tank is of fixed area, A m 2 .
5.1 Derive a mathematical model representing the change in height of this system.
6. A feed is introduced into a well stirred tank at a flowrate of F o (m3/hr) and density ρo (kg/m3).
Derive a mathematical model representing the change in volume on this system.
Tutorials: Mathematical model
1. Emihle is an engineer at Cultimex water treatement plant. In this plant, there is a chemical
storage tank that is fed by chemical A and chemical B at a flow rate w 1 and w2 respectively.
The mixture exits the tank at a flow rate of w 3 and the density of the liquid is 800kg/m 3. This
is an open cylindrical tank 2.5m tall and 2m in diameter. Bormal procedure is to fill the tank
until it reaches a nominal value of 1.75m using a contanst flow of w 1 = 120kg/min, w2 =
100kg/min and w3 = 200kg/min. At that point the inlet flow w1 is adjusted so that the level
remains constant. However on this particular day corrosion of the tank has opened up a hole
in the wall at a height of 1m, producing a leak whose volumetric flowrate q4 (m 3/min) can be
approximated by q 4=0.025 √ ( h−1 ), Where h (m)
1.1 If the tank was initially empty, how long did it take for the liquid level to reach the
corrosion point.
1.2 If mass flowrate w1, w2, w3 are kept constant indefinitely, will the tank eventually
overflow.
2. Sihle has been tasked to prepare a sugar solution by mixing 100kg of sugar into 1000L of
water into a 2000L tank. Upon finishing dissolving the sugar, he realises he has actually put
in too much sugar 150kg. He decides to dilute the solution by feeding pure water at a rate of
15kg/min, while draining the tank at the same rate. How long will it take to reach the
desired sugar concentration
3. Consider a salt solution into a mixing tank at a rate of F 0 = 15m3/hr with 30% salt. This is fed
into a mixing tank initially containing 60% salt. The exiting stream is at F = 10m 3/hr and the
density of the solution is 2000kg/m3. What will be concentration in the exiting stream after
30min when the initial salt concentration in the tank was 60%.
4. Thilo has been tasked to prepare a salt solution by mixing 250kg of salt into 1000L of water
into a 2000L tank. Upon finishing dissolving the salt, he realises he has actually put in too
much salt 150kg. He decides to dilute the solution by feeding water at a rate of 25kg/min.
what will be mass of salt in the solution after 15min.
5. Two feed streams are introduced into a well stirred tank at a flowrate of W 1 (kg/hr) and
density ρ1 (kg/m3) and W2 (kg/hr). The tank is of fixed area, A m 2 .
5.1 Derive a mathematical model representing the change in height of this system.
6. A feed is introduced into a well stirred tank at a flowrate of F o (m3/hr) and density ρo (kg/m3).
Derive a mathematical model representing the change in volume on this system.
