Question

In: Chemistry

Consider two interconnected tanks. Tank 1 initial contains 40 L (liters) of water and 160 g...

Consider two interconnected tanks. Tank 1 initial contains 40 L (liters) of water and 160 g of salt, while tank 2 initially contains 20 L of water and 250 g of salt. Water containing 20 g/L of salt is poured into tank1 at a rate of 2 L/min while the mixture flowing into tank 2 contains a salt concentration of 35 g/L of salt and is flowing at the rate of 2.5 L/min. The two connecting tubes have a flow rate of 4 L/min from tank 1 to tank 2; and of 2 L/min from tank 2 back to tank 1. Tank 2 is drained at the rate of 4.5 L/min. You may assume that the solutions in each tank are thoroughly mixed so that the concentration of the mixture leaving any tank along any of the tubes has the same concentration of salt as the tank as a whole. (This is not completely realistic, but as in real physics, we are going to work with the approximate, rather than exact description. The 'real' equations of physics are often too complicated to even write down precisely, much less solve.) How does the water in each tank change over time?

Solutions

Expert Solution

Inital Volume of Tank 1 = 20L
Initial Volume of Tank 2 = 10L

First, you have to note the inflow and outflow of water for each tank:
Tank 1: In=5 L/min Out= 5 L/min
Tank 2: In= 10 L/min Out= 9 L/min ---->(2L/min + 7L/min)

Now let p(t) = amount of salt, in grams, in Tank 1 as function of time per min
and q(t) = amount of salt, in grams, in Tank 2 as function of time per min
where at time = 0 --> p(0) = 375 grams of salt
q(0) = 185 grams of salt

Thus the derivative is:
p'(t) = the rate of change with respect to time of the amount of salt in Tank 1

q'(t) = the rate of change with respect to time of the amount of salt in Tank 2

p'(t) = (Incoming flow in Tank 1)(concentration fo salt in flow)
+ (Incoming flow from Tank 2)(Concentration of salt in Tank 2)
- (Outgoing flow from Tank 1)(concentration in Tank 1)

p'(t) = (Incoming flow)(concentration of salt in flow)

+ (Incoming flow from Tank2)[p(t)/(Volume of water in Tank 2)]
- (Outgoing flow from Tank 1)[q(t)/(Volume of water in Tank 1)]

p'(t) = (50g/L)(3L/min) + (2L/min)[p(t)/10L] - (5L/min)[q(t)/20L]

p'(t) = 150 + (2/10)p(t) - (5/20)q(t) g/min
p'(t) = 150 + (1/5)p(t) - (1/4)q(t) g/min


q'(t) = (Incoming flow)(Concentration of salt in flow)
+ (incoming flow from Tank 1)[p(t)/(Volume of water in Tank 1)]
- (Outgoing flow from Tank2)[q(t)/(Volume of water in Tank 2)]

q'(t) = (35L)(4L/min) + (5L/min)[p(t)/20L] - (9L/min)[q(t)/10L]

q'(t) = 140 + (1/4)p(t) - (9/10)q(t) g/min


Therefore p' and q' is:
p' = 150 + (1/5)p - (1/4)q g/min for Tank 1
q' = 140 + (1/4)p - (9/10)q g/min for Tank 2


Related Solutions

Course: Math Modeling A tank contains 8 L (liters) of water in which is dissolved 32...
Course: Math Modeling A tank contains 8 L (liters) of water in which is dissolved 32 g (grams) of salt. A solution containing 2 g/L of the brine flows into the tank at a rate of 9 L/min, and the well-stirred mixture flows out at a rate of 5 L/min.     A) Find an equation to model this system using A to represent the amount of salt in the tank and t to represent minutes.     B) Find the amount...
Two tanks contain a mixture of water and alcohol with tank A contain- ing 500 L...
Two tanks contain a mixture of water and alcohol with tank A contain- ing 500 L and tank B 1000L. Initially, the concentration of alcohol in Tank A is 0% and that of tank B is 80%. Solution leaves tank A into B at a rate of 15 liter/min and the solution in tank B returns to A at a rate of 5 L/min while well mixed solution also leaves the system at 10 liter/min through an outlet. A mixture...
Two tanks contain a mixture of water and alcohol with tank A contain- ing 500 L...
Two tanks contain a mixture of water and alcohol with tank A contain- ing 500 L and tank B 1000L. Initially, the concentration of alcohol in Tank A is 0% and that of tank B is 80%. Solution leaves tank A into B at a rate of 15 liter/min and the solution in tank B returns to A at a rate of 5 L/min while well mixed solution also leaves the system at 10 liter/min through an outlet. A mixture...
1. Three tanks are interconnected non-interactively. The fluid enters the first tank at the flow rate...
1. Three tanks are interconnected non-interactively. The fluid enters the first tank at the flow rate q and the third leaves the tank at flow rate q3. The speed of the fluid coming out of the third tank is fixed by means of a pump. It is maintained. According to this, a) Mathematically the effect of the change in the velocity of the fluid entering the first tank on the liquid height in the third tank Please express. (The areas...
1. A tank initially contains 200 liters of saltwater, with a concentration of 2g/L salt. Saltwater...
1. A tank initially contains 200 liters of saltwater, with a concentration of 2g/L salt. Saltwater with a concentration of 5g/L flows into the tank at 2L/min and the well-mixed solution flows out of the tank at the rate of 4L/min. (a) Set up, but do NOT solve, the initial value problem whose solution will be the VOLUME OF LIQUID in the tank as a function of time (b) Solve the IVP from the previous part and find the volume...
A tank initially contains 10 liters of clean water (with no salt in it). Every minute,...
A tank initially contains 10 liters of clean water (with no salt in it). Every minute, a solution containing one liter of water and one gram of salt is added to the tank, and two liters of fluid are drained from the tank. The fluid in the tank is mixed continuously so that the salt is distributed evenly throughout the water in the tank. How many minutes does it take for the tank to contain 2 grams of salt?
A tank contains 70 kg of salt and 2000 L of water. Water containing 0.4kg/L of...
A tank contains 70 kg of salt and 2000 L of water. Water containing 0.4kg/L of salt enters the tank at the rate 16L/min. The solution is mixed and drains from the tank at the rate 4L/min. A(t) is the amount of salt in the tank at time t measured in kilograms. (a) A(0) =  (kg) (b) A differential equation for the amount of salt in the tank is  =0=0. (Use t,A, A', A'', for your variables, not A(t), and move everything...
A 500-liter tank initially contains 200 liters of a liquid in which 150 g of salt...
A 500-liter tank initially contains 200 liters of a liquid in which 150 g of salt have been dissolved. Brine that has 5 g of salt per liter enters the tank at a rate of 15 L / min; the well mixed solution leaves the tank at a rate of 10 L / min. -Find the amount A (t) grams of salt in the tank at time t. -Determine the amount of salt in the tank when it is full.
A tank contains 2340 L of pure water. A solution that contains 0.07 kg of sugar...
A tank contains 2340 L of pure water. A solution that contains 0.07 kg of sugar per liter enters a tank at the rate 9 L/min The solution is mixed and drains from the tank at the same rate. (a) How much sugar is in the tank initially? (kg) (b) Find the amount of sugar in the tank after ?t minutes. amount =   (kg) (your answer should be a function of ?t) (c) Find the concentration of sugar in the solution...
A tank contains 2140 L of pure water. A solution that contains 0.01 kg of sugar...
A tank contains 2140 L of pure water. A solution that contains 0.01 kg of sugar per liter enters a tank at the rate 6 L/min. The solution is mixed and drains from the tank at the same rate. (a) How much sugar is in the tank initially? (b) Find the amount of sugar in the tank after t minutes. (c) Find the concentration of sugar (kg/L) in the solution in the tank after 51 minutes.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT