Two tanks are set up, each containing a saline solution. Tank #1 contains 500 li

Question

Two tanks are set up, each containing a saline solution. Tank #1 contains 500 liters of solution, with salinity at 0.2 kg of salt per liter. Tank #2 contains 400 liters of solution, with salinity at 0.35 kg of salt per liter. The solution is pumped from Tank #1 to Tank #2 at a rate of 2 liters per minute, and simultaneously pumped through a different pipe from Tank #2 to Tank #1 also at a rate of 2 liters per minute. Assume that both mixtures are well stirred. There is no fluid being pumped into or out of either tank to or from any external sources.

Let ??1 be the amount of salt in Tank #1 and ??2 be the amount of salt in Tank #2. Determine how much salt will be in each tank after 10 minutes.

Explanation / Answer

Let a and b be amount of solution in Tank #1 and Tank #2 respectively. Let c and d be initial amount of salt per litre in Tank #1 and Tank #2 respectively.

Let w1 be the initial amount of salt in Tank #1 and w2 be the amount of salt in Tank #2.

Note: the values of w1 and w2 will change after every minute.

Then, according to question,

a = 500;    b = 400;    c = 0.2;    d = 0.35;   w1 = 100 kg;    w2 =140 kg

After one minute,

The total amount of solution in both the tanks is same as before i.e., 500 litres and 400 litres resp but salinity will be changed. The new amount of salt in each of Tank #1 and Tank #2 is given as follows -

w1 = (a-2)*c+2*d = 100.3 kg
w2 = (b-2)*d+2*c = 139.7 kg
   

Also, new salinity in Tank #1, say c1 and Tank #2, say d1 is given as follows -

c1 = w1/a = 0.2006 kg per litre
d1 = w2/b = 0.3492 kg per litre

After second minute,

Again the values of w1 and w2 will be updated as follows -

w1 = (a-2)*c+2*d = 100.5973 kg
w2 = (b-2)*d+2*c = 139.4027 kg

salinity in Tank #1 and Tank #2 will also be updated as follows -

c2 = w1/a = 0.2012 kg per litre
d2 = w2/b = 0.3485 kg per litre

Similarly proceeding in the same way upto 10 times, we have the amount of salt w1 in Tank #1 and w2 in Tank #2 after 10 minutes.

w1 = 102.8814 kg

w2 = 137.1186 kg

c10 = 0.2058 kg per litre

d10 = 0.3428 kg per litre

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