Consider a closed system of three well-mixed brine tanks....

Consider a closed system of three well-mixed brine tanks. Tank 1 has volume 20 gallons, tank 2 has volume 15 gallons, and tank 3 has volume 4 gallons. Mixed brine flows from tank 1 to tank 2, from tank 2 to tank 3, and from tank 3 back to tank 1. The flow rate between each pair of tanks is 60 gallons per minute. At time zero, tank 1 contains 28 lb of salt, tank 2 contains 11 lb of salt, and tank 3 contain no salt. Solve for the amount (lb) of salt in each tank at time t (minutes). Also determine the limiting amount (as t ? ?) of salt in each tank. (Solve the problem by using Laplace Transform)

Transcript

00:01 In this question given that tank 1 has volume is equal to 20 gallons, tank 2 volume is equal to 15 gallons and tank 3 volumes is equal to 4 gallons.

00:41 Now we have to find out the limiting amount of each tank and solve the amount of salt in each tank at time.

00:51 So now we have to find out the value of amount and limiting amount.

01:10 So now it started with the question.

01:15 Let x1 t, x2 t, x3 t are the amount of salt in tank one, to 3.

01:50 So the amount governing by the system of differential equation, so x1 dash is equal to minus 60 divided by 20 x1 plus 60 divided by 4 x3.

02:45 So this is equal to minus 3 x1 plus 15 x3.

02:55 Now for a 10.

02:56 X2 dash is equal to minus 60 divided by 15 x2 plus 60 divided by 20 x1 and this value is equal to minus 4 x2 plus 3 x1 it also written in the form of 3x1 minus 4 x2 minus 4 x2 now x3 dash is equal to minus 60 divided by 4 x2 x3 plus 60 divided by 15 x2 so this value is equal to minus 15 x3 plus 4 x2 it also written in the form of 4 x2 minus 15 x3 now with the initial conditions x1 0 is equal to 28 x2 x2 is equal to 208 x2 is equal to 11 and x3 is equal to 0 with the help of this then write this is a metric form so that's why x dash is equal to minus 3 0 15 3 minus 4 0 minus 15 and x 0 0 minus 15 and x 0 is equal to 28 11 and 0 now to find ident values we use lambda so minus 3 minus lambda this is 3 this is 0 and this is 15 next term is minus 4 minus lambda 0 4 minus 15 minus lambda is equal to 0 now lambda lambda plus 4, lambda plus 13 is equal to 0.

05:50 That's why lambda 1 is equal to 0, lambda 2 is equal to 4, and lambda 3 is equal to minus 13...

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