Ninety kilograms of sodium nitrate is dissolved in 110 kg of water. When the dissolution is complete (at time t = 0), pure water is fed to the tank at a constant rate ṁ(kg/min), and solution is withdrawn from the tank at the same rate. The tank may be considered perfectly mixed.

(a) Write a total mass balance on the tank and use it to prove that the total mass of liquid in the tank remains constant at its initial value.

(b) Write a balance on sodium nitrate, letting x(t,ṁ) equal the mass fraction of NaNO₃ in the tank and outlet stream. Convert the balance into an equation for dx/dt and provide an initial condition.

(c) On a single graph of x versus t, sketch the shapes of the plots you would expect to obtain for ṁ = 50 kg/min, 100 kg/min, and 200 kg/min. (Don’t do any calculations.) Explain your reasoning, using the equation of Part (b) in your explanation.

(d) Separate variables and integrate the balance to obtain an expression for x(t,ṁ). Check your solution. Then generate the plots of x versus t for m_ ˆ 50 kg/min, 100 kg/min, and 200 kg/min and show them on a single graph. (A spreadsheet is a convenient tool for carrying out this step.)

(e) If ṁ = 100 kg/min, how long will it take to flush out 90% of the sodium nitrate originally in the tank? How long to flush out 99%? 99.9%?

(f) The stream of water enters the tank at a point near the top, and the exit pipe from the tank is located on the opposite side toward the bottom. One day the plant technician forgot to turn on the mixing impeller in the tank. On the same chart, sketch the shapes of the plots of x versus t you would expect to see with the impeller on and off, clearly showing the differences between the two curves at small values and large values of t. Explain your reasoning.