Question

Consider a cylindrical storage tank with surface area A which contains a liquid at depth y:

At time t = 0, the tank is empty (y = 0 when t = 0). Liquid is supplied to the tank at a sinusoidal rate Qin =3Qsin2 (t) and withdrawn from the tank as: 𝑄𝑜𝑢𝑡 = 3(𝑦 − 𝑦𝑜𝑢𝑡) 1.5 if 𝑦 > 𝑦𝑜𝑢𝑡 𝑄𝑜𝑢𝑡 = 0 if 𝑦 ≤ 𝑦𝑜𝑢𝑡 

Please note that both 𝑄𝑖𝑛 and 𝑄𝑜𝑢𝑡 have units m3 /h. 

The mass balance for this system can be written: (𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑢𝑚𝑒) = (𝑖𝑛𝑓𝑙𝑜𝑤) − (𝑜𝑢𝑡𝑓𝑙𝑜𝑤) or: 𝑑(𝐴𝑦) 𝑑𝑡 = 𝑄𝑖𝑛 − 𝑄𝑜𝑢𝑡 Assume that A = 1,250 m2 , Q = 450 m3 /h, yout = 10 m.

Write a program that solves the differential equation from t = 0 to t = 250 h using: 

• the Classical 4th -order Runge-Kutta method using a step size of 1 h 

• the Euler’s method using a step size which closely approximates the solution found the Classical 4th -order Runge-Kutta method (you must find the step size which closely approximates the Runge-Kutte solution) 

• the built-in method ode45 

• Plots the liquid height y versus time t for all three solutions. Your plot should show each curve using a different color. Add a title, axes labels, and a legend.