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A brine solution (a mixture of salt and water) begins to flow at a constant rate of 𝑉¤ brine = 6 L min−1 into a cylindrical tank of diameter 𝐷 = 1.5 m that initially holds 𝑉0 = 1000 L of pure water. The solution inside the tank is kept well stirred and flows out of the tank with a velocity 𝑣 = 𝐾 ℎ, where 𝐾 = 0.05 s−1 and ℎ is the height of the liquid level, through an outlet of cross-sectional area 𝐴0 = 25 cm2 . The concentration of salt in the brine entering the tank is 𝑐brine = 0.1 kg L−1 . (a) Derive a differential equation that governs the height of the liquid level in the tank with time. State associated the initial condition. What is the steady-state height of the liquid level in the tank? [5 marks] (b) Solve, either analytically or numerically, for the variation of the height of the liquid level in the tank with time. [5 marks] (c) Derive the differential equation that governs the variation of the brine concentration in the tank with time. [5 marks] (d) Determine, either analytically or numerically, the brine concentration in the tank with time, and plot the concentration of the brine in the tank from time 0 to 24 hrs
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A brine solution (a mixture of salt and water) begins to flow at a constant rate of
𝑉¤
brine = 6 L min−1
into a cylindrical tank of diameter 𝐷 = 1.5 m that initially holds
𝑉0 = 1000 L of pure water. The solution inside the tank is kept well stirred and flows
out of the tank with a velocity 𝑣 = 𝐾 ℎ, where 𝐾 = 0.05 s−1 and ℎ is the height of the
liquid level, through an outlet of cross-sectional area 𝐴0 = 25 cm2
. The concentration
of salt in the brine entering the tank is 𝑐brine = 0.1 kg L−1
.
(a) Derive a differential equation that governs the height of the liquid level in the tank
with time. State associated the initial condition. What is the steady-state height
of the liquid level in the tank? [5 marks]
(b) Solve, either analytically or numerically, for the variation of the height of the liquid
level in the tank with time. [5 marks]
(c) Derive the differential equation that governs the variation of the brine
concentration in the tank with time. [5 marks]
(d) Determine, either analytically or numerically, the brine concentration in the tank
with time, and plot the concentration of the brine in the tank from time 0 to 24 hrs
Answer ( 1 )
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