Answer:
a. The time required for the tank to empty halfway is presented as follows;
b. The time it takes for the tank to empty the remaining half is presented as follows;
The total time 't', is presented as follows;
Explanation:
a. The diameter of the tank = D₀
The height of the tank = H
The diameter of the orifice at the bottom = D
The equation for the flow through an orifice is given as follows;
v = √(2·g·h)
Therefore, we have;
Where;
P₁ = P₂ = The atmospheric pressure
z₁ - z₂ = dh (The height of eater in the tank)
A₁·v₁ = A₂·v₂
v₂ = (A₁/A₂)·v₁
A₁ = π·D₀²/4
A₂ = π·D²/4
A₁/A₂ = D₀²/(D²) = v₂/v₁
v₂ = (D₀²/(D²))·v₁ = √(2·g·h)
The time, 'dt', it takes for the water to drop by a level, dh, is given as follows;
dt = dh/v₁ = (v₂/v₁)/v₂·dh = (D₀²/(D²))/v₂·dh = (D₀²/(D²))/√(2·g·h)·dh
We have;
The time for the tank to drop halfway is given as follows;
The time required for the tank to empty halfway, t₁, is given as follows;
(b) The time it takes for the tank to empty completely, t₂, is given as follows;
The time it takes for the tank to empty the remaining half, t₂, is presented as follows;
The total time, t, to empty the tank is given as follows;