Unsteady Flowing Orifice
- 1. UNSTEADY FLOWING ORIFICE (Physical Application) Differential Equations Group 6 Presentation By: Genesis Bernaldez Marfel Obciana Teto Estelito
- 2. (a) In hydraulics , it has been shown that the discharge Q or rate of change of the volume of a liquid through a small opening or orifice located in a tank and under a head h is given by the formula,
- 3. Where: C – is the dimensionless constant known as the “ coefficient of discharge ” (C ≈ 0.60) Ao – is the area of the orifice Q – is the discharge or flow h – is the head of liquid or the vertical distance from the orifice to the liquid surface (m) d(Vol) – is the elemental volume taken at a distance of h above he orifice g – is the acceleration of gravity
- 4. (b) If the outflow or discharge through the orifice is replaced by an equivalent inflow into the tank, the flow condition is said to be “ steady ”. However, if there is no inflow, the liquid surface drops and the flow through the orifice varies since Q = f(h). This flow condition is known as “ unsteady ”. The problem here is the determination of the time for the liquid surface to drop from one elevation to another elevation. See the Figure on the next slide.
- 6. (c) Writethe EQ from (a) in the form where the (-) sign implies that the volume of the liquid in the tank decreases with the time, t.
- 7. Integration from to gives where d(Vol) = A dh and A = f(h).
- 8. Finally, after an interchange of the limits, we get REMARK : It is also possible to attain unsteady flow conditions if at time, t=0, there is a constant inflow, Q 1 ≠ Qo. This case, however, will not be treated here.
- 9. EXAMPLE PROBLEM 1 A cylindrical tank 1.5m in diameter and 3m high stands with its axis vertical. The tank is full of water and has a 10cm diameter orifice (C = 0.60) located at the bottom. Determine the time required to empty the tank of its contents through the orifice.
- 11. NOTE: In this problem the sectional area A of the tank at any elevation the water surface is constant. SOLUTION:
- 12. EXAMPLE PROBLEM 2 A tank is in the shape of a frustum of a pyramid having a square section. The upper base is 3m x 3m, and the lower base is 1.5m x 1.5m with a height of 3m. Determine the time required to lower down the water surface 2m through a 10cm by 10cm square orifice located at the bottom. Assume the tank to be initially full of water and use C = 0.60.
- 15. (b) Use EQ from (a), that is,