Analysis of Water Jet from an Orifice
Problem Statement
A jet of water issuing from 5mm diameter orifice working under a head of 2.0m, was found to travel horizontal and vertical distances of 2.772m and 1m respectively. If Cc = 0.61, determine discharge.
Given Data
| Diameter of orifice (d) | 5 mm = 0.005 m |
| Head (H) | 2.0 m |
| Horizontal distance traveled (x) | 2.772 m |
| Vertical distance traveled (y) | 1 m |
| Coefficient of contraction (Cc) | 0.61 |
| Acceleration due to Gravity (g) | 9.81 m/s² |
1. Determining Cross-sectional Area of the Jet
Calculate the cross-sectional area of the orifice:
a = (π × d²) / 4
a = (π × 0.005²) / 4
a = (π × 0.000025) / 4
a = 0.0000785 / 4
a = 1.96 × 10-5 m²
2. Calculating the Time of Flight
Using the vertical distance formula for free fall:
y = (1/2) × g × t²
1 = (1/2) × 9.81 × t²
t² = 1 / (0.5 × 9.81)
t² = 1 / 4.905
t² = 0.2039
t = 0.4516 seconds
3. Determining the Initial Velocity
Using the horizontal distance formula:
x = v × t
2.772 = v × 0.4516
v = 2.772 / 0.4516
v = 6.138 m/s
Theoretical velocity under the given head:
vtheoretical = √(2gh)
vtheoretical = √(2 × 9.81 × 2)
vtheoretical = √39.24
vtheoretical = 6.264 m/s
4. Determining the Velocity Coefficient
Calculate velocity coefficient (Cv):
Cv = Actual velocity / Theoretical velocity
Cv = 6.138 / 6.264
Cv = 0.98
5. Determining Discharge Coefficient and Discharge
Calculate discharge coefficient (Cd):
Cd = Cc × Cv
Cd = 0.61 × 0.98
Cd = 0.5978
Calculate discharge (Q):
Q = Cd × a × √(2gh)
Q = 0.5978 × 1.96 × 10-5 × √(2 × 9.81 × 2)
Q = 0.5978 × 1.96 × 10-5 × 6.264
Q = 0.5978 × 1.96 × 10-5 × 6.264
Q = 7.34 × 10-5 m³/s
Q = 0.0734 liters/s
Conclusion
In this analysis of a water jet from an orifice, we determined:
1. Jet Characteristics: A 5mm diameter orifice under a 2.0m head produces a jet with an actual velocity of 6.138 m/s.
2. Flow Coefficients: The velocity coefficient (Cv) is 0.98, and when combined with the given contraction coefficient (Cc) of 0.61, yields a discharge coefficient (Cd) of 0.5978.
3. Discharge: The calculated discharge from the orifice is 7.34 × 10-5 m³/s or 0.0734 liters per second.
4. Verification: This discharge value confirms that small orifices can produce precise, controlled water jets for various applications such as irrigation, cooling systems, or laboratory experiments.
