Water Level in a Circular Tank after Draining
Problem Statement
A circular tank with a diameter of 1.25 m contains water up to a height of 5 m. An orifice of 50 mm diameter is provided at the bottom of the tank. Determine the height of water above the orifice after 1.5 minutes of drainage. (Coefficient of discharge, Cd = 0.62)
Given Data
| Diameter of Tank (D) | 1.25 m |
| Area of Tank (A) | π/4 × (1.25)² ≈ 1.227 m² |
| Diameter of Orifice (d) | 50 mm = 0.05 m |
| Area of Orifice (a) | π/4 × (0.05)² ≈ 0.00196 m² |
| Coefficient of Discharge (Cd) | 0.62 |
| Initial Head (H1) | 5 m |
| Time (t) | 1.5 minutes = 90 s |
| Final Head (H2) | ? |
| Acceleration due to Gravity (g) | 9.81 m/s² |
Calculation
Using the principle of continuity for the draining tank:
-Q dt = A dh
With the orifice discharge given by Q = Cd · a · √(2gh), separating the variables and integrating from the initial head (H1) to the final head (H2) over time, we obtain:
Substituting the given values:
Simplifying the constants gives:
Knowing that √5 ≈ 2.236, we solve for √H2:
Finally, squaring both sides:
Detailed Explanation
The solution involves applying the principle of mass conservation (continuity) to the draining process of a tank. As water exits through the orifice, the water level decreases, and the rate of decrease is related to the outflow discharge.
- The discharge Q through the orifice is calculated using Torricelli’s law modified by the coefficient of discharge (Cd).
- The negative sign in the continuity equation indicates that as the water exits (flow rate Q), the water height h decreases.
- By separating the variables (h and t) and integrating, an expression linking time with the square roots of the initial and final heads is obtained.
- Finally, substituting the known values allows the determination of the final water head (H2) after the specified time.
Physical Meaning
The derived equation and its components represent the underlying physics of the draining process:
- Tank Area (A): A larger tank area implies that for a given discharge, the water level drops slower.
- Orifice Area (a): A smaller orifice restricts the outflow, causing a slower decrease in the water head.
- Coefficient of Discharge (Cd): This empirical factor adjusts for real-world deviations from ideal fluid behavior, accounting for viscous and contraction losses.
- Square Root Dependency (√h): The term √(2gh) in the discharge equation originates from the energy conversion of potential energy (water height) into kinetic energy (velocity of outflow).
- Integration of dh: The integration over the water height captures the continuous change in water level as the tank drains.
Together, these components illustrate how the water level evolves over time based on the interplay between the tank’s geometry and the orifice’s characteristics.
Conclusion
After 1.5 minutes of drainage, the height of water above the orifice in the tank is approximately 4.15 m.
