Analysis of Two Identical Orifices
Problem Statement
For the two orifices shown in the figure below, determine Y2 such that X2 = (3X1)/4.
Given Data
| Height of water in first tank (H1) | 2 m |
| Total height of first tank (Y1) | 10 – 2 = 8 m |
| Height of water in second tank (H2) | 10 – Y2 m |
| Required condition | X2 = (3X1)/4 |
| Total height of tanks | 10 m |
| Required | Y2 value |
1. Establishing the Relationship Between Velocity Coefficients
The coefficient of velocity for the first orifice is defined as:
Similarly, for the second orifice:
Since the two orifices are identical, their coefficients of velocity must be equal:
Therefore:
2. Substituting Known Values
We know:
- H1 = 2 m
- Y1 = 10 – 2 = 8 m
- H2 = 10 – Y2 m
- X2 = (3X1)/4
Substituting these values into our equation:
3. Solving for Y2
Rearranging the equation:
Using the quadratic formula:
Since Y1 = 8m and Y2 = 9m would make Y2 > Y1, which is not feasible in this context, we take Y2 = 1m as our solution.
4. Verification and Physical Interpretation
Let’s verify our solution by checking if X2 = (3X1)/4 when Y2 = 1m:
For Y2 = 1m:
- H2 = 10 – Y2 = 10 – 1 = 9m
From the velocity coefficient relationship:
This confirms that X2 = (3X1)/4 when Y2 = 1m, validating our solution.
Physically, this means that the distance from the second orifice to the bottom of the second tank should be 1m to satisfy the condition that X2 = (3X1)/4.
Conclusion
In this analysis of two identical orifices, we determined:
1. Required Condition: For X2 = (3X1)/4, the value of Y2 must be 1m.
2. Solution Process: We established the relationship between velocity coefficients for identical orifices and solved a quadratic equation to find Y2.
3. Solution Analysis: While solving the quadratic equation Y22 – 10Y2 + 9 = 0 yielded two potential solutions (Y2 = 1m or Y2 = 9m), we determined that Y2 = 9m is not feasible since it exceeds Y1 = 8m.
4. Physical Interpretation: The distance from the second orifice to the bottom of the second tank should be 1m, while the height of water above this orifice will be H2 = 9m.
