Flow Rate Determination in a U-Tube
Problem Statement
In the figure, one end of a U-tube is oriented directly into the flow so that the velocity of the stream is zero at this stagnation point (point 1). Neglecting friction, determine the flow of water in the 20 cm diameter pipe. Assume the center of the pipe is taken as the datum (Z = 0) and that the pressure difference is measured using a manometric setup.
Given Data
| Pipe Diameter (d) | 20 cm = 0.2 m |
| Cross-sectional Area (A) | A = (π/4) × (0.2)² ≈ 0.0314 m² |
| Velocity at Point 1 (V₁) | 0 m/s (stagnation point) |
| Datum Head at Point 1 (Z₁) | 0 m |
| Datum Head at Point 2 (Z₂) | 0 m |
| Pressure at Point 1 (P₁) | P₁ (unknown) |
| Pressure at Point 2 (P₂) | P₂ (unknown) |
| Manometric Equation | P₁ + γ_water (y + 0.07) = P₂ + γ_water y + γ_Hg (0.07) |
| Densities and γ | γ_water = 9810 N/m³, γ_Hg = 13.6 × 9810 N/m³ |
Note: From the manometric equation, the pressure difference is determined as:
P₁ – P₂ = 13.6×9810×0.07 – 9810×0.07 ≈ 8652.4 Pa
1. Applying Bernoulli’s Equation
Bernoulli’s equation between points 1 and 2 is:
P₁/γ + V₁²/(2g) + Z₁ = P₂/γ + V₂²/(2g) + Z₂
Given V₁ = 0 and Z₁ = Z₂ = 0, the equation reduces to:
P₁/γ = P₂/γ + V₂²/(2g)
2. Expressing the Pressure Difference
Rearranging the equation:
(P₁ – P₂)/γ = V₂²/(2g)
Substituting γ = 9810 N/m³ for water:
V₂² = 2g (P₁ – P₂)/9810
3. Substituting the Pressure Difference
From the manometric equation, we have:
P₁ – P₂ ≈ 8652.4 Pa
Therefore:
V₂² = 2 × 9.81 × (8652.4) / 9810
Simplify:
V₂² = (19.62 × 8652.4) / 9810 ≈ 17.22
Thus:
V₂ ≈ √17.22 ≈ 4.15 m/s
4. Calculating the Flow Rate (Q)
Flow rate is given by:
Q = A × V₂
Substituting A ≈ 0.0314 m² and V₂ ≈ 4.15 m/s:
Q ≈ 0.0314 × 4.15 ≈ 0.13 m³/s
Q ≈ 0.13 m³/s
Physical Interpretation
In this problem, one end of a U-tube is immersed in a flow, creating a stagnation point where the velocity is zero. The pressure difference between this stagnation point and a point in the flowing stream is determined using a manometric setup.
Pressure Difference: The manometric equation accounts for the difference in heights of the fluid columns, yielding a pressure difference of approximately 8652.4 Pa.
Energy Conversion: Bernoulli’s equation shows that the pressure difference is converted into kinetic energy, which is represented by the velocity head at point 2. This gives us the exit velocity of the fluid.
Flow Rate: Finally, the velocity combined with the pipe’s cross-sectional area determines the flow rate. In this case, the calculated exit velocity of about 4.15 m/s produces a flow rate of approximately 0.13 m³/s.
Detailed Explanation for Students
Step 1: Setup and Assumptions
Identify the two points in the U-tube. Point 1 is the stagnation point where the flow velocity is zero, and point 2 is in the free stream. We take the center of the pipe as the datum, so the elevation terms cancel.
Step 2: Apply Bernoulli’s Equation
With friction neglected and both points at the same elevation, Bernoulli’s equation simplifies to a relation between pressure and velocity heads.
Step 3: Determine Pressure Difference
Use the manometric reading to calculate the pressure difference between the two points. This pressure difference is key to determining the kinetic energy (and hence velocity) at point 2.
Step 4: Calculate Velocity and Flow Rate
Substitute the known pressure difference into the simplified Bernoulli equation to solve for the velocity at point 2. Then, use the cross-sectional area of the pipe to compute the flow rate.
This systematic approach illustrates how differences in pressure, measured with a manometer, drive the conversion of potential energy into kinetic energy, resulting in a measurable flow rate.
