A venturi meter having a throat 110mm in diameter is fitted in a pipeline 220mm indiameter through which oil of specific gravity 0.9 is flowing at the rate of 0.15m^3/s. Theinlet and throat of the meter are coninected differentialy to a U-tube manometer containingmercury of specific gravity 13.6 with oil immediately above this. Startîng from Bernoulli'sequation find the coefficient of discharge for the meter if the. difference in mercury levelsis0.63m
Question
A venturi meter having a throat 110mm in diameter is fitted in a pipeline 220mm indiameter through which oil of specific gravity 0.9 is flowing at the rate of 0.15m^3/s. Theinlet and throat of the meter are coninected differentialy to a U-tube manometer containingmercury of specific gravity 13.6 with oil immediately above this. Startîng from Bernoulli'sequation find the coefficient of discharge for the meter if the. difference in mercury levelsis0.63m
Solution
To solve this problem, we will use Bernoulli's equation and the equation of continuity.
Step 1: Identify the given values
- Diameter of the throat (d1) = 110mm = 0.11m
- Diameter of the pipeline (d2) = 220mm = 0.22m
- Specific gravity of oil (Sg) = 0.9
- Flow rate (Q) = 0.15m^3/s
- Specific gravity of mercury (Sgm) = 13.6
- Difference in mercury levels (h) = 0.63m
Step 2: Calculate the area of the throat (A1) and the pipeline (A2)
- A1 = π(d1^2)/4 = π(0.11^2)/4 = 0.0095 m^2
- A2 = π(d2^2)/4 = π(0.22^2)/4 = 0.038 m^2
Step 3: Use the equation of continuity to find the velocity at the throat (V1) and the pipeline (V2)
- V1 = Q/A1 = 0.15/0.0095 = 15.79 m/s
- V2 = Q/A2 = 0.15/0.038 = 3.95 m/s
Step 4: Use Bernoulli's equation to find the pressure difference
- ΔP = 0.5 * ρ * (V1^2 - V2^2) = 0.5 * Sg * ρ_water * g * (V1^2 - V2^2) = 0.5 * 0.9 * 1000 * 9.81 * (15.79^2 - 3.95^2) = 56550.6 Pa
Step 5: Calculate the pressure difference from the manometer
- ΔP = Sgm * ρ_water * g * h = 13.6 * 1000 * 9.81 * 0.63 = 84033.6 Pa
Step 6: The coefficient of discharge (Cd) is the ratio of the actual flow rate to the theoretical flow rate. The theoretical flow rate can be calculated from the pressure difference obtained from Bernoulli's equation and the actual pressure difference is obtained from the manometer reading.
- Cd = (Actual flow rate) / (Theoretical flow rate) = ΔP (manometer) / ΔP (Bernoulli) = 84033.6 / 56550.6 = 1.486
So, the coefficient of discharge for the meter is approximately 1.486.
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