Question

While conducting flow measurement using a rectangular notch, an error of 2% in head over the notch and error of 3% in the length was observed. The percentage error in the computed discharge would be
1. +6%
2. -1%
3. -2.5%
4. Zero

Answer 1

Correct Answer - Option 1 : +6%

Concept:

Flow over a Rectangular notch:

Discharge measured through rectangular notch:

 \(Q = \frac{2}{3}{C_d}\sqrt {2g} L{H^{3/2}}\)

Where, Cd = Coefficient of discharge

L = Length of crest or sill

H = Still water head

Effect on computed discharge over a weir or notch due to error in the measurement of head and length:

 \(Q = \frac{2}{3}{C_d}\sqrt {2g} L{H^{3/2}}\)

Or, \(Q = KL{H^{3/2}}\)

Where, \(K = \frac{2}{3}{C_d}\sqrt {2g} \)

Differentiating the above equation

dQ = KH^3/2dL + KL × (3/2) H^1/2 dH

 \(\frac{{dQ}}{Q} = \frac{{dL}}{L} + \frac{3}{2}\frac{{dH}}{H}\;\)

Calculation:

Give,

\(\frac{{dH}}{H} = 2\% \)

\(\frac{{dL}}{L} = 3\% \)

So, the Percentage error in discharge,

\(\frac{{dQ}}{Q} = \frac{{dL}}{L} + \frac{3}{2}\frac{{dH}}{H}\)

\(\frac{{dQ}}{Q} = 3\% + \frac{3}{2} \times \;2\% = 6\% \)