Discharge equations over weirs as a function of water head \(H\):

• Rectangular weir: \(Q = \frac{2}{3} C_d b \sqrt{2g} \cdot H^{3/2} = K_1 H^{3/2}\) - Differentiating: \(dQ = \frac{3}{2} K_1 H^{1/2} dH\)
  • Relative percentage error: \(\frac{dQ}{Q} = \frac{3}{2} \frac{dH}{H} = 1.5\%\) (for a 1% error in \(H\)).

• Triangular weir (V-notch): \(Q = \frac{8}{15} C_d \tan\left(\frac{\theta}{2}\right) \sqrt{2g} \cdot H^{5/2} = K_2 H^{5/2}\)
  • Differentiating: \(dQ = \frac{5}{2} K_2 H^{3/2} dH\) - Relative percentage error: \(\frac{dQ}{Q} = \frac{5}{2} \frac{dH}{H} = 2.5\%\) (for a 1% error in \(H\)). Since \(2.5\% \gt 1.5\%\), an error in head measurement produces more error in discharge over a triangular weir compared to a rectangular weir.