Triangular Notch (V-Notch)

Triangular V-notch weirs are a fundamental tool in hydraulic engineering, providing a cost-effective and reliable method for measuring flow in open channels. Accurate flow measurement is essential for managing water resources, monitoring the environment, and conducting hydraulic research.

These weirs operate on the principle that the flow rate is related to the opening angle of the V-shaped notch. To ensure precise and reliable measurements, calibration is necessary. This process establishes a quantitative relationship between the weir’s geometric parameters and the corresponding flow rates, compensating for variations in hydraulic conditions and notch geometry.

Theoretical and Empirical Discharge Equations

The theoretical discharge $Q_t$Qt​ over a triangular V-notch is given by:$$\begin{array}{cccc}& Q_t=\frac{8}{15}\sqrt{2g}\tan \left(\frac{\theta }{2}\right)H^{5\mathrm{/}2}& & \text{(1)}\end{array}$$where:

• $H$ = head of water above the notch crest (m)
• $\theta$ = angle of the V-notch (degrees)
• $g$ = acceleration due to gravity (m/s²)

For a weir with a fixed angle, the term$$\begin{array}{cccc}& K_1=\frac{8}{15}\sqrt{2g}\tan \left(\frac{\theta }{2}\right)& & \text{(2)}\end{array}$$is constant.

The actual discharge $Q_a$Qa​ is related to the theoretical discharge by the coefficient of discharge $C_d$Cd​:$$\begin{array}{cccc}& Q_a=C_d{Q}_t& & \text{(3)}\end{array}$$$$\begin{array}{cccc}& C_d=\frac{Q_a}{Q_t}& & \text{(4)}\end{array}$$Substituting Equation (1) gives the actual discharge as:$$\begin{array}{cccc}& Q_a=C_d{K}_1{H}^{5\mathrm{/}2}& & \text{(5)}\end{array}$$By letting $K=C_d{K}_1$​, the discharge equation simplifies to:$$\begin{array}{cccc}& Q_a=K{H}^n& & \text{(6)}\end{array}$$where the theoretical value of the exponent $n$n is $5\mathrm{/}2$5/2.

Linearization and Determination of Constants

To determine the empirical constants $K$K and $n$n from experimental data, Equation (6) is linearized by taking the logarithm of both sides: $$\begin{array}{cccc}& \log Q_a=\log K+n\log H& & \text{(7)}\end{array}$$Plotting $\log Q_a$logQa​ versus $\log H$logH on log-log graph paper yields a straight line. From the graph, two points $(H_1,Q_{a1})$ and $(H_2,Q_{a2})$ are selected, yielding the equations:$$\begin{array}{cccc}& \log Q_{a1}=\log K+n\log H_1& & \text{(8)}\end{array}$$$$\begin{array}{cccc}& \log Q_{a2}=\log K+n\log H_2& & \text{(9)}\end{array}$$Subtracting Equation (8) from Equation (9) gives:$$\begin{array}{cccc}& \log Q_{a2}-\log Q_{a1}=n(\log H_2-\log H_1)& & \text{(10)}\end{array}$$Therefore, the exponent $n$n is calculated as:$$\begin{array}{cccc}& n=\frac{\log (Q_{a2}\mathrm{/}{Q}_{a1})}{\log (H_2\mathrm{/}{H}_1)}& & \text{(11)}\end{array}$$After finding $n$n, its value is substituted back into Equation (8) or (9) to solve for $\log K$logK and thus $K$K. Finally, the coefficient of discharge $C_d$Cd​ is found using the relation:$$C_d=\frac{K}{K_1}$$​