Figure 18. Device schematic view

Weir - Free flow ($h_1 < W$ and $h_2 \leq \frac{2}{3} h_1$)

$Q = \mu _F L \sqrt{2g} h_1^{3/2} $ [12]

Classical equation for the free flow weir ($\mu_F \simeq 0.4$).

Weir - Submerged ($h_1 < W$ and $h_2 \geq \frac{2}{3} h_1$)

$Q = \mu _S L \sqrt{2g} (h_1-h_2)^{1/2} h_2$ [13]

Classical formulation for the submerged weir.

The free-flow/submerged transition takes place for $h_2 = \frac{2}{3} h_1$

Thus,
$\mu_S = \frac{3 \sqrt{3}}{2} \mu_F$ for $\mu_F = 0.4 \Rightarrow \mu_S = 1.04$

The equivalent free-flow coefficient can be calculated:

$ \mu_{F} = \frac{Q}{L \sqrt{2g} h_{1}^{3/2}} $

It indicates the degree of submergence of the weir by comparing it to the introduced free-flow coefficient $\mu_F$. In effect, the reference coefficient of the device considered is that corresponding to the free-flow weir ($\mu_F$ close to $0.4$).

Orifice - Free flow ($h_1 \geq W$ and $h_2 \leq \frac{2}{3} h_1$)

An equation of the following type is applied:

$ Q = \mu L \sqrt{2g} \left( h_1^{3/2} - (h_1 - W)^{3/2} \right)$ [14]

This formulation is applicable to large width rectangular orifices.

The continuity towards the open-channel flow is assured when:

$\frac{h1}{W} = 1$, one then has: $\mu = \mu_F$.

Orifice - Submerged

Two formulations exist, according to whether the flow is partially submerged or completely submerged.

Partially submerged flow ($h_1 \geq W$ and $\frac{2}{3} h_1 < h_2 < \frac{2}{3} h_1 + \frac{W}{3}$)

$Q = \mu_F L \sqrt{2g} \left[ \frac{3\sqrt{3}}{2} \left( \left( h_1 - h_2 \right)^{1/2} h_2 \right) - \left(h_1 - W \right)^{3/2} \right]$ [15]

Totally submerged flow ($h_1 \geq W$ and $\frac{2}{3} h_1 + \frac{W}{3} < h_2$)

$Q = \mu` L \sqrt{2g} (h_1-h_2)^{1/2} \left[ h_2 - (h_2 - W) \right]$
$\Rightarrow$ $Q = \mu` L \sqrt{2g} (h_1-h_2)^{1/2} W$ [16]

This is the classic equation of the submerged orifice with $\mu` = \mu_S$.

The operation of the weir/orifice device is represented by the above equations and in figure 19. Whatever the conditions of pipe flow, one calculates an equivalent free-flow coefficient, corresponding to the free-flow orifice:

$C_F = \frac{Q}{L \sqrt{2g} W (h_1 - 0.5 W)^{1/2}}$

(12): Weir - Free flow
(15): Orifice - Partially submerged
(13): Weir - Submerged
(16): Orifice - Totally submerged
(14): Orifice - Free flow
Figure 19. Weir - Orifice

Equations are also available in a Matlab script file (function Qouvrage) here.