# CEM88(V) : Weir / Undershot gate (small sill elevation)

__Weir - Free-flow__

Q = µF L h13/2

__Weir - Submerged__

Q = kF µF L h13/2 **[17]**

with kF = coefficient of reduction for submerged flow.

The flow reduction coefficient is a function of and of the value a of this ratio at the instant of the free-flow/submerged transition. The submerged conditions are obtained when > a = 0.75. The law of variation of the kF coefficient has been derived from experimental results.

Let x =

If x > 0.2 kF = 1 - (1 - )b

If x £ 0.2 kF = 5x (1 - (1 - )b)

With b = -2a + 2.6

One calculates an equivalent coefficient for free-flow conditions as before.

__Undershot gate - Free-flow__

Q = L (µ.h13/2 - µ1.(h1 - W)3/2) **[18]**

It has been established experimentally that the undershot gate discharge coefficient increases with . A law of variation of µ of the following form is adopted:

µ = µo - with µo » 0.4

Hence, µ1 = µo -

In order to ensure the continuity with the open channel free-flow conditions for

= 1, we must have: µF = µo - 0.08

Hence, µF = 0.32 for µo = 0.4

__Undershot gate - Submerged
__

__ Partially submerged flow__:

Q = L [kF.µ.h13/2 - µ1(h1 - W)3/2] **[19]**

kF being the same as for open channel flow.

The following free-flow/submerged transition law has been derived on the basis of experimental results:

a = 1 - 0.14

0.4 £ a £ 0.75

In order to ensure continuity with the open channel flow conditions, the free-flow/submerged transition under open channel conditions has to be realized for a = 0.75 instead of in the weir/orifice formulation.

__ Totally submerged flow:
__

Q = L (kF.µ.h13/2 - kF1.µ1.(h1 - W)3/2) **[20]**

The kF1 equation is the same as the one for kF where h2 is replaced by h2-W (and h1 by h1-W) for the calculation of the x coefficient (and therefore for the calculation of kF1).

The transition to totally submerged flow occurs for:

h2 > a1.h1 + (1 - a1).W

with:

a1 = 1 - 0.14 i.e. a1 = a (h2 - W)

The functioning of the weir / undershot gate device is represented in figure 20. Whatever the conditions of the pipe flow, one calculates an equivalent free-flow discharge coefficient, corresponding to the classical equation for the free-flow undershot gate.

CF =

The reference coefficient introduced for the device is the classic CG coefficient of the free-flow undershot gate. It is then transformed to µ0 = CG

Remark: it is possible to get CF CG, even under free flow conditions, since the discharge coefficient increases with the ratio.

(12): Weir - Free flow(19): Undershot gate - Partially submerged(17): Weir - Submerged(20): Undershot gate - Totally submerged(18): Undershot gate - Free flow

__Figure 20. Weir - Undershot gate__