A schematic illustration of flow transfer by weir diversion between two nodes is shown in the Weir Diversion dialog. Weir diversions provide relief to the sanitary system during periods of storm runoff.



Flow over a weir is computed as:

Equation 39: QW = CW·LW[ (h+V^(2)/2g)^(a) -(V^(2)/2g)^(a)]

where: 

CW  = discharge coefficient,

 LW  = weir length (transverse to overflow),

 h  = driving head on the weir,

 V  = approach velocity, and

 a  = weir exponent, 3/2 for transverse weir and 5/3 for side-flow weirs.

Both CW and LW are input values for transverse and side-flow weirs. For side-flow weirs, ideally Cw is a function of the approach velocity therefore a representative estimates for the the Cw in this case should be made. Normally, the driving head on the weir is computed as the difference h = Yup-Yc, where Yup is the water depth on the upstream side of the weir and Yc is the height of the weir crest above the node invert. However, if the downstream depth Ydn also exceeds the weir crest height, the weir is submerged and the flow is computed as:

Equation 40: QW = CSUB·CW·LW(Yup-Yc)^(3/2)

where CSUB is a submergence coefficient representing the reduction in driving head, and all other variables are as defined above.

The submergence coefficient, CSUB, is taken from Roessert’s Handbook of Hydraulics (in German, reference unavailable) by interpolation from Table 14-6, where CRATIO is defined as:

Equation 41:  CRATIO = (Ydn-Yc)/(Yup-Yc)       (41)

and all other variables are as previously defined. The values of CRATIO and CSUB are computed automatically by EXTRAN without further need for data input. If the weir is surcharged it will behave as an orifice and the flow is computed as:

Equation 42:  Qw = Csur·Lw·(Ytop - Yc(2·g·h’)

where:  

Ytop  = distance to top of weir opening,

 h’ = maximum(Ydn,Yc), and

 Csur  = weir surcharge coefficient.

 Values of Csub as a Function of Weir Submergence

CRATIO  CSUB
1.0
0.1 0.99
0.2 0.98
0.3 0.97
0.4 0.96
0.5 0.95
0.6 0.94
0.7 0.91
0.8 0.85
0.85 0.8
0.9 0.68
0.95 0.4
1.0 0.0

The weir surcharge coefficient, Csur, is computed automatically at the beginning of surcharge. When the weir begins to surcharge, the preceding weir discharge just prior to surcharge is equated to QW in equation 35 and equation 36 is then solved for the surcharge coefficient, Csur. Thus, no input coefficient for surcharged weirs is required.

Since an orifice is modeled as equivalent pipes the same technique is used for surcharge and flooding as for circular and rectangular conduits. Weirs under surcharge are also converted to equivalent pipes and the flow in a surcharged weir (Qweir) is assumed to behave as an orifice:

Equation 43: Qweir = C·A·(2·g·H)

where: 

C = calculated equivalent-roughness pipe coefficient,

A = cross-sectional area of equivalent conduit, and

H = driving head on the weir.

A weir, like an orifice, is represented as an equivalent conduit by equating the conduit and weir discharge equations:

Equation 44: (m/n)·A·R^(2/3)·S = Cw·W·H^(3/2)  

where:

m = 1.486 for U.S. customary units and 1.0 for metric units,

n = Manning’s roughness coefficient,

A = cross-sectional area (W·H),

H = head across the weir,

R = hydraulic radius,

S = slope of the hydraulic grade line (H/L), and

W = weir length.

If R is set as the value of the hydraulic radius when the head is half way between Ycrest and Ytop and L is defined based on the Courant number then the roughness, n, can be computed as:

Equation 45

Finally, EXTRAN detects flow reversals at weir nodes which cause the downstream water depth, Ydn, to exceed the upstream depth, Yup. All equations in the weir section remain the same except that Yupand Ydn are switched so that Yup remains as the "upstream" head. Also, flow reversal at a side-flow weir causes it to behave more like a transverse weir and consequently the exponent a in equation 33 is set to 1.5.