Lecture 10
Broad Crested Weirs
I. Calibration by Energy Balance
•
•
•
The complete calibration of a broad-crested weir includes the calculation of head
losses across the structure. However, the calibration can be made assuming no
losses in the converging and throat sections, and the resulting values will usually
be very close to those obtained by the complete theoretical calibration
The procedure which is presented below is useful to illustrate the hydraulic
principles which govern the broad-crested weir characteristics, and to check the
calibration of an existing structure in the field with a programmable calculator
The simplified calibration approach does not include the calculation of the modular
limit; however, this is an important consideration in the design and operation of a
broad-crested weir because the structure is usually intended to operate under
modular flow conditions
(1) The specific energy of the flow upstream of the broad-crested weir can be set
equal to the specific energy over the crest (or sill) of the structure. The energy
balance can be expressed mathematically as follows:
hu + zu +
Vu2
V2
= hc + zu + c
2g
2g
(1)
where hu is the upstream flow depth, referenced from the sill elevation; Vu is the
average velocity in the upstream section, based on a depth of (hu + zu); zu is the
height of the sill above the upstream bed; hc is the depth over the crest where
critical flow is assumed to occur; and Vc is the average velocity in the critical flow
section over the crest.
Note that the zu term cancels from Eq. 1.
Recognizing that Q = VA, where Q is the volumetric flow rate (discharge) and A is
the area of the flow cross-section,
hc − hu +
Q2
Q2
=0
(2)
Q2 ⎛ 1
1 ⎞
hc − hu +
⎜⎜ 2 − 2 ⎟⎟ = 0
2g ⎝ A c A u ⎠
(3)
2g A c2
−
2g A u2
which can be reduced to,
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Gary P. Merkley
(2) Critical flow over the crest can be defined by the Froude number, which is equal
to unity for critical flow. Thus, the square of the discharge over the crest can be
defined as follows:
Q2 =
g A 3c
Tc
(4)
where Tc is the width of the water surface over the crest. This last equation for Q2
can be combined with the equation for energy balance to produce the following:
A 3c
hc − hu +
2 Tc
⎛ 1
1 ⎞
⎜⎜ 2 − 2 ⎟⎟ = 0
⎝ A c Au ⎠
(5)
This last equation can be solved by trial-and-error, or by any other iterative method,
knowing hu, zu, and the geometry of the upstream and throat cross-sections. The
geometry of the sections defines the relationship between hc and Ac, and between
hu and Au (important: if you look carefully at the above equations, you will see that
Au must be calculated based on a depth of hu + zu). The solution to Eq. 5 gives the
value of hc.
(3) The final step is to calculate the discharge corresponding to the value of Ac,
which is calculated directly from hc. This is done using the following form of the
Froude number equation:
Q=
g A 3c
Tc
(6)
This process is repeated for various values of the upstream flow depth, and in the
end a table of values for upstream depth and discharge will have been obtained.
From this table a staff gauge can be constructed. This simple calibration assumes
that the downstream flow level is not so high that non-modular flow exists across
the structure.
•
•
•
See the computer program listing on the following two pages
In the design of broad-crested weirs it is often necessary to consider other factors
which limit the allowable dimensions, and which restrict the flow conditions for
which the calibration is accurate
Complete details on broad-crested weir design, construction, calibration, and
application can be found in the book "Flow Measurement Flumes for Open
Channel Systems", 1984, by M.G. Bos, J.A. Replogle, y A.J. Clemmens
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//----------------------------------------------------------------------------// Broad crested weir calibration for free flow by energy balance equation.
// Written in Object Pascal (Delphi 6) by Gary Merkley. September 2004.
//----------------------------------------------------------------------------unit BCWmain;
interface
uses
Windows, Messages, SysUtils, Classes, Graphics, Controls, Forms, Dialogs,
StdCtrls, Buttons;
type
TWmain = class(TForm)
btnStart: TBitBtn;
procedure btnStartClick(Sender: TObject);
private
function NewtonRaphson(hu:double):double;
function EnergyFunction(hc:double):double;
function Area(h,b,m:double):double;
function TopWidth(h,b,m:double):double;
end;
var
Wmain: TWmain;
implementation
{$R *.DFM}
const
g =
bu =
mu =
zu =
bc =
mc =
L =
9.810;
2.000;
1.250;
1.600;
6.000;
1.250;
1.500;
//
//
//
//
//
//
//
weight/mass (m/s2)
base width upstream (m)
side slope upstream (H:V)
upstream sill height (m)
base width at control section (m)
side slope at control section (H:V)
sill length (m)
var
hu,Au,hc: double;
function TWmain.NewtonRaphson(hu:double):double;
//----------------------------------------------------------------------------// Newton-Raphson method to solve for critical depth. Returns flow rate.
//----------------------------------------------------------------------------var
i,iter: integer;
dhc,F,Fdhc,change,Ac,Tc: double;
begin
result:=0.0;
for i:=1 to 9 do begin
hc:=0.1*i*hu;
for iter:=1 to 50 do begin
dhc:=0.0001*hc;
F:=EnergyFunction(hc);
Fdhc:=EnergyFunction(hc+dhc);
change:=Fdhc-F;
if abs(change) < 1.0E-12 then break;
change:=dhc*F/change;
hc:=hc-change;
if (abs(change) < 0.001) and (hc >= 0.001) then begin
Ac:=Area(hc,bc,mc);
Tc:=TopWidth(hc,bc,mc);
result:=sqrt(g*Ac*Ac*Ac/Tc);
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Gary P. Merkley
Exit;
end;
end;
end;
end;
function TWmain.EnergyFunction(hc:double):double;
//----------------------------------------------------------------------------// Energy balance function (specific energy), equal to zero.
//----------------------------------------------------------------------------var
Ac,Tc: double;
begin
Ac:=Area(hc,bc,mc);
Tc:=TopWidth(hc,bc,mc);
result:=hc-hu+0.5*Ac*Ac*Ac*(1.0/(Ac*Ac)-1.0/(Au*Au))/Tc;
end;
function TWmain.Area(h,b,m:double):double;
//----------------------------------------------------------------------------// Calculates cross-section area for symmetrical trapezoidal shape.
//----------------------------------------------------------------------------begin
result:=h*(b+m*h);
end;
function TWmain.TopWidth(h,b,m:double):double;
//----------------------------------------------------------------------------// Calculates top width of flow for symmetrical trapezoidal shape.
//----------------------------------------------------------------------------begin
result:=b+2.0*m*h;
end;
procedure TWmain.btnStartClick(Sender: TObject);
//----------------------------------------------------------------------------// Entry point for calculations (user clicked the Start button).
//----------------------------------------------------------------------------var
i: integer;
F: TextFile;
strg: string;
Q,humin,humax: double;
begin
AssignFile(F,'BCWenergy.txt');
Rewrite(F);
Writeln(F,'
hc (m)
hu (m)
Q (m3/s)');
humin:=0.075*L;
humax:=0.75*L;
hu:=humin;
for i:=0 to 100 do begin
Au:=Area(hu+zu,bu,mu);
Q:=NewtonRaphson(hu);
strg:=Format('%12.3f%12.3f%12.3f',[hc,hu,Q]);
Writeln(F,strg);
hu:=hu+0.03;
if hu > humax then break;
end;
CloseFile(F);
end;
end.
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98
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II. Calculation of Head Loss
Throat (Control Section)
Head loss in the throat (where the critical flow control section is assumed to be
located) can be estimated according to some elements from boundary layer theory.
The equation is (Schlichting 1960):
(hf )throat
CF L Vc 2
=
2gR
(7)
where L is the length of the sill; Vc is the average velocity in the throat section; and R is
the hydraulic radius of the throat section. The values of V and R can be taken for
critical depth in the throat section. The drag coefficient, CF, is estimated by assuming
the sill acts as a thin flat plat with both laminar and turbulent flow, as shown in the
figure below (after Bos, Replogle and Clemmens 1984).
The drag coefficient is calculated by assuming all turbulent flow, subtracting the
turbulent flow portion over the length Lm, then adding the laminar flow portion for the
length Lm. Note that CF is dimensionless.
⎛m⎞
⎛m⎞
CF = CT,L − ⎜ ⎟ CT,m + ⎜ ⎟ CL,m
⎝L⎠
⎝L⎠
(8)
where CL,m is the thin-layer laminar flow coefficient over the distance m, which
begins upstream of the weir crest:
CL,m =
1.328
(9)
(Re )m
When (Re)L < (Re)m, the flow is laminar over the entire crest and CF = CL,L, where
CL,L is defined by Eq. 9.
The CT,L and CT,m coefficients are calculated by iteration from:
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Gary P. Merkley
⎡
⎛
1
k
+
0.544 CT,x = CT,x ⎢5.61 CT,x − ln ⎜
⎜ Re x CT,x 4.84x CT,x
⎢⎣
⎝
⎤
⎞
⎟ − 0.638 ⎥ (10)
⎟
⎥⎦
⎠
where x is equal to L or m, for CT,L and CT,m, respectively; Re is the Reynolds number;
and k is the absolute roughness height. All values are in m and m3/s. Below are some
sample values for the roughness, k.
Material
and Condition
Glass
Smooth Metal
Rough Metal
Wood
Smooth Concrete
Rough Concrete
Roughness, k
(mm)
0.001 to 0.01
0.02 to 0.1
0.1 to 1.0
0.2 to 1.0
0.1 to 2.0
0.5 to 5.0
The Reynolds number can be calculated as follows:
(R e ) x =
Vx
ν
(11)
where ν is the kinematic viscosity (a function of temperature); x is equal to m or L; and
V is the average velocity in the throat section for critical flow. If (Re)L is less than (Re)m,
the boundary layer over the sill is laminar only, and CF = CL,m.
The value of m can be estimated as:
m=
L⎞
ν⎛
⎜ 350,000 + ⎟
V⎝
k⎠
(12)
where the units are (m2/s)/(m/s) = m
Diverging Section
The head loss in the downstream diverging section is estimated as:
(hf )ds
ξ ( Vc − Vd )
=
2g
2
(13)
where (hf)ds is the head loss in the diverging section (m); Vc is the average velocity in
the control section, at critical depth (m/s); and Vd is the average velocity in the
downstream section (m/s), using hd referenced to the downstream channel bed
elevation (not the sill crest).
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The coefficient ξ is defined as:
log10 ⎡114.6 tan−1 ( zd / Ld ) ⎤ − 0.165
⎣
⎦
ξ=
1.742
(14)
where Ld is the length of the downstream ramp; and zd is the height of the downstream
ramp, also equal to the difference in elevation between the sill and the downstream
bed elevation. Note that the recommended value for the zd/Ld ratio is 1/6.
If the downstream ramp is not used, Ld equals zero. In this case, assume ξ = 1.2.
Converging Section
After having calculated the CF drag coefficient for the throat section, and assuming the
same roughness value for the channel and structure from the gauge location to the
control section, the upstream losses can be estimated. These are the losses from the
gauge location to the beginning of the sill.
For the section from the gauge to the beginning of the upstream ramp, the head loss is
estimated as:
(hf )gauge
⎛ CF Lgauge ⎞ Vu2
=⎜
⎟
R
u
⎝
⎠ 2g
(15)
where Lgauge is the distance from the gauge to the beginning of the upstream ramp; Vu
is the average velocity in the upstream section (at the gauge); and Ru is the hydraulic
radius at the gauge. All values are in metric units (m and m/s).
For the upstream ramp, the same equation can be used, but the hydraulic radius
changes along the ramp. Therefore,
(hf )us
1 ⎛ CF Lu ⎞ ⎛ Vu2 Vr2 ⎞
= ⎜
+
⎟
⎟⎜
2 ⎝ 2g ⎠ ⎜⎝ Ru Rr ⎟⎠
(16)
where the values of Vr and Rr, at the entrance to the throat section (top of the ramp),
are estimated by calculating the depth at this location:
hr = hc + 0.625 (hu − hc )
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III. Photographs of BCW Construction
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References & Bibliography
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