Forebay design with spillway length (14 m³/s)

Given: $Q=14.0$ m³/s; 2 penstocks of diameter 2.0 m and length 500 m; retention (detention) time 3 min; limiting velocity in the forebay 0.20 m/s.

Step 1: Volume of the forebay

The forebay stores the water for the retention time:

$$V=Q\,t=14.0\times(3\times60)=2520\ \text{m}^3$$

Step 2: Cross-sectional area from the velocity limit

$$A=\frac{Q}{v}=\frac{14.0}{0.20}=70.0\ \text{m}^2$$

Step 3: Depth of water (submergence of the penstock intake)

Velocity in each penstock: $V_p=\dfrac{Q}{2\times\frac{\pi}{4}D^2}=2.228$ m/s. Minimum submergence to avoid vortices (Gordon, with $C=0.7245$ for an asymmetrical approach, which is on the safe side):

$$S=0.7245\,V_p\sqrt{D}=0.7245\times2.228\times\sqrt{2.0}=2.28\ \text{m}$$

Water depth = $S+D+$ bottom clearance of 0.5 m $=2.28+2.0+0.5=4.78$ m, adopt $H=4.8$ m.

Step 4: Width and length

$$B=\frac{A}{H}=\frac{70.0}{4.8}=14.58\ \text{m},\qquad L=\frac{V}{A}=\frac{2520}{70.0}=36.0\ \text{m}$$

Adopt $B=15.0$ m and $L=36$ m (the volume is then $B\,L\,H=2592$ m³ $\ge V$).

The penstock length (500 m) is not needed to size the forebay by this method; it matters for the penstock head loss and the water hammer.

Step 5: Spillway of the forebay

If the turbines trip, the whole discharge must spill over the forebay spillway (side-channel type) while the water level rises by $h=0.5$ m:

$$Q=C\,L_s\,h^{3/2}\ \Rightarrow\ L_s=\frac{Q}{C\,h^{3/2}}=\frac{14.0}{1.7\times0.5^{3/2}}=23.3\ \text{m}$$

($C=1.7$ for a broad-crested weir.) Adopt $L_s=24$ m, placed along the side wall (or folded), with a spill channel to the river.

PLAN
 headrace        forebay                 penstocks
 tunnel/canal  +------------------+
 ====> \       |                  |==> penstock 1
        \ trans| trash rack       |
 ====> /       |                  |==> penstock 2
              +--+-spillway-+-flush+
                   |          |
                   v          v  to river (spillway/flushing)

SECTION
 canal bed ____ WL ~~~~~~~~~~~~~~~~~~~~~~
                  |        S (submergence)
                  |   rack  ____ crown
                  |   ####  \   penstock
 flushing  ______ |___####____\_D__
 bottom clearance 0.5 m

Answer: forebay about 36 m long x 15.0 m wide x 4.8 m deep (water), with a volume of 2520 m³, and a spillway 24 m long.