Due to the contraction at the edge of the notch or weir, the value of actual discharge is theoretically reduced. Derivation of Notch and Weir formula by immersion of velocity.

Let the discharge constant be CdÂ

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### (i) When the arrival velocity is zeroÂ

Because the thickness of the strip is very less. Hence the velocity of the fluid at each point due to the head can be taken as âˆš2gh.Â

Strip in changeÂ

Cd x B.dh x âˆš2gh ............(i)

The entire area (BxH) of the flow can be bounded by many such strips. The sum of the discharge from all these strips is equal to the total discharge from the bottom or wiper.Â

Therefore, integrating equation (i), the value of h varies from h = 0 to h = H).Â Â

### (ii) When the arrival velocity is VaÂ

In this case, the fluid reaches the weir with a velocity Va, thus the top H flowing through the notch or weir will be (H + VÂ²a/2g) or (H +Ha).

And in the first case where H = 0, the value of vertex will be VÂ²a/2g or Ha. Since the arrival velocity is the same everywhere, the limits of the integral of equation (i) will be Ha and (H+Ha).

Ha = VÂ²a/2g = arrival top

## Rectangular V Notch

The immersion of the liquid is measured more smoothly by V notch. Because for a given immersion the apex will be higher in a V notch than in a rectangular notch.Â

### (i) When the arrival velocity is zeroÂ

Let there be a vertical strip of very small thickness dh at a depth h from the free surface of the liquid, the width of this strip âˆ†OMD.

tan Q/2 = MD/MO = b/2/(H-h)

b = 2(H-h) tan Q/2

Area of â€‹â€‹strip = b x dh = 2x(H-h) tanQ/2Â

Flow Velocity of strip = 2âˆš2ghÂ

Immersion in stripÂ

dQ = Cd x Area of â€‹â€‹strip x flow Velocity of stripÂ

dQ = Cd x 2(H-h) tanQ/2 x 2âˆš2gh .........(i)

Multiple strips can be considered in the entire V notch, integrating equation (i) for the entire immersion from the notch.

Q = 90Â° for right angle notchÂ

Q/2 = 45Â°Â

tanQ/2 = tant 45Â°

Immersion for right angle notch from equation (2).

### (ii) When the arrival velocity is Va

In this case also the arrival velocity is Va so due to this the top.

Ha = VÂ²a/2g

Â VÂ² = 2gHaÂ

If the vertex acts parallel to the V notch. It is related to equation (i) Limits of integration-