---
title: "Notch discharge coefficient error"
subject: "Fluid Mechanics"
chapter: 6
key: "semester_3/fluid_mechanics:notch-coefficient-discharge-error"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/notch-coefficient-discharge-error
site: BCT Engineering
---

# Notch discharge coefficient error

Fluid Mechanics · Chapter 6: Flow measurement

- Asked in: 2075 Asoj, 8 marks
- Video: [EFFECT OF DISCHARGE OVER A NOTCH DUE TO ERROR IN HEAD](https://www.youtube.com/watch?v=fCv82fm0-2w) — e Tution
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/notch-coefficient-discharge-error
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-6

## Question

A discharge of 12 lps is passed over a 45 degree sharp-edged triangular notch under a head of 21 cm. The same discharge is passed over a sharp-crested rectangular notch of length 30 cm, the head being 7.8 cm. Calculate the coefficient of discharge of two notches. What is the magnitude of error that would cause 2 percent error in discharge in the two cases.

## Short answer

Triangular notch Cd=0.607 (45°, H=0.21 m), rectangular Cd=0.622 (L=0.30 m, H=0.078 m). For 2% discharge error, head error dH=1.68 mm (triangular, n=2.5) and 1.04 mm (rectangular, n=1.5).

## Answer

**Given:** $Q = 12\ \text{L/s} = 0.012\ \text{m}^3/\text{s}$.

#### Coefficient of discharge, triangular notch

Included angle $\theta = 45^\circ$, so $\tan(\theta/2) = \tan 22.5^\circ = 0.4142$; $H = 0.21$ m.

$$
Q = \frac{8}{15}C_d\tan\frac{\theta}{2}\sqrt{2g}\,H^{5/2}
$$

$$
\begin{aligned}
H^{5/2} &= 0.21^{2.5} = 0.020208\\
0.012 &= \frac{8}{15}(C_d)(0.4142)(4.4294)(0.020208) = 0.019774\,C_d\\
C_d &= 0.607
\end{aligned}
$$

#### Coefficient of discharge, rectangular notch

$L = 0.30$ m, $H = 0.078$ m.

$$
Q = \frac{2}{3}C_dL\sqrt{2g}\,H^{3/2}
$$

$$
\begin{aligned}
H^{3/2} &= 0.078^{1.5} = 0.021784\\
0.012 &= \frac{2}{3}(C_d)(0.3)(4.4294)(0.021784) = 0.019299\,C_d\\
C_d &= 0.622
\end{aligned}
$$

#### Error in head for 2 % error in discharge

Differentiating $Q \propto H^n$: $\dfrac{dQ}{Q} = n\dfrac{dH}{H}$.

- **Triangular notch** ($n = 5/2$): $\dfrac{dH}{H} = \dfrac{0.02}{2.5} = 0.008$, so $dH = 0.008\times 210 = 1.68$ mm.
- **Rectangular notch** ($n = 3/2$): $\dfrac{dH}{H} = \dfrac{0.02}{1.5} = 0.01333$, so $dH = 0.01333\times 78 = 1.04$ mm.

**Answer: $C_d = 0.607$ (triangular) and $0.622$ (rectangular); a 2 % error in discharge corresponds to an error in head of about 1.68 mm (triangular) and 1.04 mm (rectangular), i.e. 0.8 % and 1.33 % of the head.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which discharge formula applies for a V-notch?
   A. Q = (8/15) Cd tan(theta) sqrt(2g) H^(3/2)
   B. Q = (2/3) Cd L sqrt(2g) H^(3/2)
   C. Q = (8/15) Cd tan(theta/2) sqrt(2g) H^(5/2)
   D. Q = Cd a sqrt(2gH)
   Answer: C. A triangular notch has Q proportional to H^(5/2) with tan(theta/2) for included angle theta. (Key point: Triangular notch formula)

2. What are the computed coefficients of discharge?
   A. 0.622 (triangular) and 0.607 (rectangular)
   B. 0.607 (triangular) and 0.622 (rectangular)
   C. 0.60 for both
   D. 0.70 and 0.58
   Answer: B. Solving 0.012 = 0.019774 Cd gives 0.607; 0.012 = 0.019299 Cd gives 0.622. (Key point: Cd values)

3. How is the head error related to the discharge error?
   A. dQ/Q = dH/(nH)
   B. dQ/Q = n^2 dH/H
   C. dQ/Q = dH/H for all notches
   D. dQ/Q = n dH/H, with Q proportional to H^n
   Answer: D. Differentiating Q = K H^n gives dQ/Q = n dH/H. (Key point: Error relation)

4. For 2% error in discharge, which head errors result?
   A. 0.8 mm for both
   B. 1.68 mm (triangular), 1.04 mm (rectangular)
   C. 1.04 mm (triangular), 1.68 mm (rectangular)
   D. 4.2 mm for both
   Answer: B. Triangular n = 2.5 gives dH/H = 0.8%, so 1.68 mm; rectangular n = 1.5 gives 1.33%, so 1.04 mm. (Key point: Head error values)

## Related in Fluid Mechanics

- [Percentage error in notch discharge](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Anotch-head-error-percentage) — Fluid Mechanics, chapter 6

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