---
title: "Define energy slope"
subject: "Hydraulics"
chapter: 5
key: "semester_4/hydraulics:define-energy-slope"
source: https://www.bctengineer.com/q/semester_4/hydraulics/define-energy-slope
site: BCT Engineering
---

# Define energy slope

Hydraulics · Chapter 5: Basics of Open channel flow

- Asked in: 2082 Kartik, 0.5 marks; 2076 Baisakh, 1 marks; 2072 Magh, 1 marks; 2068 Bhadra, 1 marks (asked 4 times)
- Video: [Fluid Mechanics: Topic 7.3.1 - Energy grade line (EGL) & Hydraulic grade line (HGL)](https://www.youtube.com/watch?v=-oecDDrYfyY) — CPPMechEngTutorials
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/define-energy-slope
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-5

## Question

Define energy slope.

## Short answer

Energy slope Sf is the rate of loss of total energy head per unit channel length, Sf = hf/L = -dH/dx. It is the slope of the energy grade line. For uniform flow, Sf equals water surface and bed slopes.

## Answer

**Energy slope** ($S_f$) is the rate of loss of total energy head per unit length of channel. It is the slope of the energy grade line (EGL).
$$
S_f=-\frac{dH}{dx}=\frac{h_f}{L}
$$
where $H=z+y+\dfrac{\alpha V^2}{2g}$ is the total head and $h_f$ is the friction head loss over length $L$. It is a positive number since energy falls in the flow direction.

```text
   EGL  ------\
                 \----  slope S_f
   WSL  ---------\---   slope S_w
   Bed  -----------\--  slope S_0
```
For uniform flow, $S_f=S_w=S_0$ (all three lines are parallel). For non-uniform flow, $S_f\neq S_0$. In Manning's equation, $S$ is the energy slope: $S_f=\dfrac{n^2V^2}{R^{4/3}}$.

## Check yourself

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

1. Energy slope Sf is
   A. Rate of loss of total head per unit length
   B. Slope of the water surface only
   C. Slope of the bed
   D. Velocity head
   Answer: A. Sf = -dH/dx = hf/L. (Key point: Definition)

2. In uniform flow
   A. Sf > S0
   B. Sf = Sw = S0
   C. Sf = 0
   D. Sw = 0
   Answer: B. All three lines are parallel. (Key point: Uniform flow)

3. In Manning's equation the slope S is
   A. Water surface slope
   B. Zero
   C. Energy slope
   D. Bed slope always
   Answer: C. Sf = n^2 V^2 / R^(4/3). (Key point: Manning)

## Related in Hydraulics

- [Manning's equation, terms and slope S](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Amannings-equation-terms-and-slope) — Hydraulics, chapter 6
- [Bed, hydraulic and energy slopes](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Abed-hydraulic-energy-slopes-not-parallel) — Hydraulics, chapter 5

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