---
title: "Critical flow at maximum discharge for given energy"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:specific-energy-max-discharge-critical"
source: https://www.bctengineer.com/q/semester_4/hydraulics/specific-energy-max-discharge-critical
site: BCT Engineering
---

# Critical flow at maximum discharge for given energy

Hydraulics · Chapter 7: Energy and Momentum Principles in Open channel flow

- Asked in: 2072 Magh, 1+4 marks
- Video: [Show that flow is critical when the discharge is maximum for a given energy |Hydraulic|Prashant YT|](https://www.youtube.com/watch?v=_6Czi-Ti0RY) — Prashant YT
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/specific-energy-max-discharge-critical
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

Define specific energy. Show that the flow is critical when the discharge is maximum for the given specific energy.

## Short answer

Specific energy is the energy per unit weight measured above the bed: E = y + V²/(2g). For constant E, q = y√[2g(E−y)]. Differentiating yields maximum q at y = 2E/3 = yc, Fr=1. Thus maximum discharge for a given specific energy occurs at critical flow.

## Answer

#### Specific energy
**Specific energy** is the energy per unit weight of water at a section, measured above the channel bed:
$$
E=y+\frac{V^2}{2g}\qquad(\alpha\approx1)
$$

#### Proof: critical flow gives maximum discharge for given $E$
Consider a rectangular channel (width $b$, $q=Q/b$). With $V=q/y$:
$$
E=y+\frac{q^2}{2gy^2}\;\Rightarrow\;q=y\sqrt{2g(E-y)}
$$
For constant $E$, $q$ is a function of $y$ only. Differentiate $q^2=2gy^2(E-y)$:
$$
\frac{d(q^2)}{dy}=2g\left(2yE-3y^2\right)=0\;\Rightarrow\;y=\frac{2E}{3}
$$
At this depth the second derivative is negative, so $q$ is a **maximum**. Then
$$
q_{max}=\frac{2E}{3}\sqrt{2g\left(E-\frac{2E}{3}\right)}=\sqrt g\left(\frac{2E}{3}\right)^{3/2}
$$
The critical depth in a rectangular channel is $y_c=\sqrt[3]{q^2/g}$, i.e. $q^2=gy_c^3$. Comparing: $y=\dfrac{2E}{3}=y_c$ and $E=\dfrac32y_c=E_{min}$. Also
$$
V=\frac{q}{y}=\sqrt{\frac{g\cdot y^3}{y^2}}=\sqrt{gy}\;\Rightarrow\;Fr=1
$$
Hence, for a given specific energy, the discharge is maximum when the flow is critical. (For any section, $dQ/dy=0$ gives $Q^2T/(gA^3)=1$, the same critical-flow condition.)

```text
  y ^
  E |*            q = 0 at y = E
    |   .
    |      .        upper part: subcritical
 yc |--------*  q_max (critical flow)
    |      .        lower part: supercritical
    |   .
    |.                q = 0 at y = 0
    +--------+--------> q   (E constant)
           q_max
```

## Check yourself

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

1. Specific energy is
   A. Energy per unit weight of water measured above the channel bed, E = y + V^2/2g
   B. Energy per unit weight measured above an arbitrary datum
   C. Kinetic energy per unit volume
   D. Total energy minus pressure energy
   Answer: A. Specific energy is measured with the channel bed as datum: depth plus velocity head. (Key point: Definition of specific energy)

2. For constant E, how is the maximum discharge condition found?
   A. Set dE/dq = 0 at constant y
   B. Write q^2 = 2g y^2 (E - y) and set d(q^2)/dy = 0
   C. Set the Froude number to zero
   D. Set y = E
   Answer: B. Differentiating q^2 = 2g y^2 (E - y) with respect to y and equating to zero gives 2yE - 3y^2 = 0. (Key point: Method of proof)

3. What depth gives maximum q for given E in a rectangular channel?
   A. y = E/2
   B. y = E
   C. y = 2E/3, which equals yc
   D. y = 3E/2
   Answer: C. y = 2E/3 gives E = 1.5y, the critical-flow relation, so this depth is yc. (Key point: Result of differentiation)

4. How does the answer show that flow at q_max is critical?
   A. V = sqrt(g y), so Fr = 1
   B. V = 0 at that depth
   C. The second derivative is positive
   D. q is zero at that depth
   Answer: A. With q^2 = g y^3, V = q/y = sqrt(g y), so Fr = 1. (Key point: Confirming critical flow)

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