Critical Depth & Specific Energy Calculator for Rectangular Channels

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📘 Design Guide Sections

Critical depth is the flow depth at which specific energy is minimum for a given discharge. It separates subcritical (tranquil) from supercritical (rapid) flow. This calculator computes critical depth, minimum specific energy, critical velocity, and Froude number for rectangular channels – essential for hydraulic design, flow classification, and energy dissipator sizing.

⚠️ Why Critical Depth Analysis Matters

• Determines flow regime (subcritical vs. supercritical)
• Identifies control sections (weirs, spillways, flumes)
• Essential for gradually varied flow profile classification
• Used in channel transitions and energy dissipator design

📐 Governing Equations

Critical depth (rectangular): y_c = (q² / g)^1/3   where   q = Q/B

Minimum specific energy: E_min = 1.5 × y_c

Critical velocity: V_c = √(g × y_c)

Froude number: Fr = V / √(g × y)

Fr = 1 → critical flow, Fr < 1 → subcritical, Fr > 1 → supercritical.

🧮 Critical Depth & Specific Energy Calculator

📏 Channel bottom width B (m):

💧 Flow rate Q (m³/s):

📐 Actual flow depth y (m) – optional, to compute Fr & regime:

📈 Critical Flow Parameters

Parameter	Value
Critical depth yc (m)	--
Minimum specific energy Emin (m)	--
Critical velocity Vc (m/s)	--
Unit discharge q (m²/s)	--
Actual Froude number	--
Flow regime	--

⚙️ If actual depth is provided, Froude number and regime are shown.

📉 Specific Energy Diagram (E vs y)

Specific energy curve E(y) Critical point (y_c, E_min) Subcritical branch Supercritical branch

📊 E = y + q²/(2g·y²) — The curve shows minimum energy at critical depth.

📝 Step‑by‑Step Engineering Example

Rectangular channel: B = 5 m, Q = 15 m³/s. Compute critical depth:

• q = Q/B = 15/5 = 3.0 m²/s
• y_c = (q²/g)^1/3 = (9/9.81)^1/3 = (0.917)^0.333 = 0.97 m
• E_min = 1.5 × 0.97 = 1.455 m
• V_c = √(9.81×0.97) = √9.515 = 3.08 m/s

If actual depth y = 2.0 m, then V = Q/(B×y) = 15/10 = 1.5 m/s, Fr = 1.5/√(9.81×2) = 1.5/4.43 = 0.34 → subcritical flow.

📊 Flow Regime Classification

Froude number	Regime	Characteristics	Typical applications
Fr < 1	Subcritical	Deep, slow, downstream control	Canals, rivers, floodplains
Fr = 1	Critical	Minimum energy, unstable	Control sections, weir crests
Fr > 1	Supercritical	Shallow, fast, upstream control	Spillways, chutes, steep slopes

🛠️ Design Implications

• Channel lining: Supercritical flows require erosion‑resistant surfaces (concrete, riprap).
• Hydraulic jumps: Occur when flow transitions from supercritical to subcritical.
• Control structures: Flumes and weirs are designed to create critical flow for accurate discharge measurement.
• Bridge hydraulics: Avoid critical flow near piers to prevent instability.

❓ Frequently Asked Questions

Q: What is the physical significance of critical depth?
A> At critical depth, the specific energy is minimum for a given discharge. Any disturbance can cause flow to switch between subcritical and supercritical.

Q: Can this calculator be used for trapezoidal channels?
A> The critical depth equation for trapezoidal sections is implicit (requires iteration). This calculator is for rectangular channels only. For trapezoidal, use specific charts or iterative solvers.

Q: Why is critical depth important for flow measurement?
A> At critical flow, there is a unique relationship between depth and discharge (e.g., in Parshall flumes or sharp‑crested weirs).

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