Hydraulic Engineering Tools

Chezy Equation Calculator

Calculate open channel flow velocity using the Chezy Equation. Professional hydraulic engineering tool with step-by-step solutions, shape-based hydraulic radius calculation, material-based coefficient selection, and detailed engineering interpretations.

Chezy Equation Calculator | Open Channel Flow Velocity Calculator

Chezy Equation Calculator

Select channel shape and material, or enter values manually for full control

Input Parameters

Configure channel properties for the Chezy Equation

Hydraulic Radius (R)
m

Range: 0.01 – 100 m. Typical: 0.1 – 5 m

Chezy Coefficient (C)
Channel Material
16 materials
Selected Material
Change
—
Chezy C = —
Range: —
—
m½/s

Range: 10 – 120. Typical: 30 – 80

m/m (dimensionless)

Range: 0.00001 – 0.5. Typical: 0.0001 – 0.05

Results

Calculated flow velocity and step-by-step solution

Configure parameters and click Calculate
—
m/s
Flow Velocity (V)
Equation:
V = C × √(R × S)
Enter values to see substitution

Step-by-Step Calculation

1Configure your parameters and click Calculate

Calculation History

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The Chezy Equation

Fundamental equation for uniform flow in open channels

The Chezy Equation relates flow velocity to channel geometry and roughness:

V = C × √(R × S)
V
Mean Flow Velocity
m/s (or ft/s)
C
Chezy Coefficient
m½/s (or ft½/s)
R
Hydraulic Radius
m (or ft)
S
Energy Slope
m/m (dimensionless)

Parameter Definitions

Understanding each variable in the Chezy Equation

Flow Velocity (V)

The average velocity of water flowing through the channel cross-section.

m/sTypical: 0.3–3.0

Hydraulic Radius (R)

Ratio of cross-sectional flow area to wetted perimeter (R = A/P).

mTypical: 0.1–5.0

Chezy Coefficient (C)

Resistance coefficient accounting for channel roughness. C = R1/6/n.

m½/sTypical: 30–80

Channel Slope (S)

Energy gradient or bed slope representing the driving force for flow.

m/mTypical: 0.0001–0.05

Flow Depth (d)

Vertical distance from channel bed to water surface.

mDesign parameter

Wetted Perimeter (P)

Length of the channel boundary in contact with water.

mP = b + 2d (rect.)

Cross-sectional Area (A)

Area of the flow cross-section perpendicular to flow direction.

m²A = b × d (rect.)

Hydraulic Gradient

Rate of energy loss per unit length of channel.

dimensionlessS = hf/L

Engineering Diagrams

Cross-section diagrams with hydraulic radius formulas for each channel shape

Rectangular Channel

b (width) d A = b × d P (wetted perimeter)
Hydraulic Properties
A= b × d
P= b + 2d
R= A / P = (b × d) / (b + 2d)

Trapezoidal Channel

T (top width) b (bottom width) z:1 ↗ ↖ z:1 d A = (b+zd)d
Hydraulic Properties
A= (b + z·d) × d
P= b + 2d√(1+z²)
T= b + 2zd
z= side slope (H:V)
R= (b + z·d)·d / [b + 2d√(1+z²)]

Triangular (V-Notch) Channel

T = 2zd (top width) d z:1 ↗ ↖ z:1 A = z·d²
Hydraulic Properties
A= z × d²
P= 2d√(1+z²)
T= 2zd
z= side slope (H:V)
R= z·d / [2√(1+z²)]

Circular Channel (Partial Flow)

D y θ (central angle)
Hydraulic Properties
r= D / 2
θ= 2·arccos((r−y)/r)
A= (r²/2)(θ − sin θ)
P= r × θ
R= A / P = (r/2) × (1 − sin θ / θ)

Wide Rectangular Channel (Approximation)

b (width) — b >> d d When b >> d → R ≈ d P = b + 2d ≈ b (since b >> d) R = bd/(b+2d) → as b→∞, R → d
Hydraulic Properties — Wide Channel Approximation
A= b × d
P= b + 2d ≈ b
ExactR = bd / (b + 2d)
R≈ d (when b >> d)
Valid when width-to-depth ratio b/d > 10. Error < 5% for b/d > 20. Commonly used for rivers, estuaries, and wide floodplains.

Longitudinal Profile & Energy Slope

S = tan(θ) ≈ sin(θ) d Water Surface (Energy Line) Channel Bed Flow Direction → For uniform flow: bed slope = water surface slope = energy slope = S
Uniform Flow Conditions
S₀= bed slope
Sw= water surface slope
Sf= friction/energy slope
SS₀ = Sw = Sf = S
Under uniform flow, depth d is constant along the channel. The Chezy equation V = C√(RS) uses S as the energy slope, which equals the bed slope for uniform conditions.

Engineering Applications

Where the Chezy Equation is applied in practice

Irrigation Canals

Design of lined and unlined canal systems for agricultural water delivery

River Engineering

Flood routing, channel stability analysis, and natural river modeling

Urban Drainage

Stormwater channel design and urban flood management systems

Hydropower

Headrace and tailrace channel design for hydroelectric facilities

Wastewater Channels

Gravity flow sewer design and treatment plant channel systems

Flood Channels

Spillway design, flood relief channels, and levee systems

Navigation Canals

Ship canal design maintaining required depths for navigation

Water Treatment

Settling basin and clarifier channel hydraulic design

Typical Chezy Coefficients

Reference values for common channel materials and conditions

Material / ConditionTypical CRangeComments
Smooth Concrete80–10070–110New, finished surface
Rough Concrete60–7550–80Worn or rough-formed
Earth Channel (clean)40–5535–60Uniform, no vegetation
Earth Channel (weedy)25–4020–45Moderate vegetation
Rock (smooth)65–8555–90Blasted, relatively smooth
Rock (rough)35–5530–60Irregular, jagged surface
Gravel Bed35–5030–55Uniform gravel size
Brick Work65–8060–85Glazed or smooth brick
Steel (welded)85–10075–105Smooth welded steel
Plastic / PVC90–11080–115Very smooth surface
Natural River (clean)45–6035–65Regular alignment
Natural River (weedy)25–4020–45Moderate aquatic growth
Natural River (stony)35–5030–55Cobble or boulder bed
Lined Canal (concrete)70–9060–95Standard canal lining
Vegetated Channel20–3515–40Dense grass coverage
Corrugated Metal45–6040–65Standard corrugation

Equation Comparison

How the Chezy Equation compares to other open channel flow formulas

Chezy Equation

V = C√(RS)
  • Simple and intuitive
  • Historically significant
  • Good for preliminary design
  • C varies with R and S
  • Less precise than Manning
  • Limited modern usage

Manning Equation

V = (1/n)R2/3S1/2
  • Most widely used today
  • Extensive n-value tables
  • Well validated empirically
  • Standard in most codes
  • Empirical (SI units only)
  • n selection can be subjective

Darcy-Weisbach

V = √(8gRS/f)
  • Theoretically rigorous
  • Works for all flow regimes
  • Dimensionally consistent
  • Applicable to pipes too
  • f determination complex
  • Requires Moody diagram

Equation Derivation

Deriving the Chezy Equation from fundamental hydraulic principles

Historical Development

The Chezy Equation was developed by French engineer Antoine Chézy in 1768 while studying flow in the River Seine and the Yvette river basin.

Force Balance Analysis

Gravitational driving force: Fgravity = γ × A × L × sin(θ) ≈ γ × A × L × S
Frictional resistance: Ffriction = τ₀ × P × L

Equilibrium Condition

γ × A × L × S = τ₀ × P × L → τ₀ = γ × (A/P) × S = γ × R × S

Final Equation

τ₀ = (γ/g) × (V²/C²) → γ × R × S = (γ/g) × (V²/C²)
V = C × √(R × S)

Engineering Assumptions

  • Steady, uniform flow conditions
  • Prismatic channel (constant cross-section)
  • Fully turbulent flow (high Reynolds number)
  • Small channel slope (sin θ ≈ tan θ ≈ S)
  • Hydrostatic pressure distribution

Worked Examples

Step-by-step solutions demonstrating practical applications

1 Rectangular Concrete Channel

Given

A rectangular concrete channel with width b = 3 m, flow depth d = 1.2 m, bed slope S = 0.001, and Chezy coefficient C = 65 m½/s.

Solution

A = b × d = 3 × 1.2 = 3.6 m²
P = b + 2d = 3 + 2(1.2) = 5.4 m
R = A/P = 3.6/5.4 = 0.667 m
V = C × √(R × S) = 65 × √(0.667 × 0.001) = 65 × 0.02582
V = 1.68 m/s  |  Q = V × A = 1.68 × 3.6 = 6.05 m³/s

2 Natural River Channel

Given

A natural river with hydraulic radius R = 2.5 m, slope S = 0.0003, and Chezy coefficient C = 40 m½/s.

Solution

V = C × √(R × S) = 40 × √(2.5 × 0.0003) = 40 × 0.02739
V = 1.10 m/s

3 Stormwater Drainage Channel

Given

A trapezoidal stormwater channel with bottom width b = 2 m, side slopes 2:1, depth d = 0.8 m, slope S = 0.005, and C = 55 m½/s.

Solution

A = (b + z×d)×d = (2 + 2×0.8)×0.8 = 2.88 m²
P = b + 2d√(1+z²) = 2 + 2(0.8)√5 = 5.578 m
R = A/P = 2.88/5.578 = 0.516 m
V = 55 × √(0.516 × 0.005) = 55 × 0.0508
V = 2.79 m/s  |  Q = 2.79 × 2.88 = 8.04 m³/s

4 Irrigation Canal (Lined)

Given

A lined irrigation canal with R = 0.8 m, S = 0.0004, and C = 80 m½/s (smooth concrete lining).

Solution

V = C × √(R × S) = 80 × √(0.8 × 0.0004) = 80 × 0.01789
V = 1.43 m/s

Engineering Notes

Important considerations for practical application

Assumptions & Limitations

  • Applies only to uniform, steady flow
  • Assumes prismatic channel geometry
  • Valid for fully turbulent flow (Re > 5000)
  • Small slope approximation (S < 0.1)
  • Does not account for transient flows

Common Mistakes

  • Using C as a constant regardless of R
  • Confusing energy slope with bed slope
  • Applying to non-uniform flow conditions
  • Neglecting freeboard in channel design
  • Using wrong units for C coefficient

Best Practices

  • Verify uniform flow assumption first
  • Use field measurements when possible
  • Apply safety factors (typically 1.2–1.5)
  • Check against permissible velocities
  • Validate with Manning's equation

Design Recommendations

  • Minimum velocity: 0.6 m/s (self-cleansing)
  • Maximum for earth: 0.9–1.5 m/s
  • Maximum for concrete: 3–5 m/s
  • Add 15–30% freeboard above design depth
  • Account for future degradation

Frequently Asked Questions

Common questions about the Chezy Equation and open channel flow

The Chezy Equation is V = C√(RS), where V is the mean flow velocity, C is the Chezy coefficient, R is the hydraulic radius, and S is the energy slope. Developed by Antoine Chézy in 1768, it describes uniform flow in open channels.

Used for uniform steady flow in open channels including canals, rivers, drainage systems, and irrigation channels.

Through empirical tables based on channel material, relationship with Manning's n (C = R^(1/6)/n), field measurements, or formulas like Bazin or Kutter.

Chezy uses coefficient C, Manning uses roughness n. Related by C = R^(1/6)/n. Manning's is more widely used due to better documented n-values.

Yes, it is specifically derived for fully turbulent flow conditions, which is the typical case in open channel hydraulics.

When C is properly calibrated from field data, predictions are within ±10-15%. Using estimated C values may introduce errors of ±20-30%.

R = A/P (area/wetted perimeter). It characterizes channel efficiency — larger R means less boundary friction relative to flow area.

Rectangular: R = bd/(b+2d). Trapezoidal: R = (b+zd)d/(b+2d√(1+z²)). Triangular: R = zd/(2√(1+z²)). Use our calculator's shape selector for automatic computation.

0.3-0.6 m/s in flat rivers, 0.6-1.5 m/s in irrigation canals, 1.0-3.0 m/s in natural rivers, 1.5-4.0 m/s in stormwater channels.

C = R^(1/6)/n. This means C varies with hydraulic radius. For a given channel, as depth increases, C also increases slightly.

Understanding Open Channel Hydraulics

Educational content on hydraulic engineering fundamentals

History of the Chezy Equation

Antoine Chézy (1718–1798) developed this equation around 1768 while working on the Yvette River water supply system near Paris.

Hydraulic Engineering Fundamentals

Open channel flow is characterized by a free surface exposed to atmospheric pressure. The driving force is gravity acting on the water surface slope, balanced by frictional resistance at channel boundaries.

Channel Design Principles

Practical channel design involves selecting dimensions to convey design discharge while meeting velocity constraints. The most efficient cross-section maximizes hydraulic radius for a given area.

Unit Converter

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