Stream Discharge Formula
Reference for stream discharge formula Q = A × v giving flow rate in m³/s or CFS.
Covers Manning equation, flood prediction, and hydrology applications.
The Formula
Q = A × v
Stream discharge measures the volume of water flowing past a point per unit time. It is the product of the stream's cross-sectional area and the average water velocity.
Variables
| Symbol | Meaning |
|---|---|
| Q | Discharge (m³/s or cubic feet per second) |
| A | Cross-sectional area of the stream (m² or ft²) |
| v | Average water velocity (m/s or ft/s) |
Example 1
A stream is 4 m wide, 1.5 m deep on average, with velocity 0.8 m/s
A = 4 × 1.5 = 6 m²
Q = 6 × 0.8
Q = 4.8 m³/s (4,800 liters per second)
Example 2
A river channel is 20 ft wide, 3 ft deep, flowing at 2 ft/s
A = 20 × 3 = 60 ft²
Q = 60 × 2
Q = 120 ft³/s (cubic feet per second, or "cfs")
When to Use It
Use the stream discharge formula when:
- Monitoring river flow for flood forecasting
- Assessing water availability for agriculture or cities
- Studying watershed hydrology and erosion
- Designing bridges, dams, or irrigation systems
Key Notes
- Velocity varies with depth and position across the channel — hydrologists measure at 0.6× depth for a single-point average, or average readings at 0.2 and 0.8× depth for greater accuracy
- Manning's equation (Q = (1/n) × A × R^(2/3) × S^(1/2)) can estimate discharge from channel geometry and slope alone, without direct velocity measurement — useful when current meters are unavailable
- During floods, both cross-sectional area and velocity increase simultaneously, so discharge can rise 10× or more for a modest rise in water level — the relationship is non-linear
- Permanent gauging stations use a pre-built stage-discharge rating curve to convert water level (easy to measure automatically) into discharge (difficult to measure directly in real time)
Key Notes
- Basic formula: Q = A × v: Volumetric flow rate Q (m³/s or L/s) equals cross-sectional area A times average velocity v. For a circular pipe: A = πr². This simple form underlies all open channel and pipe flow calculations.
- Manning's equation for open channels: Q = (1/n) × A × R^(2/3) × S^(1/2): n is Manning's roughness coefficient (concrete channel: n ≈ 0.013; natural stream: n ≈ 0.03–0.05); R = A/P is the hydraulic radius (area divided by wetted perimeter); S is the channel slope. The equation predicts flow in rivers, canals, and drainage ditches.
- Continuity — flow must be conserved: Q₁ = Q₂: For incompressible flow through varying cross-sections, A₁v₁ = A₂v₂. A narrower section must carry water faster. This is why rivers speed up through narrows and slow down in wide pools.
- Units — keep consistent: 1 m³/s = 1,000 L/s = 35.3 ft³/s (cfs) ≈ 264 gallons/second. GPM (gallons per minute) is common in plumbing; cfs in US hydrology; m³/s in engineering. Convert before calculating.
- Applications: Water flow rate calculations are used in stormwater drainage design (sizing pipes and channels to handle peak rainfall), irrigation system design, river flood modeling, water supply planning (daily demand vs pipe capacity), wastewater treatment plant sizing, and hydroelectric power assessment.