Storm Sewer Hydraulics

Storm sewer hydraulics deals with the design and analysis of gravity-flow pipe systems. Because storm sewers typically flow under gravity (except when surcharged), they are analyzed as open channels using Manning’s equation.

Full-Flow Pipe Geometry and Manning’s Equation

For a circular pipe of diameter $D$ (in feet) flowing completely full under gravity (not under pressure):

Area ($A_{full}$):

$$A_{full} = \frac{\pi D^2}{4}$$

Wetted Perimeter ($P_{full}$):

$$P_{full} = \pi D$$

Hydraulic Radius ($R_{full}$):

$$R_{full} = \frac{A_{full}}{P_{full}} = \frac{D}{4}$$

Full Flow Discharge ($Q_{full}$) in USCS Units:

$$Q_{full} = \frac{1.486}{n} A_{full} R_{full}^{2/3} S^{1/2} = \frac{0.463}{n} D^{8/3} S^{1/2}$$

Full Flow Velocity ($V_{full}$) in USCS Units:

$$V_{full} = \frac{1.486}{n} R_{full}^{2/3} S^{1/2} = \frac{0.590}{n} D^{2/3} S^{1/2}$$

Where:

  • $D$ = Pipe diameter (feet)
  • $S$ = Pipe slope (ft/ft)
  • $n$ = Manning’s roughness coefficient (concrete = 0.012, corrugated metal = 0.024, PVC = 0.009 to 0.011)

Partially Full Flow Hydraulics

Most storm sewers are designed to flow partially full. NCEES uses a standard Hydraulic Elements chart (or table) relating depth ratio ($y/D$) to hydraulic property ratios.

Let $y$ represent the actual depth of flow in the pipe:

  • Depth Ratio: $d_{ratio} = y/D$
  • Area Ratio: $A/A_{full}$
  • Wetted Perimeter Ratio: $P_w/P_{full}$
  • Hydraulic Radius Ratio: $R/R_{full}$
  • Velocity Ratio: $V/V_{full}$
  • Discharge Ratio: $Q/Q_{full}$

Key Features of the Hydraulic Elements Graph:

  • Maximum Velocity: Occurs at a depth ratio $y/D \approx 0.81$, where $V \approx 1.14 \cdot V_{full}$.
  • Maximum Discharge: Occurs at a depth ratio $y/D \approx 0.94$, where $Q \approx 1.07 \cdot Q_{full}$ (due to the rapid decrease in hydraulic radius as the pipe fills and wetted perimeter increases faster than area).

Design Velocity Constraints

To ensure storm sewers remain operational and do not deteriorate:

  • Minimum Velocity (Self-Cleansing Velocity): A minimum velocity of 2.0 ft/s is required during the design storm to keep sediments suspended and prevent clogging.
  • Maximum Velocity (Erosion Limit): A maximum velocity of 15.0 ft/s (for concrete pipes; lower for softer pipe materials) is recommended to prevent abrasive wear and erosion of the pipe wall.

Worked Example: Design Capacity and Surcharge Check

A 24-inch concrete storm sewer pipe ($n = 0.012$) is laid on a $0.50\%$ slope ($S = 0.005\text{ ft/ft}$). During a design storm event, the hydrologic analysis indicates a peak flow rate of $9.5\text{ cfs}$ must be carried.

1. Calculate the full-flow capacity ($Q_{full}$) and velocity ($V_{full}$) of the pipe. 2. Determine whether the pipe is surcharged (under pressure) or flowing under gravity, and find the actual depth ($y$) and velocity ($V$) of flow.

Solution:

Step 1: Convert units and calculate full-flow properties

  • Diameter $D = 24\text{ inches} = 2.0\text{ feet}$
  • Slope $S = 0.005\text{ ft/ft}$
  • $n = 0.012$

Calculate $Q_{full}$:

$$Q_{full} = \frac{0.463}{n} D^{8/3} S^{1/2}$$

$$Q_{full} = \frac{0.463}{0.012} (2.0)^{8/3} (0.005)^{1/2}$$
  • $(2.0)^{8/3} \approx 6.3496$
  • $(0.005)^{1/2} \approx 0.07071$ $$Q_{full} = 38.583 \cdot 6.3496 \cdot 0.07071 \approx 17.32\text{ cfs}$$

Calculate $V_{full}$:

$$V_{full} = \frac{0.590}{n} D^{2/3} S^{1/2}$$

$$V_{full} = \frac{0.590}{0.012} (2.0)^{2/3} (0.005)^{1/2}$$
  • $(2.0)^{2/3} \approx 1.5874$ $$V_{full} = 49.167 \cdot 1.5874 \cdot 0.07071 \approx 5.52\text{ ft/s}$$

Step 2: Determine actual flow conditions

Since the peak design flow rate ($Q = 9.5\text{ cfs}$) is less than the full capacity ($Q_{full} = 17.32\text{ cfs}$), the pipe is flowing under gravity and is not surcharged.

Calculate the discharge ratio:

$$\frac{Q}{Q_{full}} = \frac{9.5}{17.32} \approx 0.548$$

Step 3: Use Hydraulic Elements relations (or graph) to find depth and velocity

For a discharge ratio $Q/Q_{full} \approx 0.548$, we refer to standard hydraulic elements relations for circular pipes (often read from the NCEES chart):

  • A discharge ratio of $0.548$ corresponds to a depth ratio: $$\frac{y}{D} \approx 0.53$$
  • The corresponding velocity ratio at this depth ratio is: $$\frac{V}{V_{full}} \approx 1.02$$

Calculate depth ($y$):

$$y = 0.53 \cdot D = 0.53 \cdot 2.0\text{ ft} = 1.06\text{ feet} = 12.7\text{ inches}$$

Calculate velocity ($V$):

$$V = 1.02 \cdot V_{full} = 1.02 \cdot 5.52\text{ ft/s} = 5.63\text{ ft/s}$$

Check: The velocity $5.63\text{ ft/s}$ is within the safe range ($2.0\text{ ft/s} \le V \le 15.0\text{ ft/s}$).

Technical Pitfalls

  • Diameter Units: Remember to convert the diameter from inches to feet before using the USCS Manning’s full-flow formula.
  • Slope Units: Ensure the slope is in ft/ft (e.g., $1\% = 0.01$, $0.5\% = 0.005$).
  • Discharge vs Velocity curves: Remember that the velocity ratio and discharge ratio curves are different on the Hydraulic Elements chart. For $y/D = 0.53$, the discharge ratio is $0.55$, but the velocity ratio is $1.02$. Don’t mix up the curves.