Enter the diameter of the pipe and the wetted central angle (θ) in the pipe cross-section into the calculator to estimate the flow rate. This calculator assumes a constant average velocity of 1 foot per second (1 ft/s = 0.3048 m/s).

Partial Pipe Flow Calculator

Enter any 2 values to calculate the missing variable

Related Calculators

Partial Pipe Flow Formula

A partially filled circular pipe carries liquid through only part of its cross-section. This calculator estimates discharge by finding the wetted segment area and multiplying it by an assumed average velocity of 1 ft/s.

r = \frac{D}{2}
A = \frac{r^2}{2}\left(\theta - \sin{\theta}\right)
Q = vA
Q = v \cdot \frac{r^2}{2}\left(\theta - \sin{\theta}\right)

Variable Definitions

  • Q = flow rate
  • v = average flow velocity
  • A = wetted cross-sectional area
  • D = pipe diameter
  • r = pipe radius
  • θ = wetted central angle

Important: The segment-area equation uses radians. If the angle is entered in degrees, it must be converted before applying the formula. The calculator handles the needed unit conversion internally.

How the Calculator Works

  1. Convert the pipe diameter to radius.
  2. Interpret the wetted central angle as the angle subtended by the liquid-filled arc.
  3. Compute the area of the circular segment occupied by the fluid.
  4. Multiply that area by the assumed average velocity of 1 ft/s.
  5. Convert the result to the selected output unit such as cfs, cms, L/s, or GPM.

Understanding the Wetted Central Angle

The wetted central angle is measured at the center of the pipe between the two points where the liquid surface meets the pipe wall. As the pipe fills, the angle increases from zero to a full circle.

Fill Condition Central Angle Meaning
Empty 
\theta = 0
 No wetted area and no flow.
Half Full 
\theta = \pi
 The liquid reaches the pipe centerline.
Completely Full 
\theta = 2\pi
 The entire circular area is filled.

If You Know Depth Instead of Angle

In field measurements, liquid depth is often easier to observe than the wetted angle. If depth is measured from the invert at the bottom of the pipe, you can convert depth to angle first and then use the main flow equation.

\theta = 2\cos^{-1}\left(\frac{r-y}{r}\right)

Here, y is the liquid depth. Once the angle is found, substitute it into the segment-area formula to estimate the flow area and discharge.

Example Calculation

Suppose the pipe diameter is 24 in and the wetted central angle is 90°. With the calculator’s default velocity of 1 ft/s:

r = \frac{24\ \text{in}}{2} = 12\ \text{in} = 1\ \text{ft}
\theta = 90^\circ = \frac{\pi}{2}
A = \frac{1^2}{2}\left(\frac{\pi}{2} - \sin\left(\frac{\pi}{2}\right)\right) \approx 0.2854\ \text{ft}^2
Q = 1 \cdot 0.2854 \approx 0.2854\ \text{ft}^3/\text{s}

This is approximately 128.10 GPM or 8.08 L/s.

Practical Notes

  • The output changes with geometry only, because velocity is fixed at 1 ft/s.
  • Larger diameters increase flow quickly because area scales with the square of the radius.
  • As the wetted angle increases, the wetted area increases nonlinearly.
  • This tool is useful for quick estimates, comparisons between fill levels, and checking how much of the pipe cross-section is actively carrying flow.

Limits of This Estimate

This calculator is intentionally simple. Real partially full pipe flow usually depends on more than geometry, including slope, roughness, hydraulic radius, entrance conditions, and downstream control. For design-level open-channel analysis, engineers often use Manning-based methods.

Q = \frac{1.49}{n}AR_h^{2/3}S^{1/2}
  • n = Manning roughness coefficient
  • Rh = hydraulic radius
  • S = slope

If your actual average velocity is higher or lower than 1 ft/s, the true discharge will scale proportionally with that velocity.