Backwater Calculation Notes

Last updated January 26, 2023
By Ian Story

Formulas to calculate the maximum flow in a pipe, based on backwater effects

Limit for velocity-controlled flow

$Q \leq 10A$

Limit for outlet-controlled flow

$Tailwater + Friction Loss + Velocity Head + Minor Head Losses - Velocity Head_a_b_v \leq Rim Elevation$

$Friction Loss + Velocity Head + Minor Head Losses \leq Rim Elevation - Tailwater + Velocity Head_a_b_v$

$v^2(\frac{n^2L}{0.351D^1^.^3^3} + \frac{1+k_e+k_b+k_j}{2g}) \leq Rim Elevation - Tailwater + \frac{v_a_b_v^2}{2g}$

$v \leq \sqrt{\frac{Rim Elevation - Tailwater + \frac{v_a_b_v^2}{2g}}{\frac{n^2L}{0.351D^1^.^3^3} + \frac{1+k_e+k_b+k_j}{2g}}}$

$Q \leq A\sqrt{\frac{Rim Elevation - Tailwater + \frac{v_a_b_v^2}{2g}}{\frac{n^2L}{0.351D^1^.^3^3} + \frac{1+k_e+k_b+k_j}{2g}}}$

Limit for submerged inlet-controlled flow

$Inlet Invert Elevation + Inlet Headwater Depth + Minor Head Losses - Velocity Head_a_b_v \leq Rim Elevation$

$Inlet Headwater Depth + Minor Head Losses \leq Rim Elevation - Inlet Invert Elevation + Velocity Head_a_b_v$

$cD(\frac{Q}{AD^0^.^5})^2 + Y - 0.5S + v^2\frac{1+k_b+k_j}{2g} \leq Rim Elevation - Inlet Invert Elevation + \frac{v_a_b_v^2}{2g}$

$\frac{Q^2}{A^2}(c + \frac{1+k_b+k_j}{2g}) \leq Rim Elevation - Inlet Invert Elevation + \frac{v_a_b_v^2}{2g} - Y + 0.5S$

$Q \leq A\sqrt{\frac{Rim Elevation - Inlet Invert Elevation + \frac{v_a_b_v^2}{2g} - Y + 0.5S}{c + \frac{1+k_b+k_j}{2g}}}$

Limit for unsubmerged inlet-controlled flow

$Inlet Invert Elevation + Inlet Headwater Depth + Minor Head Losses - Velocity Head_a_b_v \leq Rim Elevation$

$Inlet Headwater Depth + Minor Head Losses \leq Rim Elevation - Inlet Invert Elevation + Velocity Head_a_b_v$

$H_c + K(\frac{Q}{AD^0^.^5})^M + v^2\frac{1+k_b+k_j}{2g} \leq Rim Elevation - Inlet Invert Elevation + \frac{v_a_b_v^2}{2g}$