All articles
NetworksIntermediate
Equivalent Pipes (Dupuit): Collapse Series and Parallel Groups
Replace a chain or a twin with one pipe that has the same K in h = KQ². Choose L_eq, back-calculate D_eq, and skeletonize a network.
Updated 16 August 2026Solver360 Hydraulics9 min read
Key takeaways
- Write every pipe as h = K Q² with Darcy K.
- Series: add K. Parallel: add 1/√K.
- Pick any convenient L_eq; D_eq follows.
- Equivalents preserve head and total Q, not local velocity.
Dupuit’s equivalent-pipe method is how you thin a drawing before Hardy-Cross. A series chain or a parallel bundle becomes one pipe with the same head loss at the same total flow.
Pipe resistance
K = 8 f L / (π² g D⁵)
h = K Q |Q|
Combine
Series: K_eq = Σ K_i
Parallel: 1/√K_eq = Σ 1/√K_i
D_eq = [ 8 f_eq L_eq / (π² g K_eq) ]^{1/5}
Example: three series pipes with K = 1200, 4000, 1800 s²/m⁵. K_eq = 7000. Choose L_eq = 100 m, f_eq = 0.02 → D_eq ≈ 0.19 m. That single pipe can replace the chain inside a larger loop.
Open solver: See the chain in detailOpen solver: See the split in detailOpen solver: Drop the equivalent into a loopFrequently asked questions
The same idea works, but the parallel combination changes because h ∝ Q^1.85. Do not mix Darcy K-rules with C-factor pipes.
Keywords
equivalent pipe calculatordupuit equivalent pipeseries pipe equivalent diameterparallel pipes equivalentpipe resistance Kskeletonize water network
Related solvers
Equivalent Pipe (Dupuit)
Collapse series or parallel pipes into one equivalent diameter
Series Pipes Network
Calculate total head loss in series pipe systems
Parallel Pipes Network
Analyze flow distribution in parallel pipe systems
Hardy-Cross Looped Network
Balance looped pipe networks with iterative ΔQ corrections
More from the blog
Networks
Hardy-Cross for Looped Pipe Networks, Without the Mystery
Continuity first, then loop head balance. How ΔQ is calculated, a four-pipe worked loop, equivalent pipes, and when to move on to Newton-Raphson.
Pumps
Pumps in Series and Parallel: Combined Curves and the Duty Point
Series adds head at the same flow. Parallel adds flow at the same head. How to intersect a quadratic system curve and when unequal pumps idle.
Networks
The Three-Reservoir Problem: Junction Head and Flow Direction
Three tanks, three pipes, one unknown junction head. How to guess the piezometric level, reverse a flow, and why Newton-Raphson beats nested square roots.