System Curves & the Duty Point

Where the pump’s capability meets the system’s demand.

introfluidsdesign
⏱️ About 20 min

You don’t choose flow and head independently—the system and pump negotiate an operating point where their curves intersect.

💡
The big idea: The operating (duty) point occurs where the pump curve intersects the system curve: the pump supplies exactly the head the system requires at that flow.
🎯 By the end, you'll be able to
  • Write a system curve as static head plus Q² loss terms
  • Explain why friction/minor losses scale approximately with Q²
  • Solve for the duty point by intersecting pump and system curves
  • Interpret how changing K or static lift shifts the duty point
📎 Helpful to know first

System curve: what the piping “asks for”

A piping system requires head to (1) overcome any static elevation change and (2) overcome losses due to friction and fittings.

Many loss terms scale with velocity squared (v²). Since v ∝ Q for a given diameter, it is common to model the total loss as proportional to Q².

\[ H_{\text{sys}}(Q)=H_{\text{static}}+K_{\text{sys}}Q^2 \]
Simple system curve model: static lift plus Q² losses.
✨ Why Q² is so common in piping losses

Darcy–Weisbach gives hf ∝ f(L/D)(v²/2g), and minor losses give hm ∝ K(v²/2g). For a fixed diameter, v is proportional to Q, so both terms scale approximately as Q².

Flow rate Q Head H pump curve system curve duty point
Duty point occurs at the intersection of the pump curve and the system curve.

Finding the duty point algebraically

If you approximate both curves with simple equations, you can solve for Q directly. At the duty point, Hpump(Q) = Hsys(Q).

📝 Worked example: A pump is modeled by H_pump = 50 − 200 Q² (H in m, Q in m³/s). A system is modeled by H_sys = 20 + 100 Q². Find the duty point flow rate Q and head H.
  1. Set H_pump = H_sys: 50 − 200 Q² = 20 + 100 Q².
  2. Rearrange: 50 − 20 = 300 Q² → 30 = 300 Q².
  3. Solve for Q²: Q² = 30/300 = 0.10.
  4. Take the square root: Q = √0.10 = 0.316 m³/s (3 significant figures).
  5. Compute head at duty point using either curve: H = 20 + 100(0.10) = 30 m.
✓ Q = 0.316 m³/s, H = 30 m
✏️ Practice: Solve for the duty point flow rate when H_pump = 45 − 200 Q² and H_sys = 20 + 50 Q² (H in m, Q in m³/s). Report Q.
m^3/s
Solution
  1. Set curves equal: 45 − 200 Q² = 20 + 50 Q².
  2. Rearrange: 45 − 20 = 250 Q² → 25 = 250 Q².
  3. Solve: Q² = 25/250 = 0.10.
  4. Compute Q: Q = √0.10 = 0.316 m³/s.
⚠️ Sanity check your roots

When you solve for Q², take the positive root for physical flow. Also do a quick check by substituting back into both curves to confirm they match.

Check your understanding

1. A simple system curve is often written as H_sys = H_static + K Q² primarily because:
Friction and minor losses scale ~ v², and for fixed diameter v ∝ Q, so losses scale ~ Q².
2. The duty point is the point where:
At the operating point, the head the pump can supply equals the head the system requires.
3. If the system resistance increases (larger K_sys) while the pump curve stays the same, the duty point flow will generally:
A steeper system curve intersects the pump curve at a lower flow rate.
🔎 Related lessons
Next: NPSH & Cavitation · soon
✅ Key takeaways
  • A system curve combines static lift and Q² loss terms.
  • At the duty point, H_pump(Q) equals H_sys(Q).
  • Increasing system resistance shifts the duty point to lower flow.
  • Simple algebraic models let you solve duty points cleanly by solving for Q² first.
➡️ Once you know the duty point, you still must ensure the pump can operate there safely—especially on the suction side, where cavitation can occur.
Want to test yourself on this? Try the Chemical Aptitude test →