centrifugal pump curve equation

Centrifugal Pump Curve Equation & Mathematical Model

The performance of a centrifugal pump is represented by its H-Q curve (Head vs. Flow Rate), which can be mathematically modeled using empirical or theoretical equations. Below are the key equations used to describe pump curves.

1. Theoretical Pump Curve Equation (Stepanoff Approximation)

The head-flow relationship for a centrifugal pump is often approximated by a second-order polynomial:

$$\mathrm{<span\; style="font-size:\; 18px;">H</span>}<span\; style="font-size:\; 18px;">=</span>{\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">0</span>}}<span\; style="font-size:\; 18px;">-</span>\mathrm{<span\; style="font-size:\; 18px;">k</span>}{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}$$H=H0−kQ2

Where:

• $\mathrm{<span\; style="font-size:\; 18px;">H</span>}$H = Total head (meters or feet)

• ${\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">0</span>}}$H0 = Shut-off head (head at zero flow)

• $\mathrm{<span\; style="font-size:\; 18px;">Q</span>}$Q = Flow rate (m³/s, GPM, etc.)

• $\mathrm{<span\; style="font-size:\; 18px;">k</span>}$k = System resistance coefficient

Interpretation:

• At Q = 0 (shut-off), the pump produces maximum head (H₀).

• As flow increases, head decreases quadratically.

2. Extended Pump Curve Model (Including Efficiency & Power)

A more detailed pump performance model includes:

A. Head-Flow Curve (H-Q)

$$\mathrm{<span\; style="font-size:\; 18px;">H</span>}<span\; style="font-size:\; 18px;">=</span>\mathrm{<span\; style="font-size:\; 18px;">a</span>}<span\; style="font-size:\; 18px;">-</span>\mathrm{<span\; style="font-size:\; 18px;">b</span>}\mathrm{<span\; style="font-size:\; 18px;">Q</span>}<span\; style="font-size:\; 18px;">-</span>\mathrm{<span\; style="font-size:\; 18px;">c</span>}{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}$$H=a−bQ−cQ2

• $\mathrm{<span\; style="font-size:\; 18px;">a</span>}<span\; style="font-size:\; 18px;">,</span>\mathrm{<span\; style="font-size:\; 18px;">b</span>}<span\; style="font-size:\; 18px;">,</span>\mathrm{<span\; style="font-size:\; 18px;">c</span>}$a,b,c = Pump-specific coefficients (from manufacturer data).

B. Efficiency-Flow Curve (η-Q)

$$\mathrm{<span\; style="font-size:\; 18px;">\eta </span>}<span\; style="font-size:\; 18px;">=</span>\mathrm{<span\; style="font-size:\; 18px;">d</span>}\mathrm{<span\; style="font-size:\; 18px;">Q</span>}<span\; style="font-size:\; 18px;">-</span>\mathrm{<span\; style="font-size:\; 18px;">e</span>}{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}$$η=dQ−eQ2

• Efficiency (η) peaks at the Best Efficiency Point (BEP).

C. Power-Flow Curve (P-Q)

$$\mathrm{<span\; style="font-size:\; 18px;">P</span>}<span\; style="font-size:\; 18px;">=</span>\frac{\mathrm{<span\; style="font-size:\; 18px;">\rho </span>}\mathrm{<span\; style="font-size:\; 18px;">g</span>}\mathrm{<span\; style="font-size:\; 18px;">Q</span>}\mathrm{<span\; style="font-size:\; 18px;">H</span>}}{\mathrm{<span\; style="font-size:\; 18px;">\eta </span>}}$$P=ηρgQH

• $\mathrm{<span\; style="font-size:\; 18px;">P</span>}$P = Power (W or HP)

• $\mathrm{<span\; style="font-size:\; 18px;">\rho </span>}$ρ = Fluid density (kg/m³)

• $\mathrm{<span\; style="font-size:\; 18px;">g</span>}$g = Gravity (9.81 m/s²)

(For imperial units: $\mathrm{<span\; style="font-size:\; 18px;">P</span>}<span\; style="font-size:\; 18px;">=</span>\frac{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}<span\; style="font-size:\; 18px;">\times </span>\mathrm{<span\; style="font-size:\; 18px;">H</span>}<span\; style="font-size:\; 18px;">\times </span>\mathrm{<span\; style="font-size:\; 18px;">S</span>}\mathrm{<span\; style="font-size:\; 18px;">G</span>}}{\mathrm{<span\; style="font-size:\; 18px;">3960</span>}<span\; style="font-size:\; 18px;">\times </span>\mathrm{<span\; style="font-size:\; 18px;">\eta </span>}}$P=3960×ηQ×H×SG, where SG = specific gravity)

3. System Curve Equation

The system head curve (resistance the pump must overcome) is:

$${\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">s</span>}\mathrm{<span\; style="font-size:\; 18px;">y</span>}\mathrm{<span\; style="font-size:\; 18px;">s</span>}}<span\; style="font-size:\; 18px;">=</span>{\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">s</span>}\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">a</span>}\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">i</span>}\mathrm{<span\; style="font-size:\; 18px;">c</span>}}<span\; style="font-size:\; 18px;">+</span>{\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">f</span>}\mathrm{<span\; style="font-size:\; 18px;">r</span>}\mathrm{<span\; style="font-size:\; 18px;">i</span>}\mathrm{<span\; style="font-size:\; 18px;">c</span>}\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">i</span>}\mathrm{<span\; style="font-size:\; 18px;">o</span>}\mathrm{<span\; style="font-size:\; 18px;">n</span>}}<span\; style="font-size:\; 18px;">=</span>{\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">s</span>}\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">a</span>}\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">i</span>}\mathrm{<span\; style="font-size:\; 18px;">c</span>}}<span\; style="font-size:\; 18px;">+</span>\mathrm{<span\; style="font-size:\; 18px;">R</span>}{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}$$Hsys=Hstatic+Hfriction=Hstatic+RQ2

• ${\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">s</span>}\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">a</span>}\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">i</span>}\mathrm{<span\; style="font-size:\; 18px;">c</span>}}$Hstatic = Static head (vertical lift + pressure difference)

• $\mathrm{<span\; style="font-size:\; 18px;">R</span>}$R = Friction factor (depends on pipe length, diameter, roughness)

• $\mathrm{<span\; style="font-size:\; 18px;">Q</span>}$Q = Flow rate

Operating Point:

• The intersection of the pump curve (H-Q) and system curve (Hₛᵧₛ-Q) determines the actual flow and head.

4. Affinity Laws (For Speed/Impeller Changes)

If pump speed ($\mathrm{<span\; style="font-size:\; 18px;">N</span>}$N) or impeller diameter ($\mathrm{<span\; style="font-size:\; 18px;">D</span>}$D) changes, the new performance is scaled by:

$$\frac{{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}<span\; style="font-size:\; 18px;">=</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}<span\; style="font-size:\; 18px;">=</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">D</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">D</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}$$Q1Q2=N1N2=D1D2$$\frac{{\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}<span\; style="font-size:\; 18px;">=</span>{<span\; style="font-size:\; 18px;">(</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}<span\; style="font-size:\; 18px;">)</span>}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}<span\; style="font-size:\; 18px;">=</span>{<span\; style="font-size:\; 18px;">(</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">D</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">D</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}<span\; style="font-size:\; 18px;">)</span>}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}$$H1H2=(N1N2)2=(D1D2)2$$\frac{{\mathrm{<span\; style="font-size:\; 18px;">P</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">P</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}<span\; style="font-size:\; 18px;">=</span>{<span\; style="font-size:\; 18px;">(</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}<span\; style="font-size:\; 18px;">)</span>}^{\mathrm{<span\; style="font-size:\; 18px;">3</span>}}<span\; style="font-size:\; 18px;">=</span>{<span\; style="font-size:\; 18px;">(</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">D</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">D</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}<span\; style="font-size:\; 18px;">)</span>}^{\mathrm{<span\; style="font-size:\; 18px;">3</span>}}$$P1P2=(N1N2)3=(D1D2)3

(Used for VFD adjustments or impeller trimming.)

5. Example Calculation

Given:

• Pump curve equation: $\mathrm{<span\; style="font-size:\; 18px;">H</span>}<span\; style="font-size:\; 18px;">=</span>\mathrm{<span\; style="font-size:\; 18px;">40</span>}<span\; style="font-size:\; 18px;">-</span>\mathrm{<span\; style="font-size:\; 18px;">0.1</span>}{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}$H=40−0.1Q2 (H in m, Q in L/s)

• System curve: ${\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">s</span>}\mathrm{<span\; style="font-size:\; 18px;">y</span>}\mathrm{<span\; style="font-size:\; 18px;">s</span>}}<span\; style="font-size:\; 18px;">=</span>\mathrm{<span\; style="font-size:\; 18px;">10</span>}<span\; style="font-size:\; 18px;">+</span>\mathrm{<span\; style="font-size:\; 18px;">0.05</span>}{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}$Hsys=10+0.05Q2

Find the operating point:

1. Set $\mathrm{<span\; style="font-size:\; 18px;">H</span>}<span\; style="font-size:\; 18px;">=</span>{\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">s</span>}\mathrm{<span\; style="font-size:\; 18px;">y</span>}\mathrm{<span\; style="font-size:\; 18px;">s</span>}}$H=Hsys:

   $$\mathrm{<span\; style="font-size:\; 18px;">40</span>}<span\; style="font-size:\; 18px;">-</span>\mathrm{<span\; style="font-size:\; 18px;">0.1</span>}{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}<span\; style="font-size:\; 18px;">=</span>\mathrm{<span\; style="font-size:\; 18px;">10</span>}<span\; style="font-size:\; 18px;">+</span>\mathrm{<span\; style="font-size:\; 18px;">0.05</span>}{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}$$40−0.1Q2=10+0.05Q2

2. Solve for $\mathrm{<span\; style="font-size:\; 18px;">Q</span>}$Q:

   $$\mathrm{<span\; style="font-size:\; 18px;">30</span>}<span\; style="font-size:\; 18px;">=</span>\mathrm{<span\; style="font-size:\; 18px;">0.15</span>}{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}\text{<span style="font-size: 18px;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="font-size:\; 18px;">⟹</span>\text{<span style="font-size: 18px;">\hspace{0.25em}\hspace{0.05em}</span>}\mathrm{<span\; style="font-size:\; 18px;">Q</span>}<span\; style="font-size:\; 18px;">=</span>\sqrt{\mathrm{<span\; style="font-size:\; 18px;">200</span>}}<span\; style="font-size:\; 18px;">\approx </span>\mathrm{<span\; style="font-size:\; 18px;">14.14</span>}\text{<span style="font-size: 18px;">\hspace{0.17em}</span>}\text{<span style="font-size: 18px;">L/s</span>}$$30=0.15Q2⟹Q=200≈14.14L/s

3. Find $\mathrm{<span\; style="font-size:\; 18px;">H</span>}$H:

   $$\mathrm{<span\; style="font-size:\; 18px;">H</span>}<span\; style="font-size:\; 18px;">=</span>\mathrm{<span\; style="font-size:\; 18px;">40</span>}<span\; style="font-size:\; 18px;">-</span>\mathrm{<span\; style="font-size:\; 18px;">0.1</span>}<span\; style="font-size:\; 18px;">(</span>\mathrm{<span\; style="font-size:\; 18px;">14.14</span>}{<span\; style="font-size:\; 18px;">)</span>}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}<span\; style="font-size:\; 18px;">\approx </span>\mathrm{<span\; style="font-size:\; 18px;">20</span>}\text{<span style="font-size: 18px;">\hspace{0.17em}</span>}\text{<span style="font-size: 18px;">m</span>}$$H=40−0.1(14.14)2≈20m

Operating point: 14.14 L/s @ 20 m

6. Key Takeaways

Pump curve equation describes how head varies with flow.
System curve defines the resistance the pump must overcome.
Affinity laws predict performance at different speeds/diameters.
Best Efficiency Point (BEP) is where the pump operates optimally.

Need Help?

If you have a specific pump curve or system data, I can help calculate the operating point or recommend a pump! Just provide:

• Pump curve coefficients (or manufacturer data).

• System requirements (flow, head, pipe details).

Would you like an Excel template for these calculations?

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