Constant Iimpeller Diameter Formulas |
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\( \dfrac{ Q_1}{Q_2 } \;=\; \left( \dfrac{n_1}{n_2} \right)^1 \) (Capacity varies directly with impeller diameter and speed) \( \dfrac{ h_1}{h_2 } \;=\; \left( \dfrac{n_1}{n_2} \right)^2 \) (Head varies directly with the square of impeller diameter and speed) \( \dfrac{BHP_1}{BHP_2} \;=\; \left( \dfrac{n_1}{n_2} \right)^3 \) (Horsepower varies directly with the cube of impeller diameter and speed) |
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| Symbol | English | Metric |
| \( BHP \) = brake horsepower | \(lbf-ft\;/\;sec\) | \(J\;/\;s\) |
| \( Q \) = capacity | \(gal\;/\;min\) | \(l\;/\;min\) |
| \( n \) = pump speed | \(rpm\;/\;min\) | \(rpm\;/\;min\) |
| \( h \) = total head | \( ft \) | \( m \) |
Affinity Laws, also called fan laws or pump laws, are fundamental relationships in fluid machinery engineering that describe how the performance of centrifugal pumps, fans, and similar rotodynamic equipment varies with changes in rotational speed or impeller diameter. These laws are derived from dimensional analysis using the Buckingham \(\pi\) theorem, which identifies the key dimensionless parameters governing dynamically similar machines. They apply strictly when comparing conditions that maintain geometric similarity (identical proportions in impeller and casing design), the same fluid (constant density and viscosity), and comparable flow regimes (similar Reynolds numbers and operating points relative to best efficiency).
The affinity laws are used to estimate changes in volumetric flow rate, developed head or pressure, and power consumption resulting from changes in shaft rotation speed or impeller diameter. Under the simplifying assumption that efficiency remains constant, these proportionalities allow one to scale known performance data to a new speed or size without requiring new experimental curves.
Constant Speed Formulas |
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\( \dfrac{Q_1}{Q_2} \;=\; \left( \dfrac{d_1}{d_2} \right)^1 \) (Capacity varies directly with impeller diameter and speed) \(\dfrac{ h_1}{h_2} \;=\; \left( \dfrac{d_1}{d_2} \right)^2 \) (Head varies directly with the square of impeller diameter and speed) \( \dfrac{BHP_1}{BHP_2} \;=\; \left( \dfrac{d_1}{d_2} \right)^3 \) (Horsepower varies directly with the cube of impeller diameter and speed) |
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| Symbol | English | Metric |
| \( BHP \) = brake horsepower | \(lbf-ft\;/\;sec\) | \(J\;/\;s\) |
| \( Q \) = capacity | \(gal\;/\;min\) | \(l\;/\;min\) |
| \( d \) = impeller diameter | \( in \) | \( mm \) |
| \( h \) = total head | \( ft \) | \( m \) |
The Affinity laws are widely used in hydraulic systems, HVAC design, process industries, and fluid mechanics to select, modify, or troubleshoot rotating equipment. They enable rapid approximations for variable-speed operation or impeller adjustments to match required duty points. However, the laws are approximations and hold best for moderate changes (typically within 10–20% for speed or diameter) where efficiency remains nearly constant, Reynolds number effects are minimal, and flow patterns do not shift significantly (e.g., avoiding large deviations into stall, cavitation, or regime changes). For larger alterations or non-ideal conditions, deviations occur, and full performance testing or more advanced modeling is recommended.
NPSHr Formula |
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| \(\dfrac{ NPSH_r1}{NPSH_r2} \;=\; \dfrac{d_1}{d_2}\) (Net Positive Suction Head Required by the pump varies directly with the impeller diameter) | ||
| Symbol | English | Metric |
| \( NPSH_r \) = net positive suction head required | \(lbf\;/\;in^2\) | \(N\;/\;m^2\) |
| \( d \) = impeller diameter | \( in \) | \( mm \) |
Rule of Thumb
While not an exact representation, the following relationships have been observed with regards to changing impeller diameters.
Shaft Deflection Formula |
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| \( \dfrac{ \theta_1 }{ \theta_2 } \;=\; \dfrac{ d_1}{d_2 }\) (Shaft Deflection (runout) measured prior to changing the impeller size varies with the impeller diameter) | ||
| Symbol | English | Metric |
| \( \theta \) (Greek symbol theta) = shaft deflection | \( in \) | \( mm \) |
| \( d \) = impeller diameter | \( in \) | \( mm \) |