centrifugal pump fluid mechanics

Centrifugal pumps operate on fundamental fluid mechanics principles to convert mechanical energy into hydraulic energy. Here’s a breakdown of the key concepts:

1. Working Principle

• Centrifugal Force: The rotating impeller imparts kinetic energy to the fluid, flinging it outward radially.

• Energy Conversion: The volute casing converts kinetic energy into pressure energy (Bernoulli’s principle).

Equation (Euler’s Pump Equation):

$$\mathrm{<span\; style="font-size:\; 18px;">H</span>}<span\; style="font-size:\; 18px;">=</span>\frac{<span\; style="font-size:\; 18px;">(</span>{\mathrm{<span\; style="font-size:\; 18px;">v</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">2</span>}}<span\; style="font-size:\; 18px;">\cdot </span>{\mathrm{<span\; style="font-size:\; 18px;">u</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}<span\; style="font-size:\; 18px;">-</span>{\mathrm{<span\; style="font-size:\; 18px;">v</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">1</span>}}<span\; style="font-size:\; 18px;">\cdot </span>{\mathrm{<span\; style="font-size:\; 18px;">u</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}<span\; style="font-size:\; 18px;">)</span>}{\mathrm{<span\; style="font-size:\; 18px;">g</span>}}$$H=g(vt2⋅u2−vt1⋅u1)

Where:

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

• ${\mathrm{<span\; style="font-size:\; 18px;">v</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">2</span>}}<span\; style="font-size:\; 18px;">,</span>{\mathrm{<span\; style="font-size:\; 18px;">v</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">1</span>}}$vt2,vt1 = Tangential velocities at outlet/inlet (m/s)

• ${\mathrm{<span\; style="font-size:\; 18px;">u</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}<span\; style="font-size:\; 18px;">,</span>{\mathrm{<span\; style="font-size:\; 18px;">u</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}$u2,u1 = Impeller peripheral velocities (m/s)

2. Key Fluid Mechanics Parameters

3. Critical Fluid Phenomena

1. Cavitation

• Cause: Local pressure drops below vapor pressure → bubble formation & collapse.

• Effect: Erosion, noise, efficiency loss.

• Prevention: Ensure NPSHₐ > NPSHᵣ + safety margin.

2. Net Positive Suction Head (NPSH)

   $${\text{<span style="font-size: 18px;">NPSH</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">a</span>}}<span\; style="font-size:\; 18px;">=</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">P</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">a</span>}\mathrm{<span\; style="font-size:\; 18px;">t</span>}\mathrm{<span\; style="font-size:\; 18px;">m</span>}}<span\; style="font-size:\; 18px;">-</span>{\mathrm{<span\; style="font-size:\; 18px;">P</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">v</span>}}}{\mathrm{<span\; style="font-size:\; 18px;">\rho </span>}\mathrm{<span\; style="font-size:\; 18px;">g</span>}}<span\; style="font-size:\; 18px;">+</span>{\mathrm{<span\; style="font-size:\; 18px;">h</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">s</span>}}<span\; style="font-size:\; 18px;">-</span>{\mathrm{<span\; style="font-size:\; 18px;">h</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">f</span>}}$$NPSHa=ρgPatm−Pv+hs−hf

• ${\mathrm{<span\; style="font-size:\; 18px;">h</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">s</span>}}$hs: Static suction head

• ${\mathrm{<span\; style="font-size:\; 18px;">h</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">f</span>}}$hf: Friction losses

3. Affinity Laws

• Governs pump performance with speed/impeller changes:

  $$\frac{{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">Q</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}<span\; style="font-size:\; 18px;">=</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}<span\; style="font-size:\; 18px;">,</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">H</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}<span\; style="font-size:\; 18px;">=</span>{<span\; style="font-size:\; 18px;">(</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}<span\; style="font-size:\; 18px;">)</span>}^{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}<span\; style="font-size:\; 18px;">,</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">P</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">P</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}<span\; style="font-size:\; 18px;">=</span>{<span\; style="font-size:\; 18px;">(</span>\frac{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">1</span>}}}{{\mathrm{<span\; style="font-size:\; 18px;">N</span>}}_{\mathrm{<span\; style="font-size:\; 18px;">2</span>}}}<span\; style="font-size:\; 18px;">)</span>}^{\mathrm{<span\; style="font-size:\; 18px;">3</span>}}$$Q2Q1=N2N1,H2H1=(N2N1)2,P2P1=(N2N1)3

4. Performance Curves

• Head vs. Flow (H-Q): Shows how head decreases with increasing flow.

• Efficiency vs. Flow (η-Q): Identifies Best Efficiency Point (BEP).

• Power vs. Flow (P-Q): Indicates motor sizing requirements.

https://www.pumpsandsystems.com/sites/default/files/2021-04/pump-curve-example.png

5. Practical Considerations

• Viscosity Impact: High viscosity reduces efficiency (use correction factors).

• System Curve: Intersection with pump curve determines operating point.

• Priming: Centrifugal pumps require fluid-filled casing to start.

6. Common Problems & Solutions

Summary

Centrifugal pumps rely on centrifugal force and energy conversion principles. Key equations (Euler’s, Bernoulli’s, Affinity Laws) govern their performance, while NPSH and viscosity critically impact operation. Always match the pump to the system curve for optimal efficiency.

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