Pump Energy Equation

Overview

Engineering reference data for Pump Energy Equation in fluid mechanics.

Key Formulas

Reynolds Number

$Re = \frac{\rho v D}{\mu}$

Ratio of inertial to viscous forces — determines flow regime.

Bernoulli's Equation

$P + \frac{1}{2}\rho v^2 + \rho g h = \text{const}$

Conservation of energy for steady, inviscid, incompressible flow.

Continuity Equation

$A_1 v_1 = A_2 v_2$

Conservation of mass for incompressible flow.

Darcy-Weisbach

$\Delta P = f \frac{L}{D} \frac{\rho v^2}{2}$

Pressure drop due to friction in a pipe.

Variables

Symbol	Description	Unit
$Re$	Reynolds number	—
$\rho$	Fluid density	kg/m³
$v$	Flow velocity	m/s
$D$	Characteristic dimension	m
$\mu$	Dynamic viscosity	Pa·s
$P$	Pressure	Pa
$f$	Darcy friction factor	—

Inline Pump and Fan Analysis

For inline installations where inlet and outlet velocities are equal ($v_2 = v_1$) and elevations are identical ($h_2 = h_1$), the actual head rise simplifies to:

$h_a = \frac{p_2 - p_1}{\gamma}$

This represents the most common installation configuration in industrial applications.

Specific Work

The specific work of a pump or fan is obtained by multiplying the head rise by gravitational acceleration:

$w = h_a \cdot g$

Variable	Description	Units
$w$	Specific work per unit mass	Nm/kg, J/kg
$h_a$	Actual head rise	m
$g$	Acceleration of gravity	9.81 m/s²

Head Loss Sources

Head loss through a pump or fan is proportional to the square of volume flow ($q^2$) and results from:

• Skin friction in the blade passages
• Flow separation
• Impeller blade casing clearance flows
• Other three-dimensional flow effects

The relationship between shaft work and actual head rise is:

$h_a = h_{shaft} - h_{loss}$

Worked Examples

Water Pump Example

An inline water pump operates between 1 bar and 10 bar with water density $\rho = 1000 \text{ kg/m}^3$:

$h_{water} = \frac{(10 \times 10^5) - (1 \times 10^5)}{1000 \times 9.81} = 91.7 \text{ m water column}$

Hot Air Fan Example

An inline fan with hot air ($\rho = 1.06 \text{ kg/m}^3$) adds 400 Pa to the flow:

$h_{air} = \frac{400}{1.06 \times 9.81} = 38.5 \text{ m air column}$

Expressed as equivalent water column for U-tube manometer measurement:

$h_{water} = \frac{400}{1000 \times 9.81} = 0.041 \text{ m} = 41 \text{ mm water column}$

Note: Head units reference the density of the flowing fluid. For air distribution systems, pressure is commonly measured using water column manometers.

References

• Original Source