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

Source: engineeringtoolbox.com