Specific Work of Turbo Machines - Pumps, Compressors and Fans

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Specific work of pumps, fans, compressors and turbines

Specific work is the work per unit weight. Specific work in turbo machines as fans, pumps, compressors or turbines has the SI-units N.m/kg = J/kg = m²/s².

Specific Work of a Pump or Fan

Specific work of a pump or fan working with an incompressible fluid can be expressed as:

w = (p₂ - p₁) / ρ (1)

where

w = specific work (N.m/kg = J/kg = m²/s²)

p = pressure (N/m²)

ρ = density (kg/m³)

Specific Work of a Turbine

Specific work of a turbine with an incompressible fluid can be expressed as:

w = (p₁ - p₂) / ρ (2)

Specific Work of a Compressor

A compressor works with compressible fluids and the specific work for an isentropic compressor process can be expressed with the help of:

p₁ v₁^κ = p₂ v₂^κ (3)

where

v = volume (m³)

κ = c_p / c_v - ratio of specific heats (J/kg.K)

Specific work:

w = κ / (κ -1) R T₁ [( p₂ / p₁)^((κ-1)/κ) - 1] (4)

where

R = individual gas constant (J/kg.K)

T = absolute temperature (K)

Specific Work of a Gas Turbine

A gas turbine expands a compressible fluid and the specific work can be expressed as:

w = κ / (κ -1) R T₁ [1 - ( p₂ / p₁)^((κ-1)/κ)] (5)

Head in Turbomachines

The specific work can on basis of the energy equation be expressed with the head as:

w = g h (6)

where

h = head (m)

g = acceleration of gravity (m/s²)

Transformed to express head:

h = w / g (7)

Example - Specific Work of a Water Pump

A water pump works between 1 bar (1 10⁵ N/m²) and 10 bar (10 10⁵ N/m²). The specific work can be calculated with (1):

w = (p₂ - p₁) / ρ = ( (10 10⁵ N/m²) - (1 10⁵ N/m²) ) / (1000 kg/m³)

= 900 N.m/kg

Dividing by acceleration of gravity the head can be calculated using (7):

h_water = (900 N.m/kg) / (9,81 kg/s²)

= 91,74 (m) water column

Example - Specific Work of an Air Compressor

An air compressor works with air at 20 ^oC compressing the air from 1 bar absolute (1 10⁵ N/m²) to 10 bar (10 10⁵ N/m²). The specific work can be expressed with (4):

w = κ / (κ -1) R T₁ [( p₂ / p₁)^((κ-1)/κ) - 1]

= ( (1,4 J/kg.K)/ (1,4 - 1 J/kg.K) ) (286,9 J/kg. K) (273 + 20 K) [( (10 10⁵ N/m²) / (1 10⁵ N/m²) )^{((1,4 - 1 J/kg.K)/(1,4 J/kg.K))} - 1 ]

= 274200 N.m/kg

where

κ_air = 1,4 (J/kg.K) - ratio of specific heat air

R_air = 286.9 (J/kg. K) - individual gas constant air