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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 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 with an incompressible fluid can be expressed as:
w = (p1 - p2) / ρ (2)
A compressor works with compressible fluids and the specific work for an isentropic compressor process can be expressed with the help of:
p1 v1κ = p2 v2κ (3)
where
v = volume (m3)
κ = cp / cv - 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)
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)
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)
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):
hwater = (900 N.m/kg) / (9,81 kg/s2)
= 91,74 (m) water column
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
κair = 1,4 (J/kg.K) - ratio of specific heat air
Rair = 286.9 (J/kg. K) - individual gas constant air
Dividing by acceleration of gravity the head can be calculated using (7):
hair = (274200 N.m/kg) / (9,81 kg/s2)
= 27951 (m) air column
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