p in / ρ + v in 2 / 2 + g h in + E shaft
= p out / ρ + v out 2 / 2 + g h out + E loss (1b)
p = static pressure (Pa, (N/m 2 ))
ρ = density (kg/m 3 )
v = flow velocity (m/s)
g = acceleration of gravity (9.81 m/s 2 )
h = elevation height (m)
E shaft = net shaft energy per unit mass for a pump, fan or similar (J/kg)
E loss = hydraulic loss through the pump or fan (J/kg)
The energy equation is often used for incompressible flow problems and is called the Mechanical Energy Equation or the Extended Bernoulli Equation .
The mechanical energy equation for a turbine - where power is produced - can be written as:
p in / ρ + v in 2 / 2 + g h in
= p out / ρ + v out 2 / 2 + g h out + E shaft + E loss (2)
E shaft = net shaft energy out per unit mass for the turbine (J/kg)
Equation (1) and (2) dimensions are
According to (1) more loss requires more shaft work to be done for the same rise of output energy. The efficiency of a pump or fan process can be expressed as:
η = (E shaft - E loss ) / E shaft (3)
The efficiency of a turbine process can be expressed as:
η = E shaft / (E shaft + E loss ) (4)
The mechanical energy equation for a pump or fan (1) can also be written in terms of energy per unit volume by multiplying (1) with the fluid density - ρ :
p in + ρ v in 2 / 2 + γ h in + ρ E shaft
= p out + ρ v out 2 / 2 + γ h out + ρ E loss (5)
γ = ρ g = specific weight (N/m 3 )
The dimensions of equation (5) are
The mechanical energy equation for a pump or a fan (1) can also be written in terms of energy per unit weight by dividing with gravity - g :
p in / γ + v in 2 / 2 g + h in + h shaft
= p out / γ + v out 2 / 2 g + h out + h loss (6)h shaft = E shaft / g = net shaft energy head per unit mass for a pump, fan or similar (m)
h loss = E loss / g = loss head due to friction (m)
The dimensions of equation (6) are
Head is the energy per unit weight .
h shaft can also be expressed as:
h shaft = E shaft / g
= E shaft / m g = E shaft / γ Q (7)
E shaft = shaft power (W)
m = mass flow rate (kg/s)
Q = volume flow rate (m 3 /s)
Water is pumped from an open tank at level zero to an open tank at level 10 ft . The pump adds four horse powers to the water when pumping 2 ft 3 /s .
Since v in = v out = 0, p in = p out = 0 and h in = 0 - equation (6) can be modified to:
h shaft = h out + h loss
h loss = h shaft - h out (8)
Equation (7) gives:
h shaft = E shaft / γ Q
= (4 hp)(550 ft lb/s/hp) / (62.4 lb/ft 3 )(2 ft 3 /s)
= 17.6 ft
Combined with (8) :
h loss = (17.6 ft ) - (10 ft)
= 7.6 ft
The pump efficiency can be calculated from (3) modified for head:
η = (( 17.6 ft) - ( 7.6 ft) ) / (17.6 ft)
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