Equation of Mechanical Energy

[Esam]

The Energy Equation is a statement of the first law of thermodynamics. The energy equation involves energy, heat transfer and work. With certain limitations the mechanical energy equation can be compared to the Bernoulli Equation.
[h=The Mechanical Energy Equation in Terms of Energy per Unit Mass]3[/h] The mechanical energy equation for a pump or a fan can be written in terms of energy per unit mass:

p_in / ρ + v_in² / 2 + g h_in + w_shaft = p_out / ρ + v_out² / 2 + g h_out + w_loss (1)
where
p = static pressure
ρ = density
v = flow velocity
g = acceleration of gravity
h = elevation height
w_shaft = net shaft energy inn per unit mass for a pump, fan or similar
w_loss = loss due to friction

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 can be written as:

p_in / ρ + v_in² / 2 + g h_in = p_out / ρ + v_out² / 2 + g h_out + w_shaft + w_loss (2)
where
w_shaft = net shaft energy out per unit mass for a turbine or similar

Equation (1) and (2) dimensions are

• energy per unit mass (ft²/s² = ft lb/slug or m²/s² = N m/kg)

[h=Efficiency]3[/h] According to (1) a larger amount of loss - w_loss - result in more shaft work required for the same rise of output energy. The efficiency of a pump or fan process can be expressed as:

η = (w_shaft - w_loss) / w_shaft (3)

The efficiency of a turbine process can be expressed as:

η = w_shaft/ (w_shaft + w_loss) (4)

[h=The Mechanical Energy Equation in Terms of Energy per Unit Volume]3[/h] The mechanical energy equation for a pump or a fan (1) can also be written in terms of energy per unit volume by multiplying (1) with fluid density - ρ:

p_in + ρ v_in² / 2 + γ h_in + ρ w_shaft = p_out + ρ v_out² / 2 + γ h_out + w_loss (5)
where
γ = ρ g = specific weight

The dimensions of equation (5) are

• energy per unit volume (ft.lb/ft³ = lb/ft² or N.m/m³ = N/m²)

[h=The Mechanical Energy Equation in Terms of Energy per Unit Weight involves Heads]3[/h] 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_in² / 2 g + h_in + h_shaft = p_out / γ + v_out² / 2 g + h_out + h_loss (6)
where
γ = ρ g = specific weight
h_shaft = w_shaft / g = net shaft energy head inn per unit mass for a pump, fan or similar
h_loss = w_loss / g = loss head due to friction

The dimensions of equation (6) are

• energy per unitweight (ft.lb/lb = ft or N.m/N = m)

Head is the energy per unit weight.
h_shaft can also be expressed as:

h_shaft = w_shaft / g = W_shaft / m g = W_shaft / γ Q (7)
where
W_shaft = shaft power
m = mass flow rate
Q = volume flow rate

[h=Example - Pumping Water]3[/h] Water is pumped from an open tank at level zero to an open tank at level 10 ft. The pump adds four horsepowers to the water when pumping 2 ft³/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

or

h_loss = h_shaft - h_out (8)

Equation (7) gives:

h_shaft = W_shaft / γ Q
= (4 hp)(550 ft.lb/s/hp) / (62.4 lb/ft³)(2 ft³/s)
= 17.6 ft

• specific weight of water - 62.4 lb/ft³
• 1 hp (English horse power) = 550 ft. lb/s

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)
= 0.58