8.4 Energy equation

Energy equation

The energy equation is not an exclusive pump system equation but an extension of the energy principle which states that energy cannot be created or destroyed but can only be converted to other forms.

Figure 8.4a Illustration of the energy equation.

The energy equation for the steady one-dimensional flow between stations I and Il becomes:

där

q = heat supplied from outside [J/kg]
I_t = internal shaft work extracted [Nm/kg]
u = internal heat [J/kg]
P = static pressure [N/m²]
ρ = density of liquid [kg/m³]
v  = flow velocity [m/s]
h = height above horizontal datum [m]

All the terms in the energy equation are calculated per kg of medium in flow.  The terms v²/2 och g•h respectively represent the kinetic and potential energy of the medium per kg. The term p/ρ has the dimension of energy and is often called pressure energy.

Shaft work can be supplied to or extracted from the liquid by pumps and turbines. If a section of pipe not including such equipment is examined, then the shaft work must be equal to zero (I_t = 0). If additional limitations are imposed such that no heat is transferred from outside (q = 0, adiabatic process) the energy equation becomes:

If the energy equation Equ. 8.4b is compared with Bernoulli Equation 8.2c it follows that:

u_II – u_I = g · hf    (Equ.8.4c)

i.e. the flow losses cause an increase in the internal heat. This increase of internal heat corresponds to an increase in temperature according to the relationship:

u_II – u_I = c_sp (T_II – T_I) = g · hf  (Equ. 8.4d)

where

c_sp = specific heat [J/kg,K]
T = absolute temperature [K]
h_f = head loss (m)