Energy presents in the form of pressure, velocity, and elevation in fluids with no energy exchange due to viscous dissipation, heat transfer, or shaft work (pump or some other device). The relationship among these three forms of energy was first stated by Daniel Bernoulli (1700-1782), based upon the conservation of energy principle.Bernoulli’s theorem states that sum of pressure head, velocity head and gravitational potential head remains constant along a streamline of a steady, incompressible, irrotational and non-viscous flow with no other energy exchange due to heat or external work. This is the energy equation and is based on the law of conservation of energy.
The fluid flow must be steady state, incompressible, irrotational, non-viscous and laminar in order to use Bernoulli's theorem. Heat transfer and workdone should be zero in flow. The Bernoulli's equation for flow in a duct/channel in a section is given as:
\[{P_1 \over \rho g} + {V_1^2 \over 2g} + Z_1= {P_2 \over \rho g} + {V_2^2 \over 2g} + Z_2\]
where P is the static pressure, V is the velocity of flow, Z is the elevation head.
(Note: For horizontal duct/channel the elevation head is same for different sections.)
The above equation is valid for ideal fluid, when we are working with real fluid the losses (i.e. due to viscosity, friction, openings in duct, bending’s in duct, heat transfer etc.) need to be taken in account for the equation to validate.