BERNOULLI’S, ENERGY AND MOMENTUM EQUATIONS CLASS NOTES
BERNOULLI’S, ENERGY AND
MOMENTUM EQUATIONS CLASS NOTES
These equations are very helpful in doing the
analysis. It increase the deep understanding.
It increases knowledge and clarity.

Bernoulli Equation
The equation comes from Newton’s law of motion. Assumptions used in deriving Bernoulli’s equation:

Flow is steady

In compressible flow

Nonviscous flow (ideal fluid with zero viscosity)

No losses and no energy exchanges
Bernoulli’s equation states that “Total head of a flowing fluid is constant. Total energy of a flowing fluid is constant. One head can change into another form”.
Various Forms Of Heads in a Fluid Flow
Sr. No. 
Type of Head 
Type of Energy 
1. 
Pressure head = p /ρg 
pressure energy= p V 
2. 
Kinetic head = v ^{2}/2 g 
Kinetic energy = m v ^{2}/2 
3. 
Potential head =h 
pressure energy = m g h 
4. 
Piezometric head=Pressure head +Potential head=p/ρg+ h 
Bernoulli’s Equation is
p / ρg + v ^{2}/2 g + h = Constant=Total head
p_{1} / ρ_{1}g + v_{1}^{2}/2g + h_{1} = p_{2} / ρ_{2}g + v_{2}^{2}/2g + h_{2}
Energy Equation
Here the total head of a flowing fluid is not constant. There are always some losses or gains. It is because of friction, heat transfer, shaft work.
Similarity Between Bernoulli’s Equation And Energy Equation
Bernoulli equation and energy equation are derived directly from the first law of thermodynamics.
Dissimilarity Between Bernoulli’s Equation And Energy Equation
The losses or gains are associated with the energy equation. There are no losses or gains in Bernoulli equation. Bernoulli equation is an ideal equation. Energy equation is a real equation of fluid flow.
APPLICATION OF BERNOULLI’S EQUATION TO AIR/gas FLOW
Bernoulli’s Equation has been derived assuming incompressible flow. It is applicable
(i) to air or gas flow with Mach number(M ≤ 0.3).
(ii) No work is done by the gas or on the gas.
(iii) No energy exchange with the surroundings
3. Momentum equation
Momentum equation is for the kinetics of flow and considers the forces acting on the flowing fluid. It is based on Newton’s second law of motion to a control volume. It is based on the principle of conservation of linear momentum. This equation is obtains resultant force on the control volume. It is equal to the net rate of momentum flux through the control surface.
Momentum equation is a vector equation. It considers the forces and the velocities. Momentum equation gives the magnitude and direction of the impact force exerted on the control volume by its solid boundary.
Forces in Momentum equation

Surface forces (Coming from the surroundings)

Impact force on the control surface which is in contact with a solid boundary (normally not known)

Pressure force on the control surface

Body forces or gravitational forces
Analysis procedure

Select a control volume.

Decide and draw coordinateaxis of the control volume.

Calculate the total force which equal to the rate of change of momentum across the control volume

Find the pressure force F_{p}=p x surface area on which pressure is acting

Determine the body force F_{B} = Weight of the control volume
Applications Of The Bernoulli And Momentum Equations

Pitot tube

Orifice meter

Venturimeter

Nozzle
Differences between Bernoulli’s and energy equations
Sr. No 
Bernoulli’s Equation 
Energy Equation 
1. 
For an incompressible and non viscous fluid 
Applicable to incompressible and compressible fluids.It is applicable to any real fluid 
2. 
This equation does not account for heat and work interactions between two points 
This equation accounts for heat and work interactions between two points 
3. 
It is not applicable across a compressor or a turbine, 
It is applicable across a compressor or a turbine, 
4. 
Equation accounts only for potential energy, kinetic energy and pressure energy 
Equation accounts for all types of energy encountered in the flow. 
5. 
applicable to all liquids. It can apply to gases as long as Mach Number is < 0.3 
It applies to all fluids under all conditions. 