BERNOULLI’S, ENERGY AND MOMENTUM EQUATIONS

 

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https://www.mesubjects.net/wp-admin/post.php?post=7509&action=edit       Cavitation

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BERNOULLI’S, ENERGY AND MOMENTUM EQUATIONS

  1. Bernoulli Equation

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

  1. Flow is steady
  2. In compressible flow
  3. Non-viscous flow (ideal fluid with zero viscosity)
  4. No losses and no energy exchanges

Further, Bernoulli’s equation states that “Total head of a flowing fluid is constant or total energy of a flowing fluid is constant. However one form of head (energy) 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

Thus Bernoulli’s Equation is

p / ρg + v 2/2 g + h = Constant=Total head

p1 / ρ1g + v12/2g + h1 = p2 / ρ2g + v22/2g + h2

Energy Equation

Here the total head of a flowing fluid is not constant. There are always some losses or gains of energy / head on account of friction, heat transfer, shaft work and work to overcome friction.

 Similarity Between Bernoulli’s Equation And Energy Equation

Both 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 and there are no losses or gains while using Bernoulli equation. Thus Bernoulli equation is an ideal equation while 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 in-compressible flow.

It is applicable to air or gas flow when Mach number(M ≤ 0.3) and also when no work is done by the gas or on the gas i.e. there is no energy interchange with the surroundings during gas flow.

  1. 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 (based on the principle of conservation of linear momentum). This equation is obtained  resultant force on the control volume is equal to the net rate of momentum flux through the control surface.

 Further, 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

  1. Surface forces (Coming from the surroundings)
  2. Impact force on the control surface which is in contact with a solid boundary (normally not known)
  3. Pressure force on the control surface
  4. Body forces or gravitational forces

Analysis procedure

  1. Select a control volume.
  2. Decide and draw coordinate-axis of the control volume.
  3. Calculate the total force which equal to the rate of change of momentum across the control volume
  4. Find the pressure force Fp=p x surface area on which pressure is acting
  5. Determine the body force FB = Weight of the control volume

 Applications Of The Bernoulli And Momentum Equations

  1. Pitot tube
  2. Orifice meter
  3. Venturimeter
  4. Nozzle

Differences between Bernoulli’s and energy equations

Sr. No Bernoulli’s Equation Energy Equation
1. For an in-compressible and non viscous fluid Applicable both for in-compressible and compressible fluids

i.e. for 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.