Flow of Fluids - Bernoulli's Theorem, Pharmaceutical Engineering, B.Pharmacy As per PCI Syllabi

Flow of Fluids

Content: Types of manometers, Reynolds numbers and its significance, Bernoulli's theorem and its applications. Energy losses, Orifice meter, Venturimeter, Pitot tube and Rotameter.

Part 1:- Types of manometers
Part 2:- Reynolds numbers and its significance
Part 3:- Bernoulli's theorem and its applications
Part 4:- Energy losses, Orifice meter, Venturimeter, Pitot tube and Rotameter.

Part 3:

Bernoulli's Theorem

  A fluid in motion is subjected to several forces, which result in the variation of the acceleration and the energies in the flow phenomenon.

  When the principles of the law of energy is applied to the flow of the fluidsthe resulting equation is a Bernoulli's Theorem.

• Consider a pump working under isothermal conditions between points A and B.
• Bernoulli's theorem statement, "In a steady state the total energy per unit mass consists of pressure, kinetic and potential energies are constant".

• At point A one kilogram of liquid is assumed to be entering at point A.

Pressure energy = P_A / gρ

Where P_A = Pressure at point A

             g = acceleration due to gravity

              ρ_A = Density of the liquid

Potential energy of a body is defined as the energy possessed by the body by virtue of its position.

Potential Energy = X_A 

Kinetic Energy of a body is defined as the enery possessed by the body by virtue of its motion.

Kinetic Energy = U_A² / 2g

Total Energy at point A =  Pressure energy + Potential energy + Kinetic Energy.

Total Energy at point A = P_AV + X_A + U_A² / 2g

 According to the Bernoulli's Theorem the total energy at point A is constant.

Total Energy at point A = P_AV + X_A + U_A² / 2g  = constant.

After the system reaches the steady state, whenever one kilogram of liquid enters at point A, another one kilogram of liquid leaves at point B.

Total energy at point B  =  P_BV + X_B + U_B² / 2g

 P_AV + X_A + U_A² / 2g + Energy added by pump = P_BV + X_B + U_B² / 2g

V is specific volume and it is reciprocal of density.

Theoretically all kinds of the energies involves in fluid flow should be accounted, pump has added certain amount of energy.

During the transport some energy is converted to heat due to frictional Forces.

Energy loss due to friction in the line = F

Energy add by pump = W

 P_A/ρ_A + X_A + U_A² / 2g - F + W = P_B/ρ_B + X_B + U_B² / 2g

The above equation is called Bernoulli's Equation.

Applications:

• Bernoulli's Theorem is applied in the measurement of the rate of fluid flow using orifice meter, venturi meter etc.
• Bernoulli's theorem is applied in the working of centrifugal pumps. In these pumps, the kinetic energy is converted into pressure head, which helps in pumping th liquids.
• It is easy to measure heights and apply them as energy terms, which is a contribution of Bernoulli's theorem.