Continuity Equation and Mechanical Energy

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As with many analyses, conservation of mass is used in fluids to find a relationship between mass flow rate and the volume flow rate.

Contents

Textbook Readings

Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 184-195.

Conservation of Mass

[1]It can be shown that the mass flow rate of a fluid is given by:

ṁ = ρVAc
Where:
ṁ = mass flow rate (kg/s)
ρ = fluid density (kg/m³)
Vavg = average fluid velocity (m/s)
Ac = cross sectional area (m²)

[2]Further analysis shows that mass flow rate and volume flow rate are related by:

ṁ = ρỼ = Ỽ/v
Where:
ṁ = mass flow rate (kg/s)
ρ = fluid density (kg/m³)
Ỽ = volume flow rate (m³/s)
v = specific volume (m³/kg)

Mechanical Energy

[3]Since mechanical energy is usually given by the kinetic energy and potential energy, and in fluids, by fluid work as well, it can be shown the fluid mechanical energy is given by:

emech = P/ρ + V2/2 + gz KJ/kg

And the change in mechanical energy by:

Δemech = (P2 - P1)/ρ + (V22 - V21)/2 + g(z2 - z1) (KJ/kg)

Mechanical Efficiency

Now mechanical efficiency can be shown to be equal to:

ɳmech = (Mechanical energy output)/(Mechanical energy input) = Emech,out/Emech,in = 1 - Eloss/Emech,out

References

"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).

  1. Textbook p.185
  2. Textbook p.186
  3. Textbook p. 193
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