Elementary Mechanics of Fluids

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Elementary Mechanics of Fluids

• CE 319 F
• Daene McKinney

Energy Equation
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Energy

• First Law of Thermodynamics
• Where
• E energy of a system (extensive property)
• Q Heat added to the system
• W Work done by the system
• Where
• Eu Internal energy of the system
• Ek Kinetic energy of the system
• Ep Potential energy of the system
• Intensive property (energy per unit mass)

Energy of a system changes as heat is added to
the system or work is done on the system
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Energy

• Kinetic energy per unit mass
• Potential energy per unit mass

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Energy Equation

• Reynolds Transport Theorem

• b e Bsys E

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Work

• Rate of Work
• Shaft work work done
  by a mechanical device which crosses the CS
• Flow work work done
  by pressure forces on the CS
• Viscous work work
  done by viscous stresses at the CS

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Flow Work

• Work occurs at the CS when a force associated
  with the normal stress of the fluid acts over a
  distance. The normal stress equals the negative
  of the fluid pressure.

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Energy Equation
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Kinetic Energy Correction Factor
For nonuniform flows, this term requires special
attention
We can modify this kinetic energy term by a
dimensionless factor, a, so that the integral is
proportional to the square of the average velocity
where
and
Kinetic energy correction factor
so
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Energy Equation for Pipe Flow
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Ex (7.35)

• Given PA 10 psi (12 in dia. pipe), PB 40 psi
  (6 in dia. pipe), Q3.92 cfs
• Find Horsepower of pump
• Solution

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EGL HGL for a Pipe System

• Energy equation
• All terms are in dimension of length (head, or
  energy per unit weight)
• HGL Hydraulic Grade Line
• EGL Energy Grade Line
• EGLHGL when V0 (reservoir surface, etc.)
• EGL slopes in the direction of flow

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EGL HGL for a Pipe System

• A pump causes an abrupt rise in EGL (and HGL)
  since energy is introduced here

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EGL HGL for a Pipe System

• A turbine causes an abrupt drop in EGL (and HGL)
  as energy is taken out
• Gradual expansion increases turbine efficiency

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EGL HGL for a Pipe System

• When the flow passage changes diameter, the
  velocity changes so that the distance between the
  EGL and HGL changes
• When the pressure becomes 0, the HGL coincides
  with the system

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EGL HGL for a Pipe System

• Abrupt expansion into reservoir causes a complete
  loss of kinetic energy there

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EGL HGL for a Pipe System

• When HGL falls below the pipe the pressure is
  below atmospheric pressure

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HW (7.8)
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HW (7.16)
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HW (7.27)
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HW (7.27)
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HW (7.41)