AA210A Fundamentals of Compressible Flow Chapter 7 - Entropy generation and transport 10/6/20 1
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AA210AFundamentals of Compressible Flow
Chapter 7 - Entropy generation and transport
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7.1 Convective form of Gibbs’ equation
Gibbs equation following a fluid particle
Energy conservation equation
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7.2 The kinetic energy equation
Take the dot product of the velocity vector and the momentum equation.
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Use continuity again
Rearrange the pressure term
and the viscous term.
The kinetic energy equation
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Various terms in the kinetic energy equation
Reversible change between kinetic and internal energy due to work of compression or expansion
irreversible loss of kinetic energy due to friction
kinetic energy transport by pressure forces.
Kinetic energy transport by viscous forces.
Kinetic energy generation by body forces.
kinetic energy transport by convection.
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7.3 Internal energySubtract the kinetic energy equation from the energy equation.
General conservation form
Consider the source term
Recall the continuity equation.
Thermodynamic work
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7.4 Viscous dissipation of kinetic energy
Expand the velocity gradient tensor into symmetric and anti-symmetric parts
Look at the dissipation term
Carry out the sums.
The dissipation can be written as a sum of squares.
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Stokes’ hypothesis
Mean normal stress
Note also
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7.5 Entropy
Internal energy
Recall the Gibbs equation
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Use the Gibbs equation to replace the left-hand-side.
Put this result into conservation equation form. Use continuity again.
Add these two equations
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Heat flux term
Linear heat conducting material
Let
Conservation equation for entropyThe source terms are always positive.
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Entropy rise in an adiabatic box stirred by a fan.
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7.6 Problems
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Sample problems from previous midterm exams
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x
y
z
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