P M V Subbarao

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Entropy Generators in Simplest Flow

• P M V Subbarao
• Professor
• Mechanical Engineering Department
• I I T Delhi

Identification of the Fourth Irreversibility
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Induction Deduction of Concepts
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Discovery of Entropy Generator
Reversible flow No vorticity generation between
x y,
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Discovery of Entropy Generator
Reversible flow No vorticity generation between
x y,
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The Concrete Part
With a vorticity generator between x y,
The Normal Shock !!!!
This equation relates the downstream Mach number
to the upstream Mach Number. It can be used to
derive pressure ratio, the temperature ratio, and
density ratio.
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Nature of Normal Shock

• The flow across the shock is adiabatic and the
  stagnation temperature is constant across a
  shock.
• The effect of increase in entropy across a shock
  will result in a decrease of stagnation pressure.

• This feasible only if

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Change in entropy for an ideal gas
Obtain a pressure temperature relation across
shock.

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Shock is possible only in supersonic flows
Infeasible
M
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General Adiabatic Flow
In general for an adiabatic Flow Field the
Stagnation Temperature is defined by the
relationship
Stagnation Temperature is constant throughout an
adiabatic flow field. T0 is also sometimes
referred to at Total Temperature T is sometimes
referred to as Static Temperature
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The Art of Reentry
The difference between Stagnation and Static
temperatures is a measure of the Kinetic Energy
of the flow Field. Largely responsible for the
high Level of heating that occurs on high speed
aircraft or reentering space Vehicles
stagnation (total) pressure Constant
throughout Isentropic flow field.
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What was Stagnation Temperature At Columbia
Breakup
Loss Of Signal at 61.2 km altitude 18.0 Mach
Number
T8 243 ?K
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How Nature Makes the Impossible Events into
Possible ?!?!?!
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Human Inventions Protection of Space Vehicle
Loss Of Signal at 61.2 km altitude 18.0 Mach
Number
T8 243 ?K
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The Ideal Simple Flow
Reversible flow No vorticity generation between
x y,
An isentropic flow.
It was not till Daniel Bernoulli published his
celebrated treatise in the eighteenth century
that the science of fluid Flow really began to
get off the ground G.I. Taylor
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Bernoulli equation

• The Bernoulli equation can be obtained as a
  scalar product of the Euler differential equation
  and a differential displacement vector.
• For steady flow cases, the differential distance
  along the particle path is identical with a
  distance along a streamline.
• Thus, the multiplication of Euler equation with
  the differential displacement , gives

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The terms in above equation must be rearranged as
follows. The first term starts with a scalar
product of two vectors, eliminating the Kronecker
delta and utilizing the Einstein summation
convention results in
For Steady flow this will be zero. The temporal
variation of kinetic entry is zero.
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Integrating above Equation results in
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Along a Stream line
For incompressible flows
which is the Bernoulli equation.
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The Greatest Natural Flow
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Barotropic Atmosphere

• The term barotropic is derived from the Greek
  baro, relating to pressure, and tropic, changing
  in a specific manner
• that is, in such a way that surfaces of constant
  pressure are coincident with surfaces of constant
  temperature or density.
• A barotropic atmosphere is one in which the
  density depends only on the pressure, ? ?(p).
• The isobaric surfaces are also surfaces of
  constant density.
• For an ideal gas, the isobaric surfaces will also
  be isothermal if the atmosphere is barotropic.

Along a isobar
The geostrophic wind is independent of height in
a barotropic atmosphere.
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Geostrophic Wind Patterns
Geostrophic wind is defined a horizontal velocity
field,
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Prediction of Simple Wind Patterns

• The knowledge of the pressure distribution at any
  time determines the geostrophic wind.
• This is true for large-scale motions away from
  the equator.
• The geostrophic wind becomes the total horizontal
  velocity to within 1015 in mid latitudes.