A General Mathematical model for A Turbo-Machine

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A General Mathematical model for A Turbo-Machine

• P M V Subbarao
• Professor
• Mechanical Engineering Department

Mathematical Tools for Development !!!
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The Superiority of Vector Parameters
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Eulers Work Equation
Euler Theory
Torque exerted by flow on blade row shaft
output torque Rate of change of Angular
momentum of fluid t
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Define, L as Angular momentum
Angular momentum is moment of linear momentum of
angular velocity, Vq
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The Boeing 747 Cruising at an Altitude of 10 km
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Turbo-jet Engine
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Even for a steady flow through a turbo-machine
Inlet rate of angular momentum
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Exit rate of angular momentum at exit
Change in Rate of angular momentum
Euler Theory
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Power
A change in Whirl Velocity of fluid can only
establish Power Exchange between fluid and rotor
in a turbo-machine !
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Conservation of Energy
First Law for Steady Flow Steady State Turbo
Machine
Avoid heat transfer across surface of a
turbo-machine.
This is known as Euler Turbo-machinery Equation.
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Read Through Euler Turbo-machinery Equation

• A Change in total enthalpy is proportional to
  change in tangential flow speed or tangential
  engine speed.
• For engines with little change in mean radius
  (inlet to exit), the change in total enthalpy is
  due to change in tangential flow velocity of the
  fluid.
• Creates a small change in enthalpy of fluid.
• For engines with large change in radius, the
  change in enthalpy is to a large degree due to
  the change in radius.
• The centrifugal/centripetal effect.
• Creates Large change in enthalpy of fluid.
• How to select a suitable type of action for a
  resource/demand.

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The Variations Principle of torque Generation
Centrifugal/ Centripetal Impulse Reaction
Power Generating Machines Radial flow turbines Pelton wheel De Laval Turbine Curtis Wheel Steam turbines Gas Turbines Kaplan Turbine
Power Consuming Machines Centrifugal pumps/compressors Axial flow pumps/ compressors
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The Variations Direction of Flow
Radial Mixed Axial
Power Generating Machines Turbine of a Turbocharger Francis Turbine Kaplan Turbine
Power Consuming Machines Centrifugal pump/ compressors Pumps fan
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MEL346 Syllabus

• Introduction turbo-machinery / history / types /
  classification.
• Euler equation for turbo-machines.
• Thermodynamics and fluid mechanics of
  turbo-machinery Mass, momentum and energy.
• Radial equilibrium theory and design of blades
  for a turbo-machine.
• Design of Hydraulic turbines.
• Axial flow turbines and compressors theory and
  design
• Centrifugal compressors Pumps
• Wind Turbines
• Micro Turbines

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References

• Fluid Mechanics, Thermodynamics of Turbomachinery
  S.L. Dixon.
• Principles of Turbomachinery R.K. TURTON.
• Handbook of Turbomachineryedited by Earl Logan,
  Jr. Ramendra Roy.
• The Design of High-Efficiency Turbomachinery and
  Gas Turbines David G Wilson T Korakianitis.
• Principles of Turbomachinery in Air Breathing
  Engines E A Baskharone.
• Steam Turbines for Modern Fossil-Fuel Power
  Plants Alexander S Leyzerovich.

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Attendance Policy

• Below 80 -- One grade less.
• Below 70 -- Two grades less.
• Below 60 -- Three grades less.

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No Break Through Just A Natural Evolution

• Knowledge of turbo-machines has evolved slowly
  over centuries without the benefit of sudden and
  dramatic breakthrough for more than 41500 yrs!
• Turbo-machines, such as windmills and
  waterwheels, are millenniums old.

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An Evolution from Water Wheel to Hydraulic Turbine

• Waterwheels, which dip their vanes into moving
  water, were employed in ancient Egypt, China, and
  Assyria.
• Waterwheels appeared in Greece in the second
  century B.C. and in the Roman Empire during the
  first century B.C.
• A seven-ft-diameter waterwheel at Monte Cassino
  was used by the Romans to grind corn at the rate
  of 150 kg of corn per hour,
• Waterwheels at Arles ground 320 kg of corn per
  hour.
• The Doomsday Book, based on a survey ordered by
  William the Conqueror, indicates the there were
  5,624 water mills in England in 1086.
• Besides the grinding of grain, waterwheels were
  used to drive water pumps and to operate
  machinery.

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• Agricola (14941555) showed by illustrations how
  water wheels were used to pump water from mines
  and to crush metallic ores in the 16th century.
• The pumps were driven by 14 waterwheels, each 12m
  in diameter, that were turned by the currents of
  the Seine.
• The undershot waterwheel, which had an efficiency
  of only 30, were used up until the end of the
  18th century.
• It was replaced in the 19th century by the
  overshot waterwheel with an efficiency of 70 to
  90.
• By 1850, hydraulic turbines began to replace
  waterwheels.
• The first hydroelectric power plant was built in
  Germany in 1891 and utilized waterwheels and
  direct-current power generation.
• However, the waterwheels were soon replaced with
  hydraulic turbines and alternating-current
  electric power.

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Evolution of Wind Turbines

• Although the use of wind power in sailing vessels
  appeared in antiquity, the widespread use of wind
  power for grinding grain and pumping water was
  delayed until
• the 7th century in Persia,
• the 12th century in England, and
• the 15th century in Holland.
• 17th century, Leibniz proposed using windmills
  and waterwheels together to pump water from mines
  in the Harz Mountains.
• Dutch settlers brought Dutch mills to America in
  the 18th century.
• This led to the development of a multiblade wind
  turbine that was used to pump water for
  livestock.
• Wind turbines were used in Denmark in 1890 to
  generate electric power.
• Early in the 20th century American farms began to
  use wind turbines to drive electricity generators
  for charging storage batteries.

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Discovery of Steam and Gas Turbines

• In the second century B.C. Hero of Alexandria
  invented rotors driven by steam and by gas, but
  these machines produced insignificant amounts of
  power.
• During the 18th and 19th centuries the
  reciprocating steam engine was developed and
  became the predominant prime mover for
  manufacturing and transportation industries.
• In 1883 the first steam turbines were constructed
  by de Laval whose turbines achieved speeds of
  26,000 rpm.
• In 1884 a steam turbine, which ran at 17,000 rpm
  and comprised 15 wheels on the same shaft, was
  designed and built by Charlie Parsons.
• The gas turbine was conceived by John Barber in
  1791, and the first gas turbine was built and
  tested in 1900 by Stolze .

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• Sanford Moss built a gas turbine in 1902 at
  Cornell University.
• At Brown Boveri in 1903, Armenguad and Lemale
  combined an axial-flow turbine and centrifugal
  compressor to produce a thermal efficiency of 3
  .
• In 1905 Holzwarth designed a gas turbine that
  utilized constant-volume combustion.
• This turbine was manufactured by Boveri and
  Thyssen until the 1930s.
• In 1911 the turbocharger was built and installed
  in diesel engines by Sulzer Brothers, and in 1918
  the turbocharger was utilized to increase the
  power of military aircraft engines.
• In 1939 the first combustion gas turbine was
  installed by Brown Boveri in Switzerland.
• A similar turbine was used in Swiss locomotives
  in 1942.
• The aircraft gas turbine engine (turbojet) was
  developed by Junkers in Germany around 1940.