A General Mathematical model for A Turbo-Machine

P M V SubbaraoProfessor

Mechanical Engineering Department

Mathematical Tools for Development !!!

The Superiority of Vector Parameters

kVjViVV zrˆˆˆ

Euler’s Work Equation

Torque exerted by flow on blade row = shaft output torque = Rate of change of Angular momentum of fluid =

Euler Theory:kVjViVV zrˆˆˆ

Define, L as Angular momentum

Angular momentum is moment of linear momentum of angular velocity, V

The Boeing 747 Cruising at an Altitude of 10 km

Turbo-jet Engine

Even for a steady flow through a turbo-machine:

Inlet rate of angular momentum :

Exit rate of angular momentum at exit:

Change in Rate of angular momentum:

Euler Theory: 1122 rVrVm

Power :

A change in Whirl Velocity of fluid can only establish Power Exchange between fluid and rotor in a turbo-machine !

This is known as Euler Turbo-machinery Equation.

First Law for Steady Flow Steady State Turbo Machine:

Avoid heat transfer across surface of a turbo-machine.

Conservation of Energy

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.

The Variations : Principle of torque Generation

Centrifugal/Centripetal

Impulse Reaction

Power Generating Machines

Radial flow turbines

Pelton wheelDe Laval Turbine Curtis Wheel

Steam turbines & Gas

TurbinesKaplan Turbine

Power Consuming Machines

Centrifugal pumps/compre

ssors

Axial flow pumps/

compressors

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

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

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.

Attendance Policy

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

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.

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.

• Agricola (1494–1555) 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.

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.

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 .

• 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.