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Turbomachinery 1

Turbomachinery

Prof. A. TuranTel: +44 161 200 3804

Email: [email protected]

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Recommended Book

Dixon, S,L, “Fluid Mechanics, Thermodynamics of Turbomachinery”

Downloadable from Library Resources databases A to Z Knovel

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Other Recommended Texts

Massey, B,S, “Mechanics of Fluid”

Rogers/Mayhew “Engineering Thermodynamics Work and Heat Transfer”

Douglas/Gasiorek/Swaffield “Fluid Mechanics”

Turbomachinery Chapters In

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Aim of the Course

A fundamental understanding of turbomachinery

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What is turbomachine?“Turbo” is of Latin origin and implies which SPINS

or WHIRLS around

A machine with rotating shaft

Stream of fluid passing through it

Change in enthalpy in the fluid

Work transferred through the rotating shaft

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Turbomachines

Turbines: Absorb energy, produce shaft powerPumps, fans, blowers, compressors: absorb shaft power and increase the enthalpy (and the pressure)

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Various Forms of Turbomachines

Full-admissionPartial-admission

Depends on the flow “ran full” the rotor or not

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Partial Admission Machines: Pelton Wheel

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Partial Admission Machines

Squirrel Cage Turbine

Flow Direction

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Full Admission Turbomachines

Depends on the Exit Flow DirectionAxial flow machine: the flow is approximately parallel to the axis of the rotor.

Radial-flow machine: the flow may travel radically inward or radically outward, but near the axis the flow follow the axial direction.

Mixed-flow machine: the flow direction is neither to the axial nor the radial direction.

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Axial Flow Turbine

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Axial Flow Compressor

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Radial Flow Turbine and Compressor

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Mixed Flow Compressor

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Turbomachinery of different sizes

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Aircraft Gas Turbines

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Thermodynamic and Fluid Dynamic Review

The first law of thermodynamics

The steady flow energy equation (SFEE)

Perfect gas T-s and h-s diagrams

Constant-pressure lines on the T-s diagram

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The First Law of Thermodynamics

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Steady Flow Energy Equation (SFEE)

.2

11

22

2 )22

( WUhUhm −=−−+• upvh +=

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A turbine passes 20 kg/s of combustion product of known mean specific heat 1130 J/kgK. What is the shaft power output if the mean inlet temperature is 1200C and mean exhaust temperature is 600C, both measured with stagnation probes?From SFEE,

So W=20×1130×(1200-600)=13.56MW

Example: Use SFEE

.

12 )( Whhm TT −=−

)()( 12

.

12 TTTT TTmCphhm −=−

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Gibbs Equation

Perfect Gas T-s and h-s Diagrams

RTp =υ

pdvduTds +=

upvh +=

Perfect gas law:

Enthalpy

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Gibbs Equation

Constant-Pressure Lines on the T-s Diagram

RTp =υ

pdvduTds +=

RdTvdppdv =+dTCdu v= dTCdh p= RCC vp +=

A perfect gas

vdpdTCvdpRdTdTCTds pv −=−+=

pp CT

sT

=δδThus:

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Constant-Pressure Lines on the T-s Diagram

pp CT

sT

=δδ

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Efficiency of Turbomachine Components

The ratio of work transfer between the actual and the ideal processesThe typical ideal process is: isentropic

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Temperature Rise as a Function of Efficiency

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Total and Static EnthalpyCompressor Turbine

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Gas Turbine Maximum Specific Power

Low pressure ratio cycle: 1, K2, K3, K4High pressure ratio cycle: 1, A3, A4Middle range pressure ratios

)1(

)(

)(

/

1,

'2,

1,

1,'

,2,

.

1,'

,2,,

−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−−=

−−=Δ−=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛•

•

CpR

T

TTp

TSTp

TSTsT

pp

TC

TTC

hhhm

W

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Maximum Specific Power of Ideal Joule cycle

( )1113

)112(1)

43

11(3

)112(1)

341(3

)12()43(

/)1(/)1(

/)1(

/)1(

/)1(

−−⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

−⎟⎠⎞

⎜⎝⎛−

⎟⎠⎞

⎜⎝⎛

−=

−−−=

−−−=−=

−−

−

−

−

γγγγ

γγ

γγ

γγ

rr

r

Ctn

PCpTP

PCpT

PPCpT

PP

CpT

TTCpT

TTCpT

TTCpTTCpWWW

For giving limiting temperature T3 and T1

0113)(

2

/)1(/)1( =−⎟⎟⎠

⎞⎜⎜⎝

⎛= −− CpT

PCpT

PddW

rr

nγγγγ

Thus13/)1(

TTPr =− γγ

))1(2/(

13 −

⎟⎠⎞

⎜⎝⎛=

γγ

TTPr

or

rPPP

PP

==1

2

4

3

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Gas Turbine Maximum Specific Power

There is an optimum pressure ratio at which maximum net power is produced.

T3=1800K

T1=300K

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Euler’s Equation

Newton’s LawThe angular momentum

changing rate about the axis equals the torque on the rotor

qinletuoutletu TdmCrdmCr =− ∫∫ 1122

PowerinputdmCrdmCrinletuoutletu =− ∫∫ 1122 ωω

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Euler’s Equation

Conditions are uniform at inlet and outlet

Power input

From the steady flow energy equation

( )1122 uu CUCUm −

( )112212 uuTT CUCUhh −=−

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Example of Using Euler’s EquationWhat is the power output (kw) of a single-stage gas turbine which takes 6kg/s of high temperature and pressure gas, passes the flow through nozzles, from which it leaves at a direction 70 degrees from that of axial, at a velocity of 975 m/s, and discharge it from the rotor without swirl (cu,2 =0)? The mean diameter of the turbine blade is 1m, and the shaft speed is 10,000 rev/min.

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Solution:( ) 11112212 uuuTT CUCUCUhh −=−=−

smU /6.5235.0602000,101 =××=π

smCCu /20.91670sin11 =°×=2

12 )/(721,479 smhh TT −=−

The steady flow energy equation can be used to find the power output

MWmhhW TT 878.26721,479)( 12 =×=×−−=

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Velocity DiagramsFlow direction at a blade stage

“Simple” velocity Diagram: Constant U and CN

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Absolute and Relative Velocities

The absolute velocity is in the reference frame of the stators or nozzles (the designation c)

The relative velocity is relative to the moving rotor blades (the designation w)

Angle conventions: The angles are given with the axial direction

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The Working or Loading Coefficient, ψ

Work coefficient is positive for turbines and negative for compressors and pumps. Turbine stage: “highly loaded” or “high-work” --> work coefficient usually above 1.5. Turbine stage: “lightly loaded” or “low-work” --> work coefficient usually below 1.0. Compressor stages: “highly loaded” --> above 0.5Compressor stages: “lowly loaded” --> below 0.3

22

)(uuC

uh uT Δ

−=Δ−

=ψ

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Velocity Diagram Fixed by Loading

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The Flow Coefficient, φ

In a simple velocity diagram the flow coefficient is constant and refers to the stage as a whole. In the general case, both cz and u vary through the stage for any stream surface. There will be different flow coefficient at rotor inlet and at rotor outlet, and the flow coefficient varies with radius.

uCz=φ

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Velocity Diagram Fixed by Loading and Flow Coefficient

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The Stage Reaction, Rn.

The ratio of the change in static enthalpy in the rotor to the total enthalpy change in the stage.

stageT

rotorst

hh

R,

,

ΔΔ

=

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The Stage Reaction, Rn.

22

22

21 CChh Tst −+Δ=Δ

)( 1,12,2

1,2,

uu

TTT

CuCuhhh

−=

−=Δ

( )1,12,2

21

22 2/1

uuT

st

CuCuCC

hhR

−−

−=ΔΔ

=

( )( )

uCC

CCuCCCC

R

uu

uu

uzuz

21

2/1

1,2,

1,2,

21,

222,

2

+−=

−−−+

−=

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Velocity Diagram Fixed by Loading, Flow Coefficient, and Reaction

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Typical Reactions

Zero --ImpulseFiftyHundred

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A Zero Reaction Velocity Diagram

It has axial stage exit flows and the reasonably high work coefficient of 2.0

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50% or Symmetrical Reaction

( )( )

uCC

CCuCCCC

R

uu

uu

uzuz

21

2/1

1,2,

1,2,

21,

222,

2

+−=

−−−+

−=

C1

W1C2

W2

C1W1

W2C2

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High Reaction

A diagram with 100% reaction has -Cu1=Cu2.

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Summary

Categorise of turbomachineryThermodynamic and fluid dynamic for turbomachinery

SFEEEuler

Velocity diagrams: loading, flow coefficient, reaction

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