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					  Theory of Steam/gas Turbines




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




Modeling of Flow in Turbomachines….
           Steam Turbine Stage Grouping
• In view of the specifics of the steam path design, all stages of a
  condensing steam turbine are divided into four groups
• Governing stages
• Stages with high pressure and low volume discharge of steam
• Intermediate stages with relatively low pressure and high
  volume discharge of steam
• Last stages in the lowest-pressure range,
• Characterized by a very high volume discharge.
• The design of turbine stages and the dimensions of turbine
  elements in the steam path depend substantially on the volume
  discharge.
• The specific volume of steam increases in a turbine
  substantially from first to last stages.
Options for Power Generation Split among the Modules
                 of a Steam Turbine
              Power Split between the modules

                              50.0
                              45.0
 Percent of generated power




                              40.0

                              35.0
                              30.0
                                                           HP
                              25.0
                                                           IP
                              20.0
                                                           LP
                              15.0
                              10.0

                               5.0
                               0.0
                                     A   B    C    D   E
                                             OEM
       Multi-Staging of Large Steam Turbine
• Consider a case of isentropic expansion from 30MPa, 6500C to
  3500C.
• Isentropic Drop in enthalpy available for conversion: 540kJ/kg.
 Pure Impulse Option:
 • Velocity of jet: 1039 m/s
 • Velocity of blade : 519.5 m/s.
 • A blade tip speed of 519.5m/s is not possible, which is pretty
    high for practical uses.
 • Compounding is achieved by using more than one set of nozzles
    & blades in a series, keyed to a common shaft.
 • The steam pressure or the jet velocity is absorbed by the
    turbine in stages.
 • Three main types of compounded impulse turbines are:
 • a) Pressure compounded,
 • b) velocity compounded and
 • c) pressure and velocity compounded impulse turbines.
             Modules – Stages - Cascades

• The number of modules & number of stages in a module are
  important elements affecting the overall cost of the power
  plant.
• Each module contains number of stages.
• Each stage is a cascade of blades.
• The capacity of a stage depends on the type of aerofoil shape
  used for blade.
• It is a blade & Stage syndrome.
                   Blades of A Cascade

• The blades of an ST are the components that receive the most
  attention.
• Basic analysis generates a design parameter called degree of
  reaction.
• An efficient HP/IP blading is the use of variable reaction for
  each stage in the blade path length instead of the constant 50
  percent reaction.
• Improvements of 1 percent and higher in module efficiency
  have been reported by using VR.
             Optimization of Blade Design

• A typical optimization process for the HP or IP turbine could
  contain more than 40 variables, such as:
• pressure drop between the stationary and rotating blades,
• enthalpy drop across the stages,
• blade-path geometry etc.
• Subject to more than 100 design constraints
             Stage Loading and Flow Coefficient

Stage Loading Coefficient: Ratio of specific stage work output
and square of mean rotor speed.


                           h0,stage        h0,stage
                                      
                              U   2
                                             rm  2


Flow Coefficient: Ratio of the axial velocity entering to the
mean rotor speed.


                      V f ,stage exit       V f ,stage exit
                                      
                           U                    rm
    Regions of Design









           
            Design Variables of a Stage
• The following are stage data:
• Steam flow rate, (kg/sec) obtained from heat cycle
  calculation of turbine plant.
• Average diameter, d(m) for a stage obtained from a
  special diagram.
• Rotational frequency, n (rps) ,given/obtained from grid
  frequency
• Tangential velocity at average diameter u (m/s)
• Isentropic enthalpy drop, (kJ/kg) obtained from h-s
  diagram.
• Flow velocity, (m/s).
• Velocity ratio
•   Degree of reaction, assumed.
•   Isentropic enthalpy drop in nozzle cascade, kJ/kg
•   Isentropic enthalpy drop in moving blade cascade, kJ/kg
•   Specific volume before the nozzle cascade (theoretical),
    (m3/kg.
•   Specific volume before the moving blade cascade
    (theoretical), (m3/kg).
•   Theoretical steam velocity at nozzle exit, m/s.
•   Discharge coefficient of nozzle cascade,
•   Angle of velocity vector .
•   Exit area of nozzle cascade
                     Euler Axial Turbine Equation



                                                         Vtangential, out
   Vtangential, in


 Applying conservation of angular momentum,
 the torque, T, must be equal to the time rate of change of angular
 momentum in a stream tube that flows through the device

                       T  mVw,in rin  Vw,out rout 
                           
This is true whether the blade row is rotating or not.
                   Axial Flow Fluid Machines




The power, P of a fluid Machine

         P  T  mU Vw,in  Vw,out   mh0,in  h0,out 
                                        
Two Dimensional Theory of Turbines
                  a0         Station : 0

      Va0

                                    Station : 1
       b1                     Va1
        a1
                             Vr1
                                      Ub
                              Ub

                              a2      S2ation : 3
            Vr2              b2
                       Va2
        Ub
Axial Turbine Stator Exit/Rotor Inlet Velocity Triangle


                            Va1
               Va1    Vw1

                                  Vr1
                                              Vw1
                Vf1               b1    Vf1

Va0

      Vf0
                  U
  a2         a1       b2         b1
Va2                        Va1        Vr1
              Vr2

      Va1: Inlet Absolute Velocity
      Vr1: Inlet Relative Velocity
      Vr2: Exit Relative Velocity
      Va2:Exit Absolute Velocity

      a1: Inlet Nozzle Angle.
      b1: Inlet Blade Angle.
      b2: Exit Blade Angle.
      a2: inlet Nozzle Angle (next stage).
     Impulse-Reaction Axial Flow Turbine

       Vw2                             Vw1
                        U
        a2         a1       b2             b1
      Va2                        Va1            Vr1
                    Vr2




P  T  mU Vw,in  Vw,out   mh0,in  h0,out 
                               

                                      
P  T  mU Vw,in  Vw,out  mh0,in  h0,out 
                            
Impulse-Reaction turbine : Group Behaviour of
                  Aerofoils
• This utilizes the principle of impulse and reaction.
• There are a number of rows of moving blades attached to
  the rotor and and equal number of fixed blades attached to
  the casing.
• The fixed blades are set in a reversed manner compared
  to the moving blades, and act as nozzles.
• The fixed blade channels are of nozzle shape and there is
  a some drop in pressure accompanied by an increase in
  velocity.
• The fluid then passes over the moving blades and, as in
  the pure impulse turbine, a force is exerted on the blades
  by the fluid.
• There is further drop in pressure as the fluid passes
  through the moving blades, since moving blade channels
  are also of nozzle shape.
• The relative velocity increases in the moving blades.
        P  T  mU Vw1  Vw 2   mh00  h02 
                                   

                       h00  h01
       P  T  mU Vw1  Vw 2   mh01  h02 
                                  

                                    Va2        Va22  
P  T  mU Vw1  Vw 2   m h1   1   h2  
                           
                                    2          2  
                                                    
                                U
                a2         a1       b2         b1
                                         Va1        Vr1
          Va2        Vr2
                                    Vw21 V f21          Vw21 V f21  
P  T  mU Vw1  Vw 2   m h1  
                                                h2             
                                    2     2             2     2 
                                                                   
                               Vw21 V f21          Vw21 V f21  
      U Vw1  Vw 2    h1               h2             
                               2     2             2     2 
                                                              

                                 Vw21 V f21          Vw21 V f21  
        U Vw1  Vw 2    h1               h2             
                                 2     2             2     2 
                                                                

         Conservation of mass :
         1 A1V f 1  2 A2V f 2
Blade length is small when compared blade height.

Simplification of design: constant flow velocity

            V f 1  V f 2  1 A1   2 A2
Simplification of design: constant flow area


           A1  A2  1V f 1   2V f 2