Lecture 10

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Lecture 10 Axial compressors 2

• Blade design
• Preliminary design of a seven stage compressor
• choice of rotational speed and annulus dimension
• estimation of the number of stages and stage by
  stage design
• variation of air angles from root to tip
• Compressibility effects in axial compressors
• Problem 5.1

2
Blade design

• We want
• blade must achieve required turning at maximum
  efficiency over a range of rotational speeds
• Correlated experiments are very valuable to
  estimate performance
• tests on single blades (effect of adjacent blades
  must be estimated)
• tests on rows of blades - so called cascades
• Linear cascades (rectilinear cascade).
  Mechanically simpler than annular to build. Flow
  patterns are simpler to interpret. More frequent.
  
• Annular. Many root tip ratios would be required.
  Does not satisfactorily reproduce flow in actual
  compressor

Annular cascade
Linear cascade
3
Blade design

• For a test camber angle ?, chord c and pitch s
  will be fixed and the stagger angle ? is changed
  by the turn table.
• Pressures and velocities are measured downstream
  and upstream by traversing instruments

4
Blade design

• Measurements are recorded as pressure losses and
  deflection
• Incidence is varied by turning the turn-table
• Mean deflection and loss are computed

5
Blade design

• Selecting more than 80 of stall deflection means
  risk of stalling at part load
• Select
• Stall reached when loss is twice of minimum loss
• e is dependent mainly on s/c and a2 for a given
  cascade. Thus, data can be reduced into one
  diagram

6
Blade design

• Air angles decided from design
• Pitch/Chord from Fig. 5-26
• Blade heigth from area requirement.
• Assume h/c (methods for selecting h/c are
  discussed in section 5.9 which is less relevant
  to course)
• s, c and h/c will then be known
• The blade outlet angle is determined from air
  outlet angle and empirical rule for deviation.
• Assume camber-line shape (for instance circular
  arc)

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Rotational speed and annulus dimensions

• Compressor for low cost, turbojet
• Design point specification
• rc 4.15, m 20 kg/s, T3 1100 K
• No inlet guide vanes (a10.0)
• Annulus dimension? Assume values on
• blade tip speed range 350-450 m/s. Values close
  to 350 m/s will limit stress problems
• axial velocity range 150-200 m/s. Try 150 m/s
  to reduce difficulties associated with shock
  losses
• root-tip ratio range 0.4-0.6
• You should know that these ranges represent
  typical values that you can assume on exam.

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Annulus dimension
Continuity gives the compressor inlet area

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Annulus dimension
A single stage turbine can designed to drive
this compressor if a rotational speed of 250
rev/s is chosen (Chapter 7). N 250 rev/s gives
Considering the relatively low Ut centrifugal
stresses in the root will not be critical and the
choice of a root-tip ratio of 0.5 will be
considered a good starting point for the design.
Recall the approximative formula
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Annulus dimension

• A check on the tip Mach number gives

This is a suitable level of Mach number.
Relative Mach numbers over about 1.2-1.3 would
require supercritical (controlled diffusion
blading). Too low values would result in a low
stage temperature rise.

• The compressor exit temperature is estimated
  assuming a polytropic efficiency, ? c,polytropic,
  of 90, which gives the exit area

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Annulus dimension

• The blade height at exit can now be calculated

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Estimation of the number of stages

• Assumed polytropic efficiency gave

• Reasonable stage temperature rises are 10-30 K.
  Up to 45 is possible for high-performance
  transonic stages. Let us estimate what we could
  get in our case

• We have derived the stage temperature rise as

• No IGV

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Estimation of the number of stages

• An overestimation of the temperature rise is
  obtained for a de Haller number equal to the
  minimum allowable limit 0.72

• Which gives a rotor blade outlet angle

• Setting the work done factor ? 1.0 yields

• We could not expect to achieve the design target
  unless we use

• Since 6 is close to achievable aerodynamic
  limits, seven is a reasonable assumption!

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Design goal

• Stage 1 and stage 7 are somewhat less loaded to
  allow for
• Stage 1 highest Mach numbers occur in first
  stage rotor tip gt difficult aerodynamic design.
  Inlet distortion of flow may be substantial. Less
  aerodynamic loading may alleviate these
  difficulties
• Stage 7 it is desirable to have an axial flow
  exiting the stator of the last stage gt a higher
  deflection is necessary in this stage which may
  be easier to design for if a reduction in goal
  temperature rise is allowed for
• This gives the following design criteria
  (assuming a typical work done distribution)

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Stage-by-stage design - Stage 1

• The change in whirl velocity for the first stage
  is

• A check on the de Haller number giveswhich is
  satisfactory. The diffusion factors will be
  checked at a later stage.

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Stage-by-stage design Stage 1

• For small pressure ratios isentropic and
  polytropic efficiencies are close to equal.
  Approximate the isentropic stage efficiency to be
  0.90. This gives the stage pressure rise as

• We have finally to choose a 3. Since a 1 in stage
  2 will equal a 3 in stage 1, this will be done as
  part of the design process for the second stage.

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Stage-by-stage design Stage 2

• Since we do not know a1 for the second stage we
  need further design requirements. We will use the
  degree of reaction. The degree of reaction for
  the first stage is (a3 in first stage is 11.06
  degrees. cos(11.06) 0.981)

• Since the root-tip ratio of the first stage is
  the lowest, the greatest difficulties with low
  degrees of reaction will be experienced in the
  first stage rotor. Thus, a good margin to 0.50
  has to be accepted.

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Stage-by-stage design

• Due to the increase in root-tip ratio for the
  second stage we hope to be able to use a ? of
  0.70

• Solving the two simultaneous equations for ß 1
  and ß 2 gives

• a 1 and a 2 are then obtained from (obtaining a
  1 means that the design of the first stage is
  complete)

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Stage-by-stage design - Stage 1

• The design of the first stage is now complete

• The de Haller number in thestator is

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Stage-by-stage design Stage 2

• The stage pressure rise of stage 2 becomes

• We have finally to choose a 3. Since a 1 in stage
  3 will equal a 3 in stage 2, this will be done as
  part of the design process for the third stage.

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Stage-by-stage design Stage 3

• Due to the further decrease in root-tip ratio to
  the third stage we hope to be able to use a ? of
  0.50

• Solving the two simultaneous equations for ß 1
  and ß 2 gives ß 151.24 and ß 228.00. This
  gives a to low de Haller number which can be
  dealt with by reducing the temperature increase
  over the stage to 24K. Repeating the calculation
  gives

• which is produces an ok de Haller number. a 1
  and a 2 are obtained from symmetry which is
  obtained when ? 0.50.

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Stage-by-stage design - Stage 2

• The design of the second stage is now complete

• The de Haller number in thestator is

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Stage-by-stage design Stage 3

• The stage pressure rise of stage 3 becomes

• We have finally to choose a 3. Since a 1 in stage
  4 will equal a 3 in stage 3, this will be done as
  part of the design process for the fourth.

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Stage-by-stage design Stage 4,5 and 6

• We maintain ? of 0.50 for stage 4, 5 and 6

• Since ? is 0.83 for the remaining stages, the
  stages 4, 5 and 6 are all designed with the same
  angles. Again the stage temperature rise is
  reduced to 24 K to maintain the de Haller number
  at an high enough number. Solving the two
  equations give

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Stage-by-stage design - Stage 3

• The design of the third stage is now complete

• The de Haller number in thestator is

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Stage-by-stage design stage 4,5,6

• The stage pressure rise and exit temperature and
  pressure of stages 4,5,6 become

• Stage 4, 5 and 6 are repeating stages, except for
  the stator outlet angle of stage 6 which is
  governed by the design of stage 7.

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Stage-by-stage design - Stage 4,5,6

• The design of stages 4,5 and 6 are now complete

• The de Haller number for thestators are

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Stage-by-stage design stage 7

• We maintain ? of 0.50 for stage 7

• The pressure ratio of the seventh stage is set by
  the overall requirement of an rc 4.15. The
  stage inlet pressure is 3.56 bar, which gives the
  required pressure ratio and temperature increase
  according to

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Stage-by-stage design - Stage 7

• The design of stage 7 is now complete

• Exit guide vanes can be incorporated to
  straighten flow before it enters the burner

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Annulus shape

• The main types of annulus designs exists
• Constant mean diameter
• Constant outer diameter
• Constant inner diameter
• Constant outer diameter
• Mean blade speed increases with stage number
• Less deflection required gt de Haller number will
  be greater
• U1 and U2 will not be the same!!! It will be
  necessary to use

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Variations from root to tip

• Calculate angles at root, mid and tip using the
  free vortex design principle, i.e.
• Cwr constant
• The requirement is satisfied at inlet since Cw 0
  (no IGV)
• Blade speed at root, mean and tip are

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Variations from root to tip

• Cw velocities are computed using the whirl
  velocity at the mid, Cw2,m 76.9 m/s together
  with the free vortex condition.
• Cwr constant
• The root and tip radii will have changed due to
  the increase in density of the gas. Compute the
  exit area according to

• Which gives the blade height, root and tip radii

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Variations from root to tip

• The rotor exit tip and root radii are assumed to
  be the average of the stage exit and inlet radii

• The free vortex condition gives

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Variations from root to tip

• The stator inlet angles a 2,r , a 2,m , a 2,t
  and rotor exit angles ß 2,r , ß 2,m , ß 2,t are
  obtained from

• Note the necessary radial blade twist for air
  and blade angles to agreee
• The reaction at the root is 0.697. The high
  value at the mean radius ensured a high enough
  value at the root
• Where do the highest stator Mach numbers occur
  ??

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Compressibility effects

• Fan tip Mach numbers of more than 1.5 are today
  frequent in high bypass ratio turbofans
• At some free-stream Mach number a local Mach
  number exceeding 1.0 will occur over the blade
• This Mach number is called the criticalMach
  number Mcr.
• A turbulent boundary layer will separate if the
  pressure rise across the shock exceeds that for
  a normal shock with an upstream Mach number of
  1.3 (Schlichting 1979). Keep this in mind!!!

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Effect of Mach number on losses

• As the Mach number passes the critical Mach
  number
• minimum loss increases and the range of
  incidence for which losses are acceptable is
  drastically reduced
• Simplified shock model
• The turning determines the Mach number at
  station B (Prandtl-Meyer relations - Appendix
  A.8 - you may skip that section). The more
  turning the higher the Mach number
• Shock loss Normal shock loss taken at
  averageMach number at A and B.
• Why less loaded first stage in example
• Less loaded first stage gt less turning gt
  reduced shock losses

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Supercritical design - Mrel gt 1.3

• Recall lecture 5

• How would you design a blade operating at Mach
  number 1.6 remembering that
• A turbulent boundary layer will separate if the
  pressure rise across the shock exceeds that for
  a normal shock with an upstream Mach number of
  1.3 (Schlichting 1979)

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Supercritical blading
Concave portion gt supercritical
diffusion Concave section after entrance region
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Supercritical blading
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Learning goals

• Know how to determine a multistage compressor
  design for a certain specification.
• This includes making assumptions design
  parameters
• Have an understanding of how compressor design
  must be adjusted for high Mach number effects.