Multi-Stage Axial-Flow Compressor Design Analysis
Frederick Avyasa Smith
MECE E4304: Turbomachinery
Prof. Dr.P.Akbari
December 8th, 2014
2
Table of Contents
Introduction ................................................................................................................................. 3
Analysis ......................................................................................................................................... 4 Section A: Relative Mach Number ............................................................................................................................ 4 Section B: Annulus Dimensions................................................................................................................................. 6 Section C: Number of Stages ........................................................................................................................................ 9 Section D: Initial Design Parameters .................................................................................................................. 11 Section E: Final Design Parameters ..................................................................................................................... 15 Section F: Hub-To-Tip Flow....................................................................................................................................... 22
Conclusion .................................................................................................................................. 29
Appendix ..................................................................................................................................... 32
References .................................................................................................................................. 36
3
Introduction It is the goal of this analysis to design a multi-stage axial-flow compressor. It is
noted that this is a preliminary analysis. All assumptions will be explained and justified.
The desired design parameters for the compressor can be found in the table below:
Table 1 Summary of Initial Design Parameters for Proposed Axial-Flow Compressor
There are several assumptions that must be made initially before the preliminary analysis
takes place. The working fluid will be air. Because this compressor is axial, air enters at
normal atmospheric conditions. General properties of atmospheric air will be utilized. A
modest axial velocity will be chosen, and it will be constant across the compressor. By
holding axial velocity as constant the design procedure can be simplified. The compressor
will have no inlet guide vanes to reduce weight and noise. Furthermore, a repeated stage
assumption will not be made thus allowing for more options when the aerodynamic design
of stages is considered. Work done factors through the compressor must b provided to
account for the error in stage temperature rise calculations. This error stems from axial
velocity not being constant, and varying from blade hub to tip. It is not until around the
fourth stage that axial velocity will achieve a fixed profile. Reasonable values for the work
done factor of a multi-stage-axial-flow compressor are chosen for the first, second, third,
and ongoing stages. However, for preliminary analysis the assumption is made that axial
velocity remains constant radially along the blade. Work done factor and constant axial
velocity radially will both be utilized. Finally, we will consider processes through the
compressor to be reversibly adiabatic. Therefore all calculation that are bases on isentropic
processes will be valid. In addition the ideal gas law will be heavily utilized because air is
the working fluid. Initial parameters for this axial-flow compressor are summarized in the
table below:
Name Value
CompressorPressureRatio 4.15
AirMassFlowRate 20kg/s
CompressorPolytropicEfficiency 0.9
BladeTipSpeedofFirstStage 355.3m/s
Hub-to-TipRatioofFirstStage 0.5
DesignParameters
4
Table 2 Summary of Initial Assumptions for Proposed Axial-Flow Compressor
The analysis of the axial compressor will be broken into seven sections labeled A through F.
Assumptions for each section will be listed along with explanations of used calculations. A
summary of all found data will be provided at the end of the analysis. The code that will be
used for repetitive calculations that apply to the stages will be provided in the appendix.
This preliminary analysis will be heavily based on concepts, methods, and calculation from
the textbook Principles of Turbomachinery by Seppo A. Korpela. [1]
Analysis Section A: Relative Mach Number
First, Mach number relative to the tip will be explored in order to limit the losses in
the compressor. The relative Mach number is high at the tip because of large relative
velocity. The flow can be transonic without impairing the performance of the compressor.
However, this is not the case for supersonic flow. Supersonic flow implies a large relative
Mach number, which will cause shock losses at the tip. Therefore it is imperative that the
flow remains only transonic. It is sufficient to do a check only on Stage 1 because this is
where relative flow will be the highest in the compressor. This is due to the fact that in
stage 1 the inlet airflow is axial and the radius of the blade is the largest. This is not the case
for the remainder of the stages because the stators have the same effect as inlet guide
vanes and the area of the compressor decreases with each stage. Thus, the blade rad ius also
shrinks. By using trigonometry on the velocity diagrams of stages 2-7 at the tip it can be
Name Value
AmbientStagnationPressure 101.3kPa
AmbientStagnationTemptrature 288K
ConstantSpecificHeatofAir 1005J/kg-K
SpecificHeatRatioofAir 1.4
IdealGasConstatforAir 287J/kg-K
AxialVelocityofAir 150m/s
WorkDoneFactorStage1 0.98
WorkDoneFactorStage2 0.93
WorkDoneFactorStage3 0.88WorkDoneFactorforOngoingStages 0.83
InitialAssumptions
5
seen that the relative velocity will be less than in Stage 1. Stagnation speed of sound is first
found using ambient air properties.
πͺππ= βππΉπ»ππ
= πππ.πππ
π
π = 1.4
π = 287 π½
ππβπΎ
ππ1= 288πΎ
From the stagnation speed of sound relative stagnation Mach number can be found. Note
that the relative velocity is found using trigonometry from the first stageβs velocity diagram
at the tip.
π΄πππΉ=
πΎπ
πͺππ
= π. ππ
πΆπ1= 340.17
π
π
π1 = βππ₯2 + ππ‘ = 385.66
π
π
ππ₯ = 150π
π
ππ‘ = 355.3 π
π
Next static temperature at the inlet can be found using ambient air properties and the
absolute velocity at the inlet. Note that because the air enters axially the absolute velocity is
equal to the axial velocity.
π»π = π»ππβ
π½ππ
ππͺπ
= πππ.ππ²
ππ1= 288πΎ
π1 = 150π
π
πΆπ = 1005 π½
ππβπΎ
Finally the relative Mach number at the tip can be found.
π΄ππΉ= π΄πππΉ
βπ»ππ
π»π
= π.ππ
ππ1π = 1.13
6
ππ1= 288πΎ
π1 = 276.8πΎ
Thus by calculating a relative Mach number at the tip of 1.16 it is confirmed that the flow is
transonic which is okay. This is confirmed from the book Advances in Gas Turbine
Technology by Roberto Biolla and Ernesto Benini. This reference states that a typical value
for the inlet relative Mach number at the tip is 1.3. [2]
Section B: Annulus Dimensions
In Section B the annulus dimensions of the compressor will be determined at the
inlet and outlet. In order to calculate these values for this preliminary analysis a mean-
radius value shall be utilized through the compressor. It is imperative to utilize this
parameter because blade velocity, along with other velocities and angles, vary from hub to
tip. By using a mean-radius value one can get an average idea of how the flow is behaving
through a stage. Furthermore, if mean-radius is used along with the concept that rotational
speed of the compressor remains constant, blade speed at the mean radius will be constant
throughout the compressor as well. The mean-blade speed shall be heavily used
throughout this analysis in later sections. By using the mean-radius, annulus area of the
compressor can be calculated. Calculations for finding annulus dimension are illustrated in
the rest of the section. First annulus dimensions at the inlet will be calculated. Ultimately
annulus area will be utilized to find the radius of the hub and tip. Static pressure is the first
parameter to be determined.
π·π
π·ππ
= (π»π
π»ππ
)
ππβπ
β π·π = ππ,πππ.ππ·π
ππ1= 101.3πππ
ππ1= 288πΎ
π1 = 276.8πΎ
π = 1.4
From static pressure, static density can be found by using the Ideal Gas Law.
π·π = πππΉπ»π β ππ = π. ππππ
ππ
π1 = 88,175.9ππ
π = 287 π½
ππβπΎ
7
π1 = 276.8πΎ
Annulus area can then be calculated using the equation for mass flow rate. Because the
velocity in the equation is normal to the area that will be determined, axial velocity will be
utilized in this relationship.
οΏ½ΜοΏ½ = πππ½ππ¨π β π¨π = π. ππππ
π1 = 1.11ππ
π3
ππ₯ = 150π
π
οΏ½ΜοΏ½ = 20ππ
π
Finally, from annulus area radius at the hub and tip can be determined. The hub-to-tip ratio
of the first stage will be utilized.
π¨π = πππ β ππ
π = πππ β (π.πππ)π = π. πππππ β ππ = π. πππ
πβ
ππ‘
= 0.5
ππ = π.π(ππ + ππ) β ππ = π. πππ
ππ‘ = 0.40π
ππ = π.π(ππ + ππ) = π. πππ
ππ‘ = 0.40π
πβ = 0.20π
Because mass flow rate is constant via conservation of mass one is able to find the annulus
area at the exit. The only missing parameter is the density. Remember that axial velocity is
constant. The density can be found by using isentropic process equations and the overall
pressure ratio of the compressor.
πππ
ππ1
= 4.15 β πππ= 420,395ππ
ππ1= 101.3πππ
The stagnation temperature at the end of the compressor will now be determined.
However, in order to relate the compressor stagnation pressure rise to the compressor
stagnation temperature rise an assumption must be made. It will be assumed that
polytropic efficiency is equal to stage efficiency because the stage temperature rise in an
axial compressor is small.
8
π»ππ
π»ππ
= (π·π π
π·ππ
)
πβππΌππ
β π»ππ= πππ.πππ²
πππ
ππ1
= 4.15
π = 1.4
ππ = 0.9
ππ1= 288πΎ
From the exit stagnation temperature static temperature can be found. However it is noted
that the absolute velocity at the exit of the compressor cannot be found initially. It is only
after all the stage parameters have been determined that this value can be found. The exit
velocity will be used here in order to calculate the static temperature at the exit. However,
please refer to section E in order to see how exit velocity is found.
π»π = π»ππβ
π½ππ
ππͺπ
β πππ.πππ²
πππ= 452.48πΎ
πΆπ = 1005π½
ππβπΎ
π3 = 171.05π
π
The annular dimensions for the exit can now be found in a similar manner as the inlet.
Static pressure will first be found using isentropic process equations.
π·π
π·ππ
= (π»π
π»ππ
)
ππβπ
β π·π = πππ,ππππ·π
πππ= 420,395ππ
ππ = 437.92πΎ
πππ= 452.48πΎ
π = 1.4
Static density can now be found using the Ideal Gas Law.
π·π = πππΉπ»π β ππ = π.ππππ
ππ
ππ = 374,934ππ
ππ = 437.92πΎ
9
π = 287 π½
ππβπΎ
Because the mass flow rate is constant it can be used to find the annular area at the exit.
οΏ½ΜοΏ½ = πππ½π π¨π β π¨π = π.πππππ
ππ = 2.98ππ
π3
ππ₯ = 150π
π
οΏ½ΜοΏ½ = 20ππ
π
Previously mean-radius was calculated and can now be utilized to give a relationship
between the radiuses of the hub and tip. Remember that the preliminary analysis is based
on constant mean-radius. In conjunction with the known exit area radius at the hub and tip
can be calculated.
π¨π = πππ
π β πππ
π β πππ= π. πππ
ππ = 0.5(ππ‘π+ πβπ
) = 0.30 β ππ‘π= 0.60 β πβπ
ππ = π.π(πππ+ πππ
) = π.ππ β πππ= π. ππππ
πβπ= 0.26π
Section C: Number of Stages
In Section C the number of stages needed to achieve the proper pressure rise will be
determined. A number of assumptions will need to be made in order to calculate the
number of stages. The first stageβs parameters will be heavily utilized . First the mean-blade
speed will be calculated using the tip radius and speed at the inlet of the compressor. By
using the concept of constant rotational speed in the compressor a relationship can be
made.
πΌπ = πππ β π = πππ.πππππ
π
ππ‘ = 355.3π
π
ππ‘ = 0.40π
πΌπ = πππ = πππ.πππ
π
Ξ© = 887.78πππ
π
ππ = 0.30π
10
The relative flow angle will be calculated at the inlet using the mean-blade speed and
trigonometry.
πππ§ π·π =βπΌπ
π½π
β π·π = βππ.ππΒ°
ππ = 266.28π
π
ππ₯ = 150π
π
Next the De Haller Number shall be used to calculate the relative flow angle after the rotor.
The De Haller Number shall be utilized to make sure that the flow does not diffuse
excessively and cause stalling. The De Haller Number states that the ratio between the
relative velocity after the rotor to before the rotor should be kept above 0.72. Thus 0.73
will be used in this analysis to determine the relative flow angle after the rotor. The De
Haller Number can be expressed in terms of flow angles.
ππ¨π¬ π·π
ππ¨π¬ π·π
= π. ππ β π·π = βππ.ππΒ°
π½1 = β60.63Β°
In order to determine the number of stages needed it will be initially assumed that the
temperature rise per stage is equal. To get a clearer picture on the average stage
temperature rise the work done factor will be utilized. Reasonable values for the wor k
done factor are already known. By averaging these four values one will achieve a more
accurate temperature rise per stage. However, it must also b assumed that the compressor
does not have a large number of stages. If the compressor had a large amount of stages the
average work done factor would be very low. It is reasonable to assume a small amount of
stages because of the nature of axial-compressors. It is known that axial compressors raise
the pressure from each stage slightly, and with this comes high efficiencies. Taking this into
consideration and the dimensions of the compressor one would not assume a large amount
of stages are necessary to produce an overall pressure ratio of 4.5. This average stage
temperature rise is illustrated below.
βπ»ππππ=
πππππΌππ½π
πͺπ
(πππ§ π·π β πππ§ π·π) = ππ.πππ²
πππ£π =0.905
ππ = 266.28π
π
11
ππ₯ = 150π
π
πΆπ = 1005π½
ππβπΎ
π½1 = β60.62Β°
π½2 = β47.78Β°
Finally, the number of stages can be calculated by using the average stage temperature rise.
π»ππ
π»ππ
= π +βπ»ππππ
π»ππ
β π = π.ππ β π
πππ= 452.48πΎ
ππ1= 288πΎ
βππππ£π= 24.27πΎ
Section D: Initial Design Parameters In Section D the design of the stages will be explored. Stage 1 will initially be designed
using the flow angles that have been previously calculated. This means that the flow angles
before and after the rotor are fixed using the De Haller Number when it is set to 0.73. By
doing this one can be sure that the flow will not diffuse excessively. The actual stage
temperature rise can properly be calculated using the exact work done factor. In addition
the static and stagnation pressures/temperatures will be determined. Furthermore, all
flow angles will be calculated along with the degree of reaction at the mean-radius. Stage
temperature rise will be the first to be explored using the equation for actual stage
temperature rise.
βπ»π =πππΌππ½π
πͺπ
(πππ§ π·π β πππ§ π·π) = ππ.πππ²
π1 =0.98
ππ = 266.28π
π
ππ₯ = 150π
π
πΆπ = 1005π½
ππβπΎ
π½1 = β60.62Β°
π½2 = β47.78Β°
Stagnation temperature after the stator can now easily be calculated using subtraction.
βπ»π = π»ππβ π»ππ
β π»ππ= πππ.πππ²
12
βππ = 26.26πΎ
ππ1= 288πΎ
Stagnation pressure after the stator is determined using the equation relating the stage
stagnation pressure ratio to the stage stagnation temperature ratio. It is again assumed that
polytropic efficiency is equal to stage efficiency.
[π·ππ
π·ππ
]
πβππ
= π + πΌππ (π»ππ
π»ππ
β π) β π·ππ= πππ,ππππ·π
ππ3= 314.27πΎ
ππ1= 288πΎ
ππ1= 101.3πππ
ππ‘π‘ = 0.9
Next the absolute flow angle after the rotor can be found using simple trigonometry.
πΎππ= π½π πππ§ π·π = πππ.ππ
π
π
ππ₯ = 150π
π
π½2 = β47.78Β°
πΌπ = π½ππ+ πΎππ
β π½ππ= πππ.ππ
π
π
ππ = 266.28π
π
ππ’2= 165.32
π
π
πππ§ πΆπ =π½ππ
π½π
β πΆπ = ππ.ππΒ°
ππ₯ = 150π
π
ππ’2= 100.95
π
π
In order to find the static parameters in the first stage the velocity at the exit of the stage
must be known. An assumption will be made in order to obtain this exit velocity. A
reasonable value of 160π
π will be initially assumed in order to further explore the design of
the first and remaining stages. It is noted that this exit velocity will become the inlet
velocity for Stage 2. This concept and the exit velocity assumption will be further explained
in Section E. To be able to check that the flow does not diffuse between the rotor and stator
the velocity leaving the rotor must be found. It can be found by using simple trigonometry.
13
π½π = βπ½ππ + π½ππ
π = πππ.πππ
π
ππ₯ = 150π
π
ππ’2= 100.95
π
π
By using the De Haller Number one can see that the flow does not diffuse excessively.
π½π
π½π
β₯ π.ππ β π. ππ β₯ π.ππ
π3 = 160π
π
π2 = 180.81π
π
The blade angle after the stator can be found using trigonometry.
ππ¨π¬ πΆπ =π½π
π½π
β πΆπ = ππ.ππΒ°
ππ₯ = 150π
π
π3 = 160π
π
Finally, the static parameters can be found using the exit velocity and an isentropic process
equation.
π»π = π»ππβ
π½ππ
ππͺπ
= πππ.πππ²
ππ3= 314.27πΎ
π3 = 160π
π
πΆπ = 1005π½
ππβπΎ
π·π
π·ππ
= (π»π
π»ππ
)
ππβπ
β π·π = πππ,ππππ·π
ππ3= 133,510ππ
ππ3= 314.27πΎ
π3 = 301.53πΎ
π = 1.4
14
The initial values for the static and stagnation temperatures/pressures of the first stage are
summarized below. The initial absolute and relative flow angles are also included in the
following tables:
Table 3 Initial Static and Stage Temperatures/Pressures of First Stage Before Iterative Process
Table 4 Initial Relative and Absolute Flow Angles of First Stage Before Iterative Process
To conclude Section D the degree of reaction at the mean-radius of the first stage will be
calculated. It is important to explore degree of reaction especially in the first several stages
to ensure there is no excessive diffusion at the root. Blade velocity varies greatly along a
long blade from hub-to-tip. This means that even if a desirable degree of reaction is
achieved at the mean-radius it may be to low at the hub, thus causing losses. It will be
assumed that a Free Vortex Design applies, as it is widely used in axial flow machines. Thus
it is assumed that each part of the blade section does the same amount of work. Blade
speed is low at the hub thus requiring greater diffusion in order to achieve the same
amount of work as the rest of the blade. Using the Free Vortex assumption will simplify the
process of calculating degree of reaction and allow the equation below to be utilized.
πΉ = π β π.ππ½π
πΌπ
(πππ§ πΆπ + πππ§ πΆπ) = ππ.ππ%
ππ₯ = 150π
π
ππ = 266.28π
π
πΌ2 = 33.94Β°
πΌ1 = 0Β°
Po(Pa) P(Pa) To(K) T(K)
AtInlet 101,300 88,175.90 288 276.8
AtExit 133,510 115,512 314.27 301.53
InitialStaticandStageTempratures/PressuresofFirstStage
Ξ²(Β°) Ξ±(Β°)
BeforeRotor -60.62 0
AfterRotor -47.78 33.94
AfterStator n/a 20.36
InitialFlowAnglesofFirstStage
15
It is noted that for the preliminary analysis a degree of reaction for the first stage, which
contains a long blade, is 81.04%. This seems to be a reasonable value. Thus, when
designing the rest of the stages another assumption will be made in regards to degree of
reaction. It will be reasonable to assume a degree of reaction of 70% for the second stage
and 50% for the remaining stages. Remember that the degree of reaction must be the
highest in the first stage because of the length of the blade. As the compressor shrinks from
inlet to outlet the blade shrinks, thus the variation in blade speed shrinks. A higher degree
of reaction in the first stage at the mean radius will prevent excessive diffusion at the hub.
Thus a smaller degree of reaction will be utilized for shorter blades because the flow at the
hub does not have to diffuse much greater than the rest of the blade. This is again because
the variation in blade speed is smaller in comparison to the first stage.
Section E: Final Design Parameters Section E will describe the final design of the seven stages in this preliminary analysis. Note
that the calculations used to initially find the parameters in the first stage will be used
throughout the remainder of the stages. Also, the method of finding the parameters will be
similar. For each stage the exit velocity will become the inlet velocity for the next stage.
This essentially means that absolute velocity after the stator will equal absolute velocity
coming into the next rotor. Absolute flow angles will also be equal. However, the
assumption of repeating stages throughout the compressor will not be used. Remember
that for the first stage the flow angle relationships before the rotor and after the rotor were
set using the De Haller Number. For Stage 2 through 7 the flow angle relationships will be
set using the degree of reaction. Remember that degree of reaction for the second stage is
70% and 50% for the remaining stages. Using the degree of reaction equation one can
relate absolute flow angles before and after the rotor. Just like in the first stage the exit
velocities of each stage will initially be assumed. Not that because Stage 4 through 7 have
identical parameters in terms of work done factor and degree of reaction the same exit
velocity will be assume for all to simplify the design process. The desired pressure ratio of
the compressor is known thus allowing for an iterative process to take place. Initially a
modest exit velocity of 160π
π was set to avoid diffusion within the first stage. From the
assumed exit velocities of the stages the absolute flow angles can easily be determined
using trigonometry and the known constant axial velocity. By varying the exit velocities,
16
thus varying the absolute flow angles one can increase/decrease diffusion betwee n stages
and increase/decrease the overall pressure ratio of the compressor. By increasing the exit
velocity diffusion is decreased thus decreasing the possibility of stalling. However, this
additionally lowers the overall pressure ratio of the compressor. Therefore limiting
diffusions and stalling throughout the compressor while also achieving the overall pressure
ratio binds the iterative process. These diffusions are checked by using the De Haller
Number. Furthermore, the iterative process takes into the consideration that by producing
slight pressure rises steadily per stage high compressor efficiencies can be achieved. Large
spikes in pressure rises are a probable sign of excessive diffusion. Design parameters for
Stages 1 through 7 can be seen in Table 7 at the end of the conclusion. This includes
pressures, temperatures, and flow angles. The velocity triangles of all the stages are
illustrated in the figures below. All calculations used for stage analysisβs can be found in the
appendix.
Stage 1
Figure 1a Velocity Diagram of Flow in Stage 1 Before the Rotor
-300
-200
-100
0
100
200
300
0 20 40 60 80 100 120 140 160
Velocity(m
/s)
Velocity(m/s)
BeforeRotor
V1
W1
Um
17
Figure 1b Velocity Diagram of Flow in Stage 1 After the Rotor
Figure 1c Velocity Diagram of Flow in Stage 1 After the Stator
Stage 2
Figure 2a Velocity Diagram of Flow in Stage 2 Before the Rotor
-300
-200
-100
0
100
200
300
0 20 40 60 80 100 120 140 160
Velocity(m/s)
Velocity(m/s)
BeforeRotor
V1
W1
Um
18
Figure 2b Velocity Diagram of Flow in Stage 2 After the Rotor
Figure 2c Velocity Diagram of Flow in Stage 2 After the Stator
Stage 3
Figure 3a Velocity Diagram of Flow in Stage 3 Before the Rotor
-300
-200
-100
0
100
200
300
0 20 40 60 80 100 120 140 160
Velocity(m/s)
Velocity(m/s)
BeforeRotor
V1
W1
Um
19
Figure 3b Velocity Diagram of Flow in Stage 3 After the Rotor
Figure 3c Velocity Diagram of Flow in Stage 3 After the Stator
Stage 4-7
Figure 4a Velocity Diagram of Flow in Stages 4-7 Before the Rotor
-300
-200
-100
0
100
200
300
0 20 40 60 80 100 120 140 160
Velocity(m/s)
Velocity(m/s)
BeforeRotor
V1
W1
Um
20
Figure 4b Velocity Diagram of Flow in Stages 4-7 After the Rotor
Figure 4c Velocity Diagram of Flow in Stages 4-7 After the Stator
To conclude this section plots of pressure ratio across the compressor and pressure rise
are shown.
21
Figure 5 Relationship of Pressure Ratio to Stage Number which Shows Pressure Ratio Decrease Per Stage
Figure 6 Relationship of Pressure Rise to Stage Number which Shows Stage Pressure Rise Increase Per Stage
The importance of these plots shall be later discussed in the conclusion of this analysis.
0
10000
20000
30000
40000
50000
60000
70000
1 2 3 4 5 6 7
StagePressureRise(Pa)
StageNumber
StagePressureRisevsStageNumber
StagePressureRisevsStageNumber
22
Section F: Hub-To-Tip Flow
Lastly, in Section F absolute and relative flow angle variations from hub-to-tip will be
explored. To simplify this process only the first and third stages shall be explored. The Free
Vortex Design assumption will again be utilized. It is noted that a number of velocities and
flow angles have been calculated while designing the first and third stages. These values,
which are used in this analysis, can be found in the appendix. The first stageβs analysis
takes place below:
Stage 1
At Tip Using the Free Vortex Design assumption a relationship between blade speed and velocity
in the direction of the blade can be illustrated.
πΌππ½ππ= πΌππ½ππ
β π½ππ= ππ.ππ
π
π
ππ = 266.28π
π
ππ’π= 101.00
π
π
ππ‘ = 355.3π
π
Trigonometry can be used to find the flow angles on velocity triangles that represent tip
flow characteristics.
πππ§ πΆπ =π½ππ
π½π
β πΆπ = ππ.ππΒ°
ππ’π‘= 75.75
π
π
ππ₯ = 150π
π
πππ§ π·π =βπΌπ
π½π
β π·π = βππ.ππΒ°
ππ‘ = 266.28π
π
ππ₯ = 150π
π
πΎππ= π½ππ
β πΌπ = βπππ.πππ
π
ππ’π‘= 75.75
π
π
ππ‘ = 266.28π
π
πππ§ π·π =πΎππ
π½π
β π·π = βππ.ππΒ°
23
ππ₯ = 150π
π
ππ’π‘= 279.55
π
π
At Hub
Blade speed at the hub can be found using the constant rotational speed of the compressor.
πΌπ = πππ β πππ.πππ
π
πβ = 0.20π
Ξ© = 887.78πππ
π
Using the Free Vortex Design assumption
πΌππ½ππ= πΌππ½ππ
β π½ππ= πππ.ππ
π
π
ππ = 266.28π
π
ππ’π= 101.00
π
π
πβ = 177.57π
π
Using trigonometry on velocity triangles that represent hub flow characteristic the flow
angles can be found.
πππ§ πΆπ =π½ππ
π½π
β πΆπ = ππ.ππΒ°
ππ’β= 151.57
π
π
ππ₯ = 150π
π
πππ§ π·π =βπΌπ
π½π
β π·π = βππ.ππΒ°
πβ = 177.57π
π
ππ₯ = 150π
π
πΎππ= π½ππ
β πΌπ = βππ.πππ
π
ππ’β= 151.57
π
π
πβ = 177.57π
π
πππ§ π·π =πΎππ
π½π
β π·π = βπ. ππΒ°
ππ₯ = 150π
π
24
ππ’β= β26.00
π
π
It is noted that checking diffusion at the hub by using the De Haller Number indicates a high
possibility of excessive diffusion. However, this is satisfactory for a preliminary design. The
De Haller relationship is shown below at the hub.
ππ¨π¬ πΆπ
ππ¨π¬ πΆπ
= π. ππ β π.ππ β― π. ππ
πΌ2 = 45.30Β°
πΌ1 = 0Β°
ππ¨π¬ π·π
ππ¨π¬ π·π
= π. ππ β π.ππ β― π. ππ
π½2 = β9.83Β°
π½1 = β49.81Β°
Below a summation of flow angles of the first stage from hub-to-tip can be found:
Table 5 Relative and Absolute Flow Angles in Stage 1 From the Hub to the Tip
Stage 3
Next, analysis of the third stage will take place. Annulus area, and the assumption of mean -
radius will be utilized in order to find the blade speed at the hub and tip. Using the
parameters from the designed third stage, annulus area can first be calculated.
π·π = πππΉπ»π β ππ = π. ππππ
ππ
π1 = 145,406.80ππ
π1 = 324.39πΎ
π = 287 π½
ππβπΎ
Because the mass flow rate is constant it can be used to find the annular area at the exit.
οΏ½ΜοΏ½ = πππ½π π¨π β π¨π = π.πππππ
ππ = 1.52ππ
π3
Hub Mean Tip
Ξ±1(Β°) 0 0 0
Ξ±2(Β°) 45.3 33.95 26.79
Ξ²1(Β°) -49.81 -60.62 -67.11
Ξ²2(Β°) -9.83 -47.77 -61.78
FlowAnglesofStage1FromHubtoTip
25
ππ₯ = 150π
π
οΏ½ΜοΏ½ = 20ππ
π
Next mean-radius and annulus area can be utilized to find the radiuses at the hub and tip.
π¨π = πππ β ππ
π β ππ = π. πππ
ππ = 0.5(ππ‘ + ππ) = 0.30 β πβ = 0.60 β ππ‘
ππ = π.π(ππ + ππ) = π. ππ β ππ = π. πππ
ππ‘ = 0.37π
Finally the radiuses can be used to calculate the blade speeds at the hub and tip.
πΌπ = πππ β πππ.ππ
π
ππ‘ = 0.37π
Ξ© = 887.78πππ
π
πΌπ = πππ β πππ.πππ
π
πβ = 0.23π
Ξ© = 887.78πππ
π
Now that the preliminary parameters have been determined the flow angles for the third
stage can be determined.
At Tip Before Rotor Using the Free Vortex Design assumption
πΌππ½ππ= πΌππ½ππ
β π½ππ= ππ.ππ
π
π
ππ = 266.28π
π
ππ’π= 83.14
π
π
ππ‘ = 329.6π
π
Using trigonometry on velocity triangles that represent tip flow characteristic the flow
angles can be found.
πππ§ πΆπ =π½ππ
π½π
β πΆπ = ππ.ππΒ°
ππ’π‘= 67.17
π
π
ππ₯ = 150π
π
26
πΎππ= π½ππ
β πΌπ = βπππ.πππ
π
ππ’π‘= 67.17
π
π
ππ‘ = 329.6π
π
πππ§ π·π =πΎππ
π½π
β π·π = βππ.ππΒ°
ππ₯ = 150π
π
ππ’π‘= β262.43
π
π
At Tip After Rotor
Using the Free Vortex Design assumption
πΌππ½ππ= πΌππ½ππ
β π½ππ= πππ.ππ
π
π
ππ = 266.28π
π
ππ’π= 183.14
π
π
ππ‘ = 329.6π
π
Using trigonometry on velocity triangles that represent tip flow characteristic the flow
angles can be found.
πππ§ πΆπ =π½ππ
π½π
β πΆπ = ππ.ππΒ°
ππ’π‘= 67.17
π
π
ππ₯ = 150π
π
πΎππ= π½ππ
β πΌπ = βπππ.πππ
π
ππ’π‘= 147.96
π
π
ππ‘ = 329.6π
π
πππ§ π·π =πΎππ
π½π
β π·π = βππ.ππΒ°
ππ₯ = 150π
π
ππ’π‘= β181.64
π
π
At Hub Before Rotor Using the Free Vortex Design assumption
πΌππ½ππ= πΌππ½ππ
β π½ππ= πππ.ππ
π
π
27
ππ = 266.28π
π
ππ’π= 83.14
π
π
πβ = 203.35π
π
Using trigonometry on velocity triangles that represent hub flow characteristic the flow
angles can be found.
πππ§ πΆπ =π½ππ
π½π
β πΆπ = ππ.ππΒ°
ππ’β= 108.86
π
π
ππ₯ = 150π
π
πΎππ= π½ππ
β πΌπ = βππ.πππ
π
ππ’β= 108.86
π
π
πβ = 203.35π
π
πππ§ π·π =πΎππ
π½π
β π·π = βππ. ππΒ°
ππ₯ = 150π
π
ππ’β= β94.48
π
π
At Hub After Rotor
Using the Free Vortex Design assumption
πΌππ½ππ= πΌππ½ππ
β π½ππ= πππ.π
π
π
ππ = 266.28π
π
ππ’π= 183.14
π
π
πβ = 203.35π
π
Using trigonometry on velocity triangles that represent hub flow characteristic the flow
angles can be found.
πππ§ πΆπ =π½ππ
π½π
β πΆπ = ππ.ππΒ°
ππ’β= 239.82
π
π
ππ₯ = 150π
π
28
πΎππ= π½ππ
β πΌπ = ππ.πππ
π
ππ’β= 239.82
π
π
πβ = 203.35π
π
πππ§ π·π =πΎππ
π½π
β π·π = ππ. ππΒ°
ππ₯ = 150π
π
ππ’β= 34.46
π
π
Below a summation of flow angles of the third stage can be found:
Table 6 Relative and Absolute Flow Angles in Stage 3 from the Hub to the Tip
Furthermore, plots of the fluid deflection from hub-to-tip can be found below:
Figure 7 Relationship Between Flow Deflection and Blade Height in Stage 1 that S hows Decrease in Flow Deflection as the Radius of the Blade Becomes Larger
Hub Mean Tip
Ξ±1(Β°) 35.97 29 24.12
Ξ±2(Β°) 57.97 50.68 44.61
Ξ²1(Β°) -32.21 -50.68 -60.25
Ξ²2(Β°) 13.66 -29 -50.45
FlowAnglesofStage3FromHubtoTip
29
Figure 8 Relationship Between Flow Deflection and Blade Height in Stage 3 that Shows Decrease in Flow Deflection as the Radius of the Blade Becomes Larger
The plots show an increase in needed air deflection from hub-to-tip, thus implying a
reduction of blade twist from hub to tip. This is a clear indicator of the Free Vortex Design
and the effect that occur on long blades. This concludes Section F.
Conclusion In conclusion this preliminary design is acceptable. The design approach taken was devised
heavily on limiting losses throughout the compressor through excessive diffusion. It is
noted that a preliminary analysis can be approached with a variety of methods. These
methods vary with what the designer or customer feels are the most important parameters.
The desired pressure ratio of the compressor has been achieved with also abiding by the
De Haller Number. In addition to this the plots of pressure rise and pressure ratio across
the compressor are a simple indicator of an efficient reasonable preliminary design.
Pressure ratio should decrease per stage while stage pressure rise increases steadily.
Remember that large spikes in pressure rise are usually an indicator of excessive diffusion.
All found design parameters per stage are summarized below:
30
Table 7a Summary of Several Devised Design Parameters for Stages 1 Through 7
Table 7b Summary of Devised Design Parameters Continued
Table 7c Summary of Devised Design Parameters Continued
Stage# V1(m/s) V2(m/s) V3(m/s) Ξ±1(Β°)
1 150.00 180.84 152.90 0.00
2 152.90 198.58 171.50 11.183 171.50 236.73 171.05 29.00
4 171.05 237.45 171.05 28.725 171.05 237.45 171.05 28.72
6 171.05 237.45 171.05 28.72
7 171.05 237.45 171.05 28.72
SummaryofEstimatedValuesforEachStage
Stage# Ξ²1(Β°) Ξ±2(Β°) Ξ²2(Β°) Ξ±3(Β°)
1 -60.62 33.95 -47.77 11.18
2 -57.63 40.94 -42.23 29.003 -50.68 50.68 -29.00 28.72
4 -50.82 50.82 -28.72 28.725 -50.82 50.82 -28.72 28.72
6 -50.82 50.82 -28.72 28.72
7 -50.82 50.82 -28.72 28.72
SummaryofEstimatedValuesforEachStage(Cont.)
Stage# P1(Pa) P3(Pa) P3/P1 ΞP(Pa) Po1(Pa) Po3(Pa)
1 88175.90 116999.36 1.33 28823.46 101300.00 133508.23
2 116999.36 145406.80 1.24 28407.44 133508.23 169687.553 145406.80 181389.98 1.25 35983.18 169687.55 209381.14
4 181389.98 221116.16 1.22 39726.18 209381.14 253072.565 221116.16 266404.14 1.20 45287.98 253072.56 302608.08
6 266404.14 317645.56 1.19 51241.42 302608.08 358381.82
7 317645.56 375235.44 1.18 57589.89 358381.82 420791.17
SummaryofEstimatedValuesforEachStage(Cont.)
31
Table 7d Summary of Devised Design Parameters Continued
Stage# Po3/Po1 ΞPo(Pa) T1(K) T3(K) To1(K) To3(K) ΞTo(K)1 1.32 32208.23 276.81 302.63 288.00 314.26 26.262 1.27 36179.32 302.63 324.39 314.26 339.03 24.763 1.23 39693.59 324.39 347.79 339.03 362.34 23.32
4 1.21 43691.42 347.79 370.19 362.34 384.74 22.405 1.20 49535.52 370.19 392.59 384.74 407.14 22.406 1.18 55773.74 392.59 414.99 407.14 429.55 22.40
7 1.17 62409.34 414.99 437.39 429.55 451.95 22.40Poe/Po1 4.15
SummaryofEstimatedValuesforEachStage(Cont.)
32
Appendix
V1(m/s) 150 To1(K) 288 Vx(m/s) 150
Ξ²1(Β°) -60.62 Po1(Pa) 101300 Um(m/s) 266.28
Ξ» 0.98 Cp(J/Kg-K) 1005
P1(Pa) 88,175.90 Ξ·tt 0.9T1(K) 276.81 Ξ³ 1.4
W1(m/s) Ξ²2(Β°) Wu2(m/s) W2(m/s) Vu2(m/s) V2(m/s) Ξ±2(Β°) V3(m/s)-ASSUMED DHCheck>0.72305.6224 -47.7742 -165.277 223.1963 101.0027 180.8357 33.95442 152.9 0.84551884
Ξ±3(Β°) ΞTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/P1 ΞPo(Pa) R(%)11.17691 26.26351 314.2635 133508.2 302.6325 116999.4 424.8289 32208.23363 81.03448567
DesignParametersofStage1
Input
V1(m/s) 152.9 To1(K) 314.2635064 Vx(m/s) 150Ξ±1(Β°) 11.17691 Po1(Pa) 133508.2336 Um(m/s) 266.28
Ξ» 0.93 Cp(J/Kg-K) 1005P1(Pa) 116,999.36 Ξ·tt 0.9T1(K) 302.6324566 Ξ³ 1.4
R(%) 70
Vu1(m/s) Wu1(m/s) W1(m/s) Ξ²1(Β°) Ξ±2(Β°) DHCheckΞ±>0.72 Vu2(m/s)29.63798 -236.642 -280.178 -57.6307 40.94273195 0.769968667 130.13
DHCheckΞ²>0.720.723025386
V2(m/s) Wu2(m/s) W2(m/s) Ξ²2(Β°) V3(m/s)-ASSUMED DHCheck>0.72 Ξ±3(Β°)198.5795 -136.15 -202.575 -42.229 171.5 0.86363392 28.99813
ΞTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΞPo(Pa)24.76208 339.0256 169687.6 324.3926 145406.8026 1.270989412 36,179.32
Input
DesignParametersofStage2
V1(m/s) 171.5 To1(K) 339.0255838 Vx(m/s) 150Ξ±1(Β°) 28.99813 Po1(Pa) 169687.5514 Um(m/s) 266.28
Ξ» 0.88 Cp(J/Kg-K) 1005P1(Pa) 145,406.80 Ξ·tt 0.9T1(K) 324.3926236 Ξ³ 1.4R(%) 50
Vu1(m/s) Wu1(m/s) W1(m/s) Ξ²1(Β°) Ξ±2(Β°) DHCheckΞ±>0.72 Vu2(m/s)83.13994 -183.14 -236.728 -50.681 50.680971 0.724459269 183.1401
DHCheckΞ²>0.720.724459269
V2(m/s) Wu2(m/s) W2(m/s) Ξ²2(Β°) V3(m/s)-ASSUMED DHCheck>0.72 Ξ±3(Β°)236.7283 -83.1399 -171.5 -28.9981 171.05 0.722558355 28.725ΞTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΞPo(Pa)
23.31609 362.3417 209381.1 347.7854 181389.9797 1.233921644 39,693.59
DesignParametersofStage3
Input
33
V1(m/s) 171.05 To1(K) 362.3416704 Vx(m/s) 150Ξ±1(Β°) 28.725 Po1(Pa) 209381.1424 Um(m/s) 266.28
Ξ» 0.83 Cp(J/Kg-K) 1005P1(Pa) 181,389.98 Ξ·tt 0.9
T1(K) 347.7854005 Ξ³ 1.4R(%) 50
Vu1(m/s) Wu1(m/s) W1(m/s) Ξ²1(Β°) Ξ±2(Β°) DHCheckΞ±>0.72 Vu2(m/s)82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723
DHCheckΞ²>0.720.720361435
V2(m/s) Wu2(m/s) W2(m/s) Ξ²2(Β°) V3(m/s)-ASSUMED DHCheck>0.72 Ξ±3(Β°)237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725ΞTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΞPo(Pa)
22.40134 384.743 253072.6 370.1867 221116.1559 1.208669295 43,691.42
DesignParametersofStage4
Input
V1(m/s) 171.05 To1(K) 384.7430127 Vx(m/s) 150Ξ±1(Β°) 28.725 Po1(Pa) 253072.5578 Um(m/s) 266.28
Ξ» 0.83 Cp(J/Kg-K) 1005P1(Pa) 221,116.16 Ξ·tt 0.9
T1(K) 370.1867428 Ξ³ 1.4R(%) 50
Vu1(m/s) Wu1(m/s) W1(m/s) Ξ²1(Β°) Ξ±2(Β°) DHCheckΞ±>0.72 Vu2(m/s)82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723
DHCheckΞ²>0.720.720361435
V2(m/s) Wu2(m/s) W2(m/s) Ξ²2(Β°) V3(m/s)-ASSUMED DHCheck>0.72 Ξ±3(Β°)237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725ΞTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΞPo(Pa)
22.40134 407.1444 302608.1 392.5881 266404.1359 1.195736452 49,535.52
DesignParametersofStage5
Input
34
Equation used to calculate design parameters can be found below:
πππ§ πΆπ =π½ππ
π½πβ π½ππ
, Trigonometry of Velocity Diagrams
πΎππ= πΌπ β π½ππ
, Trigonometry of Velocity Diagrams
πΎπ = ββπΎππ
π + π½ππ, Trigonometry of Velocity Diagrams
πππ§ π·π =πΎππ
π½πβ π·π, Trigonometry of Velocity Diagrams
πΉ = π β π.ππ½π
πΌπ
(πππ§ πΆπ + πππ§ πΆπ) β πΆπ, Degree of Reaction
π½π
π½πβ₯ π. ππ, De Haller Number
V1(m/s) 171.05 To1(K) 407.1443549 Vx(m/s) 150
Ξ±1(Β°) 28.725 Po1(Pa) 302608.0823 Um(m/s) 266.28
Ξ» 0.83 Cp(J/Kg-K) 1005
P1(Pa) 266,404.14 Ξ·tt 0.9
T1(K) 392.588085 Ξ³ 1.4
R(%) 50
Vu1(m/s) Wu1(m/s) W1(m/s) Ξ²1(Β°) Ξ±2(Β°) DHCheckΞ±>0.72 Vu2(m/s)82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723
DHCheckΞ²>0.720.720361435
V2(m/s) Wu2(m/s) W2(m/s) Ξ²2(Β°) V3(m/s)-ASSUMED DHCheck>0.72 Ξ±3(Β°)237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725ΞTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΞPo(Pa)
22.40134 429.5457 358381.8 414.9894 317645.5553 1.184310152 55,773.74
DesignParametersofStage6
Input
V1(m/s) 171.05 To1(K) 429.5456972 Vx(m/s) 150Ξ±1(Β°) 28.725 Po1(Pa) 358381.8238 Um(m/s) 266.28
Ξ» 0.83 Cp(J/Kg-K) 1005P1(Pa) 317,645.56 Ξ·tt 0.9
T1(K) 414.9894273 Ξ³ 1.4R(%) 50
Vu1(m/s) Wu1(m/s) W1(m/s) Ξ²1(Β°) Ξ±2(Β°) DHCheckΞ±>0.72 Vu2(m/s)82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723
DHCheckΞ²>0.720.720361435
V2(m/s) Wu2(m/s) W2(m/s) Ξ²2(Β°) V3(m/s)-ASSUMED DHCheck>0.72 Ξ±3(Β°)237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725ΞTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΞPo(Pa)
22.40134 451.947 420791.2 437.3908 375235.4437 1.174142048 62,409.34
Input
DesignParametersofStage7
35
πΎπ
πΎπβ₯ π. ππ, De Haller Number
πππ§ πΆπ =π½ππ
π½πβ π½ππ
, Trigonometry of Velocity Diagrams
πΎππ= π½ππ
β πΌπ, Trigonometry of Velocity Diagrams
πΎπ = ββπΎππ
π + π½ππ, Trigonometry of Velocity Diagrams
πππ§ π·π =πΎππ
π½πβ π·π, Trigonometry of Velocity Diagrams
βπ»π =π#πΌπ π½π
πͺπ
(πππ§ π·π β πππ§ π·π), Actual Stage Temperature Rise
βπ»π = π»ππβ π»ππ
β π»ππ, Subtraction
[π·ππ
π·ππ
]
πβπ
π= π + πΌππ (
π»ππ
π»ππ
β π) β π·ππ, Stage Pressure Ratio from Stage Temperature Ratio
π½π = βπ½ππ + π½ππ
π, Trigonometry of Velocity Diagrams
π½π
π½πβ₯ π. ππ, De Haller Number
ππ¨π¬ πΆπ =π½π
π½πβ πΆπ , Trigonometry of Velocity Diagrams
π»π = π»ππβ
π½ππ
ππͺπ , Static Temperature from Stagnation Temperature
π·π
π·ππ
= (π»π
π»ππ
)
π
πβπβ π·π , Static to Stagnation Pressure Ratio from Static to Stagnation
Temperature Ratio
36
References [1] Korpela, S. A. Principles of Turbomachinery. Hoboken, N.J.: Wiley, 2011.
[2] Roberto Biollo and Ernesto Benini (2011). State-of-Art of Transonic Axial Compressors,
Advances in Gas Turbine Technology, Dr. Ernesto Benini (Ed.), ISBN: 978-953-307-611-9,
InTech, Available from: http://www.intechopen.com/books/advances-in-gas-turbine-
technology/state-of-art-of-transonic-axial- compressors