turbo final 2

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.

𝑪𝒐𝟏= √𝝀𝑹𝑻𝒐𝟏

= 𝟑𝟒𝟎.𝟏𝟕𝒎

𝒔

𝜆 = 1.4

𝑅 = 287 𝐽

𝑘𝑔−𝐾

𝑇𝑜1= 288𝐾

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 stage’s velocity diagram

at the tip.

𝑴𝒐𝟏𝑹=

𝑾𝟏

𝑪𝒐𝟏

= 𝟏. 𝟏𝟑

𝐶𝑜1= 340.17

𝑚

𝑠

𝑊1 = √𝑉𝑥2 + 𝑈𝑡 = 385.66

𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠

𝑈𝑡 = 355.3 𝑚

𝑠

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.

𝑻𝟏 = 𝑻𝒐𝟏−

𝑽𝟏𝟐

𝟐𝑪𝒑

= 𝟐𝟕𝟔.𝟖𝑲

𝑇𝑜1= 288𝐾

𝑉1 = 150𝑚

𝑠

𝐶𝑝 = 1005 𝐽

𝑘𝑔−𝐾

Finally the relative Mach number at the tip can be found.

𝑴𝟏𝑹= 𝑴𝒐𝟏𝑹

√𝑻𝒐𝟏

𝑻𝟏

= 𝟏.𝟏𝟔

𝑀𝑜1𝑅= 1.13

6

𝑇𝑜1= 288𝐾

𝑇1 = 276.8𝐾

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.

𝑷𝟏

𝑷𝒐𝟏

= (𝑻𝟏

𝑻𝒐𝟏

)

𝝀𝝀−𝟏

⇒ 𝑷𝟏 = 𝟖𝟖,𝟏𝟕𝟓.𝟗𝑷𝒂

𝑃𝑜1= 101.3𝑘𝑃𝑎

𝑇𝑜1= 288𝐾

𝑇1 = 276.8𝐾

𝜆 = 1.4

From static pressure, static density can be found by using the Ideal Gas Law.

𝑷𝟏 = 𝝆𝟏𝑹𝑻𝟏 ⇒ 𝝆𝟏 = 𝟏. 𝟏𝟏𝒌𝒈

𝒎𝟑

𝑃1 = 88,175.9𝑃𝑎

𝑅 = 287 𝐽

𝑘𝑔−𝐾

7

𝑇1 = 276.8𝐾

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.

�̇� = 𝝆𝟏𝑽𝒙𝑨𝟏 ⇒ 𝑨𝟏 = 𝟎. 𝟏𝟐𝒎𝟐

𝜌1 = 1.11𝑘𝑔

𝑚3

𝑉𝑥 = 150𝑚

𝑠

�̇� = 20𝑘𝑔

𝑠

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.

𝑨𝟏 = 𝒓𝒕𝟐 − 𝒓𝒉

𝟐 = 𝒓𝒕𝟐 − (𝟎.𝟓𝒓𝒕)𝟐 = 𝟎. 𝟏𝟐𝟎𝒎𝟐 ⇒ 𝒓𝒕 = 𝟎. 𝟒𝟎𝒎

𝑟ℎ

𝑟𝑡

= 0.5

𝒓𝒎 = 𝟎.𝟓(𝒓𝒕 + 𝒓𝒉) ⇒ 𝒓𝒉 = 𝟎. 𝟐𝟎𝒎

𝑟𝑡 = 0.40𝑚

𝒓𝒎 = 𝟎.𝟓(𝒓𝒕 + 𝒓𝒉) = 𝟎. 𝟑𝟎𝒎

𝑟𝑡 = 0.40𝑚

𝑟ℎ = 0.20𝑚

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.

𝑃𝑜𝑒

𝑃𝑜1

= 4.15 ⇒ 𝑃𝑜𝑒= 420,395𝑃𝑎

𝑃𝑜1= 101.3𝑘𝑃𝑎

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

𝑻𝒐𝒆

𝑻𝒐𝟏

= (𝑷𝟎 𝒆

𝑷𝒐𝟏

)

𝝀−𝟏𝜼𝒑𝝀

⇒ 𝑻𝒐𝒆= 𝟒𝟓𝟐.𝟒𝟖𝑲

𝑃𝑜𝑒

𝑃𝑜1

= 4.15

𝜆 = 1.4

𝜂𝑝 = 0.9

𝑇𝑜1= 288𝐾

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.

𝑻𝒆 = 𝑻𝒐𝒆−

𝑽𝟑𝟐

𝟐𝑪𝒑

⇒ 𝟒𝟑𝟕.𝟗𝟐𝑲

𝑇𝑜𝑒= 452.48𝐾

𝐶𝑝 = 1005𝐽

𝑘𝑔−𝐾

𝑉3 = 171.05𝑚

𝑠

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.

𝑷𝒆

𝑷𝒐𝒆

= (𝑻𝒆

𝑻𝒐𝒆

)

𝝀𝝀−𝟏

⇒ 𝑷𝒆 = 𝟑𝟕𝟒,𝟗𝟑𝟒𝑷𝒂

𝑃𝑜𝑒= 420,395𝑃𝑎

𝑇𝑒 = 437.92𝐾

𝑇𝑜𝑒= 452.48𝐾

𝜆 = 1.4

Static density can now be found using the Ideal Gas Law.

𝑷𝒆 = 𝝆𝒆𝑹𝑻𝒆 ⇒ 𝝆𝒆 = 𝟐.𝟗𝟖𝒌𝒈

𝒎𝟑

𝑃𝑒 = 374,934𝑃𝑎

𝑇𝑒 = 437.92𝐾

9

𝑅 = 287 𝐽

𝑘𝑔−𝐾

Because the mass flow rate is constant it can be used to find the annular area at the exit.

�̇� = 𝝆𝒆𝑽𝒙 𝑨𝒆 ⇒ 𝑨𝒆 = 𝟎.𝟎𝟒𝟓𝒎𝟐

𝜌𝑒 = 2.98𝑘𝑔

𝑚3

𝑉𝑥 = 150𝑚

𝑠

�̇� = 20𝑘𝑔

𝑠

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.

𝑨𝒆 = 𝒓𝒕𝒆

𝟐 − 𝒓𝒉𝒆

𝟐 ⇒ 𝒓𝒉𝒆= 𝟎. 𝟐𝟔𝒎

𝑟𝑚 = 0.5(𝑟𝑡𝑒+ 𝑟ℎ𝑒

) = 0.30 ⇒ 𝑟𝑡𝑒= 0.60 − 𝑟ℎ𝑒

𝒓𝒎 = 𝟎.𝟓(𝒓𝒕𝒆+ 𝒓𝒉𝒆

) = 𝟎.𝟑𝟎 ⇒ 𝒓𝒕𝒆= 𝟎. 𝟑𝟑𝟕𝒎

𝑟ℎ𝑒= 0.26𝑚

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 stage’s 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.

𝑼𝒕 = 𝒓𝒕𝛀 ⇒ 𝛀 = 𝟖𝟖𝟕.𝟕𝟖𝒓𝒂𝒅

𝒔

𝑈𝑡 = 355.3𝑚

𝑠

𝑟𝑡 = 0.40𝑚

𝑼𝒎 = 𝒓𝒎𝛀 = 𝟐𝟔𝟔.𝟐𝟖𝒎

𝒔

Ω = 887.78𝑟𝑎𝑑

𝑠

𝑟𝑚 = 0.30𝑚

10

The relative flow angle will be calculated at the inlet using the mean-blade speed and

trigonometry.

𝐭𝐚𝐧 𝜷𝟏 =−𝑼𝒎

𝑽𝒙

⇒ 𝜷𝟏 = −𝟔𝟎.𝟔𝟐°

𝑈𝑚 = 266.28𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠

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.

𝐜𝐨𝐬 𝜷𝟏

𝐜𝐨𝐬 𝜷𝟐

= 𝟎. 𝟕𝟑 ⇒ 𝜷𝟐 = −𝟒𝟕.𝟕𝟖°

𝛽1 = −60.63°

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.

∆𝑻𝒐𝒂𝒗𝒈=

𝝀𝒂𝒗𝒈𝑼𝒎𝑽𝒙

𝑪𝒑

(𝐭𝐚𝐧 𝜷𝟐 − 𝐭𝐚𝐧 𝜷𝟏) = 𝟐𝟒.𝟐𝟕𝑲

𝜆𝑎𝑣𝑔 =0.905

𝑈𝑚 = 266.28𝑚

𝑠

11

𝑉𝑥 = 150𝑚

𝑠

𝐶𝑝 = 1005𝐽

𝑘𝑔−𝐾

𝛽1 = −60.62°

𝛽2 = −47.78°

Finally, the number of stages can be calculated by using the average stage temperature rise.

𝑻𝒐𝒆

𝑻𝒐𝟏

= 𝟏 +∆𝑻𝒐𝒂𝒗𝒈

𝑻𝒐𝟏

⇒ 𝒏 = 𝟔.𝟕𝟖 ≈ 𝟕

𝑇𝑜𝑒= 452.48𝐾

𝑇𝑜1= 288𝐾

∆𝑇𝑜𝑎𝑣𝑔= 24.27𝐾

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.

∆𝑻𝒐 =𝝀𝟏𝑼𝒎𝑽𝒙

𝑪𝒑

(𝐭𝐚𝐧 𝜷𝟐 − 𝐭𝐚𝐧 𝜷𝟏) = 𝟐𝟔.𝟐𝟔𝑲

𝜆1 =0.98

𝑈𝑚 = 266.28𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠

𝐶𝑝 = 1005𝐽

𝑘𝑔−𝐾

𝛽1 = −60.62°

𝛽2 = −47.78°

Stagnation temperature after the stator can now easily be calculated using subtraction.

∆𝑻𝒐 = 𝑻𝒐𝟑− 𝑻𝒐𝟏

⇒ 𝑻𝒐𝟑= 𝟑𝟏𝟒.𝟐𝟕𝑲

12

∆𝑇𝑜 = 26.26𝐾

𝑇𝑜1= 288𝐾

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.

[𝑷𝒐𝟑

𝑷𝒐𝟏

]

𝝀−𝟏𝝀

= 𝟏 + 𝜼𝒕𝒕 (𝑻𝒐𝟑

𝑻𝒐𝟏

− 𝟏) ⇒ 𝑷𝒐𝟑= 𝟏𝟑𝟑,𝟓𝟏𝟎𝑷𝒂

𝑇𝑜3= 314.27𝐾

𝑇𝑜1= 288𝐾

𝑃𝑜1= 101.3𝑘𝑃𝑎

𝜂𝑡𝑡 = 0.9

Next the absolute flow angle after the rotor can be found using simple trigonometry.

𝑾𝒖𝟐= 𝑽𝒙 𝐭𝐚𝐧 𝜷𝟐 = 𝟏𝟔𝟓.𝟑𝟐

𝒎

𝒔

𝑉𝑥 = 150𝑚

𝑠

𝛽2 = −47.78°

𝑼𝒎 = 𝑽𝒖𝟐+ 𝑾𝒖𝟐

⇒ 𝑽𝒖𝟐= 𝟏𝟎𝟎.𝟗𝟓

𝒎

𝒔

𝑈𝑚 = 266.28𝑚

𝑠

𝑊𝑢2= 165.32

𝑚

𝑠

𝐭𝐚𝐧 𝜶𝟐 =𝑽𝒖𝟐

𝑽𝒙

⇒ 𝜶𝟐 = 𝟑𝟑.𝟗𝟒°

𝑉𝑥 = 150𝑚

𝑠

𝑉𝑢2= 100.95

𝑚

𝑠

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 160𝑚

𝑠 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

𝑽𝟐 = √𝑽𝒙𝟐 + 𝑽𝒖𝟐

𝟐 = 𝟏𝟖𝟎.𝟖𝟏𝒎

𝒔

𝑉𝑥 = 150𝑚

𝑠

𝑉𝑢2= 100.95

𝑚

𝑠

By using the De Haller Number one can see that the flow does not diffuse excessively.

𝑽𝟑

𝑽𝟐

≥ 𝟎.𝟕𝟐 ⇒ 𝟎. 𝟖𝟖 ≥ 𝟎.𝟕𝟐

𝑉3 = 160𝑚

𝑠

𝑉2 = 180.81𝑚

𝑠

The blade angle after the stator can be found using trigonometry.

𝐜𝐨𝐬 𝜶𝟑 =𝑽𝒙

𝑽𝟑

⇒ 𝜶𝟑 = 𝟐𝟎.𝟑𝟔°

𝑉𝑥 = 150𝑚

𝑠

𝑉3 = 160𝑚

𝑠

Finally, the static parameters can be found using the exit velocity and an isentropic process

equation.

𝑻𝟑 = 𝑻𝒐𝟑−

𝑽𝟑𝟐

𝟐𝑪𝒑

= 𝟑𝟎𝟏.𝟓𝟑𝑲

𝑇𝑜3= 314.27𝐾

𝑉3 = 160𝑚

𝑠

𝐶𝑝 = 1005𝐽

𝑘𝑔−𝐾

𝑷𝟑

𝑷𝒐𝟑

= (𝑻𝟑

𝑻𝒐𝟑

)

𝝀𝝀−𝟏

⇒ 𝑷𝟑 = 𝟏𝟏𝟓,𝟓𝟏𝟐𝑷𝒂

𝑃𝑜3= 133,510𝑃𝑎

𝑇𝑜3= 314.27𝐾

𝑇3 = 301.53𝐾

𝜆 = 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.

𝑹 = 𝟏 − 𝟎.𝟓𝑽𝒙

𝑼𝒎

(𝐭𝐚𝐧 𝜶𝟐 + 𝐭𝐚𝐧 𝜶𝟏) = 𝟖𝟏.𝟎𝟒%

𝑉𝑥 = 150𝑚

𝑠

𝑈𝑚 = 266.28𝑚

𝑠

𝛼2 = 33.94°

𝛼1 = 0°

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

β(°) α(°)

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 160𝑚

𝑠 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 analysis’s 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


-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



Stage 3


-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



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 stage’s 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.

𝑼𝒎𝑽𝒖𝒎= 𝑼𝒕𝑽𝒖𝒕

⇒ 𝑽𝒖𝒕= 𝟕𝟓.𝟕𝟓

𝒎

𝒔

𝑈𝑚 = 266.28𝑚

𝑠

𝑉𝑢𝑚= 101.00

𝑚

𝑠

𝑈𝑡 = 355.3𝑚

𝑠

Trigonometry can be used to find the flow angles on velocity triangles that represent tip

flow characteristics.

𝐭𝐚𝐧 𝜶𝟐 =𝑽𝒖𝒕

𝑽𝒙

⇒ 𝜶𝟐 = 𝟐𝟔.𝟕𝟗°

𝑉𝑢𝑡= 75.75

𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠

𝐭𝐚𝐧 𝜷𝟏 =−𝑼𝒕

𝑽𝒙

⇒ 𝜷𝟏 = −𝟔𝟕.𝟏𝟏°

𝑈𝑡 = 266.28𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠

𝑾𝒖𝒕= 𝑽𝒖𝒕

− 𝑼𝒕 = −𝟐𝟕𝟗.𝟓𝟓𝒎

𝒔

𝑉𝑢𝑡= 75.75

𝑚

𝑠

𝑈𝑡 = 266.28𝑚

𝑠

𝐭𝐚𝐧 𝜷𝟐 =𝑾𝒖𝒕

𝑽𝒙

⇒ 𝜷𝟐 = −𝟔𝟏.𝟕𝟖°

23

𝑉𝑥 = 150𝑚

𝑠

𝑊𝑢𝑡= 279.55

𝑚

𝑠

At Hub

Blade speed at the hub can be found using the constant rotational speed of the compressor.

𝑼𝒉 = 𝒓𝒉𝛀 ⇒ 𝟏𝟕𝟕.𝟓𝟕𝒎

𝒔

𝑟ℎ = 0.20𝑚

Ω = 887.78𝑟𝑎𝑑

𝑠

Using the Free Vortex Design assumption

𝑼𝒎𝑽𝒖𝒎= 𝑼𝒉𝑽𝒖𝒉

⇒ 𝑽𝒖𝒉= 𝟏𝟓𝟏.𝟓𝟕

𝒎

𝒔

𝑈𝑚 = 266.28𝑚

𝑠

𝑉𝑢𝑚= 101.00

𝑚

𝑠

𝑈ℎ = 177.57𝑚

𝑠

Using trigonometry on velocity triangles that represent hub flow characteristic the flow

angles can be found.

𝐭𝐚𝐧 𝜶𝟐 =𝑽𝒖𝒉

𝑽𝒙

⇒ 𝜶𝟐 = 𝟒𝟓.𝟑𝟎°

𝑉𝑢ℎ= 151.57

𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠

𝐭𝐚𝐧 𝜷𝟏 =−𝑼𝒉

𝑽𝒙

⇒ 𝜷𝟏 = −𝟒𝟗.𝟖𝟏°

𝑈ℎ = 177.57𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠

𝑾𝒖𝒉= 𝑽𝒖𝒉

− 𝑼𝒉 = −𝟐𝟔.𝟎𝟎𝒎

𝒔

𝑉𝑢ℎ= 151.57

𝑚

𝑠

𝑈ℎ = 177.57𝑚

𝑠

𝐭𝐚𝐧 𝜷𝟐 =𝑾𝒖𝒉

𝑽𝒙

⇒ 𝜷𝟐 = −𝟗. 𝟖𝟑°

𝑉𝑥 = 150𝑚

𝑠

24

𝑊𝑢ℎ= −26.00

𝑚

𝑠

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.

𝐜𝐨𝐬 𝜶𝟏

𝐜𝐨𝐬 𝜶𝟐

= 𝟎. 𝟕𝟐 ⇒ 𝟎.𝟕𝟎 ≯ 𝟎. 𝟕𝟐

𝛼2 = 45.30°

𝛼1 = 0°

𝐜𝐨𝐬 𝜷𝟏

𝐜𝐨𝐬 𝜷𝟐

= 𝟎. 𝟕𝟐 ⇒ 𝟎.𝟔𝟓 ≯ 𝟎. 𝟕𝟐

𝛽2 = −9.83°

𝛽1 = −49.81°

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.

𝑷𝟏 = 𝝆𝟏𝑹𝑻𝟏 ⇒ 𝝆𝟏 = 𝟏. 𝟓𝟐𝒌𝒈

𝒎𝟑

𝑃1 = 145,406.80𝑃𝑎

𝑇1 = 324.39𝐾

𝑅 = 287 𝐽

𝑘𝑔−𝐾

Because the mass flow rate is constant it can be used to find the annular area at the exit.

�̇� = 𝝆𝒆𝑽𝒙 𝑨𝒆 ⇒ 𝑨𝒆 = 𝟎.𝟎𝟖𝟓𝒎𝟐

𝜌𝑒 = 1.52𝑘𝑔

𝑚3

Hub Mean Tip

α1(°) 0 0 0

α2(°) 45.3 33.95 26.79

β1(°) -49.81 -60.62 -67.11

β2(°) -9.83 -47.77 -61.78

FlowAnglesofStage1FromHubtoTip

25

𝑉𝑥 = 150𝑚

𝑠

�̇� = 20𝑘𝑔

𝑠

Next mean-radius and annulus area can be utilized to find the radiuses at the hub and tip.

𝑨𝒆 = 𝒓𝒕𝟐 − 𝒓𝒉

𝟐 ⇒ 𝒓𝒕 = 𝟎. 𝟑𝟕𝒎

𝑟𝑚 = 0.5(𝑟𝑡 + 𝑟𝑒) = 0.30 ⇒ 𝑟ℎ = 0.60 − 𝑟𝑡

𝒓𝒎 = 𝟎.𝟓(𝒓𝒕 + 𝒓𝒉) = 𝟎. 𝟑𝟎 ⇒ 𝒓𝒉 = 𝟎. 𝟐𝟑𝒎

𝑟𝑡 = 0.37𝑚

Finally the radiuses can be used to calculate the blade speeds at the hub and tip.

𝑼𝒕 = 𝒓𝒕𝛀 ⇒ 𝟑𝟐𝟗.𝟔𝒎

𝒔

𝑟𝑡 = 0.37𝑚

Ω = 887.78𝑟𝑎𝑑

𝑠

𝑼𝒉 = 𝒓𝒉𝛀 ⇒ 𝟐𝟎𝟑.𝟑𝟓𝒎

𝒔

𝑟ℎ = 0.23𝑚

Ω = 887.78𝑟𝑎𝑑

𝑠

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


⇒ 𝑽𝒖𝒕= 𝟔𝟕.𝟏𝟕

𝒎

𝒔

𝑈𝑚 = 266.28𝑚

𝑠

𝑉𝑢𝑚= 83.14

𝑚

𝑠

𝑈𝑡 = 329.6𝑚

𝑠

Using trigonometry on velocity triangles that represent tip flow characteristic the flow


𝐭𝐚𝐧 𝜶𝟏 =𝑽𝒖𝒕

𝑽𝒙

⇒ 𝜶𝟏 = 𝟐𝟒.𝟏𝟐°

𝑉𝑢𝑡= 67.17

𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠

26


− 𝑼𝒕 = −𝟐𝟔𝟐.𝟒𝟑𝒎

𝒔

𝑉𝑢𝑡= 67.17

𝑚

𝑠

𝑈𝑡 = 329.6𝑚

𝑠

𝐭𝐚𝐧 𝜷𝟏 =𝑾𝒖𝒕

𝑽𝒙

⇒ 𝜷𝟏 = −𝟔𝟎.𝟐𝟓°

𝑉𝑥 = 150𝑚

𝑠

𝑊𝑢𝑡= −262.43

𝑚

𝑠

At Tip After Rotor



⇒ 𝑽𝒖𝒕= 𝟏𝟒𝟕.𝟗𝟔

𝒎

𝒔

𝑈𝑚 = 266.28𝑚

𝑠

𝑉𝑢𝑚= 183.14

𝑚

𝑠

𝑈𝑡 = 329.6𝑚

𝑠

Using trigonometry on velocity triangles that represent tip flow characteristic the flow


𝐭𝐚𝐧 𝜶𝟐 =𝑽𝒖𝒕

𝑽𝒙

⇒ 𝜶𝟐 = 𝟒𝟒.𝟔𝟏°

𝑉𝑢𝑡= 67.17

𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠


− 𝑼𝒕 = −𝟏𝟖𝟏.𝟔𝟒𝒎

𝒔

𝑉𝑢𝑡= 147.96

𝑚

𝑠

𝑈𝑡 = 329.6𝑚

𝑠

𝐭𝐚𝐧 𝜷𝟐 =𝑾𝒖𝒕

𝑽𝒙

⇒ 𝜷𝟐 = −𝟓𝟎.𝟒𝟓°

𝑉𝑥 = 150𝑚

𝑠

𝑊𝑢𝑡= −181.64

𝑚

𝑠

At Hub Before Rotor Using the Free Vortex Design assumption


⇒ 𝑽𝒖𝒉= 𝟏𝟎𝟖.𝟖𝟔

𝒎

𝒔

27

𝑈𝑚 = 266.28𝑚

𝑠

𝑉𝑢𝑚= 83.14

𝑚

𝑠

𝑈ℎ = 203.35𝑚

𝑠



𝐭𝐚𝐧 𝜶𝟏 =𝑽𝒖𝒉

𝑽𝒙

⇒ 𝜶𝟏 = 𝟑𝟓.𝟗𝟕°

𝑉𝑢ℎ= 108.86

𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠


− 𝑼𝒉 = −𝟗𝟒.𝟒𝟖𝒎

𝒔

𝑉𝑢ℎ= 108.86

𝑚

𝑠

𝑈ℎ = 203.35𝑚

𝑠

𝐭𝐚𝐧 𝜷𝟏 =𝑾𝒖𝒉

𝑽𝒙

⇒ 𝜷𝟏 = −𝟑𝟐. 𝟐𝟏°

𝑉𝑥 = 150𝑚

𝑠

𝑊𝑢ℎ= −94.48

𝑚

𝑠

At Hub After Rotor



⇒ 𝑽𝒖𝒉= 𝟐𝟑𝟗.𝟖

𝒎

𝒔

𝑈𝑚 = 266.28𝑚

𝑠

𝑉𝑢𝑚= 183.14

𝑚

𝑠

𝑈ℎ = 203.35𝑚

𝑠



𝐭𝐚𝐧 𝜶𝟐 =𝑽𝒖𝒉

𝑽𝒙

⇒ 𝜶𝟐 = 𝟓𝟕.𝟗𝟕°

𝑉𝑢ℎ= 239.82

𝑚

𝑠

𝑉𝑥 = 150𝑚

𝑠

28


− 𝑼𝒉 = 𝟑𝟒.𝟒𝟔𝒎

𝒔

𝑉𝑢ℎ= 239.82

𝑚

𝑠

𝑈ℎ = 203.35𝑚

𝑠

𝐭𝐚𝐧 𝜷𝟐 =𝑾𝒖𝒉

𝑽𝒙

⇒ 𝜷𝟐 = 𝟏𝟑. 𝟔𝟔°

𝑉𝑥 = 150𝑚

𝑠

𝑊𝑢ℎ= 34.46

𝑚

𝑠

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

α1(°) 35.97 29 24.12

α2(°) 57.97 50.68 44.61

β1(°) -32.21 -50.68 -60.25

β2(°) 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) α1(°)

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# β1(°) α2(°) β2(°) α3(°)

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 ΔP(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


31

Table 7d Summary of Devised Design Parameters Continued

Stage# Po3/Po1 ΔPo(Pa) T1(K) T3(K) To1(K) To3(K) ΔTo(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


32

Appendix

V1(m/s) 150 To1(K) 288 Vx(m/s) 150

β1(°) -60.62 Po1(Pa) 101300 Um(m/s) 266.28

λ 0.98 Cp(J/Kg-K) 1005

P1(Pa) 88,175.90 ηtt 0.9T1(K) 276.81 γ 1.4

W1(m/s) β2(°) Wu2(m/s) W2(m/s) Vu2(m/s) V2(m/s) α2(°) V3(m/s)-ASSUMED DHCheck>0.72305.6224 -47.7742 -165.277 223.1963 101.0027 180.8357 33.95442 152.9 0.84551884

α3(°) ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/P1 ΔPo(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) 150α1(°) 11.17691 Po1(Pa) 133508.2336 Um(m/s) 266.28

λ 0.93 Cp(J/Kg-K) 1005P1(Pa) 116,999.36 ηtt 0.9T1(K) 302.6324566 γ 1.4

R(%) 70

Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DHCheckα>0.72 Vu2(m/s)29.63798 -236.642 -280.178 -57.6307 40.94273195 0.769968667 130.13

DHCheckβ>0.720.723025386

V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DHCheck>0.72 α3(°)198.5795 -136.15 -202.575 -42.229 171.5 0.86363392 28.99813

ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΔPo(Pa)24.76208 339.0256 169687.6 324.3926 145406.8026 1.270989412 36,179.32

Input



λ 0.88 Cp(J/Kg-K) 1005P1(Pa) 145,406.80 ηtt 0.9T1(K) 324.3926236 γ 1.4R(%) 50


DHCheckβ>0.720.724459269

V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DHCheck>0.72 α3(°)236.7283 -83.1399 -171.5 -28.9981 171.05 0.722558355 28.725ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΔPo(Pa)

23.31609 362.3417 209381.1 347.7854 181389.9797 1.233921644 39,693.59


Input

33


λ 0.83 Cp(J/Kg-K) 1005P1(Pa) 181,389.98 ηtt 0.9

T1(K) 347.7854005 γ 1.4R(%) 50


DHCheckβ>0.720.720361435


22.40134 384.743 253072.6 370.1867 221116.1559 1.208669295 43,691.42


Input


λ 0.83 Cp(J/Kg-K) 1005P1(Pa) 221,116.16 ηtt 0.9

T1(K) 370.1867428 γ 1.4R(%) 50


DHCheckβ>0.720.720361435


22.40134 407.1444 302608.1 392.5881 266404.1359 1.195736452 49,535.52


Input

34

Equation used to calculate design parameters can be found below:

𝐭𝐚𝐧 𝜶𝟏 =𝑽𝒖𝟏

𝑽𝒙⇒ 𝑽𝒖𝟏

, Trigonometry of Velocity Diagrams

𝑾𝒖𝟏= 𝑼𝒎 − 𝑽𝒖𝟏


𝑾𝟏 = −√𝑾𝒖𝟏

𝟐 + 𝑽𝒙𝟐, Trigonometry of Velocity Diagrams

𝐭𝐚𝐧 𝜷𝟏 =𝑾𝒖𝟏

𝑽𝒙⇒ 𝜷𝟏, Trigonometry of Velocity Diagrams

𝑹 = 𝟏 − 𝟎.𝟓𝑽𝒙

𝑼𝒎

(𝐭𝐚𝐧 𝜶𝟐 + 𝐭𝐚𝐧 𝜶𝟏) ⇒ 𝜶𝟐, Degree of Reaction

𝑽𝟏

𝑽𝟐≥ 𝟎. 𝟕𝟐, De Haller Number

V1(m/s) 171.05 To1(K) 407.1443549 Vx(m/s) 150

α1(°) 28.725 Po1(Pa) 302608.0823 Um(m/s) 266.28

λ 0.83 Cp(J/Kg-K) 1005

P1(Pa) 266,404.14 ηtt 0.9

T1(K) 392.588085 γ 1.4

R(%) 50


DHCheckβ>0.720.720361435


22.40134 429.5457 358381.8 414.9894 317645.5553 1.184310152 55,773.74


Input


λ 0.83 Cp(J/Kg-K) 1005P1(Pa) 317,645.56 ηtt 0.9

T1(K) 414.9894273 γ 1.4R(%) 50


DHCheckβ>0.720.720361435


22.40134 451.947 420791.2 437.3908 375235.4437 1.174142048 62,409.34

Input


35

𝑾𝟐

𝑾𝟏≥ 𝟎. 𝟕𝟐, De Haller Number

𝐭𝐚𝐧 𝜶𝟐 =𝑽𝒖𝟐

𝑽𝒙⇒ 𝑽𝒖𝟐


𝑾𝒖𝟐= 𝑽𝒖𝟐

− 𝑼𝒎, Trigonometry of Velocity Diagrams

𝑾𝟐 = −√𝑾𝒖𝟐

𝟐 + 𝑽𝒙𝟐, Trigonometry of Velocity Diagrams

𝐭𝐚𝐧 𝜷𝟐 =𝑾𝒖𝟐

𝑽𝒙⇒ 𝜷𝟐, Trigonometry of Velocity Diagrams

∆𝑻𝒐 =𝝀#𝑼𝒎 𝑽𝒙

𝑪𝒑

(𝐭𝐚𝐧 𝜷𝟐 − 𝐭𝐚𝐧 𝜷𝟏), Actual Stage Temperature Rise

∆𝑻𝒐 = 𝑻𝒐𝟑− 𝑻𝒐𝟏

⇒ 𝑻𝒐𝟑, Subtraction

[𝑷𝒐𝟑

𝑷𝒐𝟏

]

𝝀−𝟏

𝝀= 𝟏 + 𝜼𝒕𝒕 (

𝑻𝒐𝟑

𝑻𝒐𝟏

− 𝟏) ⇒ 𝑷𝒐𝟑, Stage Pressure Ratio from Stage Temperature Ratio

𝑽𝟐 = √𝑽𝒙𝟐 + 𝑽𝒖𝟐

𝟐, Trigonometry of Velocity Diagrams

𝑽𝟑

𝑽𝟐≥ 𝟎. 𝟕𝟐, De Haller Number

𝐜𝐨𝐬 𝜶𝟑 =𝑽𝒙

𝑽𝟑⇒ 𝜶𝟑 , Trigonometry of Velocity Diagrams

𝑻𝟑 = 𝑻𝒐𝟑−

𝑽𝟑𝟐

𝟐𝑪𝒑 , Static Temperature from Stagnation Temperature

𝑷𝟑

𝑷𝒐𝟑

= (𝑻𝟑

𝑻𝒐𝟑

)

𝝀

𝝀−𝟏⇒ 𝑷𝟑 , 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

turbo final 2

Documents