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Turbo Final 2

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Frederick (Avy) Smith

This document analyzes the design of a multi-stage axial-flow compressor. It breaks the analysis into several sections: 1) Evaluating the relative Mach number at the tip of the first stage to ensure transonic but not supersonic flow. 2) Calculating the annulus dimensions at the inlet and outlet using assumptions about mean radius, constant mass flow rate, and overall pressure ratio. 3) The analysis makes simplifying assumptions to allow preliminary design calculations while noting areas for more detailed future analysis.

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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: AnnulusDimensions.................................................................................................................................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 Compressor Pressure Ratio 4.15 Air Mass Flow Rate 20 kg/s Compressor Polytropic Efficiency 0.9 Blade Tip Speed of First Stage 355.3 m/s Hub-to-Tip Ratio of First Stage 0.5 Design Parameters
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 radius also shrinks. By using trigonometry on the velocity diagrams of stages 2-7 at the tip it can be Name Value Ambient Stagnation Pressure 101.3 kPa Ambient Stagnation Temptrature 288 K Constant Specific Heat of Air 1005 J/kg-K Specific Heat Ratio of Air 1.4 Ideal Gas Constat for Air 287 J/kg-K Axial Velocity of Air 150 m/s Work Done Factor Stage 1 0.98 Work Done Factor Stage 2 0.93 Work Done Factor Stage 3 0.88 Work Done Factor for Ongoing Stages 0.83 Initial Assumptions
5 seen that the relative velocity will be less than in Stage 1. Stagnation speed of sound is first found using ambient air properties. 𝑪 𝒐 𝟏 = √ 𝝀𝑹𝑻 𝒐 𝟏 = 𝟑𝟒𝟎. 𝟏𝟕 𝒎 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. 𝑴 𝒐 𝟏 𝑹 = 𝑾 𝟏 𝑪 𝒐 𝟏 = 𝟏. 𝟏𝟑 𝐶 𝑜1 = 340.17 𝑚 𝑠 𝑊1 = √ 𝑉𝑥 2 + 𝑈𝑡 = 385.66 𝑚 𝑠 𝑉𝑥 = 150 𝑚 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. 𝑻 𝟏 = 𝑻 𝒐 𝟏 − 𝑽 𝟏 𝟐 𝟐𝑪 𝒑 = 𝟐𝟕𝟔. 𝟖𝑲 𝑇𝑜1 = 288𝐾 𝑉1 = 150 𝑚 𝑠 𝐶 𝑝 = 1005 𝐽 𝑘𝑔−𝐾 Finally the relative Mach number at the tip can be found. 𝑴 𝟏 𝑹 = 𝑴 𝒐 𝟏 𝑹 √ 𝑻 𝒐 𝟏 𝑻 𝟏 = 𝟏. 𝟏𝟔 𝑀 𝑜1 𝑅 = 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. 𝑷 𝟏 𝑷 𝒐 𝟏 = ( 𝑻 𝟏 𝑻 𝒐 𝟏 ) 𝝀 𝝀−𝟏 ⇒ 𝑷 𝟏 = 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. 𝑷 𝟏 = 𝝆 𝟏 𝑹𝑻 𝟏 ⇒ 𝝆 𝟏 = 𝟏. 𝟏𝟏 𝒌𝒈 𝒎 𝟑 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. 𝒎̇ = 𝝆 𝟏 𝑽 𝒙 𝑨 𝟏 ⇒ 𝑨 𝟏 = 𝟎. 𝟏𝟐𝒎 𝟐 𝜌1 = 1.11 𝑘𝑔 𝑚3 𝑉𝑥 = 150 𝑚 𝑠 𝑚̇ = 20 𝑘𝑔 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. 𝑨 𝟏 = 𝒓 𝒕 𝟐 − 𝒓 𝒉 𝟐 = 𝒓 𝒕 𝟐 − ( 𝟎. 𝟓𝒓 𝒕) 𝟐 = 𝟎. 𝟏𝟐𝟎𝒎 𝟐 ⇒ 𝒓 𝒕 = 𝟎. 𝟒𝟎𝒎 𝑟ℎ 𝑟𝑡 = 0.5 𝒓 𝒎 = 𝟎. 𝟓( 𝒓 𝒕 + 𝒓 𝒉) ⇒ 𝒓 𝒉 = 𝟎. 𝟐𝟎𝒎 𝑟𝑡 = 0.40𝑚 𝒓 𝒎 = 𝟎. 𝟓( 𝒓 𝒕 + 𝒓 𝒉) = 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. 𝑃𝑜 𝑒 𝑃𝑜1 = 4.15 ⇒ 𝑃𝑜 𝑒 = 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 𝑻 𝒐 𝒆 𝑻 𝒐 𝟏 = ( 𝑷 𝟎 𝒆 𝑷 𝒐 𝟏 ) 𝝀−𝟏 𝜼 𝒑 𝝀 ⇒ 𝑻 𝒐 𝒆 = 𝟒𝟓𝟐. 𝟒𝟖𝑲 𝑃𝑜 𝑒 𝑃𝑜1 = 4.15 𝜆 = 1.4 𝜂 𝑝 = 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. 𝑻 𝒆 = 𝑻 𝒐 𝒆 − 𝑽 𝟑 𝟐 𝟐𝑪 𝒑 ⇒ 𝟒𝟑𝟕. 𝟗𝟐𝑲 𝑇𝑜 𝑒 = 452.48𝐾 𝐶 𝑝 = 1005 𝐽 𝑘𝑔−𝐾 𝑉3 = 171.05 𝑚 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. 𝑷 𝒆 𝑷 𝒐 𝒆 = ( 𝑻 𝒆 𝑻 𝒐 𝒆 ) 𝝀 𝝀−𝟏 ⇒ 𝑷 𝒆 = 𝟑𝟕𝟒,𝟗𝟑𝟒𝑷𝒂 𝑃𝑜 𝑒 = 420,395𝑃𝑎 𝑇𝑒 = 437.92𝐾 𝑇𝑜 𝑒 = 452.48𝐾 𝜆 = 1.4 Static density can now be found using the Ideal Gas Law. 𝑷 𝒆 = 𝝆 𝒆 𝑹𝑻 𝒆 ⇒ 𝝆 𝒆 = 𝟐. 𝟗𝟖 𝒌𝒈 𝒎 𝟑 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. 𝒎̇ = 𝝆 𝒆 𝑽 𝒙 𝑨 𝒆 ⇒ 𝑨 𝒆 = 𝟎. 𝟎𝟒𝟓𝒎 𝟐 𝜌𝑒 = 2.98 𝑘𝑔 𝑚3 𝑉𝑥 = 150 𝑚 𝑠 𝑚̇ = 20 𝑘𝑔 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. 𝑨 𝒆 = 𝒓 𝒕 𝒆 𝟐 − 𝒓 𝒉 𝒆 𝟐 ⇒ 𝒓 𝒉 𝒆 = 𝟎. 𝟐𝟔𝒎 𝑟 𝑚 = 0.5(𝑟𝑡 𝑒 + 𝑟ℎ 𝑒 ) = 0.30 ⇒ 𝑟𝑡 𝑒 = 0.60 − 𝑟ℎ 𝑒 𝒓 𝒎 = 𝟎. 𝟓(𝒓 𝒕 𝒆 + 𝒓 𝒉 𝒆 ) = 𝟎. 𝟑𝟎 ⇒ 𝒓 𝒕 𝒆 = 𝟎. 𝟑𝟑𝟕𝒎 𝑟ℎ 𝑒 = 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. 𝑼 𝒕 = 𝒓 𝒕 𝛀 ⇒ 𝛀 = 𝟖𝟖𝟕. 𝟕𝟖 𝒓𝒂𝒅 𝒔 𝑈𝑡 = 355.3 𝑚 𝑠 𝑟𝑡 = 0.40𝑚 𝑼 𝒎 = 𝒓 𝒎 𝛀 = 𝟐𝟔𝟔. 𝟐𝟖 𝒎 𝒔 Ω = 887.78 𝑟𝑎𝑑 𝑠 𝑟 𝑚 = 0.30𝑚
10 The relative flow angle will be calculated at the inlet using the mean-blade speed and trigonometry. 𝐭𝐚𝐧 𝜷 𝟏 = −𝑼 𝒎 𝑽 𝒙 ⇒ 𝜷 𝟏 = −𝟔𝟎. 𝟔𝟐° 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑥 = 150 𝑚 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 𝜷 𝟏 𝐜𝐨𝐬 𝜷 𝟐 = 𝟎. 𝟕𝟑 ⇒ 𝜷 𝟐 = −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 work 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. ∆𝑻 𝒐 𝒂𝒗𝒈 = 𝝀 𝒂𝒗𝒈 𝑼 𝒎 𝑽 𝒙 𝑪 𝒑 ( 𝐭𝐚𝐧 𝜷 𝟐 − 𝐭𝐚𝐧 𝜷 𝟏) = 𝟐𝟒. 𝟐𝟕𝑲 𝜆 𝑎𝑣𝑔 =0.905 𝑈 𝑚 = 266.28 𝑚 𝑠
11 𝑉𝑥 = 150 𝑚 𝑠 𝐶 𝑝 = 1005 𝐽 𝑘𝑔−𝐾 𝛽1 = −60.62° 𝛽2 = −47.78° Finally, the number of stages can be calculated by using the average stage temperature rise. 𝑻 𝒐 𝒆 𝑻 𝒐 𝟏 = 𝟏 + ∆𝑻 𝒐 𝒂𝒗𝒈 𝑻 𝒐 𝟏 ⇒ 𝒏 = 𝟔. 𝟕𝟖 ≈ 𝟕 𝑇𝑜 𝑒 = 452.48𝐾 𝑇𝑜1 = 288𝐾 ∆𝑇𝑜 𝑎𝑣𝑔 = 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. ∆𝑻 𝒐 = 𝝀 𝟏 𝑼 𝒎 𝑽 𝒙 𝑪 𝒑 ( 𝐭𝐚𝐧 𝜷 𝟐 − 𝐭𝐚𝐧 𝜷 𝟏) = 𝟐𝟔. 𝟐𝟔𝑲 𝜆1 =0.98 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝐶 𝑝 = 1005 𝐽 𝑘𝑔−𝐾 𝛽1 = −60.62° 𝛽2 = −47.78° Stagnation temperature after the stator can now easily be calculated using subtraction. ∆𝑻 𝒐 = 𝑻 𝒐 𝟑 − 𝑻 𝒐 𝟏 ⇒ 𝑻 𝒐 𝟑 = 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. [ 𝑷 𝒐 𝟑 𝑷 𝒐 𝟏 ] 𝝀−𝟏 𝝀 = 𝟏 + 𝜼 𝒕𝒕 ( 𝑻 𝒐 𝟑 𝑻 𝒐 𝟏 − 𝟏) ⇒ 𝑷 𝒐 𝟑 = 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. 𝑾 𝒖 𝟐 = 𝑽 𝒙 𝐭𝐚𝐧 𝜷 𝟐 = 𝟏𝟔𝟓. 𝟑𝟐 𝒎 𝒔 𝑉𝑥 = 150 𝑚 𝑠 𝛽2 = −47.78° 𝑼 𝒎 = 𝑽 𝒖 𝟐 + 𝑾 𝒖 𝟐 ⇒ 𝑽 𝒖 𝟐 = 𝟏𝟎𝟎. 𝟗𝟓 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑊𝑢2 = 165.32 𝑚 𝑠 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝟐 𝑽 𝒙 ⇒ 𝜶 𝟐 = 𝟑𝟑. 𝟗𝟒° 𝑉𝑥 = 150 𝑚 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 160 𝑚 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 𝑽 𝟐 = √𝑽 𝒙 𝟐 + 𝑽 𝒖 𝟐 𝟐 = 𝟏𝟖𝟎. 𝟖𝟏 𝒎 𝒔 𝑉𝑥 = 150 𝑚 𝑠 𝑉𝑢2 = 100.95 𝑚 𝑠 By using the De Haller Number one can see that the flow does not diffuse excessively. 𝑽 𝟑 𝑽 𝟐 ≥ 𝟎. 𝟕𝟐 ⇒ 𝟎. 𝟖𝟖 ≥ 𝟎. 𝟕𝟐 𝑉3 = 160 𝑚 𝑠 𝑉2 = 180.81 𝑚 𝑠 The blade angle after the stator can be found using trigonometry. 𝐜𝐨𝐬 𝜶 𝟑 = 𝑽 𝒙 𝑽 𝟑 ⇒ 𝜶 𝟑 = 𝟐𝟎. 𝟑𝟔° 𝑉𝑥 = 150 𝑚 𝑠 𝑉3 = 160 𝑚 𝑠 Finally, the static parameters can be found using the exit velocity and an isentropic process equation. 𝑻 𝟑 = 𝑻 𝒐 𝟑 − 𝑽 𝟑 𝟐 𝟐𝑪 𝒑 = 𝟑𝟎𝟏. 𝟓𝟑𝑲 𝑇𝑜3 = 314.27𝐾 𝑉3 = 160 𝑚 𝑠 𝐶 𝑝 = 1005 𝐽 𝑘𝑔−𝐾 𝑷 𝟑 𝑷 𝒐 𝟑 = ( 𝑻 𝟑 𝑻 𝒐 𝟑 ) 𝝀 𝝀−𝟏 ⇒ 𝑷 𝟑 = 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. 𝟓 𝑽 𝒙 𝑼 𝒎 ( 𝐭𝐚𝐧 𝜶 𝟐 + 𝐭𝐚𝐧 𝜶 𝟏) = 𝟖𝟏. 𝟎𝟒% 𝑉𝑥 = 150 𝑚 𝑠 𝑈 𝑚 = 266.28 𝑚 𝑠 𝛼2 = 33.94° 𝛼1 = 0° Po(Pa) P(Pa) To(K) T(K) At Inlet 101,300 88,175.90 288 276.8 At Exit 133,510 115,512 314.27 301.53 Initial Static and Stage Tempratures/Pressures of First Stage β(°) α(°) Before Rotor -60.62 0 After Rotor -47.78 33.94 After Stator n/a 20.36 Initial Flow Angles of First Stage
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 160 𝑚 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 between 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) Before Rotor 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) Before Rotor 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) Before Rotor 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) Before Rotor 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) Stage Number Stage Pressure Rise vs Stage Number Stage Pressure Rise vs Stage Number
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. 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒕 𝑽 𝒖 𝒕 ⇒ 𝑽 𝒖 𝒕 = 𝟕𝟓. 𝟕𝟓 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 101.00 𝑚 𝑠 𝑈𝑡 = 355.3 𝑚 𝑠 Trigonometry can be used to find the flow angles on velocity triangles that represent tip flow characteristics. 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜶 𝟐 = 𝟐𝟔. 𝟕𝟗° 𝑉𝑢 𝑡 = 75.75 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟏 = −𝑼 𝒕 𝑽 𝒙 ⇒ 𝜷 𝟏 = −𝟔𝟕. 𝟏𝟏° 𝑈𝑡 = 266.28 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝑾 𝒖 𝒕 = 𝑽 𝒖 𝒕 − 𝑼 𝒕 = −𝟐𝟕𝟗. 𝟓𝟓 𝒎 𝒔 𝑉𝑢 𝑡 = 75.75 𝑚 𝑠 𝑈𝑡 = 266.28 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟐 = 𝑾 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜷 𝟐 = −𝟔𝟏. 𝟕𝟖°
23 𝑉𝑥 = 150 𝑚 𝑠 𝑊𝑢 𝑡 = 279.55 𝑚 𝑠 At Hub Blade speed at the hub can be found using the constant rotational speed of the compressor. 𝑼 𝒉 = 𝒓 𝒉 𝛀 ⇒ 𝟏𝟕𝟕. 𝟓𝟕 𝒎 𝒔 𝑟ℎ = 0.20𝑚 Ω = 887.78 𝑟𝑎𝑑 𝑠 Using the Free Vortex Design assumption 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒉 𝑽 𝒖 𝒉 ⇒ 𝑽 𝒖 𝒉 = 𝟏𝟓𝟏. 𝟓𝟕 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 101.00 𝑚 𝑠 𝑈ℎ = 177.57 𝑚 𝑠 Using trigonometry on velocity triangles that represent hub flow characteristic the flow angles can be found. 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜶 𝟐 = 𝟒𝟓. 𝟑𝟎° 𝑉𝑢ℎ = 151.57 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟏 = −𝑼 𝒉 𝑽 𝒙 ⇒ 𝜷 𝟏 = −𝟒𝟗. 𝟖𝟏° 𝑈ℎ = 177.57 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝑾 𝒖 𝒉 = 𝑽 𝒖 𝒉 − 𝑼 𝒉 = −𝟐𝟔. 𝟎𝟎 𝒎 𝒔 𝑉𝑢ℎ = 151.57 𝑚 𝑠 𝑈ℎ = 177.57 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟐 = 𝑾 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜷 𝟐 = −𝟗. 𝟖𝟑° 𝑉𝑥 = 150 𝑚 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 𝜶 𝟏 𝐜𝐨𝐬 𝜶 𝟐 = 𝟎. 𝟕𝟐 ⇒ 𝟎. 𝟕𝟎 ≯ 𝟎. 𝟕𝟐 𝛼2 = 45.30° 𝛼1 = 0° 𝐜𝐨𝐬 𝜷 𝟏 𝐜𝐨𝐬 𝜷 𝟐 = 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. 𝑷 𝟏 = 𝝆 𝟏 𝑹𝑻 𝟏 ⇒ 𝝆 𝟏 = 𝟏. 𝟓𝟐 𝒌𝒈 𝒎 𝟑 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. 𝒎̇ = 𝝆 𝒆 𝑽 𝒙 𝑨 𝒆 ⇒ 𝑨 𝒆 = 𝟎. 𝟎𝟖𝟓𝒎 𝟐 𝜌𝑒 = 1.52 𝑘𝑔 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 Flow Angles of Stage 1 From Hub to Tip
25 𝑉𝑥 = 150 𝑚 𝑠 𝑚̇ = 20 𝑘𝑔 𝑠 Next mean-radius and annulus area can be utilized to find the radiuses at the hub and tip. 𝑨 𝒆 = 𝒓 𝒕 𝟐 − 𝒓 𝒉 𝟐 ⇒ 𝒓 𝒕 = 𝟎. 𝟑𝟕𝒎 𝑟 𝑚 = 0.5( 𝑟𝑡 + 𝑟𝑒) = 0.30 ⇒ 𝑟ℎ = 0.60 − 𝑟𝑡 𝒓 𝒎 = 𝟎. 𝟓( 𝒓 𝒕 + 𝒓 𝒉) = 𝟎. 𝟑𝟎 ⇒ 𝒓 𝒉 = 𝟎. 𝟐𝟑𝒎 𝑟𝑡 = 0.37𝑚 Finally the radiuses can be used to calculate the blade speeds at the hub and tip. 𝑼 𝒕 = 𝒓 𝒕 𝛀 ⇒ 𝟑𝟐𝟗. 𝟔 𝒎 𝒔 𝑟𝑡 = 0.37𝑚 Ω = 887.78 𝑟𝑎𝑑 𝑠 𝑼 𝒉 = 𝒓 𝒉 𝛀 ⇒ 𝟐𝟎𝟑. 𝟑𝟓 𝒎 𝒔 𝑟ℎ = 0.23𝑚 Ω = 887.78 𝑟𝑎𝑑 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 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒕 𝑽 𝒖 𝒕 ⇒ 𝑽 𝒖 𝒕 = 𝟔𝟕. 𝟏𝟕 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 83.14 𝑚 𝑠 𝑈𝑡 = 329.6 𝑚 𝑠 Using trigonometry on velocity triangles that represent tip flow characteristic the flow angles can be found. 𝐭𝐚𝐧 𝜶 𝟏 = 𝑽 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜶 𝟏 = 𝟐𝟒. 𝟏𝟐° 𝑉𝑢 𝑡 = 67.17 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠
26 𝑾 𝒖 𝒕 = 𝑽 𝒖 𝒕 − 𝑼 𝒕 = −𝟐𝟔𝟐. 𝟒𝟑 𝒎 𝒔 𝑉𝑢 𝑡 = 67.17 𝑚 𝑠 𝑈𝑡 = 329.6 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟏 = 𝑾 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜷 𝟏 = −𝟔𝟎. 𝟐𝟓° 𝑉𝑥 = 150 𝑚 𝑠 𝑊𝑢 𝑡 = −262.43 𝑚 𝑠 At Tip After Rotor Using the Free Vortex Design assumption 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒕 𝑽 𝒖 𝒕 ⇒ 𝑽 𝒖 𝒕 = 𝟏𝟒𝟕. 𝟗𝟔 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 183.14 𝑚 𝑠 𝑈𝑡 = 329.6 𝑚 𝑠 Using trigonometry on velocity triangles that represent tip flow characteristic the flow angles can be found. 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜶 𝟐 = 𝟒𝟒. 𝟔𝟏° 𝑉𝑢 𝑡 = 67.17 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝑾 𝒖 𝒕 = 𝑽 𝒖 𝒕 − 𝑼 𝒕 = −𝟏𝟖𝟏. 𝟔𝟒 𝒎 𝒔 𝑉𝑢 𝑡 = 147.96 𝑚 𝑠 𝑈𝑡 = 329.6 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟐 = 𝑾 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜷 𝟐 = −𝟓𝟎. 𝟒𝟓° 𝑉𝑥 = 150 𝑚 𝑠 𝑊𝑢 𝑡 = −181.64 𝑚 𝑠 At Hub Before Rotor Using the Free Vortex Design assumption 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒉 𝑽 𝒖 𝒉 ⇒ 𝑽 𝒖 𝒉 = 𝟏𝟎𝟖. 𝟖𝟔 𝒎 𝒔
27 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 83.14 𝑚 𝑠 𝑈ℎ = 203.35 𝑚 𝑠 Using trigonometry on velocity triangles that represent hub flow characteristic the flow angles can be found. 𝐭𝐚𝐧 𝜶 𝟏 = 𝑽 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜶 𝟏 = 𝟑𝟓. 𝟗𝟕° 𝑉𝑢ℎ = 108.86 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝑾 𝒖 𝒉 = 𝑽 𝒖 𝒉 − 𝑼 𝒉 = −𝟗𝟒. 𝟒𝟖 𝒎 𝒔 𝑉𝑢ℎ = 108.86 𝑚 𝑠 𝑈ℎ = 203.35 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟏 = 𝑾 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜷 𝟏 = −𝟑𝟐. 𝟐𝟏° 𝑉𝑥 = 150 𝑚 𝑠 𝑊𝑢ℎ = −94.48 𝑚 𝑠 At Hub After Rotor Using the Free Vortex Design assumption 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒉 𝑽 𝒖 𝒉 ⇒ 𝑽 𝒖 𝒉 = 𝟐𝟑𝟗. 𝟖 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 183.14 𝑚 𝑠 𝑈ℎ = 203.35 𝑚 𝑠 Using trigonometry on velocity triangles that represent hub flow characteristic the flow angles can be found. 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜶 𝟐 = 𝟓𝟕. 𝟗𝟕° 𝑉𝑢ℎ = 239.82 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠
28 𝑾 𝒖 𝒉 = 𝑽 𝒖 𝒉 − 𝑼 𝒉 = 𝟑𝟒. 𝟒𝟔 𝒎 𝒔 𝑉𝑢ℎ = 239.82 𝑚 𝑠 𝑈ℎ = 203.35 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟐 = 𝑾 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜷 𝟐 = 𝟏𝟑. 𝟔𝟔° 𝑉𝑥 = 150 𝑚 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 Shows 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 Flow Angles of Stage 3 From Hub to Tip
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 11.18 3 171.50 236.73 171.05 29.00 4 171.05 237.45 171.05 28.72 5 171.05 237.45 171.05 28.72 6 171.05 237.45 171.05 28.72 7 171.05 237.45 171.05 28.72 Summary of Estimated Values for Each Stage Stage # β1(°) α2(°) β2(°) α3(°) 1 -60.62 33.95 -47.77 11.18 2 -57.63 40.94 -42.23 29.00 3 -50.68 50.68 -29.00 28.72 4 -50.82 50.82 -28.72 28.72 5 -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 Summary of Estimated Values for Each Stage(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 169687.55 3 145406.80 181389.98 1.25 35983.18 169687.55 209381.14 4 181389.98 221116.16 1.22 39726.18 209381.14 253072.56 5 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 Summary of Estimated Values for Each Stage(Cont.)
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 26.26 2 1.27 36179.32 302.63 324.39 314.26 339.03 24.76 3 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 22.40 5 1.20 49535.52 370.19 392.59 384.74 407.14 22.40 6 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 22.40 Poe/Po1 4.15 Summary of Estimated Values for Each Stage(Cont.)
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 0.9 T1(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 DH Check>0.72 305.6224 -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 Design Parameters of Stage 1 Input V1(m/s) 152.9 To1(K) 314.2635064 Vx(m/s) 150 α1(°) 11.17691 Po1(Pa) 133508.2336 Um(m/s) 266.28 λ 0.93 Cp(J/Kg-K) 1005 P1(Pa) 116,999.36 ηtt 0.9 T1(K) 302.6324566 γ 1.4 R(%) 70 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 29.63798 -236.642 -280.178 -57.6307 40.94273195 0.769968667 130.13 DH Check β > 0.72 0.723025386 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>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 Design Parameters of Stage 2 V1(m/s) 171.5 To1(K) 339.0255838 Vx(m/s) 150 α1(°) 28.99813 Po1(Pa) 169687.5514 Um(m/s) 266.28 λ 0.88 Cp(J/Kg-K) 1005 P1(Pa) 145,406.80 ηtt 0.9 T1(K) 324.3926236 γ 1.4 R(%) 50 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 83.13994 -183.14 -236.728 -50.681 50.680971 0.724459269 183.1401 DH Check β > 0.72 0.724459269 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>0.72 α3(°) 236.7283 -83.1399 -171.5 -28.9981 171.05 0.722558355 28.725 ΔTo(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 Design Parameters of Stage 3 Input
33 V1(m/s) 171.05 To1(K) 362.3416704 Vx(m/s) 150 α1(°) 28.725 Po1(Pa) 209381.1424 Um(m/s) 266.28 λ 0.83 Cp(J/Kg-K) 1005 P1(Pa) 181,389.98 ηtt 0.9 T1(K) 347.7854005 γ 1.4 R(%) 50 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723 DH Check β > 0.72 0.720361435 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>0.72 α3(°) 237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725 ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΔPo(Pa) 22.40134 384.743 253072.6 370.1867 221116.1559 1.208669295 43,691.42 Design Parameters of Stage 4 Input V1(m/s) 171.05 To1(K) 384.7430127 Vx(m/s) 150 α1(°) 28.725 Po1(Pa) 253072.5578 Um(m/s) 266.28 λ 0.83 Cp(J/Kg-K) 1005 P1(Pa) 221,116.16 ηtt 0.9 T1(K) 370.1867428 γ 1.4 R(%) 50 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723 DH Check β > 0.72 0.720361435 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>0.72 α3(°) 237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725 ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΔPo(Pa) 22.40134 407.1444 302608.1 392.5881 266404.1359 1.195736452 49,535.52 Design Parameters of Stage 5 Input
34 Equation used to calculate design parameters can be found below: 𝐭𝐚𝐧 𝜶 𝟏 = 𝑽 𝒖 𝟏 𝑽 𝒙 ⇒ 𝑽 𝒖 𝟏 , Trigonometry of Velocity Diagrams 𝑾 𝒖 𝟏 = 𝑼 𝒎 − 𝑽 𝒖 𝟏 , Trigonometry of Velocity Diagrams 𝑾 𝟏 = −√𝑾 𝒖 𝟏 𝟐 + 𝑽 𝒙 𝟐 , Trigonometry of Velocity Diagrams 𝐭𝐚𝐧 𝜷 𝟏 = 𝑾 𝒖 𝟏 𝑽 𝒙 ⇒ 𝜷 𝟏, Trigonometry of Velocity Diagrams 𝑹 = 𝟏 − 𝟎. 𝟓 𝑽 𝒙 𝑼 𝒎 ( 𝐭𝐚𝐧 𝜶 𝟐 + 𝐭𝐚𝐧 𝜶 𝟏) ⇒ 𝜶 𝟐, Degree of Reaction 𝑽 𝟏 𝑽 𝟐 ≥ 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 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723 DH Check β > 0.72 0.720361435 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>0.72 α3(°) 237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725 ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΔPo(Pa) 22.40134 429.5457 358381.8 414.9894 317645.5553 1.184310152 55,773.74 Design Parameters of Stage 6 Input V1(m/s) 171.05 To1(K) 429.5456972 Vx(m/s) 150 α1(°) 28.725 Po1(Pa) 358381.8238 Um(m/s) 266.28 λ 0.83 Cp(J/Kg-K) 1005 P1(Pa) 317,645.56 ηtt 0.9 T1(K) 414.9894273 γ 1.4 R(%) 50 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723 DH Check β > 0.72 0.720361435 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>0.72 α3(°) 237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725 ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΔPo(Pa) 22.40134 451.947 420791.2 437.3908 375235.4437 1.174142048 62,409.34 Input Design Parameters of Stage 7
35 𝑾 𝟐 𝑾 𝟏 ≥ 𝟎. 𝟕𝟐, De Haller Number 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝟐 𝑽 𝒙 ⇒ 𝑽 𝒖 𝟐 , Trigonometry of Velocity Diagrams 𝑾 𝒖 𝟐 = 𝑽 𝒖 𝟐 − 𝑼 𝒎, Trigonometry of Velocity Diagrams 𝑾 𝟐 = −√𝑾 𝒖 𝟐 𝟐 + 𝑽 𝒙 𝟐 , Trigonometry of Velocity Diagrams 𝐭𝐚𝐧 𝜷 𝟐 = 𝑾 𝒖 𝟐 𝑽 𝒙 ⇒ 𝜷 𝟐, Trigonometry of Velocity Diagrams ∆𝑻 𝒐 = 𝝀# 𝑼 𝒎 𝑽 𝒙 𝑪 𝒑 ( 𝐭𝐚𝐧 𝜷 𝟐 − 𝐭𝐚𝐧 𝜷 𝟏), Actual Stage Temperature Rise ∆𝑻 𝒐 = 𝑻 𝒐 𝟑 − 𝑻 𝒐 𝟏 ⇒ 𝑻 𝒐 𝟑 , Subtraction [ 𝑷 𝒐 𝟑 𝑷 𝒐 𝟏 ] 𝝀−𝟏 𝝀 = 𝟏 + 𝜼 𝒕𝒕 ( 𝑻 𝒐 𝟑 𝑻 𝒐 𝟏 − 𝟏) ⇒ 𝑷 𝒐 𝟑 , Stage Pressure Ratio from Stage Temperature Ratio 𝑽 𝟐 = √𝑽 𝒙 𝟐 + 𝑽 𝒖 𝟐 𝟐 , Trigonometry of Velocity Diagrams 𝑽 𝟑 𝑽 𝟐 ≥ 𝟎. 𝟕𝟐, De Haller Number 𝐜𝐨𝐬 𝜶 𝟑 = 𝑽 𝒙 𝑽 𝟑 ⇒ 𝜶 𝟑, Trigonometry of Velocity Diagrams 𝑻 𝟑 = 𝑻 𝒐 𝟑 − 𝑽 𝟑 𝟐 𝟐𝑪 𝒑 , Static Temperature from Stagnation Temperature 𝑷 𝟑 𝑷 𝒐 𝟑 = ( 𝑻 𝟑 𝑻 𝒐 𝟑 ) 𝝀 𝝀−𝟏 ⇒ 𝑷 𝟑, 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

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Turbo Final 2

  • 1. Multi-Stage Axial-Flow Compressor Design Analysis Frederick Avyasa Smith MECE E4304: Turbomachinery Prof. Dr.P.Akbari December 8th, 2014
  • 2. 2 Table of Contents Introduction.................................................................................................................................3 Analysis .........................................................................................................................................4 Section A: Relative Mach Number............................................................................................................................4 Section B: AnnulusDimensions.................................................................................................................................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. 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 Compressor Pressure Ratio 4.15 Air Mass Flow Rate 20 kg/s Compressor Polytropic Efficiency 0.9 Blade Tip Speed of First Stage 355.3 m/s Hub-to-Tip Ratio of First Stage 0.5 Design Parameters
  • 4. 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 radius also shrinks. By using trigonometry on the velocity diagrams of stages 2-7 at the tip it can be Name Value Ambient Stagnation Pressure 101.3 kPa Ambient Stagnation Temptrature 288 K Constant Specific Heat of Air 1005 J/kg-K Specific Heat Ratio of Air 1.4 Ideal Gas Constat for Air 287 J/kg-K Axial Velocity of Air 150 m/s Work Done Factor Stage 1 0.98 Work Done Factor Stage 2 0.93 Work Done Factor Stage 3 0.88 Work Done Factor for Ongoing Stages 0.83 Initial Assumptions
  • 5. 5 seen that the relative velocity will be less than in Stage 1. Stagnation speed of sound is first found using ambient air properties. 𝑪 𝒐 𝟏 = √ 𝝀𝑹𝑻 𝒐 𝟏 = 𝟑𝟒𝟎. 𝟏𝟕 𝒎 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. 𝑴 𝒐 𝟏 𝑹 = 𝑾 𝟏 𝑪 𝒐 𝟏 = 𝟏. 𝟏𝟑 𝐶 𝑜1 = 340.17 𝑚 𝑠 𝑊1 = √ 𝑉𝑥 2 + 𝑈𝑡 = 385.66 𝑚 𝑠 𝑉𝑥 = 150 𝑚 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. 𝑻 𝟏 = 𝑻 𝒐 𝟏 − 𝑽 𝟏 𝟐 𝟐𝑪 𝒑 = 𝟐𝟕𝟔. 𝟖𝑲 𝑇𝑜1 = 288𝐾 𝑉1 = 150 𝑚 𝑠 𝐶 𝑝 = 1005 𝐽 𝑘𝑔−𝐾 Finally the relative Mach number at the tip can be found. 𝑴 𝟏 𝑹 = 𝑴 𝒐 𝟏 𝑹 √ 𝑻 𝒐 𝟏 𝑻 𝟏 = 𝟏. 𝟏𝟔 𝑀 𝑜1 𝑅 = 1.13
  • 6. 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. 𝑷 𝟏 𝑷 𝒐 𝟏 = ( 𝑻 𝟏 𝑻 𝒐 𝟏 ) 𝝀 𝝀−𝟏 ⇒ 𝑷 𝟏 = 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. 𝑷 𝟏 = 𝝆 𝟏 𝑹𝑻 𝟏 ⇒ 𝝆 𝟏 = 𝟏. 𝟏𝟏 𝒌𝒈 𝒎 𝟑 𝑃1 = 88,175.9𝑃𝑎 𝑅 = 287 𝐽 𝑘𝑔−𝐾
  • 7. 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. 𝒎̇ = 𝝆 𝟏 𝑽 𝒙 𝑨 𝟏 ⇒ 𝑨 𝟏 = 𝟎. 𝟏𝟐𝒎 𝟐 𝜌1 = 1.11 𝑘𝑔 𝑚3 𝑉𝑥 = 150 𝑚 𝑠 𝑚̇ = 20 𝑘𝑔 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. 𝑨 𝟏 = 𝒓 𝒕 𝟐 − 𝒓 𝒉 𝟐 = 𝒓 𝒕 𝟐 − ( 𝟎. 𝟓𝒓 𝒕) 𝟐 = 𝟎. 𝟏𝟐𝟎𝒎 𝟐 ⇒ 𝒓 𝒕 = 𝟎. 𝟒𝟎𝒎 𝑟ℎ 𝑟𝑡 = 0.5 𝒓 𝒎 = 𝟎. 𝟓( 𝒓 𝒕 + 𝒓 𝒉) ⇒ 𝒓 𝒉 = 𝟎. 𝟐𝟎𝒎 𝑟𝑡 = 0.40𝑚 𝒓 𝒎 = 𝟎. 𝟓( 𝒓 𝒕 + 𝒓 𝒉) = 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. 𝑃𝑜 𝑒 𝑃𝑜1 = 4.15 ⇒ 𝑃𝑜 𝑒 = 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. 8 𝑻 𝒐 𝒆 𝑻 𝒐 𝟏 = ( 𝑷 𝟎 𝒆 𝑷 𝒐 𝟏 ) 𝝀−𝟏 𝜼 𝒑 𝝀 ⇒ 𝑻 𝒐 𝒆 = 𝟒𝟓𝟐. 𝟒𝟖𝑲 𝑃𝑜 𝑒 𝑃𝑜1 = 4.15 𝜆 = 1.4 𝜂 𝑝 = 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. 𝑻 𝒆 = 𝑻 𝒐 𝒆 − 𝑽 𝟑 𝟐 𝟐𝑪 𝒑 ⇒ 𝟒𝟑𝟕. 𝟗𝟐𝑲 𝑇𝑜 𝑒 = 452.48𝐾 𝐶 𝑝 = 1005 𝐽 𝑘𝑔−𝐾 𝑉3 = 171.05 𝑚 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. 𝑷 𝒆 𝑷 𝒐 𝒆 = ( 𝑻 𝒆 𝑻 𝒐 𝒆 ) 𝝀 𝝀−𝟏 ⇒ 𝑷 𝒆 = 𝟑𝟕𝟒,𝟗𝟑𝟒𝑷𝒂 𝑃𝑜 𝑒 = 420,395𝑃𝑎 𝑇𝑒 = 437.92𝐾 𝑇𝑜 𝑒 = 452.48𝐾 𝜆 = 1.4 Static density can now be found using the Ideal Gas Law. 𝑷 𝒆 = 𝝆 𝒆 𝑹𝑻 𝒆 ⇒ 𝝆 𝒆 = 𝟐. 𝟗𝟖 𝒌𝒈 𝒎 𝟑 𝑃𝑒 = 374,934𝑃𝑎 𝑇𝑒 = 437.92𝐾
  • 9. 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. 𝒎̇ = 𝝆 𝒆 𝑽 𝒙 𝑨 𝒆 ⇒ 𝑨 𝒆 = 𝟎. 𝟎𝟒𝟓𝒎 𝟐 𝜌𝑒 = 2.98 𝑘𝑔 𝑚3 𝑉𝑥 = 150 𝑚 𝑠 𝑚̇ = 20 𝑘𝑔 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. 𝑨 𝒆 = 𝒓 𝒕 𝒆 𝟐 − 𝒓 𝒉 𝒆 𝟐 ⇒ 𝒓 𝒉 𝒆 = 𝟎. 𝟐𝟔𝒎 𝑟 𝑚 = 0.5(𝑟𝑡 𝑒 + 𝑟ℎ 𝑒 ) = 0.30 ⇒ 𝑟𝑡 𝑒 = 0.60 − 𝑟ℎ 𝑒 𝒓 𝒎 = 𝟎. 𝟓(𝒓 𝒕 𝒆 + 𝒓 𝒉 𝒆 ) = 𝟎. 𝟑𝟎 ⇒ 𝒓 𝒕 𝒆 = 𝟎. 𝟑𝟑𝟕𝒎 𝑟ℎ 𝑒 = 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. 𝑼 𝒕 = 𝒓 𝒕 𝛀 ⇒ 𝛀 = 𝟖𝟖𝟕. 𝟕𝟖 𝒓𝒂𝒅 𝒔 𝑈𝑡 = 355.3 𝑚 𝑠 𝑟𝑡 = 0.40𝑚 𝑼 𝒎 = 𝒓 𝒎 𝛀 = 𝟐𝟔𝟔. 𝟐𝟖 𝒎 𝒔 Ω = 887.78 𝑟𝑎𝑑 𝑠 𝑟 𝑚 = 0.30𝑚
  • 10. 10 The relative flow angle will be calculated at the inlet using the mean-blade speed and trigonometry. 𝐭𝐚𝐧 𝜷 𝟏 = −𝑼 𝒎 𝑽 𝒙 ⇒ 𝜷 𝟏 = −𝟔𝟎. 𝟔𝟐° 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑥 = 150 𝑚 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 𝜷 𝟏 𝐜𝐨𝐬 𝜷 𝟐 = 𝟎. 𝟕𝟑 ⇒ 𝜷 𝟐 = −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 work 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. ∆𝑻 𝒐 𝒂𝒗𝒈 = 𝝀 𝒂𝒗𝒈 𝑼 𝒎 𝑽 𝒙 𝑪 𝒑 ( 𝐭𝐚𝐧 𝜷 𝟐 − 𝐭𝐚𝐧 𝜷 𝟏) = 𝟐𝟒. 𝟐𝟕𝑲 𝜆 𝑎𝑣𝑔 =0.905 𝑈 𝑚 = 266.28 𝑚 𝑠
  • 11. 11 𝑉𝑥 = 150 𝑚 𝑠 𝐶 𝑝 = 1005 𝐽 𝑘𝑔−𝐾 𝛽1 = −60.62° 𝛽2 = −47.78° Finally, the number of stages can be calculated by using the average stage temperature rise. 𝑻 𝒐 𝒆 𝑻 𝒐 𝟏 = 𝟏 + ∆𝑻 𝒐 𝒂𝒗𝒈 𝑻 𝒐 𝟏 ⇒ 𝒏 = 𝟔. 𝟕𝟖 ≈ 𝟕 𝑇𝑜 𝑒 = 452.48𝐾 𝑇𝑜1 = 288𝐾 ∆𝑇𝑜 𝑎𝑣𝑔 = 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. ∆𝑻 𝒐 = 𝝀 𝟏 𝑼 𝒎 𝑽 𝒙 𝑪 𝒑 ( 𝐭𝐚𝐧 𝜷 𝟐 − 𝐭𝐚𝐧 𝜷 𝟏) = 𝟐𝟔. 𝟐𝟔𝑲 𝜆1 =0.98 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝐶 𝑝 = 1005 𝐽 𝑘𝑔−𝐾 𝛽1 = −60.62° 𝛽2 = −47.78° Stagnation temperature after the stator can now easily be calculated using subtraction. ∆𝑻 𝒐 = 𝑻 𝒐 𝟑 − 𝑻 𝒐 𝟏 ⇒ 𝑻 𝒐 𝟑 = 𝟑𝟏𝟒. 𝟐𝟕𝑲
  • 12. 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. [ 𝑷 𝒐 𝟑 𝑷 𝒐 𝟏 ] 𝝀−𝟏 𝝀 = 𝟏 + 𝜼 𝒕𝒕 ( 𝑻 𝒐 𝟑 𝑻 𝒐 𝟏 − 𝟏) ⇒ 𝑷 𝒐 𝟑 = 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. 𝑾 𝒖 𝟐 = 𝑽 𝒙 𝐭𝐚𝐧 𝜷 𝟐 = 𝟏𝟔𝟓. 𝟑𝟐 𝒎 𝒔 𝑉𝑥 = 150 𝑚 𝑠 𝛽2 = −47.78° 𝑼 𝒎 = 𝑽 𝒖 𝟐 + 𝑾 𝒖 𝟐 ⇒ 𝑽 𝒖 𝟐 = 𝟏𝟎𝟎. 𝟗𝟓 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑊𝑢2 = 165.32 𝑚 𝑠 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝟐 𝑽 𝒙 ⇒ 𝜶 𝟐 = 𝟑𝟑. 𝟗𝟒° 𝑉𝑥 = 150 𝑚 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 160 𝑚 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. 13 𝑽 𝟐 = √𝑽 𝒙 𝟐 + 𝑽 𝒖 𝟐 𝟐 = 𝟏𝟖𝟎. 𝟖𝟏 𝒎 𝒔 𝑉𝑥 = 150 𝑚 𝑠 𝑉𝑢2 = 100.95 𝑚 𝑠 By using the De Haller Number one can see that the flow does not diffuse excessively. 𝑽 𝟑 𝑽 𝟐 ≥ 𝟎. 𝟕𝟐 ⇒ 𝟎. 𝟖𝟖 ≥ 𝟎. 𝟕𝟐 𝑉3 = 160 𝑚 𝑠 𝑉2 = 180.81 𝑚 𝑠 The blade angle after the stator can be found using trigonometry. 𝐜𝐨𝐬 𝜶 𝟑 = 𝑽 𝒙 𝑽 𝟑 ⇒ 𝜶 𝟑 = 𝟐𝟎. 𝟑𝟔° 𝑉𝑥 = 150 𝑚 𝑠 𝑉3 = 160 𝑚 𝑠 Finally, the static parameters can be found using the exit velocity and an isentropic process equation. 𝑻 𝟑 = 𝑻 𝒐 𝟑 − 𝑽 𝟑 𝟐 𝟐𝑪 𝒑 = 𝟑𝟎𝟏. 𝟓𝟑𝑲 𝑇𝑜3 = 314.27𝐾 𝑉3 = 160 𝑚 𝑠 𝐶 𝑝 = 1005 𝐽 𝑘𝑔−𝐾 𝑷 𝟑 𝑷 𝒐 𝟑 = ( 𝑻 𝟑 𝑻 𝒐 𝟑 ) 𝝀 𝝀−𝟏 ⇒ 𝑷 𝟑 = 𝟏𝟏𝟓, 𝟓𝟏𝟐𝑷𝒂 𝑃𝑜3 = 133,510𝑃𝑎 𝑇𝑜3 = 314.27𝐾 𝑇3 = 301.53𝐾 𝜆 = 1.4
  • 14. 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. 𝟓 𝑽 𝒙 𝑼 𝒎 ( 𝐭𝐚𝐧 𝜶 𝟐 + 𝐭𝐚𝐧 𝜶 𝟏) = 𝟖𝟏. 𝟎𝟒% 𝑉𝑥 = 150 𝑚 𝑠 𝑈 𝑚 = 266.28 𝑚 𝑠 𝛼2 = 33.94° 𝛼1 = 0° Po(Pa) P(Pa) To(K) T(K) At Inlet 101,300 88,175.90 288 276.8 At Exit 133,510 115,512 314.27 301.53 Initial Static and Stage Tempratures/Pressures of First Stage β(°) α(°) Before Rotor -60.62 0 After Rotor -47.78 33.94 After Stator n/a 20.36 Initial Flow Angles of First Stage
  • 15. 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 160 𝑚 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. 16 thus varying the absolute flow angles one can increase/decrease diffusion between 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) Before Rotor V1 W1 Um
  • 17. 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) Before Rotor V1 W1 Um
  • 18. 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) Before Rotor V1 W1 Um
  • 19. 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) Before Rotor V1 W1 Um
  • 20. 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. 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) Stage Number Stage Pressure Rise vs Stage Number Stage Pressure Rise vs Stage Number
  • 22. 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. 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒕 𝑽 𝒖 𝒕 ⇒ 𝑽 𝒖 𝒕 = 𝟕𝟓. 𝟕𝟓 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 101.00 𝑚 𝑠 𝑈𝑡 = 355.3 𝑚 𝑠 Trigonometry can be used to find the flow angles on velocity triangles that represent tip flow characteristics. 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜶 𝟐 = 𝟐𝟔. 𝟕𝟗° 𝑉𝑢 𝑡 = 75.75 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟏 = −𝑼 𝒕 𝑽 𝒙 ⇒ 𝜷 𝟏 = −𝟔𝟕. 𝟏𝟏° 𝑈𝑡 = 266.28 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝑾 𝒖 𝒕 = 𝑽 𝒖 𝒕 − 𝑼 𝒕 = −𝟐𝟕𝟗. 𝟓𝟓 𝒎 𝒔 𝑉𝑢 𝑡 = 75.75 𝑚 𝑠 𝑈𝑡 = 266.28 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟐 = 𝑾 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜷 𝟐 = −𝟔𝟏. 𝟕𝟖°
  • 23. 23 𝑉𝑥 = 150 𝑚 𝑠 𝑊𝑢 𝑡 = 279.55 𝑚 𝑠 At Hub Blade speed at the hub can be found using the constant rotational speed of the compressor. 𝑼 𝒉 = 𝒓 𝒉 𝛀 ⇒ 𝟏𝟕𝟕. 𝟓𝟕 𝒎 𝒔 𝑟ℎ = 0.20𝑚 Ω = 887.78 𝑟𝑎𝑑 𝑠 Using the Free Vortex Design assumption 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒉 𝑽 𝒖 𝒉 ⇒ 𝑽 𝒖 𝒉 = 𝟏𝟓𝟏. 𝟓𝟕 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 101.00 𝑚 𝑠 𝑈ℎ = 177.57 𝑚 𝑠 Using trigonometry on velocity triangles that represent hub flow characteristic the flow angles can be found. 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜶 𝟐 = 𝟒𝟓. 𝟑𝟎° 𝑉𝑢ℎ = 151.57 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟏 = −𝑼 𝒉 𝑽 𝒙 ⇒ 𝜷 𝟏 = −𝟒𝟗. 𝟖𝟏° 𝑈ℎ = 177.57 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝑾 𝒖 𝒉 = 𝑽 𝒖 𝒉 − 𝑼 𝒉 = −𝟐𝟔. 𝟎𝟎 𝒎 𝒔 𝑉𝑢ℎ = 151.57 𝑚 𝑠 𝑈ℎ = 177.57 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟐 = 𝑾 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜷 𝟐 = −𝟗. 𝟖𝟑° 𝑉𝑥 = 150 𝑚 𝑠
  • 24. 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 𝜶 𝟏 𝐜𝐨𝐬 𝜶 𝟐 = 𝟎. 𝟕𝟐 ⇒ 𝟎. 𝟕𝟎 ≯ 𝟎. 𝟕𝟐 𝛼2 = 45.30° 𝛼1 = 0° 𝐜𝐨𝐬 𝜷 𝟏 𝐜𝐨𝐬 𝜷 𝟐 = 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. 𝑷 𝟏 = 𝝆 𝟏 𝑹𝑻 𝟏 ⇒ 𝝆 𝟏 = 𝟏. 𝟓𝟐 𝒌𝒈 𝒎 𝟑 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. 𝒎̇ = 𝝆 𝒆 𝑽 𝒙 𝑨 𝒆 ⇒ 𝑨 𝒆 = 𝟎. 𝟎𝟖𝟓𝒎 𝟐 𝜌𝑒 = 1.52 𝑘𝑔 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 Flow Angles of Stage 1 From Hub to Tip
  • 25. 25 𝑉𝑥 = 150 𝑚 𝑠 𝑚̇ = 20 𝑘𝑔 𝑠 Next mean-radius and annulus area can be utilized to find the radiuses at the hub and tip. 𝑨 𝒆 = 𝒓 𝒕 𝟐 − 𝒓 𝒉 𝟐 ⇒ 𝒓 𝒕 = 𝟎. 𝟑𝟕𝒎 𝑟 𝑚 = 0.5( 𝑟𝑡 + 𝑟𝑒) = 0.30 ⇒ 𝑟ℎ = 0.60 − 𝑟𝑡 𝒓 𝒎 = 𝟎. 𝟓( 𝒓 𝒕 + 𝒓 𝒉) = 𝟎. 𝟑𝟎 ⇒ 𝒓 𝒉 = 𝟎. 𝟐𝟑𝒎 𝑟𝑡 = 0.37𝑚 Finally the radiuses can be used to calculate the blade speeds at the hub and tip. 𝑼 𝒕 = 𝒓 𝒕 𝛀 ⇒ 𝟑𝟐𝟗. 𝟔 𝒎 𝒔 𝑟𝑡 = 0.37𝑚 Ω = 887.78 𝑟𝑎𝑑 𝑠 𝑼 𝒉 = 𝒓 𝒉 𝛀 ⇒ 𝟐𝟎𝟑. 𝟑𝟓 𝒎 𝒔 𝑟ℎ = 0.23𝑚 Ω = 887.78 𝑟𝑎𝑑 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 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒕 𝑽 𝒖 𝒕 ⇒ 𝑽 𝒖 𝒕 = 𝟔𝟕. 𝟏𝟕 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 83.14 𝑚 𝑠 𝑈𝑡 = 329.6 𝑚 𝑠 Using trigonometry on velocity triangles that represent tip flow characteristic the flow angles can be found. 𝐭𝐚𝐧 𝜶 𝟏 = 𝑽 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜶 𝟏 = 𝟐𝟒. 𝟏𝟐° 𝑉𝑢 𝑡 = 67.17 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠
  • 26. 26 𝑾 𝒖 𝒕 = 𝑽 𝒖 𝒕 − 𝑼 𝒕 = −𝟐𝟔𝟐. 𝟒𝟑 𝒎 𝒔 𝑉𝑢 𝑡 = 67.17 𝑚 𝑠 𝑈𝑡 = 329.6 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟏 = 𝑾 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜷 𝟏 = −𝟔𝟎. 𝟐𝟓° 𝑉𝑥 = 150 𝑚 𝑠 𝑊𝑢 𝑡 = −262.43 𝑚 𝑠 At Tip After Rotor Using the Free Vortex Design assumption 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒕 𝑽 𝒖 𝒕 ⇒ 𝑽 𝒖 𝒕 = 𝟏𝟒𝟕. 𝟗𝟔 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 183.14 𝑚 𝑠 𝑈𝑡 = 329.6 𝑚 𝑠 Using trigonometry on velocity triangles that represent tip flow characteristic the flow angles can be found. 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜶 𝟐 = 𝟒𝟒. 𝟔𝟏° 𝑉𝑢 𝑡 = 67.17 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝑾 𝒖 𝒕 = 𝑽 𝒖 𝒕 − 𝑼 𝒕 = −𝟏𝟖𝟏. 𝟔𝟒 𝒎 𝒔 𝑉𝑢 𝑡 = 147.96 𝑚 𝑠 𝑈𝑡 = 329.6 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟐 = 𝑾 𝒖 𝒕 𝑽 𝒙 ⇒ 𝜷 𝟐 = −𝟓𝟎. 𝟒𝟓° 𝑉𝑥 = 150 𝑚 𝑠 𝑊𝑢 𝑡 = −181.64 𝑚 𝑠 At Hub Before Rotor Using the Free Vortex Design assumption 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒉 𝑽 𝒖 𝒉 ⇒ 𝑽 𝒖 𝒉 = 𝟏𝟎𝟖. 𝟖𝟔 𝒎 𝒔
  • 27. 27 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 83.14 𝑚 𝑠 𝑈ℎ = 203.35 𝑚 𝑠 Using trigonometry on velocity triangles that represent hub flow characteristic the flow angles can be found. 𝐭𝐚𝐧 𝜶 𝟏 = 𝑽 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜶 𝟏 = 𝟑𝟓. 𝟗𝟕° 𝑉𝑢ℎ = 108.86 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠 𝑾 𝒖 𝒉 = 𝑽 𝒖 𝒉 − 𝑼 𝒉 = −𝟗𝟒. 𝟒𝟖 𝒎 𝒔 𝑉𝑢ℎ = 108.86 𝑚 𝑠 𝑈ℎ = 203.35 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟏 = 𝑾 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜷 𝟏 = −𝟑𝟐. 𝟐𝟏° 𝑉𝑥 = 150 𝑚 𝑠 𝑊𝑢ℎ = −94.48 𝑚 𝑠 At Hub After Rotor Using the Free Vortex Design assumption 𝑼 𝒎 𝑽 𝒖 𝒎 = 𝑼 𝒉 𝑽 𝒖 𝒉 ⇒ 𝑽 𝒖 𝒉 = 𝟐𝟑𝟗. 𝟖 𝒎 𝒔 𝑈 𝑚 = 266.28 𝑚 𝑠 𝑉𝑢 𝑚 = 183.14 𝑚 𝑠 𝑈ℎ = 203.35 𝑚 𝑠 Using trigonometry on velocity triangles that represent hub flow characteristic the flow angles can be found. 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜶 𝟐 = 𝟓𝟕. 𝟗𝟕° 𝑉𝑢ℎ = 239.82 𝑚 𝑠 𝑉𝑥 = 150 𝑚 𝑠
  • 28. 28 𝑾 𝒖 𝒉 = 𝑽 𝒖 𝒉 − 𝑼 𝒉 = 𝟑𝟒. 𝟒𝟔 𝒎 𝒔 𝑉𝑢ℎ = 239.82 𝑚 𝑠 𝑈ℎ = 203.35 𝑚 𝑠 𝐭𝐚𝐧 𝜷 𝟐 = 𝑾 𝒖 𝒉 𝑽 𝒙 ⇒ 𝜷 𝟐 = 𝟏𝟑. 𝟔𝟔° 𝑉𝑥 = 150 𝑚 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 Shows 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 Flow Angles of Stage 3 From Hub to Tip
  • 29. 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. 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 11.18 3 171.50 236.73 171.05 29.00 4 171.05 237.45 171.05 28.72 5 171.05 237.45 171.05 28.72 6 171.05 237.45 171.05 28.72 7 171.05 237.45 171.05 28.72 Summary of Estimated Values for Each Stage Stage # β1(°) α2(°) β2(°) α3(°) 1 -60.62 33.95 -47.77 11.18 2 -57.63 40.94 -42.23 29.00 3 -50.68 50.68 -29.00 28.72 4 -50.82 50.82 -28.72 28.72 5 -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 Summary of Estimated Values for Each Stage(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 169687.55 3 145406.80 181389.98 1.25 35983.18 169687.55 209381.14 4 181389.98 221116.16 1.22 39726.18 209381.14 253072.56 5 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 Summary of Estimated Values for Each Stage(Cont.)
  • 31. 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 26.26 2 1.27 36179.32 302.63 324.39 314.26 339.03 24.76 3 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 22.40 5 1.20 49535.52 370.19 392.59 384.74 407.14 22.40 6 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 22.40 Poe/Po1 4.15 Summary of Estimated Values for Each Stage(Cont.)
  • 32. 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 0.9 T1(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 DH Check>0.72 305.6224 -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 Design Parameters of Stage 1 Input V1(m/s) 152.9 To1(K) 314.2635064 Vx(m/s) 150 α1(°) 11.17691 Po1(Pa) 133508.2336 Um(m/s) 266.28 λ 0.93 Cp(J/Kg-K) 1005 P1(Pa) 116,999.36 ηtt 0.9 T1(K) 302.6324566 γ 1.4 R(%) 70 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 29.63798 -236.642 -280.178 -57.6307 40.94273195 0.769968667 130.13 DH Check β > 0.72 0.723025386 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>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 Design Parameters of Stage 2 V1(m/s) 171.5 To1(K) 339.0255838 Vx(m/s) 150 α1(°) 28.99813 Po1(Pa) 169687.5514 Um(m/s) 266.28 λ 0.88 Cp(J/Kg-K) 1005 P1(Pa) 145,406.80 ηtt 0.9 T1(K) 324.3926236 γ 1.4 R(%) 50 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 83.13994 -183.14 -236.728 -50.681 50.680971 0.724459269 183.1401 DH Check β > 0.72 0.724459269 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>0.72 α3(°) 236.7283 -83.1399 -171.5 -28.9981 171.05 0.722558355 28.725 ΔTo(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 Design Parameters of Stage 3 Input
  • 33. 33 V1(m/s) 171.05 To1(K) 362.3416704 Vx(m/s) 150 α1(°) 28.725 Po1(Pa) 209381.1424 Um(m/s) 266.28 λ 0.83 Cp(J/Kg-K) 1005 P1(Pa) 181,389.98 ηtt 0.9 T1(K) 347.7854005 γ 1.4 R(%) 50 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723 DH Check β > 0.72 0.720361435 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>0.72 α3(°) 237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725 ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΔPo(Pa) 22.40134 384.743 253072.6 370.1867 221116.1559 1.208669295 43,691.42 Design Parameters of Stage 4 Input V1(m/s) 171.05 To1(K) 384.7430127 Vx(m/s) 150 α1(°) 28.725 Po1(Pa) 253072.5578 Um(m/s) 266.28 λ 0.83 Cp(J/Kg-K) 1005 P1(Pa) 221,116.16 ηtt 0.9 T1(K) 370.1867428 γ 1.4 R(%) 50 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723 DH Check β > 0.72 0.720361435 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>0.72 α3(°) 237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725 ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΔPo(Pa) 22.40134 407.1444 302608.1 392.5881 266404.1359 1.195736452 49,535.52 Design Parameters of Stage 5 Input
  • 34. 34 Equation used to calculate design parameters can be found below: 𝐭𝐚𝐧 𝜶 𝟏 = 𝑽 𝒖 𝟏 𝑽 𝒙 ⇒ 𝑽 𝒖 𝟏 , Trigonometry of Velocity Diagrams 𝑾 𝒖 𝟏 = 𝑼 𝒎 − 𝑽 𝒖 𝟏 , Trigonometry of Velocity Diagrams 𝑾 𝟏 = −√𝑾 𝒖 𝟏 𝟐 + 𝑽 𝒙 𝟐 , Trigonometry of Velocity Diagrams 𝐭𝐚𝐧 𝜷 𝟏 = 𝑾 𝒖 𝟏 𝑽 𝒙 ⇒ 𝜷 𝟏, Trigonometry of Velocity Diagrams 𝑹 = 𝟏 − 𝟎. 𝟓 𝑽 𝒙 𝑼 𝒎 ( 𝐭𝐚𝐧 𝜶 𝟐 + 𝐭𝐚𝐧 𝜶 𝟏) ⇒ 𝜶 𝟐, Degree of Reaction 𝑽 𝟏 𝑽 𝟐 ≥ 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 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723 DH Check β > 0.72 0.720361435 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>0.72 α3(°) 237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725 ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΔPo(Pa) 22.40134 429.5457 358381.8 414.9894 317645.5553 1.184310152 55,773.74 Design Parameters of Stage 6 Input V1(m/s) 171.05 To1(K) 429.5456972 Vx(m/s) 150 α1(°) 28.725 Po1(Pa) 358381.8238 Um(m/s) 266.28 λ 0.83 Cp(J/Kg-K) 1005 P1(Pa) 317,645.56 ηtt 0.9 T1(K) 414.9894273 γ 1.4 R(%) 50 Vu1(m/s) Wu1(m/s) W1(m/s) β1(°) α2(°) DH Check α > 0.72 Vu2(m/s) 82.20768 -184.072 -237.45 -50.8235 50.82350882 0.720361435 184.0723 DH Check β > 0.72 0.720361435 V2(m/s) Wu2(m/s) W2(m/s) β2(°) V3(m/s)-ASSUMED DH Check>0.72 α3(°) 237.4502 -82.2077 -171.05 -28.725 171.05 0.720361435 28.725 ΔTo(K) To3(K) Po3(Pa) T3(K) P3(Pa) Po3/Po1 ΔPo(Pa) 22.40134 451.947 420791.2 437.3908 375235.4437 1.174142048 62,409.34 Input Design Parameters of Stage 7
  • 35. 35 𝑾 𝟐 𝑾 𝟏 ≥ 𝟎. 𝟕𝟐, De Haller Number 𝐭𝐚𝐧 𝜶 𝟐 = 𝑽 𝒖 𝟐 𝑽 𝒙 ⇒ 𝑽 𝒖 𝟐 , Trigonometry of Velocity Diagrams 𝑾 𝒖 𝟐 = 𝑽 𝒖 𝟐 − 𝑼 𝒎, Trigonometry of Velocity Diagrams 𝑾 𝟐 = −√𝑾 𝒖 𝟐 𝟐 + 𝑽 𝒙 𝟐 , Trigonometry of Velocity Diagrams 𝐭𝐚𝐧 𝜷 𝟐 = 𝑾 𝒖 𝟐 𝑽 𝒙 ⇒ 𝜷 𝟐, Trigonometry of Velocity Diagrams ∆𝑻 𝒐 = 𝝀# 𝑼 𝒎 𝑽 𝒙 𝑪 𝒑 ( 𝐭𝐚𝐧 𝜷 𝟐 − 𝐭𝐚𝐧 𝜷 𝟏), Actual Stage Temperature Rise ∆𝑻 𝒐 = 𝑻 𝒐 𝟑 − 𝑻 𝒐 𝟏 ⇒ 𝑻 𝒐 𝟑 , Subtraction [ 𝑷 𝒐 𝟑 𝑷 𝒐 𝟏 ] 𝝀−𝟏 𝝀 = 𝟏 + 𝜼 𝒕𝒕 ( 𝑻 𝒐 𝟑 𝑻 𝒐 𝟏 − 𝟏) ⇒ 𝑷 𝒐 𝟑 , Stage Pressure Ratio from Stage Temperature Ratio 𝑽 𝟐 = √𝑽 𝒙 𝟐 + 𝑽 𝒖 𝟐 𝟐 , Trigonometry of Velocity Diagrams 𝑽 𝟑 𝑽 𝟐 ≥ 𝟎. 𝟕𝟐, De Haller Number 𝐜𝐨𝐬 𝜶 𝟑 = 𝑽 𝒙 𝑽 𝟑 ⇒ 𝜶 𝟑, Trigonometry of Velocity Diagrams 𝑻 𝟑 = 𝑻 𝒐 𝟑 − 𝑽 𝟑 𝟐 𝟐𝑪 𝒑 , Static Temperature from Stagnation Temperature 𝑷 𝟑 𝑷 𝒐 𝟑 = ( 𝑻 𝟑 𝑻 𝒐 𝟑 ) 𝝀 𝝀−𝟏 ⇒ 𝑷 𝟑, Static to Stagnation Pressure Ratio from Static to Stagnation Temperature Ratio
  • 36. 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