eISSN: 2577-8242 FMRIJ

Fluid Mechanics Research International Journal
Research Article
Volume 2 Issue 3

A 2D aerodynamic design of subsonic axial compressor stage

Mdouki Ramzi
Laboratory of Energetic and Turbomachinery, Mechanical engineering department, Cheik Larbi Tebessi University, Algeria
Received: May 02, 2018 | Published: June 28, 2018

Correspondence: Mdouki Ramzi, Mechanical engineering department, Tebessa university, Constantine Avenue, Algeria,12002, Tel 00216698443836, Email

Citation: Ramzi M. A 2D aerodynamic design of subsonic axial compressor stage. Fluid Mech Res Int. 2018;2(3):138‒142. DOI: 10.15406/fmrij.2018.02.00030

Abstract

The purpose of this work is to provide a method for the design of an axial flow compressor stage. This latter represents the front stage of a multistage compressor of industrial gas turbine. The proposed 2D design approach is based on the mean line concept which assumes that mean radius flow conditions prevailing at all other radial stations. The different conservation equations of fluid mechanic; mass, momentum, and energy with ideal gas state equation are applied in conjunction with the NASA empirical relations to compute both incidence and deviation angles at design condition. A FORTRAN computer program was implemented with the inputs such as mass flow rate, tip speed, pressure ratio, ambient temperature and pressure under larger flow coefficient and high reaction ratio. Design process begins with calculation of the channel geometric form from inlet to outlet stage. Thermodynamic properties of the working fluid are determined at rotor inlet, stator outlet and the intermediate station. Detailed geometry of the cascade; chord, pitch, camber and stagger angle is identified for both rotor and stator. The generation of the blade coordinates is performed on the basis of NACA65 profile with circular mean camber line.

Keywords: design, meanline design, axial compressor, stage, NACA65, circular camber

Introduction

The necessity of industrial gas turbine power plants increases with the rise demand of electricity. It represents one of the most important plants in the energetic engineering field. Generally, gas turbine incorporates three main parts: compressor, combustion chamber and turbine. These parts forms the Brayton thermodynamic cycle used in either electricity production in residential and industrial areas or thrust generation in the aviation field. Compressor has the primordial role because high efficiency of the gas turbine is strongly connected to the pressure ratio delivered by the compressor. A lot of works were carried out in the field of design and analysis of axial compressors. However, to understand all details and ideas, from these works, leading to design the axial compressor and give the detailed geometric form of blading, hub, casing and its different parts, a tedious work will be devoted. The purpose to model compressor stage, on the basis of empirical correlations and thermodynamic relations, is to provide the detailed geometry in order to use it in the analysis process and avoiding the CFD approach which consumes a long time during the calculation in the design operation. In fact, this paper represents a tentative in this field for designing an axial compressor stage using the one-dimension mean line approach based on constant outer diameter design.

Axial compressor stage

Before starting the calculation, different input parameters are required to carry out the design of axial compressor stage.1,2 These parameters could be classified through three sets:

Main specifications
The compressor stage that will be calculated is qualified constant outer diameter (COD). The stage is designed without inlet guide vane and delivers a pressure ratio π=1.2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHap aCcqGH9aqpcaaIXaGaaiOlaiaaikdaaaa@3B72@ with a mass flow rate m=170kg/s. Among the main non-dimensional design parameters which are used to give the overall shape of velocity triangles; flow coefficient ϕ= Ca/U =0.7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzcqGH9aqpjuaGdaWcgaGcbaqcLbsacaWGdbGaamyyaaGcbaqcLbsa caWGvbaaaiabg2da9iaaicdacaGGUaGaaG4naaaa@40E5@ , and reaction ratio Λ= ( h 2 h 1 )/ ( h 03 h 01 ) =0.6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHBo atcqGH9aqpjuaGdaWcgaGcbaqcfa4aaeWaaOqaaKqzGeGaamiAaKqb aoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqzGeGaeyOeI0IaamiAaK qbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaaGccaGLOaGaayzkaaaa baqcfa4aaeWaaOqaaKqzGeGaamiAaKqbaoaaBaaajeaibaqcLbmaca aIWaGaaG4maaWcbeaajugibiabgkHiTiaadIgajuaGdaWgaaqcbasa aKqzadGaaGimaiaaigdaaSqabaaakiaawIcacaGLPaaaaaqcLbsacq GH9aqpcaaIWaGaaiOlaiaaiAdaaaa@563B@ . The diffusion factor for both rotor and stator equal to 0.5.

Detailed specifications
The following Table 1 shows the detailed specification used to carry out the preliminary design calculations for the axial flow compressor stage.

Detailed specifications

Rotor

Stator

Aspect ratio AR

2.5

3.5

Tip clearance ε/c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyTduMaai4laiaadogaaaa@39A8@

0.02

0.02

Relative thickness t/c

0.1

0.1

Axial velocity ratio AVR

0.99

0.99

Blockage factor BF

0.98

0.98

Table 1 Input parameters for detailed specifications

Inlet specifications
In fact, the first stage blades, being the longest, are the most highly stressed. Therefore, the security threshold considered by the designer for which the stress problems are not usually critical in the sizing of the annulus is the tip speed of around 350m/s. The same considerations of the stress problems lead to choose relatively low hub-tip ratio in order to decrease the blade stresses; rth=0.5. Concerning the stagnation conditions at inlet, the ambient air temperature T0=288K and ambient pressure P0=1.01325bar. The absolute inlet air angle is set to 15 degrees.

Design stage

A multistage axial compressor as its name indicates, it consists of several stages. Each stage includes two main parts: a moving part called rotor and stationary part named stator.3 Upon both rotor and stator, a set of blades attached to the drum for the latter and on the casing for the former. Concerning the inlet guide vane, it is not regarded as a part of the first stage and it is treated separately. The transfer of energy is assured by the rotor and the stator contributes to transform the kinetic energy to pressure potential energy.4 The total enthalpy remains constant in the stator since there is not a work given to fluid. Figure 1 shows a sketch of typical compressor stage with T-S diagram.6 Figure 2 illustrates the cascade geometry.7

Figure 1 T-S diagram of the compression process in the stage.

Figure 2 Cascade geometry.

Inlet geometry calculation
With the aim to calculate the inlet geometry, the velocity Ca1 at stage inlet should be known. In our calculation this velocity is unknown; therefore, an iterative process must be used. By giving an initial value for Ca and using the equations cited in the following algorithm, the calculation is performed and repeated until convergence is reached. The inlet flow velocity and inlet geometry are calculated.8

Algorithm:

  1. Assume Ca
  2. Calculate the following parameters: C, at, Mt, M, P, T, A, rt, rh, rm, Um
  3. Calculate C a new = U m ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadoeacaWGHbqcfa4damaaBaaajqwaa+FaaKqzadWdbiaa d6gacaWGLbGaam4Daaqcba2daeqaaKqzGeWdbiabg2da98aacaWGvb qcfa4aaSbaaKazba4=baqcLbmacaWGTbaajeaybeaajugibiabew9a Mbaa@4999@
  4. If | C a new Ca |err MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaabdaGcbaqcLbsacaWGdbGaamyyaKqba+aadaWgaaqcbasa aKqzadWdbiaad6gacaWGLbGaam4DaaWcpaqabaqcLbsapeGaeyOeI0 Iaam4qaiaadggaaOGaay5bSlaawIa7aKqzGeGaeyizImQaamyzaiaa dkhacaWGYbaaaa@4986@  then C a new MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadoeacaWGHbqcfa4damaaBaaajeaibaqcLbmapeGaamOB aiaadwgacaWG3baal8aabeaaaaa@3D6C@ is adopted
  5. Else go to step 2 and repeat the process until convergence

Where :
C, at, Mt, A, rt, rh, rm, Um represent the axial velocity, tip speed of sound, tip Mach number, cross section area, tip radius, hub radius, mean radius and blade speed at mean radius, respectively.

Exit geometry calculation
Adopting the type of compressor stage COD where the tip radius is kept constant and after the estimation of the inlet geometry, it is able now to estimate the channel geometric form at exit of the stage. According the compressor stage pressure ratio π, the total pressure and total temperature at exit are expressed as:
P t 3 =πP t 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb aeaaaaaaaaa8qacaWG0bqcfa4damaaBaaajqwaa+FaaKqzadGaaG4m aaWcbeaajugibiabg2da9iabec8aWjaadcfapeGaamiDaKqba+aada WgaaqcKfaG=haajugWaiaaigdaaSqabaaaaa@46E4@                                                             (1)
T t 3 =T t 1 [ 1+( π ( γ1/γ ) 1 )/ η stg ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadsfacaWG0bqcfa4aaSbaaKqaGeaajugWaiaaiodaaSqa baqcLbsacqGH9aqpcaWGubGaamiDaKqbaoaaBaaajeaibaqcLbmaca aIXaaaleqaaKqbaoaadmaakeaajugibiaaigdacqGHRaWkjuaGpaWa aeWaaOqaaKqzGeWdbiabec8aWLqbaoaaCaaaleqajeaibaWcpaWaae WaaKqaGeaajugWa8qacqaHZoWzcqGHsislcaaIXaGaai4laiabeo7a NbqcbaYdaiaawIcacaGLPaaaaaqcLbsapeGaeyOeI0IaaGymaaGcpa GaayjkaiaawMcaaKqzGeWdbiaac+cacqaH3oaAjuaGdaWgaaqcbasa aKqzadGaam4CaiaadshacaWGNbaaleqaaaGccaGLBbGaayzxaaaaaa@5EE5@                       (2)

Using the following algorithm, we can find the geometry at exit stage.

  1. Calculate Ca at exit C a 3 =C a 1 AV R r AV R s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadoeacaWGHbqcfa4aaSbaaKqaGeaajugWaiaaiodaaSqa baqcLbsacqGH9aqpcaWGdbGaamyyaKqbaoaaBaaajeaibaqcLbmaca aIXaaaleqaaKqzGeGaamyqaiaadAfacaWGsbqcfa4aaSbaaKqaGeaa jugWaiaadkhaaSqabaqcLbsacaWGbbGaamOvaiaadkfajuaGdaWgaa qcbasaaKqzadGaam4CaaWcbeaaaaa@4D53@
  2. Assume Um3
  3. Calculate the following parameters at exit: C, Mt, M, A, rt, rh, rm
  4. Calculate U m new = 3.14N r m / 30 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadwfacaWGTbqcfa4damaaBaaajeaibaqcLbmacaWGUbGa amyzaiaadEhaaSqabaqcLbsapeGaeyypa0tcfa4damaalyaakeaaju gibiaaiodacaGGUaGaaGymaiaaisdacaWGobGaamOCaKqbaoaaBaaa jeaibaqcLbmacaWGTbaaleqaaaGcbaqcLbsacaaIZaGaaGimaaaaaa a@4A22@
  5. If | U m new Um |err MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaabdaGcbaqcLbsacaWGvbGaamyBaKqba+aadaWgaaqcbasa aKqzadWdbiaad6gacaWGLbGaam4DaaWcpaqabaqcLbsapeGaeyOeI0 Iaamyvaiaad2gaaOGaay5bSlaawIa7aKqzGeGaeyizImQaamyzaiaa dkhacaWGYbaaaa@49C2@  then U m new MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadwfacaWGTbqcfa4damaaBaaajeaibaqcLbmapeGaamOB aiaadwgacaWG3baal8aabeaaaaa@3D8A@ is adopted
  6. Else go to step 3 and repeat the process until convergence

where N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob aaaa@3758@ , η stg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeE7aOLqbaoaaBaaajeaibaqcLbmacaWGZbGaamiDaiaa dEgaaSqabaaaaa@3D40@ and err MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadwgacaWGYbGaamOCaaaa@397D@  represent the revolutions per minute, the stage efficiency and the convergence error, respectively.

Computation of air angles and kinematic properties
We note that the dimensionless velocity triangles and the blade shape required to achieve them are totally determined by the flow coefficient ϕ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHvpGzcaGGSaaaaa@396D@  loading coefficient ψ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHipqEaaa@38C3@  and the reaction ratio Λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqqHBoataaa@386A@ (Table 2). Therefore, all angles and velocities may be expressed as functions of ϕ,ψand Λ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacqaHvpGzcaGGSaGaaGPaVlabeI8a5PGaaeyyaiaab6gacaqG KbGaaeiiaKqzGeGaeu4MdWKaaiOlaaaa@42E5@

Geometric properties

Rotor inlet (1)

Station rot_stat (2)

Stator outlet (3)

Absolute flow angle α ( o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySdeMaaiikamaaCaaaleqabaGaai4BaaaakiaacMcaaaa@3A88@

15

41.18

15

Relative flow angle β ( o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOSdiMaaiikamaaCaaaleqabaGaai4BaaaakiaacMcaaaa@3A8A@

49.25

28.97

-

Tip radius rt(m)

0.808

0.808

0.808

Mean radius rm(m)

0.639

0.653

0.668

Hub radius rh(m)

0.404

0.448

0.489

Absolute velocity C(m/s)

93.59

122.86

97.87

Relative velocity W(m/s)

138.5

105.07

-

Table 2 Geometric and kinematic properties

Computation of static and total properties
The relationship between stagnation and static properties of the flow are expressed as follows:

For temperature;
T= T 0 C 2 / 2 c p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub Gaeyypa0JaamivaKqbaoaaBaaajeaibaqcLbmacaaIWaaaleqaaKqz GeGaeyOeI0scfa4aaSGbaOqaaKqzGeGaam4qaKqbaoaaCaaaleqaje aibaqcLbmacaaIYaaaaaGcbaqcLbsacaaIYaGaam4yaKqbaoaaBaaa jeaibaqcLbmacaWGWbaaleqaaaaaaaa@479E@                                                                   (3)
For pressure;
P= P 0 ( T/ T 0 ) γ γ1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGqb Gaeyypa0JaamiuaKqbaoaaBaaajeaibaqcLbmacaaIWaaaleqaaKqb aoaabmaakeaajuaGdaWcgaGcbaqcLbsacaWGubaakeaajugibiaads fajuaGdaWgaaqcbasaaKqzadGaaGimaaWcbeaaaaaakiaawIcacaGL PaaajuaGdaahaaWcbeqcbasaaSWaaSaaaKqaGeaajugWaiabeo7aNb qcbasaaKqzadGaeq4SdCMaeyOeI0IaaGymaaaaaaaaaa@4D26@ (4)

The computed parameters are summarized in the following Table 3

Properties

Rotor inlet (1)

Station rot_stat (2)

Stator outlet (3)

Static temperature T(K)

283.63

293.7

300.02

Static pression P(Pa)

96052.35

113842.04

115079.99

Static Mach number M

2.77

3.57

2.81

Static enthalpie H(KJ/kg)

284912,42

295024.24

301373

Total temperature Tt(K)

288

304.77

304.77

Total pression Pt(Pa)

101325

124347

121590

Total Mach number Mt

2.75

3.5

2.79

Total enthalpie Ht(KJ/kg)

289296

306149

306149

Table 3 The computed parameters

Balding/ blades design
To calculate the blade angles, an iterative procedure is required with assuming an initial value for the blade camber angle. Using this value of the blade camber angle, the design incidence angle and the design deviation angle are computed. From these calculated values of incidence and deviation, a new value of camber angle is obtained and compared with the assumed value. This process is repeated until convergence.

Pitch chord or solidity
The knowledge of the value of solidity (σ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaaiikaiabeo8aZjaacMcaaaa@3982@ or the pitch chord (1/σ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaiikaiaaigdacaGGVaGaeq4WdmNaaiykaaaa@3AF0@  is necessary in the calculation of both incidence and deviation. There are several approaches to determine the pitch chord of the cascade. Among these approaches, the McKenzie method which is expressed in term of static pressure ratio Cp as following:
s/c=1/σ=9( 0.567 C p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb Gaai4laiaadogacqGH9aqpcaaIXaGaai4laiabeo8aZjabg2da9iaa iMdajuaGdaqadaGcbaqcLbsacaaIWaGaaiOlaiaaiwdacaaI2aGaaG 4naiabgkHiTiaadoeajuaGdaWgaaqcbasaaKqzadGaamiCaaWcbeaa aOGaayjkaiaawMcaaaaa@4A3B@                                                                (5)
C p =1 ( W 2 / W 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb qcfa4aaSbaaKqaGeaajugWaiaadchaaSqabaqcLbsacqGH9aqpcaaI XaGaeyOeI0scfa4aaeWaaOqaaKqbaoaalyaakeaajugibiaadEfaju aGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaaaOqaaKqzGeGaam4vaKqb aoaaBaaajeaibaqcLbmacaaIXaaaleqaaaaaaOGaayjkaiaawMcaaK qbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaaaa@4BB5@    (6)

Design incidence
Incidence angle represents the difference between the inlet flow angle and the inlet blade angle at the leading edge. For a given cascade, experimental tests lead to obtain the variation of the loss coefficient with respect to the incidence. Over a wide range of incidence angle, it exists a reference value of incidence angle which corresponds to the minimum loss. This value is called design incidence. NASA SP36 empirical relations are the most widely used to predict design incidence. Using Lieblein approach, the pressure loss at twice the minimum loss defines the operability range located between positive stall incidence and negative stall incidence. Outside this region, the blade remains under stall phenomena. The method for calculating the optimum incidence for different profiles shape such as thickness, camber and solidity is guided by the following correlation:
i D = K sh K δt i 010 +nφ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGPb qcfa4aaSbaaKqaGeaajugWaiaadseaaSqabaqcLbsacqGH9aqpcaWG lbqcfa4aaSbaaKqaGeaajugWaiaadohacaWGObaaleqaaKqzGeGaam 4saKqbaoaaBaaajeaibaqcLbmacqaH0oazcaWG0baaleqaaKqzGeGa amyAaKqbaoaaBaaajeaibaqcLbmacaaIWaGaaGymaiaaicdaaSqaba qcLbsacqGHRaWkcaWGUbGaeqOXdOgaaa@5099@                                                                             (7)
where : K sh MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaKqaGeaajugWaiaadohacaWGObaaleqaaaaa@3B4D@ and K δt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaKqaGeaajugWaiabes7aKjaadshaaSqabaaaaa@3C06@ are thickness and shape correction factors, respectively.
K sh MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaKqaGeaajugWaiaadohacaWGObaaleqaaaaa@3B4D@ differs whether the blade is DCA, NACA65-series or C-series. For example, K sh =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaKqaGeaajugWaiaadohacaWGObaaleqaaKqzGeGaeyyp a0JaaGymaaaa@3D9D@  for NACA65 profile.

K δt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaKqaGeaajugWaiabes7aKjaadshaaSqabaaaaa@3C06@ is calculated as a function of cascade thickness:
K it =0.0214+19.17( t/c )122.3 ( t/c ) 2 +312.5 ( t/c ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaKqaafaajug4aiaadMgacaWG0baaleqaaKqzGeGaeyyp a0JaeyOeI0IaaGimaiaac6cacaaIWaGaaGOmaiaaigdacaaI0aGaey 4kaSIaaGymaiaaiMdacaGGUaGaaGymaiaaiEdajuaGdaqadaGcbaqc fa4aaSGbaOqaaKqzGeGaamiDaaGcbaqcLbsacaWGJbaaaaGccaGLOa GaayzkaaqcLbsacqGHsislcaaIXaGaaGOmaiaaikdacaGGUaGaaG4m aKqbaoaabmaakeaajuaGdaWcgaGcbaqcLbsacaWG0baakeaajugibi aadogaaaaakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGa aGOmaaaajugibiabgUcaRiaaiodacaaIXaGaaGOmaiaac6cacaaI1a qcfa4aaeWaaOqaaKqbaoaalyaakeaajugibiaadshaaOqaaKqzGeGa am4yaaaaaOGaayjkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmaca aIZaaaaaaa@6873@    (8)
i 010 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGPb qcfa4aaSbaaKqaGeaajugWaiaaicdacaaIXaGaaGimaaWcbeaaaaa@3BB5@ is the zero camber minimum loss incidence for 10% thickness NACA65-series:
i 010 =( 0.03250.0674σ )+( 0.002364+0.0913σ ) α 1 +          ( 1.64  10 5 2.38  10 4 σ ) α 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi aadMgajuaGdaWgaaqcbasaaKqzadGaaGimaiaaigdacaaIWaaaleqa aKqzGeGaeyypa0tcfa4aaeWaaOqaaKqzGeGaaGimaiaac6cacaaIWa GaaG4maiaaikdacaaI1aGaeyOeI0IaaGimaiaac6cacaaIWaGaaGOn aiaaiEdacaaI0aGaeq4WdmhakiaawIcacaGLPaaajugibiabgUcaRK qbaoaabmaakeaajugibiabgkHiTiaaicdacaGGUaGaaGimaiaaicda caaIYaGaaG4maiaaiAdacaaI0aGaey4kaSIaaGimaiaac6cacaaIWa GaaGyoaiaaigdacaaIZaGaeq4WdmhakiaawIcacaGLPaaajugibiab eg7aHLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaey4kaS cakeaajugibiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaKqbaoaabmaakeaajugibiaaigdacaGGUaGaaG OnaiaaisdacaqGGaGaaGymaiaaicdajuaGdaahaaWcbeqcbasaaKqz adGaeyOeI0IaaGynaaaajugibiabgkHiTiaaikdacaGGUaGaaG4mai aaiIdacaqGGaGaaeymaiaabcdajuaGdaahaaWcbeqcbasaaKqzadGa eyOeI0IaaGinaaaajugibiabeo8aZbGccaGLOaGaayzkaaqcLbsacq aHXoqyjuaGdaqhaaqcbasaaKqzadGaaGymaaqcbasaaKqzadGaaGOm aaaaaaaa@8937@                 (9)
n is the slope of the variation in incidence with camber angle φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAaaa@3843@ . NACA SP36 curve fits give the below expression:
n=( 0.0630.02274σ )+( 0.0035+0.0029σ ) α 1          ( 3.79  10 5 +1.11  10 5 σ ) α 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi aad6gacqGH9aqpjuaGdaqadaGcbaqcLbsacqGHsislcaaIWaGaaiOl aiaaicdacaaI2aGaaG4maiabgkHiTiaaicdacaGGUaGaaGimaiaaik dacaaIYaGaaG4naiaaisdacqaHdpWCaOGaayjkaiaawMcaaKqzGeGa ey4kaSscfa4aaeWaaOqaaKqzGeGaeyOeI0IaaGimaiaac6cacaaIWa GaaGimaiaaiodacaaI1aGaey4kaSIaaGimaiaac6cacaaIWaGaaGim aiaaikdacaaI5aGaeq4WdmhakiaawIcacaGLPaaajugibiabeg7aHL qbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaeyOeI0cakeaa jugibiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaKqbaoaabmaakeaajugibiaaiodacaGGUaGaaG4naiaa iMdacaqGGaGaaGymaiaaicdajuaGdaahaaWcbeqcbasaaKqzadGaey OeI0IaaGynaaaajugibiabgUcaRiaaigdacaGGUaGaaGymaiaaigda caqGGaGaaGymaiaaicdajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0 IaaGynaaaajugibiabeo8aZbGccaGLOaGaayzkaaqcLbsacqaHXoqy juaGdaqhaaqcbasaaKqzadGaaGymaaqcbasaaKqzadGaaGOmaaaaaa aa@83E6@                       (10)

Design deviation
Deviation angle is the difference between the outlet blade angle and exit flow angle at trailing edge. It arises from the effect of boundary layer growth on the suction side towards the trailing edge pushing the streamline away from the airfoil surface. A very similar correlation to that of the reference incidence angle is used to calculate the design deviation angle:
δ D = K sh K δt δ 010 +mφ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaamiraaWcbeaajugibiabg2da9iaa dUeajuaGdaWgaaqcbasaaKqzadGaam4CaiaadIgaaSqabaqcLbsaca WGlbqcfa4aaSbaaKqaGeaajugWaiabes7aKjaadshaaSqabaqcLbsa cqaH0oazjuaGdaWgaaqcbasaaKqzadGaaGimaiaaigdacaaIWaaale qaaKqzGeGaey4kaSIaamyBaiabeA8aQbaa@5206@                                                          (11)
For deviation, the shape correction factor K sh MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaKqaGeaajugWaiaadohacaWGObaaleqaaaaa@3B4D@ is used with the same formulation as in reference incidence. The thickness correction factor K δt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaKqaGeaajugWaiabes7aKjaadshaaSqabaaaaa@3C06@ is expressed with another form:
K δt =0.0142+6.172( t/c )+36.61 ( t/c ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGlb qcfa4aaSbaaKqaGeaajugWaiabes7aKjaadshaaSqabaqcLbsacqGH 9aqpcaaIWaGaaiOlaiaaicdacaaIXaGaaGinaiaaikdacqGHRaWkca aI2aGaaiOlaiaaigdacaaI3aGaaGOmaKqbaoaabmaakeaajuaGdaWc gaGcbaqcLbsacaWG0baakeaajugibiaadogaaaaakiaawIcacaGLPa aajugibiabgUcaRiaaiodacaaI2aGaaiOlaiaaiAdacaaIXaqcfa4a aeWaaOqaaKqbaoaalyaakeaajugibiaadshaaOqaaKqzGeGaam4yaa aaaOGaayjkaiaawMcaaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaa aaaa@5A30@                           (12)
δ 010 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaaGimaiaaigdacaaIWaaaleqaaaaa @3C6C@ is the deviation angle based on 10% thickness blades.
δ 010 =( 0.0443+0.157σ )+( 0.02090.01865σ ) α 1 +          ( 0.0004+0.00076σ ) α 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajugibi abes7aKLqbaoaaBaaajeaibaqcLbmacaaIWaGaaGymaiaaicdaaSqa baqcLbsacqGH9aqpjuaGdaqadaGcbaqcLbsacqGHsislcaaIWaGaai OlaiaaicdacaaI0aGaaGinaiaaiodacqGHRaWkcaaIWaGaaiOlaiaa igdacaaI1aGaaG4naiabeo8aZbGccaGLOaGaayzkaaqcLbsacqGHRa WkjuaGdaqadaGcbaqcLbsacaaIWaGaaiOlaiaaicdacaaIYaGaaGim aiaaiMdacqGHsislcaaIWaGaaiOlaiaaicdacaaIXaGaaGioaiaaiA dacaaI1aGaeq4WdmhakiaawIcacaGLPaaajugibiabeg7aHLqbaoaa BaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaey4kaScakeaajugibi aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaKqbaoaabmaakeaajugibiabgkHiTiaaicdacaGGUaGaaGimai aaicdacaaIWaGaaGinaiabgUcaRiaaicdacaGGUaGaaGimaiaaicda caaIWaGaaG4naiaaiAdacqaHdpWCaOGaayjkaiaawMcaaKqzGeGaeq ySde2cdaqhaaqcbasaaKqzadGaaGymaaqcbasaaKqzadGaaGOmaaaa aaaa@7FB7@  (13)
The variable m represents the deviation slope factor:
m= m / σ b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb Gaeyypa0tcfa4aaSGbaOqaaKqzGeGabmyBayaafaaakeaajugibiab eo8aZLqbaoaaCaaaleqajeaibaqcLbmacaWGIbaaaaaaaaa@400E@                                                                             (14)
b is given by:
b=0.9655+2.538  10 3 α 1 +4.221  10 5 α 1 2 1.3  10 6 α 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGIb Gaeyypa0JaaGimaiaac6cacaaI5aGaaGOnaiaaiwdacaaI1aGaey4k aSIaaGOmaiaac6cacaaI1aGaaG4maiaaiIdacaqGGaGaaGymaiaaic dajuaGdaahaaWcbeqcbasaaKqzadGaeyOeI0IaaG4maaaajugibiab eg7aHLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaey4kaS IaaGinaiaac6cacaaIYaGaaGOmaiaaigdacaqGGaGaaGymaiaaicda juaGdaahaaWcbeqcbasaaKqzadGaeyOeI0IaaGynaaaajugibiabeg 7aHTWaa0baaKqaGeaajugWaiaaigdaaKqaGeaajugWaiaaikdaaaqc LbsacqGHsislcaaIXaGaaiOlaiaaiodacaqGGaGaaGymaiaaicdaju aGdaahaaWcbeqcbasaaKqzadGaeyOeI0IaaGOnaaaajugibiabeg7a HTWaa0baaKqaGeaajugWaiaaigdaaKqaGeaajugWaiaaiodaaaaaaa@6DAB@    (15)

For 65-series blades :
m =0.173.33  10 4 ( 10.1  α 1 ) α 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGTb GbauaacqGH9aqpcaaIWaGaaiOlaiaaigdacaaI3aGaeyOeI0IaaG4m aiaac6cacaaIZaGaaG4maiaabccacaqGXaGaaeimaKqbaoaaCaaale qajeaibaqcLbmacqGHsislcaaI0aaaaKqbaoaabmaakeaajugibiaa igdacqGHsislcaaIWaGaaiOlaiaaigdacaqGGaGaeqySdewcfa4aaS baaKqaGeaajugWaiaaigdaaSqabaaakiaawIcacaGLPaaajugibiab eg7aHLqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaaaa@55A3@                                    (16)

Computation of blade angles
The calculation of the blade angles follows the steps of this program:

  1. Guess φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAaaa@3843@  
  2. Calculate i D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGPb qcfa4aaSbaaKqaGeaajugWaiaadseaaSqabaaaaa@3A4F@
  3. Calculate δ D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaqcbasaaKqzadGaamiraaWcbeaaaaa@3B06@
  4. Calculate φ 1 = β 1 i D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAjuaGdaWgaaqcbasaaKqzadGaaGymaaWcbeaajugibiabg2da9iab ek7aILqbaoaaBaaajeaibaqcLbmacaaIXaaaleqaaKqzGeGaeyOeI0 IaamyAaKqbaoaaBaaajeaibaqcLbmacaWGebaaleqaaaaa@4658@ and φ 2 = β 2 δ D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAjuaGdaWgaaqcbasaaKqzadGaaGOmaaWcbeaajugibiabg2da9iab ek7aILqbaoaaBaaajeaibaqcLbmacaaIYaaaleqaaKqzGeGaeyOeI0 IaeqiTdqwcfa4aaSbaaKqaGeaajugWaiaadseaaSqabaaaaa@4711@
  5. Calculate φ new = φ 1 φ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAjuaGdaWgaaqcbasaaKqzadGaamOBaiaadwgacaWG3baaleqaaKqz GeGaeyypa0JaeqOXdOwcfa4aaSbaaKqaGeaajugWaiaaigdaaSqaba qcLbsacqGHsislcqaHgpGAjuaGdaWgaaqcbasaaKqzadGaaGOmaaWc beaaaaa@4954@
  6. If | φ new φ |ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaabdaGcbaqcLbsacqaHgpGAjuaGpaWaaSbaaKqaGeaajugW a8qacaWGUbGaamyzaiaadEhaaSWdaeqaaKqzGeWdbiabgkHiTiabeA 8aQbGccaGLhWUaayjcSdqcLbsacqGHKjYOcqaH1oqzaaa@4873@  then φ new MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiabeA8aQLqba+aadaWgaaqcbasaaKqzadWdbiaad6gacaWG LbGaam4DaaWcpaqabaaaaa@3D7B@ is adopted
  7. Else go to step 2 and repeat the process until convergence

Blade profile design
Low subsonic flow regime in the corresponding compressor stage allows to use NACA65-series profile. The circular arc camber line is commonly used as an effective camber line for NACA65-series blades to provide a meaningful definition of leading and trailing edge blade angles. Generally, blade profile is constructed by superimposing prescribed thickness distribution upon the designed camber line. NACA65 thickness distribution is chosen on circular arc mean camber line with 10% maximum relative thickness t/c. The relations used to build the mean camber line and the thickness distributions are obtained from Angier.9 The Figure 3 illustrates the geometric details concerning the circular arc camber line.

Figure 3 Circular arc mean camber line.9

The radius of the mean camber line depends on the blade chord and the camber angle φ according to the following relation:
R c =c/2sin( φ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamOua8aadaWgaaWcbaWdbiaadogaa8aabeaak8qacqGH9aqpcaWG JbGaai4laiaaikdacaWGZbGaamyAaiaad6gapaWaaeWaaeaapeGaeq OXdOgapaGaayjkaiaawMcaaaaa@4243@                                                                      (17)
The coordinates of the mean camber line are then determined from:
{ x c = x 0 + jc/ ( n1 ) y c = y 0 + R c 2 ( x c x 0 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiqaaK qzGeabaeqakeaajugibiaadIhajuaGdaWgaaqcbasaaKqzadGaam4y aaWcbeaajugibiabg2da9iaadIhajuaGdaWgaaqcbasaaKqzadGaaG imaaWcbeaajugibiabgUcaRKqbaoaalyaakeaajugibiaadQgacaWG JbaakeaajuaGdaqadaGcbaqcLbsacaWGUbGaeyOeI0IaaGymaaGcca GLOaGaayzkaaaaaaqaaKqzGeGaamyEaKqbaoaaBaaajeaibaqcLbma caWGJbaaleqaaKqzGeGaeyypa0JaamyEaKqbaoaaBaaajeaibaqcLb macaaIWaaaleqaaKqzGeGaey4kaSscfa4aaOaaaOqaaKqzGeGaamOu aSWaa0baaKqaGeaajugWaiaadogaaKqaGeaajugWaiaaikdaaaqcLb sacqGHsisljuaGdaqadaGcbaqcLbsacaWG4bqcfa4aaSbaaKqaGeaa jugWaiaadogaaSqabaqcLbsacqGHsislcaWG4bqcfa4aaSbaaKqaGe aajugWaiaaicdaaSqabaaakiaawIcacaGLPaaajuaGdaahaaWcbeqc basaaKqzadGaaGOmaaaaaSqabaaaaOGaay5Eaaaaaa@6E42@                                              (18)

Where x0 and y0 are the coordinates of the origin of the curvature radius; n isthe number of nodes used upon suction side and pressure side and j is the order of the node.
The half thickness shape yt may be used from the tabulated base profile data to calculate the coordinates of the profile points on both suction and pressure side8
{ x u = x c y t sin φ c y u = y c + y t cos φ c Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiqaaK qzGeabaeqakeaajugibiaadIhajuaGdaWgaaqcbasaaKqzadGaamyD aaWcbeaajugibiabg2da9iaadIhajuaGdaWgaaqcbasaaKqzadGaam 4yaaWcbeaajugibiabgkHiTiaadMhajuaGdaWgaaqcbasaaKqzadGa amiDaaWcbeaajugibiGacohacaGGPbGaaiOBaiabeA8aQLqbaoaaBa aajeaibaqcLbmacaWGJbaaleqaaaGcbaqcLbsacaWG5bqcfa4aaSba aKqaGeaajugWaiaadwhaaSqabaqcLbsacqGH9aqpcaWG5bqcfa4aaS baaKqaGeaajugWaiaadogaaSqabaqcLbsacqGHRaWkcaWG5bqcfa4a aSbaaKqaGeaajugWaiaadshaaSqabaqcLbsaciGGJbGaai4Baiaaco hacqaHgpGAjuaGdaWgaaqcbasaaKqzadGaam4yaaWcbeaaaaGccaGL 7baaiiaajugibiab=L6axbaa@69F4@                                         (19)
{ x l = x c + y t sin φ c y l = y c y t cos φ c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiqaaK qzGeabaeqakeaajugibiaadIhajuaGdaWgaaqcbasaaKqzadGaamiB aaWcbeaajugibiabg2da9iaadIhajuaGdaWgaaqcbasaaKqzadGaam 4yaaWcbeaajugibiabgUcaRiaadMhajuaGdaWgaaqcbasaaKqzadGa amiDaaWcbeaajugibiGacohacaGGPbGaaiOBaiabeA8aQLqbaoaaBa aajeaibaqcLbmacaWGJbaaleqaaaGcbaqcLbsacaWG5bqcfa4aaSba aKqaGeaajugWaiaadYgaaSqabaqcLbsacqGH9aqpcaWG5bqcfa4aaS baaKqaGeaajugWaiaadogaaSqabaqcLbsacqGHsislcaWG5bqcfa4a aSbaaKqaGeaajugWaiaadshaaSqabaqcLbsaciGGJbGaai4Baiaaco hacqaHgpGAjuaGdaWgaaqcbasaaKqzadGaam4yaaWcbeaaaaGccaGL 7baaaaa@67C1@                                             (20)

φ c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHgp GAjuaGdaWgaaqcbasaaKqzadGaam4yaaWcbeaaaaa@3B3D@ is the local slope of the camber line.

Stage geometry

The detailed geometry of the cascade blades of both rotor and stator is given. This geometry corresponds to the blade section at mean radius. The different geometric properties; blade chord length (c), cascade pitch (s), mean blade height (h), number of blades (N), and the different blade angles; incidence angle (i), deviation angle (δ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacqaH0o azcaGGPaaaaa@3944@ , inlet blade angle ( φ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacqaHgp GAdaWgaaWcbaGaaGymaaqabaGccaGGPaaaaa@3A4D@ , outlet blade angle ( φ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacqaHgp GAdaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@3A4E@ and the camber angle (θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacqaH4o qCcaGGPaaaaa@3955@ are shown in the following Table 4

 

c(m)

s(m)

h(m)

N

i ( o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaamyAaiaacIcadaahaaWcbeqaaiaac+gaaaGccaGGPaaaaa@39D7@

δ ( o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqiTdqMaaiikamaaCaaaleqabaGaai4BaaaakiaacMcaaaa@3A8E@

φ 1 ( o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOXdO2aaSbaaSqaaiaaigdaaeqaaOGaaiikamaaCaaaleqabaGa ai4BaaaakiaacMcaaaa@3B97@

φ 2 ( o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOXdO2aaSbaaSqaaiaaikdaaeqaaOGaaiikamaaCaaaleqabaGa ai4BaaaakiaacMcaaaa@3B98@

φ ( o ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqOXdOMaaiikamaaCaaaleqabaGaai4BaaaakiaacMcaaaa@3AA6@

Rotor

0.152

0.189

0.382

13

-2.35

5.99

51.6

22.97

28.64

Stator

0.0969

0.118

0.34

18

-3.07

7.64

7.64

7.35

36.92

Table 4 Geometry cascade at mean radius

In the below Figure 4 we illustrate the geometry of the blades in 2D linear cascade where the rotor and stator are represented with red and blue, respectively. The axial distance between the stator and rotor is calculated from the following expression
Δz=0.2 c rot MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHuo arcaWG6bGaeyypa0JaaGimaiaac6cacaaIYaGaam4yaKqbaoaaBaaa jeaibaqcLbmacaWGYbGaam4BaiaadshaaSqabaaaaa@41F7@                                                            (21)

Figure 4 2D Geometry cascade at mean radius of the stage.

Conclusion

This work is aimed to present a methodology for an axial compressor stage design using the mean line approach with constant outer diameter configuration COD. A FORTRAN code is developed and implemented to reach this preliminary design based on common thermodynamics and aerodynamics principles. The different properties at three stations; rotor inlet, stator outlet and intermediate face, are calculated. The detailed geometry of the cascade based on the NACA65 profile with circular arc camber line is obtained using NASA empirical relations. The obtained geometry will be exploited as input data in the analysis process to predict axial compressor stage performance curves, using either one dimensional mean streamline method or CFD approach, and also in the structured analysis.

Acknowledgements

None.

Conflict of interest

Authors declare there is no conflict of interest in publishing the article.

References

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