Analytical Modeling of Characteristic Maps of the SR-30 Turbojet Engine

Analytical Modeling of Characteristic Maps of the SR-30 Turbojet Engine

Joaquin Valencia Bravo*, Frederick Just Agosto, David Serrano Acevedo, Marco Menegozzo Department of Mechanical Engineering

University of Puerto Rico -Mayagüez Mayagüez, Puerto Rico

Orlando Ruiz Quinones

Writing Systems Engineer, Hewlett Packard Inc.

Aguadilla, Puerto Rico

AbstractA crucial step in the mathematical modeling of a gas turbine engine is the characteristic map development of its main components, the compressor and the turbine. Maps can be obtained from manufacturers, although they are generally not available to the user. Furthermore, the construction of these maps through experiments is expensive. Analytical models prove to be economically more advantageous. Compared to other methods, they require little experimental data to be validated. These models are based on physical principles, so the results are more accurate compared to those obtained through numerical simulations or the use of neural networks. This approach allows more flexibility to make changes to the performance characteristics of engine components. These changes may be achieved by modifying certain geometric measurements, this quality could be used in diagnostic and/or engine control systems. In this work the characteristic map of the SR-30 turbojet engine components is analytically modeled. Geometric dimensions and properties of the working fluid were taken as data. This is an one- dimensional model in which it is analyzed the aerodynamics and thermodynamics parameters in each station of the centrifugal compressor and axial turbine. Design point calculations are used to estimate the values of working fluid parameters such as velocity, pressure, temperature, enthalpy, and entropy function. Results of this work are compared with those obtained by other authors. For greater reliability in these results, an experimental validation is necessary.

Keywords Turbojet engine; characteristic map; analytical model; compressor; turbine

1. INTRODUCTIONTurbomachinery component performance maps constitute an essential input to the development of engine performance models. Maps may be obtained from experimental procedures, but this can result in an expensive endeavor. Furthermore, experiments might not completely describe the full range of interest, and the use of extrapolationis required. Manufacturers usually developproduct component characteristic maps but this information is rarely distributed to customers. In addition to experimental methods, there are other alternatives in

   obtaining performance maps. Kurzke [1] used data from published maps to reproduce component maps through commercial programs such as Smooth_C [2] and Smooth_T [3]. Lazzaretto et al. [4] developed artificial machine maps using generalized maps taken from literature with appropriate scaling techniques. Results were validated with test measurements from real plants. Many authors have used neural networks approaches. Gustafson el al. [5] obtained steady-state compressor performance maps from correlations of transient compressor maps. The learned correlations were expected to provide information for predicting stall and surge inception. Yu et al. [6] predicted an axial compressor’s performance map based on neural networks. Training was performed with experimental results provided from manufacturers. Ghorbanian et al. [7] used different types of neural networks to simulate the performance maps of an axial compressor; data was obtained from a compressor rig test. Jiang et al. [8] developed an analytical model for the centrifugal compressor based on first principles where energy transfer is taken into consideration. The dynamic performance including losses was determined from the compressor geometry. In addition incident and friction losses were modeled while the clearance, backward and volute losses were considered constants for all off-design conditions.Witkowski et al. [9] developed a one- dimensional analysis for the SR-30 gas turbine components used in education and research settings. Based on design point performance, the flow through the radial compressor and the one stage axial turbine was computed. The corresponding initial conditions of the flow along with the dimensions for each component were used. Losses were estimated using Stodola’s equation for slip factor calculations. These calculations were only presented for a design speed of 78000 rpm. Studies in developing the component maps of a SR-30 turbine were performed by various authors taking into consideration the design point analysis developed by Witkowski et al. [9]. May et al. [11] described a procedure to obtain the velocity triangles for the characteristic maps of the SR-30 gas turbine. An

   iterative calculation in Excel was used, but the mathematical analysis to achieve this goal was not presented. Resulting maps were plotted using the Smooth- C and Smooth-T commercial softwares. The model has been used to monitor health operation of the engine. Based on the mean-line analysis, Léonard et al. [10] developed the compressor characteristic curves. The results were stored in a line format to prevent numerical interpolation problems. These authors did not use a map for the turbine, instead, it was characterized using Stodola’s ellipsis equations. Compressor and turbine losses were considered in the calculation of the velocity triangles. Oh et al. [11] tested most of the loss models previously published in the literature. They found an optimum set of empirical loss models for a reliable performance prediction of centrifugal compressors. May et al. [12] cites these loss correlations for the calculation of the compressor map. In radial flow machines, like centrifugal compressors, the angular momentum imparted to the flow is reduced by a factor known as the slip factor. von Backström [13] developed a method that unifies the trusted centrifugal impeller slip factor prediction methods of Busemann [14], Stodola [15], Stanitz [16], Wiesner [20], Eck [17] and Csanady (reported by Dixon [18]) in one equation. This equation, which is used in the present work, may be adjusted for specifically constructed families to improve the accuracy of the prediction.There is no compressor map or turbine map available for the SR-30 turbojet engine that has been validated using experimental data.

   The present work develops an analytical model of the SR- 30’s component maps. A one-dimensional approach is applied to perform the calculation at the design point performance. The final goal is to produce tabulated functions that may be introduced into an engine model. In addition, interpolation will be applied to obtain values of the dependent parameters such as the referred mass flow rate m , the pressure ratio PR and the efficiency .

2. ANALYTICAL MODELIn this section, the analytical model of the component maps is described. The design point and off design performance are conducted taking into account the limits of the engine operation. Design point calculations are based on iterative procedures. This consist on assuming a value of the axial velocity component, then, performing the calculations required to find the aero-thermodynamic properties of the fluid and finally recalculating the assumed velocity. Iteration is finished when the difference between the initial guess and the calculated axial flow velocity is within the imposed tolerance. As a result, the value obtained for the remaining fluid parameters will be found. Compressor map and turbine map models are described separately. The properties of the working fluid in a gas turbine engine have a powerful impact pon its performance. The working fluid, which passes through the compressor, is assumed as dry air and the working fluid that passes through the turbine is assumed as the product of combustion; the mixture of dry air and kerosene fuel. Water content due to humidity is neglected. The correlations used for the fluid properties were taken from Walsh et al. [19]. The equations

   required to find the energy losses in the centrifugal compressor was obtained from Oh et al. [11], and the loss coefficients in the turbine, were taken from Dixon [18] and Cohen et al. [20]. Moreover, the thermodynamic relations used are derived from the Mollier diagram found in Dixon [18].

   1. Compressor MapThe SR-30 turbojet engine uses a centrifugal compressor. The analysis for the flow through the compressor is based in the one-dimensional steady flow analysis, in which according to Dixon [18], the velocity and density are regarded as uniform across each section of

      a duct or passage. By continuity, the mass flow rate ( m )

      through the compressor is given by

      m 1Ca1 A1 2I Ca2I A2I 2DCa2D A2D 2Ca2 A2

      (1)

      In equation (1), , Ca and A denote the air density, air

      axial velocity and the cross-sectional area respectively. The subscripts 1, 2I, 2D and 2 denote stations corresponding to the entrance to impeller, exit from impeller, entrance to vaned diffuser and the exit from vaned diffuser, respectively. These stations can be visualized in Fig. 1.

      Figure 1. Nomenclature of stations used in the compressor

      Compressor maps will be developed from data points obtained from calculations of the performance parameters such as: mass flow rate m , rotational speed N, pressure ratio PRc and isentropic efficiency c. For this purpose, velocity triangles are calculated initially at the design point. At this point, the mass flow rate and the rotational speed are m 0.297 kg/s and N = 78000 rpm, respectively. In addition, the geometric data and input conditions published by Witkowski et al. [9] are used.

      The first velocity triangle, depicted in Fig.2, is calculated at the impeller inlet. According to this triangle, the relation between the velocity vectors is given by

      Figure 2. Velocity triangle at impeller inlet

      V1 C1 U1

      (2)

      where V1 and C1 are the relative and absolute velocity of the air, respectively. The term U1 denotes the impeller eye velocity vector or tangential velocity vector. It is assumed that the air enters the impeller eye in the axial

      direction, i.e.,

      C1 Ca1 . The angle between the relative

      velocity and the tangential velocity is 1=57. At the impeller inlet (known also as eye or inducer), the effective diameter is defined according to

      D 0.5 D2 D2 1 t r

      (3)

      where Dt is the diameter at the impeller eye tip and Dr is the diameter at the impeller eye root. The annulus area of impeller eye is

      A D2 D2 1 4 t r

      (4)

      U1 D1N

      (5)

      U1 D1N

      (5)

       

      The mean tangential speed at the inducer may be calculated by

      Figure 3. Flow diagram of the iterative procedure at the impeller inlet

      h h 1 C 2 1 01 2 1

      (6)

      h h 1 C 2 1 01 2 1

      (6)

       

      Equations required to calculate Ca1, are presented. The static enthalpy p may be obtained by

      In addition, velocities at impeller eye tip (Ut=DtN) and at impeller eye root (Ur=DrN) may be found. It is important to check that the velocities at the impeller eye tip

      where

      h01

      is obtained from a polynomial function

      should be less than the speed of sound (Mt< 1) to avoid breakdown of flow and excessive pressure loss. To calculate the axial component of the absolute velocity Ca1, a trial and error process will be performed. Fig.3 shows the iterative procedure followed to obtain the parameters at compressor inlet or station 1, according to this flow diagram, the density is calculated on the basis of the known stagnation temperature and pressure. From the continuity equation an initial estimate of the axial velocity Ca1 is obtained. With this value and with the help of thermodynamic relations from the Mollier diagram, the density is recalculated, and the actual velocity is determined from the continuity equation. If the assumed and calculated velocities do not coincide, it is necessary to iterate until convergence is reached.

      dependent on the temperature T01, i.e. h01=fh(T01). The static pressure p1 is given by

      p p1 01 01 1 e R

      (7)

      Here, p01 is the total pressure. The terms 01 and 1 are functions dependent on the temperature T01 and T1, i.e. 01

      =f (T01) and 1 =f (T1). The air density may be calculated from the equation of state for a perfect gas, thus

      p1 1 RT 1

      (8)

      where R is the gas constant for the air and T1 may be found as a function of the enthalpy p, i.e. T1 = fh (p). Finally, the axial velocity Ca1 is given by

      m Ca1 1 A1

      (9)

      Additional calculations are considered to obtain the

      relative velocity V1, V

      C2 U 2

      and the absolute Mach

      number, M

      1abs

      C1

      1 1 1

      RT .

      The velocity triangle at the impeller tip is shown in Fig.4.

      Figure 4. Velocity triangle at impeller outlet

      According to Fig. 4, the air leaves the impeller tip with an absolute velocity C2I. This velocity has a tangential or whirl component Cw2I and a small radial component Cr2I , they are related by

      C C2 C2 2I w2I r 2I

      (10)

      Under ideal conditions the whirl component will be equal to the impeller tip speed U2I . Due to the relative eddies in the impeller blade passage the impeller exit flow angle differs from the impeller exit blade angle; a measure of deviation can be calculated using the concept of a slip factor. There are many correlations predicitng the slip factor. Von Backström [13] describes a method that unifies the centrifugal impeller slip factor (). The prediction method of Dixon [18]), that uses one equation, will be used. This equation is given by

      1 1 1 F 1 cos0.5 o Se

      (11)

      where Fo is a correction constant corresponding to the compressor. This value is adjusted to improve the accuracy of the prediction. The blade angle at impeller exit is denoted by . The ratio given l/Se is known as the blade row solidity. The blade length l and blade spacing Se are

      Figure 5. Flow diagram of the iterative procedure at the impeller outlet

      According to the diagram shown in Fig. 5, static and stagnation pressures and temperatures have to be determined. Thus, the total pressure p02I is given by

      p02I p01PR2I

      (13)

      T

      T

       

      In equation (13), p01 is given as data and PR2I is the pressure ratio. Initially PR2I is guessed and then calculated. The stagnation temperature T02I may be obtained from a correlation function of the stagnation enthalpy h02I, i.e. T02I f h02I . The enthalpy h02I is given by

      determined with

      l = re-ri / cos and

      Se = 2 re / nb . The

      2I 2I

      2I 2I

       

      h h02I h01 02Is 01 2I

      (14)

      h h h02I h01 02Is 01 2I

      (14)

       

      terms ri and re are the radius of the blade leading edge and blade trailing edge, respectively and the number of blades is denoted by nb.The area at impeller exit, A2I, and the blade

      where h01 may be obtained from a correlation in function of the stagnation temperature T01 which is given as

      speed U2I are calculated with

      A2 I D2 I b2 I

      and U D N

      data. The term 2I denotes the efficiency that is initially

      respectively. The terms D2I and b2I are the impeller tip diameter and the axial width at impeller exit respectively. In terms of the slip factor and the blade tangential speed, the whirl component of the absolute velocity Cw2I are

      guessed and finally calculated after some iterations. The stagnation enthalpy at an isentropic process h02Is is obtained by the correlation function h02Is fh T02Is . In the same

      way, the temperature T02Is may be given by the correlation

      determined with

      function

      T02Is f

      02Is . The term 02Is is an entropy

      T

      T

       

      Cw2I U2I

      (12)

      Cw2I U2I

      (12)

       

      p 02 Is 01 R ln 02 I p 01

      (15)

      p 02 Is 01 R ln 02 I p 01

      (15)

       

      function obtained by the following expression

      Some designers consider the value of the radial component C2I the same as the inlet axial velocity Ca1 (Cohen et al.[20]), but, in the current model it is obtained with an estimated value based on equation (10). The final value of the radial component is determined following an iteration procedure which is shown in Figure 5.

      where the entropy function 01 is obtained using the correlation function 01 f T01 . The static temperature

      T2I is given in terms of the static enthalpy pI with the

      correlation function

      T2I f pI . The enthalpy pI is

      T

      T

       

      calculated with the expression

      h h 1 C 2 2 I 02 I 2 2 I

      (16)

      The static pressure p2I is given by

      p p02 I 2 I 02 I 2 I e R

      (17)

      here the entropy functions 02I and 2I are given in term of the temperatures T02I and T2I respectively, i.e.

      Figure 6. Velocity triangle at the vaned diffuser inlet

      In Fig. 6, only the absolute velocity C2D and its components are present. The relative velocity vanishes because there is no moving element. To determine the

      02 I f

      T02 I and f T

      .The density and the

      whirl component Cw2D of C2D, the conservation of the

      2I 2I

      radial component of absolute velocity are calculated using

      angular momentum within the vaneless diffuser can be

      p

      RT

      and C

      m A

      respectively. The Mach

      assumed, i.e. Cw2D r2D = Cw2I r2I . Thus, the whirl

      2I 2I 2I

      r 2I

      2I 2I

      component of the velocity is given by

      numbers should be calculated to ensure that the flow does not exceed the velocity of sound. The performance parameters are also affected by losses that occur inside the compressor. The correlations used for the compressor losses were taken from Oh et al. [22] and Jiang et al. [8].

      C r C w2 I 2 I w2 D r 2 D

      (20)

      The area and absolute velocity are obtained from A2D =

      The pressure ratio and isentropic efficiency may be given

      2r2D b2D and C

      C 2 C 2

      respectively. The iterative

      2 D r 2 D w2 D

      1 h h PR euler int 1 2 I cpT01

      (18)

      1 h h PR euler int 1 2 I cpT01

      (18)

       

      by

      and

      procedure performed here is similar to that followed at the impeller exit. No further energy is supplied to the air after it leaves the impeller, so that neglecting the effect of friction, the angular momentum must be constant (Cohen et al. [20]). The above implies that no work is done in the vaneless diffuser and the process is adiabatic. Thus, the following relation is obtained

      heuler hint 2I h act

      (19)

      h h 02I 02D

      (21)

      heuler hint 2I h act

      (19)

      h h 02I 02D

      (21)

      where, heuler is the ideal specific enthalpy change of the working fluid which depends on the slip factor and the tangential velocity of the blades at the impeller tip. Thus

      2 I

      2 I

       

      int

      int

       

      heuler U 2 . The term h represents the summation

      of all internal losses in the impeller such as incidence loss, blade loading loss, skin friction loss and clearance loss,

      thus h = h + h + h + h . The actual specific

      The pressure ratio and isentropic efficiency is calculated in the same way as in the last stage. Although here, hintI is substituted by hintVD which correspond to the losses in the impeller and the vaneless diffuser, as follows: hintVD =

      hintI +hvld.

      Finally, calculations of the velocity triangle at the diffuser output are performed. Here, the whirl component of the velocity, is found using conservation of angular

      int i bld

      sf cl

      momentum, which yields Cw2 = Cw2D r2D/ r2. The area and

      enthalpy denoted by hact, is the sum of the ideal specific

      the absolute velocity (magnitude) are A2 = 2 r2 b2 and

      enthalpy change and the parasitic loss correlations

      C C 2 C 2

      respectively. The iterative procedure to

      2 a 2 w2

      (hact=heuler+hprs). The parasitic losses, hprs, are assumed to be the sum of the disc friction loss, recirculation loss and the leakage loss, i.e. hprs= hdf +

      hrc +hlk.

      The velocity triangle at the vaned diffuser inlet is depicted in Fig. 6.

      obtain the velocities and the thermodynamic states are similar to the previous stations. Assumptions done for the station of vaneless diffuser may be applied for the vaned diffuser, thus h02D = h02. In the calculations of the pressure ratio and the isentropic efficiency, the losses through the entire compressor stage (i.e., impeller, vaneless diffuser and vaned diffuser) have to be considered. These losses denoted by hintT may be obtained by

      hintT=hintVD+hiD+hsfD, where hintVD represents the internal losses in the impeller and vaneless diffuser, hiD is the incidence loss in the diffuser, and hsfD is the skin friction loss in the diffuser.

      The velocity triangles corresponding to the off-design performance are constructed taking into account the mass

      flow rate and rotational speed in the range of m = 0 – 0.5 kg/s and N =50000-90000rpm, respectively. In the off-design calculations it is assumed that the absolute inlet velocity remains axial and the design point value of the slip factor was constant for all off-design conditions.

   2. Turbine Map

   The SR-30 is a single stage axial turbine composed of one row of nozzles and one rotor. Figure 7 shows the nomenclature of the stages employed in the calculations; N refes to the nozzle and R the rotor.

   Figure 7. Nomenclature of stations used for the turbine

   Ainley and Mathieson correlations [18] were used to calculate the energy losses. These estimated losses will be useful in obtaining characteristic parameters such as the mass flow rate m , the rotational speed N, the pressure ratio PRt and the efficiency t. The operating conditions used in establishing the design point velocity triangles were: m = 0.3 kg/s and N=78000rpm. A mean annulus diameter is assumed in the calculations. This approach is valid when the ratio of the tip radius rt to the root radius rr is low (rt/rr

   m r t

   m r t

    

   = 1:38). The turbine considered in this work meets this low aspect ratio. Moreover, we can consider the mean diameter equal in all stations because the difference between the station mean diameters are negligible. Thus, the effective mean diameter may be given by D = 0.5 D2 + D2 . With the

   m r t

   m r t

    

   effective diameter and the rotational speed, the mean blade speed is U = DmN. The mean annulus area is calculated by A = D2 – D2 / 4 .

   Geometric dimensions and inlet conditions were

   Figure 8. Velocity diagram of the axial turbine stage

   Fig.9 shows the iterative procedure to calculate Ca3. This procedure is similar to the calculations made at compressor inlet.

   Figure 9. Flow diagram of the iterative procedure at the nozzle inlet

   Initially, the velocity Ca3 is guessed. Then, the static temperature is found using a correlation in terms of the

   static enthalpy p, i.e. T = f h . The enthalpy p is

   extracted from Witkowski et al. [9]. By using the geometric measurements of the turbine, a tip clearance of 0.54mm was obtained. It is assumed that the stator blades have the same shape as the rotor blades. The fluid through the turbine is the air-fuel combustion products.

   Fig. 8 depicts the velocity configurations at each station in this turbine. In a single-stage turbine the velocity at

   obtained by

   3 T 3

   h = h – 1 C 2 3 03 2 3

   (22)

   h = h – 1 C 2 3 03 2 3

   (22)

    

   turbine inlet C3 will be axial (Cohen et al. [20]), thus, C3 = Ca3 (Ca3 is the axial component of the velocity at turbine inlet or station 3). The value of this velocity will be taken as constant for all off-design conditions.

   where: h03 = fh(T03, FAR) and FAR is the fuel-air ratio. The

   static pressure is obtained using

   p p03 3 03 3 e R

   h h 1 C2 (25) (23) 4 N 04 N 2 4 N

    

   where 03=f(T03) and 3=f(T3). Similarly to previous cases, the density is calculated with 3=p3/(RT3). From the continuity equation, the axial velocity Ca3 is calculated by Ca3 = m / (3 Am ) . Finally, the corresponding mach number

   The static pressure may be calculated by

   p p03 4 N 03 4 Ns e R

   (26)

   3

   3

    

   isdetermined using M C

   3abs

   RT3 .

   Where,

   03 f T03 , FAR,

   4 Ns f T4 Ns , FAR, and

   Fig. 8 shows the velocity triangle at the nozzle outlet or station 4N. It is assumed that the nozzle angle (4N) and the angular momentum at outlet remains constant. Therefore, for all off-design conditions the deviation of flow across the nozzle remains constant, thus 4N=constant. As a result, the relationship between the whirl component Cw4N and the axial component Ca4N (notation shown in Fig.8) will be constant according to the relation.

   tan1 C 4 4 N C w N a 4 N

   (24)

   Since no work is done (adiabatic process) by the gas in the nozzle, the stagnation enthalpy must be constant over the annulus at the outlet. Thus, h04N = h03. Fig.10 shows the flow chart used for the iterative procedure to obtain the axial component of the absolute velocity Ca4N.

   T4 Ns fT h4 Ns , FAR. The enthalpy under isentropic process

   h4Ns is found using

   C 2 h4 Ns h4 N 4 N N 2

   (27)

   where the total loss coefficient in the nozzle is denoted by

   N. Neglecting the effect of friction, the critical pressure ratio is

   p03 1 1 p 2 c

   (28)

   If P03/P4N<P03/Pc, then the nozzle is not choked, otherwise, it is chocked. The density 4N and the axial flow velocity Ca4N are as follow 4N=p4N/(RT4N)

   and Ca4N = m&/ (4N Am ) . The absolute velocity and

   relative flow angle are

   V C

   U 2 C2

   and

   4N=tan-1[(Cw4N-U4)/Ca4N].

   4 N w4 N

   4 a 4 N

   Figure 10. Flow diagram of the iterative procedure at the nozzle outlet

   According to this flowchart, the absolute velocity Ca4N is guessed. Using this value and the angle 4N, the velocity component Cw4N is obtained (Cw4N=Ca4Ntan4N). The

   The velocity triangle at rotor outlet or station 4 is depicted in Fig. 8. The flow diagram of the iterative procedure at this station is presented in Fig. 11.

   absolute velocity C4N is given by

   C = C 2

   + C 2

   . The

   4N a4N w4N

   static temperature T4N is found in term of the enthalpy, i.e.

   T f h

   . The static enthalpy h4N is given by

   Figure 11. Flow diagram of the iterative procedure at the rotor outlet

   4 N T 4 N

   According to this diagram, the exit angle 4 is constant V 2 4 nd it is calculated from the equations recommended by h4 s h4 R 2

   (34)

   According to this diagram, the exit angle 4 is constant V 2 4 nd it is calculated from the equations recommended by h4 s h4 R 2

   (34)

    

   a

   Ainley and Mathieson (Dixon [18]), thus

   where R is the total loss coefficient in the turbine rotor.The static pressure p4 was calculated with

   4 11.5 1.154 cos1 4s s e

   (29)

   4 11.5 1.154 cos1 4s s e

   (29)

    

   In equation (29), is the throat width, s is the pitch and e is the mean radius of curvature of the blade suction surface. Equation (29) is for low outlet Mach numbers, i.e. 0 <

   here

   04 f

   T04

   p p4 04 04 4 e R

   (35)

   p p4 04 04 4 e R

   (35)

    

   ,FAR

   and

   4 f

   T4 ,FAR . The isentropic

   M4<0.5. If

   M4 1

   the angle 4 is calculated using

   efficiency is determined using

   4 cos1 s. More details about this angle can be found in Dixon [18]. Initially, the axial component of the absolute velocity Ca4 is guessed using the equality Ca4=Ca4N. The whirl component of the absolute velocity Cw4 may be evaluated by

   h03 h04 t h h 03 04ss

   (36)

   The density and the axial velocity component are calculated using equations similar to the previous cases. In addition, the mach number M4 may be found by

   Cw4 Ca4 tan 4 U4

   (30)

   Cw4 Ca4 tan 4 U4

   (30)

    

   M4 = V4

   p> RT4 . After doing the design point performance

   The absolute and relative velocity can be found similarly as in the other stations. The static temperature is found by the

   analysis, the off-design performance is obtained varying the speed and the mass flow rate. The speed was varied from 50000rpm to 90000rpm and the mass flow rate was

   4

   4

    

   correlation T f

   T

   by

   h4 ,FAR . The static enthalpy h4 is given

   varied until the design value was reached (0.3kg/s). The deviation of flow across the nozzle is assumed as constant at all off-design conditions. Because 4N is constant the losses across the nozzle remain constant for all off-design conditions.

   C 2 h4 h04 4 2

   (31)

   C 2 h4 h04 4 2

   (31)

    

   where h04 is calculated by

   C 4 N 4 h04 h03 UCa 4 N tan a4 tan 2U C a 4 N Ca 4 N

   (32)

   In equation (32), the stagnation enthalpy h03 is obtained by

3. RESULTS
   1. Compressor MapResults of pressure, temperature, absolute velocity and Mach number in the behavior of the design point are shown in Table I. The values presented here correspond to each station in the compressor. The analysis was generated

      T

      T

       

      h03

      f T03

      ,FAR .Using the stagnation enthalpy h04, the

      under the rotational speed of 78000rpm with a mass flow rate of 0.297kg/s. The abbreviation, PW, means present

      h

      h

       

      stagnation temperature is computed with

      The stagnation pressure is found from

      T04 f

      T04 ,FAR.

      work. The table shows that the highest stagnation pressure is reached at the impeller outlet (P0 = 323kPa).Thepressure at the diffuser outlet is higher than the impeller inlet due to irreversibility. The compression work done by the impeller is accompanied by an increase in temperature, thus, the stagnation temperature at the impeller outlet is 446K. This temperature remains constant through the diffuser. Static

      p p04 03 PRt

      (33)

      p p04 03 PRt

      (33)

       

      where PRt is the total pressure ratio which is guessed

      initially and then calculated by PRt=(03-04ss)/R . The entropy function 03 is obtained using 03 f T03 ,FAR .

      The stagnation entropy function 04ss is given by

      pressure and static temperature gradually increase through the compressor. Thus, its maximum values are reached at the diffuser outlet, these are: 249kPa and 429K. The Mach number corresponding to the absolute velocity is maximum, and less than one, at the impeller outlet.

      4

      4

       

      04 ss 4 ss R ln p04 p . The static entropy function 4ss

      is obtained from the correlation

      4ss f T4ss ,FAR

      where the static temperature T4ss is calculated using the

      T

      T

       

      correlation T4ss f h4ss ,FAR . The static enthalpy h4ss is

      given by h4ss=h4s-(T4/T4N)(h4N-h4Ns). The static isentropic enthalpy h4s is found using

      Parameter	Impeller inlet		Impeller outlet		Diffuser inlet		Vaned diffuser outlet
      	1		2I		2D		2
      	Ref. [9]	PW	Ref. [9]	PW	Ref. [9]	PW	Ref. [9]	PW
      P0 (kPa)	100	100	323	323	319	307	269	287
      T0 (K)	300	300	445	446	445	446	445	446
      P (kPa)	93.6	93.6	167	183	200	191	269	249
      T (K)	294	294.4	379	380	390	390	430	429
      C (m/s)	106	106.3	366	366	335	337	176	188
      Mabs	0.31	0.31	0.87	0.94	0.86	0.85	0.42	0.45

      Parameter	Impeller inlet		Impeller outlet		Diffuser inlet		Vaned diffuser outlet
      	1		2I		2D		2
      	Ref. [9]	PW	Ref. [9]	PW	Ref. [9]	PW	Ref. [9]	PW
      P0 (kPa)	100	100	323	323	319	307	269	287
      T0 (K)	300	300	445	446	445	446	445	446
      P (kPa)	93.6	93.6	167	183	200	191	269	249
      T (K)	294	294.4	379	380	390	390	430	429
      C (m/s)	106	106.3	366	366	335	337	176	188
      Mabs	0.31	0.31	0.87	0.94	0.86	0.85	0.42	0.45

       

      Table I. Design pointperformance for the compressor

      The results obtained correlate well with Witkowski et al.[9]. Calculations at impeller inlet, 1, are practically the same as those presented in [9]. Result differences in locations 2I, 2D, 2 are due to the correlations considered in the present work. The highest percent difference, corresponding to the static pressure at the impeller outlet, obtained when compared to [9] was 9.6%. This is most likely due to the difference in the energy loss and slip factor calculations. In the present work, correction factors were used in the energy loss calculations. The slip factor formula used in the present work was different from that used in Witkowskiet al. [9]. Furthermore, the coefficient F0 of the slip factor used was adjusted to obtain an appropriate value.

      By varying the mass flow rate (0-0.55 kg/s) and the rotational speed (50,000 rpm – 90,000 rpm), the off-design performance was obtained. Fig. 12 shows the effect of variation of mass flow and rotational speed on the pressure ratio.

      Figure 12. Compressor map: Pressure ratio vs mass flow rate

      Solid curve lines shown in Fig. 12 are the rotational speed, these are indicated by dimensionless values that vary from 0.64 (50,0000 rpm) to 1.15 (90,000 rpm). These speed lines are obtained by joining pairs of points formed by the pressure ratio and the mass flow rate. The ends of each speed line cincide with the minimum and maximum mass flow rate. Joining imaginarily the minimum mass flow of each speed line it is obtained the surge line, this line is nearly at the left of the maximum efficiency line. Surging is associated with a sudden drop in delivery pressure, and with violent aerodynamic pulsation which is

      transmitted throughout the whole machine. The line joining the maximum mass flow points for each rotational speed is called the choking line, that is, no further increase in mass flow is possible beyond this line.

      Figure 13 shows the effect of variation of the rotational speed and the mass flow rate on the isentropic efficiency. Curved lines represent the rotational speed, indicated by dimensionless values. Speed lines values are the same as in Fig. 12. The operating point of maximum efficiency on each speed line is approximately the same. Ideally, the compressor should run on the curve connecting these points.

      Figure 13. Compressor: Efficiency vs mass flow rate

      In Fig. 12, eleven dashed lines intersect the speed lines at a single point, these are called beta lines. Information on mass flow, pressure ratio and efficiency is extracted from each of these points. This information is taken in tabular form, like Tables II, III and IV. These tables may be introduced in a SR-30 turbojet engine model, this procedure is presented in another work. Values of mass flow rate, pressure ratio and isentropic efficiency of the compressor are shown in tables II, III and IV, respectively. The first column of these tables shows the enumeration of beta lines. In the second row of each table, the percentage values of the compressor rotational speeds are shown.

      Table II. Tabulated values of the compressor mass flow rate.

      	Rotational speed, N (%)
      	64	69	74	79	85	90	95	100	105	110	115
      0	0.11	0.12	0.14	0.16	0.18	0.19	0.21	0.23	0.25	0.26	0.28
      1	0.12	0.13	0.15	0.17	0.19	0.21	0.23	0.25	0.27	0.29	0.31
      2	0.13	0.15	0.17	0.19	0.21	0.23	0.25	0.27	0.29	0.31	0.33
      3	0.15	0.17	0.19	0.21	0.23	0.25	0.28	0.30	0.32	0.34	0.36
      4	0.17	0.19	0.21	0.23	0.26	0.28	0.30	0.32	0.35	0.37	0.39
      5	0.19	0.21	0.24	0.26	0.28	0.31	0.33	0.35	0.38	0.40	0.42
      6	0.22	0.24	0.27	0.29	0.31	0.34	0.36	0.38	0.40	0.43	0.45
      7	0.26	0.28	0.30	0.32	0.34	0.36	0.39	0.41	0.43	0.45	0.47
      8	0.30	0.32	0.34	0.35	0.37	0.39	0.41	0.43	0.45	0.47	0.49
      9	0.33	0.35	0.37	0.38	0.40	0.42	0.44	0.45	0.47	0.49	0.50
      10	0.37	0.38	0.40	0.41	0.43	0.44	0.46	0.47	0.49	0.50	0.52

      Table III. Tabulated values of the compressor pressure ratio

      	Rotational speed, N (%)
      	64	69	74	79	85	90	95	100	105	110	115
      0	1.68	1.68	1.68	1.67	1.66	1.63	1.60	1.56	1.50	1.44	1.37
      1	1.81	1.82	1.82	1.81	1.80	1.77	1.73	1.68	1.62	1.55	1.46
      2	1.98	1.99	1.99	1.97	1.95	1.92	1.88	1.82	1.74	1.65	1.55
      3	2.16	2.17	2.17	2.16	2.14	2.10	2.05	1.98	1.90	1.79	1.66
      4	2.37	2.38	2.38	2.37	2.34	2.30	2.24	2.16	2.06	1.93	1.78
      5	2.62	2.63	2.62	2.60	2.57	2.52	2.45	2.37	2.25	2.11	1.93
      6	2.90	2.90	2.89	2.87	2.83	2.77	2.70	2.60	2.46	2.30	2.09
      7	3.21	3.21	3.19	3.17	3.12	3.06	2.98	2.88	2.75	2.57	2.27
      8	3.56	3.55	3.53	3.50	3.45	3.39	3.30	3.20	3.07	2.86	2.40
      9	3.95	3.94	3.91	3.88	3.83	3.77	3.69	3.59	3.42	3.05	2.51
      10	4.38	4.37	4.34	4.30	4.26	4.19	4.11	3.96	3.67	3.19	2.65

      Table IV. Tabulated compressor efficiency values.

      	Rotational speed, N (%)
      	64	69	74	79		85	90	95	100	105	110	115
      0	75.5	77.7	78.5	78.2	76.9	74.6	71.3	67.0	61.5	54.7	0.46
      1	75.4	77.7	78.7	78.6	77.5	75.5	72.5	68.5	63.3	56.7	0.49
      2	76.2	78.1	79.0	78.8	77.7	75.7	72.8	68.9	63.8	57.3	0.49
      3	76.4	78.3	79.1	79.0	78.1	76.2	73.5	69.8	64.9	58.6	0.51
      4	77.9	79.6	79.3	79.2	78.3	76.5	74.0	70.3	65.6	59.4	0.53
      5	77.6	79.0	79.5	79.3	78.3	76.7	74.2	70.8	66.3	60.4	0.53
      6	78.3	79.3	79.6	79.3	78.3	76.7	74.4	71.2	66.8	61.1	0.53
      7	79.0	79.5	79.6	79.2	78.3	76.8	74.8	72.1	68.5	63.5	0.54
      8	79.4	79.6	79.5	79.0	78.1	76.9	75.1	72.9	70.0	64.9	0.53
      9	79.5	79.5	79.3	79.0	78.2	77.2	75.9	74.0	70.7	63.2	0.51
      10	79.3	79.3	79.1	78.8	78.2	77.3	76.1	73.8	70.0	60.2	0.49

      Figure 14 compares the results obtained in the present work with those presented by May et al. [12]. The last one, indicated in the legend as paper. Three speed lines of the compression ratio versus mass flow rate graph were compared. The dimensionless values of these rotational speeds are 0.7 (54,000 rpm), 1 (78,000 rpm), and 1.1 (90,000 rpm). Results show a better match as speed increases. In the present work, when the compressor operates near the lowest rotational speed, the choking point is reached with a lower mass flow rate. As speed increases, results of speed lines between the two works match better.

      Figure 14. Comparison of results for the compressor map.

   2. Turbine Map

   Table V shows comparative results between the present work, indicated by PW, and those presented by Witkowski et al. [9], indicated as Ref. [9]. Results are obtained in the design-point performance, that is, at a rotational speed of 78,000 rpm and a mass flow rate of 0.3 kg/s. As shown in the table, static and stagnation pressure decrease across each turbine location. The temperature is highest at the stator inlet. At this location, in the present work, the stagnation temperature is 973K while the static temperature is 967K. The Mach number reaches its maximum value at the stator output, this value is less than 1 (near the transonic region, M=1) in both results. If M=1,choking will occur. Both results match closely at locations 3 and 4N. It was evidenced that Witkowski et al. [9] has a mistake in the results of static pressure and static temperature, at location

   3. A great difference between both results is observed in the absolute and relative velocities, at location 4. This may be due to the tip clearance value used to calculate the tip clearance loss. In the present work a value of 0.54 mm was used, this value was obtained by measurement carried out in the compressor of the experimental module. In Witkowski et al. [9] a value of 2.44 mm was used.

   Table V. Design point performance for the turbine

   Parameter	Stator inlet		Stator outlet		Rotor outlet
   	3		4N		4
   	Ref. [9]	PW	Ref. [9]	PW	Ref. [9]	PW
   P0 (kPa)	250	250	234	229	120	120
   T0 (K)	973	973	973	973	841	855
   P (kPa)	94	244	128	128	115	106
   T (K)	294	967	834	841	832	827
   C (m/s)	114	114	557	550	141	250
   Mabs	0.19	0.19	0.98	0.97	0.25	0.37

   Off-design performance results are shown in Figures 15 and 16. Figure 15 shows the isentropic efficiency and mass flow rate against the work parameter, DH/T = (h4 – p)/T3. In Fig. 15, each line represents the rotational speed. Colors identify the value of each of these speeds. As the speed increases, the maximum efficiency is reached at a higher value of the working parameter. For each speed, the efficiency remains approximately constant above a certain value of the working parameter. This behavior is due to the fact that the loss coefficients have a small variation in a wide range of incidence. Furthermore, this approximately constant efficiency is greater as the rotational speed increases.

   Figure 155. Isentropic efficiency against work parameter.

   In Fig. 16 it is observed that for each line of rotational speed, the maximum mass flow increases as the working parameter increases. An increase in mass flow occurs until choking conditions are reached. For all speed lines, choking occurs at the same mass flow rate and at a value of the working parameter greater than 100J-kg/K.

   Table VI. Tabulated values of the turbine mass flow rate.

   DHT	Rotational speed, N (%)
   	64	69	74	79	85	90	95	100	105	110	115
   10	0.17	0.18	0.19	0.19	0.20	0.21	0.22	0.22	0.23	0.24	0.24
   25	0.21	0.22	0.22	0.22	0.23	0.23	0.24	0.24	0.25	0.25	0.26
   40	0.24	0.25	0.25	0.25	0.25	0.25	0.26	0.26	0.27	0.27	0.27
   55	0.27	0.27	0.27	0.27	0.27	0.27	0.27	0.28	0.28	0.28	0.28
   70	0.29	0.29	0.29	0.28	0.28	0.28	0.29	0.29	0.29	0.29	0.29
   85	0.30	0.30	0.30	0.29	0.29	0.29	0.29	0.29	0.29	0.30	0.30
   100	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30
   115	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30
   130	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30
   140	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30
   150	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.30

   DHT	Rotational speed, N (%)
   	64	69	74	79	85	90	95	100	105	110	115
   10	74.4	72.8	71.1	69.2	67.3	65.4	63.4	61.4	59.5	57.5	0.56
   25	78.3	78.4	78.2	77.8	77.2	75.8	75.7	74.8	73.9	72.8	0.72
   40	76.5	77.4	78.0	78.3	78.4	78.2	78.2	78.0	77.5	77.0	0.76
   55	73.6	75.2	76.3	77.2	77.8	78.4	78.4	78.5	78.5	78.4	0.78
   70	70.6	72.6	74.2	75.4	76.4	77.7	77.7	78.1	78.4	78.6	0.79
   85	67.5	69.9	71.9	73.5	74.8	76.7	76.7	77.3	77.8	78.2	0.78
   100	64.4	67.2	69.4	71.4	73.0	75.4	75.3	76.2	76.9	77.5	0.78
   115	62.3	64.4	67.0	69.2	71.1	74.0	74.0	75.0	75.8	76.6	0.77
   130	62.3	64.3	66.1	67.7	69.2	72.4	72.4	73.7	74.7	75.6	0.76
   140	62.3	64.3	66.1	67.7	69.2	71.7	71.7	72.8	74.0	75.0	0.76
   150	62.3	64.3	66.1	67.7	69.2	71.7	71.7	72.8	73.8	74.7	0.75

   DHT	Rotational speed, N (%)
   	64	69	74	79	85	90	95	100	105	110	115
   10	74.4	72.8	71.1	69.2	67.3	65.4	63.4	61.4	59.5	57.5	0.56
   25	78.3	78.4	78.2	77.8	77.2	75.8	75.7	74.8	73.9	72.8	0.72
   40	76.5	77.4	78.0	78.3	78.4	78.2	78.2	78.0	77.5	77.0	0.76
   55	73.6	75.2	76.3	77.2	77.8	78.4	78.4	78.5	78.5	78.4	0.78
   70	70.6	72.6	74.2	75.4	76.4	77.7	77.7	78.1	78.4	78.6	0.79
   85	67.5	69.9	71.9	73.5	74.8	76.7	76.7	77.3	77.8	78.2	0.78
   100	64.4	67.2	69.4	71.4	73.0	75.4	75.3	76.2	76.9	77.5	0.78
   115	62.3	64.4	67.0	69.2	71.1	74.0	74.0	75.0	75.8	76.6	0.77
   130	62.3	64.3	66.1	67.7	69.2	72.4	72.4	73.7	74.7	75.6	0.76
   140	62.3	64.3	66.1	67.7	69.2	71.7	71.7	72.8	74.0	75.0	0.76
   150	62.3	64.3	66.1	67.7	69.2	71.7	71.7	72.8	73.8	74.7	0.75

    

   Table VII. Tabulated values of the turbine efficiency.

   Figure 166. Mass flow rate against workparameter.

   In Fig. 17, results of the present work were compared with those presented by May et al. [12], the latter named in the legend as paper.For this comparison, the relationship between mass flow rate and compression ratio was used. In this relationship, graphs of speed lines corresponding to 78,000 rpm are shown. The solid blue line is the one obtained in the present work while the red dashed line is the one obtained in May et al. Graphics are similar only in shape.The difference in the numerical values between both graphs is great. This is due to the fact that the maximum flow taken into account in May et al. [12] is close to 0.22 kg/s, while in this work 0.3 kg/s was considered. The value of 0.3 kg/s is more reasonable, since it represents the combustion mass flow that would be the sum of the air mass flow coming from the compressor plus the fuel mass flow entering the combustion chamber.

   Figure 177. Comparison of results for the turbine map.

   Tabulated values of mass flow rate and isentropic efficiency of the turbine are shown in tables VI and VII, respectively. The first column of these tables shows the values of the working parameter (DH/T or simply DHT). The second row of each table shows the percentage values of the rotational speeds of the turbine, these values are the same as for the compressor.

4. CONCLUSION

In this work, a one-dimensional analytical model of the compressor and turbine maps was made. The unified slip factor formula recommended by von Backström [13] was used. This formula was used in obtaining the whirl component of the absolute flow velocity at the outlet of the compressor impeller. In the compressor map calculations, the internal and parasitic loss correlations selected by Oh et al. [22] were used. In both maps calculations, iteration loops were used to find the axial component of the absolute flow velocity. Many equations were obtained from the Mollier diagram of the corresponding component. Finally, tabulated forms of the compressor and turbine maps were obtained. The results of the design point calculations for each component were compared with that presented by Witkowski et al. [9], a significant match was observed. Off-design performance results were compared with those obtained by May et al. [12]. These match well in the case of the compressor, turbine results showed similar behavior with a significant numerical values differing. This is most likely due to fact that May et al. [12] used a maximum flow rate of 0.22 kg/s while the calculation performed used a value of 0.3 kg/s. This value represents the combustion mass flow which would be the sum of both the air mass flow and the fuel mass flow entering the combustion chamber.

In order to further improve the modeling of the component maps, the validation should be developed by using experimental data or with maps obtained from manufacturers. In addition as built construction details should be used when ever available.

ACKNOWLEDGEMENTS:

The authors would like to acknowledge the interest and encouragement of Dr. Allan Valponi. Views express in this paper are those of the authors, and neither of persons acknowledge herein nor of any funding source nor of their institutions.

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