Highly Loaded Cooled Turbines

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Reduced-OrderThrough-FlowDesignCodeforHighlyLoaded,CooledAxialTurbinesCONFERENCEPAPERJUNE2013DOI:10.1115/GT2013-95469

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1 Copyright 2013 by ASME

REDUCED-ORDER THROUGH-FLOW DESIGN CODE FOR HIGHLY LOADED, COOLED AXIAL TURBINES

Majed Sammak

Div. of Thermal Power Engineering

Dept. of Energy sciences Lund University

SE-221 00 LUND, Sweden Tel: +46 76 236 3637

[email protected]

Marcus Thern Div. of Thermal Power

Engineering Dept. of Energy sciences

Lund University SE-221 00 LUND, Sweden

Tel: +46 222 41 12 [email protected]

Magnus Genrup Div. of Thermal Power

Engineering Dept. of Energy sciences

Lund University SE-221 00 LUND, Sweden

Tel: +46 222 92 77 [email protected]

ABSTRACTThe development of advanced computational fluid dynamic codes for turbine design does not substitute the importance of mean-line codes. Turbine design involves mean-line design, through-flow design, airfoil design, and finally 3D viscous modeling. The preliminary mean-line design continues to play an important role in early design stages. The aim of this paper was to present the methodology of mean-line designing of axial turbines and to discuss the computational methods and procedures used. The paper presents the Lund University Axial Turbine mean-line code (LUAX-T). LUAX-T is a reduced-order through-flow tool that is capable of designing highly loaded, cooled axial turbines. The stage computation consists of three iteration loops cooling, entropy, and geometry iteration loop. The stage convergence method depends on whether the stage is part of the compressor turbine (CT) or power turbine (PT) stages, final CT stage, or final PT stage. LUAX-T was developed to design axial single-and twin-shaft turbines, and various working fluid and fuel compositions can be specified. LUAX-T uses the modified Ainley and Mathieson loss model, with the cooling computation based on the m*-model. Turbine geometries were established by applying various geometry correlations and methods. The validation was performed against a test turbine that was part of a European turbine development program. LUAX-T validated the axial PT of the test turbine, which consisted of two stages with rotational speed 13000 rpm. LUAX-T showed good agreement with the available performance data on the test turbine. The paper presented also the mean-line design of an axial cooled twin-shaft turbine. Design parameters were kept within limits of current practice. The total turbine power was 109 MW, of which the CT power was 55 MW. The CT was designed with two stages with a rotational speed of 9500 rpm, while the PT had two stages with a rotational speed of 6200 rpm. The total

cooling mass flow was calculated to 31 kg/s, which corresponds to 23 % of compressor inlet mass flow. LUAX-T proved capable of designing uncooled and cooled turbines. Keywords: mean-line, turbine design, cooled turbines, methodology, design loops, LUAX-T.

NOMENCLATURE A Area [m2] B Axial chord [m] C Absolute velocity [m/s] Ca Axial velocity [m/s] Cm Meridional velocity [m/s] Cr Radial velocity [m/s] Cu Tangential velocity [m/s] CC Combustion chamber CFD Computational fluid dynamic Cp Specific heat ratio [kJ/kgK] CP Pressure recovery coefficient [-] CT Compressor turbine f Fuel-to-air ratio h, H Enthalpy [kJ/kg] HONC, h/c Aspect ratio [-] HONB Axial aspect ratio [-] HTC Heat transfer coefficient [W/m2K] I Rothalpy [kJ/kg] LUAX-T Lund University Axial Turbine M Mach number [-] m Mass flow [kg/s] m* Dimensionless mass flow [-] N Rotational speed [rpm] OTDF Overall temperature distribution factor

Proceedings of ASME Turbo Expo 2013: Turbine Technical Conference and Exposition GT2013

June 3-7, 2013, San Antonio, Texas, USA

GT2013-95469

Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/15/2014 Terms of Use: http://asme.org/terms


p Pressure [bar] PR Pressure ratio [-] P.S Pressure side PT Power turbine Q Tuning factor r Radius [m] RTDF Radial temperature distribution factor S Pitch [m] SONB Spacing-to-axial chord [-] SONC Spacing-to-chord [-] S.S Suction side T Temperature [C or K] TEONO Trailing edge thickness-to-chord [-] TIT Turbine inlet temperature [C] TONC Thickness-to- chord [-] U Blade speed [m/s] V Velocity [m/s] Wpump Pump work [kW] W Relative velocity [m/s] YP Profile losses [-] YS Secondary losses [-] YTE Trailing edge losses [-] YCL Clearance losses [-] YF Film losses [-] Z Zweifel number [-] Greek symbols rad Heat transfer coefficient by radiation [kJ/kgK] Lean angle [degree] Cooling effectiveness [-] Ratio of specific heat [-] Efficiency [%] Stream line angle [degree] c coolant injection angle [degree] Flow coefficient [-] Stage loading coefficient [-] Reaction degree [-] Density [kg/m3] Subscripts a Air b Blade c Coolant c,in Coolant in c,out Coolant out in Inlet f Fuel g Gas mix Mixing o Total out Outlet p Pressure rel Relative rtr Rotor s Static

str Stator sw Swirl th Throat TT Total-to-total TS Total-to-static

INTRODUCTION Efficient turbine design comprises preliminary mean-line design, through-flow design, airfoil design and 3D viscous modeling. Despite development of advanced computational fluid dynamic (CFD) codes, mean-line design continues to be an important tool in designing turbines. Mean-line design is essentially applied during the early stages of turbine design when geometries, velocities and angles are not known. The mean-line design assumes that there is a mean streamline along the turbine and that the flow conditions on the streamline are representative of the entire turbine. In mean-line design, the major turbine geometries such as blade radius, blade height and inlet and outlet blade angles are determined. In this stage, the turbine annulus and chord, spacing and shape of the blade are also established. Turbine design is an iterative process and it is common to shift from through-flow or airfoil design stage back to the preliminary mean-line design. The mean-line design requires that many factors be considered and weighed against each other to achieve a competitive turbine design. Cooling is an important factor in turbine design and must be considered in the early stages. Cooling affects not only the entropy generation in the turbine but also the generated turbine geometry. Some commercial codes are available for calculating and performing turbine mean-line design [1-4] but not all of them consider cooling during the design process. Mean-line design methodology is not commonly discussed in the literature, although many papers review the methods and correlations used in axial turbine design [5-7]. The present paper discussed the one-dimensional mean-line design methodology for axial turbines. The developed numerical model was based on the conservation of mass, momentum and energy. The paper presented also the Lund University Axial Turbine mean-line code (LUAX-T). LUAX-T was developed to calculate single- and twin-shaft turbines. Various working fluids and fuel compositions can be also specified in LUAX-T. Cooling was estimated at the beginning of the turbine computation, thereby incorporating its influence in the turbine design.

METHOD LUAX-T calculations comprise of two main calculation procedures, stage convergence calculations and stage design calculations, (Figure 1). In the stage convergence calculations the stage convergence method is determined depending on the stage location in the turbine. Stage design calculations contain stator and rotor calculations. Figure 1 showed LUAX-T calculations structure where turbine design parameters are specified prior to computation. These parameters are number of stages, rotational speed, turbine inlet conditions and other design parameters. Furthermore in the case of twin-shaft turbines; the compressor turbine power must be specified.



Calculations proceed then to stage convergence calculations and stage design calculations.

To,Po,m,Cm

Start

TotalStages

Yes No

EndTurbineStage

FirstStage

InletConditions

InletConditions

Stator

Rotor

Stage

Yes

No

StageCalculations

Figure 1. Calculations process.

Stage convergence calculations The calculation starts by identifying the turbine stage in order to choose the convergence loop. The convergent procedures were divided into three; depending on the turbine stage (Figure 2). Turbine stages are part of the compressor turbine (CT)1 or power turbine (PT)2, final CT stage or final PT stage. Stage convergence begins with flow coefficient () loop convergence where, during this loop, the stator outlet angle is adjusted. If the stage is part of the CT or PT, the stage is converged when stage loading coefficient () convergence is reached. The initial value of the stage loading coefficient () is specified and the () loop converged by adjusting the rotor outlet pressure. The final CT stage is converged by adjusting the rotor outlet pressure until the calculated CT power matches the specified CT power. The final PT stage is converged when the calculated PT diffuser pressure matches the specified outlet diffuser pressure. The diffuser pressure is calculated by defining the pressure recovery coefficient (CP). The CP is

1 Compressor turbine is the high pressure turbine. 2 Power turbine is the low pressure turbine.

defined as the actual pressure rise in the diffuser to the isentropic pressure rise, (Equation 1). In the case of hot-end drive turbines CP is around 0.65 while in cold-end drive turbines CP is around 0.8 [8, 9].

p1 p2

1 CP 1 k12 M12k

k1 1

1

Figure 2. Stage convergence routines.

Stage design calculations The turbine stage in LUAX-T was divided into several calculation stations. This structure provides flexibility in transferring data between stations and also provides stability in calculation. Each computed stage consists of stator and rotor station. The stator calculation stations are stator inlet (1), stator outlet prior to stator cooling mixing (2), and stator outlet after stator cooling mixing (2mix). The rotor stations are rotor inlet (2gap), rotor outlet prior to rotor blade cooling mixing (3), rotor outlet after mixing rotor blade cooling (3mix), and next-stage stator inlet after mixing of rotor disc cooling (3gap). These calculation stations are presented in Figure 3. Stations 2 and 2mix and also stations 3 and 3mix are physically the same station in the turbine. They are treated differently in the model to take into account the changes in thermodynamic properties and geometries due to cooling mixing. By defining the first stage stator inlet conditions, the stator inlet static properties, geometries, velocities and angles are computed. Inlet conditions are total inlet temperature, total inlet pressure, mass flow and meridional velocity. The first stage stator inlet properties are therefore defined and the computations forced to the later turbine stages. The computation of stator and rotor are similar, but the rotor inlet properties have to be converted to the relative coordinate system.



Figure 3. Turbine calculation stations. The coordination is then transferred back to the stationary system prior to the next stage. Stage calculation comprises three main iterations loop, cooling, entropy and geometry loop. Placing the cooling iteration loop as the outer loop provides stability to the computation. Outer cooling loop has the advantage of including cooling in the stage design. The calculations start by computing the required cooling mass flow. Aerodynamic losses are calculated in the entropy loop and stage geometry is computed in the geometry loop. The stage thermodynamic properties are adjusted later, after coolant mixing. Once the thermodynamic properties and geometries of the turbine stages are defined, the overall turbine performance is determined. Figure 4 shows the calculation process in the stator and rotor. Stator calculation: Stator calculation is started by defining the stator outlet pressure or the stage reaction degree (). In the cooling iteration loop, the initial value of stator cooling is specified or estimated by defining a value for cooling efficiency and overall temperature distribution factor (OTDF). Once the cooling mass flow and cooling conditions are identified, the power loss due to heat transfer from gas to coolant is defined (station 2 in Figure 3). Bear in mind that after cooling the blade the coolant will be ejected outside the blade and mix with the gas flow. Thus mixing conditions has to be calculated with preserving mass and energy balance. The entropy iteration loop is started by defining an initial value for total entropy losses. The stator outlet conditions, area and velocities are hence defined. The stator geometry calculations are performed in the geometry iteration loop. In the geometry loop, the stator outlet hub radius is specified, after which the stator outlet tip radius, stator height, hub-to-tip ratio and stator axial chord are computed. Stator hub and tip hade angle are also calculated. Then, specification of stator gap length and rotor hub radius will provide rotor tip radius, rotor height, rotor hade angles, rotor axial chord, rotor aspect ratio and rotor outlet hub radius. Eventually the stator stream line angle () is determined. A new value of stator tip radius is calculated and computations are repeated until geometry iteration loop convergence is reached. Additional geometries are calculated prior to calculating stator aerodynamic losses. These geometries are stagger angle, solidity (S/c), Zweifel number, number of blades and other geometry relations. Once the aerodynamic losses are calculated a new value for stator outlet entropy is computed.

Figure 4. Stage calculations.

The computations are repeated until the entropy iteration loop reaches convergence. The cooling mass flow should then be recalculated because the stator geometry is completely defined. If the estimated stator mass flow differs from the calculated value, the calculations must be repeated until the cooling mass flow iteration loop converges. The next step in stator calculations is to determine the conditions after cooling mixing (station 2mix in Figure 3). The cooling flow is extracted from the compressor at different pressures. The extracted cooling mass flow causes power losses, but the coolant is injected to the main gas stream after cooling the blades. The stator outlet mass flow should therefore be corrected after cooling mixing. The stator total enthalpy after mixing is determined from mass and energy conservation. In order to calculate the other mixing conditions, either mixing pressure or density is specified. In the case of estimating the mixing static pressure, the mixing mass flow calculates a new value of mixing pressure. A relationship between pressure and mass flow is established through the conservation of mass and momentum (Equation 2). mp

m C .

M 1 2



A new value for coolant mass flow is calculated from mass conservation, so a new value of mixing static pressure is obtained. If the calculated mixing static pressure differs from the estimated value, calculations are repeated until convergence is reached. If, on the other hand, the mixing density is estimated, the meridional velocity is computed from the continuity equation, enabling coolant mass flow to be defined. The computation then proceeds in the same way to the mixing pressure iteration loop. Stator total conditions are identified after mixing. Prior to the rotor inlet, the main gas flow passes through an axial gap (station 2gap in Figure 3). At the axial gap, the duct area changes because of rotor disk cooling injection and cavity purge. The stator outlet conditions therefore differ from the rotor inlet conditions. Main gas mass flow, rotor inlet conditions and rotor inlet area have to be adjusted due to rotor disk cooling. The rotor inlet conditions are then calculated by repeating cooling mixing iteration loop. Rotor calculations: The stator gap outlet conditions are the rotor inlet conditions (station 3 in Figure 3). Prior to rotor calculations, the gas conditions are converted to the relative coordinate system. Rotor inlet relative velocities, relative flow angles and relative total conditions are computed. Rotor inlet rothalpy (I) is also determined. The rothalpy is constant across the rotor and defined in Equation 3.

I h w

2 U2 h

12C

U C 3 The rotor outlet area is computed using the rotor relative inlet properties. The AN2 is then calculated and compared with maximum specified AN2. AN2 is defined as annulus area of rotor blade multiplied by rotational speed squared. AN2 is proportional to the blade root stress at a given rotor hub radius. If the calculated AN2 exceeds the maximum defined limit, the rotor tip radius should be reduced to obtain the maximum AN2. The rotor calculations are similar to the stator, where cooling; entropy and geometry loops should converge for a completed rotor design (Figure 4). The rotor cooling mass flow is estimated after defining the radial temperature distribution factor (RTDF) and cooling efficiency. The rotor coolant involves a large radius change where it is injected at low radius and then pumped radially outward on the rotor. The coolant is passed through pre-swirl nozzles (swirl generator) in the blade root to swirl the coolant so that coolant temperature reduction is obtained. Converting velocities to the relative coordination system will promote coolant temperature reduction and less coolant will be pumped. The stage power also decreases due to pump work required to overcome the coolant radius change (Equation 4).

W m U U C 4 Entropy iteration loop begins with estimating total entropy change over rotor. Once rotor outlet pressure is defined, the rotor outlet conditions, velocities and angles are computed. Like the stator calculations, the rotor geometry is calculated later. Geometry ratios and total rotor aerodynamic losses are

finally computed before calculating a new value for rotor outlet entropy. Following a new value of cooling mass flow is calculated and checked with the estimated value. Prior to calculating main gas flow conditions after cooling mixing (station 3mix in Figure 3), relative conditions and velocities should be converted to the stationary coordinate system. The main gas flow conditions at the rotor gap are calculated in the same way as for the stator gap (station 3gap in Figure 3). The rotor gap outlet conditions are the inlet conditions for the next stage. Throat calculations: Throat conditions at stator and rotor are also calculated. The maximum gas mass flow occurs at the throat when this is choked and so the throat Mach number is unity. In order to calculate throat conditions, entropy generation from the blade leading edge to the throat and rothalpy are used. The losses from leading edge to throat are assumed to be 20 % to 30% of total blade losses. The pressure at throat is calculated from Equation 5. Throat computation is an iterative process since the speed of sound at the throat is a function of gas composition. p,p 1

k 12 M

/

5

After designing the stage the calculations proceed to the next stage where the downstream stages of the turbine obtain their inlet conditions from the upstream stages. Once the thermodynamic and aerodynamic properties of the entire turbine stages are computed, turbine performance can be calculated. Performance parameters such as stage work, loading coefficient, flow coefficient, and reaction degrees are calculated. Reduced-order through-flow: The span-wise variation of flow properties (reaction degree, static pressure, flow angle, etc.) and velocity triangle closure are calculated with the method by Dejc and Trojanovskij [10]. It is important to introduce limitations for hub reaction at the initial design since it affects several design parameters. In the Dejc and Trojanovskij original reference, there are three factors K1, K2 and K3 taking into account the effects due to stream line curvature, lean and end-wall hade, respectively. The stream line curvature is hard to calculate in a one-dimensional calculation. The radius of curvature is therefore set to infinity in this work. Dejc and Trojanovskij introduced a tuning factor (Q) and provide some recommendations as a function of hub-to-tip ratio. The current version of the code uses Equation 6 for the case of a straight vane (1=const.). CC

C

C

1 1 rr

.

6



Equation 6 is written in the original tangential angle reference plane. The second part of the exponent is the effect due to lean angle (). The method by Dejc and Trojanovskij does not replace the conventional through-flow approach. However it introduces a convenient way of introducing three-dimensional effects into the one-dimensional calculations, hence the name reduced order through-flow.

Thermodynamic properties The composition of the main gas stream is changed after cooling mixing. The fuel-to-air ratio is used to compute the new composition of the main gas flow. The fuel-to-air ratio (f) is defined as the ratio of the mass of fuel and mass of air present during combustion (Equation 7).

mm 7 A new value for fuel-to-air ratio has to be derived after cooling mixing in order to calculate a new gas flow composition. Figure 5 shows the fuel-to-air ratio after cooling mixing. The new fuel-to-air ratio after cooling mixing is derived in Equation 8.

1 1 8

cm

am

fm

inm outm

Figure 5. Fuel-to-air ratio.

The main gas flow properties are based on a semi-perfect gas state model. The gas specific heat capacity is temperature dependent, which gives good calculation accuracy. Using real gases with pressure dependence of specific heat capacity will introduce more complexity to the equations of state, with little improvement in accuracy. The state model is based on the NASA SP-273 polynomials. It uses a set of Cp(T) coefficients that have been empirically developed by NASA. The coefficients work over the temperature range from 223 K to 5000 K and are divided into two groups, the first group for calculations below 1000 K and the second for calculations from 1000 K to 5000 K. The module uses 273.15 K as reference temperature and 1.01325 bar as reference pressure. The LUAX-T mean-line code also makes it possible to change main gas composition, so turbines with different working fluid and different cooling composition can be modeled. Such turbines are oxy-fuel turbines with CO2 as working fluid, humid turbines with high air water content, and turbines with steam cooling.

Geometry calculation The turbine calculations are based on a cylindrical polar coordinate system with axial, tangential and radial axes [11]. Axial velocity (Ca), radial velocity (Cr), tangential velocity (Cu), meridional velocity (Cm), absolute velocity (C) and absolute velocity projection on the axial plane are shown in Figure 6. The velocity relationships are derived from velocity triangles and are used later in the geometry calculations (Equation 9).

C C tan, C C coscos sin cos cos

C C C coscos sin cos sin

9

Figure 6.Turbine coordinate system.

LUAX-T geometry calculations are based on continuity equation and geometry relation models. The blade axial aspect ratio (h/c) is calculated by polynomials developed by Abianc [12], (Figure 7). The blade aspect ratio is a function of blade hub-to-tip ratio and whether the blade is stator or rotor. Moustapha et al. [13, 14] state that blade inlet angle and design incidence (i) are functions of inlet and outlet flow angles. Hade angle, taper angle and stream line angle () are also calculated. The hade angle is the inclination of the gas channel at hub and tip in both stator and rotor, and taper angle is the vertical inclination of the blade profile (Figure 8). Designing the blade with some taper angle is common practice to ease the stresses in the blade root. The streamline angle () is the angle between the meridional (Cm) and axial velocity (Ca) and determined from Equation 10 [15]. The factor 0.8 takes into account the gap between the stator and rotor.

tan r rB B 0.8 10



Figure 7. Axial aspect ratio as function of hub-to-tip ratio.

Figure 8. Stream line angles.

The stagger angle is a function of blade inlet and outlet angles and is calculated using polynomials developed by Kacker and Okupuu [16]. According to Mamaev [17], the pitch-to-chord ratio (S/c) or solidity is function of exit Mach number, inlet and exit flow angles and trailing edge thickness. The number of blades is determined eventually from the calculated (S/c). Addressing losses in the turbine requires identifying a number of geometry relations. Ainley et al. [5] developed a model for calculating opening-to-pitch ratio. The model states that opening-to-pitch ratio (O/S) is function of gas flow inlet and outlet angle, exit Mach number and whether the blade is shrouded. Thickness-to-chord ratio (t/c) is calculated on the other hand using a model developed by Kacker et al.[16].

Loss calculations The most widely-used method for predicting losses in axial turbines is the modified method developed originally by Ainley and Mathieson (AM) [5]. AM models has been modified by Dunham and Came (DC) [6, 7, 13], Kacker and Okapuu (KO) [13, 16] and Moustapha and Kacker (MK) [13, 14] . The AM-

DC-KO-MK model presented new correlations for the profile (Yp) and secondary losses (YS) at off-design conditions. The model was significantly more successful than the original Ainley and Mathieson model because it includes correlations for all the components of loss, and for both design and off-design conditions. More recently, the profile and secondary losses in AM-DC-KO-MK have been further developed by Benner et al. [18, 19]. The present paper employed the Benner-modified loss model of AM-DC-KO-MK. The calculated secondary losses showed good agreement with secondary losses calculated by Denton, which correspond to the original Ainley and Mathieson secondary losses multiplied by 0.375 [20]. Trailing edge losses (YTE) were calculated by model developed by Kacker and Okupuu [16, 21]. The clearance loss (YCL) model differs between shrouded and unshrouded blades. Dunham and Came modified the original Ainley and Mathieson expression for calculating clearance losses in shrouded blades [7, 13]. In the case of unshrouded blades, Kacker and Okapuu considered a relationship for calculating clearance losses [13, 16]. Film losses (YF) occur due to turbine cooling. The coolant flow is injected to the main gas stream and causes aerodynamic mixing losses. Film cooling is implemented in the first stage where blades are exposed to very high temperatures. The mixing model is based on Hartsels mixing loss model [22]. Hartsel mixing employs a two-dimensional mixing model where the coolant and main stream are mixed under a constant static pressure. The Hartsel mixing loss is presented in Equation 11.

M

1

2

cos ] 11

Hartsel mixing losses are a function of the square of gas flow Mach number, the coolant mass flow fraction, the ratio of coolant and gas temperature and coolant and gas velocity (Figure 9).

Figure 9. Hartsel mixing model.

The total losses Ytot are therefore:

Y Y Y Y Y Y 12 Film losses along the blade profile are calculated via a generic rectangular velocity distribution around the blade. The velocity distribution was developed by Denton [23, 24] to generate

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

12

14

Hub-to-tip ratio [-]

Axi

al a

spec

t rat

io [-

]

StatorRotor



blade loss coefficient. Figure 10 shows the generic velocity distributions for first rotor.

Figure 10. Generic velocity distributions for turbine blades, rotor 1.

The coolant is injected to the outside of the blade through holes in the surface of the blade and in the trailing edge. The trailing edge mixing model of two streams is implemented. The model assumes mixing under constant area and pressure. The calculation of mixed out conditions obtained from conservation of mass, momentum and energy [24], (Figure 11).

Figure 11. Trailing edge mixing model.

The total enthalpy of the mixed flow was calculated from the energy balance over the selected control volume:

H, m H, m H,

m 13

The momentum and impulse equations are:

C, m C, m C,

m 14

C, , , 15 The mixing process is an iterative process since the mixing pressure is unknown. Mixing density is calculated from the estimated mixing pressure and mixing static enthalpy (Equation 13). The mixing mass flow is calculated from the continuity equation since the mixing process takes place under

constant area. The calculated total mass flow is compared to the obtained total mass flow from mass balance. If the obtained total mass flow differs the mixing pressure is recalculated and calculations are repeated until convergence reached.

Cooling calculation The turbine cooling mass flow is computed from the m*-model. Holland [25] and Barry [26] presented the m*-model, which was originally based on the standard blade approach adopted by Hall [27]. The m*-model identified the main factors affecting the blade cooling. The m*-model described the heat transfer from the hot gases through the blade metal toward the coolant. The cooling model was defined by employing three dimensionless parameters. Cooling effectiveness () can be considered as dimensionless blade temperature. Cooling effectiveness is a measure of the amount of cooling required to maintain a particular metal temperature [13], (Equation 16).

T TT T 16 Cooling efficiency () is a measure of coolant utilization. It is defined as the ratio between the actual heat removed from the hot gases to the maximum heat that could be removed [13], (Equation 17).

T, T,T T, 17

Dimensionless mass flow (m is a measure of the coolant mass flow. It is defined as the ratio of potential of cooling fluid extraction to the potential of mainstream heating [13], (Equation 18).

m m CHTC A 18 The entry turbine gas temperature profile has a temperature peak at the middle of the profile and lower temperatures at tip and hub. In order to account for temperature profile, the overall temperature distribution factor (OTDF) and the radial temperature distribution factor (RTDF) are used. The OTDF is used to design the stator while RTDF is used for rotor design. The OTDF and RTDF are defined as the ratio of the difference between the gas peak and mean temperature to the combustor temperature rise. Hence the gas peak temperature is calculated at the first stage stator and rotor. The temperature distribution factor works well for first turbine stage but at the entry to the next stage the radial temperature profile become flatter. The OTDF is set to 0.1 while the RTDF is set to 0.05 [8].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

800

Axial Chord [-]

Vel

ocity

dis

tribu

tion

[m/s

]

P.S

S.S



VALIDATION AND RESULTS A test turbine was chosen for LUAX-T validation. The turbine chosen was part of a European turbine development program. The test turbine was a non-aero derivative uncooled gas turbine and was developed for industrial and marine application. It was a twin-shaft gas turbine, where the CT was a radial turbine while the PT was an axial turbine. LUAX-T validated the axial PT of the test turbine, which consisted of two-stage rotations at 13000 rpm. The test turbine design specification is shown in Table 1.

Table 1.Test turbine design specification. Performance data min kg/s 11.7 TIT C 935 PRTT - 3.18

pin bar 3.45 Min - 0.193

Tables 2 and 3 show the available thermodynamic conditions at entry and exit of test PT stages. Tables 4 and 5 show the simulated results from LUAX-T.

Table 2. Test turbine thermodynamic conditions, Stage 1. Stator in Stator out Rotor in Rotor out

To C 935.00 935.00 935.00 830.90 Ts C 928.90 875.90 877.60 824.10

To,rel C - - 882.90 884.30 po bar 3.45 3.36 3.36 2.03 ps bar 3.36 2.59 2.61 1.96

po,rel bar - - 2.67 2.60 M - 0.20 0.64 0.63 0.22

Table 3. Test turbine thermodynamic conditions, Stage 2. Stator in Stator out Rotor in Rotor out

To C 830.90 830.90 830.90 712.70 Ts C 825.10 767.80 769.40 702.40


po,rel bar - - 1.52 1.47 M - 0.20 0.69 0.68 0.29

Table 4. Test turbine thermodynamic LUAX-T results, Stage 1.

Stator in Stator out Rotor in Rotor out To C 935.00 935.00 935.00 817.31 Ts C 928.58 874.57 876.67 809.23


po,rel bar - - 2.79 2.71 M - 0.19 0.60 0.59 0.23

Table 5. Test turbine thermodynamic LUAX-T results, Stage 2. Stator in Stator out Rotor in Rotor out

To C 817.31 817.31 817.31 676.04 Ts C 809.23 750.37 752.28 663.22


po,rel bar - - 1.61 1.55 M - 0.23 0.66 0.65 0.30

The thermodynamic results shown are total and static temperatures and pressure, relative total temperature and pressure and Mach number. LUAX-T showed very good correspondence with the thermodynamic data available from the test PT. The annulus test turbine is presented in Figure 12.

Figure 12. Test turbine annulus PT.

The performance results from LUAX-T are shown in Table 6. The total-to-total efficiency and total-to-static efficiency were considered. The LUAX-T stage performance results showed good correspondence with the tested turbine.

Table 6. Test turbine performance. Stage 1 Stage 2 Test turbine LUAX-T Test turbine LUAX-T

TS % 84.73 84.53 85 83.50 TT % 89.57 89.97 91.69 90.45

Cooled turbine LUAX-T was used to design a conventional twin-shaft cooled gas turbine. Due to lack of validation data from commercial cooled gas turbines, design parameters were kept within acceptable limits [8]. The total turbine power was 109 MW, of which the CT power was 55 MW. The CT consisted of two stages with rotational speed 9500 rpm. The PT was designed in two stages with rotational speed 6200 rpm. The parametric data is shown in Table 7.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0.22

0.24

0.26

0.28

0.3

0.32

0.34

Axial length [m]

Rad

ius

[m]



Table 7. Twin-shaft cooled turbine inlet conditions. minlet kg/s 100 TIT C 1400 PRTT - 23 Cm m/s 90

Preliminary mean line of the twin-shaft turbine was calculated and the results are shown in Tables 8, 9 and 10. Table 8 shows the main geometric relationships. Solidity (S/C), hub-to-tip ratio (rhub/rtip), aspect ratio (h/C) and Zwiefel number are shown in Table 8.

Table 8. Gas turbine geometry relations. CT PT Stage 1 Stage 2 Stage 3 Stage 4

S/c - 0.93/0.82 0.83/0.82 0.91/0.81 0.76/0.63 rhub/rtip - 0.84/0.84 0.81/0.77 0.72/0.69 0.64/0.62

h/c - 0.98/1.5 1.5/2.1 1.8/2.8 3.0/3.9 Z - 0.81/0.87 0.91/0.95 0.84/0.78 0.82/0.77

The gas turbine performance is shown in Table 9. Stage loading (), flow coefficient (), stage pressure ratio and reaction degree were calculated. The cooled stages were the first two CT stages while the PT stages remained uncooled. The cooling effectiveness of the cooled turbine is shown in Table 9. The total cooling mass flow was calculated to 31 kg/s, which represented 23 % of compressor inlet mass flow.

Table 9. Gas turbine performance. CT PT Stage 1 Stage 2 Stage 3 Stage 4

- 1.35 1.1 1.4 1.36 - 0.34 0.46 0.36 0.46

PR - 1.95 1.94 2.14 2.6 p - 0.39 0.39 0.45 0.45

/ % 58/46 34/17 - - The turbine profile (Yp), secondary (YS), clearing (YCL) and trailing edge (YTE) losses were calculated and are shown in Table 10. The first stage film losses (YF) are also shown in Table 10

Table 10. Gas turbine losses. CT PT

Stage 1 Stage 2 Stage 3 Stage 4 YP % 2.83/3.26 1.67/1.58 2.82/3.45 2.88/2.23 YS % 2.47/2.73 2.43/2.06 1.83/1.48 1.49/1.24YCL % 0/6 0/3.32 0/4.87 0/3.00 YTE % 0.55/0.82 0.43/0.51 0.26/0.48 0.30/0.42 YF % 0.82/0.72 0/0 0/0 0/0

Ytotal % 6.7/13.6 4.5/7.3 4.9/10.29 4.67/6.96 The twin-shaft gas turbine annulus is shown in Figure 13.

Figure 13. Twin-shaft gas turbine annulus.

Results showed that LUAX-T can design uncooled as well as cooled turbines. LUAX-T has been used to design oxy-fuel turbines where the main working fluid consists of CO2 [28, 29], Graz oxy-fuel turbine [30] and humid turbines [31]. Furthermore LUAX-T was used for parametric studies, as described in Noor et al. [32].

CONCLUSION The paper discussed the development of the Lund University Axial Turbine mean-line code (LUAX-T). The code is a reduced-order through-flow tool that can be used to design highly loaded, cooled axial turbines. The aim of this paper was to show the computational methods and procedures for mean-line designing. The stage computation consists of three iteration loops cooling, entropy and geometry loops. The stage convergence varies according to whether the stage is part of a CT or a PT, final CT stage or final PT stage. LUAX-T is capable of design single- and twin-shaft turbines. Furthermore, different working fluids and fuel compositions can be specified in LUAX-T. LUAX-T validations were performed against a test turbine that was a part of a European turbine development program. The chosen test turbine was a twin-shaft gas turbine where its CT is a radial turbine while the PT is an axial turbine. The validation was performed on the PT, which consisted of two-stage rotations at 13000 rpm. LUAX-T showed good correspondence with available performance data from the test PT. In the paper, the mean-line design for an axial twin-shaft turbine was also presented. Design parameters were kept within acceptable practice limits. The total turbine power was 109 MW, of which the CT power was 55 MW. The CT was designed with two stages at a rotational speed of 9500 rpm, while the PT had two stages with a rotational speed of 6200 rpm. The total cooling mass flow was calculated to 31 kg/s, the equivalent of 23 % of compressor inlet mass flow. LUAX-T proved capable of designing uncooled and cooled turbines. LUAX-T was also used to design oxy-fuel and humid turbines.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.2

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0.4

0.5

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0.7

0.8

Axial length [m]

Radi

us [m

]



ACKNOWLEDGMENT This research has been funded by the Swedish Energy Agency, Siemens Industrial Turbomachinery AB, Volvo Aero Corporation and the Royal Institute of Technology through the Swedish research program TURBOPOWER. The support of which is gratefully acknowledged. The development of LUAX-T would not been possible without the work done by David Olsson, Jonas Svensson and Bjrn Nyberg.

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