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Axial ﬂow, multi-stage turbine and compressor models Jean-Michel Tournier, Mohamed S. El-Genk * Institute for Space and Nuclear Power Studies and Chemical and Nuclear Engineering Department, University of New Mexico, Albuquerque, NM, USA

a r t i c l e

i n f o

Article history: Received 8 December 2008 Received in revised form 1 August 2009 Accepted 12 August 2009 Available online 29 September 2009 Keywords: Gas turbo-machine Axial-ﬂow turbine and compressor design Helium Noble gas binary mixtures Closed Brayton Cycle

a b s t r a c t Design models of multi-stage, axial-ﬂow turbine and compressor are developed for high temperature nuclear reactor power plants with Closed Brayton Cycle for energy conversion. The models are based on a mean-line through-ﬂow analysis for free-vortex ﬂow, account for the proﬁle, secondary, end wall, trailing edge and tip clearance losses in the cascades, and calculate the geometrical parameters of the blade cascades. The effects of the mean-stage work coefﬁcient, ﬂow coefﬁcient and stage reaction on the design and performance of helium turbine and compressor are investigated. The results compare favorably with those reported for 6 stages helium turbine and 20 stages helium compressor. Also presented and discussed are the results of parametric analyses of a 530-MW helium turbine, and a 251MW helium compressor. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Gas turbo-machines are widely used today throughout the world for power generation and in mechanical drives, marine and aircraft engines [1]. Natural gas ﬁred commercial power plants use axial-ﬂow turbo-machines in a Closed Brayton Cycle (CBC) for electricity generation at high thermal efﬁciency, with a growing capacity of 30 GW per year worldwide. Recently, there is an interest in using CBC turbo-machine for energy conversion in future High Temperature Reactor (HTR) power plants. In these plants, the graphite-moderated, helium-cooled nuclear reactor heat source is coupled either directly or indirectly to a CBC for electricity generation. HTR power plants are being investigated in the USA, Europe, Russia, Japan and South Africa for electricity generation at high thermal efﬁciency (>48%), and for providing process heat to a host of industrial applications that include the co-generation of hydrogen using thermo-chemical processes [2–5]. The design and optimization of HTR plants requires developing detailed design and performance models of the axial-ﬂow turbomachines for CBC, which is the focus of this work. Very little work has been reported on the subject, and the results of trade studies of HTR power plants have often used simple thermodynamic models with simplifying assumptions such as constant turbine and compressor polytropic efﬁciencies (e.g., [4,6]). In addition to the plant performance optimization, detailed turbo-machine models are needed to investigate the effects of changing the reactor temperature, type and molecular weight of the CBC working ﬂuid, and the

* Corresponding author. Tel.: +1 505 277 5442; fax: +1 505 277 2814. E-mail address: [email protected] (M.S. El-Genk). 0196-8904/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2009.08.005

rotation speed of the shaft on the turbo-machine size and plant performance [7]. Thus, the objective of this work is to develop detailed design and performance models of gas turbo-machine for CBC in HTR nuclear power plants. The developed models build on the extensive knowledge gained in the design of open cycle systems and aircraft engines, except that those of interest in this paper are for helium gas or binary mixtures of He–Xe and He–N2 as CBC working ﬂuids in HTR plants. These models are based on a mean-line through-ﬂow analysis for free-vortex ﬂow [8–10]. They account for the proﬁle, secondary, end wall, trailing edge and tip clearance losses in the cascades [11,12], and calculate the geometrical parameters of the blade cascades. Empirical correlations for the various losses are developed based on the reported data in the literature, and used to select the values of the reaction, ﬂow, and loading coefﬁcients for optimizing the blades cascades and the performance of the compressor and turbine. The properties of the helium and binary mixtures of helium with other noble gases are incorporated into the present models as function of temperature up to 1200 K and pressure up to 20 MPa [13,14]. The predictions of the models for the multistage, axial-ﬂow turbine and compressor are validated using reported data for HTR power plant with helium working ﬂuid [15]. The developed models could be used to perform preliminary design optimization, and investigate the effects of design changes, the type of working ﬂuid and the shaft rotational speed on the polytropic efﬁciency, number of stages in and size of the axial-ﬂow turbine and compressor in HTR plants. Parametric analyses performed in this work investigated the effects of the mean-stage work coefﬁcient, reaction and ﬂow coefﬁcient on the design and performance of multi-stage helium turbine and compressor operating at 3600 rpm.

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

17

Nomenclature A Aroot Atip b C CL Deq Ft h H HTE i _ m M Ma N Nsp nst O P r rm R Re1C Re2C S S tmax tTE T U ~ V Vx Vh ~ W W max _ W Y Z ZTE

cross-sectional ﬂow area (m2) cross-sectional area of rotor blade root (m2) cross-sectional area of rotor blade tip (m2) maximum camber of blade (m) actual chord length of blade (m) blades lift coefﬁcient based on mean vector velocity, Eqs. (41) and (78) equivalent diffusion ratio, Wmax/W2 (>1) tangential loading parameter, Eq. (30) enthalpy per unit mass (J/kg) height of blades (m) boundary-layer shape factor, dTE =hTE , Eq. (64) incidence angle at blades leading edge (°) mass ﬂow rate of working ﬂuid through turbo-machine (kg/s) molecular weight (kg/mol) gas Mach number number of blades in cascade pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ^ rot j3=4 stage speciﬁc speed, x A2 V x =jDh number of rotor stages in turbo-machine throat width between blades in cascade (m) pressure (Pa) radius (m) average radius of blade (m), 0.5 (rhub + rtip) stage reaction Reynolds number at the blade leading edge, (q1W1C)/l1 Reynolds number at the blade trailing edge, (q2W2C)/l2 Pitch or distance between blades in cascade (m) entropy per unit mass (J/kg K) maximum blade thickness (m) thickness of blades trailing edge (m) temperature (K) rotor tangential velocity (m/s), rx gas absolute velocity vector (m/s) gas meridional velocity component (m/s) gas tangential velocity component (m/s) gas relative velocity vector with respect to rotor wheel ~ (m/s), ð~ V UÞ peak velocity on suction surface of blade (m/s) rate of mechanical work (W) pressure loss coefﬁcient of a blades cascade position of maximum camber, measured from blade leading edge (m) spanwise penetration depth between primary and secondary loss regions (m)

Greek symbols angle between ~ V and meridional plane (°) b blade angle relative to meridional plane (°) c ratio of speciﬁc heat capacities

a

2. Models description Fig. 1 illustrates the basic conﬁguration of an axial-ﬂow turbine with inlet, exit and inner stages. Each stage has a cascade of stationary blades, Inlet Guide Vanes (IGV) or Stator (S), which increases the swirl (tangential) velocity of the gas in the direction of rotation. The cascade of rotating blades, or Rotor (R), absorbs the gas swirl velocity and converts it into torque for the rotating shaft. An exit guide vanes cascade (EGV) following the last turbine stage removes any residual swirl velocity and converts the gas kinetic energy into an increase in exit static pressure (Fig. 1). For nearly constant axial ﬂow velocity, the annular ﬂow area increases from inlet to outlet to accommodate the decreases in the gas pres-

C d d* ^ rot Dh b DP

DU

g h h

j k K

l q r rB s u /

U

v x

blade circulation parameter (dimensionless) deviation angle at blade trailing edge (°) boundary layer displacement thickness (m) total enthalpy change per rotor stage (J/kg) total pressure loss (Pa) kinetic energy loss coefﬁcient turbo-machine polytropic efﬁciency blade camber (or turning) angle (°) boundary-layer momentum thickness (m) peak-to-valley surface roughness (m) ^rot j=U 2 stage loading (work) coefﬁcient, jDh stage boss ratio, rhub/rcas coolant dynamic viscosity (kg/m s) density (kg/m3) blade cascade solidity (C/S) maximum centrifugal stress of rotor blade (Pa) blades clearance gap (m) stage ﬂow coefﬁcient ðV x =UÞ ~ and meridional plane (°) angle between W blades stagger angle measured from axial direction (°) blade angle measured from the chord line (°) shaft angular speed (radians/s)

Subscript/superscript h tangential or ‘‘whirl” component AM loss model of Ainley and Mathieson [11] B metallic blade C compressor cas casing of turbo-machinery EW end wall losses ex exit gap gap losses hub hub of impeller in inlet LE leading edge of blades m median location of annular ﬂow passage p proﬁle losses R rotating frame of reference rot rotor s secondary losses sta stator T turbine TC tip clearance losses TE trailing edge of blades tip tip of impeller x, z axial component 0 inlet or leading edge of stator blades 1 inlet or leading edge of rotor blades 2 exit or trailing edge of rotor blades

sure and density (Fig. 1). The velocity triangles at the leading and trailing edges of a turbine rotor blades cascade (Fig. 2) are used in the aerodynamic design and analysis of turbo-machines. Fig. 3 illustrates the basic conﬁguration of an axial-ﬂow compressor with inlet, exit and inner stages. Each inner stage has a cascade of stationary blades, or Stator (S), which decreases the swirl (tangential) velocity of the gas in the direction of rotation. As in the turbine, the EGV removes the swirl velocity and converts the gas kinetic energy into an increase in exit static pressure (Fig. 3). The multi-stage compressor is also designed for nearly constant axial ﬂow velocity, thus, the annular ﬂow area decreases from inlet to outlet to accommodate the increases in the gas pressure and density (Fig. 3). Typical velocity triangles at the leading and trailing

18

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

First stage Inlet gas flow

IGV

Stage nst

2nd stage

R

S

R

S

R

S

Exit stage

R

EGV

Exit gas flow

Rotor ω

{0}

{1}

{2}

Turbine shroud / casing (stationary) Fig. 1. A schematic of a multi-stage axial-ﬂow turbine.

V1

α1

φ2

W1 φ1

α2

α1

V2

U1 x

α2

W2

V1

U2

x

x

(b) Trailing edge

V2

φ2

U1

U2

(a) Leading edge

φ1

W1

W2

x

(a) Leading edge

(b) Trailing edge

Fig. 2. Typical velocity triangles for the turbine rotor blades.

Fig. 4. Typical velocity triangles for the compressor rotor blades.

edges of a compressor rotor cascade are shown in Fig. 4a and b. The smaller gas ﬂow turning angle in the compressor limits the separation of the boundary layer due to the positive pressure gradient. The developed compressor and turbine models are based on a mean-line analysis for free-vortex ﬂow along the blades [8–10]. With constant axial ﬂow velocity, Vx and mean-line blade radius, rm, both the hub and tip radii vary from one stage to the next. The models assume constant mean ﬂow coefﬁcient, /m, and uniform loading (work) coefﬁcient, km and reaction, Rm in all stages, blades aspect ratio, H/C = 1.7 for the compressor stators, and 1.4 for all other blades (Figs. 5 and 6). In addition, the maximum thickness ratio, tmax/C = 0.2 and 0.1 for the turbine and compressor

blades. The trailing edge thickness ratio, tTE/S is taken 0.02 for all turbine cascades, except the EGVs [11,12], and 0.00046 for all compressor cascades, and the blades tip clearance for all blades s = 1 mm. The turbine blades are assumed shrouded, with 2 tip seals, the compressor blades are unshrouded, and for all compressor blades and turbine EGVs the relative position of the maximum camber, Z/C = 0.5. Semi-empirical relationships are used to determine the thermodynamic and transport properties of the working ﬂuid as functions of both temperature and pressure from 300 K to 1400 K and up to 20 MPa [13,14]. The mean blade tangential velocity, Um, the mean-line blade radius, rm and the axial ﬂow velocity, Vx are calculated by the

First stage Inlet gas flow

IGV

Stage nst

Second stage

R

S

R

S

R

S

Exit stage

R

EGV

Exit gas flow

Rotor ω

{0} {1} {2} Compressor shroud / casing (stationary) Fig. 3. A schematic of a multi-stage axial-ﬂow compressor.

19

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

Φ β1

φ1

i

tan /1 ¼

W1

km =2 Rm

um

tan a1 ¼ tan /1 þ

;

;

Vx Vx ; W1 ¼ cos a1 cos /1 km =2 þ Rm 1 ; tan a2 ¼ tan /2 ; tan /2 ¼

V1 ¼

um

O

tmax

Z

Vx V2 ¼ ; cos a2

ð2aÞ

um

Vx W2 ¼ : cos /2

ð2bÞ

The total pressure losses in the stator and rotor cascades are given as:

C

Cx

1

um

b sta ¼ Y sta ð P b 1 P1 Þ; DP

b rot ¼ Y rot ð P b R P2 Þ: and D P 2

ð3Þ

The thermodynamic properties at station {1} are then calculated as: S = C/σ φ2

δ

β2

bo DP b1 ¼ P b sta ; P

W2

Φ φ1

W1

i

^R ¼ h1 þ 0:5W 2 h 1 1 O tmax

Cx

Z

^1 0:5 V 2 : and h1 ¼ h 1

ð4Þ

With the total pressure and enthalpy at station {1} known, the total temperature and entropy are obtained using property subroutines, ^ 1 Þ and b ^ 1 Þ. Since S1 ¼ b b1; h b1; h b 1 ¼ Tð P S 1 ¼ Sð P S 1 by deﬁnition, i.e. T and h1 is known from Eq. (4), the static temperature and pressure at station {1} are calculated using the property subroutines: T1 = T(h1, S1) and P1 = P(h1, S1). All other thermodynamic properties are calculated as functions of T1 and P1. The total enthalpy and entropy in the rotating frame, at the rotor inlet {1} are calculated as:

Fig. 5. A turbine blades cascade.

β1

^1 ¼ h ^o h

ð5Þ

The total temperature and pressure in the rotating frame are R ^R ; b b R ¼ Tðh obtained using the property subroutines, T 1 1 S1Þ R R bR ^ b and P 1 ¼ Tðh1 ; S 1 Þ. The thermodynamic properties at the rotor exit station {2} are then given by [16]:

bR DP bR ¼ P b rot ; P 2 1

C

and b S R1 ¼ S1 :

^R ¼ h ^R h 2 1

bR; and S2 ¼ b S R2 ¼ S P 2

^R : h 2

ð6Þ

The static thermodynamic properties at the rotor exit station {2} are calculated as: S = C/σ

φ2 δ

W2

^rot þ 0:5 V 2 V 2 ; h2 ¼ h1 þ Dh 1 2

T 2 ¼ Tðh2 ; S2 Þ

and P2 ¼ Pðh2 ; S2 Þ: β2 Fig. 6. A compressor blades cascade.

ð7Þ

The annular ﬂow area at each station {i} is then obtained from the mass balance as:

_ qi V x Þ: Ai ¼ m=ð turbo-machine models. The absolute and relative gas ﬂow angles at the inlet and exit of the blade cascades in each stage (Figs. 2–6) are determined from the given loading, ﬂow coefﬁcient and the velocity triangles, and either the mean-line stage reaction Rm (IGVs and inner stages) or the exit ﬂow angle, a = 0° (EGVs). The annular ﬂow area, A, and the boss ratio of the stages, K, are determined from the ﬂow mass balance, and the total enthalpies are obtained from the energy balance. 2.1. Axial-ﬂow turbine model Since the change in the total enthalpy across each rotor is ^ rot ¼ W=ðn _ _ < 0, while that across each stator is nil, the axDh st mÞ ial ﬂow velocity and mean blades radius and speed are given by: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ^rot =km j; rm ¼ U m =x; and V x ¼ W x ¼ u U m : U m ¼ jDh ð1Þ m With all ﬂow conditions known at station {0}, the mean ﬂow angles and velocities at stations {1} and {2} (Figs. 1 and 2) are calculated as:

ð8Þ

The boss ratio and casing and hub radii are given by:

Ki ¼

1 Ai =ð4pr 2m Þ ; 1 þ Ai =ð4pr 2m Þ

r cas ¼

2r m ; 1 þ Ki

r tip ¼ r cas s

and r hub ¼ Ki r cas :

ð9Þ

The principal tensile stress at the root of the rotor blades, due to the centrifugal forces, assuming a linear taper, is given by [17]:

rB ¼

qB x2 A 1 Atip 1 1 1 1þ 2p 3 1 þ r hub =r tip Aroot

:

ð10Þ

The turbine model uses a typical blade taper ratio, Atip/Aroot = 0.40; the effect of this parameter, however, is benign. The radial variations of the reaction, work and ﬂow coefﬁcients, and Mach numbers along the blades are calculated for a free-vortex ﬂow [9,10], ensuring that none of the operation and design lim^ its are exceeded. For a free-vortex ﬂow, oh=or ¼ 0, oVx/or = 0, and rVh is constant at all stations. The local reaction and ﬂow and loading coefﬁcients are then given by:

20

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

r 2 m 1 RðrÞ ¼ ð1 Rm Þ ; r r 2 m kðrÞ ¼ km : r

uðrÞ ¼ um

r m ; r

K Re ¼ ð11Þ

The maximum ﬂow and loading coefﬁcients and minimum reaction all occur at the hub, and a positive reaction at the hub imposes a minimum value on the stage boss ratio of:

1 2 K Kmin ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 : 1 Rm

ð12Þ

The local velocities at the blades leading and trailing edges and the maximum Mach number are then calculated and the polytropic efﬁciency is given [18] as:

hin hex gT ¼ n : f n1 ðP in =qin Pex =qex Þ

ð13Þ

nS ¼

hin hex ; ðPin =qin P ex =qSex Þ

n¼

lnðPin =Pex Þ ; lnðqin =qex Þ

lnðPin =Pex Þ : lnðqin =qSex Þ

ð14Þ

The exit density and enthalpy for an isentropic expansion are calculated as: S ex

q ¼ qðPex ; Sin Þ and

S hex

for Re2C < 2 105 ;

;

for Re2C 2 105 :

ð20Þ

The proﬁle loss coefﬁcient is multiplied by a factor 0.914 to obtain Y 0p for zero trailing edge thickness [12]. The proﬁle loss coefﬁcient of the Ainley–Mathieson loss system, Y 0p;AM , was developed for vanes and blades having a trailing edge thickness to pitch ratio, tTE/ S = 0.02 [11], and included the trailing edge losses. The factors Kp and Yshock in Eq. (19) are identical to those introduced by Kacker and Okapuu [12] to account for the gas compressibility. The Mach number correction factor, Kp in Eq. (19) is calculated as:

K p ¼ 1 K 2 ð1 K 1 Þ;

K 1 ¼ 1; for Ma2 < 0:2;

K 1 ¼ 1 1:25 ðMa2 0:2Þ for Ma2 > 0:2;

and

ð21Þ

K 2 ¼ ðMa1 =Ma2 Þ :

S

nS nS 1

and K Re ¼ 1:0;

!0:575

2

The non-ideal correction factor f, and exponents n and nS are given by:

f ¼

2 105 Re2C

¼ hðPex ; Sin Þ:

ð15Þ

Mallen and Saville [19] give a simpler expression as:

hin hex gT ¼ : hin hex þ ðSex Sin Þ ðT in T ex Þ= lnðT in =T ex Þ

ð16Þ

2.2. Pressure loss coefﬁcient in turbine blades The turbine model developed in this work incorporates the latest reﬁnements proposed by Benner et al. [20,21] of Kacker and Okapuu’s model [12]. Reynolds and Mach numbers are based on the relative gas ﬂow velocities and the total pressure loss coefﬁcient is given as:

1:75 q1 W 21 rhub 3 hub ; for Mahub > 0:4 ð22aÞ Ma1 0:4 1 2 q2 W 2 rtip 4

Y shock ¼ 0;

for Mahub 0:4 1

ð22bÞ

Also, the incident Mach number, always higher at the hub radius than at the midspan radius, is related to the mean incident Mach number:

8 2 < 5:716 rhub 10:85 rhub þ 6:153; when Mahub r tip r tip 1 ¼ : 1:0; Ma1 when

r hub r tip

0:95

r hub r tip

> 0:95; ð23aÞ

(b) For a nozzle (stator):

8 2 < 4:072 rhub 6:644 rhub þ 3:705; when Mahub rtip r tip 1 ¼ : 1:0; Ma1 when

r hub r tip

0:8

r hub r tip

> 0:8: ð23bÞ

Y 0p;AM ,

The proﬁle loss coefﬁcient, by Ainley and Mathieson [11], an interpolation between the results of two sets of cascade tests (b1 = 0 and b1 = /2), is given by:

ð17Þ

Benner et al. proposed a loss scheme for the breakdown of the proﬁle and secondary losses as:

ðY p þ Y s Þ ð1 Z TE =HÞ Y 0p þ Y 0s :

Y shock ¼

(a) For a reaction stage (rotor):

For pressure ratios and inlet pressures up to 6.0 and 10 MPa, and binary mixtures of noble gases and of helium and nitrogen, the polytropic efﬁciencies predicted by Eqs. (13) and (16) are within 0.1% of each other.

Y ¼ ðY p þ Y s Þ þ Y TE þ Y TC :

The shock losses occur at a relatively low average inlet Mach number, due to the local ﬂow acceleration at the highly curved leading edges. These losses, appearing in Eq. (19), are calculated as [12]:

Y 0p;AM ¼

h

i t =C K m b1 =/2 b b max ðb1 ¼0Þ ðb1 ¼a2 Þ ðb1 ¼0Þ Y p;AM þ 1 1 Y p;AM Y p;AM : 0:2 /2 /2

ð18Þ

ð24Þ

The proﬁle loss coefﬁcient, based on recent turbine cascade data [22], is given by:

The exponent Km is given by Zhu and Sjolander [22] as: Km = +1, when tmax/C < 0.2, and Km = 1, when tmax/C > 0.2. The results of Ainley and Mathieson using a comprehensive testing program of cascades with b1 = 0 and tmax/C = 0.2 are correlated by the relation (Fig. 7):

h i Y 0p ¼ 0:914 K in Y 0p;AM K p þ Y shock K Re :

ð19Þ

The factor Kin = 2/3, used by Kacker and Okapuu [12], underpredicts the proﬁle losses for blade rows with axial inﬂow. Zhu and Sjolander [22] introduced a higher correction, Kin = 0.825 for IGVs. For reaction blades, Kin = 2/3 is still used. Zhu and Sjolander have also introduced a Reynolds number correction factor based on their recent blade cascade data, as:

ðb ¼0Þ

1 Y p;AM ¼AþBrþ

1

r

D : Cþ

r

ð25Þ

when /2 6 63:2 , the coefﬁcients A, B, C and D are correlated as function of the trailing edge (TE) relative gas ﬂow angle as:

21

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

The coefﬁcient A and the exponent n in Eq. (27) are given as:

Yp,AM for β1 = 0

0.10 φ2 = 80

0.08

o

(a) When r1 r1 min :

75

0.06

70

n ¼ 1:524 104 /22 0:031 /2 þ 2:992;

65

A ¼ 5:407 103 /2 0:19642;

when /2 > 60 ;

A ¼ 2:91 103 /2 þ 0:30260;

when /2 60 :

0.04

60

0.02

50 40

Ainley and Mathieson [11] Present correlation, Eq. (25)

0 0.3

0.5

1

(b) When r < r

0.7 0.9 S / C Ratio

A ¼ 5:58 10 B ¼ 1:553 10

5 3

C ¼ 8:54 10 D ¼ 4:83 10

5

/22

2

þ 1:03 10

3

/22 2:32 10 /2 þ 0:238;

ð26aÞ

ð26bÞ

C ¼ 2:23 104 /22 þ 4:96 102 /2 2:548; D ¼ 5:39 105 /22 1:57 102 /2 þ 0:958:

The results of Ainley and Mathieson for a cascade with b1 = /2 and tmax/C = 0.2 are also well correlated by (Fig. 8):

1 1 ¼ Y p;min þ A r r

min

n :

ð27Þ

In this correlation, the optimum solidity for minimum losses is given by:

r

when /2 > 60 ;

n ¼ 3:271 103 /22 0:3010 /2 þ 9:023;

when /2 < 60 ;

A ¼ 9:240 103 /22 1:2067 /2 þ 40:04;

when /2 > 60 ;

A ¼ 2:701 103 /22 0:2456 /2 þ 5:909;

when /2 60 :

when /2 > 60 ; 6

¼ 8:63 10

/32

ð28aÞ 4

þ 9:68 10

/22

These results, for cascades with a maximum blade thickness, (tmax/ C) = 0.2, are extended to different thicknesses using the last factor on the right-hand side of Eq. (24).

2

3:76 10

/2 þ 1:272;

when /2 60 :

The minimum value of the loss coefﬁcient is then given as:

Y p;min ¼ 0:280 1:0

1

r

ð28bÞ

:

2.2.1. Spanwise penetration depth and secondary loss coefﬁcient The spanwise penetration depth (ZTE) of the separation line between the primary and the secondary loss regions, appearing in Eq. (18), is given [20] by:

2 Z TE 0:10 jF t j0:79 d þ 32:7 : ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ H H cos /1 = cos /2 ðH=CÞ0:55

In this expression, the tangential loading parameter, Ft is given by:

Ft ¼ 2

S cos2 ð/m Þ ½tanð/1 Þ þ tanð/2 Þ : C cos U

min

tanð/m Þ ¼

1 ½tanð/1 Þ tanð/2 Þ : 2

ð31Þ

d ¼

d 0:0463x : ¼ 8 ðq1 W 1 x=l1 Þ0:2

ð32Þ

0:038 þ 0:41 tanhð1:2d =HÞ Y 0s ¼ F AR pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : cos U ðcos /1 = cos /2 Þ ðC cos /2 =C x Þ0:55

ð33Þ

The aspect ratio factor FAR is a function of the blade aspect ratio as:

Hainley & Mathieson [11] Present model, 75 Eq. (27)

φ2 = 70

when H=C > 2:0

ð34aÞ ð34bÞ

o

60 55 50 40

0.7 0.9 S / C Ratio

when H=C < 2:0

F AR ¼ 1:36604 ðC=HÞ;

65

0.5

ð30Þ

The mean velocity vector angle is given by:

F AR ¼ ðC=HÞ0:55 ; 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.3

ð29Þ

The reference length, x, in Eq. (32) is taken as half the blade axial chord. The secondary loss coefﬁcient in Eq. (18) is given by [21]:

min

Yp,AM for β1 = φ2

In Eq. (29), the boundary layer displacement thickness at the inlet endwall, d*, is [23]:

¼ 5:14 104 /22 þ 5:48 102 /2 0:798;

min

1

:

/22 þ 2:83 104 /2 2:93 102 :

B ¼ 4:02 105 /22 þ 6:94 103 /2 0:282;

r

/22

ð28dÞ

/2 þ 8:02 102 ;

A ¼ 2:44 104 /22 4:33 102 /2 þ 1:92;

1

min

/2 0:275;

When /2 > 63.2°, these coefﬁcients are correlated as:

ðb1 ¼a2 Þ Y p;AM

2

1:5731 /2 þ 54:85;

n ¼ 1:174 10

1.1

Fig. 7. Turbine blades proﬁle loss coefﬁcient for b1 = 0 and tmax/C = 0.2 [11].

5

1

ð28cÞ

1.1

Fig. 8. Turbine blades proﬁle loss coefﬁcient for b1 = /2 and tmax/C = 0.2 [11].

2.2.2. Trailing edge losses The trailing edge (TE) kinetic energy losses are expressed in terms of the ratio of trailing edge thickness to the throat opening of the cascade. Kacker and Okapuu [12] have expressed these losses in terms of the kinetic energy loss coefﬁcient, DUTE, for axial entry nozzles (b1 = 0) and impulse blades (b1 = /2). The difference lies in the thicknesses of the proﬁle boundary layers at the trailing edge of the blades. The impulse blades, with their thick boundary layers, have lower trailing edge losses. For blades other than the two types above, the coefﬁcient for the trailing edge kinetic energy loss is interpolated as:

22

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

DUTE

h i b b ðb ¼0Þ ðb ¼a Þ ðb ¼0Þ : ¼ DUTE1 þ 1 1 DUTE1 2 DUTE1 /2 /2

y

ð35Þ

θ

For an axial entry nozzle: ðb1 ¼0Þ DUTE ¼ 0:59563

tTE O

2

t TE 2:2796 103 : þ 0:12264 O ð36aÞ

b

For an impulse blading:

χ1

2 t TE t TE ðb1 ¼a2 Þ DUTE ¼ 0:31066 þ 0:065617 1:4318 103 : O O ð36bÞ

ð37Þ

2.2.3. Tip clearance losses The tip clearance (leakage) loss coefﬁcient, YTC, in the turbine blades cascade is calculated using the approach of Yaras and Sjolander [24] as:

Y TC ¼ Y tip þ Y gap ;

ð38Þ

s

cos2 /2 C L1:5 ; Y tip ¼ 1:4K E r H cos3 /m pﬃﬃﬃﬃﬃ CL C : Y gap ¼ 0:0049K G r H cos /m

and

ð39Þ ð40Þ

The theoretical blade lift coefﬁcient, CL, is given by [11]:

CL ¼

2

r

cosð/m Þ ½tanð/1 Þ þ tanð/2 Þ :

ð41Þ

seff cos2 /2 0:37 ¼ C 1:5 1:4K E r L : 0:47 H cos3 /m

ð42Þ

2.3. Turbine cascade geometry The turbine blade proﬁle follows the parabolic-arc camberline. However, the EGVs are designed using the compressor blades model. Knowledge of (Z/C), the maximum camber (b/C), and the blade angles measured from the chord line (v1 and v2) provides four relations that uniquely deﬁne the parabolic-arc proﬁle (Figs. 5 and 9). The blade camber (or turning) angle is expressed as, h = v1 + v2 = b1 + b2, and the maximum camber is given by [8]:

b ¼ C

ﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h 2 3 i 1 þ ð4 tan hÞ2 CZ CZ 16 1 4 tan h

:

ð43Þ

The blade angles, with respect to the chord line, are given by:

tanðv1 Þ ¼

b=C ; Z=C 1=4

and tanðv2 Þ ¼

b=C : 3=4 Z=C

ð44Þ

The blade stagger angle with respect to the axial direction (Figs. 5 and 9) is expressed as:

or U ¼ ðb2 v2 Þ:

ð45Þ

The number of blades in each row is determined using an empirical model for the optimum solidity, based on matching the Ainley– Mathieson [11] minimum proﬁle loss coefﬁcients in Figs. 7 and 8. The optimum solidity for the axial-ﬂow entry nozzles (b1 = 0°) is given by [9]:

ðb1 ¼0Þ 1

r

¼ 0:427 þ

opt

2 90 /2 90 /2 : 58 93

ð46Þ

Similarly, for impulse blading (b1 = /2):

ðb1 ¼/2 Þ 1

r

opt

/ / ¼ 0:224 þ 0:575 þ 2 1 2 : 90 90

ð47Þ

The optimum solidity for the turbine’s actual blade row, interpolated in the same manner used for the proﬁle loss coefﬁcient by Ainley and Mathieson (Eq. (24)), is given by:

1

For mid-loaded turbine blades, KE = 0.5 and KG = 1.0, and for frontor aft-loaded blades, KE = 0.566 and KG = 0.943 [24]. Eq. (39) is for unshrouded blades cascade. For shrouded blades [25], the same expression developed for unshrouded blades could be used by replacing the tip clearance with an effective clearance value: seff = s/(n)0.42, and reducing the losses by 21.3%. Thus, the expression used for the tip leakage losses in a shrouded blades cascade is:

Y tip

U ¼ ðv1 b1 Þ;

ropt

¼

ðb1 ¼0Þ 1

r

opt

" ðb1 ¼/2 Þ ðb1 ¼0Þ # b b 1 1 : þ 1 1 / / r r 2

2

opt

ð48Þ

opt

The turbine blades are designed for zero incidence angle (i = 0), and the deviation angle at the trailing edge is calculated using a recent correlation by Zhu and Sjolander [22] as:

d ¼ 17:3

ð1=rÞ0:05 ð/1 þ b2 Þ0:63 cos2 ðUÞ ðt max =CÞ0:29 ð30 þ 0:01b2:07 1 Þ tanhðRe2C =200; 000Þ

:

ð49Þ

The blade angles are calculated as: b1 = /1 i and b2 = /2 d, and the blade camber angle is given by: h = b1 + b2. Because the turning angle of the turbine blades is relatively large, the turbine blade parameters are very sensitive to the value of the maximum camber position, (Z/C), which might not be known accurately. The present turbine model uses the values recommended by Kacker and Okapuu [12] and correlated in this work (Fig. 10) as:

o

Y TE

n oc=ðc1Þ 1 c1 Ma22 1D1UTE 1 1 2 ¼ : c=ðc1Þ 2 Ma 1 1 þ c1 2 2

x

C

Fig. 9. Blade proﬁle with a parabolic-arc camberline.

Blade Stagger Angle, Φ ( )

The kinetic energy loss coefﬁcient DUTE is converted to a pressure loss coefﬁcient using the following relationship [10]:

χ2

Z

80 β 2 = 80

o

60 75

40

70 65

20

60 Kacker & Okapuu [12] Present model, Eq. (50)

55 50

0 -30 -20 -10 0 10 20 30 40 50 60 o Inlet Blade Angle, β1 ( )

Fig. 10. Stagger angle for the turbine blade cascades [12].

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

U ¼ A þ B b1 þ C b21 :

ð50Þ

The coefﬁcients A, B and C are functions of b2, and are given by:

A ¼ 0:710 þ 0:2880 b2 þ 6:93 103 b22 ;

The polytropic efﬁciency is calculated using the relationship [18] for non-ideal gases, as:

gC ¼

B ¼ 0:489 0:0407 b2 þ 4:27 104 b22 ;

f

n n1

ð51Þ

ðPex =qex Pin =qin Þ : hex hin

C ¼ 3:65 103 þ 2:66 105 b2 þ 8:33 108 b22 :

The correction factor f, and exponents n and nS are given by:

Eq. (50) is used to determine the blade angles with respect to the chord line as:v1 ¼ U þ b1 and v2 ¼ b2 U. The location of maximum camber is calculated by simultaneously solving Eqs. (44a) and (44b). Finally, the number of blades in the cascade is obtained as:

f ¼

N ¼ INT

2pr m r H : C H

ð52Þ

The blades pitch and chord are calculated as: S ¼ ð2prm Þ=N and C ¼ rS: Iterations are performed until a convergence of the deviation angle is achieved.

8 !1:18 9 1

ð53Þ

b m ¼ 0:5 þ jRm 0:5j; and Km = 0.80 gives the ﬂow coefﬁcient for a R mean stall margin of 20%. With all ﬂow conditions known at station {0} (the inlet of the stator blades cascade of the compressor stage), the mean ﬂow angles and velocities at stations {1} and {2} are calculated as:

tan /1 ¼

Rm þ km =2

um

;

tan a1 ¼

1

um

tan /1 ;

Vx Vx ; W1 ¼ V1 ¼ cos a1 cos /1 Rm km =2 1 ; tan a2 ¼ tan /2 ; tan /2 ¼

um

Vx V2 ¼ ; cos a2

ð54aÞ

um

Vx W2 ¼ : cos /2

ð54bÞ

The total pressure losses in the stator and rotor cascades of the compressor are given as:

b sta ¼ Y sta ð P b o Po Þ; DP

b rot ¼ Y rot ð P b R P 1 Þ: and D P 1

ð55Þ

The thermodynamic properties at the leading and trailing edges of the rotor blades are calculated using Eqs. (4)–(7), and the annular ﬂow areas, boss ratios and radii at each station {i} are calculated using Eqs. (8) and (9). The radial variations of the reaction, work and ﬂow coefﬁcients, Mach numbers and the de Haller ratios along the blades are calculated for a free-vortex ﬂow [8,9], to ensure that operation and design limits are not exceeded. The maximum ﬂow and loading coefﬁcients and the minimum reaction all occur at the hub. A positive reaction at the hub, and loading and ﬂow coefﬁcients <1.0 for surge stability, requires that:

!1

K Kmin ¼

2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ

1 MAX 1 Rm ; km ; um

:

ð56Þ

ð57Þ

S

n¼

nS nS 1

hex hin ; ðPex =qSex Pin =qin Þ

lnðPex =Pin Þ ; lnðqex =qin Þ

nS ¼

lnðPex =Pin Þ : lnðqSex =qin Þ

ð58Þ

The exit density and enthalpy for an isentropic compression are calculated as:

qSex ¼ qðPex ; Sin Þ; and hSex ¼ hðPex ; Sin Þ:

ð59Þ

Mallen and Saville [19] give the following simpler expression for the polytropic efﬁciency:

2.4. Axial-ﬂow compressor design model Aungier [8] recommended using a de Haller limit of 0.70 to represent the stall condition for compressor blades cascade with tmax/ C = 0.10, and developed an empirical relationship for this limit, which is used to obtain the mean-line ﬂow coefﬁcient as a function of the loading coefﬁcient, the reaction and the stall margin ð1 K m Þ, as:

23

gC ¼

hex hin ðSex Sin Þ ðT ex T in Þ= lnðT ex =T in Þ : hex hin

ð60Þ

For pressure ratios up to 6.0, exit pressures up to 10 MPa, and binary mixtures of noble gases and of helium and nitrogen, Eqs. (57) and (60) are within 0.1% of each other. 2.5. Pressure loss coefﬁcient in compressor blades In this Section, Reynolds and Mach numbers are based on the relative gas ﬂow velocities. The total pressure loss coefﬁcient in the compressor cascade is given as:

Y ¼ Y p þ Y s þ Y EW þ Y TC :

ð61Þ

Lieblein [26] expressed the blade-proﬁle pressure loss coefﬁcient as:

2 h2 r cos /1 2HTE Yp ¼ 2 C cos /2 cos /2 3HTE 1 3 h2 rHTE 1 : C cos /2

ð62Þ

The boundary-layer momentum thickness at the blade outlet, h2, is given [27] as:

o h2 h ¼ 2 fM fH fRe : C C

ð63Þ

The boundary layer trailing-edge shape factor, HTE, the ratio of the boundary layer displacement thickness to the momentum thickness, h2, is expressed as:

HTE ¼ HoTE nM nH nRe :

ð64Þ

The values of ho2 and HoTE are for inlet Mach numbers, Ma1 < 0.05, no contraction in the height of the ﬂow annulus, H, an inlet Reynolds number, Re1C = 106 and hydraulically smooth blades. Based on the experimental data of Koch and Smith [27] at these conditions, the boundary-layer momentum thickness at the blade outlet is correlated accurately as:

ho2 6:713 103 ¼ 2:644 103 Deq 1:519 104 þ : C 2:60 Deq

ð65Þ

The shape factor for the boundary layer trailing-edge is correlated as:

24

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

HoTE ¼

dTE ho2

The cascade throat area is assumed to occur at one-third of the axial chord, thus:

¼ ð0:91 þ 0:35 Deq Þ n

4

6 o 1 þ 0:48 Deq 1 þ 0:21 Deq 1 :

ð66Þ

1 Athroat ¼ A1 ðA1 A2 Þ: 3

ð75bÞ

The gas density at the throat is calculated as: A value of HoTE ¼ 2:7209 is used when Deq > 2.0. For conditions other than nominal, Koch and Smith developed charts for determining the correction factors fM ; fH ; fRe in Eq. (63) and nM ; nH ; nRe in Eq. (64). The correction factor for inlet Mach number is correlated as:

fM ¼ 1:0 þ ð0:11757 0:16983 Deq Þ Man1 ;

ð67Þ

n ¼ 2:853 þ Deq ð0:97747 þ 0:19477 Deq Þ:

ð68Þ

The correction factor for the ﬂow area contraction is given by:

fH ¼ 0:53

H1 þ 0:47: H2

ð69Þ

The chart presented by Koch and Smith for the Reynolds correction factor is well approximated using the approach proposed by Aungier [9]. He introduced the critical blade chord Reynolds number, Recr = 100 C/j, above which the effect of roughness become signiﬁcant. When Re1C < Recr, the Reynolds correction factor is expressed as:

fRe ¼

8 0:166 106 > < Re ; 1C

for Re1C 2 10 ; ð70aÞ

fRe ¼

0:5 > : 1:30626 2105 ; for Recr < 2 105 : Recr

ð70bÞ

nM ¼ 1:0 þ 1:0725 þ Deq 0:8671 þ 0:18043 Deq Ma1:8 1 : ð71Þ The correction factor for the ﬂow area contraction is calculated as:

H1 nH ¼ 1:0 þ 1:0 0:0026 D8eq 0:024 : H2

ð72Þ

The correction factor for the inlet Reynolds number is given by:

¼

!0:06 106 ; when Re1C < Recr ; Re1C !0:06 106 ; when Re1C Recr : Recr

W1 t max 1 þ K3 þ K 4 C W2 C sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 cos /1 ðsin /1 K 1 rC Þ2 þ : Athroat qthroat =q1

ð73Þ

ð74Þ

In Eq. (74), the contraction ratio is given by:

t max Athroat Athroat ¼ 1:0 K 2 r : cosð0:5ð/1 þ /1 ÞÞ C A1

ð75cÞ

r 1m V h1 r2m V h2

rW 1 ðr1m þ r2m Þ=2

¼ ðtan /1 tan /2 Þ

cos /1

r

:

ð76Þ

In the present compressor model, this parameter reduces to the simpler expression on the right-hand side, since r1m = r2m = rm, which leads to U1m = U2m = Um. The secondary ﬂow loss coefﬁcient is given by the correlation proposed by Howell [28] as:

Y s ¼ 0:018 r

cos2 /1 C 2L : cos3 /m

ð77Þ

The theoretical compressor blade lift coefﬁcient, CL, is expressed as:

2

r

cosð/m Þ ½tanð/1 Þ tanð/2 Þ :

ð78Þ

The mean velocity vector angle is given by:

ð79Þ

Based on a modiﬁed Howell’s model [28], Aungier [8] developed the following expression for calculating the end wall loss coefﬁcient, as:

Y EW ¼ 0:0146

2 C cos /1 : H cos /2

ð80Þ

The tip clearance (leakage) loss factor, YTC is calculated [24] as:

Y TC ¼ Y tip þ Y gap ;

ð81Þ

s

cos2 /1 C 1:5 and Y tip ¼ 1:4K E r L ; H cos3 /m C pﬃﬃﬃﬃﬃ C L = cos /m : Y gap ¼ 0:0049K G r H

ð82Þ ð83Þ

For mid-loaded compressor blades (Z/C = 0.5), KE = 0.5 and KG = 1.0. 2.6. Compressor cascade geometry The compressor blade proﬁle is also a parabolic-arc camberline [8] (Fig. 9). The blade stagger angle, with respect to the axial direction (Figs. 6 and 9), is expressed as:

and

The equivalent diffusion ratio, Deq is given by [27]:

Deq ¼

tanð/m Þ ¼ 0:5½tanð/1 Þ þ tanð/2 Þ :

for Recr 2 105 ;

Typical ratios of the blade chord to surface roughness are: C/j = 10,000 to 20,000. When Re1C > 106 and C/j = 104, Recr = 106, and fRe = 1. The correction factor for the inlet Mach number is accurately ﬁtted by:

nRe ¼

C ¼

CL ¼

When Re1C > Recr, the friction losses are controlled by the surface roughness and the Reynolds correction factor may be expressed as:

The obtained constants in these equations from the experimental data of Koch and Smith are: K1 = 0.2445, K2 = 0.4458, K3 = 0.7688 and K4 = 0.6024. The dimensionless blade circulation parameter in Eqs. (74) and (75c) is given by:

5

0:5 > : 1:30626 2105 ; for Re1C < 2 105 : Re1C

8 0:166 106 > < Re ; cr

2

Max1 qthroat tan /1 ¼1 1 Athroat K 1 rC : q1 cos /1 1 Ma2x1

ð75aÞ

U ¼ ðb1 v1 Þ;

or U ¼ ðb2 þ v2 Þ:

ð84Þ

The location of the maximum camber is restricted to 0.25 < Z/ C < 0.75, and the blade camber angle for a compressor cascade is restricted to: h = b1 b2 < 90°. For the IGVs, b1 < b2 and the same proﬁle and camberline are used, except for reversing the ﬂow direction. In this case, Eqs. (43), (44), and (84) can be used for these blades, with the angles h, v1 and v2, and the parameter (b/C) being negative. For example, for axial-compressor EGVs parameters: b1 = 40°, b2 = 0°, Z/C = 0.40, r = 1.0 and tmax/C = 0.20, Eqs. (43), (44), and (84) give h = 40°, b/C = 0.07792, v1 = 27.45°, v2 = 12.55°, and U = 12.55°. This blade proﬁle can be inverted for IGVs as follows: b1 = 0°, b2 = 40°, Z/C = 0.60, r = 1.0 and tmax/C = 0.20, and Eqs. (43), (44), and (84) give h = 40°, b/C = 0.07792, v1 = 12.55°, v2 = 27.45°, and U = 12.55°.

25

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

0.10

o

mum incidence and deviation angles for the selected design point are used to calculate the blades LE and TE angles, b1 and b2. The optimum incidence design angle, i, is calculated [8] as:

o

φ1 = 52.9 , φ2 = 36.5 , (tmax/ C) = 0.10

(Yp + Ys)

0.08

"

0.06

i ¼ U b1 þ 3:6K t þ 0:3532h

0:25 # Z ðrÞ0:650:002h : C

ð87Þ

0.04 The blade thickness correction factor is given by:

0.02 (Yp + Ys)min

σopt = 1.01

= 0.027

0

0

0.5

1.0 1.5 2.0 Blade solidity, σ

2.5

0:28 t max 0:1 þ ðtmax =CÞ0:3 K t ¼ 10 : C

3.0

Fig. 11. Effect of solidity on the pressure loss coefﬁcient for the compressor blades.

Optimum Solidity, (σ)opt

The deviation angle at the trailing edge of the compressor blades is calculated using a modiﬁed correlation by Howell [28], as:

d ¼ ðK sh K 0t 1Þ do þ

2.5 55

o

o

50

o

45

2.0

o

40

o

30

o

25

1.5

o

20

o

15

1.0

0 0

K 0t ¼ 6:25 o

φ1- φ2 = 10

0.5 tmax / C = 0.10

10

20

40

50

2 t max t max þ 37:5 : C C

1:67þ1:09r /1 do ¼ 0:01r/1 þ 0:74r1:9 þ 3r : 90

60

o

Exit Flow Angle, φ2 ( ) Fig. 12. Optimum solidity of compressor blades cascade with tmax/C = 0.10.

The present compressor model assumes a solidity, r = 1.10 for the IGVs, and calculates the optimum solidity for the other cascades using a correlation developed in this work for minimizing the sum of the proﬁle and the secondary ﬂow loss coefﬁcients (Yp + Ys). The model assumes Athroat/A1 = 1, qthroat/q1 = 1, fM = fH = fRe = 1 and nM = nH = nRe = 1. Thus, (Yp + Ys) becomes a function of /1, /2, r and (tmax/C). In the compressor model, tmax/C = 0.10, and for each pair of ﬂow angle values {/1, /2} the model identiﬁes the optimum solidity for minimizing (Yp + Ys) (Fig. 11). The developed correlation, shown in Fig. 12, gives:

ðrÞopt ¼ A þ B ð/2 Þn ;

ð89Þ

ð90Þ

The base zero-camber deviation angle, do is calculated [8] as:

Eq. (85)

30

0:92ðZ=CÞ2 0:002b2 h pﬃﬃﬃﬃ pﬃﬃﬃﬃ : 1 0:002h= r r

A negative sign of the second term in the numerator is used for the IGVs and a positive sign is used for all other compressor blades. The blade thickness correction factor is expressed as:

o

35

ð88Þ

ð85Þ

where,

A ¼ 0:0197 þ 0:042231 ð/1 /2 Þ; B ¼ expf13:427 þ ð/1 /2 Þ ð0:33303 0:002368 ð/1 /2 ÞÞg; n ¼ 2:8592 0:04677 ð/1 /2 Þ: ð86Þ Eq. (85) is valid for 0 6 /2 6 55°, 10° 6 /1 6 65°, and 10° 6 /1 /2 6 60°. For the GTHTR300 compressor stage design for an HTR power plant with direct CBC [15]: km = 0.31, Rm ¼ 0:55, and the ﬂow coefﬁcient, /m = 0.5342 is determined (Eq. (53)) for a 20% stall margin. The relative ﬂow angles on the rotor are /1 = 52.9° and /2 = 36.5°, and the turning angle (/1 /2) = 16.4°. For these conditions, Eq. (85) gives an optimum cascade solidity of 1.01 (Fig. 12), in agreement with the recommended value of 1.0 in the literature [17]. The minimum in Fig. 11 is relatively shallow, so that a range of solidity values could give a minimum pressure loss coefﬁcient. A low solidity reduces the number of blades in the cascade and decreases the proﬁle losses. The number and aspect ratio (H/C) of the blades are ultimately determined using mechanical and vibration analyses, which are beyond the scope of this paper. The opti-

ð91Þ

To resolve the coupling between the pressure loss coefﬁcients, incidence and deviation angles, and the geometrical parameters, iterative procedures are used, assuming initially very small incidence and deviation angles, to calculate the blade camber angle as: h = /1 /2 + d i. Eq. (87) is used to calculate the blades stagger angle, U; (b/C), v1 and v2 are calculated using Eqs. (43) and (44); and the blade angles: b1 = U + v1, b2 = U v2 are determined by Eq. (84). The updated incidence angle, i = (/1 b1), is then used to determine the deviation angle using Eq. (89), and the number of blades in the cascade is given by Eq. (52). The blades pitch and chord are also calculated. Iterations are continued until a convergence in the values of the incidence and deviation angles is achieved. 3. Turbine design and model validation The turbine stages are designed for a speciﬁc speed between 0.5 and 1.0 [29] and a mean-radius reaction of 50% [9]. The present turbine model uses a blade tip clearance of 1 mm, a maximum mean blades speed of 400 m/s, a minimum blade height of 10 cm to reduce the tip clearance losses, and a maximum boss ratio of 0.91 in all stages. In order to maintain an acceptable throat area, the selected ﬂow coefﬁcient ensures ﬂow and blade angles 670°. The centrifugal stress in the Ni-based super-alloy blades of the turbine’s ﬁrst-rotor stage is kept 6126 MPa. In HTR power plants, this turbine stage operates at 61123 K. The present turbine model determines the smallest number of turbine stages to satisfy all design requirements. Results are compared in Table 1 to those reported for the helium turbine in the GTHTR300 HTR power plant [15]. A work coefﬁcient of 1.407 is used to match the reported mean blades speed, and a ﬂow coefﬁcient of 0.433 is used to match the reported inlet tip radius. The rotor blades for the GTHTR300 helium turbine have a ﬁnned shroud with 2 radial tip seals. For a shaft rotational speed of 3600 rpm, the determined number of stages in the turbine is 6, and the turbine polytropic efﬁciency is 92.3%, compared to the reported values of 6 stages and 92.8%. The calculated geometrical blade parameters are in excellent agreement with the reported values in Table 1.

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

peak turbine efﬁciency decreases as km increases. When km = 1.2, the peak turbine polytropic efﬁciency is 93.4%, compared to 92.9% and 92.3% when km = 1.4 and 1.6, respectively. For work coef-

0.4 0.5 0.6 Stage Mean Reaction, ℜm

0.82

=

8 1.

m

He turbine (nst = 6, ℜm = 0.5)

0.4 0.5 0.6 0.7 0.8 0.9 Mean Flow Coefficient, ϕm

140

φ = 70 o 2

1.0

Λmax = 0.91

λm = 1.8

1.6 1.4 1.2 1.0

120 100 80 60

First stage rotor He turbine (nst = 6, ℜm = 0.5)

40 0.3

0.4 0.5 0.6 0.7 0.8 0.9 Mean Flow Coefficient, ϕm

1.0

Fig. 15. Calculated centrifugal stresses in ﬁrst-rotor blades of 530 MW helium turbine.

94

He turbine (nst = 6, ℜm= 0.5)

1. 0

93 92

1.

2 1.4

1.

90 0.3

6 .8 =1

91

λm

Peak efficiency Λmax = 0.91 Present design

0.4 0.5 0.6 0.7 0.8 0.9 Mean Flow Coefficient, ϕm

1.0

Fig. 16. Calculated polytropic efﬁciency of 530 MW helium turbine.

He turbine (nst = 6, ℜm= 0.5) λ

1.88 1.87 1.86

m

=7 o 0

Polytopic Efficiency, ηT (%)

Fig. 13. Effect of mean stage reaction on the polytropic efﬁciency of helium turbine.

6 1.

Fig. 14. Maximum stage boss ratio for 530 MW helium turbine.

Turbine Pressure Ratio 0.7

4 1.

λ

o

He turbine λm = 1.407, φm = 0.434, nst = 6

2

70

Present design

91.9 0.3

0.84

1.89

92.1 92.0

0.86

0.80 0.3

92.3 92.2

1.

0.88

φ2=

The effects of the mean-radius stage work (loading) coefﬁcient, km, ﬂow coefﬁcient, um and reaction, Rm , on the design and performance of a helium turbine for HTR plants are investigated. The helium turbine delivers 530 MW of shaft mechanical power and operates at a shaft rotational speed of 3600 rpm. Helium from the HTR enters the turbine at 1123 K and 6.88 MPa, and ﬂow rate of 441.8 kg/s (Table 1). The turbine has 6 rotor stages followed by exit guide vanes, the stages have a speciﬁc speed between 0.5 and 1.0 [29] for high efﬁciency, and are designed for a stage work coefﬁcient, km = 1.407, and a ﬂow coefﬁcient, /m = 0.434. The turbine polytropic efﬁciency is maximum at a stage reaction of 50% (Fig. 13), which is used for all turbine stages. As shown in Fig. 14, the maximum stage boss ratio increases as the ﬂow coefﬁcient increases, but decreases as the work coefﬁcient increases. For a given stage work coefﬁcient, the boss ratio and number of blades in the cascades increase and the height of the annular ﬂow passages decreases as um increases. The maximum boss ratio of 0.91 in Figs. 14–17, represents a blade height of 9 cm and 120 blades in the ﬁrst stage cascade. The principal tensile stress at the root of the ﬁrst-rotor blades (Eq. (10)), which is proportional to the annular ﬂow area, or inversely proportional to the gas axial velocity: pﬃﬃﬃﬃﬃﬃ rB / ðum U m Þ1 / km =um , decreases rapidly with increasing the ﬂow coefﬁcient but increases with the square root of km (Fig. 15). For a given km, the turbine polytropic efﬁciency increases as the ﬂow coefﬁcient, um increases, peaks, then decreases with further increase in the ﬂow coefﬁcient (Fig. 16). The locus line indicated in Fig. 16 with solid, upward triangular symbols shows that the

0

1.2

1.85 1.0

1.84 0.3

=1 .8

1.6 1.4

2

3.1. Parametric analyses

1.

φ

6 377.0 1.00 0.9819 0.855 1.078 82 15.07 82 15.53 1.88 1.861 92.3

0.91

0.90

o

6 377 1.00 0.98 0.855 1.078 82 15.0 80 15.6 1.68 1.87 92.8

0.92

0

Number of stages Blades speed at mean radius (m/s) Blades mean radius (m) First-stage ﬂow area (m2) First stage boss ratio First-stage blade tip radius (m) No of stator blades in ﬁrst stage First-stage stator blade height (cm) Number of rotor blades in ﬁrst stage First-stage rotor blade height (cm) Length of turbine stages (m) Static pressure ratio Turbine polytropic efﬁciency (%)

0.94

=7

Reported [15]

441.8 1123 6.88 530 3600 Predicted

φ2

Mass ﬂow rate (kg/s) Inlet temperature (K) Inlet pressure (MPa) Mechanical power (MW) Shaft speed (rpm) Performance parameters

Stage Boss Ratio, Λmax

Input parameters

Centrifugal Stress, σB (MPa)

Table 1 Design parameters of GTHTR300 helium turbine.

Polytropic Efficiency, ηT (%)

26

Peak efficiency Λmax = 0.91 Present design

0.4 0.5 0.6 0.7 0.8 0.9 Mean Flow Coefficient, ϕm

1.0

Fig. 17. Pressure ratio of the 530 MW helium turbine.

27

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

230

1.0

220

0.9

210 1.0

0.8 1.2 1.4 1.6 1.8 Stage Mean Work Coefficient, λm

Fig. 18. Axial ﬂow velocity and maximum shroud radius of 530 MW helium turbine at peak polytropic efﬁciency.

ﬁcients of 1.2, 1.4 and 1.6, the turbine peak efﬁciency occurs at stage ﬂow coefﬁcients of 0.553, 0.639 and 0.707, respectively (Fig. 16). For a helium mass ﬂow rate of 441.8 kg/s, total inlet temperature of 1123 K and turbine mechanical power of 530 MW, the total exit temperature of the turbine is constant = 893.7 K, but the exit pressure depends on the pressure losses in the turbine stages. For design work coefﬁcients of 1.0–1.8, the turbine pressure ratio varies from 1.846 to 1.88 (Fig. 17), for polytropic efﬁciencies of 90.8–93.6% (Fig. 16). At mean-stage work and ﬂow coefﬁcients of 1.4 and 0.6, the turbine’s pressure ratio is 1.856 and the polytropic efﬁciency of 92.8% is near optimum. pﬃﬃﬃﬃﬃﬃ The mean blades radius, rm ¼ U m =x / 1= km , decreases but the pﬃﬃﬃﬃﬃﬃ axial ﬂow velocity through the turbine, V x ¼ um U m / um = km , increases slowly as the work coefﬁcient increases (Fig. 18). The latter is due to the rapid increase in the mean ﬂow coefﬁcient with increasing km along the locus of peak turbine polytropic efﬁciency (Fig. 16). As a result, the annular ﬂow area through the turbine decreases as km increases. The combination of lower mean blades radius and lower annular ﬂow area with increasing km decreases the turbine shroud radius (Fig. 18) and the volume of the stages, which is true when the turbine is designed at the peak polytropic efﬁciency. 4. Compressor design and model validation The results of the present model are compared in Table 2 with those reported for an HTR plant helium compressor [15]. The comTable 2 Design parameters of GTHTR300 helium compressor.

4.1. Parametric analyses The effects of km and design surge margin on the performance of a helium compressor are investigated. The compressor has 20 rotor stages, consumes 251 MW at a shaft rotational speed of 3600 rpm (Table 2). The helium working ﬂuid enters the compressor at 301 K, 3.52 MPa, and ﬂow rate of 449.7 kg/s. For a mean stage reaction of 55%, the mean ﬂow coefﬁcient (Eq. (53)) is calculated for mean stall margins of 10%, 20% and 30% (Fig. 19). For a given stall margin, the gas velocities W1 and V2 in the stages are nearly constant, but um increases rapidly with km. The gas ﬂow angles decrease rapidly with increasing the stage work coefﬁcient; a limiting gas ﬂow angle of 70° occurs at low km. The mean blades velocity, Um is inverpﬃﬃﬃﬃﬃﬃ sely proportional to km ; and decreases rapidly as the mean work coefﬁcient increases (Fig. 19). When Um > 400 m/s, the centrifugal stresses in the compressor rotor blades are excessive, limiting the mean work coefﬁcient to 0.20. The mean blades radius, rm = Um/x and the blades tip radius in the ﬁrst stage decrease as the work coefﬁcient increases (Fig. 20). The largest stage boss ratio,

1.0 0.8

400

0.4

300

0.2

φ = 70 o 1

Axial Velocity, Vx (m/s)

250

100 0.7

0.3 0.4 0.5 0.6 Work Coefficient, λm

30% surge margin 20% surge margin 10% surge margin

200 150 100

1.3 Λ

1

=Λ

mi n

1.1

0.9 He compressor (nst = 20, ℜm = 0.55) 0.7

50 0.1

o

20 300.0 0.796 0.5090 0.880 0.847 5.98 92 10.18 7.22 70 10.14 3.53 1.981 90.43

70

20 299.2 0.794 0.5133 0.880 0.852 6.0 94 10.2 7.8 72 10.1 3.80 2.0 90.5

200

He compressor (nst = 20, ℜm = 0.55)

Fig. 19. Flow coefﬁcient and blades velocity of the helium compressor stages.

=

Number of stages Blades speed at mean radius (m/s) Blades mean radius (m) First-stage ﬂow area (m2) First stage boss ratio First-stage blade tip radius (m) First-stage stator blade chord (cm) No. of stator blades in ﬁrst stage First-stage stator blade height (cm) First-stage rotor blade chord (cm) Number of rotor blades in ﬁrst stage First-stage rotor blade height (cm) Length of compressor stages (m) Static pressure ratio Compressor Polytropic efﬁciency (%)

0.2

1

Reported [15]

449.7 301 3.52 251 3600 Predicted

500

φ

Mass ﬂow rate (kg/s) Inlet temperature (K) Inlet pressure (MPa) Mechanical power (MW) Shaft speed (rpm) Performance parameters

min

0.6

0 0.1

Input parameters

600

Λ =Λ 1

30% surge margin 20% surge margin 10% surge margin

Blades Velocity, Um (m/s)

1.1

0.2

Rm

0.3 0.4 0.5 0.6 Work Coefficient, λm

0.5 0.7

1st Stage Tip Radius, Rtip (m)

240

1.2

pressor stages are designed for a speciﬁc speed of 1.0–2.0 [29], a blade tip clearance of 1 mm [15], minimum blade height of 6.7 cm for a tip clearance ratio s/H 6 1.5%, and maximum stage boss ratio of 0.93. The compressor stages have a 55% mean-radius reaction [8]. The calculated parameters for the helium compressor are compared in Table 2 to those reported for the GTHTR300 HTR plant’s compressor. A work coefﬁcient of 0.31 is used to match the reported mean blades speed, and a ﬂow coefﬁcient of 0.5342 is used to ensure a 20% stall margin. For a shaft rotational speed of 3600 rpm, the number of compressor stages is 20, and the polytropic efﬁciency is 90.43%, compared to the reported values of 20 stages and 90.5% [15]. The calculated geometrical blade parameters in the ﬁrst stage are in excellent agreement with reported values (Table 2).

Flow Coefficient, φm

250

1.3

Ma xim um Me sh rou an dr bla ad de ius sr ad ius

Radius (m)

Axial Velocity, Vx (m/s)

260

Fig. 20. Axial ﬂow velocity and blades tip radius of 251 MW helium compressor.

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

0.94

30% surge margin 20% surge margin 10% surge margin

11

0.92

10

0.90 Λ

m in

0.88

=

9

0.86

Λ

1

8 7

He compressor (nst = 20, ℜm = 0.55)

6 0.1

0.2

0.3 0.4 0.5 0.6 Work Coefficient, λm

0.84 0.82 0.7

Last Stage Boss Ratio, Λ

12

φ = 70 o 1

Rotor Blade Height, HB (cm)

K, and the shortest rotor blades occur in the last stage of the compressor. The low molecular weight helium requires relatively short compressor blades (<11 cm) and high boss ratios >0.85 (Fig. 21). The rotor blades height in the last stage of the compressor ﬁrst decreases as km increases, reaching a minimum, then increases. The minimum blade height, however, occurs at a higher stage work coefﬁcient than the maximum value of the boss ratio (Fig. 20). The minimum blades height minimizes the volume of the compressor blades and maximizes the polytropic efﬁciency, as shown later. The optimum solidity (r = C/S) of the last rotor cascade in the compressor for minimizing (Yp + Ys) (Eqs. (85) and (86)) varies from 0.7 to 1.5, and increases almost linearly as the stage work coefﬁcient increases (Fig. 22). Since the blade chord length (C) is proportional to the blade height, the number of blades in the last rotor cascade initially increases as km increases, reaching a maximum, then decreases with further increase in km (Fig. 22). This trend is true for all other cascades of the axial-ﬂow compressor. The compressor

Compressor Pressure Ratio

28

1.99

10%

1.98

20%

sur

ge

ma rg

in

30%

1.97

=Λm

in

1.96

Present design

1.95 1.94 0.1

Λ1

o

φ

=

70

1

0.2

He compressor (nst = 20, ℜm = 0.55)

0.3 0.4 0.5 0.6 Work Coefficient, λm

0.7

Fig. 24. Calculated pressure ratio of 251 MW helium compressor.

with the highest polytropic efﬁciency has the largest number of blades in the cascades (Figs. 22 and 23). At the peak efﬁciency, the compressor cascades have the largest number of shortest blades, and hence the lowest total blades volume. The calculated compressor peak polytropic efﬁciencies of 90.8%, 90.44% and 90% for stall margins of 10%, 20% and 30% occur at stage work coefﬁcients of 0.32, 0.30 and 0.25, respectively (Fig. 23). For design stall margins of 10% to 30%, the compressor pressure ratio varies from 1.941 to 1.985 (Fig. 24), and the polytropic efﬁciency from 88.1% to 90.8% (Fig. 23). For a 20% stall design margin and km = 0.18, the polytropic efﬁciency is 88.3%, and the total pressure losses through the helium compressor stages are 0.440 MPa. For the same stall design margin, the peak efﬁciency of 90.4% and the peak pressure ratio of 1.981 occur at a work coefﬁcient of 0.31, and total pressure losses of only 0.369 MPa. 5. Summary and conclusions

100

1.4 m

Λ

=

80

1.2

1

Λ

90

1.0

70

60 0.1

30% surge margin 20% surge margin 10% surge margin

o

70

0.2

0.8 0.6 0.7

0.3 0.4 0.5 0.6 Work Coefficient, λm

Rotor Cascade Solidity, σ

1.6

in

110

φ1 =

Blades in Rotor

Fig. 21. Rotor blades height and last stage boss ratio of 251 MW helium compressor.

91.0

10 %

90.5 20 %

90.0

sur ge

ma rgi n

30 %

89.5

Present design Λ

88.0 0.1

=7 1

88.5

0

o

89.0

φ

Polytropic Efficiency, ηC (%)

Fig. 22. Number of blades and last rotor solidity of 251 MW helium compressor.

0.2

=

Λ

in m

1

He compressor (nst = 20, ℜm = 0.55)

0.3 0.4 0.5 0.6 Work Coefficient, λm

0.7

Fig. 23. Calculated polytropic efﬁciency of 251 MW helium compressor.

To satisfy the need for detailed design and performance models for noble gases turbo-machines for HTR plants with CBC, this work developed multi-stage, axial-ﬂow turbine and compressor models. The developed models, based on a mean-line through-ﬂow analysis for free-vortex ﬂow along the blades, account for proﬁle, secondary, end wall, trailing edge and tip clearance (leakage) losses in the cascades, and calculate the geometrical parameters of the blade cascades as functions of the ﬂow conditions, mean-line ﬂow coefﬁcient, work coefﬁcient and stage reaction. An empirical expression is developed to determine the optimum solidity of the compressor blades cascade for minimizing the sum of the proﬁle and secondary pressure loss coefﬁcients. The developed models are validated successfully using reported performance and hardware data of the helium compressor and turbine of the GTHTR300, HTR power plant, at a shaft speed of 3600 rpm. The determined number of stages in the helium turbine of 6 at a polytropic efﬁciency of the turbine of 92.3%, are in agreement with the reported values of 6 stages and 92.8%. For 20% stall design margin in the helium compressor, the calculated minimum number of stages of 20 at a polytropic efﬁciency of 90.43% for the compressor also compares well with the reported values of 20 stages and 90.5%. The calculated geometrical blade parameters for both the helium turbine and compressor are also in excellent agreement with the reported values for the HTR plant turbo-machine. Results of parametric analyses of the 6 stages, 530 MW helium turbine show that the peak polytropic efﬁciency occurs at a mean stage reaction of 50%, and that increasing the stages work coefﬁcient decreases the turbine stages boss ratio, the shroud diameter and the rotor centrifugal stress, as well as the peak polytropic efﬁciency of the turbine. Presented results for a 251 MW helium compressor with 20 stages show that for a given stall margin, increasing the cascades mean-ﬂow work coefﬁcient increases the

J.-M. Tournier, M.S. El-Genk / Energy Conversion and Management 51 (2010) 16–29

mean ﬂow coefﬁcient and blades cascade solidity, but decreases the blades centrifugal stresses and shroud diameter. Results demonstrated that the present axial-ﬂow turbine and compressor models are versatile tools for performing preliminary design optimization of noble gases turbo-machines for HTR plants and natural-gas commercial power plants for electricity generation using CBCs. These models could also be used to investigate the impacts of changing the turbine and compressor design, the type of working ﬂuid, the shaft rotational speed, and the CBC loop pressure on the polytropic efﬁciency, size, and the number of stages in and volume of the turbine and the compressor. When integrated into a thermodynamic model of the HTR plant, the resulting plant model could be used to conduct performance optimization and investigate the effects of using direct or indirect CBC and changing the type and molecular weight of the CBC working ﬂuid on the performance of the power plant. Acknowledgments This research is funded by the Institute for Space and Nuclear Power Studies. References [1] Walsh PP, Fletcher P. Gas turbine performance. 2nd ed. Fairﬁeld, NJ: Blackwell Science Ltd. and ASME Press; 2004 [chapter 1, p. 1–60, chapter 8, p. 345–66]. [2] Current status and future development of modular high temperature gas cooled reactor technology, Report No. IAEA-TECDOC-1198. Vienna, Austria: International Atomic Energy Agency; 2001. p. 74–106, 154–67. [3] Koster A, Matzner HD, Nicholsi DR. PBMR design for the future. Nucl Eng Des 2003;222:231–45. [4] Vasyaev AV, Golovko VF, Dimitrieva IV, Kodochigov NG, Kuzavkov NG, Rulev VM. Substantiation of the parameters and layout solutions for an energy conversion unit with a gas-turbine cycle in a nuclear power plant with HTGR. Atomic Energy 2005;98(1):21–31. [5] INEEL. Next generation nuclear plant – pre-conceptual design report, Report No. INEEL/EXT-07-12967 Rev. 1. ID: Idaho National Laboratory; 2007. [6] Oh CH, Moore RL. Brayton cycle for high-temperature gas-cooled reactors. Nucl Technol 2005;149:324–36. [7] El-Genk MS, Tournier J-M. Noble gas binary mixtures for gas-cooled reactor power plants. Nucl Eng Des 2008;238:1353–72. [8] Aungier RH. Axial-ﬂow compressor – a strategy for aerodynamic design and analysis. New York, NY: ASME Press; 2003 [chapter 6], p. 118–52, 221–4. [9] Aungier RH. Turbine aerodynamics – axial-ﬂow and radial-inﬂow turbine design and analysis. New York, NY: ASME Press; 2006. p. 69–79.

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