Write a thermodynamic report based on the essay attached

Int.J. Applied Thermodynamics, Vol.4 (No.3) 127

Int.J. Applied Thermodynamics, ISSN 1301-9724 Vol.4, (No.3), pp.127-133, September2001

Thermodynamic Model of the Loss Factor Applied to Steam Turbines

Alejandro ZALETA-AGUILAR∗ Faculty of Mechanical Engineering, University of Guanajuato

Prolong. Tampico S/N, Salamanca, Gto. - Mexico Tel: +52 (464) 80911, Fax: +52 (464) 72400

E-mail: [email protected]

Javier ROYO and Antonio VALERO CIRCE (Research Center for Power Plant Efficiency)

Maria de Luna 3, 50015 Zaragoza - Spain E-mail: [email protected]

Abstract

Erosion, roughness, steam path damage, etc., are factors that reduce power capacity in a steam turbine. Any power loss occurring locally in intermediate stages of a steam turbine results in more available energy in the downstream stages, this effect is well known as the Loss Factor (Salisbury, 1974; Stodola, 1927; Husain, 1984). Currently, the Loss Factor is been calculated by graphical methods (Cotton, 1996). In this work a new thermodynamic expression for the Loss Factor (LF) is introduced, in order to improve applications to evaluate malfunctions in the first and intermediate stages of steam turbines. The new thermodynamic expression for the Loss Factor, is based on Second Law Analysis; and concepts like the internal parameter θ, and the dissipation temperature Td; (Royo, 1992). An Example of a steam turbine in a conventional power plant of 158 MW is analyzed by comparing a classical graphical method (ASME/ANSI PTC-6, 1970; and Cotton, 1993), and the proposed expression of the Loss Factor (LF). Special emphasis is made on the thermoeconomical deviations that could arise by an imprecise application of the Loss Factor Method, during an energy audit of the steam turbine internal parts.

Keywords: Loss factor, steam turbine malfunctions

∗ Author to whom all correspondence should be addressed.

1. Introduction

Any power loss occurring at the first or intermediate stages in a turbine section results in more available energy for all downstream stages, It is because the non-parallelism in isobars (known as the Reheat Effect) increases the energy available of the downstream stages where a part of this power lost can be recovered. It is convenient to multiply local power loss (first or intermediate stages) by a Loss Factor (LF) that accounts for the increased power by the following stages (Salisbury, 1974; Stodola, 1927; Cotton, 1993). There are two important application of the Loss Factor applied to steam turbines:

a) Energy audits (turbine out of service during an overhaul).

b) On-line monitoring and acceptance test (turbine operating). In an overhaul, a steam turbine energy

audit is a good way for determining internal energy losses in stage components like nozzle, bucket, seals, leaks, end-packings, etc. (affected by solid particles, erosion, roughness, damage in the steam path, etc.), giving a good reference to develop an optimum maintenance and rehabilitation program.

When the turbine starts operating, the managers in the power plants are very interested

Int.J. Applied Thermodynamics, Vol.4 (No.3) 128

in implementing on-line monitoring systems, in order to account for heat transfer rates, power generated, and fuel-impact cost due to malfunctions in the components of the plant (Zaleta et. al., 1999). The hardware of these on- line monitoring systems is based on modern field instrumentation (pressures transmitters, temperature, and mass flow meters, etc.), data acquisition processes, and very fast computers.

Software codes for steam turbine energy- audit and on-line monitoring systems include thermodynamic models (one of them is the Loss Factor method) and algorithms, for processing data and translate it in thermoeconomic information to managers.

In this paper a new thermodynamical model of the Loss Factor is introduced, in order to implement it into the algorithms for steam

turbine energy audit, and for on-line monitoring systems.

According to Figure 1, the apparent loss of power capacity (∆hintermediate) occurring locally in an intermediate stage, represents only a lower global effect (∆hend point). It is, as referred above, due to the non-parallelism in isobars increases the energy available of the downstream stages

The Loss Factor, defined in Eq. (1), is typically represented in a Mollier Chart, as shows Figure 1.

teintermedia

point end

h h

LF ∆ ∆

= − (1)

Currently LF is calculated by published graphical methods (ASME/ANSI PTC-6, 1970; and Cotton, 1993). Figures 2 and 3 show the graphics typically available in the literature to

h

s

∆∆∆∆ h i n t e r m e d i a t e

E f f e c t i v e P o w e r C h a n g e

∆∆∆∆ h e n d p o i n t

I n t e r m e d i a t e P o w e r C h a n g e S e e t h e i s o b a r n o n -

p a r a l l e l i s m s

L o s s F a c t o r ( L F ) =

∆∆∆∆ h e n d p o i n t

∆∆∆∆ h i n t e r m e d i a t e

N o m i n a l e x p a n s i o n l i n e

E x p a n s i o n l i n e w i t h a i n t e r m e d i a t e s t a g e m a l f u n c t i o n

A v a i l a b l e e n e r g y ( i s e n t r o p i c e n t h a l p y d r o p )

T e m p e r a t u r e p r e s s u r e

Figure 1. Scheme of the Loss Factor (LF) Effect When a Malfunction Occurs in a First or

Intermediate Stage of the Steam Turbine.

0.0 0.2 0.4 0.6 0.8 1.0 0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

P / P

L o s s F a c t o r

EXHAUST LOSS

1000 F INITIAL TEMPERATURE

90% AVERAGE STAGE EFFICIENCY

P = SECTION EXHAUST PRESSURE

P = PRESSURE AT WHICH LOSS OCCURS

EXHAUST

LOSS

0 200 400 600 800 1000 1200 0.40

0.50

0.60

0.70

0.80

0.90

1.001.00

Temperature at Which Loss Occurs (°

L o s s F a c t o r

EXHAUST PRESSURE 3" Hg A

90% AVERAGE STAGE EFFICIENCY

(a) (b)

Figure 2. Published Graphical Methods to Determine Loss Factor for (a) HP and IP Sections, and (b) for LP sections (Cotton, 1993)

Int.J. Applied Thermodynamics, Vol.4 (No.3) 129

40 50 60 70 80 90 100 110 120 130 140 150 0.3

0.4

0.5

0.6

0.7

Crossover Pressure at Full Prim ary (Psia)

L o s s F a c to r

Reheat Tem perature at 1000 °F

Reheat Tem perature at 1050 ° F

Figure 3. Published Graphical Methods to Determine Loss Factor vs Crossover Pressure for Reheat- ST Sections (ASME/ANSI PTC-6).

determine LF. These methods require to know specific data like pressure ratio, temperature at which loss occurs, or crossover1 pressures.

2. The Proposed Thermodynamic Model

The graphical models to determine the Loss Factor, shown in Figure 2 and 3, could be characterized numerically and introduced into the algorithm programs. However under certain conditions it will be unpractical and imprecise.

In order to improve the LF model and make it more suitable for a wide range of steam turbine evaluations, in this work the thermodynamic behavior of the Loss Factor (LF) is analyzed and a new model is proposed.

The model (shown in Figure 4) assumes an adiabatic expansion process, and it uses definitions of the Internal Parameter θ, and the concepts of the Dissipation Temperature Td , in accordance with previous works of Royo (1992), and other existing arrangements made by Ishida (1996), and Bejan (1994). For this model the Internal Parameter θ2 , in K units Eq.(2), is defined as the slope between inlet (i) and outlet (j) conditions of the expansion process:

ji

ji ij ss

hh − −

=θ (2)

and the Dissipation Temperature Td, Eq.(3), in K units, is defined as the slope generated for the changes in thermodynamic properties of the 1 Duct that feed steam to the LP section 2 2 Kinetic and Potential terms can be include in enthalpy as h=hstatic+v2/2+gz

expansion line end-point (j), when a malfunction occurs.

i

j j,d ds

dh T = (3)

m m

Inlet Conditions

Outlet Conditions

(hi, si) (hj, sj)

W Q=0 (adiabatic)

EXPANSION PROCESS

(a)

Inlet Conditions

Outlet Conditions

(hi, si)

(hj, sj)

h

s (si - sj)

(hi,- hj) Isobar

(δδδδsj)

(δδδδhj)

Nominal expansion line

Expansion Line with a malfunction

Slope of the Dissipation Temperature

(b)

Figure 4. Schematic Definition of the Parameters Considered for an Adiabatic Expansion Process (a) Control Volume, (b) Expansion Line.

Int.J. Applied Thermodynamics, Vol.4 (No.3) 130

h

s

∆∆∆∆ h 2

∆∆∆∆ h 3 L o s s F a c t o r ( L F )

= ∆∆∆∆ h 2 ∆∆∆∆ h 3

N o m i n a l e x p a n s i o n l i n e

E x p a n s i o n l i n e w i t h a n i n t e r m e d i a t e m a l f u n c t i o n

θ 1 2

θ 2 3 ≅ θ 2 ' 3 '

T 2 2 ´ = T d , 2

T d , 3

1

2

3

2 '

3 '

Figure 5. Scheme of the Loss Factor (LF) Model using the Internal Parameter θ, and the Dissipation Temperature Td.

1 2 3 4 5 0.80

0.85

0.90

0.95

1.00

High Pressure Stages

H P - S T L o s s F a c t o r

NBS Steam TableNBS Steam Table

New Thermodinamical ModelNew Thermodinamical Model

Classic Graphical ModelClassic Graphical Model

8 9 10 11 12 13 14 15 16 17 0.40

0.50

0.60

0.70

0.80

0.90

1.001.00

Intermediate and Low Pressure St

L o s s F a c t o r

Classic Graphical MethodClassic Graphical Method

New Thermodynamical ModelNew Thermodynamical Model

NBS Steam TableNBS Steam Table

Intermediate Low Pressure

(a) (b)

Figure 6. Comparison of the Loss Factor Model between the Proposed Thermodynamic Model of LF, the classical Graphical Model and Reference Values from NBS Steam Tables, for (a) High Pressure Section, and (b) for Intermediate and Low Pressure Section.

Given that pressures at intermediate stage(p2 ≈p2’), depend strongly on the mass flow rate, when a malfunction occurs with a constant mass flow rate (

• m ), then pressure remain

approximately constants (see Cooke, 1984 for a wide explanation of Stodola´s Ellipse). According to a previous definition the slope of Td on isobaric conditions is equal to the instant temperature Tj,

j p,mi

j d Ts

h T

j

= ∂ ∂

= • (4)

The ratio δh/δs depends on the kind of process. In this case the partial derivatives at p=const should be applied according to the Stodola´s Ellipse.

Figure 5 could represent a schematic expansion line of a steam turbine when a

malfunction appears in an intermediate pressure section (IP) and it discharges to a Low Pressure Section (LP). According to Spencer et. al. (1974) the slope of the expansion line in the Low Pressure Sections (LP) remains approximately constant (θ23 ≅ θ2´3´) even if Intermediate Pressure (IP) develops a malfunction. Where thermodynamic conditions at point 2' represent the expansion line end-point at which loss occurs (upstream), and point 3' represents the expansion line end-point of the downstream stages after. Under an adequate handling of the previous definition, Eqs.(1)-(4), applied to the model sketched in Figure 5, they can be expressed:

 

  

 =

2

3

dh dh

LF

.ttancons dsds dhdh

ss hh

32

32

32

32 23 =−

− =

− −

=θ

Int.J. Applied Thermodynamics, Vol.4 (No.3) 131

constp2 2

2 2dloss,d

2 T

ds dh

TT =

= 

  

 ==

constp3 3

3 3dend,d

3 T

ds dh

TT =

= 

  

 ==

by multiplying and dividing θ23 by dh2,

( )

( ) 2

3

2d2 32

2 32

23

dh ds

T 1

LF1

dh 1

dsds

dh 1

dhdh

−

− =

 

  

 −

 

  

 −

=θ

by multiplying and dividing by dh3,

( )LF T 1

T 1

LF1

dh dh

dh ds

T 1

LF1

3d2d3

3

2

3

2d

23

 

  

 −

− =

 

  

  

  

 −

− =θ

the expression of the Loss Factor (LF) can be re- defined as follows:

   





   





Τ θ

−

Τ θ

− =

∂ ∂

=

3

23

2

23

2

3

p,m 1

1

h h

LF j!

(5)

for non-differential cases of LF, it can be expressed as:

   





   





θ −

θ −

=

3

23

2

23

T 1

T 1

LF (6)

where θ23 is evaluated at nominal steam turbine conditions, 2T and 3T are represented by the average mean logarithmic temperature expressed

as )T/Tln(

)TT( T

2´2

2´2 2

− = and

)T/Tln( )TT(

T 33́

33́ 3

− = ,

respectively (Bejan, 1994).

TABLE I. STEAM PROPERTIES FOR EACH STAGE IN A 158 MW STEAM TURBINE Inlet Condition Outlet Condition Nominal Parameters

Section STAGE Pressure

bars Temperature

°C Pressure

bars Temperature

°C Mass Flow

kg/sec Power KW

HP-ST 1 124.10 538 99.29 507 128.9 6695 2 99.29 507 81.72 476 128.9 7045 3 81.72 476 65.37 444 128.9 7494 4 65.37 444 52.3 413 128.9 7195 5 52.30 413 41.84 382 128.9 7195

6 41.84 382 33.47 353 128.9 6895 7 30.13 538 23.54 502 120.9 8819 8 23.54 502 18.18 465 120.9 8995 9 18.18 465 13.88 426 120.9 9558

10 13.88 426 10.51 388 116 8904 11 10.51 388 7.673 346 116 9444 12 7.673 346 5.366 304 110.2 9231

IP-ST

13 5.366 304 3.600 258 110.2 9744 14 T 3.600 258 1.977 195 50.8 6144 15 T 1.977 195 0.999 132 49.53 5991 16 T 0.999 132 0.467 80 45.63 5466 17 T 0.467 80 0.194 59 45.63 5572 18 T 0.194 59 0.076 41 45.63 5307 14 G 3.600 258 1.977 193 50.8 6381 15 G 1.977 193 0.999 134 49.53 5530 16 G 0.999 134 0.467 80 45.63 5731 17 G 0.467 80 0.194 59 45.63 5519

LP-ST

18 G 0.194 59 0.076 41 45.63 5307 Mechanical Power Loss −2240 Generator Power Loss −2990

Total (kW): 158,932

Int.J. Applied Thermodynamics, Vol.4 (No.3) 132

TABLE II. INTERNAL PARAMETER θ AND DISSIPATION TEMPERATURES TD IN STEAM TURBINE STAGES TO OBTAIN LF VALUE AND CALCULATION ERROR.

Section Stage θ θ θ θ [K] Td,loss [K] Td,end [K] LF

(Proposed model)

Proposed Model

Error*%

LF (Graphical

model)

Graphical Model

Error*% 1 −3988 779.7 625.6 0.8292 0.0130 0.8097 2.358 2 −3833 749.3 625.6 0.8580 0.0105 0.8404 2.066 3 −3917 716.8 625.6 0.8903 0.0084 0.874 1.829 4 −4215 685.7 625.6 0.9237 0.0054 0.9089 1.606 5 −4113 655 625.6 0.9610 0.0025 0.9508 1.073

HP-ST

6 0 625.6 625.6 1 0 1 0 7 −4001 774.9 313.64 0.4480 −0.0086 0.4833 −7.869 8 −4004 738.1 313.64 0.4667 −0.0140 0.4959 −6.257 9 −3882 698.9 313.64 0.4901 −0.0208 0.5131 −4.739

10 −3770 660.5 313.64 0.5152 −0.0264 0.5352 −3.912 11 −3669 619.3 313.64 0.5453 −0.0348 0.5657 −3.779 12 −3670 576.6 313.64 0.5799 −0.0443 0.6053 −4.440

IP-ST

13 −3629 531.1 313.64 0.6231 −0.0529 0.6563 −5.385 14 −3570 468.1 313.64 0.6966 −0.0709 0.6966 −6.265 15 −3590 404.5 313.64 0.7934 −0.0937 0.7934 −5.180 16 −3323 352.65 313.64 0.8989 0.0753 0.8989 −1.281 17 −2847 332.49 313.64 0.9489 0.0406 0.9489 0.994

LP-ST

18 0 313.64 313.64 1 0 1 0

*respect to value obtained by using NBS Steam Tables

3. Study Case

To show the main features and easiness of the application of the proposed method, a 158 MW conventional steam turbine is analyzed. This turbine has three sections High Pressure (HP), Intermediate Pressure (IP), and Low Pressure (LP); sections respectively, with the following characteristics:

• High Pressure Section (HP) with 6 Impulse Stages.

• Intermediate Pressure Section (IP) with 4 Impulse Stages, 3 Reaction Stages.

• Low Pressure Section (LP) with 5 Reaction Stages in double compound . By using manufacturer information, it was

possible to determine pressure ratios, efficiencies, and nominal operating conditions (at pitch nozzle-bucket conditions) for each stage in the turbine sections, (TABLE I). From these data it was possible to evaluate threes different way for obtaining LF: i) by graphical method (Figures 2 and 3), ii) by new thermodynamical model of LF (eq.

5), and iii) by evaluating directly LF3 from eq.(1),

when a efficiency change is simulated,

3 This value of LF is a reference to compare the discrepancy of the methods (Figs. 2, and 3; and eq. (5).

using NBS steam tables (It is considered as the expected value at real conditions).

Information provided in TABLE II, allows to demonstrate that the proposed thermo- dynamical model for LF, Eq. (5), is more accurate and practical than the graphical methods. Figures 6 also shows the discrepancy of each method.

4. Conclusions

Procedures on energy auditing for all internal parts of the 158 MW turbine, as given by Cotton (1996), were followed. Final results on this energy audit are shown in Figure 7, where recovered power due to maintenance activities (aprox. 6.2 MW recovered) at the different turbine stages is shown. This figure also shows a discrepancy index, in percentage about 1.4 - 4 % when a graphical method of LF is compared with respect a simulated value. Such differences, represents almost 0.18 MW of uncertain audited power in this steam turbine due to LF used method. Nevertheless Proposed Method is as much about 0.02- 0.1 % of error with respect to calculated value of the steam tables. It is shown that the method proposed will provide a more accurate and practical way to determine the Loss Factor. This method, coupled with a good recording of field parameters, will provide a more reliable way to determine the impact of power loss in turbines.

Int.J. Applied Thermodynamics, Vol.4 (No.3) 133

P o w e r R e s t o r e d D u e t o M a i n t e n a n c e A c t i v i t i e s

1 . 4 0 % 2 % 4 % 2 . 6 0 % 0 %2 . 2 0 % 0

0 . 2

0 . 4

0 . 6

0 . 8

1

1 . 2

1 . 4

1 . 6

R ou

gh ne

ss in

bu ck

et s

R ou

gh ne

ss in

N oz

zl es

In te

rs ta

ge S

ea ls

R oo

t S ea

ls

E ro

si on

E nd

P ac

ki ng

s

M W

M W % D i s c r e p a n c y

Figure 7. Results of a Typical Steam Turbine Audit, and Comparison in Percentage % of

Discrepancy Occurred when Graphical Methods of LF are Used with respect a Simulated Value of LF.

Nomenclature

∆hend point Enthalpy Changes at Expansion Line End Point Conditions

∆hintermediate Enthalpy Changes at Intermediate Expansion Line Conditions

G Generator Side H Enthalpy HP High Pressure Section IP Intermediate Pressure Section LF Loss Factor LP Low Pressure Section P Pressure Pexhaust Pressure at Steam Turbine Exhaust

Condition Ploss Pressure at which Loss Occurs Q Heat Flow S Entropy ST Steam Turbine θ Internal Parameter T Temperature T Referred to Turbine Side Td Dissipation Temperature W Shaft Work

References

ASME/PTC-6, 1970, “Simplified Performance Test of Steam Turbines”, ed. by The American Society of Mechanical Engineers, ASME, N.Y, USA.

Bejan, A., 1994, “Entropy Generation Through Heat and Fluid Flow”, pp. 7-14. ed. Wiley and Sons, USA.

Cooke, D.H. 1984, “On prediction of Off-Design Multistage Turbine Pressure by Stodola Ellipse”, ASME Book 84-JPGC-GT-14, USA.

Cotton K.C., 1993, “Evaluating and Improving Steam Turbine Performance”, ed. by Gilson Works and Cotton Fact, Inc. NY, USA.

Husain, 1984, “Steam Turbines Theory and Design”, ed. Mc Graw Hill, USA. Ishida, M. and Chuang, C.,1996, “Energy Quality Degradation”, In ECOS 96, ed. by P. Alvfors, L. Eidensten, G. Svedbern and J. Yan, pp. 9-16.Stockholm, Sweden, 1996.

Royo J., 1992, “Las Ecuaciones Características”, Doctoral Thesis, University of Zaragoza. Spain, also “The Dissipation Temperature” in ECOS’97 Stockholm, Sweden.

Salisbury, 1974, “Steam Turbines and their Cycles”, ed. Robert E. Krieger Publishing, NY, USA.

Spencer, R.C., Cotton K.C. and Cannon, C.N. 1974, “A Method for Predicting the Performance of Steam Turbine-Generator, 16,500 kW and Larger”, ASME Power Division, Paper No. 62- WA-209, USA.

Stodola, 1927, “Steam and Gas Turbines”, ed. Mc Graw Hill.

Zaleta-Aguilar, A, Gallegos-Muñoz A., Valero A., and Royo J., 1999, “Improvement of the Exergoeconomic ‘Fuel-Impact’ Analysis for Acceptance Tests in Power Plants”, ASME- WAM 99, AES-Vol. 39, Nashville, TN, USA.