Theory for Turbomachinery Degradation and Monitoring Tools

Magnus Genrup

June 2003

Licentiate Thesis Department of Heat and Power Engineering

Lund Institute of Technology Lund University, Sweden

www.vok.lth.se

© Magnus Genrup 2003 ISSN 0282-1990 ISRN LUTMDN/TMHP--03/7014--SE Printed in Sweden Lund 2003

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Abstract The revenue from a power plant is strongly dependent of the life cycle cost. Today, when the power producing company’s role has shifted from a protected market into a deregulated market, the need for tools to monitor the investments has increased. These tools are described in this thesis for both the gas turbine and the steam turbine. This thesis will give a thorough description of the state-of-the-art model-based gas turbine flow path analysis system. The underlying mechanisms for degradation will also be described together with some remedying actions. The primary intention of the thesis is to provide guidance for the user of the plant on how a model-based system works. The information is presented in general terms since it is impossible to cover all gas turbine configurations in such a rather short text. The tools presented here have different levels of sophistication, from the most simple to state-of-the-art heat and mass balance programs. The achievable level is dependent on the amount of knowledge about the specific engine type. The highest level of sophistication is reserved for systems delivered by the manufacturing companies (OEM’s). This level of monitoring system requires detailed propriety turbine data, which disqualifies third-party systems. A system delivered by an OEM is in general more costly, but the additional know-how is indeed a valuable commodity. The prediction capability is normally a weak spot in a third party system since the performance deck is a well-guarded tool and not available outside the OEM. The overall objective of this thesis is to show how degradation can be assessed with standard calculation tools. This includes mapping of different model-based monitoring tools, as well as description of the mechanisms of several aging phenomena. The modelling tools are thoroughly described, which makes it possible for the reader to develop a program or, at least, evaluate different systems. With the knowledge of the underlying degradation mechanisms, and the possibility of including these in a condition monitoring system, the potential for improving the operation economics is significant. The availability of a plant can be increased if early warnings can be obtained. Also the cost of secondary replacement parts, in the case of component breakdowns, can be entirely avoided.

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Contents

1 Introduction ...............................................................................................1-1 1.1 Background .............................................................................................................1-1 1.2 Objective .................................................................................................................1-2 1.3 Limitations ..............................................................................................................1-3 1.4 Methodology ...........................................................................................................1-3 1.5 Outline of the thesis.................................................................................................1-4 1.6 Acknowledgements .................................................................................................1-4

2 Flow Path Deterioration ...........................................................................2-5 2.1 Air filtration.............................................................................................................2-6 2.2 Compressor degradation..........................................................................................2-7

2.2.1 Increased tip clearances...................................................................................2-8 2.2.2 Profile changes ..............................................................................................2-10 2.2.3 Index of compressor sensitivity to fouling....................................................2-13 2.2.4 Compressor washing .....................................................................................2-13

2.3 Turbine degradation ..............................................................................................2-16 2.4 Causes for turbine deterioration ............................................................................2-18

2.4.1 Surface roughness .........................................................................................2-19 2.4.2 Tip clearance .................................................................................................2-20 2.4.3 Trailing edge thickness..................................................................................2-21 2.4.4 Leading edge thickness .................................................................................2-24 2.4.5 Profile changes ..............................................................................................2-26 2.4.6 Secondary- and cooling air............................................................................2-26

3 Tools..........................................................................................................3-29 3.1 Traditional Flow Path Analysis.............................................................................3-29

3.1.1 Measured data ...............................................................................................3-29 3.2 Modelbased evaluation tools.................................................................................3-32

3.2.1 Compressor flow ...........................................................................................3-32 3.3 Component Performance.......................................................................................3-37

3.3.1 Inlet System...................................................................................................3-37 3.3.2 Compressor section performance ..................................................................3-38 3.3.3 Combustion chamber section ........................................................................3-41 3.3.4 Turbine section..............................................................................................3-43

3.4 Baseline or off-design modelling ..........................................................................3-47 3.4.1 Background curves (multi-shaft units) ..........................................................3-47

3.5 Performance program............................................................................................3-52 3.5.1 Component maps or CHIC............................................................................3-52 3.5.2 Component matching ....................................................................................3-55 3.5.3 Simplified models .........................................................................................3-56

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4 Axisymmetric through-flow calculation................................................4-59 4.1 Introduction ...........................................................................................................4-59 4.2 Radial equilibrium equation ..................................................................................4-61

4.2.1 Radial equilibrium equation integration........................................................4-66 4.2.2 Analysis case .................................................................................................4-67 4.2.3 Finite differences...........................................................................................4-68 4.2.4 Integration constant .......................................................................................4-69 4.2.5 Target pressure method .................................................................................4-71 4.2.6 Streamline relocation iteration ......................................................................4-71

4.3 Loss modelling ......................................................................................................4-72 4.3.1 Profile losses .................................................................................................4-73 4.3.2 Incidence losses.............................................................................................4-74 4.3.3 Secondary losses ...........................................................................................4-74 4.3.4 Trailing edge loss ..........................................................................................4-76 4.3.5 Tip clearance loss ..........................................................................................4-76

4.4 Supersonic conditions at the throat .......................................................................4-78 4.5 State calculation at supersonic conditions.............................................................4-79 4.6 State calculation at subsonic conditions................................................................4-80 4.7 Secondary deviation ..............................................................................................4-81

5 Steam Turbines........................................................................................5-85 5.1 Steam Turbine Deterioration.................................................................................5-85

5.1.1 Deposits .........................................................................................................5-86 5.1.2 Surface roughness .........................................................................................5-89 5.1.3 Sealing leakages ............................................................................................5-89

5.2 Simplified models .................................................................................................5-90 5.2.1 Condenser model...........................................................................................5-94

5.3 Analysis methods ..................................................................................................5-95 5.3.1 Maximum capability .....................................................................................5-96 5.3.2 Pressure-Flow Characteristics .......................................................................5-96 5.3.3 Cylinder efficiency........................................................................................5-99 5.3.4 Test Code Performance Test .........................................................................5-99 5.3.5 Valve leakage test..........................................................................................5-99

6 Conclusions ............................................................................................6-101

7 Summary of papers ...............................................................................7-103

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Nomenclature Latin

A Area [m2]

b Distance between adjacent streamlines [m] or axial chord [b]

c Velocity [m/s], blade chord [m] or coefficient [-]

C Coefficient [-]

CT Turbine constant

CS Blade surface length [m]

Cp Diffuser pressure recovery coefficient [-]

cp Specific heat capacity [kJ/(kg×K)]

D Diffusion factor [-] or diameter [m]

d Leading edge diameter [m]

e Blade (suction side) curvature downstream the throat [m]

F Force [N]

Fγ Non-work factor [-]

FN Turbine flow number/capacity

H Boundary layer form factor [-]

h Enthalpy [kJ/kg] or height [m]

I Rothalpy [kJ/kg]

i Incidence [°]

K Coefficient [-]

KC Combustor section cold pressure loss coefficient [-]

KH Combustor section hot pressure loss coefficient [-]

ks Surface roughness

LHV Lower heating value [kJ/kg]

lSP Profile backbone length [m]

M Mach number [-]

m Meridional

m Mass flow [kg/s]

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n Rotational speed [min-1 or s-1]

o Throat opening [m]

p Pressure [Pa or bar]

P Power [kW]

PF Power factor [-]

Q Heat flux [kJ/s]

R Gas constant [kJ/(kg×K)]

r Radius [m]

rc Radius of curvature [m]

Re Reynolds number [-]

RH Relative humidity [- or %]

s Entropy [kJ/(kg×K)] or blade spacing/pitch [m]

S Blade spacing/pitch [m or mm] or apparent power [kW]

T Temperature [K or °C]

U Heat transfer coefficient [kW/(m2×K)]

u Blade velocity [m/s]

v Specific volume [m3/kg]

V Velocity [m/s]

V Volumetric flow [m3/s]

w Relative velocity [m/s]

W Relative velocity [m/s]

X Loss distribution factor (meridional plane) [-] or Parson number [-]

x Volumetric fraction [-]

Y Pressure loss coefficient [-]

Z Zweifel number [-]

z Axial direction Greek

α Flow angle [°], theta exponent [-] or bleed coefficient [-]

β Relative flow angle [°]

γ Quasi-orthogonal angle

∆ Difference

δ Boundary layer thickness [mm]

δ* Displacement thickness [mm]

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ε Tip clearance [m] or over- and under turning angle [°]

ζ Energy loss coefficient [-] or vorticity

η Efficiency [-]

θ Tangential direction

κ Isentropic exponent [-]

Λ Reaction [-]

λ Blockage [-]

ν Velocity ratio [-]

ξ Stagger angle [°]

ρ Density [kg/m3]

τ Tip clearance [-]

φ Streamsurface lean angle [°] or blockage [-]

χ Overboard leakage factor [-]

ψ Stage loading [-] or stream function [-] Index

0 Total- or stagnation state

1 Vane inlet

2 Vane outlet/Rotor inlet

3 Rotor outlet

BM Bellmouth

c Cooling air

CC Combustion chamber

CW Cooling water

F Fuel

f Loss

LE Leading edge

m Meridional

REF Reference

s Isentropic

TE Trailing edge

th Throat

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Engine station numbering

0 or AMB Ambient condition

10 Bellmouth inlet

20 Before compressor blading (immediately upstream of the IGV)

30 After compressor blading (immediately downstream of the OGV)

305 Station within the compressor diffuser

31 Compressor diffuser outlet or combustor section inlet

40 Combustor section outlet

405 First turbine stator throat

41 First turbine stator outlet

415 Pseudo turbine inlet (N.B. no physical station)

42 Turbine inlet (N.B. no physical station)

44 High turbine outlet

45 Low turbine stator inlet . . .

49 Last turbine outlet

50 Turbine diffuser inlet

60 Turbine diffuser outlet

70 Stack

Abbreviations

AMDC+

KO+MK Ainley-Mathieson-Dunham-Came+Kacker-Okapuu+Moustapha-Kacker

AVDR Axial velocity density ratio (streamtube convergence/divergence)

COT Combustor outlet temperature [K or °C]

DOD Domestic object damage

FOD Foreign object damage

OEM Original equipment manufacturer

RIT Rotor inlet temperature [K or °C]

SAS Secondary air system

SCM Streamline curvature model/method

SOT Stator outlet temperature [K or °C]

TIT Turbine inlet temperature [K or °C]

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To Ellen and Elliott

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1 Introduction When investing in a power plant, the life cycle cost (LCC) is of great importance, especially after the deregulation of the electricity market. The LCC, including fuel and maintenance costs, as well as the capital investment cost, is the single most important factor in determining whether a project is successful or not. How the relation between efficiency and investment cost is valued in a specific project depends on several factors and is indeed complicated. By choosing parameters such as first cost and efficiency, a weight matrix is obtained. The choice is, of course, dependent on market parameters, e.g. electricity and fuel price.

1.1 Background The role of the utility companies has changed since the electricity market has been deregulated. Earlier, the main objective was to produce electricity and the utility companies often had long-term contracts with the government. The profitability was hence subordinated. The new actors on the market, i.e. owners of Merchant Power Plants (MPP’s), do not have the same engineering culture or tradition as the utility companies. Therefore, the need of tools for monitoring and optimisation has experienced a continuously growing interest from the MPP owners. Gas turbines were traditionally installed as low merit plants with just a low number of fired hours and there was simply no point in focusing on operational economy. This was due to the fact that the impact from the fuel cost was rather small. Things changed, however, with the increased number of base power combined cycles. The changeover from peak lopping to base power units increased the number of fired hours dramatically, from basically nothing to perhaps eleven months of operation per year. This change also led to the demand for powerful monitoring tools. There are a number of different monitoring tools on the market today. The more traditional tools are heat and mass balance programs, but also Kalman filter-based technologies are common — especially within the aero industry. During the last decade, artificial neural networks (ANNs) have been introduced as an additional aid to handle monitoring of engines. The incentive for using monitoring tools is to be able to detect faulty components in the system and hence increased production costs and availability. Another important feature of these tools is the capability of generating early warning before component failure, and thereby also avoiding costly production outages and secondary replacement parts. Additionally, the ability to identify faulty components will assist the maintenance planning since parts can be ordered in advance, and therefore the outage period can be shortened. This type of “on condition” maintenance instead of “time scheduled” maintenance will reduce the cost since

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unnecessary replacements can be avoided. The more traditional maintenance approach for gas turbines, i.e. time scheduled maintenance, is to introduce equivalent operating hours (EOH) as seen in equation (1-1). These are used for estimating both creep and low cycles (e.g. a start) where the latter is assessed as time. A typical value of time between overhauls (TBO) for an industrial type of engine is 40 000 EOH, but may be somewhat shorter for aeroderivatives. A set of hot end blading and a combustion section are typically designed to last for this time period. That means that a base load unit with just a few starts can operate for 4.5 years with the same set of hot end parts. A peak lopping unit, on the other hand, may be below one tenth of that figure. fuel firing starts start load rateEOH OH F F n F F= ⋅ ⋅ + ⋅ ⋅ (1-1) Where: EOH Equivalent operating hours OH Actual operating hours Ffuel Factor depending on fuel type Ffiring Factor depending on firing temperature1 Nstarts Number of starts2 Fstart Number of equivalent hours per start Fload rate Factor depending on load rate

1.2 Objective One of the general objectives of this thesis has been to investigate the mechanisms for gas and steam turbine performance degradation. Also the most common flow path analysis tools are investigated in more depth. More specific objectives are:

• Identify the mechanisms for gas turbine performance degradation and study the impact on the flow path.

• Investigate tools for flow path monitoring. A wide range of tools, basically from hand

calculation to artificial neural networks, has been studied.

• Presentation of the streamline curvature throughflow approach

• Introduction to steam turbine degradation and monitoring

1 Typically unity at base load and 10 at peak load. This figure may even be below unity at part load. 2 A start is normally defined as when the engine lights and the exhaust temperature has exceeded a certain value.

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1.3 Limitations The work presented in this thesis has the following limitations:

• The presented material is focused on industrial and aero-derivative gas turbines with axial flow components.

• The flow path degradation is assumed to be a certain change in geometry (e.g. increased

trailing edge thickness) rather than the metallurgical aspects (e.g. oxidation). • The calculation methods used are one- and two-dimensional (1-D, 2-D) and quasi three-

dimensional tools (Q3-D) are also used.

• Only commercial throughflow codes were used (SC90T3 and AXCAD4).

• The Ainley and Mathieson (and higher) loss model was generally used, where applicable.

• The steam turbine section is held rather brief.

1.4 Methodology This work is based on a literature survey with the purpose of putting together different theories for degradation phenomena. The necessary equations for modelling aging phenomena are also presented, especially for gas turbine components. The chosen approaches are selected according to their applicability for the specific problem. The well-regarded and widely used loss model-based on Ainley and Mathieson was the preferred choice in this work for the turbine section. The presented equations can be used to make quantitative estimations when evaluating the impact from degradation. Theory from the open literature is supplemented by the author’s own experiences in this field, and from gas turbine performance testing and evaluation. This implies an in-depth understanding for the practical use of the theoretical models when applied on gas turbines. The present author has also tested different types of commercially available throughflow calculation tools. These tools are standard turbomachinery tools and were used as general tools to model component aging. The streamline curvature method (SCM) is presented in general terms in chapter four. The presented model-based system in chapter three is based on state-of-the-art heat and mass balance models. These types of models are widely used today and will probably be used for a long time in the future. Lower order methods are also presented in the text, but to a lesser extent.

3 Available from PCA Engineers 4 Available from Concepts NREC

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A dedicated heat-balance model for steam turbine cycles was developed by the author to model off-design performance. The same heat-balance model was used for creating data to model faulty conditions when training ANN’s.

1.5 Outline of the thesis Chapter one gives a brief introduction to the subject. In chapter two, the degradation mechanisms are presented for axial compressors and axial turbines. Chapter three is an in-depth description of different flow path analysis tools. Chapter four is a summary of the SCM throughflow calculation method. In chapter five, a brief introduction to steam turbine monitoring is presented. In chapter six, the overall conclusions are presented. Chapter seven is a summary of the papers that are included in this thesis.

1.6 Acknowledgements This work is supported by Swedish Gas Center and the Swedish Energy Agency, within the framework of the research program “Thermal Processes for Electric Power Production”. My supervisors, Professor Tord Torisson and Assistant Professor Mohsen Assadi, are greatly acknowledged for their support and guidance in this work. Many former colleagues and brilliant engineers in the gas turbine business, from whom I learned so much, are not forgotten. Doctors in spe Pernilla Olausson, Ulf Engdar and Jaime Arriagada are greatly acknowledged for all assistance in numerous matters – Thanks! A special thanks goes to my mother Gunilla and Bert, who made this work possible.

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2 Flow Path Deterioration The performance of a brand new unit will start to deteriorate as soon as the engine is turned for the first time. This section will address this issue and discuss the mechanisms for changes in performance. There are basically two types of deterioration where the first is more abrupt and the second is more gradual. The first is typically due to foreign object damage (FOD) or domestic object damage (DOD), while the latter is typically due to surface roughness and clearances. The next logical step is to divide the deterioration into recoverable and non-recoverable. The process of recovering the performance is typically a compressor soak wash or a turbine nutshell wash, depending on the nature of the problem. On-line washing may prolong the time between soak washes, but may be a risky business. The argument for not doing on-line washing is concern about the secondary air system and blade cooling air passages. One of the arguments for choosing a heavy duty or an industrial type instead of an aero-derivative is, in most cases, the better degradation characteristics for the more rugged types. Another benefit that favours for the industrial type is longer service intervals and hence, lower life cycle cost. The list of arguments for and against a certain type can be very long and one parameter that may be important for one operator may not be an issue for another operator.

Figure 2-1. Showing typical deterioration behaviour of a gas turbine [1].

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2.1 Air filtration

The inlet filter system is very important since most “normal” problems enter the unit from the ambient air. The need for filtration is easily recognised when considering that a gas turbine may ingest more than half a ton of air per second. The amount of carried foreign matter may be considerable in rural areas and industrial zones. The filter system is the major contributor to the inlet pressure drop and the filter section pressure drop is in the order of 5-6 mbar. Today it is common to have self-cleaning air filters that use compressed air to clean the filter segments. The conventional or static filter type is also used today. The advent of HEPA5 technology in this field really reduced compressor-fouling issues and one can now operate a turbine for a long time without washing. The HEPA technology and/or three-stage filtration has higher pressure losses6 but the net impact on performance is still positive. It is important to recognise that the filter system really makes the difference between a successful turbine operation or not. Air filtration is important since particles 20µm and above are considered to cause erosion, whereas particles below 20µm do not. A typical gas turbine filtration system has efficiency above 99.94 % in this size range.

Figure 2-2. Example of filtration system performance, after ref [2].

It is also of utmost importance to prevent unwanted matter to enter the cooling system. The potential risk of over-heating a clogged passage is apparent. The reason for this is, besides reduced flows, that the heat transfer may be degraded. The “worst” air-borne particles are coal dust, cement dust and fly ash, since they tend to sinter inside e.g. blades.

5 High Efficiency Particulate Air 6 One common misunderstanding when evaluating the impact from inlet pressure losses is to use the correction for ambient pressure. This is incorrect since the inlet pressure drop only affects the inlet and not the turbine heat drop.

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2.2 Compressor degradation

An axial flow compressor is extremely sensitive to fouling and the performance starts to deteriorate as soon as the engine is started for the first time. The mechanism for fouling is particles and some adherence effect. Fortunately, washing may restore the fall in performance due to fouling. The adherence effect may be caused by the normal humidity in the air and oil leaks from bearing number 1. The air humidity effect is due to bellmouth condensation that is caused by the depression in the bellmouth. Depending on the inlet Mach number7, the static temperature and pressure fall to certain levels that may be below the saturation line. This means that the excessive water vapour in the air must be condensed. If this amount of condensate is about right, it will start to act as glue for the particles that enter the compressor. The ingested particles may be as small as 2µm and normal filtration efficiency for this size range is 92 - 96%. There are a couple of interesting conclusions that may be drawn from this. First, if we assume a cold climate, then the absolute humidity is low and hence there are small amounts of condensable water. This means that the rate of fouling will be reduced since there is not enough condensed water to produce the adherence effect. The second consideration is if the amount of water is high, as in a warm humid tropical day, then the amount of condensed water is sufficient to create some kind of on-line washing effect. This effect is also described in reference [3] and [28], but from the performance point of view. The condensation process will release the latent heat for the condensed water vapour and increase the temperature before the inlet guide vane. This means that a choked compressor will lose some pumping capacity due to the increased temperature. This phenomenon is actually a rather annoying problem when a performance test is carried out in a warm humid day. The reason for this is that the ambient temperature is measured where the velocity is low in the air intake section before the filters, because most of the test codes stipulate that the velocity must be below 20 m/s [4] if static probes are to be used. Another reason is the reluctance to put any temporary instrumentation behind the filter barrier. Stalder [5] investigated the effect of the amount of ingested water on power-loss for a unit. The test is based on 14 periods, each 70-72 hours, and Stalder also evaluated the accumulated ingested water.

Figure 2-3, Power loss versus accumulated amount of ingested

water, based on 14 periods of each 70-73 hours [5].

7 The velocity just upstream the inlet guide vane is typically in the range of Mach 0.5 to 0.55 for a modern compressor.

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The other adherence effect is due to oil leakage in the bearing number 1. This lubrication oil gets drawn into the flow path caused by the low pressure upstream the first rotor. This problem is two-folded since, besides the glue effect, it will form coke deposits in the hot end of the compressor. This kind of deposit may be very difficult to remove by normal water washing. It may even be necessary to use special chemicals to remove the deposits when the compressor is opened during an overhaul. If a stage, at some point in the compressor, gets increased loss level, the pressure falls and the temperature rises. All other preceding stages will then operate at a mismatched condition. The work input will normally be decreased in the succeeding stages since the meridional8 velocity component will increase. The increased loss is due mainly to incidence and the increased temperature level will reduce the Mach number effects, hence somewhat reducing the impact from incidence. Therefore, in order to fully evaluate the loss in one stage, all preceding stages must be taken into account. The fouling affects both efficiency and the pumping capacity of the compressor. The reduction in capacity is typically 1.6 times the drop in efficiency. The drop in capacity is a much more severe problem for a single shaft unit since it will not increase its speed level for compensation of lost capacity. An empirical function for the rate of reduction in capacity is given in [14], but the maximum level of reduction and the time constant must be known a priori. The equation shows the typical behaviour of fouling where the drop rate is rather high immediately after a soak wash:

( )1 B tm A em

⋅∆= − (2-1)

The constant A in the equation above is the maximum reduction and the constant B sets the rate of reduction. It must be emphasized that a higher speed level will compensate for the reduction in capacity for a multi-shaft unit. The present author once tested the capacity characteristics of a unit immediately after multiple soak washes, and then repeated the test after approximately 4 operating hours to find a reduction in capacity of approximately 2%. However, it is hard to draw any conclusions from this since no information is available regarding the amount of unwanted matter in the ambient air. The reason for reduction in performance due to fouling is normally increased blade surface roughness and changes in profile shape. The major reasons for unrecoverable deterioration are normally:

• Increased tip clearances • Profile changes

2.2.1 Increased tip clearances

The tip clearance will have a very strong influence on the individual stages’ pressure raise capability. The Koch method [7] for determination of maximum pressure rise capacity shows

8 Meridional velocity is defined as: 2 2

m a rc c c= +

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the impact from various compressor parameters like the clearance. The Koch method is very well-regarded and widely used in the industry. The Koch method is based on the equation:

, ,, , ,Re

p p pp MAX p D

p D p D p D z

Tip clearance

C C CC C

C C Cε ∆

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ⋅ ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (2-2)

The solution of the preceding equation is straightforward when solving for maximum9 capability and the second factor with index ε is taken from the figure below:

Figure 2-4, Tip clearance correction on pressure rise capability [6].

The choice of staggered spacing as ordinate is clever since it is essentially constant over the load range (only small changes in deviation). The lost pressure rise capability will naturally reduce the surge margin of the compressor. Reference [9] reports a 15% reduction in pressure ratio when the tip clearance increased from 1% chord to 3.5% chord. The impact on the efficiency is more complicated to show since the Koch and Smith procedure is solved iteratively. In this model, the secondary loss is the combined effect of the end-wall and tip-clearance loss. The procedure is quite involved and only fragments of the procedure will be presented here. Reference [7] gives excellent coverage on this topic together with an example of how to use the Koch and Smith procedure.

9 Stage stalling pressure raise

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The method is based on the correlation for the free stream efficiency relative to the efficiency with end-wall effects:

( )( )

* *1

' 1h t

matchh t

hK

h

δ δηη υ υ

− += ⋅

− + (2-3)

Where δ is the displacement thickness and υ is the defect in the blade force due to the presence of the wall boundary layer. Both of these are taken from figures in the Koch and Smith model [7] and the details of how to do it will not be presented here. To give some flavour of it, the value of the displacement thickness (δ) is a function of the ratio of the pressure ratio coefficient and the maximum ditto, and the tip clearance divided by the staggered spacing. The actual pressure rise coefficient is a function of the efficiency, blade speed and change in tangential component. Simpler approximations like 2 points drop in stage efficiency for an increase in rotor clearance of 1% height will normally give similar results [13].

2.2.2 Profile changes

It is also possible to introduce surface roughness into the profile loss calculation. Until now, little has been said about the profile loss calculation and the procedure will be briefly described. The Koch and Smith procedure is an evolved form of Lieblein’s correlation, where the equivalent diffusion factor and corrections are used to correlate the profile wake momentum thickness and form factor:

( )max 2 1, , Re,c

f C C M AVDRH

θ ⎫=⎬

⎭ (2-4)

The Cmax/C2 value is replaced by the empirical diffusion factor at low incidence (design) levels. The diffusion factor correlates the change in velocity very well for design condition and has been widely used since the 1950’s. The diffusion factor is primarily used during the design to set the number of profiles in a stage. There are other definitions [6] but the following form is the most common:

2

1 1

12

CCDC C

θ

σ∆

≈ − + (2-5)

The calculated displacement thickness and shape factor is then used to calculate the profile loss coefficient:

( )2

13

2 2

2

2 3 1cos2cos cos

1cos

H HK

c Hc

αθ σα α θ σ

α

−⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎧ ⎫⎝ ⎠ ⎛ ⎞−⎨ ⎬⎜ ⎟

⎝ ⎠⎩ ⎭

(2-6)

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The loss at other incidences, other than design, may be calculated by introducing a factor which takes into account the velocity distribution at off-design conditions. The Reynolds number correction in the equation (2-4) can be taken from the figure below:

Figure 2-5, Momentum thickness ratio versus Reynolds number and surface roughness [8].

To further illustrate the influence of surface quality, reference [9] reports decreased profile efficiency from 98% at ks/s=0.3⋅10-3 to 88% at ks/s=5⋅10-3 for a NACA 65-family profile. The individual stage characteristics are used to get the overall compressor characteristics in a compressor stage-matching program. This kind of 1-D program stacks the individual stages’ characteristics into the whole compressor characteristics. Another possibility is to use a two-dimensional SCM throughflow program directly and calculate in two dimensions. Singh et al [10] took a simpler approach and, instead of using Koch and Smith, they used simpler relations for roughness and clearance. They also investigated the effects of increased leading- edge bluntness and loss of blade chord due to erosion. It is important to recognise the effect of the end-wall boundary layers when modelling compressor performance. The effect is basically two-fold, the first is the end-wall blockage itself and the second is the resulting peaky velocity profile. This is virtually impossible to calculate accurately with a 1-D program, and one has to use a throughflow program to calculate the end-wall blockage with relevant precision. The peaky velocity profile also reduces the work input due to increased mid-span velocity. At first glance, the net impact on the work level seems to match the mid-span theoretical value, and hence cancels out the effect. This is not true and the work level is reduced as the profile gets more peaked. The cure for this in the early days was to introduce the work factor [11], [34]. This factor has the value of unity at the first stage and reduces to approximately 0.86 well inside the flow path. Another possibility for calculating blockage is to use the old rule of thumb [11]: 0.5 percent blockage per row until 4 percent is reached. This figure, however, seems low for a modern highly-loaded compressor and values in the range of 8 – 10 percent is not uncommon [12]. The normal approach in a throughflow program is to use a modified correlation from Schlictling10 [13] and calculate the momentum thickness:

0.53.4 0.2 4

1 0.016m mc C c dmθ ν−= + ∫ (2-7)

10 Boundary Layer Theory, early German edition

2-12

The use of the preceding equation is somewhat stretched and was originally used as a correlation for turbulent flows over flat plates. The blockage is then calculated by assuming a shape factor (H) of 0.7 [13]. The true characteristic of the blockage is some kind of repeating pattern, and one should use all presented methods with caution, so as not to render a faulty calculation. The leading edge thickness or bluntness is indeed important for transonic stages but will also have an effect on incidence losses on sub-sonic stages. Erosion problems in industrial turbines are not a severe problem since most units have effective inlet filtration systems. Increased trailing edge thickness should not be so much of an aerodynamic problem as a structural one. The reason for this is simply that the wake behind a profile is dominated by the thick suction side boundary layer. Reference [13] provides the equation for introducing the trailing edge thickness into the loss calculation:

( )

( )( )

2222

2122

2 22

2

1 cos 1 *cos 1 2cos 1 * *cos 1 *

*sin 11 *

TE

TETE

TE

H

K HH

H

β θ δβ β θ δ θβ θ δ

θβθ δ

⎧ ⎫⎪ ⎪

+ − ⋅ −⎪ ⎪⎧ ⎫⎛ ⎞ ⎪ ⎪⎪ ⎪= − − ⋅ − −⎨ ⎬⎨ ⎬⎜ ⎟− ⋅ −⎝ ⎠ ⎪ ⎪⎪ ⎪⎩ ⎭

⎡ ⎤⎪ ⎪− −⎢ ⎥⎪ ⎪− ⋅ −⎣ ⎦⎩ ⎭

(2-8)

The compressor performance is crucial for the whole unit’s performance. A reduction in compressor isentropic efficiency of 1 %-unit may give a reduction in output by some 2.5%. The presented figure is valid for a two-shaft unit on nominal firing temperature in both cases.

36

38

40

42

44

46

48*10 3

Sha

ft P

ower

Del

iver

ed [k

W]

-6 -4 -2 0

Delta Compr Capacity [%]

Delta Compr Capacity [%] = 0 ... -5 Delta Compr Efficiency [%] = 0 ... -5

0.365

0.3675

0.37 0.3725

0.375

0.3775

0.38

0.3825

Dotted Lines = Thermal Efficiency

d_W_HPC = 0d_W_HPC = -0.5d_W_HPC = -1d_W_HPC = -1.5d_W_HPC = -2d_W_HPC = -2.5d_W_HPC = -3d_W_HPC = -3.5d_W_HPC = -4d_W_HPC = -4.5d_W_HPC = -5

Figure 2-6, Output and efficiency versus decreased compressor

capacity and efficiency for a two-shaft unit, produced with GasTurb.

2-13

The figure above is produced with the commercially available gas turbine program GasTurb 9 Pro. These types of plots are, however, very engine-dependent and it is necessary to produce this type of for each engine.

2.2.3 Index of compressor sensitivity to fouling

The fouling sensitivity of a compressor can be described by the Index of Compressor Sensitivity to Fouling (ISF). This method was put forward by Tarabrin et. al. [14] and the ISF is defined as:

62

3

10

1

p AVG

HUBTIP

TIP

m c TISF

rDr

−⋅ ⋅ ∆= ⋅

⎧ ⎫⎛ ⎞⎪ ⎪⋅ −⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

(2-9)

The equation shows that a high loading and a small channel dimension have an injurious effect on the fouling sensitivity. The derivation is lengthy and the reader is referred to the original source for further information. Some of the assumptions made during the derivation are based on rather old “rules of thumb” and may not be up-to-date. One example is use of the old expression for solidity from Howell, which may be incorrect for modern low-aspect ratio blading. Tarabrin et. al. also compare three different engines in their paper, showing that small channel dimensions and highly loaded compressors are more sensitive to fouling. In the original ISF formula above, stage load appears in the numerator and it may be impractical to use that figure. It may be convenient to re-write the original equation to:

1

22

3

* 10

1

stagesn

HUBTIP

TIP

mISFrDr

π −⋅= ⋅

⎧ ⎫⎛ ⎞⎪ ⎪⋅ −⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭

(2-10)

The modified equation will provide the same information as the original, but is somewhat easier to evaluate. The work by Aker and Saravanamutto [15] shows, contrary to Tarabrin, that a larger unit is more sensitive than a smaller unit. They also indicate that the loading is the true governing parameter and the geometry is of second order. Both sources compare the same engines, Solar Centaur and General Electric LM2500 (with twice the stage loading of the Centaur), with contradictory results as already mentioned. The reason for this may very well be the empirical assumption for the solidity.

2.2.4 Compressor washing

The simple cure to compressor fouling is washing. Washing may be either on-line or off-line, where the first is carried out when the unit is in operation and the latter is carried out when the unit is in cold condition. The off-line, or soak wash, is the most effective one but the unit must be shut down and cooled off. The cooling period is dependent on the individual type of

2-14

engine, and varies from a very short period for an aero-derivative to 8 hours for a heavy industrial design. The important thing is that the temperature must be below 100°C everywhere inside the unit due to the risk of high stress concentrations. This non-production duration may be unacceptable for some operators. The on-line washing was introduced to solve this problem. It is indeed important to recognise that the on-line wash cannot replace the off-line wash, only extend the operation periods between off-line washes. There are risks involved with on-line washing; clogging of the secondary system and high temperature corrosion. The problems occur due to the fact that the on-line method cleans the first (and normally most fouled stages) without draining solved deposits out from the compressor. Instead, the solved deposits will follow the air through the engine and may appear in a more critical location, like the cooling passage in a turbine blade. Typically, the on-line wash must be carried out once a day to prevent larger amounts of deposits from passing through at one time. If one on-line wash is missed, then a soak wash must be performed before any further on-line washing. The soak wash is typically carried out at purge speed, but can be significantly improved by introducing variable speed. The principle of variable speed is that the amount of drain through the bellmouth is higher at lower speeds. This means that normally relative more-fouled front stages can be washed with the drain leaving the engine through the bellmouth instead of downstream drain points. The optimal procedure seems to be when injection is started at barring speed and the acceleration slope is set so that half the fluid is injected before purge speed.

Figure 2-7. Typical fouling distribution [16].

The figure above also shows the idea behind on-line washing. This method is only effective on the first stages, which normally are the most fouled. There are many suppliers of washing equipment with different types of wash units. Some advocate for high-pressure units and others for low-pressure units. However, no technology seems superior to the other. But one important design feature is capacity. If the unit is too small compared to the gas turbine, the wash unit may be incapable of removing the injected detergent from the compressor and hence, leave a sticky surface. One can also argue if it is necessary to heat the wash/rinse fluid or not since it generally prolongs the wash cycle quite

2-15

substantially. The cost of a designated wash unit may be rather high, so the use of a commercial general-purpose industrial high-pressure wash unit may be an attractive alternative. This may also reduce the wash cycle duration since the commercial units generally are furnished with on-line diesel-fired heating units, and thus eliminate the need to waiting for the internal heater. Another inherent advantage is that there is no limitation to a certain volume of e.g. rinsing fluid. It is possible to rinse as long as necessary until all traces of dirt and detergent are removed. Despite the use of wash skids, an old-fashioned hand wash of the IGV, rotor 1, bellmouth and struts may be the difference between a successful wash or not. There are several different nozzle configurations depending on the manufacturer of the wash unit. The figure below shows the configuration by one of the major companies:

Figure 2-8, Different nozzle configurations, courtesy of Turbotect Ltd.

2-16

2.3 Turbine degradation

The global turbine (or expander) performance is set by its efficiency and capacity. The latter is important since it more or less sets the cycle pressure ratio. The turbine swallowing capacity or simply capacity is defined as:

40

040

REF REFm COT RFNp R

κκ

⋅ ⋅=

⋅ (2-11)

This expression can be used to write the pressure ratio as:

( )30

020constantfuelm m COT

pπ

+≈

⋅ (2-12)

The constant in the preceding equation is the lumped effect of the turbine capacity and combustion section pressure drop. The relative impact can be shown after logarithmic differentiation:

40 020

40 020

12

m pCOT FNm COT FN p

ππ

∆ ∆∆ ∆ ∆≈ + − − (2-13)

The preceding equation shows that it is the flow passing through the turbine, and the firing temperature to a lower extent, that sets the units pressure ratio. If the capacity changes, however, the pressure ratio will change at the same relative magnitude, e.g. an increase of 10% turbine capacity will reduce the cycle pressure ratio by 10%. It is not uncommon for new designs to have capacity errors, which may in some cases be in the order of 10 per cent. It is therefore quite common, even for competent designs, to re-stagger the turbine during prototype testing. The un-cooled turbine stage efficiency can be written as:

( )

1

2 23 2 3 2

1 3

12

reheat

R NT T

w c T Th h

ζ ζη

−

−

⎧ ⎫⋅ + ⋅ ⋅⎪ ⎪= +⎨ ⎬−⎪ ⎪

⎩ ⎭

(2-14)

The loss coefficients (ζ) in the preceding equation are the energy loss coefficients for the nozzle (N) and the rotor (R), respectively. The energy loss coefficient is defined as the difference between the actual and the ideal endpoints. The energy loss coefficient for the nozzle is defined as:

2 2, 2 2,2

02 2 2 2s s

N

h h h hh h c

ζ− −

= =−

(2-15)

2-17

The same equation applies for the rotor if the absolute enthalpy and velocity are replaced with the corresponding relative ones. The energy loss coefficient can be translated into row efficiency with the relation:

1 11N or R N or R

N or R

η ζζ

= ≈ −+

(2-16)

The stage efficiency for a cooled stage is more complicated:

1 1

, ,1 1

T

out outin in p in out c c p c out

in c

power pumping power disc windage

p pm T c m T cp p

κ κκ κ

η− −

→ →

− −=

⎡ ⎤⎧ ⎫ ⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪ ⎪ ⎪⎢ ⎥⋅ ⋅ − + ⋅ ⋅ −⎨ ⎬ ⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎪ ⎪⎢ ⎥⎩ ⎭ ⎩ ⎭⎣ ⎦

∑

(2-17)

The power term in the equation above is normally evaluated with a through flow program or an advanced mid-span program. The pumping power term is the work input to the rotor cooling air as it passes through the disk and the blade. The pumping work is normally defined or calculated as: 2

2 1 ,1pumping work unit mass u u cθ≈ − ⋅ (2-18) The second term is the tangential momentum at the disk inlet at some radius. The pumping work can be reduced if this term is positive11. This is one of the reasons for using a swirl generator before the first rotor.

Figure 2-9. Cooling air swirl generator

The disk windage term is normal friction, which gives a break effect on the turbine disks [64]. The disk cavities are normally purged with cooling air to avoid increased temperature levels. The amount of cavity purging is typically in the order of 0.5 –1.0 % per disc [29].

11 The effective cooling air temperature into the disk can be minimized by minimizing the relative vector into the turbine disk since: 2 2

0, 0 2 2RELp p

c wT Tc c

= − +

2-18

Since the profile losses are associated with the profile boundary layers, there is another connection between losses and the capacity. Losses will reduce the pressure and hence the density, but the boundary layer blockage will also reduce the effective flow area. Separated flows in the turbine will have a powerful blockage effect. The flow (non-separated) through a row can be described with the equation [17]:

( ) 0 0

psss

th th ss psm c o u dy u dyδδ

ρ δ δ ρ ρ= ⋅ ⋅ − + + ⋅ + ⋅∫ ∫ (2-19)

By introducing the displacement thickness (δ*), it is possible to re-write the equation to yield:

*1th thm c oo

φ

δρ

=

⎧ ⎫= ⋅ ⋅ ⋅ −⎨ ⎬⎩ ⎭

∑ (2-20)

In the preceding equation, the bracket represents the blockage factor (φ). The blockage factor can be solved with standard methods and the reader is referred to standard fluid mechanics textbooks for further reference. Reference [17] provides an empirical equation for the blockage factor as a function of the profile loss: 1 0.56 PYφ = − ⋅ (2-21)

2.4 Causes for turbine deterioration

Typical causes for deteriorations are:

• Surface roughness

• Tip clearance

• Trailing edge thickness

• Leading edge thickness (off-design)

• Profile changes

• Secondary- and cooling-air The widely used AMDC+KO+MK model [21, 24] has been used, together with others, to explain the impact on the turbine performance. The reason for introducing other loss models is that the mentioned model sometimes is insufficient when modelling certain aging effects. The AMDC+KO+MK model is a total pressure loss model, and it may be convenient to recalculate the total pressure loss into entropy changes:

0 0

01 01

ln 1 p psR p p

⎛ ⎞∆ ∆∆= − − ≈ −⎜ ⎟

⎝ ⎠ (2-22)

2-19

The used loss model is further described under section 4. Recalculation between total pressure loss coefficients (Y) and energy loss coefficients (ζ) is possible with the equation:

2

12MY κζ

⎧ ⎫⋅≈ +⎨ ⎬

⎩ ⎭ (2-23)

The equation above is only valid, from a rigorous point of view, for sub-sonic flow. The corresponding equation for supersonic cases can be found in [18].

2.4.1 Surface roughness

Surface roughness increases profile losses through increased dissipative shear work in the boundary layer. The widely used AMDC+KO+MK12 [24] model is not suitable in this case since the surface roughness is not a parameter in this model. Instead, the Traupel method [62] has been used to display the impact from surface roughness.

Figure 2-10. Influence for surface roughness [62].

The original intension of the model/figure (2-10) is to use the Rχ as a factor for the profile loss in the equation (N.B. curve “a” in the figure 2-10 is valid for turbines): ,0p R M p h Cζ χ χ ζ ζ ζ= + + (2-24) One additional feature in the Traupel model is the loss due to end-wall surface roughness. Traupel provides two sets of equations where the first is valid for shroudless types of turbines:

11

1

1cos

f sa

m

c lD l

δζα

⎛ ⎞= +⎜ ⎟

⎝ ⎠ (2-25)

12 Ainley-Mathiesson-Dunham-Came + Kacker-Okapuu + Mousthapa-Kacker

2-20

This is simply added to the calculated secondary loss and the impact is largest when the swirl component is large (cosine for the outlet angle in the denominator). Reference [18] provides a modified version for the second set for shrouded gas turbine expander designs:

( )cos

f aa

cl

δζ

α= (2-26)

The AMDC-KO-MK model will only take into account the Reynolds number and not the surface roughness itself. The surface roughness is assumed to have a certain value in the AMDC-KO-MK loss model. It is also possible to quantify the impact from localised sections of increased surface roughness. Denton13 [19] shows that the entropy generation14 in the blade can be estimated with the expression:

( ) ( )

31

0

2cos

SS d S

REF ref

Staggered spacing

C VC d x CS V

ζα

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠∑ ∫ (2-27)

The expression shows that the generated loss is proportional to velocity at the power of three. The velocity at the suction side is always higher than at the pressure side, hence higher loss generation on the suction side. Some 80 percent of the profile loss is generated on the suction side. The dissipation coefficient15 is normally based on displacement thickness as characteristic length.

2.4.2 Tip clearance

Increased tip clearance, with its associated loss, can be modelled by the AMDC-KO-MK system. In this model, the loss is calculated with the equation:

20.78 2

32

cos'cos

LTC

m

Ainley loading parameter

Cc kY Bh c s c

ββ

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

(2-28)

Where:

( )0.42' kknumber of strips

= (2-29)

The previous is valid for turbines with shrouds (B=0.37) and the following equation is used for shroudless turbines:

( ),0

10.93cos

tipT T

T T m out

rr h

η τη β

−

−

⎛ ⎞⎛ ⎞∆ ∆= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(2-30)

13 Commonly referred to as the Denton U3-method 14 Entropy loss coefficient 15 Typically in the order of 0.002

2-21

The preceding equation is solved iteratively, but equation (3-15) can be used for turbines without shrouds if B=0.47 is used. The figure below shows that this type of correlation is valid for various types of configurations (e.g. casing treatment, squilers, tip blowing).

Figure 2-11. Tip clearance loss [20]. Equation (2-30) gives an influence factor of approximately 3 for a normal turbine design while the figure above gives an influence factor of 2.

2.4.3 Trailing edge thickness

The trailing edge thickness, or more correctly, increased thickness, will affect the machine in two ways. Blade losses will increase when the wake gets wider and the Borda-Carnot shock will increase. The second effect is that the throat will increase and hence have bigger swallowing capacity. The design level of trailing edge thickness is set by structural limitations. The thickness is typically 1 mm for uncooled blades and in the order of 2-3 mm for a cooled blade. The loss due to increased trailing edge thickness can be assessed with the loss model:

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Trailing edge thickness/Throat thickness (TE/O)

∆Ete

Axial entry nozzle, α′in

=0Impulse blading, α′

in=α

out

Figure 2-12. Trailing edge energy loss, after [21].

2-22

The two curves are for axial entry nozzle and impulse type of blading; interpolation is made with the expression [21]:

' '

IMP IMP NOZZLE

in inTE TE TE TE

out out

E E E Eα αα α

⎛ ⎞∆ = ∆ + ∆ − ∆⎜ ⎟

⎝ ⎠ (2-31)

Transformation to pressure loss coefficient is done with the following expression (assumes that ∆ETE is the energy loss coefficient based on ideal or isentropic velocity) [21]:

12

2

1 11 1 12 1

11 12

OUTTE

TE

OUT

ME

YM

κκκ

κ

−−⎧ ⎫⎛ ⎞−⎪ ⎪+ − −⎨ ⎬⎜ ⎟− ∆⎪ ⎪⎝ ⎠⎩ ⎭=

−⎛ ⎞− +⎜ ⎟⎝ ⎠

(2-32)

Another common method is to evaluate the base pressure in the wake and calculate the loss. This method is described in reference [19] and will not be repeated here. During the design phase, increasing the throat can reduce the relative impact of the trailing edge thickness. This can be realised by study figure (2-11), but there are other prevailing limitations. The outlet angle is more or less set by the ratio between the throat and the pitch. If the relative impact from the trailing edge is to be minimised while maintaining the same angle, the pitch must be increased. If the pitch is increased, the aerodynamic loading will be higher. This is normally not a desirable feature and is counteracted by increasing the blade chord. Increasing the chord will lead to increased secondary losses. This can be realised by studying the Zweifel number, which is some gauge of the blade loading in terms of the tangential aerodynamic force divided by the ideal:

22 1 22 cos tan tansZ

bα α α⎛ ⎞= −⎜ ⎟

⎝ ⎠ (2-33)

The Zweifel number is rather old (1945) but is still well-regarded. The level of the Zweifel number has increased over the years from 0.8 in 1945 to 1.0-1.2 for modern designs. The physical interpretation is that the parameter is a measure of the likelihood of suction side diffusion. A higher value normally means lower efficiency, whilst a low number leads to much wet surface in the turbine. One problem with turbines is if some portion of the blade trailing edge is lost. Then capacity will be increased as the throat moves into the passage. This may introduce additional problems with the unit’s control algorithm, and the firing temperature will be reduced since the expansion line is shortened.

2-23

Figure 2-13, showing the increased capacity when material is lost [28].

The theory behind this model is not fully rigorous, but is still very useful from an engineering perspective. This model was developed for steam turbines but there is nothing that indicates that it should not be valid for a gas turbine. K.C. Cotton [28] presents this method in his recognised textbook, “Evaluating and Improving Steam Turbine Performance.” The classic Cotton book was originally focused on large GE low-reaction utility turbines. There is nothing, however, that indicates that it should not work for any turbine regardless of reaction level.

Figure 2-14, showing the increased capacity versus missing profile

section and swirl angle16, based on backbone length [28]. The figure above clearly shows that a high swirling stage is more sensitive to changes in capacity than a low swirling stage. The state-of-the-art design philosophy is to use a highly loaded low reaction stage. This means that the flow angle out from the first vane may be as high as 75°. The reasons for this approach are basically twofold; firstly, higher loaded17 stages in the highest temperature level means fewer cooled stages, and secondly, a reduction in reaction from say 50 percent to 20 percent may give a lower relative temperature of some 16 Based on the tangential direction. 17 Typical loading for a 1st stage is in the order of 2.0 while middle stages are normally in the range of 1.6 – 1.0.

2-24

30°C. It is also possible to show that a lower reaction gives less root stagger [21] and hence lowers fixation stresses [22].

2.4.4 Leading edge thickness

Decreased leading edge thickness will increase the sensitivity for incidence. When the profile is approached by an angle deviating from the design, the stagnation point will be moved away from the physical training edge. This will give an “overspeed” effect due to the big curvature at the leading edge, and hence, zones of diffusing flow after the front section [23]. This “overspeed” may also introduce unwanted compressibility effects in the front section of the profile.

Figure 2-15. The effect of ±1° incidence on profile velocity distribution [20].

This effect is amplified as the leading edge gets thinner and hence higher profile- and secondary-losses with eroded trailing edges.

2-25

−500 0 5000

0.02

0.04

0.06

0.08

0.1

0.12

x′

∆φ2

Profile loss

−0.4 −0.3 −0.2 −0.1 0 0.1 0.20

1

2

3

4

5

6

x′′

Ys/Y

s,de

sign

Secondary loss

Figure 2-16. Incidence losses, based on reference [24].

Where:

( )21.6 '

,cos'cos

eff

inin in design

outi

dxs

α α αα

−− ⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

(2-34)

1.5 0.3' '

'

cos''cos

i

in in in

in out out

dxc

α α αα α α

−⎛ ⎞− ⎛ ⎞= ⎜ ⎟ ⎜ ⎟− ⎝ ⎠⎝ ⎠

(2-35)

The off-design profile loss is a function of leading edge diameter (d), turning, and effective incidence18. The same is true for the secondary loss multiplier. Tests in GE’s aerodynamic lab show similar results [28]:

Figure 2-17. Efficiency loss versus incidence angle [28].

18 Positive incidence at higher pressure ratio (load) or lower speed and vice-versa.

2-26

2.4.5 Profile changes

A profile shape change affects the velocity distribution around the profile. The profile section loading is determined by the curvature distribution in the individual section. When a profile is designed, the most important feature is to achieve continuous curvature (2nd derivative) since discontinuities will increase the profile loss. When the shape of a profile changes, the velocity distribution will be changed. An especially undesirable feature is suction side diffusion after the throat. The exact amount of acceptable diffusion varies between companies, as does the turbulence level in some cases. Transonic turbines are even more sensitive, since a shock wave interaction may separate the boundary layer with high losses and blockage.

Figure 2-18. Example of passage Mach number distribution.

2.4.6 Secondary- and cooling air

Changes in the secondary air system may be either a severe problem with over-heated sections or low efficiency/high blockage due to inlet of cooling air into the flow path. Hot components that are not receiving sufficient cooling will soon be beyond repair. Increased cooling airflow, regardless of origin, will have a deleterious effect on performance. The lost performance is caused mainly by gas/air not doing work, and the spoiling effect when the low momentum cooling air is mixed into the high momentum main flow. There are established principles for when cooling air is assumed to do work or not. Reference [29] gives an exhaustive description and all the details will not be repeated here. The mixing loss can be calculated according to Hartsel [25]:

0,20

0 0

1 2 cos2

cc cc

Tp m WMp m T W

κ φ⎛ ⎞∆

= − + −⎜ ⎟⎝ ⎠

(2-36)

Shapiro originally derived the preceding equation for entropy generation. This equation must be integrated over the profile surface in order to get the loss due to mixing. It is therefore,

2-27

from a numerical sense, a rather complicated task to calculate the integrated loss for a specific profile. Wei [18] approached this problem using a generic velocity distribution in the manner after Denton [19]. This approach is optimal for neither 1-D mid-span calculation nor 2-D throughflow calculation environments, and should preferably only be used in very detailed models. Sir Horlock [26] describes a method to solve equation (2-36). The method is fully described in the mentioned reference and only some features are described here. Equation (2-36) is used to calculate19:

0

0

cp mKp m

∆= − (2-37)

It is now possible to introduce the relation between pressure loss and efficiency change [26]:

1' 1 1stg stg

stgo o

K PRK p p

κκ

η η κκ

−∆ ⎧ ⎫−= ≈ −⎨ ⎬∆ ⎩ ⎭∑

(2-38)

It is now possible to evaluate the relation:

'stg c

stg

mKm

ηη∆

= − (2-39)

Values of the influence coefficient (K’) can be found in several sources and the figure below is based on [22]:

0 0.5 1 1.5 2 2.5 3−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Cooling flow [%]

Effc

ienc

y re

duct

ion

[%]

PlatformFirtreeShroud

0 0.5 1 1.5 2 2.5 3−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Cooling flow [%]

Effc

ienc

y re

duct

ion

[%]

ShowerheadTrailing edgeTip ejection

Figure 2-19. Cooling air efficiency spoiling exchange ratios, based on [22].

19 Or more correctly, equation (2-36) is used to evaluate K in equation (2-37).

2-28

It should be noted that the sign of the coefficient for trailing edge ejection might be negative (hence positive impact) in some cases, due to better/less negative base pressure in the wake [19]. Typical reasons for increased losses and lost work due to increased secondary air are:

• Worn central casing sealing with massive flow through rotor 1 rim

• Diaphragm leakage (combustion section and turbine vane no. 1)

• Worn and/or damaged showerhead

• Wider trailing edge ejection slit due to increased trailing edge thickness

• Increased tip ejection flow due to rubbing

• Introduction of new cooling features in an old engine (e.g. platform cooling) The secondary air system (SAS) is normally calculated with a dedicated flow network program. There are different levels of sophistication for SAS codes and some take into the account added heat during the passage through the engine. Seals are normally calculated as Fanno functions while channels in rotating parts are calculated as solid body flow. The radial pressure gradient in the main flow path can have a powerful effect on the cooling air calculation.

Figure 2-20. Example of secondary air system [36].

3-29

3 Tools

3.1 Traditional Flow Path Analysis

Traditional flow path analysis tools have been used on industrial units ever since the mid- 1970’s, and they were essentially based on the individual companies’ rig evaluation programs. These types of codes were used on mainframe computers lacking user-friendly interfaces and were normally operated by designated people in the calculation departments. The advent of PC technology and user-friendly environments made the use of aerothermal codes possible even for non-experts e.g. station engineers. There are also a lot of third-party companies acting on the market today with varying success. The quality of third-party systems may very well be within the systems delivered by the OEM. There are, however, limitations in some third-party systems since the details of the secondary air system are a well-kept secret in the different OEM’s. The same is also true for an array of engine parameters such as loss and geometry parameters.

Most of the methods described in this section are applicable for performance testing, both “in situ” and production pass-of testing as well as monitoring.

3.1.1 Measured data

The measured data is the most important feature in the whole flow path analysis system. A modern industrial gas turbine is generally furnished with most necessary instruments. This means that one can evaluate quite a lot of engine parameters contrary to a flying gas turbine with its sparse instrumentation. A modern gas turbine is also more or less always controlled by a DCS system simply for cost reasons. This means that all parameters are available in an electronic form and it may be possible to install some kind of communication between the control system and the database. So there is in most cases plenty of data to process in the analysis system. One should always remember, however, that the measured parameters are operational measuring points and not calibrated code type of equipment. There are typically two problems with the measured parameters. The first problem is the fuel flow into the turbine and the second one is the exhaust temperature. The gas flow meter is typically a turbine wheel meter with some kind of flow computer. The origin of the problem is the calculated density since the meter measures volumetric flow. This density is in most cases calculated according to AGA-NX19 or a similar method from measured pressure and temperature in the meter. This method is very sensitive to measuring

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errors/uncertainty in each of the measured parameters. One nice thing, however, is if the meter is used by the gas supplier to measure the consumption, then the measuring quality is probably high. Some plants are using throttling devices to measure gas flow to the turbine. There are some advantages with this method, like the long-term stability if the geometry and the surface roughness stays the same over time. One drawback is the poor turndown ratio of the throttling device since half flow only generates 25% pressure drop of the full flow. It is not very likely that testing is performed below this flow level. However, if the flow device is designed with sufficient pressure drop at nominal flow, the part load should not be a major problem. It is possible to implement the calculation methods in ISO-5167 or corresponding ASME method into the flow path system. The general method implemented in the VDU is the simple equation:

REFTconstant pTGAS

REF

pmp

= ∆ (3-1)

The constant in the equation above includes gas compressibility, gas composition, and velocity profile. The latter is fairly constant over the flow range and the error is some 5‰ at 10% of nominal flow. This means that the method is usable if the compressibility and composition issue is solved. The necessary algorithms and coefficients for various methods (AGA NX-19, GERG) may be found in [27]. The preferred method for various gas compositions is the one of Soave-Redlich-Kwong but it is more complicated to implement. The use of general compressibility diagrams should be avoided since the uncertainty is very high. The uncertainty figure is typically in the order of 2 percent. One very attractive way of avoiding the compressibility issue is to use a coriolis force meter. This type of device measures mass flow directly and may be calibrated to very low uncertainty figures. One drawback, however, is the fact that all gas turbine test codes (ISO 2314 and ASME PTC-22) are quite old and nothing regarding this type of device is mentioned. However, this type of device is excellent for monitoring purposes. The exhaust gas temperature is complicated to measure in a way that is consistent over long time periods. The reason for this is that both the circumferential- and the radial-profile may change. Some gas turbine manufacturers measure the airflow directly instead of using a heat balance. This would not be a problem if high coverage rakes were used as operational instruments. One manufacturer recognises this problem and uses 16 probes with 3 span-wise locations for operation. Type K thermocouples (T/C) should always be avoided since they are prone to aging phenomena like short range ordering. The preferred T/C today is type N and one should always use premium grade quality. The measured electrical output is normally very accurate and not necessarily worse than test code type of calibrated instruments. Typically, the plant is furnished with class 0.2% uncertainty measuring transformers (10kV→110V and 5kA→1A). These are normally tested prior to delivery and the protocols may be available. The same transformers are used for performance/acceptance testing. The ASME PTC-22 requires calibration protocols and the normal procedure may be insufficient. Typical measuring uncertainty is in the order of 0.35% and 0.3% for operational instruments and test code type, respectively. This indicates that any uncertainty problems with power output, when used for monitoring purposes, should not be expected.

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The following list is typical for a single shaft unit: Measured parameter Operational ISO 2314 ASME PTC-22 Remarks Ambient temperature 1 ≥2

±0.5K Sufficient number to

reach less than 1F from traversed mean value

Total probes if dynamic component exceeds 0.5K

Ambient pressure 1 1 ±33Pa

1 ±0.04%

Relative humidity 1 1 ?

1 ±0.002

PTC-22: uncertainty figure for specific

humidity

Compressor inlet temperature

1 Same as ambient Same as ambient

Compressor inlet pressure

1×∆p 4×∆p 1 if <20 m/s

4×∆p ISO/PTC-22: Add dynamic component

Compressor discharge temperature

1 Design dependant -

Compressor discharge pressure

1 4 -

Exhaust temperature

≥16 ≥4 ±3K

Sufficient Maximum uncertainty for calculated firing temp.

PTC-22: uncertainty figure for calculated

firing temp less than 4 K

Exhaust pressure

1×∆p 4×∆p 4×∆p

Shaft speed

1/shaft only multi-shaft units

- -

Generator output

1 IEC-46 PTC-22 Page 13-14

Fuel flow

1 1 ±1%

1 ASME code

Lower heating value (Gas)

Gas supplier Calculate from sample

Cromatograph in situ ±0.4%

The typical resulting test uncertainty for an ISO test is in the order of 0.5 – 0.7 % on power output and in the order of 1.1 – 1.5 % on thermal efficiency. The resulting test uncertainty is calculated according to Gauss-LaPlace for a 95% confidence interval. The resulting uncertainty also includes errors from indirect parameters, i.e. parameters which influence factors not equal to unity. This uncertainty may sometimes be used as the test tolerance depending on the contract. This procedure is in most cases not accepted in the US (ASME test codes) and uncertainty is completely separated from the test tolerance, which is considered a commercial issue. The following lines are from Ken Cotton’s book “Improving and evaluating Steam Turbine Performance” [28]: “The translation of test uncertainty into tolerances, which are then used to modify test results, have no logical basis and are unfair to the user. Uncertainties in test codes are usually stated at a 95% confidence interval. The ±1% test uncertainty means that there is a 95% probability that the “true” value falls within 1% of the measured value. If the measured value falls 1% short of guarantee, then the likelihood that the true value is better than guarantee is 2.5% and the likelihood that the true value is poorer than guarantee is 97.5%. Permitting any test tolerances rewards the manufacturer with odds that are overwhelmingly in his favour.”

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3.2 Modelbased evaluation tools

3.2.1 Compressor flow

The backbone in the whole system is the determination of the mass flow at all relevant engine stations. There are basically four different approaches for calculating compressor mass flow:

• Heat balance

• Bellmouth depression

• Exhaust gas composition

• First nozzle swallowing capacity The first and second methods are the most widely used and most accurate under normal circumstances.

3.2.1.1 Heat Balance

The compressor mass flow is calculated with a simple heat balance. The calculation is based on measured parameters as well as on engine characteristics.

( )( ) ( )

5010

50 101F CC F SHm LHV h h P Q

mh h hχ

ηχ χ

⋅ + − − −=

⋅ + − −∑

∑ ∑ (3-2)

The equation above must be solved iteratively since both combustion efficiency and exhaust gas enthalpy are based on the compressor flow to varying extents. The inlet enthalpy is based on the total temperature at the compressor inlet and is calculated with standard thermodynamic relations:

( )10 10 0, , ,generic air compositionh f T RH p= (3-3)

The exhaust enthalpy is a more complicated since it is a function of the combustion process as well as the air composition.

( )50 50 ,, , fuel composition, ,air compositionF air CCh f T m m= (3-4)

The shaft power is calculated by adding the alternator and gear losses to the measured electrical output:

, ,SH GEN loss GEN loss GEARP P P P= + + (3-5)

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The gear loss is composed of the idle loss (60%) and the load dependency is 40%. It is normally satisfactory to assume a linear relation [29]:

, , 0.6 0.4DES SHloss GEAR loss GEAR DES

SH

PP PP

⎧ ⎫⎛ ⎞⎪ ⎪= +⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

(3-6)

The alternator loss is, on the other hand, strongly load dependent. The normal practice is to use the manufacturer’s generator efficiency curve for the generator loss. These curves are normally produced for two power factors (0.8 and 1.0), but the following equation [30] can be used for recalculation.

1 0.0186 lnGEN RATEDPF

RATED

P PFKPF S PF

⎛ ⎞= + ⋅ ⋅ ⎜ ⎟⋅ ⎝ ⎠ (3-7)

The gas turbine mechanical losses are calculated solely on an empirical basis and all losses are normally lumped into one loss. It is common practice to establish the actual mechanical loss during engine prototype testing. That figure will probably last through the life-span for that turbine type. The bearing losses are calculated in the design phase, but the actual loading for axial bearings may be different from the original design value. The loss for a hydrodynamic bearing can be calculated with the equation [29]:

3.95 1.75 0.4 3, 0.0432 7.93 10f bearing OILP D n D n Vυ −= ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ (3-8)

It is therefore convenient to introduce the simple relation for the mechanical losses:

1.75

,f mechP const n= ⋅ (3-9)

The constant in the equation above is the mechanical loss at the design speed divided by the design speed raised to the power of 1.75. The two previous equations clearly show that the mechanical loss is constant for a single shaft power generation machine.

The overboard leakage is normally in the range of 0.5-1.0% of the compressor inlet flow. This air is used for preventing lube oil from leaking out from the bearing chambers. This air normally leaves the control boundary through the oil mist separator and fan. The enthalpy of the overboard leakage is in most cases a function of relative temperature lift: ( )30 20 20h h h hχ χ= − + (3-10) All thermal radiation is assumed to originate from the combustor section. Even though this assumption is not true from a strict rigorous point of view, it considerably simplifies matters like definitions of turbine efficiency. The radiation loss is proportional to the absolute flame temperature raised to the power of 4, but it is common practice to assume that it is a fixed fraction of the fired heat. The combustion efficiency can be determined in different ways. The most common is to use an empirical function of firing temperature. Another possibility is to measure UHC (Unburned

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Hydro Carbons) and CO in the exhaust gases and calculate the energy flow confined in the UHC and CO:

( ) 5

, 10.1 45.48 101 EXH dry

CCF

m CO UHCm LHV

η−⋅ ⋅ + ⋅ ⋅

= −⋅

(3-11)

The combustor efficiency should be very close to unity (99.99%), for a well-designed combustion system at full load. Typical CO levels are in the range of 25 ppmv with modern combustor sections. The part-load combustion efficiency is strongly determined by the matching of the whole engine. A single-shaft turbine with IGV’s controlling the airflow can maintain a high fuel-to-air ratio at part load, whilst a multi-shaft engine has to reduce the firing temperature. There are gas turbines with combustor by-pass systems that can maintain a high firing temperature at part load. The compressor mass flow is calculated with an iterative method shown in the figure. There are other approaches to solve the equations system, but the sequential approach seems to be the most widely used. The general practice is to use a nested loop method if the number of loops is lower than five, and to use a matrix method if the number is higher than five. The indicated start values are typical but it is also possible to implement a start value for the compressor flow. This is normally done for multi-shaft units by using the relation between referred parameters. The procedure makes use of the correlation between referred power and referred flow. This method will also work on a single-shaft unit (fixed physical speed) if some auxiliary parameter is introduced (e.g. ambient temp), but the accuracy is questionable. The accuracy of this method is strongly dependant on the exhaust temperature measurement. The exhaust temperature profile may change due to changes in the flow path and combustor section. If the operational probes measure the temperature at one radial position, then the section is sensitive to changes in temperature profile. If one assumes that the bulk or mass flow weighted average temperature is the same but with a different span-wise profile, the measured value at the measuring section will give a different value. This profile is not the same even on engines of the same type. This is the reason for some companies to set the firing temperature to its nominal value at the site. During a performance test, the engine is furnished with rakes in the exhaust. This measurement is supposed to give the true mass flow weighted average. The difference between this and the operational instruments is in some way used to adjust the control system in order to operate at the nominal firing temperature. This difference is crucial for evaluating the flow through the engine in e.g. the flow path analysis system. If this profile changes, the evaluated mass flow will be wrong and it may be quite tricky to track down the problem. Typical exchange rates are 1.2 to 1.5° SOT/1° exhaust temperature and -0.10 to -0.18/1° exhaust temperature. Another problem has already been mentioned and that is the maximum firing temperature controller. The firing temperature controller limits the

ok

h50 start value (typically 1.1 times T50)

ηCC start value (typically 0.9999)

solve flow eq. (eq 3-1)

COT-calculation (combustor model)

ηCC

ηCC

h50

ok

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maximum firing temperature at a certain value and may have different levels of sophistication. If the profile changes, then the maximum firing temperature will be different from the nominal. This change may be positive or negative and the result is either shorter hot section component life or lost production revenues. The measured fuel flow is indeed important and small errors will propagate through the whole set of calculations. Fortunately, this parameter may easily be replaced by an iteratively calculated fuel flow. If the exchange rate between the measured fuel flow and the high turbine swallowing capacity is known, it is possible to get a converged solution in a few iterations. See further details in the proceeding section.

3.2.1.2 Bell mouth depression

This method of determination of the compressor flow is widely used and is straightforward without iterations. One important advantage is that it is independent of the measured exhaust temperature. However, it is necessary to calibrate the flow function for the specific engine during either a production pass-off test or a DIN 1943/ASME PTC22 field test. The bell mouth method is preferred for test bed evaluation by some engine manufacturers. The key issue here is the calibration of the bell mouth. Some OEM’s have used very detailed traverses and/or CFD calculations to calibrate the flow function. The pressure taps require special attention when using CFD based “calibration” since the tap shape is important. The flow can be calculated with the following expression:

( )

2 1

1010

10 1010

1 2 1 11

EFF BM BMA p p pmR p pT

κκ κκ

κ

+⎧ ⎫⎛ ⎞ ⎛ ⎞⋅ ∆ ∆⋅ ⎪ ⎪= − − −⎨ ⎬⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

(3-12)

The effective area is basically the physical area minus the boundary layer displacement thickness. The effective area is also taking into account that the flow is not isentropic, but this effect is one order of magnitude less than the boundary layer displacement thickness. The effective area is usually correlated against the relative depression:

10

BMEFF

pA fp

⎛ ⎞∆= ⎜ ⎟

⎝ ⎠

Assuming that the gas constant and the cp/cv ratio remain constant, it is possible to express the preceding equations as:

10 10

10 10

BMm T pfp p⋅ ⎛ ⎞∆

≈ ⎜ ⎟⎝ ⎠

(3-13)

This lumped function may be expressed as a third-order polynomial derived from tests of the actual gas turbine. The present author tested the function above once, derived for a specific gas turbine from two other gas turbines of the same type. The biggest error was in the order of 0.7% compared to the ISO 2314 heat balance. These tests were actually substantially better than the

3-36

requirements stated in ISO 2314 and the tolerances were in the order of 0.5 – 0.6% mass flow. The reason for this rather low figure is that the specific engine is a three-shaft design, and it is possible to evaluate the flow with three more or less independent heat balances.

3.2.1.3 Exhaust Gas Composition

This method is based on the measured content of a specific component in the exhaust gas flow, normally the O2 or CO2 concentrations. If both measurements are available, then cross- correlation may be carried out. The theory behind this method is, loosely stated, that a certain fuel composition requires a certain fuel-to-air ratio to produce a measured content of e.g. oxygen. This method is theoretically sound but great caution regarding the measurement quality should be taken. This method has been applied with success for high accuracy prototype testing. However, the equipment used (with its calibration and measuring engineers) is much more sophisticated than what will be found in a normal power plant. This method has been used with limited success by third party organisations and the “normal” failure is the measuring uncertainty. It is important to recognise, however, how the measuring uncertainty affects parameters like firing temperature. A figure of 0.25 %-units uncertainty should be recognised as a normal figure for field tests. The word field test is highlighted because it is a rather common misunderstanding to apply laboratory type of figures in the field. The expected resulting uncertainty on firing temperature is in the order of some ± 20 K. This is the reason for the reluctance to use this method. Third party organisations should be very careful since they generally lack sufficient information about the secondary air system to check the high turbine swallowing capacity. It should be noted that even if the component is measured as “dry”, the amount of H2O is in the order of 0.85 %-volume (corresponds to saturation at 5 °C).

3.2.1.4 First nozzle swallowing capacity

This method is actually not an independent method, but it can be used as a target for an iterative method. The principle is based on the fact that the first stage nozzle will be choked at and above a certain load level. This base value must be obtained from measurements on actual engines since calculated figures can be quite flawed. This value is normally within a narrow band ± 2 %, due to manufacturing tolerances. The previously mentioned figure is only valid for new and clean condition blading. This method is sometimes used for setting the firing temperature control function after e.g. combustion chamber change. Very detailed information about the secondary air system is needed since the swallowing capacity is a gauge of the throat in the first nozzle. Therefore, it is necessary to calculate the part of the flow that actually goes through the throat. This information is normally a well-kept secret and not available outside the individual manufacturing company. It should be noted that the swallowing might change due to turbine ageing in either way.

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3.3 Component Performance

The purpose of the flow path analysis system is normally to provide the operator with information required for checking the engine condition. The operator needs to evaluate component performance for assessing losses in performance. The question may be as simple as to reconciliation of a lost power to e.g. compressor performance. The importance of accurate measurements cannot be stressed too much, and this process is indeed important since one can use the reconciliation process to check the overall measurement. There are a few things that deserve to be mentioned. For example, is a certain pressure measured as a static or total pressure? This question is highly relevant when comparing with the baseline but not so important if the purpose is trending. Another factor is the placement of the measuring points. It is impossible to be sure that e.g. the measured compressor discharge pressure really is the total pressure just downstream of the outlet guide vane. It may include partial or complete compressor diffuser pressure loss. Some companies use static wall pressure taps and apply corrections. The reason for this is that they are concerned about the probe structural integrity. This procedure is used even in production pass-off testing in some companies. The structure and method in this section is also suitable for both factory production pass-off and field tests.

3.3.1 Inlet System

The purpose of the inlet air system is to provide the correct quantity and quality of air to the compressor. The air is filtered in the filter system with an associated pressure drop. The pressure drop is always measured due to the potential risk of filter icing. One favourable way of monitoring filter fouling or icing is to introduce some kind of fouling factor. It may under certain circumstances be hard to separate a fouling problem from an icing problem. The fouling process is normally slow and may be separated from icing by checking the pressure drop time derivative. There is, however, one circumstance when a filter may quickly get severely fouled and that is a phenomenon that is called “filter saturation”. This is always associated with very high ambient relative humidity. The fouling factor may be as simple as evaluation of the pressure drop characteristics according to the equation [29]:

2

0

0 0

filter airp m Tp p

α⎛ ⎞∆

= ⎜ ⎟⎜ ⎟⎝ ⎠

(3-14)

The equation above is based on the fact that the dynamic head is proportional to the capacity squared [29]:

2

00

0 0

m Tp pp p

⎛ ⎞−∝ ⎜ ⎟⎜ ⎟

⎝ ⎠ (3-15)

The other inlet components (duct and silencing baffles) are generally not prone to deterioration.

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3.3.2 Compressor section performance

The pumping capacity and efficiency set the global compressor performance. These two parameters can be evaluated with the standard definition of efficiency and referred mass flow. The system boundary must be defined before comparison with baseline data. There is a freedom to chose a system or component boundary as long as the baseline is defined in the same way. The compressor is always preceded by an inlet bellmouth, where the air is smoothly accelerated before the IGV. One can choose to include the bellmouth in the evaluated compressor efficiency, but it is preferable to separate them. The bellmouth total pressure drop (p20-p10) is calculated as a normal adiabatic duct and the normal value should be in the order of 0.5 – 0.6 kPa. This figure can be used for evaluating a pressure drop coefficient for all operating conditions as long as the parameter is based on the design conditions (pressure drop, mass flow, temperature and pressure).

Figure 3-1. Compressor state line.

The total-to-total isentropic efficiency is defined as:

030, 020,

030 020

sC s

h hh h

η−

=−

(3-16)

00p

0p

0 10 20

2,20

2mc

030p

31p031p

30p

30 31

230

2c

h

s

00p

0p

0 10 20

2,20

2mc

030p

31p031p

30p

30 31

230

2c

h

s

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The isentropic end-point should be rigorously evaluated using the integrated form of Gibb’s 2nd statement:

( )3,

2

32

2

1 lnsT

pT

pc T dT R s conv levelT p

⎡ ⎤⎧ ⎫⎛ ⎞⎪ ⎪⎢ ⎥− ⋅ − ≤⎨ ⎬⎜ ⎟⎢ ⎥⎝ ⎠⎪ ⎪⎩ ⎭⎣ ⎦

∫ (3-17)

All parameters in equation (3-17) are known except for the isentropic end-point temperature. The enthalpy is calculated with the standard gas table. Simpler approaches with the ideal gas assumption should be avoided and there is no point in using them in a computer-based system. The polytropic efficiency is used in some cases and there is an inherent advantage since it is not dependent on the pressure ratio itself. The compressor polytropic efficiency is evaluated with the equation:

( )

3

2

3

2,

ln

1C p T

pT

pRp

c T dTT

η

⎛ ⎞⋅ ⎜ ⎟

⎝ ⎠=

∫ (3-18)

The denominator in the equation (3-18) can be solved using a standard gas table. The polytropic efficiency is sometimes calculated using a perfect gas approximation. This procedure is ideally suited for implementation into the control system, but there is no point in introducing it to a separate system. The ideal version of the equation above is:

3

3,

3

2

1 ln

lnC p

pp

TT

κκ

η

⎛ ⎞−⋅ ⎜ ⎟

⎝ ⎠≈⎛ ⎞⎜ ⎟⎝ ⎠

(3-19)

The correct way of defining the average κ (kappa) is based on the isentropic process between the compressor inlet and outlet. A simpler or iteration-free method is to base κ on the average temperature in the compressor. Although this is not correct from a fully rigorous point of view, it will still increase the accuracy compared to using a fixed value for κ (e.g. 1.4). The pumping capacity is determined by comparing the actual compressor pumping capacity with the expected capacity. In order to do that on an “apple to apple” basis the flow must be recalculated to some reference condition. There are two common ways of calculating referred flow. The first one is based on an unconventional theta exponent, while the second is based on identical velocity triangles (Mach number):

* 2020

20

mmαθ

δ⋅

= (3-20)

3-40

The value of the theta exponent is normally evaluated with the performance program by keeping the same velocity triangles at varying ambient conditions. There is no standard value for the theta exponent but it is normally in the range of 0.46 – 0.54. The second method is simpler in the case of a compressor because the gas composition is known.

20 20*20

20

REF

REF

m RmR

θ κδ κ

⋅=

⋅ (3-21)

The aerodynamic speed is also evaluated with the equation:

*

2020

REF REFRnnR

κκθ

⋅=

⋅ (3-22)

As in the previous case with referred flow, one can use non-conventional theta exponents for the referred speed:

* nn αθ= (3-23)

The value of the theta exponent is again in the order of 0.46 – 0.54. The actual value is calculated in the same manner as for the flow. The reason for calculating the referred parameters is that the velocity triangles are fixed if two parameters are known. A referred parameter is often misunderstood and is only a way of calculating what will happen if the same velocity triangles are kept at different ambient conditions. There are cases where the power absorbed by the compressor has to be evaluated. One typical case is a twin-shaft engine in which the compressor turbine work matches the compressor work. The compressor work is then used to calculate the outlet state from the driving turbine since the inlet is known from the combustor chamber and secondary air system (SAS) calculation. This work balance is adjusted for the gas generator mechanical losses. The compressor work is calculated with the expression: ( )20 30 20COMPN F m h hγ= ⋅ ⋅ − (3-24) The “non-work” factor (Fγ) in the equation above is used for allowing secondary flows to be taken out of the compressor at different enthalpy levels. This factor is defined as:

( ) 30

30 20

1h h

Fh h

γγ

γ ⋅ −= −

−∑

(3-25)

The factor (Fγ) requires detailed knowledge about the secondary air system. However, in the case of a gas generator turbine, most of the cooling air is extracted from the compressor discharge and hence at full enthalpy. The amount of cooling air to the power turbine is normally limited to cavity purging and rim sealing. The actual amount is determined by the specific gas turbine type but is normally between 0.25 – 0.5% per disc face [29]. This figure is

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strongly dependent on the technology level. The enthalpy of this air may be estimated by assuming a pressure factor of approximately 1.3 [29] compared to the pressure in the turbine and the evaluated compressor polytropic efficiency. It is quite common for twin-spool gas turbines to have a common air manifold to the power turbine and hence only one exhaust enthalpy. This will considerably simplify the calculation of the “non-work” factor in the case of a two-spool engine. The compressor outlet diffuser is generally not amendable to deterioration. As already mentioned, the diffuser may or may not be included in the measured outlet pressure. The diffuser is treated like a normal adiabatic duct. It may be necessary to evaluate the pressure drop if the value is to be subtracted from the measured discharge pressure when downstream components are evaluated.

3.3.3 Combustion chamber section

The combustion system is the direct source for many hot-end problems in a gas turbine. The combustion chamber may be quite tricky to monitor in terms of pressure drops and heat losses. The combustion itself is monitored by the temperature spread in the exhaust. This spread is indeed important since it sets the life span for all hot (and costly) components. The turbine is no blender and a faulty burner is detected with a changed temperature profile in the exhaust. It is quite common to have a maximum individual allowable deviation from the mean value before alarm and eventual trip. These levels are typically 25 K and 35 K for alarm and trip, respectively. One can imagine how this difference affects the first nozzle vane sector and preceding rotor in terms of life span. This spread should be closely monitored and any deviation from the normal level is an indication of a potential problem. A powerful complement to the spread is the NOx value, which will increase if there are hot spots in the combustion section. It is fairly easy to check if it is a real problem or a faulty thermocouple if the load is changed. If the hot or cold point is rotating with load, the cause is definitely an engine problem. If the high or low value stays in the same circumferential position, the measurement itself should be checked. It is very hard to give a general relation between an individual burner and the individual exhaust temperature probe position since it is highly dependent on the relationship between number of burners, turbine stage loading and reaction. The two latter items set the inter-stage swirl level and hence, the circumferential movement of a hot spot through the expansion process. The pressure drop is mostly evaluated because it is needed when calculating turbine capacity. The combustor pressure loss is calculated with the expression:

2

31 030 040

030 031 030

1CCC H

CC

m Tp TK Kp A p T

⎛ ⎞ ⎧ ⎫⎛ ⎞∆ ⎪ ⎪= + −⎜ ⎟ ⎨ ⎬⎜ ⎟⎜ ⎟⋅ ⎪ ⎪⎝ ⎠⎩ ⎭⎝ ⎠ (3-26)

This pressure loss parameter comprises both the cold and the hot pressure drop. The first is the normal frictional and dump pressure drop characteristics and the second is the Rayleigh-flow pressure drop due to the momentum change. The ratio between them is approximately 10:120 and it might be included in the cold loss characteristics with some reduction in accuracy. The constants in the equation above are normally derived from design data. This

20 The ratio between the hot and cold loss coefficient is typically between 4 to 5.

3-42

data is in most cases propriety design information and not known outside the individual manufacturer. The typical figure is in the range of 3 - 6% of the compressor discharge pressure. The value is dependent on the component boundary between the compressor diffuser and the combustion chamber, and the type of liner cooling. Modern convective-cooled combustion chambers are in the higher range of pressure drop due to the complex cooling passage. On the other hand, the amount of dump diffusing/mixing in the cooling holes in the conventional combustion chamber is not experienced in convective-cooled chambers. The firing temperature is calculated by using a normal heat balance, and the number of possible temperatures to be calculated increases with knowledge about the cooling air system. Without information about the cooling air system, one can only calculate the turbine inlet temperature (TIT) and not the actual combustor outlet temperature (COT). In the first case, the amount of air flowing into the compressor is assumed to take part in the combustion process. The difference in the calculation lies in the amount of air entering the combustion chamber i.e. subtraction of all cooling air, which of course requires knowledge about the cooling air amount. The true combustor outlet enthalpy (hCOT) is calculated with the equation:

31 031040

31

F CC F CC

F

m LHV h m h Qh

m mη⋅ + + ⋅ −

=+

(3-27)

( ),COTh f COT comp CC outlet= Where:

31 101

1i n

ii

m m γ=

=

⎛ ⎞= −⎜ ⎟

⎝ ⎠∑

( ) ( ),CC f COT or f CO UHCη =

3 8

2 2

2 2

4 2 2

i C HN COi

F i N COi CH i N CO

x xxh h h hM M M

=

=

⎛ ⎞= + +⎜ ⎟

⎝ ⎠∑

The actual temperature is found with the gas table since the enthalpy is a function of temperature and composition. The heat balance above could be divided with an integrated mean heat capacity value and then solved iteratively for the temperature. Both methods imply almost the same amount of calculation and neither one is superior the other. The integrated mean heat capacity value ( pc ) is derived from the enthalpy assuming semi-perfect gas:

( ) ( )1

REF REF

T T

p p pREFT T

h c T dT c c T dTT T

= =−∫ ∫

N.B. The same applies to all preceding temperatures in this section

3-43

The turbine inlet enthalpy is calculated with the equation:

10 31,

10

F CC F CCTIT ISO

F

m LHV h m h Qh

m mη⋅ + + ⋅ −

=+

(3-28)

The equation above is the definition of TIT according to ISO 2314. It is possible to increase the accuracy for the case of a two-shaft engine. The method is based on the assumption that each turbine disc requires approximately 0.5% cooling air (cavity purge and rim sealing). The suggested resulting equation is:

10 31

10

1 0.05

1 0.05F CC F PT stage CC

TITPT stage F

m LHV h m n h Qh

m n m

η⋅ + + ⋅ − ⋅ ⋅ −=

⋅ − ⋅ + (3-29)

There are other prerequisites that need attention and that is the combustion efficiency and the heat loss from the combustor section. The first is normally a function of firing temperature or measured CO and UHC; the latter is normally a fraction of fired heat. It is possible to evaluate it by measuring the heat added to the ventilation air. Both these parameters are already used in the global mass flow heat balance and they are solved (coupled) by iteration.

3.3.4 Turbine section

The turbine section is normally evaluated from the calculated combustor outlet state and the measured exhaust temperature. In the case of a two-shaft unit21, however, the temperature after the high turbine is calculated via the compressor work. The pressure is also needed for defining the outlet state. This pressure is always measured but it is not always the “correct” one since it is common practice to measure after the turbine diffuser. The diffuser could be included in the turbine efficiency and it is again an arbitrary choice. In the case of a multi-shaft unit, one has to measure the pressure between the turbines. The distance between the turbines varies depending on whether inter-turbine ducts are used or not. Usually, probes are not used in the hot section since the risk of DOD is apparent. The integrity of the probes is a problem and static wall taps are used in most cases. This introduces an additional problem since the flow may be swirling. This type of flow is balanced by a radial pressure gradient and this gradient is load-dependent. The amount of swirl is dependent on the specific engine but is fortunately not load-dependant above say 40 - 50% load. If there are struts in the inter-turbine duct, the amount of swirl is limited to 35° [23]. The outlet bulk enthalpy is calculated with the work balance:

( )20 030 020

044 TITmech compr TIT

m F h hh h

mγ

η⋅ ⋅ −

= −⋅

(3-30)

21 Same principle applies on three-shaft units

3-44

The temperature is iteratively solved from the equation above since both enthalpy and composition are known in that engine section. The mechanical loss is typically in the order of 0.5%. The number of possible inlet temperatures increases if details about the cooling air system are known. However, if no data are known, one is limited to the turbine inlet temperature (TIT) that has no physical station inside the turbine. If the COT is calculated (assuming known secondary air system), the remaining relevant temperatures (SOT, RIT22 and TIT) can be calculated with the equations:

4 040 1 030041

4 1

cool vane

cool vane

m h m hh

m m⋅ + ⋅

=+

(3-31)

51 030

51

SOT rimejectRIT

rimeject

m h m hh

m m⋅ + ⋅

=+

(3-32)

( )50

50

COT COOL COOLTIT

COOL

m h m hh

m m⋅ + ⋅

=+

∑ (3-33)

The cooled turbine efficiency is a rather complicated matter since there will be different cooling air admissions to the flow path. This problem can be solved if the efficiency is based on the TIT, since it is based on a “lumped” inlet temperature:

049,

049,

TITT s

TIT s

h hh h

η −=

− (3-34)

The measured exhaust temperature may be lower than the actual turbine bulk exhaust temperature if there are any air injections upstream of the probes (e.g. balance pistons). This is also a very convenient method since one can calculate the turbine efficiency without any great detailed knowledge about the secondary air system. The polytropic turbine efficiency is calculated in the same manner.

22 Some organisations are using relative RIT, but it is not a straightforward matter to evaluate it since detailed information regarding the velocity triangle must be known. The FPA system is definitely not suitable for such calculations and should be performed in advanced 1-D tools or SCM tools.

3-45

h

s

h

s Figure 3-2. Turbine state line.

Turbine power evaluation is based on the TIT end blading exhaust temperature and is evaluated with the equation:

( )42 049TITP m h h= ⋅ − (3-35)

There are other definitions of the temperature used to calculate work and efficiency. The other method is based on pseudo SOT (415), which is the temperature obtained when the cooling air is divided into useful air and “non-work” air for a certain work extraction. The “non-work” air is injected after the turbine. This procedure is more complicated since one cannot calculate it without detailed knowledge about the secondary air system. The turbine capacity is a gauge of the throat capacity for the first vane and is normally based on the condition immediately upstream of the vane (i.e. combustor outlet). This definition ignores film cooling from the showerhead upstream of the throat. The turbine capacity is calculated with the equation:

4 0,4

0,4

m TFN

p⋅

= (3-36)

The previous expression is simplified and the more rigorous expression is preferred:

50

40

COT REF REFm T RFNp R

κκ

⋅ ⋅=

⋅ (3-37)

This parameter is indeed the most important in the whole evaluation process. It is basically a function of most measured parameters and a deviation may indicate a corrupt set of measured data. Another important parameter is the aerodynamic speed, which has the same meaning as referred speed for compressors. The aerodynamic speed is evaluated with the equation:

4 404 04

REF REF

REF

Rn nRT T

κκ

⎛ ⎞ ⋅=⎜ ⎟⎜ ⎟ ⋅⎝ ⎠

(3-38)

3-46

Including the turbine diffuser into the efficiency definition is arbitrary. If the turbine and the diffuser are separated, the diffuser loss must be separately evaluated. The common practice is to use the diffuser loss characteristics to calculate the performance. However, the diffuser characteristic is normally not known outside the manufacturing company. The second problem is that the loss also is a function of outlet swirl angle, especially if there are struts in the diffuser and the angle is above 20 - 30°. This may cause separation and big mixing losses. This rather advanced feature is in most cases only implemented in the systems delivered by the OEM’s. There is really no point for any third party organisations to implement it since it is possible to lump the turbine and diffuser into one unit. The basis of this is that the turbine velocity triangles are set by two referred parameters, and the diffuser loss may be fully described by the outlet condition from the turbine. The swirl angle calculation algorithm is solved by iterations. See figure to the right for further information. The calculation starts with guessed values for the total pressure and density. The latter is then used for calculating the axial velocity. This value is used in the next step, together with the shaft revolution and blade angle, to calculate the outlet swirl angle. The density is then calculated using the isentropic relation between density and temperature. The guessed value is then updated with this calculated value until inner-loop convergence. The static condition after the blading is calculated with the equations:

( )2 22

49 049

1 tan

2a

p

cT T

c

α+= −

⋅ (3-39)

149

49 049049

Tp pT

κκ −⎛ ⎞

= ⋅⎜ ⎟⎝ ⎠

(3-40)

The diffuser pressure drop can now be evaluated since both the dynamic head and swirl angle are known. The diffuser loss characteristics give the value of Cp. This value is then used to calculate the actual pressure drop, which is subtracted from the guessed total pressure upstream the diffuser to yield the pressure after the diffuser. The start value is updated until the outer loop converges, i.e. the calculated pressure after the diffuser is the same as the measured one. However, it is questionable if it is worth the effort to implement the diffuser calculation into a third party system. Neither details of the diffuser map nor the last stage geometry are normally known outside the manufacturer.

( )2 2 12070

070 070

1 tan1

2a

p

cpR T c T

κκα

ρ−⎧ ⎫+⎪ ⎪= −⎨ ⎬⋅ ⋅ ⋅⎪ ⎪⎩ ⎭

12 2

2tan tan60a

nc

πα β− ⎧ ⎫⋅ ⋅= −⎨ ⎬⋅⎩ ⎭

Start values: 049p , 49ρ

49

49 49a

mcAρ

=⋅

080 ?p converged

060 050 0,diffuserp p p= + ∆

?convergedρ

Iteration scheme for diffuser calculation.

3-47

3.4 Baseline or off-design modelling

This section will deal with the baseline of various engine parameters. The previous section gave some guidance concerning how key parameters inside the flow path can be evaluated, but gave no information about how the expected value for a certain parameter could be obtained. It is possible — and fairly simple — to use more or less generic baseline curves on multi-shaft units. The reason for this lies in the relation between two referred parameters. This relation is based on the engine type’s running line. This method is not used on single-shaft (fixed speed) units since there is no such thing as a running line. It is, however, possible to introduce an auxiliary variable and find correlations between referred parameters. Individual components are always possible to normalise. There are basically three ways of establishing conventional baseline models:

• Background curves of a referred parameter versus another referred parameter • Performance program or “performance deck” • Direct comparison with component map

3.4.1 Background curves (multi-shaft units)

This method is based on the unique running line of a multi-shaft engine and will not work on a single-shaft power generation unit. The whole principle is based on the relation between any referred parameter and another referred parameter. Many misunderstandings prevail regarding referred parameters. A referred parameter is simply a parameter that is recalculated to another ambient condition whilst maintaining the same velocity triangle. If a component is normalised, then the pressure ratio and efficiency stays the same since the velocity triangles are the same. This is due to the fact that the normalised flow is related to the Mach number in the vane and the aerodynamic speed (n/√T) sets the rotor Mach23 number. Since the geometry is fixed, the flow expressed by the velocity triangles is fully defined by these two parameters. The component or stage losses are constant since the triangles are unchanged. However, there are secondary effects like the Reynolds number influence but they are in most cases negligible. If the amount of work input is the same and the losses also stay the same, the stage pressure ratio stays the same. Since e.g. the total compressor pressure ratio (π) is the product of the pressure ratio in the individual stages, the total pressure ratio is unchanged. This is valid for an un-choked compressor where the two groups (speed and flow) set all other parameters. In the case of a choked compressor, the relation between flow and pressure ratio becomes independent and one should chose the pressure ratio and the aerodynamic speed as independent variables. The same is true for turbines but it will not be repeated here. The reader is referred to reference [29] for further information. The section above dealt with referred parameters and it is closely related to why there will be relations between two of them, or why one sets all the others. First of all, the relation exists only when the power turbine is choked, which normally is the case above approximately 30% power. This is only true from a strict point of view if the turbine chokes in the first nozzle and hence negligible/minor speed influence. It is normally impossible to operate the power turbine at a fixed aerodynamic speed since the grid and the gear set the physical shaft speed. This error is probably acceptable for monitoring or trending purposes, but should not be acceptable

23 Sometimes erroneous referred to as Laval number

3-48

in a test bed. This is the main reason for choosing a water break or similar device that is independent of the grid frequency for test beds. There is a maximum size for brakes, and above a certain power the generator is the only alternative. The engine matching is very well described in reference [29] and will only be described briefly for completeness. The power turbine swallowing capacity sets the condition downstream the high turbine. Or more correctly, the power turbine sets the downstream pressure of the high turbine. The power turbine is normally choked above a certain load level and hence, a more or less fixed capacity. The high turbine swallowing capacity and efficiency are set by the high turbine pressure ratio and aerodynamic speed. The high turbine pressure ratio and hence, work, is fixed since both the capacity and efficiency (high- and power turbine) are set. The high turbine is literally “squeezed” between its nozzles and the preceding turbine. Consequently, the operating point is fully determined and the only practical way to change the power level is to change the temperature upstream the high turbine (fixed power turbine geometry). The high turbine drives the compressor at the same speed and the engine pressure ratio is set by the “pumped” air, high turbine capacity, and firing temperature. The influence from varying the ambient condition will be taken into account per definition of referred parameters.

Figure 3-3. Matching of two turbines in series.

There is no such thing as a running line for a single-shaft engine, and the reason is that it is operated at a fixed physical speed. A referred or normalised parameter can be derived in several ways; the most common ones are the Buckingham π-theorem or, assuming the same velocity triangles, based on Mach number. There is limited information about analytical methods in the open literature. One recently published method is by Volponi [31]. The analytical method is based on the assumption that an arbitrary parameter has a corresponding normalised parameter. The arbitrary parameter is assumed to be a function of the temperature, pressure, and the corrected parameter [31]: ( ), , *f T pΡ = Ρ

3-49

The full differential form yields after some algebraic manipulation [31]:

, * , * ,

1

** * *

p T T p

d d dT d dp d ddT T T dp p p d

βα

Ρ Ρ

≡

⎛ ⎞ ⎛ ⎞ ⎛ ⎞Ρ Ρ Ρ Ρ Ρ Ρ Ρ Ρ= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Ρ Ρ Ρ Ρ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

**

d d d dθ δα βθ δ

Ρ Ρ= − −

Ρ Ρ

Where:

020 020

288.15 1.01325T pθ δ= =

If the preceding equation is integrated, the general form appears:

* α βθ δΡ

Ρ = (3-41)

Where:

Table 3-1. Theta-exponents [29]

Parameter, Ρ Theta (textbook)

Theta (actual)

Delta

Temperature 1 0.97-1.03 0 Pressure 0 0-0.06 1 Power 0.5 0.5-0.6 1 Fuel flow -0.5 0.64-0.76 1 Shaft speed 0.5 0.48-0-50 0 Air flow -0.5 0.47-0.52 1

In this derivation no gas properties appeared, and if the result is compared with the more rigorous π-theorem, the similarity is obvious but the influence of gas properties is missing. The gas composition itself is not an issue and it is in many cases desirable not to evaluate it. The accuracy can be substantially increased if unconventional theta exponents are introduced. The actual engine-specific theta exponents are in most cases propriety and not known outside the manufacturer. However, if referred parameters are to be used for test bed evaluation, theta exponents must be used, otherwise rendering questionable test results. The second column in the table above shows typical theta exponents. There are different methods available to calculate theta exponents and one can chose either an approach using the performance deck or use actual engine data. The first method is based on the specific engine calculation program and the calculation model must be rigorous. The word rigorous means in this case that the full dimensionless parameters are used as a base for the maps, or e.g. a real gas 1-D turbine model instead of a turbine map. The performance is then calculated with the same set of velocity triangles under varying ambient conditions. The relevant parameters are normalised or corrected to the reference condition. There will certainly be deviations, and the RMS-value is then minimised by varying the theta exponent using a numerical method.

3-50

The largest deviation in theta exponent is the one for fuel flow, where the theoretical value is –0.5 and the actual value may be as low as –0.76. The reason for the difference may be shown using an analytical approach where the result yields [31]:

( ),40 040 ,30 030

40 40 30 40 30

11 0.64;0.762

p pFUEL c T c Tmm h h

α α− −

⋅ ⋅⎛ ⎞⎛ ⎞= + − − ∈⎜ ⎟⎜ ⎟∆ ∆⎝ ⎠⎝ ⎠

(3-42)

If one assumes that the media is an ideal gas (i.e. constant specific heat capacity), the second bracket is equal to unity. The final equation yields then:

40

12

FUELmm

α = + (3-43)

The second term in the equation above is the fuel - to - air ratio and is normally in the range of 0.02 – 0.03 and certainly below 0.07 (stochiometric). The full dimensionless group, derived with the Buckingham π-theorem is [29]:

2

FUEL CC

p

m LHV Rc D p T

ηκ

⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅

The corresponding referred parameter is (neglecting combustion efficiency and LHV changes as in previous derivation):

,* p REFFUEL REFFUEL

REF p

cm RmR c

κκθ δ

⋅=

⋅⋅ (3-44)

One other common problem area is the combustor outlet and stator outlet temperature, where errors in the order of 15 K may be introduced at cold ambient test temperatures. This magnitude of error would certainly make the whole test questionable. It is also possible to introduce a correction factor for relative air humidity (CRH) and the equation reads:

020 020

288.15 1.01325

RHMeasured parameterCorrected parameter C

T pα δ= ⋅⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(3-45)

If the procedure above is used, the results should be quite close to the ones from the performance calculation program. Typical errors are in the range of 0.2 – 0.5 percent if appropriate theta exponents are used.

3-51

3.4.1.1 Example of baseline curves

The following examples show expected curves for some parameters related to the high-pressure compressor in a three-shaft industrial unit. All figures are calculated with Dr. Kurzke’s commercially available off-design program GasTurb 9 Pro [32]. The number of relevant different baseline curves one can produce is dependent on the configuration of the specific engine. It is, however, always recommendable to use measured values from the specific engine as base for the functions.

.864

.866

.868

.87

.872

.874

.876

.878

poly

tr.H

PC

Effi

cien

cy

.842

.844

.846

.848

.85

.852

.854

.856

isen

tr.H

PC

Effi

cien

cy

1718

1920

2122

2324

HP

C In

let C

Flo

w W

25R

std

[kg/

s]

2.8 3 3.2 3.4 3.6 3.8 4 4.2

HPC Pressure Ratio

Not conv erged points are marked with a white circle.

Figure 3-4. Example of some baseline functions versus

pressure ratio for a high pressure compressor.

.864

.866

.868

.87

.872

.874

.876

.878

poly

tr.H

PC

Effi

cien

cy

.842

.844

.846

.848

.85

.852

.854

.856

isen

tr.H

PC

Effi

cien

cy

1718

1920

2122

2324

HP

C In

let C

Flo

w W

25R

std

[kg/

s]

9 10 11 12 13*103

HP Spool Speed [RPM]

Not conv erged points are marked with a white circle.

Figure 3-5. Example of some baseline functions versus

referred speed for a high-pressure compressor.

3-52

3.5 Performance program

The use of a performance program into the system omits the use of background- or baseline- curves/functions. This type of combination, where the evaluation program is used in conjunction with the performance program, is ideally suited for single-shaft fixed speed units. The performance program is frequently called a performance deck or a matching program, and all three names will be used in this text. This type of program is normally based on component characteristics like the compressor and turbine map. The use of component maps may at a first glance seem primitive, but it is indeed the best approach since each point in the maps represents unique velocity triangles. One could of course use higher order calculation models for the components and match the results from the individual model. This procedure is implemented at the expense of calculation speed.

3.5.1 Component maps or CHIC

The off-design program or calculation model used to create the maps is typically a one-dimensional tool for compressors and turbines. Two-dimensional programs are sometimes used for calculating axial compressor off-design data. This type of tool may be used since one has to use either IGV’s or bleed-off to maintain reasonable incidence levels. This type of tool will perform poorly on a multi-stage turbine when the last stage goes into the turn-up region at low load (reversing flow near the hub). Therefore 1-D tools are exclusively used for turbine off-design calculation. The state-of-the-art 1-D mid-span turbine tools are highly sophisticated with some 2-D capability. The typical 1-D turbine calculation algorithm looks like:

• Guess an outlet Mach number from the blade. • Calculate outlet state24 from inlet state, cooling and loss model25. • Calculate flow angle from continuity between throat and outlet. This procedure

requires that the state at the throat is known and the momentum conservation equation could be used. Angle correlations (e.g. Ainley) may be used if the outlet flow is sub-sonic. The calculated angle may be corrected for viscous effects like secondary flow and tip leakage.

• Determine outlet Mach number for choked throat (Mach number of unity at the throat).

• Check convergence. • Check maximum flow at the throat and update inlet flow if necessary.

This procedure is repeated for various desired speeds and pressure ratios. There are commercially available programs that are claimed to solve the flow field as a through flow program. The details are not known to this author, but one reasonable guess is that there is some blade twist function built in to program and that the mass flow is integrated from the mid-span towards the end walls. There are examples of generic span-wise distribution of reaction [63] and tangential velocity for different vortex distributions in the open literature. 24 The radial velocity component may be calculated from an assumed streamline slope based on the change in Euler diameter and axial blade chord. 25 There are different philosophies in how to smear the loss over the blade chord. Some methods only apply loss to the blade section after the throat, which may be quite erroneous if the blade has shower head cooling.

3-53

The results from this type of program are then used to generate the turbine map, which is then loaded into the performance program.

Figure 3-6. Typical turbine maps [22].

The used parameters in the figure above are simplified and they should be adjusted for gas properties in the performance program. The efficiency figures in the left diagram may be defined in several ways. There is another type of map where the information in the preceding maps is condensed into one map:

Figure 3-7. Typical compact turbine map [33].

The parameter on the x-axis is the relative speed-flow parameter, defined as: n m TpT

The compressor map is preferably derived from the compressor rig test facility, and if no specific compressor test is carried out, from the engine test. All untested values are calculated

3-54

with the calibrated (from the test) calculation model. A typical compressor map is shown in the figure below:

Figure 3-8. Typical compressor map. The effect of IGV or VSV may be taken into account in the compressor map if the IGV or the VSV are scheduled referred speed. The same is true for bleeds within the compressor but not after the compressor. In the case of a fixed-speed compressor there is no point in using the referred speed to control the IGV and VSV setting. In this case, is it custom to use multiple maps and interpolate between them. The map is normally implemented as referred flow, pressure ratio, and isentropic efficiency as a function of referred speed and beta-value. The beta-values are only an auxiliary parameter and are actually a set of lines parallel to the surge line. It is common practice to have the beta-line that “corresponds” to the surge-line as number one.

Figure 3-9. Example from SmoothC showing 20 beta-lines for clarity.

3-55

3.5.2 Component matching

The following calculation scheme is typical for single-shaft off-design calculation. The presented calculation procedure is a nested-loop approach. A matrix solver (e.g. Newton-Raphson) normally replaces this method in modern performance programs.

No

Yes

Ambient condition Tamb, pamb, RH

Chose operating point in compressor map

AMB

nT

( ), , , * 0f n T mπ η =

Cooling air (SOT & TIT) (isobaric mixing)

Combustion chamber COT

Pressure losses in the ducting and diffuser

• Power OK? • Max COT OK? • Max EGT OK?

2p m Tp p

α⎛ ⎞∆

= ⎜ ⎟⎜ ⎟⎝ ⎠

2

31 030 040

030 031 030

1CCC H

CC

m Tp TK Kp A p T

⎛ ⎞ ⎧ ⎫⎛ ⎞∆ ⎪ ⎪= + −⎜ ⎟ ⎨ ⎬⎜ ⎟⎜ ⎟⋅ ⎪ ⎪⎝ ⎠⎩ ⎭⎝ ⎠

20 30 31 31, , ,p T p m

20, mπ

( )40

40

COT COOL COOLTIT

COOL

m h m hh

m m⋅ + ⋅

=+

∑

Pressure losses in diffucer, HRSG ducts

Turbine calculation Capacity OK? ( ), ,m T f n T

pη π=

No

Swirl

Figure 3-10. Typical engine matching calculation.

3-56

These models are more or less the same in most companies and the decision to choose a nested loop or matrix method is based on the number of iteration loops. The presented method is only valid for single shaft fixed-speed units. The corresponding method for multi-shaft units is more numerically involved but will not be presented here. References [29] and [34] provide excellent coverage of this topic.

3.5.3 Simplified models

The simplified models section is rather brief and paper number 2 gives more information regarding this topic. This category of engine monitoring systems is widely spread in the gas turbine field. The simple models are more or less always based on referred parameters and the unique running line of a multi-shaft unit. This makes the use on single-shaft fixed speed turbines impractical since auxiliary parameters have to be introduced. The cost for implementing the systems in this section is very low and the calculation efforts are minimal. Two common methods are:

• Bellmouth depression versus referred speed • Polytropic compressor efficiency versus pressure ratio

An axial compressor is very sensitive to fouling and both the pumping capacity and efficiency will be reduced when the blading gets fouled. One very simple approach is to check the pumping capacity with a direct measurement. The best way of doing it is comparing the referred flow with referred speed. J.N. Scott pioneered this clever method in the 1970’s [35]. The referred flow is, per definition, a unique function of the inlet Mach number. This means that the bellmouth depression also is a function of the referred speed. The correlation between two referred parameters is described in earlier sections and will not be repeated here. Loosely stated, the engine matching will maintain more or less the same mass flow, but at a higher speed level to compensate for the lost pumping capacity. This method is straightforward and normally requires no additional instruments. The procedure to establish the baseline is as follows:

1. Soak wash the turbo set. Note that additional hand cleaning of the bellmouth, struts, IGV and 1st rotor blade may be necessary if one wants to be sure that the baseline is valid for a cleaned compressor.

2. Make readings of bellmouth pressure drop, shaft speed and ambient temperature.

Suitable load increments are in the order of 5 - 10 % load.

3. Plot the correlation or create a third order polynomial expression.

3-57

Figure 3-11. Example of baseline curve.

The actual pumping capacity is established by a single reading at some load point. Note that this point must be above bleed valve(s) closure if the air is re-injected into the air plenum. The relative pumping capacity is then calculated with the simple expression:

,

Relative flow 100 BM

BM BASE

pp∆

= ⋅∆

(3-46)

All necessary measured values should be standard operational instruments available on a modern gas turbine. However, on some old or low-cost engines, the bellmouth pressure drop may not be a standard instrument. One possible but primitive solution is to use a simple U-tube, which may be a water-filled transparent rubber tube. This possibility is limited by the inlet Mach number and accessible height under the engine. This method is suited for implementation into the control system for direct display on the operators’ terminal. The second method (polytropic compressor efficiency versus pressure ratio) is already described in a previous section and the information here is sparse. The baseline is established more or less in the same way as for the flow. The efficiency may again be plotted against any suitable referred engine parameter. However, the compressor pressure ratio seems to be the apparent choice in this case.

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.90 0.92 0.94 0.96 0.98 1.00

Referred speed

Bellm

oth

depr

essi

on

3-58

4-59

4 Axisymmetric through-flow calculation

Despite the advent of viscous commercial CFD tools in the 1980s, the S-2 throughflow is the most important tool for design and analysis of turbomachinery. The reason for this is that it is impossible to design the flow path with a CFD tool, and that the calculations are rather “costly”, or more correctly, time consuming. The designer may carry out literally hundreds of calculations of different flow path configurations before some desired optima are reached. Therefore, it is necessary to use a rapid tool to design the flow path and the most common choice is the axisymmetric throughflow calculation tool.

4.1 Introduction

There are different approaches for such methods and the most common one is the streamline curvature method (SCM). This type of code is named after the streamline curvature term in the radial equilibrium equation. One common feature for all tools at this level of sophistication is the need to use correlations for losses and outlet angles. This will imply severe limitations in the usage of such a code, and one has to use correct models when calculating the losses and angles. What the “correct” models for loss and angle are is a rather complex question, and the answer is dependent on each company’s tradition more than firm engineering practice. Today most of the successful codes (and designs) are based on decades of experience in individual companies and therefore proprietary. There is little information in the open literature, but one very good example is Denton [36] who states that, “In many applications, throughflow calculations are little more than vehicles for inclusion of empiricism in the form of loss, deviation and blockage correlations, and their accuracy is determined by the accuracy of the correlations rather than that of the numerics.” Denton also points out that this is especially true for deviation and blockage in compressors and secondary deviation in turbines. The determination of the outlet angle is indeed important in turbines since if the first vane angle is as little as 1° off the desired one, we will either have 5% too high or 5% too low turbine capacity. The impact on the cycle performance due to component matching can be rather high, and we can lose some surge margin if the turbine is too small or lose thermodynamic efficiency if it is too big (or small). The cure to this problem is to use a viscid CFD code to evaluate blockage and flow angles. It is not uncommon that even the most competent designs have to be re-staggered to adjust the turbine capacity. The open literature is somewhat limited regarding throughflow calculation despite the likelihood that all organisations designing turbomachinery are using it. There are, however, a few important contributors who should be mentioned: W. Traupel [62], C-H Wu [37], R.

4-60

Novak [38, 39], L. H. Smith [40], A. Wennerstrom [12, 41], J. Horlock [42], J. Denton [36, 43] and R. Hearsey [13, 45]. It is probably safe to say that most all throughflow codes have at least some heritage from Hearsey’s “A Revised Computer Program for Axial Compressor Design” [45] and Denton’s “Throughflow Calculations for Transonic Axial Flow Turbines” [43]. The latter is the famous paper with the “target pressure method,” solving the numerical problem with multiple choking stages in an axial flow turbine.

Figure 4-1. Typical SCM throughflow calculation grid [44].

The figure above shows a typical throughflow calculation domain. This specific case has eleven streamlines and 17 calculation stations26 or quasi-orthogonal (q-o). In each node point the momentum, energy and continuity equations are solved. Since a streamline per definition fulfills the streamfunction, the positions of the node points must move during the solution. It is possible to introduce cooling air into this calculation method, as long as the stream function is adjusted. The calculation method is iterative and the only specified items are the outer walls together with some information regarding the blading and the inlet state. There are two basic types of calculations; the first is the design case calculation and the second is the analysis or off-design mode. Both of them have their own calculation/numerical problems where one classic problem is if the solution is to be sub- or super-sonic for the design case (like in a duct). There are no such problems for the analysis mode, but instead the problem with choking appears. A single choked row is not an issue but multiple choked rows were a problem until Denton presented the target pressure method in 1978 [43]. This method has been used ever since and a few practical hints of how to use it are presented in [53].

26 Both the stator and the rotor have three internal stations

4-61

4.2 Radial equilibrium equation

The Navier-Stoke’s set of equations is reduced to Euler’s equation by assuming that the flow is inviscid and steady:

C C F p∇⋅ = − ∇a f 1ρ

(4-1)

Where:

r zCC C

r r zθ

θ∂ ∂ ∂

⋅ = + +∂ ∂ ∂

C∇ (4-2)

The corresponding set of equations in cylindrical coordinates (r, θ, z) are:

C Cr

Cr

C C Cz

Cr

F prr

r rz

rr

∂∂

+∂∂

+∂∂

− = −∂∂

θ θ

θ ρ

2 1 (4-3)

C Cr

Cr

C C Cz

C Cr

F pr z

r∂∂

+∂∂

+∂∂

+ = −∂∂

θ θ θ θ θθθ ρ θ

1 (4-4)

C Cr

Cr

C C Cz

F pzr

z zz

zz

∂∂

+∂∂

+∂∂

= −∂∂

θ

θ ρ1 (4-5)

The flow field is assumed to be axisymmetric and all terms containing derivatives in the tangential direction can be cancelled. The resulting set of equations can be written as:

C Cr

C Cz

Cr

F prr

rz

rr

∂∂

+∂∂

− = −∂∂

θ

ρ

2 1 (4-6)

C Cr

C Cz

C Cr

Fr zr∂

∂+

∂∂

+ =θ θ θθ (4-7)

C Cr

C Cz

F pzr

zz

zz

∂∂

+∂∂

= −∂∂

1ρ

(4-8)

4-62

It is convenient to write the set of equations above to “intrinsic” coordinates (n, θ, m). The transformation is rather lengthy and only the result will be presented here.

Cr

Cr

F pn

m

cn

2 2 1− = −

∂∂

θ φρ

cos (4-9)

Cr

r Cm

Fm ∂∂

=θθ

b g † (4-10)

C Cm

Cr

F pmm

mm

∂∂

− = −∂∂

θ φρ

2 1sin (4-11)

It is convenient to introduce the 1st law and Gibb’s equation into the above set of equations. The reason for this is that static pressure gradient is not a desirable feature, and it is more favourable to work with total enthalpy-, entropy-, velocity-gradients and meridional velocity. The first law and Gibb’s second equation can be written as:

2

01

2Ch h Tds dh dp

ρ= + = −

The two preceding equations can be combined to give:

01

m mdp dh Tds c dc c dcθ θρ= − − − (4-12)

Note that there is by definition no velocity component normal to the streamline. There are examples in the literature where the original forms are used. However, it is then necessary to introduce an inner iteration loop that solves for the meridional velocity. The stream-surface will only by coincidence be normal to the computing station. A computing station or a quasi-orthogonal is always specified as input to the programme and describes leading edges, trailing edges, internal stations and duct stations. The computing station may be a straight line or any other arbitrary line (e.g. a swept trailing edge). Therefore one needs to define derivatives in the l-direction along the calculation station (figure 4-2). The station angle with respect to the radial direction is named γ (gamma) and is calculated as:

tan xl

δγδ

= (4-13)

†

C Cm

C Cr

Cr

r Cm

C rm

Cr

r Cmm

m m m∂∂

+ =∂∂

+∂∂

F

HGG

I

KJJ =

∂∂

θ θ θθ

φ

θφsin

sin2

b g

4-63

Figure 4-2. Coordinate directions in the meridional plane [41].

A derivative in the l-direction can be expressed using the chain rule as:

( ) ( )cos sind dn dmdl dl n dl m n m

φ γ φ γ∂ ∂ ∂ ∂= + = − + −

∂ ∂ ∂ ∂ (4-14)

It is convenient to eliminate derivatives in the n-direction, and the expression above is re-written to yield:

( ) ( )1 tan

cosn mφ γ

φ γ∂ ∂

= − −∂ − ∂

(4-15)

The final expressions are obtained by first applying equation (4-9) with equations (4-12) and (4-14). The derivatives in the n-direction are eliminated with the equation above. The preceding equations are combined to eliminate 0h m T s m∂ ∂ − ∂ ∂ and the final expression is:

( ) ( ) ( )

( ) ( )

2

0

sin cos

sin cos

m m mm m

c

m n

d rcdc c c cc cdl m r r dl

dh dsT F Fdl dl

θθφ γ φ γ

φ γ φ γ

∂= − + − −

∂

+ − − − − −

(4-16)

The equation above is valid for stationary components (e.g. vanes and ducts) and it is possible to derive a form of it that is valid for rotating stations. A few auxiliary equations are needed to re-write the equation above with respect to a rotating blade row:

c w rθ θ= + Ω (4-17)

( )22

2 2rwI h

Ω= + − (4-18)

( )0h I r w rθ= + Ω − Ω (4-19)

4-64

Equation (4-18) is the standard way to write rothalpy and it is easy to derive the equation from the velocity triangles by defining 0I h u cθ= − . The rothalpy is unchanged along a streamline in a rotating coordinate system and hence, a very convenient property when working with rotors. If the auxiliary equations above are introduced into the radial equilibrium above, the corresponding form for a rotating coordinate system becomes:

( ) ( ) ( )

( ) ( )

2

sin cos

2 cos sin cos

m m mm m

c

m n

d r wdc c c wc cdl m r r dl

dI dsW T F Fdl dl

θθ

θ

φ γ φ γ

γ φ γ φ γ

∂= − + − −

∂

− Ω + − − − − −

(4-20)

Conversion to the stationary coordinate system is possible by setting the blade angular velocity to zero. This feature makes the latter version of the radial equilibrium the most favourable when programming it in a code. It is quite common to specify the variation in work along the radial direction when performing the design calculation. The flow angle at all node points is one outcome from the design calculation and the previous form of the radial equilibrium equation is the most useful. For off-design or analysis cases, however, the situation is the opposite and the angle is known. Therefore it is convenient to re-write the radial equilibrium equation into a form where the flow angle can be used directly. The analysis form is derived by substituting tanmw cθ β= for the tangential velocity component. The off-design or analysis form of the radial equilibrium equation is:

( ) ( ) ( ) ( )

( ) ( )

2 22 tantan1 tan sin cos

2 tan sin cos

m m m mm m

c

m m n

d rdc c c cc cdl m r r dl

dI dsc T F Fdl dl

βββ φ γ φ γ

β φ γ φ γ

∂− = − + − +

∂

− Ω + − − − − −

(4-21)

The most convenient form is again, in relative coordinates and conversions to stationary coordinate system, achieved by setting the blade speed to zero. There are two terms that have been used in the presented radial equilibrium equations without explanation. The first one (Fm) is the body force in the meridional direction. This term is a dissipative force that acts to oppose the flow, and is used to get a consistency with the meridional entropy gradient. The second one (Fn) is a blade force normal to the blade mean surface. The derivations of these terms are presented in [12], [41], [42], [45] and only the results are presented here.

4-65

Bladed regions

( )

( ) ( )

tan

cos sin sin cos tancos

sincos cos cos

m

m

n p

p

sF F Tm

r ccFr m

F F

F sF Tm

θ

θθ

θ

β

β ε β ε φ γφ γ

ββ ε ε

∂⎧ = − −⎪ ∂⎪∂⎪ =⎪ ∂⎪

⎨ ⎧ ⎫⎪ ⎪⎪ = + −⎨ ⎬⎪ −⎪ ⎪⎩ ⎭⎪∂⎪ = +⎪ ∂⎩

(4-22)

Non-bladed regions

( )

2cos

0

sin cos

m

n

m

sF Tm

Fr cc sF T

r m mθ

θ

α

α α

⎧ ∂= −⎪ ∂⎪⎪ =⎨

⎪ ∂ ∂⎪ = = −⎪ ∂ ∂⎩

(4-23)

Figure 4-3. Orientation of vectors with respect to the m, θ and n orthogonal coordinate system [41].

The final forms of the radial equilibrium equations are:

( ) ( )

( )

( )

2

2

sin cos

2 cos tan

sin cos tan sin cos

m m mm m

c

dc c cc cdl m r

d r wW dI dsw T Fr dl dl dl

sTm

θθθ θ

φ γ φ γ

γ ε

φ γ β ε β β

∂= − + −

∂

− − Ω + − −

∂⎡ ⎤+ − −⎣ ⎦ ∂

(4-24)

4-66

for the design case, or for the analysis case:

( ) ( )

( ) ( )

( )

( )

2

2

2

2

2

sin tan tan cos

tantan 2 tan sin tan coscos

sin cos tan sin cos

tantan

m mm

c

mm

mm

m

c ccm r

rc cdc r dlcdl sT

mrdI dsT c

dl dl r m

φ γ ε β φ γ

ββ ε φ β γβ

φ γ β ε β β

βε

⎧ ⎫∂⎡ ⎤− − + −⎪ ⎪⎣ ⎦ ∂⎪ ⎪

⎪ ⎪− − Ω +⎪ ⎪⎪ ⎪= ⎨ ⎬

∂⎪ ⎪⎡ ⎤+ − −⎣ ⎦⎪ ⎪∂⎪ ⎪

∂⎪ ⎪+ −⎪ ⎪∂⎩ ⎭

(4-25)

Both preceding equations are in relative coordinates and setting the blade speed to zero transforms to a stationary system. The grid is normally rather course in a through-flow calculation, especially if there are no internal calculation stations. Typically one has approximately three stations upstream the first leading edge and downstream the last stage, one station at each leading- and trailing-edge respectively, and one duct station between each row. This may introduce a numerical problem when solving the convective term. Hearsey solved this problem by introducing an analytical expression for that term [13, 45]. The derivation of the expression is indeed lengthy and consists of considerable algebra. The final equation is:

( ) ( )

( ) ( )

2

22

tan cos1 tancos 1 tan tan

sin tan1

m m m mm

c

m m m

c c c cdMm r dl m

r cc M c sMr r m R m

θθ

φ φ λφ γγ φ γ λ

φ β

∂ ∂− = − − −

∂ + ∂

∂ ∂− + + +

∂ ∂

(4-26)

There are a few problems associated with introducing the equation into the radial equilibrium equation. The first and major problem is the singularity when 2 1mM → , and the second one is simply the additional complexity. Denton points out these problems in [43] and states that, “it is not obviously more accurate than the much simple process of evaluating it from the previous iteration.” A complete set of radial equilibrium equations with the expression above is presented in reference [45].

4.2.1 Radial equilibrium equation integration

It is necessary to integrate the radial equilibrium equation (REE) in order to get the velocity profile along the calculation station. The procedure is actually not that complicated for the design case and we can re-write the radial equilibrium equation as a linear ordinary differential equation (ODE) with “non-constant” constants. The general form for the design case is:

( ) ( )2

2mm

dc A l c B ldl

+ = (4-27)

4-67

Where:

( ) ( )12 cosc

A lr

φ γ= − (4-28)

( )( ) ( )

( ) 2

sin 2 cos2

tan sin cos tan sin cos

mm

d r wc W dI dsc w Tm r dl dl dlB l

sF Tm

θθθ

θ

φ γ γ

ε φ γ β ε β β

⎧ ⎫∂− − − Ω + − −⎪ ⎪⎪ ⎪∂= ⎨ ⎬

∂⎪ ⎪⎡ ⎤− + − −⎣ ⎦⎪ ⎪∂⎩ ⎭

(4-29)

The solution to this 1st order differential equation is found by the standard “integrating factor” method:

( ) ( ) ( ) ( ) 1 1, 1 , , 1 ,

12 2

1, 1 ,1

n ni j i j i j i j

nn n A l l A l lm m ni j i j

Bc c e eA

− −+ +

−− − − −

−+= + − (4-30)

The two coefficients (A and B) are assumed to be known, or more correctly, taken from the previous loop (n-1). The integration starts with a guessed value for the middle streamline meridional velocity and is then integrated in both directions along the quasi-orthogonal. The method normally used to find the correct value will be described later in this section. This method fails if A is equal to zero, in which case some other method has to be used. This is presented in [39] and will not be dealt with here. To give an example where this may occur, one may consider the case where the streamline is a straight line and hence infinite radius of curvature.

4.2.2 Analysis case

The analysis form of the radial equilibrium equation the cannot be handled in the same way as for the design case, and the following approach is taken:

( ) ( ) ( )2

212

mm

mm m

dcc

dl

dc F l G l c J l cdl

=

= + + (4-31)

This equation is presented in [45] but provides no guidance regarding how to solve it. One possible solution is to express the velocity at the next node point along the station with a standard Taylor expansion using backward approximation [45]:

( )2

2 2 2, , 1 , ,

,

mm i j m i j

i j

dcc c l O ldl+

⎛ ⎞= + ∆ + ∆⎜ ⎟

⎝ ⎠

4-68

The derivative in the equation can be replaced with equation (5-24) to yield: ( ) ( ) ( ) , 1

2 2 2, , 2

i jm m i j m mc c F l G l c J l c l+

= + + + ∆ (4-32) Where:

( ) ( )2 2cos sin cos tan sin cos s dI dsF l T Tm dl dl

β φ γ β ε β ∂⎧ ⎫⎡ ⎤= − − + −⎨ ⎬⎣ ⎦ ∂⎩ ⎭ (4-33)

( ) ( )

( )2 sin tan tan

cos2 tan sin tan cos

mcmG l

φ γ ε ββ

ε φ β γ

∂⎧ ⎫⎡ ⎤− −⎪ ⎪⎣ ⎦ ∂= ⎨ ⎬⎪ ⎪− Ω +⎩ ⎭

(4-34)

( ) ( ) ( ) ( )2 tan tan1 tan tancos cosc

d r d rJ l

r r dl r dmβ ββ εβ φ γ

⎧ ⎫= − − −⎨ ⎬

⎩ ⎭ (4-35)

The integration starts with a guessed value for the middle streamline meridional velocity and is then integrated in both directions along the quasi-orthogonal. The method normally used to find the correct value will be described later in this section.

4.2.3 Finite differences

The partial derivatives in the radial equilibrium equation cannot be solved analytically, so a finite differences scheme must be used in order to perform the integration. Derivatives in the direction of the station are found using the forward difference (Γ denote an arbitrary parameter):

, 1 ,

, 1 ,

i j i j

i j i j

ddl l l

+

+

Γ − ΓΓ=

− (4-36)

Derivatives in the meridional direction are found using a central approximation:

, 1 , , 1 ,

, 1 , , 1 ,

12

i j i j i j i j

i j i j i j i j

ddm m m m m

+ −

+ −

⎧ ⎫Γ − Γ Γ − ΓΓ ⎪ ⎪= +⎨ ⎬− −⎪ ⎪⎩ ⎭ (4-37)

The streamline slope angle is calculated with the equation (central approximation):

1, , , 1,1 1

1, , , 1,

1 tan tan2

j i j i j i j i

j i j i j i j i

r r r rz z z z

φ φ

φ

+ −

+ −− −

+ −

⎧ ⎫⎪ ⎪⎛ ⎞ ⎛ ⎞− −⎪ ⎪= +⎜ ⎟ ⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟− −⎪ ⎪⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

(4-38)

4-69

The radius of curvature can be calculated from:

( )

( ) ( )1, 1,

2 2

1, 1,

21

i j i j

ci j i j

r m z z r r

φ φφ

− −

+ −

+ +

−∂= =

∂ − + − (4-39)

There are other possibilities for the curvature radius ― one of them is to fit a spline between the three points and then analytically evaluate the second derivative. This, however, introduces numerical instability to the calculation [46]. The radial equilibrium equation is per definition parabolic, and the anticipated ellipticity arises from the finite approximations.

4.2.4 Integration constant

To start the calculation, the initial value for the meridional velocity at the mid-streamline is guessed. This value is then used for the first integration of the radial equilibrium equation in both directions (i.e. up and down along the calculation station). This guessed value for the mid-streamline is most likely not the correct one and a deviation in mass flow will be apparent. The mass flow through the computing station is calculated according to the equation:

( )( )2 cos 1TIP

HUB

r

mr

m r c dlπ ρ φ γ λ= − −∫ (4-40)

This integral is solved with a standard numerical technique using average fluid properties for each stream tube. The density in equation (4-40) is normally calculated from the “total” density and the isentropic relation:

12 2 1

0 12m

p

c cc T

κθρ ρ

−⎧ ⎫+⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭

(4-41)

It is convenient to re-write the equation above to fit the analysis case as:

( )

12 2 1

0

1 tan1

2m

p

c

c T

κβρ ρ

−⎧ ⎫+⎪ ⎪= −⎨ ⎬⎪ ⎪⎩ ⎭

(4-42)

The necessary change in mid-streamline velocity is calculated with the equation:

( )4 40

mid

inlet eqm

m

m mc dm

dc

−−∆ = (4-43)

4-70

The derivative in the denominator in the equation above is calculated by introducing the equation for density in the integral expression for mass flow. Only the result is presented here and reference [45] provides guidance how to derive it.

21TIP

mid midHUB

rm x

m m mr

dc Mdm dmdc dc c

−= ∫ (4-44)

Where:

θif design case or c specifiedif β is specified

mx

rel

MM

M⎧

= ⎨⎩

( )

for the design case

1 for the analysis case1

mid

mid

m

mm

mm

midm

cc

dcdc

dcdl ldc dl

⎧⎪⎪⎪= ⎨⎪ ⎛ ⎞⎪ − − ⎜ ⎟⎪ ⎝ ⎠⎩

(4-45)

The final expression for the design case is:

( )21 1TIP

mid mid HUB

r

mm m r

dm M dmdc c

= −∫ (4-46)

The analysis case is more complicated and the derivative in the denominator is normally evaluated with the radial equilibrium equation. The Mach number is normally evaluated with the “Q-function”. Designing a turbomachine with the combination of known blade profiles (hence angles) and mass flow results in two branches, one sub-sonic and one super-sonic. The code cannot by itself find the correct one, and the designer must specify if the outlet velocity is sub- or supersonic. This is not a problem when working in design mode since the meridional Mach number is used for the calculation. This is more or less always subsonic, but there may be rare cases where the streamlines diverge sufficiently to provide supersonic capability in the stream-wise direction. The maximum loading condition, however, provides an effective limit to the maximum meridional velocity. It can be explained by the fact that the normal passage shock will “swing” from its normal position (T.E. → S.S.) to the maximum loading condition, which is where the shock stands from T.E. to the next T.E. Any further reduction in back-pressure will not affect the aerodynamics of the row and will only increase the profile losses. The “off-design” calculation in the analysis mode is not an issue since the pressure ratio is specified and therefore the correct branch is set by the thermodynamics.

( )0

0

m TQ f M

p A= = (4-47)

4-71

The new velocity must be relaxed to provide stability and reference [45] gives the following expression:

( )111.52

1m

nm

ccn n

m m mc c e c−

∆

−= + ∆ (4-48)

4.2.5 Target pressure method

The previous method does not work on choked stages, or more true, on multiple choked stages. The reason for this is that one specifies the mass flow and inlet state in order to seek the pressure ratio. If one studies the plot of flow versus pressure ratio, it is readily apparent that the system becomes virtually inoperable from a numerical point of view. The reason for this is that the flow rate is independent of the downstream pressure for sonic flow condition. It is possible to operate stages in heavily choked condition and the mechanism is over-expansion. This is in full analogy with a normal Prandtl-Mayer expansion which states that one can over-expand by turning of a supersonic flow. One other way of explaining it is that the flow must turn to maintain continuity. Denton [43] solved this problem by introducing the “target pressure technique”. This was indeed a major step in turbine calculation and made it possible to calculate multiple choked rows. All details, however, will not be presented here and the reader is referred to [43 and 53] for a full description of this topic. Denton’s method is based on abandoning the continuity equation at the trailing edge planes and introduces a guessed target pressure. The meridional velocity at the mid-streamline is then adjusted until the actual pressure converges to the target pressure:

( )target1

, , 2

2n nm mid m mid

p pc c

wρ−

⎧ ⎫−⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

(4-49)

When the target pressure is converged, the target pressure at a certain trailing edge plane is adjusted to improve continuity with the next downstream plane and the guessed inlet flow.

4.2.6 Streamline relocation iteration

A stream tube is bounded by two streamlines, since by definition there is no velocity component normal to a streamline, so the mass flow through the streamtube must stay the same. Another way of looking at it is to use the streamfunction, which also by definition is constant along a streamline. The common way is to define a relative streamfunction, which is zero at the hub and unity at the casing. This is valid only for the case without added cooling air and extractions. In the case of either a cooled turbine or if extractions are present, one has to adjust the mass flow or the stream function. At the first iteration, the initial streamline positions are guessed and will generally not fulfil the stated requirements. It is therefore necessary to have a routine that relocates the streamlines at all computational nodes in order to satisfy the stream function.

4-72

( ) ( )

1 at the casing

all other streamlines nodes

0 at the hubtot

mm

ψ

⎧⎪⎪= ⎨⎪⎪⎩

∑

The correct streamline positions are found by interpolation and the new streamline positions must be relaxed for stability. The detailed procedure is presented in references [13], [39], [45] and will not be repeated here. A new streamline position is calculated with the equation: ( )1 1 1n n n n

calcl l RF l l− − −= + − (4-50) Where:

( )

( )

2 2

2 2 2

1 for the design case1

16 8

1 for the analysis case1 cos

16 8

m

rel

M Rx

RF

M Rx

β

⎧⎪ − ∆⎡ ⎤⎪ + ⎢ ⎥⎪ ÷ ∆⎪ ⎣ ⎦= ⎨⎪⎪ − ∆⎡ ⎤⎪ + ⎢ ⎥÷ ∆⎪ ⎣ ⎦⎩

(4-51)

The length R∆ is defined as the length of the computing station and x∆ is defined as the meridional distance at the mid-radius.

4.3 Loss modelling

The AMDC-KO-MK can be used for modelling losses in the throughflow environment. Despite that the model is originally intended as a mid-span model, the two-dimensional version of it works very well. The general structure of the AMDC-KO-MK is to add different loss components according to the equation:

Re *S

TOT P i S TC TES

YY Y Y Y Y YY

χ⎛ ⎞

= ⋅ + + ⋅ + +⎜ ⎟⎝ ⎠

(4-52)

The preceding equation is evaluated for all streamlines, the profile-, incidence- and trailing edge loss are calculated in the exact manner as for the 1-D case. The secondary loss and tip clearance loss are local losses and a distribution scheme is required. Two basic approaches are available; span-wise mixing or loss smearing. Since most of the span-wise mixing in a turbine is driven by convective transport due to the secondary flows at the end walls, a well-calibrated smearing scheme will work fairly well. References [47, 48, 49, 50, 51 and 52] provide an in-depth treatment of span-wise mixing and the theory will not be repeated here. The general

4-73

opinion for turbines is, however, that the mixing is driven by both classic secondary flow as well as turbulent diffusion. The rather course calculation grid in a throughflow calculation makes the smearing method attractive. It is normally to introduce some kind of sub-grid if the span-wise mixing equations are to be solved.

4.3.1 Profile losses

The profile loss is basically due to shear work/dissipation in the boundary layers. The general equation is:

, 020.9143P P P i shockY K Y Y CFM=

⎛ ⎞= ⋅ + ⋅⎜ ⎟⎝ ⎠

(4-53)

In the equation above, two factors appear. The first (0.914) was introduced by Kacker and Okapuu [21] and allows the basic loss equation to be written in the form where the trailing edge loss is added. The second factor (2/3) is a vintage factor, taking into account profile development between 1951 and 1981. The basic profile loss YP,i=0 is calculated according to the equation:

'

' ' '

' 'max

, 0 , 0 , , 0 0.2

in

out

in in out in

in inP i P P P

out out

t lY Y Y Yαα

α α α α

α αα α= = = =

⎧ ⎫⎛ ⎞⎪ ⎪⎛ ⎞⎡ ⎤= + −⎨ ⎬⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭ (4-54)

The base values are taken from the figures below.

(a) (b)

Figure 4-4. Profile loss coefficients, (a) is valid for α1=0 (b) is valid for α1= −α2 [21].

The KP factor is calculated with the expression:

( )2

1 1.25 0.2 inP out

out

MK MM

⎛ ⎞= − − ⎜ ⎟

⎝ ⎠ (4-55)

The KP factor is a correction taking into account that the original profiles were tested at low subsonic velocities. These velocity levels are lower than what one should expect in a turbine and the results may be therefore conservative. The KP factor was introduced to reduce the profile losses.

4-74

The Yshock factor takes into account compressibility effects at or near the trailing edge. This correction is only applicable for design levels of incidence. The CFM factor is used to model the trailing edge compressibility effects and is calculated with the equation [21]: ( )21 60 1outCFM M= + − (4-56) The preceding equation is known to over-predict [23] the supersonic drag and reference [53] proposes the more conservative: ( ) ( )3 21 49.5 1 3.3 1out outCFM M M= + − + − (4-57)

4.3.2 Incidence losses

The incidence losses are described in section 3.3.4 Leading Edge Thickness.

4.3.3 Secondary losses The boundary layer displacement thickness and the blade chord primarily govern the span-wise extent of the influenced region [54]. Tests have shown that the influenced region was in the range of 0.35 to 0.62 of the chord when the displacement thickness over chord varied from 0.01 to 0.05 [53]. The position of the peak (within that region) is a linear function of the turning in the cascade. In reference [53], an example of a secondary loss-smearing algorithm is presented:

1. Evaluate the local secondary loss for the actual streamline:

( )

( )

,

2 2

' 3*

cos cos0.0334cos cosS loc

AMDC out outLAR

in mf const

c Accelration Loading Ainley

CY fs cδ

α αα α

⎛ ⎞≈⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞= ⋅ ⋅⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠ (4-58)

Where:

( )

1 0.25 2 21 2AR

h c h cf

h ch c

⎧ − − ≤⎪= ⎨

>⎪⎩

(4-59)

( )2 tan tan cosLin out m

Cs c

α α α= + (4-60)

1 tan tantan

2in out

mα αα − +⎧ ⎫= ⎨ ⎬

⎩ ⎭ (4-61)

4-75

The final value is then calculated:

. ,1.2 AMDCS loc S loc SY Y K= ⋅ ⋅ (4-62)

2. The calculated value from above equation is multiplied with the local secondary loss smearing factor KS:

*, ,S loc S locY Y KS= ⋅ (4-63)

The KS value can be obtained from the figure below:

Figure 4-5. Span-wise distribution of secondary loss [53].

3. Calculate the area weighted mean value for the secondary loss coefficient over the whole passage height:

*, cos

cos

SHROUD

S locHUB

S SHROUD

HUB

Y r drY

r dr

α

α

⋅ ⋅=

⋅

∫

∫ (4-64)

4. Evaluate the secondary loss coefficient at the mid-streamline using equation (4-58). This makes the total secondary loss coefficient the same, as the mid-span model would have given.

5. Introduce the ratio between the mid-span and the area weighted mean value:

,,

S MIDS rat

S

YY

Y= (4-65)

4-76

6. The corrected value for the local secondary loss coefficient is then obtained with the simple adjustment:

** *, , ,S loc S loc S ratY Y Y= ⋅ (4-66)

Example from SC90T Station 6 Stage 1 nozzle trailing-edge <----------- Angles -----------><---------- Loss coefficients ---------------> Fract.ht acos(o/s) del.Mn del.sec ang.sw Total Profile t.e. Sec`ry Clear`ce Cooling [deg] [deg] [deg] [deg] 0.00 72.01 -2.10 6.17 76.07 0.1542 0.0255 0.0239 0.1047 0.0000 0.0858 0.07 72.14 -2.07 -1.71 68.36 0.1196 0.0255 0.0237 0.0703 0.0000 0.0753 0.14 72.28 -2.04 -0.67 69.57 0.1063 0.0255 0.0237 0.0571 0.0000 0.0710 0.21 72.41 -2.01 -0.16 70.24 0.1166 0.0256 0.0235 0.0674 0.0000 0.0736 0.29 72.54 -1.98 -0.03 70.52 0.1184 0.0258 0.0232 0.0694 0.0000 0.0729 0.36 72.66 -1.97 -0.01 70.69 0.0992 0.0260 0.0229 0.0503 0.0000 0.0719 0.43 72.79 -1.95 0.00 70.84 0.0847 0.0261 0.0226 0.0360 0.0000 0.0708 0.50 72.90 -1.93 0.00 70.97 0.0877 0.0263 0.0222 0.0392 0.0000 0.0700 0.57 73.02 -1.91 -0.01 71.10 0.1099 0.0264 0.0218 0.0616 0.0000 0.0695 0.64 73.13 -1.89 -0.03 71.20 0.1232 0.0265 0.0215 0.0752 0.0000 0.0689 0.71 73.24 -1.88 -0.11 71.25 0.1139 0.0266 0.0213 0.0660 0.0000 0.0678 0.79 73.35 -1.86 -0.35 71.13 0.1108 0.0267 0.0211 0.0631 0.0000 0.0663 0.86 73.44 -1.85 -0.96 70.63 0.1242 0.0268 0.0209 0.0766 0.0000 0.0627 0.93 73.52 -1.84 -1.53 70.14 0.1508 0.0268 0.0206 0.1033 0.0000 0.0660 1.00 73.58 -1.82 5.50 77.25 0.1902 0.0270 0.0204 0.1429 0.0000 0.0742 MEAN 0.1168 0.0262 0.0222 0.0683 0.0000 0.0704

The secondary loss is also corrected for the incidence. The procedure is described under section 3.3.4, and will not be repeated here.

4.3.4 Trailing edge loss

The trailing edge loss is modelled as in section 3.3.3 and the will not be repeated here.

4.3.5 Tip clearance loss

The tip leakage loss mechanism is different depending on whether or not the blade has shroud. For the shroudless design there will be a flow from the pressure side to the suction side. This leakage flow will produce a leakage vortex that will merge with the secondary flow vortex and amplify it. The vortex will then be diffused downstream, causing both losses and over- and under-turning effects. The over- and under-turning will cause the preceding blade to work with unwanted local incidence and hence losses. In the case of a shrouded row, the leakage flow will pass over the shroud and then be mixed with the free stream flow. In the case of a shroudless blade, the effect will extend over a considerable length of the blade.

4-77

Figure 4-6. Span-wise penetration of the leakage loss [22].

In reference [55] a smearing method is presented. The details will, however, not be presented here and the reader is referred to the original source for further information. A typical result is found in the example below:

Example from SC90T

Station 11 Stage 1 rotor trailing-edge <----------- Angles -----------><---------- Loss coefficients ---------------> Fract.ht acos(o/s) del.Mn del.sec ang.sw Total Profile t.e. Sec`ry Clear`ce Cooling [deg] [deg] [deg] [deg] 0.00 62.77 0.70 0.00 -62.07 0.4023 0.0590 0.0136 0.3297 0.0000 0.0751 0.07 63.12 0.66 0.00 -62.46 0.4647 0.0505 0.0141 0.4001 0.0000 0.0731 0.14 63.46 0.60 0.00 -62.86 0.2549 0.0480 0.0148 0.1921 0.0000 0.0712 0.21 63.79 0.51 0.00 -63.28 0.1347 0.0464 0.0156 0.0727 0.0000 0.0684 0.29 64.11 0.46 0.00 -63.65 0.1085 0.0692 0.0161 0.0232 0.0000 0.0668 0.36 64.41 0.40 0.00 -64.00 0.1061 0.0429 0.0166 0.0466 0.0000 0.0657 0.43 64.70 0.35 0.00 -64.35 0.0889 0.0412 0.0172 0.0306 0.0000 0.0645 0.50 64.98 0.31 -0.01 -64.68 0.0578 0.0398 0.0177 0.0003 0.0000 0.0635 0.57 65.25 0.34 -0.02 -64.94 0.0567 0.0386 0.0181 0.0000 0.0000 0.0628 0.64 65.51 0.34 -0.07 -65.25 0.0929 0.0372 0.0184 0.0050 0.0324 0.0633 0.71 65.77 0.15 -0.25 -65.86 0.2310 0.0355 0.0184 0.0509 0.1262 0.0658 0.79 66.01 -0.33 -0.84 -67.18 0.3854 0.0340 0.0184 0.1008 0.2322 0.0710 0.86 66.24 -0.77 -2.90 -69.91 0.5037 0.0344 0.0184 0.0936 0.3573 0.0820 0.93 66.46 -0.95 -3.28 -70.69 0.6716 0.0329 0.0184 0.1033 0.5170 0.0763 1.00 66.68 -1.01 14.67 -53.02 1.1027 0.0315 0.0186 0.1529 0.8997 0.0561 MEAN 0.2690 0.0425 0.0171 0.0911 0.1184 0.0681

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4.4 Supersonic conditions at the throat

It is necessary to calculate the relation between the trailing edge plane and the throat since it is necessary to know when the row is choked, over-expanded, and for the design case, calculate supersonic deviation. It is crucial to determine if the current solution of the meridional velocity at the trailing edge plane represents a choked throat or not. The critical state at the throat is calculated by assuming M=1 at the throat. The corresponding value of meridional velocity at the trailing edge is then solved with the 1st law, continuity equation, momentum equation normal to the throat and state equations. The presented algorithm is designed for use with a standard gas or steam table. The losses are taken into account as an increase in entropy and hence, any loss model can be used. The flow at all other stations (i.e. leading edges, internal and ducts) is assumed to be sub- sonic. The outlet angle at a throat Mach number of unity can be calculated as [62]:

( ),2* * *2, 2

2

*2 2 2,

cos

cos

mth th th geo th

th th b th geo

cc o b c

p o b p TE b p b s

ρ α αα

α

⎧ ⎫⋅ ⋅ ⋅ ⋅ − − =⎨ ⎬

⎩ ⎭⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ ⋅

(4-67) Where:

12, cosgeo

os

α − ⎛ ⎞= ⎜ ⎟⎝ ⎠

(4-68)

If one uses Denton’s assumptions, the distance between the throat and the trailing edge plane is negligible compared to the blade chord. Hence, no change in AVDR is necessary to be taken into account between the two planes. b bth = 2 (4-69)

Yes

Yes

No

No

Known inlet state: h or I0 0 1,

Guess pressure at throat, pth

s s s s sth th LE TE= + ≈ +→ →0 0 0∆ ∆ Χh f p sth th th= ,b g

h h h cth th

th0 0 0

2

2, ,= = + or

I I h w uth th

th th= = + −0

2 2

2 2

M f h s c or wth th th th th= , ,b g

Mth = 1

Guess pressure at trailing edge, pTE

s s sLE LE TE2 = + →∆ h f p s

f p sLE LE LE

LE LE LE

=

=

,

,

b gb gρ

c o c bs bm

th th th,

* *

22 2

=⋅ ⋅ ⋅

⋅ ⋅ρ

ρ

outlet angle from eq. 4-67

h h cm0 1 1 12 2

112

1, , tan= + + αc h or

I h c um2 2 22 2

2 221

21 1

2= + + −, tan βc h

h hI I0 1 0 0

2 1

, ,=

=

4-79

The loss between the trailing edge and the throat can be assumed to be a certain fraction of the total loss in the stage. Denton, however, suggested that the flow could be treated as isentropic between the leading edge and the throat. There are a few cases when this assumption may be inadequate and a substantial amount of the losses are generated before the throat (e.g. shower head cooling, high inlet Mach-number, incidence). The factor X is introduced for this purpose and is most likely to be in the range 0.1 – 0.3 for “competent designs”.

4.5 State calculation at supersonic conditions

If the present solution of the REE gives a value of the meridional velocity which is higher than the calculated limiting value, the stage is over-expanded. Since no q-o is placed at the throat, it is necessary to satisfy the condition that the present solution represents a throat Mach number of unity. The exit angle varies with outlet Mach number (due to supersonic deviation) and it is not possible to use correlations like the Ainley method. Instead one has to base the outlet angle either on the work (1st law) or continuity. In any case, the solution must satisfy work, momentum and state. The proposed method is based on an iterative approach for solving the continuity equation between the known sonic state at the throat and the trailing edge. It is possible to determine the stream tube convergence/divergence by linear interpolation between the leading edge planes, or preferably by an upstream internal station and the trailing edge plane. Another solution method proposed by Denton is to use the continuity equation between the throat and the trailing edge plane, in the form of [43]:

α ρρTE

th th th

TE TE TE

o c bs c b

=⋅ ⋅ ⋅

⋅ ⋅ ⋅−cos

* *1 (4-70)

The pressure at the trailing edge is solved by iterations and the convergence criteria are the total enthalpy for stators and rothalpy for rotors. One inner loop is also necessary since the REE provides meridional velocity and not the total velocity. The total velocity is easily calculated with the standard trigonometric relation: c cTE m TE TE= , cosα . Both methods, however, give the same result since they are doing the same thing but in a somewhat different order.

yes

no

Guess pressure at the trailing edge plane, pTE

s s sTE LE= + ∆ h f p sTE TE TE= ,b g ρTE TE TEf p s= ,b g

o c b s c bth th th m⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ρ ρ* *,2 2 2

h h cm0 1 1 12 2

112

1, , tan= + + αc h or

I h c um2 2 22 2

2 221

21 1

2= + + −, tan βc h

convo c bs c b

th th th

m TE TE

=⋅ ⋅ ⋅⋅ ⋅

−ρρ2 1* *

,

conv ok?

Meridional velocity from currentREE solution (lagging one loop)

4-80

4.6 State calculation at subsonic conditions

If the present solution of the REE gives a value of the meridional velocity which is lower than the calculated limiting value, the stage is operating at subsonic conditions. In this case the flow state can be calculated in a straightforward manner. The calculated meridional velocity satisfies the continuity equation and the state is readily calculated if the flow angle is known either as input data or by correlations. If the latter is based on the Ainley-Mathieson method, one iteration loop is still required. The reason for this is that the calculated angle is a function of the outlet Mach number. The Ainley-Mathieson angle model is rather old, but still well-regarded, and gives fairly accurate results. The original model was published 1951 in graphical form. The presented equations are from [34] and are approximations to the work of Ainley and Mathieson. This method is based on interpolation between two calculated values, at a throat Mach number between 0 → 0.5 and 1.0.

α βM Mth thor o

sse= =

−= FHGIKJ −FHGIKJ0 5 0 5

1 4. . cos (4-71)

α βM Mth thor o

sf s

eos= =

− −= FHIK −FHIKFHIK1 1

1 1cos sin (4-72)

f se

se

se

se

FHIK =

FHIK

− FHIK +

FHIK

0 054

1 1 0 7422

.

.49 . (4-73)

The interpolation equation is given in reference [8] and is a linear interpolation: ( )( )0.5 1.0 0.5 0.5 1.02 1

th th th thM M th M MMα α α α< < = = == − ⋅ − − (4-74)

Islam and Sjolander proposed [56] another method for establishing deviation, rather than outlet angle, at subsonic conditions. Their method is basically an evolved form of Carter’s rule, which is widely used in axial compressor design.

( )

( )1,

1.12.253

1 2,

0.31.45 1.6422 0.22

b

b

m

sAVDRc

tc

α αδ

ξ α

⎛ ⎞ +⎜ ⎟⎝ ⎠=

⎛ ⎞ +⎜ ⎟⎝ ⎠

(4-75)

The proposed equation has the inherent advantage that the throat is no prerequisite.

4-81

4.7 Secondary deviation

The end wall secondary flow vorticity causes a phenomenon which is normally referred to as over- and under-turning. The SCM method code is not readily suited to calculate these effects in a direct manner since it is a 3-D effect. There are two different approaches to solve this problem; the first is to use a “span-wise mixing model” and the second is to use some kind of approximate model. The latter is probably the best alternative for an SCM-code since it does not include additional modifications to the code at an overlaid level. Span-wise mixing is, loosely stated, a method of distributing total enthalpy, entropy and angular momentum over the passage height. This type of method normally introduces additional complexity to a SCM code since the grid density is rather coarse (or even very coarse compared to a CFD-code), and additional node points are required to get sufficient resolution. The secondary deviation will affect the blade loading and also cause increased profile loss at the succeeding row due to incidence. There are recent examples of turbines that are designed with the equivalent to compressor end-bends to reduce the local incidence. Massardo and Satta [57] addressed this problem and introduced a rather simple method based on dimensional analysis. This method will be described in this section. The secondary deviation is defined as:

eε α α= − (4-76)

Where index e stands for the unaffected angle (i.e. as a function of throat/pitch ratio and Mach number), Lakshminarayana [58] presented the following set of equations for the secondary vorticity in intrinsic coordinates (s, n and b):

22

2 2 22s n s

c c

ws w wr r w b w b

ζ ζ ζρ µρ ρ

∂∂ ∂⎛ ⎞ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠ (4-77)

( ) 2

1n

pws s b

ρζρ

∂ ∂ ∂=

∂ ∂ ∂ (4-78)

Note: the equations were derived under certain assumptions and they will not be repeated here, see reference [58] for further information. The equations above were solved along the mean streamline in order to get the vorticity distribution. It is possible to introduce a secondary stream function if one reduces it to a 2-dimensional problem by assuming that the vorticity is constant along n. Massardo and Satta [57] then solved the stream function in the n and b plane by solving Poisson’s equation: ( ) ( )2 , , ,s ss n b s bψ ζ ζ∇ = − = − (4-79) The velocity components are then evaluated according to the well-known relations:

;u vn bψ ψ∂ ∂

= − =∂ ∂

(4-80)

4-82

The secondary deviation angle is then calculated according to the equation:

( ) 1, , tan vf s n bw

ε − ⎛ ⎞= = ⎜ ⎟⎝ ⎠

(4-81)

The numerical procedure is not trivial and the reader is referred to the original source for further information [57]. The secondary flow will cause over-turning close to the wall ( wε ) and under-turning ( mε ) at some point (zm) away from the end-wall due to the sign of the velocity component in equation (4-81). These over- and under-turning angles are passage averaged values. The model is the based on the assumption that the important parameters are:

1. Boundary layer thickness δ,

2. B.L. shape factor H12,

3. Acceleration W2/W1,

4. Blade height, h

5. Staggered spacing S’

6. Streamline curvature (replaced with turning and chord) in the S1 plane.

They then constructed a non-dimensional parameter, which is based on the list above:

( )( )

32

22

cos' 'cosSS S DP

h S hα

δ δ α δ= =

+ (4-82)

The parameter P is used as an independent parameter in the correlation, together with the turning, shape factor and acceleration. They also introduced [57]:

ln*

PPP

⎛ ⎞∆ = ⎜ ⎟⎝ ⎠

(4-83)

Where:

( )31 2 1ln * ln cos 0.155sinP f α α= − (4-84)

( ) ( ) ( )2 3

1 4.28 0.534 0.78 0.104f h S h S h S= + − + (4-85) The parameter 0mε is a function of the turning and ∆P and is defined as [57]: 0 2 3m f fε ϑ ϑ= − (4-86) The f2 and f3 are functions of the ∆P parameter.

4-83

The minimum angle is evaluated from the expression: ( )4 5 0 4 5 2 3m mf f f f f fε ε ϑ ϑ= ⋅ ⋅ = ⋅ ⋅ ⋅ − ⋅ (4-87) Where: f4 is a function of the end-wall boundary layer shape factor (H) f5 is a function of row velocity ratio (c2/c1 or w3/w2) The span-wise position for the minimum value (under-turning) is a function of the parameter P and the boundary layer thickness [57]:

( )0.817 0.0574lnmz Pδ

= + (4-88)

The maximum value (over-turning) is found with the expression [57]: 3.5w mε ε= ⋅ (4-89) In the original source [57], a span-wise smearing model is presented. No further details will be presented here and the reader is referred to the original source for further information.

Figure 4-7. Span-wise distribution of under- and over-turning [57].

4-84

5-85

5 Steam Turbines

5.1 Steam Turbine Deterioration

The steam turbine is more straightforward when it comes to degradation since it is simpler than the gas turbine. The flow path degradation can be categorised into five main root causes:

• Deposits

• Surface roughness

• Sealing leakages

• Internal leakages

• Solid particle erosion (SPE)

Figure 5-1. Example of steam turbine degradation [59].

5-86

5.1.1 Deposits

Deposits are normally caused by carry-over effects in the boiler, where unwanted matter is carried with the steam to the turbine. There are three basic mechanisms for carry-over:

• Mechanical • Vaporous • Attemperators (i.e. de-superheater sprays)

Figure 5-2. Different mechanisms for boiler carry-over [60].

These effects are always more pronounced during transient modes and it is always preferable to have a turbine by-pass when starting the unit. There is another incitement for the turbine by-pass and that is avoidance of SPE damages in the turbine. The SPE is caused by magnetite particles from the superheaters during the start. The by-pass will be effective in two ways since it will effectively avoid hot spots in the superheaters during start, and the magnetite will end up directly in the condenser instead of causing erosion problems in the turbine. Typical critical impurities are:

• Silica from make-up and condenser leakages • Copper oxides from pre-heater and condenser tubing • Chlorides from make-up and condenser leakages • Iron oxides (e.g. magnetite) from superheaters • Carbon dioxide • Sulphates • Organic and inorganic acids

The figure below shows a typical distribution of deposits in a reheat unit. Notable is the position of copper deposits. This type of deposits often leads to a reduced turbine capacity and may be hard to remove without opening the turbine.

5-87

Figure 5-3. Typical deposits distribution [60].

The amount of vaporous carry-over of a specific matter is strongly dependent of the amount of that matter in the boiler water cycle and a distribution factor. The figure below shows different distribution factors as a function of boiler pressure.

Figure 5-4. Distribution coefficients for different impurities [61].

The previous figure clearly shows that the complexity dramatically increases with increasing boiler pressure. Fortunately, nowadays it is not a problem to keep the impurity level at a low and more or less harmless level. The solubility for a specific matter is reduced as the steam is expanded through the turbine (lower pressure and temperature) and will start to deposit when it exceeds its maximum partial pressure. The effects of deposits are also demonstrated in the third paper in this thesis.

5-88

Figure 5-5. Solubility of SiO2 in superheated steam [60].

As already mentioned, the solubility is proportional to the pressure and temperature. This will make the cold end of the unit most susceptible for deposits. The same is also true at a micro level for the coldest section in the individual stages [28]. The Wilson zone is the most probable place for deposits, and deposits downstream are very seldom found. The Wilson zone is the section where the steam actually starts to condensate and is always below the saturation line. The whole theory will not be repeated here and the reader is referred to references [28, 62, 63 and 66] for further information. This sub-cooling or super saturation effect is associated with the rate (time) of expansion:

1 ppp t

∂= −

∂ (5-1)

The preceding expression is not trivial to evaluate with normal calculation tools and is preferably re-written, using the chain rule, to yield:

1 acp x ppp x t p x

∂ ∂ ∂= − = −

∂ ∂ ∂ (5-2)

This form can be evaluated with a standard finite difference method in a standard mid-span or throughflow program. The figure below shows experimental results for varying expansion rates.

Figure 5-6. Wilson lines [62].

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The Wilson region is typically, for a normal expansion line, 0.977 at a p of 10 s-1 and 0.963 at a p of 10000 s-1. Since most deposits are expected upstream the Wilson region, the possibility of using the region as a way of removing water-soluble deposits arises. If the admission temperature is lowered at a fixed pressure, then the wet region is moved upstream in the turbine blading. This may result in a very useful on-line wash for condensing turbines. The use of the attemperator spray may be sufficient if it is designed with additional capacity. Some organisations have installed special equipment for this purpose. It is indeed important to protect the turbine from water-hammering, and great caution must be taken in order to prevent it from damaging the turbine. There may be both magnetite formation and scaling if the admission temperature is sufficiently high. The main effect of deposits is, besides surface roughness, changed blade profiles and capacities. From figure 5-1 it can be seen that a cold start can restore some of the performance loss. The reason is simply that the first steam that is admitted into the turbine condensates and some kind of water wash is obtained. This may be the reason for the sometimes unexplained temporary performance losses. The combination of this effect and loosening of deposits during the cooling down period may be significant.

5.1.2 Surface roughness

Increased surface roughness is normally due to deposits but also due to aging, foreign object damage (e.g. weld bead), improper cleaning and conservation. Foreign damage occurs typically after major boiler repair and is often a very dramatic, fast process with large drops in efficiency. This is often an unnecessary, and indeed costly, type of problem. In most cases, the cure is to install a fine mesh strainer during the initial operation or even better, a full set of steam blows. The impact on efficiency is due to increased profile losses.

5.1.3 Sealing leakages

The mechanism of damaged sealing is in most cases:

• Tip rubbing • Steam flow induced vibrations • Wrong assembly

Rubbing or unwanted contact can normally be avoided if the operation manual from the supplier is followed. This is especially true for start and stop modes.

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5.2 Simplified models

The models described in this section where used for modelling the steam turbine cycle in paper number three. The model is basically an improved heat- and mass-balance calculation method. The off-design calculation models used are presented in this section. A group of individual stages is called a part turbine or turbine section. How the individual stages are divided into groups is arbitrary, but normal separations are extractions and induction points. A turbine section is normally defined by its isentropic efficiency and swallowing capacity. The efficiency of a turbine section is set by the individual stages and the reheat effect. The isentropic efficiency is defined as:

,

ss

h hh h

α ω

α ω

η −=

− (5-3)

The total to static stage efficiency is determined more or less by the stage loading and design reaction level, and other parameters are of secondary order. This is, however, only true for competent designs. There are different ways of showing this dependency but they give neither a strict definition from a physical point of view nor a practical turbine. The suggested definition [64] gives non-swirling flow after the rotor:

1' sin2000 2 1OPT

s OPT

uh

η αν

⎛ ⎞ ⋅= ≅⎜ ⎟⎜ ⎟⋅ ∆ − Λ⎝ ⎠

(5-4)

However, the true maximum efficiency occurs with few degrees of negative outlet angle. The Parson number, when dealing with turbine sections, replaces the stage loading parameter. The Parson number is defined as:

2

s

uh

Χ =∆∑ (5-5)

The Parson number gives some gauge of the average section loading and the similarity to stage loading parameter is apparent. It is very convenient to use this definition when calculating off-design efficiency since the efficiency is a parabolic function of the Parson number. It is a further simplification to correlate the efficiency versus relative Parson number. The reason for this is that the speed squared cancels out for fixed-speed turbines, and the efficiency is then only a function of relative isentropic heat drop.

,, ,

,

s dess act s des

s act

hf

hη η

⎛ ⎞∆≈ ⋅ ⎜ ⎟⎜ ⎟∆⎝ ⎠

(5-6)

The turbine swallowing capacity is also important since it sets the connection between mass flow, pressure and temperature. The model by Salisbury is based on an analogy with a series of nozzles in some kind of pipe. Work is extracted in some way after each nozzle, but this feature is not an issue in the model.

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Figure 5-7, The Salisbury turbine model [28].

It is possible to use the general flow equation on each “stage” and the general flow equation yields:

( )

2 1

1 2 2

1 1 1

21q

p p pm C Av p p

κκ κκ

κ

+⎧ ⎫⎛ ⎞ ⎛ ⎞⋅ ⎪ ⎪= ⋅ ⋅ ⋅ −⎨ ⎬⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

(5-7)

The Cq factor takes into account all factors that deviate from isentropic flow and boundary layer blockage effects. The critical pressure ratio is obtained by differentiating the equation above and setting the result to zero. As a general rule, sonic and supersonic flows are normally only found in the last stages, and in the control stage, below some 50% load. It is quite common to have supersonic flow at the tip-section of the last blade. The pressure ratio over a turbine stage is constant over the load range, except for the last stage and the control stage. This feature simplifies the equation above considerably and the following form evolves:

1,

1T rel

pm Cv

= (5-8)

The equation shows that the flow is directly proportional to the pressure and inversely proportional to the temperature27. One other important feature is, since the pressure in the condenser is fixed, all pressures in the turbine are built up from the condenser. The empirical Stodola steam cone rule has been used since the early days and the following form evolved:

( )

2 2

, , 211i j i

i j T i j T i ji i i i j

p p pm C Cp v v p p

− − −

−= ⋅ = ⋅ ⋅ − (5-9)

27 z R Tv

p=

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This equation has been widely used and reduces to equation (5-8) if pi>>pj, which normally is the case for condensing turbines. The pressure ratio over a non-reheat steam turbine train may be as high as 3500:1. Equation (5-9) will not work very well for single stages28 and/or choked stages/turbines. Beckmann solved this problem [65] and improved the results substantially for single stages. This approach is more complicated, but it better takes into account when the turbine gets choked as a function of polytropic exponent, number of stages, stage loading29 and pressure ratio.

1 1.5 2 2.5 3 3.5 40

20

40

60

80

100

Pressure ratio [−]

Mas

s flo

w [k

g/s]

Swallowing Capacity

Designx

BeckmannStodola

1 1.5 2 2.5 3 3.5 4

−2

0

2

4

Pressure ratio [−]

Diff

eren

ce [%

]

1 1.5 2 2.5 3

2000

4000

6000

Pressure ratio [−]

Slo

pe [−

]

BeckmannStodola

Figure 5-8, Comparison between the Stodola cone rule and the more advanced Beckmann model for modest loaded blades (stage pressure ratio approx 1.25).

The figure clearly shows that the Stodola steam cone rule is working very well on part turbines. Most off-design modelling is done at a level where partial turbines are calculated instead of individual stages. A partial turbine is a group of stages, typically n-stages, between two extraction/induction points. The situation is even better at higher pressure ratios like those encountered if whole cylinders are calculated.

28 Should not be used on control stages since it generally gives poor results. 29

2 2pol

v dp hu u

µη

− ∆= ≈∫

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1 1.5 2 2.5 3 3.5 40

20

40

60

80

100

Pressure ratio [−]

Mas

s flo

w [k

g/s]

Swallowing Capacity

Designx

BeckmannStodola

1 1.5 2 2.5 3 3.5 4

−5

0

5

10

15

20

Pressure ratio [−]

Diff

eren

ce [%

]

1 1.5 2 2.5 30

2000

4000

6000

8000

Pressure ratio [−]

Slo

pe [−

]BeckmannStodola

Figure 5-9, Comparison between the Stodola cone rule and the more advanced Beckmann model for a highly loaded stage (stage pressure ratio approx 3).

The capacity equation sets the pressure at the inlet of each turbine section and hence the isentropic heat drop for each section. The Parson number is determined if speed and isentropic heat drop is known, hence it is possible to calculate the efficiency. Since the outlet pressure is more or less fixed by the heat sink and condenser surface, all pressures in the turbine develops from condenser. By adopting this principle, one can show that the isentropic heat drop is more or less constant. One other beautiful thing is that the section efficiency is constant since the isentropic heat drop and speed are constant. This reasoning leads to three types of steam turbine stages:

• Control stage with varying pressure drop/efficiency

• Single stages with constant pressure drop/efficiency

• Final stage with varying pressure drop/efficiency

The control stage is more complicated to calculate since the stage, or more correctly, approximately half the arc capacity, will be heavily choked below 50% load. This is, however, dependent on the actual control stage (c-stage) design. The best approach here is to calculate the loading for each arc and then use a mass flow weighted average. The design efficiency is typically in the range of 65-7530%. 30 Includes the loss associated with the radius change after the c-stage. A small radius of streamline curvature will, if the meridional velocity is high, imply a pressure gradient to balance the centripetal acceleration.

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5.2.1 Condenser model

Condenser calculation is important when modelling steam cycle performance. The complete textbook heat transfer calculation is complicated and not practical in this lower order program level. The steam side heat transfer coefficient depends on:

• Difference between saturation temperature and tube temperature (inversely proportional to the power of 0.25).

• Relative tube position • Steam velocity • Turbulence • Non-condensable gases • Superheated steam (if any)

The water-side heat transfer is governed by the velocity, temperature and cleanliness. The exact calculation is as already mentioned, and the method developed by Heat Exchange Institute (HEI) is widely used and rather simple to implement in a program. Reference [66] gives the details and only the foundation will be given here. The heat transfer coefficient is calculated with the equation: 1 2 3 4U C C C C V= ⋅ ⋅ ⋅ ⋅ (5-10) Where:

C1 = Base value depending on tube diameter, [W/(m2 K)]

C2 = Factor depending on cooling water temperature at the inlet, [-]

C3 = Factor depending on tube material and thickness, [-]

C4 = Factor depending on tube cleanliness, [-]

V = Water velocity at the inlet of the tubes (i.e. cold condition) C1 and C3 are constants and are given in the table below:

Tube outer diameter

5/8” & 3/4” 7/8” & 1” 1” & 11/4” 13/4” & 11/2” 15/8” & 13/4” C1 2777 2705 2664 2623 2582

Tube material 304/316

Stainless steel

Admirality

Aluminium brass,

Aluminium bronze

90-10 Cu-Ni

70-30 Cu-Ni Titanium

C3 20 Gauge

0.75 1.02 1.00 0.94 0.87 0.77

18 Gauge 0.69 1.00 0.97 0.90 0.82 0.71

16 Gauge 0.63 0.96 0.94 0.85 0.77 -

The cleanliness factor (C4) is typically 0.85 for a new condenser and hence 15% additional condenser surface is available.

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The inlet cooling water temperature factor is given in the figure below:

0 5 10 15 20 25 30 350.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Cooling water temp [°C]

Tem

p co

rrec

tion

fact

or C

2 [−]

Temperature correction factor C2 [−]

Figure 5-10. Cooling water temperature correction factor, after [66].

It is convenient, for programming purposes, to use a polynomial for the cooling water temperature correction factor31:

7 4 5 3 4 2

2 , , ,

2 1,

5.64925 10 4.32349 10 6.76326 10

1.95335 10 5.78863 10CW in CW in CW in

CW in

C t t t

t

− − −

− −

= ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ +

+ ⋅ ⋅ + ⋅

The off-design heat transfer coefficient is calculated according to the equation:

2, ,

2, ,

act CW actact des

des CW des

C mU U

C m= (5-11)

5.3 Analysis methods

There are different methods of analysis available depending on the type of turbine. The biggest difference is if the expansion is wet or not. Wet expansion makes state measurements impractical (c.f. Gibbs state law) and one has to revert to other indirect methods. It is common and established practice to calculate low-pressure turbines’ efficiency by “bookkeeping.” That is, measuring a lot of parameters and evaluating how much work is produced in the HP- and IP-cylinders. Then subtract the sum from the gross output to get the contribution from the LP-cylinders. However, this method is impractical if one wants to find a specific LP-turbine with low efficiency. Instead it is better, or at least more convenient, to look for changes in stage pressure-flow characteristics. Some literature recommends that the LP-section should be throttled into a dry exhaust by e.g. using the intercept valves. This is, however, questionable,

31 N.B. Not valid in the outside of the range of the graph.

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and will only render in a completely meaningless test32. This technique will, on the other hand, work well on backpressure turbines with moderate exhaust wetness. Typical tests:

• Maximum capability

• Pressure-flow characteristics

• Cylinder efficiency

• Test code performance test

• Valve leakage test

• Condenser vacuum drop test

5.3.1 Maximum capability

This type of test is a full load test with valves wide open. The purpose of the test is to check whether the unit can produce its nominal/expected effect or not. The supplier’s correction curves are normally used for correction. It is also possible to use a generic heat balance program and the methods presented in the previous section. The incidence related corrections would, however, be missed. It is common practice for the OEM’s to use their mid-span program to produce the correction curves. For some types of turbines there are published performance information which could be used for correction [67]33. If the feedwater flow meter is in good condition, the station heat rate can be evaluated at the same time. It is, however, very hard to be sure about the flow meter (aging), and it is recommended to install a temporary flow meter in the condensate line and evaluate the feedwater flow though a heat balance around the feedwater tank. This makes the test time-consuming and costly, and it is questionable if it is worth the bother to carry it out.

5.3.2 Pressure-Flow Characteristics

The turbine capacity is already described in the previous section. These types of tools are probably the most important ones when dealing with steam turbines. The ideas are indeed not new and have been published extensively over the years. Some noteworthy references are [28, 68, 69] and the reader is referred to this work for further reference. For non-reheat units a clever parameter can be used [70]:

0EL

stg n adm

PKp T−

=⋅

(5-12)

32 The losses will be dominated by the last stage turn-up loss. The last stage will start to work as a very poor compressor and feed energy into the steam instead of extract work. The temperature increase due to this action can be estimated with the simple illustrative empirical expression: 30.79 295 339cond condT p m p∆ = ⋅ − + 33 This is the actual famous Spencer-Cotton paper/method. Their curves are still used today, and with vintage corrections, it is still a fairly accurate tool for GE’s units. This paper is a result of GE loosing an antitrust suit back in the 1960’s. This kind of information is normally not found outside the OEM.

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The derivation of this parameter can be carried out with the Buckingham pi-theorem, but a more physical approach will be used here. The electrical output of the turbine can be written as:

( ) ( )1 1 2

1

1

1 1 ...

1

I II IIIo o o

mEL mech gen ADM M

j oj

h h hP m

h

α α α

η ηα

−

+

⎧ ⎫∆ + − ∆ + − − ∆ + +⎪ ⎪

= ⋅ ⋅ ⎛ ⎞⎨ ⎬+ − ∆⎜ ⎟⎪ ⎪

⎝ ⎠⎩ ⎭∑

(5-13)

If one now assumes that the effective heat drop (bracket) and the external efficiencies stay the same, the expression above simplifies to (Willian’s law): constantEL ADMP m≈ ⋅ (5-14) If the preceding equation is combined with equation (5-8), then equation (5-12) appears. The resulting parameter is a gauge of effective flow area somewhere in the turbine.

Figure 5-11. Plot of equation 5-12 [70]. The second presented method [71] is also a rather old method and was published back in the early 1970’s. It is based on the fact that pressures inside the blading system are more or less governed by the downstream capacity. Therefore, if the possible extracted flow stays the same in the relative perspective, the ratio between two pressures inside the flow path must stay the same. However, if the capacity is changed, the pressure ratio will also change. This reasoning is only valid for sections other than the control stage and final stage. The figure on the next page gives an introduction to the method.

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Figure 5-12. Examples of pressure ratio fault analysis [71].

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5.3.3 Cylinder efficiency

The cylinder efficiency is directly measured where the steam is superheated. Comparison with expected efficiency is done by either experience or background curves in the absence of a good calculation model. There are a few nice outcomes here and one is that the IP-cylinder efficiency should stay more or less constant over the whole load range [28]. One problem is if the HP- and IP-cylinders are sharing shell (combined HP- and IP module), because if the internal sealing34 starts to leak, an increased appearing IP efficiency will be evaluated. Techniques for monitoring the sealing are presented in [28] and will not be repeated here. The procedure of how to evaluate internal efficiency is presented in many sources and will not be presented here. References [28, 72] provide in-depth information and the reader is referred to the original sources.

5.3.4 Test Code Performance Test

A Test code performance test (P/T) would be adequate in identifying faulty components. Unfortunately, a full test code P/T is a costly business and most operators try to avoid them. It is not uncommon for the only test to be carried out is the guarantee/acceptance test when the unit is new. In the US, the ASME PTC-6 test code P/T was so expensive to carry out that only a few plants were tested. The detailed procedures can be found in the test codes and will not be presented here.

5.3.5 Valve leakage test

The valve leakage test is indeed the cheapest test but it probably has the biggest economical potential. Steam flows that are bypassing the turbine will not produce work and its potential is entirely lost. It is also possible to have steam by-passing part of the expansion, and the amount of lost work is dependent on where in the process it is extracted from. Even components such as the start drains from the HP-heaters, which is flashed backwards, will reduce the available power. Installing surface temperature transmitters/probes best monitors faulty valves.

34 Commonly referred to as the N-2 packing

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6 Conclusions The overall objective of this thesis has been to show how degradation can be assessed with standard calculation tools. This includes mapping of different model-based monitoring tools, as well as description of the mechanisms of several aging phenomena. The modelling tools are thoroughly described, which makes it possible for the reader to develop a monitoring program or evaluate different tools. The methods presented are of different levels of sophistication, from the simplest ones to state-of-the-art heat and mass balance programs. The achievable level of sophistication is dependent on the amount of knowledge about the specific engine type. The highest level of sophistication is reserved for systems delivered by the manufacturing companies (OEM’s). This level of monitoring system requires detailed propriety turbine data, which disqualifies third part systems. A system delivered by an OEM is in general more costly, but the additional know-how is indeed a valuable commodity. When studying aging effects, there will be different options available. On a global perspective, the performance deck is the most useful calculation tool. However, when implementing faulty components, the component performance figures are based on the previous experience of the performance engineer. To be able to use more firm deterioration figures, it is possible to use either one-dimensional or preferably two-dimensional tools. One available two-dimensional tool is the streamline curvature method (SCM). This method is presented in detail here since this is the preferred method by the author. The reason for using a two-dimensional program is that the calculated flow capacity is predicted with greater accuracy than with a course mid-span model. This is due to the span-wise integrated flow value with blockage, which is virtually impossible with a mid-span model. The two-dimensional tool is thereby the best option when studying degradation effects of an engine. In the SCM, the magnitudes of the degradation effects are calculated. The most important mechanisms for degradation have been discussed in this thesis to show the variety of possible underlying physical reasons. Modelling of degradation phenomena is carried out with standard loss modelling tools in order to get quantitative estimations. The utilised loss models are the AMDC-KO-MK and Traupel’s, where the former is insufficient. For turbine surface roughness effects, the Traupel loss model is used instead of the AMDC-KO-MK since the latter is based on a fixed surface roughness and hence, not a parameter as e.g. the Reynolds number. As for the gas turbine, a similar investigation has been done for steam turbines, i.e. degradation or deterioration of blades. In paper number III, artificial neural networks (ANN) are introduced as a monitoring tool. The process identification in this paper was carried out with the model presented in chapter five. In general, most faults that occur in a steam turbine will only appear once or seldom. However, there are some exceptions, such as condenser

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fouling and vaporous carry-over, that occur more frequently. Solid particle erosion (SPE) is another costly but easily solved problem that appears more or less often, depending on the design of the steam plant. By using ANN for fault diagnosis it was possible to generate early warning with good accuracy. With the knowledge of the underlying degradation mechanisms, and the possibility of including these in a condition monitoring system, the potential of improving the operation economics is significant. The availability of a plant can be increased if early warning can be obtained. Also the cost of secondary replacement parts, in the case of component breakdowns, can be entirely avoided.

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7 Summary of papers Paper I: Jordal K., Assadi M. and Genrup M: Variations in Gas-Turbine Blade Life and Cost due to Compressor Fouling – A Thermoeconomic Approach. International Journal of Applied Thermodynamics, Vol. 5, No 1, pp. 37-47, March 2002. This paper presents a study of the compressor fouling impact on the lifing of the turbine section. The economical aspects of fouling and washing over the engine lifetime are also investigated. It is a complicated matter to estimate the blade lifing, and artificial neural networks (ANN) are conceivable to simplify it. One of the conclusions is that gas turbines are less sensitive to fouling if more heat resistant blading is used. This is, however, dependent on the unit’s control philosophy and load level. Paper II: Genrup M., Assadi M. and Torisson T.: A Review of Gas Turbine Flow Path Analysis – From Paper Calculation to Artificial Neural Networks, ASME GT-2002-30027, Presented at ASME Turboexpo 2002, Amsterdam, The Netherlands. In this paper, a review of the state-of-the-art monitoring systems is presented. Different types of systems are described, from running-line dependent simple tools to artificial neural networks (ANN). Model-based or heat- and mass-balance models are also described. This paper also describes the possibility to use Microsoft ExcelTM as the platform for the model- based condition-monitoring system. The necessary equations must be solved iteratively and the use of the built-in matrix solver in MS Excel is described. The use of ANN as a tool for calculating expected values/parameters is also shown in the paper. Paper III: Mesbahi E., Genrup M. and Assadi M.: Fault prediction/diagnosis and sensor validation technique for a steam power plant, submitted for publishing in “ENERGY-The International Journal” This paper demonstrates the ANN-tool’s capability for fault prediction and diagnostics, applied to a steam cycle. Data needed for training of ANN-models is simulated by a detailed heat and mass balance program, developed by the author. Different types of faults frequently observed on steam cycle components are simulated for generating training data. Results from this study showed that the ANN-model was capable of detect and identifying the faults when they were developed to only 70 percent of the predefined threshold value that generates an error message. These results show that ANN-models could be used for generation of early warnings, hence improving the maintenance planning and plant availability.

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17 Fielding L.; Turbine Design – The Effect on Axial Flow Turbine Performance of Parameter Variation, ASME Press, 2000, ISBN 0-7918-0086-5 18 Wei N.; Significance of Loss Models in Aerothermodynamic Simulation for Axial Turbines, PhD-thesis; Royal Inst of Technology, ISBN 91-7170-540-6, 2000 19 Denton J. D.; Loss Mechanisms in Turbomachines, ASME 93-GT-435, 1993 20 Hourmouziadis J. and Albrecht G,; An Integrated Aero/Mechanical Performance Approach to High Technology Design, AGARD CP 421, 1987 21 Kacker S. C. and Okapuu U.; A Mean Line Prediction Method for Axial Flow Turbine Efficiency, ASME 81-GT-58, 1981 22 Moustapha H et al; Gas Turbines – Handbook of Fluid Dynamics and Fluid Machinery, p.p. 2520-2537, J. Wiley 23 Moustapha H et al; Axial and Radial Turbines, Lecture notes, Concepts NREC, 2002 24 Moustapha S. H., Kacker S. C. and Tremblay B.; An Improved Incidence Losses Prediction Method for Turbine Airfoils, ASME 89-GT-284 25 Hartsel J. E.; Prediction of Effects of Mass-Transfer Cooling on the Blade-row Efficiency of Turbine Airfoils, AIAA Paper No 72-11, 1972 26 Horlock J. H., Watson D. T. and Jones T. V.; Limitations on Gas Turbine Performance Imposed by Large Turbine Cooling Flows, ASME 2000-GT-635 27 Physical Properties of Natural Gases, N.V. Nederlandse Gasunie, 1988 28 Cotton K. C.; Evaluating and Improving Steam Turbine Performance, Cotton Fact Inc, 1998, ISBN 0-9639955-1-0 29 Walsh P. W. and Fletcher P; Gas turbine Performance, Blackwell Science, 1998, ISBN 0-632-04874-3 30 Rules for Steam Turbine Thermal Acceptance Tests, Part 2: Method B – Wide Range of Accuracy for Various Types and Sizes of Turbines, IEC 953-1 31 Volponi A. J.; Gas Turbine Parameter Corrections, ASME 98-GT-947, Stockholm, 1998 32 GasTurb 9; Users Manual. 33 Mattingly J. D.; Elements of gas Turbine Propulsion, McGraw-Hill, 1996, ISBN 0-07-114521-4 34 Cohen H., Rodgers G. F. C., Saravanamutto H. I. H.; Gas Turbine Theory, Addison Wesley Longman Ltd., 1998, ISBN 0-582-23632-0

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35 Scott J. N.; Axial Compressor Monitoring by Measuring Air Intake Depression, Third Symposium on Gas Turbine Operation and Maintenance, National Research Council of Canada, 1979 36 Denton J. D. (ed.); Developments in Turbomachinery Design, Professional Engineering Publishing Ltd, 1999, ISBN 1-860-58237-0 37 Wu C. H.; A General Theory of Three-Dimensional Flow in Subsonic and Supersonic Turbomachines of Axial, Radial and Mixed-Flow Types, NACA TN 2604, 1952 38 Novak R. A.; Streamline Curvature Computing Procedures for Fluid Flow Problems, Trans of ASME, October 1967, 478 39 Novak R. A.; Flow Field and Performance Map Computation for Axial-Flow Compressors and Turbines, in Modern Prediction Methods for Turbomachinery Performance, AGARD-LS-83, paper 5. 40 Smith L. H. Jr; The Radial-Equilibrium Equation of Turbomachinery, Journal of Engineering for Power, January 1966 41 Wennerstrom A. J.; On the Treatment of Body Forces in the Radial Equilibrium Equation of Turbomachinery, in Traupel Festschrift, Juris-Verlag, Zurich, 1974 42 Horlock J. H.; On Entropy Production in Adiabatic Flow in Turbomachines, ASME Paper No. 71-FE-3, 1970 43 Denton J. D.; Throughflow Calculations for Transonic Axial Flow Turbines, Trans. ASME J. Engng. for Power, April 1978, 100 44 Genrup M., Thern M. and Assadi M.; Trigeneration: Thermodynamic Performance and Cold Expander Aerodynamic Design in Humid Air Turbines, ASME GT2003-38283 45 Hearsey R. M.; A Revised Computer Program for Axial Compressor Design, Vol. 1 Theory, Descriptions, and Users Instructions, NASA ARL 75-0001, Vol 1, 1975 46 Wilkinson D. H.; Stability, Convergence, and Accuracy of Two-Dimensional Streamline Curvature Methods Using Quasi-Orthogonals, ImechE 1969-70, Vol 184 Pt 3G 47 Lewis K. L.; Spanwise Transport in Axial-Flow Turbines: Part 1 – The Multistage Environment, ASME 93-GT-289 48 Lewis K. L.; Spanwise Transport in Axial-Flow Turbines: Part 2 – Throughflow Calculations Including Spanwise Transport, ASME 93-GT-290

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49 Adkins G. G. jr. and Smith L. H. jr.; Spanwise Mixing in Axial- Flow Turbomachines, ASME 81-GT-57 50 Dring R. P.; Radial Mixing in an Axial Turbine, ASME 94-GT-137 51 DeRuyck J. And Hirsch Ch.; A Radial Mixing Computation Method, ASME 88-GT-68 52 Wennerstrom A. J.; A Review of Predictive Efforts for Transport Phenomens in Axial Flow Compressors, Journal of Turbomachinery, April 1991, Vol. 113 53 Came P. M.; Streamline Curvature Throughflow Analysis of Axial-Flow Turbines, VDI Berichte Nr. 1185, 1995 54 Came P. M.; Secondary Loss Measurements in a Cascade of Turbine Blades, ImechE C33/73 55 Nikolos I. K., Douvikas, K. D. and Papailiou K. D.; Modeling of the Tip Clearance Losses in Axial Flow Machines, ASME 96-GT-72 56 Islam A. M. T. and Sjolander S. A.; Deviation in Axial Turbines at Subsonic Conditions, ASME 99-GT-26, 1999 57 Massardo A. and Satta A.; A Correlation for the Secondary Deviation Angle, ASME 85-IGT-36 58 Lakshminarayana B.; Fluid Dynamics and Heat Transfer of Turbomachinery, John Wiley and Sons Inc, 1996, ISBN 0-471-85546-4 59 VGB-Merkblatt; Wirkungsgradänderungen an Dampfturbinen – Ursachen und Gegenmassnahmen, VGB-M 114 M, 1991 60 Svoboda R. and Bodmer M.; Deposits and Corrosion in Steam Turbines, ABB Power Generation TEZ 87-20, paper presented at Ringhals in March 1987. 61 EPRI; Cycle Chemistry Guidelines for Fossil Plants: All-Volatile Treatment, EPRI TR-10541, 1996 62 Traupel W.; Thermische Turbomashinen, Springer Verlag, 1977, ISBN 3-540-07939-4 63 Leyzerovich A.; Large Power Steam Turbines, PennWell Publishing Company, 1997, ISBN 0-87814-716-0 64 Kostyuk A. and Frolov V.; Steam and Gas Turbines, Mir Publishers Moscow, 1988, ISBN 5-03-000032-1 65 Cordes G.; strömungstechnik der gasbeaufschlagen Axialturbine, Springer Verlag, 1963

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66 El-Wakil M. M.; Powerplant Technology, McGraw-Hill, 1984, ISBN 0-07-019288-X 67 Spencer R. C. and Cotton K. C., A Method for Predicting the Performance of Steam Turbine-Generators… 16 500 kW and Larger, ASME 62-WA-209, Revised 1974 68 Cotton K. C. and Schofield P.; Analysis of Changes in the Performance Characteristics of Steam Turbines, ASME 70-WA PTC 1, 1970 69 Albert P.; Steam Turbine Thermal Evaluation and Assessment, GER-4190, GE Power Systems 70 Service Information, Preventive Checks on Smaller Range of Condensing Turbo- Alternators, STAL-LAVAL, 1967 71 STAL-LAVAL, Advanced Propulsion - Machinery Operating Checks, Nov 1969, STAL-LAVAL 72 VGB; Anleitung zum Überwachen von Dampfturbinen durch Messen des inneren Wirkungsgrades, VGB R-118M

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