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Design of a Test Loop for Performance Testing of Steam
Turbines Under a Variety of Operating ConditionsThesis Proposal for Master of Science
Jonathan Guerrette
July 2009
Contents
1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Proposed Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Proposed Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Overview of Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2.1 Total Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2.2 Stage Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.3 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.4 Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.5 Non-dimensional Parameters . . . . . . . . . . . . . . . . . . . . . . 17
3.2.6 Miscellaneous Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Industry Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Turbine Performance Prediction . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4.1 Empirical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4.2 Analytical and Numerical Methods . . . . . . . . . . . . . . . . . . . 37
3.5 Measurement Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.1 Codes and Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.2 Industrial Test Loops (Dresser-Rand) . . . . . . . . . . . . . . . . . 40
3.5.3 Academic Research Labs . . . . . . . . . . . . . . . . . . . . . . . . . 40
1
3.6 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Progress to Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Test Loop Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.1 System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1.2 Component Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.3 Facility Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5 Statement of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7 Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A Derivation of Stage Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 57
A.1 Assumptions and Justifications . . . . . . . . . . . . . . . . . . . . . . . . . 57
A.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
A.2.1 Thermodynamics and Dynamics . . . . . . . . . . . . . . . . . . . . 58
A.2.2 Stage Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2
Chapter 1
Abstract
The steam turbine is one of the most widely used energy conversion devices in the world,
providing shaft power for electricity production, chemical processing, and HVAC systems.
Now there are new opportunities in growing renewable and combined cycle applications, but
more than ever before, end-users are asking for energy efficiency improvements that require
manufacturers to renew their experimentally verified design methods. Following conver-
sations with the Research and Development division at Dresser-Rand, it was proposed to
build a new steam turbine research facility at the future Rochester Energy-systems Experi-
ment Station (REES). A preliminary review of axial turbine measurement and performance
prediction has been conducted to guide the system design, which is the primary purpose of
the thesis project
To address the entire scope of modern axial turbine performance prediction would re-
quire a textbook, so this investigation was focused on empirical methods that use alge-
braic and graphical correlations of experimental data. While gas turbine experiments and
their resulting empirical correlations have continued into the 21st century, steam turbine
performance prediction has been relatively stagnant for decades. Accurate steam-specific
prediction methods require new experimental data, but there are no openly publishing lab-
oratories equipped for the necessary testing. A structured design approach is proposed for
the new steam turbine research facility, following three integrated research thrusts: Turbine
Performance Prediction, Measurement Planning, and System Modeling.
3
Chapter 2
Introduction
2.1 Motivation
Growing markets for solar thermal and combined cycle power plants, which produce low to
medium temperature working fluids, present future opportunities for steam turbines that
perform efficiently over a broad range of operating conditions. Additional opportunities for
new turbines and energy efficiency upgrades exist in shaft-driven applications like HVAC
and fossil fuel extraction. A key issue for turbine end-users is that the turbine will operate
at off-design conditions for a large portion of its life. New design and manufacturing tech-
niques for axial turbines show promise for improved efficiency at design-point and off-design
operations, but capitalizing on the current state of the turbine industry will require modern
performance prediction methods that provide more accurate and higher resolution results,
so that manufacturers can make confident predictions for their customers.
The steam turbine designer has several options in the performance prediction toolbox,
including empirical correlations, numerical techniques such as computational fluid dynamics,
and theoretical analysis, all of which are supported by experimentation. Most prominent
research institutions that study axial turbine performance are equipped with industry-
funded gas turbine laboratories. This is partly because high temperature gas turbines offer
a higher Carnot efficiency, but it is primarily because its high power density elicits financial
backing from the aircraft industry. As a result, there are very few novel methods for steam
turbine performance prediction in the open literature.
Conventional empirical correlation methods cited in recent literature were all developed
by gas turbine researchers, including Ainley and Mathieson [1], Dunham and Came [2],
4
Craig and Cox [3], and Kacker and Okapuu [4]. Their results can be applied directly to
noncondensing steam turbines, but the extension to condensing steam turbines requires a
correction factor. In their 1970 paper, Craig and Cox [3] state that numerous methods exist
for applying a correction factor to account for two-phase flow, none of which exhibits more
merit than the others. A review of the recent literature in steam turbine research reveals a
lack of deliberate correlation development. This is probably why, in 2002, Bassel [5] used
the oversimplified method recommended by Craig and Cox, that is to assume a loss of 1%
in stage efficiency per every 1% of mean stage wetness. If accurate and high resolution
steam-specific turbine performance prediction is to evolve to quantitative analysis, new and
sustained steam turbine experimentation is needed.
2.2 Proposed Objectives
The sharp contrast in academic research between steam and gas turbines and the business
opportunities expressed by Dresser-Rand necessitate the building of a steam turbine research
facility. The purpose of this thesis will be to create a design for a steam turbine test loop
that will be installed at the Rochester Energy-systems Experiment Station (REES) on the
campus of the Rochester Institute of Technology (RIT). The first aspect of the design will
be the selection of components that have an appreciable impact on the thermodynamic
performance of the test loop cycle, including, at a minimum, those labeled in Fig. 2.1.
The second aspect of the design will be selection and placement specifications for the
sensors necessary to deduce the desired performance characteristics of the steam turbine
being tested. For example, temperature, pressure, and flow measurements in the test loop
piping will be required to characterize overall steam turbine efficiency, while internal turbine
pressure and velocity sweeps would be required to determine stage efficiencies. The system
design will be verified using a simulation environment that can model the behavior of the
test loop under a range of operating conditions.
5
Figure 2.1: Schematic of a simplified steam turbine test loop
2.3 Proposed Approach
The progression from Dresser-Rand testing needs to a detailed system design will require a
structured research and design approach, so the project milestones are categorized into three
separate topic areas, or thrusts. The first thrust, Turbine Performance Prediction, is an
investigative stage that will shed light on how the results of the steam turbine testing will be
used in industry. A thorough review of axial turbine performance prediction will ensure the
final test loop design serves the needs of the modern steam turbine community. The second
thrust, Measurement Planning, includes an investigation of system-level and turbine-level
measurement practices and a sensor scheme design for the current test loop. The final thrust,
System Modeling, will use a paired simulation environment and optimization tool to generate
a final design for the test loop, including component selection. Detailed descriptions of each
thrust can be found in the Statement of Work in Chapter 5. The structured approach will
enable continuous progress from proposal submission to thesis defense.
6
Figure 2.2: Flowchart of the proposed approach.
2.4 Overview of Proposal
This proposal presents the progress-to-date on the thesis and the approach that will be taken
to complete it successfully. The status of the literature review is presented in Chapter 3.
Then, a preliminary design for the steam test loop and the selection of some components by
Brown and Dharmadasa [6] is described in Chapter 4. Chapter 5 describes the deliverables
and the structured process for completion of the final thesis project. Chapter 6 lays out
a preliminary schedule of completing the major milestones of the thesis project. Finally,
Chapter 7 lists the primary risks and concerns associated with completing the thesis.
7
Chapter 3
Literature Review
3.1 Introduction
The literature review has been conducted with the focus of predicting the performance of
a steam turbine under a variety of operating conditions. The review is organized such that
each subsequent section requires increasing knowledge of axial turbine design, which is pro-
vided by the previous sections. Section 3.2 defines some terms that are common to the axial
turbine industry and also the variables of interest to the present investigation. Methods for
axial turbine performance prediction are discussed in Section 3.4 and a detailed description
of one specific method is given. Section 3.5 covers an investigation into measurement sys-
tems, including those used in academia and those required by relevant codes and standards.
Section 3.6 discusses some different system modeling approaches. Sections 3.5 and 3.6 re-
quire additional research and writing. The literature review is in-progress and will continue
through June 2010.
3.2 Definitions
3.2.1 Total Properties
A total property is that which would result from an isentropic deceleration to zero velocity.
It is equal to the sum of the static and dynamic components. Common to turbine analyses
are the use of total enthalpy, pressure, and temperature. They are defined as follows,
h0 ≡ h+v2
2gc
8
P0 ≡ P +ρv2
2gc
T0 ≡ T +v2
2cpgc(3.1)
The expression for total temperature is derived from the one for total enthalpy assuming
constant specific heat at constant pressure. Total temperature can also be expressed in
terms of Mach number, M , and the specific heat ratio, γ, as
T0
T= 1 +
γ − 12
M2 (3.2)
3.2.2 Stage Geometry
Each stage of an axial turbine has a stationary stator row followed by a rotating rotor row.
These alternating rows can be seen in Fig. 3.1, which depicts a research turbine previously
operated by Dresser-Rand [7], with the upper housing removed.
Figure 3.1: Photograph of the internals of a 4-stage research turbine (Courtesy of Dresser-Rand[7]).
The stator blades are sometimes referred to as nozzles, guide vanes, or nozzle guide vanes
9
(NGV), because they accelerate the flow much like a nozzle and also because traditional
single stage turbines used connical shaped nozzles. The stator row is mechanically attached
to the turbine casing, leaving a gap between the blade and the shaft, which is minimized by
a diaphragm seal. The rotor blades, often referred to as buckets, are directly attached to the
shaft and are usually connected at their tips by a shroud ring. The gap between the blade
and casing or the shroud and casing is controlled by tip leakage seals. For typical blade
dimensions that will be used in this thesis, refer to Fig. 3.2. Where a single dimension type,
such as blade spacing, is common between the stator and rotor rows they will be denoted
with subscripts “N” for nozzle and “R” for rotor.
Figure 3.2: Definition of blade geometries, modified from Benner [8].
For simplified analyses, the radial velocity of fluid in each axial turbine stage is assumed
to be zero. In these analyses, referred to as 2-D analyses, the fluid flowpaths of each stage
10
are described with a single stage velocity diagram as depicted in Fig. 3.3. When a 3-D
approach is taken, multiple velocity diagrams are used at different radial stations along the
blade height. The velocity diagram of Fig. 3.3 represents the flow conditions at the “mean
Figure 3.3: Turbine stage velocity diagrams, modified from Dixon [9].
line”, or the meridional plane, of the stage. This refers to the plane at a radius equal to
the average of the hub and tip radii. The view is radial, inward toward the shaft center.
The axial direction is defined by the x-axis, and the tangential direction is defined by the
y-axis. All angles are taken as positive as they are drawn in the schematic. It was common
in traditional axial steam turbine practice to measure fluid angles from the rotational plane;
however, the axial datum convention shown in the above diagram is used in most modern
axial turbine analyses, and will be used in the analysis to follow.
Dixon [9] gives the following desciption of the axial turbine flow path:
Fluid enters the stator at absolute velocity c1 at 6 α1, measured from the
axial datum. Gas is accelerated to absolute velocity c2 at 6 α2. The rotor inlet
relative velocity, w2 at 6 α′2, is found by subtracting the blade speed, U , from
11
c2. U is found by multiplying the shaft speed, 2πN60 , and the mean blade radius,
rm. The relative flow accelerates through the rotor to velocity w3 at 6 α′3. The
absolute flow c3 and α3 is found by adding U to w3. Keep in mind that U and
w3 are defined positive in opposite tangential directions.
The velocity diagram of Fig. 3.3 applies when the turbine is operating at design condi-
tions, but additional terms are necessary to analyze off-design operation. When a turbine
operates at off-design rotational speed or mass flow rate, the flow incidence on the leading
edge of the blade αin becomes nonzero. Fig. 3.4 illustrates the nomenclature used to de-
Figure 3.4: Incidence terminology for an axial-flow turbine, modified from Benner [8].
scribe the variation of incidence relative to the blade inlet metal angle, βin, the design inlet
flow angle, αin,des, and the actual inlet flow angle, αin. The subscript in is used to denote
the blade inlet. For the stator row the inlet would be at station 1 of Fig. 3.3 and for the
rotor, the inlet would be at station 2. For the moving rotor, the relative velocity angle α′2
is substituted for the effective incidence angle ieff .
12
3.2.3 Efficiency
The isentropic efficiency of a steam turbine is traditionally expressed as
ηs =hin − houthin − houts
(3.3)
where houts is the enthalpy resulting from an isentropic expansion through the turbine
to the same exit pressure as the actual expansion. Isentropic efficiency accounts for the
aerodynamic losses of a turbine, and finding its value is a major goal of performance testing
in a steam turbine test loop. Static enthalpies are used in this expression, because the
fluid kinetic energy change across the entire turbine is assumed to be negligible. Another
important efficiency is the turbine thermal efficiency,
ηth =Wsh
hin − hout(3.4)
which accounts for rotordynamic losses in the turbine, such as friction due to bearings
and external seals (not tip or diaphragm seals). The product of thermal and isentropic
efficiencies is the overall turbine efficiency,
ηoa =Wsh
hin − hout,s= ηs · ηth (3.5)
which is the ratio of the actual shaft output power, Wsh, to the ideal maximum enthalpy
drop across the turbine. One method of determining thermal efficiency is to make the
necessary fluid and shaft measurements to find isentropic and overall efficiencies, and then
divide ηoa by ηs.
Often the greatest concern of a turbine end user is to know how the three turbine
efficiencies vary over the operating range of their particular steam power loop. In response,
manufacturers plot the efficiency performance maps against pressure ratio and flow rate.
They don’t produce these maps by running every turbine through an extensive series of tests.
Instead, they use engineering analysis to extrapolate the results of tests on a research turbine
to the appropriate off-the-shelf product. In addition to the aerodynamic/rotordynamic loss
breakdown eluded to, each of these is separated into multiple sources of loss.
A reasonable breakdown of aerodynamic losses, given by Mathis [10], separates the losses
into different control volumes: inlet, stator, rotor, diffuser, and exit. The boundaries for
13
each of these volumes depends on the particular turbine being considered. All areas of loss
are of interest to predicting overall aerodynamic performance, but the current focus will
be on stator and rotor losses. Each stage being made of a single stator and rotor pair,
these losses will impact the stage efficiency. Stage efficiencies are defined as the ratio of the
real enthalpy drop in the stage to the ideal maximum enthalpy drop, similar to the turbine
isentropic efficiency. The difference is that total enathalpy is used in single stage analysis,
because velocity changes are drastic across each stage row and kinetic energy change cannot
be assumed negligible.
Two stage efficiencies are of interest to the present research, the first being total-to-total
efficiency,
ηtt =h01 − h03
h01 − h03ss(3.6)
The appropriate states are illustrated in the Mollier diagram of Fig. 3.5. Total-to-total
Figure 3.5: Mollier diagram for a turbine stage, from Dixon [9].
14
efficiency sets the minimum ideal energy as the total ideal stage exit enthalpy, h03ss, because
it is assumed that the exit kinetic energy is used either in a subsequent stage, or for thurst
in the case of an aircraft gas turbine. The second efficiency of interest is the total-to-static,
expressed as
ηts =h01 − h03
h01 − h3ss(3.7)
Total-to-static efficiency is used for single stage stationary turbines or for the final stage of
a multiple stage stationary turbine, when the exit kinetic energy is not used. The total-
to-static efficiency will always be less than the total-to-total efficiency. Turbine researchers
often express stage efficiencies in terms of the stator exit velocity, c2, the rotor exit relative
velocity, w3, and the stage inlet and exit static enthalpies as
ηtt =
[1 +
T3T2c2
2ζN + w23ζR
2 (h1 − h3)
]−1
(3.8)
and
ηts =
[1 +
T3T2c2
2ζN + w23ζR + c2
1
2 (h1 − h3)
]−1
(3.9)
Equations 3.8 and 3.9 are derived from Equations 3.6 and 3.7 in Appendix A. The reason
why efficiency is expressed with these more complex expressions is that velocity is more
easily measured than enthalpy. Also, ζN and ζR are called enthalpy loss coefficients, and
they are the ratio of enthalpy loss to relative exit kinetic energy in the stator and rotor,
respectively. They are expressed as
ζN =h2 − h2s
2c22
(3.10)
and
ζR =h3 − h3s
2w23
(3.11)
Plotting the variation of these loss coefficients against geometries and operating parameters
enables axial turbine designers to improve performance more easily. In the early design
phase, such a plot is generated by a correlation tuned from experimental measurements.
Several correlation methods will be descibed herein.
15
Figure 3.6: P-h diagram for a turbine stage.
3.2.4 Reaction
Stages are often referred to as being of impulse, zero-reaction, or reaction type. In a
quantitative sense, the modern definition of reaction percentage is the ratio of static enthalpy
drop in the rotor row to the static enthalpy drop across the entire stage,
R =h2 − h3
h1 − h3(3.12)
The relevant static enthalpies are illustrated in Fig. 3.5 and the P-h diagram of Fig. 3.6.
In traditional steam turbine analysis, an impulse stage was defined as having zero static
pressure drop on the rotor. Such a stage would actually have a negative reaction, because
the static enthalpy at state 3 would be greater than that at state 2. Contemporary use of
the term impulse is often used to refer to a stage with R = 0, and zero-reaction will be used
to refer to such stages herein. Where this conflicts with diagrams taken from the literature,
a clarification will be given.
Often reaction is expressed in terms of turbine geometries or mass-averaed flow angles,
such as the one derived by Dixon [9].
16
Note:In the final thesis, additional expressions will be given for reaction. Also, a de-
scription will be given for how reaction is used in the design and analysis processes.
3.2.5 Non-dimensional Parameters
In addition to giving an algebraic expression for the following non-dimensional parameters,
their relavance to the present work will be described in the final thesis.
Coefficient of lift, CL
Coefficient of drag, CD
Flow coefficient, Φ
Loading coefficient, Ψ
Kinetic energy coefficient φ2
3.2.6 Miscellaneous Terms
Axial turbine vs. radial in-flow turbine
The steam turbine test loop in question will be designed to operate axial turbines, as most
steam turbines fall into this category. Axial turbines generally have an axial inlet flow to
the first stage, and an axial/tangential exit velocity from the final stage. Radial turbines
can be of the in-flow or the out-flow type. In the first case, the inlet flow is directed radially
inward toward the rotational axis, and the exit is directed axially away from the rotor disc.
Radial out-flow turbines simply reverse the flow direction.
Steam vs. gas turbine
Steam turbines use gaseous and/or mixed liquid-vapor H2O as a working fluid, and have
only a fluid expansion section. Gas turbines use a non-H2O gaseous working fluid, with
nearly zero condensation. The phrase “gas turbine” is commonly used in reference to either
the expansion section (turbine) or the entire rotor system composed of fan, compressor,
combustor, and turbine sections. Herein, the term “gas turbine” will refer to only the
expansion section.
17
Condensing vs. non-condensing steam turbines
A non-condensing steam turbine is similar to a gas turbine in that there is zero, or nearly
zero, condensation. Non-condensing turbines can be analyzed using methods similar to those
used for gas turbines, provided non-ideal and compressible gas assumptions are reasonable.
Condensing turbines have condensation in the later stages and generally operate at lower
pressures than non-condensing turbines. Whereas the exit pressure of a non-condensing
turbine might be on the order of 200 psia, that of a condensing turbine could be on the
order of 4 psia. The actual exit pressure depends on the rate at which the condenser and
pump can remove fluid from the turbine. A backup of liquid working fluid in the condenser
slows down the condensation process, and increases the turbine exit pressure.
Note: In addition to the above paragraph, the final thesis will also discuss the pros
and cons of each type of turbine, the material implications of condensing flow, and the
applications of each. Also, some condensing turbines can use wet steam, which represents a
huge advantage for solar thermal power plants, CHP applications, or waste steam recovery
where high efficiency can be the deciding factor in terms of system payback periods. This
dynamics will also be discussed in more depth in the final thesis.
Aerodynamic vs. rotordynamic performance
In short, aerodynamic losses refer to those accounted for by isentropic efficiency, whereas
rotordynamic losses are those accounted for by thermal efficiency. Aerodynamic losses
include skin friction, separation, vorticity generation, and others. Rotordynamic losses
include bearing friction and leakage over external seals, which prevent steam from escaping
to open air.
Note:The differences between these two loss types will be discussed in more detail in the
final thesis, and a figures will be given to show the different loss phenomena.
3.3 Industry Practice
Understanding Dresser-Rand’s business model is the first step in the design of the steam
turbine test loop, because they will be its first industrial user. Dresser-Rand operates
18
within a specific market niche for steam turbines that produce 100MW or less. According
to Fred Woehr, Vice President of Research and Devlopment for the Steam Turbine Business
Unit, Dresser-Rand’s business focus is to provide products (turbines, pumps, compressors,
and others) that give the best performance for the inlet/outlet flow conditions, power in-
put/output, and/or rotational speed specified by a customer. Their goal is to match their
customer’s application, as opposed to getting orders for as many of the same product as
possible, because their end-users see the value of high efficiency over the lifetime of the
turbine.
In order to select and design the most energy efficient turbine for a specific customer
application, Dresser-Rand customer engineers require specifications for the turbine oper-
ating conditions from the end-user. In order to illustrate their process of thermodynamic
performance prediction, let us consider two common customer specification scenarios. One
scenario is when a customer specifies the steam flow rate at a design point (temperature
and pressure), possibly with variations in availability for seasonal, weekly, daily, or hourly
changes. In this case, the customer wants Dresser-Rand to predict how much electric power
can be produced by a given generator. Another scenario is when a customer specifies a
horsepower for a driven product, such as a centrifugal pump, and a temperature and pres-
sure for the steam source. In this case, the customer wants Dresser-Rand to predict how
much steam is needed to drive the device. Once the customer specifications are established,
Dresser-Rand can carry out their predictions in order to select or design the right turbine.
In order to carry out their predictions, Dresser-Rand has a set of software-based selection
and design tools organized around three segments of turbine design and selection. First, for
turbines that produce 25MW or less, Dresser-Rand uses a parametric selection tool to choose
a design from standard off-the-shelf turbines. Second, for turbines that produce between
25MW and 100MW, or for more expensive turbines that are operated in remote locations
or under severe operating conditions, Dresser-Rand performs a custom design using yet
another set of software tools. Third, for turbines that require new or derivative product
development, Dresser-Rand must generate custom prediction methods based on literature
reviews and experimentation, or resort to consultants and external design tools. It is this
third segment of performance prediction that has inspired the present thesis project.
19
Note: The final thesis will elaborate on the following:
• Dresser-Rand needs updated predictions for performance in order to take advantage
of new design practices.
• Dresser-Rand indicated specific areas of interest for their research including reaction
blades and ring-and-vane nozzle designs. Address these here to set the scope of future
research. (Refer to nomenclature section as necessary)
• Dresser-Rand stated they would like to perform new product development testing at
REES, as opposed to quality or verification testing of manufactured products.
In addition to the above description of Dresser-Rand’s needs, the final thesis will give an
overview of general needs and practices in the steam turbine marketplace. It is important
to note that Dresser-Rand will not be the only user of the steam turbine test loop. Making
the business case for the test loop will require additional grants from other sources. The
author will be working closely with his advisor to determine these opportunities.
3.4 Turbine Performance Prediction
Before designing a steam turbine test loop, it is important to understand the state of the art
of turbine performance prediction, which has evolved over the past century from qualitative
guess and check to complex systems of empirical, numerical, and analytical equations.
Empirical correlation results often serve as inputs to the more complex numerical and
analytical methods. An exhaustive review of the latter methods has not been conducted,
but their relationship to the correlations will be discussed briefly herein.
3.4.1 Empirical Methods
One common purpose of an empirical correlation is to simplify the analysis of a three-
dimensional, unsteady, compressible flow problem. They do this by accounting for complex
flow behaviors with coefficients and correction factors. Although misuse of these simplified
empirical correlations can lead to poor turbine designs, in their 2008 conference paper Benini
et al. [11] state the following:
20
In spite of the remarkable advances in the field of Computational Fluid Dy-
namics, algebraic models built upon empirical loss and deviation correlations
are still one of the most reliable and effective tools to predict the performance
of gas turbine stages with reasonable accuracy, especially when low-reaction,
multi-stage architectures are considered.
It may be surprising that correlations developed a half-century ago are still relevant in
current performance prediction, but it is possible because the correlations are restructured
or retuned when a new design paradigm is believed to impact turbine performance. The
relative improvements between empiricists come from introducing new variables to the cor-
relation, which are speculated from witnessing experimental or numerical flow patterns.
Published correlations are optimized to fit the available measurement data, while particular
manufacturers may have proprietary correlations that apply to their specific line of prod-
ucts. While the algebraic expressions for each correlation differs, some common assumptions
apply to many of them.
Most empirical prediction methods fit into the category of 2D or “mean line” analy-
sis, because they assume zero radial gas velocity across a turbine blade, and they assume
the tangential and axial velocity vectors at the mean blade radius are representative of
conditions at all other radii. This is illustrated in the velocity diagram of Fig. 3.3, which
for a mean line analysis would represent the conditions for the entire blade length. If, in
addition, a method assumes axisymmetric flow about the rotor axis then it follows a 1D
analysis, since flow conditions vary only in the axial direction. A 1D analysis assumes that
the losses through a turbine stage depend solely on the mass-averaged inlet and exit condi-
tions and the physical geometry of the stage. Often in the literature, one author referring
to 2D theory and another to 1D theory are discussing the same correlation method. Unless
otherwise stated, the empirical methods discussed herein are 1D, mean line, axisymemtric
analyses. They are also limited to aerodynamic performance prediction as defined in Sec-
tion 3.2. Rotordynamic losses, including those due to friction of bearings and external seals,
will need to be researched further in order to make a complete turbine system model.
Even with the limitations imposed by the preceeding assumptions, there are several
published approaches to developing an empirical correlation. It is useful to organize these
21
approaches in order to assess their utility for the present research. Conventional approaches
will be divided according to their complexity and the abundance of their citations in recent
publications. An additional novel approach will be presented that is based on the work of
Denton [12]. Although the former is more prevalent in the literature, it is the author’s belief
that the latter will lend itself better to the accurate combination of empirical correlations
with numerical and analytical methods.
Conventional Approaches
In 1985, Sieverding [13] presented a classification of conventional empirical axial turbine
performance prediction methods into two groups. The first group, here termed Overall Pa-
rameter Models, bases turbine stage performance on overall turbine parameters and is used
in the initial design phase for the selection of the turbine design parameters. Sieverding says
the second group, here termed Detailed Parameter Models, achieves a deep understanding
of the flow, and takes into account details of the blading and of the meridional flow channel.
These two groups of empirical prediction methods will be reviewed for their relative utility
in a steam turbine test loop system model, and for their potential improvement by actual
performance testing. Because the most recent literature for both groups focuses on gas
turbines, it is also necessary to describe the application of their results to steam turbines.
Overall Parameter Models are among the oldest in the turbine community and they
are still used in the early design stages to define the turbine flowpaths. The most-cited
model of this type in textbooks is Soderberg’s correlation, which can be found in the works
of Horlock [14], Dixon [9], and Mathis [10]. Dixon and Horlock both declare that Soderberg’s
method gives turbine efficiencies with errors less than 3% over a wide range of Reynolds
number and aspect ratio when additional corrections are included to allow for tip leakage
and disc friction. Soderberg uses the enthalpy loss coefficients given in Equations 3.10
and 3.11 ζ,
Referring to the Mollier diagram in Fig. 3.5, these coefficients are the ratio of enthalpy
loss to exit kinetic energy for the stator (N for nozzle) and rotor, respectively. The enthalpy
loss coefficients are easily subtituted into the equations for total-to-total (Equation 3.8) and
22
total-to-static (Equation 3.9) isentropic stage efficiencies. The coefficients are correlated
against turbine overall parameters, such as flow coefficient and loading coefficient, for a
range of space-chord ratios and flow turning angles. Unfortunately, Soderberg’s correlation,
like other Overall Parameter Models, is unable to predict the off-design performance of a
turbine, because he assumed stator and rotor incidence angles to be zero at the mean line.
This is simply not the case when the steam flow rate or rotor speed differs from the design
value. In today’s modern designs a non-zero incidence at the mean line is even common
for the design-point, especially if the blade has twist, and the inlet metal angle varies
along the blade span. Horlock [14] cites several other Overall Parameter Models, including
those of Hawthorne, Emmert, Vavra, and Todd; however, since their approach is similar to
Soderberg’s and none have been updated by recent tests, their specific differences will not
be discussed further here.
Overall Parameter Models represent the most basic performance prediction schemes.
Their stregth lies in a simple conversion to stage efficiency, but their weakness lies in an
integrated architecture that complicates recalibration to modern design. Because they are
unable to predict off-design turbine performance, Overall Parameter Models cannot be used
for a full spectrum steam turbine test loop model; however, their continued use in the turbine
industry presents an opportunity for improvement. The next section will illustrate the
benefit of the modularity of Detailed Parameter Models, and also the increased complexity
required to meet the requirements of off-design prediction.
Detailed Parameter Models have a modular architecture that has enabled continued
improvement over the last 50 years. The model most often cited in the literature is the
pressure-loss model by Ainley and Mathieson (AM) published in 1951 [1] & [15]. It is a
prime candidate for modeling the steam turbine in a research test loop, because it is a
complete loss model for predicting design-point and off-design losses. The AM correlation
has been modified and improved by Dunham and Came (AMDC) [2], Kacker and Okapuu
(KO) [4], Moustapha et al. (MKT) [16], and most recently by Benner et al. (BSM) [17], [18],
[19], & [20]. Each researcher has improved the AM model by introducing new experimental
data or reformulation to extend the appliciability of the originial AM correlation. The AM
23
model and its extensions are designed to predict gas turbine performance, but their results
apply to all axial turbines with a gaseous working fluid, including the noncondensing stages
of a steam turbine. Additional correction factors can be applied to account for losses due to
steam condensation and these will be discussed in the Steam Condensation paragraph.
The prevalance of the AM model extensions in recent literature for gas and steam turbines
warrants the following in depth description.
AM state that whenever convenient, pressure loss components should be expressed in
terms of a loss coefficient, Y :
Y =inlet total pressure− exit total pressureexit total pressure− exit static pressure
(3.13)
One of the major advantages of pressure-loss methods is that the pressure terms can be
measured directly in experiments, whereas enthalpy can’t be measured directly and requires
measurement of two independent state properties. In terms of the loss coefficient defined in
Equation 3.13 and the stator and rotor sections defined in the velocity diagram of Fig. 3.3,
their respective total loss coefficients are
YN,total =P01 − P02
P02 − P2
=P01 − P02
12ρ2c2
2
(3.14)
and
YR,total =P02rel − P03rel
P03rel − P3
=P02rel − P03rel
12ρ3w2
3
(3.15)
Notice that stator losses are defined in terms of the absolute velocities while rotor losses are
defined in terms of the relative velocities. AM separate the total pressure-loss coefficient
for both the rotor and stator stage sections into several loss components as follows
Ytotal = YP + YS + YTC (3.16)
Kacker and Okapuu modified the formulation as
Ytotal = YP f(Re) + YS + YTC + YTE (3.17)
24
where YTE is a trailing edge loss coefficient. Although the literature separates loss compo-
nents along clear lines of distinction, they aren’t physically separate. Summed component
loss models simplify the flow so that correlations are made simpler. After all, the point of
using an empirical correlation is to simplify the overall turbine analysis.
The analysis is made simpler by the fact that pressure-loss correlations are most often
derived from experiments performed on rectilinear, not radial, stages that employ a cascade
of 2-D blade profiles extending from a planar wall in a wind tunnel as shown in Fig. 3.7.
The experiments adhere strongly to the assumption of 2-D flow at the mean line, since
there are no centrifugal forces to accelerate the flow toward either the “tip” or the “hub”
of the blades. The working fluid is compressed air, and calculations are performed using
nondimensional parameters in order to generalize the results. For a complete description of
cascade testing, please refer to Chapter 3.5.
Figure 3.7: Low-pressure turbine cascade wind tunnel at Carleton University, Ottawa [21].
One way to translate the pressure-loss coefficient to stage efficiency is to first convert
it to an enthalpy-loss coefficient, which can then be substituted directly into Equations 3.8
and 3.9. Horlock [14] gives the following expression for such a conversion
Y = ζ
[1 +
γ − 12
M22
]γ/(γ−1)
(3.18)
25
where M2 is the exit Mach number and γ is the ratio of specific heats. He states that
the derivation for a similar relationship for the rotor blades is simple, but Horlock assumes
a perfect gas in his derivation, limiting its appliciability to steam turbines. Even more
alarming is that direct substitution of cascade pressure losses into efficiency equations might
not be appropriate, because Dixon [9] states “the aerodynamic efficiency of an axial-flow
turbine is significantly less than that predicted from measurements made on equivalent
cascades operating under steady flow conditions.” Additionaly, Benner [20] states,
The loss correlations derived from cascade data are scaled or ‘calibrated’ to
reproduce stage efficiencies derived from rig or engine data...there are evidently
additional and significant loss generating mechanisms in the engine environment
that are not captured in cascade testing.
In order to understand the correlation between pressure-loss correlations gleaned from
cascade tests and actual losses in full turbine rigs, an experimental comparison would be
required. Because designing a test facility that would enable such a comparison is the object
of this thesis, the comparison will not be possible prior to its completion. This represents
one of the major barriers to success in the accurate prediction of test loop behavior. The
steps that will be taken to mitigate this risk will be discussed further in Chapter 7. To sum-
marize, the cascade experiments are important, as they allow inexpensive testing of many
blade geometries and inlet/outlet boundary conditions. They also enable reconfiguration
of instruments and isolation of each loss component (i.e. profile, secondary, tip clearance),
without dismantling a complex turbine assembly. However, the resulting correlations should
not be taken as accurate for all cascades, or turbines, under all operating conditions. With
these limitations in mind, the pressure-loss components of the most modern version of the
AM loss breakdown scheme are described below.
Profile Losses, YP - This is the primary loss mechanism due to skin friction and flow
separation in the center section of the blade. The profile loss coefficient is heavily
dependent on flow incidence, which becomes important for off-design performance
prediction. According to their method (and those that followed), profile loss is deter-
mined initially at zero incidence (design-point), as a function of inlet and exit blade
26
angles. The zero-incidence profile loss is found as an interpolation between a “nozzle
Figure 3.8: Profile loss coefficient for thickness-to-chord ratio t/c = 0.2 for “nozzle blades”(a) and “impulse blades” (b) by Ainley and Mathieson [15].
blade” (Fig. 3.8a) and an “impulse blade” (Fig. 3.8b) for a thickness-to-chord ratio
of t/C = 0.2. When the AM correlation was first derived, nozzle referred to a blade
with an inlet gas angle of αin = 0 and impulse referred to a blade with equal inlet
and outlet gas angles.
Y ′P =YP (β1=0) +
∣∣∣∣β1
α2
∣∣∣∣ (β1
α2
) [YP (β1=α2) − YP (β1=0)
]( tmax/c0.2
) β1α2
(3.19)
This form of the interpolation was given by Kacker and Okapuu [4] with the intro-
duction of the term∣∣∣ β1
α2
∣∣∣ ( β1
α2
)to allow for negative inlet angles. The original AM
interpolation had(β1
α2
)2. The multiplying factor at the end of the expression is used
to account for variations in thickness to chord ratio away from the nominal value of
0.2 in Fig. 3.8. Kacker and Okapuu defined the design-point profile losses as
YP = 0.914(
23Y ′pKP + YSHOCK
)(3.20)
KP = 1−(M1
M2
)2
[1.25(M2 − 0.2)] for M2 > 0.2
= 1 for M2 ≤ 0.2 (3.21)
YSHOCK = 0.75(M1,HUB − 0.1)1.75(RHUBRTIP
)(p1
p2
)1−(1 + γ−1
2 M21
)1−
(1 + γ−1
2 M22
)
γγ−1
(3.22)
27
where YSHOCK accounts for losses due to shocks at the blade leading edge and KP
is a correction to the AM profile losses for varying Mach number. AM conducted
their cascade tests at low subsonic velocities, not accounting for Mach number ef-
fects. According to KO, “In an accelerating flow passage, operation closer to sonic
exit velocities will tend to cause suppression of local separations and the thinning of
boundary layers. This effect is most pronounced where inlet Mach numbers are only
slightly lower than exit Mach numbers” [4]. The ratio M1,HUB/M1,MEAN is given as
an empirical function of hub-to-tip ratio for rotors and nozzles (stators) in Fig. 3.9.
M1,MEAN is the Mach number at the midspan of the blade. The Reynold’s number
Figure 3.9: Inlet Mach number ratio for nonfree-vortex turbine blades from Kacker andOkapuu [4].
correction factor is
fRe =(
Re
2x105
)−0.4
for Re ≤ 2x105
= 1.0 for 2x105 < Re < 106
=(Re
106
)−0.2
for Re ≥ 106 (3.23)
The KO extension to the AM correlation is the most recent development for design-
point profile loss calculation. The most recent correlation for off-design incidence
cascade profile losses was by Benner et al. [19] in 1997. This work is a follow-up
to Moustapha et al. [16]. Moustapha determined that the leading edge diameter, d,
was a key parameter in predicting losses for high incidence flow. Benner subsequently
28
found that d was much less critical than previously thought and that actually the
leading edge wedge angle was playing a role on off-design incidence losses. Benner
defines his incidence parameter as
χ =(d
s
)−0.05
We0.2(
cosβ1
cosβ2
)−1.4
[α1 − α1,des] (3.24)
where α1,des is the design inlet gas angle, α1 is the actual gas inlet angle, β1 is the
cascade inlet blade angle, β2 is the cascade exit blade angle, We is the leading-edge
wedge angle, and d/s is the leading edge diameter to spacing ratio. Benner defines his
loss in terms of a change to the kinetic-energy coefficient, φ, because it varies more
weakly with Mach number than does the pressure loss coefficient, Y . The kinetic-
energy coefficient is defined as the ratio of the actual exit kinetic energy to the kinetic
energy obtained in an isentropic expansion through the blade row:
φ2 =V 2
2
V 22,is
(3.25)
and Benner’s loss correlation is
∆φ2P = a8χ
8 + a7χ7 + a6χ
6 + a5χ5 + a4χ
4 + a3χ3 + a2χ
2 + a1χ (3.26)
wherea8 = 3.711x10−7, a7 = −5.318x10−6,
a6 = 1.106x10−5, a5 = 9.017x10−5,
a4 = −1.542x10−4, a3 = −2.506x10−4,
a2 = 1.327x10−3, a1 = −6.149x10−5,
for χ ≥ 0, and
∆φ2P = 1.358x10−4χ2 − 8.720x10−4χ (3.27)
for χ < 0. Moustapha et al. [16] and Benner et al. [19] give the same formula for
conversion between the kinetic energy loss coefficient and pressure loss coefficient as
Y =
[1− γ−1
2 M22
(1φ2 − 1
)]− γγ−1 − 1
1−(1 + γ−1
2 M22
)− γγ−1
(3.28)
Secondary (Endwall) Losses, YS - According to Dixon [9],
29
Secondary losses arise from complex three-dimensional flows set up as a
result of the end wall boundary layers.... There is substantial evidence that
the end wall boundary layers are convected inwards along the suction surface
of the blades as the main flow passes through the blade row, resulting in a
serious maldistribution of the flow, with losses in stagnation pressure often
a significant fraction of the total loss.
Although a mean line approach is being taken for the profile losses, the secondary loss
coefficient applies only at the end walls. Secondary losses are usually defined in terms
of the airfoil drag coeffcient, CD, and/or lift coefficient, CL, fluid velocity angles, and
the space-chord ratio, s/c.
Figure 3.10: Suction surface definition for the Benner et al. [19] loss breakdown scheme.
Multiple correlations exist for end wall losses, but the most recent method was pre-
sented by Benner et al. in a two part paper. In the first part [19] Benner redefines
the breakdown of profile and secondary losses. In order to use his correlation for
secondary losses, one must adhere to this new loss breakdown scheme. He divides the
blade surface into a “Primary” region and a “Secondary” region as shown in Fig. 3.10,
which shows a schematic picture of a typical blade suction surface oil film pattern.
The profile loss, YP , described above is redesignated Ymid, the loss encountered at the
30
midspan of the blade. The new profile loss is then
YP = Ymid
(AprimAs
)(3.29)
where Aprim is the area of the primary region projected onto the plane shown in
Fig. 3.10. This is equal to
Aprim = hCx − 2(12ZTECx). (3.30)
As is the area where the flow exhibits three-dimensional character and is equal to
As =12CxZTE . (3.31)
ZTE is called the spanwise penetration depth, denoting the distance from the endwall
at which the flow is more or less two-dimensional. The final result for profile losses is
YP = Ymid
(1− ZTE
h
). (3.32)
Benner also gives a correlation for penetration depth, ZTE , as
ZTEh
=0.10(Ft)0.79
(CR)1/2(hc
)0.55 + 32.70(δ∗
h
)2
(3.33)
CR is the convergence ratio. It accounts for acceleration of the flow through the blade
channel and is expressed in terms of the inlet and exit gas angles as
CR =cosα1
cosα2. (3.34)
Ft is the tangential loading parameter. It represents the tangential force per unit
length nondimensionalized by dynamic pressure based on the vector mean velocity.
Ft is given by Benner as
Ft = 2s
Cxcos2 αm (tanα1 − tanα2) (3.35)
where αm is the mean vector angle through the airfoil row and can be expressed as
tanαm =12
(tanα1 + tanα2) . (3.36)
The last variable in the penetration depth equation that might be difficult to determine
is δ∗, the endwall boundary layer displacement thickness. If all of these parameters
31
can be found, then the secondary losses are determined by a method from the second
part [20] of Benner’s paper. The secondary loss expression is
YS =0.038 + 0.41 tanh
(1.20 δ
∗
h
)√
cos γCR(hC
)0.55 (C cosα2Cx
)0.55
for h/C ≤ 2.0
YS =0.052 + 0.56 tanh
(1.20 δ
∗
h
)√
cos γCR(hC
) (C cosα2Cx
)0.55
for h/C > 2.0 (3.37)
Tip Clearance (Leakage) Losses, YTC - These losses occur at the tips of stator blades
at the hub-blade interface or the tips of rotor blades at the casing interface. The loss
coefficient depends on the size and nature of the interface (plain or shrouded). The
unshrouded relationship will not be given here for the sake of brevity, but for shrouded
blades, the KO expression is
YTC = 0.37c
h
(k′
c
)0.78 (CLs/c
)2 cos2 α2
cos3 αm(3.38)
where
k′ =k
N0.42seals
(3.39)
and CL is the airfoil lift coefficient and h is the blade height. The reason for the use of
CL is that traditional cascade testing used lift and drag force indicators to determine
stage losses. The KO expression is the same as the DC expression except for the
accounting for multiple seals with k′. According to Yaras and Sjolander(YS) [22], the
AMDC model substantially overestimates the losses. They classify the AM, DC, and
KO methods as “models based on momentum considerations” that were essentially
inviscid. YS state that viscous losses, which were assumed to be accounted for in pro-
file losses by the previous authors, actually increase with clearance height. YS present
a more complex system of tip clearance loss based on energy considerations, which
needs to be studied further in order to present it accurately herein. The applicability
of such a method to the present research will also be determined. There is a great
deal of literature since the 1992 YS article that needs to be reviewed in order to get a
modern view of tip clearance loss measurement and prediction. The results of a study
to determine modern techniques of measuring tip clearance losses will be presented in
the Measurement Planning section of the final thesis.
32
Trailing Edge Losses, YTE - These losses arise due to the final thickness of the trailing
edge being nonzero. The pressure and suctions surfaces cannot converge to a point
on a real blade, because the stresses generated by the high pressure gradient, high
temperature, high speed flow are too great. The losses are associated with a wake
generated by flow separation due to a low pressure pocket behind the trailing edge. In
addition to generating pressure losses, the nonuniform flow impacts the performance
of the following blade row. The original AM correlation included trailing edge losses
within the profile loss component, but the newer KO model which is presented now
gives an empirical relationship for YTE . In their scheme, trailing edge losses are found
Figure 3.11: Trailing edge energy loss coefficient correlated against the ratio of trailing edgethickness to throat opening from Kacker and Okapuu [4].
in terms of the kinetic energy coefficient and are interpolated in a similar manner to
profile losses are in Equation 3.19, i.e. ,
∆φ2TE = ∆φ2
TE(β1=0) +∣∣∣∣β1
α2
∣∣∣∣ (β1
α2
) [∆φ2
TE(β1=α2) −∆φ2TE(β1=0)
](3.40)
where ∆φ2TE(β1=α2) and ∆φ2
TE(β1=0) are correlated against t/o in Fig. 3.11. KO gave a
slightly different conversion from the kinetic energy coefficient, φ2, to the pressure-loss
coefficient, Y , than did BSM. KO gave Y = f(∆φ2), instead of BSM’s Y = f(φ2), as
follows
Y =
[1− γ−1
2 M22
(1
1−∆φ2 − 1)]− γ
γ−1 − 1
1−(1 + γ−1
2 M22
)− γγ−1
(3.41)
In order to understand the difference between the two conversions and how to use
the loss correlations given by both groups, the conversion expression will need to be
33
rederived over the course of the thesis. An important development for trailing edge
losses occurred in 1990 when Denton [23] attempted to verify an analytical relationship
for YTE with numerical results. This development will be investigated to determine
its utility in the present thesis project. Additionally, a study will be performed to
indentify modern trailing edge loss measurement techniques and the results will be
presented in the Measurement Planning section of the final thesis.
The AM, KO, MKT, BSM method presented above is the most recent available in the
open literature, but it is not necessarily the best candidate for use in steam turbine model.
At the 2008 IMECE conference, Benini et al. presented a comparison of six different Detailed
Parameter Models, based on their accuracies in design-point and off-design prediction of the
performance of a low reaction, two-stage gas turbine. The most recent method considered in
Benini et al. ’s quantitative comparison is that of Moustapha et al. Benini et al. reference
some additional pressure-loss models that will need to be addressed over the course of the
thesis, including those of Craig & Cox [3], Traupel [24], and Balje & Binsley [25]. Of
particular interest is the relatively high accuracy of the Craig & Cox method in predicting
total-to-total efficiency and stator and rotor total pressure-loss coefficients in the first stage.
Of course, the ideal Detailed Parameter Model will also accurately predict the performance
of subsequent stages.
The comprehensive applicability of Detailed Parameter Models comes at the cost of
complexity, relative to their simplified Overall Parameter Model cousins. Because pressure-
loss models are derived for gas turbine performance prediction, an additional complexity
arises in applying the results to condensing steam turbines. There are several methods to
account for two-phase flow conditions, some of which will now be described.
Steam Condensation ADD DESCRIPTION OF LOSSES
A Novel Approach
The major shortcoming of conventional methods, like AM and its extensions, is the lack of
first principles used in their derivation. The philosophy of the pressure-loss model presented
previously is that additional correction factors, correlation factors, or even loss components
34
can be added to account for new hypotheses on loss sources, or for new flow regimes of
Mach or Reynolds numbers. Continued development in this direction will lead to an overly
complex model for losses based on speculation, not on sound theory. No correlation de-
rived under such a philosophy can be general for every turbine since the overall method
(AM, AMDC, KO, etc.) is tuned by each manufacturer to match their product line. In
his 1993 International Gas Turbine Institute Scholar Lecture, Denton[12] charges that the
conventional methods are not based on a sound understanding of the flow physics. He also
explains the upshot of adhering to this philosphy in the turbine design world:
There have been many instances where a designer was unwilling to try out a
new idea because a 30-year-old loss correlation predicted that it would give no
improvement.
....
Such correlations can tell us nothing about new design features that were not
available at the time the correlation was developed.... It is the author’s view
that a good phsical understanding of the flow, and particularly of the the origins
of loss, is more important to the designer than is the availability of a good but
oversimplified correlation.
Denton proposes a novel approach, to study the detailed results of experiments and
numerical calculations to understand the fundamental physical reasons for loss in a turbine
stage. He says losses can still be divided between regions of “profile”, “endwall”, and
“leakage” components. In addition, each of these components should be divided into the
real sources of irreversibility, e.g.
1. Viscous friction in boundary layers or free shear layers.
2. Heat transfer across finite temperature differences.
3. Nonequilibrium processes such as occur in very rapid expansions or in shock waves.
In order to quantify the results, he expresses stator and rotor losses in terms of entropy
coefficients for the nozzle,
ζNs =T2∆s
h02 − h2(3.42)
35
and for the rotor,
ζRs =T3∆s
h03rel − h3(3.43)
These loss coefficients are similar to the enthalpy loss coefficients in Equations 3.10 and 3.11,
their only difference being in the numerator. This difference is easily resolved by looking at
constant pressure lines on a Mollier diagram, Fig. 3.5, whose slope is approximately equal
to static temperature for small changes in enthalpy. That is,(∂h∂s
)p
= T . Denton derived
analytical expressions for the loss coefficients in terms of the real sources of irreversibility.
If the novel approach is to be considered in the present thesis, Denton’s lecture, his
references, and his more recent works will need to be reviewed in detail in order to determine
it’s utility. In particular, Denton withholds a definitive analytical expression for endwall
losses, which represent on the order of 1/3 of total losses. On the breakdown of endwall
losses into its real flow components, he had this to say, “In all the situation is too complex
and too dependent on details of the flow and geometry for simple quantitative predictions
to be made. The main hope in the near future is that the loss can be quantified by three-
dimensional Navier-Stokes solutions, which already give good qualitative predictions of the
flow.” If a particular loss component cannot be quantified, then neither can the total loss
coefficient. This is the greatest barrier to implementing Denton’s approach in the present
steam turbine system model.
As the most widely used loss model type in the literature, pressure-loss models represent
a primary candidate for a conventional performance prediction method to be empirically
tuned by performance testing at REES. Indeed, the contemporary improvement of the 50-
year-old AM correlation by Benner et al. is a testament to its continued versatility; however,
limits to its accuracy, resolution, and ease of use in steam turbines necessitate exploration
of more novel approaches.
In 1993, Denton proposed to advance his empirical loss theory by numerical solutions of
Navier-Stokes, a not-so empirical process. As was discussed at the outset to Section 3.4, tur-
bine performance prediction requires a combination of empirical, numerical, and analytical
equations. In order to understand the continued development of both the conventional and
novel empirical approaches, it will be shown how they are implemented in more complex
numerical turbine predictions.
36
3.4.2 Analytical and Numerical Methods
Analytical
One of the most widely used analytical turbine design methods is radial equilibrium. Ac-
cording to Dixon [9], “The radial equilibrium method is based on the assumption that any
radial flow which may occur is completed within a blade row, the flow outside the row then
being in radial equilibrium.” Such an assumption is most accurate for turbines with large
hub-to-tip ratios, where the blades are too short for much radial momentum to appreciate.
Dixon describes several radial equilibrium approaches, which are divided into “indirect”
and “direct” problems. The free-vortex approach assumes that the the product of radius
and tangential velocity remains constant at all radii (rc(y) = K) and thus the axial vortic-
ity component is zero. The forced vortex approach, sometimes called solid body rotation
assumes that tangential velocity varies directly with radius (c1y = K1r and c2y = K2r). For
both of these indirect methods, the flow angle variation over the blade span is then found
through the analysis. This differs from the direct approach where the flow angle variation
is specified, and the solution of cx and cy are found. Dixon also presents an analysis for
predicting the off-design performance of a free-vortex turbine stage.
When a turbine of low hub-tip ratio is considered, Dixon recommends the more accurate
actuator disc approach, which was first used in propeller theory. The method takes the
view that the axial width of each blade row is shrunk while the space-chord ratio, the blade
angles and overall length of the machine are maintained constant. Dixon doesn’t give an
in depth derivation of the resulting governing equations, but simply presents several plots
and diagrams. For the final thesis, it might be useful to investigate this method further.
Dixon references a 1962 paper by Hawthorne and Horlock (“Actuator disc theory of the
incompressible flow in axial compressors”) several times, which could be of some use.
Note: This section requires further research for the final thesis, especially if analytical
methods are to be used for baseline turbine performance prediction. A literature review
will need to be conducted to determine the prevalence of analytical methods in recent
publications.
37
Numerical
Dixon gives a brief overview of numerical approaches for solving the momentum, energy,
and state equations. The first group of approaches is the “through-flow” category, which
includes the streamline curvature, matrix through-flow or finite difference, time-marching,
and stream function methods. Through-flow methods assume that the flow to be steady in
both the absolute and relative frames of reference, and that the flow is axisymmetric outside
the blade rows, ignoring the effects of blade wakes from upstream blade rows. Dixon lists
several authors of importance for detailed descriptions of through-flow methods: Macchi
(1985) and Smith (1966) for streamline curvature; Marsh (1968) for matrix through-flow
and Denton (1985) for time-marching. Contrary to its name, the time-marching method
is not used to solve the unsteady flow problem, but rather is marched in time toward a
convergent steady state solution.
The second group of approaches is the Computational Fluid Dynamics (CFD) category,
which are able to solve, accurately or not, three dimensional unsteady flow equations. Ac-
cording to Dixon, the use of CFD grew in the 1990’s when there was demand for a faster
solution method cost less and achieved greater resolution than the traditional axisymmetric
through-flow methods. Although CFD results are useful for investigating flow phenomena
around particular stage geometries, Horlock and Denton [26] indicate that loss predictions
by CFD are still not accurate. Dixon states, “There are many reported examples of the
successful use of CFD to improve designs but, it is suspected, many unreported failures.”
Dixon references a 2000 article by Adamczyk that could be of some use if further investi-
gation into modern CFD methods is required for the final thesis. There are also examples
in the literature of the use of empirical correlations to generate the downstream boundary
conditions for a blade row, which are then used in the CFD analysis. More examples of this
analysis technique in recent publications would further justify the development of empirical
correlations.
38
3.5 Measurement Planning
The primary motivation of a steam turbine test loop is the development of performance
prediction methods for turbine design and selection. A review of current methods of turbine
performance prediction has been presented, with a focus on empirical correlations. In order
to verify, update, and create these prediction methods, a test loop will need to capture
specific experimental data. Section 3.5 will review literature relevant to planning the sensor
selection and placement in the steam turbine test loop.
Three sets of literature present relevant examples for test loop measurement planning.
The first group, Codes and Standards, give a description of overall turbine performance
measurement, as opposed to stage performance. They detail proper placement of tempera-
ture, pressure, flow, torque, and speed sensors in the test loop, but not within the turbine
itself. The second group, Industrial Test Loops, gives information regarding turbine test
practices in the Dresser-Rand organization. For the most part, Dresser-Rand follows the
ASME Performance Test Codes (PTC) for overall turbine performance, but there are also
research reports from old turbine R&D studies that give a view into turbine internal sen-
sor placement. The third group, Academic Research Labs, describes detailed measurement
methods within the turbine, including slip-ring or radio communications between the fast-
moving rotor frame-of-reference and the stationary computer systems. Also of importance
is the vast difference between cascade and turbine testing, both of which have been used in
academia. A thorough literature review of the three groups has not been completed, but
some details of the readings to date will be presented in the proceeding sections.
3.5.1 Codes and Standards
The most applicable ASME code is Performance Test Code 6 (PTC 6) - Steam Turbines.
Other important ASME codes include PTC 19.2 - Pressure Measurement, PTC 19.3 -
Temperature Measurement, PTC 19.5 - Flow Measurement, and PTC 19.7 - Measurement
of Shaft Power.The particular details of these codes that apply to the design of the test
loop will be presented in the final thesis. ASTM is source for turbine standards; however,
their standards primarily apply to material testing and failure testing, and not so much to
39
performance testing.
3.5.2 Industrial Test Loops (Dresser-Rand)
In addition to Dresser-Rand’s documented experience with sensor placement in research
turbines, they presently conduct R&D experiments for their gas compressors. Jay Koch is
a contact at Dresser-Rand’s Olean, NY location that should be consulted for further infor-
mation in this realm. In addition, Fred Woehr of Dresser-Rand’s Steam Turbine Division
should be consulted to learn about their current steam turbine test facilities.
3.5.3 Academic Research Labs
Figure 3.12: High-pressure turbine cascade wind tunnel at Carleton University, Ottawa[21].
Fig. 3.12 shows the high speed cascade wind tunnel at Carleton University in Ottawa.
40
As mentioned in Section 3.4, Cascades are widely used by gas turbine empiricits to isolate
specific loss components. They also provide a much simpler test setup than a full-rig turbine
with a rotating shaft. The Isentropic Compression Tube Annular Cascade Facility at the
Von Karman Institute (VKI) represents a hybrid between a rotating full-rig and a stationary
cascade. It uses compressed air to 1.5 stage turbine. A full stage with stator and rotor is
followed by a single stator stage, used to investigate the effects of an upstream stage on
stator flow patterns.
In the final thesis, this subsection will give a thorough description of the measure-
ment techniques used in cascades, full-rigs, and any other variation found in axial turbine
academia. Specifically, this section will look at what measurements are required to deter-
mine stage-to-stage losses and calculate stage efficiencies using the methods described in
the performance prediction section. Of particular interest are particle imaging velocimetry
(PIV) and laser anemometry, which are two methods of determining fluid velocity without
inhibiting the flow. Slip-ring telemetry and/or radio communications may also be required
to communicate data between the rotating frame of reference and stationary computers.
These will also be discussed further in the final thesis.
3.6 System Modeling
One of the major building blocks of a Rankine cycle system model is an Equation of State
(EOS) calculator. FluidProp [27] is a versatile EOS solver in that it can calculate properties
for most liquids and gases used in the process industry. It also includes a tool called GasMix
where the user can create their own mixtures of gases. FluidProp meets the EOS needs
for the present work, an EOS for liquid water and steam, with its IF-97 tool. This tool
follows the requirements of the 1997 Industrial Formulation published by the International
Association for the Properties of Water and Steam (IAPWS). Each of the FluidProp tools is
contained with in a dynamic link library (dll) file that are easily accessed from programming
environments like Fortran, Maple, Matlab, Visual C++, Modelica, Visual Basic, and even
Excel. So far the author has proven the functionality of FluidProp in a free version of MS
Visual C++, Matlab, and Excel.
41
Journal articles by Colonna and Van Putten of Delft University show promise in terms
of organizing a simulation environment for the steam turbine test loop. These articles and
others will need to be reviewed in more depth for the final thesis. It will be important to
decide if a causal or an acausal model is most appropriate for this application. A causal
model has specific user inputs and computational outputs, while an acausal model allows a
user to specify any combination of variables that defines a determinant system and computes
the remaining variables as outputs. Programming an acausal model would require more
work if a traditional programming language such as C++ or Matlab is used, but Modelica,
a programming language released in 1997, has an architecture designed just for creating
such models. An open source compiler and development environment are available online,
but the author had some difficulty in implementation.
42
Chapter 4
Progress to Date
4.1 Test Loop Design
4.1.1 System Design
Figure 4.1: Process diagram of a potential steam turbine test loop.
43
Fig. 4.1 shows the process diagram for a steam turbine test loop. The loop is designed to
operate the steam turbine under either condensing or non-condensing conditions. The capa-
bility for both types of operation is required in order to use condensation as an independent
variable.
Variable shaft loading will be accomplished with a dynamometer. The type of dy-
namometer (e.g. DC, AC, or Eddy Current) will be decided in the final design of the test
loop. Given constant steam inlet and flowrate conditions, the rotational speed of the turbine
can be controlled by applying variable torque. Of course, under real operating conditions
the rotational speed will affect the pressure drop and flow rate.
A back pressure valve will provide control of turbine outlet pressure. In a real steam
power cycle, a back pressure valve is prohibitive to power production, but for a test loop it
may be used to maintain single-phase exit conditions.
The Condenser will reject heat from the condensing steam to the cooling loop. The
cooling loop will include at least one heat sink, such as a pond or a cooling tower, but the
cooling loop is not of detailed interest to this thesis. At least for the preliminary design,
the change in cooling water temperature across the condenser and its flowrate will be input
variables to the system model. The condensate pump might be necessary for condensing
turbine testing where the turbine exit pressure is much lower than the inlet pressure, and
especially if it is below atmospheric. The boiler feedwater pump will pressurize the water
to the desired boiler pressure. Determining the necessity of one or two pumps will require
accurate pump performance curves, including minimum net positive suction head. In either
case, the pumping system will need to operate the turbine through the desired pressure
operating range.
There will also need to be provisions for makeup water, since some leakage from the sys-
tem is to be expected, but leakage will need to be minimized in order to meet requirements
of ASME Performance Test Codes (PTC). Excessive leakage could cause inaccuracies when
assuming the turbine mass flow rate is equal to the liquid flow rate measured somewhere
else in the system. The makeup water will need to be analyzed and filtered in a separate
water treatment system in oder to protect the steam turbine. After processing, the makeup
water will need to be either throttled or pressurized for mixing with the condensate in the
44
supply tank. Another possibility not depicted here is an in-line water treatment system so
the water is sampled each time it passes through the loop.
The natural gas fired boiler will supply superheated steam at the desired temperature
and pressure for the particular test. Although the fuel flow rate will be controlled in order
to set the exit temperature of the superheater, the boiler system model for the preliminary
design of the test loop will use the heat output of the boiler as the model input.
The desuperheater will reduce the steam temperature to the desired turbine inlet value
by spraying unheated supply water into the steam. The unheated supply water is at ap-
proximately the same pressure as the steam leaving the boiler. The response of the turbine
inlet temperature to a desuperheater is much faster than it is to decreasing the boiler fuel
flow rate, since the boiler has a large heat capacitance of heated water and steam. There
are multiple types of desuperheaters that could be used, each with certain benefits like low
cost, low pressure drop, or small size. Selection of an appropriate superheater is within the
scope of this thesis.
The inlet control valve will provide control of the turbine inlet pressure and the mass
flow rate for the loop. Setting the mass flow rate through the turbine will be critical to
getting the approriate performance point of the turbine.
4.1.2 Component Selection
Turbine
Selecting the first turbine for research testing is a critical step, because the final selection
will guide the rest of the system design decisions. One candidate is a 4-stage turbine
located at Dresser-Rand in Wellsville that was last used for seals testing in 2004, but is now
disassembled. Dresser-Rand will be heavily involved with selecting the turbine, which will
have multiple sets of stator and rotor rows in order to test different flowpath geometries. The
simulation environment generated over the course of the thesis should be robust enough to
model any particular set of internal set of geometries selected. The test loop should also be
flexible enough to accomodate other turbines in the future, from other industry, government,
or research partners.
Selection of the following components will be discussed in more detail in the final thesis.
45
• Condenser
• Pump
• Boiler
• Dynamometer
4.1.3 Facility Planning
The following figures show the preliminary design of the Rochester Energy-systems Ex-
periment Station (REES) REES would provide a location for any type of energy systems
research that comes to RIT in the future. The current effort is focused on the steam turbine
test loop depicted in Fig. 4.2. The sizes of the components in the loop are representative of
the boiler, turbine, condenser, pump, and supply tank that would be in the final design.
Figure 4.2: Preliminary design of the physical test loop with pipe connections in a 50’x100’test area [6].
46
Figure 4.3: Exploded view of REES showing steam turbine test loop.
4.2 System Modeling
Although the process diagram for the test loop preliminary design was given in Fig. 4.1, a
simplified model for a Rankine cycle as depicted in Fig. 2.1 was used to create a simulation
in Matlab. The primary purpose of the simulation was to test FluidProp [27], an EOS tool
reviewed in Section 3.6. The simulation was successful in that the results were identical to
those of a spreadsheet calculation with table lookups in Excel.
47
Chapter 5
Statement of Work
The end product of this thesis will be a design for a steam turbine test loop, whose primary
goal will be to develop performance prediction schemes for turbine selection and design
tools. The broad spectrum of performance prediction touched on in Chapter 3 and the large
scale of the laboratory facility described in Chapter 4 necessitate a structured approach to
optimizing the system design. The design process will follow a structured approach along
the three different thrusts that were introduced in Chapter 2. The action items for each of
these thrusts are decribed in the following paragraphs.
1. Turbine Performance Prediction - Perform a critical review of conventional per-
formance prediction methods for axial turbines, focused primarily on empirical cor-
relation methods. Research the novel entropy loss correlation system proposed by
Denton [12] to determine if it is feasible for use in a steam turbine system model. Re-
viewing these two groups of models will provide knowledge of how the general turbine
community would use the test loop. In addition, it will be critical to consult with
Dresser-Rand to get an accurate picture of how they derive prediction methods that
are used in turbine selection and design.
2. Measurement Planning - Perform a comprehensive review of conventional and novel
axial turbine testing methods, including relevant codes and standards and academic
literature. Develop a set of minimum sensor requirements for obtaining the exper-
imental data necessary for verifying/updating conventional performance prediction
methods. Develop a set of desired sensors for obtaining the experimental data neces-
sary for extending the novel correlation. Perform an error analysis to ensure optimal
48
selection and placement of loop sensors and internal turbine sensors.
Figure 5.1: Model architecture for the Simulation Environment and Parametric Design Tool
3. System Modeling - Create a steam turbine test loop simulation tool that will sim-
ulate the expected response of the test loop to available control inputs. Create a
preliminary system design of the test loop that can be used as an input to the sim-
ulation environment. Conduct a parametric study of the critical design parameters
to arrive at an acceptable solution. A possible configuration for the simulation envi-
ronment and optimazation tool is illustrated in Fig. 5.1. The “Cost” output of the
relevant components does not nessarily represent monetary cost. The optimizing pa-
rameters will need to be selected over the course of the thesis, but examples include
measurement uncertainty (σmeas), empirical correlation accuracy (σcorr), operating
cost($/hr), and test loop versatility to test a variety of turbines and operating condi-
tions (∆OC). Once the parameters are quantified, appropriate relative weighting will
need to be assigned.
The parametric design process will be used to arrive at a final system design, selected
from several alternatives generated with different weighting schemes. The final design
49
will be used as an input to the simulation environment to generate mock sensor out-
puts that will be compared to the results of different turbine performance prediction
methods. This comparison process will be similar to that used in the real test loop,
once it has been built. The ability to perform such a comparison will prove the va-
lidity of the final design for the steam turbine test loop. Upon building the test loop,
the simulation environment could be used as a tool to calibrate the control system or
to generate more accurate performance models for any component in the loop.
The purpose of this thesis project is not to develop a new performance prediction
method, or to build an actual steam turbine test loop. The scope of this project ends
with the completion of a design for a test loop that enables new devlopment of steam tur-
bine performance prediction. The large scale and complexity associated with the design of
a steam turbine test loop represents a full scope for a Masters thesis project. This design
has not been contracted to an engineering firm, because they could easily overlook the the
complexities associated with meeting the needs of industry and the research community.
In addition to completion of the thesis project, the author will seek publication for a
conference paper and a journal article. The likely conference is the 2010 ASME International
Mechanical Engineering Congress and Exposition (IMECE). The abstract will probably be
due in March 2010, and a full-length draft in June. These dates are reflected in the schedule
of Table 6.1. Some candidate journals include the ASME Journal of Turbomachinery; ASME
Journal of Engineering for Gas Turbines and Power; and Proceedings of the Institution of
Mechanical Engineers, Part A: Journal of Power and Energy. As the nature of the present
research reveals itself and more knowledge is gained about each particular journal, their
relative compatibilities can be determined.
50
Chapter 6
Schedule
Table 6.1: Preliminary schedule of Thesis Deliverables
51
Chapter 7
Risks
Scope
Traditional theses for a Master of Science in Mechanical Engineering often combine two of
the following methods: simulation or numerical analysis, analytical analysis, and design of
and experimentation on a laboratory system. Because of the large design scope of a steam
turbine test loop, this project will cover only design and simulation of the laboratory system.
The design scope is sufficiently large and will need to be managed using a Function-Means
diagram to map out the potential combinations of test loop components. It is evident from
the literature review that a reasonable steam turbine model will require much effort, so it
will be important to consciously manage the level of detail and modeling accuracy sought
for each component in the test loop. A Component-Complexity diagram will be used to
map the relative complexities sought for each component to ensure realistic goals.
Class Workload
Pogress has been made on the literature review over the Summer 2009 quarter, but it has
not been sustained through Fall. Although the course materials for Ideal Flows, Intro to
Control Systems, and Fluid Mechanics of Turbomachinery are relevant to the thesis, they
demand significant time. It will be important to recommence the research effort during the
Winter quarter in order to stay on schedule.
52
Industrial Partners
Although there is minimal information in the open literature on steam turbine performance
prediction, there is a trove of data available in industry. The author will need to commu-
nicate effectively with Dresser-Rand and utilize them as a valuable source of information.
It has been difficult in the past to get Dresser-Rand to provide timely information for this
project, so it will be the partial responsibility of the author to convince them of its value.
This will require in depth knowledge of their current design practices and future testing
needs.
Correlation Compatibility
It was mentioned in Section 3.4 that there are additional correlating parameters to relate
cascade correlations to full-rig performance prediction that aren’t available in the open
literature. If there is to be any possibility of using the conventional models descirbed in the
literature review, these parameters will need to be determined. Dresser-Rand is a possible
source for such information, although there is no indication that they have used cascade
tests in the past. If it is not possible to determine the correlating parameters, another
prediction method that applies to full-rigs, such as that of Craig and Cox [3], will need to
be used.
53
Bibliography
[1] D.G. Ainley and G.C.R. Mathieson. A method of performance estimation for axial-flowturbines. Aeronautical Research Council Reports & Memoranda, (2974), 1951.
[2] J. Dunham and P.M. CAME. Improvements to the ainley-mathieson method of turbineperformance prediction. Journal of Engineering for Power, 92(3):252 – 256, 1970.
[3] H.R.M. Craig and H.J.A. Cox. Performance estimation of axial flow turbines. Proc.Inst. Mech. Engrs, 185:407–424, 1970-1971.
[4] S.C. Kacker and U. Okapuu. Mean line prediction method for axial flow turbine effi-ciency. Journal of engineering for power, 104(1):111 – 119, 1982.
[5] Wageeh Sidrak Bassel and Arivaldo Vicente Gomes. A metastable wet steam turbinestage model. Nuclear Engineering and Design, 216, 2002.
[6] James Brown and Neranjan Dharmadasa. unpublished Senior Design Project. May2009.
[7] Fred Woehr, May 2009.
[8] Michael William Benner. The Influence of Leading-Edge Geometry on Profile andSecondary Losses in Turbine Cascades. PhD thesis, Carleton University, 2003.
[9] S.L. Dixon. Fluid Mechanics and Thermodynamics of Turbomachinery. Elsevier-Butterworth-Heinemann, Boston, 5th edition, 2005.
[10] David M. Mathis. Handbook of Turbomachinery, chapter Fundamentals of TurbineDesign. M. Dekker, New York, 2nd edition, 2003.
[11] Ernesto Benini, Giovanni Boscolo, and Andrea Garavello. Assessment of loss correla-tions for performance prediction of low reaction gas turbine stages. In Proceedings ofIMECE2008, pages 177–184. ASME, 2008.
[12] J. D. Denton. Loss mechanisms in turbomachines. Journal of Turbomachinery, 115,1993.
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[13] C.H. Sieverding. Axial turbine performance prediction methods. In Thermodynamicsand Fluid Mechanics of Turbomachinery, volume 97 of Series E, pages 737 – 784.NATO Advanced Study Institute, 1985.
[14] J. H. Horlock. Axial Flow Turbines: fluid mechanics and thermodynamics. Butter-worths, London, 1966.
[15] D.G. Ainley and G.C.R. Mathieson. An examination of the flow and pressure lossesin blade rows of axial-flow turbines. Aeronautical Research Council Reports & Memo-randa, (2891), 1951.
[16] S. H. Moustapha, S. C. Kacker, and B. Tremblay. An improved incidence losses pre-diction method for turbine airfoils. Journal of Turbomachinery, 112(2):267–276, 1990.
[17] M. W. Benner, S. A. Sjolander, and S. H. Moustapha. An empirical prediction methodfor secondary losses in turbines—part i: A new loss breakdown scheme and penetrationdepth correlation. Journal of Turbomachinery, 128(2):273–280, 2006.
[18] M. W. Benner, S. A. Sjolander, and S. H. Moustapha. An empirical prediction methodfor secondary losses in turbines—part ii: A new secondary loss correlation. Journal ofTurbomachinery, 128(2):281–291, 2006.
[19] M. W. Benner, S. A. Sjolander, and S. H. Moustapha. Influence of leading-edge geom-etry on profile losses in turbines at off-design incidence: Experimental results and animproved correlation. Journal of Turbomachinery, 119:193–200, 1997.
[20] M. W. Benner, S. A. Sjolander, and S. H. Moustapha. The influence of leading-edgegeometry on secondary losses in a turbine cascade at the design incidence. Journal ofTurbomachinery, 126:277–287, 2004.
[21] S.A. Sjolander. Mae research, July 2009. http://research.mae.carleton.ca/Research Pages/Research Turbomachinery.html.
[22] M.I. Yaras and S.A. Sjolander. Prediction of tip-leakage losses in axial turbines. Journalof Turbomachinery, 114, 1992.
[23] J.D. Denton and L. Xu. The trailing edge loss of transonic turbine blades. Journal ofTurbomachinery, 112, 1990.
[24] W. Traupel. Termische Turbomaschinen. Springer, 4th edition, 2000.
[25] O.E. Balje and R.L. Binsley. Axial turbine performance evaluation. part a - loss-geometry relationship and part b - optimization with and without constraints. Journalof Engineering for Power, 80, 1968.
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[26] J.H. Horlock and J.D. Denton. A review of some early design practice using computa-tional fluid dynamics and a current perspective. Journal of Turbomachinery, 127(1):5–13, 2003.
[27] P. Colonna and T.P. van der Stelt. FluidProp: a program for the estimation of thermo-physical properties of fluids. Energy Technology Section, Delft University of Technology,The Netherlands (http://www.FluidProp,com), 2004.
[28] R. Shaw. Efficiency predictions for axial-flow turbines. Int. Jounal for MechanicalSciences, 8, 1966.
56
Appendix A
Derivation of Stage Efficiency
A.1 Assumptions and Justifications
1. Flow conditions at mean radius fully represent flow at all other radii. This is reason-
able if the blade height to mean radius ratio is small as in the early stages of an axial
Turbine
2. Radial velocities are zero
3. Constant axial velocity at all interstage zones
4. Flow is invariant in the circumferential direction. This means that flow characteristics
are the same from nozzle to nozzle or bucket to bucket in the same stage
5. For steam, conditions are in the vapor region only. This is a first take only, and will
need to be revisited in future analyses.
6. Compressible gas
7. Steady flow conditions at the turbine inlet and outlet
8. Constant elevation between stages and from inlet to outlet
9. The turbine is adiabatic
57
A.2 Analysis
A.2.1 Thermodynamics and Dynamics
Before defining the efficiency of a turbine, it is necessary to review some fundamental
engineering concepts. Consider the 1-D flow of a fluid through a control volume between
states 1 and 2. The first law of thermodynamics accounts for changes in the internal energy
of the fluid by
E2 − E1 =∫ 2
1(dQ− dW )
and for an infinitesimal change of state,
dE = dQ− dW (A.1)
The second law of thermodynamics states that for a system passing through a cycle involving
heat exchanges, ∮dQ
T≤ 0
If all the processes in the cycle are reversible then dQ = dQR and the equality holds true:∮dQRT
= 0
Entropy is then defined as ∫ 2
1
dQ
T= S2 − S1
For an incremental change of state
dS = mds =dQRT
(A.2)
For a system of mass m undergoing a reversible process (i.e. s1 = s2), dQ = dQR = mTds
and dW = dWR = mpdv. In the absence of motion, gravity and other effects, Equation A.1
becomes
Tds = du+ pdv
and with h = u+ pv, then dh = du+ pdv + vdp, we arrive at
Tds = dh− vdp (A.3)
58
This equation will be useful in defining turbine stage efficiency later in the analysis.
The continuity equation for uniform, steady flow for the stage depicted in Fig. 3.3 is
ρ1A1cx1 = ρ2A2cx2 = ρ3A3cx3
Since cx1 = cx2 = cx3 from Assumption 3,
ρ1A1 = ρ2A2 = ρ3A3 (A.4)
Conserving energy across the stage yields
Q− W = m
[(h3 − h1) +
12
(c2
3 − c21
)+ g (z3 − z1)
]With Assumptions 8 and 9, and the definition for total enthalpy in Equation 3.1, the energy
equation simplifies to
W =W
m= (h01 − h03) (A.5)
Figure A.1: Control volume for a generalized turbomachine, copied from Dixon pp.29 [9].
Now consider the general turbine of Fig. A.1 where the vector sum of the moments
of all external forces acting on the system is equal to the time rate of change of angular
momentum of the system, or
τA = md
dt(rcθ) (A.6)
Dixon states that for one-dimensional steady flow
τA = m (r2cθ2 − r1cθ1) (A.7)
which states “the sum of the moments of the external forces acting on fluid temporarily
occupying the control volume is equal to the net time rate of efflux of angular momentum
59
from the control volume” [9]. For a turbine rotor running at angular velocity, Ω, the rate
at which the fluid does work on the rotor is
τAΩ = m (U1cθ1 − U2cθ2) (A.8)
The specific work done on the fluid, according to Euler’s turbine equation, is
W =W
m=τAΩm
= (U1cθ1 − r2cθ2) > 0 (A.9)
So for a stage rotating at constant speed, we can say
W =W
m= (h01 − h03) = (U1cθ1 − U2cθ2) (A.10)
This definition for the specific work of a stage, W , will be used in the stage efficiency
analysis which follows.
A.2.2 Stage Efficiency
Referring to the Mollier Diagram of Fig. 3.5, the maximum power output from a stage would
be obtained by a total enthalpy drop from h01 to h03ss. So, we define the total-to-total stage
efficiency, ηts, as the ratio between the actual work output and this ideal work output when
operating to the same back pressure:
ηtt =h01 − h03
h01 − h03ss(A.11)
According to Dixon [9], flow conditions are identical at entry and exit to a normal stage
(i.e. c1 = c3 and α1 = α3). What he means is that it can be assumed that for any stage,
except the first, it is beneficial to have the same total flow turning in each stage if all stages
have the same potential to absorb the flow energy. This assumption cannot be used for
the first stage where the entry angle is nearly zero (axial flow), and the exit flow will have
some tangential component. Because multistage turbines are of interest, the case Dixon
described will be covered first, and a derivation given by Shaw [28] for the second case will
follow. If in addition to equal entry and exit velocities it is assumed that c3ss = c3, the
total-to-total efficiency becomes
ηtt =h1 − h3
h1 − h3ss
60
=h1 − h3
(h1 − h3) + (h3 − h3s) + (h3s − h3ss)(A.12)
Now, from Equation A.3, the slope of a constant pressure line on a Mollier diagram can
be expressed as(∂h∂s
)p
= T . Thus for a finite change of enthalpy in a constant pressure
process, ∆h ∼= T∆s. Therefore,
h3s − h3ss∼= T3 (s3s − s3ss)
h2 − h2s∼= T2 (s2 − s2s) (A.13)
From Fig. 3.5, s3s − s3ss = s2 − s2s. So,
h3s − h3ss =T3
T2(h2 − h2s) (A.14)
Non-dimensional enthalpy-loss coefficients for the stator nozzle and rotor bucket are then
defined in terms of their exit kinetic energies as
Stator : ζN = 2(h2−h2s)c22
Rotor : ζR = 2(h3−h3s)w2
3(A.15)
Substituting Equations A.14 and A.15 into A.12 yields
ηtt =h1 − h3
h1 − h3 + T32T2
c22ζN + 1
2w23ζR
ηtt =
[1 +
T3T2c2
2ζN + w23ζR
2 (h1 − h3)
]−1
(A.16)
Alternatively, the total-to-static stage efficiency, ηts, is used when the stage exit velocity
is not recovered, such as in the final stage of a multistage turbine. This efficiency is defined
as
ηts =h01 − h03
h01 − h3ss(A.17)
Following the same line of assumptions as for ηtt, the following derivation can be performed,
ηts =h1 − h3
h1 − h3ss + 12c
21
ηts =h1 − h3
(h1 − h3) + (h3 − h3s) + (h3s − h3ss) + 12c
21
61
ηts =
[1 +
T3T2c2
2ζN + w23ζR + c2
1
2 (h1 − h3)
]−1
(A.18)
As stated above, Shaw [28] gave a similar derivation except that he did not assume equal
stage inlet and exit velocities. As a result, his expressions for total-to-total and total-to-
static efficiencies can be used for any stage in the turbine. Instead of using the enthalpy
ratio presented in Equation A.14, he expresses the entropy relationship as
h3s − h3ss
T3s
∼= (s3s − s3ss) = (s2 − s2s) ∼=h2 − h2s
T2(A.19)
Using the nomenclature of the stage velocity diagram of Fig. 3.3 and assuming there is no
work done in the nozzle we can express the specific work of Equation A.10 as
W =W
m= (h01 − h03) = U (cθ2 + cθ3) (A.20)
Using the specific work, the denominator of the original total-to-total expression (Equa-
tion A.11) becomes
(h01 − h03ss) = W + (h3 − h3s) +T3s
T2(h2 − h2s) (A.21)
Using the enthalpy loss coefficients of Equations A.15, Shaw’s final expression for ηtt is
ηtt =W
W + ζR(w2
3/2)
+ T3sT2ζN(c2
2/2) (A.22)
In order to take advantage of the blade loading coefficient and velocity ratios, Shaw divided
all terms by U2 and assumed T3s = T2 to give
ηtt =W/U2
W/U2 + (ζR/2) (w3/U)2 + (ζN/2) (c2/U)2 (A.23)
Shaw uses the following expression for total-to-static efficiency
ηts =h01 − h03
h01 − h03ss + c23ss/2
(A.24)
and assumes c3ss = c3 to derive
ηts =W/U2
W/U2 + (ζR/2) (w3/U)2 + (ζN/2) (c2/U)2 + 12 (c3/U)2 (A.25)
As a reminder, the blade loading coefficient is
W
U2=(cθ2U
+cθ3U
)(A.26)
62
The challenge now lies in determining the enthalpy-loss coefficients ζN and ζR for the op-
erating range of interest for a particular turbine. Some methods are layed out in Section 3.4.
The velocity values U , c2, w3, and c3 are determined during the preliminary turbine design
stages.
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