Gas Turbine Parameter Corrections 3030410757, 9783030410759

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Allan J. Volponi

Gas Turbine Parameter Corrections

Gas Turbine Parameter Corrections

Allan J. Volponi

Gas Turbine Parameter Corrections

Allan J. Volponi Pratt & Whitney West Simsbury, CT, USA

ISBN 978-3-030-41075-9 ISBN 978-3-030-41076-6 https://doi.org/10.1007/978-3-030-41076-6

(eBook)

© Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to the late Louis A. Urban, my long-time friend, colleague, and mentor, who introduced me to my first corrected parameter.

Preface

Parameter corrections are a familiar and ubiquitous practice in describing interrelationships within the gas turbine engine’s flow path. A frequent usage is to remove dependency on engine inlet ambient conditions (primarily temperature and pressure) to allow succinct relationships to be developed and employed in a variety of applications. Nevertheless, the genesis of these corrections, and in particular, how they can be derived from simple principles, is not general knowledge among many of those who use them on a day-to-day basis. This book is intended to help fill that gap. With the exception of a single chapter on humidity corrections, this text concerns itself with what is commonly referred to as Standard Day corrections of gas turbine gas path parameters. This type of correction has been in use for over five decades, and as such is common currency in the engineering community engaged in a variety of activities, most notably, engine performance specifications and calculations, control law development, modeling and engine diagnostics, prognostics and health management, to name a few. Due to its longevity and common knowledge across a diverse gas turbine engineering community, it is not the intent of this work to explore the many uses of these familiar corrections, but rather to explore their history, provide an exhaustive catalog of common (and some not so common) corrections, and most importantly to provide a mathematical framework for the derivation of these important normalization factors. Little reference to application will be made. A large part of this work will be devoted to the derivation of these corrections with the intent that this will serve as a handy desk reference to the engineering practitioners utilizing these corrections in their everyday activities. West Simsbury, CT, USA

Allan J. Volponi

vii

Abstract

This book concerns itself with what is commonly referred to as Standard Day corrections of gas turbine gas path parameters. Due to its longevity and common knowledge across a diverse gas turbine engineering community, it is not the intent of this work to explore the uses of these familiar corrections, but rather to explore their history, provide an exhaustive catalog of common corrections, and most importantly to provide a mathematical framework for the derivation of these important normalization factors. Little reference to application will be made. We include here, a short Preface and Introduction describing the layout of the book. Keywords Introduction · Structure · Chapters · Description

ix

Introduction

It is well known that, at a given operating point, gas turbine gas path parameters such as inter-stage pressures, temperatures, and flows will vary with ambient conditions. As such, it is common practice to normalize these parameters for ambient temperature, pressure (and to a lesser degree, humidity) before performance calculations are preformed, especially in cases where performance is to be trended over a long period of time when it is likely that these conditions will vary. The normalization methodology is commonly referred to as Standard Day corrections. Like many engineering practices, it is not a (mathematically) precise method, but rather an approximation that is sufficient for the purpose under which it is used, which is primarily to remove the variation induced solely by the ambient environment in which the gas turbine operates. These corrections are pervasive, in the sense that they have been applied in virtually every gas turbine engineering activity ranging from design and development of components and sub-systems, control strategies and schedules, performance calculations as well as engine monitoring, trending, and health management activities. In short, their existence and routine application are well known and well utilized [1–4]. As such, we will not venture into the broad field of applications exploiting these corrections. The purpose of this book is to establish a (quasimathematical) framework from which to establish these corrections and explore certain interrelationships. The use of the term, quasi-mathematical, is intentional in that the formal mathematical manipulations that will be used in our discussion arise from certain stated relationships, some thermodynamic, some not, that will be assumed to be the axioms from which our results will be derived. We will come back to this point several times throughout this presentation. With the increased use of computer simulations, the need for such correction calculations has decreased in recent time over what was needed decades ago when analytical methods prevailed and formed the only path to explore performance assessment and manipulate data. Nevertheless, the use of parameter corrections is still an important consideration especially in real-time engine modeling, control scheduling and strategy, and by researchers and practitioners who do not possess the specialized, and often proprietary, engine simulations that are the property of xi

xii

Introduction

engine designers and manufacturers. For these reasons, knowledge of parameter corrections is still an important, if not vital, factor in supporting engineering research and practice. Reference to corrected parameters and their usage can be found from many sources, books, papers and the like, albeit incompletely. This book seeks to put in one place a compendium of all known corrections (and some perhaps not so well known) as well as to provide a mathematical foundation for their derivation. The presentation begins in Chap. 1 with an historical perspective containing a short summary of dimensional analysis and the use of Buckingham’s π Theorem as a motivation for how corrected parameters originally evolved. This methodology is a bit tedious and not well suited for deriving parameters of interest, especially those involving time derivatives. It is, however, the genesis of what we know today as Standard Day corrections and is included for completeness. Chapter 2 introduces a more general mathematical foundation to overcome some of the shortcomings of pure dimensional analysis and stipulates a handful of intuitive axioms (assumptions) to allow derivations to ensue armed with only an elementary knowledge of thermodynamics and Newton’s calculus. Using these principles, Chap. 3 derives the corrections for most common gas path parameters with Chap. 4 extending to the time derivatives of those parameters. Chapter 5 explores some additional corrections that follow easily from the derivations of Chaps. 3 and 4. In Chap. 6, the assumption of constant specific heats is relaxed to refine the corrections to several of the most common gas path parameters. The tone of the text changes in Chap. 7 with the introduction of empirical methods for further refinement, and we conclude our derivations in Chap. 8 with a discussion of the effects of humidity on performance parameters. The Appendix summarizes all of the corrections derived in this book. Lastly, a few comments on symbolic notation. The symbol ()) appears throughout the text in many of the derivations. This symbol means which implies or implying. Since it is understood that the corrections are by their nature an approximation, we will use the symbol approximately equal () interchangeably with mathematically equal (¼) throughout our presentation as a matter of convenience.

Contents

1

Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3

Common Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Corrected Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Corrected Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Corrected Air Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Corrected Fuel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Corrected Net Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Corrected Rotational Speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Corrected Horsepower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Corrected Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Corrected Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Corrected Metal Temperature Rate . . . . . . . . . . . . . . . . . . . . . . . 3.11 Corrected Fuel Air Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Thrust Specific Fuel Consumption . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

15 16 18 19 20 21 23 24 25 25 25 27 28

4

Time Derivative Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Correction of Horsepower and Acceleration Rate of Change . . . . 4.2 Correction of Torque Rate of Change . . . . . . . . . . . . . . . . . . . . . 4.3 Correction of Temperature Rate of Change . . . . . . . . . . . . . . . . . 4.4 Correction of Pressure Rate of Change . . . . . . . . . . . . . . . . . . . . 4.5 Correction of Air Flow Rate of Change . . . . . . . . . . . . . . . . . . . 4.6 Correction of Fuel Flow Rate of Change . . . . . . . . . . . . . . . . . . 4.7 Correction of Net Thrust Rate of Change . . . . . . . . . . . . . . . . . . 4.8 Correction of Fuel Air Ratio (FAR) Rate of Change . . . . . . . . . . 4.9 Correction of Thrust Specific Fuel Consumption (TSFC) Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Summary of Rate of Change Corrections . . . . . . . . . . . . . . . . . .

. . . . . . . . .

29 30 32 33 34 35 36 37 38

. 39 . 39

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Contents

5

Additional Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Correction of Time Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Correction of Metal Temperature . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Correction of Static Temperature . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

41 41 42 44

6

Refinements to Common Corrections . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Speed Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Pressure Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Airflow Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Horsepower Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Torque Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Acceleration Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Fuel Flow Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

47 48 50 53 55 58 58 59

7

Empirical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

8

Humidity Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Spool Speed Humidity Correction . . . . . . . . . . . . . . . . . . . . . . . 8.3 Air Flow Humidity Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 ΔTemperature Humidity Correction . . . . . . . . . . . . . . . . . . . . . . 8.4.1 ΔTotal  Static Temperature Correction . . . . . . . . . . . . . 8.4.2 Δ Enthalpy Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Δ Temperature Correction . . . . . . . . . . . . . . . . . . . . . . . 8.5 Horsepower Humidity Correction . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Torque Humidity Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Acceleration Humidity Correction . . . . . . . . . . . . . . . . . . . . . . . 8.8 Fuel Flow Humidity Correction . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Thrust Humidity Correction (Turbojet) . . . . . . . . . . . . . . . . . . . . 8.10 Pressure Humidity Correction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Temperature Humidity Correction . . . . . . . . . . . . . . . . . . . . . . . 8.12 Numerical Quantification of Humidity Effects . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

71 71 75 75 77 77 78 78 79 80 80 81 81 83 84 85

Appendix: Summary of Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Nomenclature

Symbol A αX cp cpa, d cpa, w cpw cv δ EGT EPR ηb FAR Fn g h HP γ I ISA J LHV Mn μX N N_ P

Area Theta (θ) exponent for gas path parameter X Specific heat at constant pressure Specific heat of dry air at constant pressure (same as cp) Specific heat of wet air at constant pressure Specific heat of water vapor at constant pressure Specific heat at constant volume Pressure ratio P/PISA Exhaust gas temperature Engine pressure ratio Burner efficiency Fuel–air ratio Net thrust Gravitational acceleration constant Enthalpy Horse power Ratio of specific heats Moment of inertia International Standard Atmosphere Mechanical equivalent of heat Fuel lower heating value Mach number Humidity correction factor for gas path parameter X Rotor speed Rotor acceleration Pressure (total) xv

xvi

Ps PISA PLA Q R Ra, d Ra, w Rw r ρ SH TSFC T Ts Tm TISA τ Θ S v V wa wf war

Nomenclature

Pressure (static) Sea level ISA pressure Power lever angle Torque Gas constant Gas constant of dry air (same as R) Gas constant of wet air Gas constant of water vapor Radius Density Specific humidity Thrust specific fuel consumption Gas temperature (total) Gas temperature (static) Metal temperature Sea level ISA temperature Time constant Temperature ratio T/TISA Entropy Velocity Volume Air mass flow Fuel mass flow Water to air ratio

Subscript a c f d w s m HD

Air Corrected (parameter) Fuel Dry Wet Static Metal property Hot day

Chapter 1

Historical Perspective

Abstract This chapter provides some historical perspective contributing to the use of gas path parameter corrections. It largely involves the use of dimensional analysis and the so-called π theorem due to Edgar Buckingham, an American physicist working at the U.S. National Bureau of Standards in the early twentieth century. This chapter briefly describes the π theorem principle and illustrates the concept with several examples. The reduction in complexity obtained by applying this principle to yield equivalent functional relationships involving new variables (that are rational functions of the fundamental variables) was the driving force that ultimately formed the parameter correction process in common use today. Keywords Buckingham · π-theorem · Dimensional analysis · Historical · Procedure · Complexity reduction · Dimensionless products · Examples

There is an important historical perspective that originally gave rise to the type of corrections we will be considering that bears consideration. It is sometimes referred to as dimensional analysis1 and has its roots in the early part of the twentieth century with the work of Buckingham [5]. We will begin with a brief description of his work as a precursor to our formal development.2 Before the advent of digital computers, engineering calculations were tediously performed by hand with the use of simple devices such as slide rules and the like. Under these conditions, it is advantageous (for time and effort) to reduce the complexity of any calculation that needed to be performed. Reducing the number of independent variables in physical interrelationships is one way to reduce complexity. This is at the core of Buckingham’s work and resulted in a relationship that has come to be known as Buckingham’s π theorem. The following description is taken largely from [6].

1

See [15]. The interested reader may wish to consult [13, 14] for the use of non-dimensional parameters in gas turbine propulsion and thermodynamics. 2

© Springer Nature Switzerland AG 2020 A. J. Volponi, Gas Turbine Parameter Corrections, https://doi.org/10.1007/978-3-030-41076-6_1

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2

1 Historical Perspective

Let us assume that we have a physical relationship to be represented by n physical quantities Q1, Q2, . . ., Qn. Without any loss of generality, we may write this relationship as f 1 ðQ1 , Q2 , . . . , Qn Þ ¼ 0

ð1:1Þ

Since physical quantities are measured in physical units (feet, seconds, lbs, degrees, etc.) or some algebraic combination of units, it can be shown that the above general relationship must have the form X

MQb11 Qb22 . . . Qbnn ¼ 0

ð1:2Þ

where M is a dimensionless number and each of the products being summed must have the same units (otherwise you could not add them together). If we divide both sides by any one term in the sum, the expression takes the form X

NQa11 Qa22 . . . Qann þ 1 ¼ 0

ð1:3Þ

where each product in the sum is a dimensionless quantity. If we represent each of these terms by πi, then we may rewrite this equation as f 2 ðπ 1 , π 2 , . . . , π n Þ ¼ 0

ð1:4Þ

Loosely stated, the Buckingham π Theorem declares that the number of dimensionless products πi necessary for Eq. (1.4) to represent Eq. (1.1) is n  k, where k is the number of fundamental dimensions used to define the variables. To illustrate this principle, let us consider the problem of determining the distance travelled by a free falling object (in a vacuum) and derive a relationship for this distance (without the aid of Mr. Newton’s calculus) using dimensional analysis and Buckingham’s π Theorem. We will assume that the variables under consideration for this problem are given in the Table below, where the units are length (L), mass (M), and time (T). Parameter S W g t

Description Distance traveled Weight of object Gravitational constant Time

Units L M L T2 L T2 T

We will assume the following general relationship holds: f ðS, W, g, t Þ ¼ 0

ð1:5Þ

1 Historical Perspective

3

We have four parameters and three unique fundamental dimensions (L, M, T). By Buckingham’s theorem, we have 4  3 ¼ 1 π terms as follows: π 1 ¼ KSa W b t c gd )     M 0 L0 T 0 ¼ ðLa Þ M b Lb T 2b ðT c Þ Ld T 2d ) ¼ Laþbþd M b T c2ðbþdÞ ) b¼0

ð1:6Þ

aþbþd ¼0 c  2b  2d ¼ 0 From the last three equations, we find that a ¼ d c þ 2a ¼ 0 ) a ¼ c=2

ð1:7Þ

Since g is a constant, we can take d ¼ 1,3 and then we may conclude that a ¼ 1 b¼0

ð1:8Þ

c ¼ 2a ¼ 2 ) π 1 ¼ KS1 W 0 t 2 g ¼ K

2

gt S

Now since π 1 is dimensionless, it must be a constant (as is K ), therefore S¼

  K gt 2 π1

ð1:9Þ

Experimentation would suggest that the constant K/π 1 ¼ 1/2, providing the wellknown relationship from Physics 101 without the aid of Calculus. An interesting side note, is that the dimensional analysis suggests that the mass (M) of the object has no bearing on the interrelationship of the freely falling body, a fact that was demonstrated experimentally and dramatically by Galileo in Pisa, circa 1590, and later by the American astronaut Col. David R Scott on the lunar surface in 1971! So how does this help us with correcting gas turbine parameters? We will illustrate this with another simple example from [2]. Suppose that we have a single spool turbojet engine having the following gas path parameters:

3 This choice is arbitrary since any value will still provide the same result. The reader can easily verify that choosing d ¼ 2, for instance, will result in the same relationship noted in Eq. (1.9).

4

1 Historical Perspective

Parameter N T1 P1 v D R Fn

Description Spool speed (rpm) Ambient temperature Ambient pressure Aircraft velocity Compressor diameter Gas constant Net Thrust

Units 1/T deg M/(LT2) L/T L L2/(T2deg) M L/T2

where T ¼ Time, deg ¼ temperature (degrees), M ¼ Mass, and L ¼ Length. The total number of fundamental dimensions in this case is 4. We assume that we have the following relationship: Fn ¼ f ðN, T 2 , P2 , v, D, RÞ

ð1:10Þ

This implies an implicit form f ðFn, N, T 2 , P2 , v, D, RÞ ¼ 0

ð1:11Þ

involving seven quantities. By virtue of the π-theorem, we may express the relationship in terms of 7 – 4 ¼ 3 dimensionless quantities as follows: ϕðπ 1 , π 2 , π 3 Þ ¼ 0 where π 1 ¼ NPa2 T b2 Dc Rd , π 2 ¼ vPe2 T f2 Dg Rh ,

ð1:12Þ

n r π 3 ¼ F n Pk2 T m 2D R

The πi terms are dimensionless and hence their units must be equal to L0M0deg0T0. For example a  2 d   1 M L b c deg L ¼ L0 M 0 deg0 T 0 T LT 2 T 2 deg ¼ Lcþ2da M a degbd T 12a2d

π1 ¼

which implies 0¼a

ð1:13Þ

0 ¼ c þ 2d  a 0¼bd 0 ¼ 1 þ 2a þ 2d which implies

a¼0 b ¼ d ¼ 1=2 c ¼ 1

This yields a parameter of the form

1 Historical Perspective

5

 pffiffiffiffiffiffiffiffi N= L RT 2

ð1:14Þ

Repeating this process for the other dimensionless quantities, yields the following relationship   F N v ¼ f 1 pffiffiffiffiffiffiffiffi , pffiffiffiffiffiffiffiffi P2 A D RT 2 RT 2

ð1:15Þ

If we assume a constant geometry (A ¼ const, D ¼ const) we obtain     F N F N ¼ f 2 pffiffiffiffiffi , Mn ) ¼ f 3 pffiffiffi , Mn P1 δ T2 θ

ð1:16Þ

This would imply that the appropriate (and well known) corrections for net thrust pffiffiffi and spool speed are F/δ and N= θ respectively. Now strictly speaking, we have only demonstrated these relationships for a single spool turbojet engine with constant geometry and thus would have to repeat the type of dimensional analysis for the specific engine under consideration (if different). What is needed is a more general approach to demonstrate that these corrections can be applied in general to any given gas turbine configuration. With this in mind, we will offer a more general definition of what a corrected parameter should represent and derive a general form for the correction.

Chapter 2

Mathematical Framework

Abstract This chapter explores the construction of a mathematical framework to enable the direct derivation of Standard Day corrections for an arbitrary gas path parameter. The fundamental approach is in contrast with the historical methodology of dimensional analysis (briefly) mentioned in Chap. 1. This is accomplished by establishing a formal definition of a corrected parameter along with a short list of properties that corrected parameters are assumed to possess. These collectively form a set of axioms for our mathematical system from which formal derivations can ensue, using only fundamental thermodynamic principles and elementary calculus. This framework will provide the basis for the discussions contained in all subsequent chapters of this book except for Chap. 8, which deals exclusively with the effects of humidity on performance. Keywords Corrected parameter · Properties · Definition · Axioms · Standard Day Correction · Mathematical framework · Notation · Delta · Theta · Engine stations · Hot Day · Reference conditions · Assumptions · Specific heat · Local Mach No. · Corrections of parameter square root · Corrections of parameter raised to arbitrary power

In this chapter we will develop a mathematical framework to allow derivation of any corrected gas path parameter independent of gas turbine engine type or configuration.1 To accomplish this we will introduce some terminology and propose a set of fundamental axioms that will be utilized in the derivations appearing throughout the remaining chapters of this book. For any gas path parameter X, the equivalent corrected parameter will be denoted by Xc throughout this discussion. In general, a change in the engine inlet conditions T2 (inlet total temperature) and P2 (inlet total pressure) will be accompanied by an attendant change in any downstream gas path parameter X. A corrected parameter Xc would be constant regardless of the change in inlet condition and represents the

1

We will assume fixed geometry.

© Springer Nature Switzerland AG 2020 A. J. Volponi, Gas Turbine Parameter Corrections, https://doi.org/10.1007/978-3-030-41076-6_2

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2 Mathematical Framework

value the parameter X would have at some fixed reference inlet condition. This reference condition can be whatever you like, however, it is common practice to select the International Standard Atmosphere (ISA) standard day conditions (T2 ¼ 288.15 deg K, P2 ¼ 101325.353 Pa, or alternatively, T2 ¼ 459.67 deg R, P2 ¼ 14.696 psia) for this purpose. Thus, we can assume without loss of generality that a corrected parameter quantity Xc is one that satisfies the following implicit relationship. Axiom 1 :

X ¼ f ðX c , T 2 , P 2 Þ

ð2:1Þ

The interpretation of Eq. (2.1) requires some explanation. Its intent is that a gas path parameter X will vary from its corrected form Xc only on account of changes in ambient condition as given by the inlet temperature and pressure. Implicit in this statement is that something is being held constant. For example, if power condition were to change (e.g., through a change in Power Lever Angle, PLA), then that would certainly affect X as well. Indeed, so would Xc. So in some sense, we are assuming from this implicit definition that the state of the engine has been held fixed. But what does this mean? Whatever it is, it needs to be independent of the engine control system, otherwise corrected parameters would not be independent of engine model, configuration and method of control. So we can rule out things like Engine Pressure Ratio (EPR) or Thrust, etc. What is certain is that we need for the thermodynamic and aerodynamic condition of the engine to be fixed. Since the gas turbine is an air breathing machine and air is the elemental fluid whose properties are being measured by the gas path parameters that we wish to normalize (i.e., correct), we will assume that local Mach numbers along the gas path have remained fixed. Consider, for example, Airflow (wa), which can be seen to be solely a function of geometry (fixed), Mach number, and (Total) Temperature and Pressure, i.e., pffiffiffiffiffi rffiffiffi wa T 2 γ ¼ R AP2

Mn 1þ

¼ f ðT 2 , P2 Þ

γ1 2 2 Mn

¼ f ðM n Þ ) wa ¼ f ðM n , T 2 , P2 Þ 2ðγþ1 γ1Þ ð2:2Þ

With Mn being held fixed, airflow will be governed only by total temperature (T2) and pressure (P2) as needed. Similar arguments can be given for other gas path parameters. Thus it would appear that holding local Mach numbers constant to be a reasonable condition to preserve a fixed engine state. We will apply this assumption liberally throughout this discussion and make other simplifying approximations (e.g., averaging) where needed. Since the corrections we are deriving should only be considered approximations themselves and not precise physical quantities, we can afford ourselves these conveniences. Equation (2.1) will be our working definition for what a corrected quantity represents. Through formal differentiation, it follows that

2 Mathematical Framework

 dX ¼

∂X ∂T 2

9



 P2 ¼ const X c ¼ const

dT 2 þ

∂X ∂P2



 dP2 þ

T 2 ¼ const X c ¼ const

∂X ∂X c

 P2 ¼ const T 2 ¼ const

dX c implying

      ∂X=X ∂X=X ∂X=X dX dT 2 dP2 dX c ¼ þ þ X ∂T 2 =T 2 P2 ¼const T 2 ∂P2 =P2 T 2 ¼ const P2 ∂X c =X c P2 ¼const X c X c ¼ const

X c ¼ const

T 2 ¼ const

If we make the (brash) assumption that the partials are approximately constant (the third is clearly unity by definition) we have dX dT dP dX dX dX dT dP a 2þb 2þ c ) c ¼ a 2b 2 ) X X T2 P2 Xc Xc T2 P2 dX c dX dθ dδ a b )  X θ δ Xc ln ðX c Þ ¼ ln ðX Þ  ln ðθÞa  ln ðδÞb )

Xc 

X θ a δb

where

θ¼

T2 , T ISA

δ¼

P2 PISA

ð2:3Þ

where T ISA ¼ 518:67,

PISA ¼ 14:696 or

ðT ISA ¼ 288:15,

PISA ¼ 101325:353Þ

Equation (2.3) yields a familiar form and provides a simple formula from which to calculate an approximate standard day parameter correction. We show here θ and δ for both British units of measure as well as for SI units, however it clearly does not matter which units are used as long as numerator and denominator have the same units. Xc provides an approximate value for the parameter X that would be seen, under the same operating state but on a standard day atmospheric condition, i.e. TISA temperature and PISA pressure. Of course, the reference condition can be something other than ISA conditions, e.g., a hot tropical day, for example, if we desired to approximate, say Exhaust Gas Temperature (EGT), at a worse case condition. The reference condition can also refer to an engine station (location) other than the inlet (station 2) for various applications. For example, we might be interested in correcting quantities local to an engine module, (e.g. the High Pressure Compressor, HPC, in a twin spool engine), to the engine station representing the inlet of the module under consideration, (e.g., station 2.5 for the HPC). There are various notations that are common in the industry to denote corrected parameters that refer to particular stations to which they are being corrected, however (unfortunately) there also appears that there is no universal notational convention. For example, one

10

2 Mathematical Framework

popular notation is to use the subscript “c” followed by the station number to denote a corrected parameter and the reference condition. If we were considering the high spool speed N2 corrected to the inlet (station 2) we would denote its corrected form as N2c2. If we were correcting this quantity to the inlet of the HPC (station 2.5), we would use the notation, N2c25. Relative to Eq. (2.3), we would carry over the station identifier to the parameters θ and δ as well. For example, if we consider the classical correction for spool speed N2 ffi, N 2c2  pffiffiffiffi θ2

N2 N 2c25  pffiffiffiffiffiffi θ25

where

θ2 

T2 T ISA

and

θ25 

T 25 T ISA

ð2:4Þ

Within this document, unless otherwise stated (explicitly), we will assume that the reference location is station 2, the inlet of the engine. Thus, within the context of this document, the appearance of θ without an explicit subscript will mean θ 2. Thus, a corrected parameter maintains the same units and is calculated by means of theta and delta exponent corrections. We might mention at this juncture, that Eq. (2.3) is not only approximate for the reasons already alluded to, but in addition, some gas path parameters such as fuel flow (wf), for example, are also (typically) corrected for humidity and fuel heating value. The impact of changes in viscosity with altitude (Reynolds effects) or changes in the gas composition and their impact on gas path parameters, for the most part, will not be considered in this presentation. Humidity corrections will be addressed in Chap. 8. The values for a and b in Eq. (2.3) will, in general, vary with engine type and cycle,2 however, there are some values which might be considered standard common corrections and are approximations which are commonly used in practice for all gas turbines. Chapter 3 will address these and derive many of the common gas path parameters and their classical standard day corrections. Before leaving this topic, we might add that there are circumstances when the reference inlet condition might be something other than standard day conditions. This was alluded to above in the discussion regarding EGT. If we defined the reference condition to be a hot tropical day temperature, say 35  C (THD), we could calculate what EGT would be at that condition (EGTHD) if we had at our disposal a measurement of EGT taken at ambient condition T2 as follows by temperature extrapolation. EGT HD ¼

EGT θ θ2 HD

where θ2 

T2 T ISA

and

θHD 

T HD T ISA

ð2:5Þ

Next, we consider a few more (reasonable) axioms that we will assume for corrected parameters. Given the relationship in Eq. (2.3) above, it would be desirable if certain arithmetic relationships hold in corrected parameter space, in particular,

2

This will be partially addressed in Chap. 7 on Empirical Methods.

2 Mathematical Framework

11

that the corrected value of a sum is the sum of corrected values and likewise that the corrected value of a product is the product of corrected values. In symbols: Axiom 2 :

Z ¼ X þ Y ) Zc ¼ Xc þ Y c

Axiom 3 :

Z ¼ XY ) Z c ¼ X c Y c

ð2:6Þ

These are very sensible assumptions to add as axioms in that if a physical relationship exists as simple as a sum or product between gas path parameters the same relationship should hold as well at standard day conditions. By the same reasoning, we can generalize to any rational function, i.e., if g is a rational function (i.e., consists of sums, products and powers), then Axiom 4 :

Z ¼ gðX 1 , X 2 , . . . , X n Þ ) Z c ¼ gðX 1c , X 2c , . . . , X nc Þ

ð2:7Þ

The axioms in Eqs. (2.6) and (2.7) represent an additional constraints since they are not directly derivable (through mathematical manipulation) from our implicit definition (Eq. (2.1)) of a corrected parameter. They are, however, reasonable constraints as far as applying it to thermodynamic gas turbine parameters. We invoke Buckingham’s argument in their defense. Since physical quantities have units of measure, their interrelationships must be rational functions in order to maintain the suitableness and consistency of any resulting unit of measure, i.e. they must have the form of Eq. (1.2). What Eq. (2.7) (and Eq. 2.6 as a special case) dictate is that if there exists a physical relationship between parameter Z and parameters X1, X2, . . ., Xn, (demanded by physics) then the same interrelationship must also hold at any ambient environment and hence at any specific reference environment such as standard day conditions. This is a very reasonable assumption to add since our corrected quantities must have physical meaning beyond just an abstract mathematical formulation. Equations 2.1, 2.6, and 2.7 (combined in Eq. 2.8 below) are sufficient to allow us to derive reasonable approximations (in view of Eq. 2.3) for most gas turbine gas path parameter corrections. Corrected Parameter Axioms X ¼ f ðX c , T 2 , P2 Þ ðlocal Mach constantÞ Z ¼ X þ Y ) Zc ¼ Xc þ Y c Z ¼ XY ) Z c ¼ X c Y c

ð2:8Þ

Z ¼ gðX 1 , X 2 , . . . , X n Þ ) Z c ¼ gðX 1c , X 2c , . . . , X nc Þ for any rational function g Along with these basic assumptions we will make other simplifying assumptions, in particular, the assumption that specific heats are constant. This will aid in the derivation of the classical corrections and will be stated clearly at the onset of a specific derivation when it is used. In Chap. 6 this assumption will be relaxed in order to explore possible refinements in the correction. For the remainder of this

12

2 Mathematical Framework

Fig. 2.1 Typical Gas Path Parameter Locations

book we will, for the most part, use a twin spool turbofan engine as the example application. For the most part, its usage will be primarily to designate engine station numbers that will be referenced in the gas path parameter symbols used throughout the text. A typical configuration3 is depicted in Fig. 2.1 above. Throughout this discourse, we will use the subscript “c” to denote a corrected parameter (as you see in Eq. 2.8 above). We end this Chapter with a couple of observations that follow from the axioms that will be used in the derivations in subsequent Chapters. The first (which may be obvious to some) is that the correction of the square root of a parameter is the square root of the corrected parameter, or in symbols pffiffiffiffi pffiffiffiffiffi X ¼ Xc ð2:9Þ c

This fact follows from axiom 3 through formal manipulation. Consider   ðXY Þc ¼ X c Y c ) X 2 c ¼ X c X c Now let Z ¼ Therefore

pffiffiffiffi X then Z2 ¼ X. pffiffiffiffi pffiffiffiffi   X Xc ¼ Z2 c ¼ ZcZc ¼ X c

3

c

Reprinted with permission from the National Aeronautics and Space Administration.

2 Mathematical Framework

This implies that

13

pffiffiffiffi pffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi X ) X Xc Xc ¼ Xc ¼ c c p ffiffiffi ffi   pffiffiffiffiffi X X pffiffiffiffic ¼ pffiffiffiffiffi c Xc X c 1 ¼ pffiffiffiffiffi ffi X c=ðpffiffi XÞ

c

This final equation is of the form A¼

1 ) A2 ¼ 1 ) A ¼ 1 A

pffiffiffiffi pffiffiffiffiffi Since X c and X c are, in our context, gas path physical quantities, A must be 1, thereby proving pffiffiffiffi pffiffiffiffiffi X ¼ Xc c

An extension of this result would consider a general exponent power of the parameter, i.e. Xβ for an arbitrary real valued exponent, β. By a similar argument, as used above for the case β ¼ 0.5, we can show 

Xβ

 c

¼ X βc

ð2:10Þ

We begin by considering     Y ¼ X β ¼ X β1 X ) Y c ¼ X β c ¼ X β1 c X c Therefore

 β X X βc Xβ X β1 c  ¼ X c ¼  β1 c )  βc ¼  β1 β1 Xc X X X c c c

Now since this last relationship must hold for arbitrary β, then these ratios must be constant, i.e. 

X βc X β1 c    ¼ const ¼ Xβ c X β1 c

Since the relationship is valid for all real β, it must be valid for β ¼ 1 which implies that the value of const ¼ 1, thereby providing the solution seen in Eq. (2.10).

Chapter 3

Common Corrections

Abstract This chapter provides the Standard Day corrections for the most common gas path parameters. These corrections are sometimes referred to as classical corrections in that, the θ and δ exponent values that are derived, are the well-recognized values that appear in the literature. The parameters considered include, Rotor Speed, Temperature, Pressure, Air Flow, Fuel Flow, Horsepower, Torque, Acceleration, Thrust, Metal Temperature Rate, Fuel Air Ratio (FAR), and Thrust Specific Fuel Consumption (TSFC). The corrections are derived in detail from the definitions, axioms, and assumptions that were proposed in Chap. 2 along with simple thermodynamic relationships relevant to the parameter under consideration. In order to avoid any circular reasoning, parameters are derived in a particular order, wherein only results from already derived parameters are utilized in a given derivation. Keywords Classical corrections · Rotor Speed · Temperature · Pressure · Air flow · Fuel Flow · Horsepower · Torque · Acceleration · Thrust · Metal Temperature Rate · Fuel Air Ratio (FAR) · Thrust Specific Fuel Consumption (TSFC) · Theta · Delta · International Standard Atmosphere · Specific heat · Local Mach No. · Ratio of specific heats

In this Chapter, we will attempt to supply some rationale for the corrections of commonly used gas path parameters. Most will be motivated from simple thermodynamic relationships and simplifying assumptions and do not require an extensive knowledge of either thermodynamics or gas turbine operation. Where applicable we will attempt to supply the required definitions. The classical parameter corrections for inter-stage temperatures, pressures, flow, force and power will be derived (in approximation) in the form of Eq. (2.2), i.e., Xc


X θa δb

where

θ¼

T2 , T ISA

δ¼

P2 PISA

using only the axioms stated in Chap. 2. Other simplifying hypotheses such as assuming ideal gas properties, constant specific heats, etc. are also invoked in the © Springer Nature Switzerland AG 2020 A. J. Volponi, Gas Turbine Parameter Corrections, https://doi.org/10.1007/978-3-030-41076-6_3

15

16

3 Common Corrections

Table 3.1 Common gas turbine parameter corrections Parameter Rotor Speed

Symbol N

a 0.5

b 0

Airflow

wa

0.5

1

Fuel Flow

wf

0.5

1

Corrected parameter N c ¼ pNffiffiθ pffiffi wac ¼ waδ θ w w wfc ¼ θafδ ¼ pffiffif

Thrust

Fn

0

1

F nc ¼ Fn δ

Horse Power

HP

0.5

1

ffiffi HPc ¼ pHP θδ

Torque

Q

0

1

Qc ¼ Qδ

Temperature

T

1

0

T c ¼ θTa ¼ Tθ

Pressure

0

1

Pc ¼ Pδ

Acceleration

P N_

0

1

Metal Temp Rate

T_ m

0.74

0.8

θδ

_ N_ c ¼ Nδ

T_ m T_ mc ¼ θ0:74 δ0:8

derivations to arrive at the classical and familiar standard day corrections. Parameters, and their corrections, that are addressed in this Chapter are summarized above in Table 3.1. The most extensively used and perhaps the simplest correction of all is the correction for temperatures, and is the first one which we shall consider.

3.1

Corrected Temperature

As a general rule, it seems reasonable that engine inlet temperature (T2 ) should have a direct effect on downstream temperatures in the engine’s gas path. It seems natural, for example, that for a given operating condition, increasing T2 would in turn increase T2.5, which in turn would increase T3, etc. In order to motivate this concept, we need to consider the T-S diagram. To be definite, let us consider the compression induced by the Fan (inner diameter) and the Low Pressure Compressor (LPC) from station 2 to station 2.5 of our sample engine appearing (figuratively) in Fig. 3.1. Without loss of generality, an isentropic compression has been assumed. Figure 3.1 depicts a compression from P2 to P2.5 starting at different (inlet) temperatures (T2 and T2*). At this point we will note two properties of the T-S diagram which we will state without proof. (The reader can derive these for him/herself, armed only with the elementary laws of thermodynamics.) The relationships are 1. The curves of constant pressure are monotonically increasing with entropy S 2. The constant pressure curves diverge from one another with increasing entropy S These properties suggest that the  quantities are larger in magnitude than their non-starred counterparts and that (T2.5  T2) < (T2.5*  T2*). What we will now show, however, is that the ratio of T2.5 over T2 is approximately equal to T2.5* over T2* !

3.1 Corrected Temperature

17

P T

2.5* 2.5

P 2*

2

S Fig. 3.1 Compression T-S diagram

To accomplish this we need one more relationship from basic thermodynamics (which follows from the definition of entropy and the perfect gas law). dh dP R T P

dS ¼

ð3:1Þ

where S, h, T, P, R denote entropy, enthalpy, temperature, pressure and the gas constant, respectively. Integrating Eq. (3.1) between states 2 and 2.5 and assuming constant specific heat, yields, Z2:5

Z2:5 Z2:5 dP dT dP 0¼ ¼ cp R dS ¼ P T P 2 2 2 2 2     T 2:5 P2:5 ¼ cp ln  R ln T2 P2 )  R=cp T 2:5 P2:5 ¼ T2 P2 Z2:5

dh R T

Z2:5

ð3:2Þ

Likewise, integrating between 2* and 2.5* with the same assumption on specific heats, provides T 2:5 * ¼ T 2*



¼ ) T 2:5 * ¼

P2:5 P2

R=cp

T 2:5 T2

T 2:5 T ¼ 2:5 θ ðT 2 =T 2 *Þ

ð3:3Þ

18

3 Common Corrections

as required. By using the same arguments from station 2.5 to 3, we have that T 3* T T * T * T * T3 T ¼ 3 ) 3 ¼ 2:5 ¼ 2 ) T 3 * ¼ ¼ 3 T 2:5 * T 2:5 T3 T 2:5 T2 θ ðT 2 =T 2 *Þ

ð3:4Þ

and so forth. Therefore, we have in general Tc ¼

T θ

ð3:5Þ

The next most common gas path parameter that immediately comes to mind is pressure, which we consider next.

3.2

Corrected Pressure

The pressure changes experienced at various stations throughout the engine’s gas-path are the effect of either compressions or expansions resulting from the action of the engine’s turbomachinery. For purposes of motivating the correction for pressure, we can consider without loss of generality, a compression process; the argument for expansion pressures would be similar. Let us consider, for example, the pressure P2.5 at the exit of the LPC. This pressure is related to temperature and pressure by virtue of the following relationship T 2:5 ¼ T2

 γ1 P2:5 γη P2

where η ¼ polytropic efficiency

ð3:6Þ

γ ¼ cp=cv ¼ ratio of specific heats Thus, since T2.5/θ ¼ constant implies T2.5/T2 ¼ constant, we have that P2.5/ P2 ¼ constant, which in turn implies that P2.5/δ is constant, which establishes the correction at station 2.5. Note, that implicit in this argument is that the ratio of specific heats γ and efficiency η do not change. Moving downstream to station 3 we have that   γη   γη T 3 =θ γ1 P3 T 3 γ1 ¼ ¼ ¼ const: P2:5 T 2:5 T 2:5 =θ therefore P3 P P ¼ 3 2:5 ¼ const δ P2:5 δ

ð3:7Þ

Thus, we proceed downstream and establish in general that P/δ ¼ constant as required. Pc ¼

P δ

ð3:8Þ

3.3 Corrected Air Flow

3.3

19

Corrected Air Flow

Consider air flow wa (in g/s) moving with uniform velocity v through a pipe having cross chapteral area A (m2). Then     wa ¼ ρAv ¼ density g=m3  area m2  velocityðm=sÞ

ð3:9Þ

For an ideal gas, we also know that Ps ¼ ρRT, from which it follows that wa P ¼ ρv ¼ s v A RT s

ð3:10Þ

where the subscripts s and t denote static and total quantities respectively. From the definition of Mach number Mn M n ¼ v=

pffiffiffiffiffiffiffiffiffiffi γRT s

ð3:11Þ

we obtain the relationship pffiffiffiffiffiffiffiffiffiffi wa Ps M n γRT s ¼ A RT s

pffiffiffiffiffi  rffiffiffiffiffi rffiffiffi wa T t Ps Tt γ M ¼ Pt Ts n R APt

)

ð3:12Þ

We know, however, that for an isentropic process Tt γ1 2 Mn ¼1þ 2 Ts

and

 γ Pt γ  1 2 γ1 Mn ¼ 1þ 2 Ps

ð3:13Þ

Substituting these terms into Eq. (3.12), we obtain pffiffiffiffiffi rffiffiffi   γþ1 wa T t γ γ  1 2 2ð1γÞ Mn 1 þ Mn ¼ R 2 APt

ð3:14Þ

Multiplying both sides of Eq. (3.14) by ( θ/Tt)1/2/(δ/Pt) to get the proper units (g/s) yields rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi  γþ1 γ ðT t =θÞ  wa θ A γ  1 2 2ð1γÞ Mn ¼ ¼ constant Mn 1 þ δ 2 R ðPt =δÞ

ð3:15Þ

for Mn ¼ constant, thus providing a corrected quantity.1 wa c

1

pffiffiffi wa θ ¼ δ

ð3:16Þ

T/θ and P/δ are corrected parameters and hence constant by virtue of the previous derivations.

20

3 Common Corrections

3.4

Corrected Fuel Flow

Consider the (simplified) energy equation bounding the combustor     BTU ¼ ηb qwf ¼ wa þ wf Δh ¼ wa þ wf ðh5  h4 Þ sec

ð3:17Þ

where h denotes enthalpy, q denotes the heating value of the fuel and η denotes the adiabatic efficiency of the combustor. If we assume that (at a given operating point) efficiency and fuel heating value are constant, then if we take logs of both sides of Eq. (3.17) and differentiate, we obtain   dwf d wa þ wf dðΔhÞ   ¼ þ Δh wf wa þ wf ! ! wf dwf wa dwa ð3:18Þ   ¼  þ  w wf wa þ wf wa þ wf a     cp3 T 3 dT 3 cp4 T 4 dT 4 þ  Δh T4 Δh T3 A little algebraic manipulation on Eq. (3.18) gives !       dwf cp3 T 4 dT 3 cp4 T 4 dT 4 wa wa dwa   ¼ þ  ð3:19Þ wa þ wf wf wa Δh T4 Δh T3 wa þ wf Now, at a given operating point, we can assume that the following relationships hold:

T3 θ

pffiffiffi wa θ ¼ const δ T4 and ¼ const θ P3 ¼ const δ

) ) )

dwa dT dP þ 1=2 3  3 ¼ 0 wa T3 P3 dT 3 dT 2 dT 4 ¼ ¼ T3 T2 T4 dP3 dP2 ¼ P3 P2

ð3:20Þ

Thus Eq. (3.19) can be re-written as,     dwf dP2 wa þ wf cp4 T 4 dT 2 wa þ wf cp3 T 3 dT 2 dT 2 ¼  1=2 þ  wf P2 T2 wa Δh T2 wa Δh T2  

 wf cp4 T 4 cp3 T 3 dP dT 2   1=2 ¼ 2þ 1þ P2 wa Δh Δh T2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ð3:21Þ x

dP dT ¼ 2þx 2 P2 T2 wf ) x ¼ const δθ

3.5 Corrected Net Thrust

21

Furthermore, if we make the assumption that specific heats are constant, i.e. cp3 ¼ cp4 then the exponent x reduces to 1/2 + Fuel-Air Ratio ¼ (approximately) 1/2 and we obtain the classical normalization wfc ¼

3.5

wf pffiffiffi δ θ

ð3:22Þ

Corrected Net Thrust

Expressions for the net thrust produced by a gas turbine engine will depend upon the configuration of the engine, i.e., whether it is a turbojet or a turbofan, mixed flow or non-mixed flow, as well as an expression for ram drag as a function of Mach number. To keep the derivation as simple as possible, we will assume a static thrust expression, i.e., Mach ¼ 0, no ram drag and, we will assume for simplicity an engine configuration which is that of a mixed flow turbofan. It should be noted that derivation that follows can be repeated (in spirit) with more complicated expressions for net thrust for other engine types and flight conditions to arrive at similar representations for corrected thrust. The thrust produced by this engine depends on the momentum of the air ejected from the exhaust nozzle as well as the net force (due to pressure) acting across the total area of the nozzle. In symbols F n ¼ const wa 9 v9 þ A9 ðP9  Ps9 Þ

ð3:23Þ

where station 8 ¼ Mixing plane in the exhaust station 9 ¼ Engine exhaust exit plane wa 9 ¼ wa 8 ¼ :Total airflow at nozzle A9 ¼ :Nozzle Area P9 ¼ :Nozzle exit total pressure Ps9 ¼ P2 ¼ :Nozzle exit static pressure ðambientÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v9 ¼ const ðh9  hs9 Þ ¼ :Velocity at exhaust nozzle h9 ¼ :Total ðstagnationÞ enthalpy at exit hs9 ¼ :Static enthalpy at exit The expression for the exit velocity follows directly from the definition of stagnation (total) enthalpy. Similarly, there is an expression for the velocity at station 8 (mixer) involving total and static enthalpies at station 8. The flow from station 8 to 9 (through a nozzle) we will assume to be adiabatic, i.e., no work done, no heat added. By conservation of energy, the total enthalpies are equal (h8 ¼ h9). If we

22

3 Common Corrections

denote the change in enthalpy (h9  hs9) ¼ (h8  hs9) ¼ Δh , then we may rewrite Eq. (3.23) as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F n ¼ const wa 8 v8 const Δh þ A9 ðP9  Ps9 Þ

ð3:24Þ

Taking logs of both sides of Eq. (3.24) and differentiating, produces pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dF n ðconst wa8 Þ const Δh dwa8 1 dΔh ¼ þ Fn 2 Δh Fn wa8    

A ðP  Ps9 Þ dA9 P9 dP9 P2 dP2 þ 9 9 þ  Fn A9 P9  P2 P9 P9  P2 P2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    

ðconst wa8 Þ const Δh dwa8 h8 dh8 hs9 dhs9 ¼ þ  wa8 2Δh h8 2Δh hs9 Fn 2 6 6 6 A9 ðP9  Ps9 Þ 6 6 þ 6 Fn 6 6 6 4

3

7 7 7     7 P9 dP9 P2 dP2 7 þ  7 P9  P2 P9 P9  P2 P2 7 7 7 5

dA9 A9 |{z} zero ðfixed geometryÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    

cp9 T s9 dT s9 cp8 T 8 dT 8 dF n ðconst wa8 Þ const Δh dwa8 ¼ þ  Fn wa8 2Δh T 8 2Δh T s9 Fn ð3:25Þ    

A9 ðP9  Ps9 Þ P9 dP9 P2 dP2 þ  Fn P9  P2 P9 P9  P2 P2 Now, at a given engine operating point (steady state), the following conditions hold pffiffiffiffiffi wa8 θ8 ¼ constant δ8 T8 ¼ constant θ2 T s9 ¼ constant θ2 P8 ¼ constant δ2 P9 ¼ constant δ2

) ) ) ) )

dwa8 1 dT 8 dP8 þ  ¼0 2 T8 wa8 P8 dT 8 dT 2  ¼0 T8 T2 dT s9 dT 2  ¼0 T s9 T2 dP8 dP2  ¼0 P8 P2 dP9 dP2  ¼0 P9 P2

3.6 Corrected Rotational Speeds

23

Substituting these values into Eq. (3.25), we obtain 2 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   7 dF n ðconst wa8 Þ const Δh 6 6dP8 þ cp8 T 8  cp9 T s9  1 dT 2 7 ¼ 4 P8 2 T2 5 Fn 2Δh 2Δh Fn |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} x

A ðP  Ps9 Þ dP2 þ 9 9 Fn P2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðconst wa8 Þ const Δh dP8 A ðP  Ps9 Þ dP2 dT 2 ¼ þx þ 9 9 Fn Fn P8 T2 P |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl ffl} 2 y1

ð3:26Þ

y2

This yields dF n dP dT dP dT ¼ ð y1 þ y2 Þ 2 þ y1 x 2 ¼ 2 þ y1 x 2 Fn P2 T2 P2 T2 Fn ) ¼ constant δ 2 θ 2 y1 x

ð3:27Þ

Since the specific heats at station 8 and 9 are approximately equal (since T8
T9s), we may assume a common specific heat cp. In this case, the theta exponent x reduces to 0 (since Δh ¼ cpΔT ). Thus, we arrive at the classical correction for thrust, namely Fn c ¼

3.6

Fn δ

ð3:29Þ

Corrected Rotational Speeds

Tangential velocity is related to rotational speed (rpm) by the radius of the object (compressor or turbine blade) in question (see Fig. 3.2). It is also related to acoustic velocity by Mach number and the square root of temperature. Mathematically, we have v ¼ rN ¼ M n

pffiffiffiffiffiffiffiffiffi γRT

ð3:30Þ

where v is the velocity in m/s, r is the radius (m), N is the rotational speed (rad/s), γ and R are constants ( ratio of specific heats and the gas constant respectively), and Mn is Mach number. Taking logs and differentiating Eq. (3.30), we obtain

24

3 Common Corrections

Fig. 3.2 Rotational speed

v

r

N

dN dM n 1 dT 1 dT ¼ ¼ and since Mach ¼ constant þ N 2 T 2 T Mn dN 1 dθ  ¼0 ) N 2 θ N ) pffiffiffi ¼ constant θ Therefore N N c ¼ pffiffiffi θ

3.7

ð3:31Þ

Corrected Horsepower

Horsepower required (or developed) by a compressor (or turbine) is given by HP / ðwa ΔhÞ ) HP ¼ const ðwa ΔhÞ ∴

HP ¼ const ðwa cpΔT Þ

ð3:32Þ

Invoking axiom 3, this relationship must also hold for corrected quantities, i.e. HPc ¼ const wa c cpΔT c pffiffiffi wa θ ΔT cp ¼ const δ θ const wa cpΔT pffiffiffi ¼ δ θ Therefore HP HPc ¼ pffiffiffi δ θ

ð3:33Þ

3.10

3.8

Corrected Metal Temperature Rate

25

Corrected Torque

The correction for torque follows from corrections for speed and horsepower, since horesepower is the product of torque and speed. It follows (from axiom 3) that a corrected horsepower should be a product of corrected torque and corrected speed, i.e. HPc ¼ Qc N c HP N ¼ pffiffiffi ¼ Qc pffiffiffi θ δ θ ) HP Qc ¼ Nδ Thus, Qc ¼

3.9

Q δ

ð3:34Þ

Corrected Acceleration

From Newton’s Law, Torque (force) ¼ Moment of Inertia (I)  Acceleration. In symbols Q ¼ I N_

ð3:35Þ

Therefore by axiom 3, we must have Qc ¼ I N_ c

ð3:36Þ

Q ¼ I N_ c δ N_ ¼I ) δ _ N N_ c ¼ δ

ð3:37Þ

This implies

3.10

Corrected Metal Temperature Rate

The (time) rate of change of a metal temperature is typically modeled as a first order lag heat transfer between the gas path temperature T and the metal temperature Tm. In symbols,

26

3 Common Corrections

1 T_ m ¼ ðT  T m Þ τ mcp where time constant τ ¼ HA

ð3:38Þ

The McAdams correlation for turbulent flow over a flat plate [7] is given by Nu ¼ 0:023ð Re Þ0:8 ðPrÞ0:4

ð3:39Þ

where Nu, Re, and Pr are the Nusselt’s, Reynold’s and Prandtl’s numbers respectively which are defined as follows: Nu ¼

HD , κ

Re ¼

ρvD , μ

Pr ¼

cpμ κ

ð3:40Þ

Substituting and re-arranging terms we can obtain the following form: H ¼ 0:023

 0:8   κ ρvD cp μ 0:4 D μ κ

κ 0:6 cp0:4 ρ0:8 v0:8 μ0:4 D0:2   0:023κ0:6 cpμ0:2 ρ0:8 v0:8 ¼ cp0:6 μ0:6 D0:2   0:023 cpμ0:2 ρ0:8 v0:8 ¼ D0:2 Pr0:6 ¼ 0:023


0:027

ð3:41Þ

cpμ0:2 ρ0:8 v0:8 D0:2

where the final approximation is obtained by noting that Prandtl’s number is approximately constant in the range of interest [7]. At this juncture we may recall that mass flow wa ¼ ρAv was corrected by square root of theta over delta. Combining these observations, we have wa wa c δ ¼ pffiffiffi ¼ ρv A θ

ð3:42Þ

ðρvÞ0:8 ¼ ðwa c Þ0:8 δ0:8 θ0:4

ð3:43Þ

Therefore,

Using the approximation at the end of Eq. (3.41) to define the reference condition H* and noting that D ¼ D*, we can write the ratio

3.11

Corrected Fuel Air Ratio

27

  0:2  0:8 cp* μ* ρ*v* cp μ ρv   0:2 pffiffiffi0:8 θ cp* 1 ¼ δ cp θn   cp* 0:40:2n 0:8 ¼ θ δ cp

H* ¼ H

ð3:44Þ

where we have assumed that the ratio μ*/μ is solely a function of temperature (θ). Now using Eq. (3.38) to define T_ m * ¼ T_ mc in a similar manner, we can form the appropriate ratio and use Eq. (3.44) to simplify the expression.    T_ m c H* cp T c  T m c ¼ H cp* T  T m T_ m ¼ θ 0:40:2n δ0:8 ¼θ

ðT=θÞ  ðT m=θÞ T  Tm

0:60:2n 0:8

δ

ð3:45Þ

) T_ m c ¼

_

θ

Tm 0:6þ0:2n 0:8 δ

The value of n appearing in the theta exponent can be determined experimentally by correlating viscosity with temperature from which it is observed that n varies from 0.8 at low temperature to 0.6 at high temperature. These values provide a theta exponent ranging from 0.76 to 0.72, respectively. The average value of 0.74 is recommended, thus providing the requisite correction. T_ mc ¼

3.11

_

θ

Tm 0:74 0:8 δ

ð3:46Þ

Corrected Fuel Air Ratio

This follows simply from definition and axiom 3, i.e., FAR ¼

wf wa

ð3:47Þ

28

3 Common Corrections

and wf wf pffiffiffi wf c w δ θ FARc ¼ ¼ pffiffiffi ¼ a wac wa θ θ δ

ð3:48Þ

Thus FARc ¼

3.12

FAR θ

ð3:49Þ

Thrust Specific Fuel Consumption

Likewise for TSFC, we have wf pffiffiffi wf TSFC δ θ TSFC ¼ ) pffiffiffi ¼ ¼ TSFCc Fn Fn θ δ

ð3:50Þ

TSFC TSFC c ¼ pffiffiffi θ

ð3:51Þ

This concludes the derivation of the classical corrections for the common gas path parameters contained in Table 3.1. We will return to these in Chaps. 6 and 7 to explore some refinements to the theta and delta exponents. In the next Chapter we consider the corrections to be applied to time derivatives of these common parameters beyond acceleration and metal temperature, which were derived above.

Chapter 4

Time Derivative Corrections

Abstract This chapter provides the Standard Day corrections for the time rate of change, (i.e., time derivative) for the gas path parameters encountered in Chap. 3. The parameters considered include, Acceleration, Temperature, Pressure, Air Flow, Fuel Flow, Horsepower, Torque, Thrust, Fuel Air Ratio (FAR), and Thrust Specific Fuel Consumption (TSFC). As in the previous chapter, corrections are derived in detail from the definitions, axioms, and assumptions that were proposed in Chap. 2 along with simple thermodynamic relationships relevant to the parameter under consideration. In order to avoid any circular reasoning, parameters are derived in a particular order, wherein only results from already derived parameters are utilized in a given derivation. Keywords Time-derivative corrections · Rate of change · Temperature · Pressure · Air Flow · Fuel Flow · Horsepower · Torque · Acceleration · Thrust · Fuel Air Ratio (FAR) · Thrust Specific Fuel Consumption (TSFC) · Theta · Delta · Specific heat · Local Mach No. · Ratio of specific heats

In this chapter we will extend the classical parameter corrections to include the correction of the time derivatives of a gas turbine parameter. Again, we will rely only on our axioms from Chap. 2 and basic thermodynamic relationships. The astute reader may have already observed, from our classical correction results from the preceding Chapter, that the corrected parameter of a time derivative is not equal to the time derivative of the corrected parameter, i.e., d X_ c 6¼ ðX c Þ dt The familiar counter-example is the time derivative of (spool) speed, i.e. acceleration. We already know from Table 3.1 that

© Springer Nature Switzerland AG 2020 A. J. Volponi, Gas Turbine Parameter Corrections, https://doi.org/10.1007/978-3-030-41076-6_4

29

30

4 Time Derivative Corrections

  _ N_ d N d _N c ¼ N 6¼ p ffiffiffi ¼ pffiffiffi ¼ ðN c Þ δ dt dt θ θ Thus, we cannot simply say, for example, that     _ € € c ¼ d N_ c ¼ d N ¼ N ðfalseÞ N δ dt dt δ In fact, we will demonstrate that the true relationship for the time derivative of acceleration (under the assumptions of our axioms) is as follows: pffiffiffi € θ N € Nc ¼ 2 δ The derivation is a bit involved and not as direct as those encountered in Chap. 3. In fact we will first show that the time derivative correction for acceleration is closely linked to the time derivative correction for horsepower. More precisely, we will show that _ _ c ¼ HP HP δ2

4.1

pffiffiffi € θ N € if and only if N c ¼ 2 δ

Correction of Horsepower and Acceleration Rate of Change

We begin with the definition of horsepower as the product of torque and rotational speed and Newton’s law, i.e. _ HP ¼ QN ¼ I NN

ð4:1Þ

where I represents the moment of inertia. Differentiating Eq. (4.1) (with respect to time) and applying the derivative of a product form, we obtain     _ ¼ I NN € þ N_ N_ ¼ I NN € þ N_ 2 HP

ð4:2Þ

Invoking axiom 2 and 3 for a sum and a product, (which will be used repeatedly in the course of this chapter), the relationship must hold at a standard day conditions, namely that  2   2 N_ N € _ € _ HPc ¼ I N c N c þ N c  I N c pffiffiffi þ δ θ

! ð4:3Þ

4.1 Correction of Horsepower and Acceleration Rate of Change

31

Dividing the Eq. (4.2) by δ2 and subtracting it from the Eq. (4.3), yields !  2   _ _ € _2 HP N NN N N _ c € c pffiffiffi þ  2  2 ) HP I N δ δ2 δ δ θ pffiffiffi       _ € € θ N NN N N N _ c  HP  I N €c p €c p ffiffi ffi ffiffi ffi pffiffiffi ) HP   ¼ I N δ2 δ2 δ2 θ θ θ pffiffiffi    _ € N θ N € _ c  HP  I p ffiffiffi N c  2 HP ) δ2 δ θ pffiffiffi    _ € θ HP N € _ HPc  2  IN c N c  2 δ δ

ð4:4Þ

Clearly, one solution to Eq. (4.4) is for both sides to equal zero which would yield the proposition _ _ c ¼ HP HP δ2

if and only if

pffiffiffi € €c ¼ N θ N δ2

Furthermore, since Eq. (4.4) must hold for any ambient condition, i.e. for all θ and δ. We also have that _ c ¼ const 1 HP IN c ¼ const 2 € c ¼ const 3 N since they are corrected quantities. Therefore we can re-write Eq. (4.4) as pffiffiffi    _ € θ HP N const 1  2 ¼ const 2 const 3  2 ) δ δ pffiffiffi _ € θ HP N ¼ const  const const þ const 1 2 3 2 δ2 δ2 pffiffiffi _ € θ HP N ¼ const 4 þ const 2 2 for all θ, δ 2 δ δ

ð4:5Þ

Thus for any δ and any two thetas, θ(1) and θ(2) we must have pffiffiffiffiffiffiffi _ € θð1Þ HP N ¼ const 4 þ const 2 δ2ffiffiffiffiffiffiffi δ2 p _ € θð2Þ HP N ¼ const 4 þ const 2 2 δ δ2

and ð4:6Þ

32

4 Time Derivative Corrections

Subtracting these two expressions from each other yields 0 ¼ const 2

€ N

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi! € θð2Þ θð1Þ N  δ2 δ2

ð4:7Þ

Since Eq. (4.7) must hold for arbitrary δ and any two thetas, θ(1) and θ(2), we have that € N

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffi € θð2Þ € θ θð1Þ N N ¼ ¼ const ) ¼ const δ2 δ2 δ2

for all θ, δ

ð4:8Þ

Therefore we have our time derivative of acceleration correction. pffiffiffi € €c ¼ N θ N δ2

ð4:9Þ

Invoking Eq. (4.5), we have the time derivative correction for HP, namely _ _ c ¼ HP HP δ2

4.2

ð4:10Þ

Correction of Torque Rate of Change

Since HP ¼ QN, we have by Eq. (4.10) HP QN Q N HPc ¼ pffiffiffi ¼ pffiffiffi ¼ pffiffiffi ¼ Qc N c θδ θδ δ θ

ð4:11Þ

Differentiating with respect to time, we obtain _ ¼ NQ _ þ N Q_ ) HP _ c ¼ N_ c Qc þ N c Q_ c HP

ð4:12Þ

pffiffiffi pffiffiffi _ _ HP Q_ θ _ N Q_ NQ N Q_ θ N_ Q p ffiffi ffi ¼ N ¼ þ ¼ þ þ N c Qc c δ δ δ2 δ2 δ2 δ2 θ δ2

ð4:13Þ

implying that

Now subtracting Eq. (4.13) from Eq. (4.12), we obtain, yields

4.3 Correction of Temperature Rate of Change

33

pffiffiffi pffiffiffi     _ HP Q_ θ Q_ θ _ _ _ HPc  2 ¼ Qc  2 N c ) 0 ¼ Qc  2 N c δ δ δ pffiffiffi  _Q θ ) Q_ c  2 ¼0 δ

ð4:14Þ

This provides the requisite correction: pffiffiffi _ _Qc ¼ Q θ δ2

4.3

ð4:15Þ

Correction of Temperature Rate of Change

Since tangential velocity ν is related to rotational speed by the radius of the object, r, we have that v ¼ rN ¼ M n

pffiffiffiffiffiffiffiffiffiffi γRT s

ð4:16Þ

If we assume local Mach number and ratio of specific heats are constant, then since pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T γ1 2 M n  const ) rN ¼ M n γRT s ¼ M n γ R const T ¼1þ Ts 2 pffiffiffiffi ¼ const T

ð4:17Þ

Differentiating Eq. (4.17) with respect to time, we obtain 1 N_ ¼ const pffiffiffiffi T_ ) T _N T_ T_ 1 1 ¼ const pffiffiffiffiffiffiffiffi pffiffiffi ¼ const pffiffiffiffiffi pffiffiffi δ Tc δ θ T=θ δ θ

ð4:18Þ

But, the first expression in Eq. (4.18) must also hold for corrected parameters by our axiom assumptions,1 therefore 1 N_ c ¼ const pffiffiffiffiffi T_ c ) Tc N_ 1 ¼ N_ c ¼ const pffiffiffiffiffi T_ c δ Tc

1

The correction of a square root is the square root of the correction : see Eq. (2.9).

ð4:19Þ

34

4 Time Derivative Corrections

Dividing Eq. (4.19) by (4.18), we have 1¼

T_ c T_ ffiffi p

ð4:20Þ

δ θ

This provides the correction we seek, i.e. T_ T_ c ¼ pffiffiffi δ θ

4.4

ð4:21Þ

Correction of Pressure Rate of Change

Consider, as we did in Chap. 3, the pressure ratio across the Fan (inner diameter) and the Low Pressure Compressor (LPC) from station 2 to station 2.5 of our sample engine, i.e. P25 ¼ P2



T 25 T2

γη γ1

) ln ðP25 Þ  ln ðP2 Þ ¼

γη ½ ln ðT 25 Þ  ln ðT 2 Þ λ1

ð4:22Þ

Differentiating we obtain P_ 25 γη T_ 25 ¼ since P_ 2 ¼ 0 ¼ T_ 2 implying that P25 γ  1 T 25   T_ 25 P25 γη P_ 25 ¼ γ1 T 25

ð4:23Þ

Likewise, this relationship (Eq. 4.23) must hold for corrected parameters as well (axiom 3),   T_ 25c P25c _P25c ¼ γη γ1 T 25c

ð4:24Þ

Therefore, from Eq. (4.24) we can obtain pffiffiffi pffiffiffi     T_ 25 P25 θ θ_ θ T_ 25 P25 γ η γη p ffiffi ffi ¼ ¼ P 25 γ  1 δ2 γ  1 δ θ δ T 25 T 25 δ2   T_ 25c P25c γη ¼ γ1 T 25c

ð4:25Þ

4.5 Correction of Air Flow Rate of Change

35

This gives us the required correction at station 2.5, i.e., pffiffiffi _ _P25c ¼ P25 θ δ2

ð4:26Þ

Moving further downstream in the engine and proceeding in the same manner, we have  γγ1η P3 T3 γη ¼ ) ln ðP3 Þ  ln ðP25 Þ ¼ ½ ln ðT 3 Þ  ln ðT 25 Þ ) γ1 P25 T 25     P_ 3 P_ 25 T_ 3 T_ 25 P_ P_ T_ 3c T_ 25c γη γη  ¼   ) 3c  25c ¼ ) ð4:27Þ P3 P25 γ  1 T 3 T 25 P3c P25c γ  1 T 3c T 25c  

T_ 3c T_ 25c P_ γη  P_ 3c ¼ P3c þ 25c γ  1 T 3c T 25c P25c Multiplying by

pffiffiffi θ=δ yields

pffiffiffi   pffiffiffi  

T_ 3 T_ 25 θ P_ 3 P_ 25 θ γη   ¼ ) δ P3 P25 δ γ  1 T 3 T 25 pffiffiffi pffiffiffi 0 _ 1 T3 T_ 25 P_ 3 θ P_ 25 θ pffiffiffi pffiffiffi C δ2  δ2 ¼ γη B Bδ θ  δ θ C ) @ P3 P25 T 25 A γ  1 T3 δ δ θ θ pffiffiffi  

_P3 θ _ _ T 3c T 25c P_ 25c γη ¼ P  þ ¼ P_ 3c 3c γ  1 T 3c T 25c P25c δ2

ð4:28Þ

Thus, the same relationship holds for station 3, i.e. pffiffiffi _ _P3c ¼ P3 θ δ2

ð4:29Þ

Repeating the same argument as we move downstream, we can conclude in general that pffiffiffi P_ θ P_ c ¼ 2 δ

4.5

ð4:30Þ

Correction of Air Flow Rate of Change

Starting with the definition of horsepower as being proportional to airflow (wa) times the change in enthalpy (Δh), i.e.

36

4 Time Derivative Corrections

HP / ðwa ΔhÞ ) HP ¼ const ðwa ΔhÞ ∴ HP ¼ const ðwa cp ΔT Þ )   HPc ¼ const wa c cp ΔT c ) _ c ¼ const w_ a c cp ΔT c þ const wa c cp ΔT_ c HP

ð4:31Þ

and _ wa cp ΔT_ w_ a cp ΔT HP ¼ const þ const 2 2 δ δ2 δ

_ _ c ¼ HP and if Since we have already demonstrated that HP δ2 ΔT ¼ T  T2 then ΔT_ ¼ T_  T_ 2 ¼ T_ and ΔT_ c ¼ T_ c  T_ 2 ¼ T_ c , we obtain const w_ a c cp ΔT c þ const wa c cp ΔT_ c ¼ const

wa cp ΔT_ w_ a cp ΔT þ const 2 δ δ2

ð4:32Þ

Cancelling like constants yields w_ ΔT w ΔT_ w_ a c ΔT c þ wa c ΔT_ c ¼ a 2 þ a 2 ) δ pffiffiffi δ   T T2 wa θ _ w T_ w_ a T c ¼ 2 ðT  T 2 Þ þ a2  þ w_ a c δ θ θ δ δ

ð4:33Þ

Thus pffiffiffi pffiffiffi     T T2 wa θ T_ wa θ T_ w_ a θ T T 2 pffiffiffi ¼ 2 pffiffiffi w_ a c   þ þ δ δ θ δ δ θ θ θ θ θ δ    w_ θ T T 2  ¼0 ) w_ a c  a2 θ θ δ   w_ θ w_ a c  a2 ¼ 0 δ

) ð4:34Þ

providing the required correction w_ a c ¼

4.6

w_ a θ δ2

ð4:35Þ

Correction of Fuel Flow Rate of Change

From basic energy balance, we have the following relationship   wf ηb LHV  wg ðΔhÞ  wa þ wf cpΔT ¼ wa ð1 þ FARÞcpΔT  wa cpΔT

ð4:36Þ

4.7 Correction of Net Thrust Rate of Change

37

where ηb is burner efficiency, LHV is fuel lower heating value and FAR is fuel air ratio. Then differentiating with respect to time, we have   w_ f ηb LHV ¼ w_ a cpΔT þ wa cpΔT_ ¼ cp w_ a ΔT þ wa ΔT_ ) pffiffiffi   w_ f w_ a θ ΔT wa θ ΔT_ p ffiffi ffi η LHV ¼ cp þ b δ δ θ δ2 δ2 θ   ¼ cp w_ a c ΔT c þ wa c ΔT_ c ¼ wfc ηb LHV

ð4:37Þ

Cancelling constant values we obtain our correction, i.e., w_ fc ¼

4.7

wf δ2

ð4:38Þ

Correction of Net Thrust Rate of Change

In general, we can assume F n ¼ wg

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi const cp ΔT  wT const γ R Mn T a

ð4:39Þ

where T a ¼ ambient temperature wT ¼ total airflow wAD ¼ bypass airflow wg ¼ wT  wAD þ wf ΔT ¼ T s  T a Therefore



pffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔT_ T_ a F_ n ¼ const cp w_ g ΔT þ wg pffiffiffiffiffiffiffi  const γRMn w_ T T a þ wT pffiffiffiffiffi 2 2 Ta ΔT



pffiffiffiffiffiffiffi 1 pffiffiffiffiffi 1 T_ a ΔT_ ¼ const w_ g ΔT þ wg pffiffiffiffiffiffiffi  const w_ T T a þ wT pffiffiffiffiffi ) 2 2 Ta ΔT " rffiffiffiffiffiffiffi# rffiffiffiffiffiffiffi pffiffiffi pffiffiffi w_ g θ ΔT 1 wg θ ΔT_ θ F_ n θ pffiffiffi ¼ const 2 þ θ 2 δ δ δ2 θ δ ΔT " rffiffiffiffiffi rffiffiffiffiffi# pffiffiffi θ w_ T θ T a 1 wT θ T_ a pffiffiffi  const þ 2 δ 2 T θ a δ θδ



pffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffi 1 T_ ac ΔT_ c ffi  const w_ Tc T ac þ wTc pffiffiffiffiffiffiffi ¼ const w_ gc ΔT c þ wgc pffiffiffiffiffiffiffiffi 2 2 T ac ΔT c

38

4 Time Derivative Corrections

Everything on the right hand side of the last equation is in terms of corrected quantities and hence constant at an arbitrary operating point and so pffiffiffi F_ n θ ¼ const δ2

ð4:40Þ

which provides our correction, i.e. pffiffiffi _ _F nc ¼ F n θ δ2

4.8

ð4:41Þ

Correction of Fuel Air Ratio (FAR) Rate of Change

By definition, we have FAR ¼

wf wa

ð4:42Þ

By axiom 3, we must have the same relationship for corrected quantities, i.e. wf wf pffiffiffi wfc FAR w δ θ FARc ¼ ¼ pffiffiffi ¼ a ¼ θ wa c wa θ θ δ

ð4:43Þ

Equation (4.16) provides the correction for FAR. If we differentiate Eq. (4.42), we obtain _ ¼ F AR

w_ f wa  wf w_ a w2a

ð4:44Þ

pffiffiffi Dividing Eq. (4.44) by δ= θ yields pffiffiffi wf w_ θ w_ f wa θ  pffiffiffi a2 _ w_ fc wa c  wfc w_ a c F AR δ2 δ δ θ δ _ c pffiffiffi ¼ ¼ ¼ F AR 2 w2a c w θ δ θ a δ2

ð4:45Þ

4.10

Summary of Rate of Change Corrections

39

Proving that _ AR _ c ¼ Fp ffiffiffi F AR δ θ

4.9

ð4:46Þ

Correction of Thrust Specific Fuel Consumption (TSFC) Rate of Change

Likewise we know wf pffiffiffi wf TSFC δ θ TSFC ¼ ) pffiffiffi ¼ ¼ TSFCc Fn Fn θ δ

ð4:47Þ

Differentiating Eq. (4.47) w_ f F n  wf F_ n ) Fn 2 pffiffiffi wf F_ w_ f F n θ  pffiffiffi n 2 2 δ _ w_ fc F nc ¼ wfc F_ n c TSFC δ δ θ δ _ c ¼ ¼ ¼ TSFC 2 δ Fn F 2n c δ2 _ ¼ TSFC

ð4:48Þ

yielding the required correction _ c¼ TSFC

4.10

_ TSFC δ

ð4:49Þ

Summary of Rate of Change Corrections

As a result of these derivations, there is an interesting observation that can be made. We list below the time derivative corrections derived in this Chapter:

40

4 Time Derivative Corrections

  pffiffiffi   pffiffiffi

_ θ d SHP θ d SHP pffiffiffi ¼ ðSHPc Þ ¼ 2 δ δ dt dt δ δ θ  pffiffiffi pffiffiffi   pffiffiffi

_ θ d _Qc ¼ Q θ ¼ θ d Q ¼ ðQ Þ δ dt δ δ dt c δ2   pffiffiffi  pffiffiffi pffiffiffi

θ d wa θ θ d w_ a θ ¼ w_ a c ¼ ðw Þ ¼ δ dt δ δ dt a c δ2   pffiffiffi   pffiffiffi

T_ θ d T θ d _T c ¼ p ffiffiffi ¼ ¼ ðT Þ δ δ dt θ dt c δ θ  pffiffiffi pffiffiffi   pffiffiffi

P_ θ θ d P θ d ¼ ðP Þ ¼ P_ c ¼ δ dt δ δ dt c δ2   pffiffiffi   pffiffiffi

_ θ d N _N c ¼ N ¼ θ d p ffiffiffi ¼ ðN Þ δ δ dt δ dt c θ  pffiffiffi pffiffiffi   pffiffiffi

€ θ θ d N_ θ d _  N € Nc ¼ N ¼ ¼ 2 δ δ dt dt c δ δ pffiffiffi   pffiffiffi

 w_ f θ d wf θ d  pffiffiffi ¼ wf c w_ f c ¼ 2 ¼ δ dt δ θ δ dt δ pffiffiffi pffiffiffi   pffiffiffi

F_ θ θ d Fn θ d ðF Þ ¼ F_ n c ¼ n 2 ¼ δ δ dt dt n c δ δ pffiffiffi   pffiffiffi

_ θ d TSFC θ d TSFC _ p ffiffi ffi TSFC c ¼ ¼ ¼ ðTSFCc Þ δ dt δ δ dt θ pffiffiffi   pffiffiffi

_ θ d FAR θ d AR _ c ¼ Fp ffiffiffi ¼ F AR ¼ ðFARc Þ δ dt θ δ dt δ θ _ c¼ SHP

From the list above, a simple relationship seems to emerge, that for a gas path parameter X, we have X_ c ¼

pffiffiffi θ d ðX Þ δ dt c

ð4:50Þ

We have established that Eq. (4.50) holds for temperatures, pressures, speeds, flows, power, i.e., all common gas path parameters of interest. We have not, however, formally demonstrated this relationship, for an arbitrary parameter X, as resulting from the axioms of Chap. 2. As such, we will use Eq. (4.50) as a conjectured result except for the parameters noted in this Chapter. For the latter, this relationship provides a simple and quick method for obtaining the form of the time derivative correction given the knowledge of the fundamental parameter correction. We will also make use of this conjecture in the next Chapter as we explore the correction of other ancillary parameters of interest.

Chapter 5

Additional Corrections

Abstract This chapter provides the Standard Day corrections for several additional parameters not already considered in previous chapters. These include Time Constants, Metal Temperature, and Static Temperature. The derivations proceed as before, employing only the definitions, axioms, assumptions, and derived results from preceding chapters. It should be noted that an additional assumption is employed in the derivation for Metal Temperature, i.e., the conjectured result given in Eq. (4.50). Keywords Corrections · Time constants · Metal temperature · Static temperature · Theta · Delta · Specific heat · Local Mach No. · Ratio of specific heats · Linear system

In this Chapter we will explore the corrections for three additional parameters, namely, (a) time constants, (b) metal temperature, and (c) static temperature. Again we will use the axioms of Chap. 2 and additionally, make use of the conjectured relationship given in Eq. (4.50) of the last Chapter.

5.1

Correction of Time Constants

Suppose we consider a first order linear system with time constant τ, where X is an arbitrary gas path parameter and Xss represents the steady state value of X after transient. Mathematically, this is represented as follows: 1 X_ ¼ ðX  X ss Þ τ

ð5:1Þ

It is a fundamental assumption that this general relationship will hold (in form) at ISA conditions, therefore

© Springer Nature Switzerland AG 2020 A. J. Volponi, Gas Turbine Parameter Corrections, https://doi.org/10.1007/978-3-030-41076-6_5

41

42

5 Additional Corrections

1 X_ c ¼ ðX c  X ssc Þ τc

ð5:2Þ

where the subscript c denotes corrected parameter. It is reasonable to assume that the time constant value τc may be different than the time constant value τ, since certainly for temperatures (X ¼ T), it is known that τ will vary with (air) flow which in turn varies with ambient temperature & pressure (τ # as wa " as θ2#). From Chap. 2, we know the general form of a corrected parameter to be (Eq. 2.2) Xc ¼

X θ δ

a b

Applying the relationship from Eq. (4.50) we obtain X_ c ¼

pffiffiffi pffiffiffi   θ d θ X_ _ ðX Þ ) X c ¼ δ dt c δ θ a δb

ð5:3Þ

Therefore pffiffiffi  pffiffiffi    θ X_ θ_ X X ss 1 _c ¼ 1 X ¼ ðX  X ss Þ  ¼ X ) δ θa δb δ τc θa δb θa δb τc

ð5:4Þ

A little manipulation yields pffiffiffi θ1 1 ðX  X ss Þ ¼ ðX  X ss Þ δ τ τc

ð5:5Þ

which provides the correction we seek  τc ¼

5.2

 δ pffiffiffi τ θ

ð5:6Þ

Correction of Metal Temperature

At this juncture we will derive a correction for metal temperatures using results from Chap. 3 and the conjectured relationship, Eq. (4.50), for an arbitrary time derivative correction. From Chap. 3, (Eq. 3.46), we know that the requisite correction for the time derivative of metal temperature Tm is as follows: T_ mc ¼

_

θ

Tm 0:74 0:8 δ

5.2 Correction of Metal Temperature

43

If we use this relationship in conjunction with Eq. (4.50), we have the following: pffiffiffi pffiffiffi   T_ m T_ θ d θ d Tm _ ¼ T mc ¼ ðT mc Þ ¼ ¼ a0:5m bþ1 a b 0:74 0:8 δ dt δ dt θ δ θ δ θ δ

ð5:7Þ

where a and b are the required correction exponents to standardize the metal temperature. Equating the left and right hand sides of this equation provides the two equations 0:74 ¼ a  0:5

and

0:8 ¼ b þ 1 ) a ¼ 1:24

and

b ¼ 0:2

This provides a correction for metal temperature as T mc ¼

T m δ0:2 θ1:24

ð5:8Þ

Since gas temperature is corrected (classically) by θ, i.e. T/θ, we see that there is an additional multiplicative factor αm as follows T mc ¼

 0:2 T m δ0:2 δ Tm T  αm m  θ θ θ θ1:24

ð5:9Þ

The magnitude of this multiplicative factor will depend on the flight condition (altitude and Mach no.) and for ISA conditions will vary as illustrated in the following Fig. 5.1. The corrections for Metal Temperatures and Spool Speeds have implications in gas turbine engine modeling, in particular with simple piecewise linear state representation models that are used in real-time (on-board) applications. In these types of models, the Metal Temperatures and Spool Speeds typically form the state parameters in the model which in its simplest formulation has the following basic form x_ ¼ AΔx þ BΔu Δy ¼ CΔx þ DΔu The term Δ indicates delta from steady state condition. The terms A, B, C, and D are matrices of partial derivatives, Δx is a vector of state parameters (combination of Metal Temperatures (Tm) and Spool Speeds, N1 and N2), Δu is a vector of input parameters (e.g. Fuel Flow, Bleed and variable geometry commands) and Δy is a vector of measured (and un-measured) dependent engine gas path parameters (e.g. gas temperatures, pressures, flows, etc). To minimize storage and model size, these elements are typically chosen to be standard day corrected parameters. To solve these equations, at each time step, integration (of the state variables) must be performed and hence the corrected parameter derivatives must be un-corrected, integrated, and the resultant (raw) states re-corrected, i.e.

44

5 Additional Corrections

Multiplicative Factor 1.1

Mach 0.2

1.08

Mach 0.3

1.06

Mach 0.4

1.04

Mach 0.5

1.02

Mach 0.6

(Delta/Theta)^0.2

1

Mach 0.7

0.98

Mach 0.8

0.96

Mach 0.9

0.94 0.92

αm

0.9 0.88 0.86 0.84 0.82 0.8 0.78 0.76 0.74

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

Altitude Fig. 5.1 Multiplicative factor αm

T_ mc ! θ0:74 δ0:8

5.3

Z

1 ! N ! pffiffiffi ! N c θ Z δ0:2 ! T_ m ! ! T m ! 1:24 ! T mc θ

N_ c ! δ ! N_ !

Correction of Static Temperature

We will now explore to what degree, if any, a static temperature’s correction differs from that of a total temperature. We start with the familiar relationship between static and total temperature and their relationship to local Mach no., i.e. T γ1 2 Mn ¼1þ Ts 2

ð5:10Þ

If we assume, (as we do for the classical corrections in Chap. 3) that the ratio of specific heats, γ, is constant and that local Mach No. is constant, then we see that

5.3 Correction of Static Temperature

45

dT s dT T T T ) s ¼ ¼ T c ¼ const ) T sc ¼ s ¼ T θ Ts θ θ

ð5:11Þ

Under these (classical) assumptions, the correction for static and total temperature is the same. We will now relax some of the assumptions and simplifications. Returning to Eq. (5.10), if we take logs of both sides and differentiate, we obtain     γ1 2 1 2 2M n γ1 dT dT s d 1 þ 2 M n 2 dM n þ 2 M n dγ  ¼ ¼ 2 2 T Ts 1 þ γ1 1 þ γ1 2 Mn 2 Mn ¼

1 2 n M 2n ðγ  1Þ dM M n þ 2 M n dγ

ð5:12Þ

2 1 þ γ1 2 Mn

Under our axiomatic assumption that local Mach No. is constant, this reduces to " # M 2n γ dT dT s dγ dγ   ¼α ¼  2 T γ γ Ts M 2 1 þ γ1 n 2 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl ffl}

ð5:13Þ

α

The multiplicative factor, α, will vary with the absolute level of local Mach No and the ratio of specific heats. Figure 5.2 below depicts how this factor varies. This plot portrays the multiplicative factor as a function of local Mach no., wherein the solid line is the average factor (across varying γ) and the dashed lines 0.6000

0.5000

Average Factor

0.4000

0.3000

0.2000

0.1000

0.0000

0

0.2

Fig. 5.2 Multiplicative factor α

0.4 0.6 Local Mach no.

0.8

1

46

5 Additional Corrections

indicate 3 sigma range. For static air, i.e. Mach ¼ 0, the factor is zero and static temperature equals static temperature as expected. For most commercial gas turbines, in areas where temperature measurements are typical, the local Mach no. is approximately in the range of 0.3–0.4, giving α a range of values between 0.06 and 0.10. For illustration, let us assume a local Mach No. value, in that range, of 0.35, which gives an average value of α ¼ 0.08. Thus dT dT s dγ  ¼ 0:08 T γ Ts

ð5:14Þ

In addition, we can approximate γ as a function of Ts as γ  1.8191(Ts)0.041 (with R2 ¼ 0.9866), which implies dγ dT  0:041 s γ Ts

ð5:15Þ

Therefore, we have the approximation dT dT s dγ dT dT dT dT   :99672 s ) s  0:08  0:00328 s ) T λ T Ts Ts Ts Ts dT  1:0033 T

ð5:16Þ

Now if αT represents the theta exponent for T, (classically αT ¼ 1),1 then dT s dT dθ ¼ 1:0033αT  1:0033 T θ Ts

ð5:17Þ

from which it follows T sc 

Ts θ1:0033αT

ð5:18Þ

The effect of relaxing our classical assumption, clearly, has very little impact for low local Mach numbers. If we consider much higher Mach numbers (around 0.8) the theta exponent increases to approximately 1.016αT which is a bit more significant. In the next Chapter we will continue to derive some refinements to the classical corrections given in Chap. 3 by relaxing some of our assumptions, in particular the assumption that specific heats are constant.

1 We will explore the use of non-unity theta exponent corrections for Temperatures in Chap. 7 on empirical methods.

Chapter 6

Refinements to Common Corrections

Abstract This chapter provides some refinement to the Standard Day corrections for most of the common gas path parameters considered in Chap. 3. In particular, the parameters considered will include, Rotor Speed, Pressure, Air Flow, Fuel Flow, Horsepower, Torque, and Acceleration. The refinements will stem from the relaxation of our previous assumption that specific heats, cp and cv, are constant and do not change with temperature. Allowing specific heats (and their ratio γ ¼ cp/cv) to vary, as they naturally do, offers theta exponents that are slightly different than the classical values provided in Chap. 3. The derivations follow the same strategy as in Chap. 3 and abide by the constraints imposed by our definitions, axioms, and previous results. Keywords Refinements · Corrections · Rotor speed · Pressure · Air flow · Fuel flow · Horsepower · Torque · Acceleration · Theta · Delta · Specific heat variation · Local Mach No. · Ratio of specific heats

This chapter explores some refinements to the classical corrections given in Chap. 3. This may provide some additional accuracy in applications where corrected parameters are being employed. The level of improvement would depend on a number of factors including the particular application at hand, thus it is not possible to estimate the actual level of improvement a-priori. The chapter, however, will provide the correction enhancements and the practitioner can evaluate their impact on an application by application basis. Improvement may be possible if we relax some of our assumptions, in particular, the assumption of constant specific heats, cp and cv. To explore this possibility, we begin by re-considering the correction for rotational spool speed as a first example.

© Springer Nature Switzerland AG 2020 A. J. Volponi, Gas Turbine Parameter Corrections, https://doi.org/10.1007/978-3-030-41076-6_6

47

48

6 Refinements to Common Corrections

6.1

Speed Correction

Tangential velocity is related to rotational speed (rpm) by the radius of the object (compressor or turbine blade) in question. It is also related to acoustic velocity by Mach number and the square root of temperature. In symbols v ¼ rN ¼ M n

pffiffiffiffiffiffiffiffiffiffi γRT s

ð6:1Þ

where v is the velocity in m/s, r is the radius (m), N is the rotational speed (rad/s), γ, and R are the ratio of specific heats and the gas constant respectively, and Mn is local Mach number. Taking logs and differentiating, we obtain from Eq. (6.1)     dr dN dM n 1 dγ dR dT s dN 1 dγ dT s þ þ ¼ þ þ ¼ þ ) r N 2 γ R N 2 γ Mn Ts Ts

ð6:2Þ

But since T γ1 2 dT dT s ¼1þ M n  const )  Ts 2 T Ts

ð6:3Þ

  dN 1 dγ dT  þ N 2 γ T

ð6:4Þ

Hence

Since γ ¼ cp/cv we have that dγ dcp dcv ¼  γ cp cv

ð6:5Þ

We also recall R ¼ cp  cv and therefore 0¼

    dR dcp dcv cp dcp cv dcv ¼  ¼  ) R cp  cv cp  cv cp  cv cp cp  cv cv   dcp cv dcv 1 dcv dcv ¼ ¼ < since γ > cp cp cv γ cv cv

ð6:6Þ

Combining with Eq. (6.5) we obtain dγ dcp dcv ¼