011801 1 by Mike Mentzos

A Simple Parametric Model for the Analysis of Cooled Gas Turbines S. Can Gülen Principal Engineer GE Energy, 1 River Road, Building 40-412, Schenectady, NY 12345 e-mail: can.gulen@ge.com

A natural gas fired gas turbine combined cycle power plant is the most efficient option for fossil fuel based electric power generation that is commercially available. Trade publications report that currently available technology is rated near 60% thermal efficiency. Research and development efforts are in place targeting even higher efficiencies in the next two decades. In the face of diminishing natural resources and increasing carbon dioxide emissions, leading to greenhouse gas effect and global warming, these efforts are even more critical today than in the last century. The main performance driver in a combined cycle power plant is the gas turbine. The basic thermodynamics of the gas turbine, described by the well-known Brayton cycle, dictates that the key design parameters that determine the gas turbine performance are the cycle pressure ratio and maximum cycle temperature at the turbine inlet. While performance calculations for an ideal gas turbine are straightforward with compact mathematical formulations, detailed engineering analysis of real machines with turbine hot gas path cooling requires complex models. Such models, requisite for detailed engineering design work, involve highly empirical heat transfer formulations embedded in a complex system of equations that are amenable only to numerical solutions. A cooled turbine modeling system incorporating all pertinent physical phenomena into compact formulations is developed and presented in this paper. The model is fully physics-based and amenable to simple spreadsheet calculations while illustrating the basic principles with sufficient accuracy and extreme qualitative rigor. This model is valuable not only as a teaching and training tool, it is also suitable to preliminary gas turbine combined cycle design calculations in narrowing down the field of feasible design options. 关DOI: 10.1115/1.4001829兴

Introduction

As is well-known from decades worth of theoretical analysis and field experience, the performance of a gas turbine 共GT兲, defined by specific work output and thermal efficiency, is driven by the two key Brayton cycle parameters: maximum cycle temperature and cycle pressure ratio 共PR兲. The ideal air-standard Brayton cycle thermodynamics are readily amenable to simple mathematical formulation and can be found in virtually all textbooks on thermodynamics and, specifically, on gas turbines 关1,2兴. An ideal GT would be constructed of materials that can withstand the highest possible operating temperatures without a need for cooling. In reality, available materials have operating temperature limits well below the temperature of the hot gases expanding in the turbine section of a modern GT, and hence require cooling. Since the early days of GT development, the components exposed to the highest temperature environment, i.e., the turbine hot gas path 共HGP兲 components have been cooled with air drawn from the compressor. The resulting flow network renders simple models describing the uncooled cycle performance of limited value for the investigation of optimal GT design parameters. The best example to illustrate this deficiency is the behavior of the GT cycle thermal efficiency as a function of PR. For the ideal Brayton cycle, thermal efficiency is a function of PR only and increases monotonically with increasing PR as opposed to a real GT with cooling, where the thermal efficiency reaches a maximum and diminishes with a further rise in PR due to increasing cooling flow needed to maintain the turbine HGP metal temperatures within their operational limits. The increase in these so-called “parasitic” Contributed by the International Gas Turbine Institute 共IGTI兲 of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received April 9, 2010; final manuscript received April 19, 2010; published online September 27, 2010. Editor: Dilip R. Ballal.

flows is a direct result of increasing compressor extraction and discharge temperatures commensurate with higher compression ratios. There are myriad ways to expand the ideal Brayton cycle formulas to accommodate turbine HGP cooling requirements. It is relatively straightforward to model the requisite flow network and provide a numerical solution to the resulting system of heat and mass balance equations. But the heart of the problem is an accurate assessment of requisite cooling flows to maintain the HGP metal temperatures. Complex heat transfer models are required to provide a solution to multiple problems associated with stationary and rotating stage components 共i.e., airfoils that are referred to as stators and rotors, respectively兲, and wheel spaces between individual stages and the parts of the turbine shaft exposed to the hot gas flow. An elegant and comprehensive mathematical treatment of these problems that can easily be translated into cycle calculations is not available. The great bulk of the past research efforts can be divided into CFD and experimental studies 共quite frequently a combination of the two兲. A comprehensive theoretical treatment of the gas turbine aerodynamics and heat transfer can be found in Ref. 关3兴. Consulting a recent article on the state-of-the-art 共SOA兲 of gas turbine heat transfer provides more specific details 关4兴. A quick glance at the cited works reveals the difficulty in translating the enormous research and field test data into formulations, let alone readily integrating them into cycle performance calculations. The most widely used approach to tackle this problem is the effectiveness curve method 关5,6兴. A series of papers by El-Masri 关7–9兴 described the application of a more detailed thermodynamic analysis of the cooled turbine stage to GT simple cycle and combined cycle 共CC兲 performance calculations. The theoretical framework for rigorous aerothermodynamic analysis of cooled gas turbine stages, established in cited references, is available in a commercial software tool for engineering calculations 关10兴. Bol-

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bustion” GT with a turbine section cooled by sensible heat transfer from the turbine casing; 2. A bottoming Brayton cycle, which is an “external combustion” GT with a heat exchanger utilizing the heat rejected from the turbine of the topping GT to increase the temperature of the working fluid.

Fig. 1 Air-cooled GT as a “combined” Brayton–Brayton cycle

land and his co-workers 关11,12兴 used El-Masri’s work as a starting point for their models to investigate novel GT cycle concepts. Horlock and his co-workers 关13–15兴 published several articles that describe the basic thermodynamics of GT cooling and investigate its impacts on cycle performance. References in those articles can be consulted for a comprehensive look at the past studies. Young and Wilcock 关16兴 developed a formal framework for modeling the air-cooled gas turbine stage, wherein they relied upon a modified version of an earlier method 关17兴 to establish the cooling flow rates. The latter also formed the basis for the analytical model by Torbidoni and Massardo 关18兴, which is amenable to conceptual cycle studies. More recently, Chiesa and Macchi 关19兴 used a very detailed one-dimensional cooled turbine stage analysis algorithm to investigate different GT cooling schemes and rank them with respect to combined cycle performance. Clearly, a significant body of work on the subject of cooled turbine stage modeling is available. What is missing or scarcely available so far is a simple parametric model that can be used to evaluate, as well as illustrate the key design and optimization aspects of the cooled turbine cycle performance. A notable exception that is known to the author is Traupel’s 关20兴 treatment of the cooled turbines in Sec. 2.5 共pp. 78兲 of his book 共in fact, the explanation in the second paragraph of Dr. Traupel’s foreword to his cited work is the inspiration for the current paper兲. Such a model should be amenable to easy implementation in a spreadsheet for rapid and large-scale analysis of GT and CC performance as a function of key design parameters. While it is obvious that a simple model cannot replace a full-blown design tool, it should be able to qualitatively represent the inherent performance trade-off features reasonably accurately. Furthermore, the model should be sufficiently grounded in the actual physical phenomena with minimal or no need for polynomial curve-fits and “fudge factors” so that the constituent formulas are compact and meaningful to illustrate the key principles clearly and unambiguously. A model that fits the bill, as described above, is developed and presented in the current paper. The primary unit system of the study described in this paper is the U.S. customary system. Conversions to SI are provided in the text, tables, and figures. Equations, however, are in their original form. Common conversion factors are provided in the nomenclature where they first appear.

Cooled GT Model

The core concept of modeling is adopted from the work of Khodak and Romakhova 关21兴. The system is shown schematically in Fig. 1. The conventional, cooled gas turbine Brayton cycle, which is the “base” system, is envisioned as a 共hypothetical兲 combined cycle comprising two Brayton cycles as follows. 1. A topping Brayton cycle, which is a standard “internal com011801-2 / Vol. 133, JANUARY 2011

The working fluid is air and gas 共after combustor兲 in the topping cycle and air only in the bottoming cycle. The mass flow rate of the bottoming GT is exactly equal to the total cooling airflow of the conventional air-cooled 共base兲 GT. The air mass flow rate of the topping GT is equal to the combustor inlet airflow of the base GT. Both GT cycles have the same compressor PR as the base GT. In other words, each turbine cycle has the same compressor discharge temperature 共CDT兲, which is the air inlet temperature to the combustor of the topping cycle and the heater of the bottoming cycle. The TIT of the topping GT is the same as the TIT of the base GT. To clarify for this discussion, the combustor system is envisioned to encompass the dilution zone and the stage-1 nozzle 共S1N兲 as well. As a result, for practical purposes, the highest cycle temperature is the turbine rotor inlet temperature 共RIT兲. Also known as the firing temperature, RIT is the temperature at which the working fluid starts producing expansion work in the turbine. It is also possible to draw the control volume around the combustor system to encompass only the dilution zone so that the highest cycle temperature T3 is the TIT at the inlet to S1N 共see Fig. 2兲. The term nonchargeable in Fig. 2 signifies that, for a given RIT, changing the amount of S1N cooling flow has no impact on turbine mass flow rate 共downstream of S1N兲 and power production. In other words, that cooling flow is not chargeable to turbine net power, whereas all cooling flows downstream of S1N are chargeable, i.e., detrimental to turbine power output. In the current simple model with continuous expansion and work production immediately downstream of the combustor exit, the distinction between chargeable and nonchargeable flows is clearly not meaningful. In that sense, setting T3 to RIT 共i.e., method I兲 is the sensible choice. Nevertheless, setting T3 to TIT 共i.e., method II兲 can be used for the analysis to estimate total turbine cooling flow, which, within the context of the simple model herein, is entirely chargeable. A typical advanced air-cooled F-Class GT has approximately 250° F 共140° C兲 temperature drop between the turbine inlet and S1N exit 关22兴. Thus, the TIT can be estimated from RIT by adding 140° C to the calculated value. For the older-technology, E-Class gas turbines with lower firing temperatures and PRⱕ 14, the appropriate adder is half of that, i.e., 125° F 共70° C兲. The TIT of the bottoming GT is determined by the amount of cooling heat rejected from the topping GT turbine. However, for the continuous expansion-cum-cooling approach adopted herein 共see Fig. 4 below兲, T3c in Fig. 1 is not relevant. The GT model per Fig. 1 is fully adequate to capture all pertinent physics of the air-cooled gas turbine system by making small changes to the ideal Brayton cycle relationships. Basic governing equations and derivations can be found in any textbook, e.g., Cohen et al. 关2兴. The goal herein is to retain the simplicity of ideal formulas for rapid and transparent calculations while improving the accuracy by incorporating the real-cycle effects via correction parameters. Calculations are done using perfect gas assumptions 共with a reference temperature T0 of 25° C for zero enthalpy兲 as follows: 1. constant air c p of 0.260 Btu/lb R 共1.089 kJ/kg K兲 and isentropic exponent of ka = 0.2857 共implying ␥ = 1.40兲 2. constant combustion gas c p of 0.322 Btu/lb R 共1.348 kJ/ kg K兲 and isentropic exponent of kg = 0.2481 共implying ␥ = 1.33兲 Turbine HGP cooling calculations use the well-known effectiveness curve method described in detail by El-Masri 关5兴 and others Transactions of the ASME

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Table 1 Air-cooled heavy-duty industrial GT „50-Hz, 3000 rpm… data from a trade publication †24‡ „for the estimation of the RIT, please refer to the text…

Compressor airflow Compressor PR Turbine RIT

kg/s °F °C kg/s °F kW kJ/kg

Turbine exhaust flow Turbine exhaust temperature Turbine output Specific work Turbine efficiency

675 17.7 2360 1294 690 1062 285,000 421.8 39.6%

627 17.0 2434 1334 640 1116 255,600 407.6 36.9%

640 18.3 2573 1412 655 1164 279,200 436.1 37.8%

678 18.2 2396 1313 692 1,071 292,000 431.0 39.8%

704 18.0 2455 1346 719 1107 312,100 443.2 39.3%

properties from the supply and return steam pressures and temperatures. Cooling steam supply at 35 bars and 371° C 共700° F兲 is assumed with 3.5 bars pressure drop and 167° C 共300° F兲 temperature rise. Equations identical to Refs. 关23,24兴 can be used for ˙ the additional useful ST work W CAC derived from the heat transferred from the GT to the RBC via CAC with a conversion effec˙ tiveness of ␧CAC = 90%. It is assumed that Q CAC is used to generate 28 bars 共400 psi 共absolute兲兲 IP steam with a 10° F 共6 ° C兲 approach subcool.

Note that the actual RIT values of production units are rarely 共if ever兲 published by OEMs. Nevertheless, some representative data is available in trade publications. One example is the RIT data of licensed turbines by one packager in Ref. 关28兴. Using PR and exhaust temperature as independent variables, the data can be represented reasonably well by a formula derived from turbine isentropic relationship, i.e., with temperatures in °F

RIT estimates from Eq. 共26兲 are listed in Table 1 and used in the calibration process described above. The published performance of the units in Table 1, i.e., specific output and net efficiency 共at the generator terminals兲, are matched by varying the model constants for listed PR and RIT values. The OEM output, efficiency, and exhaust data in Table 1, with reasonable assumptions for losses and small items, result in significant heat balance errors. Prior to data matching for model calibration, output data is adjusted 共lower by 5–10 MW兲 to ensure heat balance closure. Cooling model parameters ␤, ␸⬁, and Tb are set to 0.9, 1.0, and 1500° F 共815° C兲, respectively. Setting T3 to RIT, as illustrated in Fig. 2, for fixed values of ␦ = 0.84 and ␩c,max = 0.94, ␮c, ␾, and ␩t,max are adjusted to match individual turbine output, efficiency, and exhaust temperature. Model calibration constants are listed in Table 2. The published OEM rating data for the CC performance of the five GTs in Table 1 covers a range of 57–59.5%. The average CC efficiency is 58.2% with a scatter of ⫾1.0%. The average CC specific net power output is ⬃645 kJ/ kg 共⫾25 kJ/kg兲. In the remainder of the paper, these values will be used as a reference point for the SOA in CC performance. One can interpret this as the performance of a composite or average F-Class SOA gas turbine, which is a 1466 pps 共665 kg/s兲 machine with PR= 17.8 and RIT= 2445° F 共1340° C兲. A Monte-Carlo simulation of the model with the calibration data in Table 2 showed an error band of 共⫾1 standard deviation兲 ⫾1.5% 共points兲 for ␩GT, ⫾1.0% 共points兲 for ␩CC, ⫾24° F for the GT exhaust temperature, ⫾2.3% for the CC net output, and ⫾4.2% for the GT specific output. The obvious question that presents itself at this point is related to the future prospects for the advancement of the cooled GT technology: How much improvement in performance 共i.e., net ef-

Model Calibration

For the given RIT and PR, the model encapsulated by Eqs. 共1兲–共17兲 is expected to predict the real turbine performance reasonably well with suitable values for the key model parameters ␮c, ␾, ␦, ␤, ␸⬁, Tb, and component polytropic efficiencies ␩c and ␩t. Polytropic compressor and turbine efficiencies are strongly dependent on the aerodynamic component size and design. For a discussion and data, including predictive equations, the reader is referred to Ref. 关27兴. Herein, ␩c and ␩t are represented as simple functions of PR using the basic correlations in the cited work, for example

␩c = ␩c,max − 0.005 · ln共PRc兲

共25a兲

␩t = ␩t,max − 0.010 · ln共PRt兲 · 共1 + 冑␹兲

共25b兲

Note that the turbine efficiency function is modified to account for the detrimental effects of the coolant flow via mixing and momentum losses. The parameter ␹ in Eq. 共25b兲 is the total cooling air flow as a fraction of airflow. In spirit, the turbine efficiency correction in Eq. 共25b兲 is similar to that in Ref. 关20兴共e.g., see Eq. 2.5共23兲 on pp. 81 of the cited work.兲 The formula proposed herein is qualitative in nature and intends to provide only a sense of the real impact of the HGP cooling on turbine efficiency. It can be refined, modified or expanded via comparison 共or calibration兲 with rigorous aerothermodynamic stage design calculations. In order to determine the range of the model correction factors, five air-cooled 50-Hz 共i.e., 3000 rpm兲 heavy-duty industrial gas turbines from four different OEMs and packagers are selected from a trade publication 关24兴. Key performance data for the selected turbines, labeled as A, B, C, D, and E, are shown in Table 1.

RIT =

共Texh + 460兲 − 460 关0.882 · 共PR−0.25 − 1兲 + 1兴

共26兲

Table 2 Model calibration results „method I…

␩t,max in Eq. 共25兲 ␮c in Eq. 共1兲 ␾ in Eq. 共9兲

Sp. work 共kJ/kg兲 mc 共% of airflow兲 Qc / HC

Average

87.8% 0.0900 0.8925 413.9 10.8% 4.4%

84.7% 0.1136 0.8950 390.5 14.4% 5.9%

84.7% 0.0978 0.8996 429.0 14.6% 5.4%

87.8% 0.0869 0.8937 423.1 11.0% 4.3%

87.0% 0.0886 0.8963 430.0 11.8% 4.5%

86.4% 0.0954 0.8954 417.3 12.5% 4.9%

011801-6 / Vol. 133, JANUARY 2011

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Fig. 3 Total turbine cooling flow as a % of airflow including chargeable and nonchargeable flows „␮c = 0.113, ␤ = 0.83, ␸ⴥ = 0.9…

˙ =m ˙ c · c p,a · 共Tb − T2c兲 Q c

共5兲

T4c,i−1 +

˙ = 0, the calculations represent the ideal “unNote that for Q c cooled” gas turbine. Referring to Fig. 1, it is recognized that, for this simple model, the bottoming Brayton cycle TIT is the same as the bulk metal temperature, i.e., T3c = Tb. Simple mass balance is used to calculate air and coolant turbine flows from the base GT airflow ˙a=m ˙ GT − m ˙c m

共6兲

˙g=m ˙a+m ˙f m

共7兲

˙ exh = m ˙ GT + m ˙f m

共8兲

For a specified T3, the fuel flow of the air turbine is calculated using the energy balance around the combustor control volume ˙f= m

˙ a · 共c pg · 共T3 − T0兲 − c pa · 共T2 − T0兲兲 ␾·m 共LHV · ␩comb + c pf · 共T f − T0兲 − c pg · 共T3 − T0兲兲

共9兲

The combustor burns natural gas 共100% CH4兲 fuel with LHV of 50,000 kJ/kg, heated to 185° C 共c p of 2.5 kJ/kg K兲. A combustor efficiency of ␩comb = 99.5% is assumed. The factor ␾ accounts for the real property and composition effects. From detailed stoichiometric calculations in a heat balance simulation tool 关10兴, the value for ␾ applicable to air-methane combustion is determined as 0.89. The exhaust temperatures of the air and coolant turbines are found from the respective polytropic p-T relationships, similar to that for the compressor in Eq. 共4兲, with the turbine polytropic efficiency ␩t. The turbine pressure ratio PRt is scaled down from the compressor PR via a factor ␦ to account for the cycle pressure losses 共e.g., inlet, combustor and exhaust pressure drops兲 and the total pressure drop across S1N. A reasonable range of values for ␦ is between 0.84 and 0.9. Each turbine control volume is divided into N sections with successive adiabatic expansion and zero-work heat transfer stages, as shown in Fig. 4. For a large number of N 共say, 100兲, this provides a very good simulation of the continuous expansion plus heat rejection 共i.e., nonadiabatic air turbine兲 and expansion plus heat addition 共i.e., nonadiabatic coolant turbine兲 processes. The formulas for respective expansion plus heat transfer steps are given below: 011801-4 / Vol. 133, JANUARY 2011

T4c = T4c,i=N =

T4 = T4,i=N =

˙ ˙ −Q Q c sc ˙ c · c pa兲 N · 共m

PRt␩t·ka/N T4,i−1

␩t·kg/N

PRt

−

˙ Q c ˙ g · c pg兲 N · 共m

共10兲

共11兲

Coolant and air turbine thermal work outputs are calculated by summing up respective compressor and turbine works 共minus the cooling heat rejection for the coolant turbine兲 ˙ =m ˙ ˙ a · c pa · 共T1 − T2兲 + m ˙ g · c pg · 共T3 − T4兲 − Q W g c

共12兲

˙ + 共Q ˙ −Q ˙ 兲 ˙ =m ˙ c · c pa · 共T1 − T4c兲 − Q W c cac c sc

共13兲

The exhaust temperature of the combined gas turbine system is found from the enthalpy balance of mixing coolant and air turbine exhaust flows Texh =

˙ c · c pa · 共T4c − T0兲 ˙ g · c pg · 共T4 − T0兲 + m m + T0 ˙ g · c pg + m ˙ c · c pa兲共m

共14兲

Total GT shaft and generator outputs are evaluated with the assumed mechanical efficiency of ␩mech = 99.0% and generator efficiency of ␩gen = 98.9% ˙ ˙ ˙ ˙ =W W GT shaft · ␩gen = 共Wc + Wg兲 · ␩mech · ␩gen

共15兲

Finally, the key performance parameters, i.e., specific turbine output and net turbine efficiency, are calculated using standard definitions ˙ W GT ˙ GT m

共16兲

˙ ˙ W W GT GT = HC m ˙ f · LHV

共17兲

wsp =

␩GT =

The exergetic efficiency concept by Gülen and Smith 关23兴 is used to evaluate the RBC performance to arrive at the overall CC plant performance. This concept starts from the second law dictated thermodynamic fact that the maximum useful power that can Transactions of the ASME

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Fig. 4 Schematic description of turbine expansion model

be generated by a RBC is exactly equal to the total exergy of the GT exhaust gas. This, however, would only be possible if the RBC is a Carnot engine. Today’s SOA technology utilizing the most advanced component designs and economically feasible aggressive cycle design parameters can achieve only a fraction of that, given by the following relationship 关23兴 with Texh in °F ␧BC = 0.2441 + 0.0746 ·

冉冊

Texh Texh − 0.00279 · 100 100

˙ ˙ exh · eexh W RBC = ␧BC · m

共18兲共19兲

Note that Eq. 共19兲 gives the net RBC output, which is equal to the ST generator output minus about 1.5% for boiler feed and condensate pumps. The exhaust gas exergy in Eq. 共19兲 is a fluid property, which is a function of gas temperature and composition. A reasonable approximation for NG-burning gas turbines, valid between 480° C and 870° C, is given below 共with Texh in °F兲 eexh = 0.001628 · T1.60877 exh

共20兲

As explained in detail in Ref. 关23兴, exergy calculations are based on a reference dead state of 1.0 atm and 15° C 共the exergy associated with the latent heat of water vapor in the GT exhaust gas is ignored兲. The advanced RBC defined by Eq. 共18兲 is associated with the following SOA design features 关23兴: 1. 41 mbar 共1.2 in.兲 Hg condenser pressure and a suitably large ST exhaust annulus 2. advanced ST with SOA section efficiencies 3. steam temperatures commensurate with the GT exhaust temperature up to ⬃1100° F 共600° C兲 at highest possible steam pressures achievable in a drum-type boiler 4. a large 3PRH HRSG for the highest possible steam production within the limits of economic feasibility The RBC performance predicted by Eqs. 共18兲 and 共19兲 is in relatively good agreement with advanced cycle RBC performances reported in trade publications such as Ref. 关24兴. The net CC efficiency for an N ⫻ 1 configuration, i.e., N GT-HRSG trains and a single ST, is given by Journal of Engineering for Gas Turbines and Power

␩CC =

˙ ˙ ˙ ˙ ˙ +W N · 兵W GT RBC + WSC + WCAC其 − Waux N · HC

共21兲

The auxiliary power consumption of the CC power plant is assumed to be equal to 3% of the RBC output given by Eq. 共19兲. This is approximately in line with typical “rating” assumptions used in trade publications, such as Ref. 关24兴. Depending on the actual site and ambient conditions and the type of heat rejection system, as well as the auxiliary load scope, it can be as much as 4% or higher 关25兴. Realistically, it is necessary to also account for the fuel compression power expenditure, which is a function of the NG pipeline pressure 共highly site specific兲 and compressor PR, especially for PR⬇ 20 and higher. The following formula is used herein for that purpose 关25兴共with fuel flow rate in pps兲: ˙ ˙ fuel · 共80 · ln共PR兲 − 225兲 W fcomp = m

共22兲

The additional useful ST work derived from the heat transferred ˙ from the GT to the RBC via cooling steam W SC is a function of the conversion effectiveness of ␧SC. This is a function of optimal system design and integration within the limits imposed by mechanical and economic design criteria. Using the G and H-Class turbine models in a commercially available heat balance software 关26兴, this conversion rate is calculated as ⬃80%

冉冊

˙ =␧ ·Q ˙ · 1 − T0 W SC SC sc ¯T cs

共23兲

The mean-effective cooling steam temperature is evaluated from cooling steam supply and return conditions as ¯T = hret − hsup cs sret − ssup

共24兲

Note that due to the significant pressure loss between cooling steam supply and return, the temperature calculated from Eq. 共24兲 should be interpreted as an indicative mean-effective temperature, which is lower than the true mean-effective temperature of the process by ¯v · ⌬p / ⌬s 共about 65° C兲. The cooling steam supply and return enthalpies and entropies are evaluated using ASME 1967 JANUARY 2011, Vol. 133 / 011801-5

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Table 1 Air-cooled heavy-duty industrial GT „50-Hz, 3000 rpm… data from a trade publication †24‡ „for the estimation of the RIT, please refer to the text…

Compressor airflow Compressor PR Turbine RIT

kg/s °F °C kg/s °F kW kJ/kg

Turbine exhaust flow Turbine exhaust temperature Turbine output Specific work Turbine efficiency

675 17.7 2360 1294 690 1062 285,000 421.8 39.6%

627 17.0 2434 1334 640 1116 255,600 407.6 36.9%

640 18.3 2573 1412 655 1164 279,200 436.1 37.8%

678 18.2 2396 1313 692 1,071 292,000 431.0 39.8%

704 18.0 2455 1346 719 1107 312,100 443.2 39.3%

Model Calibration

␩c = ␩c,max − 0.005 · ln共PRc兲

共25a兲

␩t = ␩t,max − 0.010 · ln共PRt兲 · 共1 + 冑␹兲

共25b兲

RIT =

共Texh + 460兲 − 460 关0.882 · 共PR−0.25 − 1兲 + 1兴

共26兲

Table 2 Model calibration results „method I…

␩t,max in Eq. 共25兲 ␮c in Eq. 共1兲 ␾ in Eq. 共9兲

Sp. work 共kJ/kg兲 mc 共% of airflow兲 Qc / HC

Average

87.8% 0.0900 0.8925 413.9 10.8% 4.4%

84.7% 0.1136 0.8950 390.5 14.4% 5.9%

84.7% 0.0978 0.8996 429.0 14.6% 5.4%

87.8% 0.0869 0.8937 423.1 11.0% 4.3%

87.0% 0.0886 0.8963 430.0 11.8% 4.5%

86.4% 0.0954 0.8954 417.3 12.5% 4.9%

011801-6 / Vol. 133, JANUARY 2011

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Fig. 5 GT efficiency „at the generator terminals… as a function of compressor PR; RIT = 2462° F „1350° C…. CAC from T2 to 600° F „315° C…. Cycle improvements are cumulative.

ficiency and electric power兲 and by what means? In the remainder of the paper, using the current model calibrated to the SOA F-Class technology, answers to these questions will be explored. Note that the emphasis in the discussion below is solely on the thermodynamic cycle aspects. Mechanical and/or economic feasibility issues are expected to present significant hurdles to the eventual realization of the thermodynamic potential in the foreseeable future. A comprehensive discussion of those considerations is beyond the scope of the current paper. The following are three known paths to GT and CC performance improvement. 1. thermodynamic design optimization 共i.e., selection of optimal Brayton cycle PR and TIT兲 2. better component aerothermodynamic design 共e.g., compressor and turbine polytropic efficiencies兲

3. advanced materials and cooling technology, e.g., TBC, effusion or transpiration cooling, and others 共i.e., lower cooling flows兲 This is illustrated by Figs. 5 and 6 for the SOA F-Class GTs represented by the average calibration constants from Table 2 with RIT= 2462° F 共1350° C兲 and a range of compressor PR values. For the same GT, the CC efficiency and specific power output are calculated for a range of compressor PR and RIT values. The CC results are shown in Figs. 7–9. The GT and CC performance trends displayed in these plots clearly illustrate the diminishing returns on GT and CC performance from increasing the cycle PR. Increased cooling flow penalty due to higher compressor discharge temperatures is particularly pronounced for GT specific

Fig. 6 GT specific power output „at the generator terminals… as a function of compressor PR; RIT= 2462° F „1350° C…. CAC from T2 to 600° F „315° C…. Cycle improvements are cumulative.

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Fig. 7 CC net efficiency „at the generator terminals… as a function of compressor PR; RIT= 2462° F „1350° C…. CAC from T2 to 600° F „315° C…. Cycle improvements are cumulative.

output and CC net efficiency. The latter is also negatively impacted by reduced GT exhaust temperature resulting from higher PR and cooling flows. Cooling of turbine cooling air 共CAC兲 alleviates the problem to a certain extent and leads to maxima in all parameters at higher PR values. Based on the findings from Figs. 5–9, it is highly unlikely that optimal CC performance can be achieved for PR values much higher than low twenties, unless radical improvements in cooling airflow expenditure and component efficiencies are made possible by technology advances. Higher cycle PRs are also feasible for certain Brayton cycle variants, e.g., reheat combustion and closedloop external 共e.g., steam兲 cooling. These two variants will be elaborated upon further in the following paragraphs. Key takeaways can be summarized as follows. For the same levels of aerodynamic component design, materials, and cooling technologies:

1. There is an optimum PR for maximum ␩CC at a specified RIT. The optimum PR increases with increasing RIT. 2. The optimum PR for maximum ␩CC is higher than that for maximum GT specific output. 3. However, the gain in ␩CC is relatively modest between the two optima, e.g., on average about 0.20 ppt 共higher at higher RIT兲. In fact, the last observation is also supported by the current technology deployment in the field, i.e., the OEM designs of F-Class units are consistent with maximum GT specific output. This ensures a favorable economic outcome by lowering the plant specific cost with a relatively minor sacrifice in efficiency 共note that, in his Calvin Rice Memorial Lecture of 1994, Horlock 关29兴 made the same observation starting from similar premises兲. As the

Fig. 8 Relative CC net efficiency as a function of GT Brayton cycle PR and RIT. The published CC rating data is from Ref. †24‡. The origin is for the “average” state-of-the-art F-Class air-cooled GT.

011801-8 / Vol. 133, JANUARY 2011

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Fig. 9 Relative CC net efficiency and specific net power as a function of GT Brayton cycle PR and RIT. GT PR increases from 10 to 32 „from right to left… by increments of 2. The origin „0, 0… is the “average” state-of-the-art F-Class air-cooled GT.

trends in Figs. 5 and 6 suggest, at a specified RIT, optimum PR for maximum CC efficiency and GT specific power output increases with increasing component polytropic efficiency and decreasing cooling flow. The technology curves in Figs. 5–9, along with adequate cost models for the GT, as well as the RBC components, can be used in conceptual cycle trade-off studies to determine the optimal design points. The derivatives in Figs. 5–7 provide a rough guidance to the performance improvement via better component design. Similar derivatives for the key RBC hardware design parameters are also available 关23兴. Ultimately, the largest impact to the performance improvement of gas turbines in simple 共i.e., Brayton兲 or combined 共i.e., Brayton–Rankine兲 cycle configurations should come from the advances that will reduce the cooling air requirements, even when raising the cycle pressure ratio and firing temperature to levels well above today’s SOA. A technology path in that direction is external cooling, which is currently implemented in G-Class and H-Class gas turbines. In those units, cooling of selected HGP components is accomplished by using steam from the bottoming cycle. There are scant published data for the key GT parameters of the H-Class machines. Trade publications such as Ref. 关24兴 or Ref. 关28兴 list only the CC ratings. In addition to airflow and PR, the only published data for the H-Class steamcooled are the estimated TIT, CC output, and efficiency 关22,30兴. G-Class turbine’s steam cooling duty is estimated from the published data 关31,32兴. Note that the accounting for external 共steam兲 cooling in the simplified model herein is evenly distributed across the entire turbine expansion, which is different from the actual field-deployment of the technology. In G-Class machines, for in-

stance, all steam cooling is limited to upstream of the turbine inlet, whereas only the first two stages 共out of a total of four兲 are externally cooled in H-Class machines. Therefore, the cooling steam duties in the current model should be expected to overestimate actual duties. In order to estimate the impact of external 共steam兲 cooling on GT and CC performance, it is necessary to estimate the total turbine cooling flow. Thus, the model calibration described earlier is repeated by setting T3 to TIT, as illustrated in Fig. 2 共i.e., method II兲. TIT values are estimated by increasing the calculated RIT values by 250° F. Model calibration constants are listed in Table 3. In particular, at a PR of 23 and TIT of 1485° C 共⬃2700° F兲, an ˙ externally cooled gas turbine, with 30% and 60% of total Q c transferred to cooling steam, is worth 0.9 and 1.7 ppt in net ␩CC, respectively, over today’s SOA in F-Class, as predicted by the current model. The entitlement for full steam cooling at the same PR and TIT with ␧SC = 100% is about 3.5 ppt in net ␩CC. The performance entitlement of natural gas fired GTs in combined cycle configuration via reductions in HGP cooling flow is illustrated 共Fig. 10兲. Note that air-cooled turbine calculations are per method I 共calibration data in Table 2兲, whereas steam-cooled turbine calculations are per method II 共calibration data in Table 3兲. For the air-cooled turbines using compressor extraction as coolant source 共i.e., “open” loop兲, the critical path is the reduction in chargeable flows as RIT increases. For the steam-cooled turbines using steam from the bottoming cycle in a “closed” loop, the critical path is reduction in the remaining coolant 共which is air兲

Table 3 Model calibration results „method II…

␩t,max in Eq. 共25兲 ␮c in Eq. 共1兲 ␾ in Eq. 共9兲

TIT 共°C兲 TIT 共°F兲 mc 共% of airflow兲 Qc / HC

Average

89.1% 0.1572 0.9069 1432 2610 23.8% 9.6%

85.9% 0.1676 0.9085 1473 2684 26.3% 10.8%

85.9% 0.1436 0.9126 1551 2823 25.9% 9.5%

89.1% 0.1505 0.9079 1452 2646 23.8% 9.3%

88.3% 0.1466 0.9096 1485 2705 24.1% 9.2%

87.7% 0.1531 0.9091 1479 2694 24.8% 9.7%

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Fig. 10 Impact of cooling flow reduction on CC performance. External „steam… cooling is assumed to replace 30–60% of total cooling duty with εSC = 80%. The origin „0, 0… is the average state-of-the-art F-Class air-cooled GT.

flow as TIT increases. In steam-cooled turbines, particularly in H-Class, all nonchargeable flows 共ideally兲 are substituted by external cooling. In the simplified model presented herein, as explained earlier on page 3, the distinction between chargeable and nonchargeable flows is not that clear-cut. In fact, due to that very reason, method II used in steam-cooled turbine calculations is more conservative or pessimistic in performance prediction than method I used in air-cooled turbine calculations. However, it is also known that the simple constant property approach employed by the model is liable to overestimate the turbine performance. In interpreting the CC efficiencies in Fig. 10, the reader is thus advised to consider the air-cooled turbine numbers overoptimistic by about 1–2 ppt. Due to the opposing effects of these two model imperfections, however, steam-cooled turbine performance predictions are probably closer to “reality.” A well-known variation of the basic Brayton cycle is the reheat cycle, which is the basis for a commercially proven GT and referred to as “sequential combustion” by the OEM 关33,34兴. The underlying thermodynamics can be found in basic textbooks 关1兴. The model herein 共via method II in Table 2兲 is used to estimate the reheat GT performance via repetitive use of Eqs. 共9兲–共12兲 for each expansion-combustion sequence 共labeled HP and LP兲. Cooling duty of HP and LP turbines are calculated using Eqs. 共1兲–共5兲 separately for each. Reference 关33兴 indicates that the HP turbine of the reheat GT is a single impulse stage, as opposed to the four-stage LP turbine. Due to the smaller expansion ratio and shorter mechanical layout of the HP turbine, a simple scaling is applied to the cooling flow correction factor from Table 3 by considering the approximate ratio of the temperature drops in the HP turbine and overall turbine 1/3 1.35 · 共PRHP − 1兲 ␮c,HP = ␮c · T3,HP 1/3 · PRLP −1 T3,LP

冉

冊

共27兲

OEM-specified CAC heat recovery 共i.e., 12 MW in CC configuration for the 50-Hz 288 MW reheat GT兲 required cooling of compressor extraction air from ⬃1085° F to 800° F. The closest match to the published GT rating 共650 kg/s airflow with PR = 33.9 and 60–40 split between HP and LP combustor fuel flows兲 is obtained as follows: HP turbine with PR= 2, HP TIT of 1232° C 共reheat combustor inlet ⬃1000° C兲, and LP TIT of 1538° C. The 011801-10 / Vol. 133, JANUARY 2011

resulting performance is reasonably close to the OEM-published performance data quoted in Ref. 关34兴 only with the assumption of improved cooling technology 共i.e., materials, TBC, etc.兲 represented in the model by a 30° C increase in Tb 共see Table 4兲. Even with an improved cooling technology, reheat combustion cycle does not offer a significant advance over the standard Brayton cycle. It is clear that excessive increase in cooling air requirement negates the favorable impact predicted by fundamental thermodynamics. Combining external 共i.e., steam兲 cooling with reheat, on the other hand, is a viable path to significant CC performance improvement via higher PR and TIT. Assuming steam cooling to replace 30% of the total cooling load, the CC performance becomes 759 kJ/kg and 59.7%, an improvement of 1.5 ppt over the current F-Class SOA. Figure 11 shows a projection of the CC net efficiency improvement with increasing TIT 共via method II兲. Higher temperatures are enabled via reheat and closed-loop steam cooling technologies. SOA cooled turbine technology 共i.e., materials, TBC, film cooling, component efficiencies, etc.兲, as represented by the model parameters in Table 3, is assumed. For the air-cooled and steam-cooled Table 4 Air-cooled GT with reheat combustion. HP turbine pressure ratio is 2 „per Ref. †33‡…, CAC to 800° F „425° C…. Resulting LP combustor inlet temperature is 1000° C „ca. 60–40 fuel split…. Model 共method II兲

OEM Ref. 关32兴 Airflow 共kg/s兲 PR HP TIT 共°C兲 LP TIT 共°C兲 Exhaust Temp. 共°C兲 GT output 共kJ/kg兲 GT efficiency 共%兲 CC output 共kJ/kg兲 CC efficiency 共%兲 Tb 共°C兲 mc 共% of airflow兲 Qc / HC

650.0 33.9 N/A N/A 616.0 443.1 38.1 652.3 58.3% N/A N/A N/A

Reheat 650.0 33.9 1232 1538 615.3 438.0 37.7 681.6 58.7% 845 29.6% 11.5%

650.0 33.9 1232 1538 595.3 409.0 36.8 641.8 57.7% 815 32.8% 12.5%

No-reheat 664.9 17.8 1480 595.4 410.0 38.0 628.0 58.2% 815 24.8% 9.7%

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Fig. 11 Air-cooled „AC-NRHT…, air-cooled with reheat „AC-RHT…, steamcooled „SC-NRHT…, and steam-cooled with reheat „SC-RHT… gas turbine based CC performance

variants with no reheat PR values of 18 and 23, respectively, are assumed. PR is increased to limit the GT exhaust temperature to 1200° F 共650° C兲. For the reheat units, HP PR and TIT values are kept at 2 and 2250° F, respectively. The overall PR is set to 33.9 for air-cooled and steam-cooled variants, but increased as needed to limit the GT exhaust temperature to 1300° F 共700° C兲, as long as the compressor discharge temperature did not exceed 650° C. Two steam cooling technology levels are considered, as represented by 30% and 60% of the total turbine cooling duty, respectively. No further optimization is attempted. As stated earlier, in terms of being close to a “real” system performance, which can only be obtained by extremely rigorous and detailed stage-bystage turbine models 共also supported by field data兲, the numbers in Fig. 11 should be considered the best-possible predictions that can be obtained from a very simple model as described herein. The results in Fig. 11 suggest that, using today’s technology in terms of cooling techniques, materials and component efficiencies, SC-RHT is the most promising GT technology for TIT values at or below 2900° F 共⬃1600° C兲. This is approximately the limit of current DLN combustion technology for single-digit NOx emissions. At higher TIT values, excessive cooling air expenditure of reheat turbines 共relative to their nonreheat counterparts兲 favors closed-loop steam cooling with no reheat. Within that emissions and combustion technology driven TIT limit, up to 4 ppt of improvement in ␩CC is conceivable. It should be emphasized that this only represents a thermodynamic potential, which may or may not be feasible, mechanically and/or economically, in the near future. Further advances in HGP materials, coating, film cooling techniques and component efficiencies are expected to enhance the potential performance improvement. Extremely high cycle pressure ratios with resulting excessive compressor discharge temperatures 共e.g., ⬃1200° F or 650° C兲 and reheat cycles with advanced cooling schemes pushing GT exhaust temperatures up to and even beyond 1300° F 共⬃700° C兲 raise the possibility of two well-known Brayton cycle enhancement options: compressor intercooling and exhaust gas recuperation. Both technologies were around for a long time so that detailed studies can be found in most textbooks on the subject 关27兴. In fact, they were incorporated into two aeroderivative GTs for land-based power generation, which are commercially available 共e.g., see Refs. 关35,36兴兲. The model herein can be easily modified Journal of Engineering for Gas Turbines and Power

to incorporate the effects of these Brayton cycle features for thermodynamic analysis. Due to limitations of space and the fact that these variants have been studied in great detail in the past, the reader is encouraged to consult available literature. Both intercooling and recuperation have so far been unable to make inroads into heavy-duty industrial GT designs due to significant difficulties associated with mechanical design challenges 共materials, size and cost兲 and RAM considerations. Until recently, 60% net CC efficiency, established by the DOE’s advance turbine systems 共ATS兲 program in 1992, was the barrier to be broken by the new generation of power plants. Sponsored by the DOE’s National Energy Technology Laboratory 共NETL兲, the more recent Vision 21 program has a goal to develop, by 2015, the core modules for a fleet of fuel-flexible, multiproduct energy plants that boost power efficiencies to more than 60% 共75% on an LHV basis for gas-fueled plants兲, emit virtually no pollutants, and with carbon sequestration release minimal or eliminate all carbon emissions. Lately, several published claims about the future of the CC plant performance at the end of the first quarter of the 21st century stated a goal of 65% with the possibility of over 70% with an air-cooled GT 关37兴. The technology curves in Figs. 10 and 11 underline the significant hurdles to be overcome in terms of cooling flow reduction in pace with increasing firing temperatures. A combination of advanced materials, including ceramic matrix composites and external cooling 共most likely via steam in a closed-loop system兲 accompanied by bottoming cycle advances is imperative to even approach the stated efficiency goals with acceptable NOx and CO emissions. Performance improvement via bottoming cycle technology advances is not considered explicitly in this paper. Today’s SOA technology reflects an exergetic conversion effectiveness of about 72% 关23兴. In other words, the net power output 共ST generator output minus power consumption of boiler feed pumps兲 of the Rankine BC is 72% of the theoretically possible maximum value, which could only be possible with an ideal Carnot engine. Each advance in Rankine bottoming cycle technology that adds one 1 percentage point to the RBC exergetic efficiency is worth 4 percentage points in ␩CC. The practical entitlement is highly unlikely to exceed 10 percentage points or 2.5 percentage points in ␩CC due to the extreme maturity of the existing technology and limitations imposed by equipment size and cost considerations 关23兴. JANUARY 2011, Vol. 133 / 011801-11

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Even more drastic limitations can be imposed by site conditions 共e.g., lack of sufficiently cold condenser cooling water, environmental regulations limiting water consumption, water discharge and/or stack gas temperatures兲. On the other hand, extreme swings in fuel prices and greenhouse gas emission caps, penalties, and other economic factors might have a positive impact to enable aggressive designs toward that practical entitlement.

Conclusions

A simple cooled turbine modeling system is developed for the analysis of heavy-duty industrial gas turbines with cooling of HGP components. The model is based on a separation of the turbine main airflow and cooling flow streams into topping and bottoming Brayton cycles. This enables the engineer to analyze the overall GT system by using relatively simple ideal Brayton cycle formulations enhanced with basic physics-based relationships to incorporate the effects of turbine HGP cooling. The model is tested and calibrated using GT rating data available from the trade publications. It is reasonably expected to predict CC efficiency within ⫾1% 共points兲, which is adequate for most front-end feasibility studies and technology curves. Using the calibrated model for the SOA F-Class air-cooled GT technology, optimal Brayton cycle PR-TIT combinations for maximizing GT specific output and CC net efficiency are investigated. Advanced GT cycles with closed-loop 共steam兲 cooling and reheat 共sequential兲 combustion are evaluated for their potential to improve CC performance with existing materials and cooling technologies. The resulting technology curves point to the reasonably achievable improvements over today’s SOA status with advancements in cooling technologies 共including turbine materials and coatings兲 and component aerodynamic efficiencies. The model is amenable to further enhancement via more realistic property calculations 共i.e., temperature dependent c p and ␥兲 and finer calibration using actual detailed engineering models or test data for a particular GT while preserving its simplicity and transparency.

Acknowledgment The author would like to thank GE Energy for permission to publish.

Nomenclature c p ⫽ constant-pressure specific heat 共Btu/lb R 共4.1868 kJ/kg K兲兲 e ⫽ specific exergy 共availability兲共Btu/lb 共2. 326 kJ/kg兲兲 h ⫽ specific enthalpy 共Btu/lb 共kJ/kg兲兲 k ⫽ isentropic exponent ˙ ⫽ mass flow rate, pps 共0.4536 kg/s兲 m P ⫽ pressure 共psi 共absolute兲共0.06895 bar兲兲 ppt ⫽ one full percentage point ˙ ⫽ heat transfer rate 共kW 共0.947817 Btu/s兲兲 Q s ⫽ specific entropy 共Btu/lb R 共kJ/kg K兲兲 T ⫽ temperature 共°F 共°C = 关°F − 32兴 / 1.8兲兲 ˙ ⫽ power 共kW or Btu/s兲 W w ⫽ specific power 共work兲共Btu/lb 共kJ/kg兲兲 Greek Symbols ␤ ⫽ ␹ ⫽ ␧ ⫽ ␦ ⫽ ␥ ⫽ ␮c ⫽ ␸ ⫽ ␾ ⫽ ␩ ⫽

exponent in cooling effectiveness equation ratio of cooling flow to airflow exergetic conversion effectiveness turbine to compressor PR ratio specific heat ratio cooling flow correction factor cooling effectiveness fuel flow correction factor efficiency

011801-12 / Vol. 133, JANUARY 2011

Subscripts a b c cac exh f g 0 sc 1, 2, 3, 4

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

air turbine blade 共or generic HGP component兲 coolant or cooling air cooling air cooling GT exhaust gas fuel gas or combustion gas reference state for zero enthalpy steam cooling GT state points 共see Fig. 1兲

Acronyms 3PRH CAC DLN DOE DS GT HC HP HRSG LP LHV NG OEM PR RAM ST SX TIT

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

three-pressure, reheat cooling air cooling dry low NOx 共combustor technology兲 U.S. Department of Energy directionally solidified gas turbine heat consumption 共gas turbine energy input兲 high pressure heat recovery steam generator 共boiler兲 low pressure lower heating value natural gas 共100% CH4 fuel兲 original equipment manufacturer GT compressor pressure ratio reliability, availability, maintainability steam turbine single crystal turbine S1N inlet total temperature; same as T3

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