THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS E. 47th $t., New York, N.Y

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1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS E. 47th $t., New York, N.Y E S The Society shall not be responsible for statements or opinions advanced in papers or discussion at meetings of the Society or of its Divisions or 0 Sections, or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal. Authorization to photocopy for internal or personal use is granted to libraries and other users registered with the Copyright Clearance Center (CCC) provided $3/article or $4/page is paid to CCC, 222 Rosewood Dr., Danvers, MA Requests for special permission or bulk reproduction should be addressed to the ASME Technical Publishing Department. Copyright 1998 by ASME All Rights Reserved Printed in U.S.A. A GAS TURBINE DYNAMIC MODEL FOR SIMULATION AND CONTROL Horacio Perez-Blanco Todd B. Henricks, P. E. Assoc. Professor, ME Department Lieutenant, Civil Engineer Corps Pennsylvania State University United States Navy 338B Reber Building NAVFACENGCOM University Park, PA Phone: Phone (814) Fax: Fax (814) ABSTRACT The useful life of gas turbines and the availability of power after I Moment of inertia (kgm2) start-up depend on their transient response. For this reason, several articles have been written on the dynamic simulation of gas turbine k Specific heat ratio systems in electrical generation, cogeneration, and marine m Mass (kg) applications. The simulations typically rely on performance maps p Pressure (Pa) and time lags extracted from manufacturer's specifications. This work was undertaken to increase the generality of turbine models over P Power (kw) what can be obtained from performance maps. The paper describes a R Radius (m) mathematical computer model developed to investigate the dynamic response of a simple single-shaft gas turbine system. The model r Degree of reaction uses design parameters normally incorporated in gas turbine design tbh Turbine blade height (m) (e.g. load coefficient, flow coefficient, and dehaller Number) as well as compressor and turbine stage geometry and compressor and tdr Turbine blade radius (m) turbine material properties. A dynamic combustion chamber model is T Temperature (K) also incorporated. Other input parameters are included to enable the U Blade tip speed (m/s) model to be adaptable to various system sizes and environments. The model was formulated in a graphical interface, and the V Velocity (m/s) results of several trials are displayed. The influence of important W Specific work (kj/kg) parameters (e.g. fuel-air ratio, IGVs, load, efficiencies) on turbine response from a "cold" start and from steady-state is studied. To gain further insights into the response, a start-up procedure similar to that Greek Symbols reported in the literature for an industrial gas turbine system is a Fraction simulated. Because of the approach used, the computer model is easily adaptable to further improvements and combined simulation of Difference turbines and control systems. r Efficiency NOMENCLATURE p Density (kg/m3) B Hub to tip ratio T Torque (Nm) chb Compressor blade height (m) cdr Compressor disk radius (m) Cp Specific heat (kj/kg*k) f Fuel to air ratio q yv S2 Load coefficient Flow coefficient Angular velocity (rad/sec) F friction Superscripts H De Haller number Rate; (kg/s for m, 1/s 2 for S2) h Enthalpy (kj/kg) Presented at the International Gas Turbine & Aeroengine Congress & Exhibition Stockholm, Sweden June 2 June 5, 1998

2 Subscripts a Axial C Compressor L Load mix Mixture (fuel-air) r Root S Stage t Tip T Turbine 0 Stagnation 1 Compressor inlet 2 Compressor outlet 3 Turbine inlet 4 Turbine outlet INTRODUCTION Transient performance of gas turbines has a strong bearing on component life and performance. Issues of thermal fatigue (Johnson et al, 1993) determine the start-up sequence, since rapid start-ups can impair turbine blade life. As hot gas is generated in the combustion chamber from a cold-start, gradual heating of the turbine components reduces transient thermal stresses. Consequently, a careful start-up procedure and synchronization is followed in power generation applications. In marine propulsion applications, system response in terms of fuel consumption for an oscillatory load is also of concern (Smith and Stammetti, 1990), calling for advanced control strategies, assessed via dynamic modeling. Inevitably, the simulation conclusions depend on the model adopted for the gas turbine (Smith, 1988), where a semiempirical model, largely based on performance data and time lags, was used. The same approach was expanded and refined in (Smith and Stammetti, 1990). Similar empirical approaches have been employed for relatively complex systems (Ahluwalia and Domenichini, 1990). A cogeneration plant consisting of a gas turbine, a boiler, and two steam turbines (high and low pressure), was simulated. The system was approximated by using the performance curves given by manufacturers, and adding necessary time lags based on manufacturer information. Time-dependent equations were integrated with Euler's method. It was concluded that dynamic simulation was a valuable tool for assessing the suitability of controls and safety measures. A more basic approach to modeling has been adopted (Hussain and Seifi, 1992). The basic thermodynamic equations for mass and energy balances were used for model formulation, as opposed to manufacturer's curves. Yet, the inertia of rotors and shafts was absent from the equations, which probably explains the need found by the authors to add time lags to this model. The model effectively predicted behavior during load perturbations at full load. More recent work focused on developing the gas turbine model based on design parameters (Facchini, 1993). Load coefficients and stage efficiencies were employed for this simulation, along with part-load losses extracted from the literature. Inertial aspects seem to have been again neglected. The need to combine basic thermodynamic equations with inertial aspects for generic models was recognized recently. In 1996, the transient equations for inlet duct pressure and mass flow rate were formulated (Bettocchi et al). Compressor and combustion chamber conditions were calculated from a transient analysis coupled with polytropic process equations. Performance maps of each component were used to obtain isentropic efficiencies. A dynamic equilibrium equation was used to calculate the acceleration of the turbine. Good comparisons with experimental performance were obtained. The present work is inserted in this context: whereas most models are semiempirical, in that they rely on time constants and performance maps, serious recent attempts to extend the generality of gas turbine models have been made. Clearly, the flexibility of a model to competently handle a wide variety of turbines is the target. Similar to previous work, the trend of formulating the basic mechanical and thermal equations for each component is followed here. In an extension of previous work, it is attempted wherever possible to include mechanical and thermal design parameters to broaden the reach of the model. The main contribution consists of modeling the compressor stage by stage, and of including thermal and mechanical inertial effects based on first-principles, rather than on empirical constants. It is realized that complex machines cannot always be represented by simple models. Hence, the limitations of this approach are clear, especially at part-load. However, with the advent of computing power and of transient simulation software, it is inefficient to restrict modeling to numerical fitting of performance curves. Establishing feasibility of design and of control and operating strategies may call in the future for models of this type. MATHEMATICAL MODEL The compressor, combustion chamber, and turbine shown in Fig. 1 are modeled based on the following assumptions: (1) Moments of inertia are estimated analytically. (2) The diameter change of each successive stage of the compressor and turbine is approximated by constant factors. (3) The compression and expansion processes are adiabatic. (4) The compressor inlet pressure is equal to the turbine outlet pressure. (5) A 16 stage compressor and a 4 stage turbine is assumed. (6) For simplicity, stage efficiencies for the compressor and turbine are constant. (7) Combustors can be modeled as open-ended cylinders of constant wall thickness. (8) The temperature drop across the turbine is divided equally among the four stages. This distribution may be altered as desired.

3 Thermodynamic model Compressor. The compressor model was developed using a design load coefficient, dehaller Number, flow coefficient, degree of reaction, and stage geometry. Starting with load coefficient, degree of reaction and tip speed, the equations used are (Bathie, 1996): Figure 1. Schematic of the three main components: compressor, combustion chamber, and turbine. Mechanical Model The turbine and compressor, along with the load, are connected to a single shaft, such that the dynamic equation for the turbine is: i T =tic +ti p +F L +I c S2 +I T S2 (1) This equation calls for the moments of inertia of the compressor and turbine, which depend on the mass of each stage and its distribution with reference to the centerline of the axis. Whereas moments of inertia can be readily determined from dynamic measurements, an analytical approximation is sufficient for the purposes of this work. This approximation is derived from the model of a rotating disk with rotating rings placed at suitable radiuses (Hattori et al, 1988), as shown in Fig 2. The moment of inertia of the blades is then added to that of the composite disk. A simple approach, consisting of using a constant factor was used to estimate the change in stage outer diameter (Fig 3). Using these parameters, the total moment of inertia was calculated for the compressor (Ic) and the turbine (IT). = (ho2 ho1) U2 R=2 1) (3) h02 hot U=(R+chb) S2 (4) Instead of the tip speed, the mean speed can be used if desired. Combining Eqs. (2), (3) and (4), the temperature rise in each compressor stage can be calculated as follows: (cdr + cbh )2 yr ç2 Toes =T01 + C (5) Tics p Successively using Eq. (5) for each stage results in the compressor exit temperature. The pressure rise in each stage is found by using the following equation: tl,, k P2s = Pts Toes k-t Toffs 6 Where the compressor isentropic efficiency, rlc, is found using the relationship between polytropic and isentropic efficiencies (Cohen et al, 1987): (2) 3/8 r I1/6 r (k -I ) P2T 1 Pi ( ^c = 1 (7) 3/8 r (L2 k Tics Pi r eh ur Figure 2. Cross-section of a compressor and turbine stage. The torque required by the compressor is found by again using degree of reaction, dehaller Number, load coefficient, flow coefficient, and stage blade tip speed: R= h2 h1 (8) h02 h ot where (h02 - h01) is: 2 V12 h 02 h 01 =[h h l + 2^ (9) Fig 3. Compressor and turbine geometrical models. ) 3

4 The dehaller number, H, and the flow and load coefficients are defined as: Also, H = VZ (10) l U (11) W= U Z (12) Combining Eqs. (9-12) yields:!.2.(1h 2 ) R=l+ 2 (13) W The stage torque is given from an energy balance on the stage: _ m' C P. (T02 Tol) Lc 92 TIc (14) Turbine. The torque imparted by the fluid to the turbine is developed starting with the work of an individual stage of the turbine: Ws =Cpmix (T3 T4)s 7lrs (1+f) (15) The torque for the turbine is obtained with an energy balance around the stage: WS (1+f) m (16) tis = Combination of Eqs. (15) and (16) yields (1+f) C (T3 - T4) rlr 'Lr = (17) where the turbine efficiency Ss found using the relationship between polytropic and isentropic efficiencies (Cohen et al, 1987): (k-t) n,s 1-1i P3 (18) T = (k-i ) 1- P4 P3 The turbine outlet temperature is calculated as: T4 = T3 (k-i) 1T P4 k J (19) P3 The load coefficient of the turbine is calculated using the temperature drop across the turbine as a parameter: T3 T4 DTs = 4 (20) C, (AT) Vr = 2 (tdr + tbh) t1r (21) Mass Flow Rate. The mass flow rate through the compressor, assuming no blockage, is given by Cohen et al (1987): f]. m=p A V=p n r1 2 1 r` Va (22) r, Making the following substitutions into Eq. (22) rt = cdr + cbh r B=r Va =0 Q the mass flow rate can be calculated as follows: m=p.it.(cdr+cbh) 2 (1 B 2 ) ^ S2 (23) where the density is calculated using the ideal gas law. Combustion Chamber. The model for the combustion chamber is developed using the thermal mass of the chamber itself, the thermal mass of the fuel-air mixture, and the thermal mass of the first stage of the turbine. A control volume analysis of the combustion chamber yields the following differential equation: C mz I' ICP 'T.2 ]+ (m f (LHV ) / l / (24) ^m 3 ^' [C P ' T03 I+ (MCP )' T03 where mcp is the thermal mass of the system, which is modeled as follows: mc P =[p C P V01L +[p C P Vol}uei-air+[p CP Jiurbine t mixture stage (25) 4

5 The pressure drop across the combustion chamber is calculated by: P3 =Pi +(1-0 ^P2 Pi) (26) where "13" is the fractional pressure drop across the combustion chamber. The complete mathematical model is comprised of Eq. (1) as the basic equation, using Eq. (14) for tc, Eq. (17) for TT, calculated values for Ic and IT, and user-input load and friction torques. The independent variable in these equations is time (t). The remaining equations in the mathematical model are indicated in Table 1. Table 1. Model Equations Variable Eq. Mass flow rate (23) Compressor outlet temperature (5) Compressor outlet pressure (6) Compressor isentropic efficiency (7) Turbine isentro is efficiency (18) Turbine outlet temperature (19) Turbine load coefficient (21) Turbine inlet temperature (24) and (25) Combustion chamber pressure drop (26) RESULTS To demonstrate the versatility of this model, several parameters were varied independently. In each case, all variables were held to their reference state (Appendix 1) except the variable under consideration. Figure 4 shows the output of the baseline run. This run is based on a start-up from "cold" conditions, at constant fuel/air ratio, with a frictional load proportional to the square of the rotational velocity. The model allows an understanding of the variables in the uncontrolled start-up scenario. As long as the turbine produces no work, T3 (peak cycle temperature) and T4 (discharge temperature) are quite close. The compressor discharge pressure increases gradually with time. The dynamics of this turbomachine is clearly reflected here: as soon as the speed increases and the air mass flow ramps up, compressor pressure and peak temperature start to increase sharply. As both T3 and P 2 increase toward the design value, the turbine power output also increases. Congruently, T 4 decreases, and the mass flow rate rapidly increases towards its design value. At steadystate conditions, the turbine develops 56.6 MW to drive the load. In most cases, the turbine accelerates gradually until a certain value of the air mass flow rate is approached. Then, the turbine accelerates quite rapidly. Whether this is a behavior to be expected under an uncontrolled start-up or not, or whether this behavior applies to all turbines needs to be determined as part of model validation. However, the present model provides an inexpensive way to test and refine different control strategies to obtain a smooth startup. Figure 4. Response of turbine to baseline conditions (fuel/air ratio = 0.013, standard atmospheric conditions, and friction load and shaft load proportional to angular velocity squared). Startup is from "cold" conditions at constant fuel/air ratio with frictional load. Inlet guide vanes (IGVs) are completely open. It can be shown that, under steady-state conditions (Table 2), the First Law of Thermodynamics is satisfied. The fuel and air input enthalpy rates equal the power output and exhaust enthalpy rate. The fuel and air represent a MW input, equal to the exhaust discharge (256.8 MW), plus the load and frictional power (61.8 MW). The thermal cycle efficiency is Table 2. Baseline simulation results Fuel-Air Ratio (input) Initial RPM 30 RPM Turbine Inlet Temperature 1265 K Turbine Outlet Temperature 680 K Rotational Velocity 3450 RPM Pressure Ratio 17.4 Time to reach steady state approx. 725 seconds Mass Flow Rate kg/s Power Output 56.6 MW Compressor Isentropic Efficiency 0.78 Turbine Isentropic Efficiency 0.93 Start-Up Runs In what follows, a given parameter was varied and the system was started up from initial conditions. The resulting steady-state speed and time to reach the steady-state speed for each case is shown in Figure 5, where the fuel-air ratio (held constant from turbine startup to steady-state) was varied with the same start-up rotational velocity of 30 RPM. Up to approximately 500 seconds, the speed is the same for all three cases (f/a = 0.012, f/a = 0.013, and f/a = 0.014). After approximately 500 seconds, it is seen that for greater fuel-to-air ratios, less time is needed to reach higher steady-state speeds.

6 r figure a. Influence of fuel to air ratio on response time and final velocity. Figure 6 demonstrates the effect of compressor polytropic (small stage) efficiency (Cohen et al, 1987) on steady-state speed. The plot shows that the rotational speed of the system increases as the compressor polytropic efficiency increases. This is the case, because the more efficient compressor requires less work from the turbine. The peak pressure increases with efficiency, increasing in turn with turbine output Figure 7. Influence of turbine polytropic efficiency on response time and rotational velocity. Variations During Steady-State Operation From the steady-state conditions, certain parameters were varied as described below. Figure 8 shows the effect of load changes. "Load - Decreased" indicates that the load factor was instantaneously reduced from 1.2 to 1.0. (The load is proportional to this factor and to the square of rotational velocity). "Load - constant" indicates no change in the load, and "Load - Increased" indicates that increase from 1.2 to 1.4. An increased load results in lower rotational speed and a decreased load results in higher rotational speed. Similar results are obtained when varying the air to fuel ratio, in that increased fuel input results in increased velocity at constant load. Closing the IGVs reduces air flow, which in turn leads the turbine to settle at a lower speed. Interestingly, limiting the airflow to a value of 50% of design, reduces the final speed to 2510 rpm, or 73% of the baseline value. This is so because both the load and the friction torque were assumed to vary with the square of the rotational speed..500 $_ n^^^^^ m II s I 8^,. ^ ^ 8 m a a^ rim. (.wends) x- Figure 6. Influence of compressor polytropic efficiency on response time and rotational velocity. Figure 7 demonstrates the effect of turbine polytropic (small stage) efficiency (Cohen et al, 1987) on steady-state speed. The plot shows that as the turbine polytropic efficiency increases, the rotational speed of the system increases. This is due to the ability of a more efficient turbine to take more work from the combustion gases and use it to drive the compressor, friction load, and shaft load. The results of varying compressor load coefficient (w) and compressor blade height, are likewise reasonable. Increasing the load coefficient means that more work is communicated per stage to the air. Hence, the peak pressure increases, resulting in a more efficient cycle, which leads to greater power output. Longer blades also result in increased power production, as the air mass flow rate increases. -x---o =p x ^oea aeaeaseo 2800 o Load-=SM j &-toad-htmwad Time (seconds) Figure 8. Influence of changes in load during steady-state operations. 6

7 Start-up procedure Start-up procedures of an actual industrial gas turbine system have been published (Rowen, 1988). These procedures are facilitated using an automatic control system and an external "start-up" device such as a diesel engine or an electric motor. Despite the lack of a model of a control system, an investigation was done to establish if these start-up procedures could be simulated. To facilitate this test, a "start-up" engine was simulated by changing the sign of the load term in Eq. (1), making it negative. The mass flow rate and fuel input were set to zero. Using a VisSim "slider" block, the load was manually varied from a more negative value to a less negative value, allowing the system to come to 100% of design (reference state) rotational speed. This exercise demonstrated that this model can be adapted to an external start-up device and control system to simulate that in Rowen (1988), or others. CONCLUSIONS A computer model was developed to investigate the dynamic response of a simple (no reheat, regeneration, or other auxiliary equipment) single-shaft gas turbine system. The model used parameters normally incorporated in gas turbine design as well as compressor and turbine stage geometries and compressor and turbine material properties. Also incorporated was a combustion chamber model. Input parameters, design parameters and dimensions were readily modified. The results show how given parameters (e.g. fuel-air ratio, load, and efficiencies)influence start-up and steady state. The chosen parameters resulted in a 26% system thermal efficiency. Further, a literature search showed that industrial gas turbines are brought online from "cold" conditions via an elaborate start-up procedure. The goal of this procedure is dual: to avoid undue thermal stresses leading to fatigue, and to allow ready synchronization to external loads. The turbine model presented in this paper can respond to similar requirements as imposed manually by the programmer. In an uncontrolled start-up, the turbine does converge quite rapidly to steady-state conditions. It is noteworthy that the peak temperature increases gradually at first, and more rapidly after the air flow is established. Most of the temperature increase is quite gradual, on the order of 50K/min. Since the turbine responds reasonably both to start-up constraints and parameter variations during steady-state operation, it would appear that this model is a worthwhile development. With improvements, mainly in modeling heat losses, it could be employed to simulate turbine/control system strategies to save resources and development time. ACNOWLEDGEMENTS The authors wish to thank the US Navy for support of MR. Hendrick's work. Also, thanks are due to Visual Solutions for a superb software, and to Cheryl Adams for formatting this manuscript. APPENDIX 1 Model Input Variables Table 2.1-Compressor Variables Material density 7000 kg/m3 Blade height 0.41 m Blade length 0.02 m Blade thickness m Disk radius 0.52 m Disk thickness 0.02 m Number of blades 62 Decrease factor 0.98 Polytropic efficiency 0.85 Table 2.2-Turbine Variables Material density 7000 kg/m' Material specific heat 582 J/kgK Blade height 0.5 m Blade length 0.02 m Blade thickness m Disk radius 0.4 m Disk thickness 0.02 m Number of blades 87 Increase factor 1.1 Polytropic efficiency 0.89 Table 2.3-Combustor Variables Combustor diameter 0.35 m Combustor length 0.7 m Number of combustors 8 Combustor wall thickness m Combustor material density 4510 kg/m3 Combustor material specific heat 582 J/kgK Pressure drop 2%

8 Input pressure Input temperature Table 2.4-Thermodynamic Variables Hussain, A., and Seifi, H., "Dynamic Modeling of a Single Shaft Gas Turbine, " IFAC Symposium on Control of Power Plants and Power Systems, March 9-11, Ratio of specific heats for air 1.4 Ratio of specific heats for combustion products Universal gas constant Specific heat of air Specific heat of combustion products Lower heating value of fuel Molecular weight of air External load factor Table 2.5-Other Variables Pa 289 K J/molK 1005 J/kgK 1100 J/kgK J/kg kg/kmole Load coefficient (Compressor) 0.3 Flow coefficient (Compressor and Turbine) 0.72 Hub-to-tip ratio (Compressor) 0.75 Friction load factor 2.0 kgm kgm2 Johnson, D., Miller, R. W., and Rowen, W. I., "Speedtronic TM Mark V Gas Turbine Control System," General Electric Publication GER- 3658C,1993. Operator's Manual, VisSim Software v. 2.0, Visual Solutions, Inc., Rowen, W. I., "Operating Characteristics of Heavy-Duty Gas Turbines in Utility Service," ASME Paper 88-GT-150, Smith, D. L., "Linear Modeling of a Marine Gas Turbine Power Plant," ASME Paper 88-GT-71, Smith, D. L., and Stammetti, V. A., "Comparative Controller Design for a Marine Gas Turbine Propulsion System," Journal of Engineering for Gas Turbines and Power, Vol. 112, April Smith, D. L. and Stammetti, V. A., "Sequential Linearization as an Approach to Real-Time Marine Gas Turbine Simulation," Journal of Engineering for Gas Turbines and Power, Vol. 112, April Turbomachinery International Handbook/l 995, Turbomachinery Publications, REFERENCES Ahluwalia, K. S., and Domenichini, R., "Dynamic Modeling of a Combined-Cycle Plant, " Journal of Engineering for Gas Turbines and Power, Vol. 112, April Bathie, W. W., "Fundamentals of Gas Turbines," 2"d Ed., John Wiley & Sons, Inc., Bettocchi, R., Spina, P. R., and Fabbri, F., "Dynamic Modeling of Single-Shaft Industrial Gas Turbine," ASME Paper 96-GT-332, June Cohen, H., Rogers, G. F. C., and Saravanamuttoo, H. I. H., "Gas Turbine Theory," 3`d Ed., John Wiley & Sons, Inc., Facchini, B., "A Simplified Approach to Off-Design Performance Evaluation of Single Shaft Heavy Duty Gas Turbines, " ASME International Gas Turbine Institute Proceedings, Vol. 8, September Hattori, T., Ohnishi, H., and Taneda, M., "Optimum Design Technique for Rotating Wheels," Journal of Engineering for Gas Turbines and Power, Vol. 110, January 1988.