Gas Turbines Part 14 pdf

Gas Turbines 314 Step 4. Calculate the possible maximal number of redundant graphs given by rr inv Max ++ =−CX Step 5. Initialize the number of initial node n i = 1 in the search and the number of assigned redundant graph n GR =0. Step 6. Calculate the possible distinct combinations of the initial nodes for each target, selecting n i nodes out of n k − 1, with n k the cardinality of set K; this means 1 1 (1)! for each tar g et node (1)!()! k i k n k n i kii n n I n nnn − − ⎛⎞ − == = ⎜⎟ −− ⎝⎠ C Step 7. Assign the orientations of the I graphs using the set inv + C for each target node including the cycle graphs (no diagonal submatrix) and constraints of the class d. Step 8. Bring up the number n GR according the assigned redundant graphs; if n GR = Max rr , end the algorithm, otherwise continue. Step 9. If n i = n k − 1, end the algorithm, on the contrary n i = n i + 1 an return to step 6. 3. Gas turbine description The GT behavior model used at this work simulates electrical power generation in a combined cycle power plant configuration with two GT, two heat recovery-steam generators and a steam turbine. At ISO conditions, the ideal power delivered for each GT generates 80 MW and the steam turbine 100MW. This model may go from cold startup to base load generation. The main components of the GT shown in Fig. 3 are: compressor C, combustion chamber CC, gas turbine section T, electric generator EG, and heat recovery HRSG. Exciter Start Motor Generator MW Compressor Combustion Chamber Gas Turbine HRSG Stack After Burners ValveAB k 16 k 9 k 18 k 6 x 15 x 12 k 11 x 18 k 14 k 10 x 23 x 25 x 17 x 16 x 6 k 5 k 17 x 8 x 10 k 8 k 9 k 19 x 26 x 1 x 14 k 15 k 2 x 11 Vel k 13 x 3 k 3 k 4 k 7 x 9 k 1 k 12 Fig. 3. Components of the Gas Turbine Application of Structural Analysis to Improve Fault Diagnosis in a Gas Turbine 315 Fig. 4. Gas Turbine Variables Interconnection The GT unit has two gas fuel control valves; the first supplies gas fuel to CC, and the second one supplies gas fuel to heat-recovery afterburners (starting a second- additional combustion at heat recovery for increasing the exhaust gases temperature). A generic compressor bleed valve extracts air from compressor during GT acceleration, avoiding an stall or surge phenomena. Also the GT unit has an actuator for the compressor inlet guide vanes, IGVs, to get the required air flow to the combustion chamber. The dynamic nonlinear model is developed in (Delgadillo & Fuentes, 1996) and it is integrated by n c = 28 constraints, n s = 19 static algebraic constraints, and n = 9 dynamic-differential constraints. Concerning the variables one can identify 27 unknown variables x i and 19 known variables k i . The generic architecture and interconnection of the GT’s components are described by the block scheme given in Fig. 4. The variables and parameters for each block of the scheme are related by the constraints described in table 3. The variables are given in Appendix 8 and the description of the functions and parameters can be consulted in (Sánchez-Parra et al., 2010). 4. Analysis of the structure for the gas turbine Considering constraints and variables of the model described in Table (3) the following sets for the graph description are identified: • The set of known variables is given by sa p c = KY Y U U∪∪ ∪ (10) with cardinality 19. The process sensors is determined by the set { } 126101112131415 ,,,,,,,, s kkkk k k k k k=Y (11) with | Y s |=9; the position transducers from actuators define the set Y a ={k 5 , k 7 , k 8 , k 16 }; the external physical variables determine the set U p = {k 3 , k 4 , k 9 }; and the control signals defines the set U c = {k 17 , k 18 , k 19 }. • There are 28 physical parameters θ i which are assumed constant in normal conditions Sánchez-Parra & Verde (2006). Gas Turbines 316 Compressor Unit, C Combustion Chamber Unit, CC c1: 0 = f (x 1 , x 6 ,k 1 ,θ 0 ) c2: 0 = f (x 3 ,k 1 ,k 2 ,k 3 ,k 5 ,θ 1 ,θ 2 ,θ 3 ) c3: 0 = f (x 3 , x 8 ,k 1 ,k 3 ,θ 4 ,θ 5 ) c4: 0 = f (k 1 ,k 3 ,k 4 ,k 6 ,θ 5 ) c5: 0 = f (x 9 ,k 1 ,k 3 ,k 6 ,k 7 ,θ 6 ) c6: 0 = f (x 3 , x 9 , x 10 ) c7: 0 = f (x 5 ,k 5 ,k 17 ,θ 25 ) d1: 0 = x 5 − dk 5 dt c8: 0 = f (x 6 , x 12 ,k 1 ,k 8 ,k 9 ,θ 7 ) c9: 0 = f (x 10 , x 12 , x 14 ) c10: 0 = f (x 6 , x 15 ,k 1 ,θ 21 ) c11: 0 = f (x 1 , x 2 , x 14 , x 15 ,θ 17 ) d2: 0 = x 2 − dx 1 dt c12: 0 = f (x 1 , x 6 , x 7 , x 10 , x 12 , x 14 ,k 6 ,θ 8 , θ 9 ,θ 17 ,θ 18 ,θ 19 ) d3: 0 = x 7 − dx 6 dt c13: 0 = f (x 13 ,k 8 ,k 18 ,θ 26 ) d4: 0 = x 13 − dk 8 dt Gas Turbine Unit, GT Heat Recovery Unit, HR c14: 0 = f (x 10 , x 12 , x 16 ,k 6, ,θ 8 ,θ 9 ,θ 18 ) c15: 0 = f (x 1 , x 17 ,k 1 ,k 10 ,θ 10 ) c16: 0 = f (x 1 , x 16 , x 17 , x 18 ,k 1 ,θ 10 ) c17: 0 = f (x 6 ,k 11 ,k 12 ) c18: 0 = f (x 6 ,k 1 ,k 10 ,k 11 ,θ 10 ) c19: 0 = f (x 19 ,k 2 ,θ 11 ) d5: 0 = x 4 − dk 2 dt c20: 0 = f (x 4 , x 8 , x 11 , x 15 , x 16 , x 18 , x 19 ,k 2 ,k 13 ,θ 20 ) c23: 0 = f (x 23 ,k 10 ,k 14 ,θ 0 ) c24: 0 = f (x 25 ,k 3 ,k 10 ,k 15 ,θ 23 ) c25: 0 = f (x 26 ,k 9 ,k 10 ,k 14 ,k 16 ,θ 24 ) c26: 0 = f (x 15 , x 23 , x 24 , x 25 , x 26 ,θ 16 ) d7: 0 = x 24 − dx 23 dt c27: 0 = f (x 15 , x 22 , x 23 , x 26 ,k 11 ,k 14 ,θ 9 , θ 16 ,θ 18 ,θ 19 ) c28: 0 = f (x 27 ,k 16 ,k 19 ,θ 27 ) d8: 0 = x 22 − dk 14 dt d9: 0 = x 27 − dk 16 dt Electric Generator Unit EG c21: 0 = f (x 20 , x 21 ,k 13 ,θ 12 ,θ 13 ,θ 14 ,θ 15 ) c22: 0 = f (x 20 , x 21 ,k 2 ,θ 22 ) d6: 0 = x 21 − dx 20 dt Table 3. GT Model Equations with the variables meaning given in the appendix • The constraints set is given by 19 static constraints and 9 state constraints which require their additional constraints (di) and known variables. Then the constraints set has cardinality 37 and is given by { } { } 1, 2, , 28 1, 2, , 9cc c dd d=… …C ∪ (12) • The unknown variables are 27 and define the set =   XXXX∪∪ (13) where the dynamic unknown variables set has cardinality 4 and is given by { } 1 6 20 23 ,, , ,xxx x=X (14) the unknown variables set which are related by static relations of cardinality |  X |=14 are Application of Structural Analysis to Improve Fault Diagnosis in a Gas Turbine 317 { } 3891011121415161718192526 ,,,,,,,,,,,,,xxxxxxxxxxxxxx=  X (15) and the differential of the state variables are { } 5,2713421242227 ,, ,, , , ,xx x x x x x x x=  X (16) with |  X |= 9. Considering the above described sets of variables and constraints, the Incidence Matrix, IM, of dimension (37 × 27) is first obtained and this is the start point of the structural analysis. Using Matlab (MATLAB R2008, 2008) the decomposed incidence matrix given in Fig. 5 is obtained. The bottom sub-matrix IM + ∈ I 30×20 is associated to G + and IM 0 ∈ I 7×7 for G 0 with G − = Ø. The diagnosticability analysis of the first part of the analysis takes into account only the over-constrained G + . The issue of the undetectability of the subgraph G 0 will be addressed in Section 5. 4.1 Redundancy of the GT structure Based on the subgraph G + , the maximum number of RG is given by |C + | −|X + | = 10. Considering the matching sequences described in the first 20 rows of Fig. 6 and concatenating these with other 10 constraints, Table 4 is obtained and the failured components which can be detected in the GT are identified. The third column indicates the variables used to detect faults involved in the respective set of constraints for each RG. One can see that some faults can be supervised using two RGs. As example faults in the component of constraint c 9 can be supervised by the graph RG 7 or RG 8 with different subsets of K. Table 4 is obtained and the failured components which can be detected in the GT are identified. 5. Diagnosticability improvement in the GT The subsystem G 0 given at the top of the matrix in Fig. 5 describes the process without redundant data and and the unique matched graph is shown in Fig. 7. It involves some of turbogenerator variables given in Table 3. Without redundant relations, it is impossible to detect a fault at the turbogenerator section with the assumed instrumentation. Giampaolo (2003) calls this subsystem, GT Thermodynamic Gas and includes the non-measured variables: compressor energy and rotor-friction energy ( x 8 , x 19 ); exhaust gases enthalpy and combustion chamber gases enthalpy ( x 18 , x 16 ); exhaust gases density x 17 , rotor acceleration x 4 and the start motor power x 11 . Thus, the main concern of this section is the identification of the unknown variables, which can be measured and converted to new known variables. So, with this the graph decomposition G 0 will be empty and the getting of the respective ARR yields by the new measurement. 5.1 Graph structure modification The oriented graph of G 0 assuming the known variables subset K is shown in Fig. 7. The absence of paths which link a subset of known variables is recognized. The unknown variables X 0 cannot be bypassed in any path and as consequence does not exist a RG. Gas Turbines 318 Fig. 5. Decomposed Incidence Matrix for the GT, where G 0 and G + are identified by blocks Application of Structural Analysis to Improve Fault Diagnosis in a Gas Turbine 319 RG ’s Used Constraints C + Known variables K RG 1 c 4 k 1 ,k 3 ,k 4 ,k 6 RG 2 c 17 ,c 18 k 1 ,k 10 ,k 11 ,k 12 RG 3 d 1 ,c 7 k 5 ,k 7 RG 4 d 4 ,c 13 k 8 ,k 18 RG 5 d 9 ,c 28 k 16 RG 6 d 8 ,c 10 ,c 17 ,c 23 ,c 25 ,c 27 k 1 ,k 9 ,k 10 ,k 11 ,k 12 ,k 14 ,k 16 ,k 19 RG 7 d 2 ,c 1 ,c 2 ,c 5 ,c 6 ,c 8 ,c 9 ,c 10 ,c 11 ,c 17 k 1 ,k 2 ,k 3 ,k 5 ,k 6 ,k 7 ,k 8 ,k 9 ,k 11 ,k 12 RG 8 d 3 ,c 1 ,c 2 ,c 5 ,c 6 ,c 8 ,c 9 ,c 12 ,c 17 k 1 ,k 2 ,k 3 ,k 5 ,k 6 ,k 7 ,k 8 ,k 9 ,k 11 ,k 12 RG 9 d 6 ,c 21 ,c 22 k 2 ,k 13 RG 10 d 7 ,c 10 ,c 17 ,c 23 ,c 24 ,c 25 ,c 26 k 1 ,k 3 ,k 9 ,k 10 ,k 11 ,k 12 ,k 14 ,k 15 ,k 16 ,k 19 Table 4. Redundant Graphs obtained from G + Fig. 6. Matching for the GT to get 10GR Gas Turbines 320 Fig. 7. Subgraph G 0 without redundant information To determine which variables of G 0 could modify this lack of detectability, paths which satisfy the RG conditions assuming new sensors has to be builded. Then, one has to search for paths between known variables which pass by the constraint c 20 . On the other hand, from the incidence matrix of the Table 5 one can identify that variable x 11 appears only in the constraint c 20 . Thus, there are not two different paths to evaluate it. To pass by c 20 the only possibility is to asume that x 11 is measurable. Taking into account physical meaning of the set X 0 , it is feasible to assume that the start motor power x 11 is known. This proposition changes the GT structure, transforming the whole structure to an over-constrained graph. In other words adding a dynamo-meter to the GT instrumentation, x 11 became a new known variable, k 20 = x 11 , and allows the construction of the redundant graph described in Table 5. One verify that estimating first the set {x 1 , x 3 , x 10 , x 12 , x 15 } by subsets of K and C + , one can estimate 11 ˆ x following the path. Thus, the relation 20 20 ˆ ()rt k k=− (17) can be used as to generate a residual and the respective ARR 11 depends on the variables set { } * 1 2 3 5 6 7 8 9 10 11 12 13 20 ,,,,,,,, , , , ,kkkkkkkkk k k k k=K (18) and the set of constraints of the turbogenerator { } 1 2 3 5 6 8 10 14 15 16 17 19 5 20 ,,,,,,,,,,,,,ccccccc c c c c c dc ∗ =C (19) Thus, any changes in the parameters and the functions involved in this set of constraints generates an inconsistent in the evaluation of the target node 20 ˆ k . 5.2 Simulation results To validate the obtained redundant relation, a change in the friction parameter Δθ 11 = 2 in c19 of the turbogenerator non linear model has been simulated. The time evolution of the Application of Structural Analysis to Improve Fault Diagnosis in a Gas Turbine 321 C 0 X + K x 4 x 19 x 17 x 16 x 18 x 8 k 20 = x 11 d5 k 2 → ⊕ c19 k 2 → ⊕ c15 x 1 k 1 ,k 10 → ⊕ c14 x 10 ,x 12 k 6 → ⊕ c16 x 1 k 1 → • • ⊕ c3 x 3 k 1 ,k 3 → ⊕ c20 x 15 k 2 ,k 13 → • • • • • → ⊕ Table 5. Matching Sequence of G 0 to get Fault Detectability Fig. 8. Residual generated by the new ARR 11 detecting friction fault at 5000s residual (17) for a fault appearing at 5000 s is shown in Fig. 8. The fast response validates the detection system. Note that during the analysis of the detection issue, any numerical value of the turbine model can be used, giving generality to this result. The values set is used for the implementation of the residual or ARR, but not in the analysis. 6. Conclusions A fault detection analysis is presented focused on redundant information of a gas turbine in a CCCP model. The study using the structural analysis allows to determine the GT’s monitoring and detection capacities with conventional sensors. From this analysis it is concluded the existence of a non-detectable fault subsystem. To eliminate such subsystem, a Gas Turbines 322 reasonable proposition is the measure of the GT’s start motor power. Considering the new set of known variables and using the structural analysis, eleven GT’s redundant relations or symptoms generation are obtained. From these relations one identified that a diagnosis system can be designed for faults in sensors, actuators and turbo-generator. Since all constraints are involved at least one time in the 10 RGs of Table 4 or in Eq. (17). This means, a diagnosis system could be designed integrating the residuals generator with a fault isolation logic which has to classify the faults. Due to space limitation it is reported here results only for a mechanical fault in the friction parameter. Using the eleven RG obtained here, one can achieve a whole fault diagnosis for any set of parameters. 7. Acknowledgement The authors acknowledge the research support from the IN-7410- DGAPA-Universidad Nacional Autóoma de México, CONACYT-101311 and Instituto de Investigaciones Eléctricas, IIE. 8. References Blanke, M., Kinnaert, M., Lunze, J. & Staroswiecki, M. (2003). Diagnosis and Fault Tolerant Control , Springer, Berlin. Cassal, J. P., Staroswiecki, M. & Declerck, P. (1994). Structural decomposition of large scale systems for the design of failure detection and identification procedure, Systems Science 20: 31–42. De-Persis, C. & Isidori, A. (2001). A geometric approach to nonlinear fault detection and isolation, IEEE Trans Aut. Control 46-6: 853–866. Delgadillo, M. A. & Fuentes, J. E. (1996). Dynamic modeling of a gas turbine in a combined cycle power plant, Document 5117, in spanish, Instituto de Investigaciones Eléctricas, México. Ding, S. X. (2008). Model-based fault diagnosis techniques, Springer. Dion, J., Commault, C. & van der Woude, J. (2003). Generic propertie and control of linear structured systems: a survey, Automatica 39: 1125–1144. Frank, P. (1990). Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy, Automatica 26(2): 459–474. Frank, P., Schreier, G. & Alcorta-Garcia, E. (1999). Nonlinear Observers for Fault Detection and Isolation , Vol. Lecture Notes in Control and Information Science 244, Springer, Berlin, pp. 399–466. Giampaolo, T. (2003). The gas turbine handbook: principles and practice, The Fairmont Press. Gross, J. & Yellen, J. (2006). Graph Theory and its applications, Vol. 1, Taylor and Francis Group. Isermann, R. (2006). Fault Diagnosis System, Springer. Korbicz, J., Koscielny, J. M., Kowalczuk, Z. & Cholewa, W. (2004). Fault Diagnosis, Springer, Germany. Krysander, M., Åslund, J. & Nyberg, M. (2008). An efficient algorithm for finding minimal over-constrained sub-systems for model based diagnosis, IEEE Trans. on Systems, Man and Cybernetics-Part A: Systems and Humans 38(1): 197–206. Application of Structural Analysis to Improve Fault Diagnosis in a Gas Turbine 323 Mason, S. J. (1956). Feedback theory- further properties of signal flow graphs, Proceedings of the I. R. E., pp. 960–966. MATLAB R2008 (2008). Toolbox Control Systems, Math-Works, Inc., Natick, Massachuesetts. Mina, J., Verde, C., Sánchez-Parra, M. & Ortega, F. (2008). Fault isolation with principal components structural models for a gas turbine, ACC-08, Seattle. Mukherjee, A., Karmakar, R. & Kumar-Samantaray, A. (2006). Bond Graph in Modeling, Simulation and Fault Identification , Taylor and Francis. Pothen, A. & Fan, C. (1990). Computing the block triangular form of a sparse matrix, Artificial Intelligence 16: 303–324. Sánchez-Parra, M. & Verde, C. (2006). Analytical redundancy for a gas turbine of a combined cycle power plant, American Control Conference-06, USA. Sánchez-Parra, M., Verde, C. & Suarez, D. (2010). Pid based fault tolerant control for a gas turbine, Journal of Engineering for Gas Turbines and Power, ASME 132(1-1): –. Venkatasubramanian, V., Rengaswamyd, R., Yin, R. & Kavuri, S. (2003a). A review of process fault detection and diagnosis: Part i: Quantitative model based methods, Computers and Chemical Engineering 27: 293–311. Venkatasubramanian, V., Rengaswamyd, R., Yin, R. & Kavuri, S. (2003b). A review of process fault detection and diagnosis; part i: Quantitative model based methods; part ii: Qualitative model and search strategies; part iii: Process history based methods, Computers and Chemical Engineering 27: 293–346. Venkatasubramanian, V., Rengaswamyd, R., Yin, R. & Kavuri, S. (2003c). A review of process fault detection and diagnosis: Part ii: Qualitative model and search strategies, Computers and Chemical Engineering 27: 313–326. Venkatasubramanian, V., Rengaswamyd, R., Yin, R. & Kavuri, S. (2003d). A review of process fault detection and diagnosis: Part iii: Process history based methods, Computers and Chemical Engineering 27: 326–346. Verde, C. & Mina, J. (2008). Principal components structured models for faults isolation, IFAC- 08, Seoul, Korea. 9. Appendix k 1 Compressor discharge pressure x 5 Compressor IGV position rate k 2 Turbogenerador rotor speed x 6 CC gas temperature k 3 Atmospheric pressure x 7 CC gas rate temperature k 4 Outlet temperature x 8 Compressor energy k 5 Compressor IGV position x 9 Compressor bleed air flow k 6 Compressor air discharge temperature x 10 Compressor outlet air flow k 7 Compressor air bleed valve position x 11 Starting motor power k 8 Gas turbine fuel gas valve position x 12 CC gas fuel flow k 9 Inlet fuel gas valves pressure x 13 GT fuel gas valve position rate [...]...324 Gas Turbines k10 Heat recovery pressure x14 CC inlet gas flow k11 Exhaust gas temperature x15 CC outlet gas flow k12 Blade path temperature (BPT) x16 CC gas enthalpy k13 Electrical generator power output x17 GT exhaust gas density k14 Heat recovery gas temperature x18 GT exhaust gas enthalpy k15 Heat recovery gas outlet temperature x19 GT energy friction losses k16 Afterburner fuel gas valve... generator power rate angle k18 GT fuel gas valve control signal x22 Heat recovery gas rate temperature k19 AB fuel gas valve control signal x23 Heat recovery gas density k20 Starting motor power x24 Heat recovery gas rate density x1 CC gas density x25 Heat recovery outlet gas flow x2 CC gas rate density x26 AB gas fuel flow x3 Compressor inlet air flow x27 AB fuel gas valve position rate x4 Turbogenerator... diagrams 334 Gas Turbines 60 Al concentration,% 50 300 hr 1000 hr 5000 hr 10000 hr 40 14 12 10 8 6 4 2 0 0 50 100 150 200 250 300 350 400 Distance, мкм Fig 7 The Al concentration profile across a coating after isothermal oxidation at high temperatures for up to 10 000 hour Ni30Co28Cr8AlY(200 мкм) Ni30Co28Cr10AlY(200 мкм) 70 70 Volume fraction β-фазы, % 60 50 40 30 20 10 0 0 20 40 60 80 100 120 140 160 180... 2Russia 3Germany 2Polzunov 1 Introduction Modern power gas turbine blades are subject to high-temperature oxidation and are protected by metal coatings of MCrALY type The major element retarding oxidation of a coating is aluminium (Al) whose percentage in a coating amounts to 6-12% A blade coatings lifetime of 25000 h is required in stationary gas turbines at operating temperatures from 900 to 1000 ºС... cobalt, chromium, aluminium, yttrium – is to be investigated Data on Al diffusion factor can be found in literature studying similar element composition, but only for three-component NiCrAl alloy [7] 326 Gas Turbines It should be noted that the models, which have been described in [5, 6] disregard the importance of a number of alloys peculiarities of Al transport from the coating to the base alloy, which... provides more accurate predictions Life Time Analysis of MCrAlY Coatings for Industrial Gas Turbine Blades (calculational and experimental approach) 327 1 Experiment, 2 Model, 3 Inverse Problem Solution, 4 Prediction Fig 1 Diagram of the calculation and experimental approach to mass transfer and life predictive of gas turbine blade coatings The calculation-experimental approach under consideration suggests... ageing the duration of which was sufficient for model parameter identification Then the experiment was continued till coating life expiration in order to confirm the validity of the proposed method 328 Gas Turbines The work investigated the degradation of two coatings with different aluminum content The coatings were applied on IN 738LC alloy samples by LPPS (Low Pressure Plasma Spray) method The isothermal... growth decreases and α-Al2O3 increases The amount of θ-Al2O3 reaches maximum after 100 hours and with further exposure the oxide starts to disappear as a result of its transformation into α-Al2O3 330 Gas Turbines Temperature, оС Time, hr 5000 100 700 10000 900 θ- + α-Al 2O3 α-Al 2O3 α-Al 2O3 950 θ- + α-Al 2O3 α-Al 2O3 α-Al 2O3 1000 α-Al 2O3 α-Al 2O3 α-Al 2O3 α-Al 2O3 α-Al 2O3 (Me 3O4) α-Al 2O3 (Me 3O4)... across the coating layer As is known, the β-phase presents a compensation reservoir for aluminum which is spent on the protective oxide film formation on a coating surface (Brady et al., 2001) 332 Gas Turbines After applying a coating and performing heat-treatments a diffusion zone is formed in the alloy The zone width, phase composition and structure depend on coating and basic metal compositions,... x4 Turbogenerator rotor speed rate θ4 Compressor air density θ11 GT rotor friction parameter θ20 GT rotor inertia Table 6 Variables and Parameter Definition of the Gas Turbine Model 12 Life Time Analysis of MCrAlY Coatings for Industrial Gas Turbine Blades (calculational and experimental approach) Pavel Krukovsky1, Konstantin Tadlya1, Alexander Rybnikov2, Natalya Mozhajskaya2, Iosif Krukov2 and Vladislav . Starting motor power k 8 Gas turbine fuel gas valve position x 12 CC gas fuel flow k 9 Inlet fuel gas valves pressure x 13 GT fuel gas valve position rate Gas Turbines 324 k 10 . exhaust gas density k 14 Heat recovery gas temperature x 18 GT exhaust gas enthalpy k 15 Heat recovery gas outlet temperature x 19 GT energy friction losses k 16 Afterburner fuel gas. recovery gas rate density x 1 CC gas density x 25 Heat recovery outlet gas flow x 2 CC gas rate density x 26 AB gas fuel flow x 3 Compressor inlet air flow x 27 AB fuel gas valve