Thermodynamic Table for Performance Calculations in Gas Turbine Engine

Transcription

1 Proceedings o the International Gas urbine Congress 003 okyo November -7, 003 DRAF IGC003okyo S-093 hermodynamic able or Perormance Calculations in Gas urbine Engine Masumi IWAI Department o Power Engineering Shenyang Institute o Aeronautical Engineering (SYIAE) No.5 North Huanghe St. Shenyang Lioning. China,0034 Phone/FAX: , masumi_iwai@hotmail.com ABSRAC he purpose o this study is to simpliy the programming process by minimizing the iteration calculation steps. o achieve the aim, one dimensional isentropic compressible-low unction table was integrated with thermodynamic table to yield a subroutine program, which can be utilized in the programming work relating with perormance calculations in gas turbine engine. he table, written in Fortran, covers or the temperature range ; 0 ~ 5000 (K), mixture strength ; air~stoichiometric, and mach number ; 0~5. he table work also or the calculation with constant speciic heat ratio;. he table was arranged to work or the ollowing three systems o units; () SI unit, () metric unit and (3) British unit to be chosen. he program is hoped to be utilized through the GSJ i it is permitted. NOMENCLAURE A sectional area (cm ) a air/uel ratio at stoichiometric Cp speciic heat (kj/ K) at constant pressure Cv speciic heat (kj/ K) at constant volume Cpm =h/ (kj/ K) F uel/air ratio G mass low rate (kg/sec) g gravity (m/sec ) V H proportion o hydrogen h enthalpy (kj/kg) M molecular weight (kg/kg-mol) Ma molecular weight o air (kg/kg-mol) Mn Mach number P pressure (MPa) Pr relative pressure G /APt (kg/seck)/cm MPa Qs G /APs (kg/seck)/cm MPa R gas constant (kj/kg K) temperature (K) u internal energy (kj/kg) V velocity (m/sec) velocity(normalized) (m/sec)/k entropy unction (kj/kg K) relative uel/air ratio speciic heat ratio density ( kg/m 3 ) universal gas constant (kj/kmol K) ND no. o unit systems CK constant CJK conversion actor = Subscripts a air p combustion products t, s, m total, static, mean st stoichiometric INRODUCION One dimensional isentropic compressible-low unctions are oten required to be handled in the programming work relating to perormance estimation, data analyzing, and cycle calculation program etc. engaged with turbomachineries. It is rather tedious part o job to solve the required low unction each time during the main programming work. he subroutine program was studied intending to simpliy these programming processes. here are many thermodynamic tables published, however ew have the polynomial coeicients to be utilized or programming work. Chappell s table has the polynomial coeicients together with thermodynamic tables, and the coeicients are reproduced also in the IGI publication(99). his study based on Chappell s coeicients or the temperature range rom 50 to 00 K. For the temperature range less than 50 K, it was obtained based on Keenan s table(945) or air, and on JANAF table(985) or mixture. For the temperature range higher than 00 K, both o air and mixture were obtained based on Chemkin Data Base(994) up to 5000 K. Chappell s table has no relative pressure; Pr and relative volume; Vr, thereor they were deined and obtained in this study to be convenient or perormance calculation. he thermodynamic table covers the temperature range rom 0 to 5000 K, and air to stoichiometric mixture. With the thermodynamic table, the low unction table was arranged to cover the Mach number up to 5. he accuracy was checked by comparing the table obtained with the several other tables published having various units o systems, and it was considered to be acceptable or the purpose o this study. Dissociation was not taken into account in this study, however the eects were checked briely with GE table(954). Copyright (c) 003 by GSJ Manuscript Received on March 3, 003

2 HERMODYNAMIC ABLE DRY AIR he composition o dry air or the purpose o computing air properties was assumed by Keenan and Kays(945), as N : 78.03, O : 0.99 and Ar: 0.98 per cent by volume, to yield the molecular weight o air then as 8.970(kg/kmol). he molecular weights or the related species have been reviced since then, thereor applying the recent values to the above composition, the molecular weight o air was obtained as which is equivalent with that o real air (JSME, 985). he above composition and molecular weight o dry air were used in this study as the base to obtain the thermodynamic characteristics o air and mixture. SANDARD FUEL It has been studied that i the proportion o hydrogen contained in hydrocarbon uel (denoted as H ) is 0.39, the gas constant: R o the products o combustion becomes equal to that o air, irrespective o the mixture strength. he uel is called as 'standard uel' which is equivalent to CH.97. he standard uel has been widely used or perormance calculations in gas turbine engine to simpliy calculations, since gas constant or air: R can also be used or combustion gases with this uel. he standard uel is chosen in this study to simpliy to obtain the low characteristics. HERMODYNAMIC FUNCIONS Based on the air and uel chosen as above, the thermodynamic characteristics o air and stoichiometric gases can be obtained by using the ollowing equations respectively(hodge,955). (Cp) a =0.7546(Cp) N +0.39(Cp) O (Cp) Ar () (Cp) S =0.7064(Cp) N +0.06(Cp) Ar (Cp) HO +0.05(Cp) CO () he Chemkin Data Base presents the polynomial coeicients or thermodynamic properties o various species. For the species concerned in this study are given or the temperature range o =( )K and ( )K separately(chemkin,994). Speciic heat: Cp, enthalpy: h and entropy unction: were obtained or the related species such as N, O, Ar, H O and CO. he obtained values were checked by comparing with JANAF's table, which covers or the temperature range o (0-6000)K (JANAF,985). he enthalpy obtained rom Chemkin Data Base becomes negative value at =98.5 which is deined as 0 in JANAF, thereor the dierence were adjusted to meet with JANAF, and also it was urther adjusted to be h=0 at =0 (K), by using JANAF table. he thermodynamic characteristics o the species obtained were checked by comparing with Keenan's table in which the enthalpy were deined as h=0 at =0 and it covers or =( )R (Keenan,945). hese characteristics obtained as above or the related species, are applied into Eq.() to obtain air characteristics, and into Eq() or the stoichiometric gases. For the temperature range lower than 300 K, no polynomial coeicients published were ound, thereor the coeicients or air were obtained based on Keenan's air table, and coeicients or species were obtained based on JANAF data. hey were applied into Eq() to obtain the coeicients o stoichiometric gases or this temperature range. On the other hand, Chappell's thermodynamic table covers or the temperature range o (00-00)K with three systems o units, together also with the polynomial coeicients(chappell,974). he coeicients were also reproduced in the IGI publication, thereor the table have been widely utilized (IGI,99). Considering not to aect on the data already obtained, the Chappell's coeicients were tried to use in this study or the temperature range o (00-00)K. his table are divided into the two temperature ranges or (00-800) and (800-00)K..6 Cp (kj/kg K) Speciic Heat; Cp F= (Stoichiometric) F = 0 (Air) (K) Copyright (c)003 GSJ Manuscript Received on May 4, 003 Fig. Speciic heat characteristics

3 able hermodynamic unctions Comparing the table obtained or (0-300)K as above with the Chappell's table, the latter showed a slightly dierent characteristics or the range o (00-50)K, probably because o the very end o the temperature range studied. hereor, the table or (0-300)K is used or (0-50)K, and Chappell's or (50-800)K. he characteristics or (0-300)K table were slightly adjusted to meet with Chappell's at =50 K by changing the constants. he higher temperature range had no such problem, thereor the table ( )K are used to cover the range o ( ) K though the constants obtained or ( )K were also slightly adjusted to meet Chappell's at =00 K. With the characteristics or air and stoichiometric gases obtained as above, the characteristics or combustion gases at arbitrary mixture strength were obtained by using the ollowing equations. Cp Cp ( a)(cp Cp ) a st a Cp (3) a Cp ( a)(cp Cp ) (4) Cp st a he enthalpy: h and entropy unction were also obtained in the similar manner. he speciic heat : Cp obtained is shown in Fig.. RELAIVE PRESSURE AND VOLUME Relative pressure: Pr and relative volume: Vr can be obtained as; Pr exp ( 0 ) (5) R M 0 Vr exp ln( ) (6) R M 0 Where 0 at K assumed. Both o the Pr and Vr become.0 at =73.5 K according to the above deinition. However, Pr increases as temperature increases, but the Vr reduces, as the result the Vr becomes too small in value or the practical temperature range. o avoid this inconvenience, the Vr is redeined in this study as: Vr (7) Pr With this deinition o Vr, the value becomes convenient level in use, and also it is easy to obtain the values when Pr is known. SUB-PROGRAM ARRANGEMEN he thermodynamic table is arranged as shown in able which consists o 8 sub-unctions as; Cp, k, h, u,, Pr, Vr = (, F), Pr, Vr = (h, F),, Pr, Vr = (u, F ), h, u = (Pr, F),, h, u = (Vr, F ) he unction names o sub-programs are deined as ; () CP(, F, Cp, ) = ' obtain Cp rom ', () HF(, F, h) = ' obtain h rom ' and so on. A speciic eature o this table is, a corresponding temperature: can be obtained as a unction o h ( FH ), or as a unction o Pr ( PR ). hese unctions are eective to reduce the iteration calculations which will be required otherwise. able Accuracy check on Cp (kj/kg K) or air --- compared with Keenan s table 3

4 he most o sub-programs were checked to work satisactorily or the temperature range o (0-5000) K, rom air to stoichiometric mixture. However, the sub-program, ;PR, = (Pr) and HVR,=(Vr) ailed to yield reliable results or the temperature range less than 4 K, which aected on the unctions o ;HPR, ; UPR and VR as seen in Fig.. his was caused because the relative pressure; Pr and volume; Vr become too small or too big respectively to be handled or the low temperature range. he outputs or speciic heat: Cp, enthalpy: h and entropy unction: or air were compared with Keenan s table and the dierences between them were checked in Fig.3, ables and 3. he dierence are small enough as shown which considered to be satisactory or the purpose o this study. he same comparisons were made or combustion gases at 00 % air condition in able 4, since Keenan s table has no stoichiometric condition. he unction number are used to clariy the location o program among the sub-programs as shown in Fig.. Enthalpy; h able 3 Accuracy check or air ----Compared with Keenan s table Entropy unction; Di: (%) Cp ( kj/kg K ) Comparison with K & K table or Air K&K =[ (Cp)sy/(Cp)k- ] *00 (%) SYBL (K) Speciic heat; Cp Fig. 3 Cp compared with Keenan s table or air able 4 Cp, h compared with Keenan s table ---- or combustion products (00 % air) Enthalpy; h PR SF CP HF FH PR UF VR FU HVR HPR UPR UVR VR PRH VRH VRU PRU Fig. Interrelations among thermodynamic unctions 4

5 hermodynamic Functions Cpa C C C C 3 C 4 () h a = Cpad 0 = C C 3 C3 4 C4 5 C0 CH a = 0 Cp d = C C3 3 C4 4 C0 ln( ) C CF 3 4 Cp = ( + a )(Cp st Cp a ) (4) h = ( + a )( h st h a ) (5) () (3) he equations used or the thermodynamic unctions are shown above and the coeicients obtained are tabulated in able 5. he 4 data sources are used to obtain the coeicients as shown in Fig. 4. Speciic Heat: Cp (kj/kg K) Data Sources Stoichiometric Air (JANAF) ( Chemkin ) Chappell K & K emperature: (K) = ( + a )( st a ) (6) Cp = Cpa + h = h a + = a + Cp (7) h (8) (9) Cp = CP 0 + CP + CP + CP CP 4 4 +CP 5 5 (0) Y Fig. Data Sources hemmodynamic Functions Pr Vr Enthalpy; h Internal energy; u Entropy unction; Speci ic heat; Cp h = H 0 + H + H + H H H 5 5 () = F 0 + F + F + F F F 5 5 () (K) Fig. 5 hermodynamic Functions / Characteristics able 5 Coeicients or hermodynamic Functions (kj, kg, K) 5

6 FLOW FUNCION ABLE One dimensional isentropic compressible-low unctions can be written as the unctions o Mach number; Mn and speciic heat ratio:, thus; t Mn --- or const. S assumed var S m Mn --- or iable m Where = speciic heat ratio = speciic heat ratio corresponding to s, S m = mean speciic heat ratio deined as; h = where Cp m ( R Cp ) hese unctions may be summarized as; m {Mn, Ps/Pt, Pt/Ps, s/t, S t, A/Ach } = (Mn) (8) V {, Q, QS } = ( R, Mn) (9) where ; G G Q and Qs kg s K APt APs cm MPa he Eq.(8) is or non-dimensional parameters and Eq.(9) or dimensional which is directly aected by the value o gas constant; R. he low unction table studied consists o the 9 sub-programs as shown in Eq. (8) and (9). he table is arranged to obtain the 9 parameters, i.e. { Mach number; Mn, pressure ratio; Ps/Pt, Pt/Ps, temperature ratio; s/t, density ratio; s / t, velocity; V, volume low rate; Q, Qs, and area ratio; A/Ach } as unction o the one o those 9 parameters together with uel/air ratio; F and temperature;. he arrangement o low unction tables is shown in Fig. 6 schematically. he unction names o sub-programs are deined as; () FMN; Function o Mn (MN) () PSR; Pressure, Static, Ratio =Ps/Pt (3) PR; Pressure, otal, Ratio =Pt/Ps etc. he input ormats are shown in able 5 and output ormat are in able 6. he related 9 parameters such as AMNX, VRX,QX etc, can be obtained rom one parameter inputted in main program as; ' CALL PSR (Ps/Pt,F, ) '. SUB-PROGRAM ARRANGEMEN o obtain the 9 parameters corresponding to a parameter inputted, the Mach number corresponding to the inputted parameter has to be obtained. In the other word, the ollowing relation has to be solved. Mn = ( Ps/Pt. F, ) Obtaining Mach number; Mn as above, are sometimes not easy depending upon the value inputted. he accuracy o the Mach number obtained are checked through the ollowing manners. (Step-); he values o 9 parameters are obtained or a Mach number inputted. (Step-); he Mach number obtained or the parameters obtained in the (Step-). he accuracy is checked by comparing the Mach number obtained in (Step-) with the Mach number inputted in the (Step-). here are some diiculties existed or the range as Mach number approaches to near zero, since every parameters become so small to yield reliable values. he error will be less than 0.6% or the parameters corresponding to Mach number; Mn =0.0. hat is, the table are not able to work or the Mach number range o ; 0< Mn <0.0. he error or the velocity is shown in Fig. 7.. s ) (V ) ( t s ( t (V ) ) Fig. Flow unction table arrangement 6

7 Similar diiculties existed also at around Mn=.0. he errors in this area are 0.9% at Mn=0.999, and 0.00 (%) at Mn= Accuracy Check or Low Speed he low unction characteristics are shown in Fig. 8 in which the dimensional parameters are shown in red. he volume low rate; Q has its maximum value and area ratio; A/Ach has minimum value at Mn=.0. hat is, two Mach numbers exist or a parameter inputted or those two unctions. he two unction names are used or these sub-programs as shown in Fig. 9. hese are; Q G APt FQ (Mn, F, ) SFQ (Mn, F, ) --- or sub-sonic or super-sonic condition. Similarly, A/ Ach = ASR ( Mn, F, ) --- or sub-sonic = SASR ( Mn, F, ) --- or super-sonic condition. V (m/sec) Fig. 7 where V = (Ps/Pt) at =88.5 (K) Velocity; V (m/sec) Accuracy check or low speed Flow Function Characteristics hese unctions are also shown in Eq. (7) and Eq.(9), in Fig. 6. able 6 Input Formats Y 0 A/Ach V/SQR () Qs Ps/Pt Q (): FMN (Mn, F, ) Y= ( Mn ) (); PSR (Ps/Pt, F, ) Y= ( Ps/Pt ) (3): PR(Pt/Ps, F, ) Y= ( Pt/Ps ) (4); SR (s/t, F, ) Y= (s/t) (5); RSR ( / s t, F, ) Y= ( / s t ) at = 88.5 (K) E Mn Ps/Pt s/t s/t (6); VF ( V, F, ) Y= ( V ) (7); FQ ( Q, F, ) Y= ( Q ) SFQ ( Q, F, ) Fig. 8 Flow unction characteristics (8); FQS ( Qs ) Y= ( Qs ) (9); ASR ( A/A ch, F, ) Y =( A/ A ch ) SASR (A/A ch, F, ) 6 5 Mn vs. Q & A/Ach --Function Names-- 4 SFQ SASR able 7 Output Formats Mn 3 Y = Mn ( AMNX ), Ps/Pt ( PSPX ) Pt/Ps ( PPSX ), s/t ( SX ) / s t ( RSRX ), V (VRX) Q ( QX ), Qs ( QSX ) Mn =.0 FQ ASR Q & A/Ach A / Ach ( ASAX ) Fig. 9 Characteristics o volume low rate; Q 7

8 APPLICAION AND REMARKS he table developed consists o 8 thermodynamic sub-programs as shown in able and 9 low unction sub-programs as shown in Fig.6. he table, written in Fortran, is hoped to be a convenient tool when used as a subroutine program as the manner shown in Fig.9. It will be done simply by adding about 0 lines in the top o main program as described in able 9. FUEL/AIR RAIO Fuel/air ratio is arranged to be taken as shown in able 8 according to the value inputted. able 8 Fuel /Air ratio; F F = / a or 0 F < 0.09 = or 0.09 = a / or < F HREE SYSEMS OF UNIS he symbol; ND (stands or Number o Dimension) in the able 9, is the number to choose a system o units or SI, Metric and British unit as shown in able 0, and the CK (stands or Constant Kappa) is prepared to cope with constant caluculation. For example, ND=3, CK=.40 will yield the table or =.4 with British unit. hese unctions such as the ND and CK, will work as useul tools when numerical check is required or a reerence document written in various units o systems. Most o the parameters o low unction are non-dimensional except the three unctions shown in Eq. (9) which are shown in red lines in Fig. 8. hese dimensional parameters will be speciically useul to check data. able shows an example o the dimensional parameters or a same condition with dierent systems o units. able 9 Formats to be added into main program DEBUG C -----Main Program Format to use the Subroutine COMMON/SYIAE/RM,R,AFS,CK,ND,G,CJK,AKR COMMON/SHEN/SX,CPMX,AKMX,AKRMX / VRX,QX,QSX,ASAX C UR=8.3433! (kj/kmol K) AM=8.967! (kg/kmol) C G= CJK=4.868! (kj/kcal) C R=UR/AM! (kj/kg K) RM=R/CJK! (kcal/kg K) C C ND=! ND= or SI, or Metric,3 or British Units. C CK=0.0! or Constant k Fig. 0 Flow Chart/ Schematic able 0 hree systems o units ND 3 unit SI Metric British A cm cm in Cp, Cv, kj/kg K kcal/kg K Btu/lb R G kg/sec kg/sec lb/sec h, u kj/kg kcal/kg Btu/lb L m m t P MPa kg/cm psia K K R V m/sec m/sec t/sec kg/m 3 kg/m 3 lb/t 3 V Q, Qs ND m sec K kg sec K cm MPa m sec K kg ( sec) K kg cm ( ) cm t sec R lb sec in R psi able- Dimensional parameters or air at Mn=.0 =88.5 (K) V Q Qs ; S I ; Metric ; British

9 SUMMARY A subroutine program, written in Fortran, is developed to make the programming work easy, relating to perormance estimation, data analyzing and cycle calculation program etc. engaged with turbomachineries. he program consists o two parts, thermodynamic table and low unction table, the both are integrated together. he thermodynamic table is basically ater Chappell s table(974), o which the temperature range is extended to cover o-5000 (K) since it is required to work or the wider mach number range in the low table. It has been done with the help o Keenan and Kay s table, JANAF s table and Chemkin Data Base. he table covers or air to stoichiometric condition. Relative pressure: Pr and relative volume: Vr are also obtained. he thermodynamic table consists o 8 sub-unctions which arranged as below. Cp, k, h, u,, Pr, Vr = (, F), Pr, Vr = (h, F),, Pr, Vr = (u, F ), h, u = (Pr, F),, h, u = (Vr, F ) he low unction table is arranged to cover the mach number up to 5, by using the above thermodynamic table. he table consists o 9 sub-unctions o which 6 are or non-dimensional and 3 are or dimensional parameters as shown below. {Mn, Ps/Pt, Pt/Ps, s/t, S t, A/Ach } = (Mn) V {, Q, QS } = ( R, Mn) where ; G G Q and Qs APt APs kg s K cm MPa In the table, the unctions such as Y= (Ps/Pt, F, ), or example, are obtained where the Y denotes the above 9 parameters. he both o thermodynamic and low unction tables are arranged to work or the three systems o units such as ;. (): S I, (): Metric and (3): British unit hey are able also to work or constant. hese unctions, the three systems o units and constant, will work as a useul tool to check data numerically or documents written in various units. he non-dimensional parameters in low table will be speciically useul to check measured data. Most o text books are written with constant- such as, or example, =/ - or piston engine. On the other hand the relative pressure: Pr and relative volume Vr are used in practical work. A theoretical equation as above can be checked numerically by using the relative pressure: Pr and/or relative volume :Vr, with the constant option in this program. he result can be compared with air-cycle and/or uel/air cycle calculations in the same manner with this table. he unction o the table will best be described with the Visual Basic Version as shown in Fig., which arranged or hand calculation. he upper hal in the igure is or thermodynamic table and the lower hal or low unction table. he subroutine program is hoped to be utilized by whom interested. ACKNOWLEDGMENS he author has been looking or the chance to accomplish the table or personal use. he study was summarized in the Shenyang Institute o Aeronautical Engineering (SYIAE) in China. he author would like to acknowledge the generosity o Proessor Rui Xiao-miao, President Oice, or his encouragement to accomplish the study. he author appreciate Proessor Xu Rang-shu and also Associate Proessor Fu Li-xin, Department o Power Engineering,to provide various supports during the course o study. [Note] Essence o this study is in the sotware. It will be shared with whom interested. Please contact to the author by . Fig. VB Version or Hand Cal. RIFERENCES [] Chappell M.S. and Cockshutt E.P., 974, Gas urbine Cycle Calculations, National Research Council Canada. [] Chase M. W. Jr. et al., 985, JANAF hermochemical able, hird edition, J. Phys. Chem. Data, Vol. 4, Suppl.. [3] General Electric Co., 954, Properties o Combustion Gases, McGraw-Hill Book Co. [4] Hodge James, 955, Cycle and Perormance Estimation, Scientiic Publications. [5] IGI/ASME, 99, he Design o Gas urbine Engine, [6] JSME Data Book, 983, hermodynamic Properties o Fluids,JSME,985,MechanicalEngineeringHandbook, A-6/p.4. [7] Kee R. J. et al., 994, he Chemkin hermodynamic Data Base, Sandia Report, SAND87-85B. [8] Keenan J.H. and Kays J., 945, Gas ables, John Wiley & Sons [9] Strehlow Roger A., Fundamental o Combustion, International extbook Co. [0] Yan Jia-lu et al., 983, Combustion Gases hermodynamic ables or Hydrocarbon Fuel, Scientiic Press, Beijing China. 9