Fuel 113 (2013) 389–401

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Fuel

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Prediction of equilibrium products and thermodynamic propertiesin H2O injected combustion for CaHbOcNd type fuels

0016-2361/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.fuel.2013.05.095

⇑ Corresponding author at: Yildiz Technical University, Faculty of Naval Archi-tecture and Maritime, Department of Marine Engineering Operations, Istanbul34349, Turkey. Tel.: +1 90 212 383 29 42.

E-mail addresses: hasankayhankayadelen@hotmail.com (H.K. Kayadelen),yust@yildiz.edu.tr (Y. Ust).

Hasan Kayhan Kayadelen a,b,⇑, Yasin Ust c

a Princeton University, Department of Mechanical and Aerospace Engineering, Princeton, NJ 08544, USAb Yildiz Technical University, Faculty of Naval Architecture and Maritime, Department of Marine Engineering Operations, Istanbul 34349, Turkeyc Yildiz Technical University, Faculty of Naval Architecture and Maritime, Department of Naval Architecture and Marine Engineering, Istanbul 34349, Turkey

h i g h l i g h t s

� Effects of different amounts of H2O addition into equilibrium combustion products.� Lean and rich cases for propane, diesel, gasoline and methane fuels.� Change of Cp of and adiabatic flame temperatures.� Adaptable for future studies for any desired number of exhaust species.� Comparison with the results obtained from CHEMKIN and GASEQ and proved reliability.

a r t i c l e i n f o

Article history:Received 27 November 2012Received in revised form 13 May 2013Accepted 29 May 2013Available online 14 June 2013

Keywords:EmissionsWater emulsionWater injectionSteam injectionChemical equilibrium

a b s t r a c t

Water and steam injection at the upstream or in the combustion chamber or using fuel–water emulsionsare both well proven methods for reducing NOx emissions and improving performance of many types ofinternal combustion engines. This study involves a thermodynamic simulation of water/steam injectedcombustion which can be used with any CaHbOcNd type fuel and for any combustion engine. The simu-lation determines the mole fractions of each exhaust species at chemical equilibrium according to equi-librium-constant approach and then thermodynamic properties of the exhaust gas mixture. Previousstudies about steam/water injected power systems concentrate on some particular exhaust species suchas CO and NOx and unable to predict the thermodynamic properties of working fluid accurately which isneeded for an accurate performance estimation. According to previous researchers, thermodynamic prop-erties calculated with this approach depending on the equilibrium composition are precise and candirectly be used for performance estimation of combustion engines. The model gives equilibrium molefractions, specific heat of the exhaust mixtures and adiabatic flame temperatures and results for four dif-ferent fuels and varying steam/water injection ratios are presented. Apart from being directly applicableto steam/water injected power systems, the modeling process is described elaborately in order to make itadaptable for future studies such as multi-fuel or EGR applications for any desired number of exhaustspecies. Additionally, the equilibrium composition obtained, can also be used for determination ofnon-equilibrium pollutant values using existing chemical kinetics correlations in the literature. Thisapproach provides the basic needs for the two key objectives of engine design; ‘‘performance evaluationand emission optimization’’ by providing an accurate method for steam/water injected power systems.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

This section introduces the idea of steam/water injection andstudies about steam/water injected combustion systems, predic-

tion of emissions from reciprocating and gas turbine engines andlastly the studies about obtaining the equilibrium composition.

Combustion with steam and water are both well proven meth-ods for reducing NOx emissions and improving performance ofmany types of internal combustion engines. These techniques arebeing studied for more than a hundred years and they are widelyused in gas turbines for approximately fifty years [1–15]. Thereare also many studies about introducing water/steam injectionon reciprocating engines [16–30] especially of compression igni-tion types.

Nomenclature

A reactants frequencyC specific heat (kJ/kg K)CC combustion chamberFA fuel/air ratioh specific enthalpy (kJ/kg)K equilibrium constantMW molecular weightN total number of moles of the speciess H2O injection ratio, entropyT temperaturex number of moles of injected H2Oy mole fraction

Subscriptsa airady adiabaticf fuel

i exhaust speciesin inletp pressurer reactantss steam or stoichiometric

Greek Lettersa number of carbon atomsb number of hydrogen atomsd number of oxygen atomse molar air–fuel ratio/ equivalence ratioc number of nitrogen atomsm number of moles of exhaust species

Table 2Products of wet combustion at low temperature (T < 1000).

yi SPECIES / 6 1 / > 1

1 CO2 a/e a/e�v5

2 H2O b/e + x /e(c�2a) + 0.42 + x + v5

3 N2 0.79 + d/e/2 0.79 + d/e/24 O2 0.21(1�/) 05 CO 0 v5

6 H2 0 0.42(/�1)�v5

390 H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401

Combustion with steam increases both power and efficiency ofthe injected cycle while it reduces the NOx emissions. Since thepumping work for steam/water is smaller than that of the air,the net power output produced by steam is much higher than thatof air in terms of per unit mass flow rate [31]. In addition, the spe-cific heat of superheated steam is roughly double that of air andthe enthalpy of steam is higher than the value of the air at a certaintemperature. Therefore, steam injection method is a very effectiveway to boost the net power output and increase the overall effi-ciency. Besides it reduces NOx emissions lowering the combustiontemperature. Furthermore Sahai and Cheng [32] stated that if thesteam and fuel is mixed homogeneously NOx reduction can be suc-ceeded without increasing the CO emissions. In this system for gasturbines called Cheng low NOx (CLN), power increase up to 50% isachievable without need for excess fuel [33]. As a result, thermalefficiency may be improved up to 40%, depending on steam–fuelratio and since less fuel is burnt in the gas turbine, the emissionsof CO2 will also decrease.

In case of water injection, more heat will be rejected from thecombustion chamber in comparison to the same amount of steamand the heat energy consumed to bring the injected water to steamphase will also contribute to lower the combustion chamber tem-perature which leads lower NOx emissions but may also slightlydecrease the overall efficiency.

By introducing steam injection in a gas turbine, the increase inthermal efficiency is usually between 2.5% and 15% and in the netcycle work about 5–25% depending on the rate of injection. Paepeand Dick states that by steam injection, efficiency gain of about 10points and power augmentation of about 50–70% are possible in agas turbine engine [34].

For reciprocating engines H2O is generally injected as water.Studies in the literature about water injection in diesel enginesshow that water injection reduce NOx emissions up to 60%

Table 1Curvefit coefficients for chemical equilibrium constants.

A B

K1 4.3217E�01 �1.1246E+04K2 3.1081E�01 �1.2954E+04K3 �1.4178E�01 �2.1331E+03K4 1.5088E�02 �4.7096E+03K5 �7.5236E�01 1.2421E+04K6 �4.1530E�03 1.4863E+04

[28,35–37] decreases fuel consumption up to 20% [38] and increasepower between 5% to about 21% [29,38].

Specification of pollutant emissions is very important for theengine design, optimization and calibration phase. Taking the com-plexity and cost involved conducting an experimental investiga-tion into account, the recourse to a tool of simulation offers aneffective and fast alternative to deal with the pollutant emissionsfrom combustion engines. The analytical model developed for thesimulation tool can also be integrated to the engine control andon-board diagnosis tasks. Therefore numerous authors studied onprediction of emissions for particular engine parameters. Some ofthose concerning reciprocating internal combustion engines are gi-ven in [39–44]. Apart from those some research on emission pre-diction from gas turbine engines are given in [45–49].

Water/steam injection in the working fluid of an engine is a pre-cise method because injecting an inappropriate amount of diluentmay cause other emissions to rise. Some of the studies aiming forthe prediction of emissions from steam/water injected engines areas follows:

Larbi and Bessrour [27] investigated the amount of emissions ofa water injected marine diesel engine with a numerical model

C D E

2.6727E+00 �7.4574E�05 2.4248E�093.2178E+00 �7.3834E�05 3.4465E�098.5346E�01 3.5502E�05 �3.1023E�096.4610E�01 2.7281E�06 �1.5444E�09�2.6029E+00 2.5956E�04 �1.6269E�08�4.7575E+00 1.2470E�04 �9.0023E�09

H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401 391

using the computer code CHEMKIN and compared the results withexperimentally measured values.

Ishida et al. [50] worked on prediction of NOx reduction rate dueto port water injection in a DI diesel engine comparing the exper-imental results and the analytical ones based on the two-zonemodel developed by the authors.

Hung [51] developed an analytical model to predict the effectsof injected water on NOx emissions from industrial gas turbines

Fig. 1. Equilibrium mole fractions of CO2 for differen

Fig. 2. Equilibrium mole fractions of H2O for differe

taking operating conditions, combustor geometry, type of fueland changes in ambient temperature into account.

As these studies usually concentrate on some particular emis-sions they are not able to predict the full equilibrium scheme.According to Heywood [52], Rashidi [53] and Rakopoulos et al.[54], it is a good approximation for performance estimates in en-gines to regard the burned gases produced by the combustion offuel and air as in chemical equilibrium and therefore knowledge

t steam injection ratios for / = 0.6 and / = 1.2.

nt steam injection ratios for / = 0.6 and / = 1.2.

Fig. 3. Equilibrium mole fractions of N2 for different steam injection ratios for / = 0.6 and / = 1.2.

Fig. 4. Equilibrium mole fractions of O2 for different steam injection ratios for / = 0.6 and / = 1.2.

392 H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401

of the exact gas composition inside the combustion chamber iscritical to the accurate calculation of the thermodynamic cyclemodels of internal combustion engines. The equilibrium schemealso forms the baseline for predicting the quantities of exhaust pol-lutants such as nitric oxide (NO) and carbon monoxide (CO) in akinetically controlled fashion [54,55]. For this purpose a chemical

equilibrium scheme is needed considering a certain number of spe-cies present inside the combustion chamber.

There are already some studies on the calculation of equilib-rium composition for dry combustion [53–56] and for a dual fuelstudy with H2 injection [57] but none for the steam/water injectioncase present in the available literature.

Fig. 5. Equilibrium mole fractions of CO for different steam injection ratios for / = 0.6 and / = 1.2.

Fig. 6. Equilibrium mole fractions of H2 for different steam injection ratios for / = 0.6 and / = 1.2.

H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401 393

Existing steam/water injected combustion studies usually makean assumption of complete combustion of a CaHb fuel with excessair and H2O as oxidizers, so they treat the exhaust stream as a mix-

ture of complete combustion products only comprising of CO2, H2Oand N2. This approach may lack in precision although there is suf-ficient oxygen which can completely oxidize all the fuel because of

Fig. 7. Equilibrium mole fractions of H for different steam injection ratios for / = 0.6 and / = 1.2.

Fig. 8. Equilibrium mole fractions of O for different steam injection ratios for / = 0.6 and / = 1.2.

394 H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401

the dissociations of combustion products at high temperatures. Forexample, if the temperature of a mass of carbon dioxide gas in avessel is increased sufficiently, some of the CO2 molecules dissoci-ate into CO and O2 molecules. If the mixture of CO2, CO and O2 is in

equilibrium, this means CO2 molecules are dissociating into CO andO2 at the same rate as CO and O2 molecules are recombining in theproportions required to satisfy the equation COþ 1=2O2 ¼ CO2.When hydrocarbon fuels are subjected to combustion at low

Fig. 9. Equilibrium mole fractions of OH for different steam injection ratios for / = 0.6 and / = 1.2.

Fig. 10. Equilibrium mole fractions of NO for different steam injection ratios for / = 0.6 and / = 1.2.

H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401 395

temperatures, for the rich case the major product species presentare N2, H2O, CO2, CO and H2 where for the lean case N2, H2O,CO2, and O2. But at higher temperatures (greater than about2200 K), these major species dissociate and react to form addi-tional species in significant amounts [52]. So actual combustion

reactions do not go to completion and it will be useful to developan equilibrium product composition [58] by which individual spe-cies in the burned gases react together, produce and remove eachspecies at equal rates but no net change in species compositionresults [52].

Fig. 11. Constant pressure specific heat of gas mixture for different steam injection ratios for / = 0.6 and / = 1.2.

Fig. 12. Adiabatic flame temperatures for different steam injection ratios for / = 0.6 and / = 1.2.

396 H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401

In this study the thermodynamic change of the working fluidwill be calculated for 10 main products as suggested by Ferguson[59] to get more accurate values of gas properties which may leadconsiderable changes in performance results in comparison tothose made by an assumption of complete combustion. The mod-eling procedure is given in details, so the reaction can be easilyremodeled if more reactant or product species is required and it

is possible to estimate the new properties of the working fluid tobe used in the performance estimation of any combustion engine.

In order to obtain the equilibrium compositions and thermody-namic properties, the chemical equilibrium routines of Olikara andBorman [55] presented by Ferguson [59] based on equilibrium-constant approach are modified for H2O injection in this presentwork.

H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401 397

Experiments related to this study can be conducted by using aconstant pressure batch reactor. First, the reactor should be vacu-umed and secured from any leakage into the system. Then appro-priate amounts of steam, fuel, N2 and O2 should be sent to thereactor in order to observe the effects of variable amounts of steaminjection at different pressures, inlet temperatures and equivalenceratios. It is appropriate to quantify the amounts of each reactantspecies by their partial pressures so pressure should be controlledprecisely in the system. Pressure fluctuations may occur in thereactor due to mixing so there should be enough time betweensending each reactant to the system so that the system pressureis stable. It is also very important to ensure that the steam willnot condense in the reactor during the experiment. After the igni-tion, exhaust sample can be sent to a gas chromatograph (GC) withappropriate columns to quantify each species, so that the equilib-rium scheme for a specific fuel is obtained experimentally. In thisstudy, the results obtained from the new model are compared withthe results of computer programs CHEMKIN [60] and GASEQ [61]which use minimization of Gibbs free energy approach and ele-ment potential method respectively. The comparisons will be pre-sented in Section 2.

2. The equilibrium combustion products and thermodynamicproperties

In the analytical model presented for the adiabatic combustionin this section it is assumed that all gases are ideal gases and theirenthalpies and specific heats only change with temperature. Con-sidering that for / < 3 and there are 10 constituents, the high tem-perature combustion model with the reactant H2O added isestablished taking the equilibrium constants into account andthe equilibrium products of combustion are calculated as a func-tion of temperature and equivalence ratio assuming that the airsupplied for the combustion is completely dry without any mois-ture and containing only 0.21 mol of O2 and 0.79 mol of N2. Theminor existence of Ar and CO2 is also neglected.

A further increase in accuracy can be supplied by taking Aratoms in the air into account and adding more product species. Itshould also be taken into consideration that more species will re-sult a bigger solution matrix and require more computational timeaccordingly. In special cases that the equivalence ratios are over 3,more product species will be required. The chemical equation forthe present model is given below:

e/CaHbOcNd þ ð0:21O2 þ 0:79N2Þ þ xH2O

! m1CO2 þ m2H2Oþ m3N2 þ m4O2 þ m5COþ m6H2 þ m7H

þ m8Oþ m9OHþ m10NO ð1Þ

where m1–m10 represents the number of moles for each species, a, b,c, d are the numbers of carbon, hydrogen, oxygen and nitrogenatoms present in the fuel. / is equivalence ratio, e is the molarair–fuel ratio obtained from the stoichiometric combustion of thefuel and x is the molar injection ratio of H2O which are calculatedas below respectively:

/ ¼ FAFAs

ð2Þ

e ¼ 0:21aþ b

4�c2

ð3Þ

x ¼ MWair

MWH2Os ð4Þ

where s is the steam injection ratio which stands for ms/ma, which isan important parameter for introducing the performance of thesteam/water injected cycle. In order to solve for the 10 unknown

mole numbers of Eq. (10) equations are needed which 6 of themcan be provided by the criteria of equilibrium among the productsexpressed by the following chemical relations [59]:

1=2H2 ¢ H K1 ¼ y7ffiffiffipp

=ffiffiffiffiffiy6p

c1 ¼ K1=ffiffiffipp

y7 ¼ c1ffiffiffiffiffiy6p ð5:1Þ

1=2O2 ¢ O K2 ¼ y8ffiffiffipp

=ffiffiffiffiffiy4p

c2 ¼ K2=ffiffiffipp

y8 ¼ c2ffiffiffiffiffiy4p ð5:2Þ

1=2H2 þ 1=2O2 ¢ OH K3 ¼ y9=ðffiffiffiffiffiy4p ffiffiffiffiffi

y6p Þ c3 ¼ K3

y9 ¼ c3ffiffiffiffiffiy4p ffiffiffiffiffi

y6p ð5:3Þ

1=2O2 þ 1=2N2 ¢ NO K4 ¼ y10=ðffiffiffiffiffiy4p ffiffiffiffiffi

y3p Þ c4 ¼ K4

y10 ¼ c4ffiffiffiffiffiy4p ffiffiffiffiffi

y3p ð5:4Þ

H2 þ 1=2O2 ¢ H2O K5 ¼ y2=ðy6ffiffiffiffiffiy4p ffiffiffi

ppÞ c5 ¼ K5

ffiffiffipp

y2 ¼ c5y6ffiffiffiffiffiy4p ð5:5Þ

COþ 1=2O2 ¢ COK5 ¼ y1=ðy5ffiffiffiffiffiy4p ffiffiffi

ppÞ c6 ¼ K6

ffiffiffipp

y1 ¼ c6y5ffiffiffiffiffiy4p ð5:6Þ

where the unit pressure p is in atmospheres and K1–K6 are the equi-librium constants of each reactions. Olikara and Borman have curvefitted the equilibrium constants (based on partial pressures) to JA-NAF Table data with the following expression [55]:

log Kp ¼ AlnT

1000

� �þ B

Tþ C þ DT þ ET2 ð6Þ

For the temperature range 600 < T < 4000 K A, B, C, D and E aregiven in Table 1 suggested by Ferguson [59].

There are four more equations which come from the combus-tion model atom balancing:

C e/a ¼ ðy1 þ y5ÞN ð7Þ

H e/bþ 2x ¼ ð2y2 þ 2y6 þ y7 þ y9ÞN ð8Þ

O e/cþ 2 � 0:21þ x ¼ ð2y1 þ y2 þ 2y4 þ y5 þ y8 þ y9

þ y10ÞN ð9Þ

N e/dþ 2 � 0:79 ¼ ð2y3 þ y10ÞN ð10Þ

where yi stands for the mole fractions of 10 species and N is the to-tal number of moles of the species:

yi ¼ti

Nð11Þ

N ¼X10

i¼1

ti ð12Þ

With the added N, total number of unknowns are now 11 and butwe have 10 equations so far. From the definition of mole fractionone can write the following equation which makes the total numberof unknowns and total number of equations equal.

X10

i¼1

yi � 1 ¼ 0 ð13Þ

N can be eliminated by dividing Eqs. (8)–(10) into Eq. (7) whichgive:

2y2 þ 2y6 þ y7 þ y9 � ðy1 þ y5Þe/bþ 2x

e/a¼ 0 ð14Þ

2y1 þ y2 þ 2y4 þ y5 þ y8 þ y9 þ y10 �e/cþ 0:42þ x

e/aðy1 þ y5Þ ¼ 0

ð15Þ

Table 3Typical deviations of the results of presented model from the GASEQ software and CHEMKIN software for equilibrium products of methane combustion, / = 0.6 and steaminjection 10%.

/ = 0.6 Model results GASEQ results CHEMKIN results GASEQ deviations % CHEMKIN deviations %

CO2 0.05151 0.05151 0.05151 �0.0005 �0.0005H2O 0.233847 0.23382 0.23385 0.0119 �0.0009N2 0.645502 0.64552 0.64553 �0.0027 �0.0043O2 0.068249 6.82E�02 0.06827 0.0134 �0.0305CO 3.22E�07 3.22E�07 3.27E�07 0.1355 �1.4152H2 5.34E�07 5.32E�07 5.37E�07 0.3566 �0.5799H 3.84E�09 3.83E�09 3.87E�09 0.2570 �0.7847O 3.36E�07 3.35E�07 3.39E�07 0.3574 �0.8323OH 5.46E�05 5.66E�05 5.52E�05 �3.6602 �1.0962NO 8.35E�04 8.35E�04 7.76E�04 0.0077 7.0730Cp 1.4802 1.46E+00 1.46424 1.0796 1.0782Tin 379.11 379.11 379.11 0.0000 0.0000Tady 1542.4 1542.40 1542.40 0.0000 0.0000

Table 4Typical deviations of the results of presented model from the GASEQ software and CHEMKIN software for equilibrium products of methane combustion, / = 1.2 and steaminjection 10%.

/ = 1.2 Model results GASEQ results CHEMKIN results GASEQ deviations % CHEMKIN deviations %

CO2 0.0631825 0.06311 6.306E�02 0.1148 0.1939H2O 0.2786504 0.2787 2.788E�01 �0.0178 �0.0465N2 0.5948315 0.59485 5.95E�01 �0.0031 0.0003O2 1.42E�07 1.47E�07 1.41E�07 �3.6494 0.5600CO 0.0316898 0.03177 3.18E�02 �0.2532 �0.3889H2 0.0315605 0.03148 3.14E�02 0.2551 0.4041H 4.35E�05 4.42E�05 4.33E�05 �1.6489 0.5244O 3.69E�08 3.84E�08 3.67E�08 �4.0590 0.4828OH 3.62E�05 3.81E�05 3.63E�05 �5.1316 �0.0655NO 5.39E�06 5.53E�06 5.06E�06 �2.6688 6.1258Cp 1.66E+00 1.63734 1.56827 1.0790 5.2519Tin 375.20 375.20 375.20 0.0000 0.0000Tady 1972.60 1972.60 1971.60 0.0000 0.0507

398 H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401

2y3 þ y10 �e/dþ 1:58

e/aðy1 þ y5Þ ¼ 0 ð16Þ

Substituting Eqs. (5.1)–(5.6) into Eqs. (14)–(16) and into Eq. (13) weobtain four non-linear equations and four independent variables y3,y4, y5 and y6 which can be solved by Newton–Raphson iteration toobtain the mole fractions of the gas turbine emissions at the equi-librium [59] and can be written as follows:

fj ¼ ðy3; y4; y5; y6Þ ¼ 0 j ¼ 1;2;3;4 ð17Þ

Using both Gauss–Seidel and Newton–Raphson methods togetherwould combine the speed of the first and the reliability of the latteris an efficient way which converge faster to an accurate solution[62]. To start the iterations, mole fractions of low-temperature com-bustion can be used as a first approximation which can be obtainedby from the chemical equation below:

e/CaHbOcNh þ ð0:21O2 þ 0:79N2Þ þ xH2O

! m1CO2 þ m2H2Oþ m3N2 þ m4O2 þ m5COþ m6H2 ð18Þ

For the lean and stoichiometric combustion where / 6 1, assumingthat there would be enough oxygen to oxidize all CO and H2,

m5 = m6 = 0 is a good approximation so there is four unknowns andatom balance equations are sufficient to determine the productcomposition. For the rich case / > 1, m4 = 0 can be written assumingthere is insufficient oxygen to oxidize all CO and H2 so all the oxy-gen is consumed. Writing atom balances and after some algebraicsimplifications low-temperature products of wet combustion are gi-ven in Table 2.

In the lean case, one can see that the equilibrium composition isindependent from the temperature and pressure and changes onlywith /.

However as there are five unknowns and four equations in therich case we introduce one more equation depending on the equi-librium constant for the reaction CO2 þH2 ¢ COþH2O which is gi-ven by Ferguson as a function of temperature [59]:

ln K ¼ 2:743� 1:761=t � 1:611=t2 þ 0:2803=t3 ð19Þ

where t = T/1000 and in degrees Kelvin. From the definition of equi-librium constant at constant pressure:

K ¼ v2v5

v1v6ð20Þ

and combining Eqs. (19) and (20) we now have an equation for eachunknown. Solving this linear system of equations the first approxi-mation yð1Þ3 , yð1Þ4 , yð1Þ5 and yð1Þ6 is obtained to be used in the Newton–Raphson iteration. For the iteration Eq. is expanded into a TaylorSeries neglecting the second and higher order derivatives anddefining:

Dyi ¼ y�i � yð1Þi ð21Þ

the following four equations yield for approximations to Dyi:

fj þ@fj

@y3Dy3 þ

@fj

@y4Dy4 þ

@fj

@y5Dy5 þ

@fj

@y6Dy6 ffi 0 j ¼ 1;2;3;4 ð22Þ

which can be written in the matrix form as:

f1ðyð1Þ3 ; yð1Þ4 ; yð1Þ5 ; yð1Þ6 Þf2ðyð1Þ3 ; yð1Þ4 ; yð1Þ5 ; yð1Þ6 Þf3ðyð1Þ3 ; yð1Þ4 ; yð1Þ5 ; yð1Þ6 Þf4ðyð1Þ3 ; yð1Þ4 ; yð1Þ5 ; yð1Þ6 Þ

2666664

3777775þ

Results

of

JacobienMatrix

26664

37775�

Dy3

Dy4

Dy5

Dy6

26664

37775 ffi 0 ð23Þ

H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401 399

Solving for Dyi, the second approximation for the next iteration canbe obtained as follows:

yð2Þi ¼ yð1Þi þ Dyi i ¼ 3;4;5;6 ð24Þ

The procedure is repeated until |Dyi| is reached to a specified toler-ance which leads to the values for y3, y4, y5and y6. The six depen-dent unknowns are to be found using Eqs. (5.1)–(5.6) and thusthe full equilibrium scheme including mole fractions of 10 spicesis obtained.

At constant pressure, temperature variation is effective on spe-cific heat because of the dissociations of species at high tempera-tures. The effect of temperature on mole fractions should beaccounted for the equilibrium specific heat calculation differentiat-ing Eq. (17) with respect to temperature which can be written as:

@fj

@Tþ @fj

@y3

@y3

@Tþ @fj

@y4

@y4

@Tþ @fj

@y5

@y5

@Tþ @fj

@y6

@y6

@T¼ 0 ð25Þ

The above equation can be solved with the same procedure of Eqs.(21)–(24). Results will be used in finding the specific heat of thecombustion products in Eq. which will then be used in performanceassesment of the gas turbine.

Molar specific heat, enthalpy and entropy values of each speciescan be obtained from following expressions by using curve fit coef-ficients (a1. . .an) for thermodynamic properties of (C–H–O–N) sys-tems [63]:

hoi

RuT¼ a1;i þ

a2;i

2Tþ a3;i

3T2 þ a4;i

4T3 þ a5;i

5T4 þ a6;i

Tð26Þ

cp;i

Ru¼ a1;i þ a2;i þ a3;iT

2 þ a4;iT3 þ a5;iT

4 ð27Þ

soi

Ru¼ a1;i ln T þ a2;iT þ

a3;i

2T2 þ a4;i

3T3 þ a5;i

4T4 þ a7;i ð28Þ

At constant pressure, enthalpy of the mixture change due to the dis-sociations as the mole fractions of the mixture change with temper-ature. This will change the ultimate specific heat of the gas mixturedefined as follows:

h ¼X10

i¼1

yihoi ½kJ=kmol�: ð29Þ

h ¼ 1M

X10

i¼1

yihoi ½kJ=kg�: ð30Þ

@h@T

� �P

¼ cpg¼X10

i¼1

yi

M@ho

i

@Tþ ho

i

M@yi

@T� yih

oi

M2

@M@T

ð31Þ

Using Eqs. (26)–(30), rearranging Eq. (31) gives:

@h@T

� �P

¼ cpg¼ 1

M

X10

i¼1

yicpiþ @yi

@Tho

i �MT

Myih

oi

" #½kJ=kgK� ð32Þ

where

MT ¼@M@T¼X10

i¼1

Mi@Yi

@Tð33Þ

T is the combustion temperature in Kelvin at which the mole frac-tions of each equilibrium species, yi are produced, Mi is the molec-ular weight of species i, and M is the the molecular weight of themixture as follows:

M ¼X10

i¼1

mi ¼X10

i¼1

yiMi ð34Þ

From the law of conservation of mass, the mass of the products isequal to the mass of reactants (mR). A definition can be made asfollows:

mR ¼ ma1 þmf þms ð35Þ

The total number of moles of the products can be found by dividingthe mass of reactants into the molecular weight of the combustionproducts as follows:

N ¼ mR

Mð36Þ

Lastly, the number of moles m1, m2 � � � m10 are obtained from:

ti ¼ yiN ð37Þ

3. Results and discussion

In this section the equilibrium mole fractions of adiabatic com-bustion with three different fuels are presented comparatively fordifferent equivalence ratios for varying steam injection ratios. Itshould be noted that injected steam should not be regarded asan ideal gas since it is specific heat strongly depends on pressure.Steam tables should be used for the injected H2O where Eqs.(26)–(28) which ignore the pressure effect may lead significant er-rors calculating the thermodynamic properties, adiabatic flametemperature and therefore equilibrium mole fractions. The tem-perature of unburned mixture is found by assuming a thermalequilibrium among the reactants and the pressure inside the com-bustion chamber is taken as 30 atm. which is assumed equal to thepressure of injected H2O. The temperature of injected H2O at30 atm. is 300 �C corresponding to steam phase. The air and fuelmixture before the combustion is assumed to be at 300 K and atCC pressure. Initial temperatures before combustion are obtainedfrom the thermal equilibrium of fuel–air and steam mixture.Figs. 1–10 show the effect of H2O injection on the mole fractionsof each combustion product at equilibrium for four commonlyused fuels and two different equivalence ratios corresponding totwo commonly observed lean and rich cases in combustion en-gines. Fig. 11 and Fig. 12 show the changes in constant pressurespecific heats and adiabatic flame temperature. To show the reli-ability of the model, comparisons with results of CHEMKIN andGASEQ software are given in Tables 3 and 4.

4. Conclusion

This study shows a practical and precise way to estimate theequilibrium exhaust species and thermodynamic properties ofthe working gases in steam/water injected combustion systems.Effects of diluent injection on each exhaust species for differentfuels, injection and equivalence ratios are presented and a compar-ison with some existing tools are made.

In order to test the accuracy of the model, the results are com-pared with the results of CHEMKIN and GASEQ software. Compar-isons of the model results with those of GASEQ are quite similarwhich assumes the same number of product species. The resultsfrom CHEMKIN software obtained by using GRI-Mech [64] whichconsider 53 species are in reasonable accuracy which present thereliability of the proposed model. The model can be further verifiedby experiments using a constant pressure batch reactor and a gaschromatograph with appropriate columns.

The equilibrium composition obtained from the model can beused for estimating pollutant emissions and the thermodynamicproperties of the combustion products can be used for a preciseperformance estimation where existing studies in the available lit-erature on steam/water injected combustion systems assume onlythe complete combustion products.

400 H.K. Kayadelen, Y. Ust / Fuel 113 (2013) 389–401

Furthermore, the equilibrium mole fractions are often used incombustion engineering as a reference point for chemical kinetics.Taking reaction rate constants into account, equilibrium valuesdefined by the given combustion model can be used with anychemical kinetics correlation in the literature to obtain the non-equilibrium results. Equilibrium composition obtained from themodel can also be used in parametric emission monitoring systemsin order to estimate the emissions of power systems accuratelywithout need for directly sampling the exhaust gas which is anexpensive and time consuming method.

The model is independent from the fuel chemistry and the mod-eling procedure is described in details which makes it easily adapt-able to many different studies such as alternative fuel or multi-fuelstudies and EGR applications of combustion engines with or with-out H2O injection. With its precision and speed it is a fast and effec-tive tool that can also be used for parametrical analysis of multi-zone models.

Acknowledgements

This work is derived from authors unpublished Ph.D. Thesis:‘‘Effects of water/steam injection on the thermo-economic perfor-mance and emissions of gas turbines and their optimization’’ Theauthors also wish to express their sincere gratitude to Güven Gon-ca from Yıldız Technical University and Prof. Yiguang Ju fromPrinceton University for their valuable comments on the study.

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