air fuel ratio

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Energy Conversion and Management 46 (2005) 2387–2404

Temperature and air–fuel ratio dependent specific heat ratiofunctions for lean burned and unburned mixture

M.A. Ceviz *, _I. Kaymaz

Department of Mechanical Engineering, Faculty of Engineering, University of Ataturk, Erzurum 25240, Turkey

Received 19 July 2004; received in revised form 6 December 2004; accepted 29 December 2004

Available online 19 February 2005

Abstract

The most important thermodynamic property used in heat release calculations for engines is the specific

heat ratio. The functions proposed in the literature for the specific heat ratio are temperature dependent

and apply at or near stoichiometric air–fuel ratios. However, the specific heat ratio is also influenced bythe gas composition in the engine cylinder and especially becomes important for lean combustion engines.

In this study, temperature and air–fuel ratio dependent specific heat ratio functions were derived to min-

imize the error by using an equilibrium combustion model for burned and unburned mixtures separately.

After the error analysis between the equilibrium combustion model and the derived functions is presented,

the results of the global specific heat ratio function, as varying with mass fraction burned, were compared

with the proposed functions in the literature. The results of the study showed that the derived functions are

more feasible at lean operating conditions of a spark ignition engine.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Specific heat ratio; Combustion engine; Burned and unburned mixture; Heat release

1. Introduction

Thermodynamic analysis of measured cylinder pressure data is a very powerful tool for quan-tifying combustion parameters. There are two main approaches, which are often referred to as

0196-8904/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2004.12.009

* Corresponding author. Tel.: +90 442 231 48 48; fax: +90 442 235 44 93.

E-mail address: [email protected] (M.A. Ceviz).

mailto:[email protected]

Nomenclature

ATDC after top dead centrecp specific heat at constant pressure (kJkmol�1K�1)cv specific heat at constant volume (kJkmol�1K�1)EEOC crank angle for estimated end of combustionEVO exhaust valve openh heat transfer coefficient (Wm�2K�1)IVC intake valve closureK equilibrium constantm trapped mass (kg)MBT maximum brake torque timingMFB mass fraction burnedn polytropic indexp pressure inside cylinder (kPa)Q heat (kJ)R gas constant (kJkmol�1K�1)RMSE root mean square errorT temperature (K)TDC top dead centreU internal energy (kJ)V volume (m3)W work done (kJ)

Greeksc ratio of specific heatsk air–fuel ratioh crank angle

Subscripts

hr heat releaser references sensibleb burned gasi integer crank angle locationign ignition crank angle locationu unburned gasw wall

2388 M.A. Ceviz, _I. Kaymaz / Energy Conversion and Management 46 (2005) 2387–2404

‘‘burn rate analysis’’ and ‘‘heat release analysis’’. Burn rate analysis is mainly used for determiningburn angles in the gasoline and to obtain the mass fraction burned, which is a normalized

M.A. Ceviz, _I. Kaymaz / Energy Conversion and Management 46 (2005) 2387–2404 2389

quantity. Heat release analysis is most commonly used for Diesel engine combustion studies andproduces absolute energy with units of Joules or Joules/degree [1].

Both burn rate and heat release analyses are affected by a number of factors such as assumedgas properties, wall heat transfer, end of combustion location, start of combustion location, crankangle resolution, noise reduction techniques employed and so on. Previous papers [1–3] haveinvestigated the errors associated with heat release calculations.

The most important thermodynamic property used in the heat release calculations for engines isthe specific heat ratio. Gatowski et al. [3] developed a single zone heat release model based on thefirst law of thermodynamics that has been widely used, where the specific heat ratio is representedby a linear function of the mean charge temperature. Brunt et al. [1] utilized a second order func-tion, evaluated for a spark ignition (SI) engine fuelled with C8H16. This function has been eval-uated through a multi-dimensional model, and it is the mean function of the specific heat ratiofunctions evaluated for 0.83 < k < 1.25. Egnell [4] evaluated an exponential specific heat ratiofunction and emphasized that the constants have to be chosen carefully with respect to the influ-ence of temperature and gas composition. Lanzafame and Messina [5] suggested an alternativemethod for calculation of the specific heat ratio function. The method is valid for any applicationand needs gas thermodynamic properties and mass fraction burned, directly available from exper-imental pressure measurements. Klein and Erikson [6] focused on finding a specific heat ratiomodel during the combustion process by using the mass fraction burned to interpolate the specificheats for burned and unburned mixtures.

The main objective of this paper is to examine the effects of the existing temperature dependentspecific heat ratio models on the heat release calculations and to develop specific heat ratio func-tions depending on temperature and air–fuel ratio (k) for unburned and burned mixtures by usingan equilibrium combustion model to minimize the errors in the calculations of heat release. Thefunctions for unburned and burned mixtures were developed for a wide range of air–fuel ratio,1.0 < k < 1.6, to use in lean combustion spark ignition engines.

2. First law heat release model

The first law equation for the in-cylinder charge during the non-flow (constant mass) periodbetween inlet valve closure (IVC) and exhaust valve closure (EVC) can be written for an incremen-tal crank angle interval;

dQhr ¼ dW þ dU s þ dQw ð1Þ
where dQhr = gross heat energy released due to combustion, dW = work due to piston movement,dUs = change in sensible internal energy and dQw = heat transfer from charge to cylinder wall.Further equations can be written as follows:
pdV ¼ dW ð2Þ

m:�cv:dT ¼ dU s ð3Þ

dðp:V Þm:R

¼ dT ð4Þ


Rcv

¼ c� 1 ð5Þ

Substitution of Eqs. (2)–(5) into Eq. (1) and rearranging the terms gives the usual form of the firstlaw heat release equation:

dQhr ¼c

c� 1pdV þ 1

c� 1V dp þ dQw ð6Þ

in which gamma (c) is the ratio of specific heats.The gross cumulative heat release, Qhr, is calculated by successively applying Eq. (6) over the

crank angle range from ignition to the estimated end of combustion (EEOC) and summing theheat release energy obtained from each calculation. This process is shown in Eq. (7)

Qhr ¼XEEOC

i¼ign

dQhr;i ð7Þ

Brunt et al. [1] determined the EEOC from the crank angle at which p Æ V1.15 reached a maximumvalue. Calculation of the EEOC function was started at 10� ATDC and continued to 10� beforethe EVO.

The instantaneous heat transfer coefficient (h) adapted from Woschni [7] is given by

h ¼ 0:82b�0:2ðp10�3cÞ0:8T�0:53 ð8Þ

where c = 6.18 cm for a gas exchange process and c ¼ 2:28 cmþ 0:00324 DppIVC

VV IVC

for other

processes. Dp is the instantaneous pressure difference between the firing and motoring engine atthe same crank angle. The latter is estimated by using the isentropic relation P � V c ¼ PV c

IVC

[8].

3. Mass fraction burned

One well established method was developed by Rassweiler and Withrow [9] for estimating themass fraction burned profile from cylinder pressure and volume data. In this method, the massfraction burned is given by;

MFBh ¼Pi¼0

i¼ignDpc;iPi¼Ni¼ignDpc;i

ð9Þ

where MFBh = mass fraction burned at crank angle h, Dpc = corrected pressure rise due to thecombustion, i = integer crank angle location, ign = ignition crank angle location and EEOC =crank angle for estimated end of combustion.

The corrected pressure rise due to combustion is calculated from the difference between theincremental measured pressure rise and the pressure rise corresponding to a polytropic compres-sion/expansion process, referenced to the cylinder volume at TDC:


Dpc;i ¼ ½pi � ðV i�1 � V iÞnpi�1�ðV i�1 � V rÞ ð10Þ

where n is the assumed polytropic index, V is the cylinder volume and Vr is the reference volume atTDC.

4. Equilibrium combustion model

The composition and thermodynamic properties of each combustion product must be knownthroughout the combustion period to compare the results from specific heat ratio functions. Com-position of the combustion products was determined according to the solution given by Ferguson[10]. At lower temperatures and carbon to oxygen ratios less than one, the overall combustionreaction can be written as

CaHbOcNd þ ðO2 þ 3:76N2Þ ! n1CO2 þ n2H2Oþ n3N2 þ n4O2 þ n5COþ n6H2 ð11Þ
For the case of lean mixtures, the atom balance equations are sufficient to determine the compo-sition. For rich mixtures, the equilibrium reaction
CO2 þH2 $ COþH2O and KðT Þ ¼ n2n5n1n6

ð12Þ

must be included in the composition solution.The composition of the burned gases at higher temperatures was determined according to the

model of Olikara and Borman [11] using the 10 specie combustion reaction

CaHbOcNd þ ðO2 þ 3:76N2Þ ! n1CO2 þ n2H2Oþ n3N2 þ n4O2 þ n5COþ n6H2 þ n7H

þ n8Oþ n9OHþ n10NO ð13Þ

For temperatures above 1700 K, the mixture is assumed to be at equilibrium and is frozen other-wise. Conservation of the elemental species yields four equations. The six equilibrium reactionsshown below are also required in order to obtain a composition solution. The equilibrium con-stant data were taken from Ferguson [10].

Equilibrium reaction Equilibrium constant

1

2H2 $ H K1 ¼

y7p1=2

y1=26

ð14Þ

1

2O2 $ O K2 ¼

y8p1=2

y1=24

ð15Þ

1

2H2 þ

1

2O2 $ OH K3 ¼

y9y1=24 y1=26

ð16Þ

1

2O2 þ

1

2N2 $ NO K4 ¼

y10y1=24 y1=23

ð17Þ

H2 þ1

2O2 $ H2O K5 ¼

y2y1=24 y6p1=2

ð18Þ

COþ 1

2O2 $ CO2 K6 ¼

y1y5y

1=24 p1=2

ð19Þ


The composition of the burned gases is determined from a set of simultaneous nonlinear equa-tions, which were solved by the MATLAB Optimization Toolbox [12]. After calculation of thecomposition of the burned gases, the thermodynamic properties and specific heats of each speciewere calculated by using the polynomial functions fitted from the JANAF table. The coefficientsof the polynomial functions were taken from Heywood [13].

5. Specific heat ratio functions and importance of composition effects

The specific heats are dependent on charge temperature and composition and, as such, will varyduring the engine cycle and with operating conditions [2]. Although the use of a temperaturedependent specific heat ratio function greatly reduces the potential errors in calculated the heatrelease, the dependency of the specific heat on composition is greater at the lean side of stoichi-ometric conditions.

The emergence of lean fuel engine operation as a low emission technology for both liquid andgaseous fuel engines presents a new opportunity for real time optimization. Most present gasolineengines using three way catalysts, and therefore stoichiometric operation, sacrifice 10–15% in fueleconomy [14]. Exhaust gas recirculation (EGR) under moderate load conditions may reducepumping losses without sacrificing the combustion burn rate so as to recover half of the fuel econ-omy penalty of stoichiometric operation.

Figs. 1 and 2 show the specific heat ratio against the charge temperature based on the results ofthe equilibrium combustion model for burned and unburned mixtures at various air–fuel ratios ina gasoline engine. It can be seen from these figures that the specific heat ratio decreases as the

Fig. 1. Variation of specific heat ratio with temperature and air–fuel ratio for burned mixture.

Fig. 2. Variation of specific heat ratio with temperature and air–fuel ratio for unburned mixture.


temperature increases, whereas it increases as the air–fuel ratio increases. These curves reveal thatthe relationship between specific heat ratio and temperature is almost linear and the variation withlambda is significant.

In this section, the existing and derived specific heat ratio functions are described. Thereafter,the results of the chemical equilibrium combustion model and the derived specific heat ratio func-tions for unburned and burned mixtures will be compared.

1. Function of Gatowski et al. [3]:

c ¼ c0 � K1ðT � T refÞ=1000 ð20Þ
where c0 is a reference value (1.38), K1 is a constant (0.08) and Tref is a reference temperature(300 K).
2. Function of Brunt et al. [2]:

c ¼ 1:338� 6:0� 10�5T þ 1:0� 10�8T 2 ð21Þ
where T is temperature in K.
3. Function of Egnell [4]:

c ¼ c0 � k1 expð�k2=T Þ ð22Þ
where c0 is a reference value (1.38), k1 and k2 are constants (0.2 � 900) and Tref is a reference tem-perature (300 K).
4. Derived functions of burned and unburned mixtures:The specific heat ratios for the unburned and burned mixtures were computed using the equi-

librium combustion model. The air–fuel ratio was increased from 1.0 to 1.6 in steps of 0.1, and theranges of temperature were 300–1500 K and 300–2500 K for the unburned and burned mixtures,


respectively. The data of the specific heat ratio was used to derive functions for the unburned andburned mixtures using a least squares approach and choosing the parameters as temperature (T inK) and air–fuel ratio (k). The functions of the specific heat ratio for the unburned and burnedmixtures in T and k can be written as:

Table

Coeffi

Coeffi

a1a2a3a4a5a6a7–

–

–

cu ¼ a1 þ a2T þ a3T 2 þ a4T 3 þ a5T 4 þ a6T 5 þ a7k

ð23Þ

cb ¼ b1 þ b2T þ b3kþ b4T 2 þ b5

k2þ b6T

kþ b7T 3 þ b8

k3þ b9T

k2þ b10T 2

kð24Þ

The values of the coefficients of these functions are given in Table 1.

1

cients for specific heat ratio function for unburned and burned mixtures

cients (cu) Values Coefficients (cb) Values

1.464202464 b1 1.498119965

�0.000150666 b2 �0.00011303

�7.34852e-08 b3 �0.26688898

1.55726E-10 b4 4.03642e�08

�7.6951E-14 b5 0.273428364

1.19535E-17 b6 5.7462e�05

�0.063115275 b7 �7.2026e�12

– b8 �0.08218813

– b9 �1.3029e�05

– b10 2.35732e�08

Fig. 3. Specific heat ratio for unburned mixture using equilibrium combustion model and cu function.


Since the specific heat ratio is dependent on the temperature and air–fuel ratio and MFB is notdependent on the value chosen for the constant specific heat ratio, it is possible to write for thefunction of c [5]:

Table

Variat

k

1

1.1

1.2

1.3

1.4

1.5

1.6

c ¼ MFBcb þ ð1�MFBÞcu ð25Þ
The specific heat ratios of the unburned and burned mixtures for an air–fuel ratio k = 1.2 areshown in Figs. 3 and 4, together with the corresponding c function. Table 2 summarizes the var-iation of the percent root mean square error (%RMSE) with air–fuel ratio for the unburned andburned mixtures at k = 1.2. The %RMSE is used to describe the accuracy while encompassingboth random and systematic errors. It is the percent square of the difference between a true testpoint and an interpolated test point divided by the total number of test points in the arithmeticmean. The standard definition is given by [15]:
Fig. 4. Specific heat ratio for burned mixture using equilibrium combustion model and cb function.

2

ion of %RMSE with air–fuel ratio for cu and cb at k = 1.2

cu%RMSE cb%RMSE

0.0221 0.0066

0.0121 0.0110

0.0067 0.0165

0.0068 0.0209

0.0096 0.0252

0.0123 0.0303

0.0146 0.0364

Fig. 5. Comparison between several burned mixture specific heat ratio functions.

Fig. 6. Comparison between several unburned mixture specific heat ratio functions.


%RMSE ¼ 100

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1n

Pni¼1ðyi � yiÞ

2q

1n

Pni¼1yi

ð26Þ


where n is the number of samples. It can be deduced that the derived functions are a good approx-imation for the unburned and burned mixtures since the %RMSEs are reasonably small.

Several burned specific heat ratio functions are compared for burned and unburned mixtures inFigs. 5 and 6, respectively. As can be seen from these figures, the different ways of calculating the

Fig. 7. (a) Cylinder pressure data for 100 consecutive cycles, (b) Averaged cylinder pressure data for 100 consecutive

cycles (k = 0.9966, PIVC = 71 kPa, n = 2500 rpm, MBT timing 20� BTC).


specific heat ratio give different results. It can also be seen that burned mixtures have higherspecific heat ratio values than unburned mixtures. The linear function of Gatowski et al. [3] givesthe highest specific heat ratio for burned and unburned mixtures, while the function used inEgnell�s work [4] gives a good fit for a burned mixture. For a narrow high temperature interval,it can be stated that the function of Brunt et al. [1] is close to the results of the equilibriumcombustion model for a burned mixture. However, the results of the functions proposed in theliterature are far from the results that were calculated using the equilibrium combustion modelfor an unburned mixture.

6. Experimental apparatus and procedure

The engine used in the present study is a FIAT, 1.801 dm3, carbureted, four stroke sparkignition engine. The engine is fully equipped for measurements of all operating parameters.The pressure time history was measured by a piezo-electric pressure transducer (KISTLER,6117BFD17 type) and the crankshaft degree angle sensor connected to the relevant amplifiers.A data acquisition system was used to collect the important data and store the data in a personalcomputer for off line analysis. A computer program in Q-BASIC language was written to collectthe data.

The pressure signal was fed into a charge amplifier and then to a data acquisition card linked tothe personal computer. The crank angle signal was fed into a degree maker shape channel, and theoutput was fed into the acquisition card. The acquisition card could collect data at the rate of100 kHz. To reduce the amount of high frequency noise on the pressure signal, low pass digitalfiltering was applied to all the pressure data. The cylinder pressure data is pegged by assumingthe pressure at bottom dead centre after the intake stroke is equal to the mean intake manifoldpressure [16].

The experiments have been performed at 3/4 throttle valve opening position (2500 rpm) byloading the engine with a hydraulic dynamometer and at four different air–fuel ratios (0.996,1.089, 1.216 and 1.341) after running the engine for some time until it reached steady state.For elimination of the effects of the cyclic variations, the averaged value of the signal of 100 con-secutive cycles was used to compare the change in the specific heat ratio during the combustionperiod. The measured cylinder pressure for 100 consecutive cycles and the average of the datafor maximum brake torque timing (MBT) can be seen in Fig. 7a and b.

7. Results and discussion

To verify the accuracy of the derived function, it has been implemented for calculation of theheat release for a SI engine operating at four different air–fuel ratios (0.996, 1.089, 1.216 and1.341). As the heat release calculations were performed at the combustion stroke, the variationof the parameters taken into consideration is illustrated during the combustion period in the fol-lowing figures. Fig. 8 shows the different specific heat ratio curves generated when performing aheat release calculation, and Fig. 9 shows the effects of varying specific heat ratio on the heat re-lease calculation at k = 0.996.

Fig. 8. Variation of specific heat ratio with crank angle during the combustion cycle at k = 0.996.

Fig. 9. Variation of cumulative heat release with crank angle during the combustion cycle at k = 0.996.


It can be seen from Fig. 8 that the specific heat ratio function of Gatowski et al. [3] gives thehighest values, and that of Egnell [4] follows the specific heat ratio values of the burned mixture.




The results of Brunt et al. [1] seem to close to the results of the equilibrium combustion model,using mass fraction burned, and the burned and unburned mixture specific heat ratio calculations.




Fig. 9 shows a great influence of the specific heat ratio values on the cumulative heat release,and the use of a function giving high specific heat ratio values gives a lower cumulative heat




release. The high values calculated from the function of Gatowski et al. [3] reduce the heat releasevalues. It can be seen from Figs. 8 and 9 that both the specific heat ratio and cumulative heat re-lease results obtained from the equilibrium combustion model using the burned and unburned


mixture specific heat ratios calculation of each specie are quite close to that obtained from thederived functions for burned and unburned mixture specific heat ratio calculations rather thanthe functions used in the literature.

To reveal the usage of the c function in lean combustion, the results of the specific heat andcumulative heat release calculations are compared in Figs. 10–15. It can be seen from these figures,since the functions proposed in the literature are dependent on only temperature, as the air–fuelratio increases, the results of the specific heat ratio functions deviate from those of the equilibriumcombustion model. However, the equilibrium combustion model and the c function are still fittingbecause of the temperature and air–fuel ratio dependence of the derived cu and cb functions.

It is known that as the air–fuel ratio increases, the specific heat ratio increases and the in-cylinder temperature decreases. However, the cumulative heat release decreases, conversely.The influence of temperature on the specific heat ratio is more dominant than that of the air–fuelratio (Figs. 1 and 2), and the trend of variation can be seen in the following Figs. 10–15.

Additionally, the cumulative heat release results from the Egnell function at k = 1.089, as seenin Fig. 11, are quite close to the derived c function. It can be deduced that the constants of thefunction are appropriate for an engine operating near k = 1.1.

8. Conclusions

In this study, for burned and unburned mixtures, temperature and air–fuel ratio dependent spe-cific heat ratio functions were derived by using the equilibrium combustion model and the vari-ations of gases thermodynamic properties with mean temperature. Then, the global specificheat ratio was calculated by using the variation of the mass fraction burned. The results show thatimplementation of a c = c(T,k) function reduces notably the error deriving from temperature onlydependent specific heat ratio under lean operation of engine.

The experiments performed at four different air–fuel ratios show that as the air–fuel ratio in-creases, the results of the equilibrium combustion model and the c function are in reasonableagreement. Additionally, the derived functions for burned and unburned mixtures have a greatsimplicity in the mathematical formulation and only need the global air–fuel ratio, temperatureand mass fraction burned, which can be determined from experimental pressure measurements.

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