**5. Simplified combustion modelling**

The ability to describe analytically the charge burn process in SI engines, capturing the details of the most relevant influences on this, is essential for both diagnosis of performance and control requirements. The empirical combustion models that enable just the above ability are usually called non-dimensional or zero-dimensional models, because they do not incorporate any explicit reference to combustion chamber geometry and flame front propagation. Such approaches typically output the burn rate or the MFB profile in the CA domain and require several stages of calibration by fitting real engine data to appropriate analytical functions. One of the most widely used methods in engines research is to carry out curve fits of experimental MFB curves to describe the combustion evolution via a Wiebe Function,

$$\propto \chi\_{MFB} \left( \mathcal{G} \right) = 1 - EXP \left[ -a \left( \frac{\mathcal{G} - \mathcal{G}\_{ST}}{\Lambda \mathcal{G}} \right)^{n} \right] \tag{20}$$

**5.1. Combustion modelling using the Wiebe function**

combustion to the consumption of fuel.

correlating these to measurable or inferred engine variables.

*5.1.1. Methodology*

One of the most comprehensive accounts of rationale and applications of the Wiebe function as a burn rate model is due to J. I. Ghojel, in his recent *tribute to the lasting legacy of the Wiebe function and to the man behind it, Ivan Ivanovitch Wiebe* [48]. The purpose of the original work by Wiebe was to develop a macroscopic reaction rate expression to bypass the complex chemical kinetics of all the reactions taking place in engine combustion. The result, which is typically based on a law of normal distribution representing the engine burning rate, is a very flexible function, heavily used in the last few decades to model all forms and modes of combustion, including compression and spark ignition, direct and indirect injection and homogeneous charge compression ignition combustion, with a range of liquid and gaseous fuels. An extensive survey on the implementation of the Wiebe function (as well as some mathematical modifications) has been also carried out by Oppenheim et al. [49]. Here the authors recognise the practical virtues of the function and its well-established use, but question its derivation, which is described as a *gigantic leap* from chemical kinetics of the exothermic reactions of

Premixed Combustion in Spark Ignition Engines and the Influence of Operating Variables

http://dx.doi.org/10.5772/55495

29

The aim of the investigation is to model the S-shaped MFB profile from combustion initiation (assumed to coincide with spark timing) to termination (100% MFB), using the independent parameters of the Wiebe function. The total burning angle is taken as the 0 to 90% burn interval, *Δϑ*90, preferred to the 0-99.9% interval used in other work (for example [50]) as the point of 90% MFB can be determined experimentally with greater certainty. Especially for highlydiluted, slow-burning combustion events the termination stage cannot be described accurately from experimental pressure records, as the relatively small amount of heat released from combustion of fuel is comparable to co-existing heat losses, e.g. to the cylinder walls [51]. For these reasons, using the *Δϑ*<sup>90</sup> rather than the *Δϑ*99.9 as total combustion angle should ascribe increased accuracy to the overall combustion model. It can be demonstrated that, with the above choice of total combustion angle, the efficiency factor *a* takes a unique value of 2.3026. *Δϑ*90 and *n* remain as independent parameters and curve fits to experimental MFB curves allow

The experimental data used for model calibration have been collected using the research engine described in section 4.1. All tests were carried out under steady-state, fully-warm operating conditions which covered ranges of engine speed, load, spark advance and cylinder charge dilution typical of urban and cruise driving conditions. Data were recorded using always stoichiometric mixtures. Dilution mass fraction, determined from measurements of molar concentrations of carbon dioxide as indicated in section 2.4, was varied either via an intercooled external-EGR system or adjusting the degree of valves overlap via the computerised engine-rig controller. Intake and exhaust valve timing were varied independently to set overlap intervals from -20 to +42 CA degrees. The ranges of engine variables covered in this work are the same as those reported in section 4.1. The experimental database included in excess of 300 test-points. Data collected varying the valve timing setting were kept separately

where *ϑ* is the generic CA location, *ϑST* is the CA location of combustion initiation, *Δϑ* is the total combustion angle, *a* and *n* are adjustable parameters called efficiency and form factor. Owing to its simplicity and robustness of implementation the Wiebe function represents a convenient platform for extracting signals from the noises of the sensing elements and lends itself ideally for inclusion into model-based control algorithms. Section 5.1 explores derivation, experience of use and limitations of a Wiebe function-based model of the MFB profile, originally presented by the Author in [46], and developed using experimental data from the same modern-design gasoline engine described in section 4.

A different type of combustion models are the so-called phenomenological or quasi-dimen‐ sional models. These require spatial subdivision of the combustion space into zones of different temperature and chemical composition (two zones, burned and unburned, in their simplest version) and are often used to evaluate both combustion of fuel and associated pollutant formation. Phenomenological models are based on more fundamental theoretical principles, hence they should be *transportable* between engines of different size and geometry. Neverthe‐ less, these models also require some form of calibration using real measurements. Recent published work by Hall et al. [47] and by Prucka et al. [5] offers examples of relatively simple phenomenological flame propagation and entrainment models, suited for inclusion into fastexecution ST control algorithms, to ensure optimal phasing of the 50% MFB location (i.e. 7 or 8 CA degree ATDC) and improve engine efficiency and fuel economy. In these models, the instantaneous rate of combustion is calculated fundamentally, through some modifications of the basic equation of mass continuity *dmb* / *dτ* =*ρuASb*. Section 5.2 illustrates one of these models, along with results which highlight the influence of relevant engine operating variables on combustion for a modern flexible-fuel engine.

## **5.1. Combustion modelling using the Wiebe function**

One of the most comprehensive accounts of rationale and applications of the Wiebe function as a burn rate model is due to J. I. Ghojel, in his recent *tribute to the lasting legacy of the Wiebe function and to the man behind it, Ivan Ivanovitch Wiebe* [48]. The purpose of the original work by Wiebe was to develop a macroscopic reaction rate expression to bypass the complex chemical kinetics of all the reactions taking place in engine combustion. The result, which is typically based on a law of normal distribution representing the engine burning rate, is a very flexible function, heavily used in the last few decades to model all forms and modes of combustion, including compression and spark ignition, direct and indirect injection and homogeneous charge compression ignition combustion, with a range of liquid and gaseous fuels. An extensive survey on the implementation of the Wiebe function (as well as some mathematical modifications) has been also carried out by Oppenheim et al. [49]. Here the authors recognise the practical virtues of the function and its well-established use, but question its derivation, which is described as a *gigantic leap* from chemical kinetics of the exothermic reactions of combustion to the consumption of fuel.

#### *5.1.1. Methodology*

**5. Simplified combustion modelling**

28 Advances in Internal Combustion Engines and Fuel Technologies

The ability to describe analytically the charge burn process in SI engines, capturing the details of the most relevant influences on this, is essential for both diagnosis of performance and control requirements. The empirical combustion models that enable just the above ability are usually called non-dimensional or zero-dimensional models, because they do not incorporate any explicit reference to combustion chamber geometry and flame front propagation. Such approaches typically output the burn rate or the MFB profile in the CA domain and require several stages of calibration by fitting real engine data to appropriate analytical functions. One of the most widely used methods in engines research is to carry out curve fits of experimental

> *n ST*

(20)

J J

é ù æ ö - =- -ê ú ç ÷ ê ú è ø <sup>D</sup> ë û

where *ϑ* is the generic CA location, *ϑST* is the CA location of combustion initiation, *Δϑ* is the total combustion angle, *a* and *n* are adjustable parameters called efficiency and form factor. Owing to its simplicity and robustness of implementation the Wiebe function represents a convenient platform for extracting signals from the noises of the sensing elements and lends itself ideally for inclusion into model-based control algorithms. Section 5.1 explores derivation, experience of use and limitations of a Wiebe function-based model of the MFB profile, originally presented by the Author in [46], and developed using experimental data from the

A different type of combustion models are the so-called phenomenological or quasi-dimen‐ sional models. These require spatial subdivision of the combustion space into zones of different temperature and chemical composition (two zones, burned and unburned, in their simplest version) and are often used to evaluate both combustion of fuel and associated pollutant formation. Phenomenological models are based on more fundamental theoretical principles, hence they should be *transportable* between engines of different size and geometry. Neverthe‐ less, these models also require some form of calibration using real measurements. Recent published work by Hall et al. [47] and by Prucka et al. [5] offers examples of relatively simple phenomenological flame propagation and entrainment models, suited for inclusion into fastexecution ST control algorithms, to ensure optimal phasing of the 50% MFB location (i.e. 7 or 8 CA degree ATDC) and improve engine efficiency and fuel economy. In these models, the instantaneous rate of combustion is calculated fundamentally, through some modifications of the basic equation of mass continuity *dmb* / *dτ* =*ρuASb*. Section 5.2 illustrates one of these models, along with results which highlight the influence of relevant engine operating variables

J

MFB curves to describe the combustion evolution via a Wiebe Function,

( ) 1

J

same modern-design gasoline engine described in section 4.

on combustion for a modern flexible-fuel engine.

*MFB x EXP a*

The aim of the investigation is to model the S-shaped MFB profile from combustion initiation (assumed to coincide with spark timing) to termination (100% MFB), using the independent parameters of the Wiebe function. The total burning angle is taken as the 0 to 90% burn interval, *Δϑ*90, preferred to the 0-99.9% interval used in other work (for example [50]) as the point of 90% MFB can be determined experimentally with greater certainty. Especially for highlydiluted, slow-burning combustion events the termination stage cannot be described accurately from experimental pressure records, as the relatively small amount of heat released from combustion of fuel is comparable to co-existing heat losses, e.g. to the cylinder walls [51]. For these reasons, using the *Δϑ*<sup>90</sup> rather than the *Δϑ*99.9 as total combustion angle should ascribe increased accuracy to the overall combustion model. It can be demonstrated that, with the above choice of total combustion angle, the efficiency factor *a* takes a unique value of 2.3026. *Δϑ*90 and *n* remain as independent parameters and curve fits to experimental MFB curves allow correlating these to measurable or inferred engine variables.

The experimental data used for model calibration have been collected using the research engine described in section 4.1. All tests were carried out under steady-state, fully-warm operating conditions which covered ranges of engine speed, load, spark advance and cylinder charge dilution typical of urban and cruise driving conditions. Data were recorded using always stoichiometric mixtures. Dilution mass fraction, determined from measurements of molar concentrations of carbon dioxide as indicated in section 2.4, was varied either via an intercooled external-EGR system or adjusting the degree of valves overlap via the computerised engine-rig controller. Intake and exhaust valve timing were varied independently to set overlap intervals from -20 to +42 CA degrees. The ranges of engine variables covered in this work are the same as those reported in section 4.1. The experimental database included in excess of 300 test-points. Data collected varying the valve timing setting were kept separately and used for purposes of model validation. MFB profiles at each test-point were built applying the Rassweiler and Withrow methodology to ensemble-averaged pressure traces. The FDA, 50% MFB duration (*Δϑ*50) and RBA were calculated from these curves, using a linear interpo‐ lation between two successive crank angles across 10%, 50% and 90% MFB to improve the accuracy of the calculations.

(and increasing combustion duration) for overly advanced spark ignition settings (in excess of 35 CA degrees BTDC). The net effect of engine speed (or mean piston speed) on burn duration is accounted for by an hyperbolic term *S*(*N* ). Turbulence intensity in the vicinity of the spark-plug has been shown to be directly proportional to engine speed (see section 2.2 above); hence increasing engine speed enhances the burning rate via greater combustion chamber turbulence. In truth, in a modern engine there is a number of factors, including intake and exhaust valve timing, which may have an impact on turbulence intensity. Different sources spanning across several decades continue to indicate that the main driver for in-cylinder turbulence is engine speed and that other factors actually exert only a minor influence [12, 20, 21, 46, 47, 57]. However, increasing engine speed also extends the burn process over wider CA

intervals and the effect of greater turbulence is only to moderate such extension.

*P S*

*ST SN <sup>b</sup> X x T*

0.77 1 2.06 1

Premixed Combustion in Spark Ignition Engines and the Influence of Operating Variables

0.67

*<sup>b</sup> <sup>x</sup> -*

 

13.65 <sup>1</sup> *- -*

13.65 <sup>1</sup> *- -*

13.65 <sup>1</sup> *- -*

 0.77 1 2.06 1

0.77 1 2.06 1

0.46 1 2.06 1

 

0.85

0.85

 

> 

0.77 1 2.06 *<sup>b</sup> x c d e*

*<sup>b</sup> <sup>a</sup>*

 

2004 [54] *- - -* 0.238 0.272 <sup>1</sup>

[55] *- - -* 0.233 0.664 <sup>1</sup>

1.39 <sup>1</sup>

 

 

**Table 2.** Functional expressions used in [46] to account for the influence of operating variables on the total burn

The influence of charge dilution by burned gas on combustion duration is accounted for via a

  *ST*

http://dx.doi.org/10.5772/55495

31

2

 

2

 

1

1

*ST*

*ST*

*<sup>b</sup> <sup>x</sup>* 5.69 <sup>10</sup> 0.033 <sup>1</sup> <sup>4</sup> <sup>2</sup>

*<sup>b</sup> <sup>x</sup>* 0.00033 0.0263 <sup>1</sup> <sup>2</sup>

*<sup>b</sup> <sup>x</sup>* 0.0004 0.024 <sup>1</sup> <sup>2</sup>

0.77), originally identified by Rhodes and Keck [58] and by

1

1

*ST*

*ST*

*ST ST*

*ST ST x* 

*ST ST*

*ST ST* <sup>2</sup>

**Function** *R*

**Analytical Form**

Hires et al. 1978

Csallner. 1981

Bayraktar et al.

Scharrer et al.

Vávra et al. 2004

Lindstöm et al.

**Bonatesta et al. 2010 [46]**

Galindo et al. 2011 [56]

 

[23] *- -*

[52] *- <sup>N</sup>*

Witt. 1999 [53] *- <sup>N</sup>*

2004 [45] *- <sup>N</sup>*

2005 [50] *- <sup>P</sup> <sup>S</sup>*

 

 

angle, along with similar published expressions.

power law of the function (1−2.06*xb*

0.34 1 

0.34 1 

*ST <sup>P</sup> S*

*ST <sup>P</sup> S*

1.164 <sup>1</sup>

0.8 <sup>1</sup>

 *ST* 1

The combustion process in premixed gasoline engines is influenced by a wide range of engine specific as well as operating variables. Some of these variables, such as the valve timing setting, can be continuously varied to achieve optimum thermal efficiency (e.g. by improving cylinder filling and reducing pumping losses) or to meet ever more stringent emissions regulations (e.g. by increasing the burned gas fraction to control the nitrogen oxides emissions). There is some consensus in recent SI engine combustion literature upon the variables which are *essential and sufficient* to model the charge burn process in the context of current-design SI engines [23, 45, 50, 52, 53, 54, 55, 56]. For stoichiometric combustion, in decreasing rank of importance these are charge dilution by burned gas, engine speed, ignition timing and charge density. In the present work dilution mass fraction has been of particular interest as large dilution variations produced by both valve timing setting and external-EGR are part of current combustion and emissions control strategies.

#### *5.1.2. Models derivation*

Empirical correlations for the two independent parameters of the Wiebe function, the total burn angle, *Δϑ*90 and the form factor, *n*, have been developed carrying out least mean square fits of functional expressions of engine variables to combustion duration data. Whenever possible, power law functions were used in order to minimise the need for calibration coefficients. The choice of each term is made to best fit the available experimental data.

The 0 to 90% MFB combustion angle is expressed as the product of 4 functional factors, whose influence is assumed to be independent and separable:

$$
\Delta \mathcal{B}\_{\rm 00} = k \,\, \mathcal{R} \left( \rho\_{ST} \right) \,\, \text{S(N)} \,\, X \left( \mathbf{x}\_b \right) \,\, T \left( \mathcal{B}\_{\rm ST} \right) \tag{21}
$$

The functions *S*(*N* ), *X* (*xb*) and *T* (*ϑST* ) are based upon previously proposed expression, which are reviewed, along with their modifications, in Table 2. With the best-fit numerical coefficients from reference [46], the dimensional constant *k* was determined to be 178 when density *ρST* is in kg/m3 , mean piston speed *SP* is in m/s, the dilution mass fraction is dimensionless, and the spark timing *ϑST* is in CA degrees BTDC. For the spark ignition term, *T* (*ϑST* ), a second-order polynomial fit has been preferred to the hyperbolic function (*a* + *b* / *ϑST* ) proposed by Csallner [52] and Witt [53] as it is deemed to retain stronger physical meaning, showing a turning point for very advanced spark timing settings. Advancing the spark ignition generally shortens the total burn duration as combustion is phased nearer TDC; it is expected though that excessively low initial pressure and temperature would change this trend, producing slower combustion (and increasing combustion duration) for overly advanced spark ignition settings (in excess of 35 CA degrees BTDC). The net effect of engine speed (or mean piston speed) on burn duration is accounted for by an hyperbolic term *S*(*N* ). Turbulence intensity in the vicinity of the spark-plug has been shown to be directly proportional to engine speed (see section 2.2 above); hence increasing engine speed enhances the burning rate via greater combustion chamber turbulence. In truth, in a modern engine there is a number of factors, including intake and exhaust valve timing, which may have an impact on turbulence intensity. Different sources spanning across several decades continue to indicate that the main driver for in-cylinder turbulence is engine speed and that other factors actually exert only a minor influence [12, 20, 21, 46, 47, 57]. However, increasing engine speed also extends the burn process over wider CA intervals and the effect of greater turbulence is only to moderate such extension.

and used for purposes of model validation. MFB profiles at each test-point were built applying the Rassweiler and Withrow methodology to ensemble-averaged pressure traces. The FDA, 50% MFB duration (*Δϑ*50) and RBA were calculated from these curves, using a linear interpo‐ lation between two successive crank angles across 10%, 50% and 90% MFB to improve the

The combustion process in premixed gasoline engines is influenced by a wide range of engine specific as well as operating variables. Some of these variables, such as the valve timing setting, can be continuously varied to achieve optimum thermal efficiency (e.g. by improving cylinder filling and reducing pumping losses) or to meet ever more stringent emissions regulations (e.g. by increasing the burned gas fraction to control the nitrogen oxides emissions). There is some consensus in recent SI engine combustion literature upon the variables which are *essential and sufficient* to model the charge burn process in the context of current-design SI engines [23, 45, 50, 52, 53, 54, 55, 56]. For stoichiometric combustion, in decreasing rank of importance these are charge dilution by burned gas, engine speed, ignition timing and charge density. In the present work dilution mass fraction has been of particular interest as large dilution variations produced by both valve timing setting and external-EGR are part of current combustion and

Empirical correlations for the two independent parameters of the Wiebe function, the total burn angle, *Δϑ*90 and the form factor, *n*, have been developed carrying out least mean square fits of functional expressions of engine variables to combustion duration data. Whenever possible, power law functions were used in order to minimise the need for calibration coefficients. The choice of each term is made to best fit the available experimental data.

The 0 to 90% MFB combustion angle is expressed as the product of 4 functional factors, whose

The functions *S*(*N* ), *X* (*xb*) and *T* (*ϑST* ) are based upon previously proposed expression, which are reviewed, along with their modifications, in Table 2. With the best-fit numerical coefficients from reference [46], the dimensional constant *k* was determined to be 178 when density *ρST* is

spark timing *ϑST* is in CA degrees BTDC. For the spark ignition term, *T* (*ϑST* ), a second-order polynomial fit has been preferred to the hyperbolic function (*a* + *b* / *ϑST* ) proposed by Csallner [52] and Witt [53] as it is deemed to retain stronger physical meaning, showing a turning point for very advanced spark timing settings. Advancing the spark ignition generally shortens the total burn duration as combustion is phased nearer TDC; it is expected though that excessively low initial pressure and temperature would change this trend, producing slower combustion

, mean piston speed *SP* is in m/s, the dilution mass fraction is dimensionless, and the

J

(21)

( ) ( ) ( ) ( ) <sup>90</sup> *ST b ST* D =

*kR SN X x T*

 r

influence is assumed to be independent and separable:

J

accuracy of the calculations.

30 Advances in Internal Combustion Engines and Fuel Technologies

emissions control strategies.

*5.1.2. Models derivation*

in kg/m3


**Table 2.** Functional expressions used in [46] to account for the influence of operating variables on the total burn angle, along with similar published expressions.

The influence of charge dilution by burned gas on combustion duration is accounted for via a power law of the function (1−2.06*xb* 0.77), originally identified by Rhodes and Keck [58] and by Metghalchi and Keck [59] to represent the detrimental influence of burned gas on the laminar burning velocity. The power correlation given by *X* (*xb*) was retrieved changing the dilution fraction over the range 6 to 26% via external-EGR (with fixed, default valve timing setting), to minimise the disturbing influence of valve timing on other factors such as cylinder filling. Figure 15 illustrates how the density function, *R*(*ρST* ), with power index set to 0.34, fits to data recorded over a range of engine loads between 2 and 7 bar net IMEP, at each of three engine speeds. During these load sweeps, the spark timing and level of dilution, which would normally change with load as a result of changing exhaust to intake pressure differential, were set constant.

**Figure 15 Figure 15.** Total burning angle as a function of charge density at constant spark timing and constant dilution mass fraction for three engine speeds. Adapted from reference [46].

S5 S6 S7 S8 3-4 2-3 1-2 0-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 **ALPHA (WEIGHT)**……………. **∆θd FITTING ERROR (CA) Plot (a)** The density function was introduced in [46] to account for a further observed influence of engine load on combustion duration, not fully captured by the empirical terms developed for the dilution fraction or the spark ignition setting. The same density expression has been adopted more recently by Galindo et al. [56] in a work devoted to modelling the charge burn curve in small, high-speed, two-stroke, gasoline engines using a Wiebe function-based approach.

S4 The final correlation developed for the total 0 to 90% MFB burning duration is written as:

$$
\Delta\theta\_{90} = 178 \left(\frac{1}{\rho\_{ST}}\right)^{0.34} \left(1 - \frac{1.164}{\sqrt{S\_p}}\right) \left(\frac{1}{1 - 2.06 \text{ } \text{x}\_b^{0.77}}\right)^{0.85} \left(0.00033 \text{ } \text{g}\_{ST}^2 - 0.0263 \text{ } \text{g}\_{ST} + 1\right) \tag{22}
$$

S3

S1 S2 S3 S4 S5 S6 1-2 0-1

**∆θ50 FITTING**

S7 S8 3-4 2-3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 **ALPHA (WEIGHT)**……………. **ERROR (CA)** Its accuracy was tested across the whole 300-point-wide experimental database, featuring combinations of engine variables limited by the ranges discussed in section 4.1. Part of this

**Plot (b)**

**LIST OF ACRONYMS AND ABBREVIATIONS** 

**Figure 16**

1 2 3 4 5 6 7 8 9 10 11 12

31 33 35 37 39 41 43 45 47 49 51……. .. **∆θ50 - CA degree**

4

database, including the VVT data, was used for model validation only. Values of *Δϑ*<sup>90</sup> generated using equation (22) showed a maximum prediction error of 11%, while three-

Values of form factor *n* used to build the second empirical correlation advanced in [46], have been determined by minimising the error of the least square fit to the MFB profile. A covariance

( )


1 *MFB MFB Wiebe*

*<sup>N</sup> MFB*

depending on the ultimate target of the combustion modelling exercise.

follows:

2 exp

Premixed Combustion in Spark Ignition Engines and the Influence of Operating Variables

http://dx.doi.org/10.5772/55495

33

å (23)

( )

(24)

<sup>1</sup> *<sup>b</sup>*

( )

exp

was minimised to achieve a balanced fit across the burn profile from the start of combustion and up to the 50% MFB location. As evident from the error analysis illustrated in figure 16, as the weight factor *α* increases the absolute fitting error in FDA falls, whereas the one in *Δϑ*<sup>50</sup> tends to increase at least for durations in excess of 40 CA degrees. A weight factor of 0.3 allows limiting the error in both FDA and *Δϑ*50 to within 5%, but other strategies can be followed

Using the values of *n* determined as described above, the form factor has been related to the product of functions of mean piston speed, spark timing and charge dilution mass fraction, as

> 0.45 0.35 1 1 3.46 1 1.28

> > J

In this expression the mean piston speed *SP* is in m/s, the dilution mass fraction is dimension‐ less, and the spark timing *ϑST* is in CA degrees BTDC. The choice of functional expressions of relevant engine parameters has no physical basis beyond best-fitting trends to the selected experimental data. The influence of charge density on combustion evolution was not evident in the results for the Wiebe function form factor. The values of form factor generated using equation (24) were within ±8% of those deduced from individual MFB curves for the whole range of experimental conditions used for model validation, including the data recorded with variable valve timing setting. Similar functions of the same engine variables have been developed by Galindo et al. [56], using combustion records from two high-speed 2-stroke engines, strengthening to some extent the validity of the form factor modelling approach.

Expressions for the 0-90% MFB angle and the form factor of the Wiebe function have been derived empirically to represent the burn rate characteristics of a modern-design SI engine

*P ST n x*


*S*

*5.1.3. Discussion of results and accuracy*

a

quarters of the data fell within the ±5% error bands.

regression model with a weighting power function (weight factor *α*),

database, including the VVT data, was used for model validation only. Values of *Δϑ*<sup>90</sup> generated using equation (22) showed a maximum prediction error of 11%, while threequarters of the data fell within the ±5% error bands.

Values of form factor *n* used to build the second empirical correlation advanced in [46], have been determined by minimising the error of the least square fit to the MFB profile. A covariance regression model with a weighting power function (weight factor *α*),

$$\frac{1}{N} \sum \left( \frac{MFB\_{\text{Wieb\\_}} - MFB\_{\text{exp}}}{\left(MFB\_{\text{exp}}\right)^{\alpha}} \right)^{2} \tag{23}$$

was minimised to achieve a balanced fit across the burn profile from the start of combustion and up to the 50% MFB location. As evident from the error analysis illustrated in figure 16, as the weight factor *α* increases the absolute fitting error in FDA falls, whereas the one in *Δϑ*<sup>50</sup> tends to increase at least for durations in excess of 40 CA degrees. A weight factor of 0.3 allows limiting the error in both FDA and *Δϑ*50 to within 5%, but other strategies can be followed depending on the ultimate target of the combustion modelling exercise.

Using the values of *n* determined as described above, the form factor has been related to the product of functions of mean piston speed, spark timing and charge dilution mass fraction, as follows:

$$n = 3.46 \left(\frac{1}{\sqrt{S\_p}}\right)^{0.45} \left(\frac{1}{1 + \sqrt{\mathcal{B}\_{ST}}}\right)^{-0.35} \left(1 - 1.28 \mid \mathbf{x}\_b\right) \tag{24}$$

In this expression the mean piston speed *SP* is in m/s, the dilution mass fraction is dimension‐ less, and the spark timing *ϑST* is in CA degrees BTDC. The choice of functional expressions of relevant engine parameters has no physical basis beyond best-fitting trends to the selected experimental data. The influence of charge density on combustion evolution was not evident in the results for the Wiebe function form factor. The values of form factor generated using equation (24) were within ±8% of those deduced from individual MFB curves for the whole range of experimental conditions used for model validation, including the data recorded with variable valve timing setting. Similar functions of the same engine variables have been developed by Galindo et al. [56], using combustion records from two high-speed 2-stroke engines, strengthening to some extent the validity of the form factor modelling approach.

#### *5.1.3. Discussion of results and accuracy*

4

(22)

Metghalchi and Keck [59] to represent the detrimental influence of burned gas on the laminar burning velocity. The power correlation given by *X* (*xb*) was retrieved changing the dilution fraction over the range 6 to 26% via external-EGR (with fixed, default valve timing setting), to minimise the disturbing influence of valve timing on other factors such as cylinder filling. Figure 15 illustrates how the density function, *R*(*ρST* ), with power index set to 0.34, fits to data recorded over a range of engine loads between 2 and 7 bar net IMEP, at each of three engine speeds. During these load sweeps, the spark timing and level of dilution, which would normally change with load as a result of changing exhaust to intake pressure differential, were

> **1500 rpm 2100 rpm**

> > 3-4 2-3 1-2 0-1

3-4 2-3 1-2 0-1  J

**∆θ50 FITTING ERROR (CA)**

( )

**∆θd FITTING ERROR (CA)**

**2700 rpm Increasing Engine Speed**

S1 S2 S3 S4 S5 S6 S7 S8

J

1 2.06 *ST ST*

2

S1 S2 S3 S4 S5 S6 S7 S8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

**ALPHA (WEIGHT)**…………….

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

**ALPHA (WEIGHT)**…………….

234567

123456789

D = ç ÷ ç ÷ - ç ÷ - + ç ÷

1 2 3 4 5 6 7 8 9 10 11 12

31 33 35 37 39 41 43 45 47 49 51……. .. **∆θ50 - CA degree**

 19 20 23 25 27 29 31 33 ….. **∆θd - CA degree**

0.34 0.85

**Charge Density at ST (ρST) - kg/m3**

**Figure 15.** Total burning angle as a function of charge density at constant spark timing and constant dilution mass

The density function was introduced in [46] to account for a further observed influence of engine load on combustion duration, not fully captured by the empirical terms developed for the dilution fraction or the spark ignition setting. The same density expression has been adopted more recently by Galindo et al. [56] in a work devoted to modelling the charge burn curve in small, high-speed, two-stroke, gasoline engines using a Wiebe function-based

The final correlation developed for the total 0 to 90% MFB burning duration is written as:

1 1.164 1 <sup>178</sup> <sup>1</sup> 0.00033 0.0263 1

Its accuracy was tested across the whole 300-point-wide experimental database, featuring combinations of engine variables limited by the ranges discussed in section 4.1. Part of this

set constant.

**y ~ x - 0.34**

fraction for three engine speeds. Adapted from reference [46].

30

**Plot (a)**

**Plot (b)**

r

90 0.77

*ST <sup>P</sup> <sup>b</sup> S x*

è ø ç ÷ - è ø è ø

æ ö æ ö æ ö

**LIST OF ACRONYMS AND ABBREVIATIONS** 

**0 -** 

**Figure 15** 

approach.

J

**Figure 16**

**90% Burn Angle -**

**CA degress**

40

50

60

70

80

32 Advances in Internal Combustion Engines and Fuel Technologies

Expressions for the 0-90% MFB angle and the form factor of the Wiebe function have been derived empirically to represent the burn rate characteristics of a modern-design SI engine 30

**0 -** 

**90% Burn Angle -**

**CA degress**

40

50

60

70

80

**y ~ x - 0.34**

234567

**Charge Density at ST (ρST) - kg/m3**

**1500 rpm 2100 rpm**

**2700 rpm Increasing Engine Speed**

**Figure 16 Figure 16.** Plot (a): Wiebe function fitting error in FDA (Δ*ϑd*) for different weighting factors; Plot (b): Wiebe function fitting error in Δ*ϑ*50 for different weighting factors. Adapted from reference [46].

featuring VVT. Four factors – dilution mass fraction, engine speed, ignition timing and charge density at ignition location, were used to describe the evolution of premixed SI gasoline combustion. The range of engine operating conditions examined covered a large portion of the part-load envelope and high levels of dilution by burned gas. Valve timing influences combustion duration mostly through changes in the levels of dilution and in-cylinder filling; the results show that these influences are captured by the proposed models. Other effects which might results from valve timing changes, such as the changes in turbulence level, appear to be secondary and no explicit account of these has been required [46, 47, 60]. **LIST OF ACRONYMS AND ABBREVIATIONS** 

4

of fitting a Wiebe function to an experimental MFB profile [46]. Further error analysis shows that the expected errors in the various combustion duration indicators are proportional to those in *Δϑ*90. Overestimates of the form factor increase the predicted FDA and reduce the RBA. The influence of *n* upon *Δϑ*50 is instead relatively small; this suggests the approach presented here

Premixed Combustion in Spark Ignition Engines and the Influence of Operating Variables

2.5 3 3.5 4 4.5 5

2.5 3 3.5 4 4.5 5

**Form Factor**

**Figure 17.** Parametric variation of Δ*ϑ*90 and form factor from given baseline conditions: N = 1500 rev/min, Pin = 45 kPa (corresponding to spark timing density of 2.3 kg/m3), Dilution Mass Fraction = 12.7%. Adapted from reference [46].

The sensitivities of the 0-90% MFB burn angle (*Δϑ*90) and form factor to changes in the chosen relevant engine parameters, using a typical part-load operating condition as a baseline, are shown in figure 17. The influence of increasing charge density is least significant, producing the smallest reductions in burn duration. This does not fully account for the effect of changes in engine load, which would usually produce changes in dilution level as well as charge density. Increasing engine speed or the level of dilution produces an increase in the burn angle and a reduction in the form factor. Spark timing advances show opposite effects. The average changes in *Δϑ*<sup>90</sup> are of the order of one CA degree per half per cent increase in dilution or 80 rev/min increase in engine speed. The sensitivity of form factor to engine speed and level of

**Form Factor**

**0-90 Burn Angle -**

**Total Burn Angle**

**CA deg**

1000 1500 2000 2500 3000 3500

2 3 4 5 6

**Charge Density at ST - kg/m3**

**Engine Speed - rev/min**

2.5 3 3.5 4 4.5 5

http://dx.doi.org/10.5772/55495

**Form Factor**

35

may be applicable to combustion phasing control.

**Model Prediction Ref. Condition**

0.05 0.1 0.15 0.2 0.25

**Dilution Mass Fraction**

0 6 12 18 24 30

dilution is similar to that reported in [50].

**Spark Timing - CA BTDC**

**0-90 Burn Angle -**

**CA deg**

**0-90 Burn Angle -**

**CA deg**

Average errors in FDA, RBA and *Δϑ*50, calculated using the Wiebe function with modelled inputs, were in the region of 4.5%; maximum errors across the whole database were within the 13% error bands. This magnitude of uncertainty is typical of simplified thermodynamics combustion models applied to engines with flexible controls. Importantly, the analysis has shown that errors of magnitude up to 7% would be expected due to the inherent limitations of fitting a Wiebe function to an experimental MFB profile [46]. Further error analysis shows that the expected errors in the various combustion duration indicators are proportional to those in *Δϑ*90. Overestimates of the form factor increase the predicted FDA and reduce the RBA. The influence of *n* upon *Δϑ*50 is instead relatively small; this suggests the approach presented here may be applicable to combustion phasing control.

**Figure 17.** Parametric variation of Δ*ϑ*90 and form factor from given baseline conditions: N = 1500 rev/min, Pin = 45 kPa (corresponding to spark timing density of 2.3 kg/m3), Dilution Mass Fraction = 12.7%. Adapted from reference [46].

featuring VVT. Four factors – dilution mass fraction, engine speed, ignition timing and charge density at ignition location, were used to describe the evolution of premixed SI gasoline combustion. The range of engine operating conditions examined covered a large portion of the part-load envelope and high levels of dilution by burned gas. Valve timing influences combustion duration mostly through changes in the levels of dilution and in-cylinder filling; the results show that these influences are captured by the proposed models. Other effects which might results from valve timing changes, such as the changes in turbulence level, appear

**Figure 16.** Plot (a): Wiebe function fitting error in FDA (Δ*ϑd*) for different weighting factors; Plot (b): Wiebe function

1 2 3 4 5 6 7 8 9 10 11 12

31 33 35 37 39 41 43 45 47 49 51……. .. **∆θ50 - CA degree**

123456789

 19 20 23 25 27 29 31 33 ….. **∆θd - CA degree**

4

Average errors in FDA, RBA and *Δϑ*50, calculated using the Wiebe function with modelled inputs, were in the region of 4.5%; maximum errors across the whole database were within the 13% error bands. This magnitude of uncertainty is typical of simplified thermodynamics combustion models applied to engines with flexible controls. Importantly, the analysis has shown that errors of magnitude up to 7% would be expected due to the inherent limitations

to be secondary and no explicit account of these has been required [46, 47, 60].

**y ~ x - 0.34**

234567

**Charge Density at ST (ρST) - kg/m3**

**1500 rpm 2100 rpm**

> 3-4 2-3 1-2 0-1

3-4 2-3 1-2 0-1

**∆θ50 FITTING ERROR (CA)**

**∆θd FITTING ERROR (CA)**

**2700 rpm Increasing Engine Speed**

S1 S2 S3 S4 S5 S6 S7 S8

> S1 S2 S3 S4 S5 S6 S7 S8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

**ALPHA (WEIGHT)**…………….

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

**ALPHA (WEIGHT)**…………….

30

34 Advances in Internal Combustion Engines and Fuel Technologies

**Plot (a)**

**Plot (b)**

**LIST OF ACRONYMS AND ABBREVIATIONS** 

fitting error in Δ*ϑ*50 for different weighting factors. Adapted from reference [46].

**0 -** 

**Figure 15** 

**Figure 16**

**90% Burn Angle -**

**CA degress**

40

50

60

70

80

The sensitivities of the 0-90% MFB burn angle (*Δϑ*90) and form factor to changes in the chosen relevant engine parameters, using a typical part-load operating condition as a baseline, are shown in figure 17. The influence of increasing charge density is least significant, producing the smallest reductions in burn duration. This does not fully account for the effect of changes in engine load, which would usually produce changes in dilution level as well as charge density. Increasing engine speed or the level of dilution produces an increase in the burn angle and a reduction in the form factor. Spark timing advances show opposite effects. The average changes in *Δϑ*<sup>90</sup> are of the order of one CA degree per half per cent increase in dilution or 80 rev/min increase in engine speed. The sensitivity of form factor to engine speed and level of dilution is similar to that reported in [50].

The wider applicability of the proposed equations was also tested by comparing predictions with experimental combustion duration data reported in [50] for a turbocharged SI engine of similar design (see figure 18). Importantly, the comparison was made for intake manifold pressures above 1 bar, i.e. above the load range used for model derivation. The results show that the offset between experimental and modelled combustion angles is virtually eliminated when the dilution mass fraction of the reference (baseline) conditions is assumed to be 7%, suggesting a value of only 3% as given in [50] may actually be an underestimate.

advanced by Prucka et al. [5] is similarly used for predicting best ignition timing in complex SI enginearchitectures (featuringnotonlyVVTtechnologybutalsoachargemotioncontrolvalve), but it is different in its logic as it relies on a simplified adaptation of the turbulent flame entrainment concept originally introduced by Blizard and Keck [61]. The present section introduces the reader to the first type of combustion models, which is less reliant on complex descriptionsoffuelburningrateandturbulenceregimes,andhencemorereadilyimplementable.

Premixed Combustion in Spark Ignition Engines and the Influence of Operating Variables

http://dx.doi.org/10.5772/55495

37

Simplified flame propagation models tend to use readily available measurements from sensors and output estimates of ignition timing to achieve best efficiency for given engine running

**•** spark timing and relative thermodynamic conditions (temperature, pressure and charge

**•** dilution mass fraction or charge characterization (masses of fresh air and burned gas trapped

Charge dilution should be measured or estimated through gas exchange modelling, exploiting values for intake and exhaust manifold pressure as well as available cylinder volume at the IVO and at the EVC event. The exhaust manifold temperature appears in the calculations, weakening (due to the inherent slow response time of temperature sensors) the applicability of the approach to transient operation. In Hall et al. [47] the degree of positive valve overlap is considered directly responsible for the charge characterization, while modulation of valve lift and, more in general, changes of valve opening profiles, which may directly affect the level

The in-cylinder thermodynamic conditions at the time of ignition (ST) can be calculated

( / )

( ) <sup>1</sup> / *<sup>n</sup>*

In-cylinder pressure at IVC can be assumed to be approximately equal to the intake manifold pressure; charge temperature at IVC can be expressed as the weighted average of fresh air temperature (intake temperature, *Tin*) and hot, recirculated gas temperature (exhaust temper‐

*n*

*ST IVC IVC ST P PV V* = (25)

*ST IVC IVC ST TTV V* - <sup>=</sup> (26)

*5.2.1. Assumptions and methodology*

**•** engine speed

density)

at IVC)

ature, *Tex*):

conditions. The basic inputs can be listed as follows:

**•** mass of fuel injected and ethanol/gasoline ratio

**•** equivalent ratio (or air-to-fuel ratio)

of turbulence, are not investigated.

assuming polytropic compression from IVC conditions:

**Figure 18.** 0 to 99.9% MFB burn angle model predictions compared to published data from SAE 2005-01-2106. Adapted from reference [46].

#### **5.2. Flame propagation modelling**

As discussed previously, locating the 50% MFB event at 7 or 8 CA past TDC has been correlat‐ ed to achieving best fuel efficiency in SI engine combustion. The recent work by Hall et al. [47] outlinesarelativelysimple,physics-based,quasi-dimensionalflamepropagationmodelcapable of capturing the influence of variable valve timing, ethanol/gasoline blend ratio and other parameters relevant to modern SI engines, to provide estimates of the 50% MFB location and hence phase the combustion period most appropriately with respect to TDC. The model advanced by Prucka et al. [5] is similarly used for predicting best ignition timing in complex SI enginearchitectures (featuringnotonlyVVTtechnologybutalsoachargemotioncontrolvalve), but it is different in its logic as it relies on a simplified adaptation of the turbulent flame entrainment concept originally introduced by Blizard and Keck [61]. The present section introduces the reader to the first type of combustion models, which is less reliant on complex descriptionsoffuelburningrateandturbulenceregimes,andhencemorereadilyimplementable.

#### *5.2.1. Assumptions and methodology*

Simplified flame propagation models tend to use readily available measurements from sensors and output estimates of ignition timing to achieve best efficiency for given engine running conditions. The basic inputs can be listed as follows:

**•** engine speed

The wider applicability of the proposed equations was also tested by comparing predictions with experimental combustion duration data reported in [50] for a turbocharged SI engine of similar design (see figure 18). Importantly, the comparison was made for intake manifold pressures above 1 bar, i.e. above the load range used for model derivation. The results show that the offset between experimental and modelled combustion angles is virtually eliminated when the dilution mass fraction of the reference (baseline) conditions is assumed to be 7%,

25

25

35

45

55

**0-99.9% Burn Angle - CA**

**Figure 18.** 0 to 99.9% MFB burn angle model predictions compared to published data from SAE 2005-01-2106.

As discussed previously, locating the 50% MFB event at 7 or 8 CA past TDC has been correlat‐ ed to achieving best fuel efficiency in SI engine combustion. The recent work by Hall et al. [47] outlinesarelativelysimple,physics-based,quasi-dimensionalflamepropagationmodelcapable of capturing the influence of variable valve timing, ethanol/gasoline blend ratio and other parameters relevant to modern SI engines, to provide estimates of the 50% MFB location and hence phase the combustion period most appropriately with respect to TDC. The model

65

75

1000 1500 2000 2500 3000 3500 **Engine Speed - rpm**

100 110 120 130 140 150 **Intake Pressure - kPa**

 **SAE 2005-01-2106 Wiebe model [46] (Dil=3%) Wiebe model [46] (Dil=7%)**

35

45

**0-99.9% Burn Angle - CA**

55

65

75

 **SAE 2005-01-2106 Wiebe model [46] (Dil=3%) Wiebe model [46] (Dil=7%)**

suggesting a value of only 3% as given in [50] may actually be an underestimate.

25

25

Adapted from reference [46].

35

45

**0-99.9% Burn Angle - CA**

55

65

75

0.02 0.04 0.06 0.08 0.1 0.12 0.14 **Dilution mass fraction**

36 Advances in Internal Combustion Engines and Fuel Technologies

0 5 10 15 20 25 **Spark Timing - CA BTDC**

 **SAE 2005-01-2106 Wiebe model [46] (Dil=3%) Wiebe model [46] (Dil=7%)**

**5.2. Flame propagation modelling**

**SAE 2005-01-2106 (Dil=3%) SAE 2005-01-2106 (Dil=7%) Wiebe model [46] (Dil=3%) Wiebe Model [46] (Dil=7%)**

35

45

**0-99.9% Burn Angle - CA**

55

65

75


Charge dilution should be measured or estimated through gas exchange modelling, exploiting values for intake and exhaust manifold pressure as well as available cylinder volume at the IVO and at the EVC event. The exhaust manifold temperature appears in the calculations, weakening (due to the inherent slow response time of temperature sensors) the applicability of the approach to transient operation. In Hall et al. [47] the degree of positive valve overlap is considered directly responsible for the charge characterization, while modulation of valve lift and, more in general, changes of valve opening profiles, which may directly affect the level of turbulence, are not investigated.

The in-cylinder thermodynamic conditions at the time of ignition (ST) can be calculated assuming polytropic compression from IVC conditions:

$$P\_{ST} = P\_{IV\mathbb{C}} \left( V\_{IV\mathbb{C}} / V\_{ST} \right)^n \tag{25}$$

$$T\_{ST} = T\_{\text{IVC}} \left( V\_{\text{IVC}} / V\_{\text{ST}} \right)^{n-1} \tag{26}$$

In-cylinder pressure at IVC can be assumed to be approximately equal to the intake manifold pressure; charge temperature at IVC can be expressed as the weighted average of fresh air temperature (intake temperature, *Tin*) and hot, recirculated gas temperature (exhaust temper‐ ature, *Tex*):

$$T\_{IVC} = \boldsymbol{\chi}\_b \ \boldsymbol{T}\_{\rm ex} + \left(\mathbf{1} - \boldsymbol{\chi}\_b\right) \ \boldsymbol{T}\_{\rm in} \tag{27}$$

In the latter expression, *mfc* (units of kg) should be the mass of fresh charge, which includes fresh air and fuel only, and not the total trapped mass, which include an incombustible part. If the rate from equation (30) is evaluated in the CA domain, and then integrated over the combustion period, it yields the MFB profile which can be used to identify the location at which 50% of the charge has burned, or other relevant combustion intervals. The conversion of the above rates from the time to the CA domain is carried out observing that time intervals (in seconds) and CA intervals are correlated as *dϑ* =6*N dτ*, where *N* is engine speed in rev/min.

The net heat addition rate (units of J/s) due to combustion of fuel can be expressed as:

( ) <sup>1</sup> / *net fb MFB LHV LHV fc dQ dm dx <sup>Q</sup> Q AF m d d*

( ) <sup>1</sup> <sup>1</sup> / *MFB LHV fc dP dx <sup>P</sup> dV Q AF m d V d Vd*


nomial expressions from the JANAF Thermo-chemical Tables may be used.

*d*

t

where *QLHV* (J/kg) is the lower heating value of the fuel, calculated as a weighted average in case of multi-component fuel blend. Finally, the application of the First Law of the Thermo‐

used to model the in-cylinder pressure evolution from spark ignition to combustion termina‐ tion. Within the hypothesis of ideal gas, the ratio of specific heats, *γ*, should be taken as a constant and given a reasonable average value (for example 1.35). A more realistic approach considers *γ* as a function of both composition and temperature, quantities which vary and need to be evaluated at each step of the combustion process. Temperature dependent poly‐

Commonly [62], the unburned gas density *ρu* needed in equation (28) is estimated by assuming that the unburned charge undergoes a polytropic compression process of given index *n* from

( )

The pressure input *P* in this equation would be the value calculated through the application of the First Law, as shown above. Again, a constant value of polytropic compression index can be used to simplify the calculations. The temperature in the unburned region is calculated

/ *<sup>n</sup> u ST ST*

r r 1

t


Premixed Combustion in Spark Ignition Engines and the Influence of Operating Variables

http://dx.doi.org/10.5772/55495

39

g

ë û (32)

= *P P* (33)

 t

 t

dynamics with the due assumption of ideal gas mixture yields:

t

g

t

*5.2.2. Evaluation of burning rate inputs*

ST conditions due to the expanding flame front:

The polytropic index of compression *n* does vary with engine conditions, but a constant value of 1.29 has been used to yield acceptable overall results [47]; some researchers [5] have suggested a linear fit to measured or estimated values of charge dilution mass fraction. The charge density at spark timing is calculated from pressure and temperature advancing the hypothesis of ideal gas mixture, or simply by the ratio of total trapped mass *mtot* and available chamber volume *VST* .

Following spark ignition, the combustion process commences and a thin quasi-spherical flame front develops, separating burned and unburned zones within the combustion chamber. These two regions are supposed to be constantly in a state of pressure equilibrium. The rate of burning, i.e. the rate of formation of burning products (units of kg/s), is calculated step-by-step through the application of the equation of mass continuity (1), repeated here for completeness:

$$\frac{dm\_b}{d\tau} = \rho\_u \text{ A S}\_b \tag{28}$$

In this expression *ρu* (units of kg/m3 ) is the density of the gas mixture within the unburned zone, *A* (m2 ) can be considered as a mean flame surface area and *Sb* (m/s) is the turbulent burning velocity. This velocity, in the context of simplified flame propagation models, is loosely assumed to coincide with the turbulent flame velocity. The rate of fuel consumption (rate of change of the mass of fuel burned, *mfb*) is linked to the above rate of burning through the definition of the air-to-fuel ratio (*A* / *F* ), which is assumed to be invariable in time (i.e. during the combustion process) and in space (i.e. homogeneous mixture throughout the unburned zone):

$$\frac{dm\_{\tilde{\mathcal{P}}}}{d\tau} = \left(A \,/\, F\right)^{-1} \, \frac{dm\_b}{d\tau} \tag{29}$$

Equation (28) can be rewritten to define the rate of change of mass fraction burned (here indicated as *xMFB*) in the time domain:

$$\frac{d\mathbf{x}\_{MFB}}{d\tau} = \frac{1}{m\_{fc}} \left. \frac{dm\_b}{d\tau} \right| = \frac{1}{m\_{fc}} \left. \rho\_u \right| A \text{ S}\_b \tag{30}$$

In the latter expression, *mfc* (units of kg) should be the mass of fresh charge, which includes fresh air and fuel only, and not the total trapped mass, which include an incombustible part. If the rate from equation (30) is evaluated in the CA domain, and then integrated over the combustion period, it yields the MFB profile which can be used to identify the location at which 50% of the charge has burned, or other relevant combustion intervals. The conversion of the above rates from the time to the CA domain is carried out observing that time intervals (in seconds) and CA intervals are correlated as *dϑ* =6*N dτ*, where *N* is engine speed in rev/min.

The net heat addition rate (units of J/s) due to combustion of fuel can be expressed as:

$$\frac{d\mathbb{Q}\_{\text{net}}}{d\tau} = \mathbb{Q}\_{LW} \left[ \frac{dm\_{\text{fb}}}{d\tau} \right] = \mathbb{Q}\_{LW} \left[ \left( A / F \right)^{-1} m\_{\text{fc}} \frac{d\mathbf{x}\_{\text{MFB}}}{d\tau} \right] \tag{31}$$

where *QLHV* (J/kg) is the lower heating value of the fuel, calculated as a weighted average in case of multi-component fuel blend. Finally, the application of the First Law of the Thermo‐ dynamics with the due assumption of ideal gas mixture yields:

$$\frac{dP}{d\tau} = \frac{\gamma - 1}{V} \text{ Q}\_{LHV} \left[ \left( A / F \right)^{-1} m\_{fc} \frac{d\mathbf{x}\_{MFB}}{d\tau} \right] - \frac{\gamma}{V} \frac{P}{d\tau} \frac{dV}{d\tau} \tag{32}$$

used to model the in-cylinder pressure evolution from spark ignition to combustion termina‐ tion. Within the hypothesis of ideal gas, the ratio of specific heats, *γ*, should be taken as a constant and given a reasonable average value (for example 1.35). A more realistic approach considers *γ* as a function of both composition and temperature, quantities which vary and need to be evaluated at each step of the combustion process. Temperature dependent poly‐ nomial expressions from the JANAF Thermo-chemical Tables may be used.

#### *5.2.2. Evaluation of burning rate inputs*

(1 ) *IVC b ex b in T xT x T* = +- (27)

<sup>=</sup> (28)

) is the density of the gas mixture within the unburned


The polytropic index of compression *n* does vary with engine conditions, but a constant value of 1.29 has been used to yield acceptable overall results [47]; some researchers [5] have suggested a linear fit to measured or estimated values of charge dilution mass fraction. The charge density at spark timing is calculated from pressure and temperature advancing the hypothesis of ideal gas mixture, or simply by the ratio of total trapped mass *mtot* and available

Following spark ignition, the combustion process commences and a thin quasi-spherical flame front develops, separating burned and unburned zones within the combustion chamber. These two regions are supposed to be constantly in a state of pressure equilibrium. The rate of burning, i.e. the rate of formation of burning products (units of kg/s), is calculated step-by-step through the application of the equation of mass continuity (1), repeated here for completeness:

> *u b dm A S*

burning velocity. This velocity, in the context of simplified flame propagation models, is loosely assumed to coincide with the turbulent flame velocity. The rate of fuel consumption (rate of change of the mass of fuel burned, *mfb*) is linked to the above rate of burning through the definition of the air-to-fuel ratio (*A* / *F* ), which is assumed to be invariable in time (i.e. during the combustion process) and in space (i.e. homogeneous mixture throughout the

> ( ) <sup>1</sup> / *fb <sup>b</sup> dm dm A F d d*

Equation (28) can be rewritten to define the rate of change of mass fraction burned (here

t

*MFB* 1 1 *<sup>b</sup>*

*d md m*

t

*fc fc dx dm A S*

 t

) can be considered as a mean flame surface area and *Sb* (m/s) is the turbulent

 t

*u b*

= = (30)

r

r

*b*

*d*

t

chamber volume *VST* .

38 Advances in Internal Combustion Engines and Fuel Technologies

In this expression *ρu* (units of kg/m3

indicated as *xMFB*) in the time domain:

zone, *A* (m2

unburned zone):

Commonly [62], the unburned gas density *ρu* needed in equation (28) is estimated by assuming that the unburned charge undergoes a polytropic compression process of given index *n* from ST conditions due to the expanding flame front:

$$
\rho\_u = \rho\_{ST} \left( P \;/\; P\_{ST} \right)^{\frac{1}{n}} \tag{33}
$$

The pressure input *P* in this equation would be the value calculated through the application of the First Law, as shown above. Again, a constant value of polytropic compression index can be used to simplify the calculations. The temperature in the unburned region is calculated assuming polytropic compression in a similar fashion. The influence of heat transfer on *Tu* and hence on the laminar flame speed would be (for simplicity) captured by the empirical tuning factors used within the turbulent flame speed model [47].

to unburned gas density as well as mean inlet gas speed. In 1977 Tabaczynski and co-workers developed a detailed description of the turbulent eddy burn-up process to define a semi-

In [47] the turbulent flame velocity is given as the product of laminar velocity and a turbulenceenhancement factor. In line with the speed/turbulence association discussed above, this factor

In equation (36) *a* and *b* are tuning factors which should be calibrated on each specific engine

The laminar flame velocity is a quantity which accounts for the thermo-chemical state of the combustible mixture moving into the burning zone. The most classical correlations for laminar velocity have been developed, as mentioned in section 5.1.2, by Rhodes and Keck [58] and Metghalchi and Keck [59]. They used similar power-function expressions to correlate un‐ burned gas temperature, pressure and mixture diluent fraction with laminar velocities measured in spherical, centrally-ignited, high-pressure combustion vessels using multi-

> ,0 1 2.06 *<sup>u</sup> L L b ref ref <sup>T</sup> <sup>P</sup> S S <sup>x</sup> T P*

 b

In this correlation *Tu* and *P* are unburned gas temperature and pressure during the combustion process. The reference parameters, 298 K and 1 atm respectively, represent the conditions at which the *unstretched* laminar flame velocity *S*L, 0 is calculated. *α* and *β* are functions of mixture equivalence ratio only. *S*L, <sup>0</sup> depends on type of fuel and, again, on equivalence ratio. A useful review of the most common correlations proposed in the literature can be found in Syed et al. [65]. For gasoline combustion in stoichiometric conditions the following constant values can be calculated: *α* =2.129; *β* = −0.217; *S*L, <sup>0</sup> =0.28(m/s). Recent research work by Lindstrom et al. [50], Bayraktar [54] and by the Author [42], indicates that in the context of SI engine combus‐ tion, temperature and pressure exert weaker influences on the laminar flame speed (1.03 and -0.009 are the values of *α* and *β*, proposed for stoichiometric combustion in [50] to minimise the fitting errors of laminar speed functions to experimental combustion duration data). The presence of burned gas in the unburned mixture, as explained before, causes a substantial reduction in the flame velocity due to the reduction in heating value per unit of mass of the mixture. The fractional reduction in laminar speed, represented by the term (1 −2.06*xb*

originally proposed by Rhodes and Keck [58], has been found essentially independent of

pressure, temperature, fuel type and equivalent ratio ϕ.

( ) *b L <sup>L</sup> S fS aN b S* == + (36)

Premixed Combustion in Spark Ignition Engines and the Influence of Operating Variables

http://dx.doi.org/10.5772/55495

41

( ) 0.77

(37)

0.77)

fundamental model of the turbulent burning velocity [24].

is expressed as a linear function of engine speed only [57, 65]:

component hydrocarbons and alcohols similar to automotive fuels:

a

 æ öæ ö <sup>=</sup> ç ÷ç ÷ è øè ø

(their value is 0.0025 and 3.4, respectively, in [47]).

One of the major difficulties in modelling SI engine combustion is associated with the identi‐ fication or the theoretical definition of a burning front or flame surface [22]. In [47] *A* is a mean flame surface, assumed as such to be infinitely thin and modelled as a sphere during the development stage of combustion. The volume of the burned gas sphere is calculated as the difference between total available chamber volume and unburned gas volume:

$$A = 4 \quad \pi \left( r\_{flame} \right)^2 = 4 \text{ } \pi \left( \sqrt[3]{3 \text{ } V\_b / 4 \text{ } \pi} \right)^2 = 4 \text{ } \pi \left( \sqrt[3]{3 \text{ } \left( V\_{chamber} - V\_u \right) / 4 \text{ } \pi} \right)^2 \tag{34}$$

Some time into the combustion process, the flame front will start impinging cylinder head and piston crown, progressively assuming a quasi-cylindrical shape. After a transition period, the mean flame surface can be modeled as a cylinder whose height is given by the mean combus‐ tion chamber height (clearance height) *Δh* :

$$A = \begin{pmatrix} \pi & \pi \end{pmatrix} \begin{pmatrix} r\_{\text{frame}} \\ \end{pmatrix} \ \Delta \mathbf{h} = \mathbf{2} \ \pi \ \left( \sqrt{V\_{\text{b}} / \pi \ \Delta \mathbf{h}} \right) \ \Delta \mathbf{h} = \mathbf{2} \ \pi \ \left( \sqrt{\begin{pmatrix} V\_{\text{chamber}} - V\_{\text{u}} \end{pmatrix} / \pi \ \Delta \mathbf{h}} \right) \ \Delta \mathbf{h} \tag{35}$$

Similar approaches are found in more established literature, dating back to the late 1970s. Referring to a disk-shaped combustion chamber, Hires et al. [23] advance a simple, yet effective, hypothesis of proportionality between *equivalent planar burning surface* and clearance height: *A*∝ *BΔh* , with *B* the cylinder bore. Of course, the identification of a mean flame surface becomes more challenging in the context of modern combustion chamber geometries such as pent-roof shape chambers; nevertheless, the basic concepts to be adopted in the context of simplified quasi-dimensional combustion modelling remain valid. The topic of SI combustion propagating front, where a fundamental distinction is made between cold sheet-like front, burning surface and thick turbulent brush-like front, is complex and outside the scope of the present work; the interested reader is referred to some established, specialised literature [8, 11, 12, 13, 63].

The turbulent burning or flame speed, *Sb*, embodies the dependence of the burning rate on turbulence as well as on the thermo-chemical state of the cylinder charge. In time, these dependences have been given various mathematical forms, though all indicate that in-cylinder turbulence acts as an enhancing factor on the leading, laminar-like flame regime. The first turbulent flame propagation model was advanced in 1940 by Damköhler for the so-called wrinkled laminar flame regime [64]. The turbulence burning velocity was expressed as: *Sb* =*u*' + *SL* , where *u* ' is the turbulence intensity and *SL* is the laminar burning velocity. An improvement on this model was proposed by Keck and co-workers [9, 11], showing that during the rapid, quasi-steady combustion phase the turbulent burning velocity assumes the form: *Sb* ≈*a uT* + *SL* , where *uT* is a characteristic velocity due to turbulent convection, proportional to unburned gas density as well as mean inlet gas speed. In 1977 Tabaczynski and co-workers developed a detailed description of the turbulent eddy burn-up process to define a semifundamental model of the turbulent burning velocity [24].

assuming polytropic compression in a similar fashion. The influence of heat transfer on *Tu* and hence on the laminar flame speed would be (for simplicity) captured by the empirical tuning

One of the major difficulties in modelling SI engine combustion is associated with the identi‐ fication or the theoretical definition of a burning front or flame surface [22]. In [47] *A* is a mean flame surface, assumed as such to be infinitely thin and modelled as a sphere during the development stage of combustion. The volume of the burned gas sphere is calculated as the

( ) ( ) ( ( ) ) <sup>2</sup> <sup>2</sup> <sup>2</sup>

p

p

(34)

) (35)

pp= = -

Some time into the combustion process, the flame front will start impinging cylinder head and piston crown, progressively assuming a quasi-cylindrical shape. After a transition period, the mean flame surface can be modeled as a cylinder whose height is given by the mean combus‐

difference between total available chamber volume and unburned gas volume:

4 4 3 /4 4 3 <sup>3</sup> <sup>3</sup> / 4 *Ar V flame <sup>b</sup> V V chamber u* =

*A r h V hh V V hh* = 2 2/ 2

( *flame* ) D = ( *<sup>b</sup>* D D= ) ( ( *chamber u* - DD ) /

Similar approaches are found in more established literature, dating back to the late 1970s. Referring to a disk-shaped combustion chamber, Hires et al. [23] advance a simple, yet effective, hypothesis of proportionality between *equivalent planar burning surface* and clearance height: *A*∝ *BΔh* , with *B* the cylinder bore. Of course, the identification of a mean flame surface becomes more challenging in the context of modern combustion chamber geometries such as pent-roof shape chambers; nevertheless, the basic concepts to be adopted in the context of simplified quasi-dimensional combustion modelling remain valid. The topic of SI combustion propagating front, where a fundamental distinction is made between cold sheet-like front, burning surface and thick turbulent brush-like front, is complex and outside the scope of the present work; the interested reader is referred to some established, specialised literature [8,

The turbulent burning or flame speed, *Sb*, embodies the dependence of the burning rate on turbulence as well as on the thermo-chemical state of the cylinder charge. In time, these dependences have been given various mathematical forms, though all indicate that in-cylinder turbulence acts as an enhancing factor on the leading, laminar-like flame regime. The first turbulent flame propagation model was advanced in 1940 by Damköhler for the so-called wrinkled laminar flame regime [64]. The turbulence burning velocity was expressed as: *Sb* =*u*' + *SL* , where *u* ' is the turbulence intensity and *SL* is the laminar burning velocity. An improvement on this model was proposed by Keck and co-workers [9, 11], showing that during the rapid, quasi-steady combustion phase the turbulent burning velocity assumes the form: *Sb* ≈*a uT* + *SL* , where *uT* is a characteristic velocity due to turbulent convection, proportional

 p

factors used within the turbulent flame speed model [47].

40 Advances in Internal Combustion Engines and Fuel Technologies

 p

pp

tion chamber height (clearance height) *Δh* :

p

p

11, 12, 13, 63].

In [47] the turbulent flame velocity is given as the product of laminar velocity and a turbulenceenhancement factor. In line with the speed/turbulence association discussed above, this factor is expressed as a linear function of engine speed only [57, 65]:

$$\mathcal{S}\_b = f \; \mathcal{S}\_L = \begin{pmatrix} a \ N + b \end{pmatrix} \; \mathcal{S}\_L \tag{36}$$

In equation (36) *a* and *b* are tuning factors which should be calibrated on each specific engine (their value is 0.0025 and 3.4, respectively, in [47]).

The laminar flame velocity is a quantity which accounts for the thermo-chemical state of the combustible mixture moving into the burning zone. The most classical correlations for laminar velocity have been developed, as mentioned in section 5.1.2, by Rhodes and Keck [58] and Metghalchi and Keck [59]. They used similar power-function expressions to correlate un‐ burned gas temperature, pressure and mixture diluent fraction with laminar velocities measured in spherical, centrally-ignited, high-pressure combustion vessels using multicomponent hydrocarbons and alcohols similar to automotive fuels:

$$\mathcal{S}\_{L} = \mathcal{S}\_{L,0} \left(\frac{T\_u}{T\_{ref}}\right)^{\alpha} \left(\frac{P}{P\_{ref}}\right)^{\beta} \left(1 - 2.06 \cdot x\_b^{0.77}\right) \tag{37}$$

In this correlation *Tu* and *P* are unburned gas temperature and pressure during the combustion process. The reference parameters, 298 K and 1 atm respectively, represent the conditions at which the *unstretched* laminar flame velocity *S*L, 0 is calculated. *α* and *β* are functions of mixture equivalence ratio only. *S*L, <sup>0</sup> depends on type of fuel and, again, on equivalence ratio. A useful review of the most common correlations proposed in the literature can be found in Syed et al. [65]. For gasoline combustion in stoichiometric conditions the following constant values can be calculated: *α* =2.129; *β* = −0.217; *S*L, <sup>0</sup> =0.28(m/s). Recent research work by Lindstrom et al. [50], Bayraktar [54] and by the Author [42], indicates that in the context of SI engine combus‐ tion, temperature and pressure exert weaker influences on the laminar flame speed (1.03 and -0.009 are the values of *α* and *β*, proposed for stoichiometric combustion in [50] to minimise the fitting errors of laminar speed functions to experimental combustion duration data). The presence of burned gas in the unburned mixture, as explained before, causes a substantial reduction in the flame velocity due to the reduction in heating value per unit of mass of the mixture. The fractional reduction in laminar speed, represented by the term (1 −2.06*xb* 0.77) originally proposed by Rhodes and Keck [58], has been found essentially independent of pressure, temperature, fuel type and equivalent ratio ϕ.

In [47], the exponential factors *α* and *β* are given the values 0.9 and -0.05, respectively; the unstretched laminar flame speed *S*L, <sup>0</sup> (units of m/s) is calculated through a modification of the functional expression proposed by Syed et al. [65], in which an increase of the ethanol volume fraction *x*˜ *ETH* between 0% (pure gasoline) and 100% (pure ethanol) produces a 10% increase in laminar flame speed:

$$S\_{L,0} = 0.4658 \ \left(1 + 0.1 \ \tilde{x}\_{ETH}\right) \ \ \phi^{0.3} \ \text{EXP}\left(-4.48 \ \left(\phi - 1.075\right)^{2}\right) \tag{38}$$

engine operating conditions. The *Δϑ*50 interval can be evaluated indirectly, as the interval

Premixed Combustion in Spark Ignition Engines and the Influence of Operating Variables

http://dx.doi.org/10.5772/55495

43

As expected, an increase in the diluent fraction determines a slower combustion process (i.e. wider *Δϑ*50) at all engine operating conditions, except at high engine load where the variation of the Burned Gas Fraction as a function of valve overlap is very limited. Conversely, the addition of ethanol to gasoline induces faster laminar flame speed and a stronger burning rate [65], reducing *Δϑ*<sup>50</sup> as a consequence. Increasing charge dilution necessitates more advanced theoretical ignition, whereas the ethanol content shows an opposite influence. An increase in dilution of 10 point percent requires about 25 CA degrees ignition advancement at low engine load. Increasing the ethanol content between E0 and E85 would reduce the required advance‐

**Figure 19.** Spark ignition timing (SIT) required to locate the 50% MFB event at 8 CA degrees ATDC, and achieve opti‐ mal fuel efficiency, as a function of charge diluent fraction (Burned Gas Fraction), gasoline/ethanol blend ratio and engine speed/load operating conditions. Re-printed with permission by SAGE Publications from reference [47].

The present chapter explores the evolution of the combustion process in modern-design, premixed, gasoline engines, which feature the increasingly common technology of variable valve timing. The assessment of combustion rates and duration has been performed by looking at the effects of a set of significant engine operating variables, which were studied in isolation.

between SIT and 8 CA degrees ATDC.

ment between 3 and 6 CA degrees.

**6. Conclusions**

#### *5.2.3. Discussion of results*

The flame propagation type models presented above are physics-based and hence, in principle, can be generalised to engines of different geometries. Through the definition of the laminar flame velocity, they enable accounting for two very relevant influences on combustion, i.e. the charge diluent fraction and the composition and strength of the fuel mixture. In spite of their simplicity, partly due to the range of assumptions taken, this type of control-oriented models has been demonstrated to capture the rate of combustion of modern SI engines with acceptable level of confidence. Hall et al. [47] validate the model using experimental records from a turbocharged, PFI, flexible-fuel, SI engine, the specification of which are given in table 3.


**Table 3.** Specifications of the test engine used in reference [47].

More than 500 test points were used, consisting of combinations of engine speed between 750 and 5500 rev/min, intake manifold pressure between 0.4 and 2.2 bar, ST between -12 and 60 CA degrees BTDC and valve overlap between -16 and 24 CA degrees (intake valve timing only). Four basic ethanol/gasoline blend ratios were tested, between E0 (pure gasoline) and E85 (0.85 ethanol and 0.15 gasoline, as volume fractions). In the vast majority of instances, the model predicted the duration of the interval between ST and 50% MFB (*Δϑ*50) with a maximum error of 10%. The influence of relevant engine parameters on combustion duration is sum‐ marized in figure 19. This shows the theoretical Spark Ignition Timing (SIT) which ensures optimal 50% MFB location (i.e. at 8 CA degrees ATDC), when variable amounts of charge dilution (here Burned Gas Fraction), and variable amounts of ethanol are used, at different engine operating conditions. The *Δϑ*50 interval can be evaluated indirectly, as the interval between SIT and 8 CA degrees ATDC.

As expected, an increase in the diluent fraction determines a slower combustion process (i.e. wider *Δϑ*50) at all engine operating conditions, except at high engine load where the variation of the Burned Gas Fraction as a function of valve overlap is very limited. Conversely, the addition of ethanol to gasoline induces faster laminar flame speed and a stronger burning rate [65], reducing *Δϑ*<sup>50</sup> as a consequence. Increasing charge dilution necessitates more advanced theoretical ignition, whereas the ethanol content shows an opposite influence. An increase in dilution of 10 point percent requires about 25 CA degrees ignition advancement at low engine load. Increasing the ethanol content between E0 and E85 would reduce the required advance‐ ment between 3 and 6 CA degrees.

**Figure 19.** Spark ignition timing (SIT) required to locate the 50% MFB event at 8 CA degrees ATDC, and achieve opti‐ mal fuel efficiency, as a function of charge diluent fraction (Burned Gas Fraction), gasoline/ethanol blend ratio and engine speed/load operating conditions. Re-printed with permission by SAGE Publications from reference [47].
