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Kinetic Modeling Of Combustion Characteristics Of Real Biodiesel Fuels
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Book Synopsis Kinetic Modeling of Combustion Characteristics of Real Biodiesel Fuels by :
Download or read book Kinetic Modeling of Combustion Characteristics of Real Biodiesel Fuels written by and published by . This book was released on 2009 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt: Biodiesel fuels are of much interest today either for replacing or blending with conventional fuels for automotive applications. Predicting engine effects of using biodiesel fuel requires accurate understanding of the combustion characteristics of the fuel, which can be acquired through analysis using reliable detailed reaction mechanisms. Unlike gasoline or diesel that consists of hundreds of chemical compounds, biodiesel fuels contain only a limited number of compounds. Over 90% of the biodiesel fraction is composed of 5 unique long-chain C1 and C16 saturated and unsaturated methyl esters. This makes modeling of real biodiesel fuel possible without the need for a fuel surrogate. To this end, a detailed chemical kinetic mechanism has been developed for determining the combustion characteristics of a pure biodiesel (B100) fuel, applicable from low- to high-temperature oxidation regimes. This model has been built based on reaction rate rules established in previous studies at Lawrence Livermore National Laboratory. Computed results are compared with the few fundamental experimental data that exist for biodiesel fuel and its components. In addition, computed results have been compared with experimental data for other long-chain hydrocarbons that are similar in structure to the biodiesel components.
Book Synopsis Experimental and Modeling Studies of the Characteristics of Liquid Biofuels for Enhanced Combustion by :
Download or read book Experimental and Modeling Studies of the Characteristics of Liquid Biofuels for Enhanced Combustion written by and published by . This book was released on 2009 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The objectives of this project have been to develop a comprehensive set of fundamental data regarding the combustion behavior of biodiesel fuels and appropriately associated model fuels that may represent biodiesels in automotive engineering simulation. Based on the fundamental study results, an auxiliary objective was to identify differentiating characteristics of molecular fuel components that can be used to explain different fuel behavior and that may ultimately be used in the planning and design of optimal fuel-production processes. The fuels studied in this project were BQ-9000 certified biodiesel fuels that are certified for use in automotive engine applications. Prior to this project, there were no systematic experimental flame data available for such fuels. One of the key goals has been to generate such data, and to use this data in developing and verifying effective kinetic models. The models have then been reduced through automated means to enable multi-dimensional simulation of the combustion characteristics of such fuels in reciprocating engines. Such reliable kinetics models, validated against fundamental data derived from laminar flames using idealized flow models, are key to the development and design of optimal engines, engine operation and fuels. The models provide direct information about the relative contribution of different molecular constituents to the fuel performance and can be used to assess both combustion and emissions characteristics. During this project, we completed a major and thorough validation of a set of biodiesel surrogate components, allowing us to begin to evaluate the fundamental combustion characteristics for B100 fuels.
Book Synopsis Chemical Kinetic Modeling of Biofuel Combustion by : Subram Maniam Sarathy
Download or read book Chemical Kinetic Modeling of Biofuel Combustion written by Subram Maniam Sarathy and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Bioalcohols, such as bioethanol and biobutanol, are suitable replacements for gasoline, while biodiesel can replace petroleum diesel. Improving biofuel engine performance requires understanding its fundamental combustion properties and the pathways of combustion. This study's contribution is experimentally validated chemical kinetic combustion mechanisms for biobutanol and biodiesel. Fundamental combustion data and chemical kinetic mechanisms are presented and discussed to improve our understanding of biofuel combustion. The net environmental impact of biobutanol (i.e., n-butanol) has not been studied extensively, so this study first assesses the sustainability of n-butanol derived from corn. The results indicate that technical advances in fuel production are required before commercializing biobutanol. The primary contribution of this research is new experimental data and a novel chemical kinetic mechanism for n-butanol combustion. The results indicate that under the given experimental conditions, n-butanol is consumed primarily via abstraction of hydrogen atoms to produce fuel radical molecules, which subsequently decompose to smaller hydrocarbon and oxygenated species. The hydroxyl moiety in n-butanol results in the direct production of the oxygenated species such as butanal, acetaldehyde, and formaldehyde. The formation of these compounds sequesters carbon from forming soot precursors, but they may introduce other adverse environmental and health effects. Biodiesel is a mixture of long chain fatty acid methyl esters derived from fats and oils. This research study presents high quality experimental data for one large fatty acid methyl ester, methyl decanoate, and models its combustion using an improved skeletal mechanism. The results indicate that methyl decanoate is consumed via abstraction of hydrogen atoms to produce fuel radicals, which ultimately lead to the production of alkenes. The ester moiety in methyl decanoate leads to the formation of low molecular weight oxygenated compounds such as carbon monoxide, formaldehyde, and ketene, thereby reducing the production of soot precursors. The study concludes that the oxygenated molecules in biofuels follow similar combustion pathways to the hydrocarbons in petroleum fuels. The oxygenated moiety's ability to sequester carbon from forming soot precursors is highlighted. However, the direct formation of oxygenated hydrocarbons warrants further investigation into the environmental and health impacts of practical biofuel combustion systems.
Book Synopsis Combustion Characterization and Kinetic Modeling in Reactive Flow Simulations by : Shuliang Zhang
Download or read book Combustion Characterization and Kinetic Modeling in Reactive Flow Simulations written by Shuliang Zhang and published by . This book was released on 2014 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: The primary objective of this research is to characterize fuel combustion in reactive flow simulations using advanced kinetic modeling and mechanism reduction tools. Since incorporating detailed chemical kinetic model in the realistic reactive flow simulations is a computationally challenging task due to the large size of detailed kinetic mechanism, it is of great interest to develop approaches for simplifying the kinetic models and reducing computational costs in reactive flow simulations. In this dissertation, we first extend the previously developed on-the-fly reduction approach to the characterization of complex biodiesel combustion using detailed biodiesel surrogate mechanism. Major combustion characteristics such as ignition, emission, as well as engine performance for biodiesel compared with conventional fossil fuels are studied. Although the incorporation of detailed biodiesel combustion mechanism in complex reactive flow simulation is enabled, the simulation is still highly time-consuming. To further alleviate the computational intensity, a hybrid reduction scheme coupling the on-the-fly reduction with global quasi-steady-state approximation (QSSA) is developed. The proposed hybrid reduction scheme is demonstrated in various reactive flow simulations including zero-dimensional PFR model, multidimensional HCCI engine CFD model, and realistic gas phase injector CFD simulations. A flux-based quasi-steady-state (QSS) species selection procedure is introduced to facilitate the demonstration of hybrid scheme. Finally, a novel computational framework integrating automated mechanism generation and on-the-fly reduction is proposed and implemented using a stepwise integration. The proposed framework is then demonstrated in methane oxidation case studies and shows a new way of conducting reactive flow simulation without having an actual mechanism before the simulation starts. The integration of automated mechanism generation and on-the-fly reduction is a promising technique to perform reactive flow simulations and has the potential to reduce the computational cost of the simulations. The work in this dissertation provides powerful tools and important insight for the incorporation of detailed chemical kinetics in the reactive flow simulations.
Book Synopsis Investigations into the Combustion Kinetics of Several Novel Oxygenated Fuels by : Wenyu Sun
Download or read book Investigations into the Combustion Kinetics of Several Novel Oxygenated Fuels written by Wenyu Sun and published by Springer Nature. This book was released on 2023-09-22 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, attention was paid to several novel oxygenated fuels—carbonates, polyethers and ketones. Combustion kinetic investigations were performed for typical representative compounds, including dimethyl carbonate, diethyl carbonate, cyclopentanone, 3-pentanone, 1,2-dimethoxyethane and dimethoxymethane. For experiments, suitable diagnostic techniques were used to measure the detailed speciation information of the target fuels under different conditions. For kinetic modeling, rate coefficients for crucial elementary reactions were obtained through high-level theoretical calculations. Based on that, validated kinetic models with good predictive performances were developed. On the basis of experimental measurements and model interpretations, this work highlighted two important combustion characteristics regarding the practical use: the pollutant formation and the ignition performance. Besides, the correlation between oxygen-containing functional groups and the aforementioned combustion characteristics was revealed. To reveal the potential interactions between the reaction networks of oxygenated additives and the hydrocarbon base fuels during combustion. Chemical structures of laminar premixed flames fueled by binary fuels were measured, and by changing the initial fuel compositions, the addition effects of the oxygenates on the fuel consumption and pollutant formation behaviors were explored. It was found that complicated chemical interactions do not exist in the reaction networks under the investigated conditions.
Book Synopsis Chemical Kinetic Modeling of Advanced Transportation Fuels by :
Download or read book Chemical Kinetic Modeling of Advanced Transportation Fuels written by and published by . This book was released on 2009 with total page 12 pages. Available in PDF, EPUB and Kindle. Book excerpt: Development of detailed chemical kinetic models for advanced petroleum-based and nonpetroleum based fuels is a difficult challenge because of the hundreds to thousands of different components in these fuels and because some of these fuels contain components that have not been considered in the past. It is important to develop detailed chemical kinetic models for these fuels since the models can be put into engine simulation codes used for optimizing engine design for maximum efficiency and minimal pollutant emissions. For example, these chemistry-enabled engine codes can be used to optimize combustion chamber shape and fuel injection timing. They also allow insight into how the composition of advanced petroleum-based and non-petroleum based fuels affect engine performance characteristics. Additionally, chemical kinetic models can be used separately to interpret important in-cylinder experimental data and gain insight into advanced engine combustion processes such as HCCI and lean burn engines. The objectives are: (1) Develop detailed chemical kinetic reaction models for components of advanced petroleum-based and non-petroleum based fuels. These fuels models include components from vegetable-oil-derived biodiesel, oil-sand derived fuel, alcohol fuels and other advanced bio-based and alternative fuels. (2) Develop detailed chemical kinetic reaction models for mixtures of non-petroleum and petroleum-based components to represent real fuels and lead to efficient reduced combustion models needed for engine modeling codes. (3) Characterize the role of fuel composition on efficiency and pollutant emissions from practical automotive engines.
Book Synopsis Biodiesel, Combustion, Performance and Emissions Characteristics by : Semakula Maroa
Download or read book Biodiesel, Combustion, Performance and Emissions Characteristics written by Semakula Maroa and published by Springer Nature. This book was released on 2020-07-10 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on biodiesel combustion, including biodiesel performance, emissions and control. It brings together a range of international research in combustion studies in order to offer a comprehensive resource for researchers, students and academics alike. The book begins with an introduction to biodiesel combustion, followed by a discussion of NOx formation routes. It then addresses biodiesel production processes and oil feedstocks in detail, discusses the physiochemical properties of biodiesel, and explores the benefits and drawbacks of these properties. Factors influencing the formation of emissions, including NOx emissions, are also dealt with thoroughly. Lastly, the book discusses the mechanisms of pollution and different approaches used to reduce pollutants in connection with biodiesel. Each approach is considered in detail, and diagrams are provided to illustrate the points in line with industry standard control mechanisms.
Book Synopsis Recent Advances in Detailed Chemical Kinetic Models for Large Hydrocarbon and Biodiesel Transportation Fuels by :
Download or read book Recent Advances in Detailed Chemical Kinetic Models for Large Hydrocarbon and Biodiesel Transportation Fuels written by and published by . This book was released on 2009 with total page 6 pages. Available in PDF, EPUB and Kindle. Book excerpt: N-Hexadecane and 2,2,4,4,6,8,8-heptamethylnonane represent the primary reference fuels for diesel that are used to determine cetane number, a measure of the ignition property of diesel fuel. With the development of chemical kinetics models for these two primary reference fuels for diesel, a new capability is now available to model diesel fuel ignition. Also, we have developed chemical kinetic models for a whole series of large n-alkanes and a large iso-alkane to represent these chemical classes in fuel surrogates for conventional and future fuels. Methyl decanoate and methyl stearate are large methyl esters that are closely related to biodiesel fuels, and kinetic models for these molecules have also been developed. These chemical kinetic models are used to predict the effect of the fuel molecule size and structure on ignition characteristics under conditions found in internal combustion engines.
Book Synopsis Experimental and Kinetic Modeling Study of 1-Hexanol Combustion in an Opposed-Flow Diffusion Flame by : Coleman Yue Yeung
Download or read book Experimental and Kinetic Modeling Study of 1-Hexanol Combustion in an Opposed-Flow Diffusion Flame written by Coleman Yue Yeung and published by . This book was released on 2011 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: Biofuels are of particular interest as they have the potential to reduce our dependence on petroleum-derived fuels for transportation. 1-Hexanol is a promising renewable long chain alcohol that can be used in conventional fuel blends or as a cosolvent for biodiesel mixtures. However, the fundamental combustion properties of 1-hexanol have not been fully characterized in the literature.Thus, new experimental results, consisting of temperature and concentration profiles of stable species were obtained for the oxidation of 1-hexanol generated in an opposed-flow diffusion flame at 0.101 MPa. The kinetic model consists of 361 chemical species and 2687 chemical reactions (most of them reversible). This experimental data were compared to the predicted values of a detailed chemical kinetic model proposed in literature to study the combustion of 1-hexanol. Reaction pathway and sensitivity analyses were performed to interpret the results. In addition, several improvements were investigated to optimize the proposed chemical kinetic mechanism.
Book Synopsis An Experimental and Kinetic Modeling Study of Methyl Decanoate Combustion by :
Download or read book An Experimental and Kinetic Modeling Study of Methyl Decanoate Combustion written by and published by . This book was released on 2010 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt: Biodiesel is typically a mixture of long chain fatty acid methyl esters for use in compression ignition engines. Improving biofuel engine performance requires understanding its fundamental combustion properties and the pathways of combustion. This research study presents new combustion data for methyl decanoate in an opposed-flow diffusion flame. An improved detailed chemical kinetic model for methyl decanoate combustion is developed, which serves as the basis for deriving a skeletal mechanism via the direct relation graph method. The novel skeletal mechanism consists of 648 species and 2998 reactions. This mechanism well predicts the methyl decanoate opposed-flow diffusion flame data. The results from the flame simulations indicate that methyl decanoate is consumed via abstraction of hydrogen atoms to produce fuel radicals, which lead to the production of alkenes. The ester moiety in methyl decanoate leads to the formation of low molecular weight oxygenated compounds such as carbon monoxide, formaldehyde, and ketene.
Book Synopsis Detailed Chemical Kinetic Mechanisms for Combustion of Oxygenated Fuels by :
Download or read book Detailed Chemical Kinetic Mechanisms for Combustion of Oxygenated Fuels written by and published by . This book was released on 2000 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Thermodynamic properties and detailed chemical kinetic models have been developed for the combustion of two oxygenates: methyl butanoate, a model compound for biodiesel fuels, and methyl formate, a related simpler molecule. Bond additivity methods and rules for estimating kinetic parameters were adopted from hydrocarbon combustion and extended. The resulting mechanisms have been tested against the limited combustion data available in the literature, which was obtained at low temperature, subatmospheric conditions in closed vessels, using pressure measurements as the main diagnostic. Some qualitative agreement was obtained, but the experimental data consistently indicated lower overall reactivities than the model, differing by factors of 10 to 50. This discrepancy, which occurs for species with well-established kinetic mechanisms as well as for methyl esters, is tentatively ascribed to the presence of wall reactions in the experiments. The model predicts a region of weak or negative dependence of overall reaction rate on temperature for each methyl ester. Examination of the reaction fluxes provides an explanation of this behavior, involving a temperature-dependent competition between chain-propagating unimolecular decomposition processes and chain-branching processes, similar to that accepted for hydrocarbons. There is an urgent need to obtain more complete experimental data under well-characterized conditions for thorough testing of the model.
Book Synopsis Chemical Kinetic Models for HCCI and Diesel Combustion by :
Download or read book Chemical Kinetic Models for HCCI and Diesel Combustion written by and published by . This book was released on 2008 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hydrocarbon fuels for advanced combustion engines consist of complex mixtures of hundreds or even thousands of different components. These components can be grouped into a number of chemically distinct classes, consisting of n-paraffins, branched paraffins, cyclic paraffins, olefins, oxygenates, and aromatics. Biodiesel contains its own unique chemical class called methyl esters. The fractional amounts of these chemical classes are quite different in gasoline, diesel fuel, oil-sand derived fuels and bio-derived fuels, which contributes to the very different combustion characteristics of each of these types of combustion systems. The objectives of this project are: (1) Develop detailed chemical kinetic models for fuel components used in surrogate fuels for diesel and HCCI engines; (2) Develop surrogate fuel models to represent real fuels and model low temperature combustion strategies in HCCI and diesel engines that lead to low emissions and high efficiency; and (3) Characterize the role of fuel composition on low temperature combustion modes of advanced combustion engines.
Book Synopsis Chemical Kinetic Modeling of Combustion of Automotive Fuels by : W. J. Pitz
Download or read book Chemical Kinetic Modeling of Combustion of Automotive Fuels written by W. J. Pitz and published by . This book was released on 2006 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt: The objectives of this report are to: (1) Develop detailed chemical kinetic reaction models for components of fuels, including olefins and cycloalkanes used in diesel, spark-ignition and HCCI engines; (2) Develop surrogate mixtures of hydrocarbon components to represent real fuels and lead to efficient reduced combustion models; and (3) Characterize the role of fuel composition on production of emissions from practical automotive engines.
Book Synopsis A Homogenous Combustion Catalyst for Fuel Efficiency Improvements in Diesel Engines Fuelled with Diesel and Biodiesel by : Mingming Zhu
Download or read book A Homogenous Combustion Catalyst for Fuel Efficiency Improvements in Diesel Engines Fuelled with Diesel and Biodiesel written by Mingming Zhu and published by . This book was released on 2012 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: [Truncated abstract] The ferrous picrate based homogeneous combustion catalyst has been claimed to promote diesel combustion and improve fuel efficiency in diesel engines. However, the reported fuel savings were controversial. Moreover, a lack of understanding of the working mechanism of the catalyst in diesel combustion processes so far has hindered its widespread applica-tion. On the other hand, the effect of the catalyst on biodiesel combustion has never been scientifically studied and presents a new area regarding the application of the catalyst. The present research aims to systematically study the effect of the catalyst on the fuel efficiency of diesel and biodiesel in diesel engines and to understand its working mechanisms. The specific objectives of this research include a systematic study on the effect of the catalyst on fuel efficiency and combustion characteristics of diesel and biodiesel in diesel engines, an investigation into the working mechanisms of the catalyst in diesel and bio-diesel combustion processes through both experimental and mathematical modelling studies on single droplets, and a study on the chemical kinetic effect of iron atoms on the diesel combustion processes. To accomplish these objectives, the influence of the catalyst on fuel efficiency of diesel and biodiesel as a function of engine speed, load and catalyst dosing ratio was first studied using both a laboratory small single cylinder engine and a large industrial scale four cylinders engine. Then both mathematical modelling and experi-mental studies on the combustion characteristics, including ignition delay periods, burnout time, burning rates and flame temperature, of single droplets of diesel and biodiesel were performed. Lastly, a kinetic modelling study of the effect of iron on the ignition and combustion characteristics of diesel was carried out using CHEMKIN PRO. Diesel engine tests have shown that the use of the catalyst reduced the brake specific fuel consumption (BSFC). Up to 4.2% fuel saving in the small laboratory diesel engine and up to 5.6% in the large industrial scale diesel engine under tested conditions were achieved. The reduction of the BSFC was greater at light loads. In addition, the use of the catalyst reduced the BSFC of the biodiesel up to 2.8% under the tested conditions. It has been found that the addition of the catalyst shortened the ignition delay and combustion duration of diesel and biodiesel in the engine, resulting in slightly higher peak cylinder pressures and faster heat release rates. The study of combustion characteristics of single droplets has indicated that the catalyst shortened the burnout time, increased the burning rate and flame temperature of the drop-lets of both diesel and biodiesel. At the catalyst dosing ratio of 1:10000 (by volume) in the diesel and biodiesel, the flame temperatures of the catalyst dosed droplets were about 40K-50K higher than those of the droplets without the catalyst while the burning rate was 0.05-0.1 mm2s-1 higher. With the assistance of flame emissions spectroscopy, iron atoms were detected to present in the flame of the combustion of the catalyst and it was found that the pure ferrous picrate decomposed at approximately 523K using thermogravimetric analysis...
Book Synopsis Autoignition Characteristics of Diesel Fuel and Its Surrogates by : Goutham Kukkadapu
Download or read book Autoignition Characteristics of Diesel Fuel and Its Surrogates written by Goutham Kukkadapu and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The design process for development of engines could be made faster and less expensive with the help of computations which help understanding the processes prevalent in internal combustion engines. Running engine simulations are challenging as they need to accurately capture the fluid dynamic and chemical kinetic processes that occur in an engine. A major challenge in simulating chemical kinetic processes is the complexity of the fuel chemistry: real fuels are complex mixtures whose composition determines their physical properties and reactivity. The behavior of these real fuels can be conveniently represented using simpler mixtures often called â€surrogates mixtures†that match the key properties of the real fuels. Successful modeling of the ignition of real fuel first requires the formulation of an appropriate surrogate mixture whose compositions are carefully chosen in order to best emulate the combustion properties of the targeted real fuel. Then a comprehensive chemical kinetic model developed based on the surrogate fuel is used to simulate the combustion process of the real fuel. The work presented in the current dissertation intends to systematically study the surrogate modeling of diesel fuels. The study has been conducted to understand the ignition of surrogate fuel constituents and fully blended diesel fuels. Autoignition of tetralin, 1-methylnaphthalene, iso-cetane, and n-dodecane, the constituents of diesel surrogates, are investigated in the current dissertation. Besides, ignition of binary blends of the surrogate constituents has also been studied to investigate the effects of blending on ignition when neat components are blended to formulate a surrogate fuel. Furthermore, the ignition of two fully blended research grade diesel fuels has also been conducted inorder to provide quality ignition delay data for development and validation of chemical kinetic models of kinetic fuels.
Book Synopsis Record Title to Land by : Heman White Chaplin
Download or read book Record Title to Land written by Heman White Chaplin and published by . This book was released on 1893 with total page 13 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Chemical Kinetic Modeling of Component Mixtures Relevant to Gasoline by :
Download or read book Chemical Kinetic Modeling of Component Mixtures Relevant to Gasoline written by and published by . This book was released on 2008 with total page 8 pages. Available in PDF, EPUB and Kindle. Book excerpt: Detailed kinetic models of pyrolysis and combustion of hydrocarbon fuels are nowadays widely used in the design of internal combustion engines and these models are effectively applied to help meet the increasingly stringent environmental and energetic standards. In previous studies by the combustion community, such models not only contributed to the understanding of pure component combustion, but also provided a deeper insight into the combustion behavior of complex mixtures. One of the major challenges in this field is now the definition and the development of appropriate surrogate models able to mimic the actual features of real fuels. Real fuels are complex mixtures of thousands of hydrocarbon compounds including linear and branched paraffins, naphthenes, olefins and aromatics. Their behavior can be effectively reproduced by simpler fuel surrogates containing a limited number of components. Aside the most commonly used surrogates containing iso-octane and n-heptane only, the so called Primary Reference Fuels (PRF), new mixtures have recently been suggested to extend the reference components in surrogate mixtures to also include alkenes and aromatics. It is generally agreed that, including representative species for all the main classes of hydrocarbons which can be found in real fuels, it is possible to reproduce very effectively in a wide range of operating conditions not just the auto-ignition propensity of gasoline or Diesel fuels, but also their physical properties and their combustion residuals [1]. In this work, the combustion behavior of several components relevant to gasoline surrogate formulation is computationally examined. The attention is focused on the autoignition of iso-octane, hexene and their mixtures. Some important issues relevant to the experimental and modeling investigation of such fuels are discussed with the help of rapid compression machine data and calculations. Following the model validation, the behavior of mixtures is discussed on the basis of computational results.
