Keywords

7.1 Introduction

Many combustion devices such as rocket, gas turbines, diesel, and HCCI engines achieve an operating pressure , which is much higher than the critical pressure of either the fuel or the oxidizer. For example, rocket engines using H2 as fuel can achieve a pressure excess of 100 atm [1], while the critical pressure of the fuel (H2) and oxidizer (O2) is 13 atm and 50 atm, respectively [2]. Similarly, pressure inside the combustion chamber of diesel engine increases from ~25 atm during injection to 60 atm [3, 4] after ignition, which is beyond the critical pressure of any of the components of diesel fuels. The physical and the chemical mechanisms of combustion at this pressure /temperature condition is drastically different than its low-pressure counterpart, since molecules are densely packed and may experience van der Waals interactions and caging effects under high pressure [5]. Therefore, a detailed understanding of this complex process is important for the further development of these devices.

When a fuel is sprayed into the combustion chamber, it mixes with a stream of high-pressure fluid, which is primarily the oxidizer. This molecular mixing process has a great influence on autoignition and is a strong precursor of combustion . Additionally, during combustion , the heavier fuel species first undergo chemical decomposition , i.e., pyrolysis and then oxidation . Due to the nonuniform nature of the combustion (both pyrolysis and oxidation ) reactions, highly active and fast moving radicals are generated in one location and mix with fuel and oxidizer at a different location inside the combustion chamber and significantly alter the combustion dynamics . Thus, this turbulent mixing of multiple chemical species under high pressure conditions has been an active area of research for the combustion community. However, despite this great interest, the experimental studies [6,7,8,9,10,11,12] are limited to only single or binary species mixing instead of multicomponent mixing. Additionally, most studies are based on qualitative visualization with rare quantitative analysis due to the difficulties of performing experiments at supercritical conditions , and therefore cannot be used to validate the proposed kinetic models at high-pressure conditions . Furthermore, these studies except a few [6, 13, 14] have used simple species like N2, H2, and O2 in their turbulent mixing study, which require very simple chemical kinetic models , whereas the real fuels used inside combustion chambers are rather complex ones.

Since the experiments fail to provide sufficient information regarding the complex physical (diffusion , turbulence) and chemical (reaction ) coupling of turbulent mixing, computational fluid dynamics (CFD) simulations have emerged as a potential solution to study most of the physical aspects of these problems. Among different CFD techniques available, direct numerical simulation (DNS) has proved to be a powerful tool of combustion and turbulence research [15], since it can resolve any length and timescales of the flow using high order accurate methods . Masi et al. [16] have developed a DNS model to describe the multi- species high-pressure turbulence mixing. Combining this model with the rate of single-step chemical reaction consistent with ignition prediction [17], Jossette Bellan from Jet Propulsion Lab (JPL) has recently studied both the diffusion [18] and turbulent reaction rate [19] of high pressure mixing. Additionally, Gnanaskandan et al. [5] studied the length of the potential core of round fluid jet entering high-pressure chamber using both DNS and large eddy simulations (LES) methods , and Foster et al. [20] studied the effect of Soret and Dufor cross diffusions in turbulent mixing. The pressure considered in these studies typically ranges from 60 to 100 atm.

All the CFD studies mentioned above mainly focused on the physical aspects of turbulent chemistry ; their chemical model was fairly simplistic. For example, Bellan and coworkers [16, 18, 19] only considered single-step chemical reaction for n-heptane oxidation , while Foster et al. [20] considered very simple 7-step 10 species reduced mechanism for a similar problem. However, as mentioned above, a highly coupled relation between chemical and physical properties governs the turbulent mixing phenomenon, which cannot be fully captured by the simple chemical kinetic model.

Chemical kinetics models for various hydrocarbons are well documented in the literature [21]. However, these models are only developed for low-pressure /temperature condition. Additionally, these models were developed only considering temperature dependence on the reaction rates via simple Arrhenius-type rate laws, while neglecting pressure dependence on combustion pathways; which can significantly alter the chemical properties at high pressure . As experiments are difficult to be conducted at supercritical pressure and temperature , we need a computationally feasible method , which can simulate complex combustion reactions without requiring any user input of possible reactions; so that the complete reaction network can be captured. Quantum mechanical (QM) -based ab initio methods are the best choice for accurately predicting the reactions for such systems, however, they have serious system size and simulation time limitation [22]. Since these methods solve the Schrodinger equation to estimate reaction energies and barriers, they can only be applied to small systems typically containing a couple of hundred of atoms for a shorter time scale [22]. Recently, ReaxFF reactive force field method [23] has proven to be a useful alternative of QM -based methods for combustion chemistry simulations [24,25,26,27,28,29,30,31] as it can simulate larger systems for a longer time scale .

The focus of this book chapter is to demonstrate how the ReaxFF reactive force field method can be used as a valuable tool to study combustion kinetics of fuels and fuel mixtures at supercritical condition. The chapter is organized as follows: in Sect. 7.2, we will introduce the ReaxFF method itself while mentioning some of its previous applications in combustion study. In Sect. 7.3, we will elaborate on our systems and simulation setup. Next, some observations on combustions kinetics based on our simulations will be made in Sect. 7.4; and results obtained from ReaxFF simulations will be compared with continuum simulations. Lastly, we will provide some concluding remarks mentioning the pros and cons of ReaxFF simulation for studying similar problems and challenges and opportunities regarding the development of a multiscale ReaxFF /continuum simulation capability for high-temperature /high-density pyrolysis and combustion simulations.

7.2 ReaxFF Background

Since force field based methods are computationally inexpensive and can provide a reasonable agreement with quantum mechanical simulations, they are now being used extensively to get an atomistic insight of complex physical and chemical problems. While potentials such as DREIDING [32], MM [3, 33,34,35], MM [4, 36] COMPASS [37], etc., have been used to perform atomistic scale simulations of hydrocarbon fuels, mostly to study their thermodynamic properties , due to their rigid connectivity requirement they cannot be used to simulate chemical reactions. However, potentials such as first-[38] and second-generation [39] reactive empirical bond order (REBO ), charge -optimized many-body (COMB) potential [40, 41], modified embedded atom method (MEAM ) [42] and reactive force field (ReaxFF ) [23] can dynamically simulate bond formation and bond breaking and capture chemical reactions. In this chapter, we will keep our discussion limited to the ReaxFF method and its application to high-pressure combustion .

ReaxFF adopts the concept of bond order introduced by Tersoff [43] and Brenner [38] and calculates the bond order of every pair of atoms as a function of their interatomic distances. Based on a certain cutoff value, ReaxFF finds the connectivity between atoms at each step, which enables ReaxFF to simulate dynamic bond breaking and bond formation during simulation . All the connectivity-dependent terms like bond , angle, and torsion energies are calculated based on bond order, thus nonbonded atom pairs do not contribute to these partial energies . Long-range interactions like van der Waals and Coulomb are not connectivity dependent and calculated between every pair of atoms, any excessive short-range nonbonded interactions are avoided by including a shielding term. ReaxFF calculates atomic charges using a geometry-dependent charge calculation scheme and uses electronegativity equalization method (EEM) [44] for this purpose. Additionally, for long-range interactions ReaxFF uses a cutoff distance (which is typically set to 10 Å) to reduce the computational cost. To eliminate any energy discontinuity and reduce the range of the Coulomb interactions, a seventh-order Taper function is employed [23]. Equation (7.1) shows the different energy components of ReaxFF total energy , while Fig. 7.1 demonstrates the complicated internal scheme of ReaxFF . A more detailed description of the ReaxFF energy terms can be found in the previous literature [23, 24].

Fig. 7.1
figure 1

Internal scheme of ReaxFF

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$$ E_{\text{system}} = E_{\text{bound}} + E_{\text{over}} + E_{\text{under}} + E_{{\text{lp}}} + E_{{\text{val}}} + E_{{\text{tor}}} + E_{\text{vdWaals }} + E_{{\text{coulomb}}} $$
(7.1)

where Ebond, Eover, Eunder, Elp, Eval, Etor, EvdWaals, Ecoulomb represent bond energy , over-coordination energy penalty, under-coordination stability , lone pair energy , valence angle energy , torsion angle energy , van der Waals energy , and coulomb energy , respectively.

ReaxFF force field parameters are trained against QM-calculations in describing energies and barriers for chemical reactions while solving Newton’s equation of motion to generate a dynamic description of complex reactive systems. Thus, ReaxFF -based molecular dynamics (MD) simulations are a number of magnitudes faster than QM-based simulations while approaching the accuracy of those methods . Additionally, the inclusion of polarizable charge description and bond -order-dependent 3- and 4-body terms in ReaxFF makes it uniquely applicable for both metallic and covalent systems. Thus, ReaxFF has now become a great computational tool to study many reactive systems; at the same time, it has enabled researchers throughout the world to investigate previously inaccessible computational chemistry problems. Apart from studying combustion problems, ReaxFF method has been extensively used to investigate a wide range of applications in materials [45,46,47,48,49,50] catalysts [51, 52], and other chemical systems [53,54,55,56,57].

Combustion , being a complex reactive system , requires an atomistic -level understanding of the intricate details regarding the underlying reactions and species to facilitate better engine and fuel design. To this end, the first ReaxFF combustion force field (CHO-2008) was developed by Chenoweth et al. [24] in 2008. Since then, it has been implemented in a wide range of applications in scientific community for studying pyrolysis and oxidation of variety of fuels, for example, JP-10 [25] n-dodecane [58], 1-heptane [30], n-octanol [29], toluene [59], Illinois no. 6 coal [26], 1,6-dicyclopropane-2,4-hexyne [60] and liginin [61], and many more. The recent development of GPU-enabled ReaxFF [62] has made it suitable to simulate large complex systems like coal [63] and lignin [64] pyrolysis . Apart from the hydrocarbon fuel , this description has also been used to investigate a wide range of aspects related to carbon -based materials including the oxidation of graphene [65], structural and chemical properties of graphene oxide [66], chemomechanics of crack propagation in graphene [67] and dynamics of carbon nano-onion formation [68]. For a more detailed review of this force field , we refer to the recent paper by Dontgen et al. [31].

Though ReaxFF CHO-2008 description was very good in describing the initial pyrolysis and oxidation of large hydrocarbons, where the initial combustion process is mainly dominated by the pyrolysis process, it failed to describe the chemistry of smaller hydrocarbon properly [28]. Additionally, CHO-2008 parameter set was not transferable to study mechanical properties of condensed phase carbons [69]. Furthermore, the O2 in CHO-2008 description is highly reactive , which can abstract H from hydrocarbon chain at a faster-than-expected rate . This might send us a false alarm of system initiation during the simulation . Srinivasan et al. [69] derived an improved set of ReaxFF carbon parameters for describing carbon -condensed phases and taking those parameters on board, Ashraf et al. [28] have recently published an extended version of ReaxFF combustion force field (CHO-2016), which addresses the limitations of CHO-2008 description while retaining the overall quality of CHO-2008 description for larger hydrocarbon combustion . Thus, a single force field can now be used to study not only both the condensed and the gas phase carbon , but also the combustion of any hydrocarbon irrespective of fuel size or structure . This ensures a greater transferability of ReaxFF force field to study any combustion related problems.

Chemical reactions occurring in combustion are considered ‘rare events’ in MD simulations as not every collision between molecules lead to a reaction . Additionally, reactive MD simulations are expensive as it needs to calculate the bond order at every time step, thus the simulations are limited to only a couple of nano-seconds or less, whereas the timescales associated with combustion reactions at experimental conditions are micro-second-level phenomenon. MD simulations are thus typically carried out at high temperatures and pressures to accelerate the system dynamics and capture the appropriate chemical reactions. Therefore, ReaxFF -based MD simulations are suitable for studying the complex chemistry at high-pressure condition although sometimes the pressure in ReaxFF simulation may exceed the highest possible pressure that a rocket engine can survive. However, despite this shortcoming, it is arguably the most appropriate currently available method to study the high-pressure combustion chemistry . Almost all of the ReaxFF combustion studies mentioned above were performed at very high pressure to study the kinetics of different fuels. Additionally, Ashraf et al. [27] have recently demonstrated that ReaxFF -MD simulations can also be used to study dynamic properties like ignition front speed at supercritical conditions .

In this chapter, we will demonstrate how we can use ReaxFF -MD simulations to calculate Arrhenius parameters for different hydrocarbons at very high pressure and temperature condition. Since the hydrocarbons do not typically exist in a single-component form, we will also investigate how the blending of a highly reactive hydrocarbon with a less reactive one alters the combustion properties of the mixture . This enabled us to identify pressure /temperature and mixing conditions , where the simple first-order kinetics and Arrhenius-type relation fails to prevail, indicating that more complex relationship is required to calculate mixture activation energies . Since experiments are difficult to conduct and measurements are hard to perform at very high pressure /temperature condition, these information from ReaxFF -MD simulation will certainly open up the possibility to study the mixing effects of multiple hydrocarbons at supercritical condition.

7.3 Simulation Details

Since typical transportation fuels used for combustion are highly complex mixture of various hydrocarbons, for modeling purpose, Kim et al. [70] developed two different surrogates for diesel so that various physical and chemical processes inside of a diesel engine can be replicated. The surrogate fuels are either a mixture of n-dodecane/iso-cetane/methylcyclohexane/toluene or a mixture of n-dodecane/iso-cetane/decalin/toluene, where n-dodecane is the most reactive one and toluene is the least. As such, for our study, we decided to investigate the effect of n-dodecane addition on toluene pyrolysis using the reactive molecular dynamics (RMD) simulations. We also used two different densities for the overall system , to investigate the effect of density /pressure on the pyrolysis of the mixtures. All of the MD simulations in this section were performed employing the ReaxFF method with a constant number of atoms (N) in a constant volume (V) while keeping the temperature constant (T) using a thermostat, as described by the NVT-MD ensemble. Also, we used the recently developed combustion force field by Ashraf et al. [28] in this study.

7.3.1 Single-Component System

First, to determine the applicability of the force field , toluene and n-dodecane pyrolysis have been investigated independently using homogeneous system . We placed 40 toluene molecules in two cubic periodic unit boxes of dimension 31.20 Å × 31.20 Å × 31.20 Å and 25.00 Å × 25.00 Å × 25.00 Å, where the overall system densities are 0.2 kg/dm3 and 0.4 kg/dm3 respectively. Similarly, in case of n-dodecane, we placed 40 molecules in cubic boxes of dimension 38.39 Å × 38.39 Å × 38.39 Å and 30.47 Å × 30.47 Å × 30.47 Å, respectively, to generate systems of similar densities like toluene. Next, each system is energy minimized and equilibrated using NVT simulation for 10 ps at a temperature of 1500 K. Test cases has been run beforehand to ensure that no thermal decomposition occurs at this temperature for both toluene and n-dodecane. After equilibration , 10 different initial configurations of the system were selected to perform a series of NVT-MD simulations. For each case, the system temperature was varied from 2000 to 2600 K at 100 K interval. The average initial pressure of the system for the first 5 ps of the NVT-MD simulations was in the range of 26–75 MPa. Although the use of such high temperatures and pressures is rarely seen in experiments , this is essential in MD simulations to keep the computational time within a reasonable scope. The high temperature and subsequent high pressure results in a larger number of collisions in the system and thereby reduces the reaction time (Fig. 7.2).

Fig. 7.2
figure 2

Snapshots of representative single-component system of equilibrated a toluene, and b n-dodecane with density of 0.2 kg/dm3. The carbon and hydrogen atoms are displayed in cyan and white, respectively

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During all of the simulations, the time step has been kept as 0.1 fs, which is appropriate for describing hydrocarbon reaction mechanism at high temperatures [24]. Since n-dodecane is highly reactive , a simulation time of 50 ps was enough to calculate the Arrhenius parameters . On the other hand, the toluene simulations were run for 200 ps due to the less reactive nature of toluene. The results from the simulations with 10 different starting configurations were then averaged to obtain the reactant decomposition and overall product distribution . This information is then used to get the Arrhenius parameters for individual components.

7.3.2 n-Dodecane and Toluene Mixture (Multicomponent System )

Once we validated the quality of the force field against the single-component system , we moved on to explore the effect of n-dodecane presence to the pyrolysis of toluene at high-pressure /temperature condition which is the main focus of this chapter. To do this, we performed a series of NVT-MD simulations of toluene and n-dodecane mixtures, where these two hydrocarbons are introduced in different ratios. Table 7.1 represents a summary of all the input configurations for these systems.

Table 7.1 Initial configurations of different n-dodecane and toluene mixtures
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All of the systems studied were equilibrated using NVT-MD after placement of the molecules at a temperature of 1500 K for 10 ps. Then, 10 different initial configurations of the molecules were generated and a series of NVT-MD simulation at temperatures 2000, 2100, 2200, 2300, 2400, 2500, 2600 K were carried out for all of them. The total simulation time was 200 ps with the time step being 0.1 fs. We repeated all the simulations changing the system density to 0.4 kg/dm3 to study and compare the effect of a higher density /pressure of the system .

For all the simulations, whether single-component or multicomponent, the following parameters were the same. To keep the system temperature constant, the temperature was controlled by the Berendsen thermostat using a temperature damping constant of 100.0 fs. For molecular recognition, we used a bond -order cut off of 0.3. The choice of bond -order cutoff does not alter the simulation pathway , but it is only used to identify the intermediates and products formed during the MD simulations.

7.3.3 Continuum Simulations

In addition to the ReaxFF -based simulations described above, zero-dimensional (time-evolving), continuum -scale pyrolysis simulations of the same series of test cases (see Table 7.1) are performed . The intent is to highlight the deficiency of Arrhenius-type chemical kinetics in predicting chemical evolutions (e.g., pyrolysis ) under high-pressure high temperature conditions , by comparing the results obtained from both approaches. For this purpose, the initial temperatures, density , and mole fractions of n-dodecane and toluene in these continuum simulations are set to be the same as those used in the molecular dynamics simulations as described in Table 7.1, and these simulations are performed with constant volume and constant temperature to best duplicate the same running conditions as for the ReaxFF simulations. In these simulations, a cubic equation of state is used to account for real gas effects [71], and a widely used chemical kinetic mechanism [27] with Arrhenius-type rate coefficients is used. This chemical mechanism contains 179 species and 1895 chemical reactions (forward and backward reactions counted separately), and it has been extensively validated for jet fuel surrogate components under lower pressure and temperature ranges (up to 40 bars and 1250 K) [72].

7.4 Results and Discussion

7.4.1 Kinetic Analysis of Toluene and n-Dodecane Pyrolysis as Single-Component System

For single-component systems of toluene and n-dodecane, first-order kinetics has been used to study the thermal decomposition . We used the consumption rate from the ReaxFF -MD simulations in this analysis and the number of reactant molecules has been chosen to represent the reactant concentration. From the NVT-MD simulations at different temperatures, the decomposition of toluene and n-dodecane has been found to change as a function of time and temperature . Using this information, we used integrated first-order rate law to determine the rate constant at each temperature :

$$ { \ln }\left( {N_{0} } \right){-}{ \ln }\left( {N_{t} } \right) = {\text{kt}} $$
(7.2)

where N0 is the number of molecules initially in the system and Nt is the number of molecules at any time t. At each temperature , the quantity ln(N0) − ln(Nt) has been plotted against time and the rate constant has been determined from the slope of the linear fitting of the plot. We used these rate constants from different temperatures in the Arrhenius plot of toluene and n-dodecane as shown in Fig. 7.3. The activation energy (Ea) and the pre-exponential factor (A) were calculated with the help of a linear fit of the plot and the Arrhenius equation described as

Fig. 7.3
figure 3

Arrhenius plot for calculated and fitted rate constants of a toluene, b n-dodecane as single-component systems with density ρ = 0.2 and 0.4 kg/dm3

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$$ k = A\,{ \exp }\left( { - E_{a} /{\text{RT}}} \right) $$
(7.3)

Next, we compared the values of Arrhenius parameters obtained from ReaxFF -MD simulations with their experimental counterparts, which are listed in Table 7.2. It is important to note that, the experimental values are obtained at a much lower pressure and temperature condition than ReaxFF simulations. For example, Colket et al. [73] investigated the pyrolysis of toluene at a temperature range of 1200–1850 K with a total pressure of approximately 1 MPa. According to them, the formation of benzyl radical is the most dominant reaction pathway , which is also accompanied by the pathway that leads to phenyl radical formation . The activation energies mentioned in Table 7.2 are for these two key reaction pathways.

Table 7.2 Fitted Arrhenius equation and parameters for n-dodecane and toluene
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Liu et al. [74] investigated the supercritical thermal cracking of n-dodecane at a temperature range 700–800 K and pressure of 3–4 MPa. For both toluene and n-dodecane, the results obtained in this work shows good agreement with those obtained from the experiments . Wang et al. [58] investigated the pyrolysis of n-dodecane using ReaxFF -MD simulations and the force field parameters developed by Chenoweth et al. [24] using a temperature range of 2100–3000 K. The activation energies of n-dodecane derived in the study were 63.68 and 66.14 kcal/mol for densities ρ = 0.17 and 0.33 kg/dm3. The activation energy of n-dodecane derived in this work using the recently developed force field by Ashraf et al. [28] showed a better agreement with the experimental results than those derived by Wang et al. [58] using Chenoweth et al. [24] force field . Though in our simulations pressure and temperature are much higher than their experimental counterparts, the reasonable agreement with experimental values indicates that the simple Arrhenius type relation holds good for single-component system , even at a very high pressure and temperature .

Figure 7.4 shows the evolution of toluene and dodecane molecules as a function of time for both of the densities 0.2 and 0.4 kg/dm3. At temperatures 2000, 2100, and 2200 K, the number of toluene decomposed was very low, so the simulation results are not used in calculations of the Arrhenius parameters since these would include a significant statistical uncertainty . The figures indicate the reactivity of n-dodecane as opposed to that of toluene. We observed that at 2000 K, n-dodecane began to decompose within 5 ps of the simulation time; at temperatures higher than 2000 K the decomposition began in less than 5 ps. While in the case of toluene, at temperatures 2300, 2400, and 2500 K, the decomposition does not begin before 20 ps. Only at a high temperature of 2600 K, toluene starts to decompose within 5 ps of the simulation time. The trend of decomposition initiation is observed in both 0.2 and 0.4 kg/dm3 density cases.

Fig. 7.4
figure 4

Time evolution of the number of toluene molecules in single-component systems with density a ρ = 0.2 kg/dm3 and b 0.4 kg/dm3, and n-dodecane molecules in single-component systems with density c ρ = 0.2 kg/dm3 and d 0.4 kg/dm3

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As mentioned above, experimentally, the initiation mechanism for toluene pyrolysis is dominated by decomposition of toluene into benzyl (C7H7) and hydrogen atom with activation energy 88.9 kcal/mol [71]. The hydrogen atom generated from this toluene decomposition reacts with another toluene molecule to produce another benzyl radical and hydrogen molecule . Another reaction pathway is the formation of phenyl (C6H5) and methyl radical (CH3). A similar initiation mechanism is observed in ReaxFF simulations shown in Scheme 7.1a. Figure 7.5a demonstrates the distribution of toluene and its major decomposition species at 2600 K. The decomposition of toluene starting after 10 ps is on the rise accompanied by a gradual rise of both benzyl and hydrogen molecule .

Scheme 7.1
scheme 1

Initiation reaction mechanism and their relative percentages occurred in a toluene, b n-dodecane pyrolysis in a single-component system revealed by ReaxFF –MD simulations at 2600 K

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Fig. 7.5
figure 5

Time evolution of the major species generated in the pyrolysis of a toluene, b n-dodecane in single-component systems with density ρ = 0.2 kg/dm3 and temperature T = 2600 K

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The pyrolysis of n-dodecane is dominated by radical production in ReaxFF as shown in Fig. 7.5b. This is in agreement with experiment [75]. The initiation reactions principally involve C–C bond fission to produce two alkyl radicals as shown in Scheme 7.1b. On rare occasion, we observe abstraction of hydrogen from n-dodecane. As the simulation proceeds, larger alkyl radicals undergo further decomposition , which increases the number of smaller radicals such as CH3, C2H5, C3H7 and creates hydrogen molecules and stable double-bond -containing molecules like C2H4 and C3H6.

7.4.2 Pyrolysis of Toluene and n-Dodecane as Multicomponent System

7.4.2.1 Kinetic Analysis

To investigate the change in activation energy of a less reactive fuel component due to the presence of a more reactive molecule , we analyzed a multicomponent system of n-dodecane and toluene using ReaxFF NVT-MD simulations. The rate constants associated with the decomposition of toluene in the mixture are determined using the first-order kinetics and the number of toluene molecules in (7.3). We derived the Arrhenius parameters in (7.3) from the fitted plot of these rate constants at various temperatures.

Figure 7.6 shows the activation energies (Ea) of toluene in the mixtures as a function of the percentage of n-dodecane introduced in the mixture for both ReaxFF -MD and continuum simulations. The zero-percentage result indicates the pyrolysis of toluene as a single component. As the percentage of n-dodecane increases, the activation energy of toluene decreases rapidly. According to ReaxFF -MD results, the presence of a single n-dodecane molecule during the thermal decomposition of toluene reduces the activation energy from 95.69 to 81.68 kcal/mol (14.64% reduction ). If n-dodecane molecule number is increased to 5, the activation energy reduces further to 66.78 kcal/mol. When the percentage of n-dodecane is 33.33 in the mixture (n-dodecane and toluene ratio 20:40) we see that the activation energy of toluene (52.98 kcal/mol) even drops below that of single-component n-dodecane pyrolysis (52.98 kcal/mol) with the same density . The results from continuum simulations show similar behavior. Both set of results show good qualitative agreement in predicting the mixture activation energy , when the n-dodecane concentration is low (less than 20%). This suggests the capability of ReaxFF -MD to predict the activation energy of toluene within this mixing ratio. However, the results show larger deviations when n-dodecane concentration is higher than 20%. Unlike ReaxFF -MD , the activation energy of toluene from continuum simulation in all of the mixtures remains higher than n-dodecane only system . This indicates that with the presence of higher amount of n-dodecane, the first-order kinetics-based Arrhenius method fails to predict the mixture activation energy with sufficient accuracy.

Fig. 7.6
figure 6

Activation energies of toluene at various compositions of n-dodecane and toluene mixture from ReaxFF -MD simulations and Continuum simulations. The green and blue dashed line represents the activation energy of n-dodecane calculated by ReaxFF and Continuum simulations, respectively

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7.4.2.2 Effect of Toluene Presence on the Pyrolysis of n-Dodecane

Before investigating what is causing the dramatic decrease of activation energy of toluene at the presence of n-dodecane, we want to check if the presence of toluene also affects the activation energy of n-dodecane. To get an overview , here we have compared the single-component system of n-dodecane and the mixture of 40 toluene and 40 n-dodecane molecules. The rate constants were determined in this case using (7.3) and the time evolution of only n-dodecane molecules in the mixture . We calculated the activation energy similarly using (7.3) and the linear fitting of Arrhenius plot of rate constants at different temperatures. The value of activation energy is found to be 58.56 kcal/mol, which is very close to the activation energy 60.94 kcal/mol found in Sect. 7.4.1. Since the activation energy of n-dodecane in the mixture is close to that obtained from individual n-dodecane system , we can conclude the pyrolysis of n-dodecane is mostly independent of the presence of toluene.

Figure 7.7 shows the time evolution of n-dodecane molecules in both single-component system and a 40:40 mixture of toluene and n-dodecane at 2300 K and at 2600 K. The time-dependent number of n-dodecane molecules in the system is averaged from 10 different NVT-MD simulations of the same system but with different initial configurations. The simulation time of n-dodecane in extended to 100 ps to get a better comparison of n-dodecane decomposition . The figures indicate that there is no significant change in the pyrolysis of n-dodecane even though the two systems are different. We can conclude from these results that the initiation mechanism of n-dodecane pyrolysis is mostly independent and is not affected by the presence of a less reactive fuel like toluene.

Fig. 7.7
figure 7

Time evolution the numbers of n-dodecane molecules at temperatures a T = 2300 K and b T = 2600 K

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7.4.2.3 Effect of n-Dodecane Presence on the Pyrolysis of Toluene

Although the pyrolysis of n-dodecane is mostly unaffected by the presence of toluene in the mixture , this is not the case for toluene itself. Figure 7.8 shows the number of toluene molecules decomposed in the single-component system and the mixtures with different concentrations of n-dodecane for a total simulation time of 200 ps.

Fig. 7.8
figure 8

Time evolution the numbers of toluene molecules for single-component system and in mixtures with various concentrations of n-dodecane at temperatures a T = 2100 K, b T = 2200 K, c T = 2300 K, d T = 2400 K, e T = 2500 K, and f T = 2600 K

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A preliminary analysis of the product distribution observed during the decomposition of toluene at temperatures from 2000 to 2600 K shows that as the number of n-dodecane in increased in the mixture , the number of toluene decomposed also increases, even as the number of toluene molecules introduced initially is constant (40). This indicates the existence of several simultaneous processes during the thermal decomposition of toluene. At low temperatures, since the decomposition of toluene in single-component system is very low, the role of n-dodecane plays a much more significant role in further decomposition of toluene.

The Arrhenius plots for the pyrolysis simulations of toluene and n-dodecane mixtures with increasing concentration of n-dodecane is shown in Fig. 7.9. According to this, at high temperature (2500, 2600 K) the rate constant has been mostly unaffected by the inclusion of n-dodecane in the mixture . However, at low temperatures (2000, 2100, 2200, 2300, 2400 K) the rate constants gradually increase with increasing number of n-dodecane in the mixture . The result is an overall upward shift of the Arrhenius plot at lower temperatures. Additionally, with the increase of n-dodecane molecules in the system , lower temperatures become accessible to calculate the rate constants. For example, in toluene-only case, enough toluene was not decomposed to calculate a rate constant at 2200 K, however, with the presence of n-dodecane, we were able to calculate rate constant even at 2000 K. Thus, the decrease in the slope of the plot which gives the value of—Ea/R implies lower activation energies for increasing n-dodecane number in the mixture .

Fig. 7.9
figure 9

Arrhenius plot with the fitted natural logarithm of rate constant versus the inverse of temperature for toluene as single-component system and in mixtures with various concentrations of n-dodecane at densities of ρ = 0.2 kg/dm3

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Though further investigation is required, the different mechanism of toluene and n-dodecane pyrolysis as single-component systems might explain the underlying mechanism responsible for lowering the activation energies of the mixture . Figure 7.4 shows that, at lower temperature (2000 K), roughly 15% of n-dodecane molecules decompose within the 50 ps simulation time of n-dodecane only system , while it reaches to over 90% at higher temperature (2600 K). The decomposition behavior of n-dodecane is observed in the mixtures too; the process is accompanied by the production of a large radical pool. The main species in the radical pool consists of a combination of different alkyl radicals ranging from C1–C12 initially and smaller radicals such as methyl (CH3), ethyl (C2H5), propyl (C3H7) and in the later part of the simulation , hydrogen (H) radical etc. At low temperatures, the intramolecular reaction between the small free radicals and toluene molecules cause further decomposition of toluene into benzyl radical (C7H7) and H. For example, the reaction between methyl (CH3) and toluene (C7H8) to generate methane (CH4) and benzyl has an energy barrier of only 11.1 kcal/mol [70]. With the presence of n-dodecane, such reactions with low barriers cause further decomposition of toluene in a shorter time.

7.4.2.4 Density /Pressure Effect on the Pyrolysis of Toluene and n-Dodecane Mixture

To investigate the density /pressure effect on the pyrolysis of toluene and n-dodecane mixture we performed additional simulations for the systems with a high density of 0.4 kg/dm3. The rate constants and the Arrhenius parameters are calculated based on toluene using the same method in Sect. 7.4.1. Figure 7.10a shows the comparison of the activation energies from ReaxFF simulations for density 0.2 and 0.4 kg/dm3 with increasing percentage of n-dodecane in the mixture . Overall, the results show that the activation energy for a system density of 0.4 kg/dm3 is less than that for a system density of 0.2 kg/dm3 when the percentage of n-dodecane is low in the mixture (2.34 and 11.11%). For 10 and 20 molecules of n-dodecane in the mixture , the density /pressure effect is insignificant and the activation energies for both cases are almost similar. For the high density case, we observe that the presence of only 11.11% n-dodecane in the mixture results in a surprisingly lower activation energy of toluene than that of n-dodecane as a single-component system . Figure 7.10b shows results for similar cases obtained through continuum -scale pyrolysis simulations. At higher density , the activation energy of toluene decreases sharply from 95.17 to just 62.79 kcal/mol (34%) with the addition of only one n-dodecane molecule in the toluene-only system . After that, the activation energy keeps on increasing as the percentage of n-dodecane increases. Similar to ReaxFF -Md results, when n-dodecane number in the mixture is 10 and 20, the results are very similar for the two density cases. These results are in qualitative agreement with the ReaxFF -MD results in the sense that at higher concentrations of n-dodecane in the mixture , the toluene activation energy does not show any dependency on either the amount of n-dodecane in the mixture or the overall system density .

Fig. 7.10
figure 10

Activation energies of toluene at various compositions of n-dodecane and toluene mixture from a ReaxFF -MD simulations. b Continuum simulations with two different densities . The blue and red dashed line represents the activation energy of only n-dodecane in a system density of 0.2 kg/dm3 and 0.4 kg/dm3, respectively

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These ReaxFF -MD results indicate that apparent first-order kinetics is no longer suitable for calculation of activation energy of toluene in the mixture at high density /high pressure system . We can also conclude that when the ratio of n-dodecane and toluene exceeds a certain threshold (in our case more than 5:40), the method has limitations in accurately capturing the effect of the n-dodecane in the activation energy of toluene. It is evident from the results that the decomposition of toluene follows a more complex mechanism in the mixture in high density /pressure than in low density /pressure . Therefore, in order to determine the activation energy of a less reactive component such as toluene in a mixture with a highly reactive molecule such as n-dodecane, we need to look beyond the simple first-order kinetics and the Arrhenius equation .

A complete investigation of this surprising behavior of toluene decomposition with the presence of n-dodecane is beyond the scope of this study, as the purpose of this book chapter is to demonstrate the capability of ReaxFF -MD simulation for high pressure /temperature combustion . However, we would like to propose a hypothesis for this interesting finding, which requires more rigorous studies to confirm. According to our hypothesis, the main reason for low activation energy of toluene in the presence of n-dodecane is mostly related to the very different initiation mechanism of toluene and n-dodecane decomposition , when they are studied as a single-component system . As mentioned several times in this work, n-dodecane decomposition leads to a large alkyl radical pool build up, and at high density , these radicals can easily find a toluene molecule to decompose as they are diffusion limited at that density . Since n-dodecane can generate radicals even at lower temperatures, this leads to higher toluene decomposition , which significantly effects toluene activation energy . That is why, a small amount of n-dodecane can significantly reduce toluene activation energy when simple first-order kinetics and pressure -independent Arrhenius equation is used at high density (0.4 kg/dm3). However, at a relatively lower density such as 0.2 kg/dm3 and lower number of n-dodecane in the system , active alkyl radicals has more space to move around and it takes longer time for them to find a toluene molecule to collide with as the alkyl pool build up is not that significant. However, even at low density , simple Arrhenius equation is not applicable if the initial number of n-dodecane is high as it significantly increases the number of alkyl radicals in the system . Since we used first-order kinetics to calculate the rate constant, while the toluene decomposition is mostly the secondary reaction in the system initiated by active radicals, our analysis no longer holds at this situation.

Additionally, we hypothesize that, at higher temperatures, toluene decomposition shows more temperature dependence than density /pressure dependence. Also, at high temperature , the presence of n-dodecane does not greatly affect the rate constant of toluene which is supported in Fig. 7.9 too. At higher temperatures, probably toluene in mixture decomposes in a similar manner as the toluene-only system as those high energy decomposition routes become accessible. As these routes no longer require an effective collision, toluene mainly decomposes in those routes.

As mentioned above, our hypothesis requires further investigation to confirm, however, with the aid of ReaxFF -MD simulation we were able to gain some useful information of high pressure system , which was never explored before.

7.5 Conclusions

In this work, we applied ReaxFF method to study the pyrolysis of toluene and n-dodecane as both single-component systems and mixtures subjected to high temperature and pressure conditions . A series of NVT-MD simulations for a temperature range of 2000–2600 K has been performed to investigate the decomposition of the reactant molecules and to calculate the temperature -dependent rate constants. Two different densities , 0.2 and 0.4 kg/dm3 of the system were used to investigate the effect of a higher density /pressure on the Arrhenius parameters . First-order kinetics was used to calculate the Arrhenius parameters based on the toluene decomposition and corresponding rate constants calculated at different temperatures. In case of mixtures of toluene and n-dodecane, zero-dimensional (time-evolving) continuum -scale pyrolysis simulations for the same tests cases have been performed and the results were then compared to those obtained from ReaxFF -MD .

For individual systems of toluene and n-dodecane, the activation energy and the pre-exponential factor calculated in both high- and low-density cases show good agreement with previous experimental results. For mixtures having relatively lower density (0.2 kg/dm3) of the system , we observed from both of the methods that the overall effect of n-dodecane is to reduce the activation energy of toluene. The two methods showed satisfactory correlation when the n-dodecane concentration is below 20%. Overall, for all the cases the two methods showed good qualitative agreement. In addition , we observed that as the percentage of n-dodecane in the mixture is increased beyond a threshold value (in our case >11.11%), the ReaxFF -MD simulations predict an activation energy even lower than that of n-dodecane in single-component system . This suggests that the pyrolysis of toluene follows a much more complex mechanism in the presence of a greater concentration of n-dodecane. In this case, the simple Arrhenius relation based on first-order kinetics does not hold anymore to accurately predict the activation energy of toluene. A preliminary investigation of the time evolution of major species generated by n-dodecane showed that the pyrolysis of n-dodecane is radical dominated. The large alkyl radical pool generated by the thermal decomposition of n-dodecane are highly reactive and they cause further dissociation of toluene. This played a critical role in low temperatures especially, since at low temperatures the toluene decomposition by itself is very limited.

In the higher density case (0.4 kg/dm3) even small amounts of n-dodecane had much more significant impact on the decomposition of toluene and its activation energy . This was observed in both ReaxFF -MD and continuum simulation results. Beyond a certain mixture ratio, the activation energy did not show any dependence on the number of n-dodecane in the mixture or the overall system density . We hypothesize that a different initiation mechanism is at work in these cases, which needs further studies to be confirmed.

These results demonstrate that although ReaxFF -MD is useful in determining the Arrhenius parameters for single-component systems, in case of mixtures the same methodology only applies in certain cases of densities and mixture ratios. This is also confirmed using the results of continuum -scale pyrolysis simulations. When the overall system density /pressure is high or the concentration of a highly reactive molecule in the mixture is increased, the simple first-order kinetics-based Arrhenius equation does not apply. We need to look further into the complex reaction pathways to determine the effective mixture activation energies in these cases.