Sooting tendencies of diesel fuel component mixtures follow a linear mixing rule

With the growing importance of climate change, soot emissions from engines have been receiving increasing attention since black carbon is the second largest source of global warming. A sooting tendency can be used to quantify the extent of soot formation in a combustion device for a given fuel molecule, and therefore to quantify the soot reduction benefits of alternative fuels. However real fuels are complex mixtures of multiple components. In this work, we have used experimental methods to investigate how the sooting tendency of a blended fuel mixture is related to the sooting tendencies of the individual components. A test matrix was formulated that includes sixteen mixtures of six components that are representative of the main categories of hydrocarbons in diesel (eicosane (ECO) for alkanes, isocetane (ICE) for isoalkanes, butylcyclohexane (BCH) for cycloalkanes, 1-methylnaphthalene (1MN) for aromatics, tetralin for naphthoaromatics, and methyl-decanoate (MDC) for oxygenates). Most of the mixtures contain three to five components. The sooting tendency of each mixture was characterized by yield sooting index (YSI), which is based on the soot yield when a methane/air nonpremixed flame is doped with 1000 ppm of the test fuel. The YSIs were measured experimentally. The results show that the blending behavior is linear, i.e., the YSI of the mixtures is the mole-fraction-weighted average of the component YSIs. Experimental results have shown that the sooting tendency of a fuel mixture can be accurately estimated as the linear combination of the individual components. In addition, mass density of the mixtures is also measured, and a linear blending rule is applied to test whether mixing rules exist for mass density of diesel mixtures in this study. Results also have shown that the mixing rule tested in this study is valid and mass density of a mixture can be accurately estimated from the linear combination of the individual components.


Introduction
The ongoing transition from fossil fuels to biofuels and e-fuels presents a unique opportunity to improve performance and reduce emissions [1,2]. However, the chemical space of potential fuels is enormous, so achieving this goal requires procedures that can accurately predict the properties of fuels before they are synthesized. Since real fuels are typically mixtures, mixing rules are necessary that can estimate the properties of mixtures from the properties of their individual components. This paper tests some simple mixing rules for mass density and sooting tendency of diesel fuel mixtures.
Mass density (ρ) is a fundamental physical property of a fuel that determines its energy content. It is frequently used as a target parameter when formulating fuel surrogates [3,4]. It may also impact soot formation in advanced engine concepts [5]. Mixing rules are more easily formulated for specific volume (ν), which is the inverse of ρ, since ν is a property that increases with more substance. For an ideal solution, the specific volume of the mixture νmix is given by where wj and νj are the mass fraction and the specific volume of component j, and the sum is over all the components in the solution [6]. Equation 1 is readily derived from the assumption that the total volume of the solution is the sum of the volumes of the individual molecules; see Supplemental Information (SI) A. The weighting factor is w since specific volume is a massic quantity (volume/mass); it would be mole fraction x for the corresponding molar quantity molar volume (volume/mole). Equation 1 can be re-written for the mass density of the mixture ρmix as where ρj is the mass density of component j.
Several alternative approaches can be used to estimate ρ for mixtures. A simple engineering mixing rule is where φj is the volume fraction of component j. This rule has been used in several surrogate studies [7,8]. Since ρ is a thermodynamic property, it can also be calculated with equations of state (EoSs) [9]. EoS calculations are more precise and can estimate ρ at variable temperature and pressure, but they also require extensive parameters for each species, which limits the range of molecules that can be considered. For example, the National Institute of Standards and Technology (NIST) REFPROP software supports less than 130 hydrocarbons and is limited to mixtures with 20 or fewer components [10].
Sooting tendency is a laboratory-scale property that indicates how readily a fuel produces soot particles due to its chemical composition [11]. Soot emissions are the second largest source of global warming [12], and they contribute to ambient fine particulates that cause millions of deaths each year [13]. Alternative fuels can greatly ameliorate these problems given that soot formation depends strongly on fuel composition [14]. Sooting tendency is traditionally measured with the ASTM D1322 smoke point test [15], which measures the height of a wickburner flame at the point when soot first breaks through the tip. More recently, we have developed an alternative definition based on the soot yield in fueldoped flames, which we call yield sooting index (YSI) [16]. YSI requires orders of magnitude less sample volume than smoke point (100 μL versus 10 mL), which makes it useful for developing new fuel molecules. We have applied it to custom-synthesized alternative fuels such as dioxolanes and polyoxymethylene ethers where the available sample volumes were insufficient to measure smoke point [17,18].
Sooting tendency is a derived combustion property, not a fundamental physical property like ρ, so mixing rules for it are necessarily empirical. Several studies have found good agreement between linear mixing rules and measured smoke points [e.g., 19,20]. Therefore, we propose that the yield-based sooting tendency of a mixture YSImix is given by where xj and YSIj are the mole fraction and the sooting tendency of component j. Mole fraction is used as the weighting factor since the YSI protocol adds a fixed mole fraction to the base flame. We have used Eq. (4) previously [21], but this is the first study to systematically evaluate it.
In this study the linear mixing rules were tested for mixtures representative of diesel fuels. First, we defined a set of mixtures containing various proportions of six components that represent the main chemical families in diesel fuel. The mixture matrix followed a formal experimental design to ensure that it sampled the full chemical space possible with six components. The ρ of these mixtures were measured, then the results were fit to Eq. (2). Similarly, the YSIs of the mixtures were measured, then fit to Eq. (4).

Mixture Design
In this study, we prepared sixteen mixtures of six components that are representative of the main categories of hydrocarbons in diesel fuels: eicosane (ECO) for alkanes, isocetane (ICE) for isoalkanes, butylcyclohexane (BCH) for cycloalkanes, 1methylnaphthalene (1MN) for aromatics, tetralin (TET) for naphthoaromatics, and methyldecanoate (MDC) for oxygenates. Fig. 1 shows the chemical structures of these components. The first five components are pure hydrocarbons that were selected from the surrogate fuel palette defined in Ref. [3]. The last component (MDC) is an oxygenate that represents the methyl esters found in biodiesel fuels [31]. MDC was selected because it has a diesel-like carbon number (C11), and detailed chemical kinetic mechanisms are available for it [32].
The compositions of the mixtures followed a modified 2IV 6−2 fractional factorial experimental design [29] to ensure that the experiments sampled the full chemical space possible with six components, and that they systematically tested all possible non-linear interactions between the components. To illustrate how this works, Fig. 2 shows the mole fractions (x) of each component in each mixture. (This figure is intended to illustrate the structure of the mixture matrix; SI B provides more legible tables of the mixture composition.) Cells where x ≠ 0 are highlighted in blue. If we consider the columns for TET and MDC, the first tetrad of mixtures (numbers 1 to 4) contains neither of these components, the second tetrad (5 to 8) contains TET but not MDC, the third tetrad (9 to 12) contains MDC but not TET, and the fourth tetrad (13 to 16) contains both components. Thus, the difference between the second and first tetrads indicates the effect of TET in the absence of MDC, the difference between the third and first tetrads indicates the effect of MDC in the absence of TET, and the difference between the fourth and first tetrads indicates the effect of both components together; if this latter effect equals the sum of the first two, then there is no nonlinear interaction between TET and MDC. The experimental design guarantees that this same structure of tetrads applies to all other pairs of components.

Mixture Preparation
SI C lists the sources and purities of the mixture components.
The mixtures were prepared gravimetrically by adding each component to 2 mL vials with septa in the caps. ECO is a solid at room temperature, but it dissolved readily in the other components. The mass of each component was measured with a Mettler Toledo NewClassic MF analytical balance (0.1 mg precision), mass fractions were calculated from these masses, and then mole fractions were calculated from the known molecular mass of each component. SI B lists the final mass fractions and mole fractions. The uncertainties in these values are estimated to be ±0.001.

Density Measurements
Mass density was measured by dispensing 50 μL of each mixture into an empty vial with a syringe (Hamilton Gastight 1705), then determining the mass of the dispensed liquid with the same balance used for mixture preparation. Each density was measured six times and then averaged. The systematic uncertainties are estimated to be 0.005 g/mL based on densities of pure compounds compared to literature values, and the random uncertainties are estimated to be 0.009 g/mL based on the standard deviation of measuring densities six times. Overall, we estimate that the uncertainty in the measured mixture density is ±1%.

YSI Measurements
Sooting tendencies were measured using a yieldbased approach we developed previously [16]. The procedure used in this study is described in [21]. It consisted of three steps: (1) 1000 ppm of MDC, 1MN, and each mixture (mix) were separately doped into the fuel of a methane/air flame; (2) the maximum soot concentration was measured in each flame with lineof-sight spectral radiance (LSSR); and (3) these concentrations were rescaled into a yield sooting index (YSI) defined by: This rescaling removes sources of systematic uncertainty such as errors in the methane and air flowrates. Furthermore, it allows the new results to be quantitatively compared with a database that contains measured YSIs for hundreds of organic compounds [22]. We typically use n-heptane and toluene as the endpoint species (e.g., [21]), but in this case that would have required a large extrapolation since the YSIs of TET and especially 1MN are much larger than toluene (TET ≈ 400 and 1MN ≈ 650 vs. toluene ≈ 170). Therefore, 1MN and MDC were chosen as upper and lower endpoints specific to this study that were guaranteed to bracket the diesel mixtures. SI E shows a schematic diagram of the LSSR apparatus and SI F gives details of the specific burner [23,24]. The liquid diesel mixtures were injected into the gas-phase CH4/N2 fuel mixture with a syringe pump.
SI G lists the liquid-phase flowrates corresponding to 1000 ppm in the gas-phase for each mixture, and the property values [25] that were used to calculate them. Each mixture was injected for 600 s and the LSSR signal was averaged from 300 to 600 s; SI H shows that the initial 300 s is adequate for the mixture to equilibrate with the walls of the fuel line and burner. The fuel lines were heated to 125 ℃ and above, and the burner was heated to 170 ℃. SI I shows that at these temperatures the vapor pressures [28] of all the components were higher than their partial pressures. SI J shows that the LSSR signals of ECO, the least volatile component, increased linearly with the dopant mole fraction, which experimentally confirms that the components were not condensing in the fuel delivery system. Isooctane was included in each measurement set as an internal standard; SI K shows that the values were consistent over time and agreed with previous measurements [22]. SI L shows that the syringe pump stepper motor performed accurately.
Each YSI was measured three times and then averaged. The systematic uncertainty in YSI is estimated to be ±1%, mainly due to uncertainties in ρ (which is needed to calculate the dopant flowrate). The random uncertainties were estimated to be ±3% based on the standard deviation of the individual measurements. Overall, we estimate that the uncertainty in the measured mixture YSI is ±4%.

Mass Density
SI M lists the mass densities ρ measured for each of the 16 diesel-like mixtures, and the specific volumes ν (= 1/ρ) calculated from them. The first objective of this study was to determine whether the linear mixing rules, Eqs. (1) and (2), can fit this data.
Eq (1) can be written in matrix form as where νmix is a 16 × 1 column vector that contains the measured specific volumes for each mixture, w is a 16 × 6 matrix that contains the mass fraction of each component in each mixture, and νcomp is a 6 × 1 column vector that contains the specific volume of each component. We can regard νmix as a known from the measurements (see SI M), w as a known from the mixture preparation (see SI B), and νcomp as an unknown. Eq. (6) does not have an exact solution since there are more mixtures than components and the νmix contain random measurement errors, but multiple linear least-squares regression can be used to find the νcomp that most closely solves it. We refer to the optimized component values in this solution as νopt. This analysis was performed, and the results are listed in Table 1 (as densities).   (1), and the ρopt (horizontal axis). The agreement is very good, which shows that the functional form of the linear blending rule can fit the measured data. Table 1 also lists mass densities ρlit for the components from the literature ( [26] - [28]). Fig. 4 plots the ρopt versus the ρlit. For the components that are liquids at room temperature (ICE, BCH, 1MN, TET, and MDC) the agreement is very good, which further validates the linear blending rule. On the other hand, the agreement is poor for ECO, which is a solid at room temperature. This shows that the solid phase ρ is not an appropriate value for predicting the ρ for a solid dissolved in a liquid. However, Fig 4 also shows a value of ρlit calculated for ECO using the group contribution method (GCM) of Mathieu and Bouteloup [25], and in this case the agreement with the ρopt is good. The GCM does not explicitly apply to a specific phase, but since it is extrapolating data measured for liquids to ECO, it generates predictions that appear to be appropriate for solids dissolved into liquids.

Sooting Tendency (YSI)
SI N lists the YSIs measured for each of the 16 diesel-like mixtures. The second objective of this study was to determine whether the linear mixing rule, Eq. (4), can fit this data.
Eq. (4) can be written in matrix form as = where YSImix is a 16 × 1 column vector that contains the measured YSI for each mixture, x is a 16 × 6 matrix that contains the actual mole fraction of each component in each mixture, and YSIcomp is a 6 × 1 column vector that contains the YSI of each component. Again, we regard YSImix as a known from the measurements (see SI N), x as a known from the mixture preparation (see SI B), and YSIcomp as an unknown. The YSIopt obtained by solving Eq. (7) with linear regression are listed in Table 2.    (vertical axis) to the predicted YSImix using the linear blending rule, Eq. (4), and the YSIopt (horizontal axis). The agreement is very good, which shows that the functional form of the linear blending rule can fit the measured data. Table 2 also lists the sooting tendencies YSIGCM predicted for the components with the GCM from [30] (see SI D). Fig. 6 plots the YSIopt versus the YSIGCM. The agreement is reasonable for all components, which further validates the linear blending rule.

Conclusions
In this study we prepared sixteen mixtures that contain six components representative of diesel fuels. The mixture compositions followed a formal experimental design to ensure they sampled the full chemical space possible with six components. Mass densities and sooting tendencies were measured for each mixture. Analysis of the results showed that the mass densities of diesel-like mixtures can be accurately estimated with a linear mixing rule for specific volume (the inverse of mass density) weighted by mass fraction. The group contribution method from [25] can be used to estimate appropriate component values for compounds that are solids at room temperature. Similarly, the sooting tendencies of diesel-like mixtures can be accurately estimated with a linear mixing rule weighted by mole fraction.

Supplementary material
Supplemental material is available.