An assessment of surrogate fuel using Bayesian multiple kernel learning model in sight of sooting tendency

Abstract

An integrated and systematic database of sooting tendency with more than 190 kinds of fuels was obtained through a series of experimental investigations. The laser-induced incandescence (LII) method was used to acquire the 2D distribution of soot volume fraction, and an apparatus-independent yield sooting index (YSI) was experimentally obtained. Based on the database, a novel predicting model of YSI values for surrogate fuels was proposed with the application of a machine learning method, named the Bayesian multiple kernel learning (BMKL) model. A high correlation coefficient (0.986) between measured YSIs and predicted values with the BMKL model was obtained, indicating that the BMKL model had a reliable and accurate predictive capacity for YSI values of surrogate fuels. The BMKL model provides an accurate and low-cost approach to assess surrogate performances of diesel, jet fuel, and biodiesel in terms of sooting tendency. Particularly, this model is one of the first attempts to predict the sooting tendencies of surrogate fuels that concurrently contain hydrocarbon and oxygenated components and shows a satisfying matching level. During surrogate formulation, the BMKL model can be used to shrink the surrogate candidate list in terms of sooting tendency and ensure the optimal surrogate has a satisfying matching level of soot behaviors. Due to the high accuracy and resolution of YSI prediction, the BMKL model is also capable of providing distinguishing information of sooting tendency for surrogate design.

Graphical abstract

Keywords

sooting tendency / yield sooting index / Bayesian multiple kernel learning / surrogate assessment / surrogate formulation

1 Introduction

Soot emissions from commercial fuels in practical combustion devices, such as internal combustion engines, gas turbines, and furnaces, are proved to result in seriously adverse effects on both human health and environment [1–4]. Besides, soot particles can also bring risks to combustion systems. For example, soot formation of jet fuels in gas turbines leads to obvious radiation, which can heat and stress the combustor liner [5]. Moreover, soot may be one of the bottlenecks for the improvement of combustion strategies of automotive fuels, because some advanced combustion strategies (i.e., gasoline direct injection) tend to increase soot emissions while contributing to increasing efficiency [6]. According to Refs. [1,7,8], soot is known to be formed through several key steps such as fuel pyrolysis-oxidation, polycyclic aromatic hydrocarbons (PAHs) formation and growth, nucleation, surface growth, coagulation, oxidation etc., which are closely related to fuel properties and especially fuel composition. As a result, sooting tendency is regarded as one of the most important features of both conventional and alternative fuels and should get more attention and deeper understanding.

To get insight into the chemical processes (e.g., soot formation) during fuel combustion and provide theoretical support for advanced combustion strategies, computational simulations need to be performed with reliable kinetic models in fundamental and/or engine levels. However, petroleum-derived fuels have very complex compositions [9], which makes it intractable to build such intricate kinetic models. In addition, some biofuels, for example biodiesel fuels, consist of major components with high boiling points, increasing the difficulty for gas-phase experimental investigations and leading to obstacles for model validation [10]. Moreover, chemical mechanisms are limited for some major components with larger molecules in real fuels. As a result, surrogate fuels are applied to provide a modeling approach to simulate the combustion processes of the target fuel. According to Refs. [9–11], a surrogate fuel is a blend of several pure components and is constructed to match certain properties of the target fuel. The contents and proportions of surrogate fuels are usually designed according to the constraints of these parameters, such as density, viscosity, volatility, surface tension, H/C ratio, molecular weight, adiabatic flame temperature, ignition delay time, and so on.

However, although sooting tendency is one of the most important fuel features, matching sooting behaviors is often ignored in surrogate design. A limited number of works have been performed to explore the use of sooting tendency as an input constraint parameter for surrogate formulation and experimentally validate the matching degree of sooting behaviors of the target fuel and its surrogates. Violi et al. [12] produced a six-component surrogate to match the distillation curve and sooting propensity of a practical JP-8 fuel. This was one of the earliest works that took the sooting propensity into account during surrogate formulation. Based on Ref. [12], Eddings et al. [13] developed two surrogate blends of six pure hydrocarbons (n-octane, n-dodecane, n-hexadecane, xylenes, decalin and tetralin) for Jet-A fuel with the consideration of fuel volatility and sooting index. Smoke point (SP) and threshold soot index (TSI) were used as the indicators to represent sooting tendencies of the target fuel and its surrogates in Ref. [13]. The calculation of TSI was performed according to [14]

(1)\mathrm{T}\mathrm{S}\mathrm{I}=a\left({\displaystyle \frac{{{\displaystyle \mathrm{M}\mathrm{W}}}_f}{\mathrm{S}\mathrm{P}}}\right)+b,

where a and b are constants for any given experimental setup, MW_f is the fuel molecular weight, and SP is the measured smoke height. Focusing only on sooting tendency, Mensch et al. [15] devised a JP-8 surrogate, consisting of 54% iso-octane, 18% n-dodecane, and 28% 1,3,5-trimethylbenzene in molar composition, to match the TSI of the target fuel (22). For verification, the TSI of this surrogate was tested on a standard smoke point lamp and a measured value of 24 was obtained with a deviation of two TSI units from the target value, which was within the testing uncertainty. Moreover, a linear additivity rule for TSI calculation was proved using six binary mixtures with compositions having mole fractions ranging from zero to one. The linear mixing rule of TSI was first proposed in Ref. [16] as

(2){{\displaystyle \mathrm{T}\mathrm{S}\mathrm{I}}}_{\mathrm{m}\mathrm{i}\mathrm{x}}={\displaystyle {\sum }_i{{\displaystyle X}}_i}\times {{\displaystyle \mathrm{T}\mathrm{S}\mathrm{I}}}_i,

where X_i represents the mole fraction of compound i in each mixture and TSI_i is the experimental TSI value of compound i. Dooley et al. [11] designed a 4-components surrogate fuel for Jet-A POSF 4658 with several constraints including sooting tendency (TSI). The designed (or predicted) TSI value of this surrogate was 21.4, which was the same as the target value measured for POSF 4658, while the measured TSI value of the surrogate was 20.4±0.6. The performance of this surrogate in terms of sooting behavior was evaluated in wick-fed laminar diffusion flames where the soot volume fractions of the target fuel and its surrogate were quantified by laser extinction. Recently, Yu et al. [17] proposed two 5-component surrogates for a target jet fuel, with compositions inherently satisfying both physical and chemical characteristics as well as sooting tendency. A skeletal kinetic mechanism was established by considering all the surrogate components, including n-dodecane, iso-cetane, iso-octane, decalin, and toluene. The mechanism validations were performed in terms of species concentrations using the CHEMKIN-PRO software against the experimental data of a real jet fuel. It was found that the concentration of acetylene, which was regarded as one of the key precursors of soot formation, was predicted reasonably well in the validation. However, no direct comparison of sooting behavior was performed in Ref. [17]. Besides the above-mentioned works for jet fuel surrogates, few studies have been performed for diesel surrogate formulation with consideration of sooting tendency, due to the higher experimental difficulty for components of such low volatility in the diesel boiling range in traditional smoke point apparatus [6]. Szymkowicz and Benajes[18] used commercial software Reaction Workbench (Surrogate Blend Optimizer) to develop a Diesel Surrogate Fuel Library containing 18 fuel blends, with different TSI levels to match the diesel fuels of low, medium, and high sooting tendencies (TSI= 17, 31 and 48, respectively). They also provided a useful approach for independent control of the threshold soot index and cetane number for surrogate design: adjusting the volume fractions of cyclo-alkanes and aromatics to control TSI while keeping the sum of their volume fractions near to 0.3; and tuning the volume fractions of n-alkanes and iso-alkanes to control cetane number while ensuring the sum of their volume fractions close to 0.7.

In aforementioned references, TSI was often applied as the indicator of sooting tendency. However, this assessment index may not be suitable for heavily sooting fuels, such as many aromatics, whose smoke heights are too small to be measured precisely. In addition, subjective factors cannot be neglected when the experimenter observes the flame status. Therefore, McEnally et al. [19] proposed an alternative index to assess a wide range of sooting tendencies of different fuels. This index, the yield sooting index (YSI), can be obtained according to

(3)\mathrm{Y}\mathrm{S}\mathrm{I}=A\times {{\displaystyle f}}_{y,\max }+B,

where A and B are apparatus-specific constants, and f_v,max is the peak soot volume fraction along the centerline of a test flame which is doped with a certain mole fraction of the test component. The advantages of this index were described in detail in Refs. [19–21]. Later, Das et al. [6] developed the YSI measurement using a constant mass fraction basis of doping fuels, which was designed for the accurate calculation of flow rates for real fuels whose average molecular weights were difficult to be determined, providing a consistent standard for the comparison of sooting tendencies of target fuels and its surrogates. The YSI values of 15 fuel mixtures, including reference diesel fuels, jet fuels, and recently developed fuel surrogates, were obtained and compared. Two linear methods to predict the YSI of a surrogate fuel were demonstrated based on both the component mixing rule and the carbon-type mixing rule, which are expressed as

(4){{\displaystyle \mathrm{Y}\mathrm{S}\mathrm{I}}}_{\mathrm{m}\mathrm{i}\mathrm{x}}={\displaystyle {\sum }_i{{\displaystyle W}}_i}\times {{\displaystyle \mathrm{Y}\mathrm{S}\mathrm{I}}}_{\mathrm{P}\mathrm{C}},

(5){{\displaystyle \mathrm{Y}\mathrm{S}\mathrm{I}}}_{\mathrm{m}\mathrm{i}\mathrm{x}}={\displaystyle {\sum }_i{{\displaystyle W}}_i}\times ({\displaystyle {\sum }_j{{\displaystyle N}}_j})\times {{\displaystyle C}}_j,

where YSI_mix is the predicted YSI of the fuel mixture, W_i is the mass fraction of compound i, YSI_PC is the experimental YSI of each pure compound, N_j is the number of carbon atoms of type j in compound i, and C_j is the contribution of each carbon type to the YSI.

Based on this review, there are several issues and limitations for the application of sooting tendency in surrogate design. First, although sooting tendencies of pure components were widely obtained and studied, there are limited YSI or TSI data for fuel mixtures, leading to considerable difficulties for formulating and validating surrogate fuels. Second, nowadays, biofuels are of increasing interest due to their renewable sources and reduce emissions. Therefore, they have the potential to be alternatives to petroleum-based fuels [22]. Biofuels always contain lots of oxygenated contents, which show totally different sooting tendencies compared with hydrocarbon fuels. However, the previous works mentioned above mainly focused on the surrogate formulation of petroleum-based fuels (diesel and jet fuels) with the consideration of sooting tendency. Neither measured nor predicted YSI or TSI values were studied for oxygenated components or mixtures. Finally, a nonlinear relationship between soot production and fuel ratio was found experimentally in liquid fuel blends, such as n-heptane/toluene and iso-octane/toluene blends, and was proved numerically to be caused by the chemical kinetic interactions between the two components in regard to PAH and soot formation [23,24]. Obviously, the linear mixing rules of sooting indices in previous works cannot capture these interactions or possible synergetic effects, which may exist between certain fuels.

To address these concerns, many efforts have been made in this study. To begin with, an integrated YSI database for fuel mixtures with a wide range of sooting tendencies has been obtained based on a large number of experimental investigations for both hydrocarbon fuels and oxygenated blends. This database can provide systematic information of sooting tendencies for developing and validating surrogate mixtures for real fuels and support YSI as one of the important target indices or constraints for surrogate design. Next, a reliable and accurate predicting model of YSI values for fuel surrogate is proposed with the application of a machine learning method, which was developed based on the integrated YSI database. The performance of this model and that of the traditional linear additivity method are tested and compared. Finally, the application of this novel YSI predicting model is employed to evaluate and compare the sooting tendecies of surrogate candidates and their target fuels. Surrogate evaluation in terms of sooting tendency is conducted for diesel, jet fuel, and biodiesel in an efficient and low-cost way, and some suggestions for surrogate formulation are proposed.

2 Experimental setup

2.1 Test fuels

More than 190 kinds of fuels were tested to build an integrated YSI database, which can be divided into four categories.

Category 1: Fifty-eight surrogate mixtures which were once proposed and studied in the previous works were classified into Category 1. The target fuels of these existing surrogates include diesel, jet fuels, biodiesel and even some practical blends such as B30 fuels (diesel-biodiesel mixture containing 30% biodiesel by liquid volume). This category consists of binary surrogates and multi-component surrogates with component numbers ranging from 3 to 8. This category is measured not only for evaluating the sooting tendencies of existing surrogates, but also for combining the YSI values of both hydrocarbon and oxygenated fuels into one database. Considering the wide range of component types in the mixtures, the measurement for these surrogates can provide information for the effects of compositional complexity on sooting tendency.

Category 2: Ninety-two 4-component surrogates that were designed systematically based on the method proposed in Ref. [18] with four typical types of components-n-alkanes, iso-alkanes, cyclo-alkanes and aromatics were classified into Category 2. The 4 compositions had mole fractions ranging from 0 to 0.5 at a constant interval, while keeping the sum of the concentration of n-alkanes and iso-alkanes (or cyclo-alkanes and aromatics) constant with the value of 0.5. To show the blending rules in this category, the series of mixtures are listed in Table 1 as an example. For each group of species, several components were selected. For instance, toluene, ethylbenzene, m-xylene, 1,3,5-trimethylbenzene, n-propylbenzene, n-butylbenzene, and 1-methylnaphthalene were respectively chosen to represent the aromatic contents in the surrogate mixtures. The data in this category mainly concentrate on the variation of sooting tendency with systematically changing concentrations of certain components, to ensure that the proposed model capture the right YSI trends.

Category 3: Six kinds of real fuels that were available for laboratorial use in China were classified into Category 3. This category consists of commercial diesel fuels, jet fuels, and biodiesel fuels. The objectives for the test of these real fuels are to provide target values for the YSI comparison of the corresponding surrogates in Category 1 and give assessment on the matching degree in terms of sooting tendency.

Category 4: Thirty-three kinds of pure components that were included in the surrogate blends from Category 1 and 2 were classified into Category 4. These pure components constitute a complete surrogate palette, from which the sub-components can be selected to match certain properties of target fuels in the surrogate formulation process.

2.2 Burner and flames

The experimental setup is shown in Fig. 1, whose detailed information can be found in Ref. [25]. A laminar co-flow diffusion flame burner was used in this work at atmospheric pressure. The burner consists of two concentric stainless-steel tubes. The central tube (inner diameter= 10.5 mm) was used for transportation of methane (flowrate= 0.24 L/min) and a test fuel. The mass fraction of each test fuel was maintained at 1.5%. This type of doping basis was also applied by Das et al. [6] and Gao et al. [25]. The outer tube (inner diameter= 96.8 mm) was used for compressed air (flowrate= 150 L/min) as oxidizing and shielding gas.

A syringe pump (LSP01-1BH, Baoding Longer Precision Pump Co., Ltd.) was used to inject test fuels into the methane flow. A tee-junction with a sealing plug was designed to fix the syringe needle and ensure its tip to be on the central line of the fuel transfer tube. The sealing plug can be used in the temperature range from 273 K to 573 K. In the experiment, heating modules were used to heat the fuel transfer tube and the burner to 503 K and 623 K respectively. To ensure that the liquid dopants can be vaporized immediately upon injection and swept in gas phase to the burner by the CH₄ stream, the vapor pressure of decylbenzene, the least volatile component of all test fuels, was calculated based on the method used in Ref. [26]. Under the thermal condition consistent with the experiment, the vapor pressure of decylbenzene is maintained at least 180 times greater than its partial pressure in the fuel blend, proving a sufficient vaporization process of the dopants in CH₄ stream. In addition, simulations by the CHEMKIN PRO plug flow reactor model were performed to check the possibility of fuel decomposition in the heated lines (503 K) using the same fuel compositions and flowrates as those in the experiments. It was indicated that the decomposition of test dopants could be ignored under the condition inside the fuel tube.

2.3 Laser-induced incandescence (LII) measurement

Two-dimensional soot volume fraction information of test flames was obtained by using the LII method, which was applied and described in detail in Ref. [27]. The laser source was a Nd: YAG laser (QSmart850, Quantel) with a wavelength of 1064 nm and a pulse repetition rate of 10 Hz. An attenuator was arranged at laser exit to adjust the pulse energy continuously. The laser beam was shaped into a laser sheet with the thickness of 0.6 mm by a series of beam forming optics and a slot. This laser sheet passed through the flame centerline and heated local soot particles to a temperature far higher than the surrounding gases. The intensity of induced incandescence signal was acquired by an intensified CCD cameras (ICCD) (PI MAX-4:1024i). An ultraviolet lens (Nikon PF10545MF-UV) was used cooperatively with the ICCD. Moreover, a band filter (Thorlabs FB400-40) with a central wavelength of (400±8) nm and a FWHM of (40±8) nm was fixed in front of the ultraviolet lens to avoid PAH fluorescence and C2 radiation [28]. According to Ref. [27], a laser fluence of 0.24 J/cm² was selected to ensure that the LII signal was hardly related to laser fluence. A gate width of 50 ns and an intensified rate of 80 were used. The intensities of the LII signal could be transformed to soot volume fractions (f_v) via a quantitative calibration method [28]. Absolute soot volume fractions of four calibration heights (25 mm, 30 mm, 35 mm, and 40 mm) of the same test flame as Case B in Ref. [28] were measured using a laser cavity extinction technique [28]. Compared with the LII signal of the above-mentioned flame, an averaged calibration constant (K_LII) could be obtained. Detailed information of the calibration procedure can be found in Refs. [25,27]. For each type of test dopant, 100 LII images were acquired and averaged to enhance the signal-to-noise ratio.

3 YSI measurement and prediction

3.1 Experimental measurement of YSI

It is necessary to quantify sooting tendencies of different fuels using suitable indices. Based on the method proposed by Ref. [19], an apparatus-independent yield sooting index (YSI) was developed and used as a fuel characteristic in terms of sooting tendency.

As described in Ref. [25], the average soot volume fractions in a “peak region” were transformed to YSI. The “peak region” was defined as a region where the soot concentrations were in the top 5% range among all the pixels inside the flame outline. The YSI for any test fuel i is defined as

(6){{\displaystyle \mathrm{Y}\mathrm{S}\mathrm{I}}}_{\mathrm{t}}=({{\displaystyle \mathrm{Y}\mathrm{S}\mathrm{I}}}_{\mathrm{B}}-{{\displaystyle \mathrm{Y}\mathrm{S}\mathrm{I}}}_{\mathrm{A}})\times {\displaystyle \frac{{{\displaystyle f}}_{\mathrm{v},\max \mathrm{},i}-{{\displaystyle f}}_{\mathrm{v},\max \mathrm{},\mathrm{A}}}{{{\displaystyle f}}_{\mathrm{v},\max \mathrm{},\mathrm{B}}-{{\displaystyle f}}_{\mathrm{v},\max \mathrm{},\mathrm{A}}}}+{{\displaystyle \mathrm{Y}\mathrm{S}\mathrm{I}}}_{\mathrm{A}},

where A and B indicate n-heptane and tetralin with constant YSI values (YSI_A≡ 10 and YSI_B≡ 100). The average f_v in the “peak region” (f_v,max) for n-heptane and tetralin were also measured.

3.2 YSI prediction model

3.2.1 Data set preparation

The YSIs of 58 surrogate mixtures that were studied (Category 1) and 92 4-component surrogates that were designed systematically (Category 2) were combined as the data set for the YSI predicting model. The YSI values of all the samples in this data set ranged from 4.8 to 161.9. It can be seen from Fig. 2 that more than 90% of these YSIs in the data set were clustered at low values from 0 to 80, while the number of data points at larger values was markedly limited. It is worth mentioning that all the surrogate samples with YSI values higher than 80 are fuel mixtures containing methylnaphthalene. Generally, the unbalanced data structure may pose challenges to accurate prediction in the YSI range with limited samples.

As described in Section 2.1, there are 32 pure components in the surrogate palette. Therefore, for each surrogate sample in this database, its composition can be described uniquely using a 32-dimensional vector. The element position represents the component type, while the element value indicates its proportion in the fuel mixture. In this study, these 32-dimensional vectors are built and used as the inputs of the prediction model.

3.2.2 Model description

Kernel-based methods have been widely applied in statistical problems (e.g., regression and classification) [29,30]. In Ref. [21], a kernel-based method, named kernel ridge regression, was first applied to predict the sooting tendencies of a series of oxygenated fuels and showed a nice predictive performance. Based on recent extension works of Bayesian framework and the multiple kernel learning technique [31], a Bayesian multiple kernel learning model (BMKL) is used in this study. Similar to other kernel-based methods, the BMKL model can deal with nonlinear relationships in the original space by mapping data points to the feature space through a kernel function and learning a linear decision function in feature space. Compared with KRR the model in Ref. [21], the main improvement of the BMKL model is that the YSI contribution of each component can be extracted from the model operating results, which provides useful information to understand the individual and interactive effects of components on sooting tendencies. The detail of the BMKL algorithm is described as follows.

Suppose there are N independent and identically distributed training samples denoted by \{{{\displaystyle x}}_i\in {\mathrm{ℝ}}^D：I=\mathrm{1},\mathrm{...},N\}. The kernel-based decision function for a test sample x_* to predict its target output is

(7)f({{\displaystyle x}}_{*})={\mathit{a}}^{\mathrm{T}}{{\displaystyle \mathit{k}}}_{*}+e,

where the measure vector {{\displaystyle \mathit{k}}}_{*}={[k({{\displaystyle x}}_{\mathrm{1}},{{\displaystyle x}}_{*}),\mathrm{...},k({{\displaystyle x}}_N,{{\displaystyle x}}_{*})]}^{\mathrm{T}} and the kernel function k:\chi \times \chi \to ℝ computes a similarity measure determined by the chosen kernel between two samples. The N×1 vector a is the sample weight parameter vector and e is the bias. The model parameters are regarded as random variables to get a Bayesian interpretation of the model.

Based on the model structure in Ref. [32] and combining the requirement of interpretability, the similarity measure of each dimension is calculated separately and the dimension weight parameters {{\displaystyle \{{{\displaystyle b}}_d\}}}_{d=\mathrm{1}}^D is applied to each dimension in the multiple kernel model.

(8)f({{\displaystyle x}}_{*})={\mathit{a}}^{\mathrm{T}}{\displaystyle {\sum }_{d=\mathrm{1}}^D{{\displaystyle b}}_d}{{\displaystyle \mathit{k}}}_{d*}+e={\displaystyle {\sum }_{d=\mathrm{1}}^D{{\displaystyle b}}_d}({\mathit{a}}^{\mathrm{T}}{{\displaystyle \mathit{k}}}_{d*})+e,

where {{\displaystyle \mathit{k}}}_{d*}={[k({{\displaystyle \mathit{x}}}_{\mathrm{1}d},{{\displaystyle \mathit{x}}}_{*d}),\mathrm{...},k({{\displaystyle \mathit{x}}}_{Nd},{{\displaystyle \mathit{x}}}_{*d})]}^{\mathrm{T}}. Since the model is fully conjugate probabilistic, some assumptions should be made on the distribution. Before that, some notations are introduced. The N×N kernel matrix for the dth dimension is denoted by K_d. The D×N matrix of the intermediate outputs is denoted by G. The precision prior parameters in the model are represented by \mathrm{\Xi }=\{\mathit{\lambda },v,\mathit{\omega },\gamma ,\in \}, where the first four parameters are the precision priors of the variables \mathit{\Theta }=\{\mathit{a},\mathit{G},\mathit{b},e\} consecutively and the last prior is the precision prior for the target output y. The hyper-parameters are denoted by\varsigma =\{{{\displaystyle \alpha }}_{\lambda },{{\displaystyle \beta }}_{\lambda },{{\displaystyle \alpha }}_v,{{\displaystyle \beta }}_v,{{\displaystyle \alpha }}_{\omega },{{\displaystyle \beta }}_{\omega },{{\displaystyle \alpha }}_{\gamma },{{\displaystyle \beta }}_{\gamma },{{\displaystyle \alpha }}_{\in },{{\displaystyle \beta }}_{\in }\}. The distributional assumptions are introduced as follows: note that i=1,...,N, d=1,...,D, for parameters in\mathrm{\Xi },\mathit{\Theta } and the target output y,

(9){{\displaystyle \lambda }}_i\sim Gamma({{\displaystyle \lambda }}_i;{{\displaystyle \alpha }}_{\lambda },{{\displaystyle \beta }}_{\lambda }),

(10){{\displaystyle a}}_i|{{\displaystyle \lambda }}_i\sim N({{\displaystyle a}}_i;\mathrm{0},{{\displaystyle \lambda }}_i^{-\mathrm{1}}),

(11)v\sim Gamma(v;{{\displaystyle \alpha }}_v,{{\displaystyle \beta }}_v),

(12){{\displaystyle \mathit{G}}}_{di}|\mathit{a},{{\displaystyle ({{\displaystyle \mathit{K}}}_d)}}_{.i},v\sim N({{\displaystyle G}}_{di};{\mathit{a}}^{\mathrm{T}}{{\displaystyle ({{\displaystyle \mathit{K}}}_d)}}_{.i},v^{-\mathrm{1}}),

(13)\gamma \sim Gamma(\gamma ;{{\displaystyle \alpha }}_{\gamma },{{\displaystyle \beta }}_{\gamma }),

(14)e|\gamma \sim N(e;\mathrm{0},{\gamma }^{-\mathrm{1}}),

(15){{\displaystyle \omega }}_d\sim Gamma({{\displaystyle \omega }}_d;{{\displaystyle \alpha }}_{\omega },{{\displaystyle \beta }}_{\omega }),

(16){{\displaystyle b}}_d|{{\displaystyle \omega }}_d\sim N({{\displaystyle b}}_d;\mathrm{0},{{\displaystyle \omega }}_d^{-\mathrm{1}}),

(17)\epsilon \sim Gamma(\epsilon ;{{\displaystyle \alpha }}_{\epsilon },{{\displaystyle \beta }}_{\epsilon }),

(18){{\displaystyle y}}_i|e,\mathit{b},{{\displaystyle \mathit{G}}}_{.i}\epsilon \sim N({{\displaystyle y}}_i;{\mathit{b}}^{\mathrm{T}}{{\displaystyle \mathit{G}}}_{.i}+e,{\epsilon }^{-\mathrm{1}}),

where Gamma(.;\alpha ,\beta ) represents the gamma distribution with the shape parameter α and the scale parameter β, N(.;\mu ,\sum ) denotes the normal distribution with the mean vector μ and the covariance matrix Σ, and notations {{\displaystyle ({{\displaystyle K}}_d)}}_{.i} and G_.i represent the ith columns of the matrix K_d and G respectively.

For the parameter estimation problem, the variational approximation which has a high computational efficiency is applied [33]. The posterior p(\mathit{\Theta },\mathit{\Xi }|\mathit{y},{{\displaystyle \{{{\displaystyle \mathit{K}}}_d\}}}_{d=\mathrm{1}}^D) has the factorable approximation q(\mathit{\Theta },\mathit{\Xi })=q(\mathit{\lambda })q(\mathit{\alpha })q(\mathit{v})q(\mathit{G})q(\gamma )q(\mathit{\omega })q(e,\mathit{b})q(\epsilon ). Through the calculation of variations, each factor of q(\mathit{\Theta },\mathit{\Xi }) can be found to have the form of

(19)q(·)\propto \exp ({{\displaystyle E}}_{q((\mathit{\Theta },\mathit{\Xi })\backslash .)}[\log p(\mathit{y},\mathit{\Theta },\mathit{\Xi }|{{\displaystyle \{{{\displaystyle \mathit{K}}}_d\}}}_{d=\mathrm{1}}^D)]).

Because of the characteristics of conjugacy, the approximate posterior distribution of each factor is in the same probability distribution family as its prior probability distribution [34]. The approximate posterior distributions of the precision priors \mathit{\Xi } are given in Eq. (22).

(20)q(\lambda )={\displaystyle {\prod }_{i=\mathrm{1}}^{N}Gamma}\left({{\displaystyle \lambda }}_i;{{\displaystyle \alpha }}_{\lambda }+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}},{{\displaystyle \left({\displaystyle \frac{\mathrm{1}}{{{\displaystyle \beta }}_{\lambda }}}+{\displaystyle \frac{\widetilde{{{\displaystyle \alpha }}_i^2}}{\mathrm{2}}}\right)}}^{-\mathrm{1}}\right),

,

(21)q(v)=Gamma\left(v;{{\displaystyle \alpha }}_v+{\displaystyle \frac{DN}{\mathrm{2}}},{{\displaystyle \left({\displaystyle \frac{\mathrm{1}}{{{\displaystyle \beta }}_v}}+{\displaystyle \frac{{\displaystyle {\sum }_{d=\mathrm{1}}^D{\displaystyle {\sum }_{i=\mathrm{1}}^N\widetilde{{{\displaystyle r}}_{dt}^2}}}}{\mathrm{2}}}\right)}}^{-\mathrm{1}}\right),

(22)q(\gamma )=Gamma\left(\gamma ;{{\displaystyle \alpha }}_{\gamma }+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}},{{\displaystyle \left({\displaystyle \frac{\mathrm{1}}{{{\displaystyle \beta }}_{\gamma }}}+{\displaystyle \frac{\widetilde{e^2}}{\mathrm{2}}}\right)}}^{-\mathrm{1}}\right),

(23)q(\omega )={\displaystyle {\prod }_{d=\mathrm{1}}^{D}Gamma}\left({{\displaystyle \omega }}_d;{{\displaystyle \alpha }}_{\omega }+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}},{{\displaystyle \left({\displaystyle \frac{\mathrm{1}}{{{\displaystyle \beta }}_{\omega }}}+{\displaystyle \frac{\widetilde{{{\displaystyle b}}_d^2}}{\mathrm{2}}}\right)}}^{-\mathrm{1}}\right),

(24)q(\epsilon )=Gamma\left(\epsilon ;{{\displaystyle \alpha }}_{\epsilon }+{\displaystyle \frac{N}{\mathrm{2}}},{{\displaystyle \left({\displaystyle \frac{\mathrm{1}}{{{\displaystyle \beta }}_{\epsilon }}}+{\displaystyle \frac{{\displaystyle {\sum }_{i=\mathrm{1}}^N\widetilde{{{\displaystyle s}}_i^2}}}{\mathrm{2}}}\right)}}^{-\mathrm{1}}\right),

where the notation \widetilde{(·)} represents the posterior expectation, r_di=G_di-a^T(K_d)_.i and s_i=y_i-b^TG_.i-e. The approximate posterior distributions of the parameters \Theta are given in Lemma 1.

Lemma 1: The approximate posterior distribution of the sample weight parameter matrix a can be expressed as

(25)q(\mathit{\alpha })=N\left(\mathit{a};\sum (\mathit{a})\left(\tilde{v}{\displaystyle {\sum }_{d=\mathrm{1}}^D{{\displaystyle \mathit{K}}}_d\widetilde{{{\displaystyle G}}_d^{\mathrm{T}}}}\right),{{\displaystyle \left(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\tilde{\lambda })+\tilde{v}{\displaystyle {\sum }_{d=\mathrm{1}}^D{{\displaystyle \mathit{K}}}_d{{\displaystyle \mathit{K}}}_d^{\mathrm{T}}}\right)}}^{-\mathrm{1}}\right).

The approximate posterior distribution of the intermediate outputs G is

(26)q(\mathit{G})={\displaystyle {\prod }_{i=\mathrm{1}}^{N}N}\left({{\displaystyle \mathit{G}}}_{.i};\sum ({{\displaystyle \mathit{G}}}_{.i})\left(\tilde{v}\left[\begin{array}{l}{{\displaystyle ({{\displaystyle \mathit{K}}}_{\mathrm{1}})}}_{i.}\\ ⋮\\ {{\displaystyle ({{\displaystyle \mathit{K}}}_D)}}_{i.}\end{array}\right]\tilde{\mathit{a}}+\tilde{\epsilon }\left({{\displaystyle y}}_i\tilde{\mathit{b}}-\widetilde{e\mathit{b}}\right),{{\displaystyle \left(\tilde{v}\mathit{I}+\tilde{\epsilon }\widetilde{{\mathit{b}\mathit{b}}^{\mathrm{T}}}\right)}}^{-\mathrm{1}}\right)\right).

The approximate posterior distribution of the bias e and the dimension weight parameter vector b is expressed as

(27)q(e,\mathit{b})=N\left(\left[\begin{array}{l}e\\ \mathit{b}\end{array}\right];{\displaystyle \sum (e,\mathit{b})\left[\begin{array}{l}\tilde{\epsilon }{\mathrm{1}}^{\mathrm{T}}\mathit{y}\\ \tilde{\epsilon }\tilde{\mathit{G}}\mathit{y}\end{array}\right],{{\displaystyle \left[\begin{array}{l}\tilde{\gamma }+\tilde{\epsilon }N\\ \tilde{\epsilon }\tilde{\mathit{G}}\mathit{1}\end{array}\begin{array}{l}\tilde{\epsilon }{\mathbf{1}}^{\mathrm{T}}\widetilde{{\mathit{G}}^{\mathrm{T}}}\\ \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\tilde{\mathit{\omega }})+\tilde{\epsilon }\widetilde{{\mathit{G}\mathit{G}}^{\mathrm{T}}}\end{array}\right]}}^{-\mathrm{1}}}\right),

where 1 is the all-one vector {(\mathrm{1},1,\mathrm{...},\mathrm{1})}^{\mathrm{T}}\in {\mathrm{ℝ}}^N and the dimension weights are allowed to take negative values.

After estimating the parameters through the training set, the predictive distribution of the intermediate outputs G_* for a new data point x_* can be obtained by replacing the posterior distributions p(\mathit{a}|\mathit{y},{{\displaystyle \{{{\displaystyle \mathit{K}}}_d\}}}_{d=\mathrm{1}}^D) and p(v|\mathit{y},{{\displaystyle \{{{\displaystyle \mathit{K}}}_d\}}}_{d=\mathrm{1}}^D) with their approximate distributions q(a) and q(ν) respectively:

p({{\displaystyle \mathit{G}}}_{*}|\mathit{y},{{\displaystyle \{{{\displaystyle \mathit{k}}}_{d*},{{\displaystyle \mathit{K}}}_d\}}}_{d=\mathrm{1}}^D)={\displaystyle {\prod }_{d=\mathrm{1}}^{D}N}\left({{\displaystyle G}}_{d*};\mathit{\mu }{(\mathit{a})}^{\mathrm{T}}{{\displaystyle \mathit{k}}}_{d*},{\displaystyle \frac{\mathrm{1}}{\tilde{v}}}+{{\displaystyle \mathit{k}}}_{d*}^{\mathrm{T}}{\displaystyle \sum (\mathit{a}){{\displaystyle \mathit{k}}}_{d*}}\right)

, where k_d* is the corresponding kernel vector. Besides, the predictive distribution of the target output y_* can be obtained by replacing the posterior distributions p(e,\mathit{b}|\mathit{y},{{\displaystyle \{{{\displaystyle \mathit{K}}}_d\}}}_{d=\mathrm{1}}^D) and p(\epsilon |\mathit{y},{{\displaystyle \{{{\displaystyle \mathit{K}}}_d\}}}_{d=\mathrm{1}}^D) with their approximate distributions q(e,b) and q(\epsilon ) as displayed in Lemma 2.

Lemma 2: The predictive distribution of the target output of a new data point x_* is

(28)p({{\displaystyle y}}_{*}|\mathit{y},{{\displaystyle \{{{\displaystyle \mathit{K}}}_d\}}}_{d=\mathrm{1}}^D,{{\displaystyle \mathit{G}}}_{*})=N\left({{\displaystyle y}}_{*};\mathit{\mu }{(e,\mathit{b})}^{\mathrm{T}}\left[\begin{array}{l}\mathrm{1}\hfill \\ {{\displaystyle \mathit{G}}}_{*}\hfill \end{array}\right],{\displaystyle \frac{\mathrm{1}}{\tilde{\epsilon }}}+[\mathrm{1}{{\displaystyle \mathit{G}}}_{*}]{\displaystyle \sum (e,\mathit{b})}\left[\begin{array}{l}\mathrm{1}\hfill \\ {{\displaystyle \mathit{G}}}_{*}\hfill \end{array}\right]\right).

In fact, the convergence can be checked by monitoring the lower bound obtained by Jensen’s inequality.

(29)\log p(\mathit{y}|{{\displaystyle \{{{\displaystyle \mathit{K}}}_d\}}}_{d=\mathrm{1}}^D)\ge {{\displaystyle E}}_{q(\mathit{\Theta },\mathit{\Xi })}[\log p(\mathit{y},\mathit{\Theta },\mathit{\Xi })|{{\displaystyle \{{{\displaystyle \mathit{K}}}_d\}}}_{d=\mathrm{1}}^D]-{{\displaystyle E}}_{q(\mathit{\Theta },\mathit{\Xi })}[\log q(\mathit{\Theta },\mathit{\Xi })].

The flowchart of this model is displayed in Fig. 3. The matrix of compositional data for all the fuel mixtures was imported into the BMKL model through kernel function. The correlation between measured YSI and compositional data could be found through the learning process. Model testing was performed to select the optimal model parameters and ensure a good predictive capacity based on specific database.

4 Results and discussion

4.1 Verification of yield sooting index (YSI)

To verify the measured YSI, a comparison of the experimental YSI values of all common pure component samples (including n-alkanes, iso-alkanes, cycloalkanes and aromatics) in both studies in Ref. [6] and the present study were conducted. As shown in Fig. 4(a), the YSIs measured in the present study correlate extremely well with that measured in Ref. [6] in a wide sooting range. The correlation coefficient (R²) is 0.994, which indicates a convictive verification for the YSI measurement in the present study. It should be mentioned that the relative scale of the horizontal and vertical axes is different. The reason for this is that the YSIs in Ref. [6] have been scaled such that hexane ≡ 0 and benzene ≡ 100, whereas in the present study they are scaled to n-heptane ≡ 10 and tetralin ≡ 100. The key points can be obtained from the above-mentioned comparison. ① It can be indicated that the doping concentration of 1.5% by mass is still low enough to avoid the interference of the dopant to flame structure. On the other hand, the relatively higher doping concentration in the present study tends to enhance the intensity of the LII signal and be beneficial to a higher signal-to-noise ratio compared with the lower additive concentration. ② The high correlation coefficient mentioned-above reveals that YSI is a good and suitable indicator of sooting tendency, which is highly related to the inherent chemical features of test fuels and has a fairly good independence on experimental setup and condition, such as soot diagnostic techniques, geometric structure of burner, and dilution degree of inert gases.

The highly close correlation in Fig. 4 also provides a statistical approach to map the YSI values from the work of Ref. [6] (YSI _{Das 2017}) to the YSI scale in the present study (YSI-trans). The equation of the linear fit line indicates the transferring relationship expressed as

(30)\mathrm{Y}\mathrm{S}\mathrm{I}-\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}=({\mathrm{Y}\mathrm{S}\mathrm{I}}_{\mathrm{D}\mathrm{a}\mathrm{s}2017}+{41.69}^{[\pm 4.07]})/{2.92}^{[\pm 0.06]}\mathrm{.}

This transformation is meaningful to the comparison between sooting tendencies of surrogates (measured in the present study) and their target real fuels, which are not available, such as CFA diesel and Jet-A POSF4658. There are 7 common surrogate mixtures in both the studies in Ref. [6] and the present study. As an accuracy check for this transferring method, the measured YSI and the transferred YSI (both in the scale of the present study) were compared and a small average relative deviation (4.0%) was obtained.

4.2 Model testing

During model testing, 80% of the data set was randomly picked as the training data, leaving the other 20% as the testing data. The training set was used to train the BMKL learner. The testing set was applied to show the generalization ability and select suitable model parameters. This procedure was repeated 50 times.

Figure 5 shows the predicting performance of the BMKL model after 50 repetitions of the testing procedure. As indicated in Fig. 5, the predicted YSIs obtained by the BMKL model (YSI-BMKL) correlate very well with the values measured in the range from 0 to 180. As mentioned in Section 3.2, the sparse samples in the higher YSI range may lead to the difficulty in accurately predicting their YSI values. For all the surrogate samples with YSI values from 80 to 180, the average relative deviation between the values predicted by using the BMKL model and the values measured is just 2.1%. It is proved that the BMKL model can effectively overcome this challenge of sparse data and show a remarkably accurate performance for these surrogates with higher YSI values.

Furthermore, comparison between predicting performances of the BMKL model and the linear additivity (LA) method were performed based on the database in the present study. The statistical results of these two methods are listed in Table 2. Compared with the linear additivity method, the BMKL model reduced the mean and maximum deviations by 35.4% and 39.8% respectively. In addition, the mean relative error of the BMKL model is only 10.37%, indicating obvious improvement for the LA model, whose mean relative error is 15.20%. The RMSE of the BMKL model is also better. As a result, it can be confirmed that the BMKL model has a better predictive ability than the LA model

4.3 Model application for surrogate assessment in terms of sooting tendency

4.3.1 Diesel surrogate

Qian et al. [35] developed three diesel surrogates according to the constraint equations for cetane number, H/C ratio, density, viscosity, surface tension, and distillation range. The target fuel was a commercial diesel named China stage V 0# diesel, which consisted of 49.2% alkanes, 34.6% cycloalkanes and 16.2% aromatics (by mass). The compositions of three multicomponent surrogate fuels are tabulated in Table 3. The 3-component surrogate fuel consists of n-hexadecane, 2,2,4,4,6,8,8-heptamethylnonane (HMN), and 1-methylnaphthalene. Considering the fact that cycloparaffin is a typical diesel component, decalin was added to the 5-component surrogate fuel. In addition, n-octadecane was also added to match the molecule size and distillation curves. Furthermore, a 7-components surrogate fuel was formulated to match the distribution of carbon atoms of cycloparaffin and aromatics in the commercial diesel. In order to evaluate the sooting tendencies of the target diesel and these sorrogates in Ref. [35], the YSIs were measured using the method described in Section 3.1, and their predicted values were calculated by using the BMKL model in the present study. It should be noted that the China stage V 0# diesel used was bought from the same source as that in Ref. [35]. The diesel composition was tested by using ASTM D2425 and the contents of each main component were kept almost consistent with the target fuel in Ref. [35] within a 5% fluctuation. The surrogates were prepared strictly according to the component proportions in Ref. [35].

There are two main points that can be obtained from Fig. 6. ① The predicted YSIs of BMKL model matched their corresponding measured values very well for 3-component, 5-component, and 7-component surrogates with relative deviations of 1.4%, –0.8%, and 2.9% respectively. In contrast, the predicted YSIs of the linear method show a bigger relative deviations compared with the measured values for each test surrogates (3.5%, 2.4% and 5.6% 3-component, 5-component, and 7-component surrogates), which indicated that the model proposed was able to accurately predict the YSI values with a large range of component number from 3 to 7, and the performance model was better than that of the traditional linear method. The BMKL model provides a numerical approach to assess surrogate in terms of sooting tendency, which is a convenient and low-cost approach. ② The 7-component surrogate had the closest YSI value to the target fuel, with the smallest relative deviation of –8.6%. This result is easy to be understood because the 7-component surrogate meets the distribution of carbon atoms of the target diesel better than the other two surrogates with less components. However, it is interesting that the 5-component surrogate, rather than 3-component surrogate, has the worst YSI value with a relative deviation of 12.5%. As a result, it can be speculated that a higher compositional accuracy is sufficient but not necessary to match the sooting tendency for surrogate design.

4.3.2 Jet fuel surrogate

POSF 4658 is a composite Jet-A fuel of intended average properties produced by the United States Air Force. Unlike the JP-8 fuels with largely variours compositions, POSF 4658 can be regarded as a reference blend that has relatively fixed compositions [6,11]. A 2nd generation POSF 4658 surrogate fuel was developed in Ref. [11] condering the constraints of average fuel molecular weight (MW), hydrogen/carbon molar ratio (H/C), derived cetane number (DCN), and threshold sooting index (TSI). This surrogate was a 4-component mixture, consisting of 40.4% n-dodecane, 29.5% iso-octane, 7.3% 1,3,5-trimethylbenzene, and 22.8% n-propylbenzene (by mole). Compared with the 1st generation surrogate in Ref. [36], the matching of sooting behavior was introduced to the surrogate formulation method of the 2nd one, using TSI (measured by the smoke point technique) to describe sooting tendencies of the target fuel and its surrogates. In Ref. [11], the experimental TSIs of POSF 4658 and the 2nd generation sorrogate were 21.4 and 20.4 respectively, with a small relative deviation of –4.7%. In order to verify the matching level of sooting behaviors of POSF 4658 and its sorrogate, soot volume fraction distributions were aquired by the laser extinction measurements in wick-fed diffusion flames. It could be seen that the radial soot profiles showed discrepancies for either the same nondimensional height or absolute height. Further study need to be performed to investigate the possible reasons why a fairly good agreement between POSF 4658 and its sorrogate in terms of TSI leads to non-ignorable discrepancies of soot volume fraction distributions.

In the present study, the sooting tendencies of the surrogate proposed by Ref. [11] were measured using YSI, which was obtained based on the method described in Section 3.1. Considering the difficulty in obtaining the same target jet fuel as that in Ref. [11] in China, the YSI of POSF 4658 was evaluated by the transformation proposed in Section 4.1 based on the experimental data in Ref. [6]. It is observed from Fig. 7 that the measured YSI of the surrogate is markedly lower than that of the target fuel (POSF 4658), with a relatively large deviation of –14.2%. As described in Section 4.1, the transferred YSI values were calculated linearly from the experimental data in Ref. [6]. Therefore, the deviation between the transferred YSIs of the target fuel (40.1) and its surrogate (33.9) can verify the above-mentioned discrepancy of sooting tendency in another independent experiment. It is interesting to see that there exists a relatively large deviation between the target fuel and its sorrogate in terms of YSI in the present study while a good match of TSI between the target fuel and its sorrogate was found in Ref. [11]. According to the definition of TSI, the determination of TSI for the real fuel is to an extent, conditional upon an accurate determination of its average molecular weight. However, the real fuels are always complicated fuel mixtures and their empirical formulas need to be determined experimentally, which may introduce uncertainties for the calculation of molecular weights. As reported by Dooley et al. [11], the MW of POSF 4658 was 142±20 g/mol. The uncertainty of MW (approximately 30%) can lead to a high relative deviation of TSI values. Moreover, the target fuel (POSF 4658) contains approximately 30% aromatics and napthalenes by mole [11] and tends to produce a heavily sooting flame with a low smoke height which is difficult to measure precisely. The good match of TSIs in Ref. [11] may be partly incidental considering the possible uncertainties introduced by the determination of molecular weight and the measurement of smoke height of the target fuel. Compared with the TSI measurement, there is no need to know the molecular weight of test fuels during YSI defination and the YSI values are directly related to the soot volume fraction of doped flames, which can be measured quantitatively by using the LII technique. As a result, it is suggested that YSI has a big potential to accurately describe the sooting tendencies of real fuels with a higher distinguishability, which should be regarded as an important design parameter for surrogate formulation and development to match the sooting behaviors.

It can also be found from Fig. 7 that the relative deviations between measured YSI and the predicted YSI using the BMKL model for this 4-component surrogate is just 2.6%, showing the good predicting performance of the BMKL model for jet fuel surrogates. In other words, the BMKL model can be used to evaluate surrogate in terms of sooting tendency with a reasonable precision. In addition, it can be observed in Fig. 7 that the linear method can also match the measured YSI value with a small relative deviation (–4.2%), indicating that there may be no obvious cross-linked interaction between these 4 constituent components during soot formation and the YSI mixing rule can be regarded linearly for these certain components.

4.3.3 Biodiesel surrogate

Biodiesel is one of alternative oxygenated fuels, consisting of methyl or ethyl esters of long-chain fatty acids, which is produced from renewable sources such as vegetable oils or animals’ fats [37]. The biodiesel fuel has two typical features: the presence of ester moieties and one or more carbon-carbon double bonds in their moleculars. Due to the specific composition of real biodiesel, its surrogate palette includes both hydrocarbon (n-heptane, n-decane, n-hexadecane etc.) and oxygenated components (methyl butanoate, methyl octanoate, methyl decanoate etc.) [10,38–41]. Unlike the surrogates of diesel fuels and jet fuels, the assessment of sooting tendencies for biodiesel surrogates requires that the predicting model have a good capacity to capture the YSI mixing rules of the above-mentioned hydrocarbon and oxygenated components.

In Refs. [40,42,43], typical methyl esters with relatively short chains, such as methyl butanoate (MB) and methyl decanoate, were used as single component surrogate in experimental and numerical combustion studies to mainly capture the characteristics and chemical effects of the ester moiety in biodiesel. These components usually have mature and reliable chemical kinetics mechanisms, which have been developed widely. Several 2-component surrogates of biodiesel have also been proposed [41,44], usually with 50%/50% molar blend of n-alkanes and methyl esters. Recently, multi-component surrogate fuels for biodiesel have been developed to achieve a better matching precision, such as methyl decanoate, methyl-5-decenoate, and n-decane for rapeseed biodiesel [38] and methyl decanoate, n-hexadecane, methyl trans-3-hexenoate, and 1, 4-hexadiene for soybean biodiesel [1]. It should be noted that the surrogates with one or two components are always used to represent general biodiesl fuels without particular consideration of target fuel contents while multi-component surrogates aim at a certain type of biodiesl and their compositions are strongly associated with the detailed contents of the target fuel.

To evaluate the matching level of sooting tendency for different biodiesel surrogates and compare the performance of different surrogate formulation methods in matching sooting behaviors, the YSI of two real biodiesel fuels (soybean biodiesel and waste cooking oil) and several sorrogates were measured and predicted in the present study. Two types of commercial biodiesel fuels, soybean biodiesel and waste cooking oil (Yaoneng New Energy Company), were selected as the target fuels and their proportions (presented in Table 4) were determined by using gas chromatography-mass spectrometry (GC–MS) both qualitatively and quantitatively (Agilent, GC 7890A, 5975C). According to the novel surrogate formulation methodology for biodiesels proposed in Ref. [10], two 4-component surrogates were respectively formulated for two target fuels in the present study. The components and proportions of these surrogates are listed in Table 4. In addition, typical pure methyl esters and 2-component surrogates were also tested for comparison.

Figure 8 shows measured and predicted YSI values for two real biodiesel fuels and their surrogates. To make Fig. 8 readable, only the predicted YSI values of two 4-component surrogates formulated in the present study are shown. It can be found that the predicted YSIs using the BMKL model have a better matching performance to target values compared with that of the linear method for these two surrogates, which proves that the BMKL model has a good capacity to capture the YSI mixing rules of hydrocarbon and oxygenated components. As shown in Fig. 8, both the soybean surrogate (SB sur) and waste cooking oil surrogate (WB sur) underestimate the YSIs of their corresponding target fuels, with relative deviations of –12.8% and –16.6% respectively. In the contrast, the single-components and 2-component surrogates cannot capture the sooting tendencies for both biodiesel fuels at all. For example, the YSI of BS2-1 is just about half of the waste cooking oil YSI. Two conclusions can be obtained. First, compared with single and 2-component surrogates, 4-component surrogates have a more accurate matching performance in terms of sooting tendency because this formulation method takes the degree of non-saturation as one of the constraints [10], which proves to have obvious effects on soot formation [45]. In this case, the matching level increases with the increasing compositional accuracy and pertinence to the target fuel. Second, there is still some room for these two 4-component surrogates to improve the matching degree of YSI values. One effective way is to directly add YSI into the constraint group, which is described in detail in Section 4.4.

4.4 Model application for surrogate design in terms of sooting tendency

In a complete sorrogate formulation process, several key steps need to be performed carefully besides the development of chemical mechanisms [11,18,46]. First, each surrogate component should be selected from the surrogate palette, which consists of typical classes of pure substance, such as n-alkane, iso-alkane, cyc-alkane, aromatics etc. The choice of surrogate components need to take into account the contents and characteristics of target fuels. Second, based on a series of constraints for certain fuel properties and their corresponding weights, several surrogate candidates are obtained with reasonably matching levels. The estimation of each property for these fuel mixtures are performed by empirical equations. Third, experiment approaches are applied to validate the actual behaviors of surrogate candidates in fundamental combustion setups (rapid compression machine, shock tube, flow reactor, laminar flame burner etc.) and/or engines. The optimal surrogate can be selected based on the matching degree of these results to the target fuel. For example, if researchers want to ensure a good fitting between the target fuel and its surrogate in terms of soot formation, the related information, such as soot volume fraction and soot size distribution need to be measured in laboratory flames. However, the experimental tests for all the surrogate candidates not only have requirement for corresponding equipment, but also bring superabundant repetitive works, which are costly in both money and time.

It is suggested that YSI should be added to the group of constraint parameters, in order to keep similar sooting behaviors for the target fuel and its surrogates. Before the third step mentioned above, the BMKL model can be applied to predict the YSI values for all the surrogates in candidate list, eliminate the surrogates with relatively large deviations and narrow down the range of choice. It is worth mentioning that some fuel property parameters may have trade-off relationships. As a result, the BMKL model is not aimed at choosing the surrogate with the best matching level of YSI. The practical function of the model is to further narrow down the candidate list based on sooting tendency and ensure that the optimal surrogate has a satisfying matching level of soot behaviors.

To demonstrate the BMKL model application for surrogate formulation, an outspread analysis was conducted here based on the surrogate formulation process of Jet-A POSF 4658 in Ref. [11] which has been described in Section 4.3.2. Actually, according to the constraints of average MW, H/C ratio, DCN, and TSI, seven surrogate candidates that reasonably satisfied these properties of the target fuel, were obatined (shown in Table 5). Then, a 2nd generation surrogate fuel was selected from the surrogate candidate list, without explaining the reason for this choice, though. In the present study, the YSI value of each surrogate mixture on the candidate list were predicted by using the BMKL model and the results are shown in the right column of Table 5. Compared with the TSI results, there are more distinguishable differences between YSI values of the seven surrogates and their target fuel, which proves the higher resolution of the YSI predicting method using the BMKL model. Furthermore, the surrogate fuel #3 has the closest YSI value to that of the target fuel of all these candidates and this surrogate fuel is probably supposed to be the most suitable one, especially in terms of sooting tendency. In conclusion, due to the high accuracy and resolution for YSI prediction, the BMKL model can be applied to provide meaningful suggestions for surrogate selection and formulation based on more distinguishable information.

5 Conclusions

In this paper, a novel predicting model of YSI values for surrogate fuels was proposed with the application of a Bayesian multiple kernel learning model. A high correlation coefficient (0.986) between measured YSIs and predicted values with the BMKL model was obtained. Surrogate assessment in terms of sooting tendency was performed for diesel, jet fuel, and biodiesel using this convenient and low-cost approach, and some suggestions for surrogate formulation were proposed. The main conclusions drawn are as follows:

An integrated and systematic YSI database with more than 190 kinds of fuels (including pure components and mixtures) was obtained by conducting a large number of experimental investigations of both hydrocarbon fuels and oxygenated blends. This YSI database provides sufficient information of sooting tendencies for developing and validating surrogate mixtures for real fuels and supports YSI as one of the important target indices or constraints for surrogate design.

The BMKL model tends to capture the relationships between surrogate compositions and total sooting tendencies of fuel mixtures. It is found that the test results of the novel prediction model have lower values for deviation and root mean square error than those of the linear model, which indicates that the BMKL model has a reliable and accurate predictive capacity for YSI values of surrogate fuels.

The BMKL model provides a numerical approach to assess surrogate in terms of sooting tendency, which is a convenient and low-cost approach. Particularly, this model is one of the first attempts to predict the sooting tendencies of fuel mixtures that concurrently contain hydrocarbon and oxygenated components (e.g., biodiesel fuels), with a satisfying performance.

During surrogate formulation, the BMKL model can be applied to narrow down the surrogate candidate list in terms of sooting tendency and ensure that the optimal surrogate has a satisfying matching level of soot behaviors. Because of the high accuracy and resolution of YSI prediction, the BMKL model can be applied to provide distinguishing information of sooting tendency for surrogate design.

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