Correlations of the Elemental Compositions of Primary Coal

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Correlations of the Elemental Compositions of Primary Coal Tar and Char Andrew P Richards, Colson Johnson, and Thomas H Fletcher Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.9b01627 • Publication Date (Web): 30 Aug 2019 Downloaded from pubs.acs.org on August 30, 2019

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Correlations of the Elemental Compositions of Primary Coal Tar and Char Andrew P. Richards, Colson Johnson, and Thomas H. Fletcher* Department of Chemical Engineering, Brigham Young University, Provo, UT 84604, United States Keywords: Pyrolysis, coal aromaticity, NMR parameters, cross-validation, coal char, coal tar

Simulations of large-scale coal combustors rely on accurate submodels to describe the chemical and physical changes in coal during reaction. Typically, simplified empirical submodels are tuned to experimental data to reduce the computational complexity. When data are not readily available, simplifying assumptions are used, which can create inaccuracies and biases in a large simulation. One such simplifying assumption in coal research is how to describe the elemental composition of primary pyrolysis products. This paper explores several different empirical model forms to predict the dry, ash-free fractions of C, H, O, N, and S in both the char and the tar, using variables such as parent

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coal composition, reaction conditions (temperatures and particle residence times), and key coal structural parameters derived from NMR measurements to improve the treatment of coal chemistry in large simulations. These model forms were correlated to existing data using a wide range of experimental data using a cross-validation procedure. Since coal structural values can be expensive to measure, several correlations from the literature were used to estimate these values based on information from the proximate and ultimate analyses of the parent coal, including a new correlation for the coal aromaticity. These model forms were tested against a set of measured elemental compositions of tar and char to find the best fit to use in the cross-validation process. The best empirical models are presented that predict elemental composition of the coal char and tar after devolatilization.

1. INTRODUCTION Current areas of coal research have advanced understanding of the processes involved, however, little is still known about the elemental and chemical compositions of the products of coal pyrolysis. Knowing or predicting these compositions accurately can greatly influence the overall accuracy and utility of a large simulation. This paper details efforts to improve the treatment of coal chemistry in larger simulations of coal combustion and gasification using a comprehensive statistical analysis to develop better models to predict the elemental composition of the products of primary pyrolysis.


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Volatiles contain a greater fraction of hydrogen and oxygen than the char.1-3 The distribution of elements into combustion products is influenced by combustion conditions and parent fuel properties. The distribution of nitrogen and sulfur in the combustion products are of particular interest, since they can influence the NOx and SOx compositions of the flue gas.4 Simulations of large-scale combustors include submodels to describe many physical and chemical processes in combustion, including fluid and particle flow, heat transfer, mass transfer, gas and solid reactions, etc. Simplified empirical models tuned using experimental data are used to decrease the computational complexity and time requirements. Many times, there are not enough experimental data or even model dexterity to provide an accurate prediction of physical behavior, so simplifying assumptions are used. These assumptions can lead to great inaccuracies in the simulation, so these simple submodels are constantly updated to include better (or more complete) data, conditions, and mathematics to increase their accuracy and predictive capabilities. Advanced network devolatilization models such as the chemical percolation devolatilization (CPD) model,5,

6

FLASHCHAIN,7 and FG-DVC8 accurately predict

devolatilization and output some form of light gas compositional information, but they are typically too complex to run in large-scale simulations,2 leading to greater computational expenses.9 Turbulence-chemistry interactions in the gas phase add challenges which often result in the need for very simple treatments of coal pyrolysis


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products. Light gas composition information can be calculated using a light gas submodel, such as the FG model. Simple kinetic mechanisms have been used to predict the light gas composition, since the kinetic parameters can be easily derived from pyrolysis experiments.10 The large number of tar species evolved are very difficult to treat in a detailed manner in large simulations. Several different approaches have been used in boiler simulations to treat coal chemistry. The simplest approach has been to use a coal gas mixture fraction method, where the composition of the parent coal is the same as the composition of the volatiles and char.11-13 Simulations with two coal gas mixture fractions, one mixture fraction for the volatiles and one for the char, have also been made, assuming that the char is 100% carbon.14 Other approaches are to correlate char composition vs. burnout,15, 16 or to use a light gas submodel and estimate the coal (and as a result, the tar and char) as excess hydrocarbons,17-20 or to assume that the char only contains carbon.2123

Other models only treat the fate of fuel nitrogen24 for NOx formation chemistry. Some

models lump light gases and tar together and use a global volatiles combustion rate.25 Some researchers went further to try to classify the volatiles species as methane and hydrogen cyanide,26 or to assume that the tar composition was the same as that of the parent coal.27 Some approaches have attempted to use a simple tar species with a 1:1 molar ratio of carbon and hydrogen, such as acetylene or benzene.28 While these assumptions can be beneficial to reduce computational time in large simulations, these


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assumptions are not consistent with measured elemental compositions of tar and char.29, 30

In pulverized coal combustion, primary pyrolysis of a coal particle occurs soon after it is injected into a hot environment. Many of the same variables that affect primary pyrolysis yields may also affect the chemical composition of the char and tar. Different coal types have different parent coal compositions and structures, which significantly influence primary pyrolysis product yields and compositions.31 Coal type also influences the general structural makeup of the coal and tar molecules, sometimes measured by NMR solid-state spectroscopy.1,

3, 29, 32-37

The thermal environment (gas temperature,

residence time, particle heating rate, etc.) that the coal particle experiences also impacts the yields and compositions of char, tar, and light gases.38 Ko et al. used parent coal properties and a correlation for aromaticity to develop a generalized correlation for tar and volatiles yields.39, 40 Genetti et al.32 used proximate and ultimate analyses to correlate chemical structural parameters for coal, which parameters were used in the CPD model.5 In addition to the presentation of a new correlation to predict the aromaticity of coal, this paper details efforts to tune simple correlations to predict the elemental composition (C, H, O, N, S) of the primary pyrolysis products (char and tar) with the intended use in large-scale pulverized coal combustion simulations. 2. APPROACH


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A set of possible correlating variables was first determined, including parent coal composition (ultimate and proximate analysis), chemical structural parameters, heating rate, peak pyrolysis temperature, and residence time. The chemical structural parameters include those used by the CPD model (molecular weight per cluster, molecular weight per side chain, number of attachments per cluster, and the fraction of attachments that are bridges) as predicted by the Genetti correlations.32 However, it seemed logical to include an additional structural parameter, carbon aromaticity, as a possible correlating parameter. Therefore, correlations of carbon aromaticity are reviewed in Section 2.1. Elemental composition data on coal chars and tars during primary pyrolysis were then gathered (Section 2.2), and simple model forms were explored (Section 2.3). A detailed statistical cross-validation procedure for evaluating advanced model forms is described in Section 2.4. Section 2.5 details a novel model refinement procedure used to obtain the best possible correlation fit with the fewest possible parameters. The best resulting correlations are shown in Section 3, including a new correlation for carbon aromaticity. 2.1 Coal Aromaticity Coal aromaticity has been measured using NMR spectroscopy, and subsequently used in correlations of other coal properties. Several correlations have been developed for coal aromaticity based on proximate and ultimate analyses. Ko, et al.39, 40 developed a second order polynomial for a limited set of aromaticity data based on the carbon content of the parent coal:


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𝑓𝑎 = 0.830526 ― 2.008147

𝐶𝑐𝑜𝑎𝑙

𝐶𝑐𝑜𝑎𝑙 2

100

100

( ) +2.241218( )

(1)

where 𝑓𝑎 is the aromaticity and 𝐶𝑐𝑜𝑎𝑙 is the wt% carbon of the coal on a dry, mineralmatter free basis. It is unclear whether this equation attempts to correlate the total aromatic carbon contribution, sometimes called the apparent aromaticity, (𝑓𝑎), including the carbonyl contribution, or if it correlates the true or corrected aromatic carbon content (𝑓𝑎′). Gerstein, et al.41 used a simpler linear correlation of 𝑓𝑎′, also based on the carbon content of the parent coal:

(2)

𝑓𝑎′ = 0.0159(𝐶𝑐𝑜𝑎𝑙) ―0.564

Carr and Williamson42 developed a different correlation to predict the apparent aromaticity based on vitrinite reflectance:

𝑓𝑎 = ―0.027𝑅2𝑜 +0.206𝑅𝑜 +0.0570

(3)

Maroto-Valer, et al.43 proposed an updated version of a previous correlation based on the hydrogen-carbon ratio, 𝐻/𝐶, as shown in Eq 4. It is also unclear in this


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situation if this correlation refers to the apparent or the corrected aromatic carbon content.

(4)

𝑓𝑎 = 1.22 ― 0.58 ⋅ 𝐻𝑐𝑜𝑎𝑙/𝐶𝑐𝑜𝑎𝑙

Singh and Kakati studied four different expressions to predict aromaticity,44 shown in Eqs 5-8. It is also unclear whether they used the apparent or corrected aromaticity values,

𝑓𝑎1 = 1.202913 ― 0.0126 ⋅ 𝑉𝐴𝑆𝑇𝑀

(5)

𝑓𝑎2 = 1.36396 ― 0.53715 ⋅ 𝑂𝑐𝑜𝑎𝑙/𝐶𝑐𝑜𝑎𝑙 ―0.7846 ⋅ 𝐻𝑐𝑜𝑎𝑙/𝐶𝑐𝑜𝑎𝑙

(6)

𝑓𝑎3 = 1.365615 ― 0.51187 ⋅ 𝑂𝑐𝑜𝑎𝑙/𝐶𝑐𝑜𝑎𝑙 ―0.02108 ⋅ 𝑂𝑐𝑜𝑎𝑙/𝐻𝑐𝑜𝑎𝑙 ―0.78645 ⋅ 𝐻𝑐𝑜𝑎𝑙/𝐶𝑐𝑜𝑎𝑙 (7) 𝑂𝑐𝑜𝑎𝑙

𝑂𝑐𝑜𝑎𝑙

𝐻𝑐𝑜𝑎𝑙

𝑓𝑎4 = 174.4405 + 621.6823 ⋅ 𝐶𝑐𝑜𝑎𝑙 ―856.495 ⋅ 𝐻𝑐𝑜𝑎𝑙 ―629.617 ⋅ 𝐶𝑐𝑜𝑎𝑙 +9.133897 ⋅ 𝑉𝐴𝑆𝑇𝑀 (8)

where 𝑉𝐴𝑆𝑇𝑀 is the wt% volatile matter and the elemental ratios are on an atomic basis, and all values are on a dry, mineral-matter free basis. The seven aromaticity correlations from the literature (excluding Carr and Williamson) were incorporated in this analysis in two ways. First, each correlation was compared with a large set of aromaticity data. Second, coefficients for these model forms were re-fit using the current aromaticity data. This second step was to ensure that these


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literature correlations were evaluated equally. These seven correlations are shown as models 37-44 in Table S1 in the supplemental material section. The remaining models (136) were based on prior knowledge of variables that can affect coal aromaticity or other structural and chemical parameters that can be either measured by various standard tests or predicted by simple correlations in the literature (such as Genetti’s32 NMR structural parameter correlations). 2.2 Sources of Experimental Data Two separate datasets were used for this analysis: one for testing the coal aromaticity correlations and a second for testing the elemental composition analysis. There is some overlap in these two sets of data. The aromaticity models were evaluated on a set of data that includes elemental composition (proximate and ultimate analysis results) and measured and calculated chemical structure parameters from NMR analysis. Some of these data were used by Genetti et al.32 to fit chemical structure parameters for the CPD model. The data come from the following sources: Genetti, et al.,32 Solum, et al.,33 Hambly, et al.,34 Perry, et al.,29, 36

Fletcher and Hardesty,1 Watt, et al.,3 Gerschel and Schmidt,45 Cui, et al.,46 Ahmed, et al,47

Lin, et al.,48 Suggate and Dickinson,49 and Zhang, et al.50 These sources were chosen for their use of the corrected aromaticity 𝑓𝑎′. The elemental composition analysis used experimental data from literature based on several conditions:

primary tar formation (temperatures below 1100 K, to limit


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secondary tar reactions), low residence times (no hold times) and heating rates of at least 1000 K/s. Data from fluidized bed systems were not used unless accurate particle residence times were available. Frequently fluidized bed systems are reported with either an average estimated particle residence time or a fluid residence time. This can be different than a particle residence time. Compositions and yields are on a dry, ash-free basis (DAF). Table 1 lists the sources used for the elemental composition correlation analysis, along with a summary of experimental conditions for each set of data.

Table 1. Data sources used for elemental composition correlations Author(s)

Institution

Freihaut, et al.51

Gas Char/Tar Temperatures (K)

Coal Types

United Entrained Technologies flow reactor Research Center

780-1069

hvA bit, sub C, LVB

Hambly34

Brigham Young University

Drop tube reactor

820 and 1080 Char and Tar

ligA, subA, hvCb, hvAb, lvb

Perry29, 36


Drop tube reactor

895-1085

Char and Tar

Sub, hvb, mvb, lvb

Entrained flow reactor

1050

Char

Lig, sub, hvBb, hvAb, lvb

Fletcher and Sandia Hardesty1 National Laboratories

Apparatus

Tar


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Watt3


Drop tube reactor

850-1050

Char and Tar

ligA, subC, hvCb, hvAb, lvb

Parkash52

Devon Coal Research Centre

Laminar, entrained flow, atmospheric pressure reactor

820-980

Char

Sub B

Tyler53

CSIRO

Heated fluidized bed reactor

873

Tar

Bit, sub

2.3 Model Forms The first step in the development of correlations was to evaluate simple model forms based on a graphical analysis of all the variables involved. The dependent variables (normalized mass fraction of each element in either the char or the tar) were plotted against each of the proposed independent variables, which included the corrected aromaticity (𝑓′𝑎), the maximum gas temperature (in Kelvin), the residence time (in ms), the normalized volatiles yield (𝑉/𝑉𝑚𝑎𝑥, where this ratio is equal to 1 at maximum volatiles yield), the mass fraction of each element in the parent coal (including 𝐻/𝐶 and 𝑂/𝐶 ratios), the ASTM volatile matter on a dry ash-free basis, and key NMR structural parameters (𝑐0, 𝑀𝑊𝑐𝑙, 𝑀𝑊𝛿, 𝑝0, and 𝜎 + 1) as predicted by Genetti’s correlations.32 Genetti also used Gerstein’s41 correlation for aromaticity, which was included with the literature aromaticity models. This graphical approach was adopted to show the general trends of the elemental


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composition with changes in the dependent variables. Several iterations of each model form were tested based on these general trends expressed in these graphs, for each element in both the char and the tar. Each model form has a variety of variables with a mixture of polynomial, power, logarithmic, and exponential functions. Figures 1 (char) and 2 (tar) show an example of these graphs, plotted against the parent coal carbon content, which is often used as an indicator for coal rank.

Figure 1. Example of plot for the normalized mass fraction of C, H, O, N, and S in the char vs. parent coal carbon content (𝐶𝑐𝑜𝑎𝑙). Plots like these were used to identify model forms.


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Figure 2. Example of plot for the normalized mass fraction of C, H, O, N, and S in the tar vs. parent coal carbon content (𝐶𝑐𝑜𝑎𝑙). Plots like these were used to identify model forms. Plots of data like Figures 1 and 2 were used to identify several model forms for testing. A complete list of model forms explored are found in Table S4 in the appendix. Table S5 in the appendix shows the final 𝑅2 values of every model form. This table indicates which model form was used for which element. Some of these model forms are similar across all elements, but many are unique to each element.


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Figure 3. Van Krevelen plot using all coal, char, and tar data. Arrows indicate the progression of char and tar compositions with increasing residence time for three of the coals in the dataset.

Figure 3 is a Van Krevelen plot, showing the oxygen-to-carbon ratio against the hydrogen-to-carbon ratio for coals, chars, and tars. Data were plotted on a mass basis, rather than an atomic basis, as van Krevelen demonstrated.54 This figure indicates that on average, the tar seems to have a similar composition to the parent coal and the char becomes enriched in carbon but decreases in both hydrogen and oxygen content. The progression of char and tar composition is shown for three of the coals in the dataset


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(two lignites and a subbituminous coal). The final tar and char composition from these three coals were somewhat similar to each other, and had decreased amounts of both H and O, but neither char was close to 100% C. 2.4 Cross-Validation and VUQ To evaluate the goodness of fit of the tested model forms, an extensive validation and uncertainty quantification analysis was performed. Verification and validation may mean different things to different people. The definitions offered by Oberkampf and Barone55 was used in this paper. Verification refers to the process of ensuring mathematical and model implementation accuracy, and validation refers to the process of comparing model predictions to real world values. Cross-validation has been used to decrease the overall bias of mathematical models.56 The cross-validation process57 is commonly thought to have four steps: 1. The dataset is split into randomly assigned groups (typically 10 sets of roughly equal size). 2. One of the groups is left out as a test set. The remaining groups are used to train each model (i.e., to curve-fit the data for all but the test set). The resulting correlation is then evaluated using the one test set excluded from the curve fit. This process is repeated for each group, so that each group eventually acts alone as a test set.


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3. The validation metrics of all these curve fits are averaged, and the best model is chosen. A better value for a validation metric indicates that the model has a better ability to predict data using new conditions on which it was not trained. 4. The chosen model or models are finally trained on the entire dataset to produce a prediction model. To assess the fit of each cross-validation step and overall validation step, five main validation metrics were considered. (i) the L1 norm,58 which is a measure of the average absolute error over all the data points, as shown in Eq 9; (ii) the L2 or Euclidian norm,58 which gives the root mean squared error, in Eq 10; (iii) the infinity norm,58 which gives the maximum absolute error over the whole domain, shown in Eq 11; (iv) the total sum of squared errors, in Eq 12; and (v) the R2 value,59 often called the coefficient of determination, shown in Eqs 13-16.

1

𝑁

(9)

𝐿1 = ‖𝑦 ― 𝑦‖1 = 𝑁∑𝑛 = 1|𝑦𝑛 ― 𝑦𝑛| 𝐿2 = ‖𝑦 ― 𝑦‖2 =

(

1 𝑁 ∑ ( 𝑁 𝑛 = 1 𝑦𝑛

)

2

― 𝑦𝑛)

1 2

(10)

𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑦 = ‖𝑦 ― 𝑦‖∞ = 𝑚𝑎𝑥|𝑦𝑛 ― 𝑦𝑛| ,

𝑛 = 1 𝑡𝑜 𝑁

𝑁

(12)

𝑆𝑆𝐸 = ∑𝑛 = 1(𝑦𝑛 ― 𝑦𝑛)2 𝑅2 =

(

𝑆𝑥𝑦

2

)

(13)

𝑆𝑥𝑥 ⋅ 𝑆𝑦𝑦 𝑁

(11)

1

(

𝑁

)(

𝑁

)

𝑆𝑥𝑦 = ∑𝑛 = 1(𝑦𝑛𝑦𝑛) ― 𝑁 ∑𝑛 = 1𝑦𝑛 ∑𝑛 = 1𝑦𝑛

(14)


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2

(15)

2

(16)

𝑁

1

(

𝑁

)

𝑁

1

(

𝑁

)

𝑆𝑥𝑥 = ∑𝑛 = 1(𝑦𝑛)2 ― 𝑁 ∑𝑛 = 1𝑦𝑛

𝑆𝑦𝑦 = ∑𝑛 = 1(𝑦𝑛)2 ― 𝑁 ∑𝑛 = 1𝑦𝑛

In these equations, 𝑦𝑛 is the predicted value given by the proposed model, 𝑦𝑛 is the experimentally observed value, and N is the total number of points in a dataset. In this analysis, the dataset used allows for five groups for the tar oxygen and tar sulfur compositions and ten groups for the remaining elemental compositions. Fewer oxygen and sulfur compositions are reported in the literature. Oxygen is harder to measure, and is typically reported as a difference, sometimes even being a negative value. Some researchers even report a combined oxygen and sulfur composition. These combined values were not used in this analysis since they could not be separated using the provided information. Each proposed model form for each of the elemental composition correlations was tested using this cross-validation procedure. The best of these model forms were then fit against the entire dataset and the validation metrics were again calculated on the overall model fit. These validation metrics mean slightly different things during cross-validation and final training. A higher 𝑅2 value in cross-validation means that the model will give better and more accurate predictions using conditions or coals for which the model was not originally fit. Comparatively, a higher 𝑅2 value in final training means the model fits the entire suite of experimental data better.


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To train each model against the experimental data, five separate MATLAB optimizers were used to minimize the sum of squared errors: one constrained optimizer, fmincon, and an unconstrained optimizer, fminunc. The remaining optimizers were a Global Search algorithm using fmincon and a multi-start algorithm using both fmincon and fminunc. Multiple optimizers were used to increase the chances of finding a true optimal solution and obtaining a best possible fit. Each cross-validation training and testing step used a round of optimizations with each of these optimizers, as did the final training step, for each of the elements in both the char and the tar, for each tested model form. The five main statistical quantities shown in Eqs 9-16 were computed for the final correlations. In evaluating the results of the statistical analysis, 𝑅2 value was given the highest consideration, with lesser importance on the three norm values (𝐿1, 𝐿2, and 𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑦). The 𝑅2 value was taken as the most comprehensive measure of error out of the five main validation metrics, however, the remaining metrics give a deeper insight into different aspects of model error. Only correlations with the best predictive capabilities with the fewest fitted parameters were considered for presentation in this paper. To ensure the best fit with the fewest number of parameters, a modified 𝑅2 value was calculated by dividing the individual 𝑅2 values by the number of fitted coefficients, which was then used to describe each model’s utility. This 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓 value was typically higher for models with fewer


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coefficients, however consideration was taken to make sure that the 𝑅2 with fewer coefficients was within a reasonable range of the highest 𝑅2 achieved with large numbers of coefficients. A maximum of two models were selected for each element: one with the highest 𝑅2 value, and a second with the highest 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓 value. This was to find the best overall fit and the best fit with the fewest parameters. 2.5 Model Refinement The cross-validation procedure produced two models for each elemental correlation. These two models were used as a basis for a model refinement procedure, detailed in this section. This non-standard procedure was used to ensure the best predictive ability with the fewest parameters. For example, if a model with 9 fitted coefficients has an 𝑅2 value of 0.85 and a simpler model with 6 coefficients has an 𝑅2 value of 0.83, the more complex model may provide a more accurate prediction, but the simpler model will provide almost as good of a prediction with fewer coefficients. The cross validation process resulted in two model types: A. Model types with the highest 𝑅2 values, and B. Model types with the highest utility (or the highest 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓 values). Combined, these two model types would potentially form a hybridized model with the best features of the original two. The refinement procedure is illustrated in Figure 4. Each term of model type A was removed one at a time to see the change in the R2 and R2/Ncoeff values. In addition, the terms for each variable in model type A were replaced


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Page 20 of 64

by the terms containing the same variable from model type B and evaluated for new R2 and R2/Ncoeff values. An example of this model refinement procedure for a simple two-variable system (x1 and x2) is provided here. Suppose the cross-validation and final training procedure indicated that a 3rd-order polynomial model (such as in Eq 17) with 7 fitted parameters had the highest 𝑅2 value of 0.9 (this would be model type A), and a natural log-based model (such as in Eq 18) with 3 fitted parameters had an 𝑅2 value of 0.75. To find the “utility” of each model, an 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓 value is calculated, which for the polynomial model is 0.129, and for the natural log model is 0.25, meaning these two models would be chosen for the model refinement procedure. Equation 18 would be the type B model form. 𝑦1 = 𝑐1 + 𝑐2𝑥1 + 𝑐3𝑥21 + 𝑐4𝑥31 + 𝑐5𝑥2 + 𝑐6𝑥22 + 𝑐7𝑥32

(17)

y2 = c1 + c2ln (𝑥1) + 𝑐3ln (𝑥2)

(18)

Two pathways from each of these model forms were considered. The first was to take away one term at a time from the model type A (Eq 17 in this case), leading to an additional 7 equations to fit to the entire dataset—one each with a different coefficient removed, as shown in Eqs 19-25. y1.1 = 𝑐2𝑥1 + 𝑐3𝑥21 + 𝑐4𝑥31 + 𝑐5𝑥2 + 𝑐6𝑥22 + 𝑐7𝑥32

(19)

y1.2 = 𝑐1 + 𝑐3𝑥21 + 𝑐4𝑥31 + 𝑐5𝑥2 + 𝑐6𝑥22 + 𝑐7𝑥32

(20)

y1.3 = 𝑐1 + 𝑐2𝑥1 + 𝑐4𝑥31 + 𝑐5𝑥2 + 𝑐6𝑥22 + 𝑐7𝑥32

(21)

y1.4 = 𝑐1 + 𝑐2𝑥1 + 𝑐3𝑥21 + 𝑐5𝑥2 + 𝑐6𝑥22 + 𝑐7𝑥32

(22)


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Energy & Fuels

𝑦1.5 = 𝑐1 + 𝑐2𝑥1 + 𝑐3𝑥21 + 𝑐4𝑥31 + 𝑐6𝑥22 + 𝑐7𝑥32

(23)

𝑦1.6 = 𝑐1 + 𝑐2𝑥1 + 𝑐3𝑥21 + 𝑐4𝑥31 + 𝑐5𝑥2 + 𝑐7𝑥32

(24)

𝑦1.7 = 𝑐1 + 𝑐2𝑥1 + 𝑐3𝑥21 + 𝑐4𝑥31 + 𝑐5𝑥2 + 𝑐6𝑥22

(25)

The second pathway was replace the x1 terms in model type A with the x1 terms from model type B (Eq 18 in this case), resulting in Eq 26, and then switching the x2 terms between model A and B to obtain Eq 27. 𝑦2.1 = 𝑐1 + 𝑐2ln (𝑥1) + 𝑐3𝑥2 + 𝑐4𝑥22 + 𝑐5𝑥32

(26)

𝑦2.2 = 𝑐1 + 𝑐2𝑥1 + 𝑐3𝑥21 + 𝑐4𝑥31 + 𝑐5ln (𝑥2)

(27)

All 9 new equations were then fit against the entire dataset, and the model with the highest 𝑅2 value was chosen. This refinement procedure was repeated, with the new highest 𝑅2 model replacing the model in Eq 17 (i.e., a new type A model form). This process was followed either until the 𝑅2 value fell well below the original best fit value, or the number of fitted parameters decreased to the level of the highest 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓 model. This iterative refinement procedure was repeated for all 10 elemental correlations, and the best of these refined models are presented in the results section.


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Page 22 of 64

Identify Experimental Data Graphical Test (Independent variables vs dependent variables) Bad or missing data?

Yes

No Identify and narrow possible model forms Analyze Model Forms Cross-validation and final training (statistical analysis) Best R2 and R2/Ncoeff models

Model Refinement

Use variables from R2/Ncoeff model in R2 model

Remove one variable at a time from R2 model

Best R2 model

Yes

Better than previous best? No Final Correlation(s)

Figure 4. Process used to develop elemental composition correlations.


22

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Energy & Fuels

3. RESULTS AND DISCUSSION The results of the coal aromaticity analysis are presented first since aromaticity is included in some of the model forms for the correlation of the elemental composition of tars and chars. The aromaticity analysis was performed using the experimental data for all seven literature correlations, plus 44 potential correlation forms, which included a refitting of all seven literature forms. All 44 model forms are found in Table S1 in the appendix. The results of the cross-validation analysis are presented next, which was used to narrow the number of potential best-fit model forms. Finally, the results of the model refinement analysis are presented. Only the coefficients for the best-fit model forms are presented in this paper; additional details (data from intermediate steps, additional model forms, etc.) are given by Richards60 as well as in the supplemental material section. 3.1 Coal Aromaticity Correlation All seven coal aromaticity correlations from literature were tested against the aromaticity dataset, along with 44 optimized model forms. These optimized model forms also included the seven literature correlations with re-fitted parameters. Each of the additional model forms included variations of parent coal composition, ASTM volatile matter, and measured and correlated NMR structural parameters (𝑐0, 𝑀𝑊𝑐𝑙, 𝑀𝑊𝛿, 𝑝0, and 𝜎 +1), as predicted by the Genetti correlations.32 All 44 tested model forms are shown in Table S1 in the supplemental material. Key statistical best fit values (validation metrics in


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Page 24 of 64

Eqs 9-16) were calculated for each of the model forms, and the values for each of the seven literature correlations as well as the best of the additional tested model forms, referred to as the proposed model, are detailed in Table 2. This table is divided in two main sections. The first section shows statistical results using the seven literature models with original coefficients. The second section shows statistical results using the models with re-fitted coefficients (including the correlation proposed here), which were re-fitted using the aromaticity dataset. For comparison, Table S2 in the supplemental material section includes both the original literature coefficients and the re-fitted coefficients. Table S3 in the supplemental material include the complete statistical results for the final training of the aromaticity correlation analysis.

Table 2. Statistical results of coal aromaticity model analysis Original Coefficients

Fitted Coefficients

Model

L1 Norm

L2 Norm

Infinity Norm

SSE

R2

L1 Nor m

L2 Nor m

Infinity Norm

SSE

R2

Propose d

-

-

-

-

-

0.033

0.044

0.166

0.118

0.797

Ko

0.058

0.076

0.215

0.458

0.595

0.053

0.070

0.228

0.395

0.641

Gerstein

0.075

0.088

0.195

0.626

0.549

0.063

0.079

0.198

0.497

0.549

MV

0.091

0.112

0.367

1.012

0.626

0.056

0.072

0.274

0.411

0.626

SK 1

0.055

0.082

0.349

0.406

0.649

0.040

0.058

0.232

0.204

0.649

SK 2

0.052

0.070

0.231

0.394

0.716

0.047

0.062

0.201

0.304

0.724


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Energy & Fuels

SK 3

0.052

0.070

0.231

0.394

0.716

0.046

0.061

0.212

0.298

0.729

SK 4

2.75E1

4.56E1

2.74E2

1.27E5

0.057

0.034

0.046

0.176

0.129

0.777

As Table 2 shows, Singh and Kakati’s44 second and third correlations (shown in Eqs 6 and 7) performed the best out of all the literature correlations while using original parameters. Upon a re-fitting of the literature correlations, all performed better than the original correlations. This was expected as the literature correlations were not originally fit to this same dataset. Surprisingly, the worst literature correlation (Singh and Kakati no. 444) became the best literature correlation. The proposed model form, however, outperforms all the literature correlations for all five measures of fit (albeit with 9 coefficients and 4 independent variables). The proposed model form for describing coal aromaticity follows the same form as Genetti’s correlations, with the coefficient values in Table 3:

𝑓′𝑎 = 𝑐1 + 𝑐2𝐶𝑐𝑜𝑎𝑙 + 𝑐3𝐶2𝑐𝑜𝑎𝑙 + 𝑐4𝐻𝑐𝑜𝑎𝑙 + c5𝐻2𝑐𝑜𝑎𝑙 + 𝑐6𝑂𝑐𝑜𝑎𝑙 + c7𝑂2𝑐𝑜𝑎𝑙 + c8𝑉𝐴𝑆𝑇𝑀 + 𝑐9𝑉2𝐴𝑆𝑇𝑀(28)

Table 3. Aromaticity correlation coefficients c1

c2

c3

c4

c5

c6

c7

c8

c9

4.384

-8.679E2

5.352E-4

2.601E-2

-6.879E3

3.525E-3

-5.710E4

-2.666E3

5.659E-6


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Page 26 of 64

Figure 5. Parity relationship of seven aromaticity correlations from the literature using both the original and re-fit coefficients and the proposed aromaticity correlation Figure 5 shows parity plots for the coal aromaticity model form tests, showing the predictions using both the original coefficients and the re-fit coefficients. These plots are shown as visual confirmation that Singh and Kakati’s second and third correlations perform the best out of all the literature correlations while using the original coefficients. The fourth correlation performed the best using re-fit coefficients, which is interesting because of its very poor performance using the original parameters. In Figure 5 (with the purple diamonds), the values predicted by the correlation using the original parameters do not appear in the plot due to the scale of the y-axis. Even with the great fit of Singh and Kakati’s fourth correlation, the proposed model form fits the best of all.


26

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Energy & Fuels

Correlations describing coal aromaticity were analyzed primarily to add some coalspecific structural parameters to the elemental composition data used to fit the elemental correlations described in this paper. While none of the aromaticity correlations were used in the final elemental composition correlations, they can be used to limit the amount of expensive and often time-consuming NMR characterization of different coals. The aromaticity can then be used in other models or to predict other structural parameters generally only available from NMR characterizations. 3.2 Correlation of Elemental Compositions Using Cross-Validation Using the cross-validation procedure described in the approach section, ten groups of the entire dataset were used for each of the proposed model forms for carbon, hydrogen, oxygen, nitrogen, and sulfur in the char and carbon, hydrogen, and nitrogen in the tar. Five groups were used for each of the proposed model forms for oxygen and sulfur in the tar. Each of these model forms used a variation including some or all of the following variables: corrected aromaticity (𝑓′𝑎), the maximum gas temperature (in Kelvin), the residence time (in ms), the normalized volatiles yield (𝑉/𝑉𝑚𝑎𝑥, where this ratio is equal to 1 at maximum volatiles yield), the mass fraction of each element in the parent coal (including 𝐻/𝐶 and 𝑂/𝐶 ratios), the ASTM volatile matter on a dry ash-free basis, and key NMR structural parameters (𝑐0, 𝑀𝑊𝑐𝑙, 𝑀𝑊𝛿, 𝑝0, and 𝜎 +1) as predicted by Genetti’s correlations.32 Five main statistical parameters were used in this cross-validation procedure, detailed in the cross-validation section and found in Eqs 9-16. These error


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Page 28 of 64

estimates were then averaged across the five or ten group tests to find the best model forms to move on to the final training step. As an example, Table 4 shows the results of cross-validation for carbon in the tar. For carbon in the tar, 22 different model forms were tested. The experimental data were divided into ten groups for the cross-validation procedure. The average error statistics across all ten tests are in Table 3. A similar process was followed for all ten elements. The complete statistical results for all elements and model forms are found in the appendix, in Tables S5-S14.

Table 4. Averaged tar carbon cross-validation results for 22 model forms Model No.

Ncoeff

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE

22

13

0.573

0.044

0.028

0.035

0.067

0.010

4

5

0.914

0.183

0.034

0.046

0.106

0.017

5

5

0.914

0.183

0.034

0.046

0.106

0.017

6

9

0.922

0.102

0.032

0.044

0.105

0.016

7

5

0.887

0.177

0.044

0.060

0.135

0.028

8

17

0.886

0.052

0.032

0.043

0.102

0.016

23

14

0.791

0.057

0.024

0.028

0.051

0.006

34

16

0.794

0.050

0.023

0.027

0.053

0.006

25

14

0.843

0.060

0.022

0.027

0.050

0.006

26

14

0.785

0.056

0.025

0.030

0.057

0.007


28

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Energy & Fuels

28

14

0.823

0.059

0.022

0.027

0.051

0.006

29

15

0.814

0.054

0.023

0.027

0.051

0.006

35

15

0.819

0.055

0.023

0.028

0.052

0.006

36

14

0.804

0.057

0.022

0.028

0.053

0.006

30

14

0.826

0.059

0.031

0.039

0.076

0.013

31

15

0.847

0.056

0.029

0.038

0.079

0.012

37

15

0.836

0.056

0.031

0.040

0.079

0.013

38

19

0.791

0.042

0.024

0.030

0.054

0.007

21

17

0.869

0.051

0.045

0.059

0.131

0.031

2

9

0.908

0.101

0.034

0.046

0.104

0.017

3

17

0.586

0.034

0.027

0.034

0.062

0.010

33

15

0.854

0.057

0.021

0.029

0.056

0.006

A cross-validation analysis is used to find the best models to move on to the final training step by showing which models are the best at predicting “new” data (in this case, each group left out as a test set), which gives the user more confidence that the correlations can predict elemental compositions for coals or conditions for which the correlations were not fit. As this part of the analysis does not directly factor into the final result, the data and summary are included in Richards60 and the supplemental material section. The cross-validation results are also used as a check against over-fitting, which can cause a model to fit all the data very well, but not accurately predict new data.


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Page 30 of 64

3.3 Model Training Using Complete Dataset The final step in the initial cross-validation analysis was to train the model forms of interest using the complete dataset. This was an intermediary step to provide a maximum of two models to the model refinement analysis: one with the highest 𝑅2 value and a second with the highest 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓 value. The results of this complete training are briefly summarized in Section 3.4 on model refinement, and are detailed in the supplemental material. Table 5 shows an example of the results of this part of the analysis for carbon in the tar. The complete information from this section is found in Section 3.3 in the supplemental material. This includes Tables S15-S34 and Figures S1-S12.

Table 5. Summary of tar carbon model training using complete dataset Step

Model Ncoeff No.

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE

Cross21 validation, best R2

17

0.869

0.051

0.045

0.059

0.131

0.031

Final training, best R2

17

0.824

0.048

0.021

0.031

0.108

0.070

Cross5 validation, best R2/Ncoeff

5

0.914

0.183

0.034

0.046

0.106

0.017

Final training,

5

0.743

0.149

0.027

0.037

0.117

0.103

21

5


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Energy & Fuels

best R2/Ncoeff

3.4 Model Refinement Most of the previously described elemental composition correlations have many fitted parameters. Each correlation was refined by taking the model with the highest R2 value and the model with the highest 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓 value as a basis for the next iteration of models. Each new iteration had two new groups of models. The first group was formed by removing one fitted coefficient from the highest 𝑅2 model at a time, and the second group was formed by replacing one variable (𝑇𝑔𝑎𝑠,𝑚𝑎𝑥, 𝑡𝑟𝑒𝑠, 𝑉𝑛𝑜𝑟𝑚, etc.) in the highest 𝑅2 model with the corresponding variable in the highest 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓 model. After iterating on this process until none of the new models showed an increase in 𝑅2 or 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓, the best of these models is presented in the following sections, with some elements having a best overall fit and a best utility fit. The best overall fit refers to the models with the highest overall 𝑅2 values, and the best utility fit refers to the models with almost as high of 𝑅2 value with fewer coefficients. These best model(s) from each round of model refinement were included in a final cross-validation cycle. As with the initial cross-validation cycle, only the results for the best models are presented here; the remaining details are included in the supplemental material Section 3.4 and in Richards.60


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Page 32 of 64

3.4.1 Tar Carbon Starting with the base models 21 (highest 𝑅2 value) and 5 (highest R2/Ncoeff value), the correlation for carbon in the tar was refined a total of eight times. In the end, model 154 had the best 𝑅2 value had the fewest number of coefficients. The statistical values for both a cross-validation and final training step are included in Table 6.

Table 6. Tar carbon model refinement Step

Model No.

Ncoeff

R2

R2/Ncoeff L1 Norm

L2 Norm

Infinity Norm

SSE

Cross154 validation

11

0.884

0.080

0.041

0.050

0.110

0.020

Final training

11

0.825

0.075

0.022

0.031

0.115

0.070

154

Model 154 had the best overall 𝑅2 value of 0.825, which exceeded the initial crossvalidation cycle model 21 with an 𝑅2 value of 0.824, as shown in Section 3.3.1 of the supplemental material. The cross-validation process resulted in an averaged 𝑅2 value of 0.884, which means that this model is not only accurate over the whole dataset, but predicts new data very well. In addition to high 𝑅2 values, the other validation norms are also very low. Model 154 has a total of 11 coefficients and 4 independent variables,


32

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Energy & Fuels

𝐶𝑡𝑎𝑟 𝐶𝑐𝑜𝑎𝑙

= c1 + c2𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c 𝑇3

3 𝑔𝑎𝑠,𝑚𝑎𝑥

1

1 +

c4𝑇4𝑔𝑎𝑠,𝑚𝑎𝑥

+ c5𝑡𝑟𝑒𝑠 + c 𝑡2

6 𝑟𝑒𝑠

1 +

1

c7𝑡4𝑟𝑒𝑠

+ 1 + c 𝑉3

8 𝑛𝑜𝑟𝑚

+ c9𝐶𝑐𝑜𝑎𝑙 + (29)

c10𝐶2𝑐𝑜𝑎𝑙 + c11𝐶4𝑐𝑜𝑎𝑙

where 𝐶𝑡𝑎𝑟 is the dry, ash-free mass fraction of carbon in the tar, 𝐶𝑐𝑜𝑎𝑙 is the dry, ash-free mass fraction of carbon in the parent coal, 𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 is the maximum gas temperature of the reactor in Kelvin, 𝑡𝑟𝑒𝑠 is the particle residence time in ms, and 𝑉𝑛𝑜𝑟𝑚 is the normalized volatile yield (𝑉/𝑉𝑚𝑎𝑥, which is equal to 1 if a single experiment at the same conditions is performed, or between 0 and 1 for multiple experiments performed at the same temperatures but different residence times). The coefficient values for Eq 29 are found in Table 7. A parity plot showing the predictions given by the final tar carbon correlation are found in Figure 6, which also includes parity plots for the other four tar elemental correlations.


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Page 34 of 64

Figure 6. Parity plots for the finalized correlations in the tar, including both the best overall fits for all five elements and the best utility fits (highest 𝑅2 value with the fewest fitted coefficients) for oxygen, nitrogen, and sulfur. For convenience, parent coal compositions are shown as dashed lines at 1.0 on the x-axis.


34

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Energy & Fuels

Table 7. Tar correlation coefficients Coeff.

C

H

O, best R2

O, best utility

N, best R2

N, best utility

S, best R2

S, best utility

Nvar

4

4

5

5

8

7

4

4

c1

-1.059

2.356E3

-1.700E-3

-5.845E2

0.323

0.400

9.966E2

-1.994E3

c2

0.161

-1.034

3.344

0.176

4.560E-4

3.923E-4

-1.105E11

4.094E2

c3

-1.052

-2.843

-2.493E-3

-8.812E-4

-0.451

-0.163

-2.998E3

3.865E9

c4

0.614

9.306E-3

12.243

15.762

-3.370E3

-4.349E3

3.356E10

-3.171E3

c5

-0.147

-8.391E-6

-1.795E-2

-2.485E-2

6.133E3

8.781E3

-3.586E3

3.337E2

c6

8.371E-2

3.537E3

-2.165E2

-1.543E3

-3.042E3

-4.860E3

3.453E2

7.242E3

c7

-2.379E-5

-6.336E3

-1.017E3

9.242E2

3.078E2

1.623E2

7.137E3

-1.570E2

c8

-12.444

3.234E3

6.063E2

-63.059

-3.140E2

22.101

-2.129E2

c9

-3.115

-37.700

-2.907

67.100

35.669

-0.489

8.338E2

c10

0.980

0.597

63.601

-14.443

0.741

44.154

c11

-3.288E-5

-52.266

11.627

-13.794

c12

5.350

c13

15.977

As shown in Figure 6, most of the tar compositions showed an increase in carbon content over the original parent coal composition as well as a decrease in both the oxygen and sulfur compositions. On average the hydrogen and nitrogen compositions were around the parent composition. 3.4.2 Tar Hydrogen


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Page 36 of 64

Models 7 and 62 from the initial cross-validation and final training were used as base models to refine the tar hydrogen correlation. These models were refined a total of four times, with the model 157 having the best 𝑅2 value and the fewest fitted parameters. The statistical values from cross-validation and final training are included in Table 8.

Table 8. Tar hydrogen model refinement Step

Mode l No.

Ncoeff

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE

Crossvalidatio n

157

10

0.465

0.047

0.118

0.147

0.281

0.147

Final training

157

10

0.800

0.080

0.067

0.087

0.305

0.490

Model 157 had the best 𝑅2 value of 0.80 in final training, which was far higher than model 62 with an 𝑅2 value of 0.697 in the initial cross-validation and final training cycle (detailed in the supplemental material). With a final training 𝑅2 value of 0.80, this model fits all the data very well. With a cross-validation 𝑅2 value of 0.465, this model may not be the best predictor of new data, however, the other validation metrics are a little more promising with low values. Model 157 also has the added benefit of fewer fitted coefficients than model 62,


36

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Energy & Fuels

𝐻𝑡𝑎𝑟 𝐻𝑐𝑜𝑎𝑙

= c1 + c2𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c3𝑡𝑟𝑒𝑠 + c4𝑡2𝑟𝑒𝑠 + c5𝑡3𝑟𝑒𝑠 + c6𝑉𝑛𝑜𝑟𝑚 + c7𝑉2𝑛𝑜𝑟𝑚 + c8𝑉3𝑛𝑜𝑟𝑚 + c9 (30)

𝑀𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖 + c10𝑀2𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖

where 𝐻𝑡𝑎𝑟 is the dry, ash-free mass fraction of hydrogen in the tar, 𝐻𝑐𝑜𝑎𝑙 is the dry, ashfree mass fraction of hydrogen in the parent coal, and 𝑀𝛿 is the NMR structural parameter indicating the average side-chain molecular weight, as predicted by Genetti’s correlation,32

𝑀𝛿 = 4.220 ⋅ 102 ―8.647 ⋅ 𝐶𝑐𝑜𝑎𝑙 +4.639 ⋅ 10 ―2 ⋅ 𝐶2𝑐𝑜𝑎𝑙 ―8.473 ⋅ 𝐻𝑐𝑜𝑎𝑙 +1.182 ⋅ 𝐻2𝑐𝑜𝑎𝑙 +1.154 ⋅ 𝑂𝑐𝑜𝑎𝑙 ―4.340 ⋅ 10 ―2 ⋅ 𝑂2𝑐𝑜𝑎𝑙 +0.557 ⋅ 𝑉𝐴𝑆𝑇𝑀 ―6.546 ⋅ 10 ―3 ⋅ 𝑉2𝐴𝑆𝑇𝑀

(31)

where 𝑂𝑐𝑜𝑎𝑙 is the dry, ash-free mass fraction of oxygen in the parent coal and 𝑉𝐴𝑆𝑇𝑀 is the dry, ash-free ASTM volatile matter of the parent coal (typically from a proximate analysis). Coefficient values for Eq 30 are in Table 7. The parity plot showing the comparison of the predicted normalized hydrogen mass fraction in the tar to the experimentally determined normalized hydrogen mass fraction is found in Figure 6. 3.4.3 Tar Oxygen Most of the models tested for oxygen in the tar performed very well. The best overall model was 159, and the model with the best utility (highest 𝑅2 value with the fewest fitted parameters) was 161. The statistical parameters for these models are included in Table 9.


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Page 38 of 64

Table 9. Tar oxygen model refinement Step

Model Ncoeff No.

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE


11

0.839

0.076

0.096

0.113

0.201

0.090


159

11

0.843

0.077

0.082

0.097

0.229

0.322


10

0.832

0.083

0.100

0.113

0.191

0.090

Final training, best R2/Ncoeff

10

0.842

0.084

0.082

0.097

0.235

0.323

161

Model 159 had the highest 𝑅2 value of 0.843, which was slightly better than model 97 with an 𝑅2 value of 0.842 in the initial cross-validation and final training cycle (which is detailed in the supplemental material). Not only are the final training 𝑅2 values very high, but the cross-validation 𝑅2 values also are very high. This indicates that both tar oxygen correlations presented here predicted new data very well, both with a relatively small number of coefficients,


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Energy & Fuels

𝑂𝑡𝑎𝑟 𝑂𝑐𝑜𝑎𝑙

= c1 + c2𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c3𝑇2𝑔𝑎𝑠,𝑚𝑎𝑥 + c4𝑡𝑟𝑒𝑠 + c5𝑡2𝑟𝑒𝑠 + c6𝑉𝑛𝑜𝑟𝑚 + c7𝑉2𝑛𝑜𝑟𝑚 + c8𝑉3𝑛𝑜𝑟𝑚 + c9 (32)

𝑂𝑐𝑜𝑎𝑙 + c10𝑆2𝑐𝑜𝑎𝑙 + c11𝑆3𝑐𝑜𝑎𝑙

where 𝑂𝑡𝑎𝑟 is the dry, ash-free mass fraction of oxygen in the tar and 𝑆𝑐𝑜𝑎𝑙 is the dry, ashfree mass fraction of sulfur in the parent coal. Model 161 also had a high 𝑅2 value of 0.842, which is equal to that of model 97 in the initial cross-validation cycle,

𝑂𝑡𝑎𝑟

𝑂𝑐𝑜𝑎𝑙c1 + c2𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 2 𝑆𝑐𝑜𝑎𝑙 + c10𝑆3𝑐𝑜𝑎𝑙

+ c3𝑇2𝑔𝑎𝑠,𝑚𝑎𝑥 + c4𝑡𝑟𝑒𝑠 + c5𝑡2𝑟𝑒𝑠 + c6𝑉2𝑛𝑜𝑟𝑚 + c7𝑉3𝑛𝑜𝑟𝑚 + c8log10 𝑂𝑐𝑜𝑎𝑙 + c9 (33)

Model 159 is the best overall fit for tar oxygen, but 161 is the model with the best utility for tar oxygen. This means that model 159 in Eq 21 will give the most accurate results, but model 161 will give almost as accurate of results with fewer coefficients, leading to greater computational efficiency. Coefficients for Eqs 32 and 33 are in Table 7. The predictions based on these correlations are compared to the experimentally measured values, and are found in Figure 6. 3.4.4 Tar Nitrogen Using models 7 and 119 as base models, the correlation for nitrogen in the tar was refined a total of seven times. Model 166 had the highest 𝑅2 value and model 168 performed almost as well, with the highest 𝑅2 value and the fewest fitted coefficients. Statistical results for these models are included in Table 10.


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Page 40 of 64

Table 10. Tar nitrogen model refinement Step

Model Ncoeff No.

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE


13

0.760

0.058

0.094

0.113

0.197

0.085


166

13

0.747

0.057

0.105

0.139

0.388

1.250


11

0.812

0.074

0.084

0.104

0.186

0.070


11

0.747

0.068

0.105

0.139

0.436

1.252

168

Model 166 had the highest 𝑅2 value of 0.747, which was higher than model 119 with a value of 0.717 in the initial cross-validation cycle (detailed in the supplemental material),

𝑁𝑡𝑎𝑟 𝑁𝑐𝑜𝑎𝑙

c7

c8

= c1𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c2𝑇2𝑔𝑎𝑠,𝑚𝑎𝑥 + c3𝑡𝑟𝑒𝑠 + c4𝑉𝑛𝑜𝑟𝑚 + c5𝑉2𝑛𝑜𝑟𝑚 + c6𝑉3𝑛𝑜𝑟𝑚 + 𝑁𝑐𝑜𝑎𝑙 + 1 + 𝑐0 + c9

𝑀𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖 + c10𝑀2𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖 + c11𝑆𝑐𝑜𝑎𝑙 + c12𝑆2𝑐𝑜𝑎𝑙 + c13𝑉𝐴𝑆𝑇𝑀

(34)


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Energy & Fuels

where 𝑁𝑡𝑎𝑟 is the dry, ash-free mass fraction of nitrogen in the tar, 𝑁𝑐𝑜𝑎𝑙 is the dry, ashfree mass fraction of nitrogen in the parent coal, 𝑐0 is the number of stable bridges in the parent coal, as used in the CPD model, and as detailed by Genetti, et al.32 (found in Eq 35), 𝑀𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖 is the NMR structural parameter indicating the average side-chain molecular weight, as predicted by Genetti’s correlation,32 shown in Eq 31, 𝑆𝑐𝑜𝑎𝑙 is the dry, ash-free mass fraction of sulfur in the parent coal, and 𝑉𝐴𝑆𝑇𝑀 is the ASTM volatile matter fraction (on a percent dry, ash-free basis, from 0 to 100).

𝑐0 = min [0.36,max {(0.118 ⋅ 𝐶𝑐𝑜𝑎𝑙 ― 10.1),0.0}] + min [0.15,𝑚𝑎𝑥 {(0.014 ⋅ 𝑂𝑐𝑜𝑎𝑙 ― 0.175),0.0}]

(35)

Model 168 had the best utility with a high 𝑅2 value of 0.747 (only equal to model 166 due to rounding), which was also better than model 119 in the initial cross-validation cycle,

𝑁𝑡𝑎𝑟

c7

2 2 3 𝑁𝑐𝑜𝑎𝑙 = c1𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c2𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c3𝑡𝑟𝑒𝑠 + c4𝑉𝑛𝑜𝑟𝑚 + c5𝑉𝑛𝑜𝑟𝑚 + c6𝑉𝑛𝑜𝑟𝑚 + 𝑁𝑐𝑜𝑎𝑙 + c8𝑀𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖 c10

(36)

+ c9𝑀2𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖 + 𝑆𝑐𝑜𝑎𝑙 + c11𝑉𝐴𝑆𝑇𝑀

While model 166 had the highest 𝑅2 value and will yield the most accurate results, model 168 will give essentially the same results with fewer coefficients and higher computational efficiency. Both correlations show great values for all validation metrics for


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Page 42 of 64

both final training and cross-validation. This means that both correlations fit all the data well and predicted new data well. Coefficients for Eqs 34 and 36 are found in Table 7. Results from both models are shown in parity plots in Figure 6. 3.4.5 Tar Sulfur Starting with models 2 and 4, the tar sulfur correlation was refined a total of two times. The model with the best 𝑅2 value was 2 and the model with the highest utility was 172. Model 2 performed the best in the initial cross-validation analysis and was also the best in the model refinement analysis. The statistical results for models 2 and 172 are included in Table 11.

Table 11. Tar sulfur model refinement Step

Model Ncoeff No.

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE


9

0.880

0.098

0.083

0.105

0.201

0.086


9

0.765

0.085

0.121

0.151

0.358

0.779


7

0.955

0.136

0.062

0.076

0.147

0.042

Final training,

7

0.763

0.109

0.122

0.152

0.337

0.785

2

172


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Energy & Fuels

best R2/Ncoeff

Model 2 performed the best overall with a high 𝑅2 value of 0.765. This is the same as was achieved during the initial cross-validation analysis detailed in the supplemental material,

𝑆𝑡𝑎𝑟 𝑆𝑐𝑜𝑎𝑙

3 5 7 9 = c1 + c2𝑇c𝑔𝑎𝑠,𝑚𝑎𝑥 + c4𝑡c𝑟𝑒𝑠 + c6𝑉c𝑛𝑜𝑟𝑚 + c8𝑆c𝑐𝑜𝑎𝑙

(37)

where 𝑆𝑡𝑎𝑟 is the dry, ash-free mass fraction of sulfur in the tar. Model 172 performed almost as well as model 2 with a high 𝑅2 value of 0.763. This means model 172 will yield almost as accurate of results with fewer fitted coefficients,

𝑆𝑡𝑎𝑟 𝑆𝑐𝑜𝑎𝑙

4 6 = c1 + c2ln (𝑇𝑔𝑎𝑠,𝑚𝑎𝑥) + c3𝑡c𝑟𝑒𝑠 + c5𝑉c𝑛𝑜𝑟𝑚 + c7𝑆𝑐𝑜𝑎𝑙

(38)

Both tar sulfur models have great validation metrics, especially during crossvalidation. Model 172 performed particularly well with an 𝑅2 value of 0.955, which is the best of all presented models. This indicates that model 172 is very well-suited to predict new data with good accuracy. The model coefficient values for Eqs 37 and 38 are given in Table 7. The predictive capabilities of each model are shown in Figure 6. 3.4.6 Char Carbon


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Page 44 of 64

Models 6 and 8 were the basis for refining the correlation for carbon in the char, which was refined a total of six times. Model 121 had the highest 𝑅2 value, and model 126 had the best utility. The statistical results for these models are shown in Table 12.

Table 12. Char carbon model refinement Step

Model Ncoeff No.

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE


16

0.663

0.041

0.037

0.049

0.108

0.025


121

16

0.551

0.034

0.033

0.045

0.188

0.204


11

0.671

0.061

0.036

0.049

0.107

0.025


11

0.548

0.050

0.033

0.046

0.190

0.205

126


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Energy & Fuels

Model 121 had the highest 𝑅2 value at 0.551, which is higher than model 8 at 0.541 from the initial cross-validation analysis (detailed in the supplemental material),

1

𝐶𝑐ℎ𝑎𝑟 1

1

1

1

1

2 3 4 2 3 𝐶𝑐𝑜𝑎𝑙 = c1 + c2𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c3𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c4𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c5𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c6𝑡𝑟𝑒𝑠 + c7𝑡𝑟𝑒𝑠 + c8𝑡𝑟𝑒𝑠 + c9 1

1

1

1

1

4 2 3 4 2 4 + c10𝑉𝑛𝑜𝑟𝑚 + c11𝑉𝑛𝑜𝑟𝑚 + c12𝑉𝑛𝑜𝑟𝑚 + c13𝑉𝑛𝑜𝑟𝑚 + c14𝐶𝑐𝑜𝑎𝑙 + c15𝐶𝑐𝑜𝑎𝑙 + c16𝐶𝑐𝑜𝑎𝑙 𝑡𝑟𝑒𝑠

(39)

where 𝐶𝑐ℎ𝑎𝑟 is the dry, ash-free mass fraction of carbon in the char. Model 126 had a high 𝑅2 value of 0.548, which is also higher than model 8 from the initial cross-validation analysis, indicating that model 126 will yield almost as accurate predictions as model 121 with fewer coefficients and greater computational efficiency,

𝐶𝑐ℎ𝑎𝑟

1

1

1

1

1

2 4 2 3 4 𝐶𝑐𝑜𝑎𝑙 = c1 + c2𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c3𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c4𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c5𝑡𝑟𝑒𝑠 + c6𝑡𝑟𝑒𝑠 + c7𝑡𝑟𝑒𝑠 + c8𝑡𝑟𝑒𝑠 + c9 1

(40)

2 exp (𝑉𝑛𝑜𝑟𝑚) + c10𝐶𝑐𝑜𝑎𝑙 + c11𝐶𝑐𝑜𝑎𝑙

Model 126 was formed by taking out excess coefficients and combining the variables from models 6 and 8, by replacing all 𝑉𝑛𝑜𝑟𝑚 polynomial-type variables with exp (𝑉𝑛𝑜𝑟𝑚). Coefficient values for Eqs 39 and 40 are found in Table 13. The predictions from both model 121 and model 126 are shown in Figure 7. While both model 121 and 126 were the best performers and predict the carbon composition of char slightly better than the models tested in the initial cross-validation cycle, neither performed well.


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Page 46 of 64

Looking at Figure 7 (top left plot), both models appear to under-predict the compositions on both the lower end and the higher end, and over-predict the compositions in the middle. This indicates the possibility of different dependencies to predict the carbon in the char or of inaccuracies or biases in the data.

Figure 7. Parity plots for the finalized correlations in the char, including the best overall fits for all five elements and the best utility fits (highest 𝑅2 value with the fewest fitted coefficients) for carbon, nitrogen, and sulfur. For convenience, parent coal compositions are shown as dashed lines at 1.0 on the x-axis.

Table 13. Char correlation coefficients


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Energy & Fuels

Coeff.

C, best R2

C, best utility

H

O

N, best R2

N, best utility

S, best R2

S, best utility

4

4

5

5

5

5

4

4

c1

9.174E3

1.025E3

-2.609E4

7.662

17.520

0.330

-0.186

-1.090

c2

2.647

1.292

3.335E4

-4.265E-3

1.001E-5

2.056

-7.586E2

-9.211E2

c3

-4.109E2

-1.689E2

-5.592E2

-4.651

2.313E3

-2.083E-3

-4.173E2

3.572E-2

c4

8.319E2

1.117E3

2.696E4

1.160E-2

3.621

-4.066E2

-5.797E2

2.089E2

c5

9.245E2

-2.055

-5.503

-9.854E-6

-8.835E-3

9.430E2

0.492

2.483E2

c6

-3.955

2.362E2

-1.381E2

1.151E2

7.171E-6

-6.225E2

3.403E2

1.942E3

c7

4.000E2

-8.559E3

1.964E3

-1.352E2

-7.362E2

-2.110E2

-5.961E2

50.417

c8

-6.016E2

5.622E2

1.486E4

1.494E4

1.640E3

58.039

9.768E2

62.874

c9

-7.694E2

-33.779

-3.229E3

-2.231E3

-9.938E2

42.666

8.907E2

-2.029E5

c10

-2.475E2

18.491

-7.507E-6

-2.012E2

4.729E2

-0.555

1.305E4

2.671E4

c11

3.999E2

-4.095E2

4.219E3

6.199

-6.054E2

-3.538E4

1.507E4

c12

-2.843E2

-6.033E-2

1.599E2

3.100E4

c13

90.505

41.613

76.299

c14

23.009

-0.543

-1.769E5

c15

-6.063E2

9.786E4

c16

6.987E2

-1.266E4

Nvar

As Figure 7 shows, the carbon and nitrogen in the char tend to increase from the parent coal composition, the hydrogen and oxygen tend to decrease, and the sulfur tends to stay around the parent composition. 3.4.7 Char Hydrogen


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Page 48 of 64

Models 2 and 31 were used to refine the correlation for hydrogen in the char a total of three times. This led to a finalized correlation with the highest 𝑅2 value by using model 127. The statistical results for this model are in Table 14. There was no other model with better utility.


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Energy & Fuels

Table 14. Char hydrogen model refinement Step

Model Ncoeff No.

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE

Cross127 validation

11

0.831

0.076

0.055

0.067

0.134

0.046

Final training

11

0.800

0.073

0.058

0.072

0.221

0.509

127

Model 127 performed the best of all models with a high 𝑅2 value of 0.80, which is higher than model 31 at 0.753 in the initial cross-validation analysis (detailed in the supplemental material),

𝐻𝑐ℎ𝑎𝑟 𝐻𝑐𝑜𝑎𝑙

3 5 7 9 11 = c1 + c2𝑇c𝑔𝑎𝑠,𝑚𝑎𝑥 + c4𝑡c𝑟𝑒𝑠 + c6𝑉c𝑛𝑜𝑟𝑚 + c8𝐻c𝑐𝑜𝑎𝑙 + c10𝑉c𝐴𝑆𝑇𝑀

(41)

where 𝐻𝑐ℎ𝑎𝑟 is the dry, ash-free mass fraction of hydrogen in the char. Good validation metrics (very high 𝑅2 values and low values for the other metrics) for model 127 indicate that this char hydrogen correlation fit all the data well and was a good predictor for new data. This means that model 127 can be reliably used to predict the composition of hydrogen in the char from a wide variety of different coals. Coefficient values for Eq 41 are in Table 13. The predictive capabilities of this model are shown in the parity plot in Figure 7.


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Page 50 of 64

3.4.8 Char Oxygen Models 5 and 21 served as a base for refining the char oxygen correlation, which was performed a total of four times. This led to model 134 with the best overall 𝑅2 value. The statistical results for this model are shown in Table 15. No other model had better utility for oxygen in the char.

Table 15. Char oxygen model refinement Step

Model Ncoeff No.

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE

Crossvalidation

134

12

0.628

0.052

0.138

0.174

0.325

0.251

Final training

134

12

0.447

0.037

0.160

0.203

0.565

3.327

Model 134 had the highest 𝑅2 value of 0.447, which was higher than model 21 with a value of 0.438 in the initial cross-validation analysis (detailed in the supplementary material),

𝑂𝑐ℎ𝑎𝑟 𝑂𝑐𝑜𝑎𝑙

= c1𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c2𝑇2𝑔𝑎𝑠,𝑚𝑎𝑥 + c3𝑡𝑟𝑒𝑠 + c4𝑡2𝑟𝑒𝑠 + c5𝑡3𝑟𝑒𝑠 + c6𝑉𝑛𝑜𝑟𝑚 + c7𝑉2𝑛𝑜𝑟𝑚 + c8

exp (c9𝑂𝑐𝑜𝑎𝑙) + c10𝑉𝐴𝑆𝑇𝑀 + c11𝑉2𝐴𝑆𝑇𝑀 + c12𝑉3𝐴𝑆𝑇𝑀

(42)

where 𝑂𝑐ℎ𝑎𝑟 is the dry, ash-free mass fraction of oxygen in the char. Model 134 was created by removing unnecessary coefficients and replacing the variables of 𝑂𝑐𝑜𝑎𝑙 in


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Energy & Fuels

model 21 with exp (𝑂𝑐𝑜𝑎𝑙) from model 5. The values of the coefficients in Eq 42 are in Table 13. The predictive capabilities of model 134 are shown in the parity plot in Figure 7. Char oxygen is usually determined by difference, and hence has more scatter than the other elements. For this reason, the best R2 value from a curve fit is less than 0.5.

3.4.9 Char Nitrogen The correlation for nitrogen in the char was unique as model 62 had both the highest 𝑅2 value and the highest 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓 value. This model was used as the base for refinement, which was repeated a total of four times. Model 62 remained the best overall fit with model 142 as the best utility. The statistical results for these models are found in Table 16.

Table 16. Char nitrogen model refinement Step

Model Ncoeff No.

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE


14

0.600

0.043

0.077

0.097

0.185

0.097


14

0.597

0.043

0.077

0.105

0.312

1.084

62


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Page 52 of 64


10

0.682

0.068

0.076

0.090

0.156

0.079


10

0.585

0.059

0.077

0.106

0.324

1.117

142

Model 62 was still the best overall fit with a moderate 𝑅2 value for final training of 0.597 and a cross-validation 𝑅2 of 0.6. This indicates that this model is moderately accurate across all the data and will predict new data moderately well.

𝑁𝑐ℎ𝑎𝑟

c3 2 3 𝑁𝑐𝑜𝑎𝑙 = c1 + c2𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c4𝑡𝑟𝑒𝑠 + c5𝑡𝑟𝑒𝑠 + c6𝑡𝑟𝑒𝑠 + c7𝑉𝑛𝑜𝑟𝑚 c11𝑁2𝑐𝑜𝑎𝑙 + c12𝑁3𝑐𝑜𝑎𝑙 + c13𝑀𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖 + c14𝑀2𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖

+ c8𝑉2𝑛𝑜𝑟𝑚 + c9𝑉3𝑛𝑜𝑟𝑚 + c10𝑁𝑐𝑜𝑎𝑙 + (43)

where 𝑁𝑐ℎ𝑎𝑟 is the dry, ash-free mass fraction of nitrogen in the char and 𝑀𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖 is the NMR structural parameter for the average side-chain molecular weight, as predicted by Genetti’s correlation32 in Eq 31. Model 142 had a moderately high 𝑅2 value of 0.585, which is less than model 62, however, model 142 has a higher 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓, giving it greater utility. This means the predictions will be slightly less accurate, but it has fewer fitted coefficients and greater computational efficiency.


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Energy & Fuels

𝑁𝑐ℎ𝑎𝑟 𝑁𝑐𝑜𝑎𝑙

= c1𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c2𝑡𝑟𝑒𝑠 + c3𝑡2𝑟𝑒𝑠 + c4𝑉𝑛𝑜𝑟𝑚 + c5𝑉2𝑛𝑜𝑟𝑚 + c6𝑉3𝑛𝑜𝑟𝑚 + c7𝑁2𝑐𝑜𝑎𝑙 + c8𝑁3𝑐𝑜𝑎𝑙 + c9 (44)

𝑀𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖 + c10𝑀2𝛿,𝐺𝑒𝑛𝑒𝑡𝑡𝑖

The values for the coefficients in Eqs 43 and 44 are found in Table 13. Predictions using both model 62 and model 142 are shown in the parity plot in Figure 7. 3.4.10 Char Sulfur Models 6 and 21 performed the best in the initial cross-validation analysis, so they were used as a basis to refine the correlation for sulfur in the char. This was done a total of five times, with model 143 having the best overall fit and model 148 having the best utility. The statistical results for these models are detailed in Table 17.

Table 17. Char sulfur model refinement Step

Model Ncoeff No.

R2

R2/Ncoeff

L1 Norm

L2 Norm

Infinity Norm

SSE


16

0.428

0.027

0.214

0.272

0.512

0.597


143

16

0.605

0.038

0.180

0.241

0.620

4.703

Cross148 validation,

11

0.417

0.038

0.261

0.322

0.581

0.903


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Page 54 of 64

best R2/Ncoeff Final training, best R2/Ncoeff

148

11

0.605

0.055

0.182

0.241

0.601

4.702

Model 143 had the highest 𝑅2 value at 0.605, which was higher than model 21 at 0.565 in the initial cross-validation analysis detailed in the supplementary material,

𝑆𝑐ℎ𝑎𝑟 𝑆𝑐𝑜𝑎𝑙

1

= c1𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c 𝑇2 1


1 + c10𝑉2𝑛𝑜𝑟𝑚 + c11𝑉3𝑛𝑜𝑟𝑚 + c12𝑉4𝑛𝑜𝑟𝑚

+ c3𝑇3𝑔𝑎𝑠,𝑚𝑎𝑥 + c4𝑇4𝑔𝑎𝑠,𝑚𝑎𝑥

+ c13𝑆𝑐𝑜𝑎𝑙 + c

2 14𝑆𝑐𝑜𝑎𝑙

+ c5𝑡𝑟𝑒𝑠 + c 𝑡2 1

6 𝑟𝑒𝑠

1 + c7𝑡3𝑟𝑒𝑠 + c8𝑡4𝑟𝑒𝑠

+ c9𝑉𝑛𝑜𝑟𝑚 + (45)

+ c15𝑆3𝑐𝑜𝑎𝑙 + c16𝑆4𝑐𝑜𝑎𝑙

where 𝑆𝑐ℎ𝑎𝑟 is the dry, ash-free mass fraction of sulfur in the char.

Model 148 has a

very close 𝑅2 value to model 143 (0.605 with rounding), meaning the predictions from model 148 are almost as accurate with fewer coefficients and greater computational efficiency.

𝑆𝑐ℎ𝑎𝑟 𝑆𝑐𝑜𝑎𝑙

1

= c1𝑇𝑔𝑎𝑠,𝑚𝑎𝑥 + c 𝑇3 1

c9𝑆2𝑐𝑜𝑎𝑙


+ c3𝑡𝑟𝑒𝑠 + c 𝑡2

4 𝑟𝑒𝑠

1 + c5𝑡4𝑟𝑒𝑠

+ c6exp (c7𝑉𝑛𝑜𝑟𝑚) + c8𝑆𝑐𝑜𝑎𝑙 + (46)

+ c10𝑆3𝑐𝑜𝑎𝑙 + c11𝑆4𝑐𝑜𝑎𝑙

Coefficients for Eqs 45 and 46 are found in Table 13. Predictions for models 143 and 148 are shown in a parity plot in Figure 7. As with oxygen, sulfur values are not


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Energy & Fuels

always measured separately, and are often either lumped in with the oxygen compositions or the sulfur types (pyritic, organic, etc.) are not distinguished. This can lead to inaccuracies in experimental data that are then passed along to the fitted model. 4. SUMMARY AND CONCLUSIONS The intent of this paper was to provide correlations of the elemental compositions of coal tar and char after devolatilization. This research should help improve treatments of chemistry in simulations of coal combustion and gasification. This paper first compared various model forms to predict the corrected coal aromaticity, including several found in other articles. While some of those found in literature performed well against the experimental dataset, the proposed model discussed in this paper out-performed all the literature models for every measured statistical value, even after re-fitting the literature model coefficients using the same dataset. These results demonstrate that the proposed aromaticity correlation accurately predicts the corrected coal aromaticity given an input of the daf mass fractions of C, H, O, and ASTM volatile matter. The dataset used included coal ranks ranging from lignites to anthracites. Several sets of elemental composition data for tars and chars were obtained from the literature. Most of the tar and char carbon values increased over the parent coal composition, meaning the tar and char become enriched in carbon. This indicates that the light gas has a lower carbon content than both the char and the tar. The hydrogen in the char tended to increase over parent coal composition, and on average, the hydrogen in


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Page 56 of 64

the tar stayed around the parent coal composition. The light gas hydrogen content likely decreases from parent coal composition. Both the tar and char oxygen content decreases compared to parent coal composition, indicating that the light gas species are enriched in oxygen. Nitrogen in the char mostly increases and the nitrogen in the tar generally stays about the same as the parent coal, meaning that the light gas species likely increase in nitrogen composition. Similar to the oxygen, sulfur content in both the char and the tar decreases. This means that the light gas species tend to be enriched in sulfur. A cross-validation procedure coupled with a model refinement procedure was used to develop a set of correlations describing the elemental compositions of primary coal tar and the corresponding char. Consideration was given to the model form that gave the best overall fit as well as model forms with the best utility, meaning the highest possible 𝑅2 value with the fewest fitted coefficients. The trends observed experimentally are reproduced by the correlations. All the correlations use such inputs as the reaction conditions (maximum gas temperature, particle residence time, and a normalized volatile yield), and parent coal structural and compositional information (element mass fractions, ASTM volatiles yield, and NMR structural parameters such as the number of stable bridges and the average molecular weight of the side chains). Through careful statistical calculation, these correlations may reasonably be used to estimate the dry, ash-free mass fraction of each element in the char


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Energy & Fuels

and the tar, and may be applied to simulations of coal combustion to more accurately predict chemical interactions of various combustion products. While the tar correlations show good agreement with the experimental data with 𝑅2 values of at least 0.75, most of the char correlations fall below that, with the carbon and oxygen correlations at 𝑅2 values of approximately 0.55 and 0.45, respectively. The char correlations might be improved in the future by fitting them to new and better experimental data. In addition, a careful analysis of the consistency of the experimental data may show possible outliers or bad data, which upon the removal of these bad data could improve the fit and prediction capabilities of the correlations.

ASSOCIATED CONTENT Supporting Information. The following files are available free of charge. Data tables detailing all results from the initial cross-validation cycle and the model refinement analysis, as well as figures showing statistical trends with respect to number of model coefficients. (PDF) AUTHOR INFORMATION Corresponding Author *Email: [email protected].


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Page 58 of 64

Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. All authors contributed equally.

ACKNOWLEDGMENT This material is based upon work supported by the Department of Energy, National Nuclear Security Administration, under Award Number DE-NA0002375. NOMENCLATURE Variable

Description

𝑓𝑎

Apparent aromaticity.

𝑓′𝑎

True or corrected aromaticity (corrected to exclude carbonyl bonds).

𝑀𝑐𝑙

Average molecular weight per aromatic cluster.

𝑀𝛿

Average molecular weight per side chain.

𝑝0

Fraction of intact bridges.

(𝜎 + 1) 𝑐0 𝑉𝐴𝑆𝑇𝑀

Coordination number. Number of stable bridges. Dry, ash-free volatile matter yield from ASTM standard test (presented in proximate analyses of parent coal).


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Energy & Fuels

𝐶

Dry, ash-free mass fraction of carbon.

𝐻

Dry, ash-free mass fraction of hydrogen.

𝑂

Dry, ash-free mass fraction of oxygen.

𝑁

Dry, ash-free mass fraction of nitrogen.

𝑆

Dry, ash-free mass fraction of sulfur.

𝑋

Dry, ash-free mass fraction of any element (CHONS).

𝑎, 𝑏, 𝑐, …

Fitted model coefficients.

𝑇𝑔𝑎𝑠,𝑚𝑎𝑥

Maximum reactor gas temperature, measured in Kelvin.

𝑡𝑟𝑒𝑠

Particle residence time, measured in milliseconds.

𝑉𝑛𝑜𝑟𝑚

Normalized volatiles yield: 𝑉/𝑉𝑚𝑎𝑥, has a value of 0 to 1. This is calculated as the current volatiles yield over the maximum volatiles yield. Only multiple experiments performed at the same conditions but different residence times or sampling heights have a value other than 1.

𝑁𝑐𝑜𝑒𝑓𝑓

Total number of fitted coefficients of a given model.

𝑁𝑣𝑎𝑟

Total number of independent variables of a given model.

𝐿1

The L1 norm, described in Eq 9. This gives the average absolute error of a given dataset.

𝐿2

The L2 or Euclidian norm, described in Eq 10. This norm gives the rootmean-square error (RMSE) of a given dataset.

𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑦

The infinity norm, described in Eq 11. This norm gives the maximum absolute error of a given dataset.

𝑆𝑆𝐸

The sum of squared errors, given in Eq 12. This is the value that is optimized for the cross-validation and final training steps.

𝑅2

Sometimes called the coefficient of determination, this value measures the goodness-of-fit of an optimization or regression. It has a value ranging from 0 to 1, with 0 being the worst possible (or no) fit and 1 being a perfect fit. This quantity is described in Eqs 13-16.


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𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓

Page 60 of 64

A modified 𝑅2 calculation used to measure model “utility.” This is to avoid over-fitting. A model with fewer coefficients but a similar 𝑅2 value will have a higher 𝑅2/𝑁𝑐𝑜𝑒𝑓𝑓 value.

𝑦

Predicted or model value.

𝑦

Observed or experimental value (from literature)

𝑁

Statistics, the total number of points in a given dataset.

𝑛

Statistics, used as both a subscript and variable, i.e. data point “n” in a dataset.

Subscripts 𝑖

A product (char or tar) composition, used in conjunction with 𝑋.

𝑖0

A parent coal composition, used in conjunction with 𝑋.

𝑐ℎ𝑎𝑟

A quantity from the char.

𝑡𝑎𝑟

A quantity from the tar.

𝑐𝑜𝑎𝑙

A quantity from the parent coal.

𝐺𝑒𝑛𝑒𝑡𝑡𝑖

NMR structural parameters calculated using Genetti’s original correlations.32 This subscript is used in conjunction with 𝑀𝑐𝑙, 𝑀𝛿, 𝑝0, and (𝜎 + 1).

𝑟𝑒 ― 𝑓𝑖𝑡

NMR structural parameters calculated using a re-fit of Genetti’s original correlations (using NMR data from the aromaticity analysis). This subscript is used in conjunction with 𝑀𝑐𝑙, 𝑀𝛿, 𝑝0, and (𝜎 + 1).

𝑚𝑒𝑎𝑠

Measured NMR structural parameters. This subscript is used in conjunction with 𝑀𝑐𝑙, 𝑀𝛿, 𝑝0, and (𝜎 + 1).

𝑛

Statistics, used as both a subscript and variable, i.e. data point “n” in a dataset.

∞

The infinity norm, described in Eq 11.

REFERENCES


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Correlations of the Elemental Compositions of Primary Coal

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