Ind. Eng. Chem. Res. 2009, 48, 1683–1693
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Attribute-Based Modeling of Resid Structure and Reaction Darin M. Campbell,† Craig Bennett,‡ Zhen Hou,‡ and Michael T. Klein*,†,‡ Department of Chemical Engineering, UniVersity of Delaware, Newark, Delaware 19711, and Department of Chemical and Biochemical Engineering, Rutgers UniVersity, Piscataway New Jersey 08854
An attribute-based approach to modeling resid structure and reaction is described. Monte Carlo simulation of feedstock structure casts the modeling problem in molecular terms. This technique samples probability density functions (PDFs) for the attributes of the resid molecular structures to construct a representative molecular sample whose properties are compared against measured properties. Optimization methods are used to minimize the weighted sum of squares difference, and the final set of PDF parameters are the mathematical representation of resid structure. Subsequent reaction models can be based on the molecular attributes of the PDFs. The latter model, which is called the “Attribute Reaction Model” (ARM), provides a large reduction in the number of reaction equations and, therefore, solution time. The combined structural and reaction model provides a very good representation of laboratory data for four resids. Introduction The goal of molecular-level chemical kinetics models is to predict the dependence of the molecular composition on the reaction conditions. This allows use of structure-property correlations in the estimation of end-use and other properties of the molecular mixture. There are several challenges to be overcome for heavy hydrocarbon mixtures: not only are the mixtures complex, but the molecules within the mixtures are complex. There will often be thousands of “multifunctional” component species. To fix ideas, modern analytical chemistry suggests that on the order of 50 000 species can be identified and therefore could be used to describe a petroleum resid on a molecular basis. Traditional deterministic reaction models will comprise one differential equation for each species, and the numerical burden of solving 50 000 simultaneous ordinary differential equations (ODEs) is beyond the upper limit of what is presently considered practical. The goal of a resid modeling approach then would be to represent both the molecules and their chemistry with fewer balance equations. Our recent work has explored hybrid strategies to represent the molecular structures in heavy hydrocarbon mixtures and their reactions, both in terms of discrete molecules and probability density functions (PDFs) for molecular attributes. Our resulting approach, which we call Attribute Reaction Modeling (ARM), samples the information from a heavy hydrocarbon analytical analysis (HHAA) (e.g., extended SARA/PONA/Z number characterization) to construct a probabilistic structural representation and an associated reaction model in terms of the attributes for each molecule. The savings in reaction modeling computer processing unit (CPU) time is enormous. After reaction of the attributes, the molecules can be reassembled by sampling the PDFs for the attributes and, in turn, the HHAA * To whom correspondence should be addressed. Tel.: 732-445-4453. E-mail:
[email protected]. † Department of Chemical Engineering, University of Delaware. ‡ Department of Chemical and Biochemical Engineering, Rutgers University.
table can be repopulated. This hybrid approach allows the modeler to maintain the properties information of the full molecular sample (e.g., the 50 000 molecules) while modeling the kinetics in terms of a few hundred ODEs. The goal of the present review is to summarize the two main elements of this approach: modeling the molecular composition of heavy hydrocarbon feeds and their associated reaction paths. Resid Structural Models A key challenge in developing a molecule-based model for petroleum resids is that analytical techniques will not provide the identities and concentrations of the individual molecules; instead, they will gather information about the indirect structural attributes of the molecules. To construct a molecular representation of resid, it then is necessary to transform quantitative information from these analytical tests about the attributes into a molecular representation of the mixture. This modeling approach has been published in detail elsewhere and, therefore, only the key points will be highlighted here. Probability Density Functions (PDFs) for Resid Structural Attributes. A petroleum molecule can be viewed as a juxtaposition of structural attributes. For resids, the natural attributes will be the type of molecule (e.g., a saturates, aromatics, resins, and asphaltenes (SARA) category). Within these categories, paraffins will have their lengths and branching degree as attributes; for aromatics, the attributes are the number of aromatic rings, number of naphthenic rings, the number of alkyl sidechains, and so on. From a statistical view, it would be desirable to organize the information about these attributes into a probability distribution function (PDF). The PDF is a function (e.g., lognormal, γ, χ2, exponential) that provides the quantitative probability p of finding a value x or less of a given attribute. By sampling the attribute PDFs, the values of the structural attributes can be determined for an individual molecule, and a complete set of attributes would define each molecule. Each PDF is characterized by a set of parameters (R, β, γ, etc.). Trauth and co-workers3-5 used PDFs to model resid attributes, and they placed special emphasis on the χ2 distribution,
10.1021/ie8012314 CCC: $40.75 2009 American Chemical Society Published on Web 01/23/2009
1684 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009
which is a special case of the γ distribution where the standard deviation equals one-half of the mean. Irrespective of the choice of the continuous PDF, the discrete nature of real feedstocks composed of attributes with discrete integer values renders it necessary to transform these continuous distributions into discrete distributions. This requires division of the distribution into intervals and evaluation of a representative value for each interval. The attributes of the molecules in a complex feedstock all are also of finite values. Therefore, it is necessary to truncate the distributions at some physically reasonable value. A truncation criterion may be set by specifying the minimum contribution that each new interval must make to the cumulative distribution, expressed on a fractional basis. Trauth found that a value of 0.01 worked well.3 PDF Sampling. Using the information in the resid attribute PDFs requires a sampling protocol. One attractive technique is Monte Carlo sampling, where random numbers are matched to the PDFs and the combinations of the attributes provide the molecular representation. The process consists of building a molecular ensemble whose properties are compared against the experimentally measured values. Clearly there exists an optimal sample size. On the one hand, the number of sample molecules must be small enough to allow for time-efficient optimization. On the other hand, if the sample size is too small, the large variation in the objective function may make different sets of PDF parameters statistically indistinguishable. Therefore, a compromise between the desired accuracy and the time requirements must be established. Petti et al.4 determined that a sample size of 10 000 optimally balanced the CPU demand and the desired accuracy of the objective function. Resid molecules will generally comprise multiple attributes, each of which being characterized by a PDF. The sequence of PDF sampling thus becomes important. Clearly, before the detailed attributes of a resid molecule can be specified, the basic type of molecule must first be determined. Therefore, the first PDF sampled is the compound class distribution. For instance, when building a petroleum resid molecule, the first determination is whether the molecule is a paraffin, a naphthenic, an aromatic, or an asphaltene. After the molecule type has been determined, the PDFs that define the detailed attributes for that molecule type can, in principle, be sampled in any order, to specify the molecular structure. However, because certain attribute values may be dependent on other attribute values, ordering the sampling of the PDFs will be beneficial. These notions of conditional probability will allow the use of a given PDF to be dependent on the outcome of sampling a previous PDF. Monte Carlo Optimization. The Monte Carlo-generated ensemble of 10 000 or more molecules conveys a set of mixture properties. The most useful properties are those that can also be measured experimentally. This allows calculation of a quantitative measure of the model likelihood, and optimization of the PDF parameters, by placing the Monte Carlo construction procedure inside an optimization loop, allows specification of the final optimal resid representation and associated PDF parameters. The χ2 statistic is a typical objective function that is used to optimize a representative feed to an actual feed: N
χ ) 2
∑ i)1
[
E PM i (R, β, γ) - Pi σi
]
2
(1)
The numerator is the square of the difference between the model prediction P(R, β, γ) (i.e., the properties of the computer
molecules) and the experimentally determined properties (PE). The denominator is a weighting factor that is equal to the standard deviation of the experimentally determined value. This objective function can be modified easily for any particular analytical information on a feed. This objective function is minimized using a global optimization routine where the PDF parameters are varied and a representative set of molecules is built for each case. The minimized value of the χ2 parameter in eq 1 can be used with the degrees of freedom to obtain an estimate of the model likelihood. The iterative nature of these composition modeling methods renders them completely general and not limited to any particular suite of analytical tests. The objective function to be minimized can accept any test for which there is a reasonable structure property correlation. However, the precision of the output, i.e., the uncertainty of the resid composition model, is directly dependent on the number and precision of these analytical inputs. A Petroleum Resid Example. Trauth’s experimental and modeling work3-5 provides both a point of illustration and comparison. His approach was able to capture successfully many key analytical properties of petroleum resid, but CPU time issues during optimization led him to limit the functional forms of the PDFs that could be used. Modifications introduced herein allowed the consideration of a broader base of PDFs, and the discussion to follow then compares the results of the two approaches using Trauth’s dataset.3-5 In Trauth’s approach, the χ2 distribution was used to model most attributes. The exponential distribution was used to model the sidechain length, and the γ distribution was used to model the number of sidechains. This approach was extended in the present work so that a γ distribution was used to model all the attributes. The extra parameter in the γ distribution requires more CPU time for optimization but allows greater flexibility to minimize the objective function. Therefore, special emphasis in the discussion below is placed on the comparison of the χ2 PDF used by Trauth’s group and the γ distribution used herein. Trauth’s characterization data3-5 for the four petroleum resids (Hondo, Maya, Arabian Light, and Arabian Heavy), which are shown in Table 1, allow illustration. Key experimental points will be summarized only briefly, as full details are available elsewhere.3-5 The elemental analyses had an experimental standard deviation of 2% (relative) for the H/C ratio, as determined by replicate runs. Sulfur was assumed to have the same reproducibility. The vapor-pressure osmometry (VPO) estimates of molecular weight (Mn) were made using a UIC Corporation Model 070 vapor pressure osmometer. Toluene was used as the solvent for the saturate, aromatic, and resin fractions, as well as the complete resid. Nitrobenzene was used as the solvent for asphaltenes in an effort to reduce agglomeration.6 Replicate experiments indicated that the relative standard deviation of the value of Mn was 5%. The proton nuclear magnetic resonance (NMR) experiments were performed on a Bruker Model AM250 instrument. The spectrum was divided into four regions: Haromatic (6.0-9.5 ppm), HR (2.3-4.1 ppm), Hnaphthenic+methylene (1.0-2.3), and Hmethyl (0.1-1.0 ppm). Because of the overlap of peaks, only the aromatic and R peaks were considered, because information on the other peaks was considered to be less reliable. Replicate experiments indicated absolute standard deviations of 0.01 and 0.02 in the fractional content of aromatic and R protons, respectively. All SARA (saturates, aromatics, resins, asphaltenes) separations had a total recovery that was well within 1% of the initial
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 1685 Table 1. Measured and Modeled Analytical Properties of Four Petroleum Resids measured
χ2
γ
measured
921 1.47 6.3 9 6 1.0 16.2 61.1 21.7 0 9 91
881 1.45 6.8 11 8 1.0 14.6 62.0 22.4 1 8 92
944 1.46 3.2 9 7 2.6 22.9 62.5 12.0 1 11 88
10.8
2.3
Hondo molecular weight, Mn (g/mol) H/C sulfur content (wt %) HR (%) Haromatic (%) paraffin naphthenic aromatics + resin asphaltene heavy gas oil, HGO vacuum gas oil, VGO resid
862 1.47 6.9 13 9 1.3 12.0 63.7 23.0 1 7 92
objective function
objective function
842 1.42 4.0 9 11 1.6 14.7 77.9 6.0 1 9 91
957 1.41 3.0 11 8 1.4 21.1 71.4 6.2 2 9 89
949 1.46 3.2 10 7 1.3 23.0 62.4 13.3 2 10 87
6.8
0.5
Arabian Heavy
792 1.33 7.5 8 11 1.0 15.7 76.6 6.6 0 11 88
842 1.42 4.0 8 11 0.9 15.0 77.9 6.2 1 9 91
5.2
0.2
resid. The saturates fraction contained both paraffinic and naphthenic compounds. Because no simple and inexpensive technique for the separation of these saturate compounds was available, it was assumed that 10% of the saturates fraction was paraffinic. Furthermore, other than size, no differences could be identified between the resin and the aromatic classes, so they were considered to be a single compound class. Simulated distillations provided boiling point distributions categorized according to the following boiling point ranges: heavy gas oil (HGO) (610-800 °F), vacuum gas oil (VGO) (800-1000 °F), and resid (1000+ °F). None of the resids presented here contained any material that boiled below 610 °F. The experimental standard deviation for each of the boiling fractions was 1% (absolute). These measurements were used to estimate the PDF parameters for six unique resid attributes: the number of aromatic rings, the number of thiophenic rings, the number of naphthenic rings, the number of alkyl sidechains, the length of alkyl sidechains, and the number of sidechain S atoms. The main comparison with Trauth’s work is the form of the PDF. Because petroleum resid is defined only by a minimum boiling point, skewed PDFs were chosen to model the various structural attributes. The γ distribution allows for the greatest flexibility, because there are three parameters with which to work. However, the extra parameter introduces an extra time requirement for optimization. This led Trauth to model only the most sensitive structural attribute (number of sidechains) using a γ distribution. For the length of alkyl sidechains, he used an exponential distribution, and all other attribute PDFs were modeled using χ2 distributions. To increase the flexibility of the optimization, and concurrently reduce the CPU demand, Trauth’s model was enhanced to eliminate various elements of the structure routine. Trauth’s framework was based on Neurock’s Monte Carlo simulation of petroleum asphaltenes.7 In this simulation, each molecule
γ
Maya
Arabian Light molecular weight, Mn (g/mol) H/C sulfur content (wt %) HR (%) Haromatic (%) paraffin naphthenic aromatics + resin asphaltene heavy gas oil, HGO vacuum gas oil, VGO resid
χ2
622 1.33 5.3 10 8 2.5 22.5 61.0 15.0 21 20 58
563 1.35 5.5 13 10 2.4 23.3 58.1 16.3 12 26 61
582 1.35 5.3 13 8 2.4 19.8 59.7 18.1 19 20 61
25.8
6.3
had an x coordinate, a y coordinate, a group number, an atom type, and a bond number for each atom and a bond type for each bond in the molecule. Although this information can be useful for two-dimensional graphical representation and certain quantum calculations, neither of these functions was critical to the successful structure model. By removing this information, the time needed to create a large sample (100 000 molecules) was reduced by a factor of ∼30. This reduction in the CPU time requirement allowed for modeling each structural attribute with a γ distribution, which, in turn, allowed greater flexibility and improved the ability to model all of the analytical properties. Thus, the essential comparison between Trauth’s work and the present results is the comparison between χ2 and γ PDFs. The optimization exploited additional physical constraints. For example, each PDF had a physical mimimum. For instance, if an aromatic structure is being built, there must be at least one aromatic ring. Therefore, only two γ PDF parameters and one χ2 or exponential PDF parameter needed to be optimized. To account for the minimum boiling point criteria, conditional probability was invoked for the PDFs for the alkyl sidechains of the naphthenic fractions and the aromatic/resin fractions. A relatively simple conditional probability plan that required a minimum number of sidechains for one- and two-ring compounds was implemented, and a single minimum alkyl sidechain length was imposed on each of these sidechain distributions, to meet the initial boiling point criteria. No conditional probability constraints were implemented for compounds with more than four rings in the present work, whereas Trauth considered conditional probability for five- and six-ring compounds. These simplifications allowed for the description of alkyl sidechain length with only 5 distributions, as opposed to as many as 36 distributions in the prior work. The construction algorithm used the following logic. For each molecule, the molecular type (e.g., paraffin, naphthenic, aromatic/ resin, or asphaltene) was determined first. After the molecular
1686 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 Table 2. Optimal PDF Parameters for Four Resids, Using Mostly χ2 Distributions To Represent the Structural Attributesa mean structural attribute
distribution typeb
minimum
Hondo
Maya
Arabian Light
Arabian Heavy
3 3 2 1 3 3
20 1 0 1 1 0
30.0* 9.4 2.5 (1.1) 19.3 4.7 3.5
30.0* 11.6 2.9 (5.1) 17.7 8.1 3.8
30.0* 13.7 2.2 (3.4) 17.9 6.6 1.4
30.0* 8.4 1.9 (4.5) 7.5 4.5 2.4
3 3
2 0
paraffin length number of naphthenic rings number of sidechains alkyl chain length number of aromatic rings number of naphthenic rings on the aromatic core asphaltene degree of polymerization number of S atoms
4.4* 6.8
a The presence of an asterisk (*) indicates that the parameter was not optimized. given in parentheses is the standard deviation); and 3, χ2.
7.1* 2.5 b
2.5* 5.2
5.0* 3.9
Legend of distribution types: 1, exponential; 2, gamma (value
Table 3. Optimal PDF Parameters for Four Resids Using Gamma Distributions To Model Each Structural Attributea mean structural attribute paraffin length number of naphthenic rings number of sidechains alkyl chain length number of aromatic rings number of naphthanic rings on an aromatic core asphaltene degree of polymerization number of thiophenic rings fraction of sidechains with sulfur
distribution typeb
minimum
Hondo
Maya
Arabian Light
Arabian Heavy
3 2 2 2 2 2 2 2
20 1 0 1 1 0 2 0
30.0* 2.36 (0.29) 5.41 (0.90) 8.78 (9.64) 6.07 (0.61) 3.03 (0.40) 3.41 (1.64) 1.48 (1.42) 0.197
30.0* 7.93 (6.33) 5.2 (5.51) 9.68 (10.82) 7.58 (3.27) 2.12 (3.36) 2.19 (2.67) 1.65 (1.13) 0.148
30.0* 3.14 (1.10) 3.08 (1.65) 18.16 (9.79) 7.55 (1.07) 2.71 (2.01) 3.37 (0.20) 2.01 (0.90) 0.326
30.0* 2.3 (0.53) 1.81 (0.61) 8.41 (0.60) 2.99 (3.93) 1.06 (4.87) 3.91 (2.31) 1.78 (1.16) 0.516
a The presence of an asterisk (*) indicates that the parameter was not optimized. given in parentheses is the standard deviation); and 3, χ2.
type was selected, the value of each attribute needed to specify the molecule was determined by Monte Carlo sampling of the PDFs. The values of the ring attributes (number of aromatic rings, number of thiophenic rings, and number of naphthenic rings) were determined before the sidechains, because of the conditional probability dependence that the number of sidechain carbons has on the ring structure of a molecule. The PDF parameters were optimized using a simulated annealing program and the objective function of eq 1. For each step of the optimization, approximately 100 000 molecules were constructed. This number is larger than the corresponding number that Petti et al. used,4 because of the increased speed of the present construction algorithm. The results from Trauth’s χ2 optimization and the new γ PDF results are also presented in Table 1. The optimal PDF parameters are shown in Tables 2 and 3 for the χ2 and γ PDFs, respectively. Both models predicted the feedstock properties reasonably well, but the data reveal a clear better fit with the γ PDFs. Inspection of Table 1 reveals the fidelity of the structural models. The predicted aromatic ring core sizes are similar for both models. However, the average number of rings for naphthenic molecules is significantly less for the structure model using γ distributions. The γ distributions indicate values of the standard deviation for most structural attributes that are significantly different from χ2 distributions (where the standard deviation is one-half of the mean). This extra degree of freedom for the optimization greatly enhances the ability to match experimental data. For the χ2 model, the predictions of the H/C ratio for Hondo and Arabian Heavy are within one standard deviation of the experimentally determined property. However, the H/C ratio is greater than one standard deviation less than the experimental value for Maya and Arabian Light. The underprediction of the
b
Legend of distribution types: 1, exponential; 2, gamma (value
H/C ratio for Maya can be rationalized by the large fraction of the aromatic and asphaltene molecules. To increase the H/C ratio for the feedstock, the H/C ratio of these two fractions would need to be increased. This can be accomplished by increasing the number of naphthenic rings or number of sidechains on an aromatic core. However, increasing these would also increase the molecular weight and the number of R protons, while decreasing the number of aromatic protons. Because the molecular weight and the fraction of R protons are already slightly overpredicted for Maya, it is likely that increasing the H/C ratio in such a manner would not improve the overall predictions of the analytical properties. Similarly the H/C ratio of the Arabian Light cannot be increased without adversely affecting other properties. For the γ PDF model, the H/C ratio is predicted to within one experimentally determined standard deviation for each of the four resids. The extra parameter allows for tighter control of both the aromatic weight percent and the H/C ratio, so that it is possible to model each of the four resids to within one experimental standard deviation. The predictions of the sulfur weight percent for the χ2 model all deviate by more than one standard deviation. However, only the predictions for Arabian Light and Hondo deviate by more than two standard deviations. The high sulfur content for Arabian Light can be rationalized by the fact that the molecular weight for that feedstock is underpredicted. By adding more sulfur moieties, the model was able to more closely match the experimentally determined molecular weight. Conversely, the molecular weight for Hondo is overpredicted. Therefore, by having too few sulfur moieties, the molecular weight was not too greatly overpredicted. As previously mentioned, using the γ distributions improved the model results, because the sulfur weight percent was
Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 1687
predicted to within one experimental standard deviation for each of the four resids. Perhaps the property for which an accurate prediction is most important is the fraction of R hydrogens. This is because the fraction of R protons is most directly related to the number of alkyl sidechains, which are key in the conversion of resid to lighter products. Both models predict the R proton fraction fairly well, although the γ model predictions are slightly better, particularly for Hondo. The χ2 model predicted the fraction of R protons for Maya and Arabian Light to within one experimental standard deviation. However, the predictions for Arabian Heavy and Hondo are closer to two experimental standard deviations away from the measured values. The fraction of R protons for Hondo is underpredicted by the χ2 model. The number of R hydrogens can be increased by increasing either the number of alkyl sidechains or the number of naphthenic rings on the aromatic cores. However, increasing either of these also results in an increase in molecular weight, which is already overpredicted in the χ2 model for Hondo. Conversely, the fraction of R protons for Arabian Heavy is overpredicted, while the molecular weight is underpredicted for the χ2 model. The extra flexibility afforded by the γ distribution improved the ability to model Maya and Hondo and resulted in no change for Arabian Light and Arabian Heavy. For the γ model, all of the properties are predicted to within one standard deviation, except those for Arabian Heavy. Only one of the χ2 model predictions of the aromatic proton fractions was within one standard deviation, whereas all of the predictions of the γ model fell within the one-standard-deviation confidence interval. The greater flexibility of the γ distribution significantly improved the model of the fraction of aromatic and R protons. This is because the shape of the PDF for both number of alkyl sidechains and number of naphthenic rings can be varied without greatly affecting other key properties, such as molecular weight and relative weight fraction of aromatics. For the various compound class weight percents, the predictions of both models are generally within one standard deviation for each compound class. There is a slight improvement using a γ model for these properties, because the few compound classes that are not well-represented with the χ2 model fall within one standard deviation for the γ model. The χ2 model overpredicts the amount of naphthenics in Hondo. This is likely due to the high H/ C ratio and high asphaltene content of the resid. Because the asphaltene fraction has a low H/C ratio, the model compensates by increasing the size of the naphthenic molecules. This, in turn, also causes a slight overprediction of the molecular weight for Hondo. The χ2 model also overpredicted the aromatic weight percent for Maya while also underpredicting the asphaltene weight percent. Because of the fact that all these fractions have common distributions, with the exception of the number of unit sheets, this is easily understood. To increase the weight fraction of asphaltenes while decreasing the weight fraction of aromatics, the asphaltenes must have more unit sheets. At this point, however, the molecular weight is already slightly overpredicted, so that an increase in the number of unit sheets would cause a larger error in the prediction of molecular weight. The modeled versus experimentally determined molecular weights for the four resids are also listed in Table 1. The χ2 model fit only Maya to within one standard deviation of the experimentally determined value, although the other resids are just slightly outside of the confidence limits. The γ model
predicts all of the molecular weights to within one experimental standard deviation except for Arabian Heavy. However, the prediction for Arabian Heavy is improved, compared to that for the χ2 model. The χ2 model overpredicts the molecular weight for Hondo. As discussed previously, this is likely due to the high H/C ratio and the high asphaltene content. The χ2 model also underpredicts the molecular weight for Arabian Light and Arabian Heavy. The underprediction of molecular weight for Arabian Light is likely due to the fact that it has a large aromatic hydrogen content and a small R-hydrogen content. Larger molecules could be constructed with high aromatic proton content and low R-proton content by increasing the size of the aromatic cores. However, this would also cause a corresponding decrease in the H/C ratio, which is already being underpredicted. The underprediction of molecular weight for Arabian Heavy is partly due to the large amount of HGO and VGO that is present in this feedstock. Furthermore, this feedstock has a very high naphthenic concentration, as well as a very low H/C ratio. Therefore, the aromatic molecules must have very few naphthenic and alkyl sidechains to compensate for the high H/C ratio of the naphthenics fraction. The lack of alkyl sidechains and naphthenic rings on the aromatic cores limits the molecular weight of aromatic molecules and contributes to the underprediction of molecular weight. The predictions of the boiling fractions (HGO, VGO, and resid) were significantly better for the γ model than for the corresponding χ2 model. The prediction of the HGO weight percent is very similar between the two models, except for the Arabian Heavy resid. The Arabian Heavy is an unusual resid, because it contains ∼40 wt % gas oil (∼20 wt % HGO and 20 wt % VGO). The χ2 model prediction was well below the one-standard-deviation confidence limit for Arabian Heavy. The reason for this underprediction can be attributed to the large amount of low boiling (
