# Reliability Analysis of HHV Prediction Models for Organic Materials Using Bond Dissociation Energies

^{*}

## Abstract

**:**

^{−1}. Further investigation reveals significant variations in the bond dissociation energies of carbon atoms within organic compounds, resulting in a range of energy outputs from −414.30 to −275.34 kJ mol

^{−1}per mole of carbon atoms. The presence of oxygen atoms in organic compounds has a negative impact on the magnitude of combustion heat, with values ranging from 131.1 to 207.17 kJ mol

^{−1}. The combustion mechanism imposes certain constraints, leading to the equation HHV

_{g}= −31.34·[C] − 144.44·[H] + 10.57·[O] for organic compounds. Based on the parameter sensitivity analysis, the coefficient associated with carbon mass fraction exhibits a significantly greater impact on result prediction accuracy, demonstrating a sensitivity value of 92.65%. The results of further analysis indicate that empirical correlations involving the mass fractions of the elements N and S in lignocellulosic materials may be prone to over-fitting, with sensitivity indices of 1.59% and 0.016%, respectively.

## 1. Introduction

## 2. Model Development

_{c}H

_{h}O

_{o}can generally be described using the following reaction

_{m}refers to the higher heating value on a molar basis, at standard conditions (i.e., 100 kPa and 25 °C). H

_{rnx,i}represents total bond dissociation energies of a substance i, whereas D

_{i,m.}signifies the BDE value of the chemical bond m with respect to component i. The value of H

_{rnx,}

_{H2O}should include the molar heat of vaporization for water (40.7 kJ mol

^{−1}).

_{rnx,}

_{O2}, H

_{rnx,C}

_{O2}, and H

_{rnx,}

_{H2O}remain constant, with the results of 498.0 kJ mol

^{−1}, 1607.0 kJ mol

^{−1}, and 969.8 kJ mol

^{−1}, respectively. Taking CH

_{4}as an example, the total BDEs of CH

_{4}are approximately 1657.2 kJ mol

^{−1}by adding the average BDE values of four C-H bonds, i.e., $\overline{D}\left(\mathrm{H}-\mathrm{C}\right)\times 4$. Then, the evaluation of Equation (2) yields −892.8 kJ mol

^{−1}for the combustion heat of CH

_{4}, close to its actual value of −890.53 kJ mol

^{−1}.

_{A}and α

_{B}are constants characterizing radicals A and B. The value α

_{H}for the H atom is definitely equal to unity, that is, α

_{H}= 1. α

_{m}stands for the smaller of the latter, i.e., min(α

_{A}α

_{B}). Then, the typical values of D

_{H}, D

_{C}, and D

_{O}were, respectively, 215.95 kJ mol

^{−1}, 173.68 kJ mol

^{−1}, 77.32 kJ mol

^{−1}, and α

_{C}= 1.20, α

_{O}= 2.95 [21]. The reliability of Equation (3) has been thoroughly discussed and demonstrated through a comparison between the calculated and experimental values of bond dissociation energy, as reported by Tiggelen et al. in 1965 [21]. The discrepancies between calculated and experimental values are within a range of 8.37~12.56 kJ mol

^{−1}, which can be considered negligible compared to the total value H

_{rxn}(C

_{c}H

_{h}O

_{o}) of organic materials.

_{c}H

_{h}O

_{o}can be determined using

_{C,m}stands for the smaller of the latter, i.e., min(α

_{C}, α

_{R}), for C bonds in organic matter. For the organic compounds with established chemical structures, the carbon bonds have been simplified to C-C, C-H, and C-O, with corresponding bond numbers being determined through statistical analysis. For example, in the case of C-O bonds, α

_{C,m}can be expressed as min(α

_{C},α

_{O}), with a value of 1.2. Similar connotations are attributed to other symbols. The first term on the right-hand side of Equation (4) can be determined by aggregating the outcomes. Analogous procedures are executed for the bonds involving oxygen and hydrogen.

_{rnx,}

_{C}is the average of α

_{C}D

_{C}/α

_{C,m}for element C in an organic compounds, and similar definitions apply to H

_{rnx,}

_{H}and H

_{rnx,}

_{O}. Based on the assumption proposed above, the values of H

_{rnx,}

_{C}, H

_{rnx,}

_{H,}and H

_{rnx,}

_{O}can be determined with the numbers of single bonds. By applying the rearrangement of Equation (5) to Equation (2), an analytical model for estimating combustion heat can be derived as

## 3. Model Validation

_{m}values calculated using Equations (6)–(9) and theoretical values using 154 species of organic molecules with specific chemical structures [22]. The theoretical values of combustion heats are calculated based on the standard molar enthalpies of formation as documented in the scientific literature. This method has been widely accepted and yields only minor discrepancies with the experimental values. The reliability of the verification results can be ensured with a sufficient number of data samples, and errors caused by human factors can be eliminated through the use of a unified data source.

_{rnx}

_{,C}, H

_{rnx}

_{,H}, and H

_{rnx}

_{,O}values. The HHV

_{m}values and coefficients φ

_{C}, φ

_{H}, and φ

_{O}are determined using Equations (6)–(9) as the basis. Both calculating formulae Equations (2) and (6) were employed as illustrated in Figure 1. The organic molecules comprise aliphatic, alicyclic and aromatic hydrocarbons as well as aliphatic alcohols, phenols, ethers, aldehydes, ketones, steroids, lactones, and so on. The predicted heating values exhibit high consistency with the theoretical values and small standard errors, thereby confirming the reliability of estimated combustion heat using bond dissociation energies.

_{rxn}(C

_{c}H

_{h}O

_{o}) of organic materials.

## 4. Results and Discussion

_{rnx,}

_{C}, H

_{rnx,}

_{H}, and H

_{rnx,}

_{O}in Equation (5) correspond to the contributions of individual elemental atoms toward the calculated bond dissociation energy values of an organic compound. These parameters are contingent upon the chemical structure of the compound. For instance, for carbon atoms in methane (CH

_{4}), the value of α

_{C,m}is consistently equal to 1, thereby resulting in an H

_{rnx}

_{,C}value of 208.42 kJ mol

^{−1}. In contrast, the carbon bond in methanol (CH

_{3}OH) comprises three C-H bonds and one C-O bond, thereby enabling the determination of the H

_{rnx}

_{,C}value as 199.73 kJ mol

^{−1}. As depicted in Figure 3, the H

_{rnx}

_{,C}value varies in the range of 178.63 to 208.42 kJ mol

^{−1}.

_{rnx}

_{,C}value increases with a decrease in average α

_{C,m}value. As per Equation (3), the value of α

_{C,m}can be either 1.2 or 1, resulting in a theoretical range for H

_{rnx},

_{C}of 173.68 to 208.42 kJ mol

^{−1}. The maximum H

_{rnx},

_{C}value is equivalent to that of CH

_{4}, as depicted in Figure 3. However, the predicted minimum value is slightly lower than the experimental values, and no actual organic compound exhibits it. This is due to the indispensability of C-H bonds in the organic matter’s C bonds.

_{rnx,C}values display an inverse correlation with increasing CHO values. A higher value of the CHO index indicates a greater degree of carbon oxidation in organic matter, leading to a reduction in the proportion of C-H bonds within organic compounds’ C bonds. This results in the higher values of the average α

_{C,m}of organic materials, due to the higher min(α

_{C}, α

_{O}) value compared to the min(α

_{C}, α

_{H}) value. In view of Equation (4), it is inevitable that the H

_{rnx,C}value will decrease. From a chemical mechanism perspective, the breaking of C-O bonds is more facile than that of C-H bonds.

_{rnx,}

_{O}values demonstrate a limited range of fluctuation, ranging from 190.07 to 209.09 kJ mol

^{−1}across a set of 154 organic compounds, as depicted in Figure 4. The observed results align with the theoretical predictions, demonstrating consistency between theory and experiment. As the data samples do not contain any O-O bonds, the α

_{O,m}value would be 1.2 or 1. When all oxygen bonds are either C-O or C=O bonds, the theoretical lower limit value is found to be consistent with the experimental data, i.e., α

_{O}D

_{O}/α

_{C}.

_{O,m}value, however, would not be uniformly equal to 1 since the oxygen bonds in organic matters are not exclusively composed of O-H bonds. Theoretical maximum values of H

_{rnx,}

_{O}can be deduced when an oxygen atom is bonded to both a hydrogen and a carbon, suggesting that alcohols are the most likely candidates in this group. Then, the upper limit value of H

_{rnx}

_{,O}can be expressed as $\frac{{\alpha}_{\mathrm{O}}{D}_{\mathrm{O}}}{2}\left(\frac{1}{{\alpha}_{\mathrm{C}}}+\frac{1}{{\alpha}_{\mathrm{H}}}\right)$, where the calculated results align with the statistical findings from the experiments.

_{H,m}value will consistently remain equal to 1. Consequently, this leads to H

_{rnx,}

_{H}= D

_{H}. Thus, the value of H

_{rnx,}

_{H}remains a constant of 215.95 kJ mol

^{−1}.

_{C}, φ

_{H,}and φ

_{O}, along with the CHO values, are illustrated in Figure 5. In accordance with the expression of Equation (6), the parameters φ

_{C}, φ

_{H,}and φ

_{O}characterize the energy output per mole of an elemental atom in relation to the higher heating values of organic substances. The φ

_{H}value has been determined as a constant of −144.44 kJ mol

^{−1}, as H

_{rnx,}

_{O2}and H

_{rnx,}

_{H2O}in Equation (8) are constants. Based on the empirical model presented in Equation (3), it is deduced that D

_{H}≈ 0.5·H

_{rnx}(H

_{2}). Considering Equation (8) and the hydrogen combustion reaction, it can be inferred that φ

_{H}is approximately equal to 0.5·HHV

_{m}(H

_{2}). The combustion heat of H

_{2}is approximately −142.30 kJ mol

^{−1}[5], which closely aligns with the φ

_{H}.

_{C}is observed to range from −394.50 to −275.34 kJ mol

^{−1}and exhibits a significant decrease with an increase in CHO index. The higher CHO index value indicates a reduced number of carbon atoms participating in the oxidation reaction during the combustion process. Accordingly, the contribution of carbon to the calorific value of combustion is reduced.

^{−1}for parameter H

_{rnx}

_{,C}, it is theoretically anticipated that the coefficient φ

_{C}would fall within the range (−414.30, −275.34] kJ mol

^{−1}. The decrease in energy released via carbon oxidation is remarkable as the H

_{rnx}

_{,C}value increases. Therefore, achieving the minimum of φ

_{C}is theoretically impossible.

_{rnx,}

_{C}= D

_{C}. Then, the estimated value of φ

_{C}can be determined as −414.30 kJ mol

^{−1}, slightly exceeding the heat of combustion for elemental carbon (−393.5 kJ mol

^{−1}). The observed phenomenon is consistent with the fact that C-C bonds in carbon substances exhibit greater stability compared to those present in organic materials. The actual value of φ

_{C}is typically significantly higher than the lower limit value, owing to the fact that the cleavage of C-H bonds requires a greater amount of energy absorption in comparison to C-C bonds.

_{O}would be between 131.13 and 169.17 kJ mol

^{−1}, in light of the theoretical range [190.07, 209.09] kJ mol

^{−1}of the H

_{rnx}

_{,O}value. This is consistent with the statistical results as shown in Figure 5. Notably, Equation (9) denotes the difference in bond dissociation energy between O bonds in organic compounds and those in molecular oxygen O

_{2}.

_{2}produces the free atom O. (3) The carbon and hydrogen atoms directly combine with oxygen atoms to produce carbon dioxide and water. The energy required for the formation of free O atoms in organic compounds is evidently higher compared to that needed for molecular oxygen.

_{O}> 0 demonstrates that the presence of oxygen in organic compounds has a negative impact on their combustion heat, which is consistent with statistical findings documented in the previous scholarly literature [4,5,8,23]. However, the current mathematical fittings occasionally adhere to the fundamental chemical principle regarding the role of element O in combustion heat.

_{m}≈ −437.81·ν

_{O2}with a high R

^{2}value of 0.9988 for 154 different species of organic compounds. This can be further substantiated through theoretical deduction based on the proposed theoretical Equations (6)–(9). By utilizing the aforementioned hypothesis and Equation (3), Equation (6) can be rewritten as follows:

_{rnx}

_{,O2}− 4α

_{O}D

_{O}) can be computed as −414.32 kJ mol

^{−1}. Equation (10) suggests that the heat released during the combustion process mainly depends on the bond recombination of atomic oxygen in oxygen molecules, regardless of changes in bond dissociation energies for atomic carbon, hydrogen, and oxygen in organic matter before and after reaction. This elegantly clarifies the correlation between combustion calorific value and oxygen consumption in terms of bond dissociation energy, providing a clear perspective.

_{rnx,i}and HHV

_{m}values. Consequently, empirical correlations are frequently established based on mass fractions of elemental composition. For high-molecular polymers with the chemical formula C

_{c}H

_{h}O

_{o}, we define [C], [H], and [O] as the mass fractions of carbon, hydrogen, and oxygen elements, respectively. Then, we have [C] = 12c/M

_{s}, [H] = h/M

_{s}, and [O] = 16o/M

_{s}, where M

_{s}= (12c + h+16o). The higher heating values on a mass basis HHV

_{g}can be expressed as HHV

_{m}/M

_{s}. On the basis, Equation (6) can be rewritten as

_{C}∈(−414.30, −275.34], φ

_{H}= −144.44, and φ

_{O}∈[131.13, 169.17]. The fitting result has been listed as Equation (12) in Table 2, demonstrating a significantly high correlation coefficient of R

^{2}= 0.9969.

_{C}. The sensitivity values of φ

_{C}and φ

_{O}are calculated as 92.65% and 7.35%, respectively. Consequently, prior research has occasionally established a correlation between the higher heating value and carbon fractions using the equation HHV

_{g}= a·[C] + b in the literature [25,26,27]. The correlation for the organic compounds in the present work is listed as Equation (13) in Table 2. The correlation coefficient exhibits a strong positive relationship with a value of 0.96.

_{c}H

_{h}O

_{o}N

_{n}S

_{s}, the combustion process can be expressed as

_{N}= 3·H

_{rnx}

_{,N}− 0.5·H

_{rnx}

_{,N2}, φ

_{S}= 4·H

_{rnx}

_{,S}− H

_{rnx}

_{,SO2}, and the values of H

_{rnx}

_{,N2}and H

_{rnx}

_{,SO2}are equivalent to 946 kJ mol

^{−1}and 1072.36 kJ mol

^{−1}, respectively. The maximum values of H

_{rnx}

_{,N}and H

_{rnx}

_{,S}can be determined as α

_{N}D

_{N}and α

_{S}D

_{S}. The bond dissociation energies (BDE) of the N-H bond and S-H are reported as 390.80 kJ mol

^{−1}and 339.00 kJ mol

^{−1}, respectively [19,20]. According to Equation (3), the values of H

_{rnx}

_{,N}and H

_{rnx}

_{,S}should be below 174.85 kJ mol

^{−1}and 123.05 kJ mol

^{−1}, respectively. Subsequently, due to the aforementioned constrained condition, Equation (22) can be effectively fitted using a genetic algorithm in the 1stOpt 10.0 software as

_{g}= −33.71·[C]−144.44·[H] + 12.62·[O] + 3.68·[N]−18.13·[S] R

^{2}= 0.9505

_{rnx}

_{,N}and H

_{rnx}

_{,S}have really small sensitivity index values, which means it is okay to ignore the mass fractions of element N and element S when predicting the higher heating values of lignocellulosic materials. This finding aligns with the comparative outcome illustrated in Figure 9. The results of parameter sensitivity analysis critically depend on the numerical distribution of variables, which should be noted. Lignocellulosic biomass are essentially high-molecular polymers, mainly involving cellulose, hemicellulose, and lignin. According to the statistical data, lignocellulosic materials exhibit the highest carbon (C) content, followed by oxygen (O). The hydrogen (H) content is comparatively lower, while nitrogen (N) and sulfur (S) elements are present minimally or negligibly [28]. Therefore, Figure 10 implies that there is a potential risk of over-fitting when establishing empirical correlations involving the mass fractions of elements N and S in lignocellulosic materials. The efficacy of Equation (11) in predicting the performance of biomass materials and municipal solid wastes can be attributed to this observation.

## 5. Conclusions

_{2}(−144.4 kJ mol

^{−1}). Further investigation reveals significant variations in the bond dissociation energies (BDEs) of carbon atoms within organic compounds, resulting in a range of energy outputs from −414.30 to −275.34 kJ mol

^{−1}per mole of carbon atoms. The presence of oxygen atoms in organic compounds has a negative impact on the magnitude of combustion heat, with values ranging from 131.1 to 207.17 kJ mol

^{−1}.

_{g}= −31.34·[C] − 144.44·[H] + 10.57·[O]. Based on the parameter sensitivity analysis, the coefficient associated with carbon mass fraction exhibits a significantly greater impact on the prediction accuracy of results, demonstrating a sensitivity value of 92.65%. Further analysis suggests that empirical correlations involving the mass fractions of elements N and S in lignocellulosic materials may be susceptible to over-fitting.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Comparison between the HHVs predicted using Equations (2) and (6) with theoretical values.

**Figure 8.**The comparison between the heating values calculated using Equation (12) and the experimental data.

**Figure 10.**The sensitivity indices of parameters H

_{rnx}

_{,C}, H

_{rnx}

_{,O}, H

_{rnx}

_{,N}, and H

_{rnx}

_{,S}for Equation (23).

Bond | H-H | C-C | O-O | C=C | C≡C | O=O ^{a} | H-C | H-O | C-O | C=O | C=O ^{b} |
---|---|---|---|---|---|---|---|---|---|---|---|

kcal/mol | 104.2 | 83.0 | 35.0 | 146.0 | 210.0 | 119.0 | 99.0 | 111.0 | 85.0 | 178.0 | 192.0 |

kJ/mol | 436.1 | 347.4 | 146.5 | 611.0 | 878.9 | 498.0 | 414.3 | 464.5 | 355.7 | 744.9 | 803.5 |

^{a}The exact value of O=O bond enthalpy in O

_{2.}

^{b}The exact value of C=O bond enthalpy in CO

_{2.}

No. | Authors | Years | Correlation |
---|---|---|---|

Equation (12) | Present study | 2023 | HHV_{g} = −31.34·[C] − 144.44·[H] + 10.57·[O] |

Equation (13) | Present study | 2023 | HHV_{g} = −34.52·[C] − 9.09 |

Equation (14) | Jenkins and Ebeling [14] | 1985 | HHV_{g} = −29.34·[C] − 51.74·[H] − 5.64·[O] |

Equation (15) | Sheng and Azevedo [13] | 2005 | HHV_{g} = −30.00·[C] − 68.72·[H] − 1.81·[O] |

Equation (16) | Schmidt-Rohr [4] | 2015 | HHV_{g} = −34.83·[C] − 125.40·[H] + 13.06·[O] |

Equation (17) | Tao et al. [23] | 2016 | HHV_{g} = −30.99·[C] − 149.99·[H] + 10.22·[O] |

Equation (18) | Huang and Luo [8] | 2020 | HHV_{g} = −34.47·[C] − 119.20·[H] + 11.30·[O] |

Equation (19) | Dulong [24] | 1945 | HHV_{g} = −33.91·[C] − 144.44·[H] + 18.05·[O] − 9.42·[S] |

Equation (20) | Huang and Luo [8] | 2000 | HHV_{g} = −34.43·[C] − 119.20·[H] + 11.30·[O] + 2.40·[N] − 9.30·[S] |

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**MDPI and ACS Style**

Tao, J.; Pan, L.; Yao, J.; Liu, L.; Chen, Q.
Reliability Analysis of HHV Prediction Models for Organic Materials Using Bond Dissociation Energies. *Polymers* **2023**, *15*, 3862.
https://doi.org/10.3390/polym15193862

**AMA Style**

Tao J, Pan L, Yao J, Liu L, Chen Q.
Reliability Analysis of HHV Prediction Models for Organic Materials Using Bond Dissociation Energies. *Polymers*. 2023; 15(19):3862.
https://doi.org/10.3390/polym15193862

**Chicago/Turabian Style**

Tao, Junjun, Longwei Pan, Jiajie Yao, Longfei Liu, and Qiang Chen.
2023. "Reliability Analysis of HHV Prediction Models for Organic Materials Using Bond Dissociation Energies" *Polymers* 15, no. 19: 3862.
https://doi.org/10.3390/polym15193862