# An Artificial Neural Network for the Low-Cost Prediction of Soot Emissions

^{*}

## Abstract

**:**

## 1. Introduction

_{2}[4]. Larger soot aggregates and the smaller soot particulate matter both contribute to the problem. The smaller, lighter particulates remain in the air absorbing sunlight and subsequently warming the surrounding air. On the other hand, the larger, heavier aggregates fall to the ground and absorb sunlight there. As a result, any surrounding snow or ice is melted far faster [4].

## 2. Methodology

_{1}, x

_{2}, … x

_{j}, each with a corresponding weight w

_{1}, w

_{2}, … w

_{j}, the summation function with bias, b, and the activation function f, which leads to the output. In addition, the hyperbolic tangent as well as ReLU activation functions are shown in Figure 3. As can be seen, tanh can only vary between −1 and 1. Moreover, tanh, as well as it’s derivative, are both monotonic functions. Put more explicitly, the output of a neuron with the hyperbolic tangent activation function is

_{4}species, and presence of fuel oxygenates. The present study advances upon these works with full ex-situ prediction of soot concentration using ANNs.

_{2}), carbon monoxide (CO), carbon dioxide (CO

_{2}), hydrogen (H

_{2}), water (H

_{2}O), hydroxide (OH), acetylene (C

_{2}H

_{2}), benzene (A1), and soot volume fraction (f

_{v}) are gathered. As an example, the temperature contours for the flames SM32 and SM80 are shown in Figure 5. It should be pointed out that the first eight variables are considered key variables found in any combustion system. However, acetylene and benzene are included since they are the major species which result in PAH formation and hence affect soot inception and surface reactions. The soot volume fraction is the output in the present study.

_{h}), mixture fraction history (MF

_{h}), oxygen history (O

_{2,h}), carbon monoxide history (CO

_{h}), carbon dioxide history (CO

_{2,h}), hydrogen history (H

_{2,h}), water history (H

_{2}O

_{h}), hydroxide history (OH

_{h}), acetylene history (C

_{2}H

_{2,h}), and benzene history (A1

_{h}) are calculated and considered as the input dataset. To calculate the Lagrangian histories and create the library/data-frame the following steps are taken:

- (1)
- Gathering numerical data: as mentioned, the values of different parameters (such as temperature and oxygen) for the eight flames are gathered.
- (2)
- Extracting pathlines: each pathline $p$ was computed from the following ordinary differential equation:$$\frac{d\overrightarrow{{X}_{p}}}{dt}\text{}=\text{}\overrightarrow{u}\left(\overrightarrow{{X}_{p}}\left(t\right),t\right),\text{}\overrightarrow{{X}_{p,{t}_{0}}}\text{}=\text{}\overrightarrow{{X}_{p,0}},\text{}p\text{}=\text{}1,2,\dots ,{N}_{p},$$
- (3)
- Computing histories from pathlines: The history of a variable referred to the time integration of a given variable over its entire existence; for soot that is from inception to oxidation (that is along a pathline). For instance, for a general variable (Z), the history of Z is defined as:$${Z}_{h}={{\displaystyle \int}}_{pathline}Z\left(\overrightarrow{X},\text{}t\right)\text{}dt,$$
- (4)
- (5)
- Concatenating the eight libraries/data-frames and generating one library/data-frame.

_{v}on the r-z coordinate system (see Figure 6). It should be noted that in previous works [6,17,18], the aforementioned steps, as well as the interpolation method, were comprehensively discussed. Therefore, the interested reader is referred to those studies for more information.

_{h}and H

_{2}O

_{h}, which when used in any combination with other variables would lead to the same issues. This behavior might be due to multicollinearity [45,46,47,48]. In fact, multicollinearity happens when the input variables are highly correlated (for instance, in the soot formation study, there is a strong correlation between C

_{2}H

_{2,h}and A1

_{h}) and they have a strong correlation with the output. In regression analysis, this may result in poor fitting [45,46,47,48]. In the current work, the two variables (A1

_{h}and H

_{2}O

_{h}) were simply omitted from subsequent training of the network (it should be noted that from a physical perspective, it is not that A1

_{h}is not an important driver of aromatic growth and soot formation in this case, but rather that other input sets produce better results). Upon completion of some brief testing, it was determined that using all of the remaining eight variables as input parameters showed the most promise in terms of decreasing the overall error and improving reliability. It is important to emphasize again that these inputs were histories of variables as opposed to values in a given instant of time. This hysteresis method is necessary because the timescales associated with soot formation tend to be much longer than those of combustion chemistry. Furthermore, it should be noted that the concept of tracking the time history was also used by Aceves et al. [49] for fast prediction of HCCI combustion with an ANN linked to the KIVA3V fluid mechanics code. In addition, in the work of Christo et al. [28], in order to reduce the dependency of the ANN model on the selection of training sets for modeling turbulent flames, the input parameters were integrated over a prescribed reaction time.

_{h}, MF

_{h}, H

_{2,h}) from previous work [18]. Furthermore, the repeatability of the results, where previously the exact same network would give largely varying results on several subsequent runs, was now much better. Nonetheless, variation will always be present to some degree, so it is a matter of limiting it as opposed to eliminating it. The reason this issue exists is the random nature behind the assignment of the initial weights paired with the random splitting of the entire data set into the training, validation, and testing sets (in this work, 80% of the entire dataset was randomly selected for training, 10% for validation, and 10% for testing). The randomness of the initial weights has a far smaller impact due to the fact that if the network works successfully it will always reach the global minimum. However, the data splitting does have a significant effect on the final results since the same network may be learning with different training data sets in successive runs, depending on how it is split. In this study, as an overall trend, the variability of these results decreased dramatically with the introduction of more input parameters.

## 3. Results and Discussion

_{2,h}) before data pre-processing.

_{v}(over 5 executions and based on the input/output datasets obtained from the eight flames in Table 1 (training errors)) for the networks with architectures of {15,10}, {10,5,3}, and {5,5,5,5} with the rectified linear unit activation function were 445.66, 1.3 × 10

^{4}and 165.25%, respectively. However, by using the hyperbolic tangent activation function, the error for the mentioned architectures was reduced to 73.23, 139.28, and 42.58%, respectively. Similar trends were observed for other networks considered in the grid search. Therefore, only the hyperbolic tangent function was considered after this point.

_{v}for five executions were 15.21, 36.63, 16.98, 601.2, and 26.38% (average: 139.28%). As mentioned before, the variability is caused by the random splitting of the data into training/validation/testing subsets as well as random initialization of weights and biases. In the current study, a slightly higher error along with consistent results was preferred over a very low error occurring sporadically for the purpose of practical usability.

_{h}, MF

_{h}, O

_{2,h}, CO

_{h}, CO

_{2,h}, H

_{2,h}, OH

_{h}, and C

_{2}H

_{2,h}. Then, these eight parameters were used as input variables and the soot concentrations that were predicted by the network were compared with the CFD results.

_{v}along the centerline and the streamline of maximum soot, the maximum f

_{v}, and the integrated f

_{v}in the whole domain. As shown, the average integrated soot volume fraction error over the whole domain is 8.08%.

_{v}error is 4.66%. However, for the CE flame, the errors are higher and the integrated f

_{v}error reaches 63.43%, mainly due to deviation of its dataset from the model’s dynamic range (similar observations were reported by Christo et al. [27] and Heinlein et al. [67]). For example, in the simulation of the CE flame, the fuel inlet velocity was assumed to be uniform and equal to 2.42 cm/s. For other nine flames discussed in the present work, the CFD results were based on the parabolic inlet velocity assumption and the minimum averaged fuel velocity was 4.1 cm/s (hence, the minimum fuel inlet velocity was 8.2 cm/s at the centerline, which was around 3.4 times more than the inlet velocity in the simulation of the CE flame). This degradation of accuracy, which occurs when the modelled samples are far enough outside the model’s working range, highlights the importance of broadening the training datasets to represent the input/output combinations over a wide dynamic range. It should be pointed out that the lack of data is one of the main reasons why machine-learning programs often fail to predict expected results [68]. Therefore, in future research, more flames will be simulated and validated, and more data will be added to the training dataset so that the network’s dynamic range will be expanded, making it the major advantage of this framework. It is worth mentioning that although more rigorous tests must be conducted to find the network performance, the range of errors for the CE flame is still very promising compared to many available CFD models that predict soot properties with about a one or two order-of-magnitude error.

_{v}predicted by the network reaches a peak at 17.70 ppm, while the original f

_{v}obtained from CFD peaks at 12.65 ppm (see Table 1). To present more details, it should be pointed out that, at the centerline, the soot volume fraction peaks at 1.96 and 1.91 ppm for the original and the ANN cases, respectively. In addition, there are slight and insignificant fluctuations in the ANN prediction when z is between 1.5 and 3 cm (see the red arrow in Figure 9b). It only happens in a limited small area at the boundary of the soot domain and can be due to the overfitting problem. Attempts were made to eliminate these fluctuations by changing our network architecture, but were unsuccessful.

_{v}predicted by the network and calculated by the CoFlame code are 19.78 and 14.76 ppm, respectively. For the f

_{v}along the centerline, the relative error is only 3.85% in this case.

_{v}reaches a peak at 20.32 and 16.39 ppm for the ANN and the original cases, respectively. In this figure, slight fluctuations in the ANN prediction when z is between 1.5 and 3.5 cm are also present.

_{v}along the centerline and along the streamline of maximum soot, respectively. For the SM60 flame, the network underestimates the soot concentration along the streamline of maximum soot, while it shows very good prediction along the centerline. Conversely, for the SM80.2 flame, the network underestimates the soot concentration along the centerline, while its prediction along the streamline of maximum soot is slightly shifted compared with the CFD results.

_{v}reaches a peak of 12.85 ppm and the high-soot zone becomes shorter than the same respective area in the 2D plot obtained from the numerical solution of the flame. It should be noted that in the CFD results, the maximum f

_{v}for SA flame is equal to 10.74 ppm. In addition, as shown in Figure 14 and Figure 15, the network predicts the soot concentration for the SA flame along the centerline and the streamline of maximum soot with only a slight discrepancy.

_{v}are underestimated. As shown, based on the network results, the f

_{v}peaks at 2.9 ppm. However, the maximum value of f

_{v}according to our CFD results is 6.13 ppm. Figure 14 and Figure 15 also demonstrate that the network underestimates the soot concentration for the CE flame along the centerline and the streamline of maximum soot (the values of relative errors are reported in Table 4). As mentioned above, this degradation in performance happens if the modelled compositions deviate significantly from the network’s working range.

_{v}in this study is wide, from around 0.1 to 20 ppm. In general, for the flames with low f

_{v}the absolute error was also very low (for example around ${10}^{-4}$). Conversely, the flames with high f

_{v}had absolute errors in the order of magnitude of 1. To display the graphs of absolute error in an easy-to-read way, the following equation was developed firstly:

_{v}for these flames are 0.12, 16.39, 10.74, and 6.13, respectively. By normalizing the error graphs with the peak f

_{v}, the percentage of error can be obtained:

## 4. Conclusions and Future Works

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**A schematic representation of a laminar diffusion flame along with the coordinates (r, z) and the boundary conditions used in the CoFlame code (adapted from Reference [40]).

**Figure 5.**2D contour plots of the temperature for the flames: (

**a**) SM32; (

**b**) SM80; obtained from the CoFlame code.

**Figure 7.**The probability density function (PDF) of oxygen history (O

_{2,h}): (

**a**) before data pre-processing; (

**b**) after data pre-processing.

**Figure 8.**2D contour plots of the soot volume fraction for the SY48 flame: (

**a**) computed; (

**b**) predicted by ANN without data pre-processing; (

**c**) predicted by ANN with data pre-processing.

**Figure 9.**2D soot concentration fields (ppm) for different flames: (

**a**) SM80; (

**b**) SY41; (

**c**) SY46; (

**d**) SY48; as obtained by two methods, using the original numerical solution (left image in each pair) and using the ANN prediction (right image in each pair).

**Figure 10.**Comparison of experimentally-validated CFD results and those obtained from the ANN for different flames along the centerline.

**Figure 11.**Comparison of experimentally-validated CFD results and those obtained from the ANN for different flames along the streamline of maximum soot.

**Figure 12.**2D soot concentration field of the SA flame: (

**a**) original numerical solution; (

**b**) ANN prediction.

**Figure 13.**2D soot concentration field of the CE flame: (

**a**) original numerical solution; (

**b**) ANN prediction.

**Figure 14.**Comparison of experimentally-validated CFD results and those obtained from the ANN along the centerline for the flames used to test the network.

**Figure 15.**Comparison of experimentally-validated CFD results and those obtained from the ANN along the streamline of maximum soot for the flames used to test the network.

**Figure 16.**The variations of average error versus height: (

**a**) absolute error; (

**b**) absolute error normalized with the peak soot volume fraction.

Flame Code | Fuel Composition (Volume) | Fuel Velocity (cm/s) | Air Velocity (cm/s) | Inner Diameter (mm) | Computed Peak f_{v} (ppm) |
---|---|---|---|---|---|

SM32 ^{a} | 32% C_{2}H_{4}/68% N_{2} | 35 | 35 | 4 | 0.12 |

SM40 ^{a} | 40% C_{2}H_{4}/60% N_{2} | 35 | 35 | 4 | 0.36 |

SM60 ^{a} | 60% C_{2}H_{4}/40% N_{2} | 35 | 35 | 4 | 1.68 |

SM80.2 ^{a} | 80% C_{2}H_{4}/20% N_{2} | 17.5 | 17.5 | 4 | 1.80 |

SM80 ^{a} | 80% C_{2}H_{4}/20% N_{2} | 35 | 35 | 4 | 3.21 |

SY41 ^{b} | 100% C_{2}H_{4} | 4.1 | 8.7 | 11 | 12.65 |

SY46 ^{b} | 100% C_{2}H_{4} | 4.6 | 8.7 | 11 | 14.76 |

SY48 ^{b} | 100% C_{2}H_{4} | 4.8 | 8.7 | 11 | 16.39 |

Flame Code | Fuel Composition (Volume) | Fuel Velocity (cm/s) | Air Velocity (cm/s) | Inner Diameter (mm) | Computed Peak f_{v} (ppm) |
---|---|---|---|---|---|

SA ^{a} | 100% C_{2}H_{4} | 5.06 | 13.3 | 11.1 | 10.74 |

CE ^{b} | 100% C_{2}H_{4} | 2.42 | 26.55 | 10.32 | 6.13 |

**Table 3.**Relative errors between the CFD and the predicted data by ANN for eight flames used to train the network.

Flame Code | f_{v} along the Centerline (%) | f_{v} along the Streamline of Max Soot (%) | Peak f_{v} (%) | Integrated f_{v} (%) |
---|---|---|---|---|

SM32 | 31.54 | 8.59 | 11.15 | 6.07 |

SM40 | 7.66 | 15.83 | 29.69 | 31.64 |

SM60 | 11.09 | 25.36 | 6.23 | 4.79 |

SM80.2 | 45.3 | 8.24 | 10.2 | 1.91 |

SM80 | 12.65 | 6.03 | 16.74 | 2.14 |

SY41 | 2.71 | 16.82 | 39.92 | 10.91 |

SY46 | 3.85 | 21.66 | 33.98 | 6.25 |

SY48 | 1.79 | 13.76 | 23.99 | 0.93 |

Average | 14.58 | 14.54 | 21.49 | 8.08 |

**Table 4.**Relative errors between the CFD and the predicted data by ANN for two flames used to test the network.

Flame Code | f_{v} along the Centerline (%) | f_{v} along the Streamline of Max Soot (%) | Peak f_{v} (%) | Integrated f_{v} (%) |
---|---|---|---|---|

SA | 12.07 | 3.5 | 19.64 | 4.66 |

CE | 61.22 | 73.09 | 52.79 | 63.43 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jadidi, M.; Kostic, S.; Zimmer, L.; Dworkin, S.B. An Artificial Neural Network for the Low-Cost Prediction of Soot Emissions. *Energies* **2020**, *13*, 4787.
https://doi.org/10.3390/en13184787

**AMA Style**

Jadidi M, Kostic S, Zimmer L, Dworkin SB. An Artificial Neural Network for the Low-Cost Prediction of Soot Emissions. *Energies*. 2020; 13(18):4787.
https://doi.org/10.3390/en13184787

**Chicago/Turabian Style**

Jadidi, Mehdi, Stevan Kostic, Leonardo Zimmer, and Seth B. Dworkin. 2020. "An Artificial Neural Network for the Low-Cost Prediction of Soot Emissions" *Energies* 13, no. 18: 4787.
https://doi.org/10.3390/en13184787