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Hydrocracking is a catalytic reaction process in the petroleum refineries for converting the higher boiling temperature residue of crude oil into a lighter fraction of hydrocarbons such as gasoline and diesel. In this study, a modified continuous lumping kinetic approach is applied to model the hydro-cracking of vacuum gas oil. The model is modified to take into consideration the reactor temperature on the reaction yield distribution. The model is calibrated by maximizing the likelihood function between the modeled and measured data at four different reactor temperatures. Bayesian approach parameter estimation is also applied to obtain the confidence interval of model parameters by considering the uncertainty associated with the measured errors and the model structural errors. Then Monte Carlo simulation is applied to the posterior range of the model parameters to obtain the 95% confidence interval of the model outputs for each individual fraction of the hydrocracking products. A good agreement is observed between the output of the calibrated model and the measured data points. The Bayesian approach based on the Markov Chain Monte Carlo simulation is shown to be efficient to quantify the uncertainty associated with the parameter values of the continuous lumping model.

Hydrocracking is a catalytic process in which the hydrocarbon molecules with longer chains break into lighter hydrocarbons with shorter chains. Hydrocracking units in the petroleum refineries usually feed with the heavy fractions of residual oil that are not commercially valuable and instead produces more valuable lighter fractions of hydrocarbons like gasoline and diesel. The hydrocracking process is especially important due to maximizing the use of crude oil as its resources are substantially reducing. Turning to use unconventional oil and gas reservoirs in recent years is an evidence of this importance [

Hydrocarbons cover a broad range of molecular weights with different types of organic compounds that makes it difficult to monitor the consumption/ production rates in the mixture of reactions in the hydrocracking process. Therefore, process engineers usually lump a group of organic compounds regardless of their molecular shape but based on their true boiling point (TBP) temperature having similar physio-chemical properties. Accordingly, kinetic modeling of heavy oil hydrocracking can be done based on the discrete or continuous lumping approach [

The discrete lumps kinetic model considers each lump characterized by its TBP temperature as one reactive species [

The continuous lumping kinetic model [

Both the discrete and the continuous kinetic models have unknown parameters that need to be estimated in order to calibrate the model. Regression techniques have been successfully used to minimize the misfit between the modeled and measured data. Sadeghi et al. [

However, there are some uncertainties associated with the value of the parameters that are not considered in the point estimation methods. The measurement errors, the model structural error (due to model simplifications), and errors associated with the operating conditions (like isotherm reactor assumption) are the main sources of uncertainty in the hydrocracking kinetic models that can affect the value of the estimated parameters. To address the uncertainties in the parameter estimation, a probabilistic approach can be considered. Different techniques have been applied in scientific fields to deal with uncertainties. Commonly, Monte Carlo simulations over a pre-assumed range of parameters [

Parameter estimation and uncertainty analysis by using Bayesian approach are widely used in different fields of science. Alikhani et al. [_{6}) degradation pathway. Their reaction modeling network is conceptually similar to the discrete lumps kinetic models where higher order molecules in a hierarchical degradation pathway convert to lower order molecules. Therefore, the Bayesian inference approach is assumed to be suitable for this study to obtain the confidence interval of hydrocracking model parameters.

Nonetheless, Bayesian parameter estimation framework needs to be solved by applying MCMC algorithms that usually requires a relatively larger number of model simulations. Therefore, an efficient numerical algorithm [

In this study, the continuous lumping kinetic model is slightly modified to take into consideration the temperature dependency of parameters. The Bayesian parameter estimation approach is applied to measured data obtained from a hydrocracking unit of a petroleum refinery to obtain the posterior credible intervals of the model parameters. Finally, the Monte Carlo simulation is performed by taking samples over the posterior range of estimated parameters to evaluate the confidence interval of the model output concentrations in different operating conditions.

The continuous lumping kinetic model in this study is explained in detail in [

The first-order reactivity (k) of each component can be related to its

where

where the right-hand side shows the consumption (the first term) and production (the second term) rate of the

The model is called continues because a cumulative distribution is assigned to the fraction yield of the species.

where A and B are defined as:

where

Solving the continuous model results in a distribution of the component concentrations

where

To take into consideration the temperature effect on the model parameters, an Arrhenius-type relationship [

where

The main idea of the Bayesian inference application in the parameter estimation is extracted from the work by Alikhani et al. [

The results of applying Bayes’ theorem would be obtaining the posterior distribution

In Equation (2),

where n is the total number of measured data points,

The Bayesian parameter estimation is particularly useful for the system of unknown parameters that are highly dependent on the operating conditions in which the calibrated parameter value of other studies is not suitable to be used in another study. Nonetheless, the information about the parameter values in other studies can be used to construct the prior distributions; and Bayesian approach can extract the information in the observed data set to enhance our confidence about the parameter values [

Measured data points obtained from the case study introduced in [_{3}/SiO_{2}-Al_{2}O_{3} catalyst having the density of 790 kg/m^{3}, specific surface area of 235 m^{2}/g, and porosity of 0.5. The hydrocracking products were collected (

Genetic algorithm (GA) used to obtain the point estimates of the modified continuous kinetic model parameters by maximizing the likelihood function (Equation (12)). In total, 10 model parameters plus the likelihood standard deviation treated as unknown parameters to be estimated by applying the observed measured data. The results from GA for two operating temperatures at 390˚C and 450˚C are shown in

The parameter values obtained by GA optimization were used as starting

Temp (˚C) | t (hr) | %LPG | %naphtha | %kerosene | %diesel | %VGO |
---|---|---|---|---|---|---|

390 | 0.34 | 1.35 | 1.77 | 3.45 | 4.10 | 89.33 |

390 | 0.4 | 1.55 | 2.01 | 3.91 | 4.82 | 87.71 |

390 | 0.5 | 1.80 | 2.39 | 4.73 | 5.97 | 85.11 |

390 | 0.66 | 2.29 | 3.09 | 6.05 | 7.67 | 80.90 |

390 | 1 | 3.20 | 4.06 | 8.72 | 11.25 | 72.77 |

390 | 2 | 5.61 | 5.89 | 15.15 | 19.35 | 54.00 |

410 | 0.34 | 2.37 | 3.20 | 5.40 | 5.86 | 83.17 |

410 | 0.4 | 2.87 | 4.14 | 5.84 | 7.01 | 80.14 |

410 | 0.5 | 3.33 | 5.00 | 6.73 | 8.94 | 76.00 |

410 | 0.66 | 4.97 | 5.82 | 8.84 | 11.03 | 69.34 |

410 | 1 | 6.42 | 7.98 | 12.64 | 14.75 | 58.21 |

410 | 2 | 8.81 | 9.67 | 21.23 | 25.28 | 35.01 |

430 | 0.34 | 4.81 | 5.95 | 7.26 | 8.70 | 73.28 |

430 | 0.4 | 5.85 | 6.65 | 8.25 | 10.55 | 68.70 |

430 | 0.5 | 6.01 | 8.89 | 10.51 | 12.21 | 62.38 |

430 | 0.66 | 7.61 | 11.73 | 12.23 | 14.73 | 53.70 |

430 | 1 | 9.72 | 14.99 | 16.32 | 19.62 | 39.35 |

430 | 2 | 13.05 | 19.55 | 22.84 | 28.26 | 16.30 |

450 | 0.34 | 5.81 | 11.55 | 12.59 | 15.94 | 54.11 |

450 | 0.4 | 6.92 | 13.78 | 14.30 | 16.80 | 48.20 |

450 | 0.5 | 8.31 | 16.40 | 16.62 | 18.49 | 40.18 |

450 | 0.66 | 10.02 | 19.80 | 18.82 | 21.51 | 29.85 |

450 | 1 | 14.78 | 38.42 | 13.22 | 17.03 | 16.55 |

450 | 2 | 20.61 | 51.68 | 11.15 | 13.75 | 2.81 |

points of 10 parallel Markov chains aiming at sampling 100,000 posterior set of parameter values in the Bayesian parameter estimation approach. The first 10,000 of total samples discarded due to burn-in period [

Values of 2.5^{th}, 50^{th} and 97.5^{th} percentile plus the standard deviation of each model parameter are shown in _{ }show that the value of

Parameters | Units | 2.5% tile | 50% tile | 97.5% tile | Std |
---|---|---|---|---|---|

^{5} (390˚C) | [-] | 0.20 | 2.26 | 4.88 | 1.50 |

[-] | 0.54 | 0.61 | 0.68 | 0.04 | |

hr^{−1} | 2.00 | 2.28 | 2.91 | 0.26 | |

[-] | 32.89 | 57.27 | 78.85 | 10.67 | |

[-] | 59.85 | 87.52 | 119.04 | 14.50 | |

˚C^{ −1} | −0.0005 | −0.0105 | −0.0360 | −0.0099 | |

˚C^{ −1} | 0.0075 | 0.0098 | 0.0122 | 0.0013 | |

˚C^{ −1} | 0.0047 | 0.0099 | 0.0134 | 0.0024 | |

˚C^{ −1} | 0.0001 | 0.0028 | 0.0141 | 0.0038 | |

˚C^{ −1} | −0.0427 | −0.0686 | −0.0800 | −0.0114 | |

%wt. | 0.025 | 0.028 | 0.032 | 0.002 |

This finding is in good agreement with the parameter-temperature correlation presented in Elizalde et al. [

In

To evaluate the ability of the model to meet the observed data, the Monte Carlo simulation is performed and 5000 realization parameter sets randomly sampled from the 95% posterior range. The model outputs were statistically analyzed and the 95% confidence interval of weight fraction of hydrocracking products are obtained and illustrated in

Results in

The model confidence intervals reflect the uncertainty level associated with the parameter values and can be used in decision-making step, obtaining factors of safety in designing step, and increasing the level of accuracy in the data measuring (and sampling) step; and generally, can enhance the modeling and simulation efficiency.

In this study, the parameter estimation of the hydrocracking process model is assessed. One system consists of five fractions (LPG, naphtha, kerosene, diesel, and VGO) is modeled by using continuous lumping approach. The continuous model is modified by considering the effect of reactor temperature in the parameter values.

The Genetic Algorithm and the Bayesian parameter estimation approach are applied to obtain the point estimate and the credible interval of the model parameters. A good agreement between the calibrated model output and the measured data is observed showing that the continuous lumping model is able to simulate the presented hydrocracking process. The results also show that the modified continuous lumping model considering the temperature dependency of the parameters extends the ability of the model on different operating temperature. Applying the Bayesian approach resulted in the 95% credible interval of model parameters reflecting the uncertainty associated with parameter values.

A probabilistic simulation is also performed by using the posterior range of the parameters to obtain the confidence interval of model outputs. The results show that for the studied process unit, the uncertainty associated with the measured data and the model structural error is quantitatively low. The Bayesian approach based on the Markov Chain Monte Carlo simulation is shown to be efficient to quantify the uncertainty associated with the parameter values of the continuous lumping model.

The following general conclusions obtained from the presented study:

1) Continuous lumping kinetic model was able to simulate the hydrocracking of VGOs in the range of 390˚C to 450˚C, and the residence time of up to 2 hr.

2) The model parameters were estimated by maximizing the likelihood function between the model outputs and measured data points.

3) The temperature dependency of the model parameters was successfully embedded into the continuous lumping model and the temperature dependency coefficients were estimated.

4) The uncertainty associated with the parameter values was evaluated by applying Bayesian theorem, and MCMC technique and the posterior range of parameters were obtained.

5) Monte Carlo simulations were performed to evaluate the confidence interval of hydrocracking products’ yield.

Boosari, S.S.H., Makouei, N. and Stewart, P. (2017) Application of Bayesian Approach in the Parameter Estimation of Continuous Lumping Kinetic Model of Hydrocracking Process. Advances in Chemical Engineering and Science, 7, 257-269. https://doi.org/10.4236/aces.2017.73019