Design of a VGO hydrocracking reactor under uncertainties

Most hydrocrackers are trickle-bed reactors in that the mixture of the feed and hydrogen flows downward through multiple fixed catalyst beds; additionally, slurry-phase or ebullating bed units can also be encountered [4]. The general structure of a trickle-bed hydrocracking unit is as follows. The feedstock and a part of the hydrogen are introduced to the first stage at the top of the reactor unit, and after each step, additional hydrogen is added, partly to enhance conversion and partially to quench the mixture and cool it back as the overall hydrocracking process is considered exothermic [101]. Due to this dual role of the hydrogen, which initiates cooling in a shorter term and then a mid-term heating resulting from the increased conversion, the control of the hydrocracking unit is somewhat more challenging; therefore, more sophisticated control methods such as model predictive control (MPC) are more often realized here [125].

As a complex process involving thousands of reactions, hydrocracking is often modeled using the lumping methods. These methods are in fact applicable during model-based reactor design and optimization of the hydrocracking unit [76,126].

Bhutani et al. employed such an approach to investigate an industrial hydrocracking unit, finding different ways to optimize the reactor, e.g., decreasing hydrogen makeup, or increasing kerosene or diesel yields [127]. There were a high number of decision variables present in this case, including feed and recycle flow rates whose control is not always feasible. On the contrary, Zhou et al. only varied the flow rate of the makeup hydrogen and still improved diesel and kerosene yields [128].

Nevertheless, using a lumped reaction network in reactor design involves the necessity to deal with a wide variety of uncertainties. A useful tool for this purpose is sensitivity analysis. Celse et al. compared the local (one-at-a-time) and a global approach to study the effect of various inputs and found that the results of

these are qualitatively similar and both can be used to identify model inputs responsible for the uncertainty of the output; so one can focus their attention to these variables to increase robustness [129]. In the case of a lumped reaction network, sensitivity analysis has a relatively low computational demand, with the drawback that such models involve a great deal of simplification, resulting in a broader range of uncertainties. Whereas, in the case of a detailed model, it is worth constructing a surrogate model for sensitivity analysis purposes [130,131].

Lesser sensitivity index values can also be translated to a more robust design, i.e., a parameter may still vary within the same range, but its effect on the output reduces significantly [132]. Therefore; in order to characterize reactor robustness, one would take a list of uncertain parameters with a possible effect on the output, eliminate those that can be described with lower influence on the model uncertainty (i.e., those associated with lower indices), then assess robustness regarding the remaining parameters [133]. Another model-based robustness criterion states that the deviation of the objective function caused by the variance of the uncertain parameter has to be minimal [134]. This is essentially an optimization problem. Steimel and Engell formulated this objective function as a sum of two terms. The first included the design variables that would be fixed after the realization of the system, and the second term summarized the costs of different operating scenarios, including their probabilities related to the chance that a specific uncertain condition would be actually met [135].

As process models are always affected by uncertainties (whether we acknowledge that explicitly or not), the topic has been widely discussed in the literature. The explicit depiction of uncertain parameters transforms the conventional deterministic mathematical model into a stochastic one [136].

Stochastic programming models consider the variability of possibly uncertain parameters so that we can optimize the expected (average) performance of the model based on the likeliness of these uncertain events [137]. The stochastic approach is commonly used for plant modeling (i.e., the interaction between several units) [138,139] or to model supply chains [140,141]. On the other hand, it is less often applied in designing a single reactor unit, mainly because at first glance, there is not much uncertainty in a model. The concept does appear in the

literature, e.g., Calderón and Ancheyta determined the sensitivity of a hydrocracking reactor to several uncertain parameters [142]; nevertheless, there was no apparent feedback from the sensitivity study to the initial modeling process. Moreover, Alvarez-Majmutov and Chen used a stochastic modeling approach to account for the uncertainties regarding the reactor heat balance, while also comparing the results to that of the conventional methods [143]. This could be important as the reactor temperature has a high impact on process performance.

Reaction kinetics are usually not treated as uncertain parameters because in the case of conventional reactions, we have a strong theoretical basis on the reaction mechanism available. There are exceptions, especially when multiple side reactions are present. Mukkula and Engell studied the optimal operating conditions of a pilot-scale tube reactor with the assumption that there could be a mismatch between the assumed and actual reaction mechanism, providing a real-time optimization solution that could handle the discrepancy [144]. Whereas, in the case of lumped networks the corresponding kinetic parameters are essentially obtained as a result of a parameter fitting to the experimental data. Therefore, it is worth investigating, and it might be as well worth considering the model sensitivity to the kinetics during the reactor design work. Despite that, based on our literature review, the uncertainty of lumped reaction networks usually does not get enough emphasis. In Chapter 10, I investigate how the uncertainties of the lumped reaction network affect the reactor design and how one can straightforwardly account for that.

Yet another uncertain aspect of the hydrocracking reactor model, which is not inevitably recognized in full detail, is catalyst deactivation. The formation of coke and other carbonaceous deposits on the surface of the catalyst is one of the main drawbacks of residue hydrocracking. Due to their low volatility and strong adsorption properties, these components are retained on the surface of the catalyst, blocking (fouling) the active sites and thus deactivating the catalyst [145].

Mesoporous catalysts show higher resistance to fouling [146]; nevertheless, the phenomenon cannot be neglected during the reactor design. There is also a slower process present where the metal content of the feed and the adsorbed nitrogen compounds change the surface structure (called poisoning) [147]. Because of

catalyst deactivation, the temperature of the reactor must be increased in the long-term to compensate for the activity loss and to maintain conversion, resulting in higher operating costs [148].

There are many deactivation models available in the literature, the more elementary equations describe the deactivation process as a function of Time-on-Stream (ToS), whereas the complex models also include at least the concentration of the deactivating agent [149]. In the case of using a lumped reaction network, the latter is difficult to interpret as these molecules are not addressed separately.

Therefore, catalyst deactivation is usually modeled by using an exponential decay function:

𝜑 = exp(−𝛼 ∙ 𝑇𝑜𝑆) (2.3)

The decay coefficient, α, is not necessarily constant, e.g., it can be influenced by the temperature [150]. Consequently, the application of the decay function is a powerful method with industrial applications as well [151]. On the other hand, it does not take the intrinsic kinetics into account; hence, its parameters might be uncertain.

In Chapter 10, I investigate the effect of catalyst deactivation as a form of uncertainty as well. This way, we can eliminate the necessity of using computationally expensive dynamic models and simulations to account for deactivation. Moreover, we can investigate the effect of deactivation (which, as I said, can be an uncertain process) alongside with the effects of other possible uncertain parameters using a single modeling framework.

Finally, I will point out that the application of a stochastic design method is not automatically advantageous; rather, its usefulness depends on how flexibly we can operate the designed reactor system. In order to investigate that, I compare the performance of the hydrocracking reactor designed by applying the conventional and stochastic methods and quantified the extent to which the optimal reactor operation could be maintained when exposing it to changes in the uncertain parameters.