**1. Introduction**

Hydrocarbon steam cracking is the most important process for producing industrial chemicals such as olefin and aromatics. Mathematical models for steam cracking simulation have been

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studied for several decades and various models have been developed such as SPYRO [1] and COILSIM [2]. Meanwhile, the steam cracking model can be applied to the steam furnace design and optimization [3]. As various feedstock and furnaces are used in steam cracking, a more accurate and robust model is needed.

A mathematical model is a collection of relations among variables representing certain properties of a system, using equations and inequalities. By solving a model, one can predict the values of some properties of the system, given the values of some other properties, or predict the distributions of the values of some properties in particular domains, given certain boundary conditions. To meet the scientific and engineering demands in terms of revealing the characteristics of a system in-depth, a mathematical model may involve relations of properties at different scales of the system [4], referred to as a multi-scale model. The multiscale model often consists of nonlinear equations and differential equations, and is not easy to solve. The information communication between scales is the key factor in multi-scale models [5]. In recent years, multi-scale modelling has been applied in all fields of chemical engineering such as thermodynamics, reaction engineering, polymer materials and CFD (computational fluid dynamics), among others. Multi-scale modelling is used in steam cracking models to reveal the nature of steam cracking and to establish a more accurate and robust model.

The kernel part of the steam cracking model is the reaction network. Researchers have developed various reaction models to describe the steam cracking process. There are three different types of reaction models: empirical models, molecular models and elementary reaction models. Empirical models use a large database of experimental results to regress a number of empirical correlations for the yields of the main products as a function of a number of easily measurable process variables [6]. Empirical reaction models need a large number of experiment data to render a regression. Once the feedstock or the furnaces change, the reaction model has to be re-established to obtain accurate simulation results. On the basis of empirical reaction models, molecular reaction models have been developed and frequently used [7-9], e.g., the Kumar reaction model [10]; 22 molecular reactions are involved in Kumar's reaction model: one primary reaction and 21 secondary reactions. The primary reaction is shown as eq. (1). The selective coefficients of the primary reaction (*a1~a10*) have been regressed from experimental data. If the feedstock or operating conditions change, the selective coefficients of the primary reaction should be regressed again.

$$\begin{aligned} \text{Inpathha} & \rightarrow a\_1 \text{H}\_2 + a\_2 \text{CH}\_4 + a\_3 \text{C}\_2 \text{H}\_4 + a\_4 \text{C}\_2 \text{H}\_6 + a\_5 \text{C}\_3 \text{H}\_6 + \\ & + a\_6 \text{C}\_3 \text{H}\_8 + a\_7 \text{C}\_4 \text{H}\_{10} + a\_8 \text{C}\_4 \text{H}\_8 + a\_9 \text{C}\_4 \text{H}\_6 + a\_{10} \text{C}\_4 \text{s} \end{aligned} \tag{1}$$

Nowadays, elementary reaction models have been widely used to develop a more reliable and robust mathematical model. Since the pioneering work of Rice [11-13] there has been a general consensus about the elementary reaction mechanism. As the mathematical difficulties encountered for solving the detailed kinetic models can be overcome by the development of stiff solvers [14], detailed elementary reaction networks are widely used to accurately describe chemical processes in a wide range of process conditions and feedstock. Sundaram et al. [15] established a radical reaction model for the pyrolysis of simple paraffins, olefins and their mixtures. Scharfe et al. [16] established a radical reaction model for n-hexane pyrolysis. Joo et al. [17] established a radical reaction model for industrial naphtha cracking furnaces.

studied for several decades and various models have been developed such as SPYRO [1] and COILSIM [2]. Meanwhile, the steam cracking model can be applied to the steam furnace design and optimization [3]. As various feedstock and furnaces are used in steam cracking, a more

A mathematical model is a collection of relations among variables representing certain properties of a system, using equations and inequalities. By solving a model, one can predict the values of some properties of the system, given the values of some other properties, or predict the distributions of the values of some properties in particular domains, given certain boundary conditions. To meet the scientific and engineering demands in terms of revealing the characteristics of a system in-depth, a mathematical model may involve relations of properties at different scales of the system [4], referred to as a multi-scale model. The multiscale model often consists of nonlinear equations and differential equations, and is not easy to solve. The information communication between scales is the key factor in multi-scale models [5]. In recent years, multi-scale modelling has been applied in all fields of chemical engineering such as thermodynamics, reaction engineering, polymer materials and CFD (computational fluid dynamics), among others. Multi-scale modelling is used in steam cracking models to reveal the nature of steam cracking and to establish a more accurate and robust model.

The kernel part of the steam cracking model is the reaction network. Researchers have developed various reaction models to describe the steam cracking process. There are three different types of reaction models: empirical models, molecular models and elementary reaction models. Empirical models use a large database of experimental results to regress a number of empirical correlations for the yields of the main products as a function of a number of easily measurable process variables [6]. Empirical reaction models need a large number of experiment data to render a regression. Once the feedstock or the furnaces change, the reaction model has to be re-established to obtain accurate simulation results. On the basis of empirical reaction models, molecular reaction models have been developed and frequently used [7-9], e.g., the Kumar reaction model [10]; 22 molecular reactions are involved in Kumar's reaction model: one primary reaction and 21 secondary reactions. The primary reaction is shown as eq. (1). The selective coefficients of the primary reaction (*a1~a10*) have been regressed from experimental data. If the feedstock or operating conditions change, the selective coefficients

*aa a a a*

1 2 2 4 32 4 42 6 53 6

++ +++ (1)

®+ + + + +

Nowadays, elementary reaction models have been widely used to develop a more reliable and robust mathematical model. Since the pioneering work of Rice [11-13] there has been a general consensus about the elementary reaction mechanism. As the mathematical difficulties encountered for solving the detailed kinetic models can be overcome by the development of stiff solvers [14], detailed elementary reaction networks are widely used to accurately describe chemical processes in a wide range of process conditions and feedstock. Sundaram et al. [15]

*aa aaa*

6 3 8 7 4 10 8 4 8 9 4 6 10 4

naphtha H CH C H C H C H C H C H C H C H C 's

accurate and robust model is needed.

118 Advances in Petrochemicals

of the primary reaction should be regressed again.

However, it has remained problematic to generate thousands of elementary reactions and to determine the reaction rate constants of these reactions until an automatic reaction network generation technique was studied. Today, computers are used not only to solve the simulation numerically, but also to generate the elementary reaction network, construct the model and calculate the kinetic parameters [2]. Many research groups have developed computer tools for automatically generating these mechanisms [18-23] such as RMG [24] and CRACKSIM [25]. The elementary reaction model is expected to be used in furnace design and operating condition optimization. However, the huge number of reactions in the elementary reaction network usually requires significant CPU time during simulation. Thus, the steam cracking simulation model has to be reduced before it can be used in operating condition optimization, in order to make real-time optimization (RTO) realistic.

There are multiple methods for creating model reduction. In terms of chemistry, quasi-steadystate approximation (QSSA) [26], reaction rate analysis [27] and reaction path analysis [28], among others, can be used to reduce a reaction model. In terms of mathematics, principle component analysis [29] and sensitivity analysis [30] can also be used. On the other hand, data driven methods can also be used in model reduction, for example, the black box model, a neural network [31] and PCA based ROMs [32]. Several assumptions can also be applied to help retain the mechanism at a manageable size. The most important assumption is the *μ* radical hypoth‐ esis, which assumes that bimolecular reactions can be neglected for radicals with more than five carbon atoms [33]. The latter are also referred to as *μ* radicals. Thus, the QSSA method can be applied to remove *μ* radicals from the reaction network.

A multi-scale model for the steam cracking process is established in this chapter. The multiperiod steam cracking process is also studied in the context of the established multi-scale model. Coking is an unavoidable factor during the multi-period steam cracking process. Coking increases pressure drops in the reaction tube, decreases the coefficient of heat transfer between the furnace and tube, and most importantly, increases the outer-wall temperature of the tubes. If one of the tubes in the furnace reaches the maximum allowance temperature of the tube material, the furnace must be shut down to execute a decoking process, otherwise the tube will be destroyed. The operation conditions are generally maintained constant during the long-term steam cracking process. Thus, dynamic operating conditions need to be optimized using a detailed steam cracking model in order to achieve a higher profit. Abel et al. [34] used the SQP method to solve a real-time optimization problem in the olefins production process. Tarafder et al. [35] proposed a multi-objective optimization problem in the operation and design of a styrene manufacturing process. Li et al. [36] applied an artificial neural network (ANN) hybrid model in the operation optimization of a naphtha industrial cracking furnace. Gao et al. [3] used a new parallel hybrid algorithm combining NSGA-II with SQP on multiobjective optimization for the periodic operation of the naphtha pyrolysis process. However, due to the complexity of the elementary reaction model, the researchers did not use this reaction model in the optimization problem. In this chapter, an elementary reaction model is applied to the operating condition optimization problem to obtain a more reliable optimization result. Based on this, a surrogate coke thickness model is proposed to make multi-period optimization with parallel simulation possible.

The general idea of this chapter is outlined in Figure 1. The first step in the conceptual development of a detailed molecule-based model for a complex feedstock is to determine an accurate molecular representation of the feedstock. Then, a multi-scale steam cracking model is established following the feedstock prediction. Finally, operating condition optimization of multi-period steam cracking is carried out using the established multi-scale model.

**Figure 1.** Diagram of simulation and optimization of the steam cracking process methodology

This chapter is structured as follows. Section 2 offers a detailed discussion of the establishment of the multi-scale steam cracking model; a case study for naphtha steam cracking simulation is presented. Section 3 provides the operating condition optimization model; surrogate coke thickness model and parallel simulation are used to accelerate the computing of the optimi‐ zation model. A case study of operating condition optimization is presented and the results are presented and discussed. Finally, section 4 offers conclusions from this study.
