**3.1. Multi-period optimization model for steam cracking**

Coking is an unavoidable factor in the long-term steam cracking process. During the steam cracking process, coke forms on the inner walls of the tube. With coke formation, the internal diameter of the tube decreases, pressure drops increase and outer wall temperature increases. It is generally accepted that coke forms from unsaturated hydrocarbons and aromatics [43]; on this basis, an empirical model for coke formation rate is proposed, as shown in eq. (9).

Simulation and Optimization of Multi-period Steam Cracking Process http://dx.doi.org/10.5772/60558 129

$$r\_c = \sum\_{i=1}^{9} A\_i \exp\left(-\frac{E\_i}{RT}\right) c\_i \tag{9}$$

In eq. (9): 1=C2H2; 2=C2H4; 3=C3H6; 4=1-C4H8; 5=C4H6; 6=C6H6; 7=C7H8; 8=xylene; 9=C2H3-C6H5.

**Reaction A n E**

C2H4+C2H3J ↔ C4H7J 4.18E+10 0 5.2 n-C4H10+C2H3 ↔ C4H9J

C4H7J+HJ ↔ BUT-1 5.00E+13 0 0 n-C4H10+C4H7J ↔ C4H9J

C4H7J+HJ ↔ BUT-2 5.00E+13 0 0 C4H9J+BUT-1 ↔ n-

BUT-1+CH3J ↔ C4H7J+CH4 5.46E+13 0 10.39 C4H9J+BUT-2 ↔ n-

BUT-2+CH3J ↔ C4H7J+CH4 4.82E+02 2.92 7.16 C4H9J+C3H6 ↔ n-

**Table 2.** Elementary reaction network for light hydrocarbon steam cracking.

**2.2. Case study: Naphtha steam cracking model simulation**

**3. Multi-period steam cracking optimization**

**3.1. Multi-period optimization model for steam cracking**

BUT-1+C2H5J ↔ C4H7J+C2H6 3.12E-04 4.31 3.39

125 species.

128 Advances in Petrochemicals

**(kcal/mol)**

+C2H4

+BUT-1

C4H10+C4H7J

C4H10+C4H7J

C4H10+C3H5J

A light hydrocarbon reaction model was generated using RMG, which contained 91 reactions and 26 species (Table 2). A naphtha steam cracking reaction network was generated based on the light hydrocarbon network. The naphtha reaction network contained 2424 reactions and

Thirteen sets of industrial data (Table 3) were used here to verify the established multi-scale steam cracking model. The industrial data were taken from a steam cracking furnace designed by KBR (Kellogg Brown & Root). The data were collected twice a day and each set of data was the average value of these two parallel experiments as a means for preventing any errors. The multi-scale steam cracking simulation took roughly 70s CPU time. The simulation results and the industrial data of the mass fraction of the main products are shown in Figure 5, where the x-axis represents the industrial data and y-axis represents the simulation results. All points in Figure 5 are around the 1:1 diagonal line, which shows that the error of most results were small. Thus, the established multi-scale model was defined as accurate and robust; it satisfies industrial needs and can be applied further in our study of operating conditions optimization.

Coking is an unavoidable factor in the long-term steam cracking process. During the steam cracking process, coke forms on the inner walls of the tube. With coke formation, the internal diameter of the tube decreases, pressure drops increase and outer wall temperature increases. It is generally accepted that coke forms from unsaturated hydrocarbons and aromatics [43]; on this basis, an empirical model for coke formation rate is proposed, as shown in eq. (9).

**Reaction A n E**

**(kcal/mol)**

9.30E+04 2.44 5.5

3.95E+03 2.71 12.9

3.12E-04 4.31 3.39

3.36E+12 0 12.39

1.01E-04 4.75 4.13

Steam cracking is a dynamic process, as coke grows inside the tube. However, coke formation is slow enough that we can divide the entire cracking period into a series of virtual steady state periods. Thus, the established multi-scale model can be used in each steady state period, with coke thickness updated between periods.


**Table 3.** Industrial data for feedstock, operating conditions and yields of major products.

Operation condition optimization is carried out based on the multi-period process model. Coil outlet temperature (COT) is selected as the variable to be optimized. Thus, the COT of all time periods of the multi-period model is discretized as [*COT1, COT2,..., COTn*].

The optimization model is summarized below.

$$\max \frac{1}{N} \sum\_{j=1}^{M} \sum\_{i=1}^{N} o\_i y\_{j,t}^{o} \tag{10}$$

Subject to:

$$\frac{dy\_{m,t}}{dL} = \frac{\mathbb{S}\_t}{V\_t} \sum\_i \nu\_{i,m} r\_{i,t} \quad \forall m \in \{1, 2, \cdots, M\} \quad \forall t \in \{1, 2, \cdots, N\} \tag{11}$$

$$\frac{dP\_t}{dL} = -\frac{f \cdot E\{L\} \cdot G\_t^2}{5.07 \times \rho\_t \cdot D\_{i,t} \times 10^4} \quad \forall t \in \{1, 2, \cdots, N\} \tag{12}$$

$$\frac{dT\_t}{dL} = \frac{k\_t \pi D\_o \left(T\_{w,t} - T\_t\right) - \sum\_m \Delta H\_{fm,t}^0 \cdot \frac{dy\_{m,t}}{dL}}{\sum\_m c\_{pm,t} y\_{m,t}} \quad \forall t \in \{1, 2, \dots, N\} \tag{13}$$

$$D\_{i,t+1} = D\_{i,t} - 2\Delta \delta\_{c,t} = D\_{i,t} - 2 \cdot \frac{r\_{c,t}}{\rho\_{c,t}} \quad \forall t \in \{1, 2, \cdots, N - 1\} \tag{14}$$

$$r\_{c,t} = \sum\_{l=1}^{9} A\_l \exp\left(-\frac{E\_l}{RT}\right) c\_{l,t} \quad \forall t \in \{1, 2, \dots, N\} \tag{15}$$

$$\max\_{t} T\_{w,t} \le T\_M \quad \forall t \in \{1, 2, \dots, N\} \tag{16}$$

$$\left| \left| \mathbf{COT}\_{t \ast 1} - \mathbf{COT}\_t \right| \leq \delta \quad \forall t \in \{1, 2, \dots, N - 1\} \tag{17}$$

$$
\Box \mathbf{LB} \le \mathbf{CO} \mathbf{T}\_\circ \le \mathbf{LB} \quad \forall t \in \{1, 2, \dots, N\} \tag{18}
$$

In objective function (10), *M* is the number of considered species, *N* is the period number, *ω<sup>t</sup>* (*t*=1, 2,..., *M*) are weighted fractions based on the price of each species and *yo j,t* is the mass fraction of selected species *j* in products of period *t*. Eqs. (11-13) describes mass balance, momentum balance and energy balance equations in period *t* (*t*=1, 2,..., *N*). In eq. (14), the internal diameter of period *t*+1 *Di,t*+1 equals the internal diameter of the previous period *Di,t* minus the coke layer thickness. In eq. (16), peak outer-wall temperature should not exceed the maximum temperature of the tube material. Eq. (17) shows that the adjacent COT difference is restricted to a certain region to keep the operation stable. Eq. (18) shows the upper and lower boundaries of optimization variables.

**Figure 5.** Simulation results compared with industrial data.

The optimization procedure is shown in Figure 6.
