**3.2. Surrogate coke thickness model**

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

*M N*

*j i j t <sup>y</sup> <sup>N</sup>* 1 1

= =

{ } { } *m t <sup>t</sup>*

( ) { } *<sup>t</sup> <sup>t</sup>*

4 , <sup>5</sup> 1,2 .07 , ,

2

*t it*

*pm t m t*

, ,

*E r A tc RT*

æ ö = -ç <sup>÷</sup> <sup>Î</sup>

" Î= - -å

, ,

r10

*D*

1 max

*o*

,

åå (10)

*t N*

" Î 1,2, , 1,2, , L L " Î= (11)

*t N*

{ } *<sup>m</sup>*

*m t*

0 ,

*c t*

,

 1,2, ,2 12 r

{ } *<sup>i</sup>*

*r*

exp 1,2, ,

In objective function (10), *M* is the number of considered species, *N* is the period number, *ω<sup>t</sup>*

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*

*wt M* { } *<sup>t</sup>*

*COT CO t N t t* <sup>+</sup><sup>1</sup> *T* £ d

(*t*=1, 2,..., *M*) are weighted fractions based on the price of each species and *yo*

× × " Î ´× ´ = - <sup>L</sup> (12)

*dL t N*

{ } *c t*

*N*

è ø <sup>å</sup> " <sup>L</sup> (15)

*T t NT* , ax " Σ 1,2,L,m (16)

" Î- {1,2, , 1 L - } (17)

*j,t* is the mass

*LB COT t N <sup>t</sup>* £ *UB* " Σ {1,2, , L } (18)

*t N* ,

<sup>+</sup> <sup>=</sup> - <sup>=</sup> -D <sup>×</sup> " <sup>Î</sup> <sup>L</sup> - (14)

1,2, ,

<sup>å</sup> <sup>L</sup> (13)

*t*

w

periods of the multi-period model is discretized as [*COT1, COT2,..., COTn*].

*im it*

*dP E L Gf*

( )

*dL c y*

*it it ct it*

,1 , , ,

*DD D*

*i*

=

1

9

*dy k TT dT*

*t o wt t fm t*

D ×

*D H*

*m*

d

*ct i i t*

, ,

, , ån

*m M dy <sup>S</sup> r*

*t i*

*d*

*t*

*L V* ,

*dL*

p

The optimization model is summarized below.

Subject to:

130 Advances in Petrochemicals

The most time-consuming part of the optimization is the simulation of the multi-period model. As Figure 6 shows, the only connection between adjacent periods is the coke thickness. Coke thickness is related to the feedstock component, furnace running time and operating condi‐ tions. As it has been assumed that the feedstock component is fixed between periods and only COT changes in operating conditions, as eq. (19) shows, then *dt* is the coke thickness in period *t*.

$$d\_i = f\left(\text{feed,period number}, \text{COT}\_1, \text{COT}\_2, \dots, \text{COT}\_t\right) \tag{19}$$

The accumulated coke thickness within a certain period *δd<sup>t</sup>* is assumed to only be related to the furnace running time and *COTt* (*t*=1, 2,..., *N*-1), shown as eqs. (20) and (21).

$$\begin{aligned} \delta d\_t = d\_{t+1} - d\_t = f\left(t, \text{COT}\_t\right) = a \times \text{COT}\_t^2 + b \times t^2 + c \times \text{COT}\_t + d \times \text{COT}\_t + e \times t + f \\ \forall t \in \{1, 2, \dots, N - 1\} \end{aligned} \tag{20}$$

$$d\_t = \sum\_{t=1}^{l} \delta d\_t = \sum\_{t=1}^{l} f\_t \left( t, \text{COT}\_t \right) \tag{21}$$

**Figure 6.** Optimization procedure with multi-period steam cracking simulation.

Thus, *δd<sup>t</sup>* in period *t* can be regressed using the polynomial function shown as eq. (20). Here, the coke thickness data is generated using the original multi-period simulation model and based on this, a surrogate coke thickness model is obtained through regression. The coke thickness using the surrogate model and original multi-period simulation model are shown in Figure 7. Dots in Figure 7 are coke thickness from the original multi-period simulation model and the surface is from surrogate model.

The coke thickness from the surrogate model fits well with the original model; thus, the decoupled multi-period cracking model, combined with the surrogate coke thickness model was used in the multi-period simulation. The initial coke distribution along the serial operation periods was carried out using the surrogate model. Thus, the multi-period simulation problem was decoupled into *N* sub-problems and simulated, respectively, in parallel, as shown in Figure 8.
