#### The 1-D reactor model

The steady-state continuity equation for a component *j* in the process-gas mixture over an infinitesimal volume element with cross-sectional surface area, Ω, circumference ω, and length *dz* is

- (1)

The energy equation is given by

- (2)

To calculate the wall temperature based on the process-gas temperature, the internal heat-transfer coefficient is calculated from the Dittus-Boelter equation

- (3)

The pressure equation accounting for friction and changes in momentum is given by

- (4)

The boundary conditions for the 1-D problem at the inlet *(z = 0)* are

The process-gas temperature profile, conversion, and concentration profiles can be calculated based on an imposed heat-flux profile or external tube-skin temperature profile. In this article, calculations were performed using a heat-flux profile.

Integration of the continuity equations results in the concentration profiles of all the involved components. These equations form a set of stiff ordinary differential equations, that is, the local eigenvalues differ by several orders of magnitude. In practice, the concentration changes of the radicals occur on a much smaller time scale than that of the molecules. Therefore, an algorithm developed specifically for radical reaction networks has been employed (Dente and Ranzi, 1979).

#### The 2-D reactor model

The 2-D reactor model is based on a 2-D velocity vector. A mass balance for a component *j* over an annulus with height *dz,* internal radius *r,* and external radius *r + dr* leads to the continuity equation for this component *j* in the process-gas mixture (Bird et al., 2001; Froment and Bischoff, 1990)

- (5)

Correspondingly, an energy balance leads to the energy equation

- (6)

Equation 6 considers heat transport and reaction simultaneously. The latter implies a good estimation of the simulated wall temperature. The last term in Eq. 6 is referred to as the interdiffusional energy flux by Bird et al. (2001). The origin of this term is the diffusion of chemical species. Explicitly introducing the reaction enthalpy in Eq. 6 results in Eq. 7

- (7)

Radial pressure gradients are neglected and, hence, Eq. 4 is retained.

In the model equations no terms regarding coke layer thickness have been added, because the calculations are performed for the initial coke formation rate.

For the axial velocity component *u*_{z} the von Karman profile (Davies, 1972) is used. Three zones are considered over the entire tube length: a laminar, a turbulent and a transition zone. In each zone the axial velocity is calculated using a different expression. In the laminar zone

- (8)

In the transition zone

- (9)

In the turbulent zone

- (10)

After calculating the axial velocity profile according to Eqs. 8, 9, and 10, the radial component of the velocity *u*_{r} can be deduced from the total mass balance

- (11)

The turbulent conductivity and diffusivity are calculated based on the correlation of Reichardt, corrected by Cebeci (Sundaram, 1977). The parameters have been further adjusted by Sundaram and Froment (1979). In Eqs. 12 and 13 the expressions for the turbulent conductivity *ϵ*_{H} and the turbulent diffusivity *ϵ*_{D} are given

- (12)

- (13)

where *x* represents the normalized radius, *c*_{1} = 0.828664, *c*_{2} = 0.944067, *c*_{3} = 0.020530, and the following expressions hold for *a* and *b*

The Fanning friction factor is obtained from the Prandtl equation

- (14)

In the laminar zone near the reactor wall, the conductivity equals the molecular conductivity, *λ*_{m}, and the diffusivity equals the molecular diffusion coefficient, *D*_{m}.

The boundary conditions for the 2-D problem are:

In the center of the tube *(r = 0)*

At the inner reactor wall *(r = R)*

The axial heat profile is imposed at the inner wall *(r = R)*

- (15)

This heat flux profile can be determined by calculating the radiative heat transfer in the furnace (Heynderickx and Froment, 1998). Circumferential nonuniformities due to the shadow effect in the furnace are not taken into account. The differential equations (Eqs. 5 and 7) are solved via a finite difference method. The Cranck-Nicholson method is used to calculate the differential variations in the axial direction. The differential variations in the radial direction are approximated by the second order differential of the Lagrange polynomial. For the application of the integration procedure a number of grid points has to be defined. Because of the steep gradients near the wall, a sufficiently small step size is required in that zone. The step size was varied according to a geometric progression: coarse in the core and fine near the wall. The set of differential equations for a specific variable in a radial section divided in *n* gridpoints *i* can then be rewritten in a tridiagonal format. For the concentration of component *j,* Eq. 16 is obtained

- (16)

This set of equations is solved simultaneously by the Thomas algorithm.

The implementation of a kinetic model based on a radical reaction mechanism again results in a stiff set of differential equations. The pseudo-steady-state approximation (PSSA) for the radicals cannot be used because it transforms the continuity equations for the radicals into algebraic equations. This makes it impossible to take the effect of the diffusion of radicals from neighboring zones into account, which requires a differential equation. Therefore a special solution method is used. An element, *e*_{i}, on the righthand side of Eq. 16 contains the reaction rates calculated with the applied kinetic model. It is convenient to separate this contribution into two parts, one describing the cumulative rate of formation of the *j*th component by all reactions forming this component and the other describing the cumulative rate of disappearance of the same component

- (17)

The second term on the righthand side of Eq. 17 is a linear function of the concentration of component *j* in the *i*th gridpoint of a section, neglecting the small contribution of possible recombination reactions. Equation 17 can then be rewritten

- (18)

Separation of the righthand side of Eq. 16 according to Eq. 18 by moving the second term on the righthand side to the left side of Eq. 16 solves the stiffness problem. The diagonal elements of the tridiagonal matrix increase; hence, the stability of the solution is increased.

#### Simulation conditions

An industrial ethane-cracking furnace is simulated. The energy required for the steam cracking of the ethane feed is provided by radiation burners. An identical heat-flux profile is used as input, to have a good basis for the comparison of the results obtained with both reactor models. Hence, the same amount of energy is added to the process gas. The heat-flux profile was obtained by performing a coupled simulation of the furnace and the reactor tubes, using the 1-D reactor model. For the calculation of the furnace the zone method of Hottel and Sarofim (1967) is used. This simulation method is developed by Vercammen and Froment (1980), Rao et al. (1988), and Plehiers and Froment (1989).

The furnace is divided into a number of isothermal surface and volume zones. The energy balances, containing radiative, convective, and conductive contributions, are constructed for these zones. Furnace-wall, process-gas, and tube-skin temperature profiles in the furnace are obtained by solving the energy balances. From these temperature profiles, a better estimate of the heat-flux profile is obtained, based on which a new reactor calculation is performed. With the resulting tube-skin temperature profile, a new furnace calculation is performed. This cycle is repeated until convergence is reached.

For the simulated ethane-cracking furnace, the burners are located in the sidewalls on both sides of the coils. The flue gas entering the furnace through the burners delivers energy to the reactor wall and therefore decreases in temperature. The flue gas temperature is higher in the bottom section of the furnace due to the hampered flow. In the top section there are no burners, so the flue-gas temperature profile is smoother. The nonuniformity of the flue-gas temperature results in a strongly varying heat flux to the process gas, as shown in Figure 1. At the top of the furnace the heat flux is much lower than at the bottom where there are sharp peaks.

There was no coupled simulation of furnace and reactor coils performed using the 2-D reactor model. For reasons of comparison, as explained earlier, the same heat-flux profile was used. However, the slight change in the external tube-skin temperature profile will result in a small change in the heat-flux profile when a complete simulation is performed. The operating conditions and the furnace and reactor geometry are listed in Table 1. The total hydrocarbon flow rate through one reactor coil is 3.5 tons per hour. The inlet temperature of the process gas is 873 K. During the steam cracking of ethane a steam dilution of 0.35 kg per kg feed is applied. The steam reduces the partial pressure of the hydrocarbons in the gas phase and reduces the coke formation.

Table 1. Furnace, Reactor Geometry, and Process Conditions for the Standard Ethane Cracking Furnace.* |

Furnace | |

Furnace length | 9.30 m |

Furnace height | 13.45 m |

Furnace width | 2.10 m |

Thickness refractor material | 0.23 m |

Thickness insulation material | 0.05 m |

Number of burners | 128 |

Heat input | 14.43 MW |

Reactor coil | |

Number of reactors | 4 |

Number of passes | 8 |

Reactor length | 100.96 m |

Reactor diameter (int) | 0.124 m |

Wall thickness | 0.008 m |

Ethane flow rate per reactor coil | 0.972 kg s^{−1} |

Steam dilution | 0.35 kg/kg |

CIT* | 873 K |

CIP** | 0.34 MPa |