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Tubular reactors are used in industry for important processes such as steam cracking and polymerization. Their analysis and design is frequently based on 1-dimensional models, that is, considering gradients only in the axial direction. Semiempirical correlations can approximate the average of radial concentration and temperature profiles from more-dimensional models (Sundaram and Froment, 1979), but do not provide any information on the importance and consequences of the nonuniformities for the reactor performance. The latter is of particular importance for the endothermic steam-cracking process. The trend toward high-severity cracking (Plehiers and Froment, 1991) demands higher heat fluxes, higher process-gas temperatures, and shorter residence times. Higher heat fluxes amplify the radial temperature gradients and make the 1-dimensional plug-flow model insufficient (Froment, 1992). Furthermore, a radial temperature gradient implies that the conditions prevailing at the process gas–coke interface on the one hand, and the conditions in the center of the reactor coil on the other hand may differ appreciably (De Saegher, 1996). Coke formation at the interface conditions (Sundaram et al., 1981) or at averaged conditions as calculated with a 1-dimensional model will differ. It was shown before (Heynderickx et al., 1992) that circumferential nonuniformities in flux and temperature, due to the shadow effects in the furnace, also result in nonuniform coking rates and coke layers. The radial temperature profile can have a significant effect on the calculated reactant concentrations and more in particular on those of the gas-phase radicals at the internal wall of the reactor tubes (Reyniers et al., 1994). The implementation of more dimensional models for complex reactions in a tubular reactor seems to be inevitable. Sundaram and Froment (1979, Sundaram and Froment 1980) coupled a global kinetic model for the cracking of ethane to a two-dimensional (2-D) reactor model. Their results confirm the existence of important radial-temperature gradients. For molecular species, on the other hand, they found that radial-concentration gradients are less important. Valenyi et al. (1991) and Fagley (1992) proposed a 2-D model for the cracking of ethane in a tubular reactor with laminar flow, using the simulation packet PHOENICS to solve the 2-D mass, heat, and momentum balances. The kinetic model consisted of three reactions between five molecular species. Radial temperature differences up to 200 K were found, as were important radial molecular concentration gradients. No radical species were considered by any of these authors.
In this article, the cracking of ethane in a tubular reactor is simulated both with a 1-D and a 2-D reactor model. Both reactor models are coupled to a radical kinetic model for the cracking of naphtha and ethane. It consists of 60 molecular and 68 radical species and more than 1,200 reactions (Clymans and Froment, 1984). For ordinary tubular reactors a 2-D model is sufficient because of axial symmetry, but more complex geometries such as internally finned cracking coils demand a 3-D reactor model (De Saegher et al., 1996). The effect of the radial temperature and concentration profiles on the conversion and the product distribution is studied. Also the effect of these radial gradients on the coking rate is investigated. The results obtained with a coke-formation model based on elementary reactions (Wauters and Marin, 2001), coupled to a 1-D and 2-D reactor model, are compared with reference data. The effect on the calculated run length of the reactor is determined.
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
The energy equation is given by
To calculate the wall temperature based on the process-gas temperature, the internal heat-transfer coefficient is calculated from the Dittus-Boelter equation
The pressure equation accounting for friction and changes in momentum is given by
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)
Correspondingly, an energy balance leads to the energy equation
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
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 uz 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
In the transition zone
In the turbulent zone
After calculating the axial velocity profile according to Eqs. 8, 9, and 10, the radial component of the velocity ur can be deduced from the total mass balance
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 ϵH and the turbulent diffusivity ϵD are given
where x represents the normalized radius, c1 = 0.828664, c2 = 0.944067, c3 = 0.020530, and the following expressions hold for a and b
The Fanning friction factor is obtained from the Prandtl equation
In the laminar zone near the reactor wall, the conductivity equals the molecular conductivity, λm, and the diffusivity equals the molecular diffusion coefficient, Dm.
The boundary conditions for the 2-D problem are:
In the center of the tube (r = 0)
At the inner reactor wall (r = R)
At the inlet (z = 0)
The axial heat profile is imposed at the inner wall (r = R)
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
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, ei, 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 jth component by all reactions forming this component and the other describing the cumulative rate of disappearance of the same component
The second term on the righthand side of Eq. 17 is a linear function of the concentration of component j in the ith gridpoint of a section, neglecting the small contribution of possible recombination reactions. Equation 17 can then be rewritten
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.
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.*
The cracking of ethane has been simulated with a 1-D and a 2-D reactor model. For the 2-D simulation, this results in important radial nonuniformities of the temperature in the reactor coil, as shown in Figure 2. The radial temperature profile simulated with the 2-D model shows a strong gradient near the wall, but the gradients in the center of the tube cannot be neglected, in contrast to the 1-D model. The total temperature drop from the wall to the center is about 100 K for both models. Figure 2 also shows the average process gas temperature for the 2-D simulation. The cup-mixing temperature is applied for the calculation of the average process-gas temperature for the 2-D reactor model (Bird et al., 2001)
The results of Figure 2 are in agreement with the results simulated by Froment (1992), who reported a similar radial temperature profile, but with a stronger temperature drop near the wall, for a pilot configuration with a reactor tube diameter of 0.037 m and, hence, a lower velocity.
In Figure 3 the axial profiles of the average process-gas temperatures and the wall temperatures for the two simulation models in the reactor coil are given. Although important differences exist for the temperature in a radial section, there is good agreement between the average gas temperature calculated with the 2-D reactor model and the gas temperature simulated with the 1-D reactor model up to 25 m. From this position the cracking becomes significant and is more pronounced when the radial temperature profiles are accounted for. This leads to an average temperature that is lower for the 2-D model.
The wall temperature for the 1-D simulation is slightly higher than the one simulated with the 2-D reactor model. This is due to the different calculation methods used for the wall temperatures. The Dittus-Boelter equation and the gas temperature are used in the 1-D model (see Eq. 3), while in the 2-D model heat transport and reaction are considered simultaneously (see Eq. 6). The latter implies a better simulation of the wall temperature. The higher wall temperature resulting from the 1-dimensional reactor model will have consequences for coke formation. Higher temperatures in the zone near the wall will increase the simulated coking rate.
Effect of the radial temperature profile on product distribution and conversion
The radial temperature profile has a large influence on the simulated product yields and the conversion. The weight fractions at the outlet of the reactor are given in Table 2. Except for hydrogen, the differences cannot be neglected. Especially for methane and propylene the yields calculated with the 2-D model are much higher. On the other hand, for acetylene and ethylene the 2-D model simulates a lower product yield.
Table 2. Composition Between Simulated wt. % at the Reactor Outlet of the Traditional Single Coil Reactor Simulated with the 1-D and 2-D Reactor Model
The existence of an important radial temperature profile is the main cause for the differences in product distribution obtained with both reactor models. The interpretation of radial and axial concentration profiles of the most important molecules and radicals, taking into account the radial and axial temperature profiles, will allow this to be explained. The main molecules in the kinetic model for the cracking of ethane are ethane, ethylene, propylene, and hydrogen. The main radicals are the hydrogen, methyl, ethyl, and allyl radicals.
The ethane conversion profile is shown in Figure 4. Such a profile shows an induction period corresponding with the heating of the process gases to the reaction temperature. This induction period is shorter for the 2-D model because of the radial temperature profile. Figure 3 shows that in the zone near the wall the temperature is more than 60 K higher than the average temperature and, hence, activation by C–C scission or C–H scission reactions will occur at an axial position closer to the inlet. This effect results in a higher ethane conversion obtained with the 2-D reactor model. At temperatures below 1100 K, the C–C scission of ethane with an activation energy of 368 kJ/mol is the main source of methyl radicals
The C–H scission reactions only become of some importance at much higher temperatures, because of the higher activation energy of this reaction (411 kJ/mol). The influence of the small difference between the average process-gas temperature simulated with the 2-D model and the 1-D temperature is of less importance. Once the cracking has started, the conversion of ethane increases with an almost constant slope.
In Figure 5 the ethane conversion and the ethylene yield are shown over a cross section at an axial position of 75 m. The radial temperature profile is responsible for the radial differences in conversion. The higher conversion near the wall results in a higher ethylene yield in this zone. The higher conversion in this zone will also affect the yields of other products.
As shown in Table 2, the methane yield simulated with the 2-D reactor model is higher than simulated with the 1-D reactor model. As a consequence of the radial temperature profile and the resulting increase of C–C scission reactions of ethane, a higher methyl radical concentration is obtained. H-abstraction reactions by the methyl radical lead to the formation of methane. In this case, the H-abstraction reaction of ethane is the main path to the formation of methane
The higher methane yield at the outlet is thus a direct consequence of the higher methyl radical concentration.
The lower ethylene yield simulated at the reactor outlet with the 2-D reactor model is due to diffusional effects. Near the inlet of the reactor only the process gas temperature in the zone near the wall is high enough to activate the conversion of ethane. Decomposition of the ethyl radical is generally accepted to be the main reaction responsible for the production of ethylene from ethane
The ethylene produced in the zone near the wall can diffuse to the center of the tube where the temperatures are lower. These temperatures favor secondary reactions, among which addition reactions to methyl and ethyl radicals are very important
This explains the higher yield of products such as propylene, 1-butene, and butadiene, obtained with the 2-D reactor model. For example, propylene is mainly formed by the β-scission of the C3H•-radical
and in a small amount from H-abstraction reactions by the allylic C3H5• radical:
Diffusion of ethylene, followed by addition of the methyl radical (Eq. 23) results in higher propylene formation.
The same reasoning that explains the lower ethylene yield also holds for the lower acetylene yield obtained with the 2-D reactor model. Diffusion of acetylene to the center, followed by secondary reactions, explains the lower acetylene yield. The effect is even more pronounced due to the faster secondary acetylene reactions, which decrease the acetylene yield drastically in favor of heavier products.
Radical Concentration Profiles.
The radial yield profiles for the main radicals are presented in Figures 6 to 9. Comparing these profiles with those in Figure 5 for ethylene and ethane, it is clear that the radial gradient for the radicals is much steeper than for the molecular species. The radial temperature profile results in strong radial gradients for the radicals, especially for the methyl radical and the hydrogen radical.
The combined effects of the radial temperature profile and the resulting radial yield profile for the methyl radical are the main cause for the radial nonuniformities for the heavier radicals and for the higher average radical concentration simulated with the 2-D reactor model (Figures 8 and 9). The radial average yield of the allylic radical is about two times higher when using the 2-D reactor model, and that of the methyl radical is almost 1.5 times higher. The latter is in contrast with the results for the hydrogen radical in Figure 6. The average H-radical yield simulated with the 2-D reactor model is slightly lower than the one simulated with the 1-D reactor model. The lower H2 yield simulated with the 2-D reactor model in Table 2 also suggests that fewer H-abstractions by H-radicals occur. This is due to the high activation energy of the C–H scission reaction of ethane
The temperatures are high enough to activate this reaction (Eq. 27) and increase the H-radical concentration drastically only in a small zone near the wall. This zone, however, is much smaller than the one observed for the formation of the methyl radical through Eq. 20. Taking into account the lower average temperature obtained with the 2-D reactor model, a global lower average H-radical yield is simulated.
Effect of Tube Diameter.
The existence of the radial temperature profile in a tubular reactor clearly has a strong influence on the product yields. For a given average temperature, the presence of a radial temperature profile leads to a decrease of the yield of ethylene. This implies that tubes with a smaller diameter are beneficial because of a stronger 1-D behavior. This is the case for reactor configurations such as the USC-Furnace (Ultra Selective Conversion) developed by Stone and Webster, and the Kellogg Millisecond Furnace. Consider, for example, the Kellogg Millisecond Furnace. This type of furnace, consisting of a reactor with a large number of tubes (200) of small diameter (3.5 · 10−2 m) and short length (10 m) (Orriss and Yamaguchi, 1987), is known for its high ethylene yield. The total hydrocarbon flow rate of more than 20 ton per h results in very high velocities in all of 200 parallel tubes. The high yield in ethylene of this type of reactor is generally explained by the short residence time. Indeed, short residence times reduce the disappearance of light olefins because of secondary reactions such as addition reactions. Experimental work by Ennis et al. (1975) showed that residence times lower than 0.1 s are most favorable. The results in this article indicate that not only the short residence time but also a stronger 1-D behavior is the cause for the high ethylene yield.
The simulation conditions for one of 200 tubes of a standard Millisecond Furnace configuration are summarized in Table 4. They have been chosen in order to obtain an ethane conversion similar to those just discussed. The simulation results in Table 5 show that the difference between the 1-D reactor model and the 2-D reactor model are significantly smaller than for the traditional single coil reactor, but cannot be neglected. The main reason for the differences is again the radial temperature profile, but the stronger 1-dimensional behavior reduces the radial nonuniformities. A radial temperature profile for one of the tubes of the Kellogg Millisecond Furnace is shown in Figure 10. The high velocities through the small-diameter tube reduce the total temperature drop from the wall to the center to 55 K, which is significantly smaller than for the traditional single coil.
Table 4. Reactor Geometry and Process Conditions for the Reactor Configuration in a Millisecond Furnace
Table 5. Comparison Between Simulated Weight Percentages (%) at the Reactor Outlet for the 1-D and 2-D Reactor Model for One Tube of a Millescond Furnace
In industry the Kellogg Millisecond Furnace is operated under much more severe conditions. The high heat fluxes necessary to achieve these high conversions lead to more pronounced temperature gradients, and hence to stronger effects on the yields than under the simulation conditions in Table 4. Therefore rifled tubes are used in industry (Orriss and Yamaguchi, 1987), as they promote heat transfer by means of their spirally grooved internal surface, reducing the radial temperature profile.
Effect of the radial temperature profile on coke formation
The coke formation at the internal skin of the reactor coil depends on the conditions prevailing at the process gas–coke interface. It is clear from the preceding results obtained with the 2-D reactor model that these conditions differ appreciably from the process conditions at the center of the reactor coil. In typical industrial units, the temperature at the interface can be up to 100 K higher than the average process gas temperature. This has a significant effect on the gas-phase concentrations, particularly on those of the gas phase radicals. According to Reyniers et al. (1994), gas-phase radicals generate radical sites on the cokes layer. Olefins can add to these radical sites after dehydrogenation coke is formed. Therefore the effect of the radial temperature profile and the resulting gas-phase radical concentration profile on the coking rate has been investigated for a traditional single-coil reactor. The simulation results using a fundamental kinetic coking model based on elementary reactions are studied. These results are compared with reference data calculated with the model of Plehiers (1989). The coking model of Plehiers (1989), coupled to a 1-D reactor model, has been validated for experimental as well as industrial units and was found to be very useful for the prediction of coke formation in the steam cracking of light hydrocarbon feedstocks. The run length of several industrial cracking units has been simulated, showing a good agreement between industrial and simulated results. An agreement of results obtained with the coking model based on elementary reactions on the one hand, and those obtained with the coking model of Plehiers on the other hand, is therefore a good indication for the quality of the simulation results discussed in this article. The semiempirical coking model of Plehiers considers ethylene and propylene to be the only coke precursors. These components can add to the radical sites at the coke surface and form cokes. The coking rate is a function of the wall temperature and the concentration of ethylene and propylene.
A fundamental kinetic model for coke formation based on elementary reactions has been developed for the cracking of light feedstocks by Wauters and Marin (2001). The creation of radical sites on the coke layer is explicitly accounted for by considering the abstraction of hydrogen from surface species by gas-phase radicals. The resulting radical surface species can add to unsaturated components, such as olefins and aromatics, in the gas phase. The possible coke-formation reactions can be divided into five classes:
Hydrogen abstraction by gas-phase radicals;
Substitution by gas-phase radicals at the coke surface;
Addition of a radical surface species to a gas-phase olefin and the reverse decomposition of a radical surface species in a smaller radical surface species and an olefin;
Addition of a gas-phase radical to a double bond in a surface species and the reverse decomposition of a radical surface species to an olefinic surface species and a radical;
Cyclization of a radical surface species and decyclization.
Figure 11 shows one of the reaction paths for coke-layer growth starting from propylene, hydrogen radicals, and alkyl radicals as coke precursors. The coke layer is considered to be a polyaromatic surface.
Considering each component in such a coke-formation model leads to an enormous expansion of the reaction network. To obtain a rigorous but practical model, it is, therefore, necessary to select those gas-phase components that are the most important coke precursors. The contribution of a gas-phase component to the coke formation is determined by the concentration of the component and the rate coefficients of the coking reactions in which the component is involved. In steam cracking of light feedstocks the concentration of the gas-phase radicals varies between 10−6 mol m−3 and 10−1 mol m−3 (McConnell et al., 1981). The latter was confirmed by the simulation results. For light feedstocks, the hydrogen, methyl, ethyl, and allyl radicals were found to be the most important coke precursor radicals. Acetylene, ethylene, methylacetylene, and propylene are unsaturated molecular components that favor coke formation.
The kinetics of the coke-formation reactions are determined from those of corresponding gas-phase reactions, provided the presence of a solid phase is accounted for via a correction factor based on the collision theory (Wauters and Marin, 2002). It was found that the coking rate depends not only on the concentration of molecules and on the wall temperature, but in particular on the radical concentrations. A good estimation of the concentrations of the radicals in the zone near the wall is of the utmost importance for the fundamental coking model, because their effect on the coking rate is as important as the effect of the wall temperature (Wauters and Marin, 2002).
In Figure 12, the differences between the coking-rate profiles calculated with the 1-D and 2-D reactor model, are presented and compared with the reference profile. Using a 1-D reactor model and calculating the precursor concentrations at the process-gas temperature clearly underestimates the coking rate. The wall temperature used in this simulation is the wall temperature calculated with the 1-D reactor model. The reason for the large difference between the calculated profile and the reference profile is twofold. On the one hand, the effect of the temperature for the calculation of the coking rate with the fundamental model is not as strongly affected by the temperature as the calculation of the coking rate calculated with the semiempirical model (Wauters and Marin, 2002), that is, the reference profile, due to differences in kinetics. On the other hand, the precursor concentrations are those calculated in the process gas by the 1-D reactor simulation. Table 3 shows the values of the concentrations of the main coke precursors and the temperature at z = 50 m. The concentrations calculated in the process gas with the 1-D reactor simulation are lower than those prevailing at the interface as calculated with the 2-D reactor model. The underestimation of the concentrations of the species result in a drastic underestimation of the coking rate. For example the underestimation of the methyl radical by a factor of 5 results in a reduction of the coking rate by almost 50% (Wauters and Marin, 2002).
Table 3. Calculated Temperature (K) and Concentration (kmol · m−3) of the Main Coke Precursors at the Gas–Coke Interface (z = 50 m).
Note: 1-D reactor model, precursor concentrations calculated at process-gas temperature; 1-D reactor model, precursor concentrations calculated at interface temperature; 2-D reactor model, precursor concentrations at interface.
As an approximation, the concentrations of the coke precursors at the interface temperature have been calculated by solving the corresponding mass balances for the 1-D reactor model but now at the interface temperature. The agreement between the simulated profile and the reference profile is improved, but the coking rate is now overestimated. A higher temperature increases the precursor concentrations as compared to those calculated with the 2-D model (see Table 3) and causes the overestimation of the coking rate.
As shown in Figure 12 the coking-rate profile calculated with the coking model based on elementary reactions and using the 2-D reactor model shows good agreement with the reference profile obtained from the semiempirical model of Plehiers (1989). The coke-formation profile calculated with the model based on elementary reactions coupled to the 1-D reactor model shows only small peaks, if any, at the higher tube-wall temperatures. On the other hand, when coupled to the 2-D reactor model the sharp peaks obtained for the coking-rate profile calculated with the model of Plehiers are almost perfectly predicted. The peaks simulated with the latter are a direct consequence of the peaks in the tube-wall temperature profile (Figure 2). This reasoning does not hold for the coking-rate profile simulated with the fundamental coking model and the 2-D reactor model. Here, the effect of the interface temperature on the coke precursor concentrations is incorporated via the kinetics and the concentrations of the involved reactants, in particular of the radicals. Hence, the coke precursor concentrations need to be calculated at the interface temperature. The latter can only be calculated accurately with a 2-D reactor model. The simulation results clearly show the potential of this fundamental model coupled to a 2-D reactor model and shows the importance of applying a more-dimensional reactor model.
Comparison of the simulation results using a 1-dimensional and a 2-D reactor model for ethane steam cracking indicates that important radial gradients exist, not only for the temperature but also for the molecular and in particular the radical species. The effect of the radial temperature profile on the concentration profiles and on the reaction rates of the elementary reactions considered by the kinetic cracking model makes it possible to explain the differences in simulated product yields for the two reactor models. The radial temperature profile decreases the yield for ethylene because light olefins are removed by secondary reactions, generating species with higher molecular weight such as propylene and butadiene. As a corollary this, it can also be concluded that reactors with a stronger 1-D character allow a higher ethylene yield to be obtained. This was illustrated for the reactor configuration of a Kellogg Millisecond Furnace.
The simulation results obtained with a fundamental kinetic coking model based on elementary reactions have been investigated. These results have been compared with the reference data of a validated semiempirical model. The coupling of a fundamental coke-formation model based on elementary reactions to a 1-D reactor model was found to result in large disparities compared to the reference data, mainly due to deviations in the simulated concentrations of the coke precursors. A 2-D reactor model allows us to adequately account for the coke precursor concentrations and, hence, to properly simulate the coking rates in an industrial ethane steam-cracker.
The differences noticed for both the product yields and the coking rate exceed the degree of accuracy required today. Therefore, the use of a 2-D reactor model is necessary, even if this increases the computational effort.
concentration of component j, kmol · m−3
heat capacity of component j, kJ · kmol−1 K−1
molecular diffusion coefficient, m2 s−1
internal tube diameter, m
Fanning friction factor
molar flow rate, kmol · s−1
mass flux, kg · m−2 s−1
enthalpy of the process gas, kJ · kmol−1
enthalpy of formation of component, j, kJ · kmol−1
heat of reaction k, kJ · kmol−1
molecular weight of component j, kg · kmol−1
number of grid points
number of reactions in which i and j participate
total pressure, MPa
heat flux, kJ · m−2 s−1
radial coordinate, m
radius of the bend, m
radius of the bend, m
rate of reaction k in which j participates, kmol · m−3 s−1
coking rate of reaction in which j participates, kg · m s−1
stoechiometric coefficient of the component j in the reaction k
Subscripts and superscripts
grid point in a radial section
The authors are grateful to the “Fonds voor Wetenschappelijk Onderzoek-Vlaanderen” and Technip-Benelux for financial support of the steam-cracking research. Kevin M. Van Geem holds a PhD grant of the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT-Vlaanderen).