Comparative Molecular Mechanistic Modelling of a Tubular Thermal Cracker in Two and Three Dimensions

This study aimed at the modelling of a tubular thermal cracker in two and three dimensions using 0.15 kg/h of ethane and 1.5 kg/h of nitrogen in laminar flow using a molecular mechanistic model for ethane cracking, followed by the solution and comparison of the results obtained. This was used to find the effect that spatial development had on the generated profiles. The purpose was achieved by deriving the requisite model equations from mass, energy and momentum balances consisting of nine coupled partial differential equations for each of the spatial considerations and using the finite difference numerical scheme to solve. Throughout the reactor length of 1.20 m and radius of 0.0125 m, the resulting concentration and temperature profiles were predicted. In comparison, the profiles from the two and three dimensional developments showed that even though angular variations are small in the three dimensional approach, they are still present and there consideration could be helpful in the design of tubular thermal crackers.


INTRODUCTION
By definition, cracking refers to breaking of higher boiling petroleum fractions into lower boiling lighter fractions.It is considered an endothermic reaction.There is heat supply to the parent compound (s) until they ignite and are converted to various products depending on the nature of the parent compound, cracking temperature and the type of cracking employed (Walter et al., 1988).Thermal cracking is carried out in tubular reactor system with the supply of heat externally.The tubular reactor is a conduit suitable for reactants flow with no attempt made to increase the natural degree of mixing (Rutherford, 1989).
Since the heat transfer within a tube is not instantaneous, the endothermic reaction gives rise to problems of temperature control and maintenance, as well as selectivity which are very difficult to solve.The furnace caters for the heat requirements, radiative heat transfer heats up the tube walls and heat is further conveyed to the reaction mass from the heated tube walls.The transport of heat in the tube is responsible for the variation in reaction rates in radial, axial and angular directions depending on manner of heating employed, which result in gradients being setup in these directions.Olefins production from natural gas has being reported by Holman et al. (1995) in addition to the conventional feeds being used.
In thermal cracking, some modelling studies are available in the literature like Behlolav et al. (2003) and Niaei et al. (2004).Some are based on the plug flow assumptions which are 1-dimensional model where the lateral gradients are considered negligible.As reported, Sundaram and Froment (1979) have compared the predictions of a 1-dimensional model to a 2dimensional model for a single irreversible reaction.The 2-dimensional model seems to give better predictions.A fast operating algorithm for thermal cracking furnaces had been proposed by Xu et al. (2002), for optimization of the steam crackers, which again is based upon plug flow conditions.In addition, Sundaram and Froment (1979) have compared the onedimensional model to a two dimensional model for a pipe reactor with a single molecular reaction using the laminar and turbulent flow regimes.It was observed by Sundaram and Froment (1979) that a typical radial concentration and temperature profile exists.As they observed also, the one-dimensional model gives better predictions, if an averaged Nusselt number from a twodimensional model is incorporated into the onedimensional model.In another study by Sundaram and Froment (1980), these findings have been reported to hold for 2-dimensional modelling of ethane thermal cracker for turbulent conditions.Garg and Srivastava (2006) had also reported a 2-dimensional modelling for tubular thermal cracking of ethane.
It had been proposed that the resistance in the radial direction to heat and mass transfer controls the extent of reaction and product distribution obtained from the reactor.Garg and Srivastava (2001) found that annular reactors have a distinct advantage of lower diffusion path over the tubular reactors, which also lowers the radial resistance.This also implies higher surface area which can be utilized for better heat transfer.Earlier, the radial diffusional mass transfer in a fully developed laminar flow through an annular cylindrical reactor for a first order heterogeneous reaction at the wall had been studied (Houzelot and Villermaux, 1977).
The three dimensional model seems to explain reaction systems in which non-circumferential heating is utilized.The reactor is usually a differential sized tubular conduit, upon which modelling is based using the equations of mass, energy and momentum balance.A molecular mechanistic reactions scheme for ethane cracking is usually adapted from Froment and Bischoff (1990).The 2 and 3 dimensional model from this reaction scheme seems suitable for predicting feed and products concentration as well as temperature profiles in radial, axial and angular (for 3-D) directions.
This present study is intended as an inquisitive effort for the three-dimensional modelling of a tubular thermal cracker which seems to be a largely unexplored area.Ethane was used in the development of this present study as the cracking reaction of heavier compounds involves numerous steps which increase the complexity of the resulting reactions.

METHODOLOGY Model development:
The development of these models made some inextricable assumptions without which the analyses would be profoundly complex.These assumptions are chosen to simplify the resulting PDEs within reasonable limits while avoiding oversimplification so that the whole purpose of the modelling and simulation will not be lost.The proposed model for ethane cracking in the tubular reactor is based upon the following assumptions: • The reactor operates under steady state conditions because the reactor has continuous flow that stabilizes and gives rise to steady state condition.• The flow regime is laminar due to its similarity to a laboratory scale reactor.• The wall temperature is high and constant because heat is required for the system from the outside (endothermicity of system).a : In units of s -1 or m 3 /kmol.s Other needed assumptions are made as they come up in the course of the model developments.The molecular scheme used here is presented in the Table 1, also shown are the kinetic parameters that accompany each reaction.Froment and Bischoff (1990) and Garg et al. (2006) have used similar schemes.
The reactions shown in Table 1 involve eight species in seven reactions; two of the reactions are reversible which make for five reactions in effect.The following designations are used all through the study: The reaction rates for the species are: where the rate constants are expressible as: and linearized by a truncated Taylor's Series of the form: A mass balance is written for each specie and a heat balance for the overall system.
The two-dimensional model equations: Derivations here are based on a cylindrical shell of dimensions ‫ݎ∆‬ and ‫ݖ∆‬ at a distance ‫ݖ‬ from the entrance and a distance ‫ݎ‬ from the centre.The generated mass balance for the ݅‫ݐ‬ℎ specie is: where, and the simplified energy balance: The velocity profile is: and, For a completeness of procedure the velocity profile in 3-D is: In order to create a basis for the three-dimensional developments, it is presupposed that, as opposed to the case for two-dimensions, non-circumferential heating is employed for the thermal cracker under review here which induces specie and reactor property gradations in the axial, radial and angular directions.The boundary conditions for this case are chosen bearing in mind that only one side (Fig. 1) of the reactor is at 950°C; the other diametrically opposite half only receives heat from the high temperature side.This scenario may be obtained in situations where a side of the cracker tube is inaccessible so that only the exposed and reachable part can be heated.
As a result of the foregoing explanations, the mesh for the 3D Profile covers the entire Diameter (BGFD ' ) of the tubular reactor.The mesh for the angular variations covers only a quarter of the reactor surface (CEFD) on the low temperature side since the properties of the other quarter is only a repetition of that of the first due to their symmetry.Axial variations still occur down the reactor axis.
From Diffusivity values, enthalpies of formation and heat capacity values are sourced from Perry and Chilton (1997), Himmeblau (1989) and Garg et al. (2006).

NUMERICAL SCHEME
The PDEs expressed in Eq. ( 11), ( 12), ( 14) and (15) are discretized using the finite difference implicit scheme.This is done by substituting each of the differential term in the PDEs with a finite difference expression corresponding to the order of the differential term.
For the 2-D development, the axial and radial dimensions were both divided into 10 sections, creating a mesh with 40 boundary nodal points and 81 interior mesh points: and, The desired profiles were then developed from the discretized 3D model equations and their respective boundary conditions: The equations involving concentration can be written for each of the eight species.Each of these sets of equations for each species as well as the set for temperatures, written for a given axial position can be represented in the matrix form: The equations are first discretized using the finite difference implicit scheme before being solved by choosing an appropriate grid size (equal grid numbers on both axes in this case) and employing the Gauss-Seidel Algorithm in MATLAB © .

RESULTS AND DISCUSSION
The model equations are discretized with the finite difference numerical scheme and solved a pair at a time, i.e., a concentration model expression for a particular specie alongside the temperature model equation written for the same specie; the profile development was done this way because the model equations; concentration, temperature and velocity present a case of strong coupling.The generated temperature profile from the solution of any pair of equations was employed in simplifying all subsequent solutions since one and only one temperature profile exists through the tubular thermal cracker by the operation.
The profiles are generated by breaking the composite difference equation into a sum of separate gradations, which is the sum of radial and axial gradations for the two-dimensional operation and sum of radial, axial and angular gradations for threedimensions.
The various results generated are presented in graphical and tabular forms separately for the 2 and 3dimensions as well as comparatively in some cases.For all cases, the following notation holds: For the 2-dimensional considerations, Fig. 2 shows the 2-dimensional concentration profile variation with reactor axial length for all reacting specie.As plotted, the concentration of ethane decreased from 0.00107 to 0.00016 kmol/m 3 at the reactor exit, while other species rose from zero at the inlet to their respective predicted values as expected.For instance ethane concentration increases from 0 to 0.00091 kmol/m 3 .The temperature profile variation with reactor axial length is presented The 3-dimensional concentration and temperature variations with axial position is almost similar to those of the 2-dimensional profiles, hence the plots are unnecessary.The main target being reported is to really highlight the 3-dimensional contributory effect.2 shows the concentration variation with angle at axial length of 0.6 m and radius of 0.002 m, while Fig. 4 shows the angular 3-dimensional variation with temperature at z = 0.6 m and r = 0.002 m.
The 3-D development presented a more realistic profile by taking into account the limitation on circumferential heating.It corrected this assumption and showed that different diametric segments of the reactor tube exhibit different properties.It also showed that, albeit small, angular variations exist and should be taken into account in the design of tubular thermal cracker tubes.This is pointed out in the concentration variation with radial position plots for methane and temperature as shown in Fig. 5 and 6 respectively.The percentage concentration rise of some parameters for the 2-D and 3-D profiles are also given in Table 2 to 5. It can be deduced that the radial variations are more pronounced than the angular variations for all cases considered.This means that the 3-D profiles is also able to give a smoother change in process variables with variation in process conditions.
If the purpose of a design procedure is a general overview of how the tubular thermal cracker operates, then the 2-D model should satisfy this requirement.However, if an in-depth analysis is desired then the 3-D model technique seems preferable.

CONCLUSION
The modelling and simulation of the isothermal tubular thermal cracker for conversion of ethane was carried out with two and three dimensional developments.The simulated reactant and product distribution and temperature profiles are similar to previous observations as observed by Garg et al. (2006).It was found that the concentration of all the products increased in directions of increasing temperature while that of ethane reduces in the radial, axial and angular directions.
The results obtained for the 2-D model when compared to those of the 3-D model implied that if angular gradients are ignored, inadequate profiles seem inevitable as axial symmetry is assumed.As observed, even though angular variations seem small, their existence is evident and could be put into consideration when designing a tubular thermal cracker.Furthermore the models presented serves as good predictive tools in the operation of tubular thermal crackers, which could also be utilized in designing more efficient reactors.

Fig. 2 :
Fig. 2: Concentration profile variation with reactor axial length for all components

Table 5 :
Three dimensional percentage concentration rise of reacting species from angular variation of 0-π/2 rad at z = 0.6 m and r = 0.002 m