Bart Merci

CFD simulations of steam cracking furnaces using detailed combustion mechanisms

Abstract A three-dimensional mathematical model has been developed for the simulation of flow, temperature and concentration fields in the radiation section of industrial scale steam cracking units. The model takes into account turbulence–chemistry interactions through the Eddy Dissipation Concept (EDC) model and makes use of Detailed Reaction Kinetics (DRK), which allows the detailed investigation of the flame structure. Furthermore, simulation results obtained with the EDC-DRK model are compared with simulation results obtained with a simplified model combining the Eddy Break Up (EBU)/finite rate formulation with Simplified Reaction Kinetics (SRK). When the EBU-SRK model is used, much faster fuel oxidation and products formation is predicted. The location of the peak temperature is shifted towards the burner, resulting in a smaller flame and the confinement of the combustion process into a smaller area. This is most likely because of the inherent deficiency of the simplified model to correctly predict the overall (effective) burning rate when the turbulent mixing rate and the reaction rate are comparable. It is shown that when neither the “fast-chemistry” nor the “slow-chemistry” approximation is satisfied, the overall burning rate is overpredicted. The smaller flame volumes obtained with the EBU-SRK model have important effects on the predicted temperature distribution in the furnace as well as on other significant design parameters like the refractory wall and tube skin temperatures. It is suggested that more sophisticated turbulence–chemistry interaction models like the EDC model and more Detailed Reaction Kinetics should be used for combustion modeling in steam cracking furnaces under normal firing conditions.

Heat transfer predictions with a cubic k–ε model for axisymmetric turbulent jets impinging onto a flat plate

Abstract Local heat transfer in turbulent axisymmetric jets, impinging onto a flat plate, is predicted with a cubic k – e model. Both the constitutive law for the Reynolds stresses and the transport equation for the dissipation rate e contribute to improved heat transfer predictions. The stagnation point value and the shape of the profiles of the Nusselt number are well predicted for different distances between the nozzle and the flat plate. Accurate flow field predictions, obtained with the presented turbulence model, are the basis for the quality of the heat transfer results. The influence of the nozzle–plate distance on the stagnation point Nusselt number, is also correctly captured. For a fixed nozzle–plate distance, the influence of the Reynolds number on the stagnation point heat transfer is correctly reproduced. Comparisons are made to experimental data and to results from a low-Reynolds standard k – e model [1] and the v 2 – f model [2] .

Computational Treatment of Source Terms in Two-Equation Turbulence Models

The source terms in turbulence models require careful treatment to obtain a stable discretization. The choice between implicit and explicit treatment has to be made. This can be done either on the basis of individual terms or on the basis of the exact Jacobian of the source terms. A comparison of both methods shows that the latter is generally applicable and superior to the first, approximate method with respect to convergence speed. This comes from the possibility of using the multigrid technique with the exact method, whereas this is not always possible with the approximate method. In principle, for robustness a time-step restriction for the source terms has to be introduced to prevent the turbulence quantities from becoming negative or infinitely large. An approximation of the appropriate time step is calculated. Practical results, however, indicate that the time-step restriction is not always necessary. Different two-equation turbulence models are investigated confirming the generality of the approach

Application of a New Cubic Turbulence Model to Piloted and Bluff-Body Diffusion Flames

A new two-equation turbulence model is described. It combines an algebraic, non-linear expression of the Reynolds stresses in terms of strain rate and vorticity tensor components, with a modified transport equation for the dissipation rate. Thanks to the cubic law for the Reynolds stresses, the influence on turbulence from streamline curvature is accounted for, while the increase in computational costs is small. The classical transport equation for the dissipation rate is altered, in order to bring more physics into this equation. As a result, more realistic values for the turbulence quantities are obtained. A new low-Reynolds source term has been introduced and a model parameter is written in terms of dimensionless strain rate and vorticity. The resulting model is firstly applied to the inert turbulent flow over a backward-facing step, demonstrating the quality of the turbulence model. Next, application to an inertly mixing round jet reveals that the spreading rate of the mixture fraction is correctly predicted. Afterwards, a piloted-jet diffusion flame is considered. Finally, inert and reacting flows in a bluff-body burner are addressed. It is illustrated for both reacting test cases that the turbulence model is important with respect to the flame structure. It is more important than the chemistry model for the chosen test cases. Results are compared to what is obtained by linear turbulence models. For the reacting test cases, the conserved scalar approach with pre-assumed β-probability density function (PDF) is used.

Numerical study of natural convective heat transfer with large temperature differences.

Steady‐state two‐dimensional solutions to the full compressible Navier‐Stokes equations are computed for laminar convective motion of a gas in a square cavity with large horizontal temperature differences. No Boussinesq or low‐Mach number approximations of the Navier‐Stokes equations are used. Results for air are presented. The ideal‐gas law is used and viscosity is given by Sutherland’s law. An accurate low‐Mach number solver is developed. Here an explicit third‐order discretization for the convective part and a line‐implicit central discretization for the acoustic part and for the diffusive part are used. The semi‐implicit line method is formulated in multistage form. Multigrid is used as the acceleration technique. Owing to the implicit treatment of the acoustic and the diffusive terms, the stiffness otherwise caused by high aspect ratio cells is removed. Low Mach number stiffness is treated by a preconditioning technique. By a combination of the preconditioning technique, the semi‐implicit discretization and the multigrid formulation a convergence behaviour is obtained which is independent of grid size, grid aspect ratio, Mach number and Rayleigh number. Grid converged results are shown for a variety of Rayleigh numbers.

Application of a RG hybrid RANS/LES model to swirling confined turbulent jets

A renormalization group (RG) based hybrid RANS/LES model is validated for turbulent swirling confined jets. The results are compared with the experimental data of Dellenback et al. (1988, Measurements in turbulent swirling flow through an abrupt axisymmetric expansion. AIAA Journal, 26(6), 669–681) and results for the same flows of an unsteady second-moment closure RANS simulation. A general quality/cost comparison is made between the hybrid RANS/LES and the second-moment closure simulations. In the final section, the hybrid RANS/LES result is further compared to a detached-eddy simulation, dynamic -equation LES and dynamic Smagorinsky LES for one of the flows, and the overall good quality of the RG hybrid RANS/LES model demonstrated.

Benchmark solutions for the natural convective heat transfer problem in a square cavity with large horizontal temperature differences

In this study, Benchmark solutions are derived for the problem of two‐dimensional laminar flow of air in a square cavity which is heated on the left, cooled on the right and insulated on the top and bottom boundaries. The temperature differences between the hot and cold walls are large. Neither Boussinesq nor low‐Mach number approximations of the Navier‐Stokes equations are used. The ideal‐gas law is used and the viscosity is given by Sutherlands law. A constant Prandtl number is assumed. The computational method is completely described by Vierendeels et al. Grid converged results with an accuracy of 4 up to 5 digits are obtained for different Rayleigh numbers and temperature differences.

Hybrid RANS/LES modelling with an approximate renormalization group. I: Model development

A hybrid RANS/LES model based on renormalization group (RG) calculations is presented. The result of the RG approach is a one-equation LES subgrid model in well-resolved regions of the flow, and a two-equation RANS model otherwise. The two modes of the model are linked by comparing the filter width with a length scale constructed from subgrid quantities. The one-equation subgrid model uses a transport equation for the mean rate of dissipation, in which the inverse time scale is explicitly filter width dependent. The model is free of adjustable parameters (after the truncations of the approximate RG are made) in its high-Reynolds formulation, has a clear physical interpretation and is easy to implement.

Determination of ϵ at inlet boundaries

Different methods for the determination of accurate values for the dissipation rate ϵ at the inlet boundary of a computational domain, are studied. With DNS data for a fully developed channel flow and pipe flow, it is shown that the method suggested by Rhee and Sung (2000), in which the k–ϵ turbulence model is used to compute both k and ϵ from a given velocity profile, is not reliable and can result in very poor results. The method is found to be extremely sensitive to the details of the imposed velocity profile. An alternative procedure is proposed, in which only the ϵ transport equation is employed, with given profiles for the mean velocity and the turbulence kinetic energy. This way, accurate and reliable profiles are obtained for ϵ. Another procedure, based on the turbulent mixing length, was suggested by Jones (1994). The problem. The problem is then shifted towards the determination of the mixing length at the inlet boundary of the computational domain. An expression for this mixing length is proposed in this paper, based on the mentioned DNS data. Finally, the method proposed by Rodi and Scheuerer (1985) is included for comparison reasons. The different procedures are first validated on the fully developed channel and pipe flow. Next, the turbulent flow over a backward‐facing step is considered. Finally, the influence of the inlet boundary condition for ϵ is illustrated in the application of a turbulent piloted jet diffusion flame.

Hybrid RANS/LES modelling with an approximate renormalization group. II: Applications

The hybrid RANS/LES model developed previously is extended with near-wall modifications. The model is validated, in its low-Reynolds form, for channel flow and for flow over a periodic hill. The results for the channel flow are (indirectly) compared with DES results for the same flow, and are qualitatively similar. The separated flow over the periodic hill shows good agreement with the reference LES data, although performed with about 20 times less grid points. Finally the model is applied in high-Reynolds form with wall functions to a sudden pipe expansion; good agreement with experimental data is also obtained.