Hugo A. Jakobsen

A model for turbulent binary breakup of dispersed fluid particles

Abstract Accurate predictions of particle size distributions, and therefore of the underlying processes of fluid particle breakup and coalescence are of vital importance in process design, but reliable procedures are still lacking. The present paper aims at developing a modular formulation for the turbulent particle breakup process. The model is to be included in a population balance model which is formulated such as to facilitate the direct future implementation into a full multifluid CFD model. The breakup process is described without introducing adjustable parameters. The current model is a further development of an existing model by Luo and Svendsen (AIChE J. 42 (5) (1996) 1225), which has been expanded and refined, and where an inherent weakness regarding the breakup rate for small particles and small daughter particle fragments are removed. A new criterion regarding the kinetic energy density of the colliding turbulent eddy causing breakup has been introduced. This new criterion is a novel concept describing the breakup process. The details are thoroughly discussed together with possible further modifications. The results from the new model are encouraging because the breakup rate is greatly reduced when the dispersed fluid particles are reduced in size. Further, the response to changes in system variables is reasonable and the distribution of daughter sizes vary in a reasonable way for the different collision possibilities.

Effect of viscosity on droplet-droplet collision outcome: Experimental study and numerical comparison

The influence of viscosity on droplet-droplet collision behavior at ambient conditions was studied experimentally and numerically. N-decane, monoethyleneglycol (MEG), diethyleneglycol (DEG), and triethyleneglycol were used as liquid phase providing viscosities in the range from 0.9to48mPas. Collision Weber numbers ranged approximately from 10 to 420. A direct numerical simulation code, based on the volume-of-fluid concept, was used for the simulations. Experimentally, observations of two droplet streams using a modified stroboscopic technique (aliasing method) were used to investigate the whole range of impact parameters during one experimental run. The experimental method has previously been verified for the water/air system [C. Gotaas et al., Phys. Fluids 19, 102105 (2007)]. In the present work, it was tested and validated for the n-decane/air system. Measured data agree well with those published in the literature. Well-defined regions of stretching separation and coalescence were identified, while refl...

A least squares method for the solution of population balance problems

A general framework is developed, using the least squares method (LSM), for the solution of a generalized population balance equation. The basic idea in the LSM is to minimize the integral of the square of the residual over the computational domain. The capability of the method for solving the PBE is evaluated by using case problems involving coalescence and breakage kernels having analytical solutions which allow the analysis of the method to be performed in a general way. By using the LSM to solve the PBE, the error in the properties of the distribution function depends on the order of the expansion, thus avoiding the introduction of heuristic rules to obtain sufficient accuracy in the values of a few of the physical moments. An interesting characteristic of the LSM applied to PBE is that a low number of equations are required to solve the problem.

A numerical study of the interactions between viscous flow, transport and kinetics in fixed bed reactors

Abstract A numerical method is tailored for the solution of a reduced set of model equations developed for the description of reactive flows in chemical reactors. The numerical method synthesis intends to utilize experience both in chemical engineering solving dispersion models containing complex reaction kinetics, phase equilibria and non-ideal thermodynamics, and in fluid dynamics applying classical numerical algorithms constructed for pure flow calculations. The two types of model equations involved in reactive flow simulations reflect very different physics. A modular method is therefore suggested enabling a split between the flow- and chemistry model parts. In this way more optimal solution methods can be adopted for each operator, in contrast to more traditional computational fluid dynamics (CFD) methods where all equations (and operators) are normally solved by the numerical solvers originally intended for pure flow calculations. In addition, emphasis has been put on modularity enabling the same model framework to be used both for the more traditional 1D and 2D dispersion models with simple 1D flow calculations, as well as for the more elaborate 2D and 3D reactive flow calculations. In the flow part of the algorithm a fractional-step algorithm is constructed for the simulation of dynamic reactive flows in chemical reactors. The numerical scheme is based on the compressible transport equations in the low-Mach-number limit. The method can handle real gas (and liquid) mixtures with variable density as well as constant density fluids. The velocity field is advanced using a projection scheme, which consists of a partial convection–diffusion update followed by a pressure correction step with an intermediate an-elastic filter. A variable density corrector step is implemented for variable density systems in order to couple the evolution of the density and the velocity fields. In the chemistry part of the model all scalar fields are updated using Strang-type operator-split integration steps that combine several explicit convection and semi-implicit diffusion transport operators with a suitable solver tailored for non-linear sink/source terms. Diffusion terms are discretized with a second order central difference scheme in space. Convection terms are discretized with a second order TVD scheme in space. The temperature advection is discretized using a second order upwind scheme. The chemical reaction part of the model is discretized by an implicit Euler approximation, and the resulting set of algebraic equations is solved using a Broyden subroutine. The other source terms are solved using an explicit Euler approximation. The performance and behavior of the operator-split scheme are assessed based on simulations of two industrial chemical processes (i.e. the synthesis gas and methanol production processes) performed in multi-tube fixed bed reactors. These processes are important parts of the Statoil methanol plant at Tjeldbergodden in mid Norway. Both 1D and 2D pseudo-homogeneous dispersion models (i.e. heat and mass balances) with prescribed velocity, mixture density and total pressure profiles are used describing the reactor performance. The predicted profiles for both the methanol and synthesis gas processes were in accordance with results reported in the literature. An oscillatory radial void fraction distribution was then implemented in the 2D model. It was found that the non-uniform void fraction distribution had no significant effect on the temperature and mole fraction profiles. A 1D CFD simulation was performed to evaluate the effect of the variations in velocity, pressure and mixture density on the reactor performance. The changes in these variables significantly effect the composition and conversion in the reactor. A 2D CFD model was then developed for further studies of the multi-tube fixed bed reactor for the synthesis gas process, analyzing the influence of non-uniform void fraction distributions on the flow and chemical conversion. The non-uniform void fraction distribution induces a significant reduction in axial pressure drop and a much higher fluid velocity close to the wall. However, the influence on the temperature and mole fraction profiles was hardly noticeable. Mass- and enthalpy budgets were implemented for the heat and mass quantities in the program. These budgets show that the enthalpy and both the mixture and component mass balances are fulfilled in the simulations. The numerical CFD algorithm developed in this paper has been found to be both computationally stable and accurate. Computational efficiency analysis shows that the Poisson solver requires about 75% of the total computational time. The computational time spend on the chemistry part of the model is about 15% of the total time. About 10% of the cost used on the chemistry part is spend on the chemistry solver.

The interaction between mass transfer effects and morphology in heterogeneous olefin polymerization

The interaction between mass transfer effects and morphology in heterogeneous olefin polymerization

Phase distribution phenomena in two-phase bubble column reactors

Abstract In a recent paper, Jakobsen, Sannaes, Grevskott and Svendsen (Ind. Eng. Chem. Res. 36(10) (1997) 4052–4074) presented a review of the present status on fluid dynamic modeling of vertical bubble-driven flows. Special emphasis was placed on two-phase flows in bubble column reactors. For these multiphase reactors, the averaged Eulerian multifluid models have been found to represent a trade-off between accuracy and computational efforts for practical applications. Unfortunately, in such multifluid models constitutive relations are needed to describe the phase interaction processes. It was concluded that the general picture from the literature is that time-averaged liquid velocity fields are reasonably well predicted both with steady-state and dynamic models of this type. The prediction of phase distribution phenomena, on the other hand, is still a problem, in particular at high gas flow rates. The present paper gives an overview of the pertinent constitutive relations presented in the literature aiming at a firm mechanistic prediction of the phase distribution phenomena. This includes transversal forces, steady drag forces, surface tension effects, and hydrodynamic bubble–bubble and bubble–wall interactions. Several interaction mechanisms in the turbulence fields like the so-called turbulent mass diffusion, turbulent migration, turbulent drift velocities, anisotropic turbulent drag forces, as well as the interactions between these mechanisms, and the impacts of variations in bubble size and shape distributions are discussed. Various aspects of these relations have been questioned. It is therefore the aim of this paper to compare the capabilities of the existing Eulerian multifluid modeling concepts and parameterizations. The various approaches are evaluated using an in-house 2D Euler/Euler steady-state code. There are several reasons why we choose a steady 2D model for evaluation. First, the model has the advantage of being relatively simple, thus the computational effort required for practical applications involving chemical reactions, and interfacial heat and mass transfer is feasible. Second, the dynamic axisymmetric 2D models do not give much improvement as the flow phenomena possibly missing are believed to be 3D. That is, they do not resolve the swirling motion of bubble swarms. Third, most parameterizations presented in the literature are based on, and have so far only been applied to, 2D models. The results obtained indicate that the various phase interaction parameterizations available in the literature predict very different phase distributions in bubble columns. For operating reactors these deviations will significantly influence the predicted process performance. The results presented here thus confirm the demand for improved modeling including more accurate and stable numerical solution algorithms. Low-accuracy algorithms may totally destroy the physics reflected by the models implemented.

A least-squares method with direct minimization for the solution of the breakage-coalescence population balance equation

A least-squares method with a direct minimization algorithm is introduced to solve the non-linear population balance equation that consists of both breakage and coalescence terms. The least-squares solver, direct minimization solver together with a finite difference solver are implemented for comparisons. It is shown that the coalescence term introduces a strong non-linear behavior which can affect the robustness of the numerical solvers. In the comparison with the least-squares method, the direct minimization method is proved to be capable of producing equally accurate results, while its formulation is better conditioned. In the case of a non-linear population balance equation system, the direct minimization method converges faster than the standard least-squares method.

On the solution of the population balance equation for bubbly flows using the high-order least squares method: implementation issues

Abstract The prediction of the dispersed phase distribution plays a major role in multiphase chemical reactor engineering. The population balance equation (PBE) is a well-established equation for describing the evolution of the dispersed phase. However, the numerical solution of the PBE is computation intensive and challenging. In recent literature, the high-order least squares method has been applied to solve population balance (PB) problems. The interests in the least squares technique are based on the favorable numerical properties of the method. Moreover, by adopting a spectral method for the solution of the fundamental PBE, the statistical density function is directly obtained and the problem of reconstruction of the statistical density function is avoided, as is necessary, using moment methods. Furthermore, the least squares method is based on advanced linear algebra theory and thus is associated with involved implementation issues. For this reason, in this study, the theory and detailed implementation of the least squares method to solve the PBE for bubbly flow are outlined using an illustrated example.

The Foundation of the Population Balance Equation: A Review

In dispersed multi-phase flow modeling using population balances (PBs), the dispersed phase system is considered as a population of entities of the dispersed phase distributed not only in physical space but also in an abstract property space. Different frameworks exist for the formulation of the population balance equation (PBE): (i) continuum mechanical principles, (ii) statistical Boltzmann-like equation, or (iii) probability principles. The source terms, that is, birth and death of the entities in the population, are defined from mechanistic principles. This article presents a review of the foundation of the PBE. GRAPHICAL ABSTRACT

Evaluation of the impact parameter in droplet-droplet collision experiments by the aliasing method

A new method has been developed whereby the borders between the various collision outcome regimes for droplet/droplet impact can be determined. This method, called the aliasing method, is based on a small frequency shift between the droplet generating nozzles. The droplet collision outcomes, for a fixed Weber number, can then be visually followed continuously for the whole range of impact parameters during one collision experiment. The impact parameter can be evaluated numerically using the frequency shift between the droplet trains without using geometric image evaluations of the video recordings. This frequency shift and temporal aliasing method makes it possible to determine the impact parameter significantly more accurately and efficiently, compared to previous methods. The aliasing evaluation method for determining the impact parameter can also be used together with the modulation technique in order to extend the range of We numbers investigated. The new method has been validated with experiments usi...