Jannike Solsvik

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 weighted residual methods for the solution of the pellet equations: The orthogonal collocation, Galerkin, tau and least-squares methods

Abstract In recent years a number of publications have adopted the least-squares method for chemical reactor engineering problems such as the population balance equation, fixed bed reactors and pellet equations. Evaluation of the performance of the least-squares method compared to other weighted residual methods is therefore of interest. Thus, in the present study, numerical techniques in the family of weighted residual methods; the orthogonal collocation, Galerkin, tau, and least-squares methods, have been adopted to solve a non-linear comprehensive and highly coupled pellet problem. The methanol synthesis and the steam methane reforming process have been adopted for this study. Based on the residual of the governing equations, the orthogonal collocation method obtains the same accuracy as the Galerkin and tau methods. Moreover, the orthogonal collocation method is associated with the simplest algebra theory and thus holds the simplest implementation. Another benefit of the orthogonal collocation method is that the technique is more computational efficient than the other methods evaluated. The least-squares method does not obtain the same accuracy as the other weighted residual methods. In particular, the least-squares method is not suitable for strongly diffusion limited systems that give rise to steep gradients in the pellet. On the other side, considering the spectral–element framework, the least-squares method is less sensitive to the placement of the element boundaries than observed for the orthogonal collocation, Galerkin and tau methods. The present paper outlines the algebra of the weighted residual methods and illustrate the numerical solution techniques through a simplified pellet problem.

Single Air Bubble Breakup Experiments in Stirred Water Tank

Abstract Single gas bubbles have been injected into an stirred liquid tank and the eventual breakup process of the bubbles was examined through high-speed imaging. Experimental observations of the breakup probability, breakup time, number of daughter bubbles and daughter bubble size distribution have been collected. The occurrence of non-equal-size breakup events dominated over equal-size breakup events. The frequency of binary and multiple breakup events was about equal. The largest uncertainty is associated with the determination of the breakup time because the bubbles take continuously altering deformed shapes already from the point of injection into the tank. The present experimental data do not support the standard model assumption regarding the number of daughter particles. The active breakup zone in the stirred tank was in the large velocity field close to the radial impeller. It is not evident whether the breakup events are due to time average shear or pressures and velocity fluctuations.

Bubble Coalescence Modeling in the Population Balance Framework

In the churn-turbulent bubbly flow regime with highly nonuniform bubble size distributions, bubble breakage and coalescence are important processes because they govern the bubble size distribution and consequently directly affect the interfacial mass, momentum, and heat transfer fluxes through the renewal bubble surfaces. At present, accurate prediction of bubble size distributions of dispersed gas–liquid flows by use of the population balance (PB) equation is a difficult task. The modeling of bubble breakup and coalescence rates is very complex and is based on the knowledge of collision and breakup frequencies, breakage daughter size distributions, and probability of coalescence. In this work, we focus on the coalescence phenomenon. The coalescence models are still on an empirical level and the mechanisms are not fully understood. This motivates the analysis of the suitability of the coalescence closures for the prediction of experimental data obtained from coalescence dominated gas–liquid flows. For this task, a cross-sectional averaged combined multifluid-PB model is adopted. Based on different theories for the coalescence efficiency, the simulation results show a similar trend in the prediction of the experimental data. Good prediction of the Sauter mean diameter is achieved although the shape of the bubble size distribution is not completely reproduced. The second aim of this work is to review the PB framework. Here, focus is placed on the coalescence term and the combined multifluid-PB model based on kinetic theory approach.

Viscous Drop Breakage in Liquid–Liquid Stirred Dispersions: Population Balance Modeling

The breakup of viscous drops in a turbulent regime is a challenging area of research. In this study, the population balance equation is used for modeling of the drop size distribution of emulsification systems in stirred tank. The suitability of various fluid particle breakage closures in the literature is investigated for the prediction of experimental data obtained from emulsification of silicone oils with four different viscosities. Systems with breakup of dispersed-phase drops of high viscous grades call for an improved model framework that allows the breakage closures to adjust correctly for the effect of viscosity. This is particularly required for systems of higher viscosities where the complex effects of viscous breakup are more pronounced. GRAPHICAL ABSTRACT

A multifluid-PBE model for simulation of mass transfer limited processes operated in bubble columns

Abstract Modeling of reactive dispersed flows with interfacial mass transfer limitations require an accurate description of the interfacial area, mass transfer coefficient and the driving force. The driving force is given by the difference in species composition between the continuous and dispersed phases and thus depends on bubble size. This paper shows the extension of the multifluid-PBE model to reactive and non-isothermal flows with novel transport equations for species mass and temperature which are continuous functions of bubble size. The model is demonstrated by simulating the Fischer-Tropsch synthesis operated in a slurry bubble column at industrial conditions. The simulation results show different composition and velocity for the smallest and largest bubbles. The temperature profile was independent of bubble size due to efficient heat exchange. The proposed model is particularly useful in investigating the effects of bubble size on strongly mass transfer limited processes operated in the heterogeneous flow regime.

Modeling and simulation of bubbling fluidized bed reactors using a dynamic one-dimensional two-fluid model

A dynamic one-dimensional multicomponent model for two-phase flows which includes heat- and mass transfer processes are studied in the Euler framework. The model is intended for reactive gas-solid flows in bubbling fluidized bed reactors. A model is desired that allows for a more complex description of the fluidized bed reactors (e.g. prediction of the bed expansion) relative to the conventional fluidized bed reactor models such as, e.g., Kunii-Levenspiel type of models. The model should not predict details in the flow as the two- and three-dimensional Euler two-fluid models in order to ensure reasonable simulation costs. In particular, the two- and three-dimensional Euler two-fluid models challenges the current available computational capacity for studies of reactive flows.The novel sorption-enhanced steam methane reforming (SE-SMR) technology is simulated in the bubbling bed regime. Simulation results of the one-dimensional Euler two-fluid model is compared to both a two-dimensional Euler model and a conventional fluidized bed model consisting of mass and heat balances. Furthermore, a sensitivity study to operation conditions and transport coefficients is performed for the one-dimensional Euler two-fluid model.The present simulation results reveal that the chemical process performance of the reactor is to a large extent determined by the imposed temperature in the reactor. Further, the one-dimensional Euler model provides an improvement of the simpler conventional fluidized bed reactor models by prediction of the bed expansion. Compared with the two-dimensional Euler model, cross-sectional averaging results in a significant reduction in the computational time but on the cost of loss of flow details.

A Combined Multifluid-Population Balance Model Applied to Dispersed Gas–Liquid Flows

The two-phase bubble columns are widely used for carrying out gas–liquid reactions in a variety of industrial applications. The optimal operation of many of the processes carried out in the bubble columns depends on the bubble size distribution, which determines the interfacial momentum, mass, and heat transfer fluxes through the contact area between the liquid and gas phases. In this context, the population balance equation is relevant as it can be used to quantify the amount of bubbles of different sizes present at different locations in the bubble column. The combined multifluid-population balance models use mass and momentum equations coupled with population balances for the dispersed phase, whereas the continuum mechanical equations of the two-fluid or multifluid models are used for the liquid phase. The most rigorous description is based on a kinetic theory approach with size resolution (KTAWSR). In this study, the KTAWSR model is applied to simulate bubbly flows under complex conditions, that is, significant change in the Sauter mean diameter along the axial direction of the bubble column. The model is solved using the spectral orthogonal collocation method. Both sensitivity analyzes and comparison to experimental data from the literature are used to assess the potential of the model. Fair agreement between the experimental data and model prediction is achieved. It is concluded that the model adds as a valuable tool with good potentials for the population balance community, considering both the level of details resolved with this model and its possible use in developing future breakage and coalescence models.

Evaluation of the Least-Squares Method for the Solution of the Population Balance Equation

Spectral methods are evaluated for the solution of population balance problems. Both a simplified population balance equation (PBE) with an analytical solution available and a rigorous PBE are considered in this numerical analysis. In comparison with the orthogonal collocation, tau, and Galerkin methods, the least-squares method does not produce the same favorable results. On the other hand, the least-squares method with a direct minimization solver is capable of producing equally accurate results of the model equations as the orthogonal collocation, tau, and Galerkin methods. Compared to the conventional least-squares method, the direct minimization formulation is better conditioned but does not produce a symmetric positive-definite system matrix.