One-dimensional Population Balance Equation Research Articles

Langevin dynamics simulation and collision frequency modification in population balance model of nanoparticle coagulation during simultaneous agglomeration and sintering

Coagulation of nanoparticles from the free-molecular to the transition regime under a high-temperature environment is investigated by coupling Langevin Dynamics (LD) model with agglomeration and finite-rate sintering. The effect of temperature on the evolution of particle size and structure is investigated, variations in agglomeration-sintering dynamics are obtained. The fractal dimension of agglomerates at different temperatures can be expressed as a function of the ratio between the characteristic collision time and sintering time. The overall collision frequency (β) is enhanced by the polydispersity and fractal properties of agglomerates compared to the monodisperse collision kernel functions. Finally, the classic function of β is modified based on the LD simulations and used in a one-dimensional population balance equation, which reproduces a better prediction of the evolution of particle size distribution during particle coagulation with agglomeration and sintering.

Modeling of an industrial mixing valve and electrostatic coalescer for crude oil dehydration and desalination

ABSTRACT This paper aims to present a new model for an industrial desalination system including a mixing valve and an electrostatic coalescer. This model uses a one-dimensional population balance equation in steady-state conditions. The effect of hydrodynamic, diffusion and electrostatic mechanisms on the collision or breaking of water droplets dispersed in crude oil was considered. Also, using the film drainage model, the droplet coalescence rate has been determined. The developed model can predict the outlet droplet size distribution as well as the desalination and dehydration efficiency. The simulation results showed good agreement with the industrial data. Finally, the effect of some important parameters on the electrostatic desalination process, such as mixing valve pressure drop, electric field intensity, crude oil viscosity, crude oil temperature, diluting water flow rate, and crude oil API on the efficiency of the process was analyzed. The results indicated that by increasing the crude oil temperature from 314 to 334 K, the efficiency of dehydration improves from 97.95% to 99.02%. Also, reducing crude oil API from 33 to 30 and crude oil viscosity from 4 to 2 mPa.s caused a decrease in water separation by 2.88% and 1.16%, respectively.

A numerical comparison of the method of moments for the population balance equation

We investigate the application of the method of moments approach for the one-dimensional population balance equation. We consider different types of moment closures, including polynomial (PN) closures and minimum entropy (MN) closures. Additionally, we transfer the concept of Kershaw closures (KN) from radiative head transfer to the population balance equation. Realizability-preserving schemes for the moment closures as well as for a discontinuous Galerkin discretization of the space-dependent population balance equation are discussed. The numerical examples range from spatially homogeneous cases to a population balance equation coupled with the stationary incompressible Navier–Stokes equation. A detailed numerical discussion of accuracy, order of the moment method and computational time is given. We show that the minimum entropy closures are comparable or superior to the Kershaw closures in these test cases. In addition, we demonstrate that higher-order numerical schemes can be applied to the moment equations of the population balance equation.

Symmetries of population balance equations for aggregation, breakage and growth processes

The integro-differential population balance equation describing aggregation processes was proposed almost 100 years ago. Aggregation is an important size enlargement process in many industries; the modeling and design of the process can be done using the population balance framework, however it is typically impossible to obtain analytical solutions: in almost every case a numerical solution of the equations must be obtained. In this paper, we present the developed group analysis method for the one-dimensional population balance equation for aggregation in a well-mixed batch system including a crystal growth term. The determining equations are solved, the optimal system, invariant solutions and all the reduced equations are obtained. Furthermore, finding the determining equation by use of the preliminary group classification is also considered.

Extraction of Physiological State Functions in Heterogeneous Cell Population Models

Abstract: A simple yet generic approach is presented to extract the growth and breakage kinetics from the temporal data of heterogeneous cell population. The moment form of the one-dimensional population balance equation is directly solved by a MATLAB solver for linear systems to extract the kinetics. The corresponding systems of linear equations are, however, highly underdetermined and ill-conditioned. To address this, the problem is regularized by assuming a suitable upper bound of the solution. The range between minimum and maximum possible values of the solution is discretized into several sub-intervals. The system is then solved against each sub-interval, whose values are used as lower and upper bounds in a suitable MATLAB solver and a local solution is obtained in all these ranges. The final solution is then computed by taking average of all solutions having residual norms less than a particular threshold. To validate the method, the results of the inverse technique are compared and discussed against two theoretical experiments.

A new framework for population balance modeling of spray fluidized bed agglomeration

Previous work (Hussain et al. (2013). Chemical Engineering Science, 101, 35) has pointed out that the conventional, one-dimensional population balance equation for aggregation can be expanded to accurately reproduce the results of discrete simulations of spray fluidized bed agglomeration. However, some parameters had to be imported from the discrete simulation (Monte-Carlo). The present paper shows how the expanded population balance can be run without importing parameters from the Monte-Carlo simulation. The expanded population balance still reproduces the results of Monte-Carlo simulations accurately, taking into account key micro-scale phenomena (sessile droplet drying, efficiency of collisions), but with much lower computational cost. Required input parameters are just the drying time of sessile droplets (calculated in advance), and the pre-factor of an equation that correlates particle collision frequency with fluidized bed expansion. In this way, the expanded population balance is, apart from autonomous, also (nearly) predictive. Its performance is demonstrated by comparisons with both Monte-Carlo results and experimental data for various operating conditions (binder mass flow rate, gas temperature). Despite formally being a one-dimensional expression, the expanded population balance captures additional properties, such as the number of wet particles and the number of droplets in the system, which are even difficult to measure in experiments.

Fluidized bed spray granulation: Analysis of the system behaviour by means of dynamic flowsheet simulation

Nowadays, software tools for the flowsheet simulation of industrial processes are commonly used for design, simulation, balancing, troubleshooting and optimization purposes. Most of the tools are applicable to fluid processes only and cannot be effectively used for processes which involve solids. In this contribution we want to present the conceptual design of a new system applicable for the dynamic flowsheet simulation of complex solids processes. This system is developed as an enhancement to the existing simulation program. The novel software is able to simulate the unsteady behaviour of complex circuits of granulation processes. The transient behaviour during the start-up and changing of the process or material parameters can also be examined. As flowsheet examples, a typical spray granulation process with different schemes consisting of fluidized bed granulators, screens, mill and splitters was used. The mathematical model of the fluidized bed granulator is described by a one-dimensional population balance equation and coupled with heat and mass transfer and simple fluid dynamics. Received simulation results have shown that the proposed concept of the dynamic flowsheet simulation of granulation processes can be used effectively and has the potential to be generalized for other types of solids processes.

Simulation of crystal size and shape by means of a reduced two-dimensional population balance model

Commonly, population balance modeling for crystallization processes only considers one inner variable. Usually, a variable characterizing particle size, like a sphere-equivalent diameter is employed. However, crystal structures are obviously not spherical but exhibit a complex habit. Often, the habit is even crucial for the quality of the product or for the operability of downstream filter units. To describe the transient behavior of crystals in a batch process considering two-dimensional growth and nucleation, a multidimensional population balance needs to be employed. Considering two characteristic lengths of a crystal, the standard discretization of such a system (by, e.g. finite differences) leads to quite a large model size, which may be unsuited for model-based control and parameter identification purposes. In this contribution a model reduction which is based on two steps is proposed. First, a coordinate transformation is performed to model the system not in terms of two characteristic lengths but by means of the crystal volume and a shape factor. In a second step the actual model reduction is performed by generating cross-moments for the two-dimensional representation. An ansatz for the two-dimensional crystal size distribution, which gives full flexibility in the volume coordinate but restricts the dependence in the shape factor κ, allows the closure of the system of moments. The reduced system consists of three coupled population balance equations in which all three are structurally similar to a single one-dimensional population balance equation, where growth and nucleation only are considered. Solving this reduced system allows the detailed simulation of the easily measurable volume-based number density distribution and preserves average and dispersity information on the crystal shape. The resulting model size, however, scales only linearly with the number of discretization grid points instead of the quadratic scaling for standard discretizations. Numerical results for the crystallization of potassium dihydrogen phosphate (KH2PO4) in a batch process are presented for illustration.