A new framework for solution of multidimensional population balance equations

A new framework is proposed in this work to solve multidimensional population balance equations (PBEs) using the method of discretization. A continuous PBE is considered as a statement of evolution of one evolving property of particles and conservation of their n internal attributes. Discretization must therefore preserve n+1 properties of particles. Continuously distributed population is represented on discrete fixed pivots as in the fixed pivot technique of Kumar and Ramkrishna [1996a. On the solution of population balance equation by discretization—I. A fixed pivot technique. Chemical Engineering Science 51(8), 1311-1332] for 1-d PBEs, but instead of the earlier extensions of this technique proposed in the literature which preserve 2ⁿ properties of non-pivot particles, the new framework requires n+1 properties to be preserved. This opens up the use of triangular and tetrahedral elements to solve 2-d and 3-d PBEs, instead of the rectangles and cuboids that are suggested in the literature. Capabilities of computational fluid dynamics and other packages available for generating complex meshes can also be harnessed. The numerical results obtained indeed show the effectiveness of the new framework. It also brings out the hitherto unknown role of directionality of the grid in controlling the accuracy of the numerical solution of multidimensional PBEs. The numerical results obtained show that the quality of the numerical solution can be improved significantly just by altering the directionality of the grid, which does not require any increase in the number of points, or any refinement of the grid, or even redistribution of pivots in space. Directionality of a grid can be altered simply by regrouping of pivots.     

A new discretization of space for the solution of multi-dimensional population balance equations

In this work, a novel radial grid is combined with the framework of minimal internal consistency of discretized equations of Chakraborty and Kumar [2007. A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62, 4112-4125] to solve n-dimensional population balance equations (PBEs) with preservation of (n+1) instead of 2ⁿ properties required in direct extension of the 1-d fixed pivot technique of Kumar and Ramkrishna [1996a. On the solutions of population balance equation by discretization-I. A fixed pivot technique. Chemical Engineering Science 51, 1311-1332]. The radial grids for the solution of 2-d PBEs are obtained by intersecting arbitrarily spaced radial lines with arcs of arbitrarily increasing radii. The quadrilaterals obtained thus are divided into triangles to represent a non-pivot particle in 2-d space through three surrounding pivots by preserving three properties, the number and the two masses of the species that constitute the newly formed particle. Such a grid combines the ease of generating and handling a structured grid with the effectiveness of the framework of minimal internal consistency. A new quantitative measure to supplement visual comparison of two solutions is also introduced. The comparison of numerical and analytical solutions of 2-d PBEs for a number of uniform and selectively refined radial grids shows that the quality of solution obtained with radial grids is substantially better than that obtained with the direct extension of the 1-d fixed pivot technique to higher dimensions for both size independent and size dependent aggregation kernels. The framework of Chakraborty and Kumar combined with the proposed 2-d radial grid, which offers flexibility and achieves both reduced numerical dispersion and the ease of implementation, appears as an effective extension of the widely used 1-d fixed pivot technique to solve 2-d PBEs.     

An analytical solution of population balance equations for continuous fluidized bed drying

Continuous fluidized bed drying is widely used to remove moisture or solvents from granular materials. It is known that different residence times of the wet particles may lead to a distribution of product properties, e.g. different moistures. The prediction of such moisture distributions in fluidized bed dryers is of particular interest in industrial practice. In the present study, a simple analytical approach is introduced to calculate moisture distributions at the outlet of a continuous fluidized bed dryer. The model provides an analytical solution of the simple one-dimensional population balances. It will be contrasted with a traditional model approach based on averages and with experimental investigations conducted in a lab scale fluidized bed dryer under variation of the particle and the gas flow rate. Furthermore, the moisture distributions of the dried product were estimated by single particle measurements using nuclear magnetic resonance spectroscopy. It will be demonstrated that the developed analytical approach is capable to predict such moisture distributions for continuous drying processes.     

Numerical solution of population balance equations for nucleation, growth and aggregation processes

This article focuses on the derivation of numerical schemes for solving population balance models (PBMs) with simultaneous nucleation, growth and aggregation processes. Two numerical methods are proposed for this purpose. The first method combines a method of characteristics (MOC) for growth process with a finite volume scheme (FVS) for aggregation process. For handling nucleation terms, a cell of nuclei size is added at a given time level. The second method purely uses a semi-discrete finite volume scheme for nucleation, growth and aggregation of particles. Note that both schemes use the same finite volume scheme for aggregation process. On one hand, the method of characteristics offers a technique which is in general a powerful tool for solving linear growth processes, has the capability to overcome numerical diffusion and dispersion, is computationally efficient, as well as give highly resolved solutions. On the other hand, the finite volume schemes which were derived for a general system in divergence form, are applicable to any grid to control resolution, and are also computationally not expensive. In the first method a combination of finite volume scheme and the method of characteristics gives a highly accurate and efficient scheme for simultaneous nucleation, growth and aggregation processes. The second method demonstrates the applicability, generality, robustness and efficiency of high-resolution schemes. The proposed techniques are tested for pure growth, simultaneous growth and aggregation, nucleation and growth, as well as simultaneous nucleation, growth and aggregation processes. The numerical results of both schemes are compared with each other and are also validated against available analytical solutions. The numerical results of the schemes are in good agreement with the analytical solutions.     

On the solution and applicability of bivariate population balance equations for mixing in particle phase

New benchmarks are used to test two classes of discretization methods available in the literature to solve bivariate population balance equations (2-d PBEs), and the applicability of these mean-field equations to finite size systems. The new benchmarks, different from the extensions of their 1-d counterparts, relate to prediction of kinetics of mixing in particle phase under: (i) pure aggregation of particles, called aggregative mixing, and (ii) simultaneous breakup and coalescence of drops. The discretization methods for 2-d PBEs, derived from the widely used 1-d solution methods, are first classified into two classes. We choose one representative method from each class. The results show that the extensions based on minimum consistency of discretization perform quite well with respect to both the new and the old benchmarks, in comparison with the geometrical extensions of 1-d methods. We next revisit aggregative mixing using Monte-Carlo simulations. The simulations show that (i) the time variation of the extent of mixing in finite size systems has power law scaling with the system size, and (ii) the mean-field PBEs fail to capture the evolution of mixing for reduced population of particles at long times. The sum kernel limits the applicability of PBEs to substantially larger particle populations than that seen for the constant kernel. Interestingly, these populations are orders of magnitude larger than those at which the PBEs fail to capture the evolution of total particle population correctly.     

Efficient solution of population balance equations with discontinuities by finite elements

Two refinements of Galerkin's method on finite elements were evaluated for the solution of population balance equations for precipitation systems. The traditional drawbacks of this approach have been the time required for computation of the two-dimensional integrals arising from the aggregation integrals and the difficulty in handling discontinuities that often arise in simulations of seeded reactors. The careful arrangement of invariant integrals for separable aggregation models allows for a thousandfold reduction in the computational costs. Discontinuities that may be present due to the hyperbolic nature of the system may be specifically tracked by the method of characteristics. These discontinuities will arise only from the initial distribution or nucleation and are readily identified. A combination of these techniques can be used that is intermediate in computational cost while still allowing discontinuous number densities. In a case study of calcium carbonate precipitation, it is found that the accuracy improvement gained by tracking the slope discontinuity may not be significant and that the computation speed may be sufficient for dynamic online optimization.     

Accuracy and optimal sampling in Monte Carlo solution of population balance equations

Implementation of a Monte Carlo simulation for the solution of population balance equations (PBEs) requires choice of initial sample number (N₀), number of replicates (M), and number of bins for probability distribution reconstruction (n). It is found that Squared Hellinger Distance, H², is a useful measurement of the accuracy of Monte Carlo (MC) simulation, and can be related directly to N₀, M, and n. Asymptotic approximations of H² are deduced and tested for both one‐dimensional (1‐D) and 2‐D PBEs with coalescence. The central processing unit (CPU) cost, C, is found in a power‐law relationship, , with the CPU cost index, b, indicating the weighting of N₀ in the total CPU cost. n must be chosen to balance accuracy and resolution. For fixed n, M × N₀ determines the accuracy of MC prediction; if b > 1, then the optimal solution strategy uses multiple replications and small sample size. Conversely, if 0 < b < 1, one replicate and a large initial sample size is preferred. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2394–2402, 2015     

复合肥生产中的物料衡算矩阵方程及其应用

通过对复合肥生产过程进行物料衡算,得出生产过程中物料平衡的矩阵方程。介绍物料平衡矩阵方程在复合肥生产中的应用,包括养分计算、投料配方确定、养分控制、生产成本优化。利用该矩阵方程可方便地对生产过程进行产品养分调控及生产成本的控制。     

Robust algorithms for the solution of the ideal adsorbed solution theory equations

The ideal adsorbed solution (IAS) theory has been shown to predict reliably multicomponent adsorption for both gas and liquid systems. There is a lack of understanding of the conditions which guarantee convergence for various algorithms used to solve the IAS theory equations and inconsistencies are present in the reported computational effort required for the different approaches. The original nested loop and the FastIAS technique are revisited. The resulting system of equations is highly nonlinear but both methods are shown to be robust if appropriate choices are made for the starting values of the unknown variables. New initial conditions are proposed and the resulting algorithms are compared in a consistent manner with the main methods available to solve the IAS theory equations. The algorithms are extended for the first time to all nontype I isotherms. © 2014 American Institute of Chemical Engineers AIChE J, 61: 981–991, 2015     

A new discretization of space for the solution of multi-dimensional population balance equations: Simultaneous breakup and aggregation of particles

In this work, we show that straight forward extensions of the existing techniques to solve 2-d population balance equations (PBEs) for particle breakup result in very high numerical dispersion, particularly in directions perpendicular to the direction of evolution of population. These extensions also fail to predict formation of particles of uniform composition at steady state for simultaneous breakup and aggregation of particles, starting with particles of both uniform and non-uniform compositions. The straight forward extensions of 1-d techniques preserve 2ⁿ properties of non-pivot particles, which are taken to be number, two masses, and product of masses for the solution of 2-d PBEs. Chakraborty and Kumar [2007. A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62, 4112-4125] have recently proposed a new framework of minimal internal consistency of discretization which requires preservation of only (n+1) properties. In this work, we combine a new radial grid [proposed in 2008. part I, Chemical Engineering Science 63, 2198] with the above framework to solve 2-d PBEs consisting of terms representing breakup of particles. Numerical dispersion with the use of straight forward extensions arises on account of formation of daughter particles of compositions different from that of the parent particles. The proposed technique eliminates numerical dispersion completely with a radial distribution of grid points and preservation of only three properties: number and two masses. The same features also enable it to correctly capture mixing brought about by aggregation of particles. The proposed technique thus emerges as a powerful and flexible technique, naturally suited to simulate PBE based models incorporating simultaneous breakup and aggregation of particles.     

多级PDS脱硫脱氰工艺中水平衡的计算

PDS脱硫脱氰工艺具有脱除效率高、运行稳定的优点,在煤气净化中得到广泛应用。但是在实际生产中脱硫液会在系统内累积,以致将部分脱硫液当作废液处理,不但导致操作成本上升,而且浪费PDS催化剂。在水平衡计算的基础上,通过优化操作制度,使PDS脱硫脱氰过程可以实现水平衡。     

A new update for the solution of nonlinear algebraic equations

A new symmetric update for the solution of nonlinear algebraic equations is suggested. Examples treated indicate that the update is more reliable than others.     

On the solution of population balance equations (PBE) with accurate front tracking methods in practical crystallization processes

The PBE (population balance equation) containing birth, growth, agglomeration and breakage kinetics is described by a conservation law with a moving source term. For the solution of the PBE, we compare two accurate front tracking methods such as a modified method of characteristics (MOC) and a finite difference method with the weighted essentially non-oscillatory (WENO) scheme. Both methods are applied to a potassium sulfate crystallization problem (K₂SO₄-H₂O system) with a discontinuous initial condition. Parameters of agglomeration and breakage kinetics are estimated on the basis of the experimental data of the K₂SO₄-H₂O system.Owing to moving axis along a crystal growth rate (i.e. elimination of the growth term), the modified MOC is able to provide a highly accurate solution even at discontinuous points without numerical diffusion error. However, in the case of stiff nucleation which can commonly appear in practical crystallization processes, it is necessary to adaptively determine time levels to add a mesh of the nuclei size. For solving PBEs involving agglomeration and breakage terms, the MOC can take more long computational time than the spatial discretization methods like the WENO scheme. It is pointed out that the MOC is not available to solve more than two coupled PBEs in general.WENO schemes for spatial discretization are firstly addressed in this study for the dynamic simulation of batch crystallization processes. The WENO schemes show improvements of accuracy and stability over conventional discretization methods (e.g., backward, central or common upwinding schemes). However the WENO schemes on fixed meshes show, to some extent, the numerical diffusion error near discontinuities or steep moving fronts like other finite difference methods. Hence, they require spatially-adaptive mesh techniques in order to track more accurately the moving fronts. Even though the WENO schemes are less accurate than the MOC, they are of practical use for solving complex PBEs owing to a short computational time and little limitation to use.     

Quasilinearization technique for solution of boundary layer equations

Owing to the split boundary conditions, there arise many difficulties in solving numerically the nonlinear equations describing the boundary-layer flow. It is shown in this paper that the quasilinearization technique is a useful tool for solving the ordinary differential equations, which are obtained from the boundary layer equations through a similarity transformation. With very approximate initial guesses, only four to six iterations are needed to obtain a five digit accuracy.     

Simultaneous solution of the potential equations for the metal and solution phases in cylindrical electrochemical reactors

The effect of the resistance of the metal phase on the current distribution in cylindrical electrochemical reactors taking into account the resistance of the solution phase is analyzed. A mathematical model is proposed, which assumes that the external electrode is isopotential and the electrochemical reaction on the external electrode has a low polarization resistance (di/d). Consequently the equation for the potential distribution for the metal phase of the inner electrode was solved simultaneously with the Laplace equation for the solution phase. The theoretical current density distributions are compared with previous experimental results in order to determine the predictive suitability of the model and a good agreement is observed between them. Furthermore, a comparison is made between this model and an earlier one and a slight improvement in the prediction is observed.     

Solution of population balance equations by MWR

The population balance equation for crystallization in a continuous mixed suspension and mixed product removal crystallizer accounting for the effects of arbitrary crystal breakage (an outstanding problem) has been solved by the method of weighted residuals. The trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process with a suitable weight function in the definition of the inner product. The weight function emerged from the analytical solution of multilated versions of the original population balance equation. The method used also included a Vorobyev's method of moments which is essentially a variation of the method of weighted residuals. Accurate solutions were obtained with only four problem-specific polynomials thus rating these trial functions above the more standard (and traditional) choices like Laguerre polynomials with which no satisfactory solution could be obtained even when a large number of function were employed.     

A generalized solution method for the quasi-chemical local composition equations

The work of Abusleme and Vera based on the Quasi-Chemical Theory of Guggenheim in terms of groups is extended to multicomponent multigroup mixtures. The nonlinear equations arising from the theory are rearranged in a particular way that allows them to be solved using a generalized Newton-Raphson method. Initial estimates of the values of the non-random factors are obtained with a Taylor expansion around the random values. No more than five iterations are required for typical systems. Practical applications of the method are presented for ternary group systems.     

A new wavelet-based adaptive method for solving population balance equations

A new wavelet-based adaptive framework for solving population balance equations (PBEs) is proposed in this work. The technique is general, powerful and efficient without the need for prior assumptions about the characteristics of the processes. Because there are steeply varying number densities across a size range, a new strategy is developed to select the optimal order of resolution and the collocation points based on an interpolating wavelet transform (IWT). The proposed technique has been tested for size-independent agglomeration, agglomeration with a linear summation kernel and agglomeration with a nonlinear kernel. In all cases, the predicted and analytical particle size distributions (PSDs) are in excellent agreement. Further work on the solution of the general population balance equations with nucleation, growth and agglomeration and the solution of steady-state population balance equations will be presented in this framework.