Numerical solution of the PBE

The numerical solution of the PBE often leads to errors. Some of these include discretization errors, truncation errors, round-off errors, and propagated errors. Inverse problems are particularly stiff. Experimental errors include determining when steady state has been reached, noise in the tails of the DSD, sampling and analysis errors, and uncertainties that arise when in situ measurements cannot be made. 

The same authors also presented an example of the use of the population balance equation (PBE) (distribution of biomass m) coimected to the multi-zone/CFD model. This example is in several respects relevant for the assessment of the modeling approach. The coupling of the integro-differential equation of the population balance is a numerical challenge, which can nowadays be tackled within the environment of a CFD approach, albeit without consensus on the proper closure assumptions. Still, the computational effort for the numerical solution of the population balance embedded in the multizonal model is extensive, and it is difficult to extend this approach to multiple state variables necessary for dynamic metabolic models. This is an important argument to favor the alternative method of an agent-based Lagrange-Euler approach discussed in Section 3.5. 

Analytical solutions of the PBE are only available for very simple forms of the underlying growth, breakage, and coalescence kernel functions. Therefore, numerical methods are usually employed for the solution of most PBEs [9, 27, 186, 258]. The choice of numerical methods are in general problem dependent and optimized for particular applications. 

In order to avoid the assumption of constancy of a (= 1 - P) in the PIE model, attempts have been made to calculate the concentradons of ions at the ionic micellar surface by the use of cell model, which involves the numerical solution of the Poisson-Boltzmann equation (PBE) under specific spherical cell boundary conditions. In this model, the distribution of counterions around an ionic nucellar surface is carried out in terms of coulombic interactions between the... 

Equation (2.14) must be coupled with initial conditions given for the starting time and with boundaries conditions in physical space O and in phase space O. Analytical solutions to Eq. (2.14) are available for a few special cases and only under conditions specified by some very simple hypotheses. However, numerical methods can be used to solve this equation and will be presented in Chapters 7 and 8. The numerical solution of Eq. (2.14) provides knowledge of the NDE for each time instant and at every physical point in the computational domain, as well as at every point in phase space. As has already been mentioned, sometimes the population of particles is described by just one internal coordinate, for example particle length (i.e. f = L), and the PBE is said to be univariate. When two internal coordinates are needed, for example particle volume and surface area (i.e. = (v, a)), the PBE is said to be bivariate. More generally, higher-dimensional cases are referred to as multivariate PBEs. Another important case occurs when part of the internal-coordinate vector is equal to the particle-velocity vector (i.e. when the particles are characterized not by a unique velocity field but by their own velocity distribution). In that case, the PBE becomes the GPBE, as described next. 

Because of the complicated nature of biomolecular geometries and charge distributions, the PB equation (PBE) is usually solved numerically by a variety of computational methods. These methods typically discretize the (exact) continuous solution to the PBE via a finite-dimensional set of basis functions. In the case of the linearized PBE, the resulting discretized equations transform the partial differential equation into a linear matrix-vector form that can be solved directly. However, the nonlinear equations obtained from the full PBE require more specialized techniques, such as Newton methods, to determine the solution to the discretized algebraic equation. ... 

In mathematical terms the population balance equation (PBE) is classified as a non-linear partial integro-differential equation (PIDE). In the PBE (12.308) the size property variable ranges from 0 to oo. In order to apply a numerical scheme for the solution of the equation a first modification is to fix a finite computational domain. The conventional approximation is to truncate the equation by substitution of the infinite integral limits by the finite limit value max- The function/(, r, t) denotes the exact solution of the exact equation. It might be assumed that the solution of the truncated PBE is sufficiently close to the exact equation so that the two solutions are practically equal. Hence, the solutions of both forms of the PBE are denoted by fiC, r, t). 

Different variations of the sectional methods for the solution of the governing PBE (12.308) can be obtained from different choices of numerical approximations of Bbj, Db,u Bc,i and Dc,i in terms of n (r, t). It is noted that the breakage problem is generally easier to solve than the coalescence problem because the breakage model is linear whereas the coalescence problem is non-linear. 

For very small particles (in case of ZnO <3 nm), the full solution of the exponential term is not possible within a PBE framework when explicit solvers are used. A detailed discussion why numerics become highly unstable... 

In mathematical terms the population balance equation (PBE) is classified as a non-linear partial integro-differential equation (FIDE). Since analytical solutions of this equation are not available for most cases of practical interest, several numerical solution methods have been proposed during the last two decades as discussed by Williams and Loyalka [209] and Ramkrishna [151]. 

The Least Squares Method (LSM) is a well established numerical method for solving a wide range of mathematical problems [84, 12, 150, 146]. The basic idea in the LSM is to minimize the integral of the square of the residual over the computational domain. In the case when the exact solutions are sufhciently smooth the convergence rate is exponential. In particular, the application of LSM to PBE as has been discussed by [38, 39, 36, 37]. 

This book provides a consistent treatment of these issues that is based on a general theoretical framework. This, in turn, stems from the generalized population-balance equation (GPBE), which includes as special cases all the other governing equations previously mentioned (e.g. PBE and BE). After discussing how this equation originates, the different computational models for its numerical solution are presented. The book is structured as follows. 

The approximate graphical solution is used to simplify the PBE. All terms that are seen to be less than 5% of the main components in the PBE are discarded. The choice of 5% as a cutoff criterion is based on the usual limit of 0.02 units accuracy in pH measurements. Since pH is logarithmic, the corresponding numerical difference is 10° , which is 1.05, or 5%. The difference in logarithms corresponding to 5% (or 1 part... 

The PBE describes a continuous distribution of charge inside the solvent and on the surface—any effect related to the discreteness of charge and the ion volume is ignored. In general, the PBE has to be solved numerically. Analytical solutions exist only for simple geometries and/or symmetrical electrolytes (e.g. Lyklema 1995, Chap. 3.5). Even for an isolated sphere there is no exact analytical solution. 

In order to obtain a system of algebraic equations, the PBE (12.399) and the boundary conditions must be transformed into a discrete form. In spectral methods, the solution function is approximated in terms of a polynomial solution function expansion (12.408). The differentiation of the discrete solution approximation were presented as (12.410) and (12.411). The PBE (12.399) is an integro-differential equation. Thus, appropriate quadrature rules are required for the numerical solution. Integral approximations can be presented on the form ... 

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