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The population balance modeling is regarded as a universally accepted mathematical framework for dynamic simulation of various particulate processes, such as crystallization, granulation and polymerization. This article is concerned with the application of the method of characteristics (MOC) for solving population balance models describing batch crystallization process. The growth and nucleation are considered as dominant phenomena, while the breakage and aggregation are neglected. The numerical solutions of such PBEs require high order accuracy due to the occurrence of steep moving fronts and narrow peaks in the solutions. The MOC has been found to be a very effective technique for resolving sharp discontinuities. Different case studies are carried out to analyze the accuracy of proposed algorithm. For validation, the results of MOC are compared with the available analytical solutions and the results of finite volume schemes. The results of MOC were found to be in good agreement with analytical solutions and superior than those obtained by finite volume schemes.

Pharmaceutical, chemical and food industries produce significant amount of materials in crystalline form. Crystallization is an important separation unit in these industries, and has a significant impact on plant operation and economics. Crystal size distribution is an important quality aspect of the crystalline product. The industrial crystallization process faces a major challenge for the production of crystals of predefined size distribution. Dynamic modeling of crystallization process has received notable consideration in recent time due to its various applications [

On the other hand, accurate numerical solution of the population balance equation (PBE) is a challenging task for several reasons. Numerical diffusion and instability are common problems in the numerical solutions of PBEs for seeded batch systems. Incompatibility between the initial and the boundary conditions is one reason of the aforementioned problem. The number density distribution of seeds is unlikely to be the same to that generated by nucleation process. If their values match, the first order derivative of the distribution may not be identical. This can lead to sharp discontinuities that are rapidly broadened by numerical diffusion. Another problem that is usually encountered in the solution of PBEs is the occurrence of steep moving fronts, known as source of numerical instability. This problem arises from the convective nature of growth-dominated process [

This article is organized as follows. In Section 2, the population balance modeling of batch crystallization process is briefly introduced. In Section 3, the method of characteristics is derived. This is followed by Section 4 in which the forgoing numerical technique is applied to four test problems. Finally, the concluding remarks are outlined in Section 5.

In the one-dimensional batch crystallization model, the crystal size is defined by a characteristic length l. The crystal size distribution (CSD) is depicted by the number density function

The corresponding initial and boundary conditions are given as

The symbol T denotes temperature,

where

where,

In the above equation _{1} and a_{2} are constants. The relative super saturation

where

where,

To avoid undesirable phenomena of primary nucleation which often adversely influence the crystal size distribution, seeded batches are operated in industrial batch crystallization. The secondary nucleation only produces infinitesimally small crystals in seeded batch runs. Since nuclei are produced at the minimum crystal size, we can consider a homogeneous PBE by defining the ratio of nucleation and growth terms as a left boundary condition:

Note that Equation (8) is a hyperbolic partial differential equation due to the convection term (second term on the left hand side) and is equivalent to the Equation (1).

Before applying the numerical scheme, we discretize the computational domain. As explained before, the domain of interest is the crystal length, denoted by l. Suppose M is a large integer, and let

_{i} indicate the average value of the number density in each cell

The rate of change by growth of the total number of particles in the i-th size range can be obtained by integrating Equation (8) with respect to l.

By substituting the growth rate

This gives

The application of Leibnitz formula for differentiation of integral expressions that have variable limits of integration, Equation (11) becomes.

This leads to the following semi-discrete equation:

According the product rule, the above equation further simplifies to

Thus, we get

After simplifying the above equation, it takes the form

Moreover, as described above

Any time-discretization scheme can be used to solve jointly the system in Equations (16) and (17). In our case, a simple Euler method is employed. In Equations (16) and (17), there is no convection term which could cause much numerical error and instability. Hence the solution obtained by the MOC is very accurate and stable. To overcome the nucleation problem, a new mesh of the nuclei size is added at given time levels. The system size can be kept constant by deleting the last mesh at the same time levels. As a result, all variables (

In this section, some test problems are presented for the validation of the proposed numerical schemes. The results of MOC are compared with analytical solutions and results of the finite volume scheme presented in Qamar et al. [

This test problem is taken from the article of Leonard et al. [

Equation (18) corresponds to four characteristics peaks in the initial crystal size distribution. The first expression on the right side is a narrow Gaussian, the second and third expressions represent a square step and the last expression signifies a semi-ellipse. The last expression is very challenging because it combines sudden and gradual changes in the gradient. The analytic solution of this problem for the initial profile

Lim et al. [

The crystal size and time ranges are considered as

The analytical solution is given as [

The result of test problem 1 for t = 1

The result of test problem 2 at t = 0.5

This test problem is taken from [_{3}) crystals. The nucleation rate is a function of the time-dependent concentration and growth rate is a function of both concentration and crystal size. Thus, we have to solve the coupled Equations (1)-(3). The initial size distribution is given as.

Here we consider the crystals have volume

2200 cells and the simulation time is 1000 s. For size dependent growth rate, we consider

No analytical solution exists in this case, thus the results of MOC and finite volume scheme are compared with each other.

The purpose of this test problem is to illustrate the applicability MOC for the case of discontinuous crystal growth rate. Here, the simulation of potassium sulfate (K_{2}SO_{4}∙H_{2}O) is considered. The initial seed distribution is taken as [

The size range of interest is

. Parametric value for test problem 3

Description | Symbol | Value | Units |
---|---|---|---|

Total size of distribution | 1100 | μm | |

Mesh size | 0.5 | μm | |

Simulation time | 1000 | s | |

Number of grid points | 2200 | - | |

Growth coefficient | - | ||

Growth exponent | 1.32 | - | |

Nucleation coefficient | 1/μm^{3} | ||

Nucleation exponent | 1.78 | - | |

Density of crystals | g/μm^{3} |

The result of test problem 3

Zoomed plots of the results in test problem 3

where,

Here,

and

The saturated concentration quantifying solute mass per gram of solvent is given as

The temperature profile used to maintain a constant supersaturation

The concentration balance in Equation (3) is replaced by the following equation

The numerical results at 400 grid points are shown in

The results of test problem 4

[

This work focused on the application of the method of characteristics (MOC) for solving batch crystallization models. The growth and nucleation were considered to be the dominant phenomena and breakage and nucleation were neglected. Three test problems were considered for different growth and nucleation rates. The performance and accuracy of the MOC was analyzed against the analytical solutions and the numerical solutions of finite volume schemes. Steep moving fronts or discontinuities appearing in the solutions were well captured by the MOC without any spurious oscillations and its results were found to be superior over the finite scheme results. It is therefore concluded that attention must be paid to the discretization of growth term (convection term) when devising a numerical algorithm. This could help to obtain a crystal size distribution which agrees well with the experimental one.