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 [1] . The population balance modeling is a widely accepted mathematical frame work for dynamic modeling of this process. Hulbert and Katz [2] are the main architects of population balance modeling. The approach has capability to simulate the crystallization process and to describe the evolution of each individual crystal throughout the process.

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 [1] . Several researchers in this filed have tried to develop specified algorithms for tackling these complexities. A verity of accurate and efficient numerical techniques were introduced, such as the finite difference methods [3] [4] , the method of moments [5] [6] , the method of weighted residuals [7] , the Monte Carlo method [8] , and the flux limiting high resolution finite volume schemes [9] -[11] . In this article, the method of characteristics (MOC) [1] [12] -[14] is applied to solve the PBE for simulating crystallization process. The MOC has capability to avoid numerical diffusion and instabilities in the solutions on coarse meshes and has low computational cost. These virtues encourage its applicability to industrial processes. The MOC transforms the given PBE into a system of ordinary differential equations (ODEs), which are then solved along the path line of the particles (characteristic curves). The particles are identified and located at the initial time and the population is trailed with a velocity equal to the growth rate.

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 which denotes the number of crystals per crystal length at any time t. The crystal growth rate G could be either dependent or independent of crystal size. Balancing the number of crystals in an infinitesimal interval of crystal length, a partial differential equation is obtained which explains the temporal evolution of CSD [12] [15]

The corresponding initial and boundary conditions are given as

The symbol T denotes temperature, signifies the nucleation rate of particles at minimum crystal size, is the maximum crystal size, and is a Dirac Delta function. The solute concentration in the solution obeys the mass balance

where denotes the density of crystals. The negative sign on the right hand side of Equation (3) explains the decrease of solute concentration (mass) in the solution during the process of crystallization. The rate of nucleation of nuclei of size is expressed as

where, and b are regarded as kinetic parameters, V is the total crystals volume in the system, and describes the saturated solute concentration which is depending on temperature of the solution. The crystal growth depending linearly on size may be defined as

In the above equation is growth rate constant. The exponent is a kinetic parameter and a₁ and a₂ are constants. The relative super saturation, which is a driving force for the crystallization process, is expressed as

where depends on the temperature T of the solution. A quadratic fit to the solubility data gives us

where, are regarded as constant parameters. Normally, temperature T is either considered constant (isothermal case) or a monotonically decreasing function of time (non-isothermal case). Hence either remains constant or decreases with respect to time but remains positive.

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 signifies the uniform partitioning of the given domain. For and interval length

, N_i indicate the average value of the number density in each cell, i.e..

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 in the above equation, we obtain

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 (and) are reinitialized at these time levels and the time integrator restarts.

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. [12] .

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.