Kinetic equation moment method

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

The answer to this question is mainly driven by the computational cost of solving the kinetic equation due to the large number of independent variables. In the simplest example of a 3D velocity-distribution function n t, x, v) the number of independent variables is 1 + 3 + 3 = 1. However, for polydisperse multiphase flows the number of mesoscale variables can be much larger than three. In comparison, the moment-transport equations involve four independent variables (physical space and time). Furthermore, the form of the moment-transport equations is such that they can be easily integrated into standard computational-fluid-dynamics (CFD) codes. Direct solvers for the kinetic equation are much more difficult to construct and require specialized numerical methods if accurate results are to be obtained (Filbet Russo, 2003). For example, with a direct solver it is necessary to discretize all of phase space since a priori the location of nonzero values of n is unknown, which can be very costly when phase space is not bounded. 

Figure 1.5. From the kinetic equation to macroscale models using moment methods that are based on reconstruction of the NDF.

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

Fox, R. O. 2009a Higher-order quadrature-based moment methods for kinetic equations. Journal of Computational Physics 228, 7771-7791. 

The structure and interrelationship of the batch conservation equations (population, mass, and energy balances) and the nucleation and growth kinetic equations are illustrated in an information flow diagram shown in Figure 10.8. To determine the CSD in a batch crystallizer, all of the above equations must be solved simultaneously. The batch conservation equations are difficult to solve even numerically. The population balance, Eq. (10.3), is a nonlinear first-order partial differential equation, and the nucleation and growth kinetic expressions are included in Eq. (10.3) as well as in the boundary conditions. One solution method involves the introduction of moments of the CSD as defined by... 

The analysis of batch crystallizers normally requires the consideration of the time-dependent, batch conservation equations (e.g., population, mass, and energy balances), together with appropriate nucleation and growth kinetic equations. The solution of these nonlinear partial differential equations is relatively difficult. Under certain conditions, these batch conservation equations can be solved numerically by a moment technique. Several simple and useful techniques to study crystallization kinetics and CSDs are discussed. These include the thermal response technique, the desupersaturation curve technique, the cumulative CSD method, and the characterization of CSD maximum. 

The kinetic equation (11.1) is a nonlinear integro-differential equation, general theory of which does not exist. Its known exact solutions are based on the use of operational methods with reference to a case of linear dependence K(V,w) on each of drop volumes [2]. To solve the equation (11.1) with more general kernel, the approximate methods are used - parametric methods and method of moments, and also numerical methods. Parametric methods and method of moments are based on transforming the kinetic equation into a system of equations for the moments of drop distribution over volumes. However, the resulting system of equations is, as a rule, incomplete, since, apart from the integer moments,... 

The practical application of these results shows that the method of moments provides an accurate description of the kinetics of coalescence only at its initial stage. With some restrictions imposed on the form of the kernel of the kinetic equation and on the initial distribution [6], a self-similar solution of the kinetic equation for a longer time range can be obtained. In the general case, the solution can be obtained by numerical methods. 

Let us use the method of moments to solve the kinetic equation (11.1). Then, in view of (13.135), we obtain from (11.15) the following equation for zeroth moment, which has the meaning of the number concentration of drops... 

In spite of have been proposed many approximated solutions to Boltzmann equation (including the Grad s method of 13 moments, expansions of generalized polynomial, bimodal distributions functions), however the Chapman-Enskog is the most popular outline for generalize hydrodynamic equations starting from kinetics equations kind Boltzmann (James William, 1979 Cercignani, 1988). 

To solve numerically the linearized kinetic Equations (23) with the boundary condition (33) a set of values of the velocity Cj is chosen. The collision operator Lh is expressed via the values hi(x) = h(x,Ci). Thus, Eq. (23) is replaced hy a system of differential equations for the functions hi(x), which can he solved numerically by a finite difference method. First, some values are assumed for the moments being part of the collision operator. Then, the distribution function moments are calculated in accordance with Eqs. (28)-(32) using some quadrature. The differential equations are solved again with the new moments. The procedure is repeated up to the convergence. 

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