(1)

In this model, the growth rate of recrystallized grains, is expressed by the product of an effective mobility, , and the driving force P. The grain number density is then adjusted according to the conservation equation by maintaining the constant volume fraction. The grain boundary mobility is given by [2009Pay]:

(2)

where is the effective atomic mobility for a multi-component system, which is computed directly from mobility database (C_i is the instant composition of the matrix phase and is the atomic mobility); d is the grain boundary width. A₂ is a factor related to the fraction of atomic jumps which result in a 'permanent' motion of the grain boundary in the direction of the driving force. The grain coarsening driving force P=P_C+P_Z related to the local grain boundary curvature P_C and the Zener drag force P_Z due to the pinning effect of the precipitates [1985Nes]. P_C is given by:

(3)

where R is the grain radius and γ is the grain boundary interfacial energy.

(4)

where and are the volume fraction and average radius of each precipitate phase, which can be obtained from the precipitation modeling. b is a Zener drag force factor and the most used value is b=3/2.

(b) Multi-size model

In order to track the evolution of grain size distribution (GSD), the Multi-size model is implemented based on the KWN model developed for the simulation of recrystallization (not implemented in this version) and grain coarsening processes. In the Multi-size model, the continuous grain size distribution (GSD) is divided into a large number of size classes. The program takes a simulation step at every sample time hit. To maintain both accuracy and efficiency between two adjacent simulation steps, a fifth-order Runge-Kutta scheme is used to generate an adaptive step size based on the continuity equation and the growth/coarsening rate.

The coarsening of existing grains for each size class is given by [2009Pay]:

(5)

where R_c is the critical radius and can be calculated from by , α is a constant and α = 2 for spherical grains. The effective grain boundary mobility M_eff is defined in Eq. 2. The Zener drag effect Z can be obtained from Eq. 4.

After the coarsening rate of each size class is calculated, the number of grains in each size class is re-distributed and normalized to maintain the constant volume fraction of the system. To improve the efficiency, the number of size classes is fixed in the Multi-size model and the width of each size class is automatically adjusted according to the average grain size.