Adjustable discretized population balance equations: Numerical simulation and parameter estimation for fractal aggregation and break-up

Abstract

Improved adjustable discretized population balance equations (PBEs) are proposed in this study. The authors extended an improved particle coagulation model previously developed to an adjustable geometric size interval ( q), where q is a volume ratio of class k + 1 particle to class k particle (υ k+1 /υ k = q). This model was verified with the time derivative of the zero and first moments to show mass conservation and compared with previous analytical and numerical solutions. Also, the self-preserving distribution test was conducted by using size-independent and size-dependent kernels. After direct numerical simulations (DNS), the proposed model was found to have excellent agreement with the analytical and continuous numerical solutions. In addition, this proposed model was converted to a dimensionless form to enhance computational efficiency in order to be coupled with computational fluid dynamic solutions in the future. Furthermore, a parameter estimation scheme was created to computationally determine the two key parameters, the collision efficiency ( α) and the break-up coefficient ( K B), from orthokinetic experimental data. To verify the estimation procedure the fractal aggregate orthokinetic experimental data of Li and Zhang (a primary d o of 2.8 μm in dia., D f of 2.0, a fluid strain-rate of 15 (1/s)) was used [X. Li, J. Zhang, Numerical simulation and experimental verification of particle coagulation dynamics for a pulsed input, J. Coll. Interf. Sci. 262 (2003) 149–161]. Considering fractal aggregation and break-up, two major parameters were found to be collision efficiency α of 0.3938 and aggregate break-up coefficient K B of 4.4105 using a parameter estimation scheme coupled with an improved discretized population balance equation. This parameter estimation scheme was able to compute the coefficients in the coagulation model rapidly, especially in particle systems having a fractal aggregate structure.