The Numerical Solutions And Analysis Of Kinetic Equation Of Self-preserving Coagulation

Coagulation is the main method for the removal of colloid and suspended solids of water treatment. Considering the particals coagulation are induced by Brownian motion and turbulent shear flow, and the coagulation induced by fluid shear has the self-preserving size spectrum under certain conditions the same as the Brownian coagulation. The self-preserving means the reduction of the particle size distribution of spectral function does not change after a period of time during the process of coagulation, since the self-preserving size spectrum has a more centralized particle distribution, which is conducive to be removed during the subsequent precipitation.The theory for the self-preserving particle size distribution provides an effective method for the analysis of partical evolution in coagulation system. Since there is less awareness and study of the self-preserving size distribution which occurred after a period of time during coagulation, in this paper, based on the study of spherical aggregates equation, the self-preserving size distribution coagulation equation for turbulence-induced of fractal-like aggregates will be derived by using similarity transformation, and numerical methods are used for solving the coagulation equation, including Adams predict-correction method and the32points of Gaussian-Laguerre numerical integration combined with step-by-step accelerated iterative optimization procedure, to calculate the self-preserving coagulation equation for the spherical aggregates and fractal-like aggregates under Brownian motion and turbulent shear flow.By calculating the the integral-differential equation for the fractal-like aggregates, then the numerical solutions and partical size distribution can be obtained for the coagulation by Brownian motion and simultaneous turbulent shear flow. As the values of parameter P increasing, the maximum of curves becomes higher and size distribution becomes broader, which is the same as the geometrical standard deviation. For the fractal-like aggregates, partical size distribution becomes narrower and concentrated with decreasing of fractal dimensions, whether it is the Brownian motion or fluid shear coagulation. Since the self-preserving distribution of the large particles remains unchanged with increasing time and dimensionless parameter P, the curve eventually tends to exponential distribution. Based on the particle size distribution obtained by numerical calculation, the regression relations of aggregation rate constant KBS were obtained for different fractal dimensions, they are also similar to the simplified formulas obtained under the equal-volume assumption, and the coagulation rate becomes faster as the decreasing of fractal dimensions.