A new method for the reconstruction of the particle radius distribution function from the sedimentation curve

Abstract

A new method for the reconstruction of the particle radius distribution function from the sedimentation curve is proposed. This method permits us to obtain a continuous smooth distribution function. Two approaches are compared. The first approach is based on the calculation of the second derivative from the sedimentation curve. The second one is based on the solution of the original integral equation which describes a sedimentation process. Both of these approaches can be reduced to the problem of the solution of the Fredholm integral equation of the first kind. From the theory of integral equations, it is known that this problem is ill-posed. The usual methods lead to unstable solutions and we are forced to use special regularizing algorithms. In this paper, the Tikhonov regularization method is used to stabilize the solution of the integral equation. It is shown that the accuracy of both methods is higher than the accuracy of the graphical method, but the approach based on the solution of the original integral equation gives a more stable solution than that based on the derivative. The accuracy of the new method permits us to reconstruct the fine structure of the particle radius distribution function. Such an analysis cannot be carried out with the rough bar diagram obtained from the graphical method. The new method is absolutely indispensable in technology for controlling the degree of powder fineness.