The population balance equation for the evolution of drop size distributions in fully developed turbulent flow of a liquid-liquid dispersion in a circular pipe has been solved "exactly" using spectral expansion of the self-adjoint diffusion operator with a radially varying diffusion coefficient to obtain the number density at any location in the pipe. The breakage frequency is allowed to vary with position, although the size distribution of broken fragments is assumed to satisfy a form of similarity assumed in the work of Narsimhan et al. (AIChE J. 1980, 26, 991; 1984, 30, 457) that rids it of explicit spatial dependence. Of course, insofar as numerical methods are used to calculate the spectral data (eigenvalues and eigenvectors), such an exact solution is still to be regarded as approximate. Furthermore, because the solution is expressed in terms of a transient well-mixed batch dispersion evolving by breakage, the actual number density may be obtained by any of the methods for solving population balance equations in this simpler setting. In this paper, we use the method of Kumar and Ramkrishna (Chem. Eng. Sci. 1996, 51 (8), 1311) to solve the population balance equation for a batch system. This discretization method is also incorporated into a detailed simulation of the population balance equation in combination with computational fluid dynamics using the control volume approach. The latter method is incomparably demanding with regard to memory and computational time and consequently irrelevant. Self-similar solutions are obtained by including spatial scaling from the spectral expansion and particle size scaling from the work of Sathyagal et al.