This paper investigates a mean reaction rate closure for turbulent reacting flows called the Direct Quadrature Method of Moments using the Interaction by Exchange with the Mean micromixing model (DQMoM-IEM). The method was first introduced for reacting flows by Fox (Computational Models for Turbulent Reacting Flows, Cambridge University Press, 2003). We present a systematic study that considers several important new aspects of the method. In particular we introduce a new analytic expression for the DQMoM-IEM source terms. We present a rigourous numerical investigation and discuss problems of boundedness and singularity in detail. We introduce a filter function to overcome these issues in the general case and present analytic integrals for special cases of specific terms. We extend the methodology to take advantage of these developments and show details of the implementation in a commercial computational fluid dynamics (CFD) code. We present an extensive set of numerical experiments and validation. The method is proven for a problem known from the literature which includes an isothermal dimerisation process. Experimental and transported probability density function (PDF) data compare reasonably well. The method is discussed critically and areas for further research are suggested to make the method more practical.