The bivariate population balance equation (PBE) is a mathematical framework to explain the evolution of polydisperse multiphase systems. In this work, the direct quadrature method of moments (DQMOM) is implemented in an open-source CFD code, Fluidity, for the numerical solution of bivariate PBE. This efficient numerical framework is a highly-parallelised finite element (FE) CFD code that allows for the use of mesh adaptivity on fully-unstructured meshes. Various test cases to solve spatially homogeneous bivariate PBEs with aggregation, breakage, growth and dispersion were simulated and verified against analytical solutions, resulting in excellent agreement. Benchmarking, by comparison with the Monte Carlo method solutions from the literature, with realistic kernels in a gas–liquid system for simultaneous bivariate aggregation and breakage was also performed to show the feasibility of this implementation for realistic applications. This open-source framework demonstrates its impressive potential in the case of bivariate PBE and can be exploited for the simulation of complex polydisperse multiphase systems.