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In the first part of the paper, a generalization of the turbulent diffusivity concept is considered and a generalized diffusion coefficient is introduced to account for the development of turbulent diffusivity, pressure-driven countergradient transport, and effects of chemical reactions on turbulent scalar flux. The behavior of the generalized diffusivity is numerically studied in the 1-D statistically planar case and the contributions of the aforementioned processes to the diffusivity are assessed. In the second part of the paper, the generalized diffusivity is incorporated into the Flame Speed Closure (FSC) model of premixed turbulent combustion and the extended FSC model is applied to simulate recent experiments performed using the Leeds fan-stirred bomb. The extended FSC model well predicts the speed, thickness and structure of statistically spherical, premixed, turbulent flames that expand in the bomb after spark ignition.