We study heat transport in a one-dimensional chain of a finite number N of identical cells, coupled at its boundaries to stochastic particle reservoirs. At the center of each cell, tracer particles collide with fixed scatterers, exchanging momentum. In a recent paper (Collet and Eckmann in Commun. Math. Phys. 287:1015, 2009), a spatially continuous version of this model was derived in a scaling regime where the scattering probability of the tracers is γ∼1/N, corresponding to the Grad limit. A Boltzmann-like equation describing the transport of heat was obtained. In this paper, we show numerically that the Boltzmann description obtained in Collet and Eckmann (Commun. Math. Phys. 287:1015, 2009) is indeed a bona fide limit of the particle model. Furthermore, we study the heat transport of the model when the scattering probability is 1, corresponding to deterministic dynamics. Thought as a lattice model in which particles jump between different scatterers the motion is persistent, with a persistence probability determined by the mass ratio among particles and scatterers, and a waiting time probability distribution with algebraic tails. We find that the heat and particle currents scale slower than 1/N, implying that this model exhibits anomalous heat and particle transport.