Mathematical models of tumour spheroids, proposed since the early seventies, have beengenerally formulated in terms of a single diffusive nutrient which is critical for cell replicationand cell viability. Only recently, attempts have been made to incorporate in the models the cellenergy metabolism, by considering the interplay between glucose, oxygen and lactate (or pH).By assuming glucose and lactate as the only fuel substrates, we propose a simple model for thecell ATP production which takes into account the main reactions that occur in the glycolytic andthe oxidative pathway. Under the assumption that cell death occurs when ATP production fallsbelow a critical level, we have studied the free boundary problem for the concentration of glucose,lactate and oxygen inside the spheroid viable rim. We show that the existence of a necrotic core isguaranteed for a sufficiently large size of the spheroid.