With detections of the Sunyaev–Zel'dovich (SZ) effect induced by galaxy clusters becoming routine, it is crucial to establish accurate theoretical predictions. We use a hydrodynamical N‐body code to generate simulated maps, of size 1 deg2, of the thermal SZ effect. This is done for three different cosmologies: the currently favoured low‐density model with a cosmological constant, a critical‐density model and a low‐density open model. We stack simulation boxes corresponding to different redshifts in order to include contributions to the Compton y‐parameter out to the highest necessary redshifts. Our main results are as follows. (i) The mean y‐distortion is around 4×10−6 for low‐density cosmologies, and 1×10−6 for critical density. These are below current limits, but not by a wide margin in the former case. (ii) In low‐density cosmologies, the mean y‐distortion is contributed across a broad range of redshifts, with the bulk coming from z≲2 and a tail out to z∼5. For critical‐density models, most of the contribution comes from z<1. (iii) The number of SZ sources above a given y depends strongly on instrument resolution. For a 1‐arcmin beam, there are around 0.1 sources per deg2 with y>10−5 in a critical‐density Universe, and around 8 such sources per deg2 in low‐density models. Low‐density models with and without a cosmological constant give very similar results. (iv) We estimate that the Planck satellite will be able to see of order 25 000 SZ sources if the Universe has a low density, or around 10 000 if it has critical density.