Superconducting spintronics is based on the creation of spin-triplet Cooper pairs in ferromagnetsuperconductor (F-S) hybrid junctions. Previous proposals to manipulate spin-polarized supercurrents on demand typically require the ability to carefully control magnetic materials. We, instead, propose a quantum heat engine that generates equal-spin Cooper pairs and drives supercurrents on demand without manipulating magnetic components. We consider a S-F-S junction, connecting two leads at

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... temperatures, on top of the helical edge of a two-dimensional topological insulator. Heat and charge currents generated by the thermal bias are caused by different transport processes, where electron cotunneling is responsible for the heat flow to the cold lead and, strikingly, only crossed Andreev reflections contribute to the charge current. Such a purely nonlocal Andreev thermoelectric effect injects spin-polarized Cooper pairs at the superconductors, generating a supercurrent that can be switched on-off by tuning their relative phase. We further demonstrate that signatures of spin-triplet pairing are facilitated by rather low fluctuations of the thermoelectric current for temperature gradients smaller than the superconducting gap.