A previously published kinetic model for the production of luminescence signals in quartz is used to investigate the production of thermally transferred optically stimulated luminescence (TT-OSL) signals. This paper provides a mathematical description of the thermal transfer mechanism for two different phenomena that have been observed in previously published experiments (Aitken and Smith 1988 Quat. Sci. Rev. 7 387-93). The starting point is the model proposed by Bailey (2001 Radiat. Meas. 33 17-45). The numerical values of some of the parameters are varied so that they match the experimental data. The effect caused by varying these values is investigated. The first of these phenomena takes place after storing optically bleached samples at room temperature; this involves the traps responsible for the 110 °C thermoluminescence (TL) peak of quartz acting as a refuge trap. The second phenomenon concerns OSL signals that are induced by heating the samples after the bleaching of the OSL signal and involves a putative TL peak at ∼230 °C associated with the refuge trap; specifically, the paper presents a simulation of the temperature dependence of the OSL signal measured by successively heating the quartz samples to higher temperatures up to ∼400 °C.