A new type of unglazed transpired collector (UTC) incorporating phase change materials (PCMs) is presented. A time-dependent model consisting of eight 1D PDEs is derived from basic principles. Mass, momentum and energy balance equations for air flow are derived ab initio from the general transport theorem for air movement, by including mass sources contribution. Such mass sources terms appear explicitly in the continuity equation but also in the momentum and energy equations. The model has been calibrated in two stages by comparing the computation results with results obtained by measurements. The operation and performance of the UTC is simulated by using realistic meteorological and radiometric data measured during four seasons in Bucharest, Romania (South-Eastern Europe). UTCs incorporating PCMs release heat for a longer time period than usual UTCs (without PCMs) but shorter than that of the systems consisting of UTCs coupled with separate thermal energy storage units. The time variation of the temperature rise Delta T in the UTC decreases when PCMs are incorporated and the effect is stronger in clear sky days than in cloudy and overcast sky days. Also, thicker PCM plates decrease Delta T and increase the time interval of heat releasing at the end of the day. The dependence of the UTC performance on the stability of the radiative regime is studied here for the first time in a systematic manner. The type of PCM with best performance depends on month and on the value of the sunshine number (SSN). The daily amount of heat released inside the building, Xi(day), increases by increasing the daily SSN value. The dependence of the transient coefficient of performance epsilon(COP) on the sunshine stability number (SSSN) is not obvious, neither for UTCs with PCM nor for UTCs without PCM, no matter the season is. epsilon(COP) is higher for UTCs without PCM than for UTCs with PCM. This is true for all seasons, as far as similar values of SSN or SSSN are considered. The spreading of epsilon(COP) values is larger for UTCs without PCM than for UTCs with PCM, in all seasons. The two performance indicators (Xi(day) and epsilon(cop)) induce different hierarchies among the types of PCM used in the UTC.