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The present study deals with transient thermal management using phase change materials (PCMs). These materials can absorb large amounts of heat without significant rise of their temperature during the melting process. This effect is attractive for passive thermal management, particularly where the device is intended to operate in a periodic regime, or where the relatively short stages of high power dissipation are followed by long stand-by periods without a considerable power release. Heat transfer in PCMs, which have low thermal conductivity, can be enhanced by fins that enlarge the heat transfer area. However, when the PCM melts, a layer of liquid is growing at the fins creating an increasing thermal resistance that impedes the process. The present work aims to demonstrate that performance of a latent-heat thermal management unit may be considerably affected by achieving a so-called close-contact melting (CCM), which occurs when the solid phase is approaching a heated surface, and only a thin liquid layer is separating between the two. Although CCM was extensively studied in the past, its possible role in finned systems has been revealed only recently by our group. In particular, it depends heavily on the specific configuration of the fins. In the present work, close-contact melting is modeled analytically for a geometry which includes two symmetrically inclined fins. A quasi-steady approach is used for calculating the rate of melting based on the force and energy balances. The results are expressed in terms of the time-dependent melt fraction and Nusselt number, showing their explicit dependence on the Stefan and Fourier numbers. Moreover, the approach used in the present study may be applied to other geometries in which the heated surface is not horizontal or where there are a number of heated surfaces or fins.