Internet of Things (IoT) devices often have batteries of limited capacity, which are not easily replaced or recharged. This implies very short device lifetimes, and calls for a very careful device configuration to achieve the optimal trade-off between performance and power consumption. SWIPT (Simultaneous Wireless Information and Power Transfer) deals with this problem by harvesting energy at IoT devices from the received RF signals. Studying the efficiency of SWIPT in dealing with the energy and data transfer demands of IoT nodes leads to a number of open issues. In this paper, we devise an analytical model based on stochastic geometry for a SWIPT radio access network with a dense population of IoT users. With our model, it is possible to accurately study the impact of the system parameters on the key system performance indicators, while accounting in a realistic manner for device performance, and for the statistics of time scheduling at base stations. This allows us to understand (not without some surprise) what are the most effective strategies to minimize energy consumption in a SWIPT network, and what is their potential for energy savings.