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The performance of simultaneous wireless information and power transfer (SWIPT) in downlink (DL) cell-free massive multiple-input multiple-output (MIMO) is investigated. Tight approximations to the DL harvested energy and the DL/uplink (UL) achievable rates are derived for two practical channel state information (CSI) cases by using a non-linear energy harvesting model for time-switching and power-splitting protocols. Max-min fairness-based transmit power control policies are employed to mitigate the deleterious near-far effects caused by distributed transmissions/receptions in cell-free massive MIMO. The achievable common DL energy-rate trade-off is derived, and thereby, it is shown that the proposed max-min power control guarantees user-fairness regardless of near-far effects in terms of both harvested energy and achievable rate. The benefits of user estimated DL CSI to boost the SWIPT performance are explored. These performance metrics are compared against those of the conventional co-located massive MIMO, and thereby, it is revealed that the reduction of path-losses and lower average transmit powers offered by cell-free massive MIMO can be exploited to boost the energy-rate trade-off of SWIPT at the expense of increased backhaul requirements.