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In the frame of the activities promoted and encouraged by the EUROfusion Power Plant Physics and Technology (PPPT) department aimed at developing the EU-DEMO fusion reactor, strong emphasis has been recently addressed to the whole Balance of Plant (BoP) which represents the set of systems devoted to convert the plasma generated thermal power into electricity and to deliver it to the grid. Among these systems, the Divertor Primary Heat Transfer Systems (PHTSs) are intended to feed coolant to the two main components of the Divertor assembly, namely the Plasma Facing Components (PFCs) and the Cassette Body (CB). Since the DEMO Divertor must withstand high heat flux loads together with a considerable neutron deposited power, very tight tolerances may be allowed to the coolant inlet conditions. Therefore, the design of reliable PHTSs is of the utmost importance towards the development of an EU-DEMO fusion reactor. Within this framework, a study has been jointly carried out by University of Palermo, Ansaldo Nucleare and CREATE to design the Ex-Vessel PHTSs of both the PFCs and the CB for a DEMO reactor equipped with a Helium Cooled Pebble Bed Breeding Blanket. The paper describes criteria and rationale followed with the aim to achieve simple PHTS designs based on the adoption of easy-to-manufacture main components avoiding too much extrapolation from the state-of-art technology. Results of preliminary thermal-hydraulic calculations carried out to size heat exchangers, pressurizers, piping and pumps are presented and critically discussed, with particular attention to those integration, safety and feasibility constraints that may deeply affect the design of such components. Finally, the evaluation of PHTS key parameters as total pumping power and coolant inventory is reported. [ABSTRACT FROM AUTHOR]