Fuel cell stack and method of operating a fuel cell system with such a fuel cell stack

The invention relates to a fuel cell stack comprising a plurality of fuel cells which are connected together and which each have an anode and a cathode as well as membrane arranged there-between and which are each arranged between two bipolar plates, with the anode sides of the individual fuel cells having flow fields for a gaseous fuel supplied to the fuel cells and the cathode sides of the individual fuel cells having flow fields for a gaseous oxidizing agent fed to the fuel cells and wherein the fuel and the oxidizing agent react on catalyst material within the fuel cells for the production of electrical energy with the simultaneous production of water. The anode side flow fields extend between an anode space arranged at the top in the fuel cell stack and an anode space arranged at the bottom of the fuel cell stack. A collecting container is provided in the anode space arranged at the bottom of the fuel cell stack for water which is produced during the electrochemical reaction and is present in the form of a liquid and the water which is collected there has an evaporation surface and serves for the humidification of the fuel fed to the fuel cell stack.