All of the Jumo 004's problems had nothing to do with the compressor.
Rubbish. The compressor blades were made of poor quality metals and flame outs caused by the engines shedding fan blades or blades disintegrating is widely recorded.
While the engines (Jumo 004 and BMW 003) were well designed and conceived, they certainly did have their issues and were hampered because of the state of war. Here is a British engineer's impression of the German engines based on his examination of examples captured by the British in 1944:
"Both engines were in good condition and I was impressed with the quality of the design right down to the very neat petrol engine starter mounted in the "bullet" [the Riedel starter motor] of the compressor air intake. Both engines were straight through axial compressor designs. I particularly recall the hollow air cooled alloy turbine blades of the Jumo engine, which the manufacturer had to resort to because of the effectiveness of the Allied blockade in preventing supplies of nickel from reaching Germany. The short service life of these low alloy turbine blades may have been one of the factors which led to fires in the Me 262 aircraft."
It is interesting to analyse the German engines from a technical standpoint, the Jumo and the BMW were relatively simple in design, which was a virtue in that they did work - the issues that affected them were not necessarily design related, although design compromises were made because of Germany's faltering industry that did affect their reliability. By the standards of 1945 technology, they did have a foot in the past. The Riedel starter motor was one of these. Relying and small piston engines with their own fuel supply for starting means the engines cannot be started in the air and it seems like a bit of a waste of weight. Surely an electric or even cartridge starter, like the British engines would have been a better option. The Riedel was started by a ring pull starter that was done by guys on the ground standing in front of the engines, the rung protruded out of the bullet and had to be pulled to engage the starter for each engine.
A Riedel starter motor inside the cowl of a Jumo 004 without the bullet. The pull starter cable is visible.
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Another peculiarity was the fuel management of these engines. Thrust and fuel flow was managed manually by the pilot based on instrument readings, which meant the pilot had to keep a keen eye on fuel flow and engine temp gauges to move the power levers and the zwiebel, the variable thrust cone located in the exhaust nozzle, which was driven mechanically by a long shaft that extended aft to the thrust chamber. By 1945 this was relatively simplistic because there were already mechanical and electric fuel control devices fitted to piston engined aircraft, not to mention constant speed propellers, which equated to varying thrust output, rather than controlling the zwiebel, whose operation should have been automatic and keyed into the fuel control system, rather than manually actuated by the pilot.
A Jumo 004 on display at the Musee de l'Air, Paris. Note the ring pull for the starter motor in the extremity of the nose bullet and the extension shaft halfway along the engine's flank that operated the zwiebel in the exhaust shroud. The removed oval cover on the front engine cowl was for the fuel port for the starter motor. It ran on standard 87 octane fuel.
Musee de l'Air 56
Nevertheless, the German engines were great designs.