I have that PDF on file, in the pictures you can see the impulse blading of the stators where there is no pitch on the blades except prior to the first stage and after the last. This type of axial compressor is inefficient but simple with the rotors acheiving ~80% of compression and stators just acting to guid the air flow.(reaction blading used on the HeS 30 and BMW 003D acheived roughly half of the compression at the stators) Only ~1.155:1 compresson per stage was produced, opposed to ~1.25:1 with a reaction compressor of the same time period (while centrifugal compressors could manage 2.8-4.0:1), this way the HeS-30 acheived the same compresson in 5 stages that the 004 did in 8 with resulting savings in weight, length, materials, and greater energy efficiency. The reaction compressor blades had to be machined and used thrust bearings, while the impulse blading of the 004 could be stamped, a rather inexpensive and efficient method.
I'm not sure what type of axial compressor the 003A/E used though, it seems to have been slightly more efficient (~1.18:1) than the Jumo but not by much, so possibly an impulse type as well. Encorporating a reaction type compressor (with the same number of stages) in the 003D raised thrust from 800 kp to 1,200 kp (a 50% increase!) and increased compression from ~3.2:1 to ~4.7:1 with little increase in fuel consumption or weight. (thus much better specicic fuel consumption) Just think, if the 004 had similar improvments 1300+ kp with nearly 6:1 compression should be possible!
There's also the "textbook" example of an axial compressor where most of the compression occurs at the stators: the rotors accelerating the air and the stators slowing it down, compressing in the process, and guiding it into the next stage. (like a difuser of a centrifugal compressor) I'm not sue of the name of this type of compresser though...
The BMW 003A and E did indeed use simple impulse type axial compressors. A set of compressors of the reaction type was on the Bench at subcontractor ABB Cie the Hermaso I and II.
These would have produced the BMW 003C with 900kg thrust instead of 800kg but with better fuel consumption and also the BMW 003D with 1100kg thrust BMW were also working on one of these compressors. The inpulse type has lower tollerance requirements and so the blades can be pressed rather than machined furthermore there is less axial thrust which simplifies the bearing arrangment. Production costs were totally critical to the Germans who resorted to every trick and compromise to pull out more production.
Designation of the end of war engines is rather collapsed roughly the engines were
Jumo 004A, experimental version using a high proportion of refractory metals.
Jumo 004B-1 with tinidur blades; first production version. The pre production Jumo 004B-0 was actually more reliable due to the greater skill level.
Jumo 004B-4 with hollow aircooled tunidur blades, from Oct/Nov 44 onwards, more reliable.
Jumo 004B-4 with hollow aircooled cromidur blafes, from Feb 45, an inferior alloy but more reliable in practice due to easier fabrication.
The Jumo 004B2 seems to have had a more advanced compressor that however had problems
There may or may not have been a jumo 004B-3.
The Jumo 004C was a developed Jumo 004B as was the Jumo 004D with higher thrust levels. Exact improvments are hard to discern but they would have included
1/ Better fuel control to measure the air flow rate and stop under and over dosing of fuel thus preventing flameouts or temperature excursions.
2/ Duplex injector nozzles to switch to a second nozzle at low flow rates seen at idling or high altitude to ensure proper vapurisation and complete combustion. This reduces flameout issues and improves economy.
3/ Improved compressors with better altitude performance
4 Electronoics to monitor engine temperature and adjust fuel flow accordingly, thermocouple mountings were available but no electronics was yet ready.
The fuel control mod would have reached opperational service in the begining of April, I pressume as an upgrade to the Jumo 004B however the KTB I have seen doesn't specify which engine.
The Jumo 004E was a sightly scaled up and improved Jumo 004D. However it was decided to produce the Jumo 004H instead, this was a scaled down Jumo 012 which had a much higher compression ratio and blow out valves to handle this extra ratio.