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I've always thought they are empirically derived: once you have your axial compressor phisically in a bench, you set both a rotational speed and a mass flow throught the turbomachine then you measure the pressure ratio so you can fix a point on the map. After, It will be made for several mass flows therefore defining a particular constant speed line.

It is correct?

Typical compressor working line generated using Complex Off-design Calculation
(image source: Wikimedia)

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From my textbook, Fluid Mechanics and Thermodynamics of Turbo-machinery by Dixon, it is explained that they find the results for the working line empirically. This corresponds to the line with the black dots on the diagram you provided.

Essentially, your test rig is made up of a compressor that is connected to a downstream throttle. It is the size of the throttle (in a real engine, this corresponds to the size of the nozzle) that is going to determine the test working line (called an operating line if it does not pass by the Design Operation Point). In other words, a different operating line corresponds to a different size throttle.

Then you vary the speed and measure the obtained pressure ratio. Connecting the dots will give you the working line.

ADDED DETAIL: You have to know the size of the throttle because it will give you valuable information regarding your exit configuration. This has to do with the fact that the nozzle will usually be chocked, which will fix your non-dimensional mass flow through the nozzle. But that's a whole other topic in itself!

I did my best to answer your question, sorry if it does not help, I am only a student! If you want details of the calculations for compressor maps, you should check out chapter 5 of Dixon.

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I believe what Hypatie D'Alexandrie has stated is correct. To add to this, I would like to mention that nowadays as we are getting more computational, there are few softwares like Gasturb and GSP which allows to simulate and analyze the performance of gas turbine components in any kind of configuration for in steady and transient state by assuming 0-D modelling of thermodynamic cycle i.e. by taking the average of the properties across the flow sections.

With many companies using these kind of softwares, I have heard this has made them shut down huge test rigs which were specifically designed to perform tests and also draw characteristic maps. For example- RR doing that in Derby

As far as the operation line/working line is concerned, it is essentially a collection of points on each constant speed line at which the condition of power equilibrium is satisfied i.e. the bet power output is equal to the power given at the given set of conditions, without acceleration or deceleration. The Design Point (DP) is always on the constant speed line of 100% and speed lines at left or right are representational of the spool speed lower & higher than the DP.

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