Choked Flows in Turbomachinery

[finiteparts]

The topic of choked flows comes up quite a bit on this forum and since it is not a topic that most of the members have probably studied very deep, I thought it might be an interesting topic to discuss.

Since the flow of gases within our turbine engines are compressible, the effects of variable density must be discussed.   I am not going to try to explain this as there is a very excellent video that was put out in the 60's by the National Science Foundation and the one on compressible flow is an excellent learning resource on this topic.  It can be found here:

youtu.be/45w1-lwFSzM?feature=shared

The text versions are to be found here:

web.mit.edu/hml/notes.htmlhttp://web.mit.edu/hml/notes.html

As an aside, in the video, you will see the noise change in a choked throat (for the case of a vacuum source) and I got to witness this first hand when we were cold flow testing a rig component to find the effective area verses pressure ratio.  It was really cool, well at least for all the aerodynamic nerds that were present.

To help dispel a few commonly seen misconceptions that I don't think are covered in the film, I will state them here:

1.   Choked flow doesn't mean that an absolute limit has been reached for the "physical" flow.   It means that the upstream flow is "independent" of the downstream conditions.  The inlet conditions set the physical flow that will pass through the given throat area.

2.  The correct speed of sound that needs to be used to determine if the throat area will produce a choked condition is the local speed of sound.  This is always calculated with the local static temperature.  Total temperatures are not physically 'real' temperatures, they are calculated values that assume a lossless recovery of the fluid kinetic energy into temperature.

3. Choking occurs when the local velocity meets or exceeds the local speed of sound.   The local speed of sound is given by:   a = SQRT(gamma * R * Ts)  

Ok...that is all I have for now on the basics.  I want to jump into the interesting stuff to talk about in the turbine stage.

- Chris

[finiteparts]

The discussion of choking in the NGV or the rotor exit of the turbine stage has come up several times and I wanted to cover a few fundamentals to help readers to make more educated decisions on this topic.

The first point that I would like to cover is basic velocity vectors and how torque is produced in a radial turbine stage.   The typical equation given for the specific work (work per lbm (or kg) of flow).   It is:

w_s = (U1^2 - U2m^2)/2  + (W2m^2 - W1^2)/2  + (V1^2 - V2m^2)/2

where:

    U1 is the circumferential (tip) speed of the rotor inducer                        (Stationary reference frame)
    U2 is the circumferential speed of the mean line radius of the exducer     (Stationary reference frame)

    W1 = relative velocity as the flow gets onboard the rotor at the inducer   (Rotating reference frame)    
    W2m = relative velocity of the flow as it comes out the exducer              (Rotating reference frame)

    V1 = the absolute velocity of the flow as it approaches the rotor inducer  (Stationary reference frame)
    V2m = the absolute velocity of the flow as it exits the rotor                    (Stationary reference frame)

From this equation, it can be seen that in order to increase the power output,

• (U1^2 - U2m^2)/2 - For this to be positive, the circumferential velocity at the inlet must be larger than the circumferential velocity at the exit; this is why a radial inflow turbine can make more stage power for the same mass flow as opposed to a axial turbine
  
  
• (W2m^2 - W1^2)/2 - For this to make a positive contribution to the specific power, the flow in the relative frame of reference,  passing through the rotor must accelerate, i.e W2m > W1.   This is why the area ratio across the turbine rotor decreases.
  
  
• (V1^2 - V2m^2)/2 - For this to make a positive contribution to the specific power, the absolute flow velocity at the entrance (i.e., the NGV exit flow) needs to be larger than the absolute exit flow velocity, V1 > V2m.  This is one of the reasons that the NGV is usually the flow limiting component in the turbine stage, and thus chokes first.

As can be seen here, the way to control the power output of the turbine stage that we can control with pre-existing hardware is to:

• Trim the exducer shroud to reduce the diameter and thus change the mean radius of the exducer, which would reduce the flow capacity of the stage. This is not usually too appealing to our purposes, especially since we are usually looking to increase the stage through-flow.
  
  
• Trim the exducer axially (as John has done multiples times).  This has the effect of increasing the rotor throat area, reducing the relative discharge velocity (W2m, less acceleration in the rotor passage) and also increases the absolute discharge velocity (V2m).  The two-fold effect of this is to reduce the work done by accelerating the flow through the rotor passage and also reducing the momentum exchange due to the change in absolute velocity across the rotor.
  
  
• Modify the flow entering the rotor by changing the NGV discharge angle and throat size (i.e. velocity magnitude).  This one is a bit more tricky, since it really controlled by the relative velocity magnitude.   We want to bring the flow onboard the rotor (in the relative frame of reference) with as little loss as possible, but this is confounded by the secondary flows in the rotor passage that can change the way the entering flow direction substantially.  But, we also see that in order to maximize the momentum exchange in the absolute reference frame, the absolute entrance velocity (V1) needs to be large and the absolute discharge (V2) needs to be minimized.  These seemingly conflicting requirements are balanced with the NGV discharge and rotor discharge angles.

A quick reference on the mean radius  calculations.   There are two method to get this,  one is an arithmetic mean and the second is the Root Mean Squared (rms) approach.

The arithmetic mean is simple:    (exducer shroud radius + exducer hub radius)/2                        : this is the average of the two radii

The rms method is:                     SQRT[(exducer shroud radius^2 + exducer hub radius^2)/2]     : this is splitting the exducer area into two equal areas

With the rms being the one I use mostly.

[finiteparts]

Now for the second discussion point on choosing to choke in the NGV or in the rotor, the relative reference frame really comes into play.

Because the local gas conditions depend on the other quantities in the relative reference frame, the relative critical velocity is what sets the choking in the rotor passages.  This means that the choking conditions depend on rotor speed, plus...

Due to the rotor relative reference frame, the flow within the rotating rotor passages will see a radial static pressure and temperature gradient.  Since the local acoustic speed is determined primarily from the local static gas temperature, there will exist a gradient of acoustic speeds that complicate the determination of the passage choking condition.  To complicate this further, the secondary flows can also lead to local variation in gas temperatures.

Conversely, the choking in the NGV is relatively straightforward to predict and is thus very "controllable". 

• Since you usually would like to have the highest velocity of the gas flow at the entrance to the rotor (as shown above), this is usually advantageous.
  
  
• Choking the nozzle allows for better matching with downstream components, such as the next stage in a multistage turbine. It ensures that the conditions entering the next stage are well-defined and controlled.
  
  
• Choking the nozzle helps avoid instabilities and control the expansion process more effectively. It provides a clear limit to the flow rate and prevents the system from operating in a regime where the flow is no longer controlled.
  
  
• The choked NGV is also more tolerant of fluctuations or transients that could cause instabilities such as surge or similar.
  
  
• Designers may adjust the geometry of the nozzle area to control the flow and prevent choking under normal operating conditions.  If the rotor is the controlling orifice, then modifications to change the operating line by modifications to the rotor can have more drastic impacts.
  
  
• If there exists a case where the flow must exceed the local speed of sound, the shock structure could be predictable as opposed to the rotor exducer, which would be harder to predict.
  
  
• The relative velocity in the passage changes more rapidly than the axial velocity and thus as rotor speed is increased, there is a reduction in the relative critical speed, meaning that the rotor choked condition is approached sooner for a higher rotor speed design, where the NGV has a fixed corrected flow once choked and is independent of the rotor speed.
  
  

Now, there are valid reasons to choose to rotor choking condition and that is what John is choosing to do with his clipped rotors.   Because his compressor and turbine are mismatched on flow capacity, there is a need to open up the back end as much as possible.   Since the rotor exducer, by default, will be larger in area than a properly sized NGV throat, it allows for a larger flow capacity.   Additionally, the clipping of the rotor and the reduced rotor inlet gas speeds combine to reduce the work capacity of the turbine stage, this may allow a better match of the torque curves for the two shaft components.   The larger mass flow also helps bring up the actual power of the turbine stage, since Power = mass flow X specific work.  So there are trades to be made. 


The above discussion is just to give more insight to the process of choosing how you want to design the turbine stage and where you choose to design for choke.

[finiteparts]

A final illustration of how choking in the relative frame is impacted by rotor speeds.   If you look at almost any compressor map, you will see that the speed lines drop vertically downward at different corrected mass flows.



The reason for this is that the local gas conditions at the inducer throat choke plane are changing as the rotor speed changes.   This is similar to why the rotor choking in the turbine exducer changes relative to rotor speeds. 

I was able to find two maps to show the difference in the behavior of a choked nozzle verses a choked rotor.   As can be seen, the choked nozzle collapses to a single maximum corrected mass flow, while the choked rotor is dependent on the rotor speed.   Another point of interest is that the corrected mass flow reduces as the rotor speed is increased.  This is due to the centrifugal force increasingly opposing the radially inward pressure gradient as the rotor speed increases.  If they extrapolated the curve further downward, it would not cross the mass flow = 0 at a PR =1 because the centrifugal force, it is actually zero at a PR >1 and negative flow below that till it hits mass flow = 0.





These can be found in here:   ntrs.nasa.gov/citations/19950015924

Next, I thought we could walk through the flow processes for a design with a choked nozzle as the PR is increased...more to come.

[racket]

Hi Chris

Some interesting reading :-)

I like your RMS and torque production , its sorta what I've been trying to sort out for a long time , it must be ~15 years ago when I was trying to help someone with an engine that I tried this method ...................

Firstly the power required to drive the comp, simple calc fortunately

Secondly what gas deflection required when varying blade speeds were used ( inducer/exducer differences) .

I worked on guessing a reasonable mean blade position taking into account inducer tip speed and your RMS speed to produce the mean torque position, sorta like mid blade on an axial wheel when doing calcs

Then in conjunction with the velocity triangles I found positive and negative amounts of swirl from that mean torque position , which when calculated out produced either enough or not enough overall gas deflection to produce the comps power requirement .

It was a crude way of doing things but it helped when there was no information available on radial wheels , plenty on axial ones though , so I "mutilated" that info to fit the radial :-)

Looking forward to more .

Cheers
John

[monty]

Chris,

Corrected mass flow can be deceptive. I really don't understand why people persist in using it. It's useful for comparing components, and that's about it. For design it's kind of useless.

The turbine choked exducer corrected flow graph might lead you to believe the right thing to do is run a super low pressure ratio and low speed. Of course the only thing that matters is actual mass flow, and how much pressure remains for the thrust nozzle.

It might also lead you to believe that lower rpm is better for radial turbines. This is not the case. The lower the turbine tip speed, the lower the NGV discharge speed must be to generate proper inflow incidence angles. Choked NGV condition for these turbocharger turbines isn't really ideal. The tip speed/exducer speed ratio is too small. Increasing the turbine work coeff only gets you so far for a jet. At some point you start to sacrifice (actual!!) mass flow due to turbine inlet loss. Somewhere around -20deg incidence is the limit for good inflow conditions. -30ish is usually ideal. For a power turbine stage like a turboshaft -15 or so might be acceptable, but the flow coefficients will need to be lower. The turbine will be acting more like an impulse stage, and the flow will be restricted by the vortex that forms in the inlet passage. For this to work well, the blades would need to curve back toward the NGV...not good for structural considerations.

I think there is some performance still to be had by properly matching the compressor to the turbine. Probably will need to use a larger turbine or smaller comp wheel. Of course the improvement may only be in specific fuel consumption not max thrust.

I must admit to being tempted to quit fooling with the radial turbines and just make an axial turbine specifically for this. And design for a choked NGV....

Monty

[finiteparts]

Monty,

Corrected mass flow is fundamental to making maps and is definitely something that we can't do without. The reason that people persist to use it is that if you just tried to create any map, you would need to have a map for every change in inlet temperature or pressure. To collapse all that data on a SINGLE map, you need to use corrected parameters. It is fundamental and I would say for design it is absolutely critical.

I read through the thread again and I want to make sure that I state something so that people do not go down the wrong path:

• You figure out the choking conditions to set the upper bound on where you intend to run.
• It is not suggested that the hobbyist attempt to design an engine to run in full choked NGV or rotor as the tools to understand the shock losses are complicated and likely the calculated performance will be drastically different to what happens in the real world.

Ok, now to the flow maps.   It should be stated that what matters in turbine matching is that the three things are conserved:

1. Conservation of mass:   The mass flow through the compressor, plus the fuel mass flow must be what goes through the turbine
2. Conservation of energy:  The power that is require by the compressor, plus the parasitic losses due to mechanical/aero losses must be produced by the turbine stage
3. The compressor and the turbine must turn at the same rpm (unless geared)

The maps shown only really show the corrected swallowing capacity of the turbine for a given pressure ratio.  Any assumptions on rpm or mass flow selection require additional calculations. 

I totally agree with you on the incidence, but there is a lot going on and the incidence angle is only a part of the picture.  The NGV discharge angle may be better thought of as a means to move the location of the peak efficiency in the operating space.   There is an optimal placement which gives the widest efficiency islands on a different kind of turbine map, the corrected speed vs corrected enthalpy style map.

I have been working on my engine and I will use some of the plots to illustrates this.   These are NOT corrected flows, power or rpm, because I am looking at a specific firing temperature and thus I am comparing for that inlet point.  These values are derived from selecting an input condition and then finding the physical parameters from the corrected maps that were generated by software that I have been working on. 

As with any software, it needs to be validated somehow and to that end, I have run several NASA rotors (that have available information) and it predicts the corrected behavior well.  So I feel pretty comfortable with the results thus far.  It is still a work in progress, but it gives me the ability to generate radial turbine maps, with losses included (incidence, windage, trailing edge blockage losses, etc.).  I am still working the centrifugal compressor portion that will allow me to specify a mechanical loss and then match the compressor and turbine using the above conservation "laws".

With a fixed rotor geometry and a fixed inlet set of conditions, the physical flow maps are shown below:

This case has a 66 deg NGV discharge angle (66 deg from radial).  This is the widest throat that I ran in the model.   As can be seen, the rotor discharge is the controlling area and chokes at different mass flows for different rotor speeds.


The next case has the NGV discharge angle set to 68 deg.   Similar setup, choking in the rotor discharge, just a smaller spread...


Finally, this case has the NGV discharge angle set to 72 deg.  This chokes in the NGV as they all collapse on a single mass flow value.  This is the smallest mass flow.


As can be seen, tightening up the NGV throat does exactly as would be expected; it reduces the mass flow through the stage. You can also see that by taking a larger portion of the pressure drop in the NGV (tight NGV), it chokes at a higher total to static pressure ratio (PRts).  As the NGV angle is reduced (opening up the NGV) more of the acceleration occurs in the rotor, it chokes at lower PRts.

Now if we look at the power produced across the rpm range and the total to static efficiency that the stage achieves, we get more information.  In the first case, the 66 deg NGV, we can see that the Eff_t-s island is smaller than the next image of the 68 deg NGV.   What can be seen in these three images is that the tightening up the NGV (larger angle) moves the peak efficiency island over toward the higher rpm portion of the map.

  66 deg NGV

This illustrates the main idea that I am trying to share here.   The NGV angle not only sets the mass flow through the stage, it controls were on the map the turbine runs efficiently.   Higher angles generally produce wider torque curves with lower torque peaks.  Generally, higher NGV angles produce choked conditions at much higher PRts.  Higher NGV angles generally produce their peak efficiencies at higher pressure ratios and the stage limit load is reached at higher overall PRts.

  68 deg NGV

  72 deg NGV


The other thing illustrated here is that small changes in NGV angles have relatively large effects.  There is a lot of things to consider and for my design, I am still leaning toward having the NGV throat control the flow and not the rotor exit.  Being able to get to high PR is a goal for me because I want to generate shock diamonds in a small AB one day.

I hope this was helpful,
Chris

[finiteparts]

Monty,

I also agree with you on the idea of looking into axial turbine stages. The rotor weight benefit might outweigh the lower per stage specific power.

Ford was looking into using axial turbine stages in turbochargers, as a means to reduce the polar moment of inertia and thus the turbo lag. Here is a link to the paper as a point of interest.

gasturbinespower.asmedigitalcollection.asme.org/GT/proceedings/GT1985/79382/V001T04A001/235954

They stated:

"It should be emphasized that competitive efficiencies in the axial turbines can only be obtained at moderate levels of pressure ratio or boost. At high pressure ratios the efficiency of the axial turbine will start to deteriorate because of excessively high aerodynamic stage loadings. A prohibitively expensive multi-stage axial turbine would be required to obtain good efficiency at high pressure ratios."

Now, I will stipulate that they were trying to minimize the blade count to keep the inertia as low as possible, which means they were, by default, increasing the blade loading. So, I am not sure where the tipping point in blade count is; there is a point were they get more competitive at higher PRs. They will always be lower in specific power, but at some size and flow, they become more attractive.  The lure of relatively cheap and available radial turbines is what always holds me back.

- Chris

[monty]

Chris,

I didn't mean to imply corrected numbers are totally useless. For maps and comparing components they are useful. What I was getting at is for design:

Step 1: Mission analysis (what is needed from the engine?)
Step 2: Cycle design (what can be done thermodynamically given material and component efficiency limits?)
Step 3: Assign a mass flow to meet step 1 given step 2
Step 4: Design individual components for actual mass flow and cycle numbers.

For this process, corrected flows are not even used. To my way of thinking it's just a bunch of extra steps that confuses people. So when giving numbers for a certain engine, actual mass flows are more useful (to me).

Those maps are instructive. My spreadsheet only looks at 1 design point. I set the NGV exit mach# and the angle and area follow from that. The work requirement can be met over a wide range of NGV mach#s. I have been adjusting to get the turbine inlet incidence in the desirable range. This normally tends towards the higher end of your map. For a jet this makes sense. Jets typically operate 75-100% of design speed. For a turboshaft, you might want to skew towards the middle. The interesting thing is the exducer sets the exit swirl, work and flow coefficients of the turbine. The NGV just tunes the map and inlet conditions.

I have done a cursory look at axial turbines. Especially given the expense and difficulty of obtaining the larger TF15 wheels at the moment. Unfortunately, the Delta T available from the axial wheels is not as good as the radial turbines. You would need two stages to equal a radial wheel. Especially if you want to keep exit swirl down. Of course an impulse stage could do a bit better, but with lower efficiency and a lot of exit swirl. Going to an axial wheel would mean keeping pressure ratios in the 3.8-4 range for a jet and would limit stage work in a turboshaft. Casting an NGV and axial wheel for our size engine would be about $500/piece in short runs. Then the wheel needs to be X-rayed and the components machined....so at least $1500 a set no markup, no development costs....For these small engines......it's hard to beat the radial inflow turbine. The Ford paper just confirms this.

Monty