Choking model

[RoberAgro]

Develop new equation-oriented choking model for compressors

• Choking at the inlet of the impeller
• Choking at the inlet of the splitte blades?
• Choking at the exit of the impeller?
• Choking at the inlet of the diffuser (only for vaned diffusers)

We should focus first on choking at the inlet of the impeller and vaned diffusers

• Is there something analogous to the supersonic expansion in supersonic turbines where the inlet flow angle at the inlet of the compressor has to be adapted to satisfy conservation of mass and the metal angle at the inlet of the impeller?
• Is choking related to the unique incidence phenomenon?
• Are there any relevant use cases where computing the compressor performance under choking is necessary?
• We need some cases to motivate the need to develop the model we are proposing
• Developing a meanline compressor model with choking is likely scientifically relevant and we could publish a paper describing such a modeling approach.

[RoberAgro]

Hi Lasse,

I agree with the thoughts in your note: Choking.pdf.

I think that the best way would be "Option 3 Minimize the Throat Residual" as this enables us to define a variable to that quantifies whether the compressor is choked or not (rather than the code just failing when there is no solution). I suggest naming the variable containing the minimum residual as "choking_margin".

Stations to consider
I suggest computing the choking residual at the following stations:

• Impeller inlet throat
• Vaned diffuser inlet throat

If the implementation works well I suggest to also implement a similar computation for:

• Impeller outlet throat
• Splitter blade inlet throat

Calculation method

• I suggest starting the implementation with an isentropic calculation so that the problem is a single-input optimization problem.
• As the problem appears to be convex, I think that the method scipy.optimize.minimize_scalar is guaranteed to converge in just a few iterations.
• Do you think we should:
  • Solve the optimization problem for each component at each iteration? or
  • Add the "residual minimization equation" to the overall set of residuals?
• I think that the isentropic approach is sufficient for now as it will be faster to implement, and we do not need to derive the Lagrangian problem for this "preliminary" version of the problem formulation.

[lasseband]

Then we agree. The problem is convex and will be easy to solve, as you say. I have made a dev script (dev_scripts/centrifugal_compressor/compressore_model/Choking/get_residual_minimum) that solves the problem using a root finder. My first idea was to include the residual to the overall set of equations. I suggest the following pros and cons regarding this approach:

• Good: Keeps the Equation-Oriented approach
• Bad: We will have the issue of nested finite difference
  where optimizing for each optimization is the opposite. I think it will be close to no difference in terms of computational time.

I have already implemented the first approach for the impeller inlet, and preliminary results suggest that the nested finite differences is making some issues. The code converges ok, but it depends on the relative step size, which is not very nice.

[RoberAgro]

I think we could keep the first approach (add residual to the overall set of equations) if you manage to find a step size that works well "in all cases". I think that central finite differences with a "big" relative step size of, let's say, 1e-3 could work well.

The reasons why I think we should try this approach are:

• Keeping the equation-oriented approach, which will be more similar to the final version of the code (also when we replace the mass flow rate boundary condition).
• The issue of nested finite differences will be meaningless once we implement automatic differentiation. Based on the progress Srinivas is making on differentiating the axial turbine code, I believe we will be able to differentiate the centrifugal compressor model in 1-2 weeks.

[lasseband]

I have implemented the method to check choking for both the impeller and vaned diffuser. It works well and I have no issues with convergence using a relative step for the finite difference of 1e-4.

[lasseband]

When minimizing the throat residual, the throat is not evaluated at the given conditions, but at critical condition. The mass flow rate residual ($1-\dot{m}^*/\dot{m}$) is negative when the given mass flow rate is less than the critical. However, some loss coefficients (specially choke, shock and entrance diffusion) depend on the throat condition. Problem as described in the choking pdf, the throat calculation will not have a solution, when the given mass flow rate is higher than the critical. Is there a nice way to evaluate the throat, only when the minimized throat residual is negative?

[RoberAgro]

Problem might be fixed by minimizing the smoothed absolute value of the residual

[lasseband]

I have implemented an additional choking_criterion that follows this approach and it works well. The problem converge to the right solution. However, the function is minimized through finding the zero of the derivative, and for certain initial guesses it finds the maximum in the middle of the two minima:

For this reason, I added the calculation of the second derivative, such that we can keep track whether it finds the minimum or maximum while we are getting a feel of how well it works.