Abstract

Ozone depletion and global warming are major drivers for the development of cooling systems. Accordingly, new governmental regulations for the equipment are continuously introduced. The redesign of corresponding centrifugal compressors for new refrigerants is therefore necessary and remains a very challenging task. The vicinity to the two-phase region requires accurate real gas models, and the complexity of the flow patterns a robust numerical integration of the underlying governing equations. This is particularly true for the considered low flow coefficient two-stage centrifugal compressor. Classical design and optimization methods are limited in their flexibility because of the geometry parametrization, thus affecting the development of the next generation of turbo compressors. This article describes the development and application of a fully coupled, pressure-based computational fluid dynamics (CFD) framework, incorporating a highly flexible discrete adjoint method used for redesign and optimization purposes. The maturity of the underlying CFD and of the optimization algorithm makes it possible to account for real gas and turbulence effects, as well as multiple mixing-plane stage interfaces. Accordingly, it is possible to fully exploit the major advantage of this gradient-based optimization strategy, the independency between computational effort and the number of degrees-of-freedom in the geometrical description. For a two-stage centrifugal compressor, the return channel is optimized to achieve an overall reduction in entropy generation. Although based on a single-point optimization, the resulting performance characteristic shows a consistent improvement over the whole operating range with maximum values of up to 2%. The resulting geometry is manufactured and experimentally tested on the test rig. The computationally obtained efficiency increase can be confirmed by the experimental data.

References

1.
Forsgren
,
E.
,
2020
, “Evaluation of an Economizersolution for a Heat Pump.”
2.
Brunner
,
C.
,
Moser
,
T.
,
Spillmann
,
J.
,
Gloor
,
R.
, and
Tieben
,
R.
,
2016
, “Druckluft-Kompressoren.”
3.
Casey
,
M. V.
,
Dalbert
,
P.
, and
Schurter
,
E.
,
1990
, “
Radial Compressor Stages for Low Flow Coefficients
,” IMechE Fourth European Congress, Fluid Machinery for the Oil, Petromechanical and Related Industries, The Hague, The Netherlands, May.
4.
Lettieri
,
C.
,
Baltadjiev
,
N.
,
Casey
,
M.
, and
Spakovszky
,
Z.
,
2014
, “
Low-Flow-Coefficient Centrifugal Compressor Design for Supercritical CO2
,”
ASME J. Turbomach.
,
136
(
8
), p.
081008
.
5.
Fleischli
,
B.
,
Mangani
,
L.
,
Del Rio
,
A.
, and
Casartelli
,
E.
,
2021
, “
A Discrete Adjoint Method for Pressure-Based Algorithms
,”
Comput. Fluids
,
227
, p.
105037
.
6.
Mueller
,
L.
, and
Verstraete
,
T.
,
2017
, “
CAD Integrated Multipoint Adjoint-Based Optimization of a Turbocharger Radial Turbine
,”
Int. J. Turbomach. Propul. Power
,
2
(
3
), p.
14
.
7.
Wang
,
D. X.
, and
He
,
L.
,
2010
, “
Adjoint Aerodynamic Design Optimization for Blades in Multistage Turbomachines—Part I: Methodology and Verification
,”
ASME J. Turbomach.
,
132
(
2
), p.
021011
.
8.
Walther
,
B.
, and
Nadarajah
,
S.
,
2015
, “
Adjoint-Based Constrained Aerodynamic Shape Optimization for Multistage Turbomachines
,”
J. Propul. Power
,
31
(
5
), pp.
1298
1319
.
9.
Ma
,
C.
,
Su
,
X.
, and
Yuan
,
X.
,
2016
, “
An Efficient Unsteady Adjoint Optimization System for Multistage Turbomachinery
,”
ASME J. Turbomach.
,
139
(
1
), p.
011003
.
10.
Rodrigues
,
S. S.
, and
Marta
,
A. C.
,
2018
, “
Adjoint Formulation of a Steady Multistage Turbomachinery Interface Using Automatic Differentiation
,”
Comput. Fluids
,
176
, pp.
182
192
.
11.
Vitale
,
S.
,
Pini
,
M.
, and
Colonna
,
P.
,
2020
, “
Multistage Turbomachinery Design Using the Discrete Adjoint Method Within the Open-Source Software SU2
,”
J. Propul. Power
,
36
(
3
), pp.
465
478
.
12.
Fleischli
,
B.
,
Del Rio
,
A.
,
Casartelli
,
E.
,
Mangani
,
L.
,
Mullins
,
B. F.
,
Devals
,
C.
, and
Melot
,
M.
,
2021
, “
Application of a General Discrete Adjoint Method for Draft Tube Optimization
,”
IOP Conf. Ser.: Earth Environ. Sci.
,
774
(
1
), p.
012012
.
13.
Del Rio
,
A.
,
Casartelli
,
E.
,
Fleischli
,
B.
, and
Mangani
,
L.
,
2022
, “
New Concept for Design in Turbomachinery Applications Using Full RANS Gradient Methodology
,”
ASME J. Turbomach.
,
145
(
4
), p.
041009
.
14.
Mangani
,
L.
,
Darwish
,
M.
, and
Moukalled
,
F.
,
2014
Development of a Pressure-Based Coupled CFD Solver for Turbulent and Compressible Flows in Turbomachinery Applications
,” Vol. 2B: Turbomachinery, Düsseldorf, Germany, p.
V02BT39A019
.
15.
Mangani
,
L.
,
Casartelli
,
E.
, and
Darwish
,
M.
,
2019
, “
Coupled Pressure Based CFD Solver for Turbomachinery Flows: Overview of Applications
,” 13th European Conference on Turbomachinery Fluid dynamics & Thermodynamics, Lausanne, Switzerland, April,
European Turbomachinery Society
.
16.
Hanimann
,
L.
,
Mangani
,
L.
,
Casartelli
,
E.
,
Mokulys
,
T.
, and
Mauri
,
S.
,
2014
, “
Development of a Novel Mixing Plane Interface Using a Fully Implicit Averaging for Stage Analysis
,”
ASME J. Turbomach.
,
136
(
8
), p.
081010
.
17.
Hanimann
,
L.
,
Mangani
,
L.
,
Casartelli
,
E.
,
Vogt
,
D. M.
, and
Darwish
,
M.
,
2020
, “
Real Gas Models in Coupled Algorithms Numerical Recipes and Thermophysical Relations
,”
Int. J. Turbomach. Propul. Power
,
5
(
3
), p.
20
.
18.
Hanimann
,
L.
,
2022
,
Numerical Investigation of Stationary and Instationary Two Phase Flows in Low Pressure Steam Turbines
,
Berichte aus der Strömungstechnik, Shaker Verlag
,
Düren
.
19.
Peng
,
D.-Y.
, and
Robinson
,
D. B.
,
1976
, “
A New Two-Constant Equation of State
,”
Ind. Eng. Chem. Fundam.
,
15
(
1
), pp.
59
64
.
20.
Boncinelli
,
P.
,
Rubechini
,
F.
,
Arnone
,
A.
,
Cecconi
,
M.
, and
Cortese
,
C.
,
2004
, “
Real Gas Effects in Turbomachinery Flows: A Computational Fluid Dynamics Model for Fast Computations
,”
ASME J. Turbomach.
,
126
(
2
), pp.
268
276
.
21.
Kunick
,
M.
, and
Kretzschmar
,
H. J.
,
2015
, “Guideline on the Fast Calculation of Steam and Water Properties With the Spline-Based Table Look-Up Method (SBTL),”
The International Association for the Properties of Water and Steam
,
Moscow, Russia
, Technical Report No. G13-15
22.
Post
,
P.
, and
di Mare
,
F.
,
2018
, “
Highly Efficient Euler-Euler Approach for Condensing Steam Flows in Turbomachines
,” Proceedings of Montreal, Montreal Canada, Global Power and Propulsion Society.
23.
Aràndiga
,
F.
,
2016
, “
A Nonlinear Algorithm for Monotone Piecewise Bicubic Interpolation
,”
Appl. Math. Comput.
,
272
, pp.
100
113
, Subdivision, Geometric and Algebraic Methods, Isogeometric Analysis and Refinability.
24.
Lucian
,
H.
,
Fleischli
,
B.
,
Casartelli
,
E.
,
Mangani
,
L.
,
Lehr
,
A.
, and
Weickgenannt
,
A.
,
2023
, “Adjoint Optimization of Real Gas Centrifugal Compressor,” https://www.euroturbo.eu/publications/proceedings-papers/ETC2023-317/
25.
Bücker
,
M.
,
Hovland
,
P.
,
Naumann
,
U.
, and
Norris
,
B.
,
2006
,
Automatic Differentiation: Applications, Theory, and Implementations
, 1st ed.,
Springer
,
Berlin, Heidelberg
.
26.
Asouti
,
V. G.
,
Zymaris
,
A. S.
,
Papadimitriou
,
D. I.
, and
Giannakoglou
,
K. C.
,
2008
, “
Continuous and Discrete Adjoint Approaches for Aerodynamic Shape Optimization With Low Mach Number Preconditioning
,”
Int. J. Numer. Methods Fluids
,
57
(
10
), pp.
1485
1504
.
27.
Saad
,
Y.
, and
Schultz
,
M. H.
,
1986
, “
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
,”
SIAM J. Sci. Stat. Comput.
,
7
(
3
), pp.
856
869
.
28.
Saad
,
Y.
,
2003
,
Iterative Methods for Sparse Linear Systems
, 2nd ed.,
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
.
29.
Rhie
,
C. M.
, and
Chow
,
W. L.
,
1983
, “
Numerical Study of the Turbulent Flow Past an Airfoil With Trailing Edge Separation
,”
AIAA J.
,
21
(
11
), pp.
1525
1532
.
30.
He
,
P.
,
Mader
,
C. A.
,
Martins
,
J. R. R. A.
, and
Maki
,
K. J.
,
2018
, “
An Aerodynamic Design Optimization Framework Using a Discrete Adjoint Approach With OpenFOAM
,”
Comput. Fluids
,
168
, pp.
285
303
.
31.
He
,
P.
,
Filip
,
G.
,
Martins
,
J. R. R. A.
, and
Maki
,
K. J.
,
2019
, “
Design Optimization for Self-Propulsion of a Bulk Carrier Hull Using a Discrete Adjoint Method
,”
Comput. Fluids
,
192
, p.
104259
.
32.
Böhm
,
W.
,
Farin
,
G.
, and
Kahmann
,
J.
,
1984
, “
A Survey of Curve and Surface Methods in CAGD
,”
Comput. Aided Geom. Des.
,
1
(
1
), pp.
1
60
.
33.
Luke
,
E.
,
Collins
,
E.
, and
Blades
,
E.
,
2012
, “
A Fast Mesh Deformation Method Using Explicit Interpolation
,”
J. Comput. Phys.
,
231
(
2
), pp.
586
601
.
34.
Lemmon
,
E. W.
,
Bell
,
Ian H.
,
Huber
,
M. L.
, and
McLinden
,
M. O.
,
2018
, “NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 10.0, National Institute of Standards and Technology,” https://www.nist.gov/srd/refprop
35.
Menter
,
F. R.
,
1994
, “
Two-equation Eddy-Viscosity Turbulence Models for Engineering Applications
,”
AIAA J.
,
32
(
8
), pp.
1598
1605
.
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