Abstract
The improvement of existing turbines requires time-consuming computations that often limit the number of parameters that can be optimized. To address this challenge, this study uses through-flow code, which is about 200 times faster than 3D Computational Fluid Dynamics (CFD), to optimize a two-stage axial turbine with 112 geometrical parameters using a deep learning surrogate model. The surrogate model is a Convolutional Neural Network (CNN) that predicts the flow field and performance indicators of the turbine based on an input database generated by an airfoil generation method and an in-house through-flow code. The surrogate model has two components: a flow field prediction component and a performance prediction component. The network architecture is named conv2D-conv3D, as it employs 2D convolutional layers to map the geometry to the flow field and 3D convolutional layers to extract features from the flow field and output the performance indicators. The optimal hyperparameters of the network are determined by comparing different activation functions, batch sizes, and train-data sizes. The network achieves high accuracy (\({R}^{2}> 0.93\)) even with a small fraction of the dataset (30% of data) and it is more than \({10}^{5}\) times faster than the through-flow code. A vectorized genetic algorithm is applied to the surrogate model to maximize the efficiency and power of the turbine under a constant mass flow rate constraint. The optimization results are validated by through-flow and 3D CFD simulations of the baseline and optimized turbine. The efficiency and power are increased by 0.83% and 1.02%, respectively.
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Abbreviations
- \(\dot{\mathrm{m}}\) :
-
Mass flow rate \([\frac{\mathrm{kg}}{\mathrm{s}}]\)
- \(\mathrm{h}\) :
-
Specific enthalpy \([\frac{\mathrm{J}}{\mathrm{kg}.}]\)
- \(\mathrm{I}\) :
-
Rothalpy \([\frac{\mathrm{J}}{\mathrm{kg}}]\)
- \(\mathrm{m}\) :
-
Meridional direction \(\left[\mathrm{m}\right]\)
- \(\mathrm{M}\) :
-
Mach number [−]
- \({\mathrm{S}}_{\mathrm{i}}\) :
-
Defining point coordinates \(\left[\mathrm{m}\right]\)
- \(\mathrm{q}\) :
-
Quasi orthogonal direction [m]
- V:
-
Absolute velocity \(\left[\frac{\mathrm{m}}{\mathrm{s}}\right]\)
- \({\mathrm{R}}_{\mathrm{c}}\) :
-
Streamline local radius \(\left[\mathrm{m}\right]\)
- \(\mathrm{s}\) :
-
Specific entropy \([\frac{\mathrm{J}}{\mathrm{kg}..\mathrm{K}}]\)
- \(\mathrm{U}\) :
-
Blade velocity \(\left[\frac{\mathrm{m}}{\mathrm{s}}\right]\)
- \(\mathrm{N}\) :
-
Number of blades \([-]\)
- \({\mathrm{P}}_{\mathrm{i}}\) :
-
Optional point coordinates \(\left[\mathrm{m}\right]\)
- r:
-
Section radius of the airfoil \(\left[\mathrm{m}\right]\)
- \(\mathrm{P}\) :
-
Pressure [Pa]
- 0:
-
Stagnation condition
- 1:
-
Leading edge
- 2:
-
Trailing edge
- \(\mathrm{g}\) :
-
Gauging
- tt:
-
Total-to-total
- \(\beta\) :
-
Camber line angle [∘]
- \(\upphi\) :
-
Meridional slope angle [∘]
- \(\upgamma\) :
-
Quasi orthogonal angle, stagger angle [∘]
- \(\Delta \beta\) :
-
Wedge angle [∘]
- \(\upeta\) :
-
Efficiency \([-]\)
- \(\mathrm{\alpha }\) :
-
Flow angle [∘]
- \(\uprho\) :
-
Density \([\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}]\)
References
Ashouri M, Khaleghian S, Emami A (2022) Reduced-order modeling of conductive polymer pressure sensors using finite element simulations and deep neural networks. Struct Multidisc Optim 65(5):146
Ates GC, Gorguluarslan RM (2021) Two-stage convolutional encoder-decoder network to improve the performance and reliability of deep learning models for topology optimization. Struct Multidisc Optim 63(4):1927–1950
Aungier RH (2006) Turbine aerodynamics: axial-Flow and Radial-Flow turbine design and analysis. In: ASME Press eBooks. https://doi.org/10.1115/1.802418
Catalani G, Costero D, Bauerheim M, Zampieri L, Chapin V, Gourdain N, Baqué P (2023) A comparative study of learning techniques for the compressible aerodynamics over a transonic RAE2822 airfoil. Comput Fluids 251:105759
Chaquet JM, Corral R, Fernandez A (2017) Accurate method to reproduce throughflow results with a meanline solver. in turbo expo: power for land, sea, and air. American Society of Mechanical Engineers, New York
Chen L-W, Thuerey N (2023) Towards high-accuracy deep learning inference of compressible flows over aerofoils. Comput Fluids 250:105707
Denton JD (1992) The calculation of three-dimensional viscous flow through multistage turbomachines. J Turbomach 114(1):18–26
Du Q et al (2022) Performance prediction and design optimization of turbine blade profile with deep learning method. Energy 254:124351
Duru C, Alemdar H, Baran OU (2022) A deep learning approach for the transonic flow field predictions around airfoils. Comput Fluids 236:105312
Eivazi H, Veisi H, Naderi MH, Esfahanian V (2020) Deep neural networks for nonlinear model order reduction of unsteady flows. Phys Fluids 32(10):105104
Feng Y, Song X, Yuan W, Lu H (2023) Physics-informed deep learning cascade loss model. Aerosp Sci Technol 134:108165
Hendrycks, D. and K. Gimpel, Bridging nonlinearities and stochastic regularizers with gaussian error linear units. 2016.
High-Efficiency Gas Turbines Will Play a Growing Role in the Energy Transition. 2018; Available from: https://www.ge.com/power/transform/article.transform.articles.2018.sep.high-efficiency-gas-turbines.
Hu H, Song Y, Yu J, Liu Y, Chen F (2022) The application of support vector regression and virtual sample generation technique in the optimization design of transonic compressor. Aerosp Sci Technol 130:107814
Jia R, Xia H, Zhang S, Su W, Xu S (2022) Optimal design of Savonius wind turbine blade based on support vector regression surrogate model and modified flower pollination algorithm. Energy Convers Manag 270:116247
Karimi MS, Raisee M, Salehi S, Hendrick P, Nourbakhsh A (2021) Robust optimization of the NASA C3X gas turbine vane under uncertain operational conditions. Int J Heat Mass Transf 164:120537
Karthikeyan T, Avital E, Nithya V, Abdus S (2019) Optimization of a horizontal axis marine current turbine via surrogate models. Ocean Syst Eng. https://doi.org/10.12989/ose.2019.9.2.111
Kingma, D.P. and J. Ba, Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
Li X, Zhang W (2022) Physics-informed deep learning model in wind turbine response prediction. Renewable Energy 185:932–944
Li Z, Zheng X (2017) Review of design optimization methods for turbomachinery aerodynamics. Prog Aerosp Sci 93:1–23
Li J, Wang Y, Qiu Z, Zhang D, Xie Y (2023) Fast performance prediction and field reconstruction of gas turbine using supervised graph learning approaches. Aerosp Sci Technol 14:108425
Liu T, Li Y, Jing Q, Xie Y, Zhang D (2021) Supervised learning method for the physical field reconstruction in a nanofluid heat transfer problem. Int J Heat Mass Transf 165:120684
Lui YH, Shahriar M, Pan Y, Hu C, Hu S (2022) Surrogate modeling of acoustic field-assisted particle patterning process with physics-informed encoder–decoder approach. Struct Multidisc Optim 65(11):333
Luo J, Fu Z, Zhang Y, Fu W, Chen J (2023) Aerodynamic optimization of a transonic fan rotor by blade sweeping using adaptive Gaussian process. Aerosp Sci Technol 137:108255
Martin I, Hartwig L, Bestle D (2019) A multi-objective optimization framework for robust axial compressor airfoil design. Struct Multidisc Optim 59:1935–1947
Misaka T (2020) Image-based fluid data assimilation with deep neural network. Struct Multidisc Optim 62(2):805–814
Mohammadi-Ahmar A et al (2022) Model order reduction for film-cooled applications under probabilistic conditions: sparse reconstruction of POD in combination with Kriging. Struct Multidisc Optim 65(10):283
Novak RA (1967) Streamline curvature computing procedures for Fluid-Flow problems. J Eng Power 89(4):478–490. https://doi.org/10.1115/1.3616716
Osseyran A, Giles M (2015) Industrial applications of high-performance computing: best global practices, vol 25. CRC Press, Boca Raton
Persico G, Rebay S (2012) A penalty formulation for the throughflow modeling of turbomachinery. Comput Fluids 60:86–98
Raul V, Leifsson L (2021) Surrogate-based aerodynamic shape optimization for delaying airfoil dynamic stall using Kriging regression and infill criteria. Aerosp Sci Technol 111:106555
Salviano LO et al (2021) Sensitivity analysis and optimization of a CO 2 centrifugal compressor impeller with a vaneless diffuser. Struct Multidisc Optim 64:1607–1627
Shi D, Sun L, Xie Y (2020) Off-design performance prediction of a S-CO2 turbine based on field reconstruction using deep-learning approach. Appl Sci 10(14):4999
Sobol IM (1967) On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput Math Math Phys 7(4):86–112
Tiwari P, Stein A, Lin Y-L (2013) Dual-solution and choked flow treatment in a streamline curvature throughflow solver. J Turbomach 135(4):041004
Wagner, F., A. Kühhorn, and R. Parchem. Robust Design Optimization Applied to a High Pressure Turbine Blade Based on Surrogate Modelling Techniques. in ASME Turbo Expo 2015: Turbine Technical Conference and Exposition. 2015.
Wang Y, Liu T, Zhang D, Xie Y (2021a) Dual-convolutional neural network based aerodynamic prediction and multi-objective optimization of a compact turbine rotor. Aerosp Sci Technol 116:106869
Wang Q, Yang L, Rao Y (2021b) Establishment of a generalizable model on a small-scale dataset to predict the surface pressure distribution of gas turbine blades. Energy 214:118878
Wang Q, Zhou W, Yang L, Huang K (2022) Comparison between conventional and deep learning-based surrogate models in predicting convective heat transfer performance of U-bend channels. Energy and AI 8:100140
Wang Z, Liu X, Yu J, Wu H, Lyu H (2023) A general deep transfer learning framework for predicting the flow field of airfoils with small data. Comput Fluids 251:105738
Whitney, W.J., H.J. Schum, and F.P. Behning. Cold-air investigation of a turbine for high-temperature-engine application. 4: Two-stage turbine performance. 1972.
Zhao X, Gong Z, Zhang J, Yao W, Chen X (2021) A surrogate model with data augmentation and deep transfer learning for temperature field prediction of heat source layout. Struct Multidisc Optim 64(4):2287–2306
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This paper has provided all the essential information to obtain the turbine optimization data and the deep learning surrogate model that can reproduce the results. The results can be replicated by following this information. The Python code to reproduce the results, including the conv2D-conv3D network, is also available from the corresponding author upon reasonable request.
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Esfahanian, V., Izadi, M.J., Bashi, H. et al. Aerodynamic shape optimization of gas turbines: a deep learning surrogate model approach. Struct Multidisc Optim 67, 2 (2024). https://doi.org/10.1007/s00158-023-03703-9
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DOI: https://doi.org/10.1007/s00158-023-03703-9