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Appl. Sci. 2020, 10, 4999 5 of 21 where α∗ is the low Reynolds number correction coefficient for reducing turbulent eddy viscosity, c is the constant term of shear stress tensor, α0 is the empirical constant, and F2 is the mixed function. Based on a large number of physical characteristics data of carbon dioxide, the explicit equation of Helmholtz energy equation, the improved Benedict Webb Rubin (MBWR) state equation and the extended corresponding state (ECS) model were adopted. The MBWR state equation is as follows: 9 215 ρ n 2n−17 P= anρ +exp− ρ n=1 c n=10 anρ (5) where P is pressure, ρc is critical density, and an is characteristic parameters related to temperature. In this study, x = [Tin, Pin, α1, m. , ωR] is taken as the design variable of the turbine, including: inlet temperature Tin, inlet pressure Pin, inlet air flow angle α1, mass flow rate m. , and rotating speed ωR. The real result field f obtained by 3-D CFD analysis is as follows: f = Fcfd(x) = [P, T] (6) where P is pressure fields and T is the temperature fields. According to 3-D numerical results, the pressure and temperature distribution on the blade surface can be obtained. Additionally, then the performance of turbine ψ, power p, and efficiency η can be calculated based on fields information: ψ = Fper(f) = [p, η] (7) The torque of the turbine TR is obtained by solving the torque difference on the rotor blade surface between pressure side and suction side by integral method. The formula of power is as follows: p=Tω= rPdA − rPdAω (8) RR R ps ss where r is the radius, dA is the unit surface area, subscript ps is the rotor blade pressure surface and ss is the rotor blade suction surface. The total static efficiency of the turbine is: η=p (9) T−S m. ′ · [h(Pin, Tin) − h(S(Pin, Tin), Pout)] where m. ′ is the mass flow (obtained by numerical simulation), h is the enthalpy, S is the entropy, the subscripts in and out, respectively, represent the turbine inlet and outlet. 2.3. Deep Convolutional Neural Network In Figure 1, the architecture of the two-stage deep convolutional neural network composed by two stages, stage 1 employed as field reconstruction model and stage 2 employed as performance prediction model, is described in detail. In stage 1, the field reconstruction model with deconvolutional neural network is trained by minimum the loss function lstage1 between real and predicted fields. With the predicted fields from stage 1 as input, the performance prediction model with convolutional neural network is trained by minimum the loss function lstage2 between real and predicted performance. It should be emphasized that the lstage2 backward propagate through both the convolutional and deconvolutional neural networks if reconstructed fields are the input, while the lstage2 just backpropagate through the convolutional neural network with real fields as input.PDF Image | Performance Prediction of a S-CO2 Turbine
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