Advancements in Automated Methods for Fluid-Dynamic Turbomachinery Design

Abstract

Automated design methods are emerging as a powerful tool for the fluid-dynamic design of turbomachinery components. Such automated methods integrate mathematical models of different level of sophistication with numerical optimization techniques to explore large design spaces in a systematic way. This, in turn, allows the designer to achieve higher performance gains and shorten the development time with respect to traditional design workflows based on trial-and-error. In this context, the present thesis proposes a collection of models and methods for the preliminary and aerodynamic design optimization of turbomachinery that addresses some of the limitations of the design methods currently in use. With regards to the preliminary design phase, this work proposes a design optimization method for axial turbines with any number of stages. The method is based on a new mean-line model that accepts arbitrary equations of state to evaluate the thermodynamic properties of the fluid and empirical loss models to estimate the entropy generation. In addition, the kinetic energy recovered at the exit of the last stage is predicted using a new one-dimensional annular diffuser model based on the balance equations for mass, momentum, and energy. In contrast with existing methods, the preliminary design problem was formulated as a constrained optimization problem and solved using a gradient-based algorithm. This choice of optimization method allows the designer to: (1) integrate the turbine, diffuser, and loss models in a simple way by means of equality-constraints and (2) _nd the optimal solution of multi-stage design problems with tens of design variables at a low computational cost. The preliminary design method was applied to a case study and a sensitivity analysis revealed that there exists a locus of maximum efficiency in the specific speed and diameter plane (i.e, the Baljé diagram) that can be predicted with a simple analytical expression. Concerning the aerodynamic design phase, the present work proposes a unified geometry parametrization method based on computer-aided design (CAD) for axial, radial and mixed-ow turbomachinery blades. The method uses conventional engineering parameters (e.g., chord, metal angles, thickness distribution) and it exploits the mathematical properties of non-uniform rational basis spline (NURBS) curves and surfaces to produce blades with continuous curvature and rate of change of curvature. In addition, the method provides the sensitivity of the blade coordinates with respect to the design variables by means of the complex-step method, allowing the integration of the parametrization into automated, gradient-based shape optimization workflows. The proposed parametrization also allows one to replicate the geometry of an existing blade given by scattered point coordinates by solving a two-step optimization problem. The capabilities of this reverse engineering strategy were demonstrated by replicating the geometry of eight turbomachinery blades in two and three dimensions with an accuracy comparable to the tolerances of current manufacturing technologies. Furthermore, this thesis proposes an aerodynamic design method for turbomachinery blades operating under non-ideal thermodynamic conditions. The proposed method supports the simultaneous optimization of multiple blade rows in two dimensions and it relies on a new gradient-based shape optimization framework that integrates the proposed CAD-based parametrization with a Reynolds-Averaged Navier-Stokes (RANS) solver and its discrete adjoint counterpart. The aerodynamic design method developed in this work offers three main advantages with respect to other design systems: (1) the real-gas ow solver enables the optimization of unconventional turbomachinery (e.g., organic Rankine cycle turbines, supercritical carbon dioxide compressors) in which the fluid properties deviate from ideal gas behavior, (2) the discrete adjoint solver allows the designer to evaluate the cost function gradients at a computational cost that is essentially independent of the number of design variables, which, in turn, enables the exploration of large design spaces that would be untractable with gradient-free methods, and (3) compared with mesh-based parametrization methods, the CAD-based parametrization allows the designer to impose high-level geometric constraints, such as constant axial chord length, minimum trailing edge thickness, or smooth curvature distribution in a straightforward way. In order to demonstrate the capabilities of the automated design tools developed during this project, the proposed preliminary and aerodynamic design methods are applied to design a new single-stage axial turbine operating with isobutane (R600a) that is is going to be built and tested in the EXPAND facility at the Norwegian University of Science and Technology. The preliminary design method was successfully applied to design a turbine geometry and velocity triangles that maximize the total-to-total isentropic efficiency of the turbine and satisfy the technical constraints imposed by the EXPAND facility. In addition, the aerodynamic design method was used to define stator and rotor blade shapes that minimize the entropy generation within the turbine and satisfy the design specifications established during the preliminary design phase. In particular, the gradient-based shape optimization framework was able to reduce the entropy generation by 36%, relative to the baseline geometry, which corresponds to a total-to-total isentropic efficiency increase of about 4 percentage points. Furthermore, the aerodynamic optimization did not only produce a quantitative improvement in performance, but also caused qualitative changes in the ow _eld. Most notably, the baseline stator cascade featured a trailing edge shock pattern and a shock-induced separation bubble that were eliminated as a result of the optimization.