Design and optimization of turbomachinery disks

Abstract

Turbomachines, ranging from turbopumps in liquid-propellant rocket engines to turbofan, turbojet, and turboshaft gas turbine engines in aviation, as well as steam/gas turbines and turbochargers, are essential technologies to our lives. The development of various types of turbomachinery is a fundamental aspect of an independent and prosperous country. Turbomachines are rotating devices that either add energy to a fluid or extract energy from it. The rotating blades in a turbomachine generate centrifugal forces that must be carried by structures called disks. Due to their vital function and extreme operating conditions, disks are among the most critical turbomachinery components. The optimum design of turbomachinery disks is of high practical importance. In this thesis, a turbomachinery disk design and optimization framework is developed using two-dimensional axisymmetric disk templates. The disk shapes are defined by line and arc segments. The coordinates of the defining edge and center points of these segments are expressed directly in terms of disk design parameters. This method enables the generation of disk geometries mathematically without the use of any CAD software, eliminating the time-consuming and error-prone CAD file generation and export/import steps otherwise needed in a turbomachinery disk optimization routine. With this method, a fast and robust CAD-independent optimization workflow is achieved. The literature review shows that existing studies rely heavily on CAD-based parametric models or simplified geometric representations, and they use either one-dimensional finite-difference stress analysis or two-dimensional axisymmetric finite element analysis. The one-dimensional finite difference solution is noted for its speed and flexibility. Since disk geometry and loading are axisymmetric, two-dimensional axisymmetric finite element analysis provides accurate stress solutions with highly reduced element and degree-of-freedom numbers in comparison to three-dimensional analysis. In this work, turbomachinery disk optimization is formulated as a multi-objective problem, simultaneously minimizing disk mass and maximum equivalent stress. Non-dominated Sorting Genetic Algorithm II (NSGA-II) is employed due to its effectiveness in large design spaces and yielding a set of Pareto-optimal solutions rather than a single optimum design. Two stress-evaluation methods are included: the one-dimensional finite-difference method is included due to its potential for quickly exploring large design spaces; and the two-dimensional axisymmetric finite element method is included for high-accuracy stress calculation in smaller design spaces. Disk mass is obtained using Pappus' second theorem. A new turbopump turbine disk template is developed along with another template for the well-known web type disk from literature. The templates are established with defining point coordinates expressed explicitly in terms of design parameters. The availability of point coordinates enables geometric shape feasibility constraints to be defined directly through coordinate comparisons. Design parameters of the templates are physical features that can be easily measured, such as radii, widths, and angles. Point coordinate equations are obtained systematically, beginning with dependencies involving fewer parameters and then building up to more complex relationships. Geometric relations such as tangency of lines to arcs and coincidence of points on lines and arcs are used in point coordinate calculations leading to trigonometric expressions. During optimization, the initial population is generated by randomly sampling design parameter values within the predefined bounds. The candidate disks resulting from the random sampling are evaluated for geometric shape feasibility and accepted into the population only if the resulting geometry is valid. For each valid disk, mass and maximum equivalent stress are calculated. For the completed population, the NSGA-II operations of ranking and distance calculation are performed. The parent selection for next generation and associated crossover and mutation operations are also performed. Two case studies are carried out using the developed turbomachinery disk templates. In the first, the turbopump disk template is optimized with both finite difference and finite element stress solvers under identical loading and parameter ranges. Pareto-optimal disks produced by the finite difference method are then re-evaluated with finite element solver, revealing that the finite difference method underestimates stresses for certain geometries, allowing unrealistic disks to remain on the final Pareto frontiers. By contrast, the finite element-based optimization runs yield accurate and informative Pareto frontiers. In the second case study, the turbopump and web disk templates are compared within the same optimization problem using only the finite element method as stress solver. Both templates converge to very similar Pareto-optimal geometries, showing that the web disk template effectively reduces to the turbopump disk template under defined design space and loading conditions. Both templates are deemed suitable for use in a turbopump application. Finally, a disk selected from the finite element-based Pareto frontier was manufactured using additive manufacturing and tested in a spin rig test system. The disk successfully operated at thirty thousand revolutions per minute for sixty seconds without deformation or adverse effects. The result of this physical test demonstrates that the developed design and optimization framework can produce turbomachinery disks that perform reliably under realistic operating conditions.