Turbine Passage Design Methodology to Minimize Entropy Production—A Two-Step Optimization Strategy

Abstract

Rapid aerodynamic design and optimization is essential for the development of future turbomachinery. The objective of this work is to demonstrate a methodology from 1D mean-line-design to a full 3D aerodynamic optimization of the turbine stage using a parameterization strategy that requires few parameters. The methodology is tested by designing a highly loaded and efficient turbine for the Purdue Experimental Turbine Aerothermal Laboratory. This manuscript describes the entire design process including the 2D/3D parameterization strategy in detail. The objective of the design is to maximize the entropy definition of efficiency while simultaneously maximizing the stage loading. Optimal design trends are highlighted for both the stator and rotor for several turbine characteristics in terms of pitch-to-chord ratio as well as the blades metal and stagger angles. Additionally, a correction term is proposed for the Horlock efficiency equation to maximize the accuracy based on the measured blade kinetic losses. Finally, the design and performance of optimal profiles along the Pareto front are summarized, featuring the highest aerodynamic performance and stage loading.Researchers have used Bezier curves [3] or a combination of splines and b-splines to design airfoil geometries. B-splines have been proven to be a successful tool in reducing turbomachinery loss when coupled to an optimizer. However, they have limitations. Shelton et al. [4] stated that it was difficult to induce large changes to the stagger, trailing edge wedge and leading edge angle using b-splines. They instead used b-splines as a tool to fine-tune the design of the blade. Ghaly and Mengistu [5] used Non-Uniform Rational B-Splines (NURBS) to optimize an existing turbine airfoil design in 2D-NURBS are similar to B-splines except their control points each have a weight. Their parameterization showed that NURBS required fewer points to parameterize a compressor airfoil [6], as opposed to a turbine blade [7] due to the increased curvature of the suction side. Shelton et al.[3] optimized a turbine blade under transonic conditions incorporating a stacking and sweep law. They similarly observed that it was difficult to make large changes to the stagger, wedge angle, and suction side surface angles in their parameterization using b-splines. Hasenjäger et al.[8] adopted b-splines to optimize a low aspect ratio stator blade in 3D and encountered difficulties as well, attempting to limit the number of control points needed to represent the blade surface using spline-like strategies.Bezier curves have been applied in a wide variety of turbomachinery applications because they offer designers the prospect of using less parameters while controlling the curvature by removing the need to parameterize the knot vector [9,10]. Goel et al. [11] used this type of curves to define 2D turbine airfoils using 10 and 8 parameters for the suction and pressure side respectively. Pierret and Van den Braembussche [12] employed a series of Bezier curves to model the suction and pressu...

Modelling of Real Gas Compression Using the Finite Volume Thermofluid Network Methodology and Mean-Line Approach for Centrifugal Compressors

“…In the application of the network methodology the value of the mean stagnation density in each node can be obtained with the sequence provided in Equation (11) as follows: (i) determine the entropy at the stagnation conditions using the known values of stagnation enthalpy and stagnation pressure with the aid of a suitable real gas fluid property library (such as CoolProp); (ii) determine the static enthalpy from the stagnation enthalpy and the velocity; (iii) obtain the static pressure using the known values of static enthalpy and entropy (which is the same value as at the stagnation conditions) with the aid of a suitable real gas fluid property library; (iv) determine the value of the mean stagnation density from Equation (11).…”

“…For a one-dimensional fluid control volume with a well-defined single inlet (i) and single outlet (e) Equation ( 13) can be integrated to get ( ) ( ) The simultaneous solution of ( 5) and ( 6) for each node and ( 14) for each element in the network, together with appropriate boundary conditions will yield the nodal stagnation pressures and enthalpies as well as the elemental mass flow rates. Following this the static properties can be obtained from equation (11), which can then be used to determine all the other fluid properties with the aid of a suitable real gas fluid property library. Once all the fluid property values are known for the nodes, the appropriate mean values can also be determined for the elements, which in turn can be used in the calculation of the component characteristics (…”

“…The utilization of static maps does not provide a direct simulation of the internal flows and dynamic behavior through the machines. Therefore, the approach using static maps cannot be applied when performing integrated cycle design optimization [11][12][13] which includes machine dimensions such as impeller diameters and vane angles as independent design parameters, or when the fast transient response [14] of the turbomachinery systems is to be investigated. Furthermore, the conventional scaling rules based on corrected mass flow and corrected speed on which static maps are based do not necessarily ensure similarity in the case of sCO2 [15].…”

A Study of Stopping Rules in the Steepest Ascent Methodology for the Optimization of a Simulated Process

“…The ultimate goal is either to minimize the effort required or to maximize the desired benefit. For instance, Balafkandeh et al [5] and Juangphanich et al [6] focus their efforts on optimizing operations to minimize outputs, while, Oleksy-Sobczak and Klewicka [7] and Delavar and Naderifar [8] pursue optimization of inputs to maximize results. However, companies are not only concerned with creating more products with less resources; they also have the intention of performing right operations the first time.…”