Analysis and Multi-disciplinary Optimization of Internal Coolant Networks in Turbine Blades

This paper presents the theoretical methodology, conceptual demonstration, and validation of a fully automated computer program for the inverse design and optimization of internal convectively cooled three-dimensional axial gas turbine blades. A parametric computer model of the three-dimensional internal cooling network was developed, including the automatic generation of computational grids. A boundary element computer program was written to solve the steady-state non-linear heat conduction equation inside the internally cooled and thermal barrier-coated turbine blade. A finite element algorithm was written to model an arbitrary network of internal coolant passages for the calculation of the internal heat transfer coefficients, pressure losses, local flow rates, the effects of centrifugal pumping, heating of the coolant fluid, and forced convection from the thermal model of the solid to the coolant fluid. The heat conduction and internal flow analyses were iteratively coupled to account for the heat balance between the blade and the coolant fluid. The computer-automated design and optimization system was demonstrated on the second highpressure turbine blade row of the Pratt & Whitney F100 engine. The internal cooling configuration and local heat transfer enhancements (boundary layer trip strips and pin fins) inside the three-dimensional blade were optimized for maximum cooling effectiveness and durability against corrosion and thermomechanical fatigue + Systems Engineer. Member ASME. * Professor. Director of MAIDO Program. Fellow ASME. INTRODUCTION With presently available materials such as nickel-based alloys, turbine blades cannot withstand metal temperatures in excess of approximately 1300 K [1]. Internal heat transfer enhancements, such as trip strips or turbulators, impingement cooling, banks of pin fins and miniature heat exchangers can provide significant enhancements of convection heat transfer. For example, when needed in the initial turbine stages, cooling air can be made to impinge on the leading and trailing edge internal cooling passage surfaces in order to enhance convection. Impingement cooling schemes demand large leading and trailing edge diameters and thicker blades that often cause substantially increased aerodynamic losses. Complex heat exchangers have two major drawbacks. First, they induce early transition to turbulence and greatly increase the coolant passage effective friction, while moderately increasing the convective heat transfer. Second, manufacture of such complex internal configurations requires special machining processes. The external (film) cooling techniques allow even greater inlet gas temperatures, up to 1800 K. Film cooling can produce a protective layer of cool air on the surface of the blade and it can be targeted upon specific areas of the blade that absorb the most heat, for example, shower head cooling at the leading edges. This has several drawbacks. High pressure cooling air must be bled from high-pressure compressor stages which results in external pressure losses due to a reduction in the boundary layer momentum. The efforts to push the inlet turbine gas temperatures even higher are also constrained by environmental regulatory factors. The main problem is production of NOx that starts occurring at high gas temperatures especially when mixing it with coolant ejected through the blade surface film holes. This resulted in a renewed interest in exploring possibilities for better closed-loop high-pressure internal cooling schemes. Consequently, external cooling techniques were not a major focus of this paper. Despite all of these difficulties, a considerable gain in turbine performance may be realized by improving the blade's cooling effectiveness in order to offset all of those penalties. This paper presents a portion of a dissertation on the topic of multi-disciplinary design (MDO) of cooled turbine blades [2]. It has been demonstrated [2,3] that a computer-automated system can generate realistic internally cooled turbine blade designs that improve the efficiency and output of gas turbine components and make the individual cooled blades and vanes more durable so that engine operational, repair, and warranty costs are reduced. The MDO program was developed in order to demonstrate such a system within the framework of design methodologies and computational tools that are currently used in the turbomachinery industry. The implementation of MDO exposes the flaws in our subjective design practices, drives the automation of the transfer and sharing of information among various models and across scientific disciplines, reduces the design cycle time, and ultimately, optimizes the product. PARAMETRIC MODEL OF THE INTERNAL COOLING NETWORK A parametric model for three-dimensional internally cooled and thermal barrier coated turbine blades and vanes was developed for computer-automated design and optimization. A FORTRAN program (BetaCore) was written with the intent to provide rapid and robust geometry generation of realistic turbine blades and vanes and to pass information between the parametric model and the numerical analysis programs automatically without user intervention. The program was developed so that it could be executed in a parallel batchprocessing environment by a numerical optimization algorithm. The parametric model was initially constructed to represent two-dimensional cooled turbine airfoils [4,5]. It has since been extended to three dimensional turbine blades and vanes including many of the complex features used by turbomachinery engine companies. With the execution of this geometry generation program, a set of optimization design variables (the parametric model) was used to represent a virtual (electronic) prototype of the turbine blade or vane. The optimization design variable set controlled the internal coolant passage configuration, thickness variation of the coolant passage wall, positions and thicknesses of the internal ribs, die pull angles of the ribs, internal turbulator heights, turbulator pitches, turbulator skew angles, internal impingement hole diameters, impingement hole pitches, trailing edge slot feature lengths, internal pin fin diameter, pin fin spacings, coolant supply pressures, and turbine inlet temperature. The program produced a boundary element (BEM) surface mesh that was imported directly into a heat conduction analysis code in the blade material This design methodology was successful at generating a wide range of realistic internally cooled turbine blades and vanes, while the surface meshing, grid generation, and boundary conditions were automatically mapped between the interfacial surfaces. This information was transferred between the various design, optimization, and numerical analysis tools without user intervention. A constrained hybrid optimization algorithm [6] controlled the overall operation of the system and guided the multi-disciplinary internal turbine cooling design process towards the objectives of cooling effectiveness and turbine blade durability. Design variable sets that generated an infeasible or impossible geometry were restored to a feasible shape automatically using a constraint sub-minimization. INVERSE DESIGN OF TURBINE BLADES FOR STRUCTURAL INTEGRITY AND CREEP LIFE Given a fixed external blade shape, inverse shape design was used in order to meet the creep-life of the blade and the centrifugal stress limit. Inverse design is the reverse of the ubiquitous forward design process. In cooled turbine blade design, the forward design process begins with the generation of an internal cooling scheme and then the stresses or temperatures are calculated later using some analysis tool. Inverse design works in the opposite direction. That is, having knowledge of the temperature or stress; it is possible to determine an unknown geometry that is compatible with this stress and temperature data. Inverse design is possible only if some over-specified knowledge, albeit approximate and a priori, is available in addition to what would be boundary conditions of a well-posed (analysis) problem. In the internally cooled turbine blade case, the designer must know what stresses are to be allowed in the blade. With respect to the structural requirement alone, that much is easy. It is well understood that the stress at the blade root may not exceed the yield stress, σy. But blades are also designed to have a certain life span that is severely limited by temperature-dependent creep. Therefore, the designer also needs to know something about the radial variation of the blade temperature so that the internal cooling scheme can be designed to maintain certain average creep life requirement. That is, the average creep life of the blade is specified first. Then, after computing the radial variation of the centrifugal stress limit from this requirement, the thickness of the coolant passage wall is determined so as not to exceed the centrifugal stress limit. Given the non-uniform combustor exit flow profile and the highly three-dimensional nature of the aerodynamics, the hottest gases in the turbine tend to migrate to the mid-span radius, while the centrifugal stresses in the blade are the highest at the root of the blade and decrease radially. The combined threedimensional temperature and stress environment provides that the worst creep is experienced at some critical span location between the root and the tip. Thus, a radial gas temperature profile, TG(r), from a CFD calculation will be used to approximate a radial variation of the blade’s metal temperature, Tm(r). This requires some approximate knowledge of heat transfer coefficients pre-calculated or assumed on the blade’s external surface, hG, and on the internal coolant passage surfaces, hC. A nominal blade wall thickness, tm, a nominal thermal barrier coating thickness, to, and a bulk coolant temperature, TC, are also needed. The following analytic solution for one-dimensional heat transfer can be used to calculate spanwise variation of wall temperature.