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Chapter 3 The subscript ’Tχ’ clarifies that the ideal total-to-static enthalpy drop used to calculate the efficiency is reduced by a fraction (in the range 0 < χ < 1) of the discharge kinetic energy, supposing this can be recovered downstream of the last stage. In other words, ηTχ = ηTT if χ = 1, and ηTχ = ηTS if χ = 0; the choice of a value for χ has a deep influence on the turbine design, as shown in Ref. [5]. The functional dependence of η from the work and flow coefficients Ψ and φ, and the Mach and Reynolds numbers M and Re, is expressed by the first equality in Eq. (3.2). This dependence includes also the term sh, synthetically indicating the turbine shape and accounting for the influence of geometrical parameters such as the solidity, the blade angles, the ratio of the trailing-edge thickness and of the tip clearances with respect to the blade chord, etc. From a mathematical point of view, several main aspects have to be taken into account: i) the definition of a suitable objective function to be maximized or minimized, ii) the independent variables and their range of variation within the design space, iii) the geometrical and fluid-dynamics constraints bounding the space of solutions, and iv) the algorithm to be used to search the design space for the optimal solution. In this chapter the efficiency defined in Eq. (3.2) is chosen as the objective function, with χ = 0.5. This assumption implies that a diffuser is used downstream of the last rotor, and is able to recover half of the kinetic energy of the exiting flow. Concerning the optimization strategy, a flexible and non-intrusive approach is desirable in order to leave to the user the choice of the in- dependent variables, the constraints, and the search algorithm. Therefore, zTurbo is coupled to a well known open source external optimization software Dakota [24]. An evolutionary optimization strategy, based on a single-objective genetic algorithm, is adopted in the present analysis. For a single objective optimization, the application of a gradient-based algorithm would require a lower computational effort. However, due to the possibly highly non-regular behavior of the fitness func- tion, local optima may be found during the search. Genetic algorithms span the whole design space, and have therefore the advantage of being more robust under this respect, proceeding towards the global optimum [25]. In particular, the adopted optimization method is the single objective version of the multi- objective elitist genetic algorithm proposed in [26] and implemented in [24], based on binary en- coding and dynamic memory allocation. 3.3 Centrifugal Architecture for ORC applications Some critical challenges encountered in the design of efficient vapour turbines are a consequence of the large variation of the volumetric flow rate over the expansion. The low speed of sound characterizing ORC working fluids further complicates the design of the expander, leading to su- personic regime within the flow passages [5]. In the centrifugal architecture, the fluid enters the machine close to the rotational axis, and flows outward in the radial direction (see fig. 3.1). This is advantageous primarily because it provides a natural increase of the passage area along the flow path. Another major advantage is the possibility of implementing multi-stage arrangements in a comparatively easy way [27, 28]. The counter-rotating centrifugal steam turbine was proposed by Lju ̈ngstrom in the early 20th Century and widely adopted until the Sixties [29]. The limits of the Lju ̈ngstrom turbine emerged for large-capacity machines, resulting in a maximum power capacity of about 65 MWE per unit. This is mainly due to the difficulty, for larger output, of elaborating the corresponding volumetric flow rates in a single-casing arrangement. Eventually, the require- ment of two counter-rotating electrical generators determined the success of the axial-flow concept [29]. Notably, none of these issues applies in the ORC context and, as such, several studies have discussed the potential merits of these machines in this field [5, 30, 31]. These characteristics have recently driven also the industrial interest towards centrifugal ORC turbines, which have been thus 62PDF Image | New Concepts FOR Organic Rankine Cycle Power Systems
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