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Centrifugal Turbines for ORC Applications attained. The system of equations accordingly implemented is s = s(pT,in, TT,in) hth =hT,in(pT,in,TT,in)− 1c(hth,s)2 (3.1) 2 m ̇ = ρ t h ( h t h , s ) · c ( h t h , s ) · A t h , where s is the specific entropy, and the subscript ‘th’ indicates the (static) thermodynamic conditions in the throat section. Solving the continuity equation appearing in system (3.1), the throat flow passage area Ath can be evaluated and, if the throat width is assigned, the blade height can be computed. On the contrary, if subsonic flow occurs at the outlet section, the thermodynamic conditions are obtained by solving the balance equations for the mass and the momentum in the tangential direction, between the geometric throat and the down- stream non-bladed zone, as detailed in Ref. [17]. In both cases the flow angle is calculated starting from the blade geometric angle (BDA) by applying a proper deviation correlation. The blade number is evaluated by applying the Zweifel load criterion [19], which provides the optimal solidity as a function of the flow deflection across the cascade. It should be noted that a proper selection of solidity and blade loading would need a specific aerody- namic optimization, which is beyond the intrinsic limitations of a mean-line approach. 3. Theinitialisentropicdesignrepresentsthefirstguessforaniterativeproceduretoestimate the cascade losses. Several loss-prediction methods are available within zTurbo, such as those proposed by Ainley & Mathieson [20], Craig & Cox [15], and Traupel [21]. Alter- natively, user-defined loss coefficients can be specified. The estimation of losses allows to correct the flow velocities and the blades height previously estimated (see point 2 above). The choice of a suitable model is critical, since its accuracy becomes questionable for flow conditions departing from the validity range of the method [22]. In the case at hand, the situation is further complicated by i) the fact that the machine arrangement is not axial, ii) the different fluids and thermodynamic operating conditions, iii) the onset of post-expanded and supersonic flows, and vi) the possibly low scale dimensions of the machines, that may induce a stronger interaction among different loss mechanisms (e.g., profile and tip-leakage losses) which is unlikely to be properly captured by the models. 4. A similar methodology (i.e. points 1 to 4), implementing the conservation of rothalpy, is employed for the calculation across the rotor in the rotating frame of reference. The outputs of the procedure outlined above (relative to a single stage) are: i) the velocity diagrams, ii) the meridional channel shape, and iii) the performance parameters (efficiency, loss coefficients, etc.). For multi-stage turbines, such procedure is applied stage-by-stage, assuming a value for the number of stages. The repartition of the expansion among the stages is a critical aspect, particularly when dealing with turbines elaborating large expansion ratios with few stages [5]. 3.2.2 Optimization Procedure The methodology described in §3.2.1, once implemented in the framework of an optimization pro- cedure, allows to search for the optimal machine design [5, 23]. The optimization of a turbomachin- ery is typically aimed at the maximization of its fluid-dynamic efficiency, which may be formulated in general terms as: ηTχ = f(Ψ,φ,M,Re,sh,χ) = ∆w 2 , (3.2) ∆h −χCout TS 2 61PDF Image | New Concepts FOR Organic Rankine Cycle Power Systems
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