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Energies 2020, 13, 5846 11 of 23 β = pEXP,in pEXP,out BWR = Ppmp PEXP (8) (9) When volumetric expanders are considered, another fundamental parameter is the built-in volume ratio, which is defined by dividing the exhaust volume by the intake volume: VR = Vexh (10) Vsuc The value of VR (Equation (10)) in a volumetric machine is fixed and defines the pressure inside the expander chamber when the exhaust port opens. This value can be different from the pressure imposed by the circuit at expander outlet, pEXP,out, so in the machine design, the VR should be properly defined to match the considered application. Indeed, if VR is not properly sized, it affects the indicated power Pind, which is defined in Equation (11) as the integral of the p-V diagram that reports the pressure inside the expander chamber as a function of its volume. Consequently, if Pind diminishes, the mechanical power Pmech decreases also, as it can be evaluated in Equation (12) as the difference between Pind and the power loss due to friction Ploss. Concerning this latter quantity, in SVRE, it coincides with the dry friction power due to the relative motion of the blade tip on the stator inner surface. Nv pidVi Pind = i=1 (11) tcycle Pmech = Pind − Ploss (12) m. ηvol = m. th = ρin,end Vin,end Nv ω m. (13) WF The power produced by the machine depends also on volumetric efficiency evaluated in Equation (13) as the ratio between the theoretical mass flow rate aspirated by the SVRE and the real one that enters the machine. The difference is due to the leakages that flow across the clearance gap (between the blade tip at the stator inner surface, between the rotor face and casing, and between the blade side and rotor slot). 5. Component #5 represents the electric generator that was introduced as a boundary condition in terms of the rotational speed at which the SVRE rotates as a consequence of the dynamic equilibrium on the expander shaft. If the generator is connected to the electric network (as in the experimented case), the velocity is constant. Nevertheless, if a proper inverter was installed on the expander, the revolution speed at which the expander works can be changed, constituting an important degree of freedom for expander regulation. The model is able to vary the revolution speed, thus catching this phenomenon. 6. Component #6 is equivalent to Component #5 and represents the condenser. The hot (working fluid) side is the master and the tap water for cooling the slave; the cooling capacity is set by the boundary conditions (flow rate and temperature). Similarly to the evaporator, the machine is physically represented and the Dittus–Boelter correlation [52] applies to the single-phase flow region (liquid or vapour), and the Plate, Yan, Lio, and Lin correlations [52] apply to the two-phase region. 7. Component #7 allows modeling the tank located upstream from the pump. This element allows varying the volume capability and consequently the volume of the whole plant. In the post processing, this element allows also understanding the phase of the working fluid inside the plant. WFPDF Image | Design of ORC Plant for Low-Grade Waste Heat Recovery
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