Optimal Design of a Ljungstrom Turbine for ORC Power

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Optimal Design of a Ljungstrom Turbine for ORC Power ( optimal-design-ljungstrom-turbine-orc-power )

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Int. J. Turbomach. Propuls. Power 2020, 5, 19 8 of 17 Considering the above, we can now proceed to the study of the isentropic efficiency, starting from its equation and the loss model adopted here: wout,id2 − win2 − 􏰟uout2 − uin2􏰠 ∆hs−s = 2 , (9) In order to obtain the value of wout,id, Soderberg [14] loss correlation was used: w = w 􏰪1 + ξ , (10) out,id out r And after some rearrangement, the formula of the isentropic efficiency for the i-th row, with the exception of the first one, is obtained: 2􏰟1+χ2􏰠ρr cosβout−ρr2 ρr2 ηis,i = While for the first row: maboodviefimcaetniotinosneindtphredciocdtieo,nthmeoldoeslss.cIotewffiacsiefonutnξdcthanatbtheeeavdaolupattioedn ofofrdaifnfeyreonfttlhoessambovdeemlsednotieosnneodt r pnroetdicicetaiobnlyminofdlueelsn.cIet wthaesrfeosunltds,thaaststhoewadnoipntFioignuorfed9iff. erent loss models does not noticeably influence the results, as shown in Figure 9. (a) (b) Figure 9.. IIsentropic effifficiiency:: (a) Varriiattiion offttheeisiseennttrroopiicceeffiffcicieiennccyywitithhρρrffoorrddiffifeferreennttββout and 􏰣 √ 􏰤, (11) 1+ξr,i −1+χ2 1−1+ξr,i−1+4ρr2−4ρr ρr2 ρr2 2ρr cosβout−ρr2 ρr2 1+ξr,i−1cosβout ηis,1 = 1+ξr,1 2􏰘 1 􏰙, ρr2 −1+χ 1−cosβ02 (12) The Soderberg loss coefficient is adopted for all the ensuing calculations, but different models were Int. J. Turbomach. Propuls. Power 2019, 4, x FOR PEER REVIEW 8 of 16 tested as well, such as Ainley and Mathieson [15] and Craig and Cox [16] prediction models. With little fifixed χ;; ((b)) Vaarriaiattioionnoofftthheekkinineemaatitciceeffiffcicieiennccyywitihthρρrffoorrdiffifeferreennt tχχaannddfifxixeeddββout.. r out Sinceξ dependsonthefluiddynamicandgeometricconditionsthataredifferentineachrow, Since ξr depends on the fluid dynamic and geometric conditions that are different in each row, it is not possible to choose a single value of ρ for all rows that optimizes η . This justifies the choice it is not possible to choose a single value of ρrr for all rows that optimizes ηisi,si,i. This justifies the choice ofavelocitytrianglesoptimizationbasedontheselectionoftheρ thatmaximizestheη . of a velocity triangles optimization based on the selection of the ρrr that maximizes the ηkkiin,i. As shown in Figure 10, for a fixed geometry (fixed χ and β ) and by varying the velocity, As shown in Figure 10, for a fixed geometry (fixed χ and βout)oaunt d by varying the velocity, and and consequently ξ , the maximum of the isentropic efficiency is close to the range of the maxima consequently ξr, ther maximum of the isentropic efficiency is close to the range of the maxima of the kinematic efficiency. This supports the choice of an optimization procedure based on the velocity triangles. r out

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