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Int. J. Turbomach. Propuls. Power 2020, 5, 19 7 of 17 The above assumptions suffice for a definition of kinematic efficiency. Since the relative velocity provides the only positive contribution to the Euler’s work, the absolute velocity and the peripheral velocity are to be considered as losses, and this leads to the following formula: (6) vout2 − vin2 + uout2 − uin2 ηkin = 1− 2 2 , wout − win 1−χ2·1+2ρ 2 −2ρ cosβ That can be rewritten as: ηkin,i=1−22 , (7) r r out 1−χ ·(1+4ρr −4ρr cosβout) This equation provides additional useful information because it is similar for every i-th blade ring except for the first one, where it takes the following form: 2 χ2 ρr 2− cosβ02 +1−2ρr cosβout ηkin,1 = 1− The parameters to be considered to maximize the kinematic efficiency are, therefore, ρr, χ, βout Int. J. Turbomach. Propuls. Power 2019, 4, x FOR PEER REVIEW 7 of 16 and β0. Due to the second and third assumptions it is not possible to maximize both ηkin,i and ηkin,1 with respect to ρr and χ, thus only the ηkin,i equation is considered here for the optimization. It can be easily verified that the derivative of the kinematic efficiency (7) is linearly dependent It can be easily verified that the derivative of the kinematic efficiency (7) is linearly dependent on on both χ and βout and thus it is not possible to maximize this equation with respect to these both χ and βout and thus it is not possible to maximize this equation with respect to these parameters. parameters. This conclusion is reinforced by a closer inspection of Equation (7), which is maximized This conclusion is reinforced by a closer inspection of Equation (7), which is maximized either when χ2 , (8) 1− cosβ02 either when the numerator of the second term tends to zero or the denominator to a maximum. Both the numerator of the second term tends to zero or the denominator to a maximum. Both cases leading cases leading to ideal turbine configurations: to ideal turbine configurations: • • • • Therefore, the only parameter to be considered in the velocity triangles optimization is ρr, which allowThseurseftorme,athxiemoinzleythpearkaimnemtearttiocebfeficoiensciydeforerdinffethrenvtevlaolcuiteystorfiaβnougtlaensdoχp,tiamsiszhaotiwonisnρFri,gwurheic8h. allows us to maximize the kinematic efficiency for different values of βout and χ, as shown in Figure 8. χ = 1, meets the maximum of the efficiency for ρr = cos(βout)/2, and causes the numerator to be χ = 1, meets the maximum of the efficiency for ρr = cos(βout)/2, and causes the numerator to be null. In fact, in this case uin = uout and vin = vout so that the velocity triangle is similar to that of an null. In fact, in this case uin = uout and vin = vout so that the velocity triangle is similar to that of an impulse turbine; impulse turbine; βout = 0 causes the denominator to be at its maximum, in which case win = 0 because the triangle β = 0 causes the denominator to be at its maximum, in which case w = 0 because the triangle couotllapses onto a line. in collapses onto a line. (a) (b) Figure8.Kinematicefficiency:(a)Variationofthekinematicefficiencywithρ fordifferentβ and Figure 8. Kinematic efficiency: (a) Variation of the kinematic efficiency with rρr for different oβuotut and fixed χ; (b) Variation of the kinematic efficiency with ρ for different χ and fixed β . fixed χ; (b) Variation of the kinematic efficiency withrρr for different χ and fixedoβuotut. Considering the above, we can now proceed to the study of the isentropic efficiency, starting from its equation and the loss model adopted here: h = ,, (9) PDF Image | Optimal Design of a Ljungstrom Turbine for ORC Power
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