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372 Let us now imagine the turbine as comprised of an infinite number of stages with infinitesimal pressure drops of equal efficiency. Thus the *Ts become dTs and the small-stage efficiency becomes 0s = dT/dTi [dl] (9.7) The isentropic relation for a calorically perfect gas, Equation (1.19), may be written in the form T/p (k – 1)/k = a constant. Using differentiation by parts yields dTi – [(k – 1)/k]Tp – 1dp = 0 which combined with Equation (9.7), gives dT/T = [0s (k – 1)/k] p – 1dp [dl] (9.8a) If it is assumed that the stage efficiency is constant, integration of Equation (9.8a) between the turbine inlet and exit states gives T1/T2 = (p1/p2)0s (k– 1)/k [dl] (9.8b) Thus the requirement that all stages have equal efficiency yields a temperature pressure relationship of the same form as Equation (1.19), except for the exponent. Such relationships, of the form T1/T2 = (p1/p2) (n – 1)/n are called polytropic. For example, the isentropic Equation (1.19) is polytropic with n = k. Because the use of constant small-stage efficiency yields a pressure-temperature relation of polytropic form, 0s is often called the polytropic efficiency. The exponent for the turbine expansion with constant stage efficiency is then given by (n – 1)/n = 0s(k –1)/k [dl] (9.9) If the polytropic efficiency is unity, n becomes k, and Equations (9.8) become the usual isentropic relation. Values of 0s less than 1 reduce the turbine temperature ratio, T1/T2, for a given turbine pressure ratio, p1/p2, below the isentropic value in qualitatively the same way that decreasing turbine isentropic efficiency does. A similar statement applies to turbine work. The turbine isentropic efficiency, 0t, can be expressed in terms of the polytropic efficiency by substitution in the isentropic efficiency definition: 0t = (T2/T1 – 1)/(T2i /T1 – 1) = (1/r 0s(k –1)/k – 1)/ (1/r (k –1)/k – 1) [dl] (9.10)PDF Image | Advanced Systems Steam Power Plant
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