Thermodynamics filled in class notes_Part_10

# Thermodynamics filled in class notes_Part_10 - 27 2.2...

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Unformatted text preview: 27 2.2. EXERGY So we get Ïˆ = cP (T âˆ’ To ) âˆ’ To cP ln T To âˆ’ R ln 1 + v Â· v + g (z âˆ’ zo ). 2 P Po (2.94) For T âˆ¼ To P âˆ¼ Po , we can use Taylor series to simplify this somewhat. First, we recall the general Taylor series for a log function near unity. Consider y (x) = ln x, (2.95) for x âˆ¼ 1. For a Taylor series near x = 1, we have y (x) âˆ¼ y (1) + dy dx x=1 (x âˆ’ 1) + 1 d2 y 2 dx2 x=1 (x âˆ’ 1)2 + . . . . (2.96) Now for y = ln x, we have 1 d2 y = âˆ’ 2, dx2 x 1 dy =, dx x (2.97) and thus y (1) = ln(1) = 0, dy dx d2 y dx2 = 1, x=1 x=1 = âˆ’1 (2.98) So 1 y (x) = ln x âˆ¼ 0 + (x âˆ’ 1) âˆ’ (x âˆ’ 1)2 + . . . 2 (2.99) We then expand Ïˆ via the following steps: 1 P T + RTo ln + v Â· v + g (z âˆ’ zo ), To Po 2 T kâˆ’1 1 T P + + v Â· v + g (z âˆ’ zo ), âˆ’ 1 âˆ’ ln ln To To k Po 2 = cP (T âˆ’ To ) âˆ’ cP To ln (2.100) = cP T o (2.101) = Ïˆ cP T o = 1 T âˆ’1 âˆ’ To 2 T âˆ’1 âˆ’ To 1 + v Â· v + g (z âˆ’ zo ), 2 1T âˆ’1 cP T o 2 To T âˆ’1 To 2 + ... + kâˆ’1 k P âˆ’ 1 + ... Po (2.102) 2 + ... + kâˆ’1 k P âˆ’ 1+ ... Po 1 + v Â· v + g (z âˆ’ zo ). 2 (2.103) Note that in the neighborhood of the ambient state, relative pressure diï¬€erences are more eï¬€ective than relative temperature diï¬€erences at inducing high exergy. CC BY-NC-ND. 18 November 2011, J. M. Powers. 28 CHAPTER 2. CYCLE ANALYSIS 2.3 Rankine 2.3.1 Classical The Rankine cycle forms the foundation for the bulk of power generating devices which utilize steam as a working ï¬‚uid. The ideal cycle is described by â€¢ 1 â†’ 2: isentropic pumping process in the pump, â€¢ 2 â†’ 3: isobaric heat transfer in the boiler, â€¢ 3 â†’ 4: isentropic expansion in the turbine, and â€¢ 4 â†’ 1: isobaric heat transfer in the condenser. Note that to increase cycle eï¬ƒciency one can â€¢ lower the condenser pressure (increases liquid water in turbine), â€¢ superheat the steam, or â€¢ increase the pressure during heat addition. A schematic for the Rankine cycle is shown in Figure 2.3. The T âˆ’ s plane for the Rankine 3 Turbine Boiler 4 2 Condenser Pump 1 Figure 2.3: Schematic for Rankine cycle. cycle is shown in Figure 2.4. Example 2.3 (adopted from Moran and Shapiro, p. 312) Consider steam in an ideal Rankine cycle. Saturated vapor enters the turbine at 8.0 M P a. Saturated liquid exits the condenser at P = 0.008 M P a. The net power output of the cycle is 100 M W . Find CC BY-NC-ND. 18 November 2011, J. M. Powers. ...
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