Unformatted text preview: 27 2.2. EXERGY
So we get
ψ = cP (T − To ) − To cP ln T
To − R ln 1
+ v · v + g (z − zo ).
2 P
Po (2.94) For T ∼ To P ∼ 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 ∼ 1. For a Taylor series near x = 1, we have
y (x) ∼ y (1) + dy
dx x=1 (x − 1) + 1 d2 y
2 dx2 x=1 (x − 1)2 + . . . . (2.96) Now for y = ln x, we have
1
d2 y
= − 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 = −1 (2.98) So
1
y (x) = ln x ∼ 0 + (x − 1) − (x − 1)2 + . . .
2 (2.99) We then expand ψ via the following steps:
1
P
T
+ RTo ln
+ v · v + g (z − zo ),
To
Po
2
T
k−1
1
T
P
+
+ v · v + g (z − zo ),
− 1 − ln
ln
To
To
k
Po
2 = cP (T − To ) − cP To ln (2.100) = cP T o (2.101) = ψ cP T o = 1
T
−1 −
To
2 T
−1 −
To 1
+ v · v + g (z − zo ),
2
1T
−1
cP T o
2 To T
−1
To 2 + ... + k−1
k P
− 1 + ...
Po
(2.102) 2 + ... + k−1
k P
− 1+ ...
Po 1
+ v · v + g (z − zo ).
2 (2.103) Note that in the neighborhood of the ambient state, relative pressure diﬀerences are more eﬀective than
relative temperature diﬀerences at inducing high exergy. CC BYNCND. 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 ﬂuid. The ideal cycle is described by
• 1 → 2: isentropic pumping process in the pump,
• 2 → 3: isobaric heat transfer in the boiler,
• 3 → 4: isentropic expansion in the turbine, and
• 4 → 1: isobaric heat transfer in the condenser.
Note that to increase cycle eﬃciency one can
• lower the condenser pressure (increases liquid water in turbine),
• superheat the steam, or
• increase the pressure during heat addition.
A schematic for the Rankine cycle is shown in Figure 2.3. The T − 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 BYNCND. 18 November 2011, J. M. Powers. ...
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This note was uploaded on 11/26/2011 for the course EGN 3381 taught by Professor Parksou during the Fall '11 term at FSU.
 Fall '11
 ParkSou
 Dynamics

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