Graphical Illustration of Lost, Isentropic, and Optimum Work

4. Graphical Illustration of Lost, Isentropic, and Optimum Work

The previous example illustrates the differences between isentropic, actual, and opti- mum work for a steady state steady flow process. These differences and those between the adiabatic and availability or exergetic efficiencies can now be graphically illustrated. This is done in Figure 15 that contains a representative T–s diagram for gas expansion in a turbine

from a pressure P 1 to P 2 . The solid line 1–2s represents the isentropic process, while the dashed curve 1–2 (for which the actual path is unknown) represents the actual process. The isenthal- phic curve h 1 intersects the isobaric curves P 1 at (1) and P 2 at the point K. The isentropic and actual work represent the areas that lie, respectively, under the lines 2s–K (i.e., B–2s–K–D) and 2–k (i.e., C–2–K–D). The proof follows.

For any adiabatic process, w = dh. For the isentropic process 1–2s

–w 1–2s =h 2s –h 1 ,

while for the actual process 1–2

–w 1–2 =h 2 –h 1 .

The work loss during the irreversible adiabatic process is

w 1–2s –w 1–2 =h 2 –h 2s .

Consider the relation T ds + v dP = dh which is valid for a fixed mass of simple compressible substances. At constant pressure T ds = dh, at a constant pressure P 2

2 s Tds =∫ h 2 s dh .

Consider an ideal gas as an example ( Figure 15 ). The constant temperature lines are same as constant enthalpy lines. For illustration consider the expansion process 1-2 with P 2 <P 1 ; the area under 2s-K along constant P 2 line represents the work output for isentropic process. Similarly, the area under 2–K (C–2–K–D) represents the actual work. The area under 2s–2 (i.e., B–2s–2–C), therefore, represents the difference between isentropic and non-

1 P 2 <P 1 P 0

h 1 1 ,T 1 K

h ,T

h 2s ,T 2s

2s

h 0 ,T 0 E 0 F R

B CV D Q Y

Figure 15 : Graphical illustration of availability on a T-S diagram for an expansion proc- ess.

isentropic processes. Similarly, if fluid is expanded from state (1) to dead state, (say P 2 =P 0 ,T 2 =T 0 ) then work is given by area V-0-X-Y and availability stream availability at state 1 (h 1 –h 0

– To (s 1 –s 0 )) is given by area B-E-0-V+V-0-X-Y and at state 2 by area (E-F-0-V+V-0-A-Q) thus, the area E-F-C-B-+Q-A-X-Y represents w opt . The irreversibility is given by area BEFC + QAXY.

Since the actual work is given by area 2KDC+QAXY, the consequent availability loss T 0 (s 2 – s 1 ) that is represented by the area BEFC which is smaller than the work loss area B–2s–2–C. The reason for this is that the end–state conditions are maintained identical for the availability calculations, i.e., part of the isentropic work is used to pump heat so that state 2 is

reached from the state 2s. Hence, the optimum work w opt <w 1–2s–2 so that (w opt –w 12 ) < (w 12s – w 12 ). This difference is represented by the area E–2s–2–F. The adiabatic or isentropic effi- ciency is provided by the relation

η = actual work ÷ isentropic work = (Area C2KD) ÷ (Area B2S s KD).

The availability efficiency η Avail = actual work ÷ maximum work = (Area C2KD ÷ (Area C2KD+EFCB). Similar diagrams can be created for compression processes for which η = isentropic work ÷ actual work = (Area B2sKD) ÷ (Area C2KD), and

η Avail = actual work ÷maximum work = (Area C2ID) ÷ (Area C2KD) ÷ (Area C2KD + Area EFCB).