Page 72 - Entrophy Analysis in Thermal Engineering Systems

64 Entropy Analysis in Thermal Engineering Systems

Now, substituting Eqs. (5.32)–(5.34) into Eq. (5.29), one obtains

1 γ + γ 1ÞCR γ PR
PR CR ð
η ¼ 1 γ T R (5.35)
PR
PR T R

5.3 Efficiency comparison

Table 5.2 summarizes the expressions obtained for the thermal effi-
ciency of the gas power cycles in the preceding section. The third column
in Table 5.2 gives PR as a function of CR, where appropriate. An efficiency
comparison will be made assuming air as the working gas with γ ¼1.4. The
design constraints include (i) the amount of heat supplied is the same,
and (ii) the maximum degree of volume change, CR, is identical in all
engines.
Fig. 5.7 displays the efficiency of the cycles plotted against the normal-
∗
ized heat input defined as q ¼q/(c v T 1 ) at a fixed value of CR¼21. The Otto
cycle exhibits a constant efficiency—see Eq. (5.14) and Table 5.2, and it pos-
∗
sesses the highest thermal efficiency in Fig. 5.7 for any heat input q <4.12.
∗
However, the Stirling cycle becomes the most efficient engine for q >4.12.

Table 5.2 A summary of the thermal efficiency expressions and PR-CR relations.
Cycle Efficiency PR5f(CR)

Stirling 1 CR
PR PR¼CR T R
1 γ γ
Brayton
CR 2 γ 2 γ
ð
1 PR ¼ CR=T R Þ
T R
1 γ
Otto 1 CR PR¼CR T R
Atkinson 1 γ T R CR 1 γ 1 PR¼CR γ
ð
T R T R CR γ Þ 1 γ
γ
Diesel 1 T CRð γ 1 γ 1 i γ
h
Þ
1 R PR¼CR
γ T R CR γ 1

Miller –
1 γ + γ 1ÞCR γ PR
PR:CR ð
1 γ T R
PR
PR T R