Solution

12.82
Borgnakke and Sonntag
In the Otto cycle all the heat transfer q H occurs at constant volume. It is more
realistic to assume that part of q H occurs after the piston has started its downward
motion in the expansion stroke. Therefore, consider a cycle identical to the Otto
cycle, except that the first two-thirds of the total q H occurs at constant volume and
the last one-third occurs at constant pressure. Assume that the total q H is 2100
kJ/kg, that the state at the beginning of the compression process is 90 kPa, 20°C,
and that the compression ratio is 9. Calculate the maximum pressure and
temperature and the thermal efficiency of this cycle. Compare the results with
those of a conventional Otto cycle having the same given variables.
P 3
2
4
s
5
T
2
3
v
4
s
5
s
1
s
1
v
b) P 2 = P 1 (v 1 /v 2 ) k = 90(9) 1.4 = 1951 kPa
T 2 = T 1 (v 1 /v 2 ) k-1 = 293.15(9) 0.4 = 706 K
T 3 = T 2 + q 23 /C V0 = 706 + 1400/0.717 = 2660 K
P 3 = P 2 T 3 /T 2 = 1951(2660/706) = 7350.8 kPa = P 4
T 4 = T 3 + q 34 /C P0 = 2660 + 700/1.004 = 3357 K
v 5
v 4
=
v 1
v 4
=
P 4
P 1
v
T
1
×
T
4
= 7350.8 293.15
90
×
3357
= 7.131
T 5 = T 4 (v 4 /v 5 ) k-1 = 3357(1/7.131) 0.4 = 1530 K
P 1 = 90 kPa, T 1 = 20 o C
r V = v 1 /v 2 = 7
q L = C V0 (T 5 -T 1 ) = 0.717(1530 - 293.15) = 886.2 kJ/kg
η TH = 1 - q L /q H = 1 - 886.2/2100 = 0.578
a) q 23 = (2/3) × 2100
= 1400 kJ/kg;
Std. Otto Cycle: η TH = 1 - (9) -0.4 = 0.585, small difference
s
q 34 = 2100/3 = 700 kJ/kg
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