Troels Dyhr Pedersen.indd - Solid Mechanics

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(excess air ratio 3) of air and ethanol, for the corresponding burned mixture and an
average of burned and unburned. All components are assumed to be ideal gasses.
Specific heat ratio
1,40
1,38
1,36
1,34
1,32
1,30
1,28
1,26
1,24
1,22
250 700 1150 1600 2050 2500
T [K]
γmodel γmodel
γair γair
γburned γburned
γunburned γunburned
γaverage γaverage
Figure 7: The specific heat ratio calculated for dry air, for a burned and unburned charge and the
average of burned and unburned charge. The linear relation is for the model expression.
As can be seen from the figure, simply assuming a linear relationship will result in a great
deviation at elevated temperatures, when compared to the values calculated from the
tabulated properties.
For motoring conditions with pure air, the curve for pure air will provide a better estimate
on the specific heat ratio than the linear expression. For operating conditions, the specific
heat ratio is less depending on whether the gas is burned or not, than it is on temperature.
The error of using the average value is les than one percent and hence acceptable for
calculations of heat release.
The calculation of gamma may be performed with a third order polynomial fit. The
equations in table 3 are very close approximations to the curves above, at temperatures
higher than 400 K
Table 3: Polynomial fits of specific heat ratios
Air
−4
−8
2
γ = 1. 45 − 1.
62 ⋅10
T + 5.
49 ⋅10
T
−11
− 1.
26 ⋅10
T
Unburned
−4
−8
2
γ = 1. 44 − 2.
22 ⋅10
T + 9.
39 ⋅10
T
−11
− 1.
26 ⋅10
T
Burned
−4
−8
2
γ = 1. 44 − 1.
18 ⋅10
T + 6.
96 ⋅10
T
−12
− 9.
51⋅
10 T
Average
−4
−8
2
γ
= 1. 44 − 2.
03⋅
10 T + 8.
18 ⋅10
T
−11
− 1.
11⋅
10 T
3
3
3
3