Question # The Otto-cycle engine in a Mercedes-Benz SLK230 has a compression ratio of 8.8.

Ratios, rates, proportions
ANSWERED The Otto-cycle engine in a Mercedes-Benz SLK230 has a compression ratio of 8.8.
a) What is the ideal efficiency of the engine? Use $$\displaystyle\gamma={1.40}$$
b) The engine in a Dodge Viper GT2 has a slightly higher compression ratio of 9.6. How much increase in the ideal efficiency results from this increase in the compression ratio? 2021-09-21
Step 1
a) We want to calculate the ideal efficiency of the engine when the ratio of heat capacity for the gas used is $$\displaystyle\gamma={1.40}$$ Ideal efficiency (e) of the Otto-cycle is given by equation 20.6
1) $$\displaystyle{e}={1}-{\left({\frac{{{1}}}{{{r}^{{\gamma-{1}}}}}}\right)}$$
Now we can plug these values for r and $$\displaystyle\gamma$$ into equation (1) to get the ideal efficiency e
$$\displaystyle{e}={1}-{\left({\frac{{{1}}}{{{8.8}^{{{1.40}-{1}}}}}}\right)}={0.58}={58.0}\%$$
Step 2
b) The engine in a Dodge Viper GT2 has a slightly higher compression ration $$\displaystyle{r}={9.6}$$. We want to calculate the increase in the ideal efficiency e after increasing the compression ratio. So we will use equation (1) again but we will plug the value for r by 9.6
$$\displaystyle{e}={1}-{\left({\frac{{{1}}}{{{9.6}^{{{1.40}-{1}}}}}}\right)}={0.594}={59.4}\%$$
The increse in the ideal efficiency will be given by $$\displaystyle{e}_{{{b}}}-{e}_{{{a}}}$$
Increase in $$\displaystyle{e}={59.4}\%-{58}\%={1.4}\%$$
The ideal efficiency increase as the compression ratio increase.