Question

Consider a spark ignition engine which operates on an ideal air-standard Otto cycle. It uses a fuel which produces 44 MJ/kg of heat in the engine. If the engine requires 40 mg of fuel to produce 1 kJ of work output, then the compression ratio of the Otto cycle is $____________$ (rounded off to two decimal places). For the entire cycle, use $\dfrac{c_p}{c_v} = 1.4$.

Hint:

The Otto efficiency depends only on compression ratio and $\gamma$. Using fuel data just helps to calculate the actual efficiency before applying the formula.

Answer 1

Correct Answer: 8

Step 1: Heat supplied per cycle. Fuel energy: \[ q_{in} = (40 \times 10^{-6} \, kg)(44 \times 10^{6} \, J/kg) \] \[ q_{in} = 1760 \, J = 1.76 \, kJ \]
Step 2: Work output per cycle. Given: $W_{out} = 1 \, kJ$. So, cycle efficiency: \[ \eta = \frac{W_{out}}{q_{in}} = \frac{1}{1.76} \approx 0.568 \]
Step 3: Otto cycle efficiency formula. \[ \eta = 1 - \frac{1}{r^{\gamma-1}} \] where $r =$ compression ratio, $\gamma = 1.4$. \[ 0.568 = 1 - \frac{1}{r^{0.4}} \] \[ \frac{1}{r^{0.4}} = 0.432 \Rightarrow r^{0.4} = \frac{1}{0.432} \approx 2.314 \] \[ r = (2.314)^{\tfrac{1}{0.4}} = (2.314)^{2.5} \approx 7.87 \] Final Answer: \[ \boxed{7.87} \] % Quicktip