Subtopic: Stoichiometric Fuel-Air Ratio for Octane
Question Statement:
In a combustor, gaseous Octane ($\text{C}_8\text{H}_{18}$) and air are to be burned in a stoichiometric ratio. Assuming air contains 21% $\text{O}_2$ and 79% $\text{N}_2$ by volume, the stoichiometric fuel-to-air mass ratio $(f/a)_{stoich}$ is approximately:
Options:
(A) 0.033
(B) 0.066
(C) 0.099
(D) 0.132
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.45)**
**1. Intuitive Physical Setup & Given Parameters**:
- Fuel: Octane, $\text{C}_8\text{H}_{18}$.
- Molar mass of Carbon: $M_C = 12\text{ g/mol}$, Hydrogen: $M_H = 1\text{ g/mol}$.
- Molecular weight of Octane: $M_{fuel} = 8(12) + 18(1) = 96 + 18 = 114\text{ g/mol}$.
- Apparent molecular weight of dry air: $M_{air} \approx 28.97\text{ g/mol}$.
**2. Step-by-Step Chemical Combustion Derivation**:
1. **Balanced Stoichiometric Combustion Equation**:
$\text{C}_8\text{H}_{18} + 12.5\left(\text{O}_2 + \frac{79}{21}\text{N}_2\right) \longrightarrow 8\text{CO}_2 + 9\text{H}_2\text{O} + 12.5\left(\frac{79}{21}\right)\text{N}_2$
Moles of air per mole of fuel:
$n_{air} = 12.5 \times \left(1 + \frac{79}{21}\right) = 12.5 \times \frac{100}{21} \approx 59.52\text{ moles of air}$
2. **Mass of Air per Mole of Fuel**:
$m_{air} = 59.52 \times 28.97 \approx 1724.3\text{ g}$
3. **Stoichiometric Fuel-to-Air Mass Ratio**:
$(f/a)_{stoich} = \frac{M_{fuel}}{m_{air}} = \frac{114}{1724.3} \approx 0.0661$
$\text{Air-to-Fuel Ratio } (A/F)_{stoich} = \frac{1}{0.0661} \approx 15.12$
Therefore, $(f/a)_{stoich} \approx 0.066$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** 0.033: ❌ Incorrect. Half the correct ratio.
- **(B)** 0.066: ✅ **CORRECT**. Exactly matches standard stoichiometric octane value $\approx 1/15.1 = 0.066$.
- **(C)** 0.099: ❌ Incorrect. Corresponds to overly fuel-rich mixture.
- **(D)** 0.132: ❌ Incorrect. Twice the stoichiometric ratio.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **B**
- **Exam Rule**: For standard hydrocarbon aviation fuels, $(f/a)_{stoich} \approx 0.066$ to $0.068$ ($A/F \approx 14.7 - 15.1$).
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