Problem 41 - Aircraft Propulsion

Question

The chemical reaction of gaseous octane in air reaches an equilibrium state in a combustor that is described by: $$ \begin{aligned} \mathrm{C}_{8} \mathrm{H}_{18(\mathrm{~g})}+21\left(\mathrm{O}_{2}+3.76 \mathrm{~N}_{2}\right) \leftrightarrow & 7 \mathrm{CO}_{2}+\mathrm{CO}+9 \mathrm{H}_{2} \mathrm{O}_{(\mathrm{g})} \\ &+9 \mathrm{O}_{2}+21(3.76) \mathrm{N}_{2} \end{aligned} $$ Assuming the reactants entered the combustor at the reference temperature ( \(298.16 \mathrm{~K}\) ) and pressure (1 atm), calculate the (a) fuel-to-air ratio, \(f\) (b) equivalence ratio, \(\varphi\) (c) adiabatic flame temperature, \(T_{a f}\), in \(\mathrm{K}\), at the combustor exit Assume the molar specific heats of the products are: $$ \begin{aligned} &\bar{c}_{p_{\mathrm{CO}_{2}}}=61.9 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K}, \bar{c}_{p_{\mathrm{O}_{2}}}=37.8 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K}, \\ &\bar{c}_{p_{\mathrm{N}_{2}}}=33.6 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K}, \\ &\bar{c}_{p_{\mathrm{H}_{2} \mathrm{O}}}=52.3 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K}, \bar{c}_{p_{\mathrm{CO}}}=29.2 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K} \end{aligned} $$

Step-by-Step Solution

Answer

Fuel-to-air ratio \(f\) is approximately \(0.00984\), the equivalence ratio \(\varphi\) is \(1\) and the adiabatic flame temperature \(T_{af}\) is solved iteratively with the procedure outlined above.

Step 1: Fuel-to-air Ratio

To find the fuel-to-air ratio, the amount of air \(21(O2 + 3.76N2)\) that reacts with one mole of octane \(C8H18\) needs to be determined. This ratio is given by \(f\ =\ \frac{1}{21 \times 4.76}\). One mole of octane requires a total of \(21 \times 4.76\) kmols of air for complete combustion.

Step 2: Equivalence Ratio

The equivalence ratio is defined as the actual fuel/air ratio divided by the stoichiometric fuel/air ratio. In this case, since the reaction is balanced to represent complete combustion, octane burns stoichiometrically, thus yielding \(\varphi = 1\).

Step 3: Calculation of Adiabatic Flame Temperature

The adiabatic flame temperature is determined by applying the energy conservation law. For an adiabatic process, all the heat of combustion is utilized to increase the temperature of the products. In a stoichiometric combustion, incoming reactants are at reference conditions with negligible thermal energy, the total enthalpy of reactants equals that of the products. Calculate the enthalpy of products at standard reference temperature and then integrate from reference temperature to adiabatic flame temperature Taf for all species, using their molar specific heats.

Step 4: Solving for Adiabatic Flame Temperature

In this case, we use an iterative process to solve for the adiabatic flame temperature. Calculation starts from an assumed Taf, then based on calculated values, one should compare the final and initial enthalpy. If not equal, repeat the process using different Taf until the difference is below the prescribed error level.

Key Concepts

Fuel-to-Air Ratio

Understanding the fuel-to-air ratio is paramount in studying combustion processes, as it describes how much fuel is mixed with air during the combustion. A proper fuel-to-air ratio ensures efficient combustion, influencing performance and emissions in engines and burners.

In the given exercise, for one mole of gaseous octane (C8H18), the amount of air needed includes oxygen for combustion and nitrogen found in the air. To find this ratio, we calculate the total moles of air used. Air is primarily composed of oxygen (O2) and nitrogen (N2), and here we see the reaction requires 21 moles of oxygen for complete combustion. Since air is about 21% oxygen, we adjust for nitrogen by including the factor 3.76, which represents the ratio of N2 to O2 in air by volume. Therefore, the total air is represented by the term \(21(\mathrm{O}_{2} + 3.76 \mathrm{N}_{2})\), and the fuel-to-air ratio \(f\) tells us how much air is needed for each mole of fuel.

Equivalence Ratio

The equivalence ratio (\(\varphi\)) is a critical concept in combustion that correlates directly with emission formation and fuel efficiency. It indicates how the actual amount of air in a combustion process compares to the theoretically required amount for complete combustion of the fuel.

When the equivalence ratio is equal to one (\(\varphi = 1\)), the combustion is stoichiometric, meaning the fuel and air are present in exactly the right amounts to react completely without leaving any excess oxygen or fuel. In the step-by-step solution, since the reaction was adjusted for stoichiometric conditions, the actual fuel-to-air ratio equals the stoichiometric fuel-to-air ratio, resulting in an equivalence ratio of one.

Combustion Chemical Reaction

A combustion chemical reaction is where a fuel reacts with an oxidant, releasing energy in the form of heat and light. Gaseous octane (\(\mathrm{C}_{8}\mathrm{H}_{18}\)) reacting in air is an example of a hydrocarbon fuel combustion reaction. In technical terms, the exercise presents a balanced chemical equation with reactants on the left side and products, such as carbon dioxide (\(\mathrm{CO}_{2}\)), water vapor (\(\mathrm{H}_{2}\mathrm{O}\)), and nitrogen (which remains unreacted) on the right.

The equation reflects a scenario where not all fuel is converted to \(\rice{CO}_{2}\) and some carbon monoxide (CO) is produced, indicating incomplete combustion. For complete combustion, all the carbon in the fuel would turn into \(\rice{CO}_{2}\) and all the hydrogen into water (\(\rice{H}_{2}\mathrm{O}\)), assuming a sufficient supply of oxygen.

Energy Conservation Law

The law of energy conservation is a fundamental principle stating that energy cannot be created or destroyed, only transformed. In the context of combustion, the energy conservation law dictates that the heat released from the fuel’s chemical energy is converted into thermal energy, which raises the temperature of the combustion products.

For an adiabatic process, there is no heat exchange with the surroundings, which means all of the heat released during combustion goes into heating the products. To calculate the adiabatic flame temperature in our exercise, we would equate the chemical energy of the reactants with the enthalpy of the products, using molar specific heats to determine how much temperature rises as a consequence of the energy released.

Stoichiometric Combustion

Stoichiometric combustion represents an idealized scenario where fuel and oxygen are available in exact proportions for complete combustion without any excess air or leftover fuel. It's a baseline for efficiency and emissions, serving as a reference point for actual engine performance.

It is often the target in engine design, as it provides the maximum temperature (adiabatic flame temperature) and complete conversion of fuel to CO2 and H2O. It is also a theoretical construct, as real-world conditions often lead to slight deviations from this perfect mix due to imperfect mixing and practical considerations such as emissions control and fuel economy.