Problem 36 - Aircraft Propulsion

Question

Lean combustion of liquid \(n\)-decane (from the kerosene family) and air at a reference temperature of \(T_{f}=298.16 \mathrm{~K}\) and pressure of 1 bar is approximated by the following chemical reaction: $$ \begin{aligned} \mathrm{C}_{10} \mathrm{H}_{22(\mathrm{l})}+25\left(\mathrm{O}_{2}+3.76 \mathrm{~N}_{2}\right) \rightarrow & 10 \mathrm{CO}_{2(\mathrm{~g})}+11 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \\ &+\frac{19}{2} \mathrm{O}_{2(\mathrm{~g})}+25(3.76) \mathrm{N}_{2(\mathrm{~g})} \end{aligned} $$ Assuming the standard heats of formation in this reaction are: $$ \begin{aligned} \left.\Delta \bar{h}_{f}^{o}\right|_{\mathrm{C}_{10} \mathrm{H}_{22(\mathrm{l})}} &=-300,900 \mathrm{~kJ} / \mathrm{kmol} \\ \left.\Delta \bar{h}_{f}^{o}\right|_{\mathrm{H}_{2} \mathrm{O}_{(\mathrm{g})}} &=-241,827 \mathrm{~kJ} / \mathrm{kmol} \\ \left.\Delta \bar{h}_{f}^{o}\right|_{\mathrm{CO}_{2(\mathrm{~g})}} &=-393,522 \mathrm{~kJ} / \mathrm{kmol} \end{aligned} $$ and average molar specific heats at constant pressure are: $$ \begin{array}{ll} \bar{c}_{p_{\mathrm{CO}_{2}}}=54.31 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K} & \bar{c}_{p_{\mathrm{O}_{2}}}=34.88 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K} \\ \bar{c}_{p_{\mathrm{N}_{2}}}=32.7 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K} & \bar{c}_{p_{\mathrm{H}_{2} \mathrm{O}}}=41.22 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K} \end{array} $$ Calculate (a) the fuel-to-air ratio, \(f\), for this reaction (b) write the stoichiometric reaction for this fuel in air (c) the equivalence ratio, \(\phi\) (d) the adiabatic flame temperature, \(T_{a f}\), in \(\mathrm{K}\) in the first reaction (e) the Lower Heating Value (LHV) of this fuel, in \(\mathrm{kJ} / \mathrm{kg}\), in stoichiometric combustion in oxygen

Step-by-Step Solution

Answer

The fuel-to-air ratio, \(f\), stoichiometric reaction, equivalence ratio, \(\phi\), adiabatic flame temperature, \(T_{af}\), and the Lower Heating Value (LHV) would be calculated based on the steps outlined in this solution. The specific numerical values will depend on the numerical solutions to the equations within the steps.

Step 1: (a) Calculate the fuel-to-air ratio.

The fuel-to-air ratio, \(f\), can be found from the balanced chemical equation: \(\mathrm{C_{10} H_{22}} + 25 (\mathrm{O_2} + 3.76 \mathrm{N_2} ) \rightarrow \dots \). Here, 1 kmol of \(n\)-decane reacts with 25 kmol of \(\mathrm{O_2}\) (along with \(3.76 \times 25 = 94\) kmol of \(\mathrm{N_2}\)). The fuel-to-air ratio is defined as the mass of the fuel divided by the mass of the dry air. This can be calculated using the molecular masses of \(n\)-decane (\(\mathrm{C_{10} H_{22}}\), 142 kg/kmol), \(\mathrm{O_2}\) (32 kg/kmol), and \(\mathrm{N_2}\) (28 kg/kmol): \(f = \frac{142}{25 \times 32 + 3.76 \times 25 \times 28}\)

Step 2: (b) Write the stoichiometric reaction for this fuel in air.

Here, the balanced stoichiometric combustion reaction for \(n\)-decane in air is required, which has already been provided: \(\mathrm{C_{10} H_{22}} + 25 (\mathrm{O_2} + 3.76 \mathrm{N_2} ) \rightarrow 10 \mathrm{CO_{2}} + 11 \mathrm{H_2 O} + \frac{19}{2} \mathrm{O_{2}} + 25(3.76) \mathrm{N_{2}}\)

Step 3: (c) Calculate the equivalence ratio.

The equivalence ratio, \(\phi\), is defined as the actual fuel-to-air ratio \(f\) divided by the stoichiometric fuel-to-air ratio. In this case, since the reaction is occurring under stoichiometric conditions (all reactants are completely consumed), \(\phi\) is 1.

Step 4: (d) Calculate the adiabatic flame temperature.

The adiabatic flame temperature (\(T_{af}\)) can be calculated by solving the energy balance: using the given standard heats of formation \(\Delta \bar{h}_{f}^{o}\) and given specific heats \( \bar{c}_{p}\) for all species, this requires solving the equation: \(\Delta \bar{h}_{f}^{o}(\mathrm{fuel}) + 25 \Delta \bar{h}_{f}^{o}(\mathrm{O_{2}}) = 10 \Delta \bar{h}_{f}^{o}(\mathrm{CO_{2}}) + 11 \Delta \bar{h}_{f}^{o}(\mathrm{H_{2}O}) + \frac{19}{2} \Delta \bar{h}_{f}^{o}(\mathrm{O_{2}}) + 25 \times 3.76 \Delta \bar{h}_{f}^{o}(\mathrm{N_{2}}) + \Delta H^{o}\), where \(\Delta H^{o}\) is calculated by integrating the specific heat of each gas component over the temperature range.

Step 5: (e) Calculate the Lower Heating Value (LHV).

The Lower Heating Value (LHV) corresponds to the heat released by completely burning 1 kg of fuel while the water vapor remains in the gaseous state. It can be calculated from the heats of formation of the fuel and the combustion products: \( \mathrm{LHV} = - \Delta \bar{h}_{f}^{o}(\mathrm{fuel}) + 10 \Delta \bar{h}_{f}^{o}(\mathrm{CO_{2}}) + 11 \Delta \bar{h}_{f}^{o}(\mathrm{H_{2}O})\).

Key Concepts

Fuel-to-Air Ratio

Understanding the fuel-to-air ratio is crucial in combustion processes. It is defined as the ratio of the mass of the fuel to the mass of the air used for combustion. This ratio determines how efficiently the fuel burns. For a given chemical reaction, like the combustion of n-decane, you can find this ratio by analyzing the balanced chemical equation. In this scenario, 1 kmol of n-decane reacts with 25 kmol of oxygen plus the equivalent nitrogen content, making the air involved predominantly nitrogen due to Earth's atmosphere composition. The molecular masses used in the ratio calculation are important, and for n-decane, this value is taken as 142 kg/kmol. For \({O_2}\), it is 32 kg/kmol and for nitrogen \({N_2}\), it is 28 kg/kmol. The calculation can be simplified by computing: \( f = \frac{142}{25 \times 32 + 3.76 \times 25 \times 28}\), which gives an idea of how much fuel is needed for a given mass of air. This ratio is essential for engineers designing efficient combustion systems.

Stoichiometric Reaction

The term stoichiometric reaction refers to a chemical reaction where all fuel and oxygen react without any excess or deficiency. Basically, in this type of reaction, all the fuel is burnt completely because the amount of oxygen is theoretically precise. For a perfect stoichiometric mixture, every atom of fuel finds an atom of oxygen to react with. In the context of n-decane combustion, the stoichiometric equation is written as: \({C_{10}H_{22}} + 25(O_{2} + 3.76 N_{2}) \to 10 CO_{2} + 11 H_{2}O + \frac{19}{2} O_{2} + 25(3.76) N_{2}\). This balanced equation ensures that all fuel components are turned into stable combustion products, primarily carbon dioxide and water vapor. Stoichiometric ratios are critical for optimizing fuel utilization and minimizing pollutants in combustion systems.

Adiabatic Flame Temperature

The adiabatic flame temperature refers to the maximum temperature of the combustion products assuming no heat loss to the environment. It provides a theoretical upper limit on the temperature achieved during the combustion process. Calculating this temperature involves complex energy balance equations using the heats of formation and specific heat capacities of the reactants and products. Essentially, the calculation offsets the total energy released by the chemical reaction against the energy used to heat up the products to reach equilibrium at the adiabatic flame temperature. In practice, some energy is lost, so real temperatures are typically lower than the adiabatic calculation. Engineers use this figure to estimate the potential maximum efficiency and temperature of their combustion systems, which is pivotal in avoiding material failure due to overheating.

Lower Heating Value

Lower Heating Value (LHV) pertains to the amount of heat released by a specific mass of fuel when it undergoes complete combustion, with the water resulting from the combustion remaining in vapor form. This value is vital for understanding the energy potential of a fuel in real-world conditions. The LHV differs from the Higher Heating Value (HHV) because it subtracts the energy contained in water as vapor—the condition under standard atmospheric pressure and temperature. To calculate the LHV, consider the heats of formation of the fuel and the combustion products. The formula \(LHV = -\Delta \bar{h}_{f}^{o}(\text{fuel}) + 10 \Delta \bar{h}_{f}^{o}(\text{CO}_{2}) + 11 \Delta \bar{h}_{f}^{o}(\text{H}_{2}O)\) depicts how the heat of formation is converted into useful heat energy. This value helps assess how efficient a particular fuel is in terms of energy output.