Problem 34 - Aircraft Propulsion

Question

In the combustion of gaseous hydrogen and gaseous oxygen, as described by $$ \mathrm{H}_{2(\mathrm{~g})}+\frac{1}{2} \mathrm{O}_{2(\mathrm{~g})} \rightarrow 0.95 \mathrm{H}_{2} \mathrm{O}_{(\mathrm{g})}+0.05 \mathrm{H}_{2(\mathrm{~g})}+0.025 \mathrm{O}_{2(\mathrm{~g})} $$ the reactants entered the combustion chamber at \(T_{1}=\) \(298.16 \mathrm{~K}\) and pressure of \(1 \mathrm{~atm}\). Assume: $$ \begin{aligned} &\left(\Delta \bar{h}_{f}^{0}\right)_{\mathrm{H}_{2} \mathrm{O}(\mathrm{g})}=-241,827 \mathrm{~kJ} / \mathrm{kmol} \quad \bar{c}_{p_{\mathrm{O}_{2}}}=37.8 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K} \\ &\bar{c}_{p_{\mathrm{H}_{2}}}=34.18 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K} \quad \bar{c}_{p_{\mathrm{H}_{2} \mathrm{O}}}=51.0 \mathrm{~kJ} / \mathrm{kmol} \cdot \mathrm{K} \end{aligned} $$ Calculate the (a) heat of reaction, \(Q_{R}\), in \(\mathrm{kJ}\) (b) lower heating value, LHV, in \(\mathrm{kJ} / \mathrm{kg}\) (c) adiabatic flame temperature, \(T_{a f}\), in \(\mathrm{K}\)

Step-by-Step Solution

Answer

The heat of reaction \(Q_{R}\) is -229636.5 kJ/kmol, the lower heating value (LHV) is -12757.58 kJ/kg, and the adiabatic flame temperature \(T_{af}\) is 73.74 K.

Step 1: Calculate the heat of the reaction \(Q_{R}\)

Based on the chemical reaction given, the heat of reaction, \(Q_{R}\), is calculated by multiplying the stoichiometric coefficient of the product (in this case, water vapor \(H_{2}O_{(g)}\)) by its standard enthalpy of formation \(\Delta \bar{h}_{f}^{0}\), and then adding the products of the stoichiometric coefficients and standard enthalpies of formation of the reactants. As we have been given that \(\Delta \bar{h}_{f}^{0}\) = -241827 kJ/kmol for \(H_{2}O_{(g)}\), the heat of the reaction is calculated as: \(Q_{R} = 0.95*(-241827\,kJ/kmol) = -229636.5\,kJ/kmol\)

Step 2: Calculate the Lower Heating Value (LHV)

The Lower Heating Value (LHV) is calculated by dividing the heat of reaction (\(Q_{R}\)) by the molecular weight of the compound. In this case, the molecular weight of \(H_{2}O_{(g)}\) is 18 kg/kmol, so the LHV is calculated as: \(LHV = Q_{R} / Molecular \, Weight = -229636.5 \, kJ/kmol / 18 \, kg/kmol = -12757.58 \, kJ/kg\)

Step 3: Calculate the Adiabatic Flame Temperature \(T_{af}\)

The adiabatic flame temperature (\(T_{af}\)) is the temperature that would be achieved by a flame if the process were adiabatic (i.e., no heat lost to the surroundings). This is calculated using the heat capacities (\(c_{p}\)) and the given initial temperature \(T1\). Using the equation \(T_{af} = T1 + Q_{R} / ((0.95*c_{p_{H_{2}O}} + 0.05*c_{p_{H_{2}}} + 0.025*c_{p_{O_{2}}})\), we can plug in our given values, resulting in: \(T_{af} = 298.16\,K + (-229636.5\, kJ/kmol) / ((0.95*51.0\,kJ/kmol.K) + (0.05*34.18\,kJ/kmol.K) + (0.025*37.8\,kJ/kmol.K)) = 298.16\,K - 224.42\,K = 73.74\,K\)

Key Concepts

Heat of Reaction

The heat of reaction, often represented as \( Q_R \), is a measure of the energy exchanged during a chemical reaction. It reflects the difference in the enthalpy between the reactants and products. In combustion reactions, like the one involving hydrogen and oxygen, it typically measures the energy released when a substance combusts in the presence of oxygen.

In our specific scenario, the heat of reaction is determined using the stoichiometric coefficients from the balanced chemical equation. For example, the reaction forms water vapor, which is a significant product. The standard enthalpy of formation for water vapor, \( H_2O_{(g)} \), is given as \(-241,827 \, \text{kJ/kmol}\). The heat of reaction can be calculated by multiplying this value by the stoichiometric coefficient (0.95 in this case for water vapor), yielding \( Q_R = -229,636.5 \, \text{kJ/kmol} \).

Understanding the heat of reaction is crucial as it helps in predicting how much energy will be released or absorbed during a reaction. This is important for designing safe and efficient chemical processes.

Lower Heating Value (LHV)

The Lower Heating Value (LHV) indicates the amount of heat released during the combustion of a material without accounting for the energy used to vaporize water produced in the reaction. It essentially gives you the usable energy available from the fuel.

To determine the LHV, you take the heat of reaction \( Q_R \) and divide it by the molecular weight of the substance being combusted. Here, we focus on hydrogen burning to form water vapor. With hydrogen's combustion yielding water vapor, considered as \( H_2O_{(g)} \), the molecular weight is \( 18 \, \text{kg/kmol} \).

The calculated LHV is derived as \(-12,757.58 \, \text{kJ/kg} \). Note that this value can help engineers and scientists assess the efficiency of energy systems and choose suitable fuels for various applications.

Adiabatic Flame Temperature

The adiabatic flame temperature \( T_{af} \) is the theoretical maximum temperature a flame can reach if no heat is lost to its surroundings (an adiabatic process). It provides essential insights into the efficiency of combustion processes and is an indication of the reaction's energy content.

To calculate \( T_{af} \), we use information on the specific heat capacities \( c_p \) of the various substances involved. For instance, we leverage the initial temperature, \( T_1 \), and the cumulative heat capacity of the products based on their proportions and given specific heat values. The equation used is:
\[ T_{af} = T_1 + \frac{Q_R}{(0.95 \cdot c_{p_{H_2O}} + 0.05 \cdot c_{p_{H_2}} + 0.025 \cdot c_{p_{O_2}})} \] Given values plug into this equation to derive the \( T_{af} \) as \( 73.74 \, K \). This calculation highlights the energy involved without any losses, crucial for designing combustion systems such as engines or industrial burners.