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A natural gas containing 82.0 mole \(\% \mathrm{CH}_{4}\) and the balance \(\mathrm{C}_{2} \mathrm{H}_{6}\) is burned with \(20 \%\) excess air in a boiler furnace. The fuel gas enters the furnace at \(298 \mathrm{K}\), and the air is preheated to 423 \(\mathrm{K}\). The heat capacities of the stack-gas components may be assumed to have the following constant values: $$\begin{aligned}\mathrm{CO}_{2}: & C_{p}=50.0 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K}) \\ \mathrm{H}_{2} \mathrm{O}(\mathrm{v}): & C_{p}=38.5 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K}) \\\\\mathrm{O}_{2}: & C_{p}=33.1 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K}) \\ \mathrm{N}_{2}: & C_{p}=31.3 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})\end{aligned}$$ (a) Assuming complete combustion of the fuel, calculate the adiabatic flame temperature. (b) How would the flame temperature change if the percent excess air were increased? How would it change if the percentage of methane in the fuel increased? Briefly explain both of your answers.

Step by step solution

01

Calculate the moles of fuel

First, find the number of moles of CH4 and C2H6 in one mole of the gas. Since the fuel contains 82.0 mole% CH4 and the remainder is C2H6, there are 0.82 moles of CH4 and 0.18 moles of C2H6.

02

Perform Stoichiometric Calculations

Use the chemical equations to find the number of moles of oxygen required for complete combustion. For CH4, the equation is \(CH4 + 2O2 \rightarrow CO2 + 2H20\). Thus, each mole of CH4 requires 2 moles of O2 for combustion. Similarly, the equation for C2H6 is \(2C2H6 + 7O2 \rightarrow 4CO2 + 6H20\). So, each mole of C2H6 requires 7/2 moles of O2 for combustion. Given that 20\% excess air is used, calculate the number of moles of O2 supplied. 1 mole of air is 21 mol% O2 and 79 mol% N2.

03

Calculate the stack-gas composition

Calculate the moles of each gas in the combustion products or stack gas. For complete combustion, all carbon in the fuel goes to CO2 and all hydrogen goes to H2O. Oxygen in excess of the stoichiometric requirement exits as O2. The balance of the air, N2, also exits unchanged.

04

Apply the heat balance equation

Apply the heat balance equation (adiabatic flame, no heat loss or gain, \(\sum \Delta H_{reactants} = \sum \Delta H_{products}\)). The contribution of each component to \(\Delta H_{products}\) is (number of moles) \(\times (Cp)\) \(\times (T - T_{reference})\). Equate heat produced by combustion of reactants (there is no energy change for incoming air as its components are also in the stack gas). Set this equal to the heat capacity of the stack gas multiplied by the temperature rise, and solve for the flame temperature, \(T\).

05

Discuss the effect of increasing the percentage of excess air

Increasing the percentage of excess air would decrease the flame temperature. This is because the excess air, primarily nitrogen, absorbs some of the heat generated during combustion, but does not participate in the reaction. Hence, more excess air causes more heat to be absorbed, lowering the flame temperature.

06

Discuss the effect of increasing the percentage of methane in the fuel

Increasing the percentage of methane (CH4) in the fuel would increase the flame temperature. This is because methane has a higher heat of combustion than C2H6, such that for each mole of methane burned, more heat is produced than for a mole of C2H6. Thus, greater methane percentage leads to a higher flame temperature.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Adiabatic Flame Temperature

In the context of combustion analysis, the adiabatic flame temperature is essential. It refers to the temperature a flame reaches under the assumption that no heat is lost to the surroundings. The temperature depends on several factors, such as the type of fuel used and the amount of excess air introduced.

When calculating it, one needs to apply the principle of conservation of energy. This means that the total enthalpy of reactants must equal the total enthalpy of products. During combustion, the heat from burning is used to raise the temperature of gases produced. Thus, properly determining the temperature involves knowing both the specific heat capacities of the products and the amount of thermal energy released.

• An accurate measure of the adiabatic flame temperature is critical as it influences the efficiency of processes like power generation and heating.
• Higher flame temperatures typically indicate more efficient combustion and energy release.
• Variables like air preheat and ambient temperature significantly impact the final temperature.

Stoichiometric Combustion

Stoichiometric combustion involves a perfect balance between fuel and oxygen, ensuring all fuel is burned with no leftover oxygen. In our scenario, this concept helps calculate the precise amount of oxygen required for burning methane (\(\mathrm{CH}_{4}\)) and ethane (\(\mathrm{C}_{2}\mathrm{H}_{6}\)).

For methane, the reaction can be written as: \(\mathrm{CH}_{4} + 2\mathrm{O}_{2} \rightarrow \mathrm{CO}_{2} + 2\mathrm{H}_{2}\mathrm{O}\). This equation shows that each mole of methane requires 2 moles of oxygen. Similarly, ethane needs \(\dfrac{7}{2}\) moles of \(\mathrm{O}_{2}\) for every mole of \(\mathrm{C}_{2}\mathrm{H}_{6}\), as expressed by \(2\mathrm{C}_{2}\mathrm{H}_{6} + 7\mathrm{O}_{2} \rightarrow 4\mathrm{CO}_{2} + 6\mathrm{H}_{2}\mathrm{O}\).

• In stoichiometric combustion, efficiency peaks because all fuel is used for energy production without wasting resources.
• It forms the basis for determining the right amount of excess air needed in industrial settings.
• Skillfully managing combustion around these principles ensures minimal emissions and energy losses.

Excess Air in Combustion

Using excess air in combustion impacts the system's temperature and efficiency. It refers to the amount of air supplied beyond the stoichiometric requirement. In our example, there's a 20% excess air used, implying more oxygen than is theoretically required.

While adding some excess air ensures thorough combustion by ensuring complete reaction with the fuel, too much excess air can cool down the reaction. That's because the extra nitrogen in the air, which does not react, absorbs energy without contributing to the combustion process.

• Adding excess air is a common practice in industrial combustion due to its safety cushion, preventing the formation of harmful pollutants like carbon monoxide.
• Optimal excess minimizes energy losses while ensuring full combustion.
• However, an increase in excess air leads to a lower adiabatic flame temperature, highlighting the trade-off that must be managed.

Methane Combustion

Combusting methane is a crucial concept in this analysis. Methane (\(\mathrm{CH}_{4}\)) is a simple hydrocarbon with a high energy content, making it an efficient fuel choice.

Its combustion reaction is straightforward, producing carbon dioxide (\(\mathrm{CO}_{2}\)) and water (\(\mathrm{H}_{2}\mathrm{O}\)). This reaction releases a significant amount of heat energy, contributing to a higher potential flame temperature compared to heavier hydrocarbons like ethane (\(\mathrm{C}_{2}\mathrm{H}_{6}\)).

• Methane's high heat of combustion is beneficial in generating more heat per mole, leading to higher energy efficiency when used in engines or heating systems.
• Its combustion tends to produce fewer carbon-based pollutants, making it an environmentally friendly fuel compared to other fossil fuels.
• When methane composition in fuel increases, the adiabatic flame temperature also increases, enhancing the combustion efficiency of the system.