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Chapter 13: Problem 56

For a natural gas with a molar analysis of \(85 \% \mathrm{CH}_{4}\), \(8.5 \% \mathrm{C}_{2} \mathrm{H}_{6}, 2.5 \% \mathrm{C}_{3} \mathrm{H}_{8}, 4 \% \mathrm{~N}_{2}\), determine the lower heating value, in \(\mathrm{kJ}\) per kmol of fuel and in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of fuel, at \(25^{\circ} \mathrm{C}, 1 \mathrm{~atm}\)

Short Answer

Expert verified
Lower heating value are 891,140 \text{kJ/kmol} and 51,884 \text{kJ/kg}.

Step by step solution

01

Identify Given Data

The gas composition includes: 85.0% \(\text{CH}_4\)8.5% \(\text{C}_2\text{H}_6\)2.5% \(\text{C}_3\text{H}_8\)4.0% \(\text{N}_2\)
02

Find Lower Heating Values (LHV) of Constituents

The lower heating values (LHV) of each gas at standard conditions are: \(\text{CH}_4\): 50,000 \(\text{kJ/kg}\)\(\text{C}_2\text{H}_6\): 47,000 \(\text{kJ/kg}\)\(\text{C}_3\text{H}_8\): 46,000 \(\text{kJ/kg}\)\(\text{N}_2\): 0 (non-combustible)
03

Determine Molar Mass of Each Component

Molar masses are:\(\text{CH}_4\): 16 \(\text{g/mol}\)\(\text{C}_2\text{H}_6\): 30 \(\text{g/mol}\)\(\text{C}_3\text{H}_8\): 44 \(\text{g/mol}\)\(\text{N}_2\): 28 \(\text{g/mol}\)
04

Calculate Molar Lower Heating Value (LHV_m)

Convert LHV to \(kJ/kmol\) for each component:LHV_m(\text{CH}_4): \(50,000 \text{ kJ/kg} \times 16 \text{ kg/kmol} = 800,000 \text{ kJ/kmol}\)LHV_m(\text{C}_2\text{H}_6): \(47,000 \text{ kJ/kg} \times 30 \text{ kg/kmol} = 1,410,000 \text{ kJ/kmol}\)LHV_m(\text{C}_3\text{H}_8): \(46,000 \text{ kJ/kg} \times 44 \text{ kg/kmol} = 2,024,000 \text{ kJ/kmol}\)
05

Calculate the Mixture's LHV

Using the molar percentages and molar LHVs of each component:\([\text{LHV_mixture}] = 0.85 \times 800,000 + 0.085 \times 1,410,000 + 0.025 \times 2,024,000 + 0.04 \times 0 = 891,140 \text{ kJ/kmol}\)
06

Calculate Mixture's Molar Mass

Using the molar percentages and molar masses of each component:\(\text{MM_mixture} = 0.85 \times 16 + 0.085 \times 30 + 0.025 \times 44 + 0.04 \times 28 = 17.17 \text{ kg/kmol}\)
07

Calculate the LHV per Mass of Mixture

Convert the mixture LHV to \(kJ/kg\) using the molar mass:\(\text{LHV_mass} = \frac{\text{LHV_mixture}}{\text{MM_mixture}} = \frac{891,140 \text{ kJ/kmol}}{17.17 \text{ kg/kmol}} = 51,884 \text{ kJ/kg}\)

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Key Concepts

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

Natural Gas Composition
Natural gas is a mixture of different gases. In this exercise, the natural gas includes specific percentages of methane (CH鈧�), ethane (C鈧侶鈧�), propane (C鈧僅鈧�), and nitrogen (N鈧�). Each gas contributes to the overall properties of the mixture. Methane is usually the major component, followed by smaller amounts of other hydrocarbons like ethane and propane. Nitrogen, on the other hand, is non-combustible. Understanding the composition is crucial as it affects calculations like the Lower Heating Value (LHV), which measures the energy content when the gas is burned.
Molar Mass Calculation
The molar mass of each gas in natural gas is the mass of one mole (or kmol) of that gas. To find the molar mass for a mixture, use the molar percentages and individual molar masses. For example, for methane (CH鈧�), with a molar mass of 16 g/mol, if it makes up 85% of the mixture, you multiply 0.85 by 16. Summing similar calculations for each component yields the mixture鈥檚 total molar mass. This is useful for converting energy content from kJ/kmol to kJ/kg.
Combustion Thermodynamics
Combustion thermodynamics involves understanding how different gases burn and release energy. The Lower Heating Value (LHV) is the measure of the energy released during combustion without accounting for water vapor condensation. Each gas has a specific LHV, e.g., methane (CH鈧�) has 50,000 kJ/kg. Adding these values weighted by their molar percentage in the mixture helps calculate the mixture鈥檚 overall energy content. Knowing these values ensures accurate energy calculations crucial for designing heating systems and determining fuel efficiency.
Energy Content in Fuels
The energy content in fuels like natural gas is vital for determining their efficiency and usage. The primary metric for this is the Lower Heating Value (LHV), expressed in kJ/kmol or kJ/kg. Here, the energy content is calculated for each gas component and then for the mixture. For instance, methane鈥檚 LHV in kJ/kmol is determined by multiplying its LHV in kJ/kg by its molar mass. The total LHV of the gas mixture combines these individual LHVs, giving a comprehensive measure of the fuel's energy output.
Step-by-Step Problem Solving
Step-by-step problem-solving helps break complex problems into manageable parts. For this exercise:
  • Identify the given data, such as gas composition.
  • Find lower heating values of each gas.
  • Determine each gas's molar mass.
  • Calculate the molar LHV for each component.
  • Compute the mixture鈥檚 molar LHV and molar mass.
  • Finally, convert the molar LHV to kJ/kg using the mixture鈥檚 molar mass.
Adopting this organized method simplifies the process and helps ensure accurate results.

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Most popular questions from this chapter

Why are some high-efficiency, residential natural gas furnaces equipped with drain tubes?

Propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) is burned with air. For each case, obtain the balanced reaction equation for complete combustion (a) with the theoretical amount of air. (b) with \(30 \%\) excess air. (c) with \(30 \%\) excess air, but only \(80 \%\) of the propane being consumed in the reaction.

About 234 million tires are discarded in the U.S. annually, adding to a stockpile of two to three billion scrap tires from previous years. Write a memorandum discussing the potential advantages and disadvantages of using scrap tires as a fuel. How does the heating value of scrap tires compare with the heating values of gasoline and commonly used coals? How does the ultimate analysis of scrap tires compare with the ultimate analysis of commonly used coals?

A sample of dried Appanoose County coal has a mass analysis of \(68.8 \%\) carbon, \(6.1 \%\) hydrogen \((\mathrm{H}), 8.8 \%\) oxygen \((\mathrm{O}), 1.7 \%\) nitrogen \(\left(\mathrm{N}_{2}\right), 6.6 \%\) sulfur, and the rest noncombustible ash. For complete combustion with the theoretical amount of air, determine (a) the amount of \(\mathrm{SO}_{2}\) produced, in \(\mathrm{kg}\) per \(\mathrm{kg}\) of coal. (b) the air-fuel ratio on a mass basis.

Pentane \(\left(\mathrm{C}_{5} \mathrm{H}_{12}\right.\) ) burns completely with the theoretical amount of air at \(25^{\circ} \mathrm{C}, 1\) atm, \(75 \%\) relative humidity. Determine (a) the balanced reaction equation. (b) the dew point temperature of the combustion products, at \(1 \mathrm{~atm}\). (c) the amount of water condensed, in \(\mathrm{kmol} / \mathrm{kmol}\) of fuel, if the combustion products are cooled to \(25^{\circ} \mathrm{C}\), at \(1 \mathrm{~atm}\).
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