Question

An anaerobic digester (see Chapter 14 for this technology) produces biogas at \(1.1\) bar (absolute) and \(25^{\circ} \mathrm{C}\). This gas, composed of 60 vol. \(\% \mathrm{CH}_{4}\) and 40 vol. \(\% \mathrm{CO}_{2}\), is to be compressed to 25 bar (absolute) before delivery to its end use; assume the compression to be isentropic and reversible. Calculate the temperature after compression.

Short answer

The final temperature after compression is approximately 231.47°C.

Step-by-step solution

Find the specific heat ratio for the gas mixture.

To find the specific heat ratio (\(\gamma\)) for the gas mixture, we'll use the weighted average based on the volume percentages of the gases present. To do this, we need the specific heat ratios for CH4 and CO2. For CH4, \(\gamma_{CH_4}=1.31\) For CO2, \(\gamma_{CO_2}=1.29\) Now, we'll find the weighted average: $$\gamma_{mixture}= \frac{(60\%)(1.31)+(40\%)(1.29)}{100}$$ Calculating this expression gives us: $$\gamma_{mixture} = 1.30$$

Find the initial temperature in Kelvin.

To convert the given initial temperature from Celsius to Kelvin, use the formula: $$T_{1,K} = T_{1,°C} + 273.15$$ Substituting the given value: $$T_{1,K} = 25°\mathrm{C} + 273.15 = 298.15\;\mathrm{K}$$

Calculate the final temperature using the isentropic relationship.

Now that we have the specific heat ratio for the gas mixture and the initial temperature in Kelvin, we can solve for the final temperature using the isentropic relationship: $$\frac{T_{2}}{T_{1}}=\left(\frac{P_{2}}{P_{1}}\right)^{(\gamma -1)/\gamma} $$ We are given the initial pressure, \(P_1 = 1.1\;\mathrm{bar}\), and the final pressure, \(P_2 = 25\;\mathrm{bar}\). Plugging in all the known values: $$\frac{T_{2}}{298.15\;\mathrm{K}}=\left(\frac{25\;\mathrm{bar}}{1.1\;\mathrm{bar}}\right)^{(1.30 -1)/1.30} $$ Solving for \(T_2\), we get: $$T_{2} = 298.15\;\mathrm{K}\left(\frac{25}{1.1}\right)^{0.23077} = 504.62\;\mathrm{K}$$

Convert the final temperature back to Celsius.

Finally, we'll convert the final temperature from Kelvin back to Celsius using the formula: $$T_{2,°C} = T_{2,K} - 273.15$$ Substituting the value we just calculated: $$T_{2,°C} = 504.62\;\mathrm{K} - 273.15 = 231.47°\mathrm{C}$$ The temperature after compression is approximately \(231.47°\mathrm{C}\).

Key concepts

Anaerobic Digestion

Anaerobic digestion is a biological process that breaks down organic materials using microorganisms in an oxygen-free environment. This process is vital for producing biogas, a renewable energy source, which primarily consists of methane (\(\mathrm{CH}_4\)) and carbon dioxide (\(\mathrm{CO}_2\)).
Anaerobic digestion occurs in a series of stages:

• Hydrolysis
• Acidogenesis
• Acetogenesis
• Methanogenesis

Each stage involves distinct groups of bacteria breaking down the organic matter, ultimately resulting in biogas production. The methane-rich biogas is captured and can be used as a fuel source for heating, electricity, or as a vehicle fuel. The solid byproduct left after digestion, called digestate, can be used as fertilizer due to its high nutrient content.
Anaerobic digesters are designed to optimize these biological processes by maintaining ideal conditions such as temperature, pH levels, and retention time. These systems reduce waste and greenhouse emissions, making them an integral part of sustainable waste management strategies.

Isentropic Processes

An isentropic process is a thermodynamic process that occurs at constant entropy. In practical terms, this means that no heat is transferred into or out of the system, resulting in an adiabatic process that is both reversible and dauntingly idealized.
Isentropic processes are significant in systems where compression and expansion occur, like in engines or compressors. When a gas undergoes isentropic compression, it follows a specific relationship between temperature and pressure, calculated using the formula:\[\frac{T_{2}}{T_{1}}=\left(\frac{P_{2}}{P_{1}}\right)^{(\gamma -1)/\gamma} \]where:

• \(T_1\) and \(T_2\) are the initial and final temperatures (in Kelvin).
• \(P_1\) and \(P_2\) are the initial and final pressures.
• \(\gamma\) is the specific heat ratio.

Understanding isentropic processes allows engineers to optimize thermodynamic systems for energy efficiency. By maximizing efficiency, engines and turbines can operate with reduced fuel consumption and emissions.

Specific Heat Ratios

Specific heat ratio, denoted as \(\gamma\), is the ratio of the specific heat at constant pressure to the specific heat at constant volume for a given gas:\[\gamma = \frac{C_p}{C_v}\]The specific heat ratio determines how a gas behaves under processes involving changes in temperature and pressure. It is particularly essential for calculations involving isentropic processes, where the relation between pressure and temperature changes is key.
For mixtures of gases, like those found in biogas systems, the specific heat ratio can be determined by averaging the ratios of the individual gases, weighted by their volume contributions. In our example, the specific heat ratios for methane and carbon dioxide are 1.31 and 1.29 respectively. By calculating the weighted average:\[\gamma_{mixture}= \frac{(60\%)\times(1.31)+(40\%)\times(1.29)}{100} = 1.30\]Knowing \(\gamma\) helps in predicting how energy transfers occur in thermodynamic systems and is crucial for designing efficient energy processes.