Question

Methane gas \(\left(\mathrm{CH}_{4}\right)\) enters a compressor at \(298 \mathrm{~K}, 1\) bar and exits at 2 bar and temperature \(T\). Employing the ideal gas model, determine \(T\), in \(\mathrm{K}\), if there is no change in specific entropy from inlet to exit.

Short answer

366.54 K

Step-by-step solution

- Understand Given Data

Methane (\(\mathrm{CH}_4\)) enters a compressor at 298 K and 1 bar, and exits at 2 bar with an unknown temperature \(T\). There is no change in specific entropy.

- Ideal Gas Model

Since we are applying the ideal gas model, recall that for an isentropic process involving an ideal gas, the relation between temperature and pressure is given by: \[ \left( \frac{T_2}{T_1} \right) = \left( \frac{P_2}{P_1} \right)^{(k-1)/k} \]

- Specific Heat Ratio

For methane (\(\mathrm{CH}_4\)), the ratio of specific heats (\(k\)) is approximately 1.31.

- Apply Given Values

Let \(T_1 = 298\) K, \(P_1 = 1\) bar, and \(P_2 = 2\) bar. Substitute these values into the isentropic relation: \[ \left( \frac{T_2}{298} \right) = \left( \frac{2}{1} \right)^{(1.31-1)/1.31} \]

- Simplify and Solve

Simplify the exponent first: \[ \left( \frac{2}{1} \right)^{0.31/1.31} \approx 1.23 \]Then solve for \(T_2\): \[ T_2 = 298 \times 1.23 \approx 366.54 ~ \text{K} \]

Key concepts

Isentropic Process

An isentropic process is a thermodynamic process in which the entropy of the system remains constant. This means there is no heat transfer into or out of the system, and any changes in pressure or volume are adiabatic. A perfect example is the compression of gases in isolated conditions. For an ideal gas, the isentropic process can be expressed mathematically using the relationship between temperature and pressure. Understanding this no-change-in-entropy principle is crucial when solving problems involving compressors and turbines, where efficiency maximization is often desired.

Specific Heat Ratio

The specific heat ratio, often denoted by the symbol \(k\) or sometimes \(\gamma \), is the ratio of the specific heat at constant pressure (\(C_p\)) to the specific heat at constant volume (\(C_v\)). It is a critical value in thermodynamics, particularly when dealing with ideal gases. For methane (\(\mathrm{CH}_4\)), this ratio is approximately 1.31. This ratio helps in determining the relationship between different thermodynamic properties of the gas. Knowing the specific heat ratio is essential for calculating the final temperature in processes like those involving compressors, where temperature and pressure changes occur without heat exchange.

Temperature-Pressure Relationship

In thermodynamics, the relationship between temperature and pressure of an ideal gas undergoing an isentropic process is given by the formula: \[ \left( \frac{T_2}{T_1} \right) = \left( \frac{P_2}{P_1} \right)^{(k-1)/k} \] This equation states that the ratio of the final and initial temperatures is directly related to the ratio of the final and initial pressures, raised to the power of \(\frac{k-1}{k}\), where \(k\) is the specific heat ratio. Using this formula, you can determine unknown variables such as the final temperature (\(T_2\)) given initial conditions and specific heat ratio. In the given exercise, the initial temperature and pressures were substituted into this formula to find the exit temperature after compression. Such relationships are pivotal for engineers to understand and predict outcomes in practical applications like compressors and turbines.