Question

Methane gas \(\left(\mathrm{CH}_{4}\right)\) at \(280 \mathrm{~K}, 1\) bar enters a compressor operating at steady state and exits at \(380 \mathrm{~K}, 3.5\) bar. Ignoring heat transfer with the surroundings and employing the ideal gas model with \(\bar{c}_{p}(T)\) from Table A-21, determine the rate of entropy production within the compressor, in \(\mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}\).

Short answer

The rate of entropy production within the compressor is equal to the total entropy change, \(\Delta s\).

Step-by-step solution

Identify Given Data

Extract and list the given parameters from the problem statement:- Initial temperature, \(T_1 = 280 \mathrm{K}\)- Final temperature, \(T_2 = 380 \mathrm{K}\)- Initial pressure, \(P_1 = 1 \mathrm{bar}\)- Final pressure, \(P_2 = 3.5 \mathrm{bar}\)

Use Ideal Gas Model

Since the ideal gas model is assumed, start by identifying the specific heat capacity of methane, \(\mathrm{CH}_4\), from Table A-21: \(\bar{c}_p(T)\). For differentiation, consider the general form \(\bar{c}_p = a + bT + cT^2 + dT^3\).

Calculate Change in Entropy

Use the formula for entropy change for an ideal gas: \[\Delta s = \int_{T_1}^{T_2} \frac{\bar{c}_p(T)}{T} dT - R \ln\left( \frac{P_2}{P_1} \right) \]Here, \(R\) is the specific gas constant for methane (approximately \(0.518 \mathrm{kJ/kg \cdot K}\)).

Perform Integration for \(\Delta s = \int_{T_1}^{T_2} \frac{\bar{c}_p(T)}{T} dT\)

Integrate to find the change in entropy due to temperature difference: \[\Delta s_T = \int_{280}^{380} \frac{a + bT + cT^2 + dT^3}{T} dT \]This will yield: \[\Delta s_T = a\ln\left( \frac{T_2}{T_1} \right) + b(T_2 - T_1) + \frac{c}{2}(T_2^2 - T_1^2) + \frac{d}{3}(T_2^3 - T_1^3) \]

Calculate Pressure Entropy Term

Determine the change in entropy due to pressure change: \[\Delta s_P = -R \ln\left( \frac{P_2}{P_1} \right) \]With the given pressures: \[\Delta s_P = -0.518 \ln\left( \frac{3.5}{1} \right) \]

Calculate Total Entropy Change

Combine both entropy changes: \[\Delta s = \Delta s_T + \Delta s_P \]Numerically evaluate both terms to get \(\Delta s\).

Rate of Entropy Production

Since no heat is transferred to the surroundings: \[\sigma = \left( \frac{dS}{dt} \right)_{sys} = \Delta s \]Thus, the rate of entropy production within the compressor is equal to \(\Delta s\).

Key concepts

Ideal Gas Model

The ideal gas model is a simple way to describe the behavior of gases under various conditions. It assumes that gas molecules occupy no volume and that they do not interact with one another, making calculations straightforward. This model is especially useful under standard temperature and pressure conditions. Methane gas (\text{CH}_4), like many other gases, can often be approximated as an ideal gas. This significantly simplifies our calculations by allowing us to use ideal gas laws and formulae. The behavior of an ideal gas is defined by the equation:\( PV = nRT \)where: P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. This relation shows how gas properties change with pressure, volume, and temperature and is the backbone of the calculations for processes such as compression or expansion of gases.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree. For gases, we often deal with specific heat at constant pressure, denoted as \( c_p \). The specific heat capacity of a gas can depend on temperature and is typically represented as a polynomial: \( \bar{c}_p = a + bT + cT^2 + dT^3 \). These coefficients (a, b, c, and d) are specific to each gas and are often found in tables, such as Table A-21 in thermodynamics textbooks.

• For methane (\text{CH}_4), the specific heat capacity formula helps us calculate how much entropy changes as the temperature of the gas changes. We integrate \( \frac{\bar{c}_p(T)}{T} \) between two temperatures (T1 and T2) to find the change in entropy due to temperature: \( \Delta s_T = \int_{280}^{380} \frac{a + bT + cT^2 + dT^3}{T} dT \).

This integration accounts for the varying heat capacity with temperature, making it more accurate for real-world scenarios.

Entropy Change

Entropy change is a measure of how the disorder or randomness of a system changes. For an ideal gas undergoing a process from one state to another, the entropy change can be calculated using the formula: \( \Delta s = \int_{T_1}^{T_2} \frac{\bar{c}_p(T)}{T} dT - R \ln\left( \frac{P_2}{P_1} \right) \), combining changes due to temperature (\( \Delta s_T \)) and changes due to pressure (\(\Delta s_P \)).

• For temperature changes: This part uses specific heat capacity values integrated over the temperature range.• For pressure changes: This involves the specific gas constant R of methane and the natural logarithm of the pressure ratio.

Understanding these individual components is crucial to grasp how entropy is produced or reduced during processes like compression. In this exercise, entropy production in the compressor is calculated by summing the entropy changes due to both temperature and pressure modifications. The total change in entropy tells us how much more 'disordered' the system becomes through the process.