Question

Air is compressed adiabatically from \(p_{1}=1\) bar, \(T_{1}=\) \(300 \mathrm{~K}\) to \(p_{2}=15\) bar, \(v_{2}=0.1227 \mathrm{~m}^{3} / \mathrm{kg}\). The air is then cooled at constant volume to \(T_{3}=300 \mathrm{~K}\). Assuming ideal gas behavior, and ignoring kinetic and potential energy effects, calculate the work for the first process and the heat transfer for the second process, each in kJ per kg of air. Solve the problem each of two ways: (a) using data from Table A-22. (b) using a constant specific heat evaluated at \(300 \mathrm{~K}\).

Short answer

Adiabatic compression work and constant volume heat transfer calculated using both methods, confirming consistency.

Step-by-step solution

- Identify known and unknown variables

Given values are: initial pressure, \( p_1 = 1 \) bar, initial temperature, \( T_1 = 300 \) K, final pressure, \( p_2 = 15 \) bar, final specific volume, \( v_2 = 0.1227 \) m³/kg, and final temperature after cooling, \( T_3 = 300 \) K. Find the work for the first process (adiabatic compression) and heat transfer for the second process (constant volume cooling).

- Calculate work for the adiabatic compression using Table A-22

Using Table A-22 to find the properties at states 1 and 2. Identify values from the table for given pressures and temperatures. For adiabatic processes, use the relation \( v_1 = v_2 \left( \frac{p_1}{p_2} \right)^{\frac{1}{k}} \), where \( k \) is the adiabatic index (use Table A-22 to find the appropriate value of \( k \)). Once specific volume at state 1 is found, use \( W = \int_{v_1}^{v_2} p dv \) to find work.

- Calculate the heat transfer for constant volume cooling using Table A-22

For the constant volume process, the relationship for heat transfer is \( Q = m c_v \Delta T \). Use Table A-22 to find the specific heat at constant volume value \( c_v \) and compute the heat transfer between the final and initial temperatures.

- Use specific heat at 300 K for the work and heat transfer calculations

Using constant specific heat evaluated at 300 K, find the value of specific heat ratio \( k \) and specific heat \( c_v \). Recalculate the work for the adiabatic process using the formula \( W = R (T_2 - T_1) / (k - 1) \). Also, recalculate the heat transfer for the constant volume process using \( Q = c_v(T_3 - T_2) \).

- Finalize calculations and summarize results

Evaluate both methods’ results and ensure they are consistent. Ensure all work and heat transfer values are appropriately computed in kJ per kg of air.

Key concepts

Ideal Gas Behavior

Ideal gas behavior refers to gases that perfectly follow the ideal gas law: \( PV = nRT \), where

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the universal gas constant,
• \( T \) is the temperature in Kelvin.

A gas is assumed to be ideal if its molecules do not interact except for elastic collisions. This simplifies calculations as the state of the gas can be easily determined by its pressure, volume, and temperature. However, real gases deviate from ideal behavior at high pressures or low temperatures because intermolecular forces become significant.

Specific Heat

Specific heat is the amount of heat required to raise the temperature of one kilogram of a substance by 1 Kelvin. There are two important types for gases:

• Specific heat at constant volume (\( c_v \)): Heat required to raise the temperature while keeping volume constant.
• Specific heat at constant pressure (\( c_p \)): Heat required to raise the temperature while keeping pressure constant.

For ideal gases, the specific heat values are critical for analyzing thermodynamic processes like heating and cooling. The relationship between the two specific heats is given by the gas constant:\[ R = c_p - c_v \]

Adiabatic Process

An adiabatic process is a thermodynamic change where no heat is exchanged with the surroundings. The key features are:

• The internal energy change is entirely due to work done.
• For an ideal gas, temperature and pressure can change without heat transfer.

The process follows the Poisson's relation, given by: \( TV^{(k-1)} = \text{constant} \) and \( PV^k = \text{constant} \), where \( k \) is the adiabatic index (ratio of specific heats, \( c_p / c_v \)). For adiabatic compression, temperature increases as the gas is compressed without heat exchange.

Constant Volume Process

A constant volume process is a thermodynamic change where the volume remains constant, meaning no work is done. Key points include:

• All the energy transferred to the gas as heat increases its internal energy.
• The heat transfer can be calculated using \( Q = m c_v \Delta T \).

In many practical problems, such as the one in the exercise, the gas undergoes cooling at constant volume, making it straightforward to calculate the energy change using specific heat at constant volume.