Question

Air enters a compressor operating at steady state at \(17^{\circ} \mathrm{C}\), 1 bar and exits at a pressure of 5 bar. Kinetic and potential energy changes can be ignored. If there are no internal irreversibilities, evaluate the work and heat transfer, each in \(\mathrm{kJ}\) per \(\mathrm{kg}\) of air flowing, for the following cases: (a) isothermal compression. (b) polytropic compression with \(n=1.3\). (c) adiabatic compression. Sketch the processes on \(p-v\) and \(T-s\) coordinates and associate areas on the diagrams with the work and heat transfer in each case. Referring to your sketches, compare for these cases the magnitudes of the work, heat transfer, and final temperatures, respectively.

Short answer

Isothermal: W ≈ -495 kJ/kg, Q ≈ 495 kJ/kg. Polytropic: W ≈ -345.1 kJ/kg, Q = (using poiytropic equations). Adiabatic: W ≈ -125 kJ/kg, Q = 0.

Step-by-step solution

Identify Given Data and Assumptions

The initial conditions of the air are 17°C (290 K) and 1 bar. The exit pressure is 5 bar. The processes to analyze are isothermal, polytropic with n=1.3, and adiabatic. Ignore kinetic and potential energy changes as well as internal irreversibilities.

Use Ideal Gas Law and Specific Heat Relationships

Since the problems deal with ideal gas processes, use the ideal gas law: \( pV = mRT \), as well as the specific heats relation for air: \( R = 0.287 \, \mathrm{kJ/kg.K} \), \( c_p = 1.005 \, \mathrm{kJ/kg.K} \), and \( c_v = 0.718 \, \mathrm{kJ/kg.K} \).

Work and Heat Transfer for Isothermal Compression

For isothermal compression, the temperature remains constant:- Work done: \[ W = -RT \ln \left( \frac{p_2}{p_1} \right) \]Substitute the values to calculate work: \[ W = -0.287 \, \mathrm{kJ/kg.K} \cdot 290 \, \mathrm{K} \cdot \ln \left( \frac{5}{1} \right) \approx -495 \, \mathrm{kJ/kg} \]- Heat transfer: Since it is isothermal, \[ Q = -W \approx 495 \, \mathrm{kJ/kg} \]

Work and Heat Transfer for Polytropic Compression

For polytropic compression (n = 1.3):- Work done: \[ W = \frac{(p_2V_2 - p_1V_1)}{1 - n} \]Using the polytropic equation: \[ p_1 V_1^n = p_2 V_2^n \] Thus, \[ V_2 = V_1 \left( \frac{p_1}{p_2} \right)^{1/n} \]First find\( V_2 \approx \frac{RT}{p_2}(\frac{p_1}{p_2})^{1/n}\)Substitute to calculate: \[V_2 \approx V_1(\frac{1}{5})^{1/1.3}=%V1*(1/5)^1/1.3 =1.29 m^3/kg Thus, the work becomes,\( W \= \frac{(5\cdot1.29 -1 \cdot2.87)}{(1-1.3)} \approx -345.12 \, \mathrm{kJ/kg.} \This is a simplified approach \]( Q= H-U \) and using Poiytropic temperature relation equations

Work and Heat Transfer for Adiabatic Compression

For adiabatic compression (no heat transfer):- Work done: \[ W = c_v (T_2 - T_1) \]Use the adiabatic relation: \[ T_2 = T_1 \left( \frac{p_2}{p_1} \right)^{\frac{\gamma - 1}{\gamma}} \]where \( \gamma = \frac{c_p}{c_v} \)\( T_2 = 290 \left( \frac{5}{1} \right)^{\frac{0.4}{1.4}} \approx 464 \mathrm{K} \)- Thus, \[ W = 0.718 \cdot (464 - 290) \approx 125 \mathrm{kJ/kg} \]Since adiabatic, \( Q = 0 \).

Sketch Processes on p-v and T-s Coordinates

Plot each process on pressure-volume (p-v) and temperature-entropy (T-s) diagrams.- Isothermal: horizontal line on T-s, negatively sloped curve on p-v- Polytropic: sloped line on both diagrams- Adiabatic: vertical line on T-s and steep negative curve on p-vIndicate work areas under curves for p-v, and note heat transfer for T-s.

Compare Magnitudes

The magnitude of work and heat transfer:- Isothermal compression has the largest work input and equal heat out.- Polytropic compression has moderate work and less heat output.- Adiabatic compression has the smallest work input and no heat transfer.Final temperature comparison:- Adiabatic highest, polytropic moderate, isothermal lowest.

Key concepts

Isothermal Compression

Isothermal compression is a process in which the temperature of the gas remains constant throughout. This is achieved by allowing the gas to exchange heat with its surroundings. According to the Ideal Gas Law, the relationship between pressure and volume during isothermal processes is inversely proportional. Mathematically, the work done in isothermal compression is represented by:

• Work done: \( W = -RT \text{ln}\bigg(\frac{p_2}{p_1}\bigg) \)
• 
  The heat transferred is equal to the magnitude of the work done, which means \( Q = -W \).
  R is the specific gas constant; T is the absolute temperature, and p1 and p2 are the initial and final pressures, respectively. For isothermal compression in air, \( R \) is approximately 0.287 kJ/kg.K, and it remains constant.
• Isothermal processes are typically represented as horizontal lines on a T-s (Temperature vs. Entropy) diagram and as negatively sloped curves on a p-v (Pressure vs. Volume) diagram.

Polytropic Compression

Polytropic compression combines features from both isothermal and adiabatic processes, involving both heat transfer and work done. It follows the polytropic equation \( pV^n = \text{constant} \), where n is the polytropic index. For n = 1.3:

• Work done: \( W = \frac{(p_2V_2 - p_1V_1)}{1 - n} \)
• 
  Volume change: To find \( V_2 \), use \( V_2 = V_1 \bigg(\frac{p_1}{p_2}\bigg)^{1/n} \).
• 
  Polytropic processes are represented as sloped lines on both p-v and T-s diagrams:

A higher value of n means less heat transfer but higher work requirement, while a lower value of n indicates more heat transfer and less work required.

Adiabatic Compression

Adiabatic compression involves compressing a gas without allowing any heat exchange with the surroundings. The change is typically rapid:

• Work done: \( W = c_v (T_2 - T_1) \)
• For the adiabatic process, we use \( T_2 = T_1 \bigg(\frac{p_2}{p_1}\bigg)^{\frac{\text{γ} - 1}{\text{γ}}} \), where γ is the specific heat ratio (\(\text{γ} = \frac{c_p}{c_v} \)).


• 
  Adiabatic processes are shown on a T-s diagram as vertical lines because there is no change in entropy, and on a p-v diagram as steep negative curves.
  
• 
  In adiabatic compression, the work done on the gas results in a significant temperature increase, making it distinct from the other types of compression.

Specific Heats

In thermodynamics, specific heats are critical properties that describe how much heat energy is required to change the temperature of a substance. For gases, we define specific heats at constant volume (\( c_v \)) and at constant pressure (\( c_p \)):

• Constant volume: \( c_v \) represents the energy needed to increase temperature at a constant volume.
• 
  Constant pressure: \( c_p \) represents the energy needed to increase temperature at constant pressure.
The ratio of these specific heats is represented by \( \text{γ} = \frac{c_p}{c_v} \). For air, typical values are \( c_p \text{(constant pressure)} = 1.005 \text{kJ/kg.K} \) and \( c_v \text{(constant volume)} = 0.718 \text{kJ/kg.K} \).
Specific heats play a crucial role in compression calculations, as they help determine energy changes during different thermodynamic processes.