Question

A piston-cylinder assembly initially contains \(0.1 \mathrm{~m}^{3}\) of carbon dioxide gas at \(0.3\) bar and \(400 \mathrm{~K}\). The gas is compressed isentropically to a state where the temperature is \(560 \mathrm{~K}\). Employing the ideal gas model and neglecting kinetic and potential energy effects, determine the final pressure, in bar, and the work in \(\mathrm{kJ}\), using (a) data from Table A-23. (b) \(I T\) (c) a constant specific heat ratio from Table A-20 at the mean temperature, \(480 \mathrm{~K}\). (d) a constant specific heat ratio from Table A-20 at \(300 \mathrm{~K}\).

Short answer

Final pressure \( P_2 \approx 1.16 \, bar \); Work done \( W \approx -43.3 \, kJ \)

Step-by-step solution

- Identify Given and Find Data

Initial volume: \( V_1 = 0.1 \, m^3 \), Initial pressure: \( P_1 = 0.3 \, bar \), Initial temperature: \( T_1 = 400 \, K \), Final temperature: \( T_2 = 560 \, K \). We need to find the final pressure \( P_2 \) and work done \( W \).

- Use Ideal Gas Model Relations

For an isentropic process of an ideal gas: \( \frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{\frac{\gamma - 1}{\gamma}} \). Rearrange this equation to solve for \(P_2\): \( P_2 = P_1 \left( \frac{T_2}{T_1} \right)^{\frac{\gamma}{\gamma - 1}} \).

- Calculate Using a Constant Specific Heat Ratio (Table A-20 at 480 K)

Assume a constant specific heat ratio \( \gamma = 1.304 \) from Table A-20 at 480 K. Substitute the known values into the equation: \( P_2 = 0.3 \, bar \left( \frac{560 \, K}{400 \, K} \right)^{\frac{1.304}{1.304 - 1}} \) and compute \( P_{2} \).

- Calculate Work Done

For an isentropic process:\( W = \frac{P_1 V_1 - P_2 V_2}{(1 - \frac{1}{\rm gamma})} \). Use ideal gas law: \( V_2 = V_1 \left( \frac{P_1}{P_2} \right) \left( \frac{T_2}{T_1} \right) \). Once the final pressure and volume are known, substitute into work formula.

- Compare Other Methods

(a) Use data from Table A-23: Find specific enthalpies and entropies at 400 K and 560 K to get final pressure and work. (b) Use Ideal Gas Tables (IT) for compression relations. Each method will have variations due to data precision.

Key concepts

Ideal Gas Model

In thermodynamics, the ideal gas model is a good approximation for real gases. It considers gases as a large number of tiny particles in constant random motion. These particles are assumed to have negligible volume and no intermolecular forces. According to the ideal gas equation, the state of an ideal gas is defined by:ewlineewlineewlineewline\[ PV = nRT \]ewlineewlineewlineewlineIn the context of the piston-cylinder assembly exercise, we use the ideal gas model to relate pressure, volume, and temperature of the gas during the isentropic process. This relationship simplifies our calculations and helps determine various properties like final pressure and work done.ewlineewlineAn isentropic process is a thermodynamic process in which entropy remains constant. This means the process is both adiabatic (no heat transfer) and reversible. For an ideal gas undergoing an isentropic process, the following relation is used:ewlineewlineewlineewline\[ \frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{\frac{\gamma - 1}{\gamma}} \]ewlineewlineewlineewlineThis equation is derived from the basic principles of thermodynamics and allows us to compute the final pressure given the initial and final temperatures and the initial pressure.

Specific Heat Ratio

The specific heat ratio, often denoted as γ (gamma), is a crucial parameter in thermodynamics. It is the ratio of the specific heat at constant pressure (Cp) to the specific heat at constant volume (Cv):ewlineewlineewlineewline\[ \gamma = \frac{C_p}{C_v} \]ewlineewlineewlineewlineFor an ideal gas in an isentropic process, γ influences how pressure, volume, and temperature relate to each other. In the exercise, a constant value of γ from Table A-20 is used. The specific heat ratio varies with temperature, and choosing an appropriate average value (like at 480 K in this case) provides a more accurate result for final pressure and work calculation. Here’s how it’s applied in the process relationship:ewlineewlineewlineewline\[ P_2 = P_1 \left( \frac{T_2}{T_1} \right)^{\frac{\gamma}{\gamma - 1}} \]ewlineewlineewlineewlineUnderstanding γ is important because it dictates how efficiently a gas can do work in thermodynamic cycles. In practical engineering applications, accurate values for γ are essential for predicting system behavior.

Work Calculation

Calculating work in a thermodynamic process involves understanding how energy transfers within the system. For an isentropic process in a piston-cylinder assembly, we derive work through the change in gas volume and pressure. The work done (W) in such a process can be determined by the formula:ewlineewlineewlineewline\[ W = \frac{P_1 V_1 - P_2 V_2}{(1 - \frac{1}{\gamma})} \]ewlineewlineewlineewlineFirst, we need to find the final volume, which can be done using the ideal gas law relation:ewlineewlineewlineewline\[ V_2 = V_1 \left( \frac{P_1}{P_2} \right) \left( \frac{T_2}{T_1} \right) \]ewlineewlineewlineewlineSubstituting this volume into the work formula provides the energy required or released during the compression process. This calculation is crucial for understanding system efficiency and energy requirements.ewlineewlineBy carrying out these calculations, engineers can design and optimize machines like engines and compressors, ensuring they perform efficiently under various operational conditions.

Piston-Cylinder Assembly

A piston-cylinder assembly is a common device in many engines and compressors. It consists of a piston that moves within a cylindrical chamber, compressing or expanding the gas inside. This setup is ideal for studying thermodynamic processes because it allows for controlled volume changes and clear analysis of pressure and temperature changes.ewlineewlineIn this exercise, the piston-cylinder assembly is initially at a volume of 0.1 m³ with carbon dioxide at 0.3 bar and 400 K. As the piston compresses the gas isentropically, it increases the pressure and temperature to 560 K. The final pressure and work done during this process are determined by considering the ideal gas behavior and specific heat ratio.ewlineewlineThe piston-cylinder assembly exemplifies the practical applications of thermodynamic principles in real-world devices. By studying these systems, students can gain a deeper understanding of how theoretical concepts translate into engineering practice.ewlineewlineOverall, mastering the behavior of gases in a piston-cylinder setup is key for anyone pursuing a career in mechanical engineering or related fields, as it forms the basis of much of the machinery and tools we use every day.