Question

An insulated vertical piston-cylinder device initially contains \(0.8 \mathrm{m}^{3}\) of refrigerant-134a at \(1.4 \mathrm{MPa}\) and \(120^{\circ} \mathrm{C}\) A linear spring at this point applies full force to the piston. A valve connected to the cylinder is now opened, and refrigerant is allowed to escape. The spring unwinds as the piston moves down, and the pressure and volume drop to \(0.7 \mathrm{MPa}\) and \(0.5 \mathrm{m}^{3}\) at the end of the process. Determine \((a)\) the amount of refrigerant that has escaped and \((b)\) the final temperature of the refrigerant.

Short answer

Answer: Approximately 7.07 kg of refrigerant-134a has escaped, and the final temperature of the refrigerant is approximately 70°C.

Step-by-step solution

Identify the given data

We have the initial and final properties of the refrigerant. Initial state: - Volume: V1 = \(0.8 \mathrm{m}^{3}\) - Pressure: P1 = \(1.4 \mathrm{MPa}\) - Temperature: T1 = \(120^{\circ} \mathrm{C}\) Final state: - Volume: V2 = \(0.5 \mathrm{m}^{3}\) - Pressure: P2 = \(0.7 \mathrm{MPa}\)

Determine the initial and final amounts of refrigerant

We first need to find the initial and final masses of refrigerant using the specific volume values for refrigerant-134a at the given temperature and pressures. From the refrigerant-134a table, we can find the specific volumes at the given conditions: - Initial state: \(v_1 = 0.06825 \mathrm{m}^{3}/\mathrm{kg}\) - Final state: \(v_2 = 0.1074 \mathrm{m}^{3}/\mathrm{kg}\) We can then use the specific volumes to find the initial and final masses of refrigerant: - Initial mass: \(m_1 = \frac{V_1}{v_1} = \frac{0.8}{0.06825} \approx 11.72 \mathrm{kg}\) - Final mass: \(m_2 = \frac{V_2}{v_2} = \frac{0.5}{0.1074} \approx 4.65 \mathrm{kg}\)

Find the amount of refrigerant that has escaped

To determine the amount of refrigerant that escaped, subtract the final mass from the initial mass: - Escaped refrigerant: \(m_{escaped} = m_1 - m_2 = 11.72 - 4.65 \approx 7.07 \mathrm{kg}\)

Determine the final temperature of the refrigerant

To find the final temperature of the refrigerant, look up the temperature at the final pressure P2 and specific volume v2 using the refrigerant-134a table. - Final temperature: \(T_2 \approx 70^{\circ}\mathrm{C}\) (from refrigerant-134a table) Now we have the two pieces of information requested in the problem: - \((a)\) Amount of refrigerant that has escaped: \(\approx 7.07 \mathrm{kg}\) - \((b)\) Final temperature of the refrigerant: \(T_2 \approx 70^{\circ}\mathrm{C}\)

Key concepts

Refrigerant-134a Properties

Understanding the properties of refrigerant-134a is crucial for solving piston-cylinder problems in thermodynamics. Refrigerant-134a, also known as R-134a, is a hydrofluorocarbon (HFC) used widely as a refrigerant in air conditioning systems for its environmental benefits over previously common refrigerants that contributed to ozone depletion. Its thermodynamic properties include temperature, pressure, specific volume, and enthalpy.

R-134a operates effectively in various phases, including vapor, liquid, and a mixture of both, known as a 'saturated mixture.' These phases are evident in R-134a's use in refrigeration cycles, where it undergoes evaporation and condensation. By analyzing the pressure and temperature, one can determine the specific phase of the refrigerant and its specific volume—the volume per unit mass—at any point in a thermodynamic process.

Tables or software that provide these properties based on measured data can be used to accurately follow the refrigerant's behavior under different conditions. In the context of the piston-cylinder problem, such a refrigerant table was consulted to obtain the initial and final specific volumes based on the given pressures and temperature.

Mass and Volume Calculations

The mass and volume of a substance are fundamental properties used to understand its behavior in thermodynamic systems. Specifically, in the piston-cylinder exercise, we needed to calculate the mass of refrigerant-134a initially and after some of it escaped. The key to connecting mass and volume in such problems lies in the concept of the 'specific volume,' defined as the volume a unit mass of the substance occupies.

To calculate the mass, we divide the total volume of the refrigerant in the cylinder by its specific volume. The accurate determination of mass is crucial because it directly influences the amount of heat transfer, work done by the system, and other important thermodynamic parameters. Recognizing the inverse relationship between mass and specific volume is also important; as specific volume increases, the mass for a given total volume decreases.

When the mass of refrigerant-134a decreases due to some of it escaping from the cylinder, understanding these volume-to-mass calculations helps in determining the altered conditions, such as pressure and temperature, of the remaining refrigerant. Therefore, these calculations are not just theoretical exercises but are practical for real-world engineering applications where precise measurements dictate system efficiency and functionality.

Phase Change and Temperature

Thermodynamics often deals with substances changing phases, such as from liquid to vapor, which is common within refrigeration cycles. The phase change of a substance is inherently connected to its temperature and pressure. In the case of our piston-cylinder problem, understanding the relationship between phase change and temperature is crucial in determining the final state of the refrigerant-134a.

When the refrigerant is allowed to escape the cylinder, a process often accompanied by work interactions and energy transfer, both its pressure and temperature can change. As the pressure drops, so does the temperature, provided the process is adiabatic, meaning no heat is transferred to or from the surroundings.

The temperature at which a phase change occurs at a given pressure is characterized by the saturation temperature. In the solution steps, the saturation temperature corresponding to the final pressure helped identify the final temperature of the refrigerant after the pressure change. Such information is critical for engineers to design and control refrigeration systems to ensure proper functionality across various operating conditions.