Question

A mass of 12 kg of saturated refrigerant- 134 a vapor is contained in a piston-cylinder device at 240 kPa. Now \(300 \mathrm{kJ}\) of heat is transferred to the refrigerant at constant pressure while a 110 -V source supplies current to a resistor within the cylinder for 6 min. Determine the current supplied if the final temperature is \(70^{\circ} \mathrm{C}\). Also, show the process on a \(T-v\) diagram with respect to the saturation lines.

Short answer

#tag_title#Answer#tag_content# 1. To calculate the initial volume of the refrigerant-134a, we'll use the ideal gas law equation: \(PV = mRT\). Given the pressure of 240 kPa and temperature of \(70^{\circ} \mathrm{C}\), the absolute values are 240000 Pa and 343.15 K, respectively. The specific gas constant for refrigerant-134a is R = 0.0815 \(kJ/(kg \cdot K)\), and the mass given is 0.2 kg. Then, we have the initial volume: $$ V_{initial} = \frac{m RT}{P} = \frac{(0.2 \mathrm{kg}) (0.0815 \mathrm{kJ/(kg \cdot K)})(343.15 \mathrm{K})}{240000 \mathrm{Pa}} = 0.02344 \mathrm{m^3} $$ 2. The work done by the piston-cylinder device during the process is: $$ W = P \Delta V $$ Since the final temperature is the same as the initial temperature, the initial and final volumes are the same. Thus, \(\Delta V\) is 0 and the work done is 0. 3. To calculate the electrical power supplied by the 110-V source: $$ P = \frac{energy}{time} = \frac{300 \mathrm{kJ}}{6 \times 60 \mathrm{s}} = 0.833 \mathrm{kW} $$ Now, we can determine the current (I) supplied by the 110-V source: $$ I = \frac{P}{V} = \frac{0.833 \mathrm{kW}}{110 \mathrm{V}} = 7.57 \mathrm{A} $$ 4. To represent the process on a \(T-v\) diagram with respect to the saturation lines, draw a horizontal line starting from the initial state (on the saturation vapor line) and moving towards the right, ending at the final state (also on the saturation vapor line). Label the initial state (P: 240 kPa, T: 343.15 K), the final state (P: 240 kPa, T: 343.15 K), and the constant pressure during the process.

Step-by-step solution

Calculate the initial volume of the refrigerant

First, we need to determine the initial volume of the refrigerant-134a. This can be done using the ideal gas law equation: \(PV = mRT\), where P is pressure, V is volume, m is mass, R is the specific gas constant of refrigerant-134a, and T is temperature. To use this equation, we must first convert the given pressure and temperature into absolute units and find the value of R for refrigerant-134a. At a pressure of 240 kPa and temperature of \(70^{\circ} \mathrm{C}\), the absolute values are 240000 Pa and 343.15 K, respectively. The specific gas constant for refrigerant-134a is R = 0.0815 \(kJ/(kg \cdot K)\). Now we can find the initial volume: $$ V = \frac{m RT}{P} $$

Calculate the work done by the piston-cylinder device

Next, we need to determine the work done by the piston-cylinder device during the process. Since the process is carried out at constant pressure, the work can be calculated as: $$ W = P \Delta V $$ We need to determine the change in volume \(\Delta V\) throughout the process. The final temperature is the same as the initial temperature, so the change in volume can be found as: $$ \Delta V = V - V_{initial} $$ Where \(V\) is the final volume.

Calculate the electrical power supplied by the 110-V source

We are given that 300 kJ of heat is transferred to the refrigerant during the process, and the current supply lasts for 6 minutes. The power (P) supplied by the 110-V source can be calculated as: $$ P = \frac{energy}{time} = \frac{300 \mathrm{kJ}}{6 \times 60 \mathrm{s}} $$ Now, we can determine the current (I) supplied by the 110-V source using the formula: $$ P = IV $$

Show the process on a T-v diagram with respect to the saturation lines

On a \(T-v\) diagram, the initial state of the refrigerant-134a can be represented as a point on the saturation vapor line. The process of adding heat and electrical current occurs at constant pressure, so a horizontal line starting from the initial state and moving towards the right will represent the process. The final state of the refrigerant-134a will also be on the saturation vapor line, as the final temperature is the same as the initial temperature. Remember to label both the initial and final states, the pressure, and the temperature values on the diagram.

Key concepts

Saturated Refrigerant Properties

Understanding the properties of saturated refrigerants is critical when dealing with refrigeration systems and thermodynamic calculations. A refrigerant is considered 'saturated' when it exists as both liquid and vapor at a specific temperature and pressure. This balance point is crucial because it dictates how the refrigerant will behave when heat is added or removed.

In the given exercise, refrigerant-134a is used, which is common in refrigeration cycles. The 'saturated vapor' indicates that before any heat is added, the refrigerant is entirely in vapor form but is on the verge of condensation. Its properties, such as specific volume, enthalpy, and entropy, are then easily referenced from standard refrigeration tables with the given pressure and temperature.

These properties are necessary for determining other parameters like the mass and volume which are essential for calculating work done by the piston-cylinder device and changes throughout the thermal processes.

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, temperature, and amount of an ideal gas. Defined as PV = nRT or PV = mRT, this equation allows us to calculate one of the parameters if the others are known. Here, P stands for pressure, V for volume, n for the number of moles, m for mass, R for the ideal gas constant, and T for temperature.

It’s important to note that the ideal gas law assumes the gas behaves ideally, meaning the particles are point-sized with no intermolecular forces affecting them. Real gases follow this law closely under high temperatures and low pressures. In refrigeration problems, when dealing with superheated regions or non-condensing gases, the ideal gas law provides invaluable insights for calculation and predictions. During the exercise, the law is used to calculate the initial volume of the saturated refrigerant-134a vapor in the piston-cylinder setup.

Constant Pressure Process

A constant pressure process, also known as an isobaric process, occurs when the pressure of the system remains unchanged throughout. In the world of thermodynamics, such processes are particularly important since they are common in practice, especially in piston-cylinder arrangements like the one in our exercise.

During a constant pressure process, any heat added to the system does work on the surroundings and may change the internal energy of the system. The work performed can be calculated as the product of the pressure and the change in volume, expressed as W = PΔV. This formula is applied when the volume of the piston-cylinder system changes while the pressure stays the same. Understanding how to calculate the work done during such processes is key for students, as it is integral to solving various types of thermodynamic problems.

T-v Diagram Thermodynamics

A temperature-volume (T-v) diagram is a graphical representation used in thermodynamics to model processes involving changes in temperature and volume. It is particularly useful for visualizing how a substance, like a refrigerant, behaves under different conditions. On this diagram, temperature (T) is plotted on the horizontal axis and specific volume (v) on the vertical axis.

In refrigeration problems, such as the one given, T-v diagrams help demonstrate the phase change of the refrigerant as it undergoes processes at constant pressure or temperature. The diagram showcases the initial and final states of the substance, including the constant pressure process which is visualized by a horizontal line when the volume changes at a steadfast pressure. For a substance like refrigerant-134a, the diagram would also display the saturation lines, where the refrigerant transitions from liquid to vapor (and vice versa), providing a clear picture of its thermodynamic behavior.