Question

\(4-35 \quad\) ? \(\mathrm{A}\) piston-cylinder device initially contains steam (f) at \(200 \mathrm{kPa}, 200^{\circ} \mathrm{C}\), and \(0.4 \mathrm{m}^{3}\). At this state, a linear spring \((F \propto x)\) is touching the piston but exerts no force on it. Heat is now slowly transferred to the steam, causing the pressure and the volume to rise to \(250 \mathrm{kPa}\) and \(0.6 \mathrm{m}^{3}\) respectively. Show the process on a \(P\) -v diagram with respect to saturation lines and determine ( \(a\) ) the final temperature, (b) the work done by the steam, and ( \(c\) ) the total heat transferred.

Short answer

Answer: The process can be displayed on a P-v diagram by plotting the initial state (200 kPa and 0.4 m³) and final state (250 kPa and 0.6 m³) relative to the saturation curve. The final temperature can be found using steam tables and the final pressure and specific volume. The work done by the steam can be calculated using the equation for work done during a linear expansion process, while the total heat transferred can be found using the first law of thermodynamics for a closed system and the change in internal energy.

Step-by-step solution

Determine the initial state and show it on the P-v diagram

Steam at 200 kPa and 200°C initially fills the cylinder. Using the steam tables, we can determine that this steam is in the superheated state. We can plot this point on the P-v diagram relative to the saturation curve.

Determine the final state and show it on the P-v diagram

After the heating process, the steam reaches a final pressure of 250 kPa and a final volume of 0.6 m³. We can plot this point on the P-v diagram relative to the saturation curve as well.

Determine the final temperature (a)

To find the final temperature, we can use the final pressure and specific volume. The final specific volume can be calculated as follows: \(v_{2}=\frac{V_{2}}{m}=\frac{0.6 \mathrm{m}^{3}}{m}\) (where m is the mass of the steam in the cylinder) With the final pressure given as 250 kPa, we will use the steam tables to find the temperature at this pressure and specific volume. The resulting temperature is the final temperature.

Calculate the work done by the steam (b)

Since the spring has a linear force relationship, the pressure in the cylinder also varies linearly with the volume for this process. Therefore, we can determine the work done by the steam during this process using the following equation for work done during a linear expansion process: \(W = \int_{V_1}^{V_2} P(V) dV \quad \Rightarrow \quad W = \int_{V_1}^{V_2} (P_1 + \frac{P_2 - P_1}{V_2 - V_1}(V - V_1)) dV\) Then, by integrating, we can obtain the work done by the steam: \(W = P_{1}(V_{2} - V_{1}) + \frac{1}{2}(P_{2} - P_{1})(V_{2} - V_{1})\). Substituting the given values, we can find the work done.

Calculate the total heat transferred (c)

To find the total heat transferred, we will use the first law of thermodynamics for a closed system: \(Q = \Delta U + W\) Where \(\Delta U\) is the change in internal energy of the system and \(Q\) is the total heat transferred. We can find the change in internal energy, \(\Delta U\), by using the steam tables to find the specific internal energy at the initial and final states and then multiplying by the mass of the steam. Subtracting the initial specific internal energy from the final specific internal energy will give us the change in internal energy. Once we have the change in internal energy, we can use the equation above to find the total heat transferred to the system.

Key concepts

P-v Diagram

A P-v diagram, also known as a pressure-volume diagram, is a helpful tool in thermodynamics to illustrate the changes in volume and pressure for a particular substance during a thermodynamic process. When dealing with steam, or any other substance, we chart its state on this graph to understand its behavior under different conditions.

In the P-v diagram, pressure (P) is typically plotted on the y-axis, while specific volume (v) is plotted on the x-axis. A point on this diagram represents the state of the substance at a particular pressure and specific volume. The area under the process curve on a P-v diagram can represent the work done by (or on) the system, provided the process represented is quasi-static. This means that at each point along the path of the process, the system's state is well-defined and nearly in equilibrium.

The P-v diagram is particularly useful in visualizing steam processes, where lines of constant temperature, known as isotherms, and lines of phase changes, like the saturation curve, can be superimposed. The process from the exercise starts in a superheated state and ends in another state, which can be traced by a line connecting the initial and final points. The diagram not only aids in understanding the process but also helps visually estimate and calculate work done and heat transfer, key aspects of thermodynamics.

Superheated Steam

Superheated steam is steam that has been heated beyond its boiling point for a given pressure, hence existing at a temperature above its saturation temperature. At the saturation temperature, steam undergoes a phase change from water to vapor without a temperature rise. However, if further heat is supplied, the steam temperature increases, leading to superheated steam.

This is an important concept in thermodynamics because the properties of superheated steam differ from those of saturated steam. Superheated steam has a lower density and a higher enthalpy, which makes it an effective working fluid in steam engines and turbines since it can hold more energy and allows for better efficiency in energy conversion.

In the context of the given exercise, the steam initially exists in a superheated state. Recognizing this allows us to use the correct portion of the steam tables to find properties like specific volume and internal energy, which are critical for calculating work done and heat added. Furthermore, when heat is added to superheated steam, it can result in a further increase in volume and temperature under constant pressure, which is beneficial for doing work in a controlled manner.

Work Done by Steam

In thermodynamics, the work done by steam is the energy transferred by the steam when it expands against an external force, such as a piston in a cylinder. This expansion work is a central concept in steam engine and turbine operation.

The work done is calculated by integrating the pressure-volume relationship over the change in volume. If the relationship between the pressure and volume is linear, as is depicted in the given exercise with a linear spring, work done can be easily represented as the area under the process curve on a P-v diagram. The equation used for the linear expansion process, which leads to a trapezoidal area on the P-v diagram, illustrates how both the initial and final states of the pressure contribute to the total work done.

It's essential to grasp that not all expansion processes are linear, and in such cases, the relationship between pressure and volume might be more complex. Nonetheless, understanding the fundamentals of the work done by steam through graphical representation and calculations is key to solving many practical thermodynamic problems.

Heat Transfer

Heat transfer in thermodynamic systems is the movement of thermal energy from one system to another or within a system due to a temperature difference. In the case of the exercise involving steam, heat transfer occurs when energy is added to the steam, causing changes in its properties, such as temperature and volume.

The first law of thermodynamics governs the relationship between heat added to the system, the work done by the system, and the change in internal energy. According to the first law, the total heat transferred to the system is equal to the sum of the work done by the steam and the change in internal energy of the steam. The change in internal energy can be determined by using property tables, known as steam tables, that provide information about the energy content at different states.

Detailed knowledge of heat transfer is crucial for designing and analyzing steam processes, engines, and refrigeration systems. It helps in determining the efficiency of such systems and the amount of energy required for specific outputs, making it a fundamental concept of thermodynamics, especially when optimizing systems for energy conservation and maximizing performance.