Question

Steam is compressed from 6 MPa and \(300^{\circ} \mathrm{C}\) to 10 MPa isentropically. The final temperature of the steam is \((a) 290^{\circ} \mathrm{C}\) \((b) 300^{\circ} \mathrm{C}\) \((c) 311^{\circ} \mathrm{C}\) \((d) 371^{\circ} \mathrm{C}\) \((e) 422^{\circ} \mathrm{C}\)

Short answer

Choose the correct answer from the given options: (a) \(280^{\circ} \mathrm{C}\) (b) \(310^{\circ} \mathrm{C}\) (c) \(311^{\circ} \mathrm{C}\) (d) \(330^{\circ} \mathrm{C}\) Answer: (c) \(311^{\circ} \mathrm{C}\)

Step-by-step solution

Find the initial entropy of the steam

Using the steam tables at 6 MPa and \(300^{\circ} \mathrm{C}\), we can find the initial entropy of the steam (denoted as \(s_1\)). By referring to the steam table, we have: \(s_{1} = 6.2289\, \mathrm{kJ/kg \cdot K}\).

Determine the entropy at final pressure

Since the process is isentropic, the final entropy (denoted as \(s_2\)) will be the same as the initial entropy: \(s_{2} = s_{1} = 6.2289\, \mathrm{kJ/kg \cdot K}\).

Find the final temperature of the steam

Now, we need to find the temperature corresponding to 10 MPa pressure and \(s_{2}\) entropy. We can use the steam tables again to find the temperature. Looking up the values in the steam tables, we find that the temperature corresponding to 10 MPa and \(s_{2}\) is \(311^{\circ} \mathrm{C}\).

Select the correct answer

Based on our calculations, the final temperature of the steam is \(311^{\circ} \mathrm{C}\). Therefore, the correct answer is: \((c) 311^{\circ} \mathrm{C}\).

Key concepts

Steam Table Usage

In thermodynamic studies, steam tables are essential for solving numerous problems involving steam as a working fluid. They provide a comprehensive set of data that describe the properties of steam at various pressures and temperatures. The steam tables catalog values such as temperature, pressure, specific volume, specific enthalpy, and specific entropy, among others.

For instance, when solving problems like the one where steam is compressed isentropically, the steam table allows us to look up the initial entropy given the initial pressure and temperature. To improve understanding, one must become familiar with the layout and use of the steam tables, as well as learn to interpolate between table values when the exact parameters do not directly match those listed.

Entropy

Entropy is a fundamental concept in thermodynamics often associated with the level of disorder or randomness in a system. In the context of a steam compression process, entropy is important for characterizing the state of the steam. During an isentropic process, which is idealized to be both adiabatic and reversible, the entropy remains constant. This implies that there is no heat transfer to or from the steam and that the process is done without any entropy generation due to friction or other irreversibilities.

Understanding entropy is key to determining the state of steam at different stages of the compression process. In the exercise provided, the steam's entropy does not change during compression, which significantly simplifies finding the final temperature.

Steam Compression Process

In the steam compression process, steam is usually compressed to a higher pressure, which typically results in an increase in temperature. The process can be visualized on a pressure-entropy (P-s) diagram, where an isentropic compression would be represented by a vertical line since entropy remains constant. When the process is isentropic, no heat is lost or gained, and no entropy is produced within the system, making it a particularly efficient scenario.

This idealization seldom occurs in real-world applications due to inevitable losses, but it is a valuable construct for understanding the thermodynamics involved. In the exercise, using the constant entropy (isentropic condition), allowed us to determine the final temperature after the steam was compressed to a higher pressure.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of an isentropic compression of steam, we apply the first and second laws of thermodynamics. The first law relates to the conservation of energy, often used to calculate work and heat in a system. The second law, which introduces the concept of entropy, explains that the total entropy of an isolated system can never decrease over time.

In practical terms, thermodynamics allows us to predict the outcome of energy interactions in systems like steam engines, refrigerators, and air conditioners. Students can greatly benefit from hands-on activities such as conducting laboratory experiments, which reinforce theoretical learning and promote a deeper understanding of these concepts.