Thermodynamic Analysis Of Saturated Liquid-Vapor Mixture In A Piston-Cylinder Device

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Introduction

In this comprehensive analysis, we delve into the thermodynamic behavior of a 3 kg saturated liquid-vapor mixture of water confined within a piston-cylinder device at a constant pressure of 160 kPa. This system presents a fascinating case study in thermodynamics, allowing us to explore the interplay between heat transfer, phase change, and energy balance. Initially, the mixture comprises 1 kg of liquid water and 2 kg of water vapor, setting the stage for a dynamic process as heat is introduced. Understanding the underlying principles governing this system is crucial for various engineering applications, including power generation, refrigeration, and chemical processing. Our exploration will involve detailed calculations and analysis, focusing on the initial state, the heat transfer process, and the final state of the water mixture. We will examine the properties of saturated liquid and vapor, utilizing thermodynamic tables and diagrams to gain a clear understanding of the system's behavior. This analysis not only provides insights into the specific scenario but also offers a framework for approaching similar thermodynamic problems involving phase change and heat transfer.

Initial State Analysis

The initial state of the water mixture is critical for understanding the subsequent thermodynamic processes. We begin with a 3 kg mixture at 160 kPa, consisting of 1 kg of saturated liquid and 2 kg of saturated vapor. This two-phase state is crucial, as it dictates the application of specific thermodynamic properties and relationships. To fully characterize the initial state, we need to determine the specific volume, internal energy, enthalpy, and entropy of the mixture. These properties are not simply additive; instead, we must consider the proportions of liquid and vapor phases present. The quality, denoted as 'x', is a key parameter representing the mass fraction of vapor in the mixture. In this case, the quality is calculated as the mass of vapor (2 kg) divided by the total mass (3 kg), resulting in x = 0.667. This value is pivotal in determining the mixture's properties. We will utilize steam tables, which provide detailed thermodynamic data for water at various temperatures and pressures, to find the specific volume, internal energy, enthalpy, and entropy for both saturated liquid and saturated vapor at 160 kPa. The mixture's properties are then calculated using the quality factor to weigh the contributions from each phase. This thorough initial state analysis lays the groundwork for evaluating the effects of heat transfer on the system.

To delve deeper into the initial state, we must consult steam tables to obtain the specific properties of saturated liquid and saturated vapor at 160 kPa. The steam tables provide values for specific volume (v), internal energy (u), enthalpy (h), and entropy (s) for both the saturated liquid (denoted by subscript 'f') and the saturated vapor (denoted by subscript 'g'). These values are essential for calculating the mixture's overall properties. At 160 kPa, we find the following values from the steam tables:

  • Specific volume of saturated liquid (vf): 0.001053 m³/kg
  • Specific volume of saturated vapor (vg): 1.090 m³/kg
  • Internal energy of saturated liquid (uf): 484.49 kJ/kg
  • Internal energy of saturated vapor (ug): 2519.2 kJ/kg
  • Enthalpy of saturated liquid (hf): 484.55 kJ/kg
  • Enthalpy of saturated vapor (hg): 2706.3 kJ/kg
  • Entropy of saturated liquid (sf): 1.4851 kJ/kg·K
  • Entropy of saturated vapor (sg): 7.1270 kJ/kg·K

With these values, we can now calculate the mixture's specific properties using the quality (x = 0.667). The specific volume of the mixture (v_mix) is calculated as:

v_mix = vf + x * (vg - vf)

Substituting the values, we get:

v_mix = 0.001053 m³/kg + 0.667 * (1.090 m³/kg - 0.001053 m³/kg) ≈ 0.727 m³/kg

Similarly, the specific internal energy of the mixture (u_mix) is calculated as:

u_mix = uf + x * (ug - uf)

Substituting the values, we get:

u_mix = 484.49 kJ/kg + 0.667 * (2519.2 kJ/kg - 484.49 kJ/kg) ≈ 1874.6 kJ/kg

The specific enthalpy of the mixture (h_mix) is calculated as:

h_mix = hf + x * (hg - hf)

Substituting the values, we get:

h_mix = 484.55 kJ/kg + 0.667 * (2706.3 kJ/kg - 484.55 kJ/kg) ≈ 1962.2 kJ/kg

Finally, the specific entropy of the mixture (s_mix) is calculated as:

s_mix = sf + x * (sg - sf)

Substituting the values, we get:

s_mix = 1.4851 kJ/kg·K + 0.667 * (7.1270 kJ/kg·K - 1.4851 kJ/kg·K) ≈ 5.269 kJ/kg·K

These calculations provide a comprehensive understanding of the initial state of the water mixture, setting the stage for analyzing the effects of heat transfer. The specific volume, internal energy, enthalpy, and entropy are crucial parameters for evaluating the thermodynamic changes as heat is added to the system.

Heat Transfer Process

Heat transfer is the core of this thermodynamic process. As heat is transferred to the water within the piston-cylinder device, the system undergoes a transformation, primarily manifested as a change in the water's phase composition. Since the process occurs at a constant pressure of 160 kPa, it is an isobaric process. This means the pressure inside the cylinder remains constant as the piston moves to accommodate the volume change due to the phase transition. The heat transfer will initially lead to the vaporization of the liquid water present in the mixture. As more heat is added, the liquid water will continue to evaporate, increasing the vapor fraction (quality) of the mixture. The temperature will remain constant at the saturation temperature corresponding to 160 kPa until all the liquid has vaporized. This is a key characteristic of phase change processes at constant pressure. Once all the liquid has turned into vapor, further heat transfer will result in superheating of the vapor, causing both the temperature and volume to increase. To analyze this process, we need to track the changes in the system's properties as heat is added. This involves applying the first law of thermodynamics, which relates heat transfer, work done, and changes in internal energy. The work done by the system is due to the expansion of the cylinder, which can be calculated from the pressure and volume change. The change in internal energy is related to the change in temperature and phase composition. By carefully accounting for these factors, we can determine the amount of heat required to achieve different states of the mixture, such as complete vaporization or superheated vapor.

Analyzing the heat transfer process further requires a deeper understanding of the energy balance within the system. According to the first law of thermodynamics, the heat added to the system (Q) is equal to the change in internal energy (ΔU) plus the work done by the system (W):

Q = ΔU + W

In this case, the work done is due to the expansion of the piston against the constant pressure (P) of 160 kPa. The work done can be expressed as:

W = P * ΔV

where ΔV is the change in volume. Since the process is isobaric, the pressure remains constant, simplifying the calculation of work. The change in internal energy (ΔU) can be expressed in terms of specific internal energies as:

ΔU = m * (u_2 - u_1)

where m is the total mass of the water (3 kg), and u_1 and u_2 are the specific internal energies at the initial and final states, respectively. Combining these equations, we can rewrite the first law of thermodynamics as:

Q = m * (u_2 - u_1) + P * (V_2 - V_1)

This equation is crucial for determining the heat required for different stages of the process. Initially, as heat is added, the temperature remains constant at the saturation temperature corresponding to 160 kPa. From the steam tables, the saturation temperature (T_sat) at 160 kPa is approximately 113.3 °C. During this phase change process, the added heat is used to convert the liquid water into vapor, and the internal energy increases as the water transitions from liquid to vapor. The specific enthalpy of vaporization (h_fg) at 160 kPa, which represents the heat required to vaporize 1 kg of saturated liquid, is approximately 2200.8 kJ/kg. The amount of heat required to completely vaporize the 1 kg of liquid water initially present can be calculated using the enthalpy change during phase transition. As the heat transfer continues beyond complete vaporization, the water enters the superheated vapor region, where both temperature and volume increase. In this region, the relationship between pressure, volume, and temperature is governed by the superheated steam tables. To determine the final state and the heat required, we need to specify either the final temperature or the final pressure and volume. The heat transfer process can be visualized on a T-v (temperature-specific volume) diagram or a P-v (pressure-specific volume) diagram, which provides a graphical representation of the state changes during the process. These diagrams are invaluable tools for understanding and analyzing thermodynamic cycles and processes.

Final State Determination

To determine the final state, we need additional information, such as the final temperature or the amount of heat added. Without this information, we can only speculate about the possible final states. However, we can analyze two potential scenarios: complete vaporization and superheated vapor. Scenario 1: Complete Vaporization. In this scenario, enough heat is added to completely vaporize the liquid water, resulting in a saturated vapor state. The final state would be saturated vapor at 160 kPa. Using the steam tables, we can find the specific volume, internal energy, enthalpy, and entropy for saturated vapor at this pressure. The heat required for this process can be calculated using the enthalpy change during phase transition. We would compare the initial enthalpy of the mixture to the final enthalpy of the saturated vapor to determine the heat input. Scenario 2: Superheated Vapor. If even more heat is added beyond complete vaporization, the water will enter the superheated vapor region. In this region, the temperature will rise above the saturation temperature corresponding to 160 kPa. To determine the final state in this case, we would need either the final temperature or another property, such as the final specific volume. With this information, we can consult the superheated steam tables to find the remaining properties. The heat required for this process would include the heat needed for vaporization plus the heat needed to raise the temperature of the superheated vapor. Analyzing these potential final states helps illustrate the impact of heat transfer on the system's thermodynamic properties. The path the system takes from the initial state to the final state is crucial, and understanding the heat transfer process allows us to predict the system's behavior under different conditions.

Elaborating on the final state determination, let's consider a scenario where the heat transfer continues until the water reaches a specific final state. Assume, for instance, that the water is heated until it reaches a final temperature of 250 °C at the constant pressure of 160 kPa. This final state is in the superheated vapor region, as the temperature is significantly higher than the saturation temperature (113.3 °C) at 160 kPa. To determine the final state properties, we need to consult the superheated steam tables. These tables provide thermodynamic data for water at various pressures and temperatures in the superheated region. At 160 kPa and 250 °C, we can find the following properties:

  • Specific volume (v_2): 1.4298 m³/kg
  • Internal energy (u_2): 2725.6 kJ/kg
  • Enthalpy (h_2): 2954.2 kJ/kg
  • Entropy (s_2): 7.7962 kJ/kg·K

Now, we can calculate the heat required to reach this final state using the first law of thermodynamics equation:

Q = m * (u_2 - u_1) + P * (V_2 - V_1)

First, we need to calculate the initial and final volumes. The initial volume (V_1) can be calculated using the total mass (3 kg) and the initial specific volume (v_mix ≈ 0.727 m³/kg):

V_1 = m * v_mix = 3 kg * 0.727 m³/kg ≈ 2.181 m³

The final volume (V_2) can be calculated using the total mass (3 kg) and the final specific volume (v_2 = 1.4298 m³/kg):

V_2 = m * v_2 = 3 kg * 1.4298 m³/kg ≈ 4.289 m³

Now, we can substitute the values into the heat transfer equation:

Q = 3 kg * (2725.6 kJ/kg - 1874.6 kJ/kg) + 160 kPa * (4.289 m³ - 2.181 m³)

Convert the pressure from kPa to kJ/m³ (1 kPa = 0.001 kJ/m³):

Q = 3 kg * (851 kJ/kg) + 160 kJ/m³ * (2.108 m³)

Q = 2553 kJ + 337.28 kJ ≈ 2890.28 kJ

Therefore, the heat required to heat the water from the initial state to the final state of 250 °C at 160 kPa is approximately 2890.28 kJ. This detailed calculation illustrates how the final state is determined and how the heat transfer process can be quantified using thermodynamic principles and steam tables. The analysis provides a comprehensive understanding of the energy transformations and state changes within the system.

Conclusion

In conclusion, the analysis of the 3 kg saturated liquid-vapor mixture of water in a piston-cylinder device at 160 kPa provides a comprehensive understanding of thermodynamic principles and phase change processes. By examining the initial state, we determined the key properties of the mixture using steam tables and quality calculations. The heat transfer process was analyzed using the first law of thermodynamics, considering both the change in internal energy and the work done by the system. We explored the isobaric nature of the process and the importance of the saturation temperature and enthalpy of vaporization. The determination of the final state involved considering potential scenarios such as complete vaporization and superheated vapor, and we performed detailed calculations to quantify the heat required to reach a specific final state, such as 250 °C. This analysis highlights the importance of thermodynamic tables and diagrams in understanding and predicting the behavior of thermodynamic systems. The principles and methodologies discussed here are applicable to a wide range of engineering problems involving phase change, heat transfer, and energy balance. The study underscores the significance of a systematic approach to thermodynamic analysis, involving careful consideration of system properties, processes, and boundary conditions. By mastering these concepts, engineers and scientists can effectively design and optimize systems involving thermal energy and phase transitions.