Question

Determine the phase or phases in a system consisting of \(\mathrm{H}_{2} \mathrm{O}\) at the following conditions and sketch \(p-v\) and \(T-v\) diagrams showing the location of each state. (a) \(p=5\) bar, \(T=151.9^{\circ} \mathrm{C}\). (b) \(p=5\) bar, \(T=200^{\circ} \mathrm{C}\). (c) \(T=200^{\circ} \mathrm{C}, p=2.5 \mathrm{MPa}\). (d) \(T=160^{\circ} \mathrm{C}, p=4.8\) bar. (e) \(T=-12^{\circ} \mathrm{C}, p=1\) bar.

Short answer

a) Mixture, b) Superheated vapor, c) Subcooled liquid, d) Superheated vapor, e) Solid.

Step-by-step solution

Understand the Problem

The problem requires determining the phase (liquid, vapor, or mixed) of \(\mathrm{H}_{2}\mathrm{O}\) at given conditions of pressure (\(p\)) and temperature (\(T\)).

Refer to Phase Diagrams

Use the steam tables or phase diagrams for \(\mathrm{H}_{2}\mathrm{O}\) to identify the phases at specific conditions. Pay attention to the saturation temperature and pressure at each given point.

Evaluate Each Condition

Analyze each given condition (a to e) separately. Consult the referenced diagrams or tables to determine if each condition corresponds to a liquid, vapor, or mixture phase. Draw the \(p-v \) and \(T-v \) diagrams marking each state if necessary.

Condition (a)

For \(p=5\) bar and \(T=151.9^{\circ}\mathrm{C}\): The saturation temperature for 5 bar is around 151.9°C. Since the temperature equals the saturation temperature, the system is at the phase change point (mixture of liquid and vapor).

Condition (b)

For \(p=5\) bar and \(T=200^{\circ}\mathrm{C}\): The saturation temperature at 5 bar is 151.9°C. Since 200°C > 151.9°C, the system is in the superheated vapor phase.

Condition (c)

For \(T=200^{\circ}\mathrm{C}\) and \(p=2.5\mathrm{MPa}\): 2.5 MPa = 25 bar. The saturation temperature at 25 bar is around 222°C. Since 200°C < 222°C, the system is in the subcooled liquid phase.

Condition (d)

For \(T=160^{\circ}\mathrm{C}\) and \(p=4.8\) bar: The saturation temperature at 4.8 bar is around 150.88°C. Since 160°C > 150.88°C, the system is in the superheated vapor phase.

Condition (e)

For \(T=-12^{\circ}\mathrm{C}\) and \(p=1\) bar: The saturation temperature at 1 bar is 100°C. Since -12°C < 0°C (freezing point), the system is in the solid phase (ice).

Draw Diagrams

Draw the \(p-v \) and \(T-v \) diagrams. Mark the points according to the phases determined in steps 4 to 8.

Key concepts

phase diagrams

Phase diagrams are visual representations of the different phases (solid, liquid, vapor) of a substance under varying conditions of temperature and pressure. They help us understand how a substance behaves when subjected to different environmental conditions. For water (H2O), these diagrams can be very helpful.
In a phase diagram:

• The x-axis typically represents temperature (T).
• The y-axis typically represents pressure (p).

By locating a set of pressure and temperature conditions on this graph, we can deduce whether the substance is in the solid, liquid, or vapor state. For example, the \(p\text{-}v\) and \(T\text{-}v\) diagrams are particularly useful for engineers and scientists studying thermodynamic systems.

saturation temperature

Saturation temperature is the temperature at which a substance changes phase at a given pressure. This is crucial for determining the state of water at specific conditions. If we are dealing with water,
at each pressure, there is a unique temperature where water can exist as both liquid and vapor. This is known as the boiling or condensation point.
For example:

• At a pressure of 1 bar (atmospheric pressure), the saturation temperature of water is 100°C.
• For a pressure of 5 bar, the saturation temperature is around 151.9°C.

Understanding this concept helps in determining whether the water is a subcooled liquid, superheated vapor, or a mixture of both at given conditions.

superheated vapor

Superheated vapor refers to a state where a substance exists as vapor at a temperature higher than its saturation temperature for the given pressure.
In this phase:

• The substance is entirely in the vapor state without any liquid phase present.
• The temperature is above the boiling point for the given pressure.

For instance, if water is at a pressure of 5 bar and a temperature of 200°C, it is considered superheated because 200°C is higher than the saturation temperature of 151.9°C at that pressure.

subcooled liquid

Subcooled liquid describes a liquid that is at a temperature lower than its saturation temperature for a given pressure.
In this phase:

• The water is entirely in the liquid state, with no vapor present.
• The temperature is below the boiling point for the given pressure.

For instance, at a pressure of 2.5 MPa (25 bar), the saturation temperature is around 222°C. If the temperature is 200°C at this pressure, water will be in the subcooled liquid state because 200°C is below 222°C.

solid phase

The solid phase refers to a state where the substance is in solid form, such as ice for water. This occurs when the temperature is below the freezing point at a given pressure.
For example:

• At a pressure of 1 bar, the freezing point of water is 0°C.
• If the temperature drops below 0°C, water turns into ice.

In the given exercise, consider a condition where the temperature is -12°C and the pressure is 1 bar. Here, water will be in the solid phase since -12°C is well below the freezing point.