7.3 Property diagrams involving entropy

Entropy Diagrams: Interpretation and Application

Property diagrams involving entropy give you a visual way to track what's happening during thermodynamic processes. Instead of relying purely on equations, you can see how temperature, pressure, enthalpy, and entropy relate to each other as a system moves between states. These diagrams also let you calculate work, heat transfer, and efficiency geometrically, using areas on the plot.

The two most common diagrams are the T-s diagram (temperature vs. specific entropy) and the h-s diagram (specific enthalpy vs. specific entropy, also called a Mollier diagram). Both show up constantly when analyzing power cycles, refrigeration systems, and other engineering applications.

Property Diagrams and Their Uses

T-s diagrams plot temperature on the vertical axis and specific entropy on the horizontal axis. Lines of constant pressure (isobars) are typically drawn across the diagram so you can identify the pressure at any state point.

h-s diagrams (Mollier diagrams) plot specific enthalpy on the vertical axis and specific entropy on the horizontal axis. These include both isobars and isotherms (lines of constant temperature), making them especially useful for analyzing turbines, compressors, and nozzles where enthalpy changes matter most.

Both types of diagrams let you:

• Identify the thermodynamic state of a substance as a single point on the plot
• Trace the path a system follows during processes like compression, expansion, heating, or cooling
• Locate key features: the critical point, the saturated liquid line, the saturated vapor line, and the two-phase (wet) region enclosed between them

Key Features and Processes on Property Diagrams

Several common processes have distinctive appearances on entropy diagrams:

• Isentropic processes (reversible + adiabatic) appear as vertical lines on both T-s and h-s diagrams, since entropy stays constant.
• Isothermal processes appear as horizontal lines on T-s diagrams, since temperature stays constant.
• Isobaric processes follow the constant-pressure lines already drawn on the diagram.
• Isochoric (constant volume) processes appear as curves, since specific volume is constant but temperature and entropy both change.

Two geometric relationships are critical:

• The area under a process curve on a T-s diagram represents the heat transfer during a reversible process.
• On an h-s diagram, the vertical distance between two state points gives the change in specific enthalpy, which directly relates to work in steady-flow devices like turbines and compressors.

State Properties on Entropy Diagrams

Determining State Properties

The thermodynamic state of a system is fully defined by its state properties (temperature, pressure, specific volume, specific enthalpy, specific entropy). On a property diagram, each state corresponds to a single point.

To read state properties from a T-s diagram:

1. Locate the point representing the system's state.
2. Read temperature directly from the vertical axis and specific entropy from the horizontal axis.
3. Identify which isobar (constant-pressure line) passes through or near the point to determine pressure.

To read state properties from an h-s diagram:

1. Locate the state point.
2. Read specific enthalpy from the vertical axis and specific entropy from the horizontal axis.
3. Use the isobars and isotherms drawn on the diagram to determine pressure and temperature at that state.

Quality and the Lever Rule

When a state point falls inside the two-phase region (the dome under the saturation curves), the substance is a mixture of liquid and vapor. You need one more property to describe it: quality (x), the mass fraction of vapor in the mixture.

• $x = 0$ means all saturated liquid (on the left saturation curve).
• $x = 1$ means all saturated vapor (on the right saturation curve).
• Any value between 0 and 1 means a two-phase mixture.

The lever rule lets you find quality graphically. On a T-s diagram at a given temperature (or pressure), the saturated liquid and saturated vapor states define the endpoints of a horizontal tie line. The quality is the ratio of the distance from the saturated liquid point to the state point, divided by the total distance between saturated liquid and saturated vapor:

$x = \frac{s - s_f}{s_g - s_f}$

where $s_f$ is the specific entropy of the saturated liquid and $s_g$ is the specific entropy of the saturated vapor.

For example, a two-phase mixture with $x = 0.7$ is 70% vapor by mass and 30% liquid. Its state point sits 70% of the way from the saturated liquid line to the saturated vapor line.

Thermodynamic Processes on Entropy Diagrams

Representing Processes on Property Diagrams

Each type of idealized process traces a characteristic path:

Real processes won't follow these ideal paths perfectly, but the idealized versions give you a baseline for comparison and analysis.

Isentropic Processes

An isentropic process is both reversible and adiabatic: no heat transfer occurs, and entropy does not change. Because $\Delta s = 0$, these processes plot as vertical lines on both T-s and h-s diagrams.

Isentropic processes are the idealized model for:

• Compression in compressors and pumps
• Expansion in turbines and nozzles

For example, the ideal compression and expansion steps in a Brayton cycle (gas turbine) are modeled as isentropic. In reality, friction and irreversibilities cause entropy to increase, so the actual process line tilts to the right of the vertical ideal on the diagram. The horizontal gap between the ideal and actual lines gives you a visual sense of how much irreversibility is present.

Work, Heat Transfer, and Efficiency on Entropy Diagrams

Calculating Work and Heat Transfer

Heat transfer from a T-s diagram: For a reversible process, the area under the process curve on a T-s diagram equals the heat transfer per unit mass.

• For an isothermal process, this simplifies to a rectangle: $q = T \cdot \Delta s$
• For other reversible processes, you integrate: $q = \int T \, ds$

Work from enthalpy changes: For steady-flow devices (turbines, compressors, nozzles), the work per unit mass relates directly to the change in specific enthalpy on an h-s diagram. For an ideal turbine with no heat loss:

$w = h_1 - h_2$

where $h_1$ and $h_2$ are the specific enthalpies at the inlet and outlet states.

For a closed system undergoing an isobaric process, boundary work can also be calculated as:

$w = P \cdot \Delta v$

The first law ties everything together: $\Delta u = q - w$ for a closed system, where $\Delta u$ is the change in specific internal energy. You can apply this alongside the graphical information from the diagrams to solve for unknown quantities.

Efficiency and Performance

Thermal efficiency of a power cycle equals the net work output divided by the total heat input:

$\eta_{th} = \frac{w_{net}}{q_{in}}$

On a T-s diagram, $q_{in}$ is the area under the heat-addition process curve, and $q_{out}$ is the area under the heat-rejection curve. The net work equals the area enclosed by the cycle. This is why T-s diagrams are so useful for comparing cycles: a larger enclosed area means more net work for the same heat input.

Coefficient of performance (COP) applies to refrigeration and heat pump cycles:

• For a refrigerator: $COP_R = \frac{q_L}{w_{net}}$, where $q_L$ is the heat absorbed from the cold reservoir.
• For a heat pump: $COP_{HP} = \frac{q_H}{w_{net}}$, where $q_H$ is the heat delivered to the warm reservoir.