P-V & T-S Diagrams and Efficiency

Definition

A P-V diagram is a graph in which pressure is plotted against volume to represent thermodynamic processes and cycles. The area under the curve on a P-V diagram represents the work done by or on the system.

A T-S diagram is a graph in which temperature is plotted against entropy to represent heat transfer and thermodynamic changes. The area under the curve on a T-S diagram represents heat transfer during a reversible process.

Efficiency in thermodynamics is the measure of how effectively a system converts heat energy into useful work. It is usually expressed as the ratio of useful output to energy input, such as:

$\eta = \frac{\text{Net work output}}{\text{Heat supplied}}$

for a heat engine.

Main Content

1. P-V Diagram

The P-V diagram shows the relationship between pressure and volume during a thermodynamic process. It is one of the most useful tools for analyzing work done in compression and expansion processes.
The area enclosed by a cycle on the P-V diagram represents the net work output of the cycle. A clockwise cycle generally indicates work done by the system, while an anticlockwise cycle indicates work done on the system.
For an isobaric process, pressure remains constant and the diagram is a horizontal line. For an isochoric process, volume remains constant and the diagram is a vertical line. For isothermal and adiabatic processes, the curve shape changes according to the process law.
In a piston-cylinder system, expansion increases volume and often produces work, while compression decreases volume and requires work input. This makes the P-V diagram extremely important in engine analysis.
Example: In the Otto cycle, heat addition and rejection occur at constant volume, while compression and expansion follow adiabatic paths. The P-V plot clearly shows these stages and helps determine the indicated work and efficiency.

2. T-S Diagram

The T-S diagram shows the relationship between temperature and entropy and is especially valuable in studying heat transfer and reversible thermodynamic processes.
For a reversible process, the area under the curve on a T-S diagram represents the amount of heat transferred: $Q = \int T\,dS$
Isentropic processes appear as vertical lines because entropy remains constant. Constant-temperature processes appear as horizontal lines because temperature remains constant.
The T-S diagram is particularly useful for steam power plants and refrigeration systems because it gives a clear picture of heat addition, heat rejection, and entropy change.
Example: In the Rankine cycle, the boiler process is shown as heat addition at nearly constant pressure with a temperature rise, and the condenser process is shown as heat rejection. The T-S diagram helps in locating the saturation region and understanding steam quality.

3. Efficiency of Thermodynamic Cycles

Efficiency is a measure of how much useful work is obtained from the heat supplied to a system. In power cycles, higher efficiency means better utilization of fuel and less waste heat.
Thermal efficiency of a heat engine is given by: $\eta_{th} = \frac{W_{net}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}$ where $Q_{in}$ is heat supplied and $Q_{out}$ is heat rejected.
The shape and size of P-V and T-S diagrams directly influence efficiency. A larger enclosed area on a P-V diagram means more work output, while the heat transfer pattern on a T-S diagram helps determine thermal efficiency.
For the Carnot cycle, efficiency depends only on the temperatures of the hot and cold reservoirs: $\eta = 1 - \frac{T_L}{T_H}$ This shows that no engine can be more efficient than the Carnot engine operating between the same two temperatures.
In practical systems, friction, pressure drops, heat losses, and irreversibilities reduce efficiency compared to ideal cycles. T-S diagrams are useful for identifying entropy generation and irreversibility.

Working / Process

A thermodynamic system undergoes a sequence of processes such as compression, heat addition, expansion, and heat rejection. These processes are plotted on a P-V diagram and T-S diagram to observe the complete cycle behavior.
On the P-V diagram, the work done in each process is obtained from the area under the curve, and the total net work of the cycle is found from the area enclosed by the cycle.
On the T-S diagram, heat transfer in reversible processes is represented by the area under the curve. Using the heat input and heat rejected, efficiency is calculated by comparing useful output work with energy supplied.

Advantages / Applications

P-V and T-S diagrams provide a clear visual representation of thermodynamic processes, making it easier to understand complex cycles.
They help calculate work, heat transfer, and efficiency accurately, especially in idealized and real engine analysis.
They are widely used in the study of heat engines, steam turbines, compressors, refrigerators, and internal combustion engines.
These diagrams are essential for comparing different cycles and improving the design of power plants and thermal systems.
They help identify losses and irreversibilities, which is useful for improving thermal efficiency and performance.

Summary

P-V diagrams represent pressure-volume changes and show work done by a system.
T-S diagrams represent temperature-entropy changes and show heat transfer in reversible processes.
Efficiency is the ratio of useful work output to heat input and is a key measure of cycle performance.
These diagrams are widely used to analyze and compare thermodynamic cycles in engineering.