PV and TS diagrams - Explaination in Simple Words, It's Uses, It's Discovery

 

PV and TS Diagrams:

They are the maps for navigating thermodynamic processes. PV diagram is work diagram used for mechanical design. TS diagram is heat & efficiency diagram used for energy optimization. Engineers need both to fully understand, design, and improve thermal systems like engines, refrigerators, power plants.

PV diagram is a graph with pressure (P) on the y-axis and volume (V) on the x-axis. Their relationships visualize work transfer and volume changes. It plots the path of a thermodynamic process like compression, expansion for a system usually a gas. The area under the curve represents the work done by or on the system. Work done during expansion is positive; work done during compression is negative. It's easy to see net work output of a cycle.

Area under curve shows work done,

W = ∫P dV


TS diagram is a graph with temperature (T) on the y-axis and entropy (S) on the x-axis. Their relationships visualize heat transfer and entropy changes. It tracks how temperature and entropy change during a process. The area under the curve represents the heat transferred to or from the system. Increased entropy at constant temperature indicates heat addition. It's easy to see heat addition/rejection in a cycle.

Area under curve shows heat transferred, 

Q = ∫T dS.

Why we use pv to visualize work and ts to visualize heat transfer?

It Comes Directly from the Definition of Work. Work in thermodynamics is defined as,
Work = ∫ Force · dx
= ∫ (Pressure × Area) · dx
= ∫ P · (Area × dx)
= ∫ P · dV

No other diagram gives work this directly. Think of a piston,
Force on piston = pressure × piston area
Distance moved = ΔVolume/piston area

Work = Force × Distance
= (P × A) × (ΔV/A)
= P × ΔV

Think like what the piston feels. Imagine you're the piston. You only feel pressure pushing on your face. You only measure how far you've moved by volume change.
Your effort i.e. work
= pressure × distance
= P × ΔV

PV diagram is piston's journey of pressure vs position.

During engines cycle, during compression, piston's experience high pressure pushing against it i.e. negative work. And during expansion, high pressure pushing it i.e. positive work.

We get net work as area between compression and expansion lines. PV diagram captures this perfectly.


Second law of thermodynamics defines entropy change as,
dS = δQ_rev / T
Rearranging, δQ_rev = T dS
Therefore for a reversible process,

Heat Transfer = ∫ T dS

TS diagram is about what the heat feels. 

Imagine you're a unit of heat. You care about temperature means your quality or ability to do work. You care about entropy, how dispersed or degraded you become. TS diagram is heat's passport showing where it went and how it changed. If it is added at high temperature it moves right at constant high T, it is good quality heat. If it is rejected at low temperature it moves left at constant low T, it is waste heat.

We will look at the historical development of thermodynamics.

James Watt was improving the Newcomen steam engine. The Newcomen engine was used to pump water out of mines. Watt's improvements led to the modern steam engine. Watt did not use PV diagrams explicitly. The concept of pressure-volume diagrams came later, but Watt understood the relationship between pressure, volume, and work intuitively.

How watt understood work output?

The steam engine worked by using steam pressure to push a piston. The piston was connected to a beam that did the work like pumping water. Watt measured what he could. Watt measured pressure by manometers, piston position by ruler, and time by clock. He observed that higher pressure leads piston to moves faster and longer stroke makes more water pumped.

Watt measured the pressure of steam and the distance the piston traveled. He knew that the force on the piston was proportional to the steam pressure and the area of the piston. The work done was force times distance.

Watt's employee John Southern invented the indicator diagram. This was a mechanical device that plotted the pressure in the cylinder against the volume or piston position. The area under the curve was measured to find the work done per cycle.

The indicator diagram was essentially
a PV diagram. It was used to tune and
improve engine performance. So, watt and southern understood that work was related to the area under a pressure-volume curve because:
Work = force x distance
=(pressure x area) x (volume change area)
= pressure x volume change

By plotting pressure vs. volume, the area
under the curve i.e. integral of PdV is the work.

This was a practical, empirical approach. They didn't have the full theory of thermodynamics, which came later but they knew how to measure and improve work output.

How Carnot Understood Heat and Efficiency?

Carnot performed thought experiment. He imagined perfect heat engine with no friction, no heat leaks and perfectly reversible. He realized that work produced depends not on substance like steam, air, etc. but only on temperatures of heat source and sink.

Carnot imagined an ideal heat engine, now called the carnot cycle that operates between two heat reservoirs (hot and cold). He realized that the efficiency of such an engine depends only on the temperatures of the reservoirs.

Carnot formulated that the maximum efficiency of a heat engine is given by,
1 - (T_cold /T_hot), where temperatures are in absolute scale.

Carnot cycle consists of two isothermal and two adiabatic processes. When plotted in PV coordinates, it's a closed curve. However, the efficiency is more naturally expressed in terms of temperatures and heat transfers.

Carnot did not use TS diagrams, but his cycle, when plotted in TS coordinates,
becomes a rectangle. This was realized later by Clausius and others.

Rudolf clausius built on carnot's work and the work of others like joule and lord kelvin.

He gave clausius inequality. For a cyclic process, ý (ẟQ/T) < 0, with equality for reversible cycles.

He defined a state function S such that dS = ẟQ_rev/T. For a reversible process, the total entropy change is zero.

Clausius and others realized that if you plot a cycle in TS coordinates, the area under the curve i.e. for a reversible process is the heat transferred. The net area of the cycle (if there is one) would be the net heat transfer, which for a cycle equals the net work by first law.

Why TS for Heat?

Carnot and Clausius were concerned with the efficiency of heat engines and the maximum work that can be obtained from heat. They realized that heat transfer at a higher temperature is more valuable for doing work. The concept of entropy naturally captures the quality of heat. In the TS diagram, the heat transfer in a reversible process is the area under the curve. This makes it easy to see how much heat is added or rejected at each temperature.

Practical engineer like james watt developed the indicator diagram (PV diagram) to measure and improve the work output of steam engines. He understood the relationship between pressure, volume, and work. While theoretical physicists/engineers like carnot and clausius were trying to understand the fundamental principles of heat engines. Carnot's work led to the idea that efficiency depends on temperature. Clausius formalized the second law and introduced entropy. The TS diagram emerged as a natural way to visualize heat transfer and entropy changes, which are key to the second law.

Uses :-

PV diagrams help engineers calculate work and design engines for maximum power.

TS diagrams help optimize efficiency by minimizing entropy generation and managing heat.

Power plants use TS diagrams to design stages of turbines and condensers for maximum efficiency.

Refrigeration use TS diagrams to analyze COP (Coefficient of Performance) and optimize heat exchange.

Engines use PV diagrams to estimate power output and compression ratios.

TS diagrams clearly show entropy increase due to friction/unrestrained expansion (area increases), helping identify losses.

In car engine (Otto Cycle), PV diagram shows how pressure spikes during combustion (vertical rise), pushing the piston down (expansion), work output seen as area. TS Diagram shows heat addition at nearly constant volume (entropy increase), and heat rejection at exhaust.

How to use PV and TS diagrams to design a better engine and turbine respectively?

Using PV diagram to designe better engine

Most internal combustion engines like gasoline and diesel operate on a cycle like otto, diesel, or mixed cycles. The PV diagram shows the work done by the
engine in one cycle i.e. the area enclosed by the cycle. The goal is to maximize the net work output i.e. area inside the cycle, while minimizing losses due to factors like heat loss, friction, and incomplete combustion.

We use PV diagram to increase efficiency. By increasing the compression ratio (r), the PV curve becomes steeper during compression and expansion, and the area inside the cycle increases (for the same heat input, the work output increases). However, in real engines, increasing r beyond point causes knocking in gasoline engines. So, we must use higher octane fuel or design the engine to resist knocking like by optimizing the combustion chamber shape, using turbocharging, or using variable valve timing.

We want to maximize area of the cycle for a given amount of heat input or fuel. Higher peak pressures can lead to more work but require stronger engine components.

We use PV diagram to reduce losses. The PV diagram can show if the expansion stroke is below the ideal adiabatic curve due to heat loss. We can use better insulation like ceramic coatings to reduce heat loss.

The area of the cycle can be reduced by friction indicated by a less rounded cycle or by a lower expansion curve. Using low-friction materials and lubricants can help.

If the combustion is incomplete, the pressure rise during combustion, the vertical part of the cycle is less than ideal. This can be improved by better fuel injection, air-fuel mixing, and ignition timing.

Turbocharging/Supercharging increase the pressure at the beginning of the
compression stroke, so the entire cycle is shifted to higher pressures. This increases the area inside the cycle means more work without increasing the engine displacement.

By measuring the actual PV diagram, engineers can compare the actual cycle with the ideal one and identify deviations. Then, they can adjust parameters like ignition timing, fuel injection timing, valve timing, etc. to get the desired PV curve.

Using TS Diagram to Design a Better Turbine

Turbines are designed to extract work from a flowing fluid like gas or steam. In the TS diagram, the process is usually an expansion from high pressure to low
pressure. The goal is to extract the maximum work while minimizing entropy generation which reduces efficiency.

We use TS diagram to increase efficiency. In an ideal isentropic expansion, entropy is constant i.e. vertical line in TS diagram. In real turbines, the expansion is not isentropic (entropy increases due to friction, turbulence, etc.), so the line curves to the right. The deviation from the vertical line represents losses.

The actual expansion should be as close to isentropic as possible. The isentropic efficiency of a turbine is defined as the actual work output divided by the isentropic work output. In the TS diagram, the isentropic work is the area under the isentropic line, and the actual work is the area under the actual curve which is less due to entropy increase. To improve efficiency, we need to reduce entropy generation.

This can be done by using advanced blade designs to reduce friction and turbulence. By using cooling techniques (for gas turbines) to allow higher inlet temperatures without damaging the blades. By reducing clearances between blades and casing to minimize leakage.

We use TS diagram to increase the work output.

Higher inlet temperatures allow for more work extraction i.e. area under the expansion curve in TS diagram is larger. However, materials limit the maximum temperature. The ratio of inlet pressure to outlet pressure. Higher pressure ratios lead to more work, steeper drop in temperature during expansion but also require more compression work in a gas turbine cycle. Higher inlet temperatures (T_in) and higher pressure ratios increase the work output, the area under the expansion curve in the TS diagram becomes larger. However, higher temperatures require better materials nickel-based superalloys, ceramic coatings, or active cooling and higher pressure ratios require more robust
compressor designs.

The net work output of the cycle is the area enclosed by the cycle in the TS diagram. To improve the cycle, we can use regeneration i.e. using turbine exhaust to preheat the compressed air which reduces the heat input required in TS diagram, the heat addition process shifts to the left, reducing the area of heat addition and increasing efficiency.

By using reheating i.e. expanding in multiple turbine stages with reheating in between, which increases the work output, the expansion curve in TS becomes a series of steps, increasing the area under the expansion curve. By using intercooling during compression, which reduces the work required for
compression the compression curve in TS becomes a series of steps, reducing the area under the compression curve.