# Temperature-Entropy(T-s) Diagram

Updated: Jul 7

The T-s diagram is one of the most used plots in Thermodynamics. It is used to analyze vapor power cycles, gas power cycles, and gas refrigeration cycles along with the P-v diagram. Hence it becomes really important to understand the T-s diagram.

Once you understand it, applied Thermodynamics becomes easier to interpret and analyze. I have tried to explain all the important concepts that one should know about T-s diagrams in this article. There may be more things related to T-s diagrams other than I have mentioned here but these concepts are enough for you to get started and have confidence in analyzing Thermodynamic systems.

### Why do we use P-v and T-s diagrams to describe Thermodynamic systems? Why the T-s diagram in particular is useful?

The area under the curve in the P-v diagram gives the amount of work transferred during the process and the area under the curve in the T-s diagram gives heat interaction during the process. The area under the curve in P-v and T-s Diagram (Not related to each other)

Note: T-s and P-v diagrams are not related to each other in the above figure.

The T-s plot gives the temperature and entropy values at different state points. These temperature values can be used to calculate the enthalpy at that state in the case of ideal gases because for ideal gas enthalpy is a function of temperature only (h=Cp*T). Once the enthalpy values at different state points are known it becomes easier to analyze the system and calculate heat and work interactions. If any two properties at a specific state are known, one can almost easily find all other properties.

## T-s Diagram and Different Processes/Lines

### Isothermal Line (Constant Temperature Line) Isothermal Line on T-s Diagram

The direction of these lines depends on the kind of processes they follow. If the process is part of the work producing cycle then it is drawn in a clockwise sense and if the process is part of the work consuming cycle then it is drawn in an anticlockwise sense. The same logic is true for all other processes shown below.

### Isentropic Line (Constant Entropy Line) Isentropic Line on T-s Diagram

### Isobaric Line (Constant Pressure Line)

When you analyze the thermal cycle in which the ideal gas is the working fluid, that is there is no involvement of change of phase, the constant pressure lines are drawn as follows. Isobaric Line on T-s Diagram (For ideal gases)

### Example of a cycle on T-s Diagram which consists of an isobaric and an isentropic process. Example of a cycle on T-s Diagram which consists of an isobaric and an isentropic process.

But when the thermal cycle involves a change of a phase of the working fluid, the T-s diagram used for analysis has a dome shape which is a locus of saturation points. Isobaric Line on T-s Diagram (For Two-phase System)

### Example of a cycle on T-s Diagram which consists of an isobaric and an isentropic process (For a two-phase system) Example of a cycle on T-s Diagram which consists of an isobaric and an isentropic process (For a two-phase system)

### Isochoric Line (Constant Volume Line)

They are not much used relative to other lines in the T-s diagram.

So, you don't have to worry about them.

### Isenthalpic Line (Constant Enthalpy Line)

The isenthalpic process is a special kind of process. Mostly throttling process undergoes an isenthalpic process where enthalpy at the beginning of the throttling enthalpy at the end of the throttling is the same. The throttling process is highly irreversible hence it can not be drawn on the property diagram. For convenience, it is drawn as a dotted line on the T-s diagram. Isenthalpic Line on T-s Diagram

### Example of a cycle on T-s Diagram which consists of an isenthalpic, an isobaric, and an isentropic process. Example of a cycle on T-s Diagram which consists of an isenthalpic, an isobaric, and an isentropic process.

### What does the area under the curve in an irreversible process represent?

Nothing. An irreversible process can not be drawn on the T-s diagram. Every process drawn on the T-s diagram must be reversible. If we have to represent an irreversible process on the T-s diagram then it is drawn as the dotted line as shown in the figure above.

For example, throttling is a highly irreversible process but to show it in the vapor compression cycle we draw it as a dotted line on the T-s diagram.

### What is the significance of the dome shape in the T-s diagram of the two-phase systems?

State postulate states that,

A simple compressible system is completely specified by two independent, intensive properties

This holds true for single-phase systems where only two intensive properties are enough to define the state of the system. But in the Rankine cycle during heat addition and heat rejection, the phase of the system changes. Hence to define the state third property pressure is introduced in the T-s diagram. These pressure lines are constant during phase change but they vary during sensible heat addition. When these saturation points at different pressures are joined they form a dome shape.

### What are you going to do with all this information?

First of all, have confidence that these plots are easy to draw and analyze. There is nothing to have fear. Once you get hold of these plots you can almost easily solve any problem in applied thermodynamics. I encourage you to draw and practice these diagrams until you get hold of it.