We were discussing “Rankine cycle” and “Carnot cycle” in our previous posts. We have discussed there the concept of power cycle and maximum efficient cycle in the field of thermal engineering. We have also seen various basic properties of a pure substance such as “PV diagram of a pure substance”.

Today we will see here the comparison between Rankine cycle and Carnot cycle with the help of this post.

If we recall the basics of Carnot cycle, we will secure information that in case of Carnot cycle heat addition to the system will be done at maximum temperature. Every thermodynamic cycle will have one maximum temperature. In case of Carnot cycle, heat addition will take place isothermally at maximum temperature.

Whereas if we recall the basics of Rankine cycle, we will see that heat addition to the system will not take place at maximum temperature but also heat energy will be continuously added to the system from a temperature below the maximum temperature to a temperature equal to the maximum temperature.

This is the basic reason that if we compare the Carnot cycle with any other reversible cycle, with similar value for maximum temperature, Carnot cycle will have better efficiency as compared to any other reversible cycle and we will prove this statement in this post.

### Let us see here the temperature entropy (T-S) diagram for Rankine cycle and Carnot cycle

Rankine cycle is displayed here in temperature entropy diagram by 1-2-3-4R-1 and similarly we have also shown here the Carnot cycle i.e. 1-2-3-4C-1. First we will concentrate here at Carnot cycle.
We will find out here the efficiency for a Carnot cycle.
Heat rejected= h2-h3
Let us write here the efficiency of Carnot cycle

ηC= 1-(h2-h3)/ (h1-h4C)
h2- h3= T2(s2-s3)
h1- h4C = T1(s1-s4C)
As we will have, s1= s2 and s3= s4c

Therefore for Carnot cycle, we will have efficiency
ηC= 1-T2/T1

### Let us consider now the case of Rankine cycle

We will determine the efficiency of the Rankine cycle and we will also compare the efficiency of Rankine cycle with the efficiency of Carnot cycle in this post.

If we consider the case of Rankine cycle, heat addition will not take place at constant temperature but also heat addition will take place at constant pressure in case of rankine cycle. Therefore we may not write the formula for heat addition to the system in Rankine cycle as we have written for Carnot cycle.

Or in simple words, we may not write that h1- h4C = T1 (s1-s4R)

Why we can not write above equation? Because for a Rankine cycle, heat will never added to the system at a maximum temperature but also heat energy will continuously added to the system from a temperature below the maximum temperature to a temperature equal to the maximum temperature.

So a term will be introduced here and that is mean temperature of heat addition i.e. Tm

Therefore, we can write above equation for a Rankien cycle as mentioned here
Heat addition = h1-h4C = Tm (s1-s4R)
Where Tm is termed as mean temperature of heat addition
Therefore, we will have efficiency of the Rankine cycle i.e. ηR
ηR= 1-T2/Tm
Mean temperature of heat addition will be termed as a constant temperature at which if same quantity of heat energy i.e. (h1-h4C) will be added to the system then we will have same changes in entropy i.e. (s1-s4R).

Now as we can easily observe here that mean temperature of heat addition i.e. Tm will be less than T1. Therefore Carnot cycle will have higher efficiency as compared to the Rankine cycle.

Do you have any suggestions? Please write in comment box.
We will see another topic in our next post in the category of thermal engineering.

### Reference:

Engineering thermodynamics by P. K. Nag
Engineering thermodynamics by Prof S. K. Som