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 addition = h

_{1}-h_{4C}
Heat rejected= h

_{2}-h_{3}
Let us write here the efficiency of Carnot cycle

**η**

_{C}= 1-(h_{2}-h_{3})/ (h_{1}-h_{4C})
h

_{2}- h_{3}= T_{2}(s_{2}-s_{3})
h

_{1}- h_{4C}= T_{1}(s_{1}-s_{4C})
As we will have, s

_{1}= s_{2}and s_{3}= s_{4c}
Therefore for Carnot cycle, we will have efficiency

η

_{C}= 1-T_{2}/T_{1}###
**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 h

_{1}- h_{4C}= T_{1}(s_{1}-s_{4R})
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

**i.e. Tm***mean temperature of heat addition*
Therefore, we can write above equation for a Rankien
cycle as mentioned here

Heat addition = h

_{1}-h_{4C}= T_{m}(s_{1}-s_{4R})
Where Tm is termed as mean temperature of heat
addition

Therefore, we will have efficiency of the Rankine cycle
i.e. η

_{R}
η

_{R}= 1-T_{2}/T_{m}
Mean temperature of heat addition will be termed as
a constant temperature at which if same quantity of heat energy i.e. (h

_{1}-h_{4C}) will be added to the system then we will have same changes in entropy i.e. (s_{1}-s_{4R}).
Now as we can easily observe here that mean
temperature of heat addition i.e. T

_{m}will be less than T_{1}. 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

Image courtesy: Google