Carnot Cycle : Effitiency, Working and Air Standard Analysis

Today we will discuss and analysis of the Carnot cycle. Carnot cycle is one of the air standard cycles. It is given by a French engineer, Sadi Carnot so named as Carnot cycle. This is an ideal cycle of all thermodynamic cycle because it gives maximum efficiency between two temperature difference. When we study about any other cycle, we compare it to the Carnot cycle, which will give the maximum work output between this temperature range.

As we have discussed in our last post that in air standard cycle we assume air as the working fuel but actually air with fuel use as working fluid. In Carnot cycle analysis we assume that air is used as the working fluid and it completes a cyclic process. So in Carnot cycle air standard analysis we assume a reciprocating engine in which air is filled which act as the working fluid. A heat reservoir is used at the head for Heat addition and heat rejection.  Carnot cycle is a combination of four thermodynamic processes, in which two are isothermal processes and the remaining one are isentropic processes.  An isothermal process is also known as constant temperature process in which temperature of working fluid remains constant until process done. In the isentropic process, the entropy of a system remains constant which mean the system would not emit or absorb heat until the whole process. The combination of this process makes the Carnot cycle.

Carnot Cycle:

Working:

As we have discussed Carnot cycle is the combination of two isothermal processes and two isentropic processes. In Carnot cycle heat is add or removed at constant temperature and expansion and compression done
isentropically. In the Carnot cycle, we assume a piston-cylinder device which is insulated from the surrounding. No heat can add or removed from the system. The head of the system can be replaceable during the process by the heat reservoir. At the starting, the piston is at the top or at the top dead center of the cylinder. At this position, the insulated head replaced by the higher temperature heat reservoir which adds the heat energy into the system by Zeroth law. This heat energy move or expand the piston downward. One thing notes that the heat energy added by the reservoir does not raise the temperature of the system. It only has done work by expanding piston So this process is done isothermally.

When the piston reaches mid of the cylinder, the head is replaced by the insulated head so no heat can add into the system but due to the inertia of the piston, it expands continuously by using the heat of the working fluid (Air) which down the temperature of the system. Because no extra heat is added into the system, this process has done isentropically. At this point, piston reaches its maximum downward limits or at bottom dead center.

Now the cylinder head replaced by the lower temperature heat reservoir which absorb the heat from the system. Due to this the air contracts to maintain its temperature constant. The piston moves upward or compresses the air. In this process the heat removed by the system used to compress the system and the temperature of system remains constant. So the compression done isothermically.

When the piston is at mid of the cylinder the heat reservoir at head replaced by the insulated head. So no head is removed from the system at this end. Due to the inertia of the piston, it moves continuously upward which compressed the air and increase its temperature. No heat is added or removed until this compression so this process did isentropically. At this point the piston reaches its initial condition (Top dead center), so complete a cyclic process. The working fluid (Air) also reached its initial condition so it completes a thermodynamic cycle.

Air Standard Analysis:

As we discuss in the upward paragraph that a Carnot cycle consists of four processes. These are:-

1. Isothermal Expansion:

In this process, the heat is added into the system and the piston expands isothermally. In this process, the temperature of the air remains constant.

The amount of heat added to the system is given by the isothermal process is :

Cheat addition = mRT1log(V2/V1)

           = mRT1log(r)

Where is r is compression ratio.

2. Isentropically Expansion:

In this process, the no heat is added into the system and the piston expands isentropically. In this process, the temperature of the air decreases from the initial temperature. In this process work done by the system. The work done during this isentropic process is given by

Work done =  (P2V2 – P3V3) / ( γ – 1 )

Where γ is a specific heat ratio of air.

3. Isothermally Compression:

In this process, the air is compressed and heat is removed from the system. In this process, the temperature of the air remains constant.

The heat removed by the system is given by the isothermal process is :

Q heat removed = mRT3log(V4/V3)

              = mRT1log(r)

Because V4/V3 = V2/V1 = r

4. Isentropically Compression:

In this process, the no heat is added into the system and the piston compresses the air isentropically. In this process, the temperature of air increases from the initial temperature. In this process, the work is done on the system.

The work done during this process given by isentropic process is :

Work done =  (P1V1 – P4V4) / ( γ – 1 )

Where γ is a specific heat ratio of air.

Efficiency:

The efficiency of the Carnot cycle is analyzed thermodynamically. The efficiency of any engine given by

η = Work done / Heat supplied

Work done = Heat supplied – Heat rejected

Work done =  mRT1log(r)  –  mRT3log(r)

                                                =   mRlog(r) [T1 – T3]

Because heat supplied and heat rejected isothermically which is given by

η = Work done  / Heat supplied

            =  mRlog(r) [T1 – T3]  /  mRlog(r)T1

                                               =   [T1 – T3] / T1

                                               =  1 – T3/T1

From the above equation, we see that the efficiency of the Carnot cycle increases by increasing the higher temperature of the cycle (T1) or by decreasing the lower temperature of the cycle (T3). It is impossible to make an engine work on Carnot cycle because to achieve isothermal expansion and compression the piston should run extremely slow to ensure that the air is always at the same temperature and isentropic expansion and compression should carry out as fast as possible to no exchange of heat.  

Between two fixed temperatures Carnot cycle has the maximum possible efficiency compared to other slandered cycles. The Carnot cycle does not have any real life example so it is an ideal cycle, not a practical cycle. Today we have discussed about working of Carnot cycle and analysis of Carnot cycle. If you have any query ask by commenting. If you like this article, don’t forget to share it on social networks. Subscribe our website for more informative articles. Thanks for reading it.