Carnot Cycle

How can you convert heat into mechanical energy and work?

• One can use a steady-state flow device like a turbine where the thermal energy of a flowing fluid can be converted to mechanical energy.
• Another technique is the use of a thermodynamic cycle. In this process, a working fluid like air or steam undergoes a series of state changes and finally returns to its initial state. 

The Carnot cycle is the most efficient thermodynamic cycle where an ideal gas in a frictionless piston-cylinder device undergoes the following 4 reversible successive processes:

1. The ideal gas undergoes an isothermal and reversible expansion through the addition of heat from a hot reservoir at a temperature, T_H,
2. The adiabatic and reversible expansion of the ideal gas,
3. The ideal gas undergoes an isothermal and reversible compression by discarding heat to a cold reservoir at a temperature, T_C,
4. And finally, the adiabatic and reversible compression of the ideal gas to its initial state.

This can be schematically represented as follows:

This process can also be graphed onto a P-V graph as shown below:

If we consider our system as an ideal gas in a frictionless piston-cylinder assembly, the analysis of the Carnot cycle is as follows:

Process 1 to 2 (Reversible isothermal expansion at T_H)

Let us take the 1^st law of thermodynamics for a closed system and work done by an ideal gas:

Process 2 to 3 (Reversible Adiabatic Expansion)

Once again let us take the 1^st law of thermodynamics for a closed system and work done by the ideal gas:

Process 3 to 4 (Reversible Isothermal Compression)

Once again let us take the 1^st law of thermodynamics for a closed system and work done by the ideal gas:

Process 4 to 1 (Reversible Adiabatic Compression)

Once again let us take the 1^st law of thermodynamics for a closed system and work done by the ideal gas:

Thus, if we look at the net work output of the Carnot cycle,

If we apply the first law of thermodynamics to the Carnot cycle:

The Efficiency of Carnot Cycle, η

The efficiency, η, can be derived as follows:

Since the processes 2 to 3 and 4 to 1 are reversible and adiabatic, equation 17 can be further simplified as follows:

If we combines Equations 19 and 20 we get the following equation 21:

 (21)   

The Carnot cycle is the most efficient heat engine operating between two constant heat reservoirs T_H and T_C. A heat engine is any mechanism that operates cyclically with the primary purpose to partially convert heat into work. 

Note that: W_net = Q_H - Q_C                        (22)

Using this new notation, Equations 16 and 19 combine as follows:

Substituting Equation 22 into 23 we get:

Finally, with all these new notations, we can represent the Carnot Cycle in a simpler way as shown:

Solved Examples

Example 1

A Carnot cycle receives 1000 kJ of heat at 800°C and rejects heat at 300°C. Calculate the work output.

Solution

From Equation 23:

Substituting the provided values into the above equation:

Thus:

W_net = 466 kJ

Example 2

Suppose an iceberg weighing 10¹⁰ kg drifts into the Gulf stream which has a temperature of 22°C. If we operate a Carnot heat engine using the Gulf stream as a heat source and the iceberg as a cold sink, what is the work output that could be generated while the iceberg is melting?

Solution

Assume that the temperature of the iceberg is 0°C and the heat of fusion for ice is 334,880 J/kg. Take the iceberg as the heat sink.

The amount of heat rejected to the sink is:

Q_C = (10¹⁰) (334,880) = 3.35 × 10¹⁵ J

The amount of heat absorbed can be calculated using equation 24: