# What is Example of Carnot Efficiency – Problem with Solution – Definition

Example of Carnot Efficiency – Problem with Solution. Calculate the carnot efficiency of coal-fired power plant. Compare it with real cycles of power plants. Thermal Engineering

## Carnot Cycle – Processes

In a Carnot cycle, the system executing the cycle undergoes a series of four internally reversible processes: two isentropic processes (reversible adiabatic) alternated with two isothermal processes:

1. isentropic compression – The gas is compressed adiabatically from state 1 to state 2, where the temperature is TH. The surroundings do work on the gas, increasing its internal energy and compressing it. On the other hand the entropy remains unchanged.
2. Isothermal  expansion – The system is placed in contact with the reservoir at TH. The gas expands isothermally while receiving energy QH from the hot reservoir by heat transfer. The temperature of the gas does not change during the process. The gas does work on the surroundings. The total entropy change is given by: ∆S = S1 – S4 = QH/TH
3. isentropic expansion – The gas expands adiabatically from state 3 to state 4, where the temperature is TC. The gas does work on the surroundings and loses an amount of internal energy equal to the work that leaves the system. Again the entropy remains unchanged.
4. isothermal compression – The system is placed in contact with the reservoir at TC. The gas compresses isothermally to its initial state while it discharges energy QC to the cold reservoir by heat transfer. In this process the surroundings do work on the gas. The total entropy change is given by: ∆S = S3 – S2 = QC/TC

## Carnot Cycle Efficiency

In general, the thermal efficiency, ηth, of any heat engine is defined as the ratio of the net work it does, W, to the heat input at the high temperature, QH.

Since energy is conserved according to the first law of thermodynamics and energy cannot be be converted to work completely, the heat input, QH, must equal the work done, W, plus the heat that must be dissipated as waste heat QC into the environment. Therefore we can rewrite the formula for thermal efficiency as:

Since QC = ∆S.TC and QH = ∆S.TH, the formula for this maximum efficiency is:

where:

• is the efficiency of Carnot cycle, i.e. it is the ratio = W/QH of the work done by the engine to the heat energy entering the system from the hot reservoir.
• TC is the absolute temperature (Kelvins) of the cold reservoir,
• TH is the absolute temperature (Kelvins) of the hot reservoir.

## Example: Carnot efficiency for coal-fired power plant

In a modern coal-fired power plant, the temperature of high pressure steam (Thot) would be about 400°C (673K) and Tcold, the cooling tower water temperature, would be about 20°C (293K). For this type of power plant the maximum (ideal) efficiency will be:

ηth= 1 – Tcold/Thot = 1 – 293/673 = 56%

It must be added, this is an idealized efficiency. The Carnot efficiency is valid for reversible processes. These processes cannot be achieved in real cycles of power plants. The Carnot efficiency dictates that higher efficiencies can be attained by increasing the temperature of the steam. This feature is valid also for real thermodynamic cycles. But this requires an increase in pressures inside boilers or steam generators. However, metallurgical considerations place an upper limits on such pressures. Sub-critical fossil fuel power plants, that are operated under critical pressure (i.e. lower than 22.1 MPa), can achieve 36–40% efficiency. Supercritical designs, that are operated at supercritical pressure (i.e. greater than 22.1 MPa),  have efficiencies around 43%. Most efficient and also very complex coal-fired power plants that are operated at “ultra critical” pressures (i.e. around 30 MPa) and use multiple stage reheat reach about 48% efficiency.

References:
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