4.7 The Carnot Efficiency

A general expression for the efficiency of a heat engine can be written as:

$Efficiency = \frac{Work}{{Heat Energy}_{Hot}}$

We know that all the energy that is put into the engine has to come out either as work or waste heat. So work is equal to Heat at High temperature minus Heat rejected at Low temperature. Therefore, this expression becomes:

$Efficiency= \frac{Q_{Hot} {-Q}_{Cold}}{Q_{Hot}}$

Where, Q_Hot = Heat input at high temperature and Q_Cold= Heat rejected at low temperature. The symbol (Greek letter eta) is often used for efficiency this expression can be rewritten as:

$η^{'} (%) = 1 - \frac{Q_{Cold}}{Q_{Hot}} \times 100$

The above equation is multiplied by 100 to express the efficiency as percent.

French Engineer Sadi Carnot showed that the ratio of Q_HighT to Q_LowT must be the same as the ratio of temperatures of high temperature heat and the rejected low temperature heat. So this equation, also called Carnot Efficiency, can be simplified as:

$η^{'} (%) = 1 - \frac{T_{Cold}}{T_{Hot}} \times 100 %$

Note: Unlike the earlier equations, the positions of T_cold and T_hot are reversed.

The Carnot Efficiency is the theoretical maximum efficiency one can get when the heat engine is operating between two temperatures:

The temperature at which the high temperature reservoir operates ( T_Hot ).
The temperature at which the low temperature reservoir operates ( T_Cold ).

In the case of an automobile, the two temperatures are:

The temperature of the combustion gases inside the engine ( T_Hot ).
The temperature at which the gases are exhausted from the engine ( T_Cold ).

Below is a table showing two temperature scales. The scale labeled "HOT," shows the range of temperatures for the combustion of gases in a car engine. The scale labeled "COLD," shows the range of temperatures at which gases are exhausted from the car engine.

Scales showing car engine combustion temperature ranging from 500 to 2,000 degres C and exhaust gases ranging from 25 to 150 degrees C. — Car Engine Temperatures: red indicates combustion temperatures and blue indicates exhausted temperatures.

Instructions: Look carefully at the efficiency numbers in the body of the table. How do the Hot and Cold temperatures' effect on the efficiency.

Car Engine Efficiency
Hot Columns Cold Rows	Hot 500°C	Hot 600°C	Hot 700°C	Hot 800°C	Hot 900°C	Hot 1,000°C	Hot 1,500°C	Hot 2,000°C
Cold 150°C	45	52	57	61	64	67	76	81
Cold 125°C	49	54	59	63	66	69	78	82
Cold 100°C	52	57	62	65	68	71	79	84
Cold 75°C	55	60	64	68	70	73	80	85
Cold 50°C	58	63	67	70	72	75	82	86
Cold 25°C	61	66	69	72	75	77	83	87

Answer the following questions based on the information in the Car Engine Efficiency table above.

Example

For a coal-fired utility boiler, the temperature of high pressure steam (T_hot)would be about 540°C and T_cold, the cooling tower water temperature, would be about 20°C. Calculate the Carnot efficiency of the power plant:

Solution:

Carnot efficiency depends on high temperature and low temperatures between which the heat engine operates. We are given both temperatures. However, the temperatures need to be converted to Kelvin:

$\begin{matrix} T_{Hot} = 540^{o} C + 273 = 813 K \\ T_{Cold} = 20^{o} C + 273 = 293 K \\ η = [1 - \frac{T_{Cold}}{T_{Hot}}] \times 100 % \\ η = [1 - \frac{293 K}{813 K}] \times 100 % \\ = 64 % \end{matrix}$

Practice

A solar thermal power plant uses concentrated sunlight to heat a working fluid. The high-temperature reservoir reaches 450°C, while the cooling system maintains the low-temperature reservoir at 90°C. Calculate the Carnot efficiency of this power plant. .

Step 1

Convert the high and low temperatures from Celsius to Kelvin:

$\begin{matrix} T_{Hot} = 450^{o} C + 273 \\ = 723 K \end{matrix}$

$\begin{matrix} T_{Cold} = 90^{o} C + 273 \\ = 363 K \end{matrix}$

Step 2

Determine the efficiency using the Carnot efficiency formula:

$\begin{matrix} η = [1 - \frac{T_{Cold}}{T_{Hot}}] \times 100 % \\ η = [1 - \frac{363 K}{723 K}] \times 100 % \\ = 49.8 % \end{matrix}$
From the Carnot Efficiency formula, it can be inferred that a maximum of 49.8% of the fuel energy can go to generation. To make the Carnot efficiency as high as possible, either T_hot should be increased or T_cold (temperature of heat rejection) should be decreased.

Check Your Understanding

Example

Practice

Step 1

Step 2