3.6: Carnot Efficiency

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

\[ Efficiency = \dfrac{Work}{Heat \, Energy \, (hot)} \]

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

\[ Efficiency = \dfrac{Q_{hot} - Q_{cold}}{Q_hot} \]

where \( Q_{hot} \) is the heat input at high temperature and \( Q_{cold} \) is the heat rejected at low temperature. The symbol \( \eta \) (Greek letter eta) is often used for efficiency this expression can be rewritten as:

\[ \eta ' \, (\%) = 1 - \dfrac{Q_{cold}}{Q_{hot}} * 100\% \]

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

French Engineer Sadi Carnot showed that the ratio of Q_hot to Q_cold 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:

\[ \eta ' \, (\%) = 1 - \dfrac{T_{cold}}{T_{hot}} * 100\% \]

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_h_ot).
The temperature at which the low temperature reservoir operates (T_c_old).

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

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

The following video explains how automobile engines work.

Then, why should we operate the automobiles at low efficiencies?

It is not that we cannot achieve high temperatures, but we do not have the engine materials that can withstand the high temperature. As a matter of fact, we do not let the engine gases go the maximum that they can go even now and instead try to keep the engine cool by circulating the coolant.

So we are taking the heat out of the gases (thus lowering the T_hot) and making the engine operate at cooler temperatures so that the engine is protected - but lowering the efficiency of an automobile.

Additional Information

An analogy is taxes. The more money you earn (heat), the more money is taxed (cold), leaving you with less money to take home (efficiency). However, if you could earn more money (heat) and find a way to have less taxes taken out (better engine material), you would have more money to take home (efficiency).

Figure 3.6.1 shows 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.

Figure 3.6.1. Hot and cold temperature scales in a typical car engine

Table 3.6.1 lists efficiencies as a function of hot and cold temperatures.

Table 3.6.1. Car engine efficiency (in %)

	Hot 500°C	Hot 600°C	Hot 700°C	Hot 800°C	Hot 900°C	Hot 1000°C	Hot 1500°C	Hot 2000°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

Example Problems

Example 1

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.

Answer

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 first.

\[ T_{hot} = 540 + 273 = 813 K \nonumber\]

\[ T_{cold} = 20 + 273 = 293 K \nonumber\]

Using equation 3.6.4,

\[ \eta = (1 - \dfrac{293}{813}) * 100\% = 64\% \nonumber\]

From the Carnot Efficiency formula, it can be inferred that a maximum of 64% 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.

For extra practice problems, refer to this link.