9.3 Introduction to the second law of thermodynamics: heat engines  (Page 3/10)

so that

Thus the net work done by the system equals the net heat transfer into the system, or

just as shown schematically in [link] (b). The problem is that in all processes, there is some heat transfer $Q_{\text{c}}$ to the environment—and usually a very significant amount at that.

In the conversion of energy to work, we are always faced with the problem of getting less out than we put in. We define conversion efficiency $\text{Eff}$ to be the ratio of useful work output to the energy input (or, in other words, the ratio of what we get to what we spend). In that spirit, we define the efficiency of a heat engine to be its net work output $W$ divided by heat transfer to the engine $Q_{\text{h}}$ ; that is,

Since $W=Q_{\text{h}}-Q_{\text{c}}$ in a cyclical process, we can also express this as

making it clear that an efficiency of 1, or 100%, is possible only if there is no heat transfer to the environment ( $Q_{\text{c}}=0$ ). Note that all $Q$ s are positive. The direction of heat transfer is indicated by a plus or minus sign. For example, $Q_{\text{c}}$ is out of the system and so is preceded by a minus sign.

Daily work done by a coal-fired power station, its efficiency and carbon dioxide emissions

A coal-fired power station is a huge heat engine. It uses heat transfer from burning coal to do work to turn turbines, which are used to generate electricity. In a single day, a large coal power station has $2\text{.}\text{50}\times {\text{10}}^{\text{14}}\text{J}$ of heat transfer from coal and $1\text{.}\text{48}\times {\text{10}}^{\text{14}}\text{J}$ of heat transfer into the environment. (a) What is the work done by the power station? (b) What is the efficiency of the power station? (c) In the combustion process, the following chemical reaction occurs: _{$\text{C}+{\text{O}}_2\to {\text{CO}}_2$} . This implies that every 12 kg of coal puts 12 kg + 16 kg + 16 kg = 44 kg of carbon dioxide into the atmosphere. Assuming that 1 kg of coal can provide $2\text{.}5\times {\text{10}}^6\text{J}$ of heat transfer upon combustion, how much _{${\text{CO}}_2$} is emitted per day by this power plant?

Strategy for (a)

We can use $W=Q_{\text{h}}-Q_{\text{c}}$ to find the work output $W$ , assuming a cyclical process is used in the power station. In this process, water is boiled under pressure to form high-temperature steam, which is used to run steam turbine-generators, and then condensed back to water to start the cycle again.

Solution for (a)

Work output is given by:

Substituting the given values:

Strategy for (b)

The efficiency can be calculated with $\text{Eff}=\frac{W}{Q_{\text{h}}}$ since $Q_{\text{h}}$ is given and work $W$ was found in the first part of this example.

Solution for (b)

Efficiency is given by: $\text{Eff}=\frac{W}{Q_{\text{h}}}$ . The work $W$ was just found to be ${\text{1.02 × 10}}^{\text{14}}\text{J}$ , and $Q_{\text{h}}$ is given, so the efficiency is

Strategy for (c)

The daily consumption of coal is calculated using the information that each day there is $2\text{.}\text{50}\times {\text{10}}^{\text{14}}\text{J}$ of heat transfer from coal. In the combustion process, we have _{$\text{C}+{\text{O}}_2\to {\text{CO}}_2$} . So every 12 kg of coal puts 12 kg + 16 kg + 16 kg = 44 kg of _{${\text{CO}}_2$} into the atmosphere.

Solution for (c)

The daily coal consumption is

Assuming that the coal is pure and that all the coal goes toward producing carbon dioxide, the carbon dioxide produced per day is