Heat engines

Now, let’s turn back to more traditional description of 2nd law. The 2nd law states that 100% conversion from heat to work in heat engines is impossible and there’s always a limitation. Here’s a heat engine. We have a high temperature heat source at temperature TH or T2 and a low temperature heat sink at TL or T1. Heat is extracted from the high temperature source by QH and released to the low temperature sink by QL, doing work W. That is the basic operating principle of the heat engine. Then, we want to calculate the maximum work that can be obtained from this machine. Take the machine as our system. The machine is reversible.

The reversibility is assumed to calculate the maximum work and the highest efficiency of this machine. And the machine is steady state, since the equipment itself does not change over time. And it is a closed system, since there is no material exchange. To find out the maximum work and the highest efficiency, let’s start from the generalized entropy equation. The first two things in this equation is related to the material transfer, so they are zero since the system is closed. The lost work is also zero, since the process is reversible. And the entropy change of the system is zero, since the machine is steady-state, meaning that the initial and the final state are the same.

So the equation becomes sigma delta Q over T is zero. There are two heats, QH and QL. So the equation is QH over T2 + QL over T1 is zero. So QL is -T1 over T2 times QH. Let’s go to the 1st law then. The first law states that sum of heat change and work change is the internal energy change. Since the machine itself is steady states, internal energy does not change, thus delta U is zero. Insert the previously derived a equation, QL equal to -T1 over T2 times QH. Then, the work can be expressed with QH and temperatures. By convention the work done on the system is W.

So the work done by the system is -W and it is QH times (1-T1/T2). This work is reversible work, since the machine operates reversibly, and it is the maximum work done by the system. The efficiency is generally defined as output divided by input. The output is what we want to get, so here for the heat engines, the output is the work done by the machine. The input is heat, so the efficiency of heat engine is -Wrev over QH and it is 1-(T1 over T2). So the efficiency only depends on the temperature T1 and T2. And the efficiency of 1, the 100 % conversion from heat to work is impossible since the temperature is a positive value.

We never can get absolute zero temperature. Here is the actual constitution of heat engines. Let’s start from the high temperature heat source. It’s a boiler. Starting from the point 1, the liquid boils at temperature TH. The boiler provides the heat for the vaporization. At the exit of boiler, it’s now vapor at temperature TH. It goes into turbin at temperature TH. In turbin, work is done by the vapor by its expansion. Since the turbin operates under adiabatic, steady state and reversible condition, the expansion of vapor is isoentropic. At the exit of turbin, the temperature of vapor is decreased to TL due to the expansion. Then, the vapor goes into the condenser at temperature TL.

There, the vapor becomes liquid, since condensation reaction happens in condenser at temperature TL. The latent heat is removed by the cooling water. At the exit of condenser, it is now liquid at temperature TL. Then, it goes into compressor at temperature TL. Here, the liquid are compressed. The compressor pumps the liquid isoentropically, thus at the exit of compressor, the liquid temperature is increased to TH. Then, the liquid at temperature TH go back to the boiler completing a cylcle. The energy conversion process can be usefully displayed on a set of axes such as temperature-entropy diagram. Start from point 1. It is the liquid at temperature TH. In the boiler, the liquid vaporize at constant temperature TH.

The vapor has higher entropy than the liquid, so the entropy increase from S1 to S2. Then, at turbin, the vapor expands isoentropically, in other words, at constant entropy S2, the temperature decreases TL. Next at condenser, vapor condensed into liquid at constant temperature TL. Finally, at compressor, the liquid compressed isoentropically, increasing temperature back to TH. The heat, if operated reversibly, is temperature times entropy change by definition, so the heat QH supplied at boiler, is TH times (S2-S1). It is the heat source. Likewise, the heat QL removed at condenser, is TL times (S1-S2). The work done by this heat engine is the difference between QH and QL.

So it is (TH-TL) times (S2-S1) The efficiency of heat engine is work divided by heat supply. So it is (TH-TL) times (S2-S1) divided by TH times (S2-S1). Cancel out the same thing. Then the efficiency is (TH-TL) over TH, as before.