In class last Wednesday, we discussed conservation of energy, using the example of an automotive engine. We drew a control volume around the engine, and carefully considered the energy (or energy per second) coming into the control volume and the energy (or energy per second) leaving the control volume.

We wrote in general:

And in particular, for the automotive engine we wrote:

We also noted:

Now look at FIGURE 3.3 in the text. This shows another heat engine: the steam cycle of a steam-electric power plant. This is the Rankine cycle. All coal- and nuclear-fired power plants operate on this cycle. The main components of the cycle are four:

1. The boiler: heat is transferred from a combustion process (or from a nuclear fission process) to the high-pressure water, creating high pressure steam (ie, thermal energy). The maximum temperature of the steam is about 600 degrees C in a fossil-fueled power plant, and about 300 degrees C in a nuclear-fired power plant.
2. Turbine: the high-pressure, high-temperature steam continuously expands through the turbine, doing work on the turbine blades, which turn the shaft of the turbine. The output shaft of the turbine turns the electrical generator, creating the electrical energy (or power) desired. The steam (or steam-water mixture) leaves the turbine at a very low pressure (it may be sub-atmospheric). However, the steam did not give up all of its thermal energy in the process of creating the work. There is some thermal energy remaining in the steam/water outlet of the turbine.
3. Condenser: In order to get the H2O back to its original state (ready to re-enter the boiler), thermal energy (ie, heat) is removed. In the condenser, heat is transferred from the H2O of the cycle to a cold external stream of water. The loss of heat by the cycle H2O causes it to fully condense to liquid phase. This low-pressure water of the cycle leaves the condenser. (The text shows the heat being discharged to a lake. However, this is seldom permitted these days. Rather, for most power-plants of recent and new construction in the USA, the waste heat is discharged to the atmosphere using either a wet or dry cooling tower (see pp. 303-314 in text).
4. Pump: The final step is the pumping of the water to the high pressure, ready to start the cycle over.

Let W = net work done by cycle over

Let Q_H = heat added to cycle over

the given time

Let Q_C = heat rejected from cycle

over the given time

By the FIRST LAW OF THERMODYNAMICS (ie, by the principle of conservation of energy):

And by our definition of efficiency:

where m_f = mass of fuel burned over

and HV = heating value of the fuel

We may also write:

The last equation indicates we could have a heat engine of 100% efficiency if we could reduce the amount of the rejected heat to zero!

However, nature does not permit this.

To understand this, we must consider order and disorder.

We must consider a new principle: THE SECOND LAW OF THERMODYNAMICS.

This says a spontaneous process (such as a porcelain cup falling off a table), may proceed from order to disorder (the cup breaks into pieces), though the reverse process (from disorder to order) is not permitted unless work is done (we glue to cup back together).

Some forms of energy, such as the kinetic energy of a body, have a lot of order. However, heat has a lot of disorder, since it is a manifestation of the random motions of a huge number of molecules. Thus, as a pendulum swings, and the kinetic energy of the pendulum is slowly transformed to heating of the air around the pendulum and the material of the pendulum (and thus, the pendulum eventually ceases swinging), we have an example of order to disorder. This is a spontaneous process. The reverse spontaneous process is not permitted. We cannot spontaneously proceed from the disorder of the air molecules randomly moving around the pendulum to ordered motion of the pendulum swinging again. (However, we could use the heat of the air as Q_H in a heat engine, and use the work output of this heat engine to restart the swinging of the pendulum.)

Now that we know about order and disorder, let’s go back to the heat engines in the text (ie, the steam-electric power or the generic heat engine depicted in Figure 4.19).

We start with heat, ie, Q_H. That is, we start with disorder! By the Second Law of Thermodynamics we have to end up with at least as much disorder as we started with. Thus, we have to heat up something. We need to reject heat from our heat engine to a heat sink. We must have a Q_C. We cannot build (even on paper) a 100% efficient heat engine.

OK, suppose you "buy" this logic. Heat engine efficiencies are less than 100%. The next question is, what is maximum possible efficiency a heat engine may have (on paper)?

Consider two identical blocks, though one is hotter than the order. The respective temperatures are T_H and T_C. If we put the blocks in contact, heat will flow from the hot one to the cold one. There is direction! The Second Law of Thermodynamics must be in play.

The overall disorder of the two blocks in contact must be greater than or equal to the starting disorder of the two separated blocks.

Thermodynamicists use ENTROPY (S) to express the disorder. It is a property of the system, that is, the blocks in our case. The change in the disorder (ie, entropy) of the system is expressed:

where Q is a tiny amount of heat added to the system, and T is its temperature (which is almost constant since the amount of heat

added is tiny). Of course, if our system is very large, than Q could be big, and T still would hardly change.

The best possible heat engine is one for which:

This is called a reversible heat engine. If our heat source and heat sink are huge, then their T’s are essentially constant, and we may write:

Thus:

Q_H/T_H = Q_C/T_C

Our equation for the best efficiency becomes: