LECTURE 8 NOTES, February 4, 2003

1 - Expansion and Contraction of Materials (continued)

Most substances expand as their temperature is increased and contract as the temperature is decreased. The amount a substance expands/contracts with a change in temperature is proportional to the coefficient of expansion for that material. To prevent breakage, substances which must be in contact with each other should have very similar coefficients of expansion. For example, teeth and teeth fillings. You could imagine how a hot cup of coffee, or a cold ice cream bar, could crack the bone of your teeth if the filling material did not expand or contract about the same amount as did the bone.

The thermostat in your home takes advantage of the different coefficients of brass and iron. A bimetallic strip controls the thermostat. This consists of a strip of brass in contact with a strip of iron. At normal room temperature, the two strips have the same length. However, since brass expands (or contracts) more than iron when its temperature is raised (or cooled), the bimetallic strip will bend one way or another depending on the temperature being above or below room temperature. When the temperature is lower than normal room temperature, the length of the brass will be less than the length of the iron, and so the bimetallic strip will become "U-shaped" with the brass on the topside. When the temperature is greater than normal room temperature, the length of the brass will be greater than the length of the iron and the bimetallic strip will become U-shaped with the iron part of the strip on the topside of the U.

There are lots of other similar examples of expansion and contraction with respect to temperature change, but water is an exception to the normal rules. A good example is that of a pond of water that cools and freezes at its surface. Water has the unique property that its greatest density is not when it is in its solid state as ice, but when it is liquid at 4 degrees Centigrade. So when the air temperature drops and forces the water temperature downward, the pond eventually reaches a temperature of 4 degrees Centigrade from top to bottom. As further heat is removed from the water, the temperature above the bottom of the pond, which remains at 4 degrees Centigrade, will be colder. So there will be a layer of ice at the surface of the pond, at zero degrees, but the temperature at the bottom will remain about 4 degrees and maintain itself in the liquid state.

This is important for aquatic life because if solid ice were denser than liquid water at 4 degrees Centigrade, the ice would sink to the bottom and leave no place for the living creatures to swim about. Since iso much heat msut be removed to freeze a pond to the bottom, the temperature remains near 4 degrees near the bottom and the aquatic life survives. Of course, if the pond is shallow enough, and the air temperature above the pond is low enough, the entire pond will be reduced to ice at zero degrees, and removing further heat from the ice will decrease its temperature even further.

2 - Begin Chapter 10: Available Energy

The title of chapter 10, available energy, addresses an essential question about energy. Since energy exists in all substances with finite temperature, how can that energy be put to use? For example, an iceberg has lots of internal energy, but are there practical means by which to extract that energy? Energy is needed to heat and light our houses, make cars, trains and airplanes propell themselves, power the electronics in our computers and television sets, etc.

To understand what energy is available, we need to study the second law of thermodynamics. Joule's paddle wheel apparatus converted work completely into heat, but can that process be reversed? In particular, can we extract heat from a substance and convert it all into useful work. The first law of thermodynamics says yes. However, the second law of thermodynamics says no. We will first examine how these laws apply to heat engines.

3 - Heat Engines - Discovering How the First and Second Laws of Thermodynamics Apply

The basic heat engine includes a source of heat at high temperature, a low temperature enviroment where heat is exhausted (wasted), and a mechanism to convert the rest of the heat into useful work...e.g., driving a piston which turns a wheel. The first law of thermodynamics applies. It says energy is conserved, so the energy provided as heat from the source at high temperature is equal to the energy expended as useful work plus the heat energy lost at low temperature.

The concept of an ideal engine, the most efficient possible engine, was studied in 1824 by Sadi Carnot. His studies were based on applying what is known about heat transfer, the behavior of an ideal gas and conservation of energy. Much of what he inferred predated the laws of thermodynamics that were developed later.

The basic elements of a heat engine are a cylinder filled with gas and a piston at one end. The piston moves in and out. The piston does work on some external object as the gas expands and pushes it outward. However, some work is required to recompress the gas...less work than in the expansion stroke, but work that is in some sense wasted because it is being used for operating the engine...not for driving something else. Let's envision how this works. The compression stroke of the engine results from the fact that the engine is being used to put something in motion, for example a large wheel. As that wheel continues to turn because of its rotational inertia, it is able to drive the piston back inward and compress the gas.

Carnot's ideas derived from thoughts about how a water wheel works. Water at a high level drops and is caught by buckets in a wheel which rotates and does useful work. In principle, the water wheel could be made to run backwards as well. That is, by providing work to rotate the wheel in the opposite direction, water at the low level could be hoisted to water at the high level. Hence, Carnot thought about a theoretical engine that would run both forward and backward, in analogy with the water wheel running forward and backward. Running forward, the heat engine would operate between a high temperature source and a low temperature environment. The engine does some useful work, but it also wastes some heat. Running backward as a refrigerator, work is applied to extract heat from a region and exhausts it at a higher temperature. Carnot sought the most efficient possible engine that could operate in both directions.

Carnot's ideal engine operates in four discrete steps:

  1. Heat is applied from an external high temperature source which is large enough that its temperature remains essentially constant even as it transfers energy in the form of heat. A system which has this property is called a heat reservoir. Heat flows into the cylinder causing the gas to expand as work is done by the piston moving outward. The temperature of the gas remains the same as that of the external heat source. Hence, the volume of the gas increases as the pressure decreases. This called an isothermal process because the temperature remains constant. Furthermore, with no change in the temperature of the gas, its internal energy does not change. This step results in an amount of work W(out1) based on an amount of heat applied Q(in).
  2. The external heat source is removed, but the gas continues to expand, cool and do more work as the piston moves further outward. This step involves no heat exchange with its environment. The technical term for zero heat exchange is adiabatic. However, the internal energy of the gas decreases since it is being expended in the form of work. Decreasing the internal energy is associated with a decrease in the gas temperature. Here the additional work done by the gas is W(out2) .
  3. The piston goes as far out as it can and then reverses its direction. Some of the energy expended in steps 1 and 2 (for example, in causing a wheel to rotate) is now used to push the piston inward and compress the gas. In this step, heat is removed from the cylinder at constant temperature by a low temperature reservoir in contact with the cylinder. Hence, like step 1, this is an isothermal process. Note, in this step, work is being done on the gas and being expended in the form of heat. The internal energy of the gas itself remains the same, since its temperature is fixed. In this step, an amount of heat Q(out) was expended, or wasted. Also, an amount of work W(in3) was done on the gas.
  4. The final step is where heat is no longer removed since the heat reservoir is removed. Work is still being done on the gas (and heating it up). The gas is further compressed by the the piston as it completes its inward stroke. Since the gas temperature is increased, its internal energy is increased too. After the piston goes as far as it can, step 1 begins again. No heat was gained or lost but an amount of work W(in4) was done on the gas.

Keep in mind that we have been discussing and ideal engine. A real engine would waste even more energy since it would be impossible to control the heat and work so precisely. In particular, moving parts would lose energy in the form of heat due to friction. So the four steps indicated above truly represent the best possible heat engine.

Now let's see what we learn about the efficiency of an ideal engine. The efficiency of any engine is defined as follows:

                                Net Work Done
                  Efficiency = ---------------
                                  Heat In

Based on the amounts of heat and work indicated above, applying conservation of energy gives
Net Work Done = W(out1) + W(out2) - W(in3) - W(in4) = Q(in) - Q(out).
Heat In = Q(in).
Substituting in above, it follows that
                            Q(in) - Q(out)          Q(out)
              Efficiency = ---------------- =  1 -  ------.
                                Q(in)               Q(in)

But it can also be shown that the ratio Q(out)/Q(in) = T(cold)/T(hot). Hence, the formula for the efficiency of an ideal engine becomes.

              Efficiency = 1 -  ------ .

A real engine can never have an efficiency greater than the ideal, i.e., the Carnot engine. Carnot concluded that it is impossible to make an engine with 100% efficiency since it is highly impractical, if not theoretically impossible, to operate an engine with T(cold) = 0.

The heat engine form of the second law of thermodynamics is the following: It is impossible to build a heat engine to perform mechanical work that does not exhaust heat to the surroundings.

An engine of a suitable kind running backwards is a refrigerator, as mentioned above. Work is needed to extract heat from a cold region and extract it into a hot region. The ideal refrigerator is the Carnot engine made to run backward.

The refrigerator form of the second law is as follows: It is impossible to build a refrigerator that can transfer heat from a lower temperature region to a higher temperature region without using mechanical work.

R.S. Panvini