A “guided” research problems for the student

Energy, power and the physics of the internal combustion engine.

IL 70 *** (A comprehensive short discussion of efficiency and the second law of thermodynamics).

The first law is simply an expression of the general law of conservation of energy. This law (or principle) can be expressed verbally and symbolically (mathematically) on several levels of complexity. We will give two simple verbal definitions, followed by two simple symbolic representations. Note that in the second version we have a statement of the first law of thermodynamics.

The law of conservation of energy states that:

1. Energy cannot be created (made from nothing), or destroyed (made to disappear) and that energy can be changed from one form to another (such as heat energy into mechanical work, or electrical energy into heat energy).

2. The change in the internal energy of a system equals the difference between the heat taken in by the system and the work done by the system.

For the first law:

where Q is equal to the net amount of heat flowing into a system during a given process, W is the net work done by the system, and Δ U is the change in the system’s internal energy.

The second law describes and prescribes the restrictions placed on the first law. These restrictions were suggested by such empirical evidence as the observation that no work can be done unless heat can be taken in at one temperature and exhausted at a lower temperature. In addition, we know from elementary physics that gravitational potential energy problems, work can only be done if a body from one height is taken to a lower height. Does this contradict the law of the conservation of energy?

We will also express the second law verbally first and then symbolically.

Another way to state the second law is:

There is a standard mathematical expression that relates to the second law that uses the concept of entropy, which is a complex concept that relates to the so called disorder of a system. But we will only use the idea of efficiency to demonstrate the how the first and the second laws are related.

The second law forbids that any energy transformation is 100% efficient. For example, it is not possible to convert all the heat energy in an internal combustion engine to driving the car. You already know that efficiency is defined as the energy put into the system divided by the work derived from the system. For a car engine, for example, the purpose is to transform as much of the extracted heat Q_H as possible and transform it to mechanical energy.

Look at Fig. During every cycle, energy is extracted as heat Q_H from a reservoir at temperature T_{H ,} a portion is diverted for useful work. W, and the rest is lost as Q_C. to a reservoir at a lower temperature T_C.

The first law says that

Q = W + Δ U.

Rewriting, we have: W = Q - Δ U.

Clearly, the change of the internal energy Δ U, is zero here, because the gas in the cylinder returns to its original value. From the first law then we can write for the engine

W = Q. But Q here is given by Q_H - Q_C.

Since efficiency e is given by W_in / W_{out ,} we have

(Remember, the temperature must be expressed in degrees Kelvin)

(It is interesting to note that the French engineer Sadi Carnot discovered this relationshipin 1824, 25 years before the laws of thermodynamics were established by von Helmholtz and Kelvin). This equation was applied to finding the efficiencies of steam engines before the internal combustion engine was developed. It was known that the theoretical efficiency of a steam engine was determined by the difference of the intake temperature and the exhaust temperature of the gases involved. We can write this famous equation this way:
% efficiency = 100 x ( 1 –Temp. of the exhaust gas / Temp. of the exploding gas)
The temperatures must be expressed in the Kelvin scale.

It is important to note that this is an ideal efficiency - real engines also lose some efficiency due to friction, etc., but this is above this theoretical limit. Thus, a heat engine would operate with 100% efficiency in converting heat into useful work only if the cool reservoir was at 0º K ( -273 K), and if their were no frictional losses. This impossible to achieve.