Homework 1. The conversion efficiency is a key factor in all methods of energy production. In the case of thermal engines, the maximum possible efficiency is given by the Carnot Equation: e = 1 - T_c/T_h.

The first thermal engine ever built was the Newcomen's "atmospheric engine". Here is a very good animation that shows how this engine works:
Newcomen's machine in action. Please open this web site and watch the animation. Below the figure there is a "dial". Set it in the lowest speed position, and then return to this text.

In Newcomen machine, the cylinder was places at the top of a boiler. The cycle starts when the piston is in its lowest position. The boiler valve is set open, so that when the piston moves upward, the cylinder fills with steam (by the way, in the early versions of the machine the valves were operated manually -- usually young boys were hired for doing that). Adopting the terms from the "gasoline engine vocabulary", we can call the phase when the piston moves up, and the cylinder fills with steam, the intake stroke When the piston reaches its uppermost position, the boiler valve is closed, cutting off the cylinder from the steam source. Then, another valve was opened, letting a portion of cold water to be sprayed into the cylinder. In contact with the cold water the steam condensates, creating vacuum in the cylinder (not exactly vacuum, but in the first approximation it can be thought of as vacuum). Now the atmosperic pressure drives the piston down, and when it reaches its lowermost position. We can call this phase the power stroke. After the ennd of the power stroke, a new cycle starts.

Again open the animation, and check whether the animation is consistent with the above dascription. Set the speed to the lowest value; look at the valves and their operation.

Now, what I want you to do is to calculate the thermal efficiency of this engine. For simplicity, assume that the displacement volume of the cylinder (i.e., the volume of the space between the lowermost and the uppermost piston positions) is 1 m³. Calculate the amount of thermal energy needed for generating 1 m³. of water steam. It is known that the mass of one cubic meter of water steam at normal atmospheric pressure is 0.59 kg. Making steam is a two-stage process. First, one has to heat liquid water from its initial temperature (say, 20 Celsius degrees) up to the boiling point (100 degrees). The amount of heat needed to raise the temperature of 1 kg of liquid water by 1 degree is 4.18 kJ. Next, liquid water has to be turned into vapor (note that "steam" is the technical term for "water vapor"). The amount of heat needed for changing a unit mass of liquid into vapor is called the latent heat of evaporation. Its value for water is 2270 kJ/kg. Having all that information, you can readily calculate how much thermal energy is needed to produce 1 cubic meter of steam.

Next, you have to calculate how much energy is generated in the power stroke (in other words, how much work is done by the atmospheric pressure). For simplicity, assume that the atmospheric pressure is 100,000 N/m² (in fact, it is higher, but only slightly hogher). I suggest to do the following: assume that the surface area of the piston is 1 m². So, what is the force acting on the piston from above? (since we assume that the injection of cold water changes steam into vacuum, there is no force acting from below). Next, if the surface area of the piston is 1 m², and the volume of the cylinder is 1 m³, what is the distance traveled by the piston during its downward motion? With that, you will have all that is needed to calculate the work (by the way, it is easy to show that the work output does not depend on the cross section area of the cylinder -- all that matters is its volume; in order to check that, consider a piston of a different surface area, e.g., 0.5 m², or 2 m² -- the work will always be the same).

Once you have calculated the thermal energy input, and the work output, just divide the latter by the former, multiply by 100%, and you will get the efficiency. The last thing I want you to do is to compare that efficiency with: (a) the Carnot engine efficiency using boiling water as the hot source, and 20 Celsius degrees water as the "heat sink" (but don't forget to change these temperatures to Kelvins, because the temperatures in the Carnot Equation must be in Kelvins, not in Celsius scale!); and (b) the maximum efficiency of a "real" thermal engine (if you don't remember the formula, you can readily find it in the Power Point presentation in which the efficiency of a real engine, designed to deliver maximum power, was discussed).