# Theory

The big advantage of having a theory which describes the physics of an engine is, that we know which factors affect the performance, and how much energy we can expect from the process  (always less than the theory predicts, and experience tells that a realistic value is 50 – 60% of the theoretical value).

### The Carnot efficiency

Sadly, this efficiency gives us the upper limit of a conversion efficiency η of a heat engine as a function of the operational temperature Top, and the heat sink temperature Tout, where both temperatures are given in Kelvin (i.e. degree centigrade plus 273)

η = 1-Top / Tout

In our case, with an operating temperature of 100°C, and a heat sink temperature of 30°C the efficiency η becomes 1 – 303/373 = 0.19 or 19%. Not very much but it puts the efficiency values given earlier of 15% into perspective. 15% is 79% of 19, so that the conversion efficiency within the possible limits is actually quite high.

### And, the condensing cycle:

Here, we have to look briefly at the actual going-ons in one cycle. The following figure shows four phases. The theory is described here in some detail to give the reader an insight into the physics. Fig. 1: Expansion cycle

Fig. 1 shows a closed system consisting of a boiler at temperature T0, a cylinder of length L1 with a piston, and an external condenser with a pressure pcond ≈ 0 (absolute). The cylinder is double acting, but only one stroke will be described.

Initially, the piston is at the uppermost position ”1”, and the pressure p2  below the piston is close to zero (absolute) and equal to the condenser pressure pcond, p2 = pcond. The boiler produces steam at a constant temperature T0, and a pressure p0, which is a function of the boiler temperature T0, p0 = f(T0).

No expansion (n = 1): the steam pressure p0 remains constant, and acts on the piston. The pressure difference drives the piston downwards from point “1” to “3”, and steam fills the upper part of the cylinder. The steam is then condensed and the cycle repeats itself in the opposite direction. A large pressure difference still exists at the end of the stroke, and a substantial part of the steam’s energy remains unused.

Steam expansion: In order to utilize the steam’s energy more completely, it is necessary to allow the steam to expand. In this case, Fig. 1 (i), the piston initially travels a distance L0 with steam filling the upper cylinder. At point “2”, Fig. 1(ii)), the boiler valve is closed. The steam now expands and drives the piston through a distance ΔL from point “2” to point “3”. The pressure drops from p0 at “2” to p1 > p2 at the end of the stroke, Fig. 1(iii).  Once the piston has arrived at point “3” with a pressure p1, the condenser valve for the upper part is opened, the expanded steam rushes into the condenser, and the cycle is repeated in the opposite direction, Fig. 1 (iv). The following video illustrates the steam expansion:

Fig. 2: The expansion cycle

Note that the steam valve is closed whilst the piston is at half stroke, the steam then expands.. Fig. 3 shows the efficiency as a function of the expansion ratio n = L1 / L0 and the steam temperature T0. Efficiencies range from 17.2% for n = 8 to 6.4% for  n = 1.Adiabatic and isothermal efficiencies are virtually identical. Fig. 3: Adiabatic and isothermal efficiency as function of expansion ratio n

### So, what does all that actually mean?

Well, a number of things. Firstly, we can increase the efficiency of the cycle with the expansion of steam. The efficiency increase becomes larger as the expansion ratio increases, but approaches a maximum (i.e. the Carnot efficiency) for n > 8. Also, we need to remember that expansion goes hand-in-hand with variable piston pressure, which is not so desirable from the power output point of view.

The maximum efficiency for an expansion ratio of 1:8 is 17.8%, which compares well with James Watt’s original estimate of 19.6%. Here it must be remembered that in Watt’s days, the theory of thermodynamics was in its infancy – in fact, Watt and John Southern contributed greatly in its development, see “History”.