For the purposes of explanation, the method used to calculate efficiency for the competition held at Guildford Model Engineering Society is given below. Characteristics will vary from track to track according to gradients and lengths, but the principle is the same for all. Solving the final equation will result in what might be called a track constant, which can be used for all entries in a competition at a particular track.
The GMES track has the following characteristics, starting from the footbridge by the clubhouse and going in the normal clockwise running direction:
1 in 72 down for 119 feet
Level - for 65 feet
1 in 160 down for 99 feet
1 in 125 down for 231 feet
1 in 125 up for 207 feet
Level - for 32 feet
1 in 201 down for 125 feet
1 in 124 up for 106 feet
1 in 130 up for 100 feet
Level - for 87 feet
1 in 110 up for 241 feet
A typical rolling resistance for steel wheeled trolleys on the GMES steel track has been assumed at 12 pounds per ton weight. This is for all weight behind the locomotive, that of the trolley as well as any load carried,but excluding the weight of the locomotive itself. In this way, the efficiency calculated will be a theoretical equivalent of that that would result from the use of a dynamometer. Note that in some of the events held up to 2013, locomotive weight has been inadvertently included in the load, resulting in higher efficiencies than should have been the case.
The rolling resistance figure above has been based on a known figure for a 'runaway' train at Woking some years ago, and figures given in technical articles. It was used along with measured results from OMLEC 2006/7 to verify the suitability of 12 pounds per ton.
Although 12 pounds per ton is a little on the high side for straight track, it does add an allowance for the less predictable, but higher, resistance that will be encountered on curves.
The force required to move a load along a gradient, ignoring rolling resistance, is equal to the load multiplied by the sine of the gradient. However, a very close approximation adequate for our purposes is equal to the load divided by the gradient. This force will be positive for up gradients and negative (pushing) for down gradients.
On all down gradients in this example, the pushing force from the train is sufficient to overcome rolling resistance, except for the 1 in 201 down grade, and this results in a subtraction of a pull over that part of the track, from the rolling resistance. For ease of calculation it has been assumed that the 1 in 124 up gradient can be combined with the 1 in 125 up. Thus for our purposes the track profile reduces to:
Level - for 184 feet
1 in 110 up for 241 feet
1 in 125 up for 313 feet
1 in 130 up for 100 feet
1 in 201 down for 125 feet
Thus the work required to pull a loaded train around one lap of the track is:
(Load/2240 x12 x 963) + ((Load/110) x 241) + ((Load/125) x 313) + ((Load/130) x 100) - ((Load/201) x 125) ft lbs
The first element gives the total contribution from rolling resistance for the entire track, and the remaining elements give the contributions from each of the gradients over which the locomotive is pulling the train. Remember that on the remaining down gradients the train is pushing the locomotive, and therefore no work is being done.
Solving the equation for a one pound load results in a 'track constant' of 10. For the GMES track, work done is therefore (10 x load) ft lbs per lap.
The calorific value of the coal used is assumed to be 14,500 BTU per pound and 1BTU is equivalent to 777.6 ft lbs. Efficiency is therefore:
total work done / (coal used x 14500 x 777.6)
Remember that these figures are for the GMES track only. However, the same principle can be applied to any track provided that the appropriate figures for gradient and length are put into the work calculation above.
All figures can of course be modified to use metric units, but the above shows the method used.