For t = 0.35d, d = 0.20 √ (Q / √p)
For t = 0.45d, d = 0.18 √ (Q / √p)
If p is the pressure difference in the fan in inches of water, and N the revolutions of fan,
| v = πdN/60 | ft. per sec. |
| N = 1230 √ p/d | revs. per min. |
As the pressure difference is small, the work done in compressing the air is almost exactly 5.2pQ foot-pounds per second. Usually, however, the kinetic energy of the air in the discharge pipe is not inconsiderable compared with the work done in compression. If w is the velocity of the air where the discharge pressure is measured, the air carries away w2/2g foot-pounds per ℔ of air as kinetic energy. In Q cubic feet or 0.0807Q ℔ the kinetic energy is 0.00125 Qw2 foot-pounds per second.
The efficiency of fans is reckoned in two ways. If B.H.P. is the effective horse-power applied at the fan shaft, then the efficiency reckoned on the work of compression is
η = 5.2pQ / 550 B.H.P.
On the other hand, if the kinetic energy in the delivery pipe is taken as part of the useful work the efficiency is
η2 = (5.2 pQ + 0.00125 Qw2) / 550 B.H.P.
Although the theory above is a rough one it agrees sufficiently with experiment, with some merely numerical modifications.
An extremely interesting experimental investigation of the action of centrifugal fans has been made by H. Heenan and W. Gilbert (Proc. Inst. Civ. Eng. vol. 123, p. 272). The fans delivered through an air trunk in which different resistances could be obtained by introducing diaphragms with circular apertures of different sizes. Suppose a fan run at constant speed with different resistances and the compression pressure, discharge and brake horse-power measured. The results plot in such a diagram as is shown in fig. 213. The less the resistance to discharge, that is the larger the opening in the air trunk, the greater the quantity of air discharged at the given speed of the fan. On the other hand the compression pressure diminishes. The curve marked total gauge is the compression pressure + the velocity head in the discharge pipe, both in inches of water. This curve falls, but not nearly so much as the compression curve, when the resistance in the air trunk is diminished. The brake horse-power increases as the resistance is diminished because the volume of discharge increases very much. The curve marked efficiency is the efficiency calculated on the work of compression only. It is zero for no discharge, and zero also when there is no resistance and all the energy given to the air is carried away as kinetic energy. There is a discharge for which this efficiency is a maximum; it is about half the discharge which there is when there is no resistance and the delivery pipe is full open. The conditions of speed and discharge corresponding to the greatest efficiency of compression are those ordinarily taken as the best normal conditions of working. The curve marked total efficiency gives the efficiency calculated on the work of compression and kinetic energy of discharge. Messrs Gilbert and Heenan found the efficiencies of ordinary fans calculated on the compression to be 40 to 60% when working at about normal conditions.