Now the velocity in feet per second of falling bodies is about eight times the square root of the height from which they have fallen; and the formula for determining this is—
v = c √(2gh) = 8·2c √h.
- g = 32·17 feet per second;
- h = distance fallen through by the body;
- c = a constant determined by experiment, and expressing the proportion of the actual to the theoretical velocity.
Adapting this formula to the special circumstances under which Montgolfier’s formula holds, we find that the force which drives the warm air up the flue is the force of gravity, i.e. of the excess of the weight of a column of cold air over the weight of a column of warm air of exactly the same size (represented by BC in the preceding diagram). The difference of the two weights or pressures is found by multiplying the distance from the point of escape of heated air out of the room (fire-place or elsewhere) to the point of escape into the outer air (top of chimney or other point of exit), by the difference in temperature inside and outside, and again multiplying this product by 1 ∕ 492 for degrees of Fahrenheit temperature, or 1 ∕ 273 for degrees Centigrade.
Thus omitting c for the present, we have—
v = √(2gh(t - t1)/492) = 8·2√(h(t - t1)/492)
Where t = temperature in the chimney,
t1 = temperature of the external air, and
h = height of chimney.
Example.—The chief means of ventilating a given room is by its open fire-place. The temperature in the chimney is 100° F., that of the external air 40°, and the height of the chimney 50 feet; what is the velocity with which air is leaving the room?
v = 8·2 √((100 - 40) × 50 ∕ 492)
= 20.
This gives the theoretical velocity, but the real velocity will differ from the theoretical by an amount varying from 20 to 50 per cent.