(0.7854pd^2 x 0.03604) = 0.028302pd^2 lb.
Consequently a bell which is d inches in diameter, and gives a pressure of p inches of water, will weigh 0.028302pd^2 lb. Or, if W = the weight of the bell in lb., the pressure thrown by it will be W/0.028302d^2 or 35.333W/d^2. This is the fundamental formula, which is sometimes given as p = 550W/d^2, in which W = the weight of the bell in tons, and d the diameter in feet. This value of p, however, is actually higher than the holder would give in practice. Reductions have to be made for two influences, viz., the lifting power of the contained gas, which is lighter than air, and the diminution in the effective weight of so much of the bell as is immersed in water. The effect of these influences was studied by Pole, who in 1839 drew up some rules for calculating the pressure thrown by a gasholder of given dimensions and weight. These rules form the basis of the formula which is commonly used in the coal-gas industry, and they may be applied, mutatis mutandis, to acetylene holders. The corrections for both the influences mentioned vary with the height at which the top of the gasholder bell stands above the level of the water in the tank. Dealing first with the correction for the lifting power of the gas, this, according to Pole, is a deduction of h(1 - d)/828 where d is the specific gravity of the gas and h the height (in inches) of the top of the gasholder above the water level. This strictly applies only to a flat-topped bell, and hence if the bell has a crown with a rise equal to about 1/20 of the diameter of the bell, the value of h here must be taken as equal to the height of the top of the sides above the water-level (= h'), plus the height of a cylinder having the same capacity as the crown, and the same diameter as the bell, that is to say, h=h' + d/40 where d = the diameter of the bell. The specific gravity of commercially made acetylene being constantly very nearly 0.91, the deduction for the lifting power of the gas becomes, for acetylene gasholders, 0.0001086h + 0.0000027d, where h is the height in inches of the top of the sides of the bell above the water- level, and d is the diameter of the bell. Obviously this is a negligible quantity, and hence this correction may be disregarded for all acetylene gasholders, whereas it is of some importance with coal-gas and other gases of lower specific gravity. It is therefore wrong to apply to acetylene gasholders formulæ in which a correction for the lifting power of the gas has been included when such correction is based on the average specific gravity of coal-gas, as is the case with many abbreviated gasholder pressure formulæ.
The correction for the immersion of the sides of the bell is of greater magnitude, and has an important practical significance. Let H be the total height in inches of the side of the gasholder, h the height in inches of the top of the sides of the gasholder above the water-level, and w = the weight of the sides of the gasholder in lb.; then, for any position of the bell, the proportion of the total height of the sides immersed (H - h)/H, and the buoyancy is (H - h)/H x w/S + pi/4d^2, in which S = the specific gravity of the material of which the bell is made. Assuming the material to be mild steel or wrought iron, having a specific gravity of 7.78, the buoyancy is (4w(H - h)) / (7.78Hpid^2) lb. per square inch (d being inches and w lb.), which is equivalent to (4w(H - h)) / (0.03604 x 7.78Hpid^2) = (4.54w(H - h)) / (Hd^2) inches of water. Hence the complete formula for acetylene gasholders is:
p = 35.333W / d^2 - 4.54w(H - h) / Hd^2
It follows that p varies with the position of the bell, that is to say, with the extent to which it is filled with gas. It will be well to consider how great this variation is in the case of a typical acetylene holder, as, if the variation should be considerable, provision must be made, by the employment of a governor on the outlet main or otherwise, to prevent its effects being felt at the burners.
Now, according to the rules of the "Acetylen-Verein" (cf. Chapter IV.), the bells of holders above 53 cubic feet in capacity should have sides 1.5 mm. thick, and crowns 0.5 mm. thicker. Hence for a holder from 150 to 160 cubic feet capacity, supposing it to be 4 feet in diameter and about 12 feet high, the weight of the sides (say of steel No. 16 S.W.G. = 2.66 lb. per square foot) will be not less than 12 x 4pi x 2.66 = 401 lb. The weight of the crown (say of steel No. 14 S.W.G. = 3.33 lb. per square foot) will be not less than about 12.7 x 3.33 = about 42 lb. Hence the total weight of holder = 401 + 42 = 443 lb. Then if the holder is full, h is very nearly equal to H, and p = (35.333 x 443) / 48^2 = 6.79 inches. If the holder stands only 1 foot above the water-level, then p = 6.79 - (4.54 x 401 (144 - 12)) / (144 x 48^2) = 6.79 - 0.72 = 6.07 inches. The same result can be arrived at without the direct use of the second member of the formula:
For instance, the weight of the sides immersed is 11 x 4pi x 2.66 = 368 lb., and taking the specific gravity of mild steel at 7.78, the weight of water displaced is 368 / 7.78 = 47.3 lb. Hence the total effective weight of the bell is 443 - 47.3 = 395.7 lb., and p = (35.333 x 395.7) / 48^2 = 6.07 inches. [Footnote: If the sealing liquid in the gasholder tank is other than simple water, the correction for the immersion of the sides of the bell requires modification, because the weight of liquid displaced will be s' times as great as when the liquid is water, if s' is the specific gravity of the sealing liquid. For instance, in the example given, if the sealing liquid were a 16 per cent. solution of calcium chloride, specific gravity 1.14 (vide p. 93) instead of water, the weight of liquid displaced would be 1.14 (368 / 7.78) = 53.9 lb., and the total effective weight of the bell = 443 - 53.9 = 389.1 lb. Therefore p becomes = (35.333 x 389.1) / 48^2 = 5.97 inches, instead of 6.07 inches.]
The value of p for any position of the bell can thus be arrived at, and if the difference between its values for the highest and for the lowest positions of the bell exceeds 0.25 inch, [Footnote: This figure is given as an example merely. The maximum variation in pressure must be less than one capable of sensibly affecting the silence, steadiness, and economy of the burners and stoves, &c., connected with the installation.] a governor should be inserted in the main leading from the holder to the burners, or one of the more or less complicated devices for equalising the pressure thrown by a holder as it rises and falls should be added to the holder. Several such devices were at one time used in connexion with coal-gas holders, and it is unnecessary to describe them in this work, especially as the governor is practically the better means of securing uniform pressure at the burners.
It is frequently necessary to add weight to the bell of a small gasholder in order to obtain a sufficiently high pressure for the distribution of acetylene. It is best, having regard to the steadiness of the bell, that any necessary weighting of it should be done near its bottom rim, which moreover is usually stiffened by riveting to it a flange or curb of heavier gauge metal. This flange may obviously be made sufficiently stout to give the requisite additional weighting. As the flange is constantly immersed, its weight must not be added to that of the sides in computing the value of w for making the correction of pressure in respect of the immersion of the bell. Its effective weight in giving pressure to the contained gas is its actual weight less its actual weight divided by its specific gravity (say 7.2 for cast iron, 7.78 for wrought iron or mild steel, or 11.4 for lead). Thus if x lb. of steel is added to the rim its weight in computing the value of W in the formula p = 35.333W / d^2 should be taken as x - x / 7.78. If the actual weight is 7.78 lb., the weight taken for computing W is 7.78 - 1 = 6.78 lb.
THE PRESSURE GAUGE.--The measurement of gas pressure is effected by means of a simple instrument known as a pressure gauge. It comprises a glass U- tube filled to about half its height with water. The vacant upper half of one limb is put in communication with the gas-supply of which the pressure is to be determined, while the other limb remains open to the atmosphere. The difference then observed, when the U-tube is held vertical, between the levels of the water in the two limbs of the tube indicates the difference between the pressure of the gas-supply and the atmospheric pressure. It is this difference that is meant when the pressure of a gas in a pipe or piece of apparatus is spoken of, and it must of necessity in the case of a gas-supply have a positive value. That is to say, the "pressure" of gas in a service-pipe expresses really by how much the pressure in the pipe exceeds the atmospheric pressure. (Pressures less than the atmospheric pressure will not occur in connexion with an acetylene installation, unless the gasholder is intentionally manipulated to that end.) Gas pressures are expressed in terms of inches head or pressure of water, fractions of an inch being given in decimals or "tenths" of an inch. The expression "tenths" is often used alone, thus a pressure of "six-tenths" means a pressure equivalent to 0.6 inch head of water.