Acetylene, the Principles of Its Generation and Use(5) The quant.i.ty of gas pa.s.sing through the pipe--Q cubic feet per hour.
This quant.i.ty is the product of the mean velocity of the gas in the pipe and the area of the pipe.
The only work done in maintaining the flow of gas along a pipe is that required to overcome the friction of the gas on the walls of the pipe, or, rather, the consequential friction of the gas on itself, and the laws which regulate such friction have not been very exhaustively investigated. Pole pointed out, however, that the existing knowledge on the point at the time he wrote would serve for the purpose of determining the proper sizes of gas-mains. He stated that the friction (1) is proportional to the area of rubbing surface (viz., pi_ld_); (2) varies with the velocity, in some ratio greater than the first power, but usually taken as the square; and (3) is a.s.sumed to be proportional to the specific gravity of the fluid (viz., _s_).
Thus the force (_f_) necessary to maintain the motion of the gas in the pipe is seen to vary (1) as pi_ld_, of which pi is a constant; (2) as _v^2_, where _v_ = the velocity in feet per hour; and (3) as _s_. Hence, combining these and deleting the constant pi, it appears that
_f_ varies as _ldsv^2_.
Now the actuating force is equal to _f_, and is represented by the difference of pressure at the two ends of the pipe, _i.e._, the initial pressure, viz., that at the place whence gas is distributed or issues from a larger pipe will be greater by the quant.i.ty _f_ than the terminal pressure, viz., that at the far end of the pipe where it branches or narrows to a pipe or pipes of smaller size, or terminates in a burner. The terminal pressure in the case of service-pipes must be settled, as mentioned in Chapter II., broadly according to the pressure at which the burners in use work best, and this is very different in the case of flat-flame burners for coal-gas and burners for acetylene. The most suitable pressure for acetylene burners will be referred to later, but may be taken as equal to p_0 inches head of water. Then, calling the initial pressure (_i.e._, at the inlet head of service-pipe) p_1, it follows that p_1 - p_0 = _f_. Now the cross-section of the pipe has an area (pi/4)_d^2_, and if _h_ represents the difference of pressure between the two ends of the pipe per square inch of its area, it follows that _f_ = _h(pi/4)d^2_. But since _f_ has been found above to vary as _ldsv^2_ , it is evident that
_h(pi/4)d^2_ varies as _ldsv^2_.
Hence
_v^2_ varies as _hd/ls_,
and putting in some constant M, the value of which must be determined by experiment, this becomes
_v^2_ = M_hd/ls_.
The value of M deduced from experiments on the friction of coal-gas in pipes was inserted in this equation, and then taking Q = pi/4_d^2v_, it was found that for coal-gas Q = 780(_hd/sl_)^(1/2)
This formula, in its usual form, is
Q = 1350_d^2_(_hd/sl_)^(1/2)
in which _l_ = the length of main in yards instead of in feet. This is known as Pole's formula, and has been generally used for determining the sizes of mains for the supply of coal-gas.
For the following reasons, among others, it becomes prudent to revise Pole's formula before employing it for calculations relating to acetylene. First, the friction of the two gases due to the sides of a pipe is very different, the coefficient for coal-gas being 0.003, whereas that of acetylene, according to Ortloff, is 0.0001319. Secondly, the mains and service-pipes required for acetylene are smaller, _cateria paribus_, than those needed for coal-gas. Thirdly, the observed specific gravity of acetylene is 0.91, that of air being unity, whereas the density of coal-gas is about 0.40; and therefore, in the absence of direct information, it would be better to base calculations respecting acetylene on data relating to the flow of air in pipes rather than upon such as are applicable to coal-gas. Bernat has endeavoured to take these and similar considerations into account, and has given the following formula for determining the sizes of pipes required for the distribution of acetylene:
Q = 0.001253_d^2_(_hd/sl_)^(1/2)
in which the symbols refer to the same quant.i.ties as before, but the constant is calculated on the basis of Q being stated in cubic metres, l in metres, and d and h in millimetres. It will be seen that the equation has precisely the same shape as Pole's formula for coal-gas, but that the constant is different. The difference is not only due to one formula referring to quant.i.ties stated on the metric and the other to the same quant.i.ties stated on the English system of measures, but depends partly on allowance having been made for the different physical properties of the two gases. Thus Bernat's formula, when merely transposed from the metric system of measures to the English (_i.e._, Q being cubic feet per hour, _l_ feet, and _d_ and _h_ inches) becomes
Q = 1313.5_d^2_(_hd/sl_)^(1/2)
or, more simply,
Q = 1313.4(_hd^5/sl_)^(1/2)
But since the density of commercially-made acetylene is practically the same in all cases, and not variable as is the density of coal-gas, its value, viz., 0.91, may be brought into the constant, and the formula then becomes
Q = 1376.9(_hd^5/l_)^(1/2)
Bernat's formula was for some time generally accepted as the most trustworthy for pipes supplying acetylene, and the last equation gives it in its simplest form, though a convenient transposition is
d = 0.05552(Q^2_l/h_)^(1/5)
Bernat's formula, however, has now been generally superseded by one given by Morel, which has been found to be more in accordance with the actual results observed in the practical distribution of acetylene. Morel's formula is
D = 1.155(Q^2_l/h_)^(1/5)
in which D = the diameter of the pipe in centimetres, Q = the number of cubic metres of gas pa.s.sing per hour, _l_ = the length of pipe in metres, and _h_ = the loss of pressure between the two ends of the pipe in millimetres. On converting tins formula into terms of the English system of measures (_i.e._, _l_ feet, Q cubic feet, and _h_ and _d_ inches) it becomes
(i) d = 0.045122(Q^2_l/h_)^(1/5)
At first sight this formula does not appear to differ greatly from Bernat's, the only change being that the constant is 0.045122 instead of 0.05552, but the effect of this change is very great--for instance, other factors remaining unaltered, the value of Q by Morel's formula will be 1.68 times as much as by Bernat's formula. Transformations of Morel's formula which may sometimes be more convenient to apply than (i) are:
(ii) Q = 2312.2(_hd^5/l_)^(1/2)
(iii) _h_ = 0.000000187011(Q^2_l/d^5_)
and (iv) _l_ = 5,346,340(_hd^5_/Q^2)
In order to avoid as far as possible expenditure of time and labour in repeating calculations, tables have been drawn up by the authors from Morel's formulae which will serve to give the requisite information as to the proper sizes of pipes to be used in those cases which are likely to be met with in ordinary practice. These tables are given at the end of this chapter.
When dealing with coal-gas, it is highly important to bear in mind that the ordinary distributing formulae apply directly only when the pipe or main is horizontal, and that a rise in the pipe will be attended by an increase of pressure at the upper end. But as the increase is greater the lower the density of the gas, the disturbing influence of a moderate rise in a pipe is comparatively small in the case of a gas of so high a density as acetylene. Hence in most instances it will be unnecessary to make any allowance for increase of pressure due to change of level. Where the change is very great, however, allowance may advisedly be made on the following basis: The pressure of acetylene in pipes increases by about one-tenth of an inch (head of water) for every 75 feet rise in the pipe.
Hence where acetylene is supplied from a gasholder on the ground-level to all floors of a house 75 feet high, a burner at the top of the house will ordinarily receive its supply at a pressure greater by one-tenth of an inch than a burner in the bas.e.m.e.nt. Such a difference, with the relatively high pressures used in acetylene supplies, is of no practical moment. In the case of an acetylene-supply from a central station to different parts of a mountainous district, the variations of pressure with level should be remembered.
The distributing formulae also a.s.sume that the pipe is virtually straight; bends and angles introduce disturbing influences. If the bend is sharp, or if there is a right-angle, an allowance should be made if it is desired to put in pipes of the smallest permissible dimensions. In the case of the most usual sizes of pipes employed for acetylene mains or services, it will suffice to reckon that each round or square elbow is equivalent in the resistance it offers to the flow of gas to a length of 5 feet of pipe of the same diameter. Hence if 5 feet is added to the actual length of pipe to be laid for every bond or elbow which will occur in it, and the figure so obtained is taken as the value of _l_ in formulae (i), (ii), or (iii), the values then found for Q, _d_, or _h_ will be trustworthy for all practical purposes.
It may now be useful to give an example of the manner of using the foregoing formulae when the tables of sizes of pipes are not available.
Let it be supposed that an inst.i.tution is being equipped for acetylene lighting; that 50 burners consuming 0.70 cubic foot, and 50 consuming 1.00 cubic foot of acetylene per hour may be required in use simultaneously; that a pressure of at least 2-1/2 inches is required at all the burners; that for sufficient reasons it is considered undesirable to use a higher distributing pressure than 4 inches at the gasholder, outlet of the purifiers, or initial governor (whichever comes last in the train of apparatus); that the gasholder is located 100 feet from the main building of the inst.i.tution, and that the trunk supply-pipe through the latter must be 250 feet in length, and the supplies to the burners, either singly or in groups, be taken from this trunk pipe through short lengths of tubing of ample size. What should be the diameter of the trunk pipe, in which it will be a.s.sumed that ten bonds or elbows are necessary?
In the first instance, it is convenient to suppose that the trunk pipe may be of uniform diameter throughout. Then the value of _l_ will be 100 (from gasholder to main building) + 250 (within the building) + 50 (equivalent of 10 elbows) = 400. The maximum value of Q will be (50 x 0.7) + (50 x 1.0) = 85; and the value of _h_ will be 1 - 2.5 - 1.5.
Then using formula (i), we have:
d = 0.045122((85^2 x 400)/1.5)^(1/5) = 0.045122(1,926,667)^(1/5)
= 0.045122 x 18.0713 = 0.8154.
The formula, therefore, shows that the pipe should have an internal diameter of not less than 0.8154 inch, and consequently 1 inch (the next size above 0.8154 inch) barrel should be used. If the initial pressure (i.e., at outlet of purifiers) could be conveniently increased from 4 to 4.8 inches, 3/4 inch barrel could be employed for the service-pipe. But if connexions for burners were made immediately the pipe entered the building, these burners would then be supplied at a pressure of 4.2 inches, while those on the extremity of the pipe would, when all burners were in use, be supplied at a pressure of only 2.5 inches. Such a great difference of pressure is not permissible at the several burners, as no type of burner retains its proper efficiency over more than a very limited range of pressure. It is highly desirable in the case of the ordinary Naphey type of burner that all the burners in a house should be supplied at pressures which do not differ by more than half an inch; hence the pipes should, wherever practicable, be of such a size that they will pa.s.s the maximum quant.i.ty of gas required for all the burners which will ever be in use simultaneously, when the pressure at the first burner connected to the pipe after it enters the house is not more than half an inch above the pressure at the burner furthermost removed from the first one, all the burner-taps being turned on at the time the pressures are observed. If the acetylene generating plant is not many yards from the building to be supplied, it is a safe rule to calculate the size of pipes required on the basis of a fall of pressure of only half an inch from the outlet of the purifiers or initial governor to the farthermost burner.
The extra cost of the larger size of pipe which the application of this rule may entail will be very slight in all ordinary house installations.
VELOCITY OF FLOW IN PIPES.--For various purposes, it is often desirable to know the mean speed at which acetylene, or any other gas, is pa.s.sing through a pipe. If the diameter of the pipe is _d_ inches, its cross-sectional area is _d^2_ x 0.7854 square inches; and since there are 1728 cubic inches in 1 cubic foot, that quant.i.ty of gas will occupy in a pipe whose diameter is _d_ inches a length of
1728/(_d^2_ x 0.7854) linear inches or 183/_d^2_^ linear feet.
If the gas is in motion, and the pipe is delivering Q cubic feet per hour, since there are 3600 seconds of time in one hour, the mean speed of the gas becomes
183/_d^2_ x Q/3600 = Q/(19 x 7_d^2_) linear feet per second.
This value is interesting in several ways. For instance, taking a rough average of Le Chatelier's results, the highest speed at which the explosive wave proceeds in a mixture of acetylene and air is 7 metres or 22 feet per second. Now, even if a pipe is filled with an acetylene-air mixture of utmost explosibility, an explosion cannot travel backwards from B to A in that pipe, if the gas is moving from A to B at a speed of over 22 feet per second. Hence it may be said that no explosion can occur in a pipe provided
Q/(19.7_d^2_) = 22 or more;
_i.e._, Q/_d^2_=433.4
In plain language, if the number of cubic feet pa.s.sing through the pipe per hour divided by the square of the diameter of the pipe is at least 433.4, no explosion can take place within that pipe, even if the gas is highly explosive and a light is applied to its exit.