We condense from The Engineer the following on the above subject. It being one upon which we are very frequently asked to give information, it will be of interest to many of our readers : " In order to determine the number of cubic feet of steam or air, or other gas, which will be discharged through a given orifice in a given time, it is necessary to ascertain the velocity of issue. In no other way can the problem be solved, except by experiments with vessels of known capacity, from one of which the air, steam, or gas, flows to the other. Such a solution is, for reasons on which it is not necessary to enter, practically beyond the reach of most men; and it has already been tried by many, with results which have enabled a general law to be laid down, to which law we shallcome presently. If the velocity is known all the rest follows easily enough. Let us suppose the orifice in the side of a boiler to be one inch square. A cubic foot of steam contains 1,728 cubic inches. We may suppose this cubic foot of steam all contained in a column or bar 1,728 inches long and 1 inch square. Let one end of this bar be brought opposite the orifice and the work of expulsion begun; then it is obvious, that before the whole cubic foot of steam is discharged, a column of steam 1,728 inches long must be passed through the hole. Now, if the velocity of efflux is 1,728 inches per minute, then one minute of time will be required for the escape of one foot of steam. If it have a velocity of efflux of 1,728 feet per second, then the orifice will discharge one cubic foot per second, and so on. And this law is totally independent of the pressure or weight of the steam. As the pressure increases the velocity of discharge will increase in a certain ratio to be presently explained; but the pressure will not affect the fact that the velocity of diicliarye in inches per second, multiplied by the area of the orifice in square inches, and divided by 1,728, will give the discharge in cubic feet per second. " When a discharge of water, steam, gas, or other liquid or fluid takes place through an orifice in a thin plate, a certain contraction takes place in the issuing column which reduces the amount of discharge t ilow that proper to the actual area of the orifice, but it is needless to do more than mention the fact hero. It is quite unnecessary to complicate a statement which we wish to make as simple as possi"ble, by further reference to the Vena Contracta. "Wo have said that the velocity is regulated by the pressure, but this fact only holds good for each particular fluid. Speaking comprehensively, the velocity of discharge depends on the density as well as the pressure of the fluid; the lighter the fluid the greater will be the discharge. Thus, hydrogen v/ill issue more rapidly under a given pressure through a given orifice, than will atmospheric air under the same conditions of pressure and orifice. If our readers have followed us thus far, they will be able to comprehend the nature of the law determining the velocity of discharge under given conditions of orifice and pressure. But before giving this law it may be as well to explain that any body falling freely under the influence of gravity has a progressively accelerated rate; the velocity being in England, and similar latitudes, such that 16 feet 1 inch will be traversed the first second, 48 feet 3 inches in the next second, 80 feet 5 inches in the third second, and so or. The velocity of a falling body at any distance from the point where it started, may be found by multi-plj ing the square root of the hight passed through in feet by 8,1, the product being the velocity in feet per second. Thus, a bullet has been sufiered to drop from the top of a tower 100 feet high; what is its velocity at the moment of touching the ground ? The square root of 100 is 10, and 10 multiplied by 8L gives 80'042 feet asthe velocity. Our non-mathematical readers will now be in a position to understand the law regulating the velocity of efiiux of elastic fluids, such' as steam, under pressure, which may be thus stated : Elastic fluids flow into a mcuum leith a mlocity the same as that which a hody of the same density would acquire infilling through a space equal to the hight of a column of steam or gas of the given pressure. Let us suppose that we are dealing with steam of 45 pounds on the square inch, and the orifice of discharge has one square inch of area. Let us further suppose that a column of steam stands on a valve temporarily closing the orifice. What hight must the column of steam one inch square be to weigh 45 pounds ? Avoiding fractions, nine cubic feet of such steam will weigh one pound; therefore, our column of steam one inch square must contain 9 x 45, or 405 cubic feet of steam; and multiplying 405 by 1,728, we get 699,840 as the hight in inches, or 58,320 as the hight in feet of our column of steam. (This is an approximation only. The true volume of one pound of steam at 45 pounds total pressure is 9-000216 cubic feet.) The square root of 58,320 is 241-5 nearly, and this multiplied by 8gj, or 8-042, gives 1942-14 feet per minute as the velocity with which steam of 45 pounds pressure would issue into a vacuum. " It is here necessary to explain that to avoid the introduction of a multiplicity of figures, we have omitted several frac-i.mp, and, therefore, the velocity we have given above is too low, but this in no way affects the principle of the arithmetical process we have described. Any of our readers mastering it will be able to calculate for themselves the velocity with which elastic fluids flow into a vacuum. The calculation, as we have worked it out, is, however, laborious, and for the benefit of such of our readers as understand logarithms, we give the following comprehensive rule for finding the velocity of discharge : Add 4-29 to the pressure in pounds per square inch; deduct the logarithm of this sum from the logarithm of the pressure; to one half the remainder add 3-3254, and the natural number of this sum will be the velocity in feet per second. The difference between the velocities due to any two pressures is the velocity with which steam or air will flow into the lower pressure. Thus, if the pressure in a cylinder is 20 pounds, while that in the condenser is 5 pounds, at what rate will the steam flow from the former to the latter ? The velocity proper to steam of 5 pound pressure, calculated by the last rule, is 1,552 feet per second, while that proper to 20 pounds is 1,919, and 1,919 — 1,552 gives 367 feet per second as the velocity of the exhaust. " In the earlier portion of this article we stated that the actual area of the column of discharge was less than that of the orifice through which it flowed, and it is now time to say that this fact materially modifies the results of such calculations as the foregoing. Moreover, account must be taken of the frictional resistance due to the sides of pipes or tubes through which the fluid flows. On this latter subject there is considerable diversity of opinion; the subject has been keenly discussed once in our correspondence columns, and we shall not be surprised if it be discussed again. Meanwhile we cannot better conclude this article than with the following rule, extracted from ' Bourne's Treatise on the Steam Engine,' and regarded by many engineers as one of the best yet made on the subject. It refers to the flow of steam through a straight pipe of uniform diameter, and its relation to the rules we have laid down will be readily traced : ' To the temperature of the steam in degrees Pah., add the constant 459, and multiply the square root of the sum by 60-2143; the product is the required velocity.' All enlargements and contractions, and all bends or elbows, will reduce the velocity, but there is no trustworthy formula in existence which will enable us to determine exactly how much in any of the particular cases which may suggest themselves to our readers.”
This article was originally published with the title "On the Flow of Elastic Fluids through Orifices or Pipes"