Law of Velocity.
—Next let us see what the velocity of the air is in the tube. Suppose that we have some convenient means of measuring the velocity of the air at any point, in feet per second or miles per hour, with some form of anemometer. We will have our measurements taken at the five points where we measured the pressure,—viz., at the tank, one-quarter, one-half, three-quarters and one mile from the tank. We will represent the velocities by a diagram similar to the one used for pressures. At the tank we find the air entering the tube with a velocity of 59.5 feet per second (40.6 miles per hour). We draw the vertical line M N, to represent this. At the quarter mile point the velocity is sixty-five feet per second (44.4 miles per hour) an increase in the first quarter of a mile of 5.5 feet per second. We construct the vertical line O P. At the half-mile point the velocity is 72.4 feet per second (49.4 miles per hour); at the three-quarter mile point it is eighty-three feet per second (56.8 miles per hour); and at the end of the tube, one mile from the tank, the air comes out of the tube with a velocity of 100.4 feet per second (68.5 miles per hour), about 1.7 times faster than it entered the tube at the tank. Drawing all the vertical lines to represent these velocities, and drawing a smooth curve line through the tops of our vertical lines, we have the curve of velocities, N, P, R, T, V, for all points along the tube. It is an increasing velocity and increases more rapidly as we approach the end of the tube. This is shown more clearly by drawing the straight dashed line N V.
If the fluid flowing in the tube were inelastic, like water, then the curve of velocities would be a straight horizontal line, for the water would not come out of the tube any faster than it went in. But we are dealing with air, which is an elastic fluid, and, as we stated before, it expands as the pressure is reduced and becomes larger in volume. It is this expansion that increases its velocity as it flows along the tube. It must go faster and faster to make room to expand. Since the same actual quantity of air in pounds must come out of the tube each minute as enters the tube at the other end in the same time, to prevent an accumulation of air in the tube, and since it increases in volume as it flows through the tube, it follows that its velocity must increase.
Characteristics of the Velocity Curve.
—This velocity curve is both interesting and surprising, if we have not given the subject any previous thought. It might occur to us that the air expands in volume in the tube, and we might reason from this fact that the velocity of the air would increase as it flowed through the tube, but very few of us would be able to see that the rate of increase of velocity also increases. That is to say, it gains in velocity more rapidly as it approaches the open end of the tube. If the velocity were represented on the diagram by a straight horizontal line, we should know that it was constant in all parts of the tube, which would be the case if water were flowing instead of air. If it were represented by a straight inclined line, like the dashed line N V, then we should know that the velocity increased as the air flowed along the tube, but that it increased at a uniform rate. The slope of the line would indicate the rate of increase. Neither of these suppositions represent correctly the velocity of the air at all points in the tube; this can only be done by a curved line such as we have shown. The slope of the curve at any point represents the rate of increase of velocity of the air at that point. If the curve is nearly horizontal, then we know that the velocity does not increase much; but if the curve is steep, then we know that it is increasing rapidly, the actual velocity being indicated by the vertical height of the curve above the horizontal line M U.
Use of Velocity Curves.
—Besides being interesting, a knowledge of the velocity of the air at all points in a tube is of much practical value. It gives us the time a carrier will take in going from one station to another. Usually the first questions asked, when it is proposed to lay a pneumatic tube from station A to station B, are, How quickly can you send a carrier between these points? How much time can be saved? These questions are answered by constructing a velocity curve. Since the velocity changes at every point along a tube, to get the time of transit between two points we must know the average velocity of the air between those points. We can find this approximately from our curve by measuring the height of the curve above the horizontal line M U at a large number of points, and then taking the average of all these heights; but there is a more exact and easier method by means of a mathematical formula. As such formula would be out of place here, we will not give it; suffice it to say, that the average velocity of the air between the tank and the end of the tube, in the case we have assumed, is about seventy-three feet per second (49.7 miles per hour), a little less than one-half the sum of the velocities at the two ends, and a little more than the velocity at the half-mile point. Knowing the average velocity, we can tell how long it takes for a particle of air, and it will be nearly the same for a carrier, to travel from the tank to the end of the tube, by dividing the distance in feet by the average velocity in feet per second. This we find to be one minute 12.3 seconds. Since the air moves more rapidly as it approaches the open end of the tube, a carrier will consume a greater period of time in going from the tank to the quarter mile point than in going from the three-quarter mile point to the open end. The last quarter of a mile will be covered in a little more than fourteen seconds, while the first quarter will require a little more than twenty-one seconds. This difference is surprising, and it becomes even more marked in very long tubes with high initial pressures. This explains why the service between stations located near the end of the tube is more rapid than between stations on other parts of the line.
This velocity curve shows us the velocity of the carriers at each station along the line and enables us to regulate our time-locks and to locate the man-holes and circuit-closers connected with each intermediate station. It gives us the length of the “blocks” in our “block system.” When we know the velocity and weight of our carriers, we can compute the energy stored up in them, and from this the length we need to make our air-cushions so as not to have the air too highly compressed. It would be impossible to design our apparatus properly if we did not know the laws that govern the flow of air in the tubes.
Quantity of Air Used.
—The next important fact that we learn from the velocity curve is the quantity of air that flows through the tube each second or minute. If we multiply the velocity with which the air escapes from the open end of the tube by the area of the end of the tube in square feet, we have the number of cubic feet of air at atmospheric pressure discharged from the tube per unit of time. The same quantity of air must be supplied to the tank in order to maintain a constant flow in the tube. In the present case that we have assumed, the tube is eight inches in diameter; therefore the cross-sectional area is 0.349 square foot. The velocity of the air as it comes out of the end of the tube is 100.4 feet per second; therefore about thirty-five cubic feet of air are discharged from the tube each second, or two thousand one hundred cubic feet per minute. This same amount must be supplied to the tank A in order to maintain the pressure constant, but when it is compressed so that it exerts a pressure of ten pounds per square inch, the two thousand one hundred cubic feet will only occupy a space of one thousand two hundred and fifty cubic feet, if its temperature does not change. This leads us to consider the effect of temperature changes.