Water Hammer in Pipes, Including Those Supplied by Centrifugal Pumps: Graphical Treatment

Abstract

The damage done to pipes as a result of water hammer is so serious that no engineer can afford to neglect it in the design of long pipes, particularly those under low heads. Unfortunately the problem has appeared to many to be very intricate, and indeed the mathematical treatment is so involved and so lengthy that few practising engineers would attempt to solve water-hammer problems mathematically. In many cases a purely arbitrary rule has been employed in finding the pressure rise, with little knowledge on the part of the engineer as to whether the formula fits his case or not, and thus much money may be wasted on too heavy a line, or, on the other hand, it may be so light as to be dangerous.In the following treatment of the subject, the rigid theory has first been explained and some problems have been solved, but most of the paper deals with the elastic (and correct) theory. A problem is first worked out by the arithmetical integration method in order to establish ideas and to make the subsequent argument clear. The fundamental equations giving rise to Allievi's equations must be clearly understood for a grasp of the true relationships that exist, and these have been treated in some detail, but it is safe to say that if the argument has been followed up to the end of the general equations, no difficulty will be experienced with the graphical treatment.In explaining the diagrams, very simple cases are first dealt with and the elegance of the method will make an appeal at once. The water-hammer pressure can be found for any simple pipe, and at any point on the pipe, and for any chosen gate movement, and results can be obtained with accuracy in less than thirty minutes in simple cases. For compound pipes, the solution is nearly as rapid. The pressures arising in the discharge lines from pumps have been fully discussed in the case of pumps with small inertia, and also in cases in which the inertia is large enough for consideration, when an existing pump has been employed to supply the data used.The accuracy of the method is beyond question; where intersections are at small angles the exact points are hard to determine, but this trouble may easily be avoided by changing the scales. If a slide rule is used in solving the problem mathematically the corresponding case will be encountered.