Quadratic-equation inaccuracy for water hammer

Abstract

The standard solution of a quadratic equation is applied for modeling several boundary conditions (such as orifice, valve, and pump) in the water-hammer analysis of pipelines by the method of characteristics. Numerical instabilities may result in the computer solution of the quadratic equation using the standard solution. An alternative algorithm that overcomes the problem of an unstable solution is presented in the paper. Both algorithms are compared for modeling of an in-line orifice with different head-loss coefficients for flow in opposite directions. A numerical example for the case of rapid closure of the upstream valve in a simple two-reservoir pipeline system with the in-line orifice clearly shows numerical instabilities when the standard quadratic solution for the orifice is applied. The instabilities are eliminated by using an alternative proposed solution of the quadratic equation.