NUMERICAL ANALYSIS AND OPTIMAL CONTROL OF FLUID TRANSIENTS DUE TO VALVE OPERATIONS (WATERHAMMER, HYDRAULIC TRANSIENTS)

Transient-state flow of highly compressible liquids through closed conduits with linearly elastic walls are described by a set of quasi-linear, hyperbolic, partial differential equations. Two numerical methods, Richtmyer and MacCormack methods with modifications, are introduced to analyze the transient flows due to valve operations. These two-step methods yield conditionally stable computation and second-order accuracy. The details of these methods as well as characteristics methods for a typical single pipeline system are presented and compared. It is shown that with the largest possible grid-mesh ratio these second-order methods provide better accuracy than characteristics method does for those piping system whose friction losses are relatively important and whose fluid is highly compressible. Characteristics method yields unstable computation for such a system.;Transient flow caused by valve operations can be controlled by operating the valve in a proper manner. A control system approach to determine this optimal valve control is presented also. In this approach the valve operation is treated as control input whereas the whole pipeline is the controlled system, and the unsteady flow conditions due to valve operation are the system outputs. The optimal control problem is thus formulated and solved by applying modern control theory and optimal control theory. Minimizing the selected performance measure, this obtained optimal control input leads the system responses to their desired outputs. Two performance measures are proposed, one of them includes a penalty on the step difference of valve movement while the other merely on the deviation of system responses from their desired values. One optimal feedback gain for each performance measure is determined, they are used to implement the optimal valve control system by designing either an open-loop or a close-loop system. The values of optimal feedback gain are dependent on the time-variant weighting factors. These factors are specified to assign relative importance to each of the penalty in performance measure, such that lead the transient flow to any feasible desired responses.