Method of Characteristics (MOC)

Bentley HAMMER CONNECT uses the most widely used and tested method, known as the Method of Characteristic (MOC), to solve governing equations and for unsteady pipe flow. Using the MOC, the two partial differential equations can be transformed to the following two pairs of equations:

Equations and cannot be solved analytically, but they can be expressed graphically in space-time as characteristic lines (or curves), called characteristics, that represent signals propagating to the right (C+) and to the left (C-) simultaneously and from each location in the system, as shown in the figure below.

At each interior solution point, signals arrive from the two adjacent points simultaneously. A linear combination of H and V is invariant along each characteristic if friction losses are neglected; therefore, H and V can be obtained exactly at solution points. With head losses concentrated at solution points and the assumption that friction is small, an iterative procedure is used in conjunction with MOC to advance the solution in time.

Transient modeling essentially consists of solving these equations, for every solution point and time step, for a wide variety of boundary conditions and system topologies. To obtain a general computer model like Bentley HAMMER CONNECT, the following additional capabilities are required:

Boundary conditions must also be expressed as algebraic and/or differential equations based on their physical properties. This must be done for every hydraulic element in the model and solved along with the characteristic equations.
Equations of state are incorporated to model vaporous cavitation, whereby the fluid can flash into vapor at low pressures, for example. The assumptions incorporated into Bentley HAMMER CONNECT are described in Water Column Separation and Vapor Pockets.
The length of computational reaches must be set to achieve sufficient accuracy without resulting in too small a time step and an excessively long execution time. Bentley HAMMER CONNECT automatically sets an optimal time step based on pipe lengths, wave speeds, and overall system size, so you can get your model results faster.
Friction losses are assumed to be concentrated at solution points. Different models can be implemented, ranging from steady-state to quasi-steady to unsteady (transient) friction.

Bentley HAMMER CONNECT has been used for over 15 years on a large number of water and wastewater projects, evolving during this time to add new boundary conditions while preserving ease of use and accuracy. Thus, it is a proven model with many "electron miles" and a solid track record of matching field observations (when available). It has also been used to model other fluids and tackle problems in other industry sectors, adding to its generality and confirming its robust algorithms.

A derivation of the complete equations for transient analysis (using elastic theory) is beyond the scope of this manual, but it can be found in other references, such as Almeida and Koelle (1992) and Wylie and Streeter (1993).

The derivation for incompressible flow and rigid pipe walls is provided in the next section. The derivation of the wave celerity and pressure-wave speed for compressible flow and elastic system boundaries is provided next.