HAMMER normally models air or vapor volumes as concentrated at specific points along a pipe. However, HAMMER can simulate an extended air volume if it enters the system at a local high point (via an Air Valve element, sometimes called a combination air valve or CAV).

To enable this, from the Transient Solver Calculation Options, set the Run Extended CAV field to True. HAMMER will track the extent of the air pocket and the resulting mass-oscillation and water column accelerations. HAMMER still calculates the system-wide solution using MOC and elastic theory; it uses rigid-column theory only for the pipes nearest the high point. This results in more accurate solutions, without increasing execution time.

Rigid Liquid Columns in Branches

When a sufficiently large volume of air enters a pipeline, the flow regime evolves from hydraulic transients to mass oscillations. Thus, at least in the vicinity of the air, the system may be represented by rigid-column theory in lieu of the elastic approach. Besides improved computational efficiency, the rigid approach allows for the tracing of the air-liquid interface, under simplifying assumptions, with a concomitant change in the hydraulic grade line, and also tracks momentum more accurately. A rigid column is considered in each branch adjacent to an Air Valve extending to the neighboring node which is at a lower elevation in order that the branch be sloped upward towards to the Air Valve. Furthermore, it is assumed that the liquid surface is horizontal and that each branch is terminated at its upper end by the intersection of a vertical plane through the Air Valve with the pipe.

The air pocket consists of portions in each of the branches overlying the rigid columns, with the air pocket instantaneously at constant pressure due to its low density. The neighboring node is a M-way junction, each branch of which (except for the one containing the AV) is handled by means of the elastic theory. In light of this background, we formulate the equations of motion at each neighboring node, An say, in terms of the following (2M + 4) variables: head and flow in each of the M branches, head and flow of the rigid column, and length and level of the rigid column. Correspondingly, there are (2M + 4) equations comprised of characteristics and head loss for each branch, continuity at An and at the horizontal surface, conservation of momentum for the rigid column, and column length as a function of its level. These equations are solved iteratively by postulating that all friction coefficients are small and that the flows may be reasonably approximated by values from the previous time step.

Determination of Air Pocket Properties

In the process of handling the rigid columns described above, the pressure of the overlying air is taken to be provided by the value at the previous time step. At the conclusion of each time step, there is generally a change in the level in each constituent branch which in turn leads to a variation in the total air volume. Simultaneously, depending on the mode of operation of the Air Valve as described in the Air Valve Theory section, air can enter or exit the valve freely or under compression. To determine the volume, mass, and pressure of the air in the pipeline, we solve nonlinear equations for mass flow rate through the air valve along with the isentropic gas law and (air) mass continuity. In this way, the air pocket properties are updated and employed for the succeeding time step, after due allowance for flow from any full branches as outlined below.

Liquid Transfer from Full Branches

At any instant, it is quite possible for some branches of an Air Valve to be full while others possess air volumes above the rigid columns. Consequently, as in the case of the elastic theory, ambiguity arises when flow towards the Air Valve occurs in a full branch: how does this inflow past the Air Valve distribute itself among the existing air pockets? In essence, the same rules enforced in the elastic case are also applicable in the present situation as follows: (i) For inflow into a full branch, the branch remains full. (ii) 'Over-the-top' inflow from the full branch(es) is allocated to the air volumes in adjacent branches in proportion to their sizes.

Transitions between Elastic and Inelastic Approaches

The elastic (concentrated) model is intended for when the closed conduits are filled with liquid or when there are only small air volumes present. As HAMMER is based upon the conveyance of a single liquid in a network of closed conduits flowing full, this representation is readily integrated in the program; moreover, there are no restrictions on the type or elevation of neighbor nodes. However, the elastic treatment is an oxymoron inasmuch as the air has finite volume with zero extent, so that the liquid level is constrained to be no lower than the pipe's elevation at the Air Valve location. On the other hand, the inelastic (extended) model works best for large air volumes by tracing the movement of the horizontal interface between the overlying air and the liquid in each branch adjacent to the Air Valve. In this way, the liquid level is not limited to the Air Valve's elevation as a lower bound. Such an approach is more difficult to implement and visualize. Furthermore, multiple additional constraints are imposed on neighbor nodes which must be junction elements (with no demands), lower than the Air Valve, and associated with (neighbor to) exactly one Air Valve.

Initial Model and First Transition

At the start of a run, as there is typically no air present in the system, the elastic (concentrated) representation is normally invoked. HAMMER tracks the volume of air in each branch, together with the level of the virtual horizontal liquid surface if the rigid-column approach were being applied. As soon as the transition level for any branch is reached, the rigid (extended) model is utilized in all branches. This level is chosen as being 10% of the vertical drop from the Air Valve to the adjacent interior point within the branch. By definition, at the instant that the transition level is breached in some branch, the liquid levels in the other branches are above their respective transition levels. Immediately prior to the transition, the flows in the branches should be nearly constant, whereas afterwards the level drops from the Air Valve's height to the transition level. It is crucial that the discharges and heads be properly transferred at all interior and end points of each branch in a continuous fashion. In the user notifications, there is an informational message of the form "At time step 'x' at node 'y', transition from CONCENTRATED to EXTENDED." to indicate that a transition has occurred.

Limit on Air Pocket Size

In the rigid methodology, the basic premise is that each branch pipe around the Air Valve contains a liquid column extending from the horizontal surface to the neighbor node. In the event that the air expands greatly so that the interface moves down towards the neighbor node to the verge of draining, HAMMER issues a warning message, freezes the horizontal surface at the elevation of the neighbor node, and continues to track the volume (which could conceivably exceed the branch's volume). The warning message has the form "*** WARNING: At time step 'a' at Air Valve 'y', the branch connected to node 'z' has drained."

Counter-Transition Strategy

If the rigid model is invoked to simulate a large air pocket at the Air Valve, it is possible that the volume will subsequently shrink with the liquid levels in the branches receding until they cross the transition levels. When all liquid levels are above the transition levels, the Transient Solver reverts to the elastic model with the printing of the message "At time step 'x' at node 'y', transition from EXTENDED to CONCENTRATED." in the user notifications. Such transitions can recur multiple times during a simulation. It should be observed that the instantaneous volume of the air pocket at the moment that the transition occurs is indeed variable by virtue of the criterion adopted. During the rigid (extended) phase, the flow is constant along each branch while the head is linear from the neighbor node to the horizontal surface whence it is parallel to the pipe until the peak at the Air Valve.