Question
How are pressurized conditions modelled in Collection System models?
Contents
Background Information on Pressurized Flow
Pressurized flow can be modelled in MIKE+ generalizing the Saint Venant Equations. Preissmann and Cunge, 1961 /2/, introduced the concept of a fictitious slot to allow "open channel-like" conditions on full running pipes.
The continuity equation can be rewritten to include the top slot in the Saint-Venant by means of expressing the water density term in relation to the water density (assuming that it is constant over the cross section)
Using partial differentiation, it is possible to express the continuity equation like this:
The density of the water can be approximated as:
Where:
p0 is the water density at free surface flow (kg/m3)
a0 = the speed of sound in water (m/s)
y= the water depth (m)
D = the reach diameter (m)
Now if the cross-section area is explained in terms of the excess pressure approximately as
Where the A0 is the area without excess pressure and ar is given as:
and
Er = the Young's modulus of elasticity (N/m2)
e = the reach wall thickness, (m).
Putting all these equations together we get
Assuming that the water density varies only slowly along the reach then it is possible to annulate the density variable term
Then using the that the cross-section area for water level in the slot varies as
Then the continuity equation can be recovered by selecting the slot width according to:
a is in the order of 1000 m/s for most concrete conducts.
Application to models (trick)
Now that the approximation to run pressurized flow using Saint Venant Equations has been clarified, a couple of tips and tricks will be given for modelling purposes while working with parallel pumps leading into pressure mains.
The rising mains Preissmann slot width factor is used to enable open channel conditions even under full capacity (default is 0.001). With this setting, default slot width (1% of the pipe's diameter), the slot width is reduced to an extremely narrow width (in example with a pipe diameter of 0.15 m, the slot is set to 0.0015 mm). See Fig. 1
Fig 1. Schematic depiction of a circular pipe running at full capacity and the implementation of the Preissmann slot.
This setting of a 1% of the diameter slot width can lead to extremely high-water heads in the pressurized pipes, which in occasions can be deemed unrealistic. A solution to this situation is to change the engine setting to e.g. 0.1, this will stabilize the model and prevents this extreme elasticity in the pipe. See Fig. 2
Fig. 2 MIKE 1D Engine Configuration - Pressure reaches top Slot width.
Note that the change of setting applies for the current database only, i.e. it is not changing engine's settings generally in your installation. If wanted, it has to be set as wanted in every new database.
The presence of the Preissmann's slot volume makes that pressurized pipes behave similarly as conduits with elastic walls, in example a rubber hose. This means that the flow in the rising main pipe at some downstream location is not the same as the actual pump discharge but is routed by the volume available in the slot. This is an equivalent to the pipe's cross section expansion under pressure due to pipe wall's elasticity.
In the graph below, an example of the pumped discharge of a pumping station and the flow at a few locations in the pressure mains is presented. See Fig. 3
Fig. 3 Discharge time series for pumping stations and downstream pipes in the rising main showing pipe elasticity.
In the example shown it is possible to demonstrate the downstream location increase of discharge is slower than the pump's discharge at start-up, and when the pump stops the flow in the pipeline continues for a few seconds longer. This implies a small change in the flow dynamics, while volume continuity is preserved. Also, even if the simulated velocity in the pipeline is smaller, due to the additional flow cross section area in the slot, this has not impact on the simulation's accuracy in terms of pump capacity curve.
In pressurized systems, the applicable simulation time step may be limited. Note that accurate simulation of pressure oscillations in rising mains requires time step in the order of milliseconds and different computation methods. Then with MIKE 1D approach the time step must be adjusted considering the relative size of the pump acceleration/deceleration time, length, and size of the rising mains. If the modeler is using self-adaptive dt, the lower limit should be set to 1 second. As shown in the graph above, a fixed dt of 2 seconds seems be proper for the model.
Conclusion
Actually, all pipes are more or less elastic, and the same thing happens in real sewer systems, it just happens at a lower scale. Then the “price” to be paid for this solution is a slight inaccuracy in the flow dynamics when modeling rising mains. While dealing with unexpected flow conditions in your model results there are couple of bonus tips that you can implement.
- The modeler can identify any unexpectedly high pressure by creating the result table for all nodes. Per default it shows maximum values, sorting them by size would bring the highest Water Level blow ups first.
- When testing a model, it is a good idea to save every simulated time step. This procedure gives the full picture of what is really going on in the model. This limits the length of the simulation, given that the result file grows rapidly with this kind of high saving frequency.
In MIKE+ 2025 this default value for this parameter has been changed to 0.1.
FURTHER INFORMATION AND USEFUL LINKS
[Manuals and User Guides]
MIKE 1D DHI Simulation Engine for 1D river and urban modelling
Release Notes
MIKE+ Release Note
Scientific papers and articles
Preissmann, M.A. and Cunge J.A.: Calculation of Wave Propagation on Electronis Machines. Proceedings, 9th Congress, IAHR,
Dubrovnik, pp. 656-664, 1961.