This section discusses the EXTRAN translation from nodal depths to upstream and downstream conduit depths. The linking of conduit flows and junction depths discussed previously cannot be applied without modification to every conduit for the following reasons:
- The invert elevations of conduits which join at a node may be different since stormwater and sanitary sewer systems are frequently built with invert discontinuities.
- Critical depth may occur in the conduit and thereby restrict the discharge.
- Normal depth in the conduit may control the flow.
- The conduit may be dry.
In all of these cases, or combination of cases, the flow must be computed by special techniques.
EXTRAN converts the nodal water depth to the depth of flow above the invert of the connecting conduits then classifies the conduit as having a dry, subcritical, critical upstream, or critical downstream regime based on the following criteria:
- If the flow is positive, then the conduit is assumed:
- subcritical if the invert elevation of the downstream node is equal to the invert elevation of the downstream end of the conduit, and subcritical or critical otherwise,
- critical downstream if the minimum of the critical or normal conduit depth is greater than the downstream conduit depth,
- subcritical if the maximum of the critical or normal conduit depth is less than the downstream conduit depth,
- otherwise critical downstream and the value of variable FASNH must be calculated.
- If the flow is negative, then the conduit is assumed
- subcritical if the invert elevation of the upstream node is equal to the invert elevation of the upstream end of the conduit, and subcritical or critical otherwise,
- critical upstream if the critical conduit depth is greater than the upstream conduit depth, and subcritical if the critical conduit depth is less than the upstream conduit depth.
- If the upstream depth is positive and the downstream depth is zero then the conduit is dry if the upstream HGL is less than the invert of the downstream end of the conduit. The conduit is assumed sub-critical if the invert elevation of the downstream junction is equal to the invert elevation of the downstream end of the conduit, and critical downstream otherwise.
- If the downstream depth is positive and the upstream depth is zero then the conduit is dry if the downstream HGL is less than the invert of the upstream end of the conduit. The conduit is assumed subcritical if the invert elevation of the upstream junction is equal to the invert elevation of the upstream end of the conduit, and critical upstream otherwise.
Next, based on the flow regime of the conduit, EXTRAN computes the conduit surface width and then assigns the conduit surface area to the upstream and downstream nodes according to the following criteria:
- For the normal situation in which both pipe inverts are submerged and the flow is sub-critical throughout the conduit, the surface area of that conduit is assigned equally to the two connecting junctions. The surface area is calculated as the product of the mean of the upstream and downstream top widths multiplied by half the conduit length.
- If a critical flow section is detected at the downstream end of a conduit, then the entire surface area for that conduit is assigned to the upstream node.
- If a critical section occurs at the upstream end, then the entire conduit surface area is assigned to the downstream node.
- In the case of a dry pipe (pipe inverts unsubmerged), the surface area is zero, the velocity is set equal to zero, and cross-sectional area and hydraulic radius are set to the minimum conduit depth defined in the "Modify Conduits" dialog under Job Control.
- If the pipe is dry only at the upstream end, then all surface area for the conduit is assigned to the downstream junction.
Note that adverse flow in the absence of a critical section is treated as in (1) immediately above. If a critical section occurs upstream, then all surface area for the adverse pipe is assigned downstream as in (3).
The assignment of nodal surface area, based on the top width and length of the conduit, is essential to the proper calculation of head changes computed at each node from the junction mass continuity equation. Following the surface area assignment, EXTRAN computes the current weighted average values of cross-sectional area, velocity, and hydraulic radius for each conduit.
EXTRAN also computes the depth at orifice junctions for all sump orifices that are not flowing full. The condition of the sump orifices are either critical downstream or subcritical.
Once these depth and surface area corrections are applied, the computations of head and discharge can proceed in the normal way for the current time-step. Note that any of these special situations may begin and end at various times and places during simulation. EXTRAN detects these automatically in Subroutine HEAD.
EXTRAN prints a summary of the special hydraulic cases.
EXTRAN prints the time in minutes that a conduit was:
- dry (depth less than the minimum conduit depth [ ft or m],
- critical depth upstream, and
- critical depth downstream.
It should be noted that these designations refer strictly to the assignment of upstream and downstream nodal surface area and conduit depths.
EXTRAN computes the critical and normal depths corresponding to a given flow in a conduit using the critical flow and Manning uniform flow equations, respectively. Tables of normalized values for the cross-sectional area, hydraulic radius and surface width of each conduit class are initialized to speed the computations of critical and normal depth.
The dimensionless tables for each conduit class consists of 26 numbers or 25 increments of depth for cross sectional area and topwidth. The depths range from 0.0 to 1.0 with increments of 0.04. The critical depth is calculated from setting the Froude number to 1 and finding the position of the critical depth corresponding to the conduit flow. The normal depth is calculated from Manning’s equation and finding the position of the normal depth corresponding to the conduit flow. The depths are then interpolated linearly using the two nearest depth levels.
- Partial Differential Equations
- Finite Difference Equations
- Implicit Time Weighting
- Significant Differences between EXTRAN Versions 3, 4 and 5
- Finite Difference Solution
- Convergence in EXTRAN
- Automatic Time Step Selection
- Special Conduit Flow Conditions
- Flow and Head Computation during Surcharge and Flooding
- Ground and Invert Elevations
- Flow Control or Diversion Devices
- Storage Devices, Ponds, and Lakes
- Weirs with Tide Gates
- Pump Stations
- Outfall Structures
- Boundary Conditions
- Initial Conditions
- Pit or Junction Losses
- Oscillations during a Hydraulic Jump
- Irregular Closed Conduits
- Irregular Open Channels
- Kinematic and Diffusion Wave Equations
- Special Finite Difference Approximations
- Conduit Depth Calculations