Continuity [2.4] :

In these equations,
x = distance along the channel,
t = time,
h = depth of flow,
v = average velocity of flow,
S_b = channel bottom slope,
S_f = friction slope,
g = acceleration due to gravity,
q = precipitation intensity.

The one-dimensional form of the equations is used because this form is highly suited to the presentation of the basic concepts of the numerical procedures to be discussed. Extension to irregular surfaces can be accomplished with little difficulty. Derivation of these equations is given in standard reference works (e.g. Chow, 1959, 1969; Henderson, 1966; Daily and Harleman, 1966; Grace and Eagleson, 1965, 1966), and follows from the following assumptions:

• a moderately wide flow surface (h/width << 1),
• small bottom slope (q @ sin q @ tan q ),
• uniform velocity distribution (momentum distribution coefficient is unity),
• overpressure due to vertical inflow is negligible,
• precipitation has negligible effect on flow dynamics (Eagleson, 1970) in comparison with that of gravity,
• friction slope for steady, uniform flow applies to unsteady, non-uniform flow of the same depth and average velocity.

Formulation of the dynamic wave equations is reviewed here since this has more general application than the simpler kinematic formulation. We use simpler kinematic routing methods in TRANS and RUNOFF.

For the mathematical solution of the gradually-varied unsteady flow (St. Venant) equations, we use a link-node description to represent the sewer system, in which manholes are nodes, and conveyances (like sewers) are links.

As shown in Figure below, the conduit system is idealized as a series of links (or pipes) which are connected at nodes or junctions. Links and nodes have well-defined properties which permit representation of the entire pipe network, including flow control devices. The continuity equation is solved at the node and the momentum equation along the conveyance (thus a conveyance is the smallest spatial unit in this discretization - discharge and velocity are averaged along a pipe and head is averaged over an area centered on a node and roughly halfway to the next node). The specific properties of links and nodes are summarized in the Table below.

Properties of Nodes and Links in EXTRAN

Links transmit flow from node to node, and are characterized by a discharge/head relation such as a rating curve or a weir equation Q = f(H). Properties associated with the links are roughness, length, cross-sectional area, hydraulic radius, and surface width. The last three properties are functions of the instantaneous depth of flow. The primary dependent variable in the links is the discharge, Q. The solution is for the average flow in each link.

Conceptual Representation of the EXTRAN Model.

Nodes are the storage elements of the system and correspond to manholes, sewer junctions or just the conceptual end of one conveyance and the start of another in the physical system. The variables associated with a node are volume, head, and surface area. The primary dependent variable is the head, H (elevation to water surface = invert elevation plus water depth), which is assumed to be changing in time but constant over the area assigned or covered by a node. (A plot of head versus distance along the sewer network yields the hydraulic grade line, HGL, and its dynamic variation is the DHGL.) Inflows, such as inlet hydrographs, and outflows, such as weir diversions, take place at the nodes of the idealized sewer system. The volume of the node at any time is equivalent to the water volume in the half- pipe lengths connected to any one node. The change in nodal volume during a given time step, dt, forms the basis of head and discharge calculations as discussed below.

Derivation of unsteady flow equations for sewers

where g = gravitational constant,
H = z + h = hydraulic head,
z = invert elevation,
h = water depth, and
S_f = friction (energy) slope.

The continuity equation may be manipulated to replace the second term of the above equation, using Q = AV,

In EXTRAN there are three solutions. The above equation is the basis of the ISOL = 0 solution. The momentum equation for the ISOL = 1 and ISOL = 2 solutions are derived in the following manner. The ¶ (Q²/A)/ ¶ x term in equation 3-2 is expanded as the product of Q and Q/A instead of V²/A as in the ISOL = 0 solution.

Again the continuity equation is used to substitute for the ¶ Q/¶ x term in equation 3-9. This term is inadmissible in EXTRAN since the flow is assumed to be constant in a link. The link momentum equation used by the ISOL = 1 and ISOL = 2 solutions is:

         (3-11)

The friction slope is defined by Manning's equation, i.e.

where A_s = surface area of node.

Solution of flow equation by modified Euler method

Figure 3-3. Modified Euler method for discharge based on half-step, full-step projection

 Half-step at node j: Time t+D t/2

(3-17)

Full-step at node j: Time t+D t

1. Compute half-step discharge at t+D t/2 in all links based on preceding full-step values of head at connecting junctions.
2. Compute half-step flow transfers by weirs, orifices, and pumps at time t+D t/2 based on preceding full-step values of head at transfer junction.
3. Compute half-step head at all nodes at time t+D t/2 based on average of preceding full-step and current half-step discharges in all connecting conduits, plus flow transfers at the current half-step.
4. Compute full-step discharge in all links at time t+D t based on half-step heads at all connecting nodes.
5. Compute full-step flow transfers between nodes at time t+D t based on current half-step heads at all weir, orifice, and pump nodes.
6. Compute full-step head at time t+D t for all nodes based on average of preceding full-step and current full-step discharges, plus flow transfers at the current full-step.

Numerical stability

Time-step restrictions

The modified Euler method yields a completely explicit solution in which the momentun equation is applied to discharge in each link and the continuity equation to head at each node, with implicit coupling during the time-step. Explicit methods involve fairly simple arithmetic and require little storage space compared to implicit methods. However, they are generally less stable and often require very short time-steps. Experience has indicated that the program is stable numerically when the following inequalities are met:

Links:

(3-19)

where D t = time-step, sec,
L = conduit length, ft [m];
g = gravitational acceleration, 32.2 ft/sec² [9.8 m/sec²], and
D = maximum pipe depth, ft [m].

This is a form of the Courant condition, in which the time step is limited to the time required by a dynamic wave to propagate the length of a conduit.

Nodes:

(3-20)

 where C' = dimensionless constant, determined by experience to be approx 0.1,
D H_max = maximum water-surface rise during the time-step,
A_s = corresponding surface area of the node, and
S Q = net inflow to the node (junction).

Examination of inequalities 3-19 and 3-20 reveals that the maximum allowable time-step, D t, will be determined by the shortest, smallest pipe having high inflows. Based on past experience with EXTRAN, a time-step of 10 seconds is nearly always sufficiently small to produce outflow hydrographs and stage-time traces which are free from spurious oscillations and also satisfy mass continuity under non-flooding conditions. If smaller time-steps are necessary the user should eliminate or aggregate the offending small pipes or channels. In most applications, 15 to 30 second time-steps are adequate; occasionally time-steps up to 60 seconds can be used.

Equivalent pipes

An equivalent pipe is the computational substitution of an element of the drainage system by an imaginary conduit which is in terms of steady state hydraulics similar to the element it replaces. Note by W James: A longer or shorter pipe will behave quite differently: substitution of the imaginary pipe will detune the network in a dynamic sense. The authors of EXTRAN use an equivalent pipe when it is suspected that a numerical instability will be caused by the element of the drainage system being replaced in the computation. Short conduits and weirs are known at times to cause stability problems and thus the authors replaced them by an equivalent pipe. (Orifices are automatically converted to equivalent pipes by the program; also a serious detuning of the problem, see the description below).

The equivalent pipe substitution involves the following steps. First the flow equation for the element in question is set equal to the flow equation for an "equivalent pipe." This, in effect, says that the head losses in the element and its equivalent pipe are the same. The length of the equivalent pipe is computed using the numerical stability equation 3-19. Then, after making any additional assumptions which may be required about the equivalent pipe's dimensions, a Manning's n is computed based on the equal head loss requirement. In the case of orifices, this conversion occurs internally, but in those cases where short pipes and weirs are found to cause instabilities, the user must make the necessary conversion and revise the input data set.

Special pipe flow considerations

The solution technique discussed in the preceding paragraphs cannot be applied without modification to every conduit for the following reasons. First, the invert elevations of pipes which join at a node may be different since sewers are frequently built with invert discontinuities. Second, critical depth may occur in the conduit and thereby restrict the discharge. Third, normal depth may control. Finally, the pipe may be dry. In all of these cases, or combinations thereof, the flow must be computed by special techniques. Figure 3-4 shows each of the possibilities and describes the way in which surface area is assigned to the nodes.

The options are:

1. Normal case. Flow computed from motion equation. Half of surface area assigned to each node.
2. Critical depth downstream. Use lesser of critical or normal depth downstream. Assign all surface area to upstream node.
3. Critical depth upstream. Use critical depth. Assign all surface area to downstream node.
4. Flow computed exceeds flow at critical depth. Set flow to normal value. Assign surface area in usual manner as in 1.
5. Dry pipe. Set flow to zero. If any surface area exists, assign to downstream node.

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.

EXTRAN now prints a summary of the special hydraulic cases illustrated in Figure 3-4, the time in minutes that a conduit was (1) dry (depth less than 0.0001 ft or m), (2) normal depth, (3) critical up stream, and (4) critical downstream. It should be noted that these designations refer strictly to the assignment of upstream and downstream nodal surface area.

During the calculation of conduit flow another normal flow approximation is used when all of the following three conditions occur:

1. The flow is positive. EXTRAN automatically designates the highest invert elevation as the upstream node and the lowest as the downstream node. This adjustment (if made) is now printed out by the model. Positive flow is from the upstream to the downstream node.
2. The water surface slope in the conduit is less than the conduit slope. See the section on Additional EXTRAN Solutions for more details.
3. The flow calculated from Manning's equation using the upstream cross-sectional area and hydraulic radius is less than the flow calculated by equation 3-14.

When all three conditions are met the flow is "normal." Normal flow is labeled with an asterisk in the intermediate printout. The conduit summary lists the number of minutes the normal flow assumption is used for each conduit.

 

Figure 3-4. Special hydraulic cases in EXTRAN flow calculations.

Head computation during surcharge and flooding

Theory

Another hydraulic situation which requires special treatment is the occurrence of surcharge and flooding. Surcharge occurs when all pipes entering a node are full or when the water surface at the node lies between the crown of the highest entering pipe and the ground surface.

Flooding is a special case of surcharge which takes place when the hydraulic grade line breaks the ground surface and water is lost from the sewer node to the overlying surface system. While it would be possible to track the water lost to flooding by surface routing, this is not done automatically in EXTRAN. To track water on the surface the user must 1) simulate the surface pathways as conduits, and 2) simulate the vertical pathways through manholes or inlets as conduits also. Since a conduit cannot be absolutely vertical, equivalent pipes must be used.

During surcharge, the head calculation in equations 3-17 and 3-18 is no longer possible because the surface area of the surcharged node (area of manhole) is too small to be used as a divisor. Instead, the continuity equation for each node is equated to zero

(3-21)

where S Q(t) is the sum of all inflows to and outflows from the node from surface runoff, conduits, diversion structures, pumps and outfalls.

Since the flow and continuity equations are not solved simultaneously in the model, the flows computed in the links connected to a node will not exactly satisfy equation 3-21. However, an iterative procedure is used in which head adjustments at each node are made on the basis of the relative changes in flow in each connecting link with respect to a change in head: ¶ Q/¶ H. Expressing equation 3-21 in terms of the adjusted head at node j gives

(3-22)

Solving for D H_j gives

(3-23)

This adjustment is made by half-steps during surcharge so that the half-step correction is given as

(3-24)

where H_j(t+D t/2) is given by equation 3-23 while the full-step head is computed as

(3-25)

where D H_j(t) is computed from equation 3-21. The value of the constant k theoretically should be 1.0. However, it has been found that equation 3-22 tends to over-correct the head; therefore, a value of 0.5 is used for k in the half-step computation in order to improve the results. Unfortunately, this value was found to trigger oscillations at upstream terminal junctions. To eliminate the oscillations, values of 0.3 and 0.6 are automatically set for k in the half-step and full-step computations, respectively, at upstream terminal nodes.

The head correction derivatives are computed for conduits and system inflows as follows:

Links

      (3-26)

      (3-27)

where
Dt = time-step,
A(t) = flow cross sectional area in the conduit,
L = conduit length,
n = Manning n,
m = 1.49 for U.S. customary units and 1.0 for metric units,
g = gravitational acceleration,
R = hydraulic radius for the full conduit, and
V(t) = velocity in the conduit.

System inflows

         (3-28)

      (3-28)

 Orifice, weir, pump and outfall diversions

Orifices are converted to equivalent pipes (see below); therefore, equation 3-26 is used to compute ¶Q/¶H. For weirs, ¶Q/¶H in the weir link is taken as zero, i.e., the effect of the flow changes over the weir due to a change in head is ignored in adjusting the head at surcharged weir junctions. (The weir flow, of course, is computed in the next time-step on the basis of the adjusted head.) As a result, the solution may go unstable under surcharge conditions. If this occurs, the weir should be changed to an equivalent pipe.

For pump junctions, ¶Q/¶H is also taken as zero. For off-line pumps (with a wet well), this is a valid statement since Q_pump is determined by the volume in the wet well, not the head at the junction. For in-line pumps, where the pump rate is determined by the water depth at the junction, a problem could occur if the pumping rate is not set at its maximum value at a depth less than surcharge depth at the junction. This situation should be avoided, if possible, because it could cause the solution to go unstable if a large step increase or decrease in pumping rate occurs while the pump junction is surcharged.

For all outfall pipes, the head adjustment at the outfall is treated as any other junction. Outfall weir junctions are treated the same as internal weir junctions (¶Q/¶H for the weir link is taken as zero). Thus, unstable solutions can occur at these junctions also under surcharge conditions. Converting these weirs to equivalent pipes will eliminate the stability problem.

Because the head adjustments computed in equations 3-24 and 3-25 are approximations, the computed head has a tendency to "bounce" up and down when the conduit first surcharges. This bouncing can cause the solution to go unstable in some cases; therefore, a transition function is used to smooth the changeover from head computations by equations 3-17 and 3-18 to equations 3-24 and 3-25. The transition function used is

(3-29)

where DENOM is given by

(3-30)

where D_j = pipe diameter,
y_j = water depth, and
A_sj = nodal surface area at 0.96 of full depth.

The exponential function causes equation 3-30 to converge to within two percent of equation 3-23 by the time the water depth is 1.25 times the full-flow depth.

Surcharge in multiple adjacent nodes

Use of ¶Q(t)/¶H_j in the manner explained above satisfies continuity at a single node, but may introduce a small continuity error when several consecutive nodes are surcharged. These small continuity errors combine to artificially attenuate the hydrograph in the surcharged area. Physically, inflows to all surcharged nodes must equal outflows during a time-step since no change in storage can occur during surcharge. In order to remove this artificial attenuation, the full-step computations of flow and head in surcharge areas are repeated in an iteration loop. The iterations for a particular time-step continue until one of the following two conditions is met:

1. The net difference of inflows to and outflows from all nodes in surcharge is less than a tolerance, computed every time-step, as a fraction of the average flow through the surcharged area. The fraction is input by the user.

2. The number of iterations equals a maximum set by the user.

The iteration loop has been found to produce reasonably accurate results with little continuity error. The user may need to experiment somewhat with ITMAX and SURTOL in order to accurately simulate all surcharge points without incurring an unreasonably high computer cost due to extra iterations.

Flow control devices

Options

The link-node computations can be extended to include devices which divert sanitary sewage out of a combined sewer system or relieve the storm load on sanitary interceptors. All diversions are assumed to take place at a node and are handled as inter-nodal transfers. The special flow regulation devices include: weirs (both side-flow and transverse), orifices, pumps, and outfalls. Each of these is discussed in the paragraphs below.

Storage devices

In-line or off-line storage devices act as flow control devices by providing for storage of excessive upstream flows thereby attenuating and lagging the wet weather flow hydrograph from the upstream area. The conceptual representations of a storage junction and a regular junction are illustrated in Figure 3-5. Note that the only difference is that added surface area in the amount of ASTORE is added to that of the connecting pipes. Note also that ZCROWN(J) is set at the top of storage "tank." When the hydraulic head at junction J exceeds ZCROWN(J), the junction surcharges.

An arbitrary stage-area-volume relationship may also be input, e.g. to represent detention ponds. Routing is performed by ordinary level-surface reservoir methods. This type of storage facility is not allowed to surcharge.

 Figure 3-5. Conceptual representation of a storage junction.

 Orifices

The purpose of the orifice generally is to divert sanitary wastewater out of the stormwater system during dry weather periods and to restrict the entry of stormwater into the sanitary interceptors during periods of runoff. The orifice may divert the flow to another pipe, a pumping station or an off-line storage tank.

Figure 3-6 shows two typical diversions:

1. a dropout or sump orifice, and
2. a side outlet orifice.

EXTRAN simulates both types of orifice by converting the orifice to an equivalent pipe. The conversion is made as follows. The standard orifice equation is:

(3-31)

where C_o = discharge coefficient (a function of the type of opening and the length of the orifice tube),
A = cross-sectional area of the orifice,
g = gravitational acceleration, and
h = the hydraulic head on the orifice.

Figure 3-6 Typical orifice diversions

Values of C_o and A are specified by the user. To convert the orifice to a pipe, the program equates the orifice discharge equation and the Manning pipe flow equation, i.e.,

(3-32)

where m = 1.49 for U.S. customary units and 1.0 for metric units, and
S = slope of equivalent pipe.

The orifice pipe is assumed to have the same diameter, D, as the orifice and to be nearly flat, the invert on the discharge side being set 0.01 feet [3 mm] lower than the invert on the inlet side. In addition, for a sump orifice, the pipe invert is set by the program 0.96D below the junction invert so that the orifice pipe is flowing full before any outflow from the junction occurs in any other pipe. For side outlet orifices, the user specifies the height of the orifice invert above the junction floor.

If S is written as H_s/L where L is the pipe length, H_s will be identically equal to h when the orifice is submerged. When it is not submerged, h will be the height of the water surface above the orifice centerline while H_s will be the distance of the water surface above critical depth (which will occur at the discharge end) for the pipe. For practical purposes, it is assumed that H_s = h for this case also. Thus, letting S = h/L and substituting R = D/4 (where D is the orifice diameter) into equation 3-32 and simplifying gives,

(3-33)

The length of the equivalent pipe is computed as the maximum of 200 feet (61 m) or

(3-34)

to ensure that the celerity (stability) criterion for the pipe is not violated. Manning's n is then computed according to equation 3-33. This algorithm produces a solution to the orifice diversion that is not only as accurate as the orifice equation but also much more stable when the orifice junction is surcharged.

Weirs

A schematic illustration of flow transfer by weir diversion between two nodes is shown in Figure 3-7. Weir diversions provide relief to the sanitary system during periods of storm runoff. Flow over a weir is computed by

(3-35)

where C_w = discharge coefficient,
L_w = weir length (transverse to overflow),
h = driving head on the weir,
V = approach velocity, and
a = weir exponent, 3/2 for transverse weirs and 5/3 for sideflow weirs.

Both C_w and L_w are input values for transverse weirs. For sideflow weirs, C_w should be a function of the approach velocity, but the program does not provide for this because of the difficulty in defining the approach velocity. For this same reason, V, which is programmed into the weir solution, is set to zero prior to computing Q_w.

Figure 3-7. Representation of weir diversions.

Normally, the driving head on the weir is computed as the difference h = Y₁-Y_c, where Y₁ is the water depth on the upstream side of the weir and Y_c is the height of the weir crest above the node invert. However, if the downstream depth Y₂ also exceeds the weir crest height, the weir is submerged and the flow is computed by

(3-36)

where C_SUB is a submergence coefficient representing the reduction in driving head, and all other variables are as defined above.

The submergence coefficient, C_SUB, is taken from Roessert's Handbook of Hydraulics by interpolation from Table 3-3, where C_RATIO is defined as:

 (3-37)

 and all other variables are as previously defined. The values of C_RATIO and C_SUB are computed automatically by EXTRAN and no input data values are needed.

 Table 3-3. Values of C_SUB as a function of degree of weir submergence

CRATIO	CSUB
0.0	1.00
0.10	0.99
0.20	0.98
0.30	0.97
0.40	0.96
0.50	0.95
0.60	0.94
0.70	0.91
0.80	0.85
0.85	0.80
0.90	0.68
0.95	0.40
1.00	0.00

If the weir is surcharged it will behave as an orifice and the flow is computed as:

       (3-38)

where Y_TOP = distance to top of weir opening shown in Figure 3-7,
h' = Y₁ - maximum(Y₂,Y_c), and
C_SUR = weir surcharge coefficient.

The weir surcharge coefficient, C_SUR, is computed automatically at the beginning of surcharge. At the point where weir surcharge is detected, the preceding weir discharge just prior to surcharge is equated to Q_w in equation 3-36, and equation 3-38 is then solved for the surcharge coefficient, C_SUR. Thus, no input coefficient for surcharged weirs is required.

Finally, flow reversals are detected at weir nodes which cause the downstream water depth, Y₂, to exceed the upstream depth, Y₁. All equations in the weir section remain the same except that Y₁ and Y₂ are switched so that Y₁ remains as the "upstream" head. Also, flow reversal at a sideflow weir causes it to behave more like a transverse weir and consequently the exponent a in equation 3-39 is set to 1.5.

Weirs with tide gates

Frequently, weirs are installed together with a tide gate at points of overflow into the receiving waters. Flow across the weir is restricted by the tide gate, which may be partially closed at times. This is accounted for by reducing the effective driving head across the weir according to an empirical factor published by Armco (undated):

(3-39)

where h is the previously computed head before correction for flap gate and V is the velocity of flow in the upstream conduit.

Pump stations

A pump station is conceptually represented as either an in-line lift station, or an off-line node representing a wet-well, from which the contents are pumped to another node in the system according to a programmed rule curve. Alternatively, either in-line or off-line pumps may use a three-point pump curve (head versus pumped outflow).

For an in-line lift station, the pump rate is based on the water depth, Y, at the pump junction. The step-function rule is as follows:

Pump Rate = R₁ for 0 < Y < Y₁

= R₂ for Y₁ < Y < Y₂ (3-40)

= R₃ for Y₂ < Y < Y₃

For Y = 0, the pump rate is the inflow rate to the pump junction.

Inflows to the off-line pump must be diverted from the main sewer system through an orifice, a weir, or a pipe. The influent to the wet-well node must be a free discharge regardless of the diversion structure. The pumping rule curve is based on the volume of water in the storage junction. A schematic presentation of the pump rule is shown in Figure 3-8.

The step-function rule operates as follows:

1. Up to three wet-well volumes are pre-specified as input data for each pump station: V₁ < V₂ < V₃, where V₃ is the maximum capacity of the wet well.

2. Three pumping rates are pre-specified as input data for each station. The pump rate is selected automatically by EXTRAN depending on the volume, V, in the wet-well, as follows:

Pump Rate = R₁ for 0 < V < V₁

= R₂ for V₁ < V < V₂ (3-41)

= R₃ for V₂ < V < V₃

Figure 3-8. Schematic presentation of pump diversion.

3. A mass balance of pumped outflow and inflow is performed in the wet- well during the model simulation period.

4. If the wet-well goes dry, the pump rate is reduced below rate R₁ until it just equals the inflow rate. When the inflow rate again equals or exceeds R₁, the pumping rate goes back to operating on the rule curve.

5. If V₃ is exceeded in the wet-well, the inflow to the storage node is reduced until it does not exceed the maximum pumped flow. When the inflow falls below the maximum pumped flow, the inflow "gates" are opened. The program automatically steps down the pumping rate by the operating rule of (2) as inflows and wet-well volume decrease.

A conceptual head-discharge curve for a pump is shown in Figure 3-16. When this method is used for either type of pump, an iteration is performed until the dynamic head difference between the upstream and downstream nodes on either side of the pump corresponds to the flow given on the pump curve. In other words, the pump curve replaces equation 3-14.

Outfall Structures

EXTRAN simulates both weir outfalls and free outfalls. Either type may be subject to a backwater condition and protected by a tide gate. A weir outfall is a weir which discharges directly to the receiving waters according to relationships given previously in the weir section. The free outfall is simply an outfall conduit which discharges to a receiving water body under given backwater conditions. The free outfall may be truly "free" if the elevation of the receiving waters is low enough (i.e., the end of the conduit is elevated over the receiving waters), or it may consist of a backwater condition.

 In the former case, the water surface at the free outfall is taken as critical or normal depth, whichever is less. If backwater exists, the receiving water elevation is taken as the water surface elevation at the free outfall.

Up to twenty different head versus time relationships can be used as boundary conditions. Any outfall junction can be assigned to any of the twenty boundary conditions.

When there is a tide gate on an outfall conduit, a check is made to see whether or not the hydraulic head at the upstream end of the outfall pipe exceeds that outside the gate. If it does not, the discharge through the outfall is equated to zero. If the driving head is positive, the water surface elevation at the outfall junction is set in the same manner as that for a free outfall subjected to a backwater condition. Note that even if the tide gate is closed, water can still enter and fill an empty outfall pipe as sometimes happens at the beginning of a simulation.

Initial conditions

For each conduit, EXTRAN then computes the normal depth corresponding to the initial flow. Junction heads are then approximated as the average of the heads of adjacent conduits for purposes of beginning the computation sequence. The initial volume of water computed in this manner is included in the continuity check. A more accurate initial condition for any desired set of flows may be established by letting EXTRAN "warm up" with the initial inflows and restarted using the "hot start" feature.

Initial heads at junctions with a sump orifice are increased by 0.96 times the equivalent pipe diameter of the orifice at the start of the simulation.

Numerical procedure

The solution uses the modified Euler method and a special iterative procedure for surcharged flow. The following principal steps are performed:

1. For all the physical conduits in the system, compute the following time-changing properties based on the last full-step values of depth and flow:
· Hydraulic head at each conduit end.
· Full-step values of cross-sectional area, velocity, hydraulic radius, and surface area corresponding to preceding full-step flow. This is done by calling subroutine HEAD.
· Half-step value of discharge at time t = t+ Dt/2 by modified Euler solution.
· Check for normal flow, if appropriate. Normal flow is indicated by an asterisk in the intermediate printout.
· Set system outflows and internal transfers at time t + Dt/2 by call to subroutine BOUND. BOUND computes the half-step flow transfers at all orifices, weirs, and pumps at time t = t + Dt/2. It also computes the current value of tidal stage and the half-step value of depth and discharge at all outfalls.
2. For all physical junctions in the system, compute the half-step depth at time t = t+ Dt/2. This depth computation is based on the current net inflows to each node and the nodal surface areas computed previously in step 1. Check for surcharge and flooding at each node and compute water depth accordingly.
3. For all physical conduits, compute the following properties based on the last half-step values of depth and flow (repeat step 1 for time t+ Dt/2):
· Hydraulic head at each pipe end.
· Half-step values of pipe cross-sectional area, velocity, hydraulic radius, and surface are corresponding to preceding half-steep depth and discharge
· Full-step discharge at time t+ D t by modified Euler solution.
· Check for normal flow if appropriate.
· Set system outflows and internal transfer at time t+ D t by calling BOUND.
4. For all junctions, repeat the nodal head computation of step 6 for time t+D t. Sum the differences between inflow and outflow for each junction in surcharge.
5. Repeat steps 3 and 4 for the surcharged links and nodes until the sum of the flow differences from step 4 is less than fraction SURTOL multiplied by the average flow through the surcharged area or the number of iterations exceeds parameter ITMAX.
6. Return to subroutine TRANSX for time and output data updates.

Additional EXTRAN solutions

General

Basic flow equations

The basic differential equations for the sewer flow problem come from the gradually varied, one-dimensional unsteady flow equations for open channels, otherwise known as the St. Venant or shallow water equations (Lai, 1986). For use in EXTRAN, the momentum equation is combined with the continuity equation to yield an equation to be solved along each link at each time-step,

(3-42)

where Q = discharge through the conduit,
V = velocity in the conduit,
A = cross-sectional area of the flow,
H = hydraulic head (invert elevation plus water depth), and
S_f = friction slope.

The fourth term of equation 3-42 is different from the term used in equation 3-11. The terms have their usual units. For example, when U.S. customary units are used, flow is in units of cfs. When metric units are used, flow is in m³/sec. These units are carried through internal calculations as well as for input and output.