Under dynamic wave flow routing, both SWMM 4 and 5 treat conduit entrance and exit losses as an average energy loss spread over the entire length of the conduit, analagous to the way that the friction head loss is treated. However, rather than using “average” values for hydraulic variables as is done for friction loss, the entrance loss is based on values at the upstream end of the conduit while the exit loss uses values from the downstream end. It appears that hydraulic results obtained using this approach violate the Bernoulli energy equation for steady, subcritical flow conditions where conservation of energy should apply. We illustrate this point with a simple example and suggest a modification to the code that improves the energy balance obtained.

The figure below shows a simple 3 channel system with a fixed downstream boundary condition. Each channel is rectangular, with a width of 2 ft, a maximum height of 1 ft. and a roughness value of 0.01. A steady inflow of 2 cfs is introduced at the upstream end. The steady water profile that results when there are no local losses is shown in the figure.

Because the flow is steady, the Bernoulli energy equation should be satisfied across each channel link where subcritical flow conditions occur. This equation states that the total energy head at the upstream end of the channel minus any energy losses ocurring within the channel must equal the total energy head at the downstream end of the channel. For the middle channel between nodes 2 and 3 with no entrance or exit losses this equation would be:

where H_i = elevation head at node i, V_i = velocity in channel at node i, g = acceleration of gravity, and h_f = friction head loss in the channel. The latter term can be computed from the Manning equation:

where L = channel length, n = Manning roughness, V = average velocity in the channel and R = average hydraulic radius in the channel.

For the conditions shown in Figure 1, a SWMM 5 run using dynamic wave routing with inertial terms included gives the following results:

Upstream Conditions	Downstream Conditions	Average Conditions
H2 = 4.4035 ft	H3 = 2.2681 ft	Q = 2.0000 cfs
Y2 = 0.4035 ft	Y3 = 0.2681 ft	Y = 0.3358 ft
A2 = 0.8070 ft2	A3 = 0.5362 ft2	A = 0.6716 ft2
V2 = 2.4783 ft/sec	V3 = 3.7300 ft/sec	V = 2.9780 ft/sec
L = 800 ft	n = 0.01	R = 0.2514 ft
V22/2g = 0.0954 ft	V32/2g = 0.2160 ft	hf = 2.0140 ft

And the Bernoulli equation is satisfied as follows:

LHS: 4.4035 + 0.0954 – 2.0140 = 2.485

RHS: 2.2681 + 0.2160 = 2.484

Now consider what happens when an entrance loss with coefficient value of 10 is applied to the middle conduit. The Bernoulli equation is modified to include the additional loss term on the RHS:

When this entrance loss is added to the middle channel in the current version of SWMM the resulting energy balance looks as follows:

Upstream Conditions	Downstream Conditions
Helev = 4.5070 ft	Helev = 2.2681 ft
Hveloc = 0.0604 ft	Hveloc = 0.2160 ft
Hfric = 1.3212 ft	Htotal = 2.484 ft
Hent = 0.6041 ft
Htotal = 2.642 ft

These SWMM results no longer satisfy the Bernoulli equation:

If instead we introduce an exit loss with coeffcient of 10 to the middle channel the discrepancy is even worse:

Upstream Conditions	Downstream Conditions
Helev = 4.6131 ft	Helev = 2.2681 ft
Hveloc = 0.0413 ft	Hveloc = 0.2160 ft
Hfric = 0.9057 ft	Htotal = 2.484 ft
Hexit = 2.1603 ft
Htotal = 1.588 ft

We believe that the problem lies in how the entrance/exit loss value, KV_e²/2g, is embedded into the finite difference form of the momentum equation. The latter equation includes the Manning equation for the standard friction slope term:

which is a loss per unit length along the channel, with k given by g(n/1.49)². By analogy with this term, the entrance/exit loss is also expressed as an energy slope term as follows:

where the subscript e represents either the channel entrance or exit location. When the conservation of mass and momentum equations are expressed in finite difference form, the following result for flow Q at time t+t is obtained:

where has been factored out of the expressions for h_f and h_e.

Neither SWMM 4 or 5 currently contains the (A/A_e) factor when computing the entrance/exit loss term. When this term was introduced into SWMM 5, the following Bernoulli equation results were obtained for the test problem with entrance and exit losses, respectively:

Entrance Loss Example

Upstream Conditions	Downstream Conditions
Helev = 4.4859 ft	Helev = 2.2681 ft
Hveloc = 0.0658 ft	Hveloc = 0.2160 ft
Hfric = 1.4338 ft	Htotal = 2.484 ft
Hent = 0.6577 ft
Htotal = 2.460 ft

Exit Loss Example

Upstream Conditions	Downstream Conditions
Helev = 4.9136 ft	Helev = 2.2681 ft
Hveloc = 0.0186 ft	Hveloc = 0.2160 ft
Hfric = 0.3887 ft	Htotal = 2.484 ft
Hexit = 2.1603 ft
Htotal = 2.383 ft

These results are closer to meeting an energy balance but are still not perfect. One other issue to consider is how the average area A and hydraulic radius R are computed within a conduit when computing the friction head loss. This is an old topic that has seen different approaches used within various versions of SWMM. Currently, both SWMM 4 and 5 compute A and R using a depth equal to the average of the upstream and downstream depths. One alternative method is to take the average of the A and R values computed at the depths at each end of the conduit. When this method was utilized in SWMM 5 the Bernoulli results for the two test cases were as follows:

Entrance Loss Example

Upstream Conditions	Downstream Conditions
Helev = 4.4894 ft	Helev = 2.2681 ft
Hveloc = 0.0648 ft	Hveloc = 0.2160 ft
Hfric = 1.4142 ft	Htotal = 2.484 ft
Hent = 0.6483 ft
Htotal = 2.492 ft

Exit Loss Example

Upstream Conditions	Downstream Conditions
Helev = 4.9340 ft	Helev = 2.2681 ft
Hveloc = 0.0178ft	Hveloc = 0.2160 ft
Hfric = 0.3703 ft	Htotal = 2.484 ft
Hexit = 2.1603 ft
Htotal = 2.421 ft

These results come even closer to satisfying the energy balance. However, redefining the way that SWMM computes an average A and R would likely change flow and depth results for many existing models that do not include entrance/exit losses. The subsequent confusion that would occur does not seem worth the slight improvements in energy closure achieved. Therefore it is recommended that only the (A/A_e) correction be applied to the SWMM 5 code.

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