&EPA
United States
Environmental Protection
Agency
EPA/600/C-22/016 www.epa.gov/water-research
Addendum to the
Storm Water Management Model
Reference Manual
Volume II -Hydraulics
Office of Research and Development
Center for Environmental Solutions & Emergency Response
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EPA/600/C-22/016
February 2022
Addendum to the
Storm Water Management Model
Reference Manual
Volume II - Hydraulics
by
Lewis Rossman and Michelle Simon
Office of Research and Development
Center for Environmental Solutions and Emergency Response
Cincinnati, OH 45268
Center for Environmental Solutions and Emergency Response
Office of Research and Development
U.S. Environmental Protection Agency
26 Martin Luther King Drive
Cincinnati, OH 45268
February 2022
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Disclaimer
The information in this document has been funded wholly or in part by the U.S. Environmental
Protection Agency (EPA). It has been subjected to the Agency's peer and administrative review
and has been approved for publication as an EPA document. Note that approval does not signify
that the contents necessarily reflect the views of the Agency. Mention of trade names or
commercial products does not constitute endorsement or recommendation for use.
NOTICE: This report was prepared as an account of work sponsored by an agency of the United
States Government. Neither the United States Government, nor any agency thereof, nor any of
their employees, nor any of their contractors, subcontractors, or their employees, make any
warranty, express or implied, or assume any legal liability or responsibility for the accuracy,
completeness, or usefulness of any information, apparatus, product, or process disclosed, or
represent that its use would not infringe privately owned rights. Reference herein to any specific
commercial product, process, or service by trade name, trademark, manufacturer, or otherwise,
does not necessarily constitute or imply its endorsement, recommendation, or favoring by the
United States Government, any agency thereof, or any of their contractors or subcontractors. The
views and opinions expressed herein do not necessarily state or reflect those of the United States
Government, any agency thereof, or any of their contractors.
11
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Abstract
SWMM is a dynamic rainfall-runoff simulation model used for single event or long-term
(continuous) simulation of runoff quantity and quality from primarily urban areas. This document
describes the additional modeling features that have been added into SWMM's hydraulic model
because the original publication of the SWMM Reference Manual Volume II - Hydraulics (May
2017). This addendum covers the addition of a Preissmann Slot option, new features for modeling
street runoff capture by inlet drains, a Type 5 variable speed pump, and additional Storage Curve
options to SWMM.
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Acknowledgements
Lewis A. Rossman is the lead author of this SWMM Reference Manual II - Addendum, in
collaboration with Michelle Simon and the U.S. Environmental Protection Agency (USEPA),
Cincinnati, OH.
This addendum to Storm Water Management Model Reference Manual Volume II - Hydraulics
was reviewed by Michelle Simon (USEPA), Katherine Ratliff (USEPA), Anne Mikelonis
(USEPA), Michael Tryby (USEPA), Robert Dickinson (Innovyze), Mitch Heineman (CDM
Smith), and Mike Gregory (CHI).
iv
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Table of Contents
Disclaimer ii
Abstract iii
Acknowledgements iv
Table of Contents v
1. Preissmann Slot 6
2. Street Cross-Sections 7
3. Storage Unit Geometry 8
3.1 Standard Storage Shapes 8
3.2 Functional Storage Shapes 9
3.3 Tabular Storage Shapes 10
4. Pumps 13
5. Storm Drain Inlets 16
5.1 Model Setup 17
5.2 Computational Scheme 19
5.3 Flow Capture for On-Grade Inlets 19
5.4 Flow Capture for On-Sag Inlets 25
5.5 Drop Inlets 28
6. References 29
v
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1. Preissmann Slot
Chapter 3 of the SWMM Hydraulics Reference Manual (Rossman, 2017) describes how the full
dynamic wave solution for unsteady flow in a conveyance network is computed. In that procedure,
when a network node becomes surcharged resulting in pressurized flow, the normal method of
updating the hydraulic head at the node is replaced by the so-called Surcharge Algorithm. The
assumptions of this algorithm have been questioned (Yen, 2001) and it can experience numerical
instability in the transition between open-channel and pressurized flow.
As an alternative to the surcharge algorithm, SWMM can utilize the more widely accepted
Preissmann Slot Method (Cunge and Wegner, 1964) for handling pressurized flow in closed
conduits. In this case, the conduit's cross-section is assumed to have a narrow open slot at the top
along its entire length. This permits the water level in the conduit to exceed its full depth while
only slightly increasing its flow area. It thus becomes possible to compute a surface area
contribution to the conduit's end nodes once it reaches full depth. As a result, SWMM is able to
continue to use its regular solution for updating nodal heads under all flow conditions without
having to resort to the surcharge algorithm.
In theory, the width of the slot should be determined by having the celerity of an open channel
gravity wave equal to the speed of a pressure wave affected by the compressibility of the elastic
pipe wall. This would result in a slot width wsiot equal to:
where gis the acceleration of gravity, A is the conduit's cross-sectional area when full and cis the
speed of the pressure wave. The latter quantity depends on the conduit's diameter, wall thickness,
and modulus of elasticity and typically ranges from a few hundred to several thousand ft/sec (Yen,
2001).
Some care is needed in choosing a slot width because too large a value will result in reduced
accuracy while too small a value can cause numerical instabilities. There is also the issue of
maintaining a smooth transition between almost full flow and slot flow. The choice used by
SWMM is a modified version of a formula proposed by Sjoberg (1982) and is given by:
where Wmax is the conduit's maximum width, Ymi is its full depth, and Y is depth of flow. This
equation applies to Y/YfuU values between 0.985257 and 1.7. Below this range the slot is not used,
Wsiot = gA/c2
(1)
w5/0f/W max=.0 52 3 exp (- (Y/Y&//)2 -4)
(2)
6
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while above it the slot width relative to Wmaat is fixed at 0.01 or 1 percent. The range's lower limit
was chosen so that the width computed from Equation 2 is the same as the width across a circular
pipe at that flow depth. This helps produce a smooth transition between open channel and
pressurized flow regimes.
When the slot method is employed, for computational purposes the flow depth in a closed conduit
is allowed to exceed its physical full depth. When it reaches the limit at which the slot formula
applies, the resulting water surface width is used to compute the surface area that the conduit
contributes to SWMM's regular head updating formula at its end nodes. It also contributes to the
conduit's flow area when it rises above the full depth. It is not used when computing the conduit's
hydraulic radius.
2. Street Cross-Sections
SWMM defines the geometry of a street or roadway cross-section as a special case of the irregular
channel discussed in Section 5.3 of the SWMM Hydraulics Reference Manual (Rossman, 2017).
The shape of a one-sided street cross-section is shown in Figure 1. In its most basic form it consists
of a road surface with downward slope Sx extending a distance of Tcrown to a vertical curb of height
Hcurb. To this can be added:
• an optional depressed gutter section of width W that extends to a depth "a" below the
normal curb height
• an optional backing section extending beyond the curb a distance Tback that rises at a slope
of Sback.
A two-sided street cross-section adds a mirror image of the one-sided street to the right of the street
crown with the same roadway, gutter, curb, and backing dimensions.
Tback Terown
Sback
¦a
u
w
i _ —
Street
Crown
I
s*
a
Figure 1. A one-sided street cross-section (not to scale)
Street cross-sections use the same procedures as irregular channel transects to compute tables of
flow area, top width, and hydraulic radius at 50 equally spaced increments of flow depth (for both
7
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one-sided and two-sided streets). The 50 equally spaced increments are the same as the increments
for SWMM Transects. This requires that in addition to the dimensions shown in Figure 1, a
Manning's roughness coefficient must be specified for the road surface and for the backing surface,
if present.
3. Storage Unit Geometry
SWMM's hydraulic modeling procedures require knowledge of how a storage unit's surface area
A and volume Vvary with surface depth Fabove the bottom of the unit. It is sufficient to specify
either an area or volume relationship with respect to depth because one can be derived from the
other (A = dV/dY and V = / AdY). SWMM uses surface area to describe a storage unit's shape.
One can select either from several standard shapes where A is a quadratic function of Y, from a
general power law relation between A and Yor use a tabular listing of Fand A values.
3.1 Standard Storage Shapes
SWMM supports several common storage unit shapes, listed in Table 1, whose top surface area A
can be expressed as a quadratic function of height Y:
A = a0 + axY + a2Y2 (3)
The constants ao, ai, and a2 depend on the shape's dimensions as shown in Table 1.
Dynamic wave analysis requires specification of volume V varies with depth Y. Integrating
Equation 3 over depth yields:
V = a0Y + {a^iyY2 + (a2/3)F3 (4)
Kinematic wave analysis requires specification of the depth associated with a given volume. For a
cylindrical shape: Y = V/aQ, while for paraboloid shape: Y = yj2V/a[. For the other shapes the
cubic Equation 4 is solved numerically for Fgiven Fusing the Newton-Raphson-Bisection method
over the interval [0, Yfuii\ with initial estimate Y = V/a0, convergence tolerance of 0.001 ft and
derivative given by Equation 3.
8
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Table 1. Standard storage unit shapes
Shape
Coefficients
Dimensions
Elliptical
Cylinder
f"—•
a0 = (n/A)LW
CLi — 0-2 — 0
L = major axis length
W= minor axis width
Elliptical
Paraboloid
CLq — CL^ — 0
tti = (7t/4) LW/H
L = major axis length
W= minor axis width
\ /
H= paraboloid height
y"-
a0 = (n/4)LW
L = bottom major axis length
Elliptical
V -7
ax = nWZ
W= bottom minor axis width
Cone
V. J
a2 = n(W/L)Z2
Z= side slope (run/rise) along
major axis
a0 = LW
L = bottom length
Rectangular
ax = 2{L + W)Z
W= bottom width
Pyramid
a2 = 4 Z2
Z = wall slope (run/rise) (same
for each face)
3.2 Functional Storage Shapes
SWMM's functional storage shape option uses a power law to relate surface area to depth:
A = c0 + ctf0* (5)
where co, ci, and C2 are user-supplied constants.
The surface area at a given depth is found directly from this equation. The relation between volume
[/and depth ^(required for dynamic wave analysis) is:
v = c'r+(-^h)rC2+1 <6>
To find the depth associated with a given volume (required for kinematic wave analysis) one solves
the following nonlinear equation for Y:
f(X) = V- (c0r + ^C2+1) = 0 (7)
9
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It is solved using the Newton-Raphson-Bisection method over the interval [0, Yruii\ with initial
estimate Y = V/(c0 + cx), convergence tolerance of 0.001 ft and derivative f'(Y) given by
Equation 5.
Some shapes and their coefficients that can be represented with this functional option include:
• Shapes with vertical sides and constant surface area no matter how irregular in outline,
including cylinders and rectangular prisms:
co = area of the base
Cl = C2 = 0.
• An open channel with a trapezoidal cross-section and vertical ends:
c0 = WL
c1 = 2 ZL
c2 = 1
where W= bottom width of cross-section, L = channel length, and Z= side slope.
• An open channel with a parabolic cross-section and vertical ends:
c0 = 0
cx = WLH05
c2 = 1
where W= top width, L = channel length and H = full height.
• An elliptical paraboloid:
c0 = 0
Ci =A/H
c2 = 1
where A is the surface area at height H.
3.3 Tabular Storage Shapes
The shape of a storage unit can also be defined by a Storage Curve which is a series of user-
supplied data pairs Yi, A that represent the points on a curve of surface area versus depth above the
bottom of the unit. An example of this type of curve is shown in Figure 2. It can represent natural
depressions with irregular-shaped contour intervals, spheroid storage vessels or conventional
shapes with different base sizes stacked atop one another. The first point supplied to the curve
should be the surface area of the unit's base at a depth of 0. Otherwise, it will be assumed that the
unit has zero surface area at its base. The curve will be extrapolated outwards to meet the unit's
maximum depth if need be.
10
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Storage Curve
Depth (ft)
Section View
Figure 2. Example of a storage curve and its section view
To find the area associated with a given storage depth one interpolates between data points that
bracket the depth value on the storage curve. Determining the storage volume Fat a given depth
Yis equivalent to finding the area under the storage curve from depth 0 to Y This can be done by
using the Trapezoidal Rule (Atkinson, 1989) which results in:
v=l\L(y<- ym)(a>+U+\(y - r»xA+¦4») (8)
li
-------
where n is the largest data point index with Yn < Y and A is the surface area associated with depth
Y as found from the storage curve itself. The shaded rectangles in Figure 3 illustrate how the
trapezoidal rule is applied to a storage curve to find the stored volume at a particular depth. This
procedure is the same as the widely used Average-End-Area method except that the area at the
desired depth is first interpolated from the storage curve rather than converting the original area
curve to a volume curve and interpolating directly from it.
60000
50000
40000
30000 -
20000 -
3
LI
10000 -
Storage Curve
Y, A
""
'
J
2 3 4
Depth (ft)
Figure 3. Finding the volume at a given depth for a storage curve
The depth that corresponds to a particular volume for a storage curve can be found as follows.
Using the trapezoidal rule, sum the volumes contributed by each curve segment starting from 0
until the accumulated volume Vsum exceeds the target volume V Let the data point index at the
start of this segment be denoted by /. Then the depth Fthat results in volume Fis:
Y = Yi +
Af + 2a(V - Vsum) - At
where a = {Ai+1 - A{)/(Ji+1 - Yt).
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4. Pumps
SWMM treats pumps as links that have a pre-defined relationship between flow rate Q and head
Hor some suitable surrogate. This relationship is defined by a user-supplied Pump Curve. Table 2
depicts the five types of pump curves that SWMM recognizes. Although not a requirement, a
pump's inlet node would typically be a storage node that represents a pump station's wet well. An
exception would be an inline booster pump placed inside a force main line under dynamic wave
analysis. A sixth type of pump, called an Ideal pump, does not use a pump curve but instead has
its flow rate equal to the inflow rate into its inlet node. It must be the only outflow link from its
inlet node and is used mainly for conceptional design.
A single point on a Typel or Type2 curve would typically represent an operating point for a
constant flow positive displacement pump. Additional points might represent flow rates at different
pump speeds or contributions from additional constant speed pumps running in parallel. The Type3
curve represents the characteristic curve of a centrifugal pump operating at some fixed speed,
where there is a continuous range of flows available depending on the head required. The Type4
curve could be a positive displacement pump with continuous speed control or a centrifugal pump
that lifts water to a more or less fixed elevation so that the required head depends only on the water
level at its inlet node. A Type5 pump is a variable speed version of the Type3 pump. As the pump's
impeller speed varies relative to some nominal value, flow changes in direct proportion, while
head changes in proportion to the speed squared.
Whenever a pump link is encountered in either the dynamic wave or kinematic wave methods its
new flow is found directly from its pump curve using previously computed values were for nodal
heads and volumes.
For Typel and Type2 curves, the curve is searched in stepwise fashion for the first point whose
volume or depth exceeds the volume or depth at the pump's inlet node. The pump's flow is the
flow associated with that point. For the Type3 and Type5 curves, the flow is determined by first
finding the pair of adjacent data points that bracket the difference in head between the pump's
outlet and inlet nodes and then interpolating a flow between these points for the given head
difference. A similar lookup procedure is used for the Type4 curve except that water level at the
pump's inlet node is used instead of head difference. A pump's flow is not allowed to be outside
the minimum and maximum values defined by its pump curve and is not allowed to be negative.
The Type5 pump curve is shifted depending on what relative speed setting ct> the pump is currently
operating under, where a setting of 1.0 applies to the original user-supplied curve (i.e., the rated
impeller speed). Following the pump affinity laws (Sanks et al., 1998), a point with head Hand
flow Q on the original curve becomes a>2H and a>Q, respectively, on the speed-adjusted curve. For
13
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other pump types only the flow value found from the original curve is multiplied by the speed
setting.
Table 2. Pump curves recognized by SWMM
Tvpel (Fixed/Volume)
Consists of a series of constant flow rates that
apply over a series of volume intervals at the
pump's inlet node.
*
o
Ll_
| ~
Volume
Tvpe2 (Fixed/Depth)
Similar to a Typel pump except that the fixed
flow rate levels vary over a set of depth
intervals at the pump's inlet node.
£
O
Ll_
Depth
Tvoe3 (Variable/Head)
Uses a pump characteristic curve at some
nominal impeller speed to relate flow rate and
delivered head.
"O
CO
CD
X
Flow
Tvpe4 (Variable/Depth)
A variable speed pump where flow varies
continuously with inlet node water depth.
£
o
l_l_
Depth
Tvoe 5 ("Variable/Affinity)
A variable speed version of the Type3 pump
where the pump curve shifts position when
control rules change the pump's relative
speed setting.
"O
CO
OJ
X
Flow
14
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Speed settings can be changed during a simulation by using control rules. The setting can also be
used to control pump operation based on wet well level (e.g., set a to 1 when the level is above a
startup depth and to 0 when below a shutoff depth). The adjusted pump flow is checked to ensure
it does not cause the water level at the inlet node to drop below zero over the current time step. If
the node is a storage node then the pumping rate cannot exceed Qmax where
and Qin is the most recently computed total inflow to the node, Vn is the node volume at the start
of the time step, and zlfis the current time step interval. If the inlet node is not a storage node and
dynamic wave analysis is being made, SWMM's normal head updating formula (given in Equation
3-15a in the Hydraulics Reference Manual (Rossman 2017)) is used with the current pumping rate
to estimate the inlet node head at the end of the time step. If this head is below the node's invert
elevation then the pumping rate is set equal to the node's current inflow.
Some additional computational details regarding pumps are as follows:
1. If the inlet node of a Typel (flow versus volume) pump is not a storage node then it is
assigned a virtual wet well whose volume varies linearly with depth up to the highest
volume on the pump curve at full node depth. While the normal non-storage node methods
are used to update the node's water level, the virtual wet well volume corresponding to the
node's water level is used to determine the pumping rate. Equation 10 is also used to limit
the pump flow to the maximum flow that the node can release.
2. For dynamic wave modeling:
a. Pumps do not contribute any surface area to the node-link assemblies at their inlet
and outlet nodes.
b. For Type3, Type4, and Type5 pump curves the dQ/dH term used for evaluating a
surcharged node, if the surcharge algorithm option is used, is the negative slope of
the line segment on which the pumping rate lies. For the other pump types it is zero
because their line segments have zero slope.
c. No under-relaxation is applied to consecutive pump flows at Step 3 of the iterative
solution method described in Section 3.2 of the Hydraulics Reference Manual.
3. SWMM computes the power consumed in kilowatt-hours (Kwh) by each pump over each
time step zlfas:
Q
max
— Qin + VN/At
(10)
Kwh = 0.7457(H2 - (?(At/3600)/8.814
(11)
15
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where heads Hi and H2 are in feet, flow Q is in cfs, and time step At is in seconds. The
pump's wire to water efficiency is not included in this calculation. The power consumption
totaled up over the model simulation period and reported for each pump in SWMM's
Pumping Summary Report. Also reported are the percent of time each pump is online and
operates at either the lower or upper end of its pump curve.
5. Storm Drain Inlets
Storm drain inlets are structures that convey runoff from roadway pavements into below ground
storm sewers (see Figure 4). Inlet type, sizing and spacing are normally chosen to meet limits on
the spread and depth of water across a roadway as set by local drainage agencies to maintain public
safety. The U.S. Department of Transportation Federal FTighway Administration's Urban Drainage
Design Manual (HEC-22) (FHWA, 2009) contains experimentally derived equations for
computing the amount of flow captured by different types of inlets. These equations are widely
used throughout North America and have been incorporated into SWMM's flow routing routines.
Figure 4. Examples of storm drain inlets.
Figure 5 depicts the different types of street inlet structures whose hydraulic performance is
computed using the HEC-22 procedures. In addition to these standard inlet types, a custom inlet
can also be deployed. Its performance is defined by a user-supplied rating curve of captured flow
as a function of flow depth or a diversion curve of captured flow as a function of flow upstream of
the inlet (see Figure 6). These curves are defined by a tabular listing of their data points.
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Figure 5. Standard types of curb and gutter inlets
Figure 6. Performance curve for a custom inlet
5.1 Model Setup
To add storm drain inlet modeling into SWMM, a site is represented as a dual drainage system
consi sting of both street conduits along the ground surface and sewer conduits below ground, as
shown in Figure 7. An inlet structure will divert some portion of the street flow it recei ves into a
designated node of the sewer system with the rest being bypassed to downstream street conduits.
17
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When an inlet's sewer node reaches its full depth any excess flow that causes it to flood is routed
back into the street's downstream node rather than having it leave the system as it normally would.
O Street Junction Sewer Manhole Street Ponding
- > Street Flow > Captured Flow
Figure 7. Representation of a dual drainage system
The HEC-22 procedures assume that the curb and gutter inlets shown in Figure 5 are placed in
conduits that have a Street cross-section shape as described in Section 2 above. Even if a street
conduit does not contain inlets, it should still be assigned a Street cross-section if the spread and
depth of surface water across it needs to be reported. Street cross-sections can be either single-
sided (i.e., have a single section that slopes downward from the street crown) or two-sided (i.e.,
have mirror image sloping sections on either side of the street crown) as required.
Streets can be assigned any number of a specific inlet type (e.g., grate, curb opening, combination,
or slotted drain). If the cross-section is two-sided then each side receives the replicate number of
inlets. If a street needs to use a mix of inlet types then it must be divided into separate street
conduits where each utilizes just a single type of inlet.
As shown in Figure 7, inlets can be located either on a continuous sloping section of roadway (on-
grade, sometimes referred to as a flow-by condition) or at a low point where flow tends to pool
(on-sag, sometimes referred to as a sump condition). The ITEC-22 procedures make flow capture
for on-grade inlets a function of the approach flow rate, whereas flow capture for on-sag inlets
depends on the depth of water that pools above them.
18
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Because there is no physical link in the model, such as a shared manhole, weir, or orifice, that
connects the street and sewer system to one another, there is no need to have the rim elevation of
a sewer manhole match the invert elevation of the street node that sends inlet flow into it.
5.2 Computational Scheme
To account for the flow capture and diversion provided by inlets, at each flow routing time step
SWMM adjusts the lateral inflows seen at the downstream nodes of street conduits with inlets and
at the sewer nodes designated to receive inlet flow. These lateral flows contribute to the node flow
continuity equation for dynamic wave routing and to the link flow continuity equation for
kinematic wave routing. The steps involved can be summarized as follows:
1. Compute the flow captured at each street inlet using the HEC-22 procedures for standard
inlets or table lookup for custom inlets, using current values of street flow rates and flow
depths.
2. Add each inlet's captured flow to the lateral inflow that enters the inlet's assigned sewer
node and subtract that same flow from the lateral inflow seen by the downstream node of
the inlet's street conduit.
3. Add any current overflow (i.e., flooding) that an inlet's sewer node experiences onto the
lateral inflow for the inlet's street node instead of having it leave the system as it normally
would. This allows for a two-way flow exchange between the street and sewer once the
water level in the sewer node reaches the ground elevation.
4. Apply the usual flow routing step normally taken by SWMM's routing methods.
In Step 1, because each side of a two-sided street has the same cross-section geometry and number
of inlets, flow capture is computed for only one side using half the total street flow as the approach
flow seen by its inlets. The one-sided flow capture is then doubled to determine the full flow
capture for the entire street.
5.3 Flow Capture for On-Grade Inlets
The flow capture efficiency of an inlet placed on-grade is affected by:
• inlet type and dimensions
• approach flow rate, velocity, and spread of water on the street
• street cross-slope and curb depression
• longitudinal street slope and surface roughness.
Grate Inlets
For a standard street grate located on-grade the HEC-22 equation for flow capture is:
19
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Qc=Q{R/E0+Rs (1-E0 )} (12)
where:
Qc = captured flow (cfs)
Q = approach flow (cfs)
Eo = ratio of flow over the grate's width to total flow
Rr = frontal capture efficiency
Rs = side capture efficiency
The frontal capture efficiency Rf is:
Rf = 1 — 0.09 max(0, V — V0) (13)
while the side capture efficiency is:
Rs = 1/{1 + 0.15 V18/(SxL23)} (14)
with:
V = velocity of flow over the grate (ft/sec)
Vo = velocity at which water begins to splash over the inlet (ft/sec)
Sx = street cross slope (ft/ft)
L = grate length (ft)
FHWA (2009) contains curves showing how the splash-over velocity Vo increases with increasing
grate length L for the common grate designs listed in Table 3. Table 4 contains polynomial
expressions that were fit to these curves by UDFCD (2016) that are used by SWMM. For grates
that do not conform to one of the listed designs, the splash-over velocity must be supplied by the
user.
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Table 3. Description of grate inlet types1
Grate Type
Layout
Description
P-50
Parallel bar grate with 17/x" bar spacing on center
P-50xl00
m
Parallel bar grate with 17/x" bar spacing on center and 3/s"
diameter lateral rods spaced at 4" on center
P-30
i
Parallel bar grate with lVs" bar spacing on center
Curved Vane
m
ii
Sid
e
Curved vane grate with 3 longitudinal bar and 41/i"
transverse bar spacing on center
45° Tilt Bar
ii
Sid
e
45° tilt-bar grate with 3Vi" longitudinal bar and 4"
transverse bar spacing on center
30° Tilt Bar
m
II
Side
a
30° tilt-bar grate with 3Vi" longitudinal bar and 4"
transverse bar spacing on center
Reticuline
mi*
Honeycomb pattern of lateral bars and longitudinal bearing
bars
^ee FHWA (2009) for more detailed descriptions and pictures.
Table 4. Splash-over velocity for different types of grate inlets1
Grate Type
Splash over velocity Vo(ft/s) as a function of grate length L (ft)
P-50
V0 = 2.22 + 4.03L - 0.65L2 + 0.06L3
P-50xl00
V0 = 0.74 + 2.44L - 0.27L2 + 0.02L3
P-30
V0 = 1.76 + 3.12L - 0.45L2 + 0.03L3
Curved Vane
V0 = 0.30 + 4.85L - 1.31L2 + 0.15L3
45° Tilt Bar
V0 = 0.99 + 2.64L - 0.36L2 + 0.03L3
30° Tilt Bar
V0 = 0.51 + 2.34L - 0.20L2 + 0.01L3
Reticuline
V0 = 0.28 + 2.28L - 0.18L2 + 0.01 L3
1 Source: UDFCD (2016).
21
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Curb Opening Inlets
For a curb opening inlet located on-grade, the HEC-22 equation for flow capture is:
Qc = Q{l-(l-mm(l,L/LT))ls}
(15)
L is now the length of the curb opening and Ltis the length at which complete flow capture occurs.
The latter quantity is computed as:
If L > Ltthen complete capture is obtained.
Computing Eo
The on-grade flow capture formulas for grate and curb opening inlets requires specification of E0,
the fraction of total street flow Q within a distance IV from the curb or as depicted in Figure 8, the
ratio of Qwto Q. For grates this distance is the width of the grate. For curb openings it is the width
of the depressed gutter (if present).
Lt = 0.6Q0A2Sj>-3(nSe)-°-6
0.42 c0.3
(16)
where:
Sl = longitudinal street slope (ft/ft)
n = Manning's roughness coefficient for the street surface
Se = $x + (a/W)E0
a = curb depression (ft)
W = depressed gutter width (ft)
Eo = ratio of flow over depressed gutter width to total flow
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HEC-22 bases its determination of Eo on Izzard's form of the Manning equation that relates flow
spread T to flow rate Q for a triangular cross-section. It is derived from the standard Manning
equation by integrating the hydraulic radius across successive increments of street width. The
result for US standard units is:
Q = (0.S6/ri)Sx 67Si 5T2-67 (17)
(Note: the standard Manning equation has the same form except with the constant being 0.47.)
Solving for Tas a function of O gives:
T=
Qn
O.S6S^67S°s
0.375
(18)
If the street has a uniform cross slope (a = 0 in Figure 8) then Equation 18 can be used to derive
the following expression ioxEo'.
E0 = 1 - (1 - W/T)2-67 (19)
where Tis evaluated at a given flow rate Q.
For a compound street cross-section with depressed curb (a > 0 in Figure 8), HEC-22 provides
the following equation for Ec:
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1 + ¦
Sw/$x
1 +
Sw/$x
(T/W - 1),
2.67
(20)
where Sw = + a/W. It is not possible to solve directly for Eo because Equation 18 for T(Q)
applies only to a triangular section of uniform slope.
To solve Equation 20 let Qx and Tx denote the flow and spread, respectively, across the non-
depressed triangular portion of the street's cross-section. Then the following relations apply:
QX = Q(1-E0)
Tx = T — W
T/W - 1 = Tx/W
(21)
(22)
(23)
These can be used in the following iterative procedure to find Eo for a particular flow rate Q.
1. Assume a value for Tx.
2. Use Equation 20 to compute Eo, with Tx/W substituted for T/W- 1.
3. Compute Qx from Equation 21.
4. Compute a new value for 7> using Equation 18 with Qx as the flow rate.
5. If there is negligible change in Tx then stop with the last value of Eo as the solution.
Otherwise return to Step 2.
In cases where the width of the grate is smaller than the width Wof the depressed gutter section,
Eo is adjusted by the ratio of the flow area over the grate's width to the flow area over the depressed
gutter width.
Combination Inlets
A combination inlet consists of both a grate and curb opening placed together at the same location.
Its on-grade flow capture equals that of the grate plus any flow captured by the portion of the curb
opening that extends upstream of the grate's length. The latter flow capture is computed first and
is subtracted from the approach flow used to determine the grate's flow capture.
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Slotted Inlets
The flow capture capability of a slotted inlet located on-grade is the same as that of a curb opening
inlet of equal length.
Custom Inlets
If an on-grade custom inlet is supplied with a flow diversion curve (captured flow versus approach
flow) then that curve will be used to determine its flow capture. If it uses a flow rating curve
(captured flow versus water depth) then that curve will be used in conjunction with the water
surface depth at the downstream end of the conduit containing the inlet.
5.4 Flow Capture for On-Sag Inlets
HEC-22 has flow capture efficiency for an inlet in a sag location depend on the size of the inlet's
opening and the depth of water that collects ponds on top of it at the street curb. At low flow depths
the inlet acts as a weir with
In these equations:
Cw = weir coefficient (ft0 5/sec)
Co = orifice coefficient
g = acceleration due to gravity (ft/sec2)
Lw = effective length of inlet (ft)
Ao = open area of inlet (ft2)
d = effective depth of water at the inlet (ft).
Grate Inlets
For grate inlets the following values are used in Equations 24 and 25:
(24)
while at higher depths it acts as an orifice with
Qc — CoAoyj2gd
(25)
Cw = 3.0
Co = 0.67
Lw = L+2W
Ao = LW f0
25
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d = dt - (W/2)Sw
where L is the grate's length, Wits width, fo the ratio of open area to full area, and di\% the depth
of water at the downstream node of the conduit containing the inlet. Opening area ratios for several
types of grate designs are listed in Table 5.
HEC-22 does not provide clear guidance on what depth causes a switch from weir flow to orifice
flow for grates. Therefore, it is assumed the switch occurs at a depth dwhere Equation 24 equals
Equation 25. This results in weir flow for depths below 1.79 A0/Lw and orifice flow for depths
above it.
Table 5. Open area relative to full area for grate inlets1
Grate Type
Open Area Ratio
P-50
0.90
P-50xl00
0.80
P-30
0.60
Curved Vane
0.35
45° Tilt Bar
0.17 (assumed)
30° Tilt Bar
0.34
Reticuline
0.80
Source: Chart 9B of FHWA (2009)
Curb Opening Inlets
For streets with uniform cross slope or for openings greater than 12 feet in length, the values used
in Equation 24 for weir flow are Cw= 3.0 and Lw= opening length. Otherwise, Cw= 2.3 and Lw =
L + 1.8W where L = opening length and W= width of the depressed gutter. The values used in
Equation 25 for orifice flow are C0 = 0.67 and A0 = hL where h is the height of the opening. The
effective depth d occuring at the curb opening inlet under orifice flow depends on the orientation
of the opening's throat relative to the street surface as shown in Table 6.
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Table 6. Effective depth for curb opening inlets under orifice flow
Throat Angle
Effective Depth
Horizontal
......
d = di — h/2
, i ,
Inclined
1
1.
d = dt — 0.7071(/i/2)
Vertical
' J 7
,r ^
1 ¦r'"*"
d = di
HEC-22 states that weir flow for on-sag curb openings occurs at effective depths below h while
orifice flow occurs at depths greater than 1.4h. For depths in between these SWMM uses the
following interpolation formula:
Qc ~ (1 — rfQweir TQorif (26)
where Qweir is weir flow capture at depth /?, Qorif is orifice flow capture at depth 1.4h and r =
0d-K)/(0Ah).
Slotted Inlets
For slotted inlets the variables in the weir and orifice Equations 24 and 25 are as follows:
Cw = 2.48
Co = 0.8
Lw = L
Ao = LW
di — d
where L = inlet length and W= inlet width. Weir flow holds for d < 0.2 feet while orifice flow
occurs for d > OA feet. In between these values flow capture is computed using Equation 26
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with Qweir as weir flow capture at depth 0.2, Qorit as orifice flow capture at depth 0.4 and r =
(d- 0.2)/0.2.
Custom Inlets
If an on-sag custom inlet is supplied with a flow rating curve (captured flow versus water depth),
then that curve is used to determine its flow capture. The depth supplied to the curve's lookup
table is the depth of the downstream node of the conduit containing the inlet. If the inlet was
assigned a diversion curve (captured flow versus approach flow) then that curve is used, thus
essentially treating the inlet as if it were on-grade.
5.5 Drop Inlets
Drop inlets, pictured in Figure 9, are used to drain water from roadside ditches, swales, and flat
bottom channels. SWMM allows these structures to be placed in open channels that have either an
open rectangular or trapezoidal cross-section. Model set-up for utilizing these inlets is the same as
described in Section 5.1 above and the same computational scheme applies. The methods used to
compute their flow capture efficiencies are described in the following paragraphs.
Drop Grate Inlet
Drop Curb Inlet
Figure 9. Types of channel drop inlets
Drop Grate Inlets
The flow capture equation for a drop grate inlet located on-grade is the same as for a street grate
(see Equation 12) except that the ratio Eo of flow over the grate to total cross-section flow Q is
given by:
E0 =
lA86jS^(yW)1(,:
nQ(W + 2y)0-67
(27)
where:
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side length of grate parallel to flow direction (ft)
flow depth in the channel (ft)
channel Manning's roughness coefficient
channel longitudinal slope (ft/ft)
A cross-slope 5>of 1% is assumed unless the grate extends across the entire bottom width of a
trapezoidal channel. In that case 5Hs taken as the slope of the channel's side wall.
Drop grates located on-sag use the same weir and orifice equations (24 and 25) as do street grates
with the only difference being that for weir flow the effective length of the inlet's sides (Lwin
Equation 24) is the sum of the lengths of all four sides.
Drop Curb Inlets
Flow capture for drop curb inlets is computed the same as for curb opening inlets located on sag.
The only difference is that the effective length of the opening is the total length of all four sides
and the open area is the height of the opening times the total length of all four sides.
W =
y =
n =
Sl =
6. References
Atkinson, K.E., An Introduction to Numerical Analysis (2nd ed.), John Wiley & Sons, New York
1989.
Cunge, J.A. and Wegner, M., "Numerical integration of Barre de Saint-Venant's flow equations
by means of an implicit scheme of finite differences", La Houille Blanche, Number 1, 1964.
Federal Highway Administration (FHWA), Urban Drainage Design Manual, Third Edition,
Hydraulic Engineering Circular No. 22, FHWA-NHI-10-009, U.S. Department of Transportation,
2009.
Rossman, L.A., Storm Water Management Model Reference Manual Volume II - Hydraulics,
EPA/600/R-17/111, National Risk Management Laboratory, U.S. Environmental Protection
Agency, Cincinnati, OH, 2017.
Sanks, R.L., Pumping Station Design, Second Edition, Butterworth, London, UK, 1998.
Sjoberg, A., "Sewer Network Models DAGVL-A and DAGVL-DIFF" in Urban Stormwater
Hydraulics and Hydrology, B.C. Yen, ed., Water Resources Publications, Highlands Ranch, CO,
1982.
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Urban Drainage and Flood Control District (UDFCD), "Streets, Inlets, and Storm Drains", Chapter
7 in Urban Storm Drainage Criteria Manual: Volume 7, Urban Drainage and Flood Control
District, Denver, CO, 2016.
Yen, B.C., "Hydraulics of Sewer Systems", Chapter 6 in Stormwater Collection Systems Design
Handbook, L.W. Mays editor, McGraw-Hill, NY, 2001.
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