7.4 Inlet Control Computations

In the past, inlet control analysis relied on simple nomographs or a single-value orifice or weir coefficient applied to the general orifice equation (Equation 6.6) or to the general weir equation (Equation 6.7). However, laboratory studies with field verification (FHWA, 1985) have resulted in significantly improved equations for inlet control conditions. These studies found that two forms of an equation for weir flow were applicable for various inlet configurations under unsubmerged inlet control conditions and that an equation for orifice flow was suitable for submerged inlet conditions. For the unsubmerged inlet condition, Form 1 is the most complete solution. However, the coefficients for many culvert shapes have only been developed for Form 2, which is also computationally easier for solving by hand. The equations (U.S. standard units only) are as follows:

Unsubmerged Inlet - Weir Flow

Form 1: , (7.1)

Form 2: , (7.2)

Submerged Inlet - Orifice Flow

,

where	HW	=	headwater depth above the upstream culvert invert (ft)
	D	=	full interior height of the culvert (ft)
	Hc	=	specific energy head at the critical depth (ft)
	Q	=	discharge (ft3/s)
	A	=	full cross-sectional area of the culvert barrel (ft2)
	S	=	culvert slope (ft/ft)
	Y, M, c, K	=	coefficients describing the culvert inlet conditions

(7.3)

For mitered inlets, substitute +0.7S for -0.5S in Equation 7.1 and Equation 7.3.

A transition stage occurs in the range 3.5 < Q/AD^0.5 < 4.0. There is a wide variety of inlet geometries, with the various combinations grouped and labeled with chart and scale numbers (FHWA, 2001). The chart number refers to the shape and material of the culvert and the scale number refers to the type of inlet edge. For each chart number, two to four scale numbers are possible. Table 7.2 shows the coefficients associated with various chart and scale numbers, as well as the applicable form of the equation.

Table 7.2 Chart and scale numbers with coefficients for inlet control design equations.

Chart Number	Shape & Material	Nomograph Scale	Inlet Edge Description	Equation Form	Unsubmerged		Submerged
					K	M	c	Y
1	Circular Concrete	1	Square edge with headwall	1	0.0098	2.0	0.0398	0.67
		2	Groove end with headwall		0.0018	2.0	0.0292	74
		3	Groove end projecting		0.0045	2.0	0.0317	0.69
2	Circular CMP	1	Headwall	1	0.0078	2.0	0.0379	0.69
		2	Mitered to slope		0.0210	1.33	0.0463	0.75
		3	Projecting		0.0340	1.50	0.0553	0.54
3	Circular	1	Beveled ring, 45º bevels	1	0.0018	2.50	0.0300	0.74
		2	Beveled ring, 33.7º bevels		0.0018	2.50	0.0243	0.83
8	Rectangular Box	1	30º to 75º wingwall flares	1	0.026	1.0	0.0347	0.81
		2	90º and 15º wingwall flares		0.061	0.75	0.0400	0.80
		3	0º wingwall flares		0.061	0.75	0.0423	0.82
9	Rectangular Box	1	45º wingwall flare, d = 0.083D	2	0.510	0.667	0.0309	0.80
		2	18º to 33.7º wingwall flare, d = 0.083D		0.486	0.667	0.0249	0.83
10	Rectangular Box	1	90º headwall with 3/4" chamfers	2	0.515	0.667	0.0375	0.79
		2	90º headwall with 45º bevels		0.495	0.667	0.0314	0.82
		3	90º headwall with 33.7º bevels		0.486	0.667	0.0252	0.865
11	Rectangular Box	1	3/4" chamfers, 45º skewed headwall	2	0.545	0.667	0.0505	0.73
		2	3/4" chamfers, 30º skewed headwall		0.533	0.667	0.0425	0.705
		3	3/4" chamfers, 15º skewed headwall		0.522	0.667	0.0402	0.68
		4	45º bevels, 10º-45º skewed headwall		0.498	0.667	0.0327	0.75
12	Rectangular Box, 3/4" Chamfers	1	45º nonoffset wingwall flares	2	0.497	0.667	0.0339	0.803
		2	18.4º nonoffset wingwall flares		0.493	0.667	0.0361	0.806
		3	18.4º nonoffset wingwall flares, 30º skewed barrel		0.495	0.667	0.0386	0.71
13	Rectangular Box, Top Bevels	1	45º offset wingwall flares	2	0.497	0.667	0.0302	0.835
		2	33.7º offset wingwall flares		0.495	0.667	0.0252	0.881
		3	18.4º offset wingwall flares		0.493	0.667	0.0227	0.887
16-19	C M Boxes	2	90º headwall	1	0.0083	2.0	0.0379	0.69
		3	Thick wall projecting		0.0145	1.75	0.0419	0.64
		5	Thick wall projecting		0.0340	1.5	0.0496	0.57
29	Horizontal Ellipse, Concrete	1	Square edge with headwall	1	0.0100	2.0	0.0398	0.67
		2	Groove end with headwall		0.0018	2.5	0.0292	0.74
		3	Groove end projecting		0.0045	2.0	0.0317	0.69
30	Vertical Ellipse, Concrete	1	Square edge with headwall	1	0.0100	2.0	0.0398	0.67
		2	Groove end with headwall		0.0018	2.5	0.0292	0.74
		3	Groove end projecting		0.0095	2.0	0.0317	0.69
34	Pipe Arch, 18" Corner Radius C M	1	90º headwall	1	0.0083	2.0	0.0379	0.69
		2	Mitered to slope		0.0300	1.0	0.0463	0.75
		3	Projecting		0.0340	1.5	0.0496	0.57
35	Pipe Arch, 18" Corner Radius C M	1	Projecting	1	0.0300	1.5	0.0496	0.57
		2	No bevels		0.0088	2.0	0.0368	0.68
		3	33.7º bevels		0.0030	2.0	0.0269	0.77
36	Pipe Arch, 31" Corner Radius C M	1	Projecting	1	0.0300	1.5	0.0496	0.57
		2	No bevels		0.0088	2.0	0.0368	0.68
		3	33.7º bevels		0.0030	2.0	0.0269	0.77
41-43	Arch C M	1	90º headwall	1	0.0083	2.0	0.0379	0.69
		2	Mitered to slope		0.0300	1.0	0.0463	0.75
		3	Thin wall projecting		0.0340	1.5	0.0496	0.57
55	Circular	1	Smooth tapered inlet throat	2	0.534	0.555	0.0196	0.90
		2	Rough tapered inlet throat		0.519	0.64	0.0210	0.90
56	Elliptical Inlet Face	1	Tapered inlet, beveled edges	2	0.536	0.622	0.0368	0.83
		2	Tapered inlet, square edges		0.5035	0.719	0.0478	0.80
		3	Tapered inlet, thin edge projecting		0.547	0.80	0.0598	0.75
57	Rectangular	1	Tapered inlet throat	2	0.475	0.667	0.0179	0.97
58	Rectangular Concrete	1	Side tapered, less-favorable edges	2	0.56	0.667	0.0446	0.85
		2	Side tapered, more-favorable edges		0.56	0.667	0.0378	0.87
59	Rectangular Concrete	1	Slope tapered, less-favorable edges	2	0.50	0.667	0.0446	0.65
		2	Slope tapered, more-favorable edges		0.50	0.667	0.0378	0.71

HEC-RAS automatically inserts these coefficients in the appropriate equation when the modeler specifies the chart and scale number. The headwater depth calculation is based on the discharge and the inlet geometry, with a slight correction factor for culvert slope. Tailwater conditions are not included or necessary for inlet control. The engineer should be aware that both the depth (D) and area (A) terms in the inlet control equations represent the full depth and area of the culvert, not the actual flow depth or area. The coefficients Y, M, c, and K have been found from laboratory studies on small-scale models of culverts.

Example 7.1 Analysis of a culvert under inlet control.

A 5-ft diameter concrete culvert is 100 ft long, with upstream and downstream invert elevations of 501 and 496 ft, respectively. The entrance is a beveled ring (33.7° bevels). If the outlet is under free-flow conditions, compute the headwater elevation for flows of 125 and 250 ft³/s.

Solution

Assume that n for the concrete culvert is 0.013. The culvert invert drops 5 ft over a distance of 100 ft, for a slope of 5%. Paved slopes approaching 0.5% are typically supercritical, so supercritical flow is expected in this culvert and inlet control will dominate. A value of Q/AD^0.5 must be computed to determine if the culvert is acting as a weir or an orifice. For the discharge of 125 ft³/s, the result is

Since this is less than 3.5, the upper limit for weir flow, the culvert is acting as a weir.

For the type of culvert (Chart Number 3) and entrance conditions (Scale Number 2) described, Table 7.2 is used to select the appropriate coefficients for this culvert. From Table 7.2, Form 1 is the appropriate weir flow equation, with K = 0.00018, M = 2.5, c = 0.0243 and Y = 0.83.

Form 1 of the weir equation for culverts requires that the specific energy at critical depth be included (H_c term). Critical depth and velocity may be determined by application of the Manning's equation for a circular shape, by applying nomographs for a circular shape, or from using a program such as Haestad Methods' FlowMaster to compute y_c and V_c. Using any of these methods, critical depth is found to be 3.2 ft and critical velocity is 9.42 ft/s for the discharge of 125 ft³/s. Specific energy (y + V²/2g) for critical depth is thus 4.6 ft. Applying Equation 7.1 gives

or

.

Solving for HW yields a headwater depth of 4.6 ft and a headwater elevation (HW + culvert invert elevation) of 505.6 ft.

For the discharge of 250 ft³/s, the Q/(AD)^0.5 term must be computed to determine weir flow or orifice flow. For the higher discharge the term is 5.7 > 4.0 (the lower limit for orifice flow). Therefore, orifice flow exists for the discharge of 250 ft³/s. Using the orifice equation for culverts (Equation 7.3) gives

or

.

Solving for HW gives a headwater depth of 7.80 ft and a headwater elevation of 508.80 ft.