The design equations used to develop the inlet control nomographs are based on the research conducted by the National Bureau of Standards (NBS) under the sponsorship of the Bureau of Public Roads (now the Federal Highway Administration). Seven progress reports were produced as a result of this research. Of these, the first and fourth through seventh reports dealt with the hydraulics of pipe and box culvert entrances, with and without tapered inlets (4,7 to 10) These reports were one source of the equation coefficients and exponents, along with other references and unpublished FHWA notes on the development of the nomographs. (56,57)
The two basic conditions of the inlet control depend upon whether the inlet end of the culvert is or is not submerged by the upstream headwater. If the inlet is not submerged, the inlet performs as a weir. If the inlet is submerged, the inlet performs as an orifice. Equations are available for each of the above conditions.
Between the unsubmerged and the submerged conditions there is a transition zone for which the NBS research provided by drawing a curve between and tangent to the curves defined by the unsubmerged and submerged equations. In most cases, the transition zone is the short and the curve is easily constructed.
Below are the unsubmerged and submerged inlet control design equations. Note that there are two forms of the unsubmerged equation. Form (1) is based on the specific head at critical depth, adjusted with two correction factors, Form (2) is an exponential equation similar to a weir equation. Form (1) is preferable from a theoretical standpoint, but form (2) is easier to apply and is the only documented form of equation for some of the inlet control nomographs. Either form of unsubmerged inlet control equation will produce adequate results.
Definitions:
HWi = Headwater depth above inlet control section invert, metres or feet
D = Interior height of culvert barrel, metres or feet
Hc = Specific height of culvert barrel, metres or feet
Q = Discharge, m3/s or ft3/s
A = Full cross sectional area of culvert barrel, metres2 or feet2
S = Culvert barrel slope m/m or ft/ft
K, M, c, Y = Constants from the table

Inlet Type  Description  Index Value 

Circular Concrete  Square edge with headwall  1 
Groove end with headwall  2  
Groove end projecting  3  
Circular Corrugated Metal Pipe  Headwall  4 
Mitered to slope  5  
Projecting  6  
Circular Pipe, Beveled Ring Entrance  45 deg. bevels  7 
33.7 deg. bevels  8  
Rectangular Box; Flared Wingwalls  3075 deg. wingwall flares  9 
90 or 15 deg. wingwall flares  10  
0 deg. wingwall flares (straight sides)  11  
Rectangular Box; Flared Wingwalls and Top Edge Bevel  45 deg flare; 0.43D top edge bevel  12 
1833.7 deg. flare; 0.083D top edge bevel  13  
Rectangular Box, 90deg Headwall, Chamfered / Beveled Inlet Edges  Chamfered 3/4in.  14 
Beveled 1/2in/ft at 45 deg. (1:1)  15  
Beveled 1in/ft at 33.7 deg. (1:1.5)  16  
Rectangular Box, Skewed Headwall, Chamfered / Beveled Inlet Edges  3/4" chamfered edge, 45 deg. skewed headwall  17 
3/4" chamfered edge, 30 deg. skewed headwall  18  
3/4" chamfered edge, 15 deg. skewed headwall  19  
45 deg. beveled edge, 1045 deg. skewed headwall  20  
Rectangular Box, Nonoffset Flared Wingwalls, 3/4" Chamfer at Top of Inlet  45 deg. (1:1) wingwall flare  21 
18.4 deg. (3:1) wingwall flare  22  
18.4 deg. (3:1) wingwall flare, 30 deg. inlet skew  23  
Rectangular Box, Offset Flared Wingwalls, Beveled Edge at Inlet Top  45 deg. (1:1) flare, 0.042D top edge bevel  24 
33.7 deg. (1.5:1) flare, 0.083D top edge bevel  25  
18.4 deg. (3:1) flare, 0.083D top edge bevel  26  
Corrugated Metal Box  90 deg. headwall  27 
Thick wall projecting  28  
Thin wall projecting  29  
Horizontal Ellipse Concrete  Square edge with headwall  30 
Grooved end with headwall  31  
Grooved end projecting  32  
Vertical Ellipse Concrete  Square edge with headwall  33 
Grooved end with headwall  34  
Grooved end projecting  35  
Pipe Arch, 18" Corner Radius, Corrugated Metal  90 deg. headwall  36 
Mitered to slope  37  
Projecting (FHWA 1974)  38  
Pipe Arch, 18" Corner Radius, Corrugated Metal  Projecting (Bossy 1963)  39 
No bevels  40  
33.7 deg. bevels  41  
Pipe Arch, 31" Corner Radius,Corrugated Metal  Projecting  42 
No bevels  43  
33.7 deg. bevels  44  
Arch, Corrugated Metal  90 deg. headwall  45 
Mitered to slope  46  
Thin wall projecting  47  
Circular Culvert  Smooth tapered inlet throat  48 
Rough tapered inlet throat  49  
Elliptical Inlet Face  Tapered inlet, beveled edges  50 
Tapered inlet, square edges  51  
Tapered inlet, thin edge projecting  52  
Rectangular  Tapered inlet throat  53 
Rectangular Concrete  Side tapered, less favorable edges  54 
Side tapered, more favorable edges  55  
Slope tapered, less favorable edges  56  
Slope tapered, more favorable edges  57 