INLET CONTROL EQUATIONS

A.1 INTRODUCTION

This appendix contains the inlet control equations used to develop the design charts of this publication (HDS-5). Section A.2 contains the equations for the unsubmerged and submerged inlet control equations. Section A.3 demonstrates how the Section A.2 equations are used to create dimensionless design curves for culvert shapes with coefficients (Section A.3.1) and without (Section A.3.2). Section A.4 discusses how the dimensionless design curves are used to develop the nomographs in Appendix C. Section A.5 discusses how the dimensionless design curves are used to develop the polynomial equations used in FHWA software.

A.2 INLET CONTROL EQUATIONS

The equations used to develop the inlet control nomographs in Appendix C 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). John L. French of the NBS produced seven progress reports as a result of this research. Of these, the first (NBS 1955) and fourth (NBS 1961) through seventh reports (NBS 1966a, 1966b, and 1967) dealt with the hydraulics of pipe and box culvert entrances, with and without tapered inlets. Herbert G. Bossy of the FHWA provides an excellent synthesis of the research in his paper, “Hydraulic of conventional Highway Culverts” (Bossy 1961). Additional background on the development of the equations is found in HEC-13 (FHWA 1972a) and unpublished notebooks and notes (Bossy 1963 and Normann 1974).

The two basic conditions of 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 by the headwater, the inlet performs as a weir and the unsubmerged equations are used (Section A.2.1). If the inlet is submerged by the headwater, the inlet performs as an orifice and the submerged equations are used (Section A.2.2).

Between the unsubmerged and the submerged conditions, there is a transition zone for which the NBS research provided only limited information. The transition zone is defined empirically by drawing a curve between and tangent to the curves defined by the unsubmerged and submerged equations. In most cases, the transition zone is short and the curve is easily constructed.

A.2.1 Unsubmerged Inlet Control Equations

The unsubmerged equation has two forms. 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 equations. Equations (A.1) and (A.2) apply up to about Q/(A*D^0.5) = 3.5 (1.93 SI).

A.2.2 Submerged Inlet Control Equations

The submerged equation (A.3) applies above about Q/(A*D^0.5) = 4.0 (2.21 SI). The terms are defined in Section A.2.1.

A.3 INLET CONTROL DIMENSIONLESS DESIGN CURVES

The equations in Section A.2 may be used to develop design curves for any conduit shape or size. Careful examination of the equation constants for a given form of equation reveals that there is very little difference between the constants for a given inlet configuration. Therefore, given the necessary conduit geometry for a new shape from the manufacturer, a similar shape is chosen and the constants are used to develop new design curves. The curves may be quasi- dimensionless, in terms of Q/(A*D^0.5) and HW_i/D, or dimensional, in terms of Q and HW_i for a particular conduit size. To make the curves truly dimensionless, Q/(A*D^0.5) must be divided by g^0.5, but this produces small decimal numbers. Note that coefficients for rectangular (box) shapes should not be used for nonrectangular (circular, arch, pipe-arch, etc.) shapes and vice- versa. A constant slope value of 2 percent (0.02) is usually selected for the development of design curves. This is because the slope effect is small and the resultant headwater is conservatively high for sites with slopes exceeding 2 percent (except for mitered inlets). The procedure is illustrated in Section A.3.1.

A.3.1 Elliptical Structural Plate Example

Develop a dimensionless design curve for elliptical structural plate corrugated metal culverts, with the long axis horizontal. Assume a thin wall projecting inlet. Use the coefficients and exponents for a corrugated metal pipe-arch, a shape similar to an ellipse.

From Table A.1, Chart 34, Scale 3:

• Unsubmerged: equation Form (1) with K = 0.0340 M = 1.5
• Submerged: c = 0.0496 and Y = 0.53

Unsubmerged, equation Form (1) (Equation A.1):

HW_i/D = H_c/D + 0.0340*(Q/(A*D^0.5))^1.5 – 0.5*0.02

Submerged (Equation A.3):

HW_i/D = 0.0496*(Q/(A*D^0.5))² + 0.53 – 0.5*0.02

A direct relationship between HW_i/D and Q/(A*D^0.5) may be obtained for the submerged condition. For the unsubmerged condition, it is necessary to obtain the flow rate and equivalent specific head at critical depth. At critical depth, the critical velocity head is equal to one-half the hydraulic depth.

V_c²/(2*g) = y_h/2 = A_p/2*T_p

Therefore, specific head at critical depth divided by D is:

Since the Froude number equals 1.0 at critical depth, V_c can be determuned from the Froude number equation and set equal to V_c in the continuity equation to solve for Q_c.

F_r = V_c/(g*y_h)^0.5 = 1, and V_c = Q_c/A_p = (g*y_h)^0.5, from which Q_c = A_p*(g*y_h)^0.5, or

From geometric data supplied by the manufacturer for a horizontal ellipse (Kaiser 1984), the necessary geometry is obtained to calculate H_c/D and Q_c/(A*D^0.5).

From unsubmerged equation and the above table:

For the submerged equation, any value of Q/(A*D^0.5) may be selected, since critical depth is not involved:

Note that overlapping values of HW_i/D were calculated in order to define the transition zone between the unsubmerged and the submerged states of flow. The results of the above calculations are plotted in Figure A.1. A transition line is drawn between the unsubmerged and the submerged curves. The scales are dimensionless in Figure A.1, but the figures could be used to develop dimensional curves for any selected size of elliptical conduit by multiplying: Q/(A*D^0.5) by A*D^0.5 and HW_i/D by D.

x

Figure A.1. Dimensionless performance curve for structural plate elliptical conduit, long axis horizontal, thin wall projecting entrance.

A.3.2 Dimensionless Design Charts for Culverts without Coefficientc

The dimensionless inlet control design charts provided for long span arches (Chart 52) and for circular and elliptical pipes (Chart 51) were derived using the inlet control equations in Section A.2, selected constants from Table A.1, conduit geometry obtained from various tables and manufacturer’s information (FHWA 1972b, Kaiser 1984, AISI 1983).

Some inlet configurations have no hydraulic tests. In lieu of such tests, the selected edge conditions should approximate the untested configurations and lead to a good estimate of culvert performance. In some cases, it will be necessary to evaluate the inlet edge configuration at a specific flow depth. For example, some inlets may behave as mitered inlets at low headwaters and as thin wall projecting inlets at high headwaters. The designer must apply engineering judgment in selection of the proper relationships for these major structures.

Unsubmerged Conditions. Equation (A.1) was used to calculate HW_i/D for selected inlet edge configurations. The following constants were taken from Table A.1, Chart 34 for pipe- arches, except for the 45 degree beveled edge inlet. These constants were taken from Chart 3, Scale A, for circular pipe. No constants were available from tests on pipe-arch models with beveled edges.

Geometric relationships for the circular and elliptical (long axis horizontal) conduits were obtained from Tables 4 and 7 (FHWA 1972b), respectively. Geometric relationships for the high and low profile long span arches were obtained from DP-131 (Kaiser 1984) and the results were checked against tables in AISI handbook (AISI 1983).

Submerged Conditions. Equation (A.3) was used to calculate HW_i/D for the same inlet configurations using the following constants:

In terms of Q/(A*D^0.5), all non-rectangular shapes have practically the same dimensionless curves for submerged, inlet control flow. This is not true if Q/(B*D^1.5) is used as the dimensionless flow parameter.

To convert Q/(B*D^1.5) to Q/(A*D^0.5), divide by A/(B*D) for the particular shape of interest as shown in Equation (A.6). This assumes that the shape is geometrically similar, so that A/(B*D) is nearly constant for a range of sizes.

Dimensionless Curves. By plotting the results of the unsubmerged and submerged calculations and connecting the resultant curves with transition lines, the dimensionless design curves shown in Charts 51 and 52 of Appendix C were developed. All high and low profile arches can be represented by a single curve for each inlet edge configuration. A similar set of curves was developed for circular and elliptical shapes. It is recommended that the high and low profile arch curves in Chart 52 be used for all true arch shapes (those with a flat bottom) and that the curves in Chart 51 be used for curved shapes including circles, ellipses, pipe-arches, and pear shapes.

A.3.3 Dimensionless Critical Depth Charts

Some of the long span culverts and special culvert shapes have no critical depth charts. These special shapes are available in numerous sizes, making it impractical to produce individual critical depth curves for each culvert size and shape. Therefore, dimensionless critical depth curves were developed for the shapes which have adequate geometric relationships in the manufacturer’s literature. It should be noted that these special shapes are not truly geometrically similar, and any generalized set of geometric relationships will involve some degree of error. The amount of error is unknown since the geometric relationships were developed by the manufacturers.

The manufacturers’ literature contains geometric relationships which include the hydraulic depth divided by the rise (inside height) of the conduit (y_h/D) and area of the flow prism divided by the barrel area (A_p/A) for various partial depth ratios, y/D. From Equation (A.5):

Setting y/D equal to d_c/D, it is possible to determine A_p/A and y_h/D at a given relative depth and then to calculate Q_c/(A*D^0.5). Dimensionless plots of d_c/D versus Q_c/(A*D^0.5) have been developed for the following culvert materials and shapes:

• Chart 20, corrugated metal box culverts (see Second edition HDS 5)
• Chart 44, corrugated metal arches (see Second edition HDS 5)
• Chart 53, Structural plate corrugated metal ellipses, long axis horizontal
• Chart 54, Structural plate corrugated metal arches, low and high profile

A.4 INLET CONTROL NOMOGRAPHS

The nomographs in Appendix C were developed using the equations in Section A.2 and the constants shown in Table A.1. The unsubmerged and submerged equations for a given shape, material and edge configuration were plotted using the dimensionless design curve procedures shown in Section A.3.1. A constant slope value of 2 percent (0.02) was used for the development of these design curves. This is because the slope effect is small and the resultant headwater is conservatively high for sites with slopes exceeding 2 percent (except for mitered inlets). A smooth transition was drawn by hand. This curve was the data used for constructing a nomograph. Dr. F. T. Mavis describes the process of making nomographs in “The Construction of Nomographic Charts” (Mavis 1939). Nomographs were used extensively in engineering prior to the introduction of microcomputers in the early 1980s.

In formulating inlet and outlet control design nomographs, a certain degree of error is introduced into the design process. This error is due to the fact that the nomograph construction involves graphical fitting techniques resulting in scales which do not exactly match the equations. Checks by the authors of the first edition and others indicate that all of the nomographs from HEC-5 have precisions of + 10 percent of the equation values in terms of headwater (inlet control) or head loss (outlet control). The nomographs for tapered inlets have errors of less than 5%, again in terms of headwater.

A.5 INLET CONTROL POLYNOMIAL EQUATIONS

The polynomial equations were developed to be used in software. The equations in Section A.2 with the constants shown in the tables of constants for a given shape, material and edge configuration were plotted using the dimensionless design curve procedures shown in Section A.3.1. The coordinates of selected points can be read from the curve and a best fit statistical analysis performed. A polynomial equation of the following form has been found to provide an adequate fit.

HW_i/D = A + B*[Q/(B*D^1.5)] + C*[Q/(B*D^1.5)]² + … + X*[Q/(B*D^1.5)]ⁿ + K_s*S

For fitting the polynomial equations, K_s = 0 was used for most equations so that the slope correction could be applied by the software. The flow factor can be based on A*D^0.5 rather than B*D^1.5. The constants for the best fit equations are found in the HY-8 User Manual provided with HY-8. For equations that included K_s*S, the A term is adjusted so that K_u*S = 0. HY-8 uses the polynomial equations for all shapes that have constants determined in the laboratory or by FHWA. These include:

• Table A.1 – circles, boxes and tapered inlets (NBS, Bossy 1961)
• Table A.2 – pipe-arches, ellipses, metal boxes and arches (Bossy 1961)
• Table A.3 – South Dakota DOT RCB (FHWA 2006c)
• Table A.4 – open bottom concrete boxes (Chase 1999)
• Table A.5 – embedded circular shapes (NCHRP 2011)
• Table A.6 – embedded elliptical shapes (NCHRP 2011)

For shapes without constants, HY-8 uses Chart 52 developed using the procedures of Section A.3.2.

Note From HDS-5 Section 3.1.3 Inlet Control

The original equations for computer software were generally 5th order polynomial curve-fitted equations that were developed to be as accurate as the nomograph solution (plus or minus 10%) within the headwater range of 0.5*D to 3.0*D. These equations are still being used in HY-8, but have been supplemented with a weir equation from 0.0*D to 0.5*D and an orifice equation above 3.0*D.

Sketches are shown in the HY-8 documentation and research report. Since sketches 2 and 8 show fillets, a 6-inch fillet is assumed.

Sketches 1 through 5 have the first configuration. Sketches 7 through 13 have the second.

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