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Hydraulic Loss Coefficients for Culverts (2012)

Chapter: Chapter 4 - Culvert Exit Loss

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Suggested Citation:"Chapter 4 - Culvert Exit Loss." National Academies of Sciences, Engineering, and Medicine. 2012. Hydraulic Loss Coefficients for Culverts. Washington, DC: The National Academies Press. doi: 10.17226/22673.
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Suggested Citation:"Chapter 4 - Culvert Exit Loss." National Academies of Sciences, Engineering, and Medicine. 2012. Hydraulic Loss Coefficients for Culverts. Washington, DC: The National Academies Press. doi: 10.17226/22673.
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Suggested Citation:"Chapter 4 - Culvert Exit Loss." National Academies of Sciences, Engineering, and Medicine. 2012. Hydraulic Loss Coefficients for Culverts. Washington, DC: The National Academies Press. doi: 10.17226/22673.
×
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Suggested Citation:"Chapter 4 - Culvert Exit Loss." National Academies of Sciences, Engineering, and Medicine. 2012. Hydraulic Loss Coefficients for Culverts. Washington, DC: The National Academies Press. doi: 10.17226/22673.
×
Page 28
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Suggested Citation:"Chapter 4 - Culvert Exit Loss." National Academies of Sciences, Engineering, and Medicine. 2012. Hydraulic Loss Coefficients for Culverts. Washington, DC: The National Academies Press. doi: 10.17226/22673.
×
Page 29
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Suggested Citation:"Chapter 4 - Culvert Exit Loss." National Academies of Sciences, Engineering, and Medicine. 2012. Hydraulic Loss Coefficients for Culverts. Washington, DC: The National Academies Press. doi: 10.17226/22673.
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25 4.1 Summary The methodology for estimating culvert exit loss as pre- sented in FHWA’s HDS-5 (Normann et al., 2001), which is based on conservation of energy principles, is reviewed and evaluated. An improved method that considers both conservation of energy and momentum principles is sug- gested, which is useful for culverts with channelized outlets for both submerged and unsubmerged outlet conditions. The expression utilized, although originally developed for sudden expansions in pressurized pipes, is shown through laboratory experiments to be valid for sudden-expansion (projecting) culvert outlets. A comparison of the experimen- tal data to the traditional exit loss methods and the improved method is made. The derivation of the improved method is reviewed and a design example provided. For short culverts where friction or other energy loss components are relatively small, a more accurate prediction of exit energy loss may impact culvert design. 4.2 Introduction FHWA’s HDS-5 (Normann et al., 2001) and the U.S. Army Corps of Engineers’ River Analysis System Hydraulic Refer- ence Manual (HEC-RAS) (Brunner, 2002) both recommend using the difference in culvert and downstream channel flow velocity heads to calculate culvert exit loss. The fol- lowing is a paraphrase of the discussion in HDS-5 regarding exit loss: Outlet control culvert flow capacity is calculated based on a con- servation of energy approach. The total energy differential, or driving head required to pass a given flow rate through a cul- vert barrel, is equivalent to the summation of the total energy loss, comprised of entrance loss, friction loss through the barrel, exit loss, and any other minor losses that may be applicable to the particular installation. Exit loss is expressed as the change in velocity head at the outlet of the culvert barrel. For a culvert with a sudden expansion, the exit loss is H k V V g o o p ch = −    2 2 2 4 1( )- where Ho is exit loss expressed in units of length; Vp and Vch are the pipe velocity and channel flow velocity downstream of the culvert exit, respectively; g is the gravitational acceleration con- stant; and the exit loss coefficient ko is typically equal to 1.0. This equation may overestimate exit loss and a coefficient of less than 1.0 may be appropriate, according to the Hydraulic Engineering Circular No. 14 (Thompson and Kilgore, 2006), referred to as HEC-14 which provides empirically determined coefficients for five uniquely proportioned outlet geometries. The downstream channel velocity is often assumed to be small and is neglected, in which case the exit loss is equal to the velocity head in the barrel per Equation 4-2: H k V g o o p =     2 2 4 2( )- Equations 4-1 and 4-2 likely overestimate exit loss because they do not account for a conversion of a por- tion of the kinetic energy in the pipe to potential energy in the channel, a phenomenon observable in laboratory tests and likely in the field as well. In many cases, a small hydraulic jump with three-dimensional velocity compo- nents develops at the surface near the culvert exit, suggest- ing that momentum principles may also be important in describing exit loss. Experimental exit loss data collected as part of this study indicate that with ko = 1.0, Equation 4-1 overestimates the actual exit loss by as much as 143% for the conditions tested. Neglecting the downstream channel velocity head in the exit loss calculation (Equation 4-2) and assuming ko = 1.0 results in a larger overestimation of the exit loss, up to 187%. The Borda-Carnot loss is an expression for head loss at sudden expansions in pressurized pipes, derived using both energy and momentum principles (Vennard, 1940). It is hypothesized that this expression may accurately characterize C h a p t e r 4 Culvert Exit Loss

26 head loss at culvert outlets with projecting, sudden expan- sions for pressurized and free-surface culvert flow and for submerged and unsubmerged outlet conditions. 4.3 Borda-Carnot Derivation The derivation of the Borda-Carnot energy loss equa- tion assumes steady, incompressible, turbulent flow with no change in elevation, no appreciable friction loss, and no external forces acting on the control volume besides hydro- static pressure forces (Vennard, 1940). A control volume sche- matic to accompany the derivation is provided in Figure 4-1, with upstream and downstream locations labeled 1 and 2, respectively; Location 1 is at the culvert outlet and Loca- tion 2 is located in the discharge channel a short distance downstream. The projecting culvert boundary condition is not included in this derivation. The derivation stems from one-dimensional momentum and one-dimensional energy equations. Momentum Assuming the flow area at Location 1 to be the cross-sectional flow area of the culvert exit (A1) and the hydrostatic pressure at Location 1 to be acting on the channel cross-sectional flow area (A2), the one-dimensional momentum equation, relative to the primary flow direction (x), can be expressed as F P A P A V V A V V Ax = − = ( )− ( )∑ 1 2 2 2 2 2 2 1 1 1 4 3ρ ρ ( )- where P is the hydrostatic pressure, A is the cross-sectional flow area, r is the fluid density, V is the mean flow velocity, and the subscripts 1 and 2 correspond to the upstream and downstream boundaries of the control volume, as illustrated in Figure 4-1. Substituting g/g (fluid specific weight/gravitational acceleration constant) for r and V2A2 for V1A1 (continuity) and rearranging terms yields P P V VV g 1 2 2 2 1 2 4 4 − = − γ ( )- Energy Along the centerline of the control volume between Loca- tions 1 and 2 in Figure 4-1, the applicable terms from the one-dimensional Bernoulli Energy Equation are presented in Equation 4-5: V g P V g P Ho 1 2 1 2 2 2 2 2 4 5+ = + + γ γ ( )- Isolating the (P1-P2)/g terms in Equation 4-5 and substitut- ing this expression into Equation 4-4 yields the following relationship: H V V g o = −( )1 2 2 2 4 6( )- Designating V1 as the pipe velocity (Vp) and V2 as the downstream channel flow velocity (Vch) and adding the loss coefficient ko yields the following equation, referred to here as the Borda-Carnot loss for sudden expansions: H k V V g o o p ch = −( )2 2 4( )-7 Note that the numerator in Equation 4-7 is the square of the velocity difference, whereas the numerator in Equation 4-1 is the difference in the squared velocities. Assuming ko = 1.0 in Equation 4-7 and replacing Vch with VpAp/Ach (continuity) results in the following alternate form of the Borda-Carnot loss expression, as presented by Streeter and Wylie (1979): H k V g k A A o o p o p ch = = −     2 2 2 1 4, ( )where -8 Equations 4-2 and 4-8 are identical with the exception of the way ko is defined. Equation 4-8 suggests that based on energy and momentum considerations, the exit loss is proportional to the pipe velocity head, with a ko that varies with flow condi- tions. Although this expression is not new, its application to culvert exit loss is new. Montes (1998) compared the experimentally determined open channel flow sudden-expansion exit loss results reported by Hinds (1928) and Mathaei and Lewin (1932) to values pre- dicted by Equation 4-3. Mathaei and Lewin (1932) reported a Borda-Carnot loss coefficient (ko) of approximately 1.0. This result is consistent with pressurized pipe applications. Hinds (1928) and Formica (1955) reported ko values closer to 0.8. The objective of this study was to experimentally determine the exit loss for a variety of culvert sizes and flow conditions, including pressurized culvert flow and free-surface culvert flow for submerged and unsubmerged outlet conditions. The test culvert sizes ranged from 12 to 60 in. in diameter. All culvert exit loss tests were conducted in a laboratory flume as Figure 4-1. Control volume for momentum and energy analysis.

27 shown in Figure 4-2. By varying the culvert size, the influence of the culvert-to-downstream channel flow area ratio on the exit loss was evaluated at the projecting, sudden-expansion outlet. The experimental exit loss data were compared with the predicted results from the two exit loss expressions from HDS-5 (Equations 4-1 and 4-2) and the Borda-Carnot loss expression (Equations 4-7 and 4-8). 4.4 Experimental Results Five exit loss experiments were conducted featuring four different circular pipe sizes ranging from 2 to 60 in. in diam- eter. Three of the five tests featured free-surface flow in the pipe and an unsubmerged outlet. The other two test condi- tions featured pressurized pipe flow and a submerged outlet. Each culvert or pipe was installed in a flume that was 8 ft wide by 6 ft deep by 500 ft long with a rectangular cross-section. Each test pipe length was a minimum of 10 pipe diameters from the upstream laboratory supply pipe to the pipe exit. The test pipe diameters, the pipe diameter-to-exit-channel- width ratio, and the tailwater conditions are summarized in Table 4-1. The specific range of test pipe sizes was selected to provide a wide range of expansion ratios (Ap/Ach), approximately 0.02 to 0.39. Due to momentum effects under free discharge condi- tions, the tailwater elevation was higher than the water level in the pipe. As a result, each test pipe was chained to the flume floor to keep the pipe from floating due to buoyant forces. Due to the magnitude of the buoyant forces, however, no submerged outlet tests were conducted with the two largest test pipe sizes, as the pipes could not be kept from floating. The 24-in. diameter pipe was the only pipe tested under both pressurized and free-surface flow conditions. Water was supplied to the test pipes via a 20-in. diameter supply line instrumented with a calibrated ASME flow tube, traceable to the National Institute of Standards and Technol- ogy (NIST) by weight, for flow rate quantification. A sluice gate near the downstream end of the channel was used to con- trol tailwater elevation, ensuring outlet flow control in the test pipe. All test pipes were installed with a slope of zero (horizon- tal). All pipes had projecting ends, some of which were flanged (standard 150-lb flanges). No improved end treatments were tested. Figure 4-2 shows photos of two exit loss tests, and Figure 4-3 shows a schematic overview of the test setup. The total energy in the pipe was determined two diameters upstream of the outlet in order to ensure hydrostatic pres- sure conditions at the measurement point. The total energy in the pipe was determined as follows: a pressure tap, installed (12-inch diameter pipe exit loss test) (60-inch diameter pipe exit loss test) Figure 4-2. Overview photos of exit loss testing.

28 in the invert of each test pipe approximately two pipe diam- eters upstream of the outlet, was connected to a stilling well for measuring piezometric head at the outlet. The velocity head at the measurement location was calculated based on the calculated flow depth, calculated flow rate, and flow cross- sectional area and added to the piezometric head to get total energy values. The total energy at the pipe outlet was deter- mined by subtracting the amount of energy lost due to friction between the pressure tap in the pipe and the pipe outlet. The friction loss was estimated using the Darcy-Weisbach Equa- tion and friction factors as determined by the Swamee-Jain Equation (Crowe et al., 2001) using a pipe wall roughness, ks, of 0.00016 ft (steel pipe). Surveying equipment was used to determine the channel invert profile and pipe invert elevation at the outlet. The total energy was determined in the channel (downstream of the jet expansion) at 50-ft intervals using the piezometric head measured using stilling wells connected through the channel sidewall and the calculated velocity head at each location. The energy differential between the pipe and channel flows at the outlet represents the exit loss. It was not possible to deter- mine a representative one-dimensional total energy value in the channel at the outlet based on direct measurements due to the turbulent, three-dimensional flow characteristics at the outlet. A representative total energy value in the channel at the outlet was determined by projecting the total energy grade line, as determined by the downstream channel measurement location, back to the pipe outlet. This total energy value rep- resents a theoretical water surface (with its corresponding average velocity and velocity head) consistent with a one- dimensional energy assumption. The channel friction fac- tor for each run was determined using the Darcy-Weisbach Equation and the total energy data from the downstream channel measurement locations. Test Nominal pipe diameter Actual pipe inside diameter Pipe diameter over channel width Tailwater condition at pipe exit 12-S 12 in. 12 in. 0.17 Submerged 24-S 24 in. 23.25 in. 0.32 Submerged 24-U 24 in. 23.25 in. 0.32 Unsubmerged 48-U 48 in. 47.25 in. 0.66 Unsubmerged 60-U 60 in. 59.25 in. 0.82 Unsubmerged Table 4-1. Exit loss test configurations. Figure 4-3. Plan view schematic of exit loss test setup.

29 Table 4-2 summarizes exit losses observed and other infor- mation pertinent to the aforementioned experiments. Run numbers consist of the nominal pipe diameter in inches and a one-letter indicator of the outlet submergence condition: “U” for unsubmerged and “S” for submerged. The “Pipe Re” column contains pipe flow Reynolds number values, where four times the hydraulic radius was used to represent the characteristic length. The “Pipe Fr” column contains pipe flow Froude number values for the free-surface flow tests. All tests were conducted under outlet control as indicated by either a full-pipe or subcritical flow condition (Fr < 1.0). The “Dz” column contains the height of each pipe’s invert above the channel invert. The “% Full pipe” column is the pipe flow depth divided by the inside pipe diameter and multi- plied by 100. The pipe outlet submergence parameter, Tw/D, was calculated by dividing the channel tailwater depth (Tw), measured relative to the pipe exit invert, by the pipe inside diameter (D). The flow area expansion term, Ap/Ach, is the ratio of the pipe and channel flow cross-sectional areas. The exit loss coefficients corresponding to Equations 4-1, 4-2, 4-7, and 4-8 were calculated with Ho equal to the experi- mentally determined exit loss. The calculated ko values cor- responding to each predictive method are presented for each run in the “ko Exp.” columns in Table 4-3. The corresponding theoretical exit loss coefficient values, equal to 1.0 for Equa- tions 4-1, 4-2, and 4-7 or calculated using Equations 4-7 and 4-8, are given in the “ko Theor.” columns. A comparison of the experimental and theoretical coefficient values for each method is also presented in Table 4-3 in terms of percent error. Dividing the difference between the experimental and theoretical coefficient values by the experimental value and multiplying the result by 100 determined the percent error reported in the “Error” columns. General inspection of Table 4-3 reveals that discrepan- cies between experimental and theoretical coefficient values are significantly lower for the Borda-Carnot method (Equa- tions 4-7 and 4-8) than the more traditional exit loss methods (Equations 4-1 and 4-2). The error percentages reported for Equations 4-7 and 4-8 are identical because they are different algebraic forms of the same expression. Because Equations 4-2 and 4-8 only differ by the theoretical discharge coefficient definition, the experimental coefficient values reported for Equations 4-2 and 4-8 are identical. According to a sensitiv- ity analysis, the experimental uncertainty was approximately ±2.0%, which is consistent with the percent variation between the experimental and theoretical exit loss coefficients associ- ated with the Borda-Carnot loss expression (Equations 4-7 and 4-8) in Table 4-3. The disparity between the experimental and theoretical exit loss coefficients for the more traditional meth- ods (Equations 4-1 and 4-2) is larger than can be explained based solely on experimental uncertainty. Accordingly, the Borda-Carnot loss expression (Equations 4-7 and 4-8), applied to culvert exit loss at a projecting sudden expansion such as a projecting outlet, is significantly more accurate than the more traditional energy-based methods (Equations 4-1 and 4-2) when the theoretical discharge coefficients are used. Run Q Pipe Re Pipe Fr ∆z % Full-Pipe Tw/D Ap/Ach Ho [cfs] [ft] [%] [ft] 12-S 7.06 722,450 – 2.51 100 1.99 0.0218 1.18 24-S 16.24 651,033 – 2.13 100 1.26 0.0808 0.420 24-U 13.07 827,299 0.97 2.13 69 0.75 0.0757 0.469 48-U 34.96 1,304,322 0.72 1.18 53 0.57 0.2413 0.259 60-U 62.15 2,104,402 0.52 0.728 64 0.68 0.3920 0.131 Table 4-2. Experimental data from exit loss tests. Run Equation 4-1 Equation 4-2 Equation 4-7 Equation 4-8 Ko Exp. Ko Theor. Error [%] Ko Exp. Ko Theor. Ko Exp. Ko Theor. Error [%] Ko Exp. Ko Theor. Error [%] 12-S 0.93 1.0 7.3 0.93 1.0 7.3 0.97 1.0 2.7 0.93 0.96 2.7 24-S 0.88 1.0 13.9 0.87 1.0 14.6 1.03 1.0 3.1 0.87 0.85 3.1 24-U 0.84 1.0 18.9 0.84 1.0 19.6 0.98 1.0 2.2 0.84 0.85 2.2 48-U 0.63 1.0 60.0 0.60 1.0 69.9 1.02 1.0 2.2 0.60 0.58 2.2 60-U 0.41 1.0 143.1 0.35 1.0 187.3 0.94 1.0 6.2 0.35 0.37 6.2 Ho = ko Ho = ko Ho = ko Ho = ko ko = V2p − V2ch 2g 2g 2g 2 2g , 1 − Ap Ach 2 V2p V2p Vp − Vch Error [%] Table 4-3. Experimental and theoretical exit loss coefficient comparison.

30 4.5 Example of Application The relative impact of using the Borda-Carnot approach (Equations 4-7 and 4-8) instead of the traditional methods (Equations 4-1 and 4-2) is illustrated by the following sample calculation. An 80-ft long, 48-in. diameter, circular corru- gated metal culvert is installed under a road prism on grade in a trapezoidal channel 4 ft wide at the base, with a side slope of 0.5 horizontal to 1.0 vertical (m = 0.5) and a bed or channel slope (So) of 0.0008. The culvert has a square-edged inlet with a vertical headwall and a projecting outlet. The channel has a Mannings n value of 0.025. The design discharge is 90 cfs. The upstream depth is to be calculated in order to design a road prism of sufficient height to prevent overtopping. The normal depth of 4.98 ft in the channel is calculated using Manning’s Equation: V n R Sh e= 1 42 3 1 2 ( )-9 where n is the hydraulic roughness coefficient (Mannings n), Rh is the hydraulic radius (flow area divided by the wetted perimeter) at normal depth, and Se is the slope of the total energy grade line, which equals the channel slope (So) at nor- mal depth conditions. The Froude number, corresponding to normal depth, is 0.26 as calculated by Equation 4-10: Fr Q g A T c = 3 4( )-10 In Equation 4-10, T is the width of the water surface in the channel perpendicular to the primary flow direction. A Froude number less than 1.0 indicates subcritical flow. Assuming no downstream tailwater control other than channel friction, the tailwater will be equal to the normal depth at the culvert outlet. As the normal depth is greater than the culvert rise (culvert diameter), the culvert will likely be flowing full (in the absence of any air vents) with both the inlet and outlet submerged (outlet control). Assuming an entrance loss coefficient, ke, of 0.5 (square-edged inlet with headwall, Normann et al., 2001), a Mannings n of 0.02 for the culvert, and using the Borda-Carnot expres- sion (Equation 4-8) to describe exit loss, the upstream flow depth is 6.43 ft. Using Equation 4-2 with ko = 1.0, as sug- gested in HDS-5, results in an upstream depth of 6.93 ft, which exceeds the previously obtained value by 7.8% or 6 in. Although this exact difference is unique to these design parameters, it shows that improved accuracy in exit loss quantification can impact design. It should be noted that these sample calculations are solely to illustrate application of the expressions recommended for adoption, and they do not demonstrate the complete design process as the scenario is evaluated under only one flow con- trol regime. For a detailed treatment of the design process, see HDS-5 (Normann et al., 2001). 4.6 Conclusions Although originally developed for sudden expansions in pressurized pipes, the Borda-Carnot loss expression can be used to more accurately express energy losses at projecting sudden-expansion culvert outlets (and likely non-projecting sudden-expansion outlets), relative to energy-based tradi- tional methods such as those described in HDS-5HEC-14. Exit loss is more accurately described by multiplying the pipe or culvert velocity head by an exit loss coefficient, ko, defined as ko = (1 - Ap/Ach)2 (Equation 4-8) than by the traditional exit loss coefficient, defined as ko = 1.0. This approach to exit loss quantification correlated well with laboratory tests for both pressurized and free-surface pipe flow for sudden-expansion projecting outlets, submerged or unsubmerged, where the downstream channel flow is supplied solely by the culvert discharge. The theoretical exit loss coefficient for traditional methods varied from the corresponding experimental exit loss coefficient by as much as 187%, while the same comparison for the Borda-Carnot loss expression had a maximum varia- tion of 6.2%, with most data being better than approximately 3%. This improvement in accuracy may be particularly sig- nificant for culvert design where the culvert span is similar to the channel width, such as driveway cross drains along highly channelized ditches and fish passage culverts, which often have spans just larger than the bank-full channel width. The results of this study show that when appropriately applied, the exit loss approach recommended in HDS-5 is conservative in that it overestimates exit loss at outlets with channelized discharge channels. The degree of overestima- tion is relatively large in some cases. Future exit loss research should include additional tests with submerged outlets and channelized downstream flow; the number of submerged outlet test conditions was limited by the size of the test chan- nel, pipe sizes, and buoyant forces, which caused the test pipe to float under submerged conditions. As the derivation of the Borda-Carnot expression is independent of pipe geometry, investigation of its applicability to other culvert shapes, such as buried-invert culverts and box culverts, might not be nec- essary; however, it may be valuable to evaluate this expres- sion for culvert outlets with end treatments other than the projecting sudden-expansion outlet evaluated in this study.

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TRB’s National Cooperative Highway Research Program (NCHRP) Report 734: Hydraulic Loss Coefficients for Culverts explores culvert designs that maintain natural velocities and minimize turbulence to allow migratory species to pass through the culvert barrel.

The report describes the refinement of existing hydraulic relationships and the development of new ones for analysis and design of culverts for conventional and nontraditional, environmentally sensitive installations.

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