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Publication Number: FHWA-HRT-07-036
Date: March 2007

Junction Loss Experiments: Laboratory Report

2. THEORY

Kilgore's proposed method for calculating the energy loss across an access hole has three fundamental steps.(4)

  1. Determine an initial access hole energy level, Ea1, based on "inlet controlled" flow conditions (i.e., weir and orifice) or "full flow" conditions.
  2. Adjust the initial access hole energy level for benching, inflow angle(s), and plunging flows to compute the final calculated energy level, Ea.
  3. Calculate the exit loss from each inflow pipe and estimate the energy gradeline, Ei, which will then be used to continue calculations upstream.

These three steps are illustrated in figure 3.

View alternative text

Figure 3. Diagram. Flow chart for the proposed junction loss method.

INITIAL ACCESS HOLE ENERGY LEVEL

The initial energy level in the access hole structure, Ea1, is calculated as the maximum of three possible conditions; these determine the hydraulic regime within the structure. The three conditions considered for the outlet pipe are full flow, submerged inlet control, and unsubmerged inlet control. A fourth condition, partially full outlet control, could occur but was not pursued because it was not considered to be a practical limiting condition in storm drain design.

The full flow condition is considered when the outlet pipe is flowing full. This is a common occurrence when a storm drain system is surcharged and may also occur if flow in the pipe is limited by pipe capacity. The submerged inlet control condition occurs if flow is limited by the opening in the access hole structure to the outlet pipe rather than by the pipe capacity, and the resulting water depth in the access hole is sufficiently high that flow through the opening can be analyzed by orifice flow equations. The unsubmerged inlet control condition is also limited by the opening, but the resulting water level in the access hole is lower and weir flow equations can be used. The initial estimate of energy level is taken as the maximum of the three potential values (equation 2).

Equation 2. E subscript a1 equals the maximum of E subscript a, f f, E subscript a, i c s, and E subscript a, i c u. (2)

In this equation:

Ea,ff

is estimated access hole energy level for full flow.

Ea,ics

is estimated access hole energy level for submerged inlet control.

Ea,icu

is estimated access hole energy level for unsubmerged inlet control.

The full flow computation uses velocity head, but full flow only applies when the outlet pipe is flowing full. The two inlet control estimates depend only on discharge and pipe diameter. This is important because velocity is not a reliable parameter for the following reasons:

  • In cases where supercritical flow occurs in the outlet pipe, flow in the outlet pipe (and the corresponding velocity head) are defined by the upstream condition at the access hole rather than the velocity head determining upstream conditions.
  • n the laboratory setting, velocity is not directly measured. It is calculated from depth and the continuity relationship. Small errors in depth measurement can result in large variations in velocity head.
  • Velocities produced in laboratory experiments are the result of localized hydraulic conditions, which are not necessarily representative of the velocities calculated based on equilibrium pipe hydraulics in storm drain computations.

Full Flow

In the full flow condition, discharge into the access hole is limited by surcharges in the downstream storm drain system such that the outflow pipe is flowing full. Using a culvert analogy, this is one of the potential cases of outlet control. In this case, the initial structure energy level is estimated by the equation 3.

Equation 3. E subscript a, f f equals y subscript o plus P subscript o divided by gamma (specific weight) plus V subscript o superscript 2 divided by the product of 2 and g (gravity) plus delta E subscript o c. (3)

The entrance loss assuming outlet control, ΔE oc, is calculated by equation 4.

Equation 4. Delta E subscript o c equals K subscript o times the quantity: V subscript o superscript 2 divided by the product of 2 and g. (4)

In these equations:

g

is acceleration due to gravity (meters per second squared (m/s2)).

γ

is the specific weight of water (Newtons per cubic meters (N/m3)).

yo

is outlet flow depth (m).

Po

is outlet pressure head (m).

V o 2/2g

is outlet velocity head (m).

Ko

is the entrance loss coefficient, which is 0.2 and dimensionless.

Equations 3 and 4 are used only when the outlet flow depth (yo) plus the outlet pressure head (Po) is greater than Do, where Do is the outlet pipe diameter; otherwise Ea,ff is 0.0. The outlet pipe invert elevation (zo) is the datum for this analysis, and is set to zero. Equation 3 estimates energy level directly without considering the water surface within the access hole. Defining a one-dimensional velocity head in a location where highly turbulent multidirectional flow may exist presents a challenge. Marsalek noted that the energy loss coefficient is unaffected by changes in the relative access hole diameter as long as the ratio of b over Do ranges from 2 to 6.(5) Sangster's study showed that the energy loss coefficient is primarily affected when the ratio is less than 3.(6) The reanalyzed lab data, however, does not appear to support the need for a contraction factor such as the ratio of b over Do.

Submerged Inlet Control

The inlet control calculations employ the dimensionless ratio adapted from the analysis of culverts that is referred to here as the discharge intensity (DI). It is described by the ratio of discharge to pipe dimensions, where A is the area (equation 5).

Equation 5. D subscript I equals Q divided by the following quantity: A times the square root of the product of g and D subscript o. (5)

The original submerged inlet control equation (equation 6) uses the analogy of inflow via a submerged orifice. Equation 6, however, should be limited to discharge intensities less than or equal to 1.6 because the reanalyzed data set did not include observations above this threshold.

Equation 6. E subscript a, i c s equals 1.2 times two terms. The first term is the square of Q divided by the following quantity: A times the square root of the product of g and D subscript o. The second term is D subscript o. (6)

Kilgore selected a coefficient equal to 1.2.(4) However, early lab experiments show that this coefficient should be about 1.0, instead of 1.2. Equation 7 is the revised equation.

Equation 7. E subscript a, i c s equals 1.0 times two terms. The first term is the square of Q divided by the following quantity: A times the square root of the product of g and D subscript o. The second term is D subscript o. (7)

Unsubmerged Inlet Control

The unsubmerged inlet control equation (equation 8) uses the analogy of inflow via a weir.

Equation 8. E subscript a, i c u equals 1.6 times two terms. The first term is the 0.67 power of Q divided by the following quantity: A times the square root of the product of g and D subscript o. The second term is D subscript o.(8)

The coefficients for equations 4, 6, 7, and 8 were empirically derived to achieve a best fit with the reanalyzed laboratory data. These coefficients are lower than comparable values for culverts as reported in Norman, et al.(7) Since conditions in an access hole differ from conditions at the inlet to a culvert, one would not expect an exact correspondence. What is apparent from the lab data, however, is that the change in access hole energy level is analogous to culverts.

ADJUSTMENTS FOR BENCHING, ANGLED INFLOW, AND PLUNGING INFLOW

The initial structure energy level calculated in the previous section is used as a basis for estimating additional losses for discharges entering the structure at angles other then 180°, benching configurations, and discharges entering the structure at elevations above the water depth in the access hole. Flows entering a structure from an inlet above the water surface in the access hole can be treated as plunging flows. The effects of these conditions may be estimated and applied to the initial access hole energy level using the principle of superposition. This additive approach avoids a problem experienced in other methods where unreasonable values of energy losses are obtained when a single multiplicative coefficient takes on an extreme value. The revised access hole energy level, Ea, equals the initial estimate modified by each of the three factors covered in this section, as shown in equation 9.

Equation 9. E subscript a equals the sum of E subscript a 1 plus delta E subscript B plus delta E subscript theta plus delta E subscript H. The sum can also be written as E subscript a1 plus delta E. (9)

In this equation:

EB

is additional energy loss for benching (floor configuration).

Eθ

is additional energy loss for angled inflows other than 180°.

EH

is additional energy loss for plunging flows.

E

is additional energy loss for ∆EB, ∆Eθ, and ∆EH.

As described earlier, Ea represents the level of the energy grade line (EGL) in the access hole. However, if Ea is calculated to be less than Eo (the outlet energy level), then Ea should be set equal to Eo. Designers may also wish to know the water level in the access hole. A conservative approach would be to use Ea as ya for design purposes. Traditional approaches to energy losses typically attempt to estimate all losses based on a single velocity head and for reasons described earlier the approach proposed in this paper is moving away from this strategy. The alternative proposed here is to estimate these additional energy losses as a function of the total energy losses computed between the access hole and the outlet pipe. The formulation in equation 10 expresses the additional losses, ∆E, as directly proportional to the energy loss estimated in the first step between the access hole and the outlet pipe.

Equation 10. Delta E equals C times the quantity: E subscript a 1 minus y subscript o minus the quotient of P subscript o divided by gamma minus the quotient of V subscript o superscript 2 divided by the product of 2 and g. (10)

In this equation, C is the energy loss coefficient—CB, Cθ, or CH—for benching, angled inflow, and plunging inflow, respectively. The term between the parentheses and beginning with yo is the total energy, Eo, calculated for the upstream end of the outlet pipe.

Note that the final Ea value cannot be less than Eo, and that the bracketed terms in equation 10 must be nonnegative. Another difficulty with equation 10 concerns the outlet velocity head. When the entrance condition to the outlet pipe limits flow into the pipe (inlet control) or the outlet pipe is flowing in a supercritical flow condition, the outlet velocity head may not be representative of the energy losses occurring within the access hole for reasons described earlier. The derivation of C for each adjustment is addressed in the following subsections.

Benching

The reanalyzed lab data suggests that the correction factors reported by Chang et al. and HEC 22 for benching, after adapting them to the form of equation used in this methodology, needed some adjustment.(1,2) Table 1 summarizes the suggested values for CB. A negative value indicates water depth will be reduced rather than increased.

Table 1. Values for the Coefficient CB.

Floor Configuration

Bench Submerged*

Bench Unsubmerged*

Flat (level)

-0.05

-0.05

Depressed

0.0

0.0

Half Benched

-0.05

-0.85

Full Benched

-0.25

-0.93

Improved

-0.60

-0.98

* Submerged is Ea1/Do greater than 2.5, and unsubmerged is Ea1/Do less than 1.0. Linear interpolation between the two values is used for intermediate values.

No evaluation or adjustments to these floor configurations have been made because these configurations were not included in the FHWA testing.

Angled Inflow

The effect of skewed inflows entering the structure is addressed with momentum vectors. To maintain simplicity in the method, the contributions (θj) from all of the nonplunging inflows are resolved into a single flow-weighted angle, θw (equation 11).

Equation 11. Theta subscript w equals the quotient of the summation of the product of Q subscript j and theta subscript j, divided by the summation of Q subscript j.(11)

The angles in the previous equation are measured from the outlet pipe (e.g., 180° is a straight pipe), and the summation only includes nonplunging flows as indicated by the subscript j. If all flows are plunging, θw is set to 180°. It does not matter whether the angle is defined in a clockwise or counterclockwise orientation as long as it is defined consistently. For example, if two pipes are entering the structure orthogonal to the outflow pipe, one must be designated as 90° and the other as 270° for the momentum vectors. The angled inflow coefficient is then calculated using equation 12.

Equation 12. C subscript theta equals the product of three terms. The first term is 4.5. The second term is the magnitude of the cosine of the quotient of theta subscript w divided by 2. The third term is the quotient of the summation of Q subscript j divided by Q subscript o.(12)

Note that the angled inflow coefficient approaches zero as θw approaches 180° and the relative inflow approaches zero.

Plunging Inflow

Plunging inflow is defined as inflow (pipe or inlet) where the invert of the pipe, zi, is greater than the estimated structure water depth, ya1 (taken as Ea1 as an approximation). Several approaches were attempted to describe a parameter that captures the presumed effect of plunging height on energy losses in the access hole based on defining a plunge height. A balance between simplicity and effectiveness was achieved by defining relative plunge height, Hk, for each plunging pipe, denoted by the subscript k (equation 13).

Equation 13. H subscript k equals the quotient of the difference of z subscript k minus y subscript a1 divided by D subscript o.(13)

Equation 13 was incorporated into a relationship for CH (equation 14).

Equation 14. C subscript H equals the quotient of the summation of the product of Q subscript k times H subscript k, divided by Q subscript o. (14)

Note that as the proportion of plunging flows approaches zero, CH also approaches zero. Equations 13 and 14 are limited to conditions where zk is less than 10Do. If zk is greater than 10Do, zk should be set to 10Do.

INFLOW PIPE EXIT LOSSES

The final step is to calculate the EGL in each inflow pipe. Two cases must be considered. The first case is for nonplunging inflow pipes—that is, those pipes with a hydraulic connection to the water in the access hole. Inflow pipes operating under this condition are identified when Ea is greater than zi. In these cases, the energy at the inlet, Ei, is backcalculated from the access hole energy level and the exit loss, ∆Ei (equation 15).

Equation 15. E subscript i equals E subscript a plus delta E subscript i.(15)

Since the criticality of the inflow does not influence the exit loss, the exit loss can be calculated in the traditional manner using the inflow pipe velocity head (equation 16).

Equation 16. Delta E subscript i equals K subscript i times V subscript i superscript 2 divided by the product of 2 times g. (16)

In this equation, the exit loss coefficient, Ki, equals 0.4 and is dimensionless.

As was found in examining the entrance losses, the ratio of b over Do was not a significant predictor of exit energy losses. The second case for an inflow pipe is a plunging condition. For pipes that are plunging, Ei is the EGL calculated from the inflow pipe hydraulics and will be approximately critical energy if the inlet pipe is on a subcritical slope.

The resulting energy level is used to continue computations upstream to the next access hole except when the inlet pipe is on a steep (supercritical) slope and is not submerged for its full length, in which case the hydraulics are controlled at the upstream end of the pipe.

Thus, the three-step procedure of estimating—first, entrance losses, then additional losses, and finally exit losses—is repeated at each access hole.

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The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). The hydraulics and hydrology research program at the TFHRC Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
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