## Abstract

Due to an increasing number of heavy rainfall events, the managing of urban flooding requires new design approaches in urban drainage engineering. With bidirectional coupled numerical models the surface runoff, the underground sewer flow and the interaction processes between both systems can be calculated. Most of the numerical models use a weir equation to calculate the surface to sewer flow with unsurcharged flow conditions, but uncertainties still exist in the representation of the real flow conditions. Street inlets, existing in different types, are the connecting elements between the surface and the underground system. In the present study, an empirical formula was developed based on physical model test runs to estimate the hydraulic capacity and type-specific efficiency of grate inlets with supercritical surface flow. Influencing hydraulic parameters are water depth and flow velocity upstream of the grate and, in addition, different geometrical parameters are taken into account, such as the grate dimensions or the orientation of the bars (transverse, longitudinal or diagonal). Good agreement between estimated and measured results could be proven with relative deviations less than 1%.

## INTRODUCTION

With regard to an increasing number of heavy rainfall events, potentially caused by the climate change (IPCC 2014), the managing of urban flooding is becoming more important currently (Digman *et al.* 2014). In order to localize possible inundation areas and to develop new design approaches above the ground (e.g. flood routes) instead of enlarging the classical underground drainage system, information about the surface runoff using topography and streets (major system), as well as the underground drainage system (minor system) and the interaction processes between both systems is necessary (Fratini *et al.* 2012). The coupling of above- and below-ground models leads to modelling tools (bidirectional coupled models; dual drainage models – 1D-1D as well as 1D-2D models), which are capable of representing urban flooding under extreme flow conditions (Butler & Davies 2011). 1D-1D models (one-dimensional pipe flow model and one-dimensional surface flow model) give sufficient results as long as the flow stays within the street cross section (Djordjević *et al.* 1999; Mark *et al.* 2004). State-of-the-art 1D-2D models (two-dimensional surface flow models) provide likely more realistic results when the surface flow exceeds the capacity of the street profile. Inundation areas with corresponding flow depths and flow velocities are calculated (Ettrich *et al.* 2005; Maksimović *et al.* 2009).

The exchange discharge such as sewer-to-surface or surface-to-sewer flow is calculated by empirical hydraulic formulas at linking nodes/elements. Different hydraulic conditions occur, a free inflow from the surface to the unsurcharged sewer system through street inlets (Figure 1(a)), submerged surface-to-sewer flow (Figure 1(b)) and the outflow from the sewer system to the surface (Figure 1(c)). After Rubinato *et al.* (2017) uncertainties in calculating the interaction processes still exist. Connecting elements serving as intake structures between the surface and the underground drainage system are street inlets. Those exist in different types, such as grate inlets, curb-opening inlets or combination inlets.

*et al.*2017). The discharge can be assumed with (e.g. Chen

*et al.*2007): where

*Q*is the intercepted discharge (exchange discharge surface-to-sewer flow),

_{I}*C*is an energy loss coefficient,

_{i}*w*is the weir crest width (e.g. manhole perimeter),

*g*is the gravitational acceleration and

*h*the water depth upstream of the street inlet.

_{G}Several investigations based on physical models deal with the grate efficiency by neglecting the system below, e.g. Larson (1947), Cassidy (1966), Burgi & Gober (1977), Spaliviero *et al.* (2000), Despotovic *et al.* (2005), Gómez & Russo (2005), Brown *et al.* (2009) and Guo & MacKenzie (2012). Numerical investigations were done, e.g. by Djordjević *et al.* (2013) or Lopes *et al.* (2016).

In Kemper (2018) it was found that, with supercritical surface flow conditions, the water depth and the flow velocity are the main influencing hydraulic parameters on the grate efficiency.

With 2D surface flow models, flow depths as well as flow velocities are known upstream of the grate and can be taken into account for calculating the surface to sewer flow. Almost all of the published empirical equations to calculate the grate efficiency depend on parameters that are not immediately known from the 2D surface flow model, so firstly, they need to be calculated or estimated (see ‘Empirical approaches’).

In the present study, an empirical calculation formula was developed to estimate the hydraulic grate efficiency with supercritical flow conditions by the use of results of physical model test runs.

## PHYSICAL MODEL SETUP

The measurements were done in a flume made of acrylic glass with *L _{Flume}* = 10.00 m in length and

*W*= 1.50 m in width with variable slope in longitudinal (

_{Flume}*S*) and transverse (

_{L}*S*) direction. The bottom roughness is approximately

_{T}*k*= 1.5 mm (roofing paper). The grate of street inlets can be integrated in a 1:1 scale at the downstream end of the flume at

*X*= 8.50 m (Figure 2). The total discharge

*Q*is controlled by an electromagnetic flowmeter (Optiflux 2000, Krohne, resolution: 0.1 L/s with a reproducibility of ± 0.1%) and the inlet width is limited to 1.00 m. The incoming flow is distributed uniformly over the inlet width using a tube bundle. Therefore, the available length of 8.50 m is sufficiently high to obtain uniform flow conditions upstream of the grate. In all model test runs, the water spread width

*W*is less than the total flume width

*W*= 1.50 m and a uniform triangular cross-sectional flow area is obtained (see Figure 3).

_{Flume}Six different grate geometries with transversal, longitudinal or diagonal orientated bars were tested (Figure 2). The grate width is *W _{G}* = 500 mm and the grate length between

*L*= 500 mm and 780 mm. The grate dimensions, such as opening area

_{G}*A*or bar opening width

_{0}*B*, are given in Table 1.

_{X}Type no. | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Opening area A [cm²] _{0} | 980 | 1,040 | 1,160 | 955 | 970 | 1,567 |

Bar opening width B [mm] _{X} | 36 | 24 | 25 | 31 | 26 | 34.5 |

Bar width [mm] | 32 | 12 | 12 | 32 | 24 | 34.5 |

Type no. | I | II | III | IV | V | VI |
---|---|---|---|---|---|---|

Opening area A [cm²] _{0} | 980 | 1,040 | 1,160 | 955 | 970 | 1,567 |

Bar opening width B [mm] _{X} | 36 | 24 | 25 | 31 | 26 | 34.5 |

Bar width [mm] | 32 | 12 | 12 | 32 | 24 | 34.5 |

The German guideline for designing street inlets recommends a connected catchment area of 400 m² for one inlet each (FGSV 2005). Therefore, based on KOSTRA-DWD-2010R (2017), the surface runoff approaching one inlet is given in Table 2 with different rainfall durations *D* [min] and return periods *T* [a], exemplarily for Wuppertal, a city in the western part of Germany. One hundred percent runoff on the street is assumed. Considering even extreme rainfall events within the physical model tests, the total surface runoff was varied between *Q* = 3 L/s up to *Q* = 21 L/s with Δ*Q* = 3 L/s.

T [a] | 1.0 | 2.0 | 5.0 | 10.0 | 20.0 | 30.0 | 50.0 | 100.0 |
---|---|---|---|---|---|---|---|---|

D [min] | ||||||||

5.0 | 6.93 | 9.60 | 12.93 | 15.60 | 18.13 | 19.73 | 21.60 | 24.13 |

10.0 | 5.53 | 7.20 | 9.40 | 11.07 | 12.73 | 13.73 | 15.00 | 16.67 |

15.0 | 4.58 | 5.87 | 7.60 | 8.89 | 10.18 | 10.93 | 11.91 | 13.20 |

T [a] | 1.0 | 2.0 | 5.0 | 10.0 | 20.0 | 30.0 | 50.0 | 100.0 |
---|---|---|---|---|---|---|---|---|

D [min] | ||||||||

5.0 | 6.93 | 9.60 | 12.93 | 15.60 | 18.13 | 19.73 | 21.60 | 24.13 |

10.0 | 5.53 | 7.20 | 9.40 | 11.07 | 12.73 | 13.73 | 15.00 | 16.67 |

15.0 | 4.58 | 5.87 | 7.60 | 8.89 | 10.18 | 10.93 | 11.91 | 13.20 |

The investigation program is given in Table 3. In total, 28 different combinations of longitudinal slope and discharge were investigated with each of the six grate inlets (168 combinations in total). The transverse slope was fixed to *S _{T}* = 2.5% as a standard transverse slope of streets in urban areas. The longitudinal slope was varied between

*S*= 2.5% and

_{L}*S*= 10.0% (street inlets on grade). The focus of the present study is on supercritical surface flow conditions. Therefore, high longitudinal slopes were taken into account. Smaller slopes and therefore different hydraulic conditions with subcritical flow conditions or stagnant water (e.g. sinks) were not investigated within the study. Each model setup defined by longitudinal slope, discharge and grate inlet type was repeated five times (six model runs for each combination). The final results were calculated as the mean values of the six model runs.

_{L}Longitudinal slope S _{L} | 2.5–10.0% with ΔS = 2.5% _{L} |

Transverse slope S _{T} | 2.5% |

Total discharge Q | 3 L/s–21 L/s with ΔQ = 3 L/s |

Grate inlet type | I, II, III, IV, V, VI |

Longitudinal slope S _{L} | 2.5–10.0% with ΔS = 2.5% _{L} |

Transverse slope S _{T} | 2.5% |

Total discharge Q | 3 L/s–21 L/s with ΔQ = 3 L/s |

Grate inlet type | I, II, III, IV, V, VI |

Uniform flow conditions were reached upstream of the street inlet with water spread widths less than *W _{Flume}* = 1.50 m. Thus, there is no influence due to the model limitations such as length and width. Only supercritical flow conditions occur, therefore, no backwater effects at the lower boundary of the model arise (free outflow).

*Q*and that of the water that was flowing across the grate

_{S,Lab}*Q*was measured over the time by the use of platform load cells (Single Point Load Cell Model 1260, Tedea-Huntleigh). The intercepted discharge

_{O}*Q*was calculated with:

_{I}## THEORATICAL BACKGROUND AND LITERATURE OVERVIEW

### Gutter flow

*S*[-], water depth

_{T}*h*[m] directly at the curb and water spread width

*W*[m] (Figure 3). The water spread width is: The cross-sectional flow area is:

*n*is Manning's roughness coefficient [s/m

^{1/3}]. Using the continuity equation, the mean flow velocity is calculated with

*v*=

_{m}*Q/A*. The Froude number, modified for a triangular channel is:

### Hydraulic efficiency

*Q*[L/s] whereas the total hydraulic grate efficiency

_{I}*E*is described as a percentage of the approaching flow rate with: To get information about the actual grate efficiency, the type-specific efficiency

*E*is defined within the presented study by neglecting the water already flowing alongside the grate

_{T}*Q*, which is unaffected by the grate geometry itself. Therefore, the frontal flow on the width of the grate

_{S}*Q*with

_{F}*Q*

_{F}*=*

*Q – Q*(see Figure 3) is taken into account only: The water flowing alongside the grate can be estimated with Equation (4) with

_{S}*W*=

_{S}*W*−

*W*(if

_{G}*W*>

*W*). In order to describe the type-specific efficiency, the water depth

_{G}*h*and the flow velocity

_{G}*v*as well as the Froude number

_{m,G}*Fr*are defined concerning the frontal flow

_{G}*Q*and therefore the grate width

_{F}*W*with

_{G}*v*=

_{m,G}*Q*with

_{F}/A_{G}*A*=

_{G}*h*and

_{G}·W_{G}*Fr*=

_{G}*v*/(

_{m,G}*g*·

*h*)

_{G}^{0.5}.

### Empirical approaches

*W*<

*W*). After Spaliviero

_{Flume}*et al.*(2000) the hydraulic efficiency can be calculated depending on the ratio of the total discharge

*Q*to the water depth

*h*with: where

*G*and

*β*are empirical coefficients with

*β*= 102.7,

*h*[m] is the water depth at the curb upstream of the grate inlet and

*Q*[m³/s] is the total discharge.

*G*is an empirical coefficient considering the geometry of the grate. A similar approach is given in Gómez & Russo (2005) with: where

*a*and

*b*are coefficients considering the grate geometry. Based on physical model test runs described in Burgi & Gober (1977), the guideline HEC-22 (Hydraulic Engineering Circular No. 22, Brown

*et al.*2009) contains an empirical approach to calculate the efficiency with: where

*R*is the ratio of frontal flow intercepted to total frontal flow,

_{w}*E*is the ratio of frontal flow to total discharge and

_{w}*R*is the ratio of side flow intercepted to total side flow.

_{x}The empirical equations to calculate the grate efficiency depend on parameters that are not immediately known from the 2D surface flow model and first need to be calculated or even estimated, e.g. the total surface runoff *Q*, the water depth *h* directly at the curb (therefore, the transversal slope *S _{T}* has to be known additionally) or the longitudinal slope

*S*, which has to be known to calculate the ratio

_{L}*R*, for example.

_{X}## RESULTS AND DISCUSSION

### General results

The physical model results show that with increasing discharge the (type-specific) efficiency decreases. The bypass flow *Q _{S}* increases just like the flow velocity and therefore the discharge over the grate

*Q*. Within the investigated limitations of discharge, street geometry and grate geometry, the efficiency is between

_{O}*E*≈ 0.7–1.0. The percentage of

*Q*is maximum 15%, and the percentage of

_{O}*Q*is maximum 30%. The type-specific efficiency is between

_{S}*E*≈ 0.8–1.0 for the given boundary conditions such as discharge, grate geometry and street geometry (longitudinal and transverse slope). The percentage of the lateral intercepted discharge

_{T}*Q*is negligibly small and it is

_{I,S}*Q*≈

_{I}*Q*. Depending on the discharge, Figure 4 displays the efficiency

_{I,F}*E*, the type-specific efficiency

*E*and the side flow ratio

_{T}*E*as well as the overflow ratio

_{S}*E*exemplarily for a longitudinal slope of

_{O}*S*= 7.5%. The parameters that are influencing the grate efficiency are divided into hydraulic and geometric parameters (grate geometry) and are described in detail below.

_{L}### Hydraulic parameters

Three main flow cases could be defined within the presented investigations (Kemper 2018), see Figure 5. Flow Case 1 with *E _{T}* = 1 (no overflow), Flow Case 2 with

*E*< 1 (partial overflow) and Flow Case 3 with

_{T}*E*≪ 1 (full overflow). With Flow Case 1, the total amount of the frontal discharge

_{T}*Q*is intercepted whereas with Flow Case 2 the curb side of the grate is overflowed only. Due to the model limitations Flow Case 3 was not investigated within the presented physical model test runs. The grate efficiency with full overflow across the entire grate width (Flow Case 3) is investigated, e.g. by Tellez

_{F}*et al.*(2017).

The aim of the presented study is to eliminate the parameters *S _{L}*,

*S*and

_{T}*Q*from the calculation formula (see ‘Empirical approaches’) for the type-specific efficiency (and capacity

*Q*respectively) and to take only already known hydraulic parameters from inundation simulations (2D surface models) such as flow depth or flow velocity into account.

_{I},Depending on the street geometry with longitudinal slope and transversal slope and on the surface flow *Q*, the hydraulic parameters water depth *h*, mean flow velocity *v _{m}* and water spread width

*W*can be calculated upstream of the grate inlet (see ‘Gutter flow’).

To describe the type-specific efficiency by means of an empirical formula the hydraulic parameter water spread width *W* can be neglected and the water depth *h* and mean flow velocity *v _{m}* remain as effective hydraulic parameters. With increasing frontal discharge, the water depth, as well as the flow velocity, increases. With increasing longitudinal slope, the water depth, decreases and the flow velocity, increases. With increasing transversal slope the water depth, as well as the flow velocity, increases.

The results of the physical model test runs are displayed exemplarily for one grate geometry (Type I) depending on the flow depth *h* and mean flow velocity *v _{m}* in Figure 5. The black diamonds represent the investigated hydraulic cases within the physical model test runs. The measured water depths (directly at the curb) are less than

*h*= 0.035 m (

*h*= 0.026 m) and the mean flow velocities less than

_{G}*v*= 1.6 m/s (

_{m}*v*= 1.7 m/s). The type-specific efficiency decreases with increasing water depth and flow velocity.

_{m,G}### Geometric parameters

Several geometric parameters such as grate width, grate length, opening area and orientation of the bars influence the grate efficiency. It was found that with increasing grate width the type-specific efficiency increases, but in terms of traffic not suitable. *E _{T}* increases with increasing grate length, which is especially advisable for high longitudinal slopes. The opening area insignificantly influences the type-specific efficiency when bar orientation and outer dimensions are the same. With high longitudinal slopes, diagonal or longitudinal bars are more efficient than transversal bars, as it can be seen in Figure 6. In case of flat streets, transversal bars with great opening widths in flow direction lead to the best efficiency. The influence of the opening width

*B*in flow direction was investigated in detail in Kemper & Schlenkhoff (2018).

_{X}### Calculation formula

*h*and flow velocity

_{G}*v*(mean value with regard to the width of the grate) are taken into account in the developed calculation formula. In addition, geometric parameters such as dimensions of the grate (length

_{G}*L*and width

_{G}*W*), the opening area

_{G}*A*and the orientation of the bars

_{0}*S*are considered:

*S*= parameter for the orientation of the bars (Table 4) with

*B*= width of the bar openings in flow direction. The intercepted discharge can be calculated as follows:

_{X}Orientation of the bars | S |
---|---|

Transversal bars (B > 0,03 m) _{X} | 3.6 |

Transversal bars (B < 0,03 m) _{X} | 3.8 |

Longitudinal bars | 4.1 |

Diagonal bars | 4.1 |

Orientation of the bars | S |
---|---|

Transversal bars (B > 0,03 m) _{X} | 3.6 |

Transversal bars (B < 0,03 m) _{X} | 3.8 |

Longitudinal bars | 4.1 |

Diagonal bars | 4.1 |

The formula is limited to supercritical flow conditions with no backwater effects caused by the underground drainage system, as well as the hydraulic boundary conditions of the physical model test runs with 0.006 m ≤ *h _{G}* ≤ 0.026 m and 0.66 m/s ≤

*v*≤ 1.68 m/s,

_{m,G}*S*> 0 and

*A*<

_{0}*W*·

_{G}*A*. Good agreement between estimated and measured results could be proven with relative deviations less than 1%. The limitation of the hydraulic parameters is due to the experimental setup limitations. The applicability to smaller or higher flow depths or velocities was not proven yet.

_{G}In Figure 7, the calculated results (Equation (14), *E* = *Q _{I}/Q*) are compared to the laboratory results, as well as the empirical approaches from the literature. Especially with high longitudinal slopes, differences in the available approaches arise. They occur due to different underlying boundary conditions of the physical model test runs, e.g. total discharge up to 200 L/s.

### Plausibility check and application for non-tested grate inlets

Within the plausibility check of the developed formula the proper consideration of the influencing parameters was verified. As it can be seen in Figure 8 with large grate width *W _{G}* and therefore large opening area A

_{0}(

*W*[mm] ×

_{G}*L*[mm] = 500 × 500), the type-specific efficiency is greater than with small grate width and therefore less opening area (300 × 500). The grate length

_{G}*L*and bar geometry, as well as orientation are the same (transversal bars,

_{G}*S*= 3.6). With a large grate length

*L*(500 × 780), the type-specific efficiency increases compared to a grate with less grate length and otherwise the same properties.

_{G}The physical model results of the grate inlet 300 × 500 were not considered within the development of the calculation formula. The estimated results *E _{T}* with Equation (13) show qualitatively the same behaviour like the experimental results. Even quantitatively sufficient results could be reached with relative deviations less than 5.0%.

## CONCLUSIONS

Bidirectional coupled models (dual drainage models) which consider the interaction between the underground drainage flow and the surface runoff are still employed in practice. To represent the interaction processes more realistically, information about the hydraulic capacity of the connecting elements must be known. In the presented study, the hydraulic capacity of grate inlets with supercritical surface flow and unsurcharged drainage flow conditions was investigated with physical model test runs. The type-specific efficiency was defined by taking the frontal flow on the width of the grate into account. The main effective hydraulic parameters on the grate capacity are the water depth and flow velocity when supercritical surface flow conditions exist. With increasing water depth and flow velocity, the type-specific efficiency decreases. Depending on both of the hydraulic parameters, as well as on geometrical parameters such as grate dimensions (grate length and width, opening area) and orientation of the bars (transversal, longitudinal or diagonal), an empirical calculation formula was developed to estimate the hydraulic capacity or type-specific efficiency of grates with supercritical surface flow conditions. The formula is limited to supercritical flow conditions with no backwater effects caused by the underground drainage system as well as the hydraulic boundary conditions of the physical model test runs with 0.006 m ≤ *h _{G}* ≤ 0.026 m and 0.66 m/s ≤

*v*≤ 1.68 m/s. Good agreement between estimated and measured results could be proven with relative deviations less than 1%. Additionally, the application to non-tested grates was tested. Sufficient results could be reached with relative deviations less than 5.0% to experimental results.

_{m,G}