# Experimental and Numerical Analysis of Egg-Shaped Sewer Pipes Flow Performance

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Egg-Shaped Section Definition

_{av}is the average velocity (m/s), R

_{h}is the hydraulic radius (m), S is the slope of the pipe (m/m) and n is the Manning’s roughness coefficient (s/m

^{1/3}). According to this equation, a higher hydraulic radius means a higher mean velocity, hydraulic performance, and more sediment transport capacity. The circular geometry shows the highest full-bore discharge capacity as it presents the largest hydraulic radius regarding any cross-section with the same area. Nevertheless, in low flow conditions the egg-shaped conduit has a lower hydraulic radius. Therefore, the best aspect ratio for the egg-shaped cross-section should fit a higher hydraulic radius under low flow conditions but without losing significant full-filling discharge capacity regarding the circular discharge value.

^{1/3}, resulting in a full-filling discharge capacity of 47 L/s for the 300 mm circular pipe. Dry weather flow conditions were calculated using three different rates of daily average wastewater flow to wet weather flow (1:10, 1:20, and 1:50). Assuming a certain safety margin, the full-bore discharge capacity was set to a value of Q

_{0}= 40 L/s. The resulting base-flow discharges were 4.0, 2.0, and 0.8 L/s, respectively. From the whole set of the different egg-shaped pipes analyzed, the cross-sections with the highest hydraulic radius for each low flow condition and maximal full-filling discharge are those with H/R = 3.5 and r/R = 0.7, 0.5, and 0.3, respectively (Table 1). The differences found in the hydraulic performance do not justify the commercial development of three egg-shaped pipe sets, so the cross-section with ratios H/R = 3.5 and r/R = 0.5 was chosen because it presents adequate yields in all conditions. It was found that a typical value of H/R in egg-shaped pipe design is 3.0 [12], but the cross-section with ratio H/R = 3.5 has a similar hydraulic performance and improves its momentum of inertia by 15.3%. Therefore, the egg-shaped section with equivalent target area has a total height of 385 mm, a top radius of 110 mm, and a bottom radius of 55 mm.

#### 2.2. Experimental Set-Up

#### 2.3. CFD Model

^{6}hexahedral elements.

^{1/3}) for the real egg-shaped pipe. However, the roughness in the numerical model is defined as an equivalent roughness (k

_{s}) which can be estimated as a function of n by means of the Strickler’s equation (n = k

_{s}

^{1/6}/25). Applying this equation, the value of equivalent roughness in the numerical model was set to k

_{s}= 0.729 mm.

## 3. Results

#### 3.1. Boundary Shear Stress and Centreline Velocity Profiles

_{*av}with the equation τ = ρU

_{*av}

^{2}, where ρ is the fluid density (kg/m

^{3}). The average friction velocity was calculated as U

_{*av}= (gR

_{h}S)

^{1/2}, with S the slope of the pipe (%), R

_{h}the hydraulic radius (m), and g the gravity acceleration (m/s

^{2}). The differences between experimental and output modelling shear stress were less than 10% (Table 2).

_{*c}is the centerline friction velocity, κ is the von Kármán constant, k

_{s}is the equivalent roughness (0.729 mm), and A

_{r}is a constant of integration from Prandtl’s mixing-length formulation. In open-channel flows a value of κ = 0.41 is accepted [4]. Both centerline friction velocity and constant of integration were fitted from Equation (2) using a numerical routine, resulting in a value of A

_{r}= 7.9. Figure 3b shows the visual performance of the logarithmic formula and the friction velocity U

_{*c}results. Note that the figure axes are normalized with the total height of the pipe and the value of U

_{*c}for each experiment respectively.

#### 3.2. Cross-Sectional Velocity Distributions

_{*av}and the centerline shear velocity U

_{*c}(Figure 3b):

_{*c}/U

_{*av}is the ratio of the centerline to the average shear velocity (1.02 ± 0.02 range) and z

_{0}is the hydrodynamic roughness length of the pipe wall. This term approaches the velocity profile at the logarithmic zone, as in Equation (2). Comparing both equations, the value of z

_{0}can be expressed through the relation A

_{r}= ln(k

_{s}/z

_{0})/κ, resulting in a value of z

_{0}= 0.0285 mm. Furthermore, Guo et al. [5] introduced a cubic deduction to the logarithmic equation near the water surface, which depends on the velocity-dip position from the bottom (δ). The velocity-dip position varies depending on the discharge and the secondary currents. This variable was set equal to the surface water level, as no dip-phenomenon was observed either in the numerical or experimental velocity profiles (see Figure 3a). The last term represents the reduction of the velocity distribution because of the cross-section contour, where $\mathsf{\phi}\left(y,{y}_{b}\right)$ is the velocity-defect function defined below (y

_{b}represents the pipe’s half-width coordinate):

#### 3.3. Numerical Comparison of Circular and Egg-Shaped Mean Flow Behavior

^{1/3}). The tested flow discharges were 1.5, 2.5, 5.0, 7.5, 10.0, 20.0, and 40.0 L/s, using more resolution for low-depths ratios. In order to reach uniform flow conditions at the analyzed central section, the upstream and downstream water depths were established with Manning’s Equation.

_{0}) and shear stress (τ

_{0}) were calculated with Manning’s Equation and averaged shear stress formula (τ = (gR

_{h}S)

^{1/2}), respectively. Egg-shaped cross-section pipe presented higher mean velocity and shear stress values up to a filling ratio of h/H = 0.25, which is over the design cross-section depth for combined sewer pipelines in operating condition (dry weather flow regime). For common operating filling ratios of 0.10 and 0.15, the improvement of the shear stress was 15% and 9%, respectively. Thus, for relative depths h/H < 0.25 a greater sediment transport capacity is expected in the egg-shaped cross-section than in the equivalent-area circular pipes because of the higher velocity and shear stress values. This should reduce the risk of sediment accumulation at the pipe bottom and decrease the risk of pollution associated with sediment deposits [18]. The circular cross-section had a better performance above a filling ratio of h/H = 0.25, which is outside of the range of normal operating conditions of a combined sewer network. For full-filling conditions, the performance of the egg-shaped pipe in terms of averaged shear stress was only a 5.3% lower than the equivalent circular profile.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Egg-shaped cross-section definition from variables H, R, and r with a tangent connecting top and bottom arcs and (

**b**) H/R and r/R relationships. The best egg-shaped cross-sections are highlighted with triangles.

**Figure 3.**(

**a**) Experimental and numerical comparison of velocity profiles for a filling ratio of h/H = 0.2 and 0.3 and (

**b**) results of fitting Equation (2) to all test using U

_{*c}for normalizing.

**Figure 4.**Comparison of velocity contours and relative errors for h/H = 0.2 (

**a**); 0.3 (

**b**); 0.4 (

**c**); 0.5 (

**d**). Left data from Equation (3) and right for numerical model. Velocity contours are expressed in m/s.

**Figure 5.**(

**a**) Averaged velocity and (

**b**) shear stress comparison of numerical results for circular (circles) and egg-shaped (triangles) cross-sections with Manning (continuous line) and Thormann-Franke (dashed line) formulas. Axes are normalized with the height of each conduit (H) and their full-depth mean velocity (U

_{0}) and averaged shear stress (τ

_{0}), respectively.

**Table 1.**Comparison of hydraulic radius (R

_{h}) for low flows (1:10, 1:20, and 1:50 wastewater and rainfall rates) and full-bore section discharges (Q

_{0}) conditions in egg-shaped cross-sections with best hydraulic performance. Hydraulic radius and discharges were normalized with circular cross-section values.

H/R | r/R | R_{h} _{1:10} | R_{h} _{1:20} | R_{h} _{1:50} | R_{h} _{Q0} | Q_{0} |
---|---|---|---|---|---|---|

3.5 | 0.3 | 1.038 | 1.103 | 1.193 | 0.905 | 0.934 |

3.5 | 0.5 | 1.064 | 1.125 | 1.187 | 0.897 | 0.930 |

3.5 | 0.7 | 1.078 | 1.114 | 1.132 | 0.925 | 0.949 |

**Table 2.**Experimental parameters: discharge Q (L/s), averaged velocity U

_{av}(m/s), filling ratio h/H (dimensionless), hydraulic radius R

_{h}(m), Reynolds number Re, average friction velocity U

_{*av}(m/s). Total shear stress results from the experimental methodology τ and output modelling shear stress τ

_{CFD}(N/m

^{2}) (relative errors are in parenthesis).

Test | Experimental Conditions | CFD Model | ||||||
---|---|---|---|---|---|---|---|---|

Q (L/s) | U_{av} (m/s) | h/H (-) | R_{h} (m) | Re (×10^{3}) | τ = ρU_{*av}^{2} (N/m^{2}) | τ_{CFD} (N/m^{2}) | ||

1 | 3.20 | 0.410 | 0.2 | 0.034 | 5.7 | 0.684 | 0.664 | (−2.9%) |

2 | 7.04 | 0.528 | 0.3 | 0.045 | 9.5 | 0.883 | 0.964 | (9.2%) |

3 | 13.08 | 0.582 | 0.4 | 0.057 | 13.3 | 1.121 | 1.159 | (3.4%) |

4 | 19.03 | 0.658 | 0.5 | 0.064 | 16.8 | 1.254 | 1.374 | (9.6%) |

**Table 3.**Comparison of CFD/experimental discharges with the values obtained from Guo et al.’s formula [5]. Relative errors are in parenthesis.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Regueiro-Picallo, M.; Naves, J.; Anta, J.; Puertas, J.; Suárez, J.
Experimental and Numerical Analysis of Egg-Shaped Sewer Pipes Flow Performance. *Water* **2016**, *8*, 587.
https://doi.org/10.3390/w8120587

**AMA Style**

Regueiro-Picallo M, Naves J, Anta J, Puertas J, Suárez J.
Experimental and Numerical Analysis of Egg-Shaped Sewer Pipes Flow Performance. *Water*. 2016; 8(12):587.
https://doi.org/10.3390/w8120587

**Chicago/Turabian Style**

Regueiro-Picallo, Manuel, Juan Naves, Jose Anta, Jerónimo Puertas, and Joaquín Suárez.
2016. "Experimental and Numerical Analysis of Egg-Shaped Sewer Pipes Flow Performance" *Water* 8, no. 12: 587.
https://doi.org/10.3390/w8120587