TítuloExperimental and Numerical Analysis of Egg Shaped Sewer Pipes Flow Performance

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(1)water Article. Experimental and Numerical Analysis of Egg-Shaped Sewer Pipes Flow Performance Manuel Regueiro-Picallo *, Juan Naves, Jose Anta, Jerónimo Puertas and Joaquín Suárez Universidade da Coruña, Water and Environmental Engineering Group (GEAMA), Elviña, 15071 A Coruña, Spain; juan.naves@udc.es (J.N.); jose.anta@udc.es (J.A.); jpuertas@udc.es (J.P.); jsuarez@udc.es (J.S.) * Correspondence: manuel.regueiro1@udc.es; Tel.: +34-881-105-430 Academic Editor: Peter J. Coombes Received: 11 November 2016; Accepted: 6 December 2016; Published: 9 December 2016. Abstract: A Computational Fluid Dynamics (CFD) model was developed to analyze the open-channel flow in a new set of egg-shaped pipes for small combined sewer systems. The egg-shaped cross-section was selected after studying several geometries under different flow conditions. Once the egg-shaped cross-section was defined, a real-scale physical model was built and a series of partial-full flow experiments were performed in order to validate the numerical simulations. Furthermore, the numerical velocity distributions were compared with an experimental formulation for analytic geometries, with comparison results indicating a satisfactory concordance. After the hydraulic performance of the egg-shaped pipe was analyzed, the numerical model was used to compare the average velocity and shear stress against an equivalent area circular pipe under low flow conditions. The proposed egg shape showed a better flow performance up to a filling ratio of h/H = 0.25. Keywords: CFD modeling; egg-shaped section; sewer design; shear stress; velocity distributions; water pipelines. 1. Introduction Egg-shaped pipes appear as a suitable geometry for combined sewer sewage networks. Egg-shaped conduits present higher resistance against traffic loads than conventional circular pipes. In addition, this kind of pipe also shows a better hydraulic performance in normal operation dry weather conditions of combined sewer systems, in which a high percentage of the time the flow discharge is conveyed by the lower part of the section. In these conditions, egg-shaped pipes present higher flow velocities due to their smaller wetted perimeter, reducing the sedimentation of particles and the sewer cleaning operational costs [1]. The resuspension of sewer sediments during wet weather flows is an important source of the pollution of Combined Sewer Overflows [2], and their control is one of the main objectives of the integrated urban water management in urban systems [3]. In spite of the structural and hydraulic advantages, egg-shaped pipes are not commonly used in the construction of small combined sewer systems because of their highest production costs. Nevertheless, with the evolution of production techniques such as plastic injection or extrusion, the fabrication costs of plastic egg-shaped pipes can be as competitive as circular plastic pipes. In this work we present the first stage of the collaborative OvalPipe R&D project that aims to develop a new functioning egg-shaped plastic pipe that is commercially viable and market competitive with the 300–400 mm diameter circular pipes. The first steps of the process consisted in the geometric definition and in the hydraulic analysis of the egg-shaped cross section. The egg-shaped geometry was designed with the objective of maximizing the hydraulic radius under low flow conditions and the discharge capacity under full-depth or near full-depth conditions. Once the cross-section was defined, a real-scale egg-shaped pipe was built at a laboratory facility to study its hydraulic characteristics. Water 2016, 8, 587; doi:10.3390/w8120587. www.mdpi.com/journal/water.

(2) Water 2016, 8, 587. 2 of 9. Most of the open-channel pipe flow studies were performed in circular conduits. For instance, the early studies of turbulence developed by Nezu and Nakagawa [4] proposed different formulations to describe velocity profiles in circular cross-sections. Guo et al. [5] developed new velocity distribution formulas for circular, elliptic, parabolic, and hyperbolic open-channels (hereinafter named as conic open-channels). Particle Image Velocimetry (PIV) technique was also developed to determine velocity distribution in small circular pipes [6]. Nevertheless, detailed hydrodynamic experiments for egg-shaped pipes are missing. In order to analyze the behavior of the circular and egg-shaped pipes, open-channel flow experiments were conducted with ANSYS CFX Computational Fluid Dynamics (CFD) code. To simulate the open-channel flow in closed conduits such as pipes, a two-phase flow model was developed to solve the interactions between liquid (water) and gas (air) interface [7–10]. The experimental velocity profiles and shear stress values were compared with the numerical results, following the methodology proposed in previous studies [11]. Finally, numerical results from egg-shaped and circular pipe analysis were also compared with the analytical open-channel flow Manning and Thormann-Franke equations. 2. Materials and Methods 2.1. Egg-Shaped Section Definition In the first step of the study an egg-shaped cross section was defined with an equivalent area to a 315 mm circular pipe (300 mm inner diameter). In terms of geometric construction of the egg-shaped profile, it was possible to use different families of curves and ellipses, sinusoidal functions, arcs, or specific methods. In the present work, a combination of arcs was chosen in order to build an egg-shaped cross-section (Figure 1a). This cross-section was defined from three variables: the top and bottom radii (R and r) and the total height (H). In order to define the transition curve between the upper and bottom circumferences, the construction method proposed by other authors was allowed, where the circular lateral arcs have their centers at the same height as the top circumference center [12]. Thus, the r/R and H/R ratios allow the design of a wide variety of theoretical ovoids with different aspect ratios, which potentially could be used in the field of sanitation and urban drainage engineering. A set of egg-shaped cross-sections with ratios r/R between 0.3 and 0.9 and H/R between 2.1 and 3.6 were compared with a 300 mm inner diameter circular section (Figure 1b). All the proposed theoretical ovoids have the same area as the circular pipe. The performance of the pipes was analyzed by means of the hydraulic radius in low-depth condition and the full filling discharge capacity determined with the Manning’s Equation for uniform flow:. √ Uav = R2/3 S/n h. (1). where Uav is the average velocity (m/s), Rh is the hydraulic radius (m), S is the slope of the pipe (m/m) and n is the Manning’s roughness coefficient (s/m1/3 ). 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. The hydraulic conditions to perform the analysis of the different pipe shapes were a slope S = 0.2% and a Manning’s coefficient n = 0.012 s/m1/3 , 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 Q0 = 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.

(3) Water 2016, 8, 587. of 99 33 of. condition and maximal full‐filling discharge are those with H/R = 3.5 and r/R = 0.7, 0.5, and 0.3, maximal full-filling are those with H/R = and r/R = 0.7,performance 0.5, and 0.3, respectively (Tablethe 1). respectively (Tabledischarge 1). The differences found in3.5 the hydraulic do not justify The differences found in the hydraulic performance do not justify the commercial development of commercial development of three egg‐shaped pipe sets, so the cross‐section with ratios H/R = 3.5 and three egg-shaped pipebecause sets, so itthe cross-section withyields ratiosin H/R = 3.5 and r/R 0.5found was chosen r/R = 0.5 was chosen presents adequate all conditions. It = was that abecause typical it presents adequate yields in all conditions. It was found that a typical value of H/R in value of H/R in egg‐shaped pipe design is 3.0 [12], but the cross‐section with ratio H/Regg-shaped = 3.5 has a pipe design is 3.0 [12], but the cross-section withits ratio H/R = 3.5 of hasinertia a similar performance similar hydraulic performance and improves momentum by hydraulic 15.3%. Therefore, the and improves its momentum of inertia by 15.3%. Therefore, the egg-shaped section with equivalent egg‐shaped section with equivalent target area has a total height of 385 mm, a top radius of 110 mm, target area hasradius a totalofheight of 385 mm, a top radius of 110 mm, and a bottom radius of 55 mm. and a bottom 55 mm.. (a). (b). Figure 1. (a) Egg‐shaped cross‐section definition from variables H, R, and r with a tangent connecting 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 top and bottom arcs and (b) H/R and r/R relationships. The best egg-shaped cross-sections are highlighted with triangles. highlighted with triangles. Table 1. Comparison of hydraulic radius (Rh) for low flows (1:10, 1:20, and 1:50 wastewater Table 1. Comparison of hydraulic radius (Rh ) for low flows (1:10, 1:20, and 1:50 wastewater and rainfall and rainfall rates) and full‐bore section discharges (Q0) conditions in egg‐shaped cross‐sections rates) and full-bore section discharges (Q0 ) conditions in egg-shaped cross-sections with best hydraulic with best hydraulic performance. Hydraulic radius and discharges were normalized with circular cross‐ performance. Hydraulic radius and discharges were normalized with circular cross-section values. section values. H/R 3.5 3.5 3.5. r/R H/R 0.3 3.5 0.5 3.5 0.7 3.5. r/R RhR1:10 h 1:10 1.038 0.3 1.038 0.5 1.064 1.064 1.078 0.7 1.078. RRh h1:201:20 Rh 1:50Rh 1:50 Rh Q0 1.103 1.193 1.103 1.193 0.905 1.125 1.187 1.187 1.125 0.897 1.114 1.132 1.114 1.132 0.925. R Qh0 Q0. Q0. 0.905 0.934 0.897 0.930 0.925 0.949. 0.934 0.930 0.949. 2.2. Experimental Experimental Set‐Up Set-Up 2.2. A series series of model of of anan egg-shaped pipe located at the A of experiments experimentswere werecarried carriedout outinina aphysical physical model egg‐shaped pipe located at R&D Centre of Technological Innovation in Building and Civil Engineering (CITEEC) of the University the R&D Centre of Technological Innovation in Building and Civil Engineering (CITEEC) of the of A Coruña 2). (Figure This model consisted an 11 mof long stainless egg-shaped pipe with University of (Figure A Coruña 2). This model of consisted an 11 m longsteel stainless steel egg‐shaped R = 110 mm, H = 385 mm, and r = 55 mm. At the beginning of the pipe an inlet chamber was placed, pipe with R = 110 mm, H = 385 mm, and r = 55 mm. At the beginning of the pipe an inlet chamber while a horizontal tail gate was provided to allow the adjustment of water levels and flow uniformity was placed, while a horizontal tail gate was provided to allow the adjustment of water levels and downstream of the pipe. Water was measured using ultrasonic sensors distributed along flow uniformity downstream of level the pipe. Water level wasfive measured using five ultrasonic sensors several apertures opened in the pipe. The resolution of sensors was 0.13 mm and the deviation of distributed along several apertures opened in the pipe. The resolution of sensors was 0.13 mm and ◦ ultrasonic beam 4.6 . Discharge was measured using ultrasonicusing flowmeter with an accuracy of the deviation of was ultrasonic beam was 4.6°. Discharge wasanmeasured an ultrasonic flowmeter ± 1% of measured values and registered with a data logger during each test. with an accuracy of ±1% of measured values and registered with a data logger during each test. Four experiments experimentswere wereconducted conductedatata a0.2% 0.2%slope slope with different filling ratios (h/H) of to 0.20.5. to Four with different filling ratios (h/H) of 0.2 0.5. Uniform conditions were established adjusting positionand andheight heightof ofaa downstream downstream Uniform flowflow conditions were established by by adjusting thethe position tailgate. Centerline velocity profiles were measured with a Nortek Vectrino© (Rud, Norway) Acoustic tailgate. Centerline velocity profiles were measured with a Nortek Vectrino© (Rud, Norway) Doppler Velocimeter (ADV) with an accuracy of ± 1 mm/s at a distance of 5.5 m from the inlet Acoustic Doppler Velocimeter (ADV) with an accuracy of ±1 mm/s at a distance of 5.5 m from the chamber. Water velocity was measured with a spatial resolution of 5 and 2.5 mm for measures close to inlet chamber. Water velocity was measured with a spatial resolution of 5 and 2.5 mm for measures the pipe bottom and with a sampling frequency of 25 Hz during 300 s to ensure that the measured close to the pipe bottom and with a sampling frequency of 25 Hz during 300 s to ensure that the streamwisestreamwise turbulenceturbulence intensity was within was 5% of its measured long-term average. All velocity measured intensity within 5% of its measured long‐term average.data All were de-spiked using the phase-space thresholding method [13,14]. velocity data were de‐spiked using the phase‐space thresholding method [13,14]..

(4) Water 2016, 8, 587. 4 of 9. Water 2016, 8, 587. 4 of 9. Figure 2. Schematic drawing of the physical model. Figure 2. Schematic drawing of the physical model.. 2.3. CFD Model 2.3. CFD Model Numerical simulations were performed with ANSYS CFX software (Canonsburg, PA, USA). Numerical simulations were performed with ANSYS CFX software (Canonsburg, PA, USA). This code solves the 3D Reynolds‐Averaged Navier‐Stokes (RANS) equations [15]. A two‐phase flow This code solves the 3D Reynolds-Averaged Navier-Stokes (RANS) equations [15]. A two-phase flow model was selected to simulate the interaction of the air friction with the water surface in partially model was selected to simulate the interaction of the air friction with the water surface in partially filled filled pipes (Thormann‐Franke formulation). In order to calculate the interface between both fluids, pipes (Thormann-Franke formulation). In order to calculate the interface between both fluids, ANSYS CFX ANSYS CFX uses the volume of fluid (VOF) model. In the VOF model, multi‐phase fluids share uses the volume of fluid (VOF) model. In the VOF model, multi-phase fluids share governing equations governing equations of mass and momentum conservation. The VOF model tracks the interface of mass and momentum conservation. The VOF model tracks the interface position between phases at position between phases at control volumes within the domain. For this, volume fractions are assigned control volumes within the domain. For this, volume fractions are assigned to each control volume [9]. to each control volume [9]. An unstructured (block-structured) non-uniform mesh was selected to discretize pipe geometry. An unstructured (block‐structured) non‐uniform mesh was selected to discretize pipe geometry. To avoid convergence problems at the interface between fluids (air-water), the height of the mesh To avoid convergence problems at the interface between fluids (air‐water), the height of the mesh elements was reduced progressively from 3 mm in the main fluid body to 1 mm close to the pipe wall elements was reduced progressively from 3 mm in the main fluid body to 1 mm close to the pipe wall and to the interface [16,17]. As the position of the interface varied in each case because of the water and to the interface [16,17]. As the position of the interface varied in each case because of the water level, a new grid system was necessary for each simulation. The average mesh size in the whole pipe level, a new grid system was necessary for each simulation. The average mesh size in the whole pipe was ~3 × 106 6 hexahedral elements. was ~3 × 10 hexahedral elements. Boundary conditions were set from experimental flow conditions. At the inlet of the channel, Boundary conditions were set from experimental flow conditions. At the inlet of the channel, discharge and water level were established to constant values depending on the position of each phase. discharge and water level were established to constant values depending on the position of each phase. At the outlet, the water level was also fixed. The initial condition imposed to the model was the average At the outlet, the water level was also fixed. The initial condition imposed to the model was the average velocity obtained from the experiments. Additionally, a steady state simulation in combination with velocity obtained from the experiments. Additionally, a steady state simulation in combination with the Shear Stress Transport turbulence model was selected for all cases. Wall function was set by the wall the Shear Stress Transport turbulence model was selected for all cases. Wall 1/3 function was set by the wall roughness that was established with Manning’s coefficient (n = 0.012 s/m ) for the real egg-shaped roughness that was established with Manning’s coefficient (n = 0.012 s/m1/3) for the real egg‐shaped pipe. However, the roughness in the numerical model is defined as an equivalent roughness (ks ) which pipe. However, the roughness in the numerical model is defined as an equivalent roughness (ks) which can be estimated as a function of n by means of the Strickler’s equation (n = ks 1/61/6/25). Applying this can be estimated as a function of n by means of the Strickler’s equation (n = ks /25). Applying this equation, the value of equivalent roughness in the numerical model was set to k = 0.729 mm. equation, the value of equivalent roughness in the numerical model was set to ks =s 0.729 mm. 3. Results 3. Results 3.1. Boundary Shear Stress and Centreline Velocity Profiles 3.1. Boundary Shear Stress and Centreline Velocity Profiles The shear stress over the wetted perimeter and the centerline velocity profile were used in order The shear stress over the wetted perimeter and the centerline velocity profileranging were used order to compare CFD model outputs and the flume tests measurements. Discharges fromin3.20 to to compare CFD model outputs and the flume tests measurements. Discharges ranging from 3.20 to 19.03 L/s were used, resulting in different uniform conditions of water depth and Reynolds number 19.03 L/s were used, resulting in different uniform conditions water depth and Reynolds variations. From the experimental data, total shear stress can beof expressed as a function of the number average 2 , stress 3 ). variations. FromU the experimental data, total shear can be expressed as a function of the average friction velocity with the equation τ = ρU where ρ is the fluid density (kg/m The average *av *av 2, where ρ is the fluid density (kg/m3). The average 1/2 friction velocity was U*av with the equation = ρU *av friction velocity calculated as U*av =τ (gR , with S the slope of the pipe (%), Rh the hydraulic h S) 1/2, with S the slope of the pipe (%), Rh the hydraulic friction velocity as U*av = (gR hS)2 radius (m), and gwas the calculated gravity acceleration (m/s ). The differences between experimental and output The differences between experimental and output radius (m),shear and gstress the gravity acceleration (m/s2).2). modelling were less than 10% (Table modelling shear stress were less than 10% (Table 2). The CFD model centerline profiles were compared with the ADV measurements at the The CFD of model centerline were compared with the ADV and measurements at the middle-section the pipe (Figure profiles 3a). The agreement between experimental numerical velocity middle‐section of the pipe (Figure 3a). The agreement between experimental and numerical velocity series was estimated with the root mean square (RMS). RMS < 0.076 was found to be an acceptable fit series estimated with thevertical root mean square (RMS). < 0.076 was found to be shear an acceptable for all was the cases. In addition, velocity profiles canRMS be used to obtain centerline stress as fit for all the cases. In addition, vertical velocity profiles can be used to obtain centerline shear stress as an estimation of the friction in the pipe bottom. In open‐channel flows, this value is related with the logarithm region of the vertical velocity profile (0.05–0.2 h/H) following a log‐law approach [4]:.

(5) Water 2016, 8, 587. 5 of 9. an estimation of the friction in the pipe bottom. In open-channel flows, this value is related with the Water 2016,region 8, 587 of the vertical velocity profile (0.05–0.2 h/H) following a log-law approach [4]: 5 of 9 logarithm 1 z U (z) U ( z= ) 1ln  z  + Ar U∗c  κ ln  k s   Ar U*c. κ. (2).  ks . (2). where U(z) is the centerline velocity at the height z, U*c is the centerline friction velocity, κ is the where U(z) is the centerline velocity at the height z, U*c is the centerline friction velocity, κ is the von von Kármán constant, k is the equivalent roughness (0.729 mm), and Ar is a constant of integration Kármán constant, ks is sthe equivalent roughness (0.729 mm), and Ar is a constant of integration from from Prandtl’s mixing-length formulation. In open-channel value of κis=accepted 0.41 is accepted Prandtl’s mixing‐length formulation. In open‐channel flows aflows valueaof κ = 0.41 [4]. Both[4]. Both centerline friction velocity and constant of integration were fitted from Equation (2) using centerline friction velocity and constant of integration were fitted from Equation (2) using a numerical a numerical routine, resulting in a value of A = 7.9. Figure 3b shows the visual performance of the r 3b shows the visual performance of the logarithmic routine, resulting in a value of Ar = 7.9. Figure logarithmic formula and the frictionUvelocity Note thataxes the figure axes are normalized with *c results. formula and the friction velocity *c results.U Note that the figure are normalized with the total theheight total height of the pipe and the value of U for each experiment respectively. of the pipe and the value of U*c for each *c experiment respectively. Table 2. Experimental parameters: velocity U Uavav(m/s), (m/s),filling filling ratio h/H Table 2. Experimental parameters:discharge dischargeQQ(L/s), (L/s), averaged averaged velocity ratio h/H (dimensionless), Re, average averagefriction frictionvelocity velocityUU (m/s). (m), Reynolds Reynolds number number Re, *av*av (m/s). (dimensionless),hydraulic hydraulicradius radiusRRhh (m), Total shear stress results from thethe experimental methodology τ and output modelling shear stress τCFD Total shear stress results from experimental methodology τ and output modelling shear stress 2) (relative errors are in parenthesis). τCFD2(N/m (N/m ) (relative errors are in parenthesis).. Test 1 2 3 4. Experimental Conditions Experimental Conditions Q (L/s) Uav (m/s) h/H (‐) Rh (m) Re (×103) 103 ) 1Q (L/s)3.20Uav (m/s) 0.410 h/H (-)0.2 Rh (m) 0.034 Re (× 5.7 0.2 0.3 0.034 5.7 2 3.20 7.04 0.410 0.528 0.045 9.5 0.3 0.4 0.045 9.5 3 7.04 13.08 0.528 0.582 0.057 13.3 0.4 0.5 0.057 13.3 4 13.08 19.03 0.582 0.658 0.064 16.8. Test. 19.03. 0.658. 0.5. (a). 0.064. 16.8. τ = ρU*av2 (N/m2) 2 2 τ = ρU *av (N/m ) 0.684 0.684 0.883 0.883 1.121 1.121 1.254 1.254. CFD Model CFD Model τCFD (N/m2) (N/m2 ) 0.664 τCFD (−2.9%) (−2.9%) 0.9640.664 (9.2%) 1.1590.964 (3.4%)(9.2%) 1.3741.159 (9.6%)(3.4%) 1.374. (9.6%). (b). Figure 3. (a) Experimental and numerical comparison of velocity profiles for a filling ratio of Figure 3. (a) Experimental and numerical comparison of velocity profiles for a filling ratio of h/H = 0.2 h/H = 0.2 and 0.3 and (b) results of fitting Equation (2) to all test using U*c for normalizing. and 0.3 and (b) results of fitting Equation (2) to all test using U*c for normalizing.. 3.2. Cross‐Sectional Velocity Distributions 3.2. Cross-Sectional Velocity Distributions In this section CFD model outputs were compared with the formulation of Guo et al. [5] for the In this section CFD model outputs were compared witha the formulation of Guo et al. [5] for cross‐sectional velocity distribution. Guo et al. [5] proposed simple velocity distribution model forthe cross-sectional velocity distribution. Guo et al. [5] proposed a simple velocity distribution model conic open‐channels without fitting any parameter. Their experiments were motivated by a designfor conic open-channels without fitting any parameter. Their experiments motivated by a design for fish stream‐crossing, but they suggested that this model was also validwere for self‐cleaning drainage forsystems. fish stream-crossing, they suggested model metal was also valid drainage The analyticalbut model was testedthat in this a circular pipe but for noself-cleaning laboratory data of systems. The analytical modelwere was available tested in atocircular pipe but no laboratory of non-circular non‐circular conic sections validatemetal this formulation. Followingdata the approach by conic were to validate this formulation. Following the approach Guo etconic al. [5], Guosections et al. [5], the available cross‐sectional velocity distribution (U(y,z)) in an egg‐shaped or aby generic can be calculated as a function(U(y,z)) of the averaged shear velocity *av and the centerline shearcan thegeometry cross-sectional velocity distribution in an egg-shaped or aUgeneric conic geometry velocity U*c (Figure 3b):.

(6) Water 2016, 8, 587. 6 of 9. be calculated as a function of the averaged shear velocity U*av and the centerline shear velocity U*c (Figure 3b): λU∗ av z 1 z 3 U (y, z) = ln − U∗av ϕ (y, yb ) (3) − κ z0 3 δ Water 2016, 8, 587. 6 of 9. where y and z are the cross-sectional coordinates. In the first term of the Equation (3), λ = U*c /U*av is 3 the ratio of the centerline to the averageλUshear (1.02 ± 0.02 range) and z0 is the hydrodynamic  velocity  z  1z  * av    U * av   y , y b  U ( y , z )  ln  (3)     roughness length of the pipe wall. This term κ  approaches z 3    the velocity profile at the logarithmic zone,   0  as in Equation (2). Comparing both equations, the value of z0 can be expressed through the relation y and z are the cross‐sectional coordinates. In the first term of the Equation (3), λ = U*c/U*av is Ar = ln(kwhere s /z0 )/κ, resulting in a value of z0 = 0.0285 mm. Furthermore, Guo et al. [5] introduced a cubic the ratio of the centerline to the average shear velocity (1.02 ± 0.02 range) and z0 is the hydrodynamic deduction to the logarithmic equation near the water surface, which depends on the velocity-dip roughness length of the pipe wall. This term approaches the velocity profile at the logarithmic zone, positionas from the bottom (δ). Theboth velocity-dip position depending on the the discharge in Equation (2). Comparing equations, the value ofvaries z0 can be expressed through relation and the secondary This variable was equalmm. to the surface water as no dip-phenomenon Ar =currents. ln(ks/z0)/κ, resulting in a value of set z0 = 0.0285 Furthermore, Guo et level, al. [5] introduced a cubic deduction to the equation near the water velocity surface, which depends the velocity‐dip was observed either in logarithmic the numerical or experimental profiles (see on Figure 3a). The last term position from the bottom (δ). The velocity‐dip position varies depending on the discharge the ϕ (y, y ) represents the reduction of the velocity distribution because of the cross-section contour,and where b secondary currents. This variable was set equal to the surface water level, as no dip‐phenomenon is the velocity-defect function defined below (yb represents the pipe’s half-width coordinate): was observed either in the numerical or experimental velocity profiles (see Figure 3a). The last term ( distribution " ofthe cross‐section represents the reduction of the velocity because #)contour, where 1 y 1 y 3   y , yb  is the velocity‐defect function defined below (y b represents the pipe’s half‐width coordinate): ϕ(y, y ) = − + ln 1 − 1− 1− b. yb. κ. 3. yb. (4). 3      y  1 y    ln 1 1 1         (4)they were  with velocity model and  Guo  al.’s    y b et y b distribution    3       compared with numerical results, resulting in relative errors under 8% (Figure 4). In order to evaluate. 1 (y , yb )   All test conditions were reproduced . test conditionsaccuracy, were reproduced with Guo etintegrated al.’s velocity from distribution model and they were et al. [5] the velocity All distributions the discharges the approach by Guo compared with numerical results, resulting in relative errors under 8% (Figure 4). In order to evaluate were compared with CFD model input values, which were set from experimental measurements. the velocity distributions accuracy, the discharges integrated from the approach by Guo et al. [5] were The differences between both discharges were less than 5% (Table 3). compared with CFD model input values, which were set from experimental measurements. The differences between both discharges were less than 5% (Table 3).. Table 3. Comparison of CFD/experimental discharges with the values obtained from Guo et al.’s formula Table [5]. Relative errorsofare in parenthesis.discharges with the values obtained from Guo et al.’s 3. Comparison CFD/experimental formula [5]. Relative errors are in parenthesis.. Q (L/s). h/H = h/H 0.2 = 0.2 h/H Q (L/s) h/H== 0.3 0.3 h/H =h/H 0.4 = 0.4 h/H = 0.5 h/H = 0.5 CFD/Experimental 7.04 19.03 CFD/Experimental 3.20 3.20 7.04 13.0813.08 19.03 (−4.4%) (−4.5%) 13.51 13.51 19.02 (−0.1%) Guo et al. Guo [5] et al. [5] 3.06 3.06 (−4.4%) 6.72 6.72 (−4.5%) (3.3%) (3.3%) 19.02 (−0.1%). (a). (b). Figure 4. Cont..

(7) Water 2016, 8, 587. 7 of 9. Water 2016, 8, 587. 7 of 9. (c). (d) Figure 4. Comparison of velocity contours and relative errors for h/H = 0.2 (a); 0.3 (b); 0.4 (c); 0.5 (d).. Figure 4.Left Comparison of velocity contours and relative errors for h/H = 0.2 (a); 0.3 (b); 0.4 (c); 0.5 (d). data from Equation (3) and right for numerical model. Velocity contours are expressed in m/s. Left data from Equation (3) and right for numerical model. Velocity contours are expressed in m/s. 3.3. Numerical Comparison of Circular and Egg‐Shaped Mean Flow Behavior. 3.3. Numerical Comparison of Circular and Egg-Shaped Mean Flow Behavior against a circular section Lastly, the egg‐shaped cross‐section conduit behavior was compared with an equivalent area in order to evaluate its efficiency in partially filled pipe flow. A CFD model. Lastly, the egg-shaped cross-section conduit behavior was compared against a circular section was performed for a circular pipe with an inner diameter of 300 mm, which corresponds roughly to with an equivalent area order to evaluate itssewer efficiency partially filled pipe A CFD a standard 315 mmin Polyvinyl Chloride (PVC) pipe. Ain series of simulations were flow. conducted in model both an egg‐shaped and circular cross‐section model. For each simulation the same hydraulic was performed for a circular pipe with an inner diameter of 300 mm, which corresponds roughly to conditions were used (S = 0.2%, n = 0.012 s/m1/3).sewer The tested flowAdischarges 1.5, 2.5, 5.0,were 7.5, 10.0, a standard 315 mm Polyvinyl Chloride (PVC) pipe. series ofwere simulations conducted 20.0, and 40.0 L/s, using more resolution for low‐depths ratios. In order to reach uniform flow in both an egg-shaped and circular cross-section model. For each simulation the same hydraulic conditions at the analyzed central section, the1/3 upstream and downstream water depths were conditions were used (S = 0.2%, n = 0.012 s/m ). The tested flow discharges were 1.5, 2.5, 5.0, established with Manning’s Equation. 7.5, 10.0, 20.0, and 40.0velocity L/s, using more resolution for low-depths ratios. In order to reach Flow mean and averaged shear stress results are compared in Figure 5 for circular and uniform egg‐shaped Note thatcentral the axessection, are normalized with the and height of each conduit and depths their flow conditions at pipes. the analyzed the upstream downstream water were full‐depth mean velocity (U 0) and shear stress (τ0) were calculated with Manning’s Equation and established with Manning’s Equation. averaged shear stress formula (τ = (gRhS)1/2), respectively. Egg‐shaped cross‐section pipe presented Flow mean velocity and averaged shear stress results are compared in Figure 5 for circular higher mean velocity and shear stress values up to a filling ratio of h/H = 0.25, which is over the and egg-shaped pipes. Note that axes are normalized the height each conduit design cross‐section depth for the combined sewer pipelines inwith operating conditionof(dry weather flowand their full-depth meanFor velocity shearratios stress (τ0and ) were withofManning’s Equation and regime). common(U operating of 0.10 0.15,calculated the improvement the shear stress was 0 ) andfilling 1/2 ), and stress 9%, respectively. for relative depths h/H < 0.25 a greater sediment transport capacity is averaged15% shear formulaThus, (τ = (gR S) respectively. Egg-shaped cross-section pipe presented h expected in the egg‐shaped cross‐section than inup the to equivalent‐area circular pipes because the higher higher mean velocity and shear stress values a filling ratio of h/H = 0.25,ofwhich is over the velocity and shear stress values. This should reduce the risk of sediment accumulation at the pipe design cross-section depth for combined sewer pipelines in operating condition (dry weather flow bottom and decrease the risk of pollution associated with sediment deposits [18]. The circular regime). cross‐section For common filling ratios the improvement of the shear stress was hadoperating a better performance aboveofa 0.10 fillingand ratio0.15, of h/H = 0.25, which is outside of the range 15% andof9%, respectively. for relative depths sewer h/H <network. 0.25 a greater sediment transport normal operating Thus, conditions of a combined For full‐filling conditions, the capacity performance of the egg‐shaped pipe in terms of averaged shear stress was only a 5.3% lower the of the is expected in the egg-shaped cross-section than in the equivalent-area circular pipesthan because equivalent circular profile. 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. Numerical results were also compared with the analytical open-channel flow Manning and Thormann-Franke formulas in Figure 5. The Thormann-Franke correction coefficients for egg-shaped.

(8) Water 2016, 8, 587. 8 of 9. sections were obtained in Fresenius et al. [19]. It can be observed that there is a good fit between the numerical and analytical mean velocities and averaged shear stress. Thus, the CFD 3D-RANS model reproduces the Thormann-Franke flow reduction due to air friction in the pipes. Water 2016, 8, 587. 8 of 9. (a). (b). Figure 5. (a) Averaged velocity and (b) shear stress comparison of numerical results for circular. Figure 5. (a) Averaged velocity and (b) shear stress comparison of numerical results for circular (circles) (circles) and egg‐shaped (triangles) cross‐sections with Manning (continuous line) and and egg-shaped (triangles) cross-sections with Manning (continuous line) and Thormann-Franke Thormann‐Franke (dashed line) formulas. Axes are normalized with the height of each conduit (H) (dashed line) formulas. Axes are normalized with the height of each conduit (H) and their full-depth and their full‐depth mean velocity (U0) and averaged shear stress (τ0), respectively. mean velocity (U0 ) and averaged shear stress (τ0 ), respectively.. Numerical results were also compared with the analytical open‐channel flow Manning and Thormann‐Franke formulas in Figure 5. The Thormann‐Franke correction coefficients for egg‐shaped 4. Conclusions sections were obtained in Fresenius et al. [19]. It can be observed that there is a good fit between the. Within the framework of an R&D project a newshear egg-shaped cross-sectional pipemodel for small numerical and analytical mean velocities and averaged stress. Thus, the CFD 3D‐RANS combined sewer systems was defined and analyzed. The geometric definition resulted from an analysis reproduces the Thormann‐Franke flow reduction due to air friction in the pipes. of dry-weather flow conditions in sewers. As the main source of pollution in low-flow conditions Conclusions at the bottom of pipes, egg-shaped pipes will improve the transport of solids is the4.sedimentation because this section a lower radius thanegg‐shaped standard circular pipes during dry weather Within the presents framework of anhydraulic R&D project a new cross‐sectional pipe for small flow conditions. combined sewer systems was defined and analyzed. The geometric definition resulted from an analysis flow conditionsofinthe sewers. As the pipe, main asource of pollution in low‐flow To studyof thedry‐weather hydraulic characteristics egg-shaped CFD model was developed so that conditions is the sedimentation at the bottom of pipes, egg‐shaped pipes will improve the transport the egg-shaped profile could be compared with an equivalent-area circular section. The CFD model of solids because thisofsection presentsina an lower hydrauliccross-section radius than standard circular pipes profiles during and was validated with a set experiments egg-shaped metal pipe. Velocity dry weather flow conditions. shear stress were used to compare the numerical model and the experimental results, obtaining a good To study the hydraulic characteristics of the egg‐shaped pipe, a CFD model was developed so agreement. Furthermore, the numerical velocity distributions were compared with an experimental that the egg‐shaped profile could be compared with an equivalent‐area circular section. The CFD formulation for analytic geometries resulting in a satisfactory concordance. model was validated with a set of experiments in an egg‐shaped cross‐section metal pipe. Velocity Once hydraulic characteristics of the egg-shaped cross-section were analyzed,results, a circular profilesthe and shear stress were used to compare the numerical model and the experimental pipe with an equivalent area was modeled. Several discharge conditions were simulated mainly obtaining a good agreement. Furthermore, the numerical velocity distributions were compared with for low-depth ratios. Atformulation the same time, numerical results were compared with concordance. analytical Manning and an experimental for analytic geometries resulting in a satisfactory Once the hydraulic characteristics the egg‐shaped cross‐section were analyzed, a circular pipe withpipes Thormann-Franke open-channel flow of formulas. It was proved that egg-shaped cross-section an better equivalent area was modeled. Several discharge conditions were simulated mainly low‐depth present hydraulic characteristics for dry-weather flows up to h/H = 0.25 fillingforratio in terms of At theand same time, numerical resultsvalues. were compared withof analytical Manning andthat Thormann‐ meanratios. velocities averaged shear stress The results this study suggest egg-shaped Franke open‐channel formulas. It was that egg‐shaped pipes present better cross-section pipes may flow be competitive withproved conventional circularcross‐section pipes for the design of combined hydraulic characteristics for dry‐weather flows up to h/H = 0.25 filling ratio in terms of mean velocities sewer systems.. and averaged shear stress values. The results of this study suggest that egg‐shaped cross‐section pipes may be competitive conventional the design of combined sewer systems. Acknowledgments: This with study was fundedcircular by the pipes Centreforfor the Development of Industrial Technology (CDTI) through the FEDER-INNTERCONECTA project “OvalPipe: Desarrollo de tuberías ovoides para la mejora de la Acknowledgments: This study was funded by the Centre for the Development of Industrial (CDTI)S.L.U., eficiencia las redes de alcantarillado” (Ref. ITC 20133052) powered by companies ABN Technology PIPE SYSTEMS throughEDAR the FEDER‐INNTERCONECTA “OvalPipe: Desarrolloand de tuberías la mejoraAnálisis de la EMALCSA, Bens S.A. and M. Blancoproject S.L., and by the MINECO FEDERovoides project para “SEDUNIT: de eficiencia redes de alcantarillado” (Ref. ITC 20133052) powered by companies ABN PIPE SYSTEMS S.L.U., los procesos delas acumulación, erosión y transporte de sedimentos cohesivos en sistemas de saneamiento unitario” (Ref. CGL2015-69094-R). TheS.A. authors would also like to by thank to María Bermúdez Luis“SEDUNIT: Cea for their assistance EMALCSA, EDAR Bens and M. Blanco S.L., and the MINECO and FEDERand project Análisis in reviewing the manuscript. de los procesos de acumulación, erosión y transporte de sedimentos cohesivos en sistemas de saneamiento.

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