Numerical model for a nineteenth-century hydrometric module

Numerical Model for a Nineteenth-Century Hydrometric Module Jordi Vives-Costa1; José-Juan De-Felipe2; and Esteban Pena-Pitarch3 Abstract:TheCequiaofManresaisa26-kmopenchannelconstructedinthefourteenthcenturythatconveyswaterfromtheLlobregatRiver tothecityofManresausinganelevationdifferenceof10.4m.Thechannelisstilloperationaltoday,supplyingwaterfordomestic,industrial, andagriculturalusestoanoverallpopulationof150,000people.Ahydrometricmodulewasconstructedin1864toregulatetheflowratein theCequiatounder1,000L=s.Thismodulewasdismantledin1959andiscurrentlynonoperational.Thisworkstudiedtheoperationofthe moduleanddeterminedwhetheritmettheobjectivesforwhichitwasbuilt.Themodulewasmodelednumerically.Owingtothelackof experimentalmeasuresfromthemodule,themodelwasvalidatedwithananalyticalmodelfromtheliterature,whichdemonstratedits accuracy.Themodelwascreatedbyapplyingcomputationalfluiddynamics(CFD)usingFLOW-3Dsoftware.Theresultsshowedthat thenumericalmodelreproducedthedynamicbehaviorofthehydrometricmodule(transitory),andthattheoldautomatedsystemoperated correctly,limitingtheflowtotherequiredvalue.Thenumericalmodelscanbeusedasatoolforhistoricalresearch. Authorkeywords:Computationalfluiddynamics(CFD);Three-dimensionalsimulation;Sluicegate;Weir. Cequia was conceived exclusively for agricultural purposes; however, coinciding with the beginnings of the first industrial revolution, the small waterfalls along its route to the Cardener River became a source of energy to power the machines of an incipient industry. This caused a greater demand for water resources to meet the growing consumption of energy. Soon after, the first conflicts between irrigators and industrialists for the exploitation of canal water occurred (Oliveras 1986). The disputes over the flow rate finally led to the construction of the hydrometric module in 1864, by order of Spanish Queen Isabel II. The objective of the hydrometric module was to limit the flow rate to a maximum of 1,000 L=s. Despite the construction of the hydrometric module, the controversy over the flow of the Cequia during its operation remained. This paper investigated the operation of the module and determined whether it met the objective of regulating the flow of the channel when it was operational. The original hydraulic module project was designed by engineer Jaime de Castro, who delivered a memorandum accompanied by sketches on August 10, 1863. Figs. 2and 3show the original sketches of the project from 1863. The system consisted of two preparatory relaxation vessels. In the first vessel, the width of the channel ranged from 1.5 to 2.4 m over a length of 7 m. The water then passed to a different vessel, which was 3.8 m wide and 13.4 m long. A weir with a crest length of 3.8 m and a width of 0.4 m (central weir) was located near the downstream end of the second vessel, over which, supposedly, only 1,000 L=s should pass according to the concession. To achieve this objective, an upstream weir located at the first relaxation vessel with a crest located 0.28 m above the crest level of the central weir was placed on the left wall, with a constant 30-cm head. The flow discharged from the side weir entered a manhole with a manually adjustable sluice gate. This first manhole was connected with a second manhole, which contained the regulation mechanism that operated by means of a float-operated discharge valve. These manholes and the entire regulator mechanism were protected by a hut. The structure of the hydrometric module is still standing today, with the exception of the automatic mechanism and the central weir, which were removed in the midtwentieth century, when the module fell into disuse. Fig. 4shows the Introduction TheCequiaofManresacapturesflowfromtheLlobregatRiver intheprovinceofBarcelona(Fig.1).Thediversionisobtained bymeansofadamlocatedatthemunicipalityofBalsareny;the Cequiaconveyswateralongarouteofmorethan26kmtothecity ofManresa,whereitisdistributedintwoslopes.Itfeedsmorethan over800haofirrigatedland;itisusedfordomesticandindustrial purposes(Latorre2002). Thechannelisconsideredthemostimportanthydraulicworkof thelateMiddleAgesinCatalonia(Latorre1995).Itsconstruction beganin1337andinvolvedtheconstructionof30aqueductsto negotiatetopographicirregularitiesand70bridgestomakeway forroadsandtodivertwatersfromtherainwaterrunoff.Similarly, a321-m-longundergroundtunnelwasexcavatedtoconveywater. Alltheseelementsconstituteexponentsofmedievalengineering andarchitecture(Sarret1906). Kirchneretal.(2002)arguedthathydraulicsystemscannotbe understoodoutsidetheirhistoricalframework;tounderstandwhy thehydrometricmoduleofthechannelwasdesigned,itisnecessarytoinvestigatethereasonsthatledtothisdecision.Atfirst,the 1Professor,Departamentd’EnginyeriaMinera,IndustrialiTIC,Escola PolitècnicaSuperiord’EnginyeriadeManresa,UniversitatPolitècnicade Catalunya,Av.BasesdeManresa61–73,Manresa08242,Spain(correspondingauthor).Email:[email protected].03$*%IUUQT PSDJEPSH9 2Professor,Departamentd’EnginyeriaMinera,IndustrialiTIC,Escola PolitècnicaSuperiord’EnginyeriadeManresa,UniversitatPolitècnicade Catalunya,Av.BasesdeManresa61–73,Manresa08242,Spain.ORCID: https://orcid.org/0000-0002-7758-3967 3Professor,Departamentd’EnginyeriaMecànica,EscolaPolitècnica Superiord’EnginyeriadeManresa,UniversitatPolitècnicadeCatalunya, Av.BasesdeManresa61–73,Manresa08242,Spain.ORCID:https:// orcid.org/0000-0001-5986-5728 central relaxation vessels, as well as the booth containing the manholes, where the regulation mechanism was housed. Fig. 5shows the side weir, which is protected by a grate, and the structure of the two regulation manholes. The first (upstream) manhole has an adjustable sluice gate. The second (downstream) manhole is connected to the first by a 1.15-m2orifice at the bottom. This second manhole contains the float and actuation mechanism of the automatic discharge valve. The space behind the second manhole (Fig. 5) is where the discharge duct with an internal diameter of 0.3 m was housed. Computational fluid dynamics (CFD) and the finite-volume method (FVM) have been applied many times for weir flow problems with satisfactory results (Hargreaves et al. 2007;Arvanaghi and Oskuei 2013;Namaee et al. 2014;Zeng et al. 2017). The specific case of lateral wall weirs also has been studied several times (Aydin et al. 2011;Taghavi and Ghodousi 2015;Namaee and Shadpoorian 2016), frequently using FLOW-3D simulation software for general free-surface problems, and specifically for weirs and free nappes (Sarkardeh et al. 2014;Taghavi and Ghodousi 2015). Method Geometries and Computational Domain The numeric model of the hydrometric module was constructed on the basis of a reference inertial system in which the x-axis was established in the direction of flow, the y-axis was established normal to the direction of flow and parallel to the bottom of the channel and, the z-axis was established in the direction of the gravity acceleration component. The module was composed, following the flow direction of the fluid stream, in the following sections: 1. The first section is the approach channel, with nominal characteristics equal to those of the channel, i.e., 1.5 m wide and a wall height of 2 m. For this first section, a length sufficient to stabilize the inlet flow (6 m) was chosen. 2. The second section (first relaxation chamber), subsequent to the first section, is a section with a 45° divergence widening to a section 2.4 m wide and 7 m long. A side weir is located on the left wall that discharges to the first regulation manhole. 3. The third section (second relaxation chamber) is a continuous channel with a second widening, also at 45°, gives rise to the third section of the module, which is the final part of the regulator. The third section is 3.8 m wide and 13.4 m long and houses the weir (central weir) that controls the upstream water level. This weir is just 1.8 m from the downstream 30° convergence section, and ends in the fourth section of the system, which is 1.5 m in length. The physical walls were reproduced in the geometric model as shown in Fig. 6. 4. For simulation purposes, the 1.5-m-wide fourth section was lengthened to 8 m, sufficient to stabilize the flow during simulations. In total, the model has a length of 34.4 m, a maximum width of 3.8 m and a wall height of 2 m. The average slope of the channel is 0.0004. The central weir interferes with the main flow that must pass over the weir with a variable head Hthat is a function of the flow rate. The weir is a rectangular suppressed weir with a crest length of 3.8 m (y-axis) and a height of 1 m (z-axis). The weir wall thickness is 0.4 m (x-axis). Depending on the head on the weir, H, a(Hþ1)-m water depth level is obtained in the second and third sections of the module upstream of the weir, which determines the flow of water (if Hsufficient) from the side regulating weir. Fig. 1. Location map. Fig. 2. Plan view of the project of the hydrometric module. (Adapted with permission from J. de Castro, unpublished data, 1863, CDAHCF Archives, Parc de la Sèquia, Manresa, Spain.) The side weir has a height of 1.28 m above the floor of the channel, which is 0.38 m higher than the main weir. The first regulation manhole has a rectangular section of 3 m on the x-axis and 2 m on the y-axis, with a height of 3.5 m (z-axis). This structure is equipped with a 1-m-wide sluice gate that can be adjusted in height to release the excess flow and maintain a relatively constant head that feeds the second manhole. The second manhole is connected to the first through an opening in the common wall. This second manhole has dimensions of 1.4 m (x-axis) by 2 m (y-axis) and height of 2.5 m (z-axis). The second manhole contains the prismatic flotation mechanism, whose dimensions are 1 m (x-axis) by 0.7 m (y-axis) and 0.7 m (z-axis). Fig. 7shows the right-side view of the model geometry. For the simulation, the original rocker arm float system was replaced with a rigid system of bars that connect the float to the discharge orifice cover (Fig. 8) to simplify simulation while maintaining the same function. Procedure A numerical free-surface CFD model was applied to simulate the flow rate in the hydrometric module. FLOW-3D software version 11.2 was applied to numerically solve the Navier–Stokes equations for solution domains, namely the input and output sections with nominal cross sections of the channel, the relaxation vessels, the central weir, the side weir, the two regulation manholes, a sluice gate, and a circular discharge valve. To estimate turbulence flow, a standard K-epsilon model was used. Aydin (2016) analyzed the free surface over a side weir and concluded that the volume of fluid (VOF) method is a good choice for modeling free surface flow with a suitable turbulence model such as K-epsilon and is able to simulate flow over a weir (Andersson et al. 2013). Flow conditions in the main channel upstream and downstream of a side weir can have a major influence on the behavior of the flow at the weir itself Fig. 4. Hydrometric module structure. (Image by authors.) Fig. 3. Cross-section view of the project of the hydrometric module. (Adapted with permission from J. de Castro, unpublished data, 1863, CDAHCF Archives, Parc de la Sèquia, Manresa, Spain.) (Namaee and Shadpoorian 2016). The calculated results were compared, such as pressure, velocities, and output flow rates for different inlet flows. Six simulations were carried out for inflow rates of 1,000, 1,100, 1,200, 1,300, 1,400, and 1,500 L=s (Table 1). Due to the relatively large physical dimensions of the model and the variable nature of the simulation as a result of the regulation system incorporated in the model, a fairly long simulation time was needed to obtain a suitable solution that demonstrated the model’s behavior. A total simulation time of 360 s was set for each simulation. For an input channel flow of 1,000 L=s into an empty channel, a module recharge interval of about 56 s was observed, so all six flow-rate simulations started at simulation time t¼56 s and ended at t¼360 s. The average time required to complete each simulation was around 384 h using an Intel Core i7 7700 s1151 CPU with 16 GB RAM, with a total computational time of 2,304 h for the six simulations. Mathematical and Numerical Model The filtered incompressible Navier–Stokes equations are solved to study the behavior of the flow. The principle of mass conservation in differential, nonstationary, and three-dimensional form for a given point is given by ∂ρ ∂tþ∇·ðρvÞ¼0ð1Þ For these simulations, the fluid can be considered incompressible and of constant density, both in time and in the studied domain, obtaining ∇·v¼0ð2Þ The principle of conservation of momentum, considering incompressible flow is given by ∇·ðρuvÞ¼u∇·ðρvÞþðρvÞ·∇uð3Þ Considering flow occurs at low velocity and underestimating viscous dissipation, the energy equation expressed in terms of specific enthalpy is Fig. 6. Definition of the geometry of the walls of the model. Fig. 7. Right-side view of model geometry. Fig. 5. Inside the hydrometric module building. (Image by authors.) ∂ðρeÞ ∂tþ∇·ðρveÞ¼∇·ðk∇TÞþS hð4Þ The governing equations were discretized by a structured grid arranged using the finite-volume method (Verstappen and Veldman 2003). A second-order conservative scheme was used for spatial discretization. These schemes preserve the symmetrical properties of continuous differential operators and ensure both the conservation of kinetic-energetic equilibrium and model stability (Versteeg and Malalasekera 2007). Spatial Approximation The implicit method was used to solve algebraic equations derived from the system of equations in partial derivatives. Von Neumann stability analysis [Eq. (5)] was used to check the convergence of each mesh block, where vfluid is the average speed module, Δtis the time step, and Δxis the cell size Δx≤ð2vfluidΔtÞ1=2ð5Þ Different grades of meshing were used to simulate flow on the model. In order to correctly solve the boundary layer, especially in weir, it was necessary to have a very fine mesh (Ali et al. 2017). A fine mesh was also required around the regulating mechanism, which consisted of the float and the discharge valve actuator, as well as its full potential trajectory. Therefore, a prism layer was appropriate in this area due to the low nonorthogonal corrections required for this type of elements and the relative simplicity in placing this type of control volume near the surface. As flow moves away from the weir, the relative importance of flow structures in force coefficients and velocity profiles decreases, as does mesh resolution. For the approach channel, the two central relaxation vessels, and the outlet, a mesh with cubic cells of 0.1 m on each side was arranged. The same mesh size was also used for the first regulation manhole; however, for the second manhole, which houses the float, the mesh was refined to obtain a better characterization of the turbulent phenomena around the float (Stasa, P., Kebo, V. and Kodym, O., Effect of mesh density on the accuracy of the calculation using CFD, presented at 14th SGEM GeoConf. on Informatics, Geoinformatics and Remote Sensing, 2014), with cubic cells of 0.05 m on each side. The same was done for the outlet volume around the sluice gate of the first manhole and for the volume around the discharge valve of the regulating mechanism. In addition, all the space through which the mechanism can be moved was also rendered with 0.05-m/side cells. Table 2summarizes the different meshes in the domain, the size of the cells, and the total number of cells in each region. Boundary Conditions Limiting conditions specify the flow variables or their gradients within the boundaries of the computational flow domain. An upstream boundary can be specified in a channel section where the water level is known for a given flow rate. In this case, the upstream boundary should be at a sufficient distance from the first relaxation vessel to avoid the effect of reflection (Bhajantri et al. 2006). A constant velocity input profile, u¼ðUref;0;0Þ, was imposed to set the limiting conditions. For the model output domain, the relative pressure was set to zero. Symmetrical and no-slip conditions were established for the walls and base of the channel. Relative pressure conditions of zero were imposed for the top of the domain. Results Graphical Analysis Graphical analysis was used to observe the behavior of the water velocity gradient in the hydrometric module. Graphical output from Fig. 8. Left-side view of model geometry. Table 1. Cases for different inlet flows Case Inlet flow (L=s) 1 1,000 2 1,100 3 1,200 4 1,300 5 1,400 6 1,500 Table 2. Mesh blocks used in domain Mesh block Domain area Sections involved Cell size (m) Cell units m1 Central core of canal, relaxation vessels, and output channel 2, 3, 4 0.10 158,400 m2 Approach channel 1 0.10 10,080 m3 First manhole 2 (side) 0.10 13,225 m4 Sluice gate region 2 (side) 0.05 2,800 m5 Second manhole and discharge valve region 2 (side) 0.05 16,128 m6 Region of displacement of regulating mechanism 2 (side) 0.05 6,720 FLOW-3D software showed the stabilization of the outlet flow downstream of the central weir, as well as the turbulence and hydraulic jump caused by the weir. The same phenomenon was found in the lateral weir, where velocity gradients also were notable. The water velocity decreased in the two central vessels, and the fluid accelerated in the vicinity of central weir. Turbulence was generated in the region where the hydraulic jump occurred (Babaali et al. 2015). Likewise, the maximum velocity of the water outlet was at the side discharge valve, where the maximum velocity of the entire modeled regulator occurred, whereas the velocity was lowest at the exit of the sluice gate located at the first regulation manhole. Fig. 9shows the hydrometric module rear view at t¼ 240 s with an input flow rate of 1,300 L=s, and shows the outlet water jet issuing from the discharge valve when it is activated by the ascent of the float in the control box. Numerical Analysis Output flows were simulated for (1) the weir, (2) the downstream boundary of the module, (2) the sluice gate of the first manhole, and (4) the discharge valve of the regulating mechanism. The simulated output flows were variable over time and it was necessary to extend the simulation time to obtain model behavior under steady-state flow conditions. Results obtained for the six cases in Table 1are shown in Fig. 10, detailing the behavior of the module for different inlet flows. Table 2lists the mesh blocks used in the domain. Table 3 lists the sections at which flow rates were measured. Fig. 10 shows the results achieved in the six cases studied. For Case 1, with an input flow rate to the hydrometric module of 1,000 L=s, the convergence of the flow on the central weir (section a) was obtained from t¼80 s, with a value of around 740 L=s. For the output flow in the final section of the module (section b), after the hydraulic jump, there was a logical delay, 8 s, with respect to the flow on the weir. In addition, this output flow rate was very fluctuating and did not completely stabilize over the entire simulation range. This behavior is explained by the oscillating character of the hydraulic jump between these two sections. This induced very turbulent alterations in its proximity and caused great variations of the free surface of the flow, as well as of its depth and velocities. On the other hand, the discharge flow through the sluice gate of the first manhole (section c) was also variable because the flow that enters the manhole comes from the lateral weir and causes a high level of turbulence during its fall into the manhole. A conclusion of the simulation for the input flow rate of 1,000 L=s is that the regulation mechanism is not activated throughout the entire simulation range, and consequently the output flow rate through the control section (d), which corresponds to the discharge valve, is zero. For Case 2, for an input flow of 1,100 L=s the flow on the weir (section a) stabilized from t¼91 s with a value of around 771 L=s [Fig. 10(b)]. For the control section (b), which is the final output section, the flow had an oscillating behavior; however, the dispersion was smaller than that in Case 1. The flow rate through the sluice gate (section c), was also very variable, similar to that Fig. 9. Rear view at t¼240 s and Q¼1,300 L=s. obtained in Case 1; however, the behavior was more stable from t¼135 s. The automatic regulation mechanism was actuated from t¼125 s; the flow rate across the control section (d) was zero from t¼0to t¼125 s, and from this point onward ranged between Q¼0and Q¼100 L=s, depending on the position of the gauge inside the control manhole. In this case, the mechanism worked perfectly in accordance with the concept and did not reach its limit, which was around Q¼200 L=s, as was checked subsequently in Simulations 3, 4, 5, and 6. Case study 3, which corresponded to a flow rate of 1,200 L=s, showed the maximum opening of the discharge valve of the automatic regulating system for the first time; this occurred from t¼190 s. The discharge valve started to open partially from t¼ 120 s and maintained this regulation with partial openings from this time to t¼190 s, after which time it remained fully open until the end of the simulation. From t¼190 s onward the flow rate through the discharge valve (section d) remained constant at Fig. 10. Balance of output flows for different input flows: (a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4; (e) Case 5; and (f) Case 6. Table 3. Measured flow rates through sections Measured flow rate Section Mesh involved QðaÞCentral weir m1 QðbÞOutput channel m1 QðcÞSluice gate m4 QðdÞDischarge valve m5 200 L=s. Outlet flow on the weir, QðaÞ, was around 800 L=s at a stable rate between t¼80 and t¼242 s. Between t¼242 and t¼315 s, the flow on the weir increased to 1,000 L=s, and stabilized again at 803 L=s from t¼315 s to the end of the simulation interval, t¼360 s. The same phenomenon also occurred in Simulations 4, 5, and 6. In all these cases, the discharge valve opened completely. The maximum flow peaks over the weir were induced after full opening of the discharge valve, probably because of the suction that altered the normal flow over the weir downstream at some point. The regulated flow rate crossing over the weir (section a) was around 906 L=s for Case 4. Flow behavior on the weir was very stable from t¼90 s. However, the point flow varied between t¼ 210 and t¼277 s, which resulted in a maximum of 1,050 L=s to restore stability up to t¼360 s. For an input flow rate of 1,400 L=s (Case 5) the regulated flow rate crossing section (a), above the weir, was 981 L=s, and remains very stable except for a punctual rebound in which the flow rate was a maximum of 1,115 L=s. With the regulation system operated by the float in the manhole float, it started operating from t¼85 s in a partial opening mode, with an average flow rate of around 110 L=s for section (d) until t¼273 s. From that time onward, the discharge valve opened completely and facilitated discharge of 200 L=s until the end of the simulation. The last case studied (Case 6), was obtained for an input flow of 1,500 L=s. Outcomes showed that the regulated flow was above the limit value set by the administrative concession. The flow rate above the weir (section a) was 1,070 L=s, which was slightly higher than the target flow rate, i.e., 7% higher than the target value of the regulation, 1,000 L=s, for the concession. This regulated flow rate was 71.3% of the input flow; in other words, the flow disregarded by the module that was returned to the river in this situation was about 28.7% of the flow rate absorbed by capturing the irrigation ditch. The value of the flow rate across section (b), at the outlet of the hydrometric module, was very stable with respect to the previous cases, probably due to the fact that when there is a greater depth of water in the channel, fluctuations of free surface caused by the hydraulic jump are relatively minor. The automatic regulation mechanism started to operate at t¼80 s; it operated in partial mode until t¼140 s, after which it remained at maximum speed, with the discharge valve fully open, until the end of the simulation, with an average flow in section (d) of 200 L=s. Fig. 11 compares the output flow rate results in section (a), the sluice gate, obtained for Cases 2, 4, and 6, i.e., for inlet flow rates 1,100, 1,300, and 1,500 L=s. The values of the waste flow had great variability, from a minimum of 130 L=s to a maximum of 400 L=s. Case 1, corresponding to a flow rate of 1,000 L=s, is not represented in Fig. 11 because this flow rate did not drive the automatic regulation mechanism (Fig. 10). Results of Cases 3 and 5 are not represented in order to make Fig. 11 easier to read. Fig. 12 compares the discharge valve flow rates (section d) for Models 2, 3, 4, 5, and 6, which correspond to inlet flow rates from 1,100 to 1,500 L=s in 100-L/s steps. The mechanism started partial operation between t¼80 and t¼120 s depending on the different inlet flow rates, so it opens earlier, when the flow is greater. The only case in which the regulation mechanism operated partially for the entire simulation was Case 2 for Q¼1,100 L=s, whereas for the rest of cases, the operation was partial opening to full opening from t¼130 to t¼280 s, depending on the input flow rate. When the automatic regulation system operated in total mode, a discharge capacity of 200 L=s was achieved regardless of the input flow rate. Fig. 13 compares of the flow values obtained on the weir (section a) for different inlet flows between 1,100 and 1,500 L=s in intervals of 100 L=s. The values obtained were below the maximum limit of 1,000 L=s established by the administrative concession for Cases 1, 2, 3, 4, and 5. However, for an input flow of 1,500 L=s (Case 6), there was a small surplus of 7%. This percentage is very small in relation to the input flow rate, which is 50% higher than the flow to be regulated. Oscillations appeared as peak flow rates; however, the discharge valve is fully opened after a certain interval of time, then fluctuations stabilized quickly (Fig. 13). Validation Validation was based on the correspondence of the values obtained analytically in advance of the numerical model. The analytical model described a physical phenomenon from the equations recognized by the scientific literature. Numerical results obtained for the discharge pipe outlet flow rate (section d) were compared with the calculations obtained analytically for different heads Fig. 11. Flow rate at the outlet sluice gate (section c). QðdÞ¼kdAhð2gHÞ1=2ð6Þ The Torricelli equation [Eq. (6)] was used to determine the outlet flow, Q, through the discharge valve from the head above it, H; the orifice area, S h; and a nondimensional discharge coefficient, Kd, that depends on the geometry of the orifice (Franchini and Lanza 2013;Blasone et al. 2015). The discharge coefficient for a thin-walled circular orifice has been estimated at 0.6 (White 1999). Discharge over the central weir (section a) was determined from QðaÞ¼2 3Cdð2gÞ1=2bh3=2ð7Þ where h= head on the crest; b= weir width; and Cd= dimensionless discharge coefficient which depends on the geometry of the weir (Aydin et al. 2011;Pandeyp et al. 2016;Arvanaghi and Oskuei 2013;Wu and Rajaratnam 2015). The discharge coefficient for a thin-walled rectangular weir has been estimated at 0.6 (Arvanaghi and Oskuei 2013;Turalina et al. 2015;Lin et al. 2002). Table 4compares the values obtained with the numerical simulation and those obtained analytically from Eq. (1) for section (d), corresponding to the discharge valve. Results for Cases 1 and 2 were not be relevant. For the first case the automatic mechanism was not activated and therefore the flow rate was zero, even though there was an average head of 0.2 m; Case 2 was not taken into Fig. 12. Flow rate at the discharge valve (section d). Fig. 13. Flow rate at the central weir (section a). Table 4. Flow rates obtained analytically versus model-derived flow rates in section (d) (discharge valve) Case Input flow rate (L=s) Qof discharge valve (L=s) Head on section (m) Qanalytically obtained (L=s) Difference (%) 1 1,000 0 —— — 2 1,100 51.3 0.20 84.0 −63.8 3 1,200 198.7 1.05 192.5 3.1 4 1,300 195.9 1.08 195.2 0.3 5 1,400 200.4 1.07 194.3 3.0 6 1,500 201.6 1.15 201.5 0.1