Scientific Paper / Artículo Científico  https://doi.org/10.17163/ings.n23.2020.01 pISSN: 1390-650X / eISSN: 1390-860X EXPERIMENTAL AND NUMERICAL STUDY OF THE PRESSURE OF THE WATER FLOW IN A VENTURI TUBE ESTUDIO EXPERIMENTAL Y NUMÉRICO DE LA PRESIÓN DEL FLUJO DE AGUA EN UN TUBO VENTURI
 San Luis B. Tolentino Masgo1,*,2
 Abstract Resumen The Venturi tube is a device used to measure the flow rate in different industrial processes. In the present work, a study is carried out for two cases, one experimental and another numerical of the pressure exerted by the flow of water on the walls of a Venturi tube. In the first case, five experiments with different flow rates are carried out. In the second, the flow is simulated for two types of meshes and two turbulence models, using the code COMSOL Multhiphysics 4.3. The experimental and numerical results showed that the pressures of the flow on the walls in two references identified as C and G keep their magnitude constant; in addition, the numerical profiles showed that the lowest pressure drop occurs in the wall at the inlet and outlet of the throat section. It is concluded that, the distribution of the flow pressure in the wall of the throat section has a convex profile, and the results of pressures obtained for the standard k − e turbulence model are more adjusted to the experimental data. El tubo Venturi es un dispositivo utilizado para medir el caudal en diferentes procesos de la industria. En el presente trabajo, se realiza un estudio para dos casos, uno experimental y otro numérico de la presión ejercida por el flujo de agua en las paredes de un tubo Venturi. En el primer caso, se realizan cinco experimentos con diferentes caudales. En el segundo, el flujo se simula para dos tipos de mallas y dos modelos de turbulencia, utilizando el código COMSOL Multhiphysics 4.3. Los resultados experimentales y numéricos mostraron que las presiones del flujo sobre las paredes en dos referencias identificadas C y G mantienen constante su magnitud; además, los perfiles numéricos mostraron que la menor caída de presión se presenta en la pared a la entrada y salida de la sección de la garganta. Se concluye que, la distribución de la presión del flujo en la pared de la sección de la garganta tiene un perfil convexo, y los resultados de presiones obtenidos para el modelo de turbulencia k-e estándar, se ajustan más a los datos experimentales. Keywords: Water flow, Turbulence model, Pressure, Simulation, Venturi tube. Palabras clave: flujo de agua, modelo de turbulencia, presión, simulación, tubo Venturi. 1,*Department of Mechanical Engineering, Universidad Nacional Experimental Politécnica “Antonio José de Sucre” Vice-Rectorado Puerto Ordaz, Bolívar, Venezuela. 2Group of Mathematical Modelling and Numerical Simulation, Universidad Nacional de Ingeniería, Lima, Perú. Corresponding author ): sanluist@gmail.com http://orcid.org/0000-0001-6320-6864 Received: 28-06-2019, accepted after review: 29-10-2019 Suggested citation: Tolentino Masgo, San Luis B. (2020). «Experimental and numerical study of the pressure of thewater flow in a venturi tube». Ingenius. N._ 23, (january-june). pp. 9-. doi: https://doi.org/10.17163/ings.n23.2020.01.

 is located at the outlet of the Venturi tube, and the purge valve is located at the right end of the collector. The valve for regulating the flow rate is ahead of the Venturi tube and after the pump, which is not shown in the figure. Figure 1. Venturi tube experimental equipment. The readings of the water columns are measured in millimeters.   The 3D geometry and the projection on the plane of the Venturi tube are illustrated in Figure 2, which also shows the location of the references A, B, C, D, E, F, G, H, J, K and L, places to where the eleven plastic hoses are connected. Reference A is located at the beginning of the straight section; B and C are located in the convergent section; D in the middle of the straight section of the throat; E, F, G, H, J and K are located in the divergent section; and L is located at the end of the straight section, at the outlet of the Venturi tube. The greater internal diameter of the two straight sections is 26 mm, the internal diameter of the throat is 16 mm, and the total length of the Venturi tube is 156 mm. The internal diameters of the cross sections and of the locations of the references are shown in Table 1. Figure 2. (a) 3D geometry of the Venturi tube. (b) Location of the references and dimensions of the longitudinal sections in millimeters.     Table 1. Internal diameter for each reference, and axial distance where the references are located The experimental test was carried out through the following steps: initially, both valves, the valves for flow rate regulation and for flow rate control, were opened at 100 %. Once the pump of the test bench was in operation, the control valve was closed 100 %, the air trapped in the hoses and in the collector was let out through the purge valve leaving it totally full of water; subsequently, the regulation valve was closed 100 %. Then the control valve was opened 100 %, and through the purge valve air from the local atmosphere was let in, thus allowing the formation of water columns at the established height of 140 mm as initial position, all at the same level, for the eleven hoses. The reading of 140 mm remained inside the range of 0.0-200 mm of the panel, as shown in Figure 3. Afterwards, five experimental tests were carried out increasing the opening of the regulation valve, for visually taking the piezometric readings for the range of flow rate 2, 244 × 10-4 − 3, 7 × 10-4 (m3/s). Figure 3. Initial position of the piezometric heights of 140 mm of the levels of the eleven water columns. From left to right, the first piezometric tube is connected in the reference A of the wall of the Venturi tube, the second in the reference B, and similarly, the remaining are located up to the reference L (see Figure 2).

2.2. Numerical simulation

2.2.1. Governing equations

The governing equations applied to the CFD, for an incompressible flow, in stationary conditions, and simulated for a 2D computational domain with axial symmetry, in their differential form are expressed as:

Equation of conservation of mass. (1)

Equation of conservation of linear momentum, in the axial direction. (2) (3)

where the parameters are: the density p, the axial

velocity vx and radial velocity vr, the radius r, the viscosity μ, the pressure gradients and ,and the forces in the axial direction Fx and in the radial direction Fr.

The turbulence model is coupled to the equation of linear momentum, and are semi-empirical transport equations that model the mixing and diffusion that increase because of the turbulent eddies, and are solved through the Reynolds average number Navier Stokes equation (RANS) . The initial research studies about turbulence were conducted by Kolmogorov (1941), based on the results obtained by Reynolds (1883). It is worth noting that the turbulence models standard k-e of Launder and Spalding  and standard k w of Wilcox , are employed in the present work for the simulation of the flow.

2.2.2. Computational domain and mesh

The 2D computational domain with axial symmetry shown in Figure 4 is considered, due to the symmetry

of the geometry of the Venturi tube. This simplification of the geometry from 3D to 2D contributes to reduce the number of cells in the mesh, the processing time, and the computational cost; the simplifications are very common for solids of revolution and symmetric primitive geometries. Besides, in the same figure of the 2D domain, the places where the boundary conditions are applied have been marked. Figure 4. 2D computational domain with axial symmetry in the x axis, of the Venturi tube.

Figure 5 shows the 2D meshed domain, in which two types of cells are used, quadrilateral and triangular. The domain meshed with quadrilateral cells has 19600 elements, and the domain meshed with triangular cells 18047 elements. For both cases, the mesh was refined in the regions adjacent to the walls, due to the presence of shear stresses in those regions of flow. The throat section is also shown in detail in the same

figure, where it is observed how the quadrilateral and triangular cells are distributed.

As part of a study of the numerical convergence, before obtaining the final mesh which is shown in Figure 5, the throat section was refined five times until obtaining an optimum mesh density. Such refinement

in the throat was because this is a critical section due to pressure drop in the flow. The reference D is located in the middle of the throat length

 (see Figure 2), where the final numerical result of the pressure was 44.79 (mmH2O) for the mesh with quadrilateral cells and 51.38 (mmH2O) for the mesh with triangular cells, evaluated with the standard k −e turbulence model; and the pressure was 48.53 (mmH2O) for the mesh with quadrilateral cells and 55.75 (mmH2O) for the triangular cells, evaluated with the standard k − w turbulence model; obtaining for both cases numerical convergence errors smaller than 0.01 %. The quality of the mesh was evaluated for two dimensional cells, where for the case of quadrilateral cells it was obtained a maximum element size of 0.0105 mm, a minimum element size of 4.68×10−5 mm, a curvature of 0.3, and a rate of increase of 1.3; similarly, for the case of triangular cells it was obtained a maximum element size of 3.64×10−4 mm, a minimum element size of 5.2×10−5 mm, a curvature of 0.25, and a rate of increase of 1.15. These final results indicate that the two domains meshed with quadrilateral and triangular cells are of good quality. The computational domains were discretized in the mesh platform of the code COMSOL Multiphysics version 4.3, which applies the finite element method (FEM). The boundary conditions for the pressures of the water flow applied at the inlet (reference A) and at the outlet (reference L) of the Venturi tube, are shown in Table 2. The walls of the Venturi tube are considered adiabatic. The velocity of the flow at the walls in the radial and axial direction is zero due to the presence of shear stresses. In the axial symmetry in the x axis, the velocity of the flow in the radial direction is zero. Figure 5. (a) Meshed 2D computational domain. Detail of the throat section, (b) Structured mesh with quadrilateral cells, (c) Mesh with triangular cells. Isothermal flow is considered along the entire computational domain, and for a water temperature of 24 °C the density was 997.1015 kg⁄m3 and the dynamic viscosity 0.00091135 Pa.s, where both physical parameters are set as constants for the simulation of the flow.   Table 2. Boundary conditions: inlet pressure (reference A), outlet pressure (reference L) 2.2.3. Method of computational solution and Equipment   A 2D geometry with axial symmetry and stationary flow conditions, was chosen as the option for the simulation of the isothermal flow in the COMSOL Multiphysics code. The turbulence models standard k – e and standard k − w were applied for the turbulent flow, for the domains meshed with both quadrilateral and triangular cells. A fixed value of 0.001 was determined for the relative tolerance. For the solution, the maximum number of iterations was established as 100, and the solution method parallel sparse direct solver (PARDISO) was employed. For data processing, an equipment with the following characteristics was utilized: Siragon Laptop, model M54R, Intel Core 2 Duo, two 1.8 GHz processors, and a RAM memory of 3 GB.   3. Results and discussion   3.1. Experimental results   For each experiment that was carried out, the flow rates of the water were obtained by means of the volumetric method, and the results are shown in Table 3. The Reynolds number was determined with these values of flow rate.   Table 3. Experimental data of flow rates Table 4 presents the magnitudes of the Reynolds number obtained in the references A, L and D. In reference A the Reynolds number has the same magnitude than in reference L, and this is because the Venturi tube has the same diameter. For the five experiments, the Reynolds number obtained in the references A and L was in the range 12000 Figure 6. Experimental data of readings of the piezometric heights of the water columns taken at different reference points of the walls of the Venturi tube Temperature of the water 24 . Figure 7. Piezometric heights for different levels of the water columns, corresponding to experiment 3.   On the other hand, if the pressure loss from the reference C to the reference G is analyzed, the pressure drop remains invariant even though the kinetic energy of the fluid is increased in such regions because, as shown in Table 5 and plotted in Figure 6, there is a fixed piezometric height of 139.5 mm in the reference C and of 136.0 mm in the reference G, where the pressure difference is 3.5 (mmH2O) for the five experiments that were conducted. The pressure differences between the references AD and A-L are presented in Table 6, and Figure 8 shows the behavior of both straight lines by means of the linear trend line, which has a determination coefficient R2=0.997 for the A-D pressure difference, and a value R2=0.996 for the A-L pressure difference. Both results show that there is a proportionality of pressure difference with respect to the flow rate.   Table 6. Pressure difference between the references A and D, and the references A and L, for each experiment  Figure 8. Trend lines and coefficient of determination R2.   During the operation of the pump, for taking the experimental readings there were vibrations in the test bench. Therefore, after an estimated time between four and six minutes, when the disturbances were minimum, it was proceeded to take the readings by direct observation of the level of each of the eleven water columns in the measurement unit of one millimeter, and a magnifying glass was used for amplifying the image when taking a reading of the water level located in the middle of the unit of one millimeter. The obtained experimental results do not quantify the magnitude of the pressure between each reference, because they have a separation distance. Therefore, it is of interest to quantify and know the behavior in a continuous manner of the trajectory of the pressure profile along the walls of the Venturi tube, being of greatest interest between the references C and E, because this is the place where the largest pressure drops occur. For this purpose, the flow should be simulated through CFD, and thus know what could really happen.

 3.2. Numerical results and comparison with the experimental data   The simulation of the velocity distribution of the isothermal flow in the Venturi tube is shown in Figure 9, both in the cross section and in the plane, where the increase of the velocity of the flow occurs in the throat, and the decrease in the divergent section. In the latter the contour lines of velocity acquire a parabolic profile in direction to the x axis, due to the effect of the boundary layer. The velocity of the flow is maximum in the axial symmetry of the x axis, and its magnitude decreases toward the walls of the Venturi tube, thus having a gradient of velocity in the flow field. It is worth noting that the domain of the flow shown in Figure 9, was simulated with structured mesh and quadrilateral cells, employing the standard k – e turbulence model; since they are similar, other figures of the contour lines of velocity for the results of the standard k − w turbulence model are not shown. Figure 9. Distribution of velocity (m⁄s). (a) In the cross section, and (b) Projected on the plane.   Figure 10 shows the behavior of the velocity profiles evaluated in the axial symmetry of the x axis. According to all the trajectories of the profiles, toward the end of the convergent section the flow increases its velocity, in the throat section reaches a maximum velocity in the reference D, and decreases its velocity in the divergent section. The magnitude of the velocity of the flow in the reference A is smaller with respect to the reference L, thus it is understood that the behavior of the profile of the flow velocity in the radial direction has smaller curvature in the reference A, and larger curvature in the reference L. The numerical results of the flow rates are shown in Table 7, for each turbulence model and type of mesh. The largest magnitude of flow rate was for the standard k-e turbulence model and domain meshed with quadrilateral cells, and the smallest flow rate for the standard k − w turbulence model and domain meshed with triangular cells. It should be noted that the flow rate was determined with the average velocity of the flow, using the numerical integration method. Figure 10. Profiles of velocity evaluated in the axial symmetry of the x axis, for the standard k − e turbulence model.   When the numerical flow rates presented in Table 7 are compared with the experimental values presented in Table 3, it can be seen that the greatest percentage error was 9.68 % for the standard k-e turbulence model and the domain meshed with quadrilateral cells, and 8.68 % for the mesh with triangular cells; the minimum percentage error was 1.48 % for the standard k-e and the mesh with quadrilateral cells, and 1.01 % for the standard k − w and the mesh with triangular cells, as shown in Table 8. Based on the results, it is evident that there is an influence on the numerical results of the type of mesh applied to the computational domain.   Table 7. Flow rates obtained for two turbulence models and two types of applied meshes Table 8. Percentage error of the flow rates The numerical results of Reynolds number obtained for the greatest diameter in the references A and L are shown in Table 9, and for the smallest diameter in the reference D are shown in Table 10. In the references A and L, for the five simulations, the Reynolds number was in the range 12100

 pressure drop in the throat occurs because the flow passes hrough the narrow section at a higher velocity, due to the large difference of the flow pressure between the inlet and the outlet of the Venturi tube. Figure 12. Experiment 1 of pressures of water columns, and pressure profiles evaluated at the wall of the Venturi tube. Pressure in (mmH2O): 160 mm at the inlet and 150 mm at the outlet. Figure 13. Experiment 2 of pressures of water columns, and pressure profiles evaluated at the wall of the Venturi tube. Pressure in (mmH2O): 170 mm at the inlet and 155.5 mm at the outlet. Figure 14. Experiment 3 of pressures of water columns, and pressure profiles evaluated at the wall of the Venturi tube. Pressure in (mmH2O): 179 mm at the inlet and 161 mm at the outlet. Figure 15. Experiment 4 of pressures of water columns, and pressure profiles evaluated at the wall of the Venturi tube. Pressure in (mmH2O): 190 mm at the inlet and 167 mm at the outlet. Figure 16. Experiment 5 of pressures of water columns, and pressure profiles evaluated at the wall of the Venturi tube. Pressure in (mmH2O): 199.5 mm at the inlet and 173 mm at the outlet.   The numerical results and the experimental data in the reference D are shown in Table 11. The standard k−e turbulence model showed the greatest percentage error of 5.88 % for experiment 4, for the standard k−w  it was 7.84 % for experiment 5, and values smaller than these for the remaining results, i.e. for the mesh with quadrilateral cells. On the other hand, for the mesh with triangular cells, corresponding to experiment 5, the standard k − e had a percentage error of 14.17 %, for the standard k − w it was 21.66 %, and smaller for the remaining results.   Table 11. Experimental and numerical data for the reference D, for two turbulence models and two types of applied meshes The pressure profiles for the standard k − e turbulence model, simulating the flow with the domain meshed with quadrilateral cells, and the experimental data of pressure presented in Table 5 and plotted in Figure 6, are presented in Figure 17, where it is observed that the trajectories of the profiles satisfy the validation with the experimental data. In the reference D located at the throat section, the trajectories of the profiles are convex. Although a graph similar to Figure 17 is not presented for the standard k − w turbulence model, this model also shows validity but with slightly varied margins of numerical results with respect to the standard k − e turbulence model, as shown in the results of Table 12 previously presented. From the analysis that was carried out, the numerical results are influenced by the type of mesh applied to the computational domain, and the structured mesh with quadrilateral cells provides more accurate numerical results compared to the mesh with triangular cells. Figure 17. Experimental data of pressure of water columns and pressure profiles evaluated at the walls of the Venturi tube with the standard k−e turbulence model   The pressures of the flow along the axial symmetry of the x axis, compared with the experimental data of the pressures on the walls of the Venturi tube, are shown in Figure 18. The profiles intercept with one another in the references C and G, and it is observed the evolution of the pressure drop trajectories in the left end of the reference D and of the pressure

 increase in the right end of the same reference; the trajectories of the curves are concave along the throat section. The small separations of the profiles of the experimental data are also shown, thus the magnitudes of the pressures in each of the references are some slightly greater and others slightly smaller with respect to the experimental data of the pressures on the walls. The profiles were obtained for the domain meshed with quadrilateral cells and the standard k − e turbulence model. Figure 18. Experimental data of pressure of water columns and pressure profiles evaluated at the axial symmetry (x axis) of the Venturi tube with the standard k –e turbulence model.   When the numerical results of Figure 18 are compared with Figure 17, it is evident that for the locations of the references A, B, C, D, E, F, G, H, J, K and L, the pressures tend to be perpendicular to the x axis and to the walls, forming a trajectory of curves known as isobars. However, in the places where the sections come together, in the vertices, the configuration of the trajectories of the curves has a different behavior due to the sharp variations of pressures, induced by the geometrical profile of the section of gradual contraction and gradual expansion of the Venturi tube. Figure 19, as a detail, unifies Figures 17 and 18, for the distance range 30-100 mm, showing superimposed profiles stretches for the pressures on the walls and on the x axis, which are compared with the experimental data for the references C, D, E, F and G. It is observed the pressure drops at the ends of the throat and how the curves intercept and border the experimental data. It is shown that the abrupt drops of the numerical pressures on the wall occur for the 38.67 mm position, at the beginning of the throat; and the other pressure drops occurs at the 54.49 mm position, which is located at the beginning of the divergent section. Figure 19. Experimental data for the references C, D, E, F and G, and stretches of pressure profiles evaluated at the walls and at the x axis of the Venturi tube with the standard k − e turbulence model     Table 12 shows the numerical values of the pressure drops at the wall at the ends of the throat section. It is shown that the smallest drop of numerical pressure occurs at the 38.67 mm position, for the curve 199.5- 173 mm which corresponds to experiment 5, being the magnitude of the pressure drop –23.23 (mmH2O), and since it is a negative pressure, it is evident that it is a suction pressure; similarly, for the curve 190-167 mm corresponding to experiment 4 and for the same position, the pressure was –2.08 (mmH2O).   The vertex, where the negative pressure occurs, corresponds to a very small part of an estimated radius of action of 0.2 mm, where the negative pressure is an unexpected result, because the hydrodynamic profile of the internal wall of the Venturi tube that measures the flow rate, the convergent section has an average angle of design with the purpose of avoiding negative pressures.

 Table 12. Pressure drops at the ends of the throat Similarly, it is remarked that during experiments 4 and 5 no air bubbles were observed in the region around the vertex located at the inlet of the throat section and downstream, as a sign of cavitation. Therefore, the numerical result of the negative pressure, induces to investigate with sensitive instruments to capture the possible air bubbles with dimensions imperceptible to the human eye that might be present. Therefore, it should be verified simulating the flow with different turbulence models in a future work, to determine if negative pressures appear or not, thus obtaining conclusions close to the reality of the physical phenomenon. From a comparison of the numerical results with the experimental data, it is evident that the simulation yields satisfactory results sustained by the ranges of error which are acceptable according to engineering criteria; thus the standard k − e and standard k − w turbulence models are validated. These two validated turbulence models strengthen their application in the computational fluid dynamics in the simulation of the flow in computational domains with simple or complex geometries in the field of engineering, and allow to determine the magnitude of some physical parameter that cannot be obtained through measuring instruments and analytical equations.   4. Conclutions   Based on the analyses that were carried out, for the cases of experimental and numerical study, it is concluded that: The obtained numerical flow rates for the standard k-e and standard k − w turbulence models, when compared to the five experimental data of flow rates, yielded percentage errors in the range 1.01-9.68 %. Similarly, it was determined the percentage error in the range 1.01-9.68 % for the Reynolds number. For the five experiments, the Reynolds number is in the range 12000

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