International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:04 19 Modeling and Applications of 3D Flow in Orifice Plate At Low Turbulent Reynolds Numbers Mohamed A. Siba 1, Wan Mohd Faizal Wan Mahmood 2, a, Mohd Z. Nuawi 2, b, Rasidi Rasani 2, c, and Mohamed H. 3, d* Nassir 1 Institute of Technology-Baghdad, Foundation of Technical Education, Baghdad, Iraq. moh_siba@yahoo.com 2 Department Mechanical and Material, Faculty of Engineering and Built Environment, National University of Malaysia, Selangor, 43600 Bangi, Malaysia; a wmfaizal@eng.ukm.my; b zakinuawi@eng.ukm.my; c rasidi@eng.ukm.my; d* mohamedh.nassir@taylors.edu.my Abstract-- Numerical study of turbulent flow in orifice plate within a pipe is carried out by utilizing the Navier-Stokes (N-S) equations. These equations are numerically solved using primitive variables with a finite volume method, and simulated using the based commercial CFD code ANSYS. The N-S equations are solved for flow patterns that are presented for unsteady flow of an orifice plate at different values of Reynolds number and aspect ratios. The study is performed at Reynolds numbers of 5000, 10000, and 15000 and at aspect ratios of 0.2, 0.3, and 0.5. The results demonstrate the following parameters: velocity profile, differential Pressure, and vorticity of the flowing fluid inside the pipe and around the orifice. The study also demonstrates the mechanical effects caused by the flow passed the orifice such as stress, strain, and total deformation. It is found that as Reynolds number increases the flow velocity increases, while the differential pressure shows very steep jump across the orifice. Concerning the vorticity, the images show that both Reynolds number and the aspect ratio influence the structure of the flow that passes the orifice. The mechanical properties are found to be strongly affected by both Reynolds number and aspect ratios. Index Term-- Orifice, CFD-ANSYS, Mechanical Properties, Navier-Stocks Equations 1. INTRODUCTION The solution of Navier-Stokes equations (N-V) for the flow in a pipe becomes complex as more conditions are implemented to better represent a solution such as viscosity, incompressibility, orifice type, and any other fittings added. Navier-Stokes equations (N-V) have been utilized to solve for the characteristics of the flow. The orifice plate has advantage over all other fittings as it is used not only to control the flow but also as a measuring device [1]. In the Unites States alone there are more than one million orifice device which are primarily used for flow measurements in industrial applications such as food industry, petrochemical, and oil/gas refineries. In some applications, such as pharmaceutical plants, a high accuracy of flow metering is unrelentingly required in order to control the precise chemical reactions and drugs formation processes. On the other hand, errors in flow measurement can result in significant cost losses and inefficiency repercussions. For instance, slight flow metering inaccuracies can cause enormous monetary losses in the transfer of oil and natural gas due to greater volumes involved in these transfers. The orifice is one of the most important devices that have been under continuous investigations theoretically, analytically, and experimentally despite the fact that it has pressure and pumping losses greater than similar devices [2]. The basic principle of the orifice depends on the differential pressure due to area reduction which is normally expressed by the aspect ratio, [3-5]. The orifice as a device, which is designed to be used in industry, has to follow the international standard such as ISO 5167-1 [1]. Previous studies have focused on the behavioral of the flow passed the orifice plate. The pattern of this flow could be classified as sudden contraction followed by sudden expansion. For sudden expansion, a study conducted by Cascaras et al. [6] explained the three dimensional turbulent flow. The study, which is experimental in its nature, was performed by means of a 2-D particle image velocimetry technique and found that there is a critical value of Reynolds number at which the shear layers are reattached. This condition extensively relies on the aspect ratios of 1.5 or higher. Other researchers in this field have provided valuable information about relevant topics such as solution of 2-D Navier-Stokes equations and symmetry-breaking bifurcation [7-10]. Experimentally, several studies were conducted at different conditions of aspect ratios and Reynolds numbers where laminar and turbulent flows were considered. Analytically, computational fluid dynamics (CFD) [2, 4, 5, and 11], finite element method (FEM) [12], and finite volume method (FVM) [13] are among several techniques to predict the characteristics behavioral of the flow. The experimental results and analytical simulated results are mostly in very good agreement [1]. However, there are limited studies in which the analytical analysis is not within the realm of the experimental work as in De Zilwa et al. [10] when 80% discrepancy was found. The high disagreement was attributed to poorly chosen boundary conditions and not considering the limitation of k turbulence model. The transition from 2D to 3D turbulent flow through an orifice plate depends on the boundary conditions set by either the theoretical analysis or the experimental settings. The first boundary condition is the flow velocity, ( ) which is assumed constant according to Dirichlet boundary condition, namely, ( ) constant; however it is considered variable according to Neumann boundary condition namely, ( ) constant. In a complementary process, Dirichlet and Neumann boundary conditions are taken together, i. e.,
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:04 20 ( ) ( ) constant. The other boundary conditions of pressure (inlet and outlet), wall, fluid, orifice, and orifice shadow are illustrated in Figure 1. For the purpose of computational analysis, fluid, solid regions, and materials are assigned to cell zones while boundaries, boundaries data, and internal surfaces are assigned to face zones. At the outlet, the static pressure depends on the environment surrounding the outlet and it is normally assumed to be constant. For 2D cases, the pressure could be considered as axial and radial while for cases of strong swirling cases, the flow could be considered as 3D flow. In all cases the backflow should be taken into account of the solution assuming that the backflow is normal to the boundary and for any convergent difficulties, the backflow may minimize to avoid divergence. The parameters that characterized the 2D or 3D flow were investigated experimentally and analytically using software known as CFD. The modeling are based on the Boussinesq hypothesis which was proposed in 1877 in which Reynolds stresses could be linked to the mean rate of deformation. The classical models are based on Reynolds Average Navier-Stokes equations (RANS). Within the concept of this model, CFD characterizes the flow motion in terms of one or two equations. For the one-equation solution, Fig. 1. The initial boundary conditions mild separation and recirculation are taken into account while For incompressible fluids the case in this study- the pressure for two-equation solution, the kinetic energy ( ) and at the inlet should be identified in terms of static pressure, dissipation ( ) are related to each other together with the eddyviscosity, and dynamic pressure,, according the following stress-strain relationship which, finally, constitutes equation: the - turbulence model. In Table 1 a brief description of some mathematical models with their strength and weaknesses. Table I Brief description for some mathematical models. Model Strengths Weaknesses Zero Equation ( model) [14-18] STD k- RNG k- Realizable k- Reynolds Stress Model (RSN) Economical, good for mildly complex Robust, economical, reasonably accurate, long time data collection Good for moderately complex, separating flows, swirling flows, and secondary flows Serves as RNG k- with addition of solving round-jet anomaly. Mist complete model for turbulent flow regarding history, transport, coupled anisotropy 2. MODELING AND THEORETICAL APPROACH The numerical solution sought in this study relies on the mean approach since most engineering applications for turbulent flow do not take all details into consideration but, instead, consider solutions that depend on models without considering full time-dependent flow. In principles, investigating how turbulence affects the mean flow is the primary goal. The classical models are based on Reynolds Average Navier- Stokes (RANS) equations. The following models could be classified according to the number of equations taken into consideration investigation the behavior of the flow. The following are the models that are being currently used (Table 1): zero equation model, one equation model, two equation models (k ) and (k ) model, and seven equation model So far tested moderately, lack of sub-models Deals with severe pressure gradient, strong streamline curvature, swirl and rotation. Predictions within Subjected to limitations due to isotropic eddy viscosity, predictions within Subjected to limitation due to isotropic eddy viscosity. Predictions are slightly less than Requires high computer memory due to tightly couple momentum and turbulence. Predictions are within (Reynolds Stress) model. As another approach for seeking solutions for flow behavior is based on models using the Boussinesq hypothesis where the turbulence decays unless there is shear in isothermal incompressible flow. One of findings of Boussinesq hypothesis is that turbulence increases as the mean rate of deformation increases. In this study, numerical solution of the time dependent three-dimensional Navier-Stokes equations will be presented in order to show the structure of the flow and the numerical observation of unsteady three-dimensional flows throughout a rigid wall pipe conveying fluid. Three-dimensional calculations are performed using equation determined by the following assumptions that the flow is turbulent and the fluid is Newtonian and incompressible. The commercial CFD code ANSYS, version 14.0 is customized by building in some relatively Low Reynolds
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:04 21 Number (LRN) ( ε) turbulence models. Solutions are sought for unsteady flow through an orifice plate with a diameter ratio of one half of the pipe diameter. The flow field is represented by mesh of 2,350,000 elements and global iteration required in the presented work is 1000; however, if solution is not converged, extra 1000 iteration is to be taken into account. It was shown that solutions converged after (1220) iterations and all residuals fell below error of 10-4. The parameters under investigation in this study are: velocity, differential pressure, vorticity, strain, stress, and total deformation. Simulation was taken for three Reynolds numbers Re= 5000, 10000, and 15000 at aspect ratio ( ) of 0.2, 0.3, and 0.5. There are two fundamental equations governing the flow inside and beyond the orifice: the continuity equation and force-momentum equation: Continuity equation ( ) (1) Momentum equation ( ) ( ) [ ( )] ( ) (2) The Reynolds stress term, in Eq.2 represents the nonlinear convective term in the un-averaged equation and reflects the fact that theconvective transport due to turbulent velocity fluctuations will act to enhance mixing over and above that caused by fluctuations at the molecular level. It needs to be modeled to close the system of equations. A common method employs the Boussinesq hypothesis to relate the Reynolds stresses to the mean velocity as shown in the following stress equation (3): ( ) ( ) (3) where is the turbulent viscosity. This approach has the advantage of the relatively low computational cost associated with the computation of though it assumes as an isotropic scalar quantity, which is not strictly true. The alternative approach, embodied in the Reynolds Stress Models through which transport equations are to be solved for each of the terms in the Reynolds stress tensor. In this study, three Bossinesq hypotheses and one Reynolds stress approaches are employed. The standard k-ε model is a two equation eddy viscosity turbulence model. In this model, the eddy viscosity is computed based on the turbulence kinetic energy k, and the turbulence dissipation rate ε using: ( ) ( ) *( ) + (4) ( ) ( ) *( ) + ( ) (5) where areempirical constants. For details, the k-ε model turbulence has five empirical constants,,, and with values of 0.09, 1.0, 1.3, 1.44 and 1.92, respectively. 3. METHODOLOGY 3.1. The Orifice The orifice plate is described in Figure 2. The pipe is 1.00 m in length, 0.01 m internal diameter, 0.02 m outer diameter, and 0.05 m thickness. (a) (b) Fig. 2. The Orifice (a) Mesh construction, and (b) Cross-sectional
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:04 22 3.2 Setting Theoretical Parameters A numerical solution for Navier-Stocks equations using commercial CFD code ANSYS, version 14.0 is performed for unsteady, low turbulent flow and (k ε) turbulence models. The mesh was taken at 2,350,000 elements and convergent solutions were sought such that the residuals fell below error of 10-4. The velocity, differential pressure, vorticity, strain, stress, and total deformation were measured (estimated) or identified by ANSYS at three Reynolds numbers Re = 5000, 10000, and 15000 (all turbulent) and at aspect ratio ( )of 0.2, 0.3, and 0.5. 3. THE RESULTS AND DISCUSSION Three Reynolds numbers (5000, 10000, and 15000) are chosen for this study. It is well-known that Re, where is the density, is the flow average velocity,, is the characteristic length (diameter of the pipe in this case), and is the fluid viscosity. Figure 3 shows the two-dimensional velocity profile of the flow for a typical aspect ratio of 0.2 and Re number of 5000 (Figure 3a), 10000 (Figure 3b), and 15000 (Figure 3c). It is important to note that within the frame of the conditions set for these experiments, parameters of the density, the characteristic length, and the viscosity are kept unchanged and thus Reynolds number will be a function of the velocity only. The velocity profile of flowing streamline at a given shows that the contour lines become more condensed and the swirling angle reduces as Reynolds number increases and more importantly the flow starts creating a boundary layer which becomes very distinctive as Reynolds number increases. Keep in mind that, at a certain aspect ratio and for an incompressible fluid, when Reynolds number changes, the velocity is the only parameter plays the role as other flow dynamic parameters were kept constant. Thus, as Reynolds number increases from 5000 to 10000 and 15000, the velocity contour lines become more condensed, which is indicative to higher shedding frequency of the swirling flow lines and, consequently, results in more back stream against the flow. The results of vorticity taken at same sequence of Reynolds number and aspect ratio shown in Figure 3 are demonstrated in Figure 4 a, b, and c for Re number of 5000, 10000,and 15000, respectively. These findings agree with other results of [11]. For other aspect ratios, and 0.5, the images of velocity and vorticity show same behavior. (a) (b) (c) Fig. 3. Velocity at and (a) Re = 5000, (b) 10000, and (c) 15000 (a) (b) (c) Fig. 4. Vorticity at and (a) Re = 5000, (b) 10000, and (c) 15000 pressure impacting the wall. As mentioned earlier, the velocity of recorded by ANSYS is two-dimensional velocity which can be demonstrated by Navier-Stokes equation. The axial component is the measure of the forward velocity while the radial velocity is the measure of the impact on the wall surface which turns to create the Figure 5a shows the axial velocity while the Figure 5b shows the differential pressure. The velocity and the differential pressure were tested along the testing pipe of length 1.0 meter and the orifice is mounted at the center of this pipe. The results presented in Figure 5 (a, b) were taken at same previous sequence of Reynolds number
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:04 23 and aspect ratio of 0.2. The first observation is that the maximum axial velocity appears at the position of the orifice which is expected as the orifice behaves as contraction device. The second observation is that the peak velocity reduces as Reynolds number decreases confirming that Reynolds number is a function of the velocity only in throughout this study. The static pressure for same parameters setting is shown in Figure 5b. The static pressure experiences sudden increase at the orifice plate location at 0.50 m along the testing tube. The static pressure is the highest at Reynolds number of 15000 which is the highest turbulent value in this study. At this position, the static pressure is about 30 bars while at the lowest Reynolds number of 5000, this pressure decreases to about half bar. Since most industrial establishments carries flow at much higher value of Reynolds number, the static pressure becomes problematic to the pipes and the need for perfect design becomes a must. The way that the static pressure increases at the location of the orifice depends on Reynold number through the relation of pressure gradient, The date in Figure 5b suggests that the pressure gradient increase as Reynolds number increases, or simply saying that the gradient becomes deeper with increasing Reynolds number. This result suggests again deep consideration to the mechanical properties of the pipe around the orifice location. It is also noticed that when Re increases, the location of the maximum velocity slightly shifts backwards. One possible reason for this shifting is that the strong vorticity in the flow which causes stronger reverse swirling which results in pushing the velocity in back direction. This is clearly shown in Figure 5b when the differential pressure across the orifice is significantly increased as Reynolds number increases. (a) (b) Fig. 5. The maximum axial velocity (a) and the static pressure (b) at The mechanical properties such as stress and strain were studied under the concept of fluid-structure interaction (FSI) model. The interaction between the fluid and the structure is caused by the vibration in addition to the direct interaction between the flow and the pipe. Figure 6 shows the maximum stress, stress, and total deformation taken at and Reynolds number of 5000, 10000, and 15000. When Reynolds number increases and so does the velocity, the mechanical factors show significant increase by approximately three time for Re = 10000 and about six times for Re = 15000. This dramatic increase is expected as the differential pressure increases across the orifice plate. The norm of this result agrees with other results predicted by El Drainy [11]. For, the velocity of the flow decreases by about 20% compare to, the mechanical properties significantly decreased. Only at Reynolds number of 10000, stress, strain, or total deformation shows a small amount. Unfortunately, no and Reynolds number of 5000, 10000, and15000. one has been reported such a similar behavior. Seemingly, at Re of 5000 and 15000, there are two factors play important role in determining the stress and the strain; namely the radial pressure due to its direct impact on the wall and vorticity and swirling eddies which plays its role with the viscosity which results in shear stress ( ) which exhibits maximum force at the wall. It might happen that at this specific Reynolds number, there is imbalance between these two factors that may result in this unpredicted result. As the aspect ratio is taken at, the trend of the mechanical properties back to their normal values as the maximum values appear at Reynolds number of 15000. The effect of the aspect ratio is shown in Figure 6. The bigger the orifice has the less velocity and, consequently, the less pressure and eventually the less mechanical properties. Figure 6 concludes that the minimum stress, strain, or total deformation takes place at.
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:15 No:04 24 3.5e+7 3.0e+7 2.5e+7 Max 5000 Max 10000 Max 15000 Stress (N/m 2 ) 2.0e+7 1.5e+7 1.0e+7 5.0e+6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (a) (b) (c) Fig. 6. Maximum Stress (a), Maximum Strain, (b).and Maximum Deformation (c) at Aspect Ratios of 0.2, 0.3, and 0.5 for Reynolds Number of 5000, 10000, and 15000. CONCLUSIONS In this study, computational solutions of unsteady and incompressible fluid flow around and beyond a circular plate orifice placed in the mid of a pipe were performed using a commercial CFD code ANSYS, version 14.0. Grid resolution test was conducted to insure that the numerical results obtained are independent of the spatial computational mesh resolution. The results show that the numerical solutions with a mesh of 2,350.000 elements are converged with a relative error of less that 10-4. Thereby, all the results shown in this paper have been run with this above mentioned mesh elements. The effects of Reynolds number (Re = 5000, 10000 and 15000) and aspect ratio (β=0.2, 0.3 and 0.5) on the flow characteristics are investigated in detail. This includes investigating the impacts of these parameters on the flow velocity profile, the differential pressure, and the mechanical properties such as the stress, the strain, and the total deformation. It was found that the velocity and the pressure increase as Reynolds number increases and with increasing aspect ratio; all these results are in agreement with other published studies in the literature. The maximum axial velocity experiences severe reduction as the flow passes the orifice region. The vorticity images show the flow swirling behavior before and after the orifice suggest that the flow is turbulent. The pressure across the orifice shows very steep gradient as Reynolds number increases. The maximum stress, strain, and total deformation are shown to have almost the same trend for aspect ratio of 0.2 and 0.5 for all Reynolds numbers. However, at aspect ratio of 0.3, the trend of the mechanical properties appears to be significantly different. The results show that somewhere between aspect ratio of 0.2 and 0.5, the mechanical properties, i.e. the stress and the strain, have minimum values. The authors believe that more investigations are needed to explain this abnormal behavior. REFERENCES [1] N.R. Sarker, M. M. Razzaque, and Md. KhayrulEnam, Numerical investigation on effects of deformation on accuracy of orifice meters, International Conference on Mechanical Engineering, ICME 2011, pp 1-6, Dhaka, Bangladesh, December 2011. [2] S. Manish, J. Shah, B. Joshi, K. Avtar S., C.S.R. Prasad, Shukla, S. Daya, Analysis of Flow Through an orifice mete: CFD Simulation, Chemical Engineering Science, Vol71, pp 300-309, 2012. [3] G.H. Nail, A study of 3-Dimensionalflow through orifice meters, Ph.D. Dissertation, Texas A&M University, USA, 1991. [4] E. Smith, A. Ridluan, P. Somravysin, P. Promvongee, Numerical investigationof turbulent flow through a circular orifice, KMITL Sci. Vol 8, pp43-50, 2008. [5] A. Naveenji, A., S. Kaushik, M.H. Srianananth, CFD analysis on discharge coefficient duringnonnewtonian flows through orificemeter, Int. J. Eng. Sci. Technology, Vol 2, pp3151-3164, 2010. [6] L. Casrasa and P. Giannattasio, Three Dimensional features of the turbulentflow through a planar sudden expansion, Physics of Fluids, Vol 20, pp 1-15, 2008. [7] I.J. Sobey and P.G. Drazin, Bifurcations of two-dimensional channel flow, Journal of Fluid Mechanics, Vol 171, pp 263-287, 1986. [8] M. Shapiro, D. Degani, and D. Weihs, Stability and existence of multiple solutions for viscous flow in sudden enlarged channel, Computers and Fluids, Vol 18, pp 239-258, 1990. [9] F. Durst, L.C.F. Pereira, and C. Tropea, The plane symmetric sudden-expansion flow at low Reynolds number, Journal of Fluid Mechanics, Vol 248, pp 567-581, 1993. [10] S.R.N. De Zilwa, L. Khezzar, and J.H. Whitelaw, Flow through plane sudden-expansion, International journal for Numerical Methods in Fluids, Vol 32, pp 313-329, 2000. [11] Y.A. El Drainy, K.M. Saqr, H.S. Aly, and M.N.Mohd.Jaafar, CFD analysis of Incompressible turbulent swirling flow through Zanker plate, Engineering Applications of computational Fluid Mechanics, Vol 3(4), pp 562-572, 2009. [12] C.E. Baumann and J.S. Oden, A discontinuous hp finite element for the Euler and Navier-Stokes equations, International Journal for Numerical in Fluids, Vol 31(1), pp 79-95, 1999. [13] Y. Fu-Cheng, S. Sheau-Wen, H. Tzeng-Yuan, J. Chan-Yung, and W. Chang, A study of numerical simulation applying to the design of an orifice with high-velocity water jet, Tamkang Journal of Science and Engineering, Vol 11(2), pp 145-154, 2008.
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