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Characterization of particulated flow induced erosion in elbow geometry

Quamrul H. Mazumder1 (), Khairul Hassan2, Aniket Kamble3, Venkat Teja Nallamothu1

1. University of Michigan-Flint, Flint, MI 48502, USA 2. Virtual Analysis Group, ICT Product Design & Engineering, FCA Inc, Auburn Hills, MI, USA 3. EDAG Inc., MI, USA Abstract

Particulated multiphase flow is a complex phenomenon analyzed using computational and

experimental methods for elbow geometry. The flow behavior in the elbow region is important

as the velocity of particles impacting the wall can cause severe damage to the inner surface of the

elbow. The particle impact behavior is influenced by fluid velocity, solid particle size, concentration

due to spatial distribution of the particles. Computational and mechanistic erosion models are

currently used to predict the potential failure modes and erosional damage to the wall. CFD

prediction of erosion and flow behaviors in elbow geometry at three different velocities are

presented. Velocities in the elbow region have been measured using particle image velocimetry

(PIV) in particle laden flows. PIV technique measures the instantaneous velocity field within

an illuminated plane of fluid by scattering light from particles in the fluid. The particle and flow

velocities near the wall region can provide better understanding of the fluid and wall interactions.

The predicted velocity profiles are compared with PIV results showing reasonably good

agreement. Further investigation will be performed to provide more quantitative comparison of

the velocities.

Keywords computational fluid dynamics

fluid flow

elbow

erosion

particle image velocimetry (PIV)

Article History Received: 19 January 2020

Revised: 25 March 2020

Accepted: 30 March 2020

Research Article © Tsinghua University Press 2020

1 Introduction

The flow of incompressible viscous fluids through an elbow is characterized by flow separation, secondary flow, and unsteadiness that depend on fluid properties, velocity, and the radius of curvature of the bend. When fluid such as water passes through an elbow, a radial pressure gradient is developed by the centrifugal force acting on the fluid due to changes in flow direction. The interaction between this centrifugal force and viscous force creates a strong secondary flow in the plane normal to the pipe axis and distributions of the velocity vector. This secondary flow consists of two counter-rotating vortices due to the presence of the pressure gradient. The fluid at the center of a pipe moves towards the outer side and comes back along the wall towards the inner side, and the adverse pressure gradient near the inner wall and immediately downstream of the elbow may lead to flow separation, giving rise to a large increase in pressure losses. The scouring action of these counter-rotating vortices

accelerates the magnitude of erosion in the pipe wall (Homicz, 2004).

The particle and flow velocities near the wall region can provide a better understanding of the fluid and wall interactions. The repeated particle impact on the inner wall erodes the surface, gradually decreasing the wall thickness, and compromising the safe operation of the equipment. Among various factors that cause erosion, particle impact velocity has been recognized as the most significant one. However, the particle impact behavior inside the elbow region is not well understood due to complex flow behavior. The discrete phase model (DPM) can be used to analyze the velocity and trajectories of particles.

Investigations of the flow through an elbow are of great significance in understanding the particle impact behavior. A generalized erosion prediction procedure that involves flow modeling, particle tracking, and penetration rate calculations has been developed by researchers. A number of investigations have been conducted to understand the

Vol. 3, No. 2, 2021, 100–107Experimental and Computational Multiphase Flow https://doi.org/10.1007/s42757-020-0066-2


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flow behavior in the elbow region (Homicz, 2004; Iwamoto et al., 2012; Mazumder, 2012a, 2012b; Kim et al., 2014; Liu et al., 2015; Dutta et al., 2016). LDV (laser Doppler velocimeter) and PIV (particle image velocimetry) methods have been used to improve the understanding of particulated flow behavior in elbow. LDV is a point-measurement using a photo-detector and PIV is a planar technique that uses a camera. PIV has recently matured to a reliable experimental technique used to investigate a wide variety of applications (Day et al., 2001; Day and McDaniel, 2005).

Computational fluid dynamics (CFD) is a numerical simulation method, widely used for analysis the flow in elbow or other complex geometries. Computational and mechanistic models of multiphase flow behavior in elbow have been developed to predict the potential failure modes and damage to the wall. Investigation conducted by Swienty et al. (2015) showed good agreement between experimental measurements and CFD predictions using RANS turbulent models (k–ω).

Erosion due to particulated multiphase has been analyzed using numerical simulation (CFD) and presented in this paper. The simulations were performed using a discrete phase model (DPM) in the commercial CFD software “Fluent”. The k–ω turbulence model was used as this model has provided better results for flow in geometry such as an elbow. In the DPM model water was used as the continuous phase with 50 micron anthracite solid particles as the discrete phase. The flow characteristic in the elbow was also investigated using PIV for validation and for verification of the CFD results. The analysis by CFD method is conducted for four different velocities and these results are validated by using experimental (PIV) results.

2 Model geometry

Figure 1 shows a schematic of the elbow geometry used for CFD simulation and PIV test. The inside diameter was D = 25.4 mm and a r/D ratio of 1.5 was used for a standard elbow (r = 38.1 mm). The model geometry was selected as one inch pipe is most widely used in fluid handling equipment with particulated multiphase flow application. Another consideration was the ability to conduct experimental investigation with one inch pipe in the PIV experimental set-up. The geometry was divided in three different sections: upstream and downstream straight pipe sections, and the elbow section. The mixture composition and phase velocities were defined at the inlet boundary of the upstream pipe. The upstream and downstream pipes were 100 mm in length.

3 CFD analysis

Computational fluid dynamics (CFD) provides a qualitative

Fig. 1 Elbow geometry used for CFD analysis.

as well as quantitative prediction of fluid flows by means of (a) mathematical modeling (partial differential equations (PDE)), (b) numerical methods (discretization and solution techniques), and (c) software tools (solvers, pre- and post- processing utilities). Fluid flows are governed by partial differential equations using conservation of mass, momentum, and energy. The current analysis used the discrete phase model (DPM) in a commercial CFD solver, “Fluent”, to solve the equation via domain discretization, using a control volume approach. For incompressible fluid flow, the continuity equation reduces to 0u = .

Furthermore, the momentum equation can be written as

2uρ ρu u η u p ρgt

¶=- + - +

¶ (1)

where u is the fluid velocity, ρ is the fluid density, is the dynamic viscosity, and p is the pressure

The trajectory of a discrete phase particle was determined by integrating the force balance on the particle, using Lagrangian reference frame. This force balance equates the particle inertia with the forces acting on the particle, and can be written as

( ) ( )D

dd

p Pp x

P

u g ρ ρF u u F

t ρ-

= - + + (2)

Where D ( )pF u u- is the drag force per unit particle mass

and

DD 2

1824P P

μ C ReFρ d

= (3)

u and pu are the fluid velocity and particle velocity, res-pectively; μ is the molecular viscosity of the fluid; ρ and

Pρ are fluid density and particle density respectively; g is

Q. H. Mazumder, K. Hassn, A. Kamble, et al.

102

the gravitational acceleration; and xF is a term accounting for additional forces including virtual mass force, Brownian force, Staffman’s lift force, etc.

Discrete phase model (DPM) was used for particle tracking in order to model the erosion process. The particle tracking calculations aim to determine the particle trajectory as well as its velocity before impact either on the pipes or walls. The particle impact velocity is not only important for the calculation of erosion but also important in determining the particle trajectory during its subsequent motion following impact.

Particle erosion and accretion rates can be calculated at wall boundaries and the erosion rate is defined as

particles ( )

erosionface1

( ) ( )N b vp p

p

m C d f α vR

A=

= å (4)

where pm is the mass flow rate of a particle; ( )pC d is a function of particle diameter; α is the impact angle of the particle path with the wall face; ( )f α is a function of impact angle; v is the relative particle velocity; ( )b v is a function of the relative particle velocity, and faceA is the area of the cell face at the wall. C, f, and b are default values of 1.8×10-9, 1, and 0, respectively. C, f, and b are defined as boundary conditions at the wall rather than properties of the material. Since b is the function of particle velocity, the type of particle also influences erosion. The default values for f and b were successfully used by other investigators (Edwards, 2000; Mazumder, 2016) for the eroded pipe.

3.1 CFD modeling and simulation

The turbulent model was selected based on careful con-sideration to improve accuracy of the results. The turbulent behavior in the particle fluid mixture may have greater influence on erosion phenomenon (Homicz, 2004). The k–ω turbulence model with standard wall functions was used due to their proven accuracies in solving for similar flow geometry (Swienty et al., 2015). The turbulent kinetic energy (k) and the turbulent dissipation rates (ω) were solved to determine the coefficient of turbulent viscosity (μ). In the k–ω turbulent model, the turbulent kinetic energy is calculated as

tˆk

ρk ηρku P βρηωk η kt σ

æ ö¶ ÷ç+ = - + + ÷ç ÷ç¶ è ø (5)

And the specific rate of dissipation is calculated as

( ) t2ˆω

ρω ηωρωu α P βρω k η ωt k σ

æ ö¶ ÷ç+ = - + + ÷ç ÷ç¶ è ø (6)

where β is the turbulent parameter; P̂ is the pressure; η is the dynamic viscosity; and σ is the turbulent parameter.

The work presented in this paper used four different inlet flow velocities (1 m/s, 0.923 m/s, 0.762 m/s, 0.283 m/s). The density and viscosity of water were used ρ = 998.9 kg/m3 and μ = 3.145 kg/(m·s) respectivily. The Reynolds number for inlet velocity, u = 1 m/s was

80674 4000ρuDReμ

= = >

indicating the flow to be fully turbulent (https:// en.wikipedia.org/wiki/Reynolds_number). The flow was turbulent at other two velocities (0.923 m/s and 0.762 m/s). However, for u = 0.283 m/s, Reynolds number is Re = 2283 < 2300 where the flow was laminar. Three-dimensional unstructured mesh was used in the CFD analysis using an implicit method for solving the mass and momentum equations. Tetrahedral mesh was used as it has been proven to provide reliable solution of complex three-dimensional problems (Du and Wang, 2003). The meshed elbow geometry is shown in Fig. 2. Mesh sensitivity analysis with mesh refinement were performed to improve accuracy and eliminate variation of results. The mesh sensitivity analysis was performed to eliminate mesh dependency on results. The near wall treatment of meshed elbow was performed to obtain accurate results. The meshed geometry used in the analysis had approximately 0.26 million cells. The CFD simulation parameters are listed in Table 1.

Fig. 2 Meshed elbow geometry.

Table 1 CFD simulation parameters

Number of cells 0.26 million

Type of cells Tetrahedral

Fluid Water

Fluid density 998.97 kg/m3

Viscous regime Turbulence/laminar

Fluid velocities 1 m/s, 0.923 m/s, 0.762 m/s, 0.283m/s

Particle size 50 microns

Particle rate (kg/s) 1.1574×10-5

Particle shape Rounded


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3.2 CFD results

CFD results are presented as velocity profiles and velocity vectors. The results were compared with experimental PIV results of the velocity vectors.

3.2.1 Velocity profile in elbow

The velocity profiles of four different velocities are presented in Figs. 3(a), 3(b), 3(c), and 3(d). For velocity 0.283 m/s, the simulation was performed using laminar model instead of turbulent model. No significant differences in velocity profiles were evident between laminar and turbulent models with

similar velocity profiles at all four velocities. Cross-sectional velocity contours were plotted at 10-degree

increment from inlet to outlet of the elbow. As flow enters the elbow area, the inlet angle at the start of the elbow was assumed to be zero degree, 45 degrees at the mid pint of the elbow and 90 degrees at the exit of the elbow. The angle is the relative angle to the inlet flow direction. The cross-sectional velocity contours are presented in Fig. 4 for 1.0 m/s inlet velocity. To compare the velocity profiles for four different inlet velocities, cross sectional velocity contours are presented at 45-degree location of the elbow in Fig. 5. It can be observed that the velocity at inner wall increases as the fluid moves

Fig. 3 Velocity profiles in the elbow at different velocities.

Fig. 4 Cross-sectional velocity profiles for 1.0 m/s inlet velocity.


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in the elbow. The velocity at the outer wall increases as the fluid exits the elbow region.

3.2.2 Erosion analysis

The primary reason for erosion in particulated multiphase flow is due to the presence of solid particles in the flow that impinge on the inner wall of the geometry (Mohammadi and Luo, 2010). Accurate prediction of erosion requires detailed investigation of the solid particle motion before and after impact. The effects of flow velocity and particle sizes were investigated by Badr et al. (2002) but they assumed that particle motion had negligible effect on the fluid velocity. But turbulent flow makes the particle trajectory and impact characteristics difficult to predict taking into consideration all fluid forces acting on the particle (Mahdi et al., 2013).

CFD analysis results showed location of maximum erosion at the outer wall of the elbow spread over a region as shown in Fig. 6. Particle trajectories were plotted to evaluate the particle impact locations in the elbow wall. The particle trajectories of Fig. 7 showed the particle impact location to be similar to the location of maximum erosion of Fig. 6.

By solving Eq. (4) and using the discrete phase model erosion rates for different particle velocities were obtained. The CFD prediction of erosion rates at four different velocities are presented in Table 2 and Fig. 8. CFD analysis of each condition was repeated three to five times and the average of the results is presented in Table 2. Higher erosion rates

Fig. 6 Location of maximum erosion in elbow at 1.0 m/s inlet velocity.

Fig. 7 Particle trajectories at 1.0 m/s inlet velocity.

Fig. 5 Cross-sectional velocity profiles at 45-degree location in the elbow.


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Table 2 Erosion at different velocities

Inlet velocity

Flow characteristics

CFD predicted erosion (kg/(m2·s))

1 m/s Turbulent 1.17×10-9

0.923 m/s Turbulent 6.07×10-10

0.762 m/s Turbulent 4.66×10-10

0.283 m/s Laminar 4.506×10-10

Fig. 8 CFD predicted erosion at different fluid velocities.

were observed at higher fluid and particle velocities. The solid particle trajectories presented in Fig. 7 show the location of maximum particle impact. The location of maximum erosion shown in Fig. 6 was similar to the location of maximum particle impact of Fig. 7.

4 Experimental method

A technique that utilizes particles and their images falls into the category commonly known as particle-image velocimetry, or PIV (Adrian, 1991). PIV is an optical technique that measures the instantaneous velocity field within an illuminated plane of the fluid field using light scattered from particles seeded into the fluid. The fluid with entrained particles is illuminated so that particles are visible.

Adrian (1996) reported that PIV method was widely used for various purposes and was rarely used for flow analysis in elbow. A limited number of investigations (Shiraishi et al., 2009; Kimura et al., 2010; Kalpakli, 2012; Kubo et al., 2013) were reported for PIV analysis of flow in elbows. PIV is a very useful and practical tool for conducting study of complex flow behavior due to the advantage of simultaneous measurement at multiple locations within an illuminated plane. Successive image pairs within the laser illuminated plane are captured using a high-speed digital camera. Using a pulsed laser, it can be ensured that each image pair may be obtained at effectively instantaneous time (very small relative to the transit time of the particle through the measurement region). The precise time separation between the two images of the pair is known, and by comparing the particle displacement between the two images, the local velocity is calculated at any sub-region of these images.

4.1 Experimental procedure

In the PIV method, particles in the fluid are illuminated by a sheet of light that is pulsed (Jahanmiri, 2011). The particles scatter light into a photographic lens located at 90° to the sheet, so that its in-focus object plane coincides with the illuminated slice of fluid. Images are formed on a photographic film or on a video array detector, and the images are subsequently transferred to a computer for automatic analysis. The analysis of the recorded image field is one of the most important steps in the entire process, as it couples with the image-acquisition process to determine the accuracy, reliability, and spatial resolution of the measurements. The motion of the seeding particles is used to calculate speed and direction (the velocity field) of the flow being studied.

4.1.1 Equipment details

The equipment used in the PIV experiment are a camera, laser units (including power supply unit, control unit), test specimen (elbow), water or fluid supply pump, computer including software for analysis, etc. The schematic of the PIV experimental set-up used in this work is shown in Fig. 9. Among all equipment, the laser is very important for different aspects including better images as well as its sensitivity to use for safety reasons (McIlroy et al., 2006; Kirschner and Ruprecht, 2007). A Nano-PIV series class 4 laser was used in the experiment for flow analysis in elbow. The laser system had two pulsed and Q-switched laser resonators producing infrared laser light at 1064 nm which is converted to visible 532 nm laser light by a harmonic generation assembly (HGA).

Nano-PIV series lasers are a dedicated configuration primarily for use as an illumination source for particle imaging velocimetry (PIV). The system using two indepen-dently pulsed and controlled resonators allows the generation of double pulse output with inter-pulse separation time of less than 1 nano second where required. The short pulse

Fig. 9 Schematic of the PIV system components.


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duration achieved by the electro-optical Q-switching and the laser allows the motion of the fast moving articles to be captured in the PIV image. The two beams are pre-aligned at the factory to be co-axial and exit the laser aligned orthogonal to output faceplate. The laser control section of the power supply unit (PSU) is fitted with inputs that are compatible with the timing and camera synchronization generators supplied by all the principle manufacturers of PIV software and control systems. The laser system control section contains the interface and control electronics for the two laser resonators, the harmonics and the cooling system.

The result of the PIV images provides a better understanding of the flow behavior in particle laden flow. PIV images provide both qualitative and quantitative data for comparison with CFD results. This allows the user the ability to view original and filtered flow recordings, a color- coded vector field within the PIV window, analytical graphs at a desired point within the PIV window representing fluid velocity versus frame of the recording, and an entire library of all data points processed, all frames stored, and all stills captured.

4.2 Validation and verification

Experimental PIV test methods are used to plot velocity vectors of the fluid flow for velocity 1 m/s in elbow area as shown in Fig. 10(b) that can be compared with the velocity vector obtained from CFD simulation as shown in Fig. 10(a).

The CFD analysis shows the flow to be uniformly distributed. However, in experimental situations, the restriction of pipe length connected with entrance of elbow is not a sufficient length to get stable fluid flow from the entrance of elbow.

5 Results and discussion

Computational fluid dynamics analysis was performed to predict erosion in particulated multiphase flow in elbow geometry. The CFD predicted maximum erosion was observed in the outer wall at downstream side of the elbow. Erosion is a micromechanical process that results from repeated particle impact velocities and therefore, locations with high impact velocities are susceptible to higher erosion. To validate the CFD predicted erosion results, the anticipated flow fields in the elbow area were analyzed using particle image velocimetry experimental system (PIV).

In PIV experiments, each set of image pairs results in the velocity field at one particular axial location and for each image pair the time-averaged mean velocity is estimated at each axial location. The PIV experimental results showed higher flow and particle velocities in the outer wall at downstream side of the elbow. The high velocities observed from PIV experimental results match closely to the location of maximum erosion predicted by CFD analysis.

6 Conclusions and future work

In this paper, CFD results of erosion in elbow due to particulated multiphase fluid flow are presented. The discrete phase model (DPM) was used with 50-micron anthracite particles. Turbulent flow of multiphase incompressible fluid was analyzed using k–ω turbulent model. The validation of velocity vectors from CFD method and with experimental PIV method indicates a good agreement.

At the entrance of elbow, the velocity at the inner wall was higher. However, the velocity at the outer wall was higher at the exit of the elbow. Due to different inlet velocities,

Fig. 10 Comparison of velocity vector for 1 m/s in elbow.


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the spatial distribution of velocities in the elbow cross-section was also different. Maximum erosion was observed around 45-degree location of the elbow at the outer wall of the elbow. This agrees with previous predictions and experimental observations reported by other investigators. The particle trajectories showed the locations of particle impact to be similar to the location of maximum erosion. This also validates the well accepted phenomenon that the erosion is caused by repeated particle impact in the wall. Higher erosion rates were observed at higher velocities.

The PIV experimental investigations are being conducted for further analysis of the particle and flow velocities in the elbow region. Due to limitations associated with the experimental set-up of the PIV, lower velocities were used for the study. These lower velocities may limit the abilities to accurately predict erosion and characterization of flow behaviors. The PIV test loop is being redesigned to conduct experiments at higher velocities.

References

Adrian, R. J. 1991. Particle-imaging techniques for experimental fluid mechanics. Ann Rev Fluid Mechanics, 23: 261–304.

Adrian, R. J. 1996. Bibliography of Particle Velocimetry Using Imaging Methods: 1917–1995. Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign.

Badr, H. M., Habib, M. A., Ben-Mansour, R., Said, S. A. M. 2002. Effect of flow velocity and particle size on erosion in a pipe with sudden contraction. In: Proceedings of the 6th Saudi Engineering Conference: 79–95.

Day, S. W., McDaniel, J. C. 2005. PIV measurements of flow in a centrifugal blood pump: Steady flow. J Biomech Eng, 127: 244–253.

Day, S. W., McDaniel, J. C., Wood, H. G., Allaire, P. E., Landrot, N., Curtas, A. 2001. Particle image velocimetry measurements of blood velocity in a continuous flow ventricular assist device. ASAIO J, 47: 406–411.

Du, Q., Wang, D. 2003. Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations. Int J Num Meth Eng, 56: 1355–1373.

Dutta, P., Saha, S. K., Nandi, N., Pal, N. 2016. Numerical study on flow separation in 90° pipe bend under high Reynolds number by k-ε modelling. Eng Sci Technol, 19: 904–910.

Edwards, J. K. 2000. Development, validation, and application of a three dimension, CFD-based erosion prediction procedure. Ph.D. Thesis. The University of Tulsa, Tulsa.

Homicz, G. F. 2004. Computational Fluid Dynamic simulations of pipe elbow flow. Technical Report. Sandia National Laboratories, United States.

Iwamoto, Y., Kondo, M., Minamiura, H., Tanaka, M., Yamano, H. 2012. Unsteady flow characteristics in a 90 degree elbow affected by developed, undeveloped and swirling inflow conditions. J Fluid Sci Tech, 7: 315–328.

Jahanmiri, M. 2011. Particle Image Velocimetry: Fundamentals and its Applications. Research Report 2011:03. Department of Applied Mechanics, Chalmers University of Technology Göteborg, Sweden.

Kalpakli, A. 2012. Experimental study of turbulent flows through pipe bends. Technical Report. Royal Institute of Technology, KTH Mechanics.

Kim, J., Yadav, M., Kim, S. 2014. Characteristics of secondary flow induced by 90-degree elbow in turbulent pipe flow. Eng Appl Comput Fluid Mech, 8: 229–239.

Kimura, N., Ogawa, H., Kamide, H. 2010. Experimental study on fluid mixing phenomena in T-pipe junction with upstream elbow. Nucl Eng Des, 240: 3055–3066.

Kirschner, O., Ruprecht, A. 2007. Velocity measurement with PIV in a straight cone draft tube. In: Proceedings of the 3rd German– Romanian Workshop on Turbomachinery Hydrodynamics.

Kubo, T., Ebara, S., Hashizume, H. 2013. Experimental study of influence of elbow curvature on a swirling flow generated in a three-dimensionally connected dual elbow. E-Journal of Advanced Maintenance, 5-1: 34–41.

Liu, M., Liu, H., Zhang, R. 2015. Numerical analyses of the solid particle erosion in elbows for annular flow. Ocean Eng, 105: 186–195.

Mahdi, E., Rauf, A., Ghani, S., El-Noamany, A. Pakari, A. 2013. Erosion–corrosion behavior and failure analysis of offshore steel tubular joint. Int J Electrochem Sci, 8: 7187–7210.

Mazumder, Q. H. 2012a. CFD analysis of single and multiphase flow characteristics in elbow. Engineering, 4: 210–214.

Mazumder, Q. H. 2012b. CFD analysis of the effect of elbow radius on pressure drop in multiphase flow. Model Simul Eng, 2012: 125405.

Mazumder, Q. H. 2016. S-bend erosion in particulated multiphase flow with air and sand. J Comput Multiph Fl, 8: 157–166.

McIlroy, H. M., McEligot, D. M., Schultz, R. R., Christensen, D., Pink, R. J., Johnson, R. C. 2006. PIV experiments to measure flow phenomena in a scaled model of a VHTR lower plenum. Technical Report. Idaho National Laboratory, United States.

Mohammadi, F., Luo, J. 2010. Effects of particle angular velocity and friction force on erosion enhanced corrosion of 304 stainless steel. Corros Sci, 52: 2994–3001.

Shiraishi, T., Watakabe, H., Sago, H., Yamano, H. 2009. Pressure fluctuation characteristics of the short-radius elbow pipe for FBR in the postcritical Reynolds regime. J Fluid Sci Tech, 4: 430–441.

Swienty, A., Ackl, R. A., Thamsen, P. U. 2015. Numerical prediction of flow patterns after various pipe fittings. J Mech Eng Auto, 5: 542–548.

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