development and validation of 2-d spps viii-1 viii ... · a guideline that has been widely used in...

DEVELOPMENT AND VALIDATION OF 2-D SPPS VIII-1

September 2009

VIII. DEVELOPMENT AND VALIDATION OF 2-D SPPS

Introduction

In the oil and gas industry, in order to design equipment to withstand erosive conditions or to optimize the production rate while keeping the piping system operating safely, a reliable erosion prediction tool is necessary. A wide variety of erosion prediction methods have been proposed by many investigators. These methods are either based on some simple models or just accumulated field experience. Some professional tools developed by some research institutes are also available.

A guideline that has been widely used in industry is the American Petroleum Institute (API) Recommended Practice 14E (API RP 14E, 1991). According to this guideline, less erosion is anticipated for a less dense fluid. But, this has been shown not to be true experimentally. Also, this guideline does not account for many factors affecting erosion, such as particle properties, wall material mechanical properties, impact angle and geometry. Salama and Venkatesh (1983) and Salama (2000) proposed an alternate correlation to API RP 14E. Based on the work by Salama and Venkatesh and extensive experimental data of Bourgoyne (1989), Svedeman and Arnold (1993) suggested a similar formula for predicting a threshold velocity for flow. In another study, Jordan (1998) presented a method for using the E/CRC erosion model for multiphase flow.

The major drawbacks of these models are that they are based on specific experimental conditions and do not account for variations of fluid properties such as fluid density, viscosity, and composition. Also some other important factors, such as sand size, sand shape, and multiphase flow, are not considered. Shirazi, McLaury, Shadley and Rybicki (1995) proposed a mechanistic model to address the need for accounting for the significant factors. Based on the fact that sand particles must penetrate through a fluid region in order to impinge the target wall, this model assumes that the particle moving in this region can be represented by an equivalent particle moving in the stagnation region in the direct impingement geometry. Both particles will have the same velocity when they impact the target wall. Figure 1 illustrates the flow regions around the target walls of interest. Figure 2 shows the flow in the representative stagnation region. The behavior of the particle in the stagnation region is strongly dependent on the fluid and particle properties. The impact speed of one representative particle is calculated and this information is used to calculate erosion rate. This model also assumes that the representative particle moves along the centerline of the stagnation region, with its initial velocity as the fluid upon entering the stagnation region. The initial distance between the representative particle and the target wall as this particle enters the stagnation region is called the stagnation length. In order to

VIII-2 EROSION/CORROSION RESEARCH CENTER

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calculate particle velocity along the stagnation length, the fluid velocity at each location is needed. Previously, it was assumed that the fluid velocity varies linearly from a given value at the entrance to the stagnation region to a value of zero at the wall. Later, this was improved by assuming that the change of the fluid velocity follows a so-called jet profile as illustrated in Figure 3. The jet profile represents the velocity of fluid along the centerline of a jet approaching a wall normally.

Figure 1. Flow Region of Interest in Tee and Elbow.

Figure 2. Schematic of Stagnation Region and Stagnation Length.

Flow Region of Interest

Contour of Flow Velocity (by CFD) Red: High Velocity Blue: Low Velocity

Stagnation Region

Stagnation Length

Target Wall

Representative Particle

Nozzle


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Figure 3. Particle Tracking along the 1-D Stagnation Length.

The stagnation length can be changed to incorporate the effect of geometry on erosion. Increasing the stagnation length may represent an increase in the size of the geometry. For instance, a tee with larger diameter has more fluid through which the particle must penetrate to impact the target wall. Therefore, a longer stagnation length is needed for larger diameter geometries. Changing the stagnation length may also better simulate geometries of different shapes. A particle in a tee geometry has a tendency to follow the fluid into the wall, while a particle in an elbow has greater tendency to follow the fluid around the curvature. This can be accounted for by using a larger stagnation length for the elbow geometry. By altering the stagnation length and conducting particle tracking along it, this model accounts for many key factors such as flow stream velocity, fluid viscosity, fluid density, sand size, sand density, sand shape, and flow geometry and size. Once the representative impact velocity is determined, the erosion rate is then calculated using erosion equation. In addition to the particle impact velocity, the sand shape, sand flow rate and the target material are also considered in the erosion equation.

This model has many advantages and is the core of the SPPS (Sand Production Pipe Saver), which is a set of subroutines automating calculations with an Excel interface. Its theoretical foundation provides a degree of confidence in the generated predictions when the area of interest is outside the experimental testing regime. This is not the case for many prediction tools which are essentially empirical expression developed from a set of experimental data. By comparing to several sets of experimental data in literature, this

Flow Velocity Target wall

Jet Profile

Linear Profile

Stagnation LengthParticle


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model has been examined and good agreement exists between the data and the predictions (Shirazi and McLaury, 2000).

However, this model has two significant limitations. The stagnation length is a straight line segment that is assumed to be the path of a particle when it moves across the stagnation zone. Therefore, this model only calculates the particle velocity component along the stagnation length. The second limitation is that this model does not account for the effect of turbulence for most geometries. Even though this model works well for many cases, it may fail under certain flow conditions due to these limitations. More about these limitations will be discussed later in this chapter. Since the particle tracking in this model is conducted along the one-dimensional stagnation length, it is referred to as 1-D model in the rest of this chapter. The SPPS based on this 1-D model is hence called 1-D SPPS.

A few sets of experimental data recently published by different authors are chosen to further examine the 1-D model. This data ranges from about 1.0e-5 to 1.0e+1 mil/lb, as can be seen in the following figures. The unit mil/lb represents the material thickness loss in mil (=1/1000 inch) caused by one pound of sand. Most of the data mentioned here was collected using electrical-resistance (ER) probe, which is very sensitive and can measure material thickness loss in the range of nanometers.

Reuterfors (2006) conducted a series of direct impingement tests using sand in water or air under room temperature and pressure. The erosion is measured using an ER probe placed near the discharge of the nozzle. Average sizes of the sand particles were 150 μm and 20 μm. Pyboyina (2006) collected erosion data on a 2” ID standard elbow using an ER probe. The ER probe was installed in the middle of the outer bend of the elbow at 45º. The carrier fluid was air under pressure of about 20 psig. The sand used in these tests has an average diameter of 150 μm. Evans et al. (2004) carried out a study to validate the efficacy of erosion-corrosion inhibitors in high velocity corrosive gas systems. The test loop contained 4” diameter 5D (r/D = 5) elbows and ER probes are installed where the centerline of the elbow inlet intersects with the elbow surface. The sand used in these tests has an average diameter of 150 μm. The carrier fluid is dry natural gas at about 1,000 psia. Bourgoyne (1989) built a model diverter system and studied the failure in the system due to sand erosion. Most of Bourgoyne’s tests were conducted using sand in air. The sand particles have an average diameter of 450 μm. Wall thickness loss on some standard elbows (r/D = 1.5) with 2” ID is measured.

Figure 4 shows the comparison between the above-mentioned experimental data and the prediction made by using the 1-D model. It can be seen that the 1-D model over predicts for most cases. The over-prediction factor is even greater than 1 order of magnitude for some cases. There is also one case which the 1-D model under predicts by more than 200


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times. This is the only case in which the 20 μm small sand was used. The extreme under-prediction results from the two limitations, as will be explained later in this chapter.

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01

Experimental Data (mil/lb)

Pred

ictio

n by

1-D

Mod

el (m

il/lb

)

Figure 4. Comparison of Experimental Data with Prediction by 1-D Model.

One way to address these two limitations is to use CFD (Computational Fluid Dynamics) along with particle tracking and an erosion equation. Some commercial CFD software like FLUENT has this framework built in. Users can use the built-in particle tracking function and provide parameters of their own for a built-in erosion equation provided by FLUENT. The off-the-shelf version of commercial CFD software usually requires significant user interaction to obtain reasonable erosion results for a given geometry and flow conditions. For example, Zhang et al. (2006, 2007, 2009) improved and validated the CFD-based erosion modeling procedure and utilized a different erosion model than that available in FLUENT. Error! Reference source not found. shows the comparison of the same experimental data with the prediction made by using the CFD-based model. All CFD simulations were conducted using three-dimensional geometries. The comparison shows that the prediction agrees with the experimental data very well. Notice that unlike the 1-D

Evans et al., 2004 (Gas @ 1000 psia) Reuterfors, 2006 (Water and Air) Bourgoyne, 1989 (Air) Pyboyina, 2006 (Air) Perfect Agreement

Direct impingement 20 μm sand Water at 10 m/s


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model, the CFD-based model successfully predicts the erosion rate of the case with 20 μm small sand.

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01


Pred

ictio

n by

CFD

- Bas

ed M

odel

(mil/

lb)

Figure 5. Comparison of Experimental Data with Prediction by CFD-Based Model.

Objective and Approach

By using CFD, one can certainly calculate particle impact velocity components and account for the effect of turbulence. But, this approach requires CFD software installed and tuned for erosion prediction. Compared with using a specialized tool such as the 1-D model mentioned above and used in SPPS, the CFD-based approach is much more time consuming. The users also need to be well-trained in using CFD. This makes it impractical to apply a CFD-based erosion prediction procedure to pipeline design where effects of many parameters need to be examined.

The primary goal of this work is to improve SPPS using knowledge learned from CFD-based erosion modeling. The idea is to introduce two-dimensional particle tracking into SPPS. By using 2-D particle tracking, two velocity components rather than one are calculated and being used in erosion calculation. Tracking many particles in 2-D space will also give more representative results. In addition, the effect of turbulence on particle motion




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can be easily incorporated in 2-D particle tracking. SPPS with 2-D particle tracking is referred to as 2-D SPPS in this chapter.

The improvement, namely 2-D particle tracking, is made to single-phase flow (sand particles carried by single-phase liquid or gas) first. Once the 2-D particle tracking is validated, it will be applied to multiphase flows, working together with other mechanistic models that are for predicting multiphase flow regimes and calculating flow parameters.

SPPS with 2-D particle tracking works in a way similar to the CFD-based model. There are three main steps involved as described in the next section.

Description of the 2-D Mechanistic Model

Like the 1-D model, the basic part of the 2-D model also assumes that flow in regions around the target wall of interest can be represented by the flow in an equivalent stagnation region in the direct impingement geometry. Unlike the 1-D model, the 2-D model calculates the impact velocity of many particles in the stagnation region instead of just one particle on the stagnation length. This is important since particles in the stagnation region may have different impact velocities. Particle tracking in 2-D space also makes it easier to incorporate the effect of turbulence.

Similar to the CFD-based erosion modeling procedure, the 2-D model can also be broken into 3 main steps as described below.

CFD Flow Solution

The first step is to obtain 2-D flow field information. The purpose is to feed this information to the second step of this method, which requires flow field information to calculate sand particle trajectories. The basic part of the 2-D model focuses on the flow in the stagnation region. The flow information in the stagnation region comes from CFD simulations indirectly.

In developing the basic part of the 2-D model, CFD simulations of the direct impingement geometry are conducted for a range of Reynolds numbers. The flow field information of these cases has been saved in SPPS database for each computational cell, which includes geometry dimensions, location of computational grids, fluid density and viscosity, fluid velocity components and their gradient, pressure and its gradient, turbulent kinetic energy and its gradient, turbulent kinetic energy dissipation rate and its gradient, and Reynolds stresses if Reynolds turbulence model (RSM) is used. For any flow conditions of interest, the 2-D model calculates its Reynolds number and interpolates among the pre-saved cases to obtain the flow field information.


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In the 1-D model, the type and size of geometry is accounted for by changing the stagnation length. A similar option for the basic part of the 2-D model to handle this is to change the size of the stagnation region. Yet a better way is to go beyond the concept of stagnation region by pre-saving the flow field information for different types of geometries separately. For example, the 2-D flow field information can be pre-saved for a few typical geometries, such as elbow, tee, sudden contraction, sudden expansion, straight pipe, and so on. The procedure will be the same as described in the previous paragraph. Since it is not possible to include all types of geometry in this way, one has to decide which one of the pre-saved typical geometries best resembles the actual flow geometry, then select that geometry to continue the calculation. The 2-D model in current version SPPS can handle direct impingement and standard elbow geometries.

Two-Dimensional Particle Tracking

The second step in the erosion calculation procedure is to calculate sand particle trajectories using the information provided in the first step. The subroutine for calculating particle trajectories is designed to handle any types of 2-D geometries. Many particles are tracked to generate statistically representative results. The particle impact information is recorded each time a particle hits the target wall. Typical particle impact information includes impact location, impact speed, and impact angle. This impact information is needed in the third part to calculate the erosion rate.

The particle trajectory is determined by integrating the force balance on the particle. This force balance equates the particle inertia with the forces acting on the particle (Newton’s Second Law). In Cartesian coordinates this equation can be written as

dt

dVFFFF pABPD =+++ (1)

where the left side represents the resultant (total) force per unit particle mass acting on the particle and Vp is the particle velocity. The major component of the force acting on a particle is the drag that is exerted on the particle by the fluid. The drag force, FD, depends primarily on the local relative (slip) velocity between the particle and the fluid and is given by

( )pfrD2pd

fD VV24

ReCdρ

18μF −= (2)


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where, Vf is the fluid phase velocity, Vp is the particle velocity, μf is the molecular viscosity of the fluid, ρp is the density of the particle, dp is the particle diameter, and CD is the drag coefficient. Rer is the relative Reynolds number, which is defined as

f

ppffr μ

dVVρRe

−= (3)

where ρf is the fluid density. There are additional forces on the particle, which can be included in the simulation.

These additional forces account for large pressure gradients (FP), gravity and buoyancy (FB), and added mass (FA). Equations (4) through (6) give mathematical representations of these terms.

Pressure gradient: Pρ1

23F

pP ∇−= (4)

Gravity and buoyancy: ( )

p

fpB ρ

gρρF

−= (5)

Added mass: ( )

dtVVd

ρρ

21F pf

p

fA

−= (6)

In the pressure gradient term given by Equation (4), ∇P is the local pressure gradient in the carrier fluid. The gravity and buoyancy force in Equation (5) is needed when the particles and fluid have significantly different densities and when inclusion of gravitational effect is desired. The added mass force described by Equation (6) is the force required to accelerate the fluid surrounding the particle. When relative motion between the particles and the carrier fluid occurs, fluid in the immediate vicinity of the particle must also be accelerated. This results in resistive force acting on the particle and is referred to as the “added mass” force.

There are still other forces that can be included in Equation (1). For example Suffman’s lift force due to shear, force accounts for both centrifugal and Coriolis effect which is due to rotating frames of reference.

Turbulent dispersion of particles is modeled in a stochastic approach. A Discrete Random Walk (DRW) model, or “eddy lifetime” model is applied to account for the interaction between particles and turbulent eddies. This model assumes that the particles travel through a succession of turbulent structures (eddies) that are present in the flow. Each individual eddy that the particle encounters is characterized by a Gaussian distributed random velocity fluctuation (u’, v’, and w’) and a time scale, τe, which is called eddy


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lifetime. Interaction with eddies causes particles to deviate from the trajectory as predicted by Equation (1).

The values of the random velocity fluctuations that prevail during the lifetime of the turbulent eddy are sampled by assuming that they obey a Gaussian probability distribution, so that

2u'ζu'= (7)

where ζ is a normally distributed random number, and the remainder of the right-hand side is the local rms value of the velocity fluctuations.

For k-ε model, k-ω model and their variants, the kinetic energy of turbulence is known at each point in the flow. If the turbulence is assumed to be isotropic, the values of the rms fluctuating components in Equation (7) can be obtained as

32kw'v'u' 222 === (8)

where k is the turbulent kinetic energy. If a non-isotropic turbulence model, such as Reynolds Stress Model (RSM) is used,

the values of the rms fluctuating components in Equation (7) can be computed directly from the known Reynolds stress field:

222 w'ζw';v'ζv';u'ζu' === (9)

The eddy lifetime, τe is defined as a random variation about TL:

( )rlog-Tτ Le = (10)

where r is a uniform random number between 0 and 1 and TL is the fluid Lagrangian integral time and is proportional to the particle dispersion rate. This time scale can be approximated as

εkCT LL = (11)

where ε is the local dissipation rate of turbulence kinetic energy; CL is an empirical constant and is suggested to be 0.15 for k-ε model, k-ω model and their variants, and 0.30 for Reynolds Stress Model (RSM).

The particle eddy crossing time, τcross, is defined as

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−=

pfp

epcross VVτ

L-1lnττ (12)


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where τp is the particle relaxation time, defined as

rDf

2pp

p ReC3μd4ρ

τ = (13)

where ρp is the density of the particle, dp is the particle diameter, μf is the molecular viscosity of the fluid, CD is the drag coefficient, and Rer is the relative Reynolds number defined by Equation (3).

In Equation (12), |Vf-Vp| is the magnitude of the relative velocity and Le is the eddy length scale, which is given by

ε

kCL1.5

μe = (14)

where Cμ is a turbulence model constant. The particle is assumed to interact with the fluid phase eddy over the smaller of the

eddy lifetime, τe, and the eddy crossing time, τcross. That implies that the interaction time τi is dictated by either of the following possible events:

1. The particle moves sufficiently slowly relative to the fluid so as to remain inside the eddy during the entirety of the eddy lifetime.

2. The slip velocity between the particle and the fluid is sufficiently large so that the particle crosses the eddy in the particle eddy crossing time. Therefore,

( )crossei τ,τminτ = (15)

When the interaction time is reached, a new value of the instantaneous velocity is sampled.

Erosion Calculation

The third part of the 2-D erosion model is to calculate the erosion rate using erosion equations and the impact information provided in the second part. An erosion equation takes the particle impact information as input and calculates the erosion rate caused by the corresponding sand particle impingement. This is done for each impingement recorded in the second part. Depending on the particle impact location, the calculated erosion rate is saved for the corresponding portion of the target wall. Knowing the erosion distribution on the target wall, one can easily read the maximum erosion rate, obtain the total erosion rate by integrating over the target wall, or do other types of calculation.

There are many erosion equations available in literature. Meng and Ludema (1995) have done a comprehensive review on this topic. The E/CRC erosion equation recently proposed and validated by Zhang et al. (2007) is applied in the current work and is given by,


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( )

( ) θ 5.3983θ 10.1068θ 10.9327θ 6.3283θ 1.4234θf

V θf F (HB) CER

2345

nps

0.59

+−+−=

= −

(16)

where ER is the erosion ratio, defined as the amount of mass lost by the wall material due to particle impacts divided by the mass of particles impacting; HB is the Brinell hardness; Fs is particle shape coefficient; Fs=1.0 for sharp (angular), 0.53 for semi-rounded, or 0.2 for fully rounded sand particles; θ is the particle impact angle in radians; Vp is the particle impact speed in m/s; and C and n are empirical constants, C = 2.17×10−7 and n = 2.41. The function f(θ) used in Equation (16) is for a corrosion resistance alloy. For other materials f(θ) must be determined by erosion testing. Also, the hardness function was developed for carbon steels. For other materials, the factor C must be modified based on empirical information.

Validation of the DRW (Discrete Random Walk) Model

As described in the previous section, the turbulent dispersion of particles can be modeled using a Discrete Random Walk (DRW) model. Both FLUENT 6 and the current 2-D model use the DRW model to account for the effect of turbulence on particle tracking. This model is first examined using some erosion data collected at BOT (Baker Oil Tools). More information about the data can be found in Chapter 8 of E/CRC 2007 Report.

The data was collected in 1” ID 90º sharp bend (Figure 6). Each bend consists of upstream and downstream sections, joined together at the corner ends, which are cut 45° by high axial compressive force. Wall thickness at multiple locations on both upstream and downstream sections was measured before and after the tests. Four tests were conducted as listed in Table 1. Erosion in the upstream section is very likely caused by sand particles that are driven toward the wall by turbulence since there is no significant flow re-direction. Therefore, the wall thickness loss of the upstream section is used to examine the DRW model in this work.

Table 1. BOT Test Conditions (Russell et al. 2004).

Test 1 256 μm sand, 50 ft/s, Water Test 2 25 μm sand, 50 ft/s, Water

Test 3 25 μm sand, 85.8 ft/s, Water

Test 4 256 μm sand, 85.8 ft/s, Water


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Figure 6. Geometry Tested by BOT (Russell et al. 2004).

Simulations were conducted for all four cases using FLUENT 6. The DRW model was applied to model the particle-turbulence interaction. Erosion on the upstream wall was predicted and compared with the experimental data. Figure 7 to 10 show the comparison between the CFD prediction and the experimental data for the four cases listed in Table 1. The x-axis and the plot series identify the location where the wall thickness was measured and the comparison is made. The y-axis is the ratio of the CFD prediction over the measured wall thickness loss. It can be seen from these figures that the average ratio ranges from 0.43 to 1.47. This is considered to be very good performance given so many factors that are affecting sand erosion.

Figure 7. Performance of the DRW Model (Test 1: 256 µm, 50 ft/s).

Flow

Upstream


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Figure 8. Performance of the DRW Model (Test 2: 25 µm, 50 ft/s).

Figure 9. Performance of the DRW Model (Test 3: 25 µm, 85.8 ft/s).


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Figure 10. Performance of the DRW Model (Test 4: 256 µm, 85.8 ft/s).

Particle Tracking: 2-D versus 1-D

It has been pointed out that the 1-D model is limited due to the calculation of only one velocity component and the lack of turbulent effect. This can be better understood by examining some sample CFD and experimental results. Figure 11 shows the schematic of the direct impingement geometry used in both CFD and experiments. The nozzle diameter is 8 mm and the nozzle-wall distance is 12.7 mm. In the experiment, both of the particle radial and axial velocity components at about 0.2 mm away from the target wall are measured using LDV (Laser Doppler Velocimetry). In this LDV experiment, the carrier fluid is water with average velocity of 10 m/s in the nozzle. Aluminum particles with an average size of 120 μm were used for measuring velocity components. The measured velocity components are used to calculate the velocity magnitude (speed) and the angle between the velocity vector and the target wall. Since it is so close to the wall, these values are assumed to be the actual impact speed and impact angle.


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Figure 11. Schematic of Direct Impingement Geometry.

Figure 12 shows how the impact angle changes with the radial distance, both measured using LDV and predicted by CFD. It can be seen that particles near the centerline impact at greater angles. Away from the centerline, the flow turns its direction and the particle impact angle decreases. Figure 13 shows the distribution of particle impact speed on the target wall. Both LDV data and CFD results show similar trends, namely, the lowest impact speed near the centerline and the highest impact speed at around 6 to 8 mm away from the centerline.

CFD simulations with air as the carrier fluid are also performed and the predicted particle impact angle and impact speed are displayed in Figure 14 and Figure 15, respectively. The predicted impact angle and impact speed are almost constant within the nozzle diameter and then dramatically decrease.

The 1-D model tracks one representative particle along the centerline. Depending on the flow conditions, the 1-D model may not be able to catch the maximum erosion rate. The 2-D model does not have this limitation


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0

10

20

30

40

50

60

70

80

90

0 2 4 6 8 10 12 14

Radial Distance from Nozzle Centerline (r, mm)

Impa

ct A

ngle

(deg

ree)

LDV Data CFD Result

Figure 12. Distribution of Particle Impact Angle on the Target Wall (Water, 10 m/s).

0

2

4

6

8

10

0 2 4 6 8 10 12 14


Impa

ct S

peed

(m/s

)

LDV Data CFD Result

Figure 13. Distribution of Particle Impact Speed on the Target Wall (Water, 10 m/s).


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0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5 6 7 8 9 10 11 12 13


Impa

ct A

ngle

(deg

)

Figure 14. Predicted Particle Impact Angle on the Target Wall (Air, 10 m/s).

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10 11 12 13


Impa

ct S

peed

(m/s

)

Figure 15. Predicted Particle Impact Speed on the Target Wall (Air, 10 m/s).


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Effect of Turbulence on Erosion Prediction

Figure 16 to 18 show the distribution of predicted erosion rate on the target wall for different cases. The results with and without accounting for the effect of turbulence are compared. In the case of Figure 16, the carrier fluid is water and the diameter of particles is 75 μm. For this case, it is obvious that the effect of turbulent fluctuations in the particle tracking model is significant. For the case of 300 μm particles in water as shown in Figure 17, the effects are negligible. Figure 18 shows the results for 25 μm particle carried by air. It can be seen that the effect of turbulence varies depending on both particle and fluid properties. Large particles possess more momentum and are not affected by the turbulence as much as small particles. Turbulent fluctuations in liquid flow also affect particle motion more than in gas flow.

One should also notice that the predicted maximum erosion in the liquid case is outside of the nozzle radius; while for the gas case, the predicted maximum erosion is within the nozzle radius and almost does not change with respect to the radial distance. This is consistent with what have been shown in Figures 12 to 15.

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

0 1 2 3 4 5 6 7 8 9 10 11 12 13


Eros

ion

(kg/

m2-

s)

75 μm, with turb.75 μm, without turb.

Figure 16. Effects of Turbulence and 2-D Particle Tracking (Water, 10 m/s, 75 μm Sand).


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0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

3.5E-03

4.0E-03

0 1 2 3 4 5 6 7 8 9 10 11 12 13


Eros

ion

(kg/

m2-

s)


Figure 17. Effects of Turbulence and 2-D Particle Tracking (Water, 10 m/s, 300 μm Sand).

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

1.2E-02

1.4E-02

1.6E-02

1.8E-02

0 1 2 3 4 5 6 7 8 9 10 11 12 13


Eros

ion

(kg/

m2-

s)


Figure 18. Effects of Turbulence and 2-D Particle Tracking (Air, 10 m/s, 25 μm Sand).


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Validation of the 2-D Particle Tracking in SPPS

The 2-D particle tracking subroutine developed for 2-D SPPS is first examined by comparing the particle trajectories calculated by both 2-D SPPS and FLUENT 6. Figure 19 shows the calculated trajectories of 5 particles between the nozzle and the target wall. The geometry is the same as shown in Figure 11. The flow is water at 10 m/s and the particles’ diameter is 150 µm. The flow field information used in 2-D SPPS is exported from FLUENT 6, which means that both 2-D SPPS and FLUENT 6 use the same flow field information in the trajectory calculation. Since the purpose is to examine the 2-D particle tracking subroutine, the DRW model was turned off to get rid of the effect of turbulence. The random fluctuation caused by turbulence makes it hard to compare particle trajectories. It can be seen from this figure that the particle trajectories calculated by 2-D SPPS and FLUENT 6 are almost the same. A close-up view of the same particle trajectories near the target wall is shown in Figure 20. Very small difference exists between the trajectories calculated by 2-D SPPS and FLUENT 6. Figure 21 and Figure 22 show the calculated trajectories of 25 µm sand particles. Again, the trajectories calculated by 2-D SPPS and FLUENT 6 are almost the same, which means the 2-D particle tracking subroutine for 2-D SPPS is working properly.

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 0.002 0.004 0.006 0.008 0.01 0.012

Distance from Nozzle Exit, m

Dis

tanc

e fro

m C

ente

rline

, m

2-D SPPSCFD (Fluent)Target Wall T

arge

t Wal

l


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Figure 19. Calculated Particle Trajectories (150 µm, Water, 10 m/s, No Turbulence).

Figure 20. Calculated Near-Wall Particle Trajectories (150 µm, Water, No Turbulence).

0

0.002

0.004

0.006

0.008

0.01

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Figure 21. Calculated Particle Trajectories (25 µm, Water, No Turbulence).

Figure 22. Calculated Near-Wall Particle Trajectories (25 µm, Water, No Turbulence).

Notice that the 25 µm sand particles are not able to reach the target wall if the DRW model is turned off, as shown in Figure 22. This is because the fluid flows parallel to the wall in regions very close to the wall. The small sand particles do not possess enough momentum and tend to follow the streamline. However, the fluctuations (turbulent eddies) can drive these small sand particles toward the wall, causing certain amount of erosion damage. Therefore, the DRW model has to be turned on to predict the erosion caused by the small sand. Figure 23 shows the calculated trajectories of a few 25 µm sand particles near the target wall, with the DRW model turned on. These small particles are now able to impact the wall.

In addition to particle trajectories, calculated erosion rate is also compared between 2-D SPPS and FLUENT 6 to ensure that 2-D SPPS works properly. Figure 24 demonstrates that predictions made by using 2-D SPPS agree with CFD predictions extremely well, including the case with 20 µm sand carried by water, in which the 1-D model significantly under-predicts the erosion rate.

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Figure 23. Calculated Near-Wall Particle Trajectories (25 µm, Water, With Turbulence).

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Figure 24. Calculated Erosion Rate: 2-D SPPS versus FLUENT 6.

Interpolation of the Flow Field Information

The predictions made by using 2-D SPPS shown in Figure 24 still rely on CFD simulations directly. For all 7 cases shown in this figure, 2-D SPPS uses the flow field information exported from FLUENT 6. In actual applications, however, it is not feasible to run CFD simulations and feed the flow field information to 2-D SPPS. A method is needed for 2-D SPPS to handle general cases without running CFD simulations.

For each type of geometries that 2-D SPPS can handle, namely direct impingement and standard elbow, CFD simulations for different flow conditions with a range of Reynolds numbers are run and the flow field information is exported and saved. For any particular flow conditions under which the erosion is to be calculated, its Reynolds number is first calculated and compared with the pre-saved cases. The flow field information needed for conducting calculation for this particular case is then interpolated from the pre-saved cases based on the Reynolds number.

Following this procedure, erosion rate of cases shown in Figure 4 and Figure 5 is calculated using 2-D SPPS. The results are compared with the experimental data and shown in Figure 25. The calculation uses pre-saved flow field information for direct impingement


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geometry and standard elbow geometry. No CFD simulation is run specifically for any of the cases shown in this figure. It can be seen by comparing Figures 4, 5, and 25 that the 2-D model performs much better than the 1-D model. The predictions made by using 2-D SPPS are similar to those by the CFD-based model. Both agree with the experimental data very well.

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Figure 25. Comparison of Experimental Data with Prediction by 2-D Model.

2-D SPPS for Multiphase Flow

The previous sections show that the 2-D particle tracking works very well for single-phase flows, namely flows with sand particles carried by gas or liquid. The next step is to extend the feature to multiphase flows. The basic idea is to use mechanistic models to predict flow regime and multiphase flow parameters such as annular film thickness, entrainment rate, slug characteristic, etc. Two-dimensional stagnation zone is then defined based on the calculated multiphase flow parameters. The flow in the 2-D stagnation zone is assumed to be single phase. Trajectories of multiple particles are then calculated in the 2-D stagnation zone to provide impact information, which is then used for erosion calculation.




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Figure 26 illustrates the 2-D particle tracking for annular flow in elbow geometry. The flow parameters such as annular film thickness, film velocity, gas core velocity, gas and liquid properties, are provided by mechanistic models that are available in SPPS. The pre-saved flow field information is then interpolated to provide the information needed for calculating particle trajectories in the gas core and the annular film. When any particle impacts the wall, erosion rate is calculated and stored for the corresponding location.

Figure 26. 2-D Particle Tracking in Annular Flow in Elbow.

The same procedure is being applied to slug flow (Figure 27). Mechanistic models within SPPS are utilized to calculate the slug characteristics such as the slug length, velocity, frequency, etc. It is assumed that the erosion is mainly caused by sand particles carried by the slug body (the liquid slug part of the slug unit). When the liquid slug enters the elbow, the flow in this region becomes nearly single phase or at least liquid dominated. It is then treated as single-phase flow and the 2-D particle tracking model is applied within the liquid slug. This work is still under development and results are being generated with this model.


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Figure 27. 2-D Particle Tracking in Slug Flow in Elbow.

For other types of multiphase flow such as bubbly flow, churn flow, dispersed

bubble flow, similar assumption can be made to simplify the flow. Two-dimensional stagnation zone can then be defined based on the flow parameters. The flow in the 2-D stagnation zone is then treated as single phase and the 2-D particle tracking model can be applied. This work is ongoing and more details will be available in the next annual E/CRC report.

Summary and Future Work

SPPS has been developed based on one-dimensional particle tracking. For certain cases such as small sand carried by liquid, the 1-D particle tracking may result in significant under-prediction of erosion. There are two main reasons. First, only one velocity component of sand particle is calculated in 1-D particle tracking. This may work well if the other two velocity components are relatively small. Ignoring the small velocity components does not affect the erosion calculation too much. But there are cases in which the other velocity components are comparable to or even greater than the velocity component being calculated. Using 1-D particle tracking in these cases will cause under-prediction. The second reason is


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that the 1-D particle tracking does not account for the effect of turbulence. For particles moving parallel to the wall, it is the turbulence that drives the particles toward the wall. Therefore, neglecting the effect of turbulence will also result in under-prediction of erosion for certain cases.

In order to address the limitations that 1-D particle tracking has, two-dimensional particle tracking is introduced into SPPS. Two velocity components and the effect of turbulence are considered in the 2-D particle tracking. SPPS with 2-D particle tracking has been examined for single-phase flow. Its overall performance is very close to the CFD-based model and much better than the 1-D SPPS.

The current work is to extend 2-D SPPS to multiphase flow. Basic steps are first using mechanistic models to predict flow regimes and provide fluid properties and flow characteristics, then performing 2-D particle tracking is being tailored to different flow regimes.

References

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McLaury, B.S., 1996, “Predicting Solid Particle Erosion Resulting from Turbulent Fluctuations in Oilfield Geometries”, Ph.D. Thesis, The University of Tulsa.

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