models for the calculations of the uid velocity eld and the motion of the solid particles have been estab-

lished and an erosion model was used to predict the erosion rate. The uid velocity (continuous phase)

* Corresponding author. Tel.: +966 3 860 2543; fax: +966 3 860 2949.

E-mail address: badrhm@kfupm.edu.sa (H.M. Badr).

www.elsevier.com/locate/compuid

Computers & Fluids 34 (2005) 7217420045-7930/$ - see front matter 2004 Elsevier Ltd. All rights reserved.model was based on the time-averaged governing equations of 3-D turbulent ow and the particle-tracking

model (discrete phase) was based on the solution of the governing equation of each particle motion taking

into consideration the eect of particle rebound behavior. The eects of ow velocity and particle size were

investigated for one contraction geometry considering water ow in a steel pipe. The results showed the

strong dependence of erosion on both particle size and ow velocity but with little dependence on the direc-

tion of ow. The eect of ow direction was found to be signicant only for large particle size and moderate

ow velocity. The erosion critical area was found to be the inner surface of the tube sheet (connecting the

two pipes) in the region close to the small pipe. The results also indicated the presence of a threshold velo-city below which erosion is insignicant for all particle sizes.

2004 Elsevier Ltd. All rights reserved.Numerical investigation of erosion threshold velocity ina pipe with sudden contraction

H.M. Badr *, M.A. Habib, R. Ben-Mansour, S.A.M. Said

Mechanical Engineering Department, King Fahd University of Petroleum and Minerals,

Box # 322, Dhahran 31261, Saudi Arabia

Received 14 April 2003; received in revised form 31 January 2004; accepted 27 May 2004

Available online 27 October 2004

Abstract

This paper deals with erosion prediction in a pipe with sudden contraction for the special case of two-

phase (liquid and solid) turbulent ow with low particle concentration. The pipe axis was considered ver-

tical and the ow was either in direction of gravity (downow) or against it (upow). The mathematicaldoi:10.1016/j.compuid.2004.05.010

Nomenclature

A surface areab constant dened in Eq. (13)CD drag coecientCl constant dened in Eq. (4)C1 constant dened in Eq. (6)C2 constant dened in Eq. (8)C2 constant dened in Eq. (6)d diameterDp Solid particle diameterE Erosion rate, mg/gF forceGk generation of turbulent kinetic energyg gravitational accelerationk turbulent kinetic energymp mass of individual particleNp number of particlesp pressureRep particle Reynolds numbers Sand owUj average velocity componentu uid velocity vectoruj uctuating velocity componentup particle velocityVi Flow inlet velocityVt Threshold erosional velocityup particle velocityxj space coordinatet time

Greek letters

a impact anglee dissipation rate of turbulent kinetic energyl dynamic viscosityq densityrk eective Prandtl number for kre eective Prandtl number for e

Superscripts time rate time average

722 H.M. Badr et al. / Computers & Fluids 34 (2005) 721742

1. Introduction

Erosion is one of the important problems in various gas and liquid ow passages such as ow inpipes and pipe ttings (valves, bends, elbows, ow meters, . . .etc.), ow in pumps, turbines, com-pressors and many others. Erosion may cause equipment malfunctioning (vibration, leakage,excessive energy losses, . . .etc.) and may also lead to complete failure of machine components.Accurate prediction of the rate of erosion in a specic application is one of the very complicatedproblems since it requires detailed investigation of the solid particle motion before and after im-pact. The diculty arises mainly from the fact that most ows occurring in industrial processesare turbulent which makes the particle trajectory and impact characteristics dicult to predicttaking into consideration all uid forces acting on the particle. The following literature reviewis limited to previous work done on erosion in pipes and pipe ttings.Previous erosion studies can be classied under three categories; experimental investigations,

erosion model developments, and numerical simulations. Tilly [1] presented a thorough analysisof the various parameters aecting erosion, including particle properties, impact parameters, par-ticle concentration, material temperature, and tensile stress. He also reviewed the dierent mech-anisms of erosion, which were categorized into brittle and ductile behaviors. Ru and Wiederhorn[2] presented another review of the solid particle erosion phenomena considering single and mul-tiple particle models on erosion of metals and ceramics. The signicant parameters for eroding

Subscripts

D dragf uidsl Saman liftlc localm target materialp particlepg pressure gradientvm virtual mass

H.M. Badr et al. / Computers & Fluids 34 (2005) 721742 723particles and material characteristics were also presented. Humphrey [3] reported a more compre-hensive review of the fundamentals of uid motion and erosion by solid particles. The review in-cludes a discussion of the experimental techniques and the various fundamental considerationsrelating to the motion of solid particles. An assessment of the uid mechanics phenomena thatcan signicantly inuence erosion of material surfaces by impinging particles was also presented.Because of its direct relevance to gas and oil industries, erosion of pipes and pipe ttings attracted

many researchers. Several experimental studies were conducted with the main objective being todetermine the rate of erosion in such ow passages and its relation with the other parameters in-volved in the process. Among these studies are the works by Rochester and Brunton [4], Trueand Weiner [5], Glaeser and Dow [6], Roco et al. [7], Venkatesh [8], and Shook et al. [9]. Soderberget al. [10,11] andHutchings [12,13] reported the advantages and disadvantages of such experiments.The recent experimental study by McLaury et al. [14] on the rate of erosion inside elbows andstraight pipes provided correlations between the penetration rate and the ow velocity at dierent

values of the elbow diameter and sand rate and size. Edwards et al. [15] reported the eect of thebend angle on the normalized penetration rate. The objective of most of these experimental studieswas to provide data for establishing a relationship between the amount of erosion and the physicalcharacteristics of the materials involved, as well as the particle velocity and angle of impact. Blan-chard et al. [16] carried out an experimental study of erosion in an elbow by solid particles entrainedin water. The elbow was examined in a closed test loop. Electroplating the elbow surface and pho-tographing after an elapsed period of time were carried out to show the wear pattern. The theoret-ical model developed by Rabinowicz [17] was used to calculate the volume of material removed.The results indicated that the sand particle trajectories appeared to be governed by the secondaryows and that there was no simple liquid velocity prole that can be used to calculate the particletrajectories in order to make an accurate prediction of the location of the point of maximum wear.Several erosion models/correlations were developed by many researchers to provide a quick an-

swer to design engineers in the absence of a comprehensive practical approach for erosion predic-tion. One of the early erosion prediction correlations is that developed by Finnie [18] expressingthe rate of erosion in terms of particle mass and impact velocity. In that correlation, the rate oferosion was proportional to the impact velocity squared. In a recent study, Nesic [19] found thatFinnies model overpredicts the erosion rate and presented another formula for the erosion rate interms of a critical velocity rather than the impact velocity. One of the early erosion models wasthat suggested by Bitter [20,21]. In that model, the erosion was assumed to occur in two mainmechanisms; the rst was caused by repeated deformation during collisions that eventually resultsin the breaking loose of a piece of material while the second was caused by the cutting action ofthe free-moving particles. Comparisons between the obtained correlations and the test resultsshowed a good agreement. It was concluded that cutting wear prevails in places where the impactangles are small (such as in risers and straight pipes) and it is sucient to use hard material in suchplaces to reduce erosion. Tilly [1] suggested another two-stage mechanism for explaining dierentaspects of the erosion process for ductile materials. In the rst stage, the particles indent the targetsurface, causing chips to be removed and some material to be gouged and extruded to form vul-nerable hillocks around the scar. The second stage was the one in which the particles break up onimpact causing fragments to be projected radially to produce a secondary damage. A correlationwas presented relating erosion to the energy required to remove a unit mass and the particle veloc-ity and size. The calculated values of erosion were compared with the experimental data for dif-ferent particle sizes and a reasonable agreement was found, however, the validity of the work waslimited to ductile materials and could not be generalized to include other materials. Other erosionmodels were suggested by Laitone [22], Salama and Venkatesh [23], Bourgoyne [24], Chase et al.[25], McLaury [26], Svedeman and Arnold [27], and Jordan [28].Recently, Shirazi and McLaury [29] presented a model for predicting multiphase erosion in

elbows. The model was developed based on extensive empirical information gathered from manysources, and it accounts for the physical variables aecting erosion, including uid properties,sand production rate and size, and the uid-stream composition. An important dierent featureof this model was the use of the characteristic impact velocity of the particles. The method usedfor obtaining this characteristic velocity for an elbow was an extension of a previous methodintroduced by the same authors for the case of a single-phase ow. The results from the modelwere compared with previous experimental results for elbows and were found to have a better

724 H.M. Badr et al. / Computers & Fluids 34 (2005) 721742agreement with eld failure data.

The use of computational methods in erosion prediction constitutes a combination of owmodeling, Lagrangian particle-tracking, and the use of erosion correlations. The ow model isused to determine the ow eld for a given geometry while the particle-tracking model is usedto determine the particle trajectories for solid particles released in the ow. The particle impinge-ment information extracted from the trajectories is used along with the empirical erosion equa-tions to predict the erosion rates. This model, which is sometimes called the Lagrangianapproach, requires expertise in uid dynamic modeling and a large amount of computationalwork. Boulet et al. [30] conducted numerical solutions for the turbulent ow of an airsolid sus-pension in a heated vertical pipe using EulerianEulerian and EulerianLagrangian formulations.The main task was to assess the accuracy of these two formulations, taking the experimental datareported by Tsuji et al. [31] and Jepson et al. [32] as a base for comparison. The rst part of thepipe contains developing ow with no heat transfer. In the second part of the pipe, the dynami-cally fully developed ow was heated using a heated section of the pipe (constant heat ux). Thesimulation was carried out for dierent values of mass loading with particles of 500lm diameter.The comparisons with experimental data for the dynamic features of the ow showed the sameaccuracy level for both formulations, especially for dilute ows. However, the accuracy was foundto decrease signicantly in both formulations as more particles were injected in the ow.Lagrangian models were developed by many researchers such as Lu et al. [33], Wang et al. [34],

Keating and Nesic [35] and Wallace et al. [36] who used combinations of computational uiddynamics and dierent Lagrangian particle-tracking models to predict the particle movementthrough complex geometries. Dierent computational uid dynamic packages such as PHOE-NICS [35,37] and CFX-code [38] were used to predict the uid ow eld. Wang et al. [34] devel-oped a computational model for predicting the rate of erosive wear in a 90 elbow for the twocases of sand in air and sand in water. The ow eld was rst obtained and then the particle tra-jectory and impacting characteristics were then determined by solving the equation of particle mo-tion taking into consideration all the forces including drag, buoyancy, and virtual mass eectswith the assumption of a uniform distribution of the solid particles at the starting section. Thepenetration rate was obtained using a semi-empirical relation that was previously developed byAhlert [39]. A comparison between the predicted penetration rates and the available experimentaldata showed a good agreement.In a recent study by Edwards et al. [15], an erosion prediction procedure was developed and

veried based on a CFD code combining ow eld analysis and particle-tracking for obtainingparticle impingement data. The erosion rate was then computed using the empirical relations ofAhlert [39] and applied to predict erosion in a pipe bend tting made of carbon steel. TheCFD code utilized a nite-volume multiblock approach for solving NavierStokes equationsbased on a user-dened computational model that was described by Patankar [40]. The authorsused the Lagrangian particle-tracking algorithm of the CFD code for the prediction of individualtrajectories of the dispersed phase through the ow eld.Based on the above literature search and to the best of the authors knowledge, most of the pub-

lished work on erosion in pipes focused on straight pipes and pipe ttings such as bends andelbows. Apart from the work of Nesic [19], Postlethwaite and Nesic [41] and Blatt et al. [42],the erosion process occurring in a pipe with sudden contraction or sudden enlargement wasnot considered in any previous study. Postlethwaite and Nesic [41] and Blatt et al. [42] provided

H.M. Badr et al. / Computers & Fluids 34 (2005) 721742 725experimental data for erosion inside a pipe with sudden contraction and sudden enlargement. The

present research work aims at studying the eect of uid ow parameters on the rate of erosion ina pipe contraction under conditions simulating the actual working conditions. The calculations ofthe ow pattern and solid particle motion inside the pipe contraction were performed and theavailable data in the literature were used for estimating the rate of erosion. The computationalprocedure is validated against the results of Postlethwaite and Nesic [41].

2. Problem statement and solution methodology

The ow domain consists of a straight pipe of diameter, D = 200mm, connected to a smallerpipe of diameter, d, as shown in Fig. 1 with diameter ratio, d/D = 0.5. The pipe centerline is alwaysvertical while the direction of uid ow is either vertically upward or vertically downward. Bothpipes are made of carbon steel and both are long enough to justify the assumption of fully devel-oped ow at the entrance and exit sections of the ow domain. The uid considered in this study iswater at 20C with low particle concentration such that the eect of particle motion on the uid

726 H.M. Badr et al. / Computers & Fluids 34 (2005) 721742ow eld is negligibly small.In general, the rate of erosion in tubes depends upon many parameters such as the properties of

the impacting particles, the properties of the tube material, and the other parameters of the impactprocess [1,2,43]. In this study, the main parameters aecting erosion are the ow velocity and par-ticle size and concentration. In order to predict the rate of erosion, the ow eld characteristicsand the details of the particle impact process in addition to the erosion rate correlations are re-quired. The Lagrangian particle-tracking method is used to model the erosion process and is nor-mally carried out using the following steps [36]:

(a) Predict the ow velocity eld in the domain of interest.(b) Calculate the trajectories of solid particles entrained in the uid using Lagrangian particle-

tracking calculations and then extract the particle impact data.(c) Predict the erosive wear using one of the available semi-empirical correlations.Fig. 1. Flow passage geometry for the two cases of upow and downow.

Th

Aused

Refs.

The steady state time-averaged conservation equations of mass and momentum can be writtenas

oUi oUj

2

H.M. Badr et al. / Computers & Fluids 34 (2005) 721742 7272.1.2. Conservation equations for the turbulence model

The conservation equations of the turbulence model (Reynolds [46] and Shih et al. [47]) aregiven as follows:

o qUjk o leff ok

Gk qe 5quiuj leff oxj oxj 3 qkdij 3

where dij is the Kronecker delta and le = lt + l is the eective viscosity. The turbulent viscosity,lt, is calculated using the high-Reynolds number form as

lt qClk2

e4

with Cl = 0.0845, k and e are the kinetic energy of turbulence and its dissipation rate. These areobtained by solving their conservation equations as given below.oxjqUiUj oxi oxj l oxj oxj quiuj 2

where p is the static pressure and the stress tensor quiuj is given byooxj

qUj 0 1

o op o oUi

o2.1.1. The continuity and momentum equations[44,45] and can be presented as follows.ow pattern of the continuous ow phase, the conservation equations for mass and momentumare solved. Additional transport equations for the turbulence model are also solved since the owis turbulent. The time-averaged governing equations of 3-D turbulent ow can be found in manycombination of computational uid dynamics and Lagrangian particle tracking are normallyto predict the particle movement through complex geometries [34,35,15,36]. To predict the2.1. The continuous phase modellow volume of particles is simulated. Two computational models were developed; the rst isthe continuous phase model (dealing with the prediction of the ow velocity eld) and the secondis the particle-tracking model (dealing with the prediction of particle motion). A brief discussionof the two models is presented in the following sections.is approach represents a one-way ow-to-particle coupling method that can be used whenoxj oxj rk oxi

The wall functions establish the link between the eld variables at the near-wall cells and the

dissipation rate are assigned through a specied value of k=U equal to 0.1 and a length scale, L,

down

cedumati

ing tof remoremod

728 H.M. Badr et al. / Computers & Fluids 34 (2005) 721742he grids are shown in Fig. 2a and b for the axial and radial velocity proles. The inuencening the grid on the continuous-phase velocity eld is very negligible and indicates thatmesh renement will result in negligible changes in the results of the computationalthan 0.1%. The grid independence tests were performed by increasing the number of controlvolumes from 7000 to 43,750 in two steps; 700030,000 and 30,00043,750. The results of ren-re described by Patankar [40]. Convergence is considered when the maximum of the sum-on of the residuals of all the elements for U, V, W and pressure correction equations is lessaxial direction). This ne mesh is necessary to capture the steep velocity gradients close tothe contraction. The conservation equations are integrated over every nite volume to yieldthe details of the velocity eld. The equations are solved simultaneously using the solution pro-to a minimum rectangular cell size of 0.1mm (in the radial direction) 0.15mm (in theequal to the diameter of the inlet pipe. The boundary condition applied at the exit section is thatof fully developed ow. At the wall boundaries, all velocity components are set to zero in accord-ance with the no-slip and impermeability conditions. Kinetic energy of turbulence and its dissipa-tion rate are determined from the equations of the turbulence model.

2.1.4. Solution procedureThe solution domain was rst divided into a large number of nite volumes (at least 30,000

nite volumes). All meshes are of structured type and have ne meshing at the contractioncorresponding quantities at the wall. These are based on the assumptions introduced by Launderand Spalding [48] and have been most widely used for industrial ow modeling. The details of thewall functions are provided by the law-of-the-wall for the mean velocity as given by Habib et al.[44].

2.1.3. Boundary conditionsThe velocity distribution is considered fully developed at the inlet section. Kinetic energy and its

2qooxj

qUje ooxileffre

oeoxi

C1Gk ek C

2q

e2

k6

where Gk represents the generation of turbulent kinetic energy due to the mean velocity gradientsand is given by

Gk quiuj oUjoxi 7The quantities rk and re are the eective Prandtl numbers for k and e, respectively and C

2 is given

by Shih et al. [47] as

C2 C2 C3 8where C3 is a function of the term k/e and, therefore, the model is responsive to the eects of rapidstrain and streamline curvature and is suitable for the present calculations. The model constantsC1 and C2 have the values; C1 = 1.42 and C2 = 1.68.el.

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

Axi

al v

eloc

ity, m

/s

Mesh 2, 30000 volumes

Mesh 1, 7000 volumesMesh 3, 43750 volumes

H.M. Badr et al. / Computers & Fluids 34 (2005) 721742 7292.2. Particle-tracking

The particle-tracking calculations aim to determine the particle trajectory from the moment itenters the ow domain until it leaves the small tube. Of special interest is the particle velocity(magnitude and direction) before every impact either on the pipes walls or anywhere on the tubesheet. Such impact velocity is not only important for the calculation of solid surface erosion butalso important in the determination of the particle trajectory during its subsequent course of mo-tion following impact. One of the main assumptions in this study is that the solid particles are notinteracting with each other (the particles do not collide and the motion of any particle is not inu-enced by the presence or motion of neighboring particles). Moreover, the inuence of particle mo-tion on the uid ow eld is considered very small and can be neglected. These two assumptionsare based on the condition of fairly dilute particle concentration. The same assumptions were

0.0-0.30 -0.20 -0.10 0.00 0.10 0.20 0.30

Axial distance measured from the contraction section

0.000

0.005

0.010

0.015

0.020

0.025

-2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0

Axial velocity, m/s

Rad

ial d

ista

nce,

m

Mesh 1, 7000 volumesMesh 2, 30000 volumesMesh 3, 43750 volumes

(a)

(b)

Fig. 2. Grid independence tests. (a) Comparison of axial velocity along the tube axis for dierent mesh sizes.

(b) Comparison of axial velocity along the radius at 1mm upstream of the contraction for dierent mesh sizes.

Takibe w

p

Maging f

whersever

wherBe

consotherThe

pg

owF pg qqp

rp 14

The above statement implies that the pressure does not vary signicantly over a distance of oneparticle diameter, a condition that is normally satised for reasonably small particles. Accordingly,

the pwhich acts on every volume element of the owing medium and can be written as: !uid surrounding the particle. This term can be expressed as

F vm 1

2

qqp

d

dtu up 13

Although the virtual mass force, Fvm, is only important when q > qp which is not the case in thepresent study, the virtual mass force was considered in the present calculations. The second forceis that due to pressure gradient, F , that arises from the inuence of the pressure gradient in thee b1, b2, b3 and b4 are constants that depend on the particle shape.cause of the low particle concentration assumed in the present study, the particle motion isidered non-interacting and the dominant force in Eq. (9) is the drag force [15]. Some of theforces given in Eq. (9) are of small order of magnitude and can be neglected in this study.

rst of these is the virtual mass term that takes care of the force required to accelerate theCD 24Rep 1 b1Rb2ep

b3Repb4 Rep 12e the as are constants given by Morsi and Alexander [51] for smooth spherical particles overal ranges of Re. Another equation that is frequently used for CD [52] is given bythe drag coecient, CD, are obtained from

Rep qDpjup uj

l10

CD a1 a2Rep a3R2ep

11pg sl

nus lift force (resulting from particle rotation) and the Basset history force (the force account-or the ow eld unsteadiness) have been neglected. The particle Reynolds number, Rep, anddt F Du up gqp q=qp F vm F pg F sl 9

where FD(uup) is the drag force per unit particle mass and F D 3CDlRep=4qpD2p, gqp q=qp isthe buoyancy force term, Fvm is the virtual mass term (force required to accelerate the uid sur-rounding the particle), F is the pressure gradient term and F is the Saman lift force [50]. Theng the main hydrodynamic forces into consideration, the particle equation of motion canritten as:

dumade by Lu et al. [33], Shirazi et al. [49], Edwards et al. [15], Keating and Nesic [35] and Wallaceet al. [36] in the solution of similar problems of low particle concentration (

Thopenconned ow zone. In such a case, the trajectory calculations are terminated.

2.3. The erosion model

Erosion is dened as the wear that occurs when solid particles entrained in a uid stream strikethat the particle has escaped and the trajectory calculations are then terminated.(c) Particle trappingThe trajectory calculations for some particles (normally very few particles) are terminated when

the particles get trapped in the ow eld. This is found to occur when a particle circulates in aa sure calculations of the particle trajectory are terminated at the point when it passes through anboundary (the exit section). When the particle encounters such boundary, it is consideredThe boundary conditions considered when a particle strikes a boundary surface depends on thenature of that surface and one of the following possibilities may occur:

(a) Reection via an elastic or inelastic collisionReection is the term used to describe the particle rebound o the solid boundarywith a change in

its momentum. The normal coecient of restitution denes the amount of momentum in the direc-tion normal to the wall that is retained by the particle after colliding with the boundary [54]. Thecoecient of restitution is taken as 0.9 in the present calculations for the case of reection at a wall.(b) Escape through the boundaryparticles but also due to the small pressure gradient prevailing in the ow eld. The other forcesinclude the thermophoretic force which is related to small particles suspended in a gas that hasa temperature gradient. The particles under such circumstances experience a force in the directionopposite to that of the gradient. Brownian force [53] apply for sub-micron particles. These forcesare neglected in the present study. The Samans lift force, or lift due to shear is also neglected.

2.2.1. Particle trajectoryThe particle velocity, up, is rst obtained by stepwise integration of the particle equation of mo-

tion (9) over a discrete time step. The particle trajectory is then predicted by integrating the equation

drdt up 15

where r is the position vector. The above equation is integrated in each coordinate direction topredict the trajectories of the discrete phase. During the integration, the uid phase velocity, u,is taken as the velocity of the continuous phase at the particle position. Turbulent dispersionof particles was modeled using a stochastic discrete-particle approach, Wallace et al. [36]. Thetracking for the particle is done with a step size of 0.1mm to make sure that the particle-trackingis updated in every cell in the particle path. Including all the important forces (gravity, virtualmass and pressure gradient) the particle-tracking is expected to be as accurate as the predictionof the uid ow eld.

2.2.2. Discrete phase boundary conditions

H.M. Badr et al. / Computers & Fluids 34 (2005) 721742 731face. The previous experimental results [43,55] show that the erosive wear-rate exhibits a

empirical coecients provided by various experimental erosion tests. No denitive theory of ero-sion currently exists, however, a number of qualitative and quantitative models do exist. These

wher

and 77419.7 respectively [36]. This formula is used in the present calculations of erosion rate.Using the particle-tracking model, the impingement data (impact speed and angle) were rst

compiled for all particles impacting the solid boundaries of the ow domain. The compiled datawere then used together with Eqs. (16a) and (16b) for computing the erosion rate at dierent loca-tions on the tube sheet. This part required the use of FORTRAN subroutines together with theCFD code.

3. Results and discussion

The rate of erosion in a pipe with sudden contraction has been investigated for the case whenthe diameter ratio, d/D = 1/2. The direction of ow is considered either vertically upward (againstgravity) or vertically downward (in direction of gravity) as shown in Fig. 1. Although the directionof the pipe axis has no eect on the ow pattern, it is expected to aect the particle trajectorythrough the contribution of gravity forces on the particle motion. The upstream pipe diameteris 200mm and the average velocity of the approaching ow ranges from 1.0 to 10.0m/s. The uidconsidered is water at 20C (q = 998kg/m3 and l = 103Ns/m2) which results in ow Reynolds

numE Np c

r

a 6 45 16a

E 1Np

12u2pcos

2a

c

12u2psin

2a

r

( )a > 45

16b

e c and r are the cutting wear and deformation wear coecients having the values 33316.9were described by Finnie [18] and Finnie et al. [57], Wang et al. [34], Keating and Nesic [35], Ed-wards et al. [15] and Shirazi and McLaury [29].The empirical erosion equations suggested by Neilson and Gilchrist [58] were later used by Wal-

lace et al. [36] to correlate the experimental erosion data in order to develop an erosion modelingtechnique. Wallace et al. [36] reported the following formulae that resulted in good accuracy whencompared to the experimental data:

112u2pcos

2a sin 2a 12u2psin

2a( )power-law velocity dependence. The velocity exponent ranges from 1.9 to 2.5. The results alsoindicate that the erosion rate is a function of the angle of impact. It is shown that the inuenceof the angle of impact depends greatly on the type of material being brittle or ductile. Predictionof erosion in straight pipes, elbows and tees show the strong inuence of uid properties, sand sizeand ow velocity on the rate of erosion [56,49,41].There have been many attempts in the past to express the solid particle erosion by an analytical

formula that could be used to predict erosion under any condition. The complexity of the erosionprocess and the number of factors involved made it dicult for obtaining a generally applicableequation. Almost all of the formulae generated have therefore some degree of dependence on

732 H.M. Badr et al. / Computers & Fluids 34 (2005) 721742ber (based on the diameter of the large pipe, D) ranging from 2 105 to 2 106. The solid

case.r/R =(10lm, 100lm, 200lm, 400lm) and for three values of ow velocity (1m/s, 5m/s, 10m/s) as

shown in Fig. 5ad. The results indicate that for particles of small diameter (Dp = 10lm), the ero-sion rate is negligibly small in the outer region of the annular plate (0.67 6 r/R 6 1.0) and reachesits maximum close to the entrance of the small pipe (r/R 0.5) as shown in Fig. 5a. The highestrate of erosion (E 8 107mg/g) was found when the velocity of ow is maximum (10m/s) anddecreases rapidly with the decrease of ow velocity until reaching zero value when the ow veloc-ity reaches 1m/s. Although the rate of erosion increases with the increase of particle diameter asshown in Fig. 5bd, the trend is almost the same in the four cases. However, for large particle size(Dp = 400lm), the region of negligible erosion for all ow velocities diminishes to (0.83 6r/R2 6 1.0) which is much smaller than that obtained in the case of small particle sizes. The otherinteresting feature that is common in the four gures is the absence of erosion for all particle sizesin the entire ow domain in the case of low ow velocity (1m/s). Qualitatively, such behavior is inconformity with the erosion prevention criterion established by Salama [59] in which a threshold

velocThis tube sheet has the shape of an annulus with inner radius, r/R = 0.5, and outer radius,1.0, where R = D/2. The erosion rates are obtained for four values of the particle diameterparticles are considered sand particles of spherical shape with diameters ranging from 10lm to400lm.In order to verify the accuracy of the computational scheme, the present results were compared

to the experimental data of Postlethwaite and Nesic [41]. The experimental data were obtained forcontraction ratio of d/D = 0.5. The large tube is 42.1mm diameter and has an inlet velocity of3.3m/s. Three values of particle concentration of 2%, 5% and 10%, by volume, were considered.The sand particle diameter was 430lm. Short 3-mm segments were used at the inlet region of thesudden contraction to determine the penetration rate. In the present calculations, the step heightwas divided to 1000 discs, then, data were integrated to give the results over a disc of 3mm widthhaving its inner diameter as the smaller tube. The results of the comparison are given in Fig. 3 interms of the penetration rate. The penetration rate, Pn, is calculated using the following equation[34,49]:

pn 31:536 106_s

qmAElc 17

where A is the impingement area (m2), Elc is the local erosion rate (mg/g), Np is the total numberof particles being tracked, pn is the penetration rate (mm/year), _s is the sand rate (kg/s) and qm isthe density of target material (kg/m3). The comparison shown in Fig. 3 indicates a reasonablygood agreement.

To present the obtained data in a meaningful way, a number of investigations were carried outwith the objective to determine the critical erosion areas. These investigations covered the entireranges of ow velocity and particle diameter. It was found that erosion occurs mainly in the con-traction section ABCD shown in Fig. 1 while being insignicant upstream and downstream of it.Fig. 4 shows the trajectories of a number of particles released at the same time at the inlet sectionof the ow eld when the ow velocity is 10m/s and the particle diameter is 400lm. The gureshows that almost all particle impacts occur on the at surface ABCD while impacts on the pipewalls are insignicant. Accordingly, erosion data will be presented only at section ABCD.Fig. 5 shows the variation of the local erosion rate on the tube sheet (ABCD) for the upow

H.M. Badr et al. / Computers & Fluids 34 (2005) 721742 733ity was set by the recommended practice API RP 14E for eliminating erosion. Another

1000

734 H.M. Badr et al. / Computers & Fluids 34 (2005) 721742similar criterion for the threshold velocity was developed by Salama and Venkatesh [23] for ero-sion in elbows.

10

100

0 2 4 6 8 10 12Particle volumetric percentage

Pene

trat

ion

rate

, mm

/y

Postlethwaite and Nesic [41]

Present calculations

Fig. 3. Comparison of the calculated penetration rate and the experimental data of Postlethwaite and Nesic [41].

Fig. 4. The trajectories of a number of particle released at the same time at the inlet section showing impact on the

contraction plate for the case of downow with Vi = 10m/s and Dp = 400lm.

0.0E+001.0E-072.0E-073.0E-074.0E-075.0E-076.0E-077.0E-078.0E-079.0E-07

0.5 0.7 0.9r/R

Eros

ion

rate

, mg/

g

1m/s5m/s10m/s

0.0E+00

2.0E-07

4.0E-07

6.0E-07

8.0E-07

1.0E-06

1.2E-06

0.5 0.7 0.9r/R

Eros

ion

rate

, mg/

g 1m/s5m/s10m/s

2.5E-06

3.0E-06

3.5E-06

4.0E-06

, mg/

g

1m/s5m/s10m/s 5.0E-06

6.0E-06

7.0E-06

8.0E-06

g/g

1m/s5m/s10m/s

(a) (b)

H.M. Badr et al. / Computers & Fluids 34 (2005) 721742 735The variation of the local erosion rate presented in Fig. 4 can be explained on the basis of thestreamline pattern plotted in Fig. 6a for the case when the ow velocity is 5m/s. The gure showsa recirculating ow region upstream of the contraction section and extending to the tube sheet(ABCD). An enlarged view of that region is shown in Fig. 6b. The ow velocity in this regionis very small and the presence of solid particles, if any, in such low velocity zone will cause neg-ligible erosion in accordance with Eqs. (16a) and (16b). The gure also shows that this recirculat-ing ow zone occupies the area on the annular plate between r/R 0.72 and r/R = 1.0. This isapproximately the same area characterized by negligible erosion in Fig. 4ad. Moreover, the max-imum erosion rate occurs in a region where the approaching ow has high velocity and large cur-vature. Both eects will give rise to higher particle velocity that impacts the surface of the tubesheet close to r/R 0.5. These features are conrmed by the particle trajectories given in Fig. 4that clearly shows the high intensity of particle impact on the tube sheet in the region close tor/R 0.5.The eect of particle diameter on the total rate of erosion occurring on the tube sheet (ABCD)

is presented in Fig. 7 for four dierent velocities. The strong dependence of erosion on ow velo-city is very clear in the gure. It is also clear that there is a threshold velocity, Vt, below whicherosion is insignicant. The gure also shows that the rate of erosion increases exponentially with

0.0E+00

5.0E-07

1.0E-06

1.5E-06

2.0E-06

0.5 0.6 0.7 0.8 0.9 1.0

r/R

Eros

ion

rate

0.0E+00

1.0E-06

2.0E-06

3.0E-06

4.0E-06

0.5 0.7 0.9

r/R

Eros

ion

rate

, m

(d)(c)

Fig. 5. The variation of the local erosion rate on the contraction plate (ABCD) for the case of upow: (a) Dp = 10lm,(b) Dp = 100lm, (c) Dp = 200lm, (d) Dp = 400lm.

736 H.M. Badr et al. / Computers & Fluids 34 (2005) 721742particle diameter. Fig. 8 represents the same data plotted in Fig. 7, however, the inlet ow velocityis used as abscissa instead of the particle diameter. The gure emphasizes the power law growth of

Fig. 6. (a) The streamline pattern for the case of Vi = 5m/s. (b) An enlarged view of the circulatory ow zone at the

contraction regioncase of Vi = 5m/s.

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

0 50 100 150 200 250 300 350 400

Particle diameter, m

Eros

ion

rate

, mg/

g

1m/s3m/s5m/s10m/s

Fig. 7. Eect of particle diameter on the total rate of erosion occurring on the tube sheet for dierent inlet velocities in

the case of upow.

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

0 1 2 3 4 5 6 7 8 9 10Inlet flow velocity, m/s

Eros

ion

rate

, mg/

g10 m100 m200 m400 m

Fig. 8. Eect of inlet ow velocity on the total rate of erosion occurring on the tube sheet for dierent particle diameters

in the case of upow.

0.0E+00

1.0E-07

2.0E-07

3.0E-07

4.0E-07

5.0E-07

6.0E-07

7.0E-07

8.0E-07

9.0E-07

0.5 0.6 0.7 0.8 0.9 1.0

r/R r/R

Eros

ion

rate

, mg/

g

1m/s5m/s10m/s

0.0E+00

2.0E-07

4.0E-07

6.0E-07

8.0E-07

1.0E-06

1.2E-06

0.5 0.7 0.9

Eros

ion

rate

, mg/

g

1 m/s5m/s10m/s

0.0E+005.0E-071.0E-061.5E-062.0E-062.5E-063.0E-063.5E-064.0E-06

0.5 0.7 0.9

r/R

Eros

ion

rate

, mg/

g 1m/s

5m/s

10m/s

0.0E+00

1.0E-06

2.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

7.0E-06

8.0E-06

0.5 0.6 0.7 0.8 0.9 1.0

r/R

Eros

ion

rate

, mg/

g 1m/s5m/s10m/s

(a) (b)

(d)(c)

Fig. 9. The variation of the local erosion rate on the contraction plate (ABCD) for the case of downow:

(a) Dp = 10lm, (b) Dp = 100lm, (c) Dp = 200lm, (d) Dp = 400lm.

H.M. Badr et al. / Computers & Fluids 34 (2005) 721742 737

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

0 50 100 150 200 250 300 350 400

Particle diameter

1m/s3m/s5m/s10m/s

Eros

ion

rate

, mg/

g

Fig. 10. Eect of particle diameter on the total rate of erosion occurring on the tube sheet for dierent inlet velocities in

the case of downow.

738 H.M. Badr et al. / Computers & Fluids 34 (2005) 721742the rate of erosion with the increase of inlet ow velocity. Moreover, the threshold velocity, Vt,can be approximately determined from Fig. 7 since the rate of erosion is insignicant for owvelocities below 2m/s for all particle diameters.The erosion rates obtained for the downow case are presented in Fig. 9 for the same particle

diameters and inlet ow velocities. The results are almost the same as those obtained in the upowcase except in Fig. 9d (Dp = 400lm) that shows higher rate of erosion (50% increase) at a owvelocity of 5m/s. It is quite expected that the eect of gravity on particle motion becomes signif-icant at low ow velocities. However, such eect did not inuence the rate of erosion at the lowestow velocity (1m/s) because such velocity is considerably below the threshold velocity, Vt. On theother hand, the ow velocity of 5m/s is denitely above the threshold velocity (see Fig. 8) and the0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

0 1 2 3 4 5 6 7 8 9 10Inlet flow velocity, m/s

Eros

ion

rate

, mg/

g

10 m100 m200 m400 m

Fig. 11. Eect of inlet ow velocity on the total rate of erosion occurring on the tube sheet for dierent particle

diameters in the case of downow.

eect of gravity becomes sensible. A quick comparison of the data presented in Figs. 5 and 9shows that the eect of gravity on the rate of erosion is very small in the case of high inlet owvelocity (10m/s) for all particle sizes. This can be explained based on the fact that the relative con-tribution of gravity to the motion of solid particles gets smaller with the increase of ow velocity.Fig. 10 shows the variation of the total erosion rate at the contraction section with particle size fordierent ow velocities. Although the trends are the same as in Fig. 7 the values obtained areslightly dierent especially in the case of moderate ow velocity (5m/s) and large particle size(Dp = 400lm). The same data is presented in Fig. 11, however, the inlet ow velocity is used asabscissa instead of the particle diameter. The gure shows that the threshold inlet velocity isapproximately the same as that in the case of upow (2m/s). To clearly show the eect of owdirection on the rate of erosion, Fig. 12a and b are plotted with inlet ow velocity as abscissa and

0.0E+00

1.0E-06

2.0E-06

3.0E-06

4.0E-06

5.0E-06

6.0E-06

7.0E-06

8.0E-06

0 2 4 6 8 10 12

Inlet flow velocity, m/s

Eros

ion

rate

, mg/

g

upward flowdownward flow

2.00E-05

2.50E-05

/g

(a)

H.M. Badr et al. / Computers & Fluids 34 (2005) 721742 7390.00E+00

5.00E-06

1.00E-05

1.50E-05

0 2 4 6 8 10 12Inlet flow velocity, m/s

Eros

ion

rate

, mg

upflowdownflow

(b)

Fig. 12. Eect of ow direction and inlet velocity on the total rate of erosion for the two cases of (a) Dp = 200lm, (b)

Dp = 400lm.

1m/s to 10m/s and the particle size ranged from 10lm to 400lm. In these ranges, the results

mately 2m/s below which erosion is insignicant for all particle sizes.

API production division meeting. Los Angeles, CA: 1976. p. I-1, I-27.

[6] Glaeser WA, Dow A. Mechanisms of erosion in slurry pipelines. In: Proceedings of the second international

740 H.M. Badr et al. / Computers & Fluids 34 (2005) 721742conference on slurry transportation. Las Vegas, NV: March 24, 1977. p. 13640.

[7] Roco MC, Nair P, Addie GR, Dennis J. Erosion of concentrated slurries in turbulent ow. J Pipelines

1984;4:21321.

[8] Venkatesh ES. Erosion damage in oil and gas wells, SPE paper 15183. Rocky mountain regional meeting of theAcknowledgement

The authors wish to acknowledge the support received from King Fahd University of Petro-leum and Minerals during this study.

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4. Conclusions

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Numerical investigation of erosion threshold velocity in a pipe with sudden contractionIntroductionProblem statement and solution methodologyThe continuous phase modelThe continuity and momentum equationsConservation equations for the turbulence modelBoundary conditionsSolution procedure

Particle-trackingParticle trajectoryDiscrete phase boundary conditions

The erosion model

Results and discussionConclusionsAcknowledgementReferences