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A phenomenological model for erosion of material in a horizontal slurry pipeline flow
Wear 269 (2010) 190–196
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A phenomenological model for erosion of material in a horizontal slurry pipeline flow Cunkui Huang a , P. Minev b , Jingli Luo c , K. Nandakumar d,∗ a
Alberta Innovates – Technology Futures, 250 Karl Clark Road, Edmonton, Alberta T6N 1E4, Canada Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G6, Canada Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 2G6, Canada d Gordon A. and Mary Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA b c
a r t i c l e
i n f o
Article history: Received 10 November 2009 Received in revised form 25 February 2010 Accepted 1 March 2010 Available online 7 March 2010 Keywords: Erosion models Particle impact Slurry pipeline flow
a b s t r a c t Based on the turbulent flow theory and a single particle erosion model developed by Huang et al. (2008) [9], a comprehensive phenomenological model for erosion of material in slurry pipeline flow is developed. This model captures the effects of particle shape, particle size, slurry mean velocity, pipe diameter, fluid viscosity and the properties of target material. The model shows that the erosion rate has a power-law relation with slurry mean velocity, particle size, pipe diameter, fluid viscosity and solid concentration. The erosion rate depends strongly on the slurry mean velocity and weakly on pipe diameter and fluid viscosity. The exponent of slurry mean velocity varies in a range of 2–3.575, which is consistent with most of the experiments. The model also elucidates that the effect of particle size on erosion rate depends on the particle shape, flow condition and erosion location on the periphery of a pipe. To test the model developed, a simplified version was used to compare with the experiments conducted by Karabelas. Both of them are in good agreement. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The transport of solids laden liquids in slurry pipeline is one of the key industrial processes. The erosion existing in this kind of system affects the life of equipments and hence the capital costs. Unfortunately the mechanisms of erosion in slurry pipeline flow are not well understood due to the complex nature of the process. In practice, there are two kinds of erosions: (a) erosion by liquid containing solid particles (slurry erosion) and (b) erosion by cavitating liquid [1]. First one is caused by particles’ impact with the surface of equipment; the second one results from gas or vapor bubbles’ collapse which produces stress pulses to damage the surface of target. Experimental study [2–8] shows that erosion by single particle has a power-law relation with particle’s impingement velocity and the exponent of the impingement velocity varies in a range of 2–2.5. Huang et al. [9] developed a new model for erosion by single particle. In this model, the exponent of impingement velocity depends on the shape of abrasive particles. Models for two limiting cases, viz., line cutting and area cutting were considered. This study elucidated that the exponent of impingement velocity in cutting wear has a range of 2–2.75.
∗ Corresponding author. Tel.: +1 780 492 5810; fax: +1 780 492 2881. E-mail addresses: [email protected], [email protected] (K. Nandakumar). 0043-1648/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2010.03.002
The models mentioned in the above paragraph focus on erosion by a single particle. To use these models, the particle’s impingement velocity and impingement angle must be known, while they are associated with a liquid flow field. Actually, calculating flow field and tracking particle trajectory to determine the impact location, velocity and angle are difficult and demanding tasks. Therefore, it is desired to develop a model that is suitable for calculating erosion by particles’ random impingement in pipeline flows. Hutchings [1] indicated that there is a power-law dependence of erosion on flow velocity in pipe loop tests. The exponent of flow velocity is between 0.85 and 4.5 for pipeline wears, and a value of 3 is commonly used for pump wears. Truscott [10,11] found a similar range of exponents on flow velocity in pipeline wears. More detailed experimental investigation by Karabelas [12] revealed that the exponent of flow velocity varies with the location of the pipe’s periphery. He measured at three different points of the pipe, i.e., top, middle and bottom, and found that the erosion rate increased with mean flow velocity to a power 2.32, 2.84 and 3.27 for the top, middle and bottom, respectively. Other researchers [13–15] have also obtained the exponent of flow velocity in a range of 2.1–2.6 in their pot testers. Besides the effect of flow velocity, properties of particles (e.g. shape, size, solid concentration) and target materials (hardness and ductility) also influence the erosion in pipeline flows. Studies have shown that erosion in a pipeline flow also has a power-law relation with particle size and solid concentration. For exponent of the particle size, experimental studies show that it varies in a wide range. Gupta
C. Huang et al. / Wear 269 (2010) 190–196
et al. [15] observed that the power index on the particle size was 0.291 for brass material, 0.344 for mild steel material. Elkholy’s experiments [16] gave a value of 0.616. Gandhi et al. [13] found that the exponent of particle size was about 0.85. Desale et al. [17] in their study presented a value of 1.62 and Karabelas [12] showed a value of 2.15. For the exponent of solid concentration, a range of 0.5–1.0 was proposed by different researchers [13,15,18]. In this paper, we focus on developing a new model by combining the single particle erosion model developed previously with additional information about the nature of the abrasive particles, target material and the liquid flow based on the turbulent flow theory. The new model shows that erosion rate mainly depends on slurry mean velocity. The exponent of flow velocity for pure erosion is 2–3.575. The model also demonstrates the effects of particle shape, size, density, liquid viscosity, pipe size and its properties on erosion rate. To test the model developed in this paper, a simplified version of it was used. The predictions calculated by the simplified model are consistent well with experiments [12].
191
the deformation wear, Eq. (1) is simplified as: 1+3(1−n)/4
QE =
Cmp
2+3(1−n)/2
Vp
1/4b
QE =
Bmp p
(Vp sin )
2+1/2b
1/b 1+1/4b Pn
εc
1+3(1−n)/4
+
Cmp
2+3(1−n)/2
Vp
2
(cos ) (sin )
(1−n)/4 i 3(1−n)/4 ε0 Pt Pn
3(1−n)/2
,
(1)
dp
where QE is the volume loss, mp is the particle mass, p is the particle density, Vp is the impingement velocity, is the impingement velocity, Pt and Pn are the tangential and normal cutting pressures during cutting process, respectively, ε0 is the elongation of target material, B and C are the coefficients, the exponent b can be determined by experiments and the exponent n is the shape factor of impingement particles and changes in a range of 0.5 (line cutting)–1 (area cutting). In this equation, the first and second terms of the right-hand side are the deformation wear and cutting wear, respectively. For details derivation of this equation, please refer Ref. [9]. For a slurry flow in a pipeline, particle’s lateral movement is mainly driven by the following factors: turbulence of liquid, gravitational and buoyant forces, momentum of inertia of particles and centrifugal force. The solid’s lateral movement normal to the wall, caused by the first two factors, always exists in pipeline flows. The effects produced by the last two factors mainly exist in the elbow places of a pipeline. If the curve of an elbow is not serious, the influence caused by the last two factors can be neglected. In a straight or slightly curved pipeline flow, the bulk velocities of liquid and solids are predominantly parallel to the surface of the pipe, and both of them have the same order. The random lateral velocity of solids, produced by liquid turbulence and the gravitational force, is much smaller than the solid’s main streamwise velocity, if the particles’ size is not big. Therefore, the impingement angle of particles in a straight or slightly curved pipeline flow is very small as measured from the pipe wall. Wellinger and Uetz [13] found that the impact angle was less than 5◦ in their tests. Finnie and Bitter indicated that erosion is dominated by cutting wear for ductile materials if impingement angle is small. Hence, neglecting
.
(2)
3(1−n)/2
1+3(1−n)/4
QE =
Cmp
2+3(1−n)/2
Vpx
(tan )
3(1−n)/2
,
(1−n)/4 i 3(1−n)/4 ε0 Pt Pn
(3)
dp
where Vpx is the component of particle velocity in the axial direction of a pipe. If the retardation of particle velocity (comparing with liquid velocity) is ignored, the bulk velocity of a slurry flow, U0 , can be used to replace Vp approximately. Moreover, as analyzed above, the impingement angle in a straight or slightly curve pipeline is small. tan can be replaced by approximately. Thus Eq. (3) is recast as: 1+3(1−n)/4
Many researchers such as Bitter [6,7], Sundararajan [19] and Wasp et al. [20] indicated that erosion caused by solid particles consists of two parts, cutting wear and deformation wear. The deformation wear is caused by the normal impact of solid particles. The cutting wear is caused by the oblique impact of solid particles. From a study of a single particle impacting on the surface of a target, Huang et al. [9] developed a comprehensive phenomenological erosion model which is expressed by the following equation:
3(1−n)/2
Multiplying the (cos ) to the numerator and denominator of the above equation, one has:
QEx =
2. Mechanistic model for erosion in pipeline flow
2
(cos ) (sin )
(1−n)/4 i 3(1−n)/4 dp ε0 Pt Pn
Cmp
2+3(1−n)/2 3(1−n)/2
U0
(1−n)/4 i 3(1−n)/4 ε0 Pt Pn
.
(4)
dp
The above equation gives the relation of volume loss caused by a single particle in a pipeline flow. In this equation, the impingement angle is an unknown. In the following part, we will focus on determining it. For a fully developed turbulent pipe flow, the fluctuation of velocity depends on the fluid’s properties, pipe size and roughness of inner surface of the pipe. The flow range in the pipe can be divided into a small boundary layer and a major turbulent core. In the boundary layer, the axial velocity variation with respect to the normal distance away from the wall of the pipe can be expressed by a power-law relation. In the major core part, the axial velocity has no significant change. In this study, the effect of the small boundary layer on particles’ motion when particles penetrate from the major turbulent core to the surface of pipe is ignored. Fig. 1 is a sketch of a control surface in a pipeline. From the turbulent flow theory, one knows that the shear stress on a control surface can be expressed by the following equation: = −
∂U + L u w , ∂r
(5)
where is the shear stress, is the liquid viscosity, U is a timeaveraged velocity in the axial direction, L is the density of liquid and u and w are the fluctuations of velocity in the axial and radial directions, respectively. In Eq. (5), the first term of the right-hand side is the viscous stress; the second term is the Reynolds stress. In the main core part, ∂U/∂r is small so that it can be neglected. Therefore, Eq. (5) is simplified as: ≈ L u w .
(6)
The friction coefficient is applied to express the intensity of resistance. Its definition is: Cf = . (7) 0.5L U02 Blasius and others (see page 195 of Ref. [21]) found that the friction coefficient with the Reynolds number has the following relation: a (8) Cf = m , Red where a and m are a coefficient and an exponent which depend on the Reynolds number, respectively. Here Red = L U0 D/ is the Reynolds number. D is the diameter of the pipe. Their values are a = 0.079 and m = 0.25 for Red ≤ 105 , a = 0.046 and m = 0.2 for 105 ≤ Red ≤ 106 .
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C. Huang et al. / Wear 269 (2010) 190–196
Fig. 1. A sketch of a control surface in a pipeline.
From Eqs. (6)–(8), one has: u w ≈
aU02 m 2Red
.
(9)
Because the fluctuating velocity u and w are random, they can be positive or negative in a controlled period. Therefore, there is the following inequality, |u ||w | ≥ u w .
b∗ U0 m/2
bU0
wp ≈
m/2
,
(11) 1/2
(12)
where b is a combined coefficient. For gravitational force, it plays two roles in a slurry pipe flow, i.e., causing solid concentration distribution non-uniformly in the vertical direction and producing a component of velocity in the vertical direction. The forces exerted on a particle are: gravitational force, F៝g = mp g៝ ,
(13)
buoyant force, L + (p − L ) F៝b = −mp g៝ , p
(14)
and drag force
Red
where b∗ = [2/(ak)] . Eq. (11) indicates that the liquid fluctuating velocity in the normal wall direction is directly proportional to the liquid mean velocity but inversely related to the Reynolds number with a power of m/2. To examine the validity of this relation, we compared the predictions by Eq. (11) with the experiment conducted by Laufer. In the Laufer’s experiment, Reynolds number was 5 × 105 and the maximum velocity at the center of the pipe was chosen as a reference. Wasp et al. [20] indicated that the mean velocity in a fully developed turbulent flow in a pipe is about 0.85 time of the maximum velocity. Transfer the reference velocity from the maximum velocity to the mean velocity, the relation between the fluctuating velocity in the radial direction and the mean velocity measured by Laufer was w = 0.047U0 close to wall, w = 0.031U0 at the center of the pipe. In predictions calculated by Eq. (11), we took k = 1.5 (an average value from the center to wall of the pipe) and = 10−6 m2 /s (water’s kinematic viscosity at room temperature). The calculation results were w ≥ 0.045U00.9 for a pipe with a diameter of 25 mm, w ≥ 0.036U00.9 for a pipe with a diameter of 250 mm. One sees that the fluctuating velocities in the normal wall direction predicted by Eq. (11) have the same orders as those measured by experiments for both coefficient and exponent of the mean velocity. For solid particles, their fluctuating velocity is mainly dominated by liquid’s fluctuations, but the momentum of particle’s inertia has some influence, for example, a large particle’s momentum makes the particle’s fluctuating velocity deviate from the liquid fluctuating velocity. Therefore, the particle fluctuating velocity in practice should be directly proportional to the
,
Red
(10)
Laufer (see pages 116–118 of Ref. [21]) measured a turbulent pipe flow with a Reynolds number of 5 × 105 . He found that the fluctuating velocities in three different directions increase as the radial distance from the center of the pipe increases; but near the wall region, the fluctuating velocities decrease quickly and reach zero at the wall. The fluctuating velocity in the axial direction is always larger than that in the radial direction. To determine them, a postulate, u = kw , is used approximately, where k is a coefficient. From the Laufer’s experiment, k is a function of the radial distance and varies from about 1 in the center of the pipe to about 2 near the wall regime. Combining the postulate and Eqs. (10) and (11), one has: |w |ave ≥
liquid fluctuating velocity, but inversely to the momentum of particle’s inertia in some manner. As a first order of approximation, we assume that the particle’s fluctuating velocity in the normal wall direction is proportional to the fluctuating velocity of liquid, that is:
F៝d =
dp2 L CD V៝ p − V៝ L 8
V៝ p − V៝ L ,
(15)
where g៝ is the gravity acceleration, is the solid concentration by volume, p is the density of solid particles, dp is the particle’s diameter, C is the drag coefficient and V៝ p and V៝ are the total velocities D
L
of the particles and the liquid, respectively. For the drag coefficient, it depends on the particle’s Reynolds number and is expressed by:
CD =
⎧ ⎨ 24 1 + 0.1R0.75 Rep ≤ 1000 ep ⎩
Rep
1000 ≤ Rep ≤ 2 × 10
0.44
,
(16)
5
where Rep = dp |V៝ p − V៝ L |/ is the particle’s Reynolds number. If Rep ≤ 1, the particle flow is in the Stokes regime, and the drag coefficient can be simplified as CD = 24/Rep ; if 1 ≤ Rep ≤ 1000, it is in the viscous regime; if 1000 ≤ Rep ≤ 2×105 , it is in the inertial regime or turbulent region. In most straight or slight curve pipe flows, Rep is smaller than 1000. Therefore, we only focus on the first two situations in this paper. To study the effect of gravitational force on the component of particle velocity in the vertical direction, we assume that the pipe is placed horizontally. From the balance of the forces acting on a particle in vertical direction, the particle’s settling velocity is calculated by the following equation: " = wpy
dp2 g៝ (1 − )(p − L ) 0.75 18(1 + 0.1Rep )
,
(17)
C. Huang et al. / Wear 269 (2010) 190–196
where is the viscosity of liquid. The component of settling velocity in the normal wall direction along pipe’s perimeter is: dp2 g៝ (1 − )(p − L ) cos ˛
" = wpyn
0.75 18 (1 + 0.1Rep )
,
(18)
where ˛ is the angle between the vertical line and a line normal to the pipe surface, as shown in Fig. 1. The total component of a particle velocity in the normal wall " . The particle impact angle is: direction is a sum of wp and wpyn =
+ w"
wp
pyn
U0
.
(19)
Substituting Eqs. (12) and (18) into (19) and then substituting Eq. (19) into (4), the erosion caused by a single particle in a pipeline flow is: 1+3(1−n)/4
Cmp
QE =
U02
(1−n)/4 i 3(1−n)/4 ε0 Pt Pn
dp
bU0
×
m/2
Red
+
dp2 g៝ (1 − )(p
− L ) cos ˛
3(1−n)/2
0.75 18(1 + 0.1Rep )
3wp t dp3
,
(21)
" where is the particle’s concentration by volume; wp = wp + wpyn is the total component of particle velocity in the normal wall direction. In this equation, we assume that number of particles moving toward the wall is equal to the number of particles leaving from the wall. Combining Eqs. (20) and (21), the total volume loss per unit wall area per unit time is: ER =
1+3(1−n)/4 2(1−n) 2 t dp U0
3(1−n)/4 i ε0 Pt Pn
C ∗ p
×
bU0 m/2
Red
+
dp2 g៝ (1 − )(p − L ) cos ˛ 0.75 ) 18(1 + 0.1Rep
(5−3n)/2 ,
ER =
1+3(1−n)/4 2(1−n) 2 t dp U0
3(1−n)/4 i ε0 Pt Pn
C ∗ p
(22)
where C* is a combined coefficient and equals to (C/2) × 3(1−n)/4 ( /6) . Eq. (22) captures the functional relationship showing the erosion rate caused by particle’s random motion in a pipeline flow. It is clearly complicated, but takes into account all of the important mechanisms thought to influence the erosion in a pipe.
(5−3n)/2 bU0 m/2
.
(23)
Red
In Ref. [9], two different cutting mechanisms, i.e., area cutting (n = 1) and line cutting (n = 0.5), were introduced. To show the relationship between erosion rate and different parameters clearly, erosion rates under different cutting mechanisms (n = 1 and n = 0.5) and different ranges of flow Reynolds numbers (Red ≤ 105 and 105 ≤ Red ≤ 106 ) are expressed by the following equations separately: C # p U02.875 0.125 t εi0 Pt D0.125
(20)
Eq. (20) gives the volume loss caused by a single particle. For the total volume loss on a unit inner surface area of a pipe, it is the sum of volume loss caused by all particles impacting on this area. In practice, the total impact number on this unit area depends on the particle’s local concentration. In principle, this could be coupled to a Computational Fluid Dynamics model to predict the impact velocity, angle and frequency to predict the erosion pattern. However, that adds another level of complexity and in this work, we look at a simple algebraic relation. Hence we need a further model to account for concentration effects. If particles in a liquid are dilute, the collisions between particles moving towards the wall and those leaving the wall for the inner part of the pipe can be ignored. In this case, the impact number has a linear relation with the local solid concentration. But if the particle’s concentration at near wall region is large enough so that the collisions between particles cannot be neglected, the impact number will depend on a local effective concentration, i.e., part of local concentration. The local effective concentration can be presented by a power-law approximately, namely, e = t , where t is an exponent which is determined by experiments. The impact number in a unit wall surface is: N˙ =
To understand the mechanisms for this kind of erosion, the following special cases are analyzed. First we assume that particle’s settling velocity, compared with particle’s random velocity in the normal wall direction due to turbulence, is small. In this case, the influence of the particle settling velocity on erosion rate can be ignored. Consequently, Eq. (22) is simplified as:
ER = .
193
ER =
ER =
ER =
for n = 1, m = 0.25 (Red ≤ 105 )
C # p1.375 dp U03.53 0.22 t εi0 Pt Pn0.375 D0.22
C # p U02.9 0.1 t εi0 Pt D0.1
(23a)
for n = 0.5, m = 0.25 (Red < 105 ) (23b)
for n = 1, m = 0.2 (105 < Red < 106 ) (23c)
C # p1.375 dp U03.575 0.175 t εi0 Pt Pn0.375 D0.175
for n = 0.5, m = 0.2 (105 < Red < 106 ) C#
(23d)
C ∗ b(5−3n)/2 .
where = From Eqs. (23a)–(23d), one sees that erosion rate significantly depends on the cutting mechanism (or particle’s shape), while the range of flow Reynolds number has only a slight influence. For erosion by the area cutting, erosion rate depends on particle density, slurry mean velocity, solid concentration, pipe size, fluid kinetic viscosity and properties of pipe material. In these parameters, the slurry mean velocity is a major one. Another thing one should note is that erosion rate by the area cutting is independent on the particle size. For erosion by the line cutting, there are two points which should be noted. One is the increase of the exponents of particle density and slurry mean velocity, which means that the sharper the particles, the larger the erosion rate. The other is that erosion rate depends on particle size. The two different cutting mechanisms elucidate that the exponents of the slurry mean velocity and particle size depend on the particle shape and vary from 2.875 to 3.575 for slurry mean velocity and 0–1 for particle size. In practice, particle edges are neither sharp as a line nor dull as a flat area. It should be between them (0.5 < n < 1.0). For fresh particles with sharp edges, the value of n trends to 0.5; while for particles with smooth surface, this value trends to 1. Due to lack of the data of erosion materials, we use an average value (n = 0.75) to express the shape factor of particles approximately. Thus Eq. (23) becomes: ER=
C # p1.1875 dp0.5 U03.2 0.17 t εi0 Pt Pn0.1875 D0.17
for n = 0.75, m = 0.25 (Red < 105 ) (23e)
194
ER =
C. Huang et al. / Wear 269 (2010) 190–196
C # p1.1875 dp0.5 U03.2375 0.1375 t εi0 Pt Pn0.1875 D0.1375
for n = 0.75, m = 0.2 (105 < Red < 106 )
(23f)
These two simplified equations indicate that the exponents of slurry mean velocity and particle size are 3.2 and 0.5, respectively. When we lack any data on the nature of abrasive materials, Eqs. (23e) and (23f) can be used to predict erosion rate approximately if the effect of the particle’s settling velocity caused by the gravitational force can be ignored. The assumption used in Eqs. (23a)–(23f) is that the particle’s settling velocity, compared with particle’s fluctuating velocity caused by fluid turbulence, is small. However, for particles with large settling velocity, the effect of particle settling velocity on erosion rate cannot be neglected. In the next part, we examine the other limit, i.e., particle’s settling velocity is much larger than particle’s fluctuating velocity. In this case, Eq. (22) is recast as: ER =
1+3(1−n)/4 2(1−n) 2 t dp U0
3(1−n)/4 i ε0 Pt Pn
C ∗ p
dp2 g៝ (1 − )(p − L ) cos ˛
(5−3n)/2 .
0.75 18(1 + 0.1Rep )
(24) This equation indicates that erosion rate has a quadratic relation with slurry mean velocity, a power-law relation with particle size. The power index for particle size is associated with the particle’s 0.75 1, the power index Reynolds number. For small Rep or 0.1Rep for particle size is 7 − 5n, namely, 2 for area cutting and 3.5 for line 0.75 1, the exponent of particle size cutting. For large Rep or 0.1Rep is (41 − 31n)/8, i.e., 1.25 for area cutting and 3.06 for line cutting. It should be noted that the particle’s settling velocity in a practical case should be smaller than the particle’s fluctuating velocity caused by fluid turbulence. Otherwise, particles will deposit on the bottom of a pipe. The above analysis indicates that the exponent of slurry mean velocity not only depends on particle’s lateral velocity caused by fluid turbulence, but also on the particle’s settling velocity created by the gravitational force. The two limits elucidate that the exponent of slurry mean velocity varies in a range of 2–3.575; the power-law index for particle size has a wide range. The equations derived in the above section were based on a pipeline placed horizontally. If a pipeline is set with an inclination ˇ (an angle between pipeline and horizontal surface), the component of the particle’s settling velocity in the flow streamwise direction, compared with the particle’s bulk velocity U0 , is ignorable. In this case, the equations derived for a horizontal slurry pipeline flow can be extended to a slurry pipeline flow with an inclination ˇ by only changing g in the above equations to g · cos ˇ. 3. Results and discussions The goal in this section is to test the model developed in this paper, especially the variation of erosion rate with slurry mean velocity, particle size and flow Reynolds number. Therefore, the unknown properties of target material (ε0 , Pt and Pn ) in Eq. (22) are combined into the coefficient C* . Due to lack of data of the particle shape, an average value of n = 0.75 was taken to express the effect of particle shape on erosion rate. Thus, Eq. (22) becomes:
ER = C ∗ p1.1875 dp0.5 U02 t
bU0 m/2
Red
+
dp2 g៝ (1 − )(p − g ) cos ˛ 0.75 18(1 + 0.1Rep )
1.375 . (25)
Fig. 2. Comparisons of predictions with experiments [12].
For exponent of the solid concentration, most of previous studies show that it varies in a range of 0.5–1. Gupta et al. [15] studied the erosion rate using different solid concentrations (0.15–0.45 by weight). Their results show that the exponent of solid concentration was 0.516 for brass material, 0.556 for mild steel. Karabelas [12] used the sand/water as slurry to study erosion in horizontal pipe flows. He measured the erosion rates at three different locations around the pipe’s periphery, i.e., top, middle and bottom. In his experiments, the target material was brass; the diameter of the pipe was 35.6 mm; the mean velocity of the slurry was in a range of 2.29–4.12 m/s which corresponds the Reynolds number 81524–146672; the solid concentration was either 0.3 or 0.4 by volume; the weighted mean diameter of particles was 138 . In this paper, Karabelas’ experiment results are used to compare with those calculated by Eq. (25). To test the model developed in this study, the exponent of solid concentration (0.516 for brass material) determined by Gupta et al. was used. To use Eq. (25), one has to fix the coefficients of b and C* . An easy way to do it is to use the experimental values measured at the middle part of the pipe because ˛ = 90◦ makes Eq. (25) simplify into: −0.6875m . ER = b1.375 C ∗ p1.1875 dp0.5 U03.375 0.516 Red
Taking logarithms of both sides of the above equation and using the method of least squares, we got the best values of b and C* to be 0.072 and 0.148, respectively. Fig. 2(a)–(c) shows the comparisons of predicted erosions with the experimental data given in Ref. [12]. In these figures, symbols and 䊉 express the erosion rates calculated by Eq. (25)
C. Huang et al. / Wear 269 (2010) 190–196
195
Fig. 3. Effect of particle size on erosion rate (ER) at different locations of a pipe predicted by simplified model (n = 0.75). Parameters used in predictions were the slurry mean velocity was 3 m/s, solid concentration was 30% by volume, pipe diameter was 35.6 mm and target material was brass. The inset in the figure shows the variation of exponent vs. particle size (i.e., ER ∝ dp ), where = Ln(ER)/Ln(dp).
and experimental measurements, respectively. In the experimental data, Karabelas used two different solid volume concentrations, i.e., 0.3 (correspond to data points 1, 3, 5 and 6 in the figures, counted from left to right) and 0.4 (data points 2 and 4). Fig. 2(b) shows the comparison of the predictions with experiments at the middle part of the pipe, i.e., ˛ = 90◦ . From Eq. (25), one knows that the gravitational force has no impact on erosion rate when ˛ = 90◦ . Therefore, the erosion at this point is only dominated by particle’s random motion. From Fig. 2(b), one sees that predictions and experiments are fairly consistent, as in this region it is merely a fit. Both of them show the similar tendencies, namely, erosion rate increases as the slurry mean velocity increases. To examine the effect of the gravitational force on erosion, the erosion rates at the top and bottom of the pipe were calculated using Eq. (25). For erosion at the top of the pipe, ˛ is 180◦ , i.e., the gravitational force decreases the erosion rate at the top of the pipe because the gravitational force decreases the particle’s impingement velocity in the normal wall direction that simultaneously leads to the particle’s impingement angle decrease. For erosion at the bottom of the pipe, an inverse result is produced by the gravitational force. The predictions and experimental measurements at the top and bottom of the pipe are shown in Fig. 2(a) and (c), respectively. Comparing them, one sees that both of them are in a good agreement. Fig. 2 (a)–(c) shows a good agreement between predictions and experiments, which implies that the simplified model (n = 0.75) is reasonable and can be used to study erosion in pipeline flows approximately. In the next part, this simplified model will be used to study the effect of other parameters such as particle diameter and Reynolds number on erosion rate, respectively. The impact of particle size on erosion rate has been observed in many experimental studies. To investigate the correlation between them, the simplified Eq. (25) was used. The erosion rates at the three different locations of the pipe, i.e., top, middle and bottom, were calculated and expressed by the lines with symbols , and in Fig. 3, respectively. To check the influence of particle size on erosion rate on the whole periphery of the pipe, an average value, i.e., integration of erosion rate along the whole periphery divided by the periphery, was also calculated and depicted in Fig. 3 by the line with symbol . The parameters used in these calculations were the solid concentration 0.3 by volume, slurry mean velocity 4 m/s, liquid viscosity 1 cp and pipe diameter 35.6 mm. From this figure, one sees that erosion rates at the three different locations of the pipe are different. Erosion rate at the middle of the pipe increases monotonically as the particle size increases. However, erosion rates at the top and bottom of the pipe have an inverse tendency with particle size, i.e., erosion rate at the top of the pipe almost decreases as particle size increases (note; erosion rate has a slight increase when parti-
Fig. 4. Effect of Reynolds number on erosion rate under different conditions. (a) Relationship between erosion rate and Reynolds number when pipe diameter varies, (b) relationship between erosion rate and Reynolds number as liquid viscosity changes and (c) comparison of mean erosion rates averaged on the whole periphery obtained when pipe diameter and liquid viscosity changes separately.
cle size is smaller than 150 m); while erosion rate at the bottom of the pipe increases as particle size increases. The average value of erosion rate on the whole periphery is close to that predicted at the middle part of the pipe, which means that the erosion rate measured at the middle part of the part can be used to express the average erosion rate approximately. To examine the dependence of
the erosion rate on particle size, i.e., ER ∝ dp , the variation of exponent vs. particle size is depicted in the inset of Fig. 3. One sees that the exponent is a function of particle size and varies with the location on the pipe. For the case studied, the exponent at the top of the pipe decreases as particle size increases, while increases at the bottom of the pipe as particle size increases. The value of at the bottom of the pipe changes from 0.6 to 1.42 when particle size increases from 50 to 500 m. The exponent at the middle of the pipe is a constant and has a value of 0.5. For average erosion rate on the whole periphery of the pipe, the exponent changes slightly (from 0.5 to 0.77) when particle size increases from 50 to 500 m. In practice, exponent depends not only on particle size, but also on particle shape. For particles with sharp edges (n → 0.5), the values of should be larger than those expressed in the inset of Fig. 3. Inversely, the value of should be smaller than those expressed in the inset of Fig. 3 if particles have smooth surface (n → 1.0). It should be emphasized that the predicted erosion rates in Fig. 3 were determined from Eq. (25) without adjusting the exponents on dp . The effective local value of the exponent (i.e., slope) was determined as indicated in Fig. 3 caption. Fig. 4(a)–(c) shows the effect of Reynolds number on erosion rate under different conditions. In these figures, two parameters were used to change Reynolds number. One of them was the pipe diameter; the other was the liquid viscosity. Fig. 4(a) shows the
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C. Huang et al. / Wear 269 (2010) 190–196
impact of Reynolds number (or pipe diameter) on erosion rates at the top, middle and bottom of the pipe. In this figure, the particle size, slurry mean velocity and liquid viscosity were kept constant at 138 m, 4 m/s and 1 cp, respectively. From Fig. 4 (a), one sees that the erosion rates at three different locations have the same pattern and it depends weakly on the Reynolds number (or pipe diameter), i.e., the larger the Reynolds number (or pipe diameter), the less the erosion rate. Fig. 4 (b) shows the variation of erosion rates at the top, middle and bottom of the pipe when Reynolds number increases (or liquid viscosity decreases). The particle size and slurry mean velocity used in Fig. 4(b) were the same as those used in Fig. 4(a). The diameter of the pipe used in this figure was 35.6 mm. From Fig. 4(b), one sees that erosion patterns at the three different locations are different. Erosion rate at the top and middle of the pipe decreases as Reynolds number increases (or liquid viscosity decreases), while erosion rate increases at the bottom of the pipe as Reynolds number increases (or liquid viscosity decreases). The reason causing this phenomenon is that a smaller liquid viscosity produces a larger settling velocity of particles which reduces the normal impingement velocity at the top of the pipe, while it enhances the normal impingement velocity at the bottom of the pipe. Fig. 4(c) shows the comparison of the mean erosion rates averaged on the whole periphery produced by changing the pipe diameter and liquid viscosity separately. It is noted that the mean erosion rates on the whole periphery of the pipe are same as long as the Reynolds numbers are same. 4. Conclusions Based on the turbulent flow theory and a single particle erosion model derived previously, a novel model for erosion of pipe wall material in a slurry flow has been developed. This model shows that erosion rate has a power-law relation with particle shape and size, slurry mean velocity, liquid viscosity and pipe diameter. The exponent of slurry velocity varies in a range of 2–3.575, which is consistent with most of experimental results (2–3.3). The power index for particle size depends on the particle shape, effect of gravitational force, slurry flow conditions, erosion location on the periphery of a pipe and varies over a wide range. Erosion rate with
pipe diameter and liquid viscosity has a weak relation, namely, erosion rate slightly decreases as the pipe diameter increases; while the impact of liquid viscosity on erosion rate depends on the erosion location on the periphery of a pipe. References [1] I.M. Hutchings, The Erosion of Materials by Liquid Flow, Publication No. 25, MTI, 1986. [2] I. Finnie, Erosion of surface by solid particles, Wear 3 (1960) 87–103. [3] I. Finnie, Some observation on the erosion of ductile metals, Wear 19 (1972) 81–90. [4] I. Finnie, Some reflections on the past and future of erosion, Wear 186–187 (1995) 1–10. [5] T.H. Kosel, Solid particle erosion, ASM handbook, friction, Lubrication and Wear Technology 18 (1992) 199–213. [6] J.G.A. Bitter, A study of erosion phenomena, Part I, Wear 6 (1963) 5–21. [7] J.G.A. Bitter, A study of erosion phenomena, Part II, Wear 6 (1963) 169–190. [8] J.E. Goodwin, W. Sage, G.P. Tilly, Study of erosion by solid particles, Proceedings of Institution of Mechanical Engineers 184 (1969–1970) 279–291. [9] C. Huang, P. Minev, J. Luo, S. Chiovelli, K. Nandakumar, A comprehensive phenomenological model for erosion of materials in jet flow, Powder Technology 187 (2008) 273–279. [10] G.F. Truscott, Wear in Pumps and Pipelines, BHRA Information Series No. 1, Cranfield, UK, BHRA Fluid Engineering Publication, 1980, pp. 1–22. [11] G.F. Truscott, A literature Survey on Wear on Pipelines, BHRA Information Series No. 1, Cranfield, UK, BHRA Fluid Engineering Publication, 1980, pp. 23–49. [12] A.J. Karabelas, An experimental study of pipe erosion by turbulent slurry flow. Hydrotransport-5, BHRA Fluid Engineering, Cranfield (UK), Paper E2, 1978. [13] K. Wellinger, H.S. Uetz, Scouring and blasting wear under the influence of granular solids, VDI-Forschungsheft Vol. 21, 1955 (in German). [14] W. Wiedenroth, An experimental study of wear of centrifugal pumps and pipeline components, Journal of pipelines 4 (1984) 223–228. [15] R. Gupta, S.N. Singh, V. Sehadri, Prediction of uneven wear in a slurry pipeline on the basis of measurements in a pot tester, Wear 184 (1995) 169–178. [16] A. Elkholy, Prediction of abrasion wear for slurry pump materials, Wear 84 (1983) 39–49. [17] G.R. Desale, B.K. Gandhi, S.C. Jain, Slurry erosion of ductile materials under normal impact condition, Wear 264 (2008) 322–330. [18] M. Wang, C. Huang, P. Minev, J. Luo, S. Chiovelli, K. Nandakumar, Computational fluid dynamics modelling and experimental study of erosion in slurry jet flows, International Journal of Computational Flow Dynamics 23 (2) (2009) 155–172. [19] G. Sundararajan, A comprehensive model for the solid particle erosion of ductile materials, Wear 149 (1991) 111–127. [20] E.J. Wasp, J.P. Kenny, R.L. Gandhi, Solid–Liquid Flow Slurry Pipeline Transportation, first ed., Trans Tech Publications, 1977. [21] A.J. Reynolds, Turbulent Flows in Engineering, John Wiley and Sons, 1974.
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Wear journal homepage: www.elsevier.com/locate/wear
A phenomenological model for erosion of material in a horizontal slurry pipeline flow Cunkui Huang a , P. Minev b , Jingli Luo c , K. Nandakumar d,∗ a
Alberta Innovates – Technology Futures, 250 Karl Clark Road, Edmonton, Alberta T6N 1E4, Canada Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G6, Canada Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 2G6, Canada d Gordon A. and Mary Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA b c
a r t i c l e
i n f o
Article history: Received 10 November 2009 Received in revised form 25 February 2010 Accepted 1 March 2010 Available online 7 March 2010 Keywords: Erosion models Particle impact Slurry pipeline flow
a b s t r a c t Based on the turbulent flow theory and a single particle erosion model developed by Huang et al. (2008) [9], a comprehensive phenomenological model for erosion of material in slurry pipeline flow is developed. This model captures the effects of particle shape, particle size, slurry mean velocity, pipe diameter, fluid viscosity and the properties of target material. The model shows that the erosion rate has a power-law relation with slurry mean velocity, particle size, pipe diameter, fluid viscosity and solid concentration. The erosion rate depends strongly on the slurry mean velocity and weakly on pipe diameter and fluid viscosity. The exponent of slurry mean velocity varies in a range of 2–3.575, which is consistent with most of the experiments. The model also elucidates that the effect of particle size on erosion rate depends on the particle shape, flow condition and erosion location on the periphery of a pipe. To test the model developed, a simplified version was used to compare with the experiments conducted by Karabelas. Both of them are in good agreement. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The transport of solids laden liquids in slurry pipeline is one of the key industrial processes. The erosion existing in this kind of system affects the life of equipments and hence the capital costs. Unfortunately the mechanisms of erosion in slurry pipeline flow are not well understood due to the complex nature of the process. In practice, there are two kinds of erosions: (a) erosion by liquid containing solid particles (slurry erosion) and (b) erosion by cavitating liquid [1]. First one is caused by particles’ impact with the surface of equipment; the second one results from gas or vapor bubbles’ collapse which produces stress pulses to damage the surface of target. Experimental study [2–8] shows that erosion by single particle has a power-law relation with particle’s impingement velocity and the exponent of the impingement velocity varies in a range of 2–2.5. Huang et al. [9] developed a new model for erosion by single particle. In this model, the exponent of impingement velocity depends on the shape of abrasive particles. Models for two limiting cases, viz., line cutting and area cutting were considered. This study elucidated that the exponent of impingement velocity in cutting wear has a range of 2–2.75.
∗ Corresponding author. Tel.: +1 780 492 5810; fax: +1 780 492 2881. E-mail addresses: [email protected], [email protected] (K. Nandakumar). 0043-1648/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2010.03.002
The models mentioned in the above paragraph focus on erosion by a single particle. To use these models, the particle’s impingement velocity and impingement angle must be known, while they are associated with a liquid flow field. Actually, calculating flow field and tracking particle trajectory to determine the impact location, velocity and angle are difficult and demanding tasks. Therefore, it is desired to develop a model that is suitable for calculating erosion by particles’ random impingement in pipeline flows. Hutchings [1] indicated that there is a power-law dependence of erosion on flow velocity in pipe loop tests. The exponent of flow velocity is between 0.85 and 4.5 for pipeline wears, and a value of 3 is commonly used for pump wears. Truscott [10,11] found a similar range of exponents on flow velocity in pipeline wears. More detailed experimental investigation by Karabelas [12] revealed that the exponent of flow velocity varies with the location of the pipe’s periphery. He measured at three different points of the pipe, i.e., top, middle and bottom, and found that the erosion rate increased with mean flow velocity to a power 2.32, 2.84 and 3.27 for the top, middle and bottom, respectively. Other researchers [13–15] have also obtained the exponent of flow velocity in a range of 2.1–2.6 in their pot testers. Besides the effect of flow velocity, properties of particles (e.g. shape, size, solid concentration) and target materials (hardness and ductility) also influence the erosion in pipeline flows. Studies have shown that erosion in a pipeline flow also has a power-law relation with particle size and solid concentration. For exponent of the particle size, experimental studies show that it varies in a wide range. Gupta
C. Huang et al. / Wear 269 (2010) 190–196
et al. [15] observed that the power index on the particle size was 0.291 for brass material, 0.344 for mild steel material. Elkholy’s experiments [16] gave a value of 0.616. Gandhi et al. [13] found that the exponent of particle size was about 0.85. Desale et al. [17] in their study presented a value of 1.62 and Karabelas [12] showed a value of 2.15. For the exponent of solid concentration, a range of 0.5–1.0 was proposed by different researchers [13,15,18]. In this paper, we focus on developing a new model by combining the single particle erosion model developed previously with additional information about the nature of the abrasive particles, target material and the liquid flow based on the turbulent flow theory. The new model shows that erosion rate mainly depends on slurry mean velocity. The exponent of flow velocity for pure erosion is 2–3.575. The model also demonstrates the effects of particle shape, size, density, liquid viscosity, pipe size and its properties on erosion rate. To test the model developed in this paper, a simplified version of it was used. The predictions calculated by the simplified model are consistent well with experiments [12].
191
the deformation wear, Eq. (1) is simplified as: 1+3(1−n)/4
QE =
Cmp
2+3(1−n)/2
Vp
1/4b
QE =
Bmp p
(Vp sin )
2+1/2b
1/b 1+1/4b Pn
εc
1+3(1−n)/4
+
Cmp
2+3(1−n)/2
Vp
2
(cos ) (sin )
(1−n)/4 i 3(1−n)/4 ε0 Pt Pn
3(1−n)/2
,
(1)
dp
where QE is the volume loss, mp is the particle mass, p is the particle density, Vp is the impingement velocity, is the impingement velocity, Pt and Pn are the tangential and normal cutting pressures during cutting process, respectively, ε0 is the elongation of target material, B and C are the coefficients, the exponent b can be determined by experiments and the exponent n is the shape factor of impingement particles and changes in a range of 0.5 (line cutting)–1 (area cutting). In this equation, the first and second terms of the right-hand side are the deformation wear and cutting wear, respectively. For details derivation of this equation, please refer Ref. [9]. For a slurry flow in a pipeline, particle’s lateral movement is mainly driven by the following factors: turbulence of liquid, gravitational and buoyant forces, momentum of inertia of particles and centrifugal force. The solid’s lateral movement normal to the wall, caused by the first two factors, always exists in pipeline flows. The effects produced by the last two factors mainly exist in the elbow places of a pipeline. If the curve of an elbow is not serious, the influence caused by the last two factors can be neglected. In a straight or slightly curved pipeline flow, the bulk velocities of liquid and solids are predominantly parallel to the surface of the pipe, and both of them have the same order. The random lateral velocity of solids, produced by liquid turbulence and the gravitational force, is much smaller than the solid’s main streamwise velocity, if the particles’ size is not big. Therefore, the impingement angle of particles in a straight or slightly curved pipeline flow is very small as measured from the pipe wall. Wellinger and Uetz [13] found that the impact angle was less than 5◦ in their tests. Finnie and Bitter indicated that erosion is dominated by cutting wear for ductile materials if impingement angle is small. Hence, neglecting
.
(2)
3(1−n)/2
1+3(1−n)/4
QE =
Cmp
2+3(1−n)/2
Vpx
(tan )
3(1−n)/2
,
(1−n)/4 i 3(1−n)/4 ε0 Pt Pn
(3)
dp
where Vpx is the component of particle velocity in the axial direction of a pipe. If the retardation of particle velocity (comparing with liquid velocity) is ignored, the bulk velocity of a slurry flow, U0 , can be used to replace Vp approximately. Moreover, as analyzed above, the impingement angle in a straight or slightly curve pipeline is small. tan can be replaced by approximately. Thus Eq. (3) is recast as: 1+3(1−n)/4
Many researchers such as Bitter [6,7], Sundararajan [19] and Wasp et al. [20] indicated that erosion caused by solid particles consists of two parts, cutting wear and deformation wear. The deformation wear is caused by the normal impact of solid particles. The cutting wear is caused by the oblique impact of solid particles. From a study of a single particle impacting on the surface of a target, Huang et al. [9] developed a comprehensive phenomenological erosion model which is expressed by the following equation:
3(1−n)/2
Multiplying the (cos ) to the numerator and denominator of the above equation, one has:
QEx =
2. Mechanistic model for erosion in pipeline flow
2
(cos ) (sin )
(1−n)/4 i 3(1−n)/4 dp ε0 Pt Pn
Cmp
2+3(1−n)/2 3(1−n)/2
U0
(1−n)/4 i 3(1−n)/4 ε0 Pt Pn
.
(4)
dp
The above equation gives the relation of volume loss caused by a single particle in a pipeline flow. In this equation, the impingement angle is an unknown. In the following part, we will focus on determining it. For a fully developed turbulent pipe flow, the fluctuation of velocity depends on the fluid’s properties, pipe size and roughness of inner surface of the pipe. The flow range in the pipe can be divided into a small boundary layer and a major turbulent core. In the boundary layer, the axial velocity variation with respect to the normal distance away from the wall of the pipe can be expressed by a power-law relation. In the major core part, the axial velocity has no significant change. In this study, the effect of the small boundary layer on particles’ motion when particles penetrate from the major turbulent core to the surface of pipe is ignored. Fig. 1 is a sketch of a control surface in a pipeline. From the turbulent flow theory, one knows that the shear stress on a control surface can be expressed by the following equation: = −
∂U + L u w , ∂r
(5)
where is the shear stress, is the liquid viscosity, U is a timeaveraged velocity in the axial direction, L is the density of liquid and u and w are the fluctuations of velocity in the axial and radial directions, respectively. In Eq. (5), the first term of the right-hand side is the viscous stress; the second term is the Reynolds stress. In the main core part, ∂U/∂r is small so that it can be neglected. Therefore, Eq. (5) is simplified as: ≈ L u w .
(6)
The friction coefficient is applied to express the intensity of resistance. Its definition is: Cf = . (7) 0.5L U02 Blasius and others (see page 195 of Ref. [21]) found that the friction coefficient with the Reynolds number has the following relation: a (8) Cf = m , Red where a and m are a coefficient and an exponent which depend on the Reynolds number, respectively. Here Red = L U0 D/ is the Reynolds number. D is the diameter of the pipe. Their values are a = 0.079 and m = 0.25 for Red ≤ 105 , a = 0.046 and m = 0.2 for 105 ≤ Red ≤ 106 .
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C. Huang et al. / Wear 269 (2010) 190–196
Fig. 1. A sketch of a control surface in a pipeline.
From Eqs. (6)–(8), one has: u w ≈
aU02 m 2Red
.
(9)
Because the fluctuating velocity u and w are random, they can be positive or negative in a controlled period. Therefore, there is the following inequality, |u ||w | ≥ u w .
b∗ U0 m/2
bU0
wp ≈
m/2
,
(11) 1/2
(12)
where b is a combined coefficient. For gravitational force, it plays two roles in a slurry pipe flow, i.e., causing solid concentration distribution non-uniformly in the vertical direction and producing a component of velocity in the vertical direction. The forces exerted on a particle are: gravitational force, F៝g = mp g៝ ,
(13)
buoyant force, L + (p − L ) F៝b = −mp g៝ , p
(14)
and drag force
Red
where b∗ = [2/(ak)] . Eq. (11) indicates that the liquid fluctuating velocity in the normal wall direction is directly proportional to the liquid mean velocity but inversely related to the Reynolds number with a power of m/2. To examine the validity of this relation, we compared the predictions by Eq. (11) with the experiment conducted by Laufer. In the Laufer’s experiment, Reynolds number was 5 × 105 and the maximum velocity at the center of the pipe was chosen as a reference. Wasp et al. [20] indicated that the mean velocity in a fully developed turbulent flow in a pipe is about 0.85 time of the maximum velocity. Transfer the reference velocity from the maximum velocity to the mean velocity, the relation between the fluctuating velocity in the radial direction and the mean velocity measured by Laufer was w = 0.047U0 close to wall, w = 0.031U0 at the center of the pipe. In predictions calculated by Eq. (11), we took k = 1.5 (an average value from the center to wall of the pipe) and = 10−6 m2 /s (water’s kinematic viscosity at room temperature). The calculation results were w ≥ 0.045U00.9 for a pipe with a diameter of 25 mm, w ≥ 0.036U00.9 for a pipe with a diameter of 250 mm. One sees that the fluctuating velocities in the normal wall direction predicted by Eq. (11) have the same orders as those measured by experiments for both coefficient and exponent of the mean velocity. For solid particles, their fluctuating velocity is mainly dominated by liquid’s fluctuations, but the momentum of particle’s inertia has some influence, for example, a large particle’s momentum makes the particle’s fluctuating velocity deviate from the liquid fluctuating velocity. Therefore, the particle fluctuating velocity in practice should be directly proportional to the
,
Red
(10)
Laufer (see pages 116–118 of Ref. [21]) measured a turbulent pipe flow with a Reynolds number of 5 × 105 . He found that the fluctuating velocities in three different directions increase as the radial distance from the center of the pipe increases; but near the wall region, the fluctuating velocities decrease quickly and reach zero at the wall. The fluctuating velocity in the axial direction is always larger than that in the radial direction. To determine them, a postulate, u = kw , is used approximately, where k is a coefficient. From the Laufer’s experiment, k is a function of the radial distance and varies from about 1 in the center of the pipe to about 2 near the wall regime. Combining the postulate and Eqs. (10) and (11), one has: |w |ave ≥
liquid fluctuating velocity, but inversely to the momentum of particle’s inertia in some manner. As a first order of approximation, we assume that the particle’s fluctuating velocity in the normal wall direction is proportional to the fluctuating velocity of liquid, that is:
F៝d =
dp2 L CD V៝ p − V៝ L 8
V៝ p − V៝ L ,
(15)
where g៝ is the gravity acceleration, is the solid concentration by volume, p is the density of solid particles, dp is the particle’s diameter, C is the drag coefficient and V៝ p and V៝ are the total velocities D
L
of the particles and the liquid, respectively. For the drag coefficient, it depends on the particle’s Reynolds number and is expressed by:
CD =
⎧ ⎨ 24 1 + 0.1R0.75 Rep ≤ 1000 ep ⎩
Rep
1000 ≤ Rep ≤ 2 × 10
0.44
,
(16)
5
where Rep = dp |V៝ p − V៝ L |/ is the particle’s Reynolds number. If Rep ≤ 1, the particle flow is in the Stokes regime, and the drag coefficient can be simplified as CD = 24/Rep ; if 1 ≤ Rep ≤ 1000, it is in the viscous regime; if 1000 ≤ Rep ≤ 2×105 , it is in the inertial regime or turbulent region. In most straight or slight curve pipe flows, Rep is smaller than 1000. Therefore, we only focus on the first two situations in this paper. To study the effect of gravitational force on the component of particle velocity in the vertical direction, we assume that the pipe is placed horizontally. From the balance of the forces acting on a particle in vertical direction, the particle’s settling velocity is calculated by the following equation: " = wpy
dp2 g៝ (1 − )(p − L ) 0.75 18(1 + 0.1Rep )
,
(17)
C. Huang et al. / Wear 269 (2010) 190–196
where is the viscosity of liquid. The component of settling velocity in the normal wall direction along pipe’s perimeter is: dp2 g៝ (1 − )(p − L ) cos ˛
" = wpyn
0.75 18 (1 + 0.1Rep )
,
(18)
where ˛ is the angle between the vertical line and a line normal to the pipe surface, as shown in Fig. 1. The total component of a particle velocity in the normal wall " . The particle impact angle is: direction is a sum of wp and wpyn =
+ w"
wp
pyn
U0
.
(19)
Substituting Eqs. (12) and (18) into (19) and then substituting Eq. (19) into (4), the erosion caused by a single particle in a pipeline flow is: 1+3(1−n)/4
Cmp
QE =
U02
(1−n)/4 i 3(1−n)/4 ε0 Pt Pn
dp
bU0
×
m/2
Red
+
dp2 g៝ (1 − )(p
− L ) cos ˛
3(1−n)/2
0.75 18(1 + 0.1Rep )
3wp t dp3
,
(21)
" where is the particle’s concentration by volume; wp = wp + wpyn is the total component of particle velocity in the normal wall direction. In this equation, we assume that number of particles moving toward the wall is equal to the number of particles leaving from the wall. Combining Eqs. (20) and (21), the total volume loss per unit wall area per unit time is: ER =
1+3(1−n)/4 2(1−n) 2 t dp U0
3(1−n)/4 i ε0 Pt Pn
C ∗ p
×
bU0 m/2
Red
+
dp2 g៝ (1 − )(p − L ) cos ˛ 0.75 ) 18(1 + 0.1Rep
(5−3n)/2 ,
ER =
1+3(1−n)/4 2(1−n) 2 t dp U0
3(1−n)/4 i ε0 Pt Pn
C ∗ p
(22)
where C* is a combined coefficient and equals to (C/2) × 3(1−n)/4 ( /6) . Eq. (22) captures the functional relationship showing the erosion rate caused by particle’s random motion in a pipeline flow. It is clearly complicated, but takes into account all of the important mechanisms thought to influence the erosion in a pipe.
(5−3n)/2 bU0 m/2
.
(23)
Red
In Ref. [9], two different cutting mechanisms, i.e., area cutting (n = 1) and line cutting (n = 0.5), were introduced. To show the relationship between erosion rate and different parameters clearly, erosion rates under different cutting mechanisms (n = 1 and n = 0.5) and different ranges of flow Reynolds numbers (Red ≤ 105 and 105 ≤ Red ≤ 106 ) are expressed by the following equations separately: C # p U02.875 0.125 t εi0 Pt D0.125
(20)
Eq. (20) gives the volume loss caused by a single particle. For the total volume loss on a unit inner surface area of a pipe, it is the sum of volume loss caused by all particles impacting on this area. In practice, the total impact number on this unit area depends on the particle’s local concentration. In principle, this could be coupled to a Computational Fluid Dynamics model to predict the impact velocity, angle and frequency to predict the erosion pattern. However, that adds another level of complexity and in this work, we look at a simple algebraic relation. Hence we need a further model to account for concentration effects. If particles in a liquid are dilute, the collisions between particles moving towards the wall and those leaving the wall for the inner part of the pipe can be ignored. In this case, the impact number has a linear relation with the local solid concentration. But if the particle’s concentration at near wall region is large enough so that the collisions between particles cannot be neglected, the impact number will depend on a local effective concentration, i.e., part of local concentration. The local effective concentration can be presented by a power-law approximately, namely, e = t , where t is an exponent which is determined by experiments. The impact number in a unit wall surface is: N˙ =
To understand the mechanisms for this kind of erosion, the following special cases are analyzed. First we assume that particle’s settling velocity, compared with particle’s random velocity in the normal wall direction due to turbulence, is small. In this case, the influence of the particle settling velocity on erosion rate can be ignored. Consequently, Eq. (22) is simplified as:
ER = .
193
ER =
ER =
ER =
for n = 1, m = 0.25 (Red ≤ 105 )
C # p1.375 dp U03.53 0.22 t εi0 Pt Pn0.375 D0.22
C # p U02.9 0.1 t εi0 Pt D0.1
(23a)
for n = 0.5, m = 0.25 (Red < 105 ) (23b)
for n = 1, m = 0.2 (105 < Red < 106 ) (23c)
C # p1.375 dp U03.575 0.175 t εi0 Pt Pn0.375 D0.175
for n = 0.5, m = 0.2 (105 < Red < 106 ) C#
(23d)
C ∗ b(5−3n)/2 .
where = From Eqs. (23a)–(23d), one sees that erosion rate significantly depends on the cutting mechanism (or particle’s shape), while the range of flow Reynolds number has only a slight influence. For erosion by the area cutting, erosion rate depends on particle density, slurry mean velocity, solid concentration, pipe size, fluid kinetic viscosity and properties of pipe material. In these parameters, the slurry mean velocity is a major one. Another thing one should note is that erosion rate by the area cutting is independent on the particle size. For erosion by the line cutting, there are two points which should be noted. One is the increase of the exponents of particle density and slurry mean velocity, which means that the sharper the particles, the larger the erosion rate. The other is that erosion rate depends on particle size. The two different cutting mechanisms elucidate that the exponents of the slurry mean velocity and particle size depend on the particle shape and vary from 2.875 to 3.575 for slurry mean velocity and 0–1 for particle size. In practice, particle edges are neither sharp as a line nor dull as a flat area. It should be between them (0.5 < n < 1.0). For fresh particles with sharp edges, the value of n trends to 0.5; while for particles with smooth surface, this value trends to 1. Due to lack of the data of erosion materials, we use an average value (n = 0.75) to express the shape factor of particles approximately. Thus Eq. (23) becomes: ER=
C # p1.1875 dp0.5 U03.2 0.17 t εi0 Pt Pn0.1875 D0.17
for n = 0.75, m = 0.25 (Red < 105 ) (23e)
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ER =
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C # p1.1875 dp0.5 U03.2375 0.1375 t εi0 Pt Pn0.1875 D0.1375
for n = 0.75, m = 0.2 (105 < Red < 106 )
(23f)
These two simplified equations indicate that the exponents of slurry mean velocity and particle size are 3.2 and 0.5, respectively. When we lack any data on the nature of abrasive materials, Eqs. (23e) and (23f) can be used to predict erosion rate approximately if the effect of the particle’s settling velocity caused by the gravitational force can be ignored. The assumption used in Eqs. (23a)–(23f) is that the particle’s settling velocity, compared with particle’s fluctuating velocity caused by fluid turbulence, is small. However, for particles with large settling velocity, the effect of particle settling velocity on erosion rate cannot be neglected. In the next part, we examine the other limit, i.e., particle’s settling velocity is much larger than particle’s fluctuating velocity. In this case, Eq. (22) is recast as: ER =
1+3(1−n)/4 2(1−n) 2 t dp U0
3(1−n)/4 i ε0 Pt Pn
C ∗ p
dp2 g៝ (1 − )(p − L ) cos ˛
(5−3n)/2 .
0.75 18(1 + 0.1Rep )
(24) This equation indicates that erosion rate has a quadratic relation with slurry mean velocity, a power-law relation with particle size. The power index for particle size is associated with the particle’s 0.75 1, the power index Reynolds number. For small Rep or 0.1Rep for particle size is 7 − 5n, namely, 2 for area cutting and 3.5 for line 0.75 1, the exponent of particle size cutting. For large Rep or 0.1Rep is (41 − 31n)/8, i.e., 1.25 for area cutting and 3.06 for line cutting. It should be noted that the particle’s settling velocity in a practical case should be smaller than the particle’s fluctuating velocity caused by fluid turbulence. Otherwise, particles will deposit on the bottom of a pipe. The above analysis indicates that the exponent of slurry mean velocity not only depends on particle’s lateral velocity caused by fluid turbulence, but also on the particle’s settling velocity created by the gravitational force. The two limits elucidate that the exponent of slurry mean velocity varies in a range of 2–3.575; the power-law index for particle size has a wide range. The equations derived in the above section were based on a pipeline placed horizontally. If a pipeline is set with an inclination ˇ (an angle between pipeline and horizontal surface), the component of the particle’s settling velocity in the flow streamwise direction, compared with the particle’s bulk velocity U0 , is ignorable. In this case, the equations derived for a horizontal slurry pipeline flow can be extended to a slurry pipeline flow with an inclination ˇ by only changing g in the above equations to g · cos ˇ. 3. Results and discussions The goal in this section is to test the model developed in this paper, especially the variation of erosion rate with slurry mean velocity, particle size and flow Reynolds number. Therefore, the unknown properties of target material (ε0 , Pt and Pn ) in Eq. (22) are combined into the coefficient C* . Due to lack of data of the particle shape, an average value of n = 0.75 was taken to express the effect of particle shape on erosion rate. Thus, Eq. (22) becomes:
ER = C ∗ p1.1875 dp0.5 U02 t
bU0 m/2
Red
+
dp2 g៝ (1 − )(p − g ) cos ˛ 0.75 18(1 + 0.1Rep )
1.375 . (25)
Fig. 2. Comparisons of predictions with experiments [12].
For exponent of the solid concentration, most of previous studies show that it varies in a range of 0.5–1. Gupta et al. [15] studied the erosion rate using different solid concentrations (0.15–0.45 by weight). Their results show that the exponent of solid concentration was 0.516 for brass material, 0.556 for mild steel. Karabelas [12] used the sand/water as slurry to study erosion in horizontal pipe flows. He measured the erosion rates at three different locations around the pipe’s periphery, i.e., top, middle and bottom. In his experiments, the target material was brass; the diameter of the pipe was 35.6 mm; the mean velocity of the slurry was in a range of 2.29–4.12 m/s which corresponds the Reynolds number 81524–146672; the solid concentration was either 0.3 or 0.4 by volume; the weighted mean diameter of particles was 138 . In this paper, Karabelas’ experiment results are used to compare with those calculated by Eq. (25). To test the model developed in this study, the exponent of solid concentration (0.516 for brass material) determined by Gupta et al. was used. To use Eq. (25), one has to fix the coefficients of b and C* . An easy way to do it is to use the experimental values measured at the middle part of the pipe because ˛ = 90◦ makes Eq. (25) simplify into: −0.6875m . ER = b1.375 C ∗ p1.1875 dp0.5 U03.375 0.516 Red
Taking logarithms of both sides of the above equation and using the method of least squares, we got the best values of b and C* to be 0.072 and 0.148, respectively. Fig. 2(a)–(c) shows the comparisons of predicted erosions with the experimental data given in Ref. [12]. In these figures, symbols and 䊉 express the erosion rates calculated by Eq. (25)
C. Huang et al. / Wear 269 (2010) 190–196
195
Fig. 3. Effect of particle size on erosion rate (ER) at different locations of a pipe predicted by simplified model (n = 0.75). Parameters used in predictions were the slurry mean velocity was 3 m/s, solid concentration was 30% by volume, pipe diameter was 35.6 mm and target material was brass. The inset in the figure shows the variation of exponent vs. particle size (i.e., ER ∝ dp ), where = Ln(ER)/Ln(dp).
and experimental measurements, respectively. In the experimental data, Karabelas used two different solid volume concentrations, i.e., 0.3 (correspond to data points 1, 3, 5 and 6 in the figures, counted from left to right) and 0.4 (data points 2 and 4). Fig. 2(b) shows the comparison of the predictions with experiments at the middle part of the pipe, i.e., ˛ = 90◦ . From Eq. (25), one knows that the gravitational force has no impact on erosion rate when ˛ = 90◦ . Therefore, the erosion at this point is only dominated by particle’s random motion. From Fig. 2(b), one sees that predictions and experiments are fairly consistent, as in this region it is merely a fit. Both of them show the similar tendencies, namely, erosion rate increases as the slurry mean velocity increases. To examine the effect of the gravitational force on erosion, the erosion rates at the top and bottom of the pipe were calculated using Eq. (25). For erosion at the top of the pipe, ˛ is 180◦ , i.e., the gravitational force decreases the erosion rate at the top of the pipe because the gravitational force decreases the particle’s impingement velocity in the normal wall direction that simultaneously leads to the particle’s impingement angle decrease. For erosion at the bottom of the pipe, an inverse result is produced by the gravitational force. The predictions and experimental measurements at the top and bottom of the pipe are shown in Fig. 2(a) and (c), respectively. Comparing them, one sees that both of them are in a good agreement. Fig. 2 (a)–(c) shows a good agreement between predictions and experiments, which implies that the simplified model (n = 0.75) is reasonable and can be used to study erosion in pipeline flows approximately. In the next part, this simplified model will be used to study the effect of other parameters such as particle diameter and Reynolds number on erosion rate, respectively. The impact of particle size on erosion rate has been observed in many experimental studies. To investigate the correlation between them, the simplified Eq. (25) was used. The erosion rates at the three different locations of the pipe, i.e., top, middle and bottom, were calculated and expressed by the lines with symbols , and in Fig. 3, respectively. To check the influence of particle size on erosion rate on the whole periphery of the pipe, an average value, i.e., integration of erosion rate along the whole periphery divided by the periphery, was also calculated and depicted in Fig. 3 by the line with symbol . The parameters used in these calculations were the solid concentration 0.3 by volume, slurry mean velocity 4 m/s, liquid viscosity 1 cp and pipe diameter 35.6 mm. From this figure, one sees that erosion rates at the three different locations of the pipe are different. Erosion rate at the middle of the pipe increases monotonically as the particle size increases. However, erosion rates at the top and bottom of the pipe have an inverse tendency with particle size, i.e., erosion rate at the top of the pipe almost decreases as particle size increases (note; erosion rate has a slight increase when parti-
Fig. 4. Effect of Reynolds number on erosion rate under different conditions. (a) Relationship between erosion rate and Reynolds number when pipe diameter varies, (b) relationship between erosion rate and Reynolds number as liquid viscosity changes and (c) comparison of mean erosion rates averaged on the whole periphery obtained when pipe diameter and liquid viscosity changes separately.
cle size is smaller than 150 m); while erosion rate at the bottom of the pipe increases as particle size increases. The average value of erosion rate on the whole periphery is close to that predicted at the middle part of the pipe, which means that the erosion rate measured at the middle part of the part can be used to express the average erosion rate approximately. To examine the dependence of
the erosion rate on particle size, i.e., ER ∝ dp , the variation of exponent vs. particle size is depicted in the inset of Fig. 3. One sees that the exponent is a function of particle size and varies with the location on the pipe. For the case studied, the exponent at the top of the pipe decreases as particle size increases, while increases at the bottom of the pipe as particle size increases. The value of at the bottom of the pipe changes from 0.6 to 1.42 when particle size increases from 50 to 500 m. The exponent at the middle of the pipe is a constant and has a value of 0.5. For average erosion rate on the whole periphery of the pipe, the exponent changes slightly (from 0.5 to 0.77) when particle size increases from 50 to 500 m. In practice, exponent depends not only on particle size, but also on particle shape. For particles with sharp edges (n → 0.5), the values of should be larger than those expressed in the inset of Fig. 3. Inversely, the value of should be smaller than those expressed in the inset of Fig. 3 if particles have smooth surface (n → 1.0). It should be emphasized that the predicted erosion rates in Fig. 3 were determined from Eq. (25) without adjusting the exponents on dp . The effective local value of the exponent (i.e., slope) was determined as indicated in Fig. 3 caption. Fig. 4(a)–(c) shows the effect of Reynolds number on erosion rate under different conditions. In these figures, two parameters were used to change Reynolds number. One of them was the pipe diameter; the other was the liquid viscosity. Fig. 4(a) shows the
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C. Huang et al. / Wear 269 (2010) 190–196
impact of Reynolds number (or pipe diameter) on erosion rates at the top, middle and bottom of the pipe. In this figure, the particle size, slurry mean velocity and liquid viscosity were kept constant at 138 m, 4 m/s and 1 cp, respectively. From Fig. 4 (a), one sees that the erosion rates at three different locations have the same pattern and it depends weakly on the Reynolds number (or pipe diameter), i.e., the larger the Reynolds number (or pipe diameter), the less the erosion rate. Fig. 4 (b) shows the variation of erosion rates at the top, middle and bottom of the pipe when Reynolds number increases (or liquid viscosity decreases). The particle size and slurry mean velocity used in Fig. 4(b) were the same as those used in Fig. 4(a). The diameter of the pipe used in this figure was 35.6 mm. From Fig. 4(b), one sees that erosion patterns at the three different locations are different. Erosion rate at the top and middle of the pipe decreases as Reynolds number increases (or liquid viscosity decreases), while erosion rate increases at the bottom of the pipe as Reynolds number increases (or liquid viscosity decreases). The reason causing this phenomenon is that a smaller liquid viscosity produces a larger settling velocity of particles which reduces the normal impingement velocity at the top of the pipe, while it enhances the normal impingement velocity at the bottom of the pipe. Fig. 4(c) shows the comparison of the mean erosion rates averaged on the whole periphery produced by changing the pipe diameter and liquid viscosity separately. It is noted that the mean erosion rates on the whole periphery of the pipe are same as long as the Reynolds numbers are same. 4. Conclusions Based on the turbulent flow theory and a single particle erosion model derived previously, a novel model for erosion of pipe wall material in a slurry flow has been developed. This model shows that erosion rate has a power-law relation with particle shape and size, slurry mean velocity, liquid viscosity and pipe diameter. The exponent of slurry velocity varies in a range of 2–3.575, which is consistent with most of experimental results (2–3.3). The power index for particle size depends on the particle shape, effect of gravitational force, slurry flow conditions, erosion location on the periphery of a pipe and varies over a wide range. Erosion rate with
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