Simplified model for particle collision related to attrition in pneumatic conveying

Advanced Powder Technology xxx (xxxx) xxx

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Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt

Original Research Paper

Simplified model for particle collision related to attrition in pneumatic conveying Dmitry Portnikov a,⇑, Nir Santo a, Haim Kalman a,b a b

The Laboratory for Conveying and Handling of Particulate Solids, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Aaron Fish Chair in Mechanical Engineering – Fracture Mechanics, Israel

a r t i c l e

i n f o

Article history: Received 10 July 2019 Received in revised form 7 October 2019 Accepted 25 October 2019 Available online xxxx Keywords: Particle attrition Breakage modeling Pneumatic conveying Machine functions Material functions

a b s t r a c t This paper presents a simple method for predicting particle attrition during pneumatic conveying. The model calculates the changes in the particle size during pneumatic conveying (as a result of the collisions between the particles and bend walls) by using empirical correlations for both the machine and material functions. The method does not require the use of complicated simulations such as DEM–CFD. Furthermore, the computational model was written in MATLAB, and the results agree well with the experimental results for salt particles. The computation time was very short: a few seconds for the first collision (particles passed through one bend), and below one minute for six collisions. The experimental results and parametric study show that higher bend radius ratios caused less damage to the conveyed material. Moreover, higher air velocities and larger pipe diameters caused more damage to the conveyed material. Ó 2019 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.

1. Introduction The attrition of particles in pneumatic conveying is an important phenomenon that should be considered during design or operation. When particles are being conveyed through a pipeline, they experience collisions at different velocities and angles, which result in particle breakage. Most of the damage usually occurs in bends, because the collision angles in bends are much higher and close to normal collision, compared to those in straight pipes. The ability to predict the final size of the conveyed product in a pneumatic conveying pipeline would increase the design accuracy of such systems. In previous studies, Kalman [1] and Kalman et al. [2] predicted the size reductions of particles in pneumatic conveying in experiments with various types of bends and operational conditions. The results revealed that certain flow, structural, and material parameters affect the breakage; for instance, the fluid velocity, pipeline geometry, different bend types, number of bends, and mechanical properties of particles. Therefore, determining purely empirical correlations that are able to predict the final size distribution of a product is difficult, and other possible solutions (e.g., numerical simulations) should be developed.

⇑ Corresponding author.

Currently, complicated numerical simulations such as Discrete Element Method coupled with Computational Fluid Dynamics (DEM–CFD) are commonly employed to predict the final size distribution of conveyed material. The computational breakage algorithm is usually based on the concept of two groups of comminution functions: the process functions and material functions [3]. The process functions, also known as the machine functions, define the stress conditions that the particles are subjected to during comminution; for example, the particle velocity, number of collisions, and collision angles. The material functions comprise the effects of the material properties on the comminution result. It should be noted that whereas the machine functions depend on the application and differ for pneumatic conveying and jet mills, the material functions are independent of the process, because they are pure material properties. In 2009, Kalman et al. [4] suggested for the first time a method in which the machine and material functions are combined in a computational breakage algorithm. The concept of this algorithm is shown in Fig. 1. The algorithm is initialized with a randomly defined size and strength for each particle based on empirical initial size distribution and strength distribution functions. Then, the machine functions define the stress condition for each particle (‘‘stress analysis” area in figure). The material functions (‘‘strength properties” area in figure) are inserted into the numerical simulation to define the experience of each particle. The applied force

E-mail address: [email protected] (D. Portnikov). https://doi.org/10.1016/j.apt.2019.10.028 0921-8831/Ó 2019 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.

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Nomenclature A; B; C empirical constants of fatigue function AF ; BF ; DF empirical constants of strength distribution function Aeq ; Beq empirical constants of equivalence function a; b; aa ; ba ; ca empirical constants of breakage function Ar Archimedes number [–] Bi;j cumulative mass fragment ratio [–] b mass fragment frequency ratio [–] D pipe diameter ½m Du shape parameter of particle velocity distribution [–] d particle size ½m di fragment particle size ½m dj mother particle size ½m F cumulative mass-related size distribution [–] FB breakage force ½N equivalent force ½N F eq f mass-related frequency size distribution -

N Rb R=D S up usf u50

cumulative numbered fragment ratio [–] bend radius ½m bend ratio [–] cumulative distribution [–] particle velocity ½m=s superficial air velocity ½m=s median particle velocity ½m=s

Greek letters h collision angle ½deg l air viscosity
½Pa  s 3 qa air density kg=m
 qp particle density kg=m3

Fig. 1. Computational breakage algorithm, combining material and machine functions [4].

acting on each particle is compared with its strength. If the acting force is higher than the strength, the particle breaks. The number and size of daughter particles are defined by a breakage function, and all the daughter particles are provided with an appropriate new strength based on the strength distribution function. If the acting force is lower than the strength of a particle, the particle does not break. Its strength is reduced due to fatigue, and the reduced strength is defined by a fatigue function. Then, the process resets for the next collision detection. Evidently, the simulation should be verified with a set of experiments. Brosh et al. [5] applied this algorithm in a complete DEM-CFD simulation of dilute-phase pneumatic conveying, in which both machine functions and the breakage algorithm were calculated. While the results were in good agreement with the experiments, the computation time was significantly high. Several weeks were necessary to obtain the results for a certain case with initial particle sizes, certain flow rate conditions, and a pipeline route. In that situation, the simulation of a certain design is challenging. Uzi et al. [6] managed to reduce the computation time. Although they still

defined the machine functions with DEM-CFD simulations, they were defined in a general way to provide impact velocity distribution and impact frequency functions. The computation time has been reduced significantly for the breakage calculation and it takes a few minutes. However, this still requires time to define the machine functions, because the DEM-CFD code should be performed for various particle properties and operating conditions. In this paper, we suggest an even more simplified method for the calculation of the particle breakage during pneumatic conveying. The paper is focusing on the calculation productivity of the machine functions since they are the primary factor, which are responsible for the computation time of the particle’s attrition model. The proposed model calculates the changes in the particle sizes with empirical correlations for both machine and material functions and does not require DEM–CFD simulations. Because most of the damage occurs in bends, the model calculates the breakage as a result of collisions between the particles and bend walls only. The material functions are defined based on previous empirical studies and include five empirical comminution

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functions: strength distribution [6], selection [7,8], breakage [7], fatigue [9], and equivalence [10]. The machine functions include three empirical and geometry-based functions. The first one is the particle steady-state velocity distribution function. Santo et al. [11] developed this function empirically for various materials and particle sizes. The two other functions are the impact angle distribution and particle location distribution functions. They are introduced in this paper. 2. Machine functions This section introduces all machine functions that were developed in the current attrition model. Several assumptions were made before the development. First, it was assumed that the material is conveyed in a dilute phase; there are no collisions between the particles. Second, each particle moves horizontally in a bend before the collision. Third, the particle collides only once with the bend wall without any secondary collisions inside the bend. In practice, not all particles follow this assumption. However, when the air velocity is sufficiently high, the assumption is reasonable. Furthermore, it was verified in visualization experiments conducted with a transparent bend. The captured frame sequence of the particle trajectory in the bend made by a high-speed camera is shown in Fig. 2. The figure shows a horizontal particle trajectory before the collision, a single collision with the bend wall, and a rebound-particle trajectory without any secondary collisions in the bend. 2.1. Particle steady-state velocity The first and most important machine function is the particle steady-state velocity, which defines the collision velocity. Santo et al. [11] developed this function based on extensively large numbers of experiments performed in pneumatic conveying with various materials, particle sizes, pipe diameters, and air flow rates. The velocity of each particle was measured by tracking the particles in a transparent pipeline section with a high-speed camera. They discovered that particle velocity is not constant; it has a wide distribution that can be represented with a logistic distribution function:

 up ¼ u50

Su 1  Su

1=Du ð1Þ

where up denotes the particle velocity, Su the probability to obtain a certain particle velocity, u50 the median particle velocity, and Du the distribution wideness. When Du increases, the dispersion of the distribution values decreases. Furthermore, Santo et al. [11] discovered that the ratio between the median particle velocity and superficial

Trajectory Before Collision

collision

3

air velocity is a function of three non-dimensional groups: the Archimedes number, pipe diameter ratio, and density ratio:

"   2 #0:14 qp  qa u50 D ¼ 1  0:02 Ar D2in: usf qa

ð2Þ

where usf denotes the superficial air velocity; qp and qa denote the particle and air densities, respectively; D is the internal pipe diameter and D2in: the internal diameter of a 2-in. pipe or a constant value of 52:5 mm; Ar denotes the Archimedes number, which can be calculated with the following definition:

Ar ¼

  3 d g qp  qa

l2

ð3Þ

where d is the particle size, g the gravitational acceleration, and l the dynamic viscosity of air. Note that for small particle sizes with a small Archimedes value, the velocity ratio given in Eq. (2) approaches one. Thus, for a fine powder, the particle velocity approaches the superficial air velocity. A similar behavior was found for the distribution wideness parameter Du , which is a function of two non-dimensional groups: the Archimedes number and pipe diameter ratio:

  D 0:14 Du ¼ 16:4 þ 4:8  104 Ar0:687 D2in:

ð4Þ

When the superficial air velocity, particle size, and particle and air properties are known and can be used as input parameters, the particle steady-state velocity distribution can be predicted with Eqs. (1)–(4). Thus, the specific velocity for each particle can be predicted by selecting a random number between 0 and 1 for Su . Note that Eqs. (1)–(4) apparently holds for any particulate material, particle size and pipe diameter within an error of 15% (as was reported by Santo et al. [11]). 2.2. Impact angle distribution function The collision or impact angle between the particle and bend wall is predicted based on the impact angle distribution function. By assuming that the particles move along the axial direction of the pipe (either horizontally or vertically) until the collision, the impact angle can be calculated based only on the bend geometry. This assumption is supported by experiments performed by Santo et al. [11], where they discovered that in terms of kinetic energy, the non-axial velocities have a negligible contribution on the overall three-dimensional particle velocity profile. In practice, the particles might move chaotically at small positive or negative angles in relation to the axial direction, resulting with higher and lower collision angles, respectively. However, assuming that the particles

Trajectory of Rebound Parcle

Fig. 2. Visualization of particle trajectory in transparent bend.

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are moving in the axial direction as an average is reasonable. In Fig. 3, x and y denote the location of the particle in a pipe cross section, Rb is the bend radius and D is the internal pipe diameter. Then, the impact angle ðhÞ can be calculated with the following expression:

0

1

Rb  y B C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h ¼ cos1 @ A Rb þ ðD=2Þ2  x2

ð5Þ

Here, the impact angle is defined as acute angle between the particle trajectory line and tangent line at the collision point of the bend wall. Note that this equation holds for any pipe diameter and bend curvature (bend radius). According to Eq. (5), the impact angles decrease when the bend radius increases. Previous studies have shown that when the impact angle decreases, the damage caused to the particle decreases [8,12]. Therefore, for higher bend radius ratios ðRb =DÞ, the damage caused to the conveyed material should be lower. The cumulative distribution of particle impact angles with respect to various bend ratios is shown in Fig. 4. Each curve represents the calculated impact angle of 500 particles according to Eq. (5). These particles were located homogeneously in a pipe cross section. The particle coordinates x and y were randomly chosen within the pipe cross section without any additional conditions. According to the plotted graph, the impact angles decrease with increasing bend ratio. 2.3. Location distribution function for horizontal pipeline A homogeneous distribution of particles in a pipe cross section (Fig. 4) is usually suitable for vertical conveying pipelines and fine powders or high air velocities in horizontal pipelines. However, in a horizontal conveying pipeline for other cases, the particles are usually concentrated at the pipe bottom owing to gravity. To determine the location distribution of particles in a horizontal pipeline, the particles were tracked in a transparent horizontal pipeline section with a high-speed camera. The experiments were performed with salt particles with a narrow particle size interval of +2– 2.36 mm and various superficial air velocities between 13 and

1.0

n=500 D=52.5 mm

0.9 0.8

Cumulative Distribution

4

Bend Ratio, Rb/D:

2 4 6 10

0.7 0.6 0.5 Rb/D

0.4 0.3 0.2 0.1 0.0

0

10

20

30

40

50

60

Impact Angle, θ , deg Fig. 4. Cumulative distribution of impact angles of homogeneously distributed particles for various bend ratios.

29 m=s. The Y coordinate of each particle measured from the bottom of the pipe was recorded, and the results were plotted in a cumulative distribution graph in Fig. 5. Although each experiment curve was plotted with only twenty points for clarity, it actually represents at least 200 measured particles. Evidently, all experimental curves overlap and exhibit a linear behavior to approximately 70% of the pipe diameter. It can be concluded that the examined salt particles (sizes of approximately 2 mm; conveyed with superficial air velocities of 13  29 m=s) are homogenously distributed along the Y coordinate to approximately 70% of the pipe diameter measured from the bottom of the internal pipe wall. The solid line in the graph shows the locations of 500 particles predicted with the model. The investigation of the location distribution function should be extended in further studies for various materials, particle sizes, and air velocities. Presently, the results are satisfying because the presented attrition model was validated with experiments performed with this particular material. It should be noted that a constant Y distribution does not reflect a constant volumetric distribution, because the cross-sectional area

Fig. 3. Bend geometry and particle trajectory.

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0.9

Cumulative Distribution

0.8 0.7 0.6 0.5 0.4

Z

13 20 24 29 Predicted

f ¼

0.2 0.1 0.2

f =d

3

f =d  @d

@d

ð6Þ

dmin

where N d denotes the cumulative numbered size distribution, d the particle size, dmax the maximal particle size, dmin the minimal particle size, and f the mass-related frequency distribution:

Usf, m/s

0.1

R dmax dmin

0.3

0.0 0.0

3

d

Nd ¼

Salt d=2-2.36 mm D=52.5 mm

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Normalized Coordinate, Y/D Fig. 5. Cumulative distribution of particle locations along normalized Y coordinate for various superficial air velocities.

in the pipe changes with Y. In addition, it should be emphasized that the illustrations of the particle location in a pipe cross section shown in Figs. 4 and 5 don’t reflect the particle’s concentration or material loading ratio. Each particle in the model is treated as individual without any particle-particle collision. The number of simulated particles don’t enter the bend at the same time, but they can be spread over a long time. Thus, collisions between particles may be neglected. Evidently, in real pneumatic conveying systems the collisions between the particles takes place, and the collision frequency increases as the particle’s concentration or solid loading ratio increases. However, in terms of damage to the conveyed material, as the collision frequency is higher, the damage to the material reduces because the collision velocities between particles are smaller. Therefore, by treating the particles as individual in the model, i.e. very low loading ratio, the calculated results will be at the safe side of the system design in terms of particle attrition – result with higher attrition. 3. Material functions This section introduces all material functions that were used in the presented attrition model. Most functions have been determined in previous studies. The characterization of material functions is complicated and takes time. However, those functions represent pure material impact properties and are independent of the machine functions. Hence, once the material functions are determined for a certain material, the same functions can be used not only for study of pneumatic conveying but also for other impact systems, such as jet mill. 3.1. Initial size distribution function The computational algorithm is initialized by randomly defining the size of each particle. If the conveyed material consists of mono-sized particles, a constant size is chosen. If the conveyed material consists of particles with different sizes, each particle size has to be chosen randomly according to the measured size distribution. Usually, the size distribution of the particles is measured with sieves and defined by the mass-related cumulative distribution. In this case, the mass size distribution must be transformed into a numbered size distribution before the random selection of the individual particles. If the cumulative mass size distribution ðF Þ is a continuous function, the conversion can be performed with the following equation [4]:

@F Dd @d

ð7Þ

Hence, the size of each initial particle was defined by selecting a random number between 0 and 1 for N d . In some cases, particularly when the size distribution is measured with sieves, it is more convenient to solve Eq. (6) with the discretization method. This method will be described in detail in Section 3.4.1. Nevertheless, for a sufficiently large enough number of particles, their size distribution is equal to the one originally measured. 3.2. Strength distribution function The initial strength of each particle in the presented attrition model was chosen according to the empirical model proposed by Rozenblat et al. [6]:

  S 1=DF F 2 F B ¼ AF þ BF d 1  SF

ð8Þ

where F B denotes the particle strength in terms of breakage force, SF the probability to obtain a certain strength, d the particle size; AF , BF , and DF are empirical material constants. For salt particles (used in this study), the empirical constants were determined to be: AF ¼ 6:8N; BF ¼ 3:3 MPa, and DF ¼ 5 [6]. Hence, the strength of each particle was defined according to Eq. (8) by selecting a random number between 0 and 1 for SF . 3.3. Equivalence/selection functions The selection function defines the percentage of broken particles at a certain velocity, and is developed through impact experiments with various particle sizes and velocities. In the presented attrition model, the particle velocity is predicted according to Eqs. (1)–(4). However, the decision whether particle break or not is performed by comparing the applied force during the impact with the particle strength (which is defined in terms of force). Hence, the applied force acting on a particle is required for this comparison. The equivalent function is an empirical expression

1.0 0.9

6

1

3

7

0.8

Cumulative Distribution

1.0

4

0.7 0.6

2

5

0.5 0.4 0.3 Number fragment ratio Mass fragment ratio Chosen particles

0.2 0.1 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Particle Size, mm Fig. 6. Example of fragment size prediction for a broken single salt particle. The initial particle size was 1 mm, and the impact velocity was 15 m=s.

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which leads to less attrition of the conveying product compared with the attrition of a normal collision. Therefore, the equivalent force that causes the damage should decrease accordingly, and the effect of the collision angle must be included in Eq. (9). Portnikov et al. [8] recently suggested that the ratio between the hor

izontal impact velocity at any collision angle up to the horizontal

Bend Filter Bag Feeder

Transparent Secon



impact velocity at 90 ðu0 Þ, which causes the same damage to the particles, is only a function of the collision angle and has an exponential behavior:

2'’ Horizontal Pipe

  up h þ1 ¼ 4:4exp 18 u0

Fig. 7. Schematic of test section.

that calculates the required equivalent compression force that causes an equivalent damage to the particles as in an impact event. It transforms the particle impact velocity into an equivalent compression force acting on the particle. The determination of the equivalent force is based on a comparison between the breakage probabilities due to the impact (selection function) and breakage probabilities due to the compression (strength distribution)

obtained for the same particle size. The equivalent force F eq acting on a particle during the impact with the bend wall is predicted with the following expression [10]: B

3:3Beq

F eq ¼ Aeq u0eq d

ð9Þ

where u0 denotes the horizontal particle velocity during a normal collision and d the particle size; Aeq and Beq are the empirical material constants. For a salt particle, Aeq ¼ 2:94  105 and Beq ¼ 0:53. The units of the particle velocity u0 and particle size d in Eq. (9) are ½m=s and ½m, respectively. The model for the equivalent force in Eq. (9) was developed only for a normal collision between the particle and colliding surface. However, in bends, the particles collide at angles lower than 90°,

ð10Þ

where, h denotes the collision angle (in [°]). The model is suitable for six different particulate materials, including salt particles. Substituting the expression for the horizontal impact velocity at normal collision ðu0 Þ from Eq. (10) into Eq. (9) leads to the final expression for the equivalent force, which takes into account the effect of the collision angle: 3:3Beq

F eq ¼ Aeq d

up

4:4exp h þ1 18

!Beq ð11Þ

3.4. Breakage function The breakage function defines the size distribution of the fragments (the daughter particles) due to collision and breakage of the original particle (the mother particle) at a certain velocity. Thus, the breakage function is affected by the size of the mother

particle dj , size of daughter particles ðdi Þ, and impact velocity

up . According to the experimental study of Rozenblat et al. [7], the breakage function has the following expression (after Tavares [13]):

Fig. 8. Employed computational breakage algorithm.

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a.

where a; b; aa ; ba ; and ca denote the empirical, material-dependent parameters. For salt particles, a ¼ 0:05; b ¼ 2; aa ¼ 0:6; ba ¼ 0:3, and ca ¼ 0:24.

1.0 0.9

Cumulative Distribution

0.8

Salt d=1.4-2.36 mm Usf=30 m/s R/D=2

3.4.1. Implementing the breakage function The empirical cumulative fragment ratio model is based on the particle mass/weight according to Eqs. (12)–(14). However, using the mass size distribution formulation in the computational algorithm will lead to a considerable error because the breakage algorithm is suitable for individual particles, and the determination procedure for the fragment sizes requires a random selection of individual particles. Therefore, the mass size distribution must be converted to a number size distribution. The conversion is similar to that in Section 3.1:

0.7 0.6 0.5 0.4

Number of initial particles 100 1000 2000 10000

0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

Particle Size, mm

Z

1.0 0.9

R dmax

Repeatability for 100 initial particles

Cumulative Distribution

0.7

0.5 0.4 0.3

b¼

Calculation number 1 2 3 4 5

0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

Particle Size, mm

c.

1.0 0.9

Repeatability for 2000 initial particles

0.8

Cumulative Distribution

b=d

0.4 0.3

@Bi;j Dd @d

ð16Þ

Determining an analytical solution for N might be complicated depending on the complexity of the derivative of Bi;j . Therefore, in some cases, it is more convenient to solve Eq. (15) with the discretization method. In this study, the computational breakage model was written in MATLAB. Thus, the discretization method is the preferred choice. The cumulative mass fragment ratio (Bi;j ) for each mother particle was divided into discrete portions. Because the size distribution of the fragments during the validation experiments was measured with sieves, each discrete portion represents a size interval defined by the standard sieve sizes based on the American Society for Testing and Materials (ASTM) E11 standard. Next, the cumulative numbered fragment ratio was calculated according to a discrete formulation of Eq. (15):

  3 bk =dmk  Ni ¼ P  3 j k¼1 bk =dmk

Calculation number 1 2 3 4 5

ð17Þ

where index i represents the i-th sieve size and index j the largest sieve size defined by the mother particle; dmk denotes the average particle diameter of the specific size fraction k, and bk is defined as the differential of Bi;j :

0.2

bk ¼ Bkþ1  Bk

0.1 0.0 0.0

ð15Þ

dmin

k¼1

0.5

@d

Pi

0.7 0.6

3

b=d  @d

where N denotes the cumulative numbered fragment ratio, d the fragment size, dmax the maximal fragment size (size of mother particle), dmin the minimal fragment size, and b the mass fragment frequency ratio:

0.8

0.6

3

d

N¼ dmin

b.

7

0.5

1.0

1.5

2.0

2.5

Particle Size, mm Fig. 9. Effect of number of initial particles on computational results.

a

Bi;j ¼ 1  ð1  t 10 Þ½9=ðdj =di 1Þ

ð12Þ

where Bi;j denotes the cumulative mass breakage function; t 10 and a denote the model parameters, which are functions of the mother



particle size dj and the impact velocity up , respectively:

  t 10 ¼ 1  exp aubp dj

ð13Þ

a ¼ aa ubpa dcj a

ð14Þ

ð18Þ

Hence, the cumulative numbered fragment ratio of a certain mother particle that collided with a certain impact velocity was estimated with Eqs. (17) and (18). Next, the size of the individual fragment was predicted by selecting a random number between 0 and 1 for N and interpolating in between the evaluated discrete points. This selection was repeated until the total volume of the chosen fragments reached the volume of the mother particle. When the total fragment volume was greater than the mother particle volume, the last chosen fragment was deleted, and the fragment size of the last particle was calculated according to the missing volume. An example of the procedure for a mother particle with a size of 1 mm and impact velocity of 15 m=s is shown in Fig. 6. The mass fragment ratio curve was predicted according to Eqs. (12)–(14), and the numbered fragment ratio curve was predicted according to Eqs. (17) and (18). The dashed vertical lines represent the discretization process. The numbers in the graph represent the order

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of randomly selected fragments. Furthermore, the plotted graph emphasizes the significance of predicting the fragments with the number fragment ratio rather than with the mass fragment ratio. 3.5. Fatigue function In the computational breakage algorithm, if the equivalent force acting on a particle is lower than its strength, its strength is reduced due to fatigue although it does not break. Therefore, the fatigue function should define the reduced strength of an unbroken particle. The determination of the appropriate relationship of the reduced strength consists of performing repeated impact tests with a certain velocity and size fraction. In addition, the strength of the unbroken particles is measured after each impact cycle through compression tests. According to Rozenblat et al. [9], the fatigue function has the following form:

2

31:5

1:89  1013 6 7 F Bi ¼ F Bi1 41 þ h  i2 5 B 1=3 exp Ad F C1=3 827:8  F eq Bi1

ð19Þ

where F B denotes the particle strength in terms of breakage force; index i denotes the i-th impact of the specific particle and i  1 the previous impact; F eq is the equivalent force, which is calculated based on Eq. (11), and d denotes the particle size. Additionally, the model includes two constants: 1:89  1013 and 827:8. The first value is assumed to be constant for all materials [14]. The second value represents the ratio between a theoretical fracture strength and theoretical tensile strength of the salt material [9]. Moreover, the model has three empirical material-dependent constants that are obtained through experiments. For salt particles, A ¼ 6:72  103 ; B ¼ 0:725, and C ¼ 0:333 [9]. 4. Experimental To validate the results obtained with the model, experiments were conducted with a short horizontal section of a pneumatic conveying system. The system consists of a straight, horizontal, 7 m long 2-in. pipe, an examined bend, gravity feeder, and filter bag (Fig. 7). A sample of approximately 70 g salt particles with sizes of +1.4–2.36 mm was fed into the pipeline by a gravity feeder with various superficial air velocities between 15 and 30 m=s. Furthermore, various bend radii were tested with the following bend ratios: R=D ¼ 1:5; 4:5 and 10. The particles were accelerated in the pipe toward the examined bend and collected behind the bend by a filter bag to measure the size distribution of the particles by

a.

1.0 0.9

weight. In addition, to validate the fatigue behavior, the collision tests were repeated with the same sample. The pipe length of 7 m was sufficient for the particles to reach a steady-state particle velocity in front of the bend entrance [15]. In addition, a transparent section of 0:5 m length was installed in front of the bend to measure the location distribution of the particles in the pipe cross section with a high-speed camera. 5. Results and analysis The schematic employed computational breakage algorithm is shown in Fig. 8. The computational algorithm is started by selecting the total number of particles for the analysis and randomly defining the size and strength for each particle based on the empirical initial size distribution and strength distribution functions. In addition, the superficial air velocity, pipe diameter, bend ratio, and number of bends are chosen as input parameters. By using the velocity distribution function according to Eqs. (1)–(4), a specific velocity is calculated for each particle by selecting a random number between 0 and 1 for Su . When the bend entrance was connected to a vertical pipe, each particle was randomly located within the pipe cross section geometry, as shown in Fig. 4. When the bend entrance was connected to a horizontal pipe, each particle was randomly located along the Y direction to 70% of the pipe diameter, as shown in Fig. 5. The collision angle was determined with Eq. (5) based on the x and y coordinates of each particle. The impact velocity and collision angle define the required stress conditions for the comparison with the particle strength to determine whether the particle breaks or not. For broken particles, the size distribution of the fragments was predicted based on the breakage function according to the process described in Section 3.4.1. For unbroken particles, the reduced strength was calculated with the fatigue function according to Eq. (19). For the next collision, the particle velocity, particle location, and collision angle were re-chosen randomly for each particle. However, their material properties and particle sizes remained unchanged. 5.1. Number of initial particles The model was written in MATLAB. Before the model validation, the minimal number of initial particles necessary to obtain accurate computational results was determined. Various numbers of initial particles were inserted into the computational algorithm with the same initial conditions (i.e., same material, particle size, air velocity, and bend geometry). The results are shown in Fig. 9. According to Fig. 9-a, the size distributions of the product after

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Please cite this article as: D. Portnikov, N. Santo and H. Kalman, Simplified model for particle collision related to attrition in pneumatic conveying, Advanced Powder Technology, https://doi.org/10.1016/j.apt.2019.10.028

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scattering in the computational results (see Fig. 9-b) and therefore a high inaccuracy. However, all repeated calculation results performed with 2000 initial particles (see Fig. 9-c) overlap. Hence, 2000 particles are a sufficient initial-particle number for a computation with sufficient accuracy.

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the first collision with the bend for 1000 and 10,000 initial particles are equal. Although 100 initial particles provide reasonable results as well, the repeated calculations show otherwise. The repeatability tests were performed by running the computational algorithm with the same initial conditions several times. Each calculation was performed by re-selecting the same number of initial particles. The repeated tests with 100 initial particles exhibit high

All the machine functions, as they were developed, are suitable for any material, pipe diameter and bend radius. Therefore, it is sufficient to validate the model with only one material for which all the material functions are fully determined. In our case, salt particles meet all the conditions. If one will choose to work with other material, the machine functions will remain the same without any changes. However, the material functions should be determined separately because all the empirical parameters of the material functions are material dependent. In the computation, 2000 was chosen as initial number for the particles. The calculated and experimental results are shown in Fig. 10. The plotted points in the graphs represent the experiments, and the solid curves represent the calculated results. Furthermore, both results were obtained with the same superficial air velocity

usf ¼ 30 m=s . The calculated results agree well with the experimental ones. The left graph shows the size distribution by weight for a single collision with the bend wall for different bend ratios. As expected, a higher bend ratio results in less attrition. The right graph shows the size distribution by weight for repeated collision tests performed with the same bend ðR=D ¼ 1:5Þ. Index N represents the collision number or the number of times the particles pass through the same bend. This graph presents the fatigue behavior. For the same air velocity (i.e., equal stress conditions), the conveyed product experiences a progressive damage with increasing number of times the particles pass through the bend. It is also important to note that the computation time is a few seconds for a certain case with a single collision (Fig. 10-a) and below one minute for six repeated collisions (N ¼ 6 in Fig. 10-b), which represents a very short computation time compared with that of usual DEM-CFD simulations. It should be noted that the calculations performed with six repeated collisions for the same bend represent the same behavior of real pneumatic conveying system constructed of six different bends. It is because after the first collision (first bend), whether particles break or not, the change in particle’s size and strength distributions is kept in memory for the next collision (next bend). Moreover, particle’s velocity, locations and impact angles are chosen randomly once again by the machine functions before the next collision with appropriate reduced particle size and bend geometry. Since the particle velocity model used in this study is suitable only for steady-state conditions, the current attrition model is suitable in case where the distance between the bends in real system is high enough for particles reach the steady state conditions. If the distance does not meet the requirements, the particle acceleration length and velocity distribution can be predicted by a recently developed model of Santo et al. [15]. However, this particular case does not fit the purpose of this study. For further definition of particle acceleration length and velocity development models, the reader can refer to the study of Santo et al. [15]. The experiments and the current model don’t take into account the compressibility of the air which increase the air velocity from one bend to the other in a pneumatic conveying pipeline. Although, it is easy to apply into the current model, pressure drop models over straight pipes and bends have to be applied. In addition, it should be noted that in real pneumatic conveying system, the particles can pass the bend by either roping or impact collision with the bend wall. While roping phenomenon usually

Please cite this article as: D. Portnikov, N. Santo and H. Kalman, Simplified model for particle collision related to attrition in pneumatic conveying, Advanced Powder Technology, https://doi.org/10.1016/j.apt.2019.10.028

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exists in dense phase conveying and very fine powders, impact collisions usually can be observed in dilute phase conveying. Recent experimental studies by Tripathi et al. [16] and Kalman et al. [17] supports this behavior. They measured particle velocity reduction in horizontal-horizontal and horizontal-vertical bends of dilute phase pneumatic conveying. The experiments performed by tracking the particles in various transparent bend geometries using high-speed camera. The roping was not observed due to low tested concentration of particles. Instead, secondary collisions with the bend wall were identified. Regarding that, it should be emphasized that the current attrition model refers only to the first collision because the secondary collisions, in any case, occurs with much lower velocities (coefficient of restitution) and therefore, the damage is less relevant.

5.3. Parametric study The parametric study was performed to check the effects of various operational conditions on the particle attrition. The predicted median particle size was plotted versus the times the particles pass through the bend for various superficial air velocities, pipe diameters, and bend ratios in Fig. 11. The median particle size decreases with increasing number of cycles for all studied cases. As expected, higher air velocities cause more damage to the conveyed material (see Fig. 11-a) because the higher the air velocity, the higher are the applied forces acting on the particles. According to Fig. 11-b, larger pipe diameters cause more damage to the particles for the same bend ratio. However, higher bend radius ratios cause less damage to the conveyed material (see Fig. 11-c) because higher bend radius ratios and smaller pipe diameters cause lower collision angles between the particles and bend walls.

Moreover, the bend orientation affects the attrition of particles during pneumatic conveying. Three possible bend orientations were studied: horizontal–vertical (H–V), horizontal–horizontal (H–H), and vertical–horizontal (V–H). The calculated results are shown in Fig. 12-a as the median particle size with respect to the number of cycles. For equal operational conditions, the H–V bend orientation causes less damage to the conveyed material than the H–H and V–H bend orientations. This is because the range of collision angles is much smaller for H–V than for H–H. To illustrate this behavior, Fig. 12-b and -c show schematic views of the particle movements inside H–V and H–H bends, respectively. In both figures, the concentrations and locations of the particles in the horizontal pipe cross section in front of the bend entrance are equal. The side view of the H–V bend shows that the maximal collision angle is much smaller than the maximal collision angle of the H– H bend. The situation in V-H bend is the same as in H-H bend because in a vertical pipe, the particles are distributed within the entire pipe cross section. Therefore, the damages caused by H–H and V–H bends to the conveyed material are expected to be equal, as shown in Fig. 12-a.

6. Summary and conclusions In this study, an easy to use breakage algorithm was developed. The model does not require the use of DEM-CFD simulations, because the machine functions are defined based on empirical correlations and the bend geometry. The model includes three machine functions: the particle steady-state velocity distribution, impact angle distribution, and location distribution functions. Furthermore, the model includes five material functions: the initial size distribution, strength distribution, equivalence/selection,

Please cite this article as: D. Portnikov, N. Santo and H. Kalman, Simplified model for particle collision related to attrition in pneumatic conveying, Advanced Powder Technology, https://doi.org/10.1016/j.apt.2019.10.028

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breakage, and fatigue functions. To validate the computational model, experiments were conducted in a short pneumatic conveying system with various bend ratios and particle cycle numbers. The size distributions of the broken particles predicted by the model for pneumatic conveying are in good agreement with the experimental ones. In addition, a parametric study was performed to check the effect of various operational condition parameters on the particle attrition. The following conclusions can be drawn:  The computation time was reduced significantly compared with that of a complete DEM–CFD simulation. For the first collision (particles passed through one bend), the computation time was a few seconds. For six collisions (particles passed through six bends), the computation time was below one minute.  The analysis of the number of initial particles revealed that 2000 initial particles are sufficient in the computational algorithm to obtain results with sufficient accuracy. More than 2000 initial particles can be used. However, this will increase the computation time.  Both validation experiments and parametric study revealed that higher bend radius ratios cause less damage to the conveyed product because higher bend radius ratios produce lower collision angles with the bend walls.  According to the parametric study, higher air velocities and larger pipe diameters lead to higher damage to the conveyed material.  The proposed computational model provides a powerful and simple design tool to predict the breakage of particles during pneumatic conveying.  Finally, although the computation time was reduced, efforts should be made in future studies to reduce the characterization time of the material functions.

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[2] H. Kalman, M. Hubert, E. Grant, Y. Petukhov, M. Haim, Fatigue behavior of impact comminution and attrition units, Powder Technol. 146 (2004) 1–9. [3] L. Frye, W. Peukert, Attrition of bulk solids in pneumatic conveying: mechanisms and material properties, Particul. Sci. Technol. 20 (2002) 267– 282. [4] H. Kalman, V. Rodnianski, M. Haim, A new method to implement comminution functions into DEM simulation of a size reduaction system due to particle-wall collisions, Granul Matter. 11 (2009) 253–256. [5] T. Brosh, H. Kalman, A. Levy, DEM simulation of particle attrition in dilutephase pneumatic conveying, Granular Matter. 13 (2011) 175–181. [6] Y. Rozenblat, D. Portnikov, A. Levy, H. Kalman, S. Aman, J. Tomas, Strength distribution of particles under compression, Powder Technol. 208 (2011) 215– 224. [7] Y. Rozenblat, E. Grant, A. Levy, H. Kalman, J. Tomas, Selection and breakage functions of particles under impact loads, Chem. Eng. Sci. 71 (2012) 56–66. [8] D. Portnikov, R. Peisakhov, G.O. Gabrieli, H. Kalman, Selection function of particles under impact loads: the effect of collision angle, Particul. Sci. Technol. 36 (2018) 420–426. [9] Y. Rozenblat, A. Levy, H. Kalman, I. Peyron, F. Ricard, A model for particle fatigue due to impact loads, Powder Technol. 239 (2013) 199–207. [10] Y. Rozenblat, A. Levy, H. Kalman, J. Tomas, Impact velocity and compression force relationship — Equivalence function, Powder Technol. 235 (2013) 756– 763. [11] N. Santo, D. Portnikov, I. Eshel, R. Taranto, H. Kalman, Experimental study on particle steady state velocity distribution in horizontal dilute phase pneumatic conveying, Chem. Eng. Sci. 187 (2018) 354–366. [12] A.D. Salman, C.A. Biggs, J. Fu, I. Angyal, M. Szabó, M.J. Hounslow, An experimental investigation of particle fragmentation using single particle impact studies, Powder Technol. 128 (2002) 36–46. [13] L.M. Tavares, Optimum routes for particle breakage by impact, Powder Technol. 142 (2004) 81–91. [14] T. Han, H. Kalman, A. Levy, Theoretical and experimental study of multicompression particle breakage, Adv. Powder Technol. 14 (2003) 605–620. [15] N. Santo, D. Portnikov, N. Tripathi Mani, H. Kalman, Experimental study on the particle velocity development profile and acceleration length in horizontal dilute phase pneumatic conveying systems, Powder Technol. 339 (2018) 368– 376. [16] N.M. Tripathi, N. Santo, A. Levy, H. Kalman, Experimental analysis of velocity reduction in bends related to vertical pipes in dilute phase pneumatic conveying, Powder Technol. 345 (2019) 190–202. [17] H. Kalman, N. Santo, N.M. Tripathi, Particle velocity reduction in horizontalhorizontal bends of dilute phase pneumatic conveying, Powder Technol. 356 (2019) 808–817.

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Please cite this article as: D. Portnikov, N. Santo and H. Kalman, Simplified model for particle collision related to attrition in pneumatic conveying, Advanced Powder Technology, https://doi.org/10.1016/j.apt.2019.10.028