A model for particle fatigue due to impact loads

Powder Technology 239 (2013) 199–207

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A model for particle fatigue due to impact loads Yevgeny Rozenblat a,⁎, Avi Levy a, Haim Kalman a, Isabelle Peyron b, Francois Ricard c a b c

Laboratory for Conveying and Handling of Particulate Solids, Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 84105, Israel Respiratory Centre of Excellence, GlaxoSmithKline GMS, 27000 Evreux, France Particle Generation Control & Engineering, GlaxoSmithKline R&D, Stevenage, SG1 2NY, UK

a r t i c l e

i n f o

Article history: Received 25 March 2012 Received in revised form 24 December 2012 Accepted 26 January 2013 Available online 1 February 2013 Keywords: Fatigue effect Fatigue model Particle strength

a b s t r a c t The wide use of particulate materials in industry has elicited various problems that are related to particle size variations (increase or decrease) in diverse mechanical processes. One of the main problems is the low accuracy of designing size reduction systems. The particles break as a result of continuous loads higher than their strength. The particles may also break due to repeated loads lower than their strength (a phenomenon related to fatigue). Therefore, a characterization of particle strength with static and repeated loads is crucial. The current work introduces and validates the fatigue model which development is based on the theoretical model. The modified model enables us to calculate particle strength decreases with every additional loading event. The model depends on six empirical and material related parameters, and was validated by appropriate experiments with a variety of NaCl particle sizes. Since the fatigue effect among particulate materials is a very common event, this study may enable the simulating comminution processes with higher accuracy. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Size reduction of particulate products is a common issue in variety of industries including the chemical and pharmaceutical industries. This process not only consumes a great investment of energy, but is generally very ineffective [1,2]. One of the reasons for this inefficiency is the lack of a comprehensive knowledge regarding particle strength changes due to repeated loadings. The influence of the load magnitude whether impact, compression or shear loading has been intensively studied [3–6]. However, particles subjected to impact or percussion often show an additional effect, namely repeated loadings of a particle at a loading smaller than the particle strength may decrease the strength significantly. Crushing can occur when the strength is reduced to a level lower than the applied load. Since in the majority of the comminution systems, particle breakage occurs after a number of loading events [7], there is a real necessity to study and improve the understanding of particle fatigue. Attempts of various researches to investigate particulate material fatigue indicate particle strength is dependent on the load number [8–12], but do not offer any mathematical expressions to be considered in design. Many researchers mentioned the fatigue phenomenon, and evaluated its significance and potential for designing attrition and comminution procedures [10,13,14]. However, none of them introduced appropriate model to support their findings. One of the few quantitative approaches for the particle strength changes as a consequence of repeated compression ⁎ Corresponding author. E-mail address: [email protected] (Y. Rozenblat). 0032-5910/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.01.059

and impact loadings was introduced by Han et al. [7]. This model was based on Griffith's theory of crack propagation. However, this model has a number of disadvantages which will be discussed later in the paper. The purpose of the present paper is to modify and generalize Han's fatigue model. Also, a procedure to consider fatigue effects in simulation or design of size reduction systems for repeated impacts at various velocities (for each impact) is developed. 2. Experimental In order to investigate particle fatigue behavior, a specific experimental system is required. The system has to provide either impact loads, as some of the researchers used [8,11] or compression loads as others [4,5] and be able to apply controlled cyclic loadings. 2.1. Experimental systems The fatigue experiments were performed using an impact experimental system. Cyclic loadings were achieved by refeeding the unbroken particles from the previous cycle. The definition of the particle strength is varying among different researchers including; breakage force [11,15,16], breakage energy [5,17], crushing stress [18–20], etc. Due to a number of reasons which are explained in a previous work [21], the particle strength will be expressed in terms of breakage force along the current work and thus all of the references to the particle strength are actually breakage force. Since a stress is a force divided by the applied area, the breakage results for various particle sizes are opposite for these two terms. Bigger particles will crush under higher forces, but under smaller stresses (larger contact area).

Y. Rozenblat et al. / Powder Technology 239 (2013) 199–207

In order to define the fatigue and strength reduction, the compression strength distribution was measured after each loading. Therefore, investigation of the fatigue behavior of the particles was accomplished by combining tests with both compression and impact experimental systems. 2.1.1. Impact experimental system Many different types of impact systems have been used in previous researches including a pendulum impact tester [22], a vertical impact system [3], a horizontal impact system [23], and an air gun system [24]. For the present study a horizontal impact experimental system was used [25]. In such a system, the particles are fed into an acceleration tube (length 590 mm and inner diameter 15.8 mm) by the vibrating feeder, in which the particles are accelerated by means of compressed air. The acceleration tube was designed so that the distance between the end of the tube and the target can be adjusted. At the outlet of the tube the accelerated particles collide with a rigid target within the impact chamber. The impact chamber provides a sealed environment which keeps the particles and their ricochets within the chamber. Furthermore, a part of the impact chamber walls were made of transparent fiberglass in order to allow the measurement of the particle impact velocity by a fast digital camera (CamRecord 1000, >2000 fps). To õprevent secondary breakage of the particles, the rest of the walls inside the impact chamber were covered by rubber. The impacted particles are then felled to the collection hopper. A dust filter was installed into the upper part of the impact chamber. The filter is a fiber bag that prevents dust expansion out of the system. Collected particles after each impact cycle were sieved by set of “Retsch” screens on a “W.S. Tyler Incorporated” sieving device and weighed by fractions on an electronic “M.R.C. — BB-3100” weight. The unbroken fraction of the particles (particles which remain within the feeding size interval) was reefed into the system in order to simulate the following sequential load. Scheme of the impact experimental system is shown in Fig. 1.

Feed Tank

2.1.2. Compression experimental system The literature describes a variety of compression system types including simple hydraulic press [17], ultra-micro loading system [26], Zwick texture analyzer [15], etc. A new compression system was developed and constructed to meet the requirements of the research. Full experiment system specifications and it advantages described

elsewhere [27,28], so they are only be briefly discussed here. The particle is compressed between two rigid platens until it fails (breaks). The applied force is measured by a load cell, and the displacement by a LVDT (linear variable differential transformer) system. Both of the measuring devices are controlled by the Labview program interface. By measuring the force for a sufficient number of particles [28], the particle compression strength distribution for a given particles population can be estimated. A scheme of a compression tester is presented in Fig. 2.

2.2. Test procedure As mentioned above, the application of the particles cyclic loading was performed by the impact experimental system. Each cycle was conducted under constant loading conditions, i.e. constant impact velocity. After each impact cycle, a sample of the unbroken size fraction (about 25 g) was taken in order to estimate the particle strength distribution by the compression experimental system. An extensive testing program, shown in Fig. 3, was conducted in order to investigate the fatigue phenomenon of the particulate materials. The experimental plan shown in Fig. 3 is presented per one specific material, initial (mother) size fraction and impact velocity. The series of the “fatigue” experiments begins with a predefined initial size fraction (upper-left square marked by number 1). The strength distribution of the mother fraction marked by F1 (beneath the square number 1) measured by compression system. After impacting the initial sized particles at velocity V, the sample was sieved to the various size fractions (marked 1,2,3,4,5). The compression strength distribution of each of the sieved size fractions of the particles were then measured (noted as F1′, F2′, etc.). Then, the survived particles (size fraction 1) were impacted again at the same velocity, and the impact cycle number two emerged. Impact cycles in the diagram are indicated by n. By this way, a particle initially at the upper limit of the size fraction experiences some chipping in which the particle size remains in the initial size fraction although at the lower limit is considered unbroken. However, this wrong consideration might become negligible for testing a sufficiently large amount of particles. In any case, the alternative to the statistical analysis used in this work would be single particle impact tests which require enormous time and effort.

Impact Cage Vibrating Feeder

Air Supply

AccelerationTube Target Plate

Collection Hopper

Fig. 1. The scheme of a horizontal impact tester.

Dust Filter

200

Y. Rozenblat et al. / Powder Technology 239 (2013) 199–207

201

Hydraulic Piston

4. Comparing the size distribution after each impact loading enable to detect the effect of “History” on the breakage function. 5. Analyzing the strength distribution of the survived mother size fraction (F1, F1′, F1″, F1‴, etc.) can define the effect of the strength decreasing of the survived particles due to fatigue.

LVDT

Subjects 1–4 will be researched and then published as a separate paper, while subject 5 is the purpose of current work. Measurements of the unbroken particles strength in each impact cycle will exhibit the compression particle strength behavior with each additional impact load and as the result the fatigue model can be developed. The list of all conducted experiments is presented in Table 1.

Upper Punch

3. Results The following section will interpret the experimental data acquisition, the analysis of the experimental data, and the development and validation of the fatigue model. 3.1. Experimental data analysis

Particle

Load Cell

Fig. 2. A scheme of a compression tester.

By analyzing of such experiments a number of different subjects can also be addressed: 1. Analyzing the mass fractions remains under size interval 1 defines the effect of the number of impact cycles on the breakage probability, i.e. selection function. 2. Comparing the strength distributions of the same size fractions (F2, F2′, F2″, F2‴, etc.) can define the effect of preloading (“History”) on the strength distribution. 3. Adding measures of the impact breakage probability to the above measurements for each size fraction (BP2, BP2′, BP2″, BP2‴, etc.) may elucidate the effect of the “History” on the breakage probability.

The average strength change (mean strength of the entire particle population per one specific impact cycle) of the surviving particles after consecutive impact cycles is shows in many cases an increase and then decrease, as is presented for example in Fig. 4. This phenomenon might be explained by two major factors [29,30]. The first is that due to the breakage of the weakest particles the average strength increases. The second is that each load on the particles will decrease their average strength due to fatigue and crack growth as in nonparticulate materials [31]. Fig. 4 demonstrates, on one given sample, the influence of these two factors on the average crushing force. It is clear that in order to eliminate the effect of the average strength growth and be focused on the particle fatigue, another approach, as is explained in the followings, should be used. According to Vervoorn and Austin [31], the reasons for the statistical nature of the breakage of particles are due to: 1. The distribution of strength of particles. 2. The distribution of the applied force resulting from the system load. 3. Different orientations of particles during the load on them. 4. Reduction of strength with repeated loads.

Fig. 3. Schematic diagram of the experimental plan.

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Table 1 Experiment summary. Initial size interval

Intercycle impact velocity

# of cycles

d0 [mm]

V [m/s]

n

4.1 6.5 8.6 13 4 6.2 8.5 13 7.1 8.5 10 13 13

9 7 5 3 10 10 8 4 10 8 7 5 8

3.35–4

2.36–3.35

2–2.36

1.4–2

The impact velocities of the particles in the impact system (which provide repeating loadings on the particles) are controlled, and therefore reason 2 can be disregarded. The inhomogeneous nature of the particles caused by confined air bubbles, micro cracks (in different orientations) and different impurities, reason number 3 can also be disregarded. Consequently, the change of the particle strength as a result of the cycle loading will be explained based only on reasons 1 and 4. Since the particle population strength cannot be described by a discrete value, it has to be described by a statistical distribution. A sample of a statistical expression which describes the initial particle strength distribution is shown elsewhere [21]. Each strength distribution but the initial one represents the strength of only the survived particles. Therefore, in order to eliminate the influence of the weakest particles breakage, it is required to convert each strength distribution into a portion of the survived particles out of the initial population. The only assumption for this calculation is that at each impact cycle only the weakest particles are broken. The following expression presents the required conversion equation: ′

S c ¼ Sc ð1−SI Þ þ SI

ð1Þ

where, Sc is the cumulative percentage of crushed particles for any crush force at any impact cycle–compression breakage probability (squares in Fig. 5), SI is the percentage of broken particles by impact–impact breakage probability, and Sc′ is the emerged breakage probability for any crush force within the initial crushing strength distribution (circles in Fig. 5). Since it is easier to track changes of a small set of discrete numbers rather than after the whole distribution [32], the particle strength distribution was divided into ten evenly distributed strength groups.

Average Crushing Force, Favg [N]

50 45

Each group will be represented by the average strength between its boundaries. The initial strength distribution of NaCl particles and the strength distribution after 3, 6 and 10 impact cycles including their groups' annotations are shown in Fig. 5, which represent all other similar strength distributions that were plotted after each impact cycle. Fig. 5 also shows how the measured strength distribution of the survived particles is applied to only the survived particles taking into account the particle breakage probability. In this way, it is possible to follow the strength change of each one of the 10 strength groups as shown in Fig. 6. Fig. 6 combines data presented in Fig. 5 and presents it as the average strength of the initial 10 groups as a function of the impact number. Fig. 6 illustrates only one set of an experimental conditions, in this case the initial particle size interval was 2.36–3.35 mm and the impact velocity was about 6 m/s. The particle strength changes among different strength groups (each symbol refers to a different strength group) are marked as hollow symbols. The squares connected with the solid line symbolize impact selection function, i.e. the cumulative percentage of broken particles. Fig. 6 clearly shows that the strength of all groups is decreasing over impact number progress. For instance, group 9 which is the second “strongest” group of the particle population has initially a breakage force of about 55 N and decreases to about 30 N after 10 impact cycles. A circled symbol means that the group is partially broken and the strength represented by the circled symbol is the average strength of the surviving particles from this group. After 6 impacts 67% of the initial sample weight is broken. Groups 1–6 are totally disappear, i.e. these groups are undergoes total breakage, and group 7 has an additional 7% (67– 60) of the brakeage. By observing the circled symbols, it is possible to separate between the fatigue and the breakage zones. Thus, the particles strength is decreased due to fatigue until it reaches the limit (the line between fatigue and breakage zones) and then the particle breaks. The decrease in strength of each group is clearly demonstrated in Fig. 6, with some exceptions. The dispersion of results can originate from the following reasons: distribution of applied force, different particles orientations, and different particles shapes. 3.2. Fatigue model The following section will focus on the basis of the fatigue model development, the assumptions and the constraints of the proposed model. 3.2.1. Model description In order to explain how particle strength changes due to fatigue under repeated loadings, a theoretical fatigue model, based on modify Griffith's theory [33] of crack propagation, was previously developed by Han et al. [7]. The assumptions, development description and parametric studies of the model are fully explained in [7]. In the following we briefly summarize Hans's model. The fatigue phenomenon of particulate materials is formulated by the following equation: 9−3n 8 2 > > = < P1 P ¼ Pm ⋅ 1 þ h i2 > > : exp P3 −P2 ðPÞ1=3 ;

40

′

35

Salt d0=2.36-3.35 mm

30

V=8.5 m/sec

ð2Þ

where Pm is the initial compression strength of the particles and n is the number of loading events. The dimensionless number P* indicates the ratio between the applied load and the initial compression strength as follows:

25 20 0

1

2

3

4

5

6

7

8

Impact Cycle, n Fig. 4. Change of the average crushing force with impact number.

9

P P ¼ r Pm

ð3Þ

1.0

1.0 Group 10 0.9 Group 9

Crushing Breakage Ration, Sc

Crushing Breakage Ration, Sc

Y. Rozenblat et al. / Powder Technology 239 (2013) 199–207

0.8 Group 8 0.7 Group 7 0.6 Group 6 0.5 Group 5 Salt d0=2.36-3.35 mm

0.4 Group 4 0.3 Group 3

V=6.24 m/sec n=0 SF=0 %

0.2 Group 2 0.1 Group 1

Initial Strength Distribution

0.0 10

20

30

40

50

60

70

Group 10

0.9 Group 9 0.8 Group 8 0.7 Group 7 0.6 Group 6 0.5

Salt d0=2.36-3.35 mm

0.4

V=6.24 m/sec n=3 SF=47.49 %

0.3 0.2

Measured Sample Converted Result

0.1 0.0

80

20

30

1.0

1.0 Group 10 0.9 Group 9 0.8 Group 8 0.7 0.6 Salt d0=2.36-3.35 mm

0.4

V=6.24 m/sec n=6 SF=67.82 %

0.3 0.2

Measured Sample Converted Result

0.1 0.0 10

20

30

40

50

60

40

50

60

70

80

Crushing Force, F [N]

70

80

Crushing Breakage Ration, Sc

Crushing Breakage Ration, Sc

Crushing Force, F [N]

0.5

203

0.9

Group 10 Group 9

0.8 0.7 0.6 0.5

Salt d0=2.36-3.35 mm

0.4

V=6.24 m/sec n=10 SF=82.48 %

0.3 0.2

Measured Sample Converted Result

0.1 0.0 20

30

Crushing Force, F [N]

40

50

60

70

80

Crushing Force, F [N]

Fig. 5. Presentation of the strength of the survived particles by the strength groups.

where Pr = Fr / Ar is the normal compression stress acting on a single particle. Model parameters P1, P2 and P3 are evaluated as functions of the system and material properties. P1 is the translational degree of freedom of the activated complex along the reaction and is given by: P1 ¼

kT : hω

ð4Þ

P2 shows the influence of the initial compression strength of particles on the fatigue of particles. P2 ¼ qσ m

sffiffiffiffiffiffiffiffiffiffi Ψ kTEs

ð5Þ

70

20

Fatigue 30

50 40

O

O O O

10

O

O

90 80 70 60 50 40

Breakage 20 Salt d0=2.36-3.35 mm

O

10

V=6.24 m/sec 0

0 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

Impact Cycle, n Fig. 6. Representative fatigue chart.

Impact Breakage Probability, SI [%]

30

60

Crushing Force, F [N]

100

Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Selection Function

where σm is the original tensile strength of particles and it is propor1/3 tional to Pm . P3 indicates the influence of the theoretical strength of particles on particle fatigue and is expressed as follows: sffiffiffiffiffiffiffiffiffiffi Ψ : P3 ¼ R m kTEs

ð6Þ

Han's model (Eq. (2)) enables calculation of the changes in particle strength as a result of progressive loadings. The model is theoretically based and it has physical bounds, i.e. the model result initial particle strength (Pm) value while no impacts were conducted and value zero while infinite loads were made or impact velocity was infinite. However, the model has the following disadvantages: 1. It calculates particle strength in terms of crushing stress, the estimation of which is relatively difficult [32]. 2. It is restricted to cases with constant loads (impact velocity) of all cycles. 3. Although the model parameters (P1, P2 and P3) can be calculated by means of the known material and system properties, it is practically impossible to estimate these properties. 3.2.2. Model modifications Since Han's model is the only quantitative expression we have found in the literature which describes particle strength changes during cyclic loading quantitatively, we attempt to modify it so that it is easier for practical use. In order to overcome the drawbacks listed above, Han's model was modified according to the following procedure. First, it was transformed to calculate particle strength in terms of the crushing force instead of stress terms. Since the contact area was deleted from the equation its effect will be expressed by affecting the model parameters. Accordingly,

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all the stress terms in Han's model were simply reformulated by crushing force terms. The validation of this assumption will be considered further. Eq. (7) presents the updated formulation of the fatigue model: 8 9−3n 2 > > < = P1 F ¼ Fm ⋅ 1 þ h i2 > > : exp P3 −P2 ðPÞ1=3 ; ′

ð7Þ

where dimensionless group P* is now equal to Fr / Fm. Since in our case cyclic loads on the particles are impact loads, i.e. are expressed in impact velocity terms, the loads has to be converted to force terms in order to calculate applied loads and compare them to the initial particle strength which is expressed in force terms. Eq. (8) presents the following equivalent force function (see [34] for its detailed development description).

3:3⋅Bfv

ð8Þ

where Afv and Bfv are the model parameters. Their values for different materials and a detailed evaluation explanation of the equivalence function is described elsewhere [34]. Parameters P1, P2 and P3 of Han's model (Eq. (7)) are defined by Eqs. (4)–(6), respectively. In order to calculate them one has to estimate different particle material and system properties, which are often hard to find, e.g. stress concentration factor, material crystalline volume, etc. By converting these parameters into empirical ones, the inability to estimate Han's model parameters is overcome. Parameter P1 was assumed to be constant for the various sizes of all the materials (1.89× 1013). The justification of this assumption stems from its definition (Eq. (4)). P1 is a parameter which is related to the system and environment variables (loading frequency, and temperature). Therefore, if the environment variables are preserved, parameter P1 can be assumed to be constant. According to the definition P2 and P3 (Eqs. (5) and (6)) they are interdependent. The relationship between them is demonstrated by the following expression: P3 ¼

P2 ⋅Pm q⋅σ m

ð9Þ

where Rm is the theoretical fracture strength of particle (7.275 × 109 Pa for NaCl), q is the stress concentration factor (4.7 [33]), and σm is the 1/3 original tensile strength of particles (1.87 × 106 · Fm for NaCl). Substitution of these variables (Rm, q, and σm) into Eq. (10) reveals the dependency of P3, which is presented by Eq. (10) as follows: P3 ¼ 827:8⋅P2 ⋅Fm

1

−

=3 :

C

P2 ¼ A⋅Fmc :

B

A ¼ Aa ⋅d0b :

ð10Þ

The conversion of Han's fatigue model parameters (P1, P2, and P3) into empirical ones reveals that actually only P2 is a materialdependent parameter. The rest of the model parameters become constant — P1 or dependent — P3. Consequently, the fatigue model can be rewritten as a function of only one free empirical parameter, namely P2. Eq. (11) presents the modified version of the Han's model: 9−3n 8 2 > > > > > > = < 1:89  1013 ′ F ¼ Fm ⋅ 1 þ    1=3 2 > : > 1 F > > > > exp 827:8⋅P2 ⋅Fm − =3 −P2 Feq ; :

ð11Þ

m

ð13Þ

d0=3.675 mm, V=4.1 m/sec

2.2

d0=3.675 mm, V=6.5 m/sec

2.1

d0=3.675 mm, V=8.6 m/sec d0=2.855 mm, V=4 m/sec

2.0

d0=2.855 mm, V=6.2 m/sec

1.9

d0=2.855 mm, V=8.5 m/sec d0=2.855 mm, V=13 m/sec

1.8

d0=2.18 mm, V=7.1 m/sec

1.7

d0=2.18 mm, V=8.5 m/sec d0=2.18 mm, V=10 m/sec

1.6

d0=2.18 mm, V=13 m/sec

1.5

Cc

Power model P2=A*Pm

1.4 1.3 20

The change in particle strength as a result of cyclic loading is now expressed as an only one-parameter model.

ð12Þ

Each symbol in Fig. 7 symbolize different combination of the initial particle size and the intercycle impact velocity and the solid line represents P2 model (Eq. (12)). Fig. 7 clearly shows that parameter P2 is also particle size dependent, since all the data divided into separate size groups. Moreover, according to Fig. 7, P2 is independent of the impact velocity, since there is no distinction between behaviors of experiments with different impact velocities. This conclusion is supported by Han's theoretical definition of P2 parameter, i.e. Eq. (5) is not a function of impact velocity. The summary of P2 parameter fittings statistics is presented in Table 2. Consequently, parameter P2 is a function of the initial particle strength and initial particle size. While parameter P2 is power dependent on Fm, its dependency on the initial particle size is still unknown. In order to estimate this relationship, both secondary empirical parameters A and Cc were plotted versus the initial particle size. Fig. 8 demonstrates this relationship for both secondary parameters. Each data series in Fig. 8 depicts different impact velocity. According to Fig. 8 secondary parameter Cc can be assumed to be constant with particle size. In contrast, secondary parameter A is power related to the initial particle size. The expression of the A relation by initial particle size is:

Fatigue Model Parameter, P2

B

Feq ¼ Afv ⋅V fv ⋅d0

3.2.3. Investigation of a new model parameter In order to develop a general expression for the fatigue phenomena, which will be described by Eq. (11), it is necessary to estimate the dependency of parameter P2 on material properties. For that purpose, the fatigue model was fitted to the experimental data. Since the quality of the model fittings for different series of experiments is reasonably fair, the assumption that substitution of crushing force instead of stress in Han's fatigue model is satisfied. Parameter P2 was estimated by the fatigue model (Eq. (11)) fitting for each experiment, using different sets of impact velocity and initial particle size. Based on Han's definition of parameter P2, which is an expression of the influence of the initial compression strength, it is rational to start investigation of P2 from its dependence on particle initial strength. Thus, P2 was plotted versus Fm (particle initial strength) for each experiment case (different impact velocity and initial particle size). Fig. 7 demonstrates that parameter P2 depends on Fm, since there is a clear tendentiousness in each experimental case. The relationship between P2 and Fm obeys a power law, shown by Eq. (12) with high values of R2 (coefficient of determination).

30

40

50

60

70

80

90 100 110 120 130 140 150

Initial Particle Strength, Fm [N] Fig. 7. Relationship between P2 and initial particle strength Fm.

Y. Rozenblat et al. / Powder Technology 239 (2013) 199–207 Table 2 Statistics of the P2 fittings.

3.675

2.855

2.18

Table 3 Fatigue model parameters.

V [m/s]

A

Cc

R2

Aa

Bb

Cc

4.1 6.5 8.6 4 6.2 8.5 13 7.1 8.5 10 13

0.324 0.362 0.328 0.335 0.385 0.430 0.410 0.486 0.517 0.532 0.514

0.381 0.350 0.372 0.430 0.386 0.357 0.371 0.372 0.357 0.351 0.360

0.997 0.999 1.000 0.994 0.998 0.999 1.000 0.998 0.999 0.994 0.999

6.724 × 10−3

−0.725

0.333

Combination of Eqs. (13) and (12) reveals dependency of parameter P2 on particle properties. The final formulation of P2 is: B

C

P2 ¼ Aa ⋅d0b ⋅Fmc :

ð14Þ

Introducing Eq. (14) into the fatigue model (Eq. (11)) allows us to estimate the model parameters (Aa, Bb, and Cc) by optimization and results a new semi theoretical model with empirical parameters only, which is presented in Eq. (15). 9−3n 8 2 > > > > > > 13 = < 1:89  10 ′ F ¼ Fm ⋅ 1 þ      1=3 2 > : > C −1 F > > B B C > > exp 827:8⋅Aa ⋅d0b ⋅F mc 3 − Aa ⋅d0b ⋅F mc ⋅ Feq ; : m

ð15Þ The optimization procedure estimates the parameters of Eq. (15) such that the calculated particle strength will represent the experimental results in the most accurate way. The optimization was based on the least squares method using Matlab software. Optimized parameters significantly improved the ability of the fatigue model to reflect the experimental data of the particle strength. Optimized parameters are presented in Table 3. Fig. 9 qualitatively demonstrates the fatigue model agreement with the experimental data for only one specific case (d0 = 2.855 mm and V = 6.2 m/s). The solid line demonstrates model (Eq. (15)) prediction in Fig. 9. While disconnected different symbols are the experimental average strengths correspondingly to their groups in decreasing order. Group 10 represents the strongest group of the particle population and group 1 the weakest one. Since groups 1and 2 are not survived the first impact, i.e. break after just one impact, they not appear in Fig. 9. As can be seen form Fig. 9, all the series (all strength groups) are decreasing with each

P2Fitting Parameter, A

0.55

V~4 m/sec V~6.3 m/sec V~8.5 m/sec V~13 m/sec

0.50

0.45

0.40

0.35

0.30

additional impact. Since the groups are vary in their strength, not all the group series survive up to 10 impacts (in case of experiment presented by Fig. 9). As a validation of the proposed fatigue model (Eq. (15)) the experimental results of the particle strengths decreases were compared to the calculated ones. The results are presented by the comparison chart in Fig. 10. Fig. 10 includes three different lines. Two thin lines bound the limit of the error and the thick central line depicts a 100% compliance of the model. It should be emphasized that each series, i.e. different combination of impact velocity (V) and particle size (d0), includes ten strength groups. Fig. 10 demonstrates that most of the results are within the ±20% error limits, which is reasonable considering the comprehensiveness of the model and the complexity of the measurement and results analysis. 3.3. Discrete fatigue model During the process of size reduction, loadings cycles occur at various magnitudes. If the variation between the loads is small, it is reasonable to assume constant amplitude of the loads. However, in many applications of size reduction systems the loads amplitudes are highly dissimilar. Since the previously shown fatigue model (Eq. (15)) is not applicable to loads that are not constant, it cannot be used to calculate particle strength changes of the real comminution systems. In addition, calculations performed with the average load may result in large errors. Consequently a model that can estimate particle strength after each impact event of different amplitudes is required. The fatigue model (Eq. (15)) presented in the previous section is limited to cases with constant loads, due to its dependence on the impact number and the initial particle strength Fm. In order to use the model at the inconstant amplitude cases the dependencies on the impact number and initial particle strength has to be excluded from it. As a result, the particle strength is referred to the previous cycle strength instead of initial one and the impact number (n) is deleted from the fatigue model (Eq. (15)), since now each even is a new one regardless of the “History”. Therefore, the model which calculates particle strength with constant impact velocities between the cycles will be defined as continuous fatigue model. The model which calculates the particle strength using step by step manner for the inconstant amplitude loads case now defined as discrete model. For the continuous fatigue model, parameter P2 was found to be dependent, among others, on the initial particle strength (Eq. (14)). Therefore, in the discrete case it will be dependent on the strength of 0.50

P2 Fitting Parameter, Cc

d0 [mm]

205

V~4 m/sec V~6.3 m/sec V~8.5 m/sec V~13 m/sec

0.45

0.40

0.35

0.30 0.0021 0.0024 0.0027 0.0030 0.0033 0.0036

0.0021 0.0024 0.0027 0.0030 0.0033 0.0036

Particle Size, d0, [m] Fig. 8. Parameters A and Cc correlation with particle size.

Particle Size, d0, m

206

Y. Rozenblat et al. / Powder Technology 239 (2013) 199–207

Salt, d0=2.855mm, V= 6.2m/sec

80

Experimental data Fatigue Model

Experimental data Discrete Fatigue Model

70

60 Group 10

50

Group 9 Group 8

40

Group 7 Group 6

30

Group 5 Group 4

20

Crushing Force, F [N]

Crushing Force, F [N]

70

Salt, d0=2.855mm, V=6.2m/sec

80

60 Group 10 Group 9

50

Group 8 Group 7

40

Group 6 Group 5

30

Group 4

Group 3

Group 3

20

10 0 0

2

4

6

8

10

10 0

12

1

2

3

4

Fig. 9. Particle strength groups vs. impact number, exp. data and fatigue model.

the particle at the previous impact event, while the empirical parameters of Eq. (14) (Aa, Bb, and Cc) are kept constant. The discrete fatigue model is presented in Eq. (16) as follows: 8 9−3 2 > > > > > > < = 1:89  1013 Fi ¼ F i−1 ⋅ 1 þ    −1 =     1=3 2 > > F > > B Cc B Cc > > ⋅Fi−1 3 − Aa ⋅d0b ⋅Fi−1 ⋅ F eq exp 827:8⋅ Aa ⋅d0b ⋅Fi−1 : ; i−1

ð16Þ where index i indicates the ith impact of the specific particle, and i − 1 represents the previous impact. The rest of the symbols are the same as for the continuous fatigue model. In order to examine the discrete form of the fatigue model, Eq. (16) was fitted to the experimental results with constant impact velocities. If the discrete model can predict particle strength changes with inconstant load amplitudes, then obviously it should predict particle strength after each impact cycle with constant impact velocities. Fig. 11 demonstrates (on one experimental set) the agreement of the discrete fatigue model (Eq. (16)) with the experimental data when the amplitude of the impact is kept constant. The interpretation of Fig. 11 is the same as of Fig. 9. Proposed model (Eq. (16)) was validated with experimental data of loading cycles with constant velocity in the same manner as it was described previously, i.e. by comparison chart.

V=6.5 m/sec, d0=3.675 mm

80

V=8.5 m/sec, d0=2.855 mm V=13 m/sec, d0=2.18 mm V=7.13 m/sec, d0=2.18 mm

60

V=10 m/sec, d0=2.18 mm V=6.2 m/sec, d0=2.855 mm

50

V=4.1 m/sec, d0=3.675 mm V=8.5 m/sec, d0=2.18 mm

40

9

10

Experimental data Fatigue Model-varying velocity Fatigue Model-avg. velocity

70

V=4 m/sec, d0=2.855 mm

70

8

Salt, d0=2.855mm, V= non-constant

V=13 m/sec, d0=2.855 mm

80

7

The difference between the continuous fatigue model implementation and the discrete one was found to be negligible. Accordingly, the discrete fatigue model is appropriate to describe the particle strength changes with constant velocity impacts. In order to examine the suitability of Eq. (16) with the varying impact velocity loads, additional experiments in which collision velocities were not kept constant were conducted. The results of these experiments are presented in Fig. 12. Fig. 12 consists of a separate data points which symbolize the experimental strength, solid lines which indicates varying velocity fatigue model, and dashed lines are the average velocity fatigue model. It is clear that the particle strength changes are inconsistent (Fig. 12). Moreover, the magnitude of the strength decrease is proportional to the impact velocity, i.e. as impact velocity is higher the decrease of the strength is steeper. Fig. 12 demonstrates the agreement between the discrete fatigue model and the experimental impact results. It also depicts the differences between the predicted particle strength with changing impact velocities and average one. Furthermore, the discrete fatigue model with varying velocities predicts experimental results with satisfactory accuracy, while the model with average velocity over cycles does not. Since in actual comminution processes the impact velocity range is wide, the average velocity cannot reflect all of these alterations. In the previous work of Kalman et al. [16] it was found that a different velocity at each impact results in a significantly different particle size distribution that has been resulted by using the same average velocity. Therefore, the average

±20%

V=8.56 m/sec, d0=3.675 mm

90

6

Fig. 11. Particle strength groups vs. impact number exp. data and discrete fatigue model.

Crushing Force, F [N]

Calculated Particle Strength, Fcalc [N]

100

5

Impact Cycle, n

Impact Cycle, n

30

Group 9

60

Group 8 Group 7

50

Group 6 Group 5

40

Group 4 Group 3

30

20 20

10

Group 10

Group 2

V1= 13 m/sec

Group 1

V2=6.3 m/sec V3= 10 m/sec V4=8.5 m/sec

0 0

10

20

30

40

50

60

70

80

90

Experimental Particle Strength, Fexp [N] Fig. 10. Comparison of the predicted and experimental particle strengths.

100

10 0

1

2

3

Impact Cycle, n Fig. 12. Fatigue model for the impact experiment with varying loads.

Y. Rozenblat et al. / Powder Technology 239 (2013) 199–207

value of the impact velocity between the cycles cannot be used to calculate particle strength changes if the loads at each cycle are different. Since it is only one fatigue experiment with varying impact velocities, additional impact experiments are required to fully validate the discrete fatigue model. 4. Conclusions The main purpose of this paper was to introduce a reliable, general and relatively simple fatigue model for particles. The proposed model is suitable for particle population strength changes under repeated impacts at either constant or varying impact velocities. The modified model depends on six material empirical parameters for Dead Sea salt particles. The validation of the model for different sizes showed a high accuracy in prediction of particle strength estimation. Since the model was tested only for one material, an extended experimental program with additional materials is needed in order to examine the proposed fatigue model generality. This model is a part of the comminution functions assemble whose purpose is to describe the breakage mechanism of particulate materials. A better understanding of the fatigue effect on particle strength will result in more accurate simulations of particle comminution systems, therefore making their design and optimization easier and more precise. Nomenclature Symbol

Description

Particle projection cross area Initial particle diameter Young's modulus of elasticity Breakage force or particle strength in term of force Normal compression force Average crushing force Calculated crushing force Equivalence force Experimental crushing force Normal compression force acting on single particle Original compression strength of particle Compression strength distribution after ′ number of impact loads h Planck constant, 6.626 × 10−34 index (i) Number of impacts k Boltzmann constant, 1.38 × 10−23 n Impact number P Compression stress P′ Fatigue compression strength of particles P* Dimensionless number indicating the ratio between the applied load and the initial compression strength P1, P2, P3, Afv, Bfv, Model parameters A, Aa, Bb, Cc Original compression strength of particles Pm Pr Normal compression stress q Stress concentration factor from the experiment R2 Coefficient of determination Rm Theoretical fracture strength of particle Sc Cumulative percentage of crushed particles for any crush force at any impact cycle Percentage of broken particles by impact SI Sc′ Appropriate cumulative percentage of crushed particles for any crush force within the initial crushing strength distribution SF Selection function — % of the initial particle population which was broken T Absolute temperature V Impact velocity Ψ Crystalline volume σm Original tensile strength of particles ω Compression loading frequency Ar do Es F Fr Favg Fcalc Feq Fexp Fr Fm F1′,F1″, F1‴

Units m2 m Pa N N N N N N N N N J·s – J/K – Pa Pa – – Pa Pa – Pa – – –

– K m/s m3 Pa s−1

207

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