A nonextensive statistical model of multiple particle breakage
Oscar Sotolongo-Costa, Luis Manuel Gaggero-Sager, Miguel Eduardo Mora-Ramos
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec A NONEXTENSIVE STATISTICAL MODEL OF MULTIPLEPARTICLE BREAKAGE
O. Sotolongo-Costa, L. M. Gaggero-Sager, M. E. Mora-Ramos
Facultad de Ciencias, Universidad Aut´onoma del Estado de Morelos,Av. Universidad 1001, CP 62209 Cuernavaca, Morelos, Mexico
A time-dependent statistical description of multiple particle breakage is presented.The approach combines the Tsallis non-extensive entropy with a fractal kinetic equa-tion for the time variation of the number of fragments. The obtained fragment sizedistribution function is tested by fitting some experimental reports.
Keywords : particle breakage; nonextensive statistics; time dependence
PACS numbers:
I. INTRODUCTION
The reduction of particle size can be achieved under many different conditions. In par-ticular, the multiple breakage of crystals occurs during distinct research and technologicalprocesses such as, for instance, milling or recirculation loops. It is, in fact, a very complexphenomenon in which the quantity known as fragment size distribution function (FSDF)is relevant. In consequence, the problem of finding the time-dependent FSDF (TDFSDF)corresponding to an event of solid multiple breakage, as a function of the main macroscopicmeasurable variables is of key interest in many areas.For instance, there are experimental reports on breakage rates [1], and on size distributionduring particle abrasion [2, 3]. The usual theoretical approach can be found, in a muchcomplete form, in the work by Hill and Ng [4], and more recently in the study published byYamamoto et al. [5].The aim of the present work is to provide an alternative way to determine the TDFSDFin multiple breakage processes as a function of the relevant macroscopic variables. It isbased on the use of a nonextensive statistical description, as it has been previously donein modeling the problems of fragmentation [6, 7], cluster formation [8] and particle sizedistribution [9]. The original report on this particular version of the statistics is due toTsallis [10–12], who postulated a generalized form of the entropy that among other thingsintends to account for the frequent appearance of power-law phenomena in nature.On the other hand, the time evolution of the fragment distribution is presented as obey-ing a fractal-like kinetics [13]. Then, the combination of the nonextensive statistics andthe fractal kinetics will allow to derive the expressions for the volume and characteristiclength of fragments in breakage events. The paper is organized in such a way that the nextsection contains a detailed derivation of the distribution functions. Then, the section III isdevoted to present and discuss the application of these functions in order to fit with availableexperimental data and, finally the section IV is devoted to the conclusions.
II. STATISTICAL MODEL
If progressive particle fragmentation takes place within a liquid environment, then quan-tities such as shear rate and viscosity can be some of the macroscopic variables mentionedabove. On the other hand, since the multiple particle breakage involves the effect of iner-tial impact with container walls or with another particles, it is possible to assume that thefragment mass and the shear rate must appear in the distribution function. The effect ofattrition also determines the fragment size, so we also need to take into account the influenceof viscosity. Besides, both inertial impact and viscosity depend of fragment concentration.All these variables appear in [1] as the main factors governing the FSDF.The model considers the following quantities • m → mass of the fragments. • ˙ γ = ∇ v → shear rate (gradient of the velocity) • η → liquid viscosity. • n → number of fragments per unit volume.If the basic dimensions entering this problem are mass ( M ), length ( L ), and time ( T )(all positive, as noticed); then, according to the Vaschy-Buckingham theorem [14], the lawrelating all these variables can be transformed to one which will include only one dimen-sionless variable. This provides a method for computing sets of dimensionless parametersfor the given variables even if the form of the equation is still unknown. } Let us look at this statement core closely.The dimensions involved in the variables abovelisted are: m → M ; ˙ γ → T − ; η → M L − T − ; n → L − . Accordingly, a singledimensionless quantity defined from them could be ξ = m ˙ γn / η ; (1)Thus, the problem of finding the distribution of volume (mass), or FSDF, can be formu-lated as the derivation of the distribution function of the dimensionless variable ξ . This canbe accomplished using basic Physics principles such as the Second Law of Thermodynamics,which is nothing but a maximization of the system’s entropy.However, the breakage is a phenomenon with long-range correlation among different partsof the system, and the use of the Boltzmann-Gibbs (BG) entropy is not suitable in this case.When long-range correlation is relevant, it turns out that the use of the so-called Tsallisentropy [2] reveals to be convenient. In its continuous version, the form of this entropy (inunits of the Boltzmann constant) is: S q = 1 − R p q ( ξ ) dξq − . (2)In this expression, p is the probability density function. The quantity q is known as the”degree of non-extensivity” and, in principle, can take any real value. With the use of theL’Hˆopital rule, it is possible to verify that, under the normalization condition Z p ( ξ ) dξ = 1 , the limit when q → S q must include some constraints. One of them is, precisely,the normalization condition Z A p ( ξ ) dξ = 1 , (3)which is usual in the analysis of the BG case. In the integration, the upper limit A is themaximal value of the dimensionless variable ξ , which corresponds to the maximal value ofthe mass of the fragments, M , under stationary conditions for the remaining parameters.In other words, A = M ˙ γn / η .The second constraint is not that usual. In this case, it is customary to impose thefiniteness of the so-called q -mean value, also named as the first-order q -moment: hh ξ ii q = Z A ξ p q ( ξ ) dξ, (4)and, then, the constrain condition should read hh ξ ii q = µ, / | µ | < ∞ . (5)The same limits of integration considered in (3)-(4) apply for the integral in the equation(2). Actually, integration limits should include a minimal fragment size that correspondsto the situation when the breakage process cannot yield fragments of smaller dimensions.However, in order to simplify the treatment, our approach sets the lower integration limitas zero.Under the constrains mentioned, the problem of finding the maximum of the entropy isno other than a Lagrange multipliers one. So, we define the Lagrange functional L [ p ] = S q + α (cid:18)Z A p ( ξ ) dξ − (cid:19) + β (cid:18)Z A ξ p q ( ξ ) dξ − µ (cid:19) , (6)and demand the fulfillment of δLδp = 0The result for the probability density function has the form p ( ξ ) = α q − (cid:18) qq − (cid:19) − q − [1 − β ( q − ξ ] − q − . (7)Once the Lagrange multipliers α and β are properly determined, the final expression for p ( ξ ) is: p ( ξ ) = (cid:18) − qµ (cid:19) − q " − q − − q (cid:18) − qµ (cid:19) − q ξ − q − ; (8) p ( ξ ) = (cid:18) − qµ (cid:19) − q (cid:20) − ξA (cid:21) − q − . (9) A. The Kinetic Approach
During the process of continuous breaking, the concentration of fragments of a givensize varies. Therefore, we must consider ξ = m ˙ γn / /η as a time-dependent variable. Thisdependence can be considered as if the fragments were a given species originated during theprocess. Therefore, the problem will be to determine the kinetics of the fragments.In a complex system, a very general kinetic equation for a given species can be posed inthe form of a ”fractal” differential equation [13]: dNdt ν = κN, (10)where κ is the reacting coefficient and ν is a fractional time index. The solution of theequation (10) is N = N e κt ν , (11)and defines a kind of ”Weibull kinetics”, and is a result of our conjecture about the variationof the total number of fragments. If we name V as the total volume of the system, then thedensity of the fragments of the N species is n = NV . (12)If we set n as the initial concentration, ρ the crystal density, and τ the volume of thecrystal fragment, it is possible to write ξ = ρτ ˙ γn η / , (13)and, in correspondence, the time evolution of our dimensionless variable will be -accordingto (11): ξ ( t ) = ρτ ˙ γn / e at ν η (14)Within this context, the time-dependent probability density distribution function for thevolume of the fragments can be written as: p ( τ ) = Ω e at ν [1 − Bτ e at ν ] c ; (15)where Ω = (cid:18) − qµ (cid:19) − q ρ ˙ γn η / ; (16) B = q − − q (cid:18) − qµ (cid:19) − q ρ ˙ γn η / ; (17) c = 11 − q . (18)It is possible to notice that, written in the form (14), the time-dependent probabilitydensity function explicitly depends on the macroscopic measurable variables of the system.On the other hand, the integration of (14) over the crystal volume, from zero to Γ, definesthe fraction of fragments with a volume size smaller that Γ, in the system: F (Γ) = Z Γ0 p ( τ ) dτ. (19)If the volume of the largest fragment, Γ max , is taken as the unity, one may clearly seethat F (Γ max ) = 1; because any particle in the system will have a volume smaller than themaximal one. This provides a condition for the normalization of the distribution F (1) = 1,so one obtains F ( τ ) = 1 − [1 − Bτ e at ν ] − q − q − [1 − Be at ν ] − q − q (20)The distribution (20) gives the fraction of the crystal particles that, at the time t , havea volume smaller or equal to τ .Now, if we look for a distribution in terms of the particle’s characteristic length, l , insteadof volume, we state the cubic dependence of the particle volume with l as τ = σl / dτ = σl dl . Then, p ( l, t ) = σ Ω l e κt ν [1 − σBl e κt ν ] − q . (21)Again, the length distribution function is defined as the integral of p ( l ) from zero to agiven l . A procedure analogous to that previously described leads to the normalized lengthdistribution H Μ m L p H l LH n o r m a li ze d L FIG. 1: (color online) Available data on particle size length for phosphate minerals as determinedby a commercial analyzer [15] (dots), and best static fit obtained with the use of Eqn. (28) (solidline). f ( l, t ) = 1 − [1 − σBl e κt ν ] − q − q − [1 − (cid:0) σ (cid:1) Be κt ν ] − q − q . (22) III. RESULTS AND DISCUSSION
In order to test our model we fit the results of the particle diameter distribution ofphosphate minerals reported as an outcome of a commercial particle size analyzer (see Ref.[15]). Since the experimental information is not time-depending, we chose a stationary fittingprocedure that uses the expression derived for the -normalized- distribution density in sucha way that the proposed expression will contain three adjusting parameters, { r, s, q } . Thechoice of r and s is due the presence of such unknown quantities as µ , σ , and the time-relatedexponents in the expression. Then; p ( l ) = r l (2 − q ) − q (cid:20) − s l (2 − q ) − q (cid:18) q − − q (cid:19)(cid:21) − q . (23)The normalized data and the resulting curve appear in the Fig. 1. We used the WolframMathematica ”NonLinearModelFit” package with the Finite Difference Gradient Methodand the sole restrictions r > s < < q <
2. The best fit parameters obtained are: r = 1 . s = − .
01, and q = 1 .
78, with the goodness of the fit determined by the parameter R with a value of 0 . q = 1 .
97 reported by Calboreanuet al. [9].Conti and Nienow [3] reported on the abrasion experiments in a solution with a solidphase of nickel-ammonium sulphate hexa-hydrate crystals. The procedure included stopingthe agitation at certain intervals. Then, the abrasion fragments were separated from theliquid to determine the total mass abraded and to measure size distribution. The countedparticles were grouped into a number of size ranges in order to give a clearer representationof the changes of size distribution with time. We have, for instance: 0 − µ m (range 1);36 − µ m (range 2); 73 − µ m (range 3). Assuming a constant shape factor, the totalparticle mass in each of the ranges was calculated (see both Table 1 and Figure 1 in Ref.[3]).We use the kinetic model proposed in this work for fitting the mentioned results in Ref.[3] of the time-dependent distribution of abraded particle mass (size). The expression usedto fit the different data sections is derived from (23) considering that the exact values ofthe particle size in each data point are not known and will be considered as parts of theadjustment. This leads to consider the exponents in the time-evolution law (11) as new fittingparameters, { u, v } , through the substitutions r l → r exp ( u t v ) and s l → s exp ( u t v );considering u and v as another two fitting parameters; p ( t ) = r (2 − q ) − q e u t v (cid:20) − s (2 − q ) − q (cid:18) q − − q (cid:19) e u t v (cid:21) − q . (24)The figure 2 contains the results of the fitting of the total mass of the fragments (curveand dots in black) as well as the fitting of the data corresponding to the above mentionedthree first particle size (mass) ranges appearing in Table 1 of Conti and Nienow report (inblue, red, and purple color, respectively). The fitting was carried out with the same package æ æ æ æ æ æ æà à à à à à à ò ò ò ò ò ò òì ì ì ì ì ì ì H hours L M a ss A b s o r b e d H - K g L FIG. 2: (color online) Available data on particle mass (size range) for nickel-ammonium sulphatehexa-hydrate crystals [3] as functions of abrasion time (dots); and best kinetic fit obtained withthe use of Eqn. 23, modified along the comments in the text (solid lines). Black curve (diamonds)corresponds to the measured total mass of the fragments. Mass distribution of particles withinrange 1 (blue); range 2 (red); range 3 (purple). above referred. The conditions for the fitting procedure were set as: r > s < t > u >
0, and 1 < q <
2. In all cases thecharacteristic fitting index R ranked above 0 . r = 3 . s = − . q = 1 . t = 1 . u = 0 .
26. First size range (blue,circles): r = 1 . s = − . q = 1 . t = 0 . u = 0 .
57. Second range (red, squares): r = 2 . s = − . q = 1 . t = 2 . u = 0 .
24. Third range (purple, triangles): r = 107 . s = − . q = 1 . t = 1 . u = 0 . ν -exponent here obtained constitute a validationof our hypothesis of fractal kinetics to describe the time evolution of the FSDF. On the0other hand, it is possible to notice that a good fitting can be achieved by demanding thatthe non-extensivity parameter be a non-integer with values between 1 and 2, as previouslyfound. One readily notices that, the highest values of q obtained are around 1 .
7, which isquite below the value of 2, and the value of 1 .
97 reported in Ref. [9].
IV. CONCLUSIONS
The use of a non-extensive statistical description is a correct choice for deriving thefragment size distribution function arising from processes of multiple breakage of crystalsin stirred vessels. This can be performed both under static and time-dependent conditions,with the introduction of a suitable fractal kinetics, and fractional power time evolution.The proposed distribution functions were tested by fitting available experimental data.We have shown that the statistics of multiple breakage phenomenon can be modeled via aTsallis entropy. On the other hand, we have obtained values of the magnitude of the fractaltime exponent within the range between 0 . . [1] C. A. Shook, D. B. Haas, W. H. W. Husband, and M. Small, Can. J. Chem. Engng. , 448(1978).[2] A. W. Nienow and R. Conti, Chem. Engng. Sci. , 1077 (1978).[3] R. Conti and A. W. Nienow, Chem. Engng. Sci. , 543 (1980).[4] P. J. Hill and K. M. Ng, AIChE Journal , 1600 (1996).[5] K. Yamamoto and Y. Yamazaki, Phys. Rev. E , 011145 (2012).[6] O. Sotolongo-Costa, A. H. Rodr´ıguez, G. J. Rodgers, Entropy , 172 (2000).[7] O. Sotolongo-Costa, A. H. Rodr´ıguez, G. J. Rodgers, Physica A , 638 (2000).[8] A. Calboreanu, Rom. J. Phys. , 67 (2005).[9] A. Calboreanu, E. Dimitriu, and R. Ramer, Rom. J. Phys. , 545 (2005).[10] C. Tsallis, J. Stat. Phys , 479 (1988).[11] C. Tsallis, Braz. J. Phys. , 337 (2009).[12] C. Tsallis, Introduction to Nonextensive Statistical Mechanics (Springer, 2009).[13] F. Brouers and O. Sotolongo-Costa, Physica A , 165 (2006). [14] E. Buckingham, Physical Review4