Energy and population balances in comminution process modelling based on the informational entropy

Powder Technology 167 (2006) 33 – 44 www.elsevier.com/locate/powtec

Energy and population balances in comminution process modelling based on the informational entropy Henryk Otwinowski ⁎ Department of Boilers and Thermodynamics, Faculty of Mechanical Engineering and Informatics, Czestochowa University of Technology, al. Armii Krajowej 19C, PL-42 218 Czestochowa, Poland Received 30 March 2005; received in revised form 4 April 2006 Available online 6 June 2006

Abstract The results of theoretical and experimental studies of a comminution process are presented. There are two random functions: the selection function and the breakage function in the stochastic model based on a population balance. This model enables prediction of particle size distributions of comminution products after determination of both random functions. Maximum entropy method is used in the entropy model for determining the breakage function. Two cases are analysed, based on continuous and discrete particle size distribution functions of the fed material. Apart from mass balance, the energy balance of comminution process is also used. Searched form of breakage function is determined with the application of methodology of calculus of variations. The results of experimental identification of both models are presented. The parameters that occur in the discrete form of the selection and breakage functions were the identification objects. The results of experimental investigations of quartz sand single comminution in a laboratory jet mill provided an identification base. The experimentally identified results of the entropy model confirmed the adequacy of the theoretical analysis and demonstrated the possibility of adequate prediction of particle size distributions resulting from single comminution. © 2006 Elsevier B.V. All rights reserved. Keywords: Comminution modelling; Particle size distribution; Energy balance; Population balance; Breakage functions; Informational entropy

1. Introduction

2. A review of selected methods of comminution modelling

The rapid development of the chemical, cement and mining industries as well as the power industry has focused attention on the problem of the efficiency and energy-consumption of comminution. The mechanism of comminution and the various systems of forces and stresses present in the particles of any comminuted material are very complicated. As a result, there is currently no universal theory of comminution. The extensive literature dealing with this problem presents results of research and measurements. The formulation and modification of adequate mathematical models of the phenomena occurring during the comminution process is the best way of gaining an understanding of comminution.

Technical development has made the application of highly comminuted materials increasingly necessary. The comminution of brittle solids is a highly energy-consuming process. So, in the second half of the 19th century, the first theories aimed at establishing the relationship between solid comminution effect and energy used during comminution were formed. These theories, also called hypotheses, are known by the surnames of their authors, such as: Rittinger [1], Kick [2–4], Bond [5–7] and Charles [8–10]. An alternative formulation of the energy hypotheses was proposed by Djingheuzian [11–13]. Djingheuzian's hypothesis, known as the thermodynamical theory of comminution, was developed by Guillot [14] and Mielczarek [15,16]. The main task of research into loose material comminution is to determine general laws concerning the evolution of the particle size distribution during comminution. Mathematical models predicting the particle size distribution of the comminution product

⁎ Tel.: +48 34 3250583; fax: +48 34 3250579. E-mail address: [email protected]. 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.05.011

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H. Otwinowski / Powder Technology 167 (2006) 33–44

are called stochastic or statistic models. The models, based on the population balance, are numerous and represented by a highly developed group of stochastic models. Two probability functions, namely breakage and selection, are introduced in stochastic modelling. Epstein [17], Sedlatschek and Bass [18], Broadbent and Callcott [19], Rumpf [20,21], and Gardner and Austin [22,23] are regarded as the creators of the stochastic model of comminution. The equation of the population balance is a linear differential– integral equation. Many ways of solving the population balance equation have been formed, especially over the last few years. Gaudin and Meloy [24] obtained one of the first solutions. Analytical solutions exist only for specific forms of integrands (McGrady and Ziff [25]). The latest method provides an analytical solution of the equation for an optional breakage function (Liou et al. [26]). Applying discretization methods to the balance equation is an important approach in numerical solutions (Hill and Ng [27], Kumar and Ramkrishna [28], Vanni [29], Campbell and Webb [30,31]). This has lead to the development of numerical methods over the last few years (Attarakih et al. [32], Liu and Tadé [33], Wang et al. [34], Bove et al. [35], Kostoglou and Karabelas [36], Attarakih et al. [37], Bilgili et al. [38], Sommer et al. [39]). An analysis of methods to solve the population balance equation was carried out by Kostoglou and Karabelas [40,41], Nopens et al. [42]. In the stochastic modelling of comminution the following methods are also applied: statistic moment method (Heim and Pietrzykowski [43], Heim and Olejnik [44], Kostoglou and Karabelas [40,41], Sommer et al. [39]), Maxwell–Boltzmann distribution (Mielczarek and Urbaniak [45], Pastucha et al. [46]) and Monte Carlo method (Lee and Matsoukas [47], Mishra [48], Zhao Haibo et al. [49]). The review of selected methods of comminution modelling, presented in this section, shows the importance of the subject of comminution modelling. It should also be added that the modelling methods could be divided into two main groups. The dependence between energy and comminution effect is analysed in deterministic models, while stochastic models deal with the evolution of the particle size distribution of the comminuted substance. There is a lack of universal models of comminution which take into account the energy balance and mass balance of the particle population simultaneously (Zhukov et al. [50]). 3. The population balance The modelling of the comminution process presented in this paper is similar to the inverse breakage problem — a case where the selection function is known and the breakage function is sought (Kostoglou and Karabelas [51]). The random functions, selection S( y) and breakage B(x,y), are present in the stochastic model of single comminution, which is based on the population balance. The selection function S( y) determines the comminution probability of a single particle with size y at a single loading. This function depends on the load parameters and the type of material. The function S( y) represents a mass fraction of the comminuted particles of a fed material for a single comminution of monofraction. The selection function can also be interpreted

as the rate of comminution of the particles with size y, in other words this function is a measure of mill productivity and of the material's susceptibility to comminution. The breakage function B(x,y) is a dimensionless function of two variables and determines the mass fraction of particles smaller than x created after the single comminution of particles with size y. In the theory of probability, it is the cumulative mass distribution of particles smaller than x of the product of a single comminution of particles of size y. As B(x,y) function concerns only comminuted particles, it can be defined as a conditional cumulative distribution. Function b(x, y) determines the mass fraction of comminuted particles of an elementary size class of the fed material (y,y + dy), which entered into the elementary class of the product (x,x + dx). This function represents the probability of occurrence of particles from comminuted fraction of the fed material (y,y + dy) in the size class of the product (x,x + dx). In the theory of probability, b(x,y) is a conditional density of probability of the mass distribution of particles from the product with sizes from the range (x,x + dx), formed as a result of the comminution of particles of the fed material with sizes from the range (y,y + dy). This function represents the probability of transition from y to x state. The breakage function depends on the physical and mechanical properties of the particles of the fed material. The size parameter x (and y, too) is usually taken as a linear dimension, such as an equivalent sphere diameter (Frances [52], Bilgili and Scarlett [53]). Particle volume can also be chosen as the size parameter, which makes the particle change mechanisms more apparent for physical interpretation (Kostoglou and Karabelas [51]). The (0,ymax〉 interval, where ymax = xmax, is a domain of the selection function. This means that the maximum size of particles of product xmax at the utmost can be equal to the maximum size of particles of the fed material ymax. While this is generally true, there are situations where the size of the feed and of the output material are measured in different ways that are not directly comparable (Fang and Campbell [54,55]). The breakage function is the function of two variables; a square field is its domain theoretically. The square field can be written as follows:
0 < x V xmax ð1Þ 0 < y V ymax ¼ xmax If the size of the output and feed material are measured in the same way, the domain of B(x,y) is a field of the following triangle:
0 < x V xmax ð2Þ x < y V ymax ¼ xmax B(x,y) and b(x,y) functions are connected by the relationship Z x bðx; yÞdx ð3Þ Bðx; yÞ ¼ 0

From the definition of the breakage function, the above relationship fulfils the following relationships: Z x bðx; y ¼ xÞd x ¼ 1 ð4Þ 0

H. Otwinowski / Powder Technology 167 (2006) 33–44

Bðx ¼ y; yÞ ¼ 1

ð5Þ

and condition of normalization Z y bðx; yÞdx ¼ 1

ð6Þ

0

The selection and breakage functions take the following values: 0 V SðyÞV1

for 0 < y V ymax

ð7Þ

for 0 < x V y

ð8Þ

0 V Bðx; yÞV1

0 V bðx; yÞ < þl

for 0 < x V y

ð9Þ

Bðx; yÞ ¼ 1

for x z y

ð10Þ

bðx; yÞ ¼ 0

for x z y

ð11Þ

The population balance of the fed material particles and comminution product for the (x,x + dx) class takes the form Z xmax fp ðxÞdx ¼ ½1−SðxÞ fs ðxÞdx þ dx bðx; yÞSð yÞ fs ð yÞdy x

ð12Þ where: fp(x)dx — normalized mass of particles with sizes from (x,x + dx) range in the product; fs( y)dy — normalized mass of particles with sizes from ( y,y + dy) range in fed material; S( y)fs ( y)dy — normalized mass of particles of fed material which underwent breakage with sizes from ( y,y + dy) range; b(x,y)dx S ( y)fs( y)dy — normalized mass of particles with (x,x + dx) sizes originating broken particles of fed material with ( y,y + dy) R yfrom max ¼xmax sizes; dx y¼x bðx; yÞSð yÞfs ðyÞdy — normalized mass of particles with (x,x + dx) sizes originating from all broken particles of fed material with sizes from x ≤ y ≤ ymax = xmax range; [1 − S(x)] fs(x)dx — normalized mass of unbroken particles of fed material which passed into the comminution product with sizes from (x, x + dx) range. After integrating Eq. (12) with respect to x one can obtain an equivalent equation for the cumulative particle size distribution of the comminution product Z xmax Fp ðxÞ ¼ Fs ðxÞ þ Bðx; yÞSð yÞ fs ð yÞdy ð13Þ x

where: Fs(x) — cumulated mass fraction of unbroken particles of fed material smaller than x, which are found in the product; B(x,y)S( y)fs( y)dy — cumulated mass fraction of particles smaller than x, originating from broken R x particles of fed material with sizes from ( y,y + dy) range; x max Bðx; yÞSðyÞfs ð yÞdy — cumulated mass fraction of particles smaller than x, originating

35

from all broken particles of fed material with sizes from x ≤ y ≤ ymax = xmax range. It is possible to predict the particle size distribution of the comminution product on the basis of Eq. (12) after determining the selection and breakage functions. It is possible to determine these functions in the following ways: – single comminution of samples of the narrow fraction (Austin and Bhatia [56]), – single comminution of samples containing marked fraction (Gardner and Sukanjanajtee [57], Tavares and King [58], Lebedev [59]), – single comminution of samples containing different masses of selected single fraction (for example: lack of class or double mass) (Liu and Schönert [60]), – numerical optimization and approximation on the basis of results of experimental research (Williams et al. [61], Berthiaux et al. [62], Das [63]). None of the above methods is universal. In many cases it is difficult to find an adequate marker for a single fraction. It is difficult, sometimes even impossible, to obtain samples of the narrow fraction for small sizes of particles by screening. 4. Informational entropy If polydispersed material is represented as a set of size classes, then the process of comminution relies on the transition of particles from one class to the other. This transition can proceed in different ways. The formation of the particle size distribution of a single comminution product is a random process as depicted by the theory of probability. In statistical physics the concept of entropy is strictly connected with the theory of information. According to Shannon [64], entropy is a function which measures the uncertainty of event realization. If the event is certain, then information about it is maximal and entropy is equal to zero. The higher the value of the entropy function, the higher the uncertainty of event realization and so the lower information content. In real systems the event proceeds in such a way that the lowest information content is generated: entropy increases with a decrease in information about the system. If the probability distribution is given for a continuous quantity x with p(x) density, the informational entropy equals Z þl h½ pðxÞ ¼ − pðxÞln pðxÞd x ð14Þ −l

with the fulfillment of the following normalization condition Z þl pðxÞd x ¼ 1 ð15Þ −l

The above definition of entropy is known as Shannon's definition. Informational entropy takes the following values h½ pðxÞ ¼ 0

⇔

pðxÞ ¼ 0

ð16Þ

36

H. Otwinowski / Powder Technology 167 (2006) 33–44

under the assumption that limpðxÞY0 pðxÞln pðxÞ ¼ 0 and h½ pðxÞ > 0

for

pðxÞ > 0

5. Energy balance ð17Þ

If pk denotes the discrete distribution of probabilities for n events, the informational entropy H is determined by the following equation H ¼−

n X

ð18Þ

pk ln pk

Let us assume that the energy supplied by the comminution device is transferred to all the particles of a fed material irrespective of whether they underwent comminution or not. The density of the energy absorbed by a unit of a normalized mass of particles with y size is denoted by e( y). The elementary energy taken by the mass of an elementary class ( y,y + dy) equals dEð yÞ ¼ eð yÞ fs ðyÞdy

ð22Þ

Energy taken (0,y) class equals Z y eð yÞ fs ð yÞdy Eð yÞ ¼

ð23Þ

k¼1

preserving normalization condition n X

pk ¼ 1

ð19Þ

k¼1

Quantity of informational entropy H equals zero, if any of pk probabilities equals one while all the rest equal zero. This means that the results of tests can be reliably predicted and there is no uncertainty in information. H quantity takes the maximum value when all pk are equal to each other and amount to pk ¼

1 n

ð20Þ

This border case is characterised by the maximum uncertainty and simultaneously by the maximum probability. Informational entropy H is an additive quantity in relation to the set of statistically independent events. On the basis of the Shannon's theory of information, Jaynes introduced the maximum entropy method (MEM) into statistical mechanics [65–67]. The maximum entropy method is a generalised tool for creating the pE(x) function, which approximates the unknown probability distribution p(x). This distribution describes the given physical system when experimental information is insufficient. Commonly one accessible information about system is given in the form of mean value M of certain substantial Ar(x) quantities Z hAr ðxÞiu

þl

−l

0

Total energy E0 taken by the particles of whole fed material equals Z ymax E0 ¼ Eð ymax Þ ¼ eð yÞ fs ð yÞdy ð24Þ 0

Two parts can be distinguished in an elementary mass of (y, y + dy) class: – the part comminuted with the normalized mass S( y)fs( y)dy, – the part non-comminuted with the normalized mass [1-S( y)] fs( y)dy. It is also assumed that absorbed energy is proportional to partial masses: – energy absorbed by the comminuted particles dE1 ð yÞ ¼ eð yÞSð yÞ fs ð yÞdy

ð25Þ

– energy absorbed by non-comminuted particles dE2 ð yÞ ¼ eð yÞ½1−Sð yÞ fs ð yÞdy

ð26Þ

Energy balance takes the form Ar ðxÞpðxÞdx

r ¼ 1; 2; …; M

ð21Þ

dEð yÞ ¼ dE1 ð yÞ þ dE2 ð yÞ

ð27Þ

and adequately: According to Jaynes, the distribution closest to the actual distribution, is the one which maximalizes Shannon's informational entropy as given by Eq. (14) and fulfilling the normalization condition (15) and the conditions of the known mean values (21). The advantage of MEM is that it can take into account many additional conditions, which may have physical sense and can satisfy the laws of physics. MEM allows us to find the distribution of the random function which meets the above conditions and which is closest to the actual distribution. The maximum entropy method for comminution can be formulated in the following way (Otwinowski [68]): from the all possible forms of the breakage density function, fulfilling the mass and energy balances, that form which gives the maximum value of the informational entropy, will be the most probable.

Eð yÞ ¼ E1 ð yÞ þ E2 ð yÞ

ð28Þ

E0 ¼ E01 þ E02

ð29Þ

Let e(x,y) denote the specific energy of a transition of the fed material particles from an elementary class ( y,y + dy) into the product class (x, x + dx). By e(x,y) one should understand the density of energy used to comminute a unit of a normalized mass of particles with y to x size. An elementary mass of particles of (y,y + dy) class undergoing comminution is equal to S(y)fs( y)dy. Part of this mass, equal to b(x,y)dxS( y)fs(y)dy, lands in (x, x + dx) class of product, where x < y. Hence, the second

H. Otwinowski / Powder Technology 167 (2006) 33–44

relationship for dE1(y) energy, absorbed by the comminuted particles of fed material, arises Z

y

dE1 ð yÞ ¼ Sð yÞ fs ð yÞdy

eðx; yÞbðx; yÞd x

ð30Þ

0

By comparing relationships (25) and (30), the equation connecting e( y) and e(x,y) densities is obtained (Bodziony [69]) Z

y

eð yÞ ¼

ð31Þ

eðx; yÞbðx; yÞdx 0

37

The breakage function b(x,y) from the above equation can be determined by applying the maximum informational entropy model from the following system of equations: – equation of the informational entropy Z ymax Z y bðx; yÞln bðx; yÞdxdy Z max h½bðx; yÞ ¼ − 0

ð36Þ

0

– equation of normalization Z y bðx; yÞdx ¼ 1

ð37Þ

0

Substituting Eq. (31) into Eqs. (22)–(24) leads to the following energy relationships: – energy absorbed by an elementary class ( y,y + dy) Z

y

dEð yÞ ¼ fs ð yÞdy

eðx; yÞbðx; yÞdx

ð32Þ

– energy absorbed by (0,y) class Z

y

Z fs ðyÞdy

0

y

eðx; yÞbðx; yÞdx

ð33Þ

0

– energy absorbed by the particles of whole fed material Z Eð ymax Þ ¼ E0 ¼

ymax

Z fs ð yÞdy

y

eðx; yÞbðx; yÞdx

ð34Þ

0

0

6. Entropy model of comminution The functions of particle size distribution, present in the entropy model of comminution, can be written in continuous or discrete forms. Owing to this, both forms of the functions were applied to the entropy model of single comminution.

ð38Þ

0

0

0

EðyÞ ¼

– balance of the total absorbed energy E0 Z ymax Z y bðx; yÞeðx; yÞ fs ð yÞdxdy ¼ E0

Variational calculus was used to solve the above system of equations (Forray [70]). A constrained extremum of the integral functional of the informational entropy from Eq. (36), fulfilling the additional conditions of integral type expressed by Eqs. (37) and (38), is sought in the considered case. The solution of the above variational problem for the unknown e( y) function is obtained via the Lagrange multipliers method, which is based on Lagrange–Brunacci theorem. The solution of the problem takes the following form (detailed transformations are presented in Appendix A) bðx; yÞ ¼ l1 ð yÞel2 eðx; yÞ fs ð yÞ

ð39Þ

where l1( y),l2 are Lagrange multipliers. Lagrange multiplier l1( y), present in Eq. (39), depends on y because the indirect condition is expressed by a single integral. Substitution of b(x,y) function into Eq. (37) leads to a relationship which contains Lagrange multiplier l1(y) Z y l1 ðyÞel2 eðx; yÞ fs ð yÞ dx ¼ 1 ð40Þ 0

6.1. Continuous model

Eq. (39), after taking into account Eq. (40), takes the form

The comminution of a fed material in the form of a polydispersed sample is examined in this section. The hypothesis of independent comminution of particular size classes (this means that there is no energy exchange between the size classes) has been taken into account in the considerations mentioned below. It arises from the assumed hypothesis that the effect of concurrent comminution of a system of size classes and the comminution of every class separately will be the same for the same energy absorbed. Eq. (12) expressing the density of the particle size distribution of a single comminution product can be written in the following form Z

xmax

fp ðxÞ ¼ ½1−SðxÞ fs ðxÞ þ x

el2 eðx; yÞ fs ð yÞ l2 eðx; yÞfs ð yÞ dx 0 e

bðx; yÞ ¼ R y

b(x,y) function from Eq. (39) can be used in relationship (38) for the total absorbed energy E0, obtaining Z

ð35Þ

ymax

Z

y

kðyÞel2 eðx; yÞ fs ð yÞ eðx; yÞfs ðyÞdx dy ¼ E0

ð42Þ

0

0

Because l1( y) multiplier is independent of x, it can be determined from Eq. (40) after moving it outside of the integral sign, and then after substituting it into Eq. (42) one can obtain Z

bðx; yÞSð yÞ fs ð yÞdy

ð41Þ

0

ymax

Ry 0

! el2 eðx; yÞ fs ð yÞ eðx; yÞfs ð yÞd x Ry dy ¼ E0 l2 eðx; yÞ fs ð yÞ d x 0 e

ð43Þ

38

H. Otwinowski / Powder Technology 167 (2006) 33–44

Expression (43) is a non-linear equation due to the unknown, constant Lagrange multiplier l2, which can be determined with the use of the numerical methods. 6.2. Discrete model

calculating the elements of breakage matrix B after discretization of Eq. (41) takes the form el2 eij fsj bij ¼ P m el2 eij fsj

ð48Þ

i¼jþ1

The numerical solutions of non-linear equations are used in the practical applications of the maximum entropy method. Therefore the discrete forms of the particle size distribution functions and matrix models of comminution are considered in this part of the paper. The equation of population balance of the particles (12) in the discrete form is represented by the following expression i−1 X fpi ¼ ð1−Si Þ fsi þ bij Sj fsj ð44Þ j¼1

where i — number of the size class of the product, j — number of the size class of the fed material, m — number of size classes and index m refers to the size class with the smallest size. It is assumed that the size classes are identical for the fed material and product, and the computation method for an average particle of the size classes is also identical. Eq. (44), written for m size classes, creates a system of m linear Cramer equations, which can be presented in matrix form f p ¼ ðI−SÞf s þ BSf s

ð45Þ

In Eq. (45) the individual matrix symbols denote: fs fp I S B

unicolumn matrix of the density of the particle size distribution of the fed material unicolumn matrix of the density of the particle size distribution of the comminution product unit scalar matrix diagonal matrix of selection strictly lower triangular matrix of breakage

The values of elements of bij matrix belong to 〈0,1〉 range and the sum of the elements in every column equals one m X

bij ¼ 1

for every

j ¼ 1; 2; …; m

ð46Þ

i¼jþ1

The elements of the B matrix of breakage determine the mass fraction bij of the comminuted particles of j class of the fed material, whose particles go into i class of the product. After distributing multiplication of matrix over addition, Eq. (45) takes the following form f p ¼ ½ðI−SÞ þ BSf s

ð47Þ

Analysing the above equation or Eq. (45) one must pay attention to the notation sequence of individual symbols in the matrix product, because in general the multiplication of matrixes is not commutative. The breakage function bij, present in Eq. (44), can be determined for every size class of the fed material using the maximum informational entropy method. The final formula for

where l2 is the Lagrange's multiplier and eij is the specific energy, applied to the transition of particles from j class of fed material to i class of the product. The consumed specific energy eij, present in Eq. (48), can be written in different forms depending on the assumed comminution hypothesis: – Rittinger's hypothesis   1 1 eij ¼ eij R ¼ c R − xi yj

ð49Þ

– Kick's hypothesis eij ¼ eijK ¼ cK ln

yj xi

ð50Þ

where cR, cK — constant factors which depend on material properties. The most important formal difference between Kick's and Rittinger's hypothesis is that Kick's hypothesis does not contain the absolute value of particle size: only the degree of size reduction. It also simplifies the conditions of comparison of different regimes of comminution. Rittinger's hypothesis gave better correlation for fine comminution and Kick's hypothesis for crushing. The particle size distribution of the comminution product can be determined for the given values bij from Eq. (45) assuming that the selection matrix S is known.

7. An experimental identification The experimental identification of the stochastic and entropy comminution model is aimed at the parameters occurring in the discrete forms of the breakage and selection functions. The identification was carried out on the basis of Eq. (44) for the population mass balance of particles in the discrete form. The task of estimating the unknown parameters of the random function distribution, based on the results of an experiment, is present in mathematical statistics. The statistical simulation is applied to the data analysis. It relies on the computer realization of the probabilistic mathematical model, physical phenomenon or technical object. In the narrower range the statistical simulation is often understood as a realization of the Monte Carlo method, which relies on the generation of random variables to estimate the parameters of their distribution (Nowak [71]). Assumptions and methodology of the conducted identification were as follows: – The particle size distribution of the fed material and single comminuted product are given.

H. Otwinowski / Powder Technology 167 (2006) 33–44

– The linear form of the experimentally determined selection function is assumed Sj ¼ ayj ð51Þ where a is a constant factor. – The breakage function in the stochastic model is specified by the8following experimental equation i < imin < 0; 1 bij ¼ ð52Þ : y −x ; imin V i V m j imin

– – –

– –

where imin is the number of the border class size, i.e. no particle of this class underwent fragmentation during a single comminution. The comminution tests on anthracite and coke were carried out in different types of mills: a ball mill, a roller mill and a jet mill to determine the selection and breakage functions (Otwinowski [68]). Eqs. (51) and (52) are the result of experimental data approximation. The breakage function in the model of entropy takes the form given by Eq. (48). The correctness of one of the comminution hypotheses: Rittinger's (Eq. (49)) or Kick's (Eq. (50)) is assumed. The following parameters are the subject of identification: in both models the constant a present in Eq. (51) of the selection function; additionally in the model of entropy: Lagrange's multiplier l2, present in Eq. (48) of the breakage function, and the constant of proportionality cR or cK, present in one of the comminution hypotheses, Eqs. (49) or (50). The parameters are estimated by applying the least squares method; the Monte Carlo method is applied to conduct the computer estimation. The numerical optimisation is carried out to achieve the best accordance between the calculated function – the cumulative distribution function of the product particle size distribution

39

– and the experimental dependence; this aim is realized by choosing sufficiently high values of the iteration index in numerical algorithm with the simultaneous assurance of convergence of the iterative process (Brandt [72]). The parametrical identification of the stochastic and entropy model was carried out on the basis of research on a single comminution in a laboratory jet mill with a flat metal barrier. A schematic of the experimental stand is presented in Figs. 1 and 2). The air m˙a1 from piston compressor (1) with flow rate 66 m3/h and maximum compression to 1.0 MPa was given through a reductive valve to Bendemann's nozzle (9) of the jet mill (3). The overpressure of air was changed in the 300– 900 kPa range. The thermodynamical parameters of working air: temperature and pressure were measured using thermometer (7) and springy manometer (8). Air expands to pressure about 100 kPa, what enables reaching air velocity in the nozzle mouth section comparable to sound velocity. The high velocity of air outflow caused subatmospheric pressure. Suction of particles of solid phase m˙s together with the stream of transporting air m˙a2 occurred in the intake chamber (10) and in the suction pipe (2) because of this pressure. In the intake chamber the stream of transporting air m˙a2 was connected to stream of working air m˙a1, giving stream of air m˙a. These streams were measured using rotameters (6). The two-phase mixture, created in the intake chamber, flew through the acceleration tube (11) and collided with the metal barrier (13) in the central part of milling chamber (12). After comminution the two-phase mixture m˙a + m˙p, flow into cyclone (4), where releasing of material m˙pl took place. The mixture of air containing the finest particles was sucked by the ventilating hood–industrial vacuum cleaner (5) with 510 m3 /h flow rate. The final separation of solid phase from air took place in a fabric filter of the vacuum cleaner. The geometric parameters

Fig. 1. Schematic of the experimental stand: 1—piston compressor; 2—suction pipe; 3—jet mill; 4—cyclone; 5—vacuum cleaner; 6—rotameter; 7—thermometer; 8—springy manometer; 9—Bendemann's nozzle; 10—intake chamber; 11—acceleration tube; 12—milling chamber; 13—metal barrier.

40

H. Otwinowski / Powder Technology 167 (2006) 33–44

Fig. 2. Schematic of the jet mill ejector.

of jet mill were as follows: nozzle diameter — 2.4 mm, diameter of acceleration tube — 10 mm, length of acceleration tube — 100 mm, diameter of milling chamber — 120 mm, distance between acceleration tube and barrier — 24 mm. After each conducted experiment of single comminution, a sample of material was taken to determine the particle size distribution of the comminuted product. The particle size distribution of the fed material and product were measured by an Infrared Particle Sizer manufactured by Kamika Instruments (www.kamika.com). Samples of a quartz sand 2 kg in mass, with different particle sizes were used in the research. Sauter's diameter da was taken to estimate the graining of the samples of the fed material. The Sauter's diameter of the samples of the

fed material was equal to: 255, 358 and 812 μm. Three values of working air overpressure, 300, 600 and 900 kPa, were applied to comminution of the fed material of da = 812 μm in diameter. The working air overpressure was constant and equal to 300 kPa for the remaining trials of comminution.

Fig. 3. The results of parametrical identification carried out in a laboratory jet mill with working air overpressure 300 kPa for the fed material with a Sauter's diameter da = 255 μm.

Fig. 4. The results of parametrical identification carried out in a laboratory jet mill with working air overpressure 300 kPa for the fed material with a Sauter's diameter da = 358 μm.

8. Results and discussion The results of identification carried out for the comminuted samples of the fed material with different particle sizes are presented in Figs. 3–7. The results of an experiment are marked by a dotted line, while the results of computation by a solid line. Rittinger's hypothesis was used to calculate the particle size

H. Otwinowski / Powder Technology 167 (2006) 33–44

41

Fig. 5. The results of parametrical identification carried out in a laboratory jet mill with working air overpressure 300 kPa for the fed material with a Sauter's diameter da = 812 μm.

Fig. 7. The results of parametrical identification carried out in a laboratory jet mill with working air overpressure 900 kPa for the fed material with a Sauter's diameter da = 812 μm.

distribution of the product as the most suitable for free, fine comminution. The values of the fitted parameters are presented in Table 1. The stochastic model differs from the experimental data for fine particles with sizes below 0.5 mm in every case. This may result from an insufficient number of identification parameters, it means from linear form of the breakage function. It can be seen in Figs. 5–7 that an increase in pressure leads to an increase in the accuracy of the stochastic model. The entropy model differs clearly from the experimental data only in Fig. 4. Moreover both models are relatively poor in this Figure. In this case an error probably has occurred at the stage of experiment. The values of almost all the parameters for both models are

highest (Table 1) for the highest pressure (Fig. 7). Rittinger's constant cR has an essentially constant value because all the experiments were conducted in the same mill using the same material. Analytical form of the breakage function is determined in the entropy model, so the shape of the estimated cumulative breakage functions B(x,y) for various feed classes is presented in Fig. 8. This shape is characteristic for fine materials (Ouchiyama et al. [73]). It can be seen in Fig. 8 that the form of the breakage function approaches the linear form with an increase in feed particle size, what means that the entropy model approaches the stochastic model. Many authors (Berthiaux and Dodds [74], Müller et al. [75], Shinohara et al. [76], Nikolov [77], Kotake et al. [78], Vogel and Peukert [79], Petukhov and Kalman [80]), using the stochastic model, apply different form of functions than the linear selection and breakage functions. The most common are the power functions introduced by Austin [81]. They are used to model the comminution process in various mills, such as ball mill (Yekeler et al. [82], Deniz [83]), circular fluid energy mill (Nair [84]), jet mill (Gommeren et al. [85]), stirred media mill (Wang and Forssberg [86]), hammer mill (Austin [87]) and millclassifier systems also (Kis et al. [88]).

Table 1 The values of the identification parameters for the stochastic and entropy models Feed Overpressure Values of parameters da, kPa Stochastic model Entropy model μm a, μm− 1 cR, J · m · kg− 1 l2, J− 1·kg a, μm− 1

Fig. 6. The results of parametrical identification carried out in a laboratory jet mill with working air overpressure 600 kPa for the fed material with a Sauter's diameter da = 812 μm.

255 358 812 812 812

300 300 300 600 900

0.0481 0.0306 0.0443 0.0502 0.0511

0.0329 0.0229 0.0382 0.0228 0.0409

480.72 480.23 481.25 480.94 481.43

− 0.8051 − 3.4852 − 7.7559 − 9.7898 − 1.4169

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H. Otwinowski / Powder Technology 167 (2006) 33–44

breakage function is an argument of the constrained extremum of an integral functional with two indirect conditions. The obtained form of the breakage function was applied to the equation of the population balance. This leads to the prediction of the particle size distribution of the single comminution product with an additional assumption concerning the form of the selection function. – The results of identification show that the entropy model enables the theoretical prediction of the variation of particle size distribution in ground material with a wide range of milling parameters. – Future work needs to be done to identify other forms of the selection function for different mills and materials. 10. Lists of symbols Fig. 8. The shape of the estimated cumulative breakage functions B(x,y) for various feed classes.

Sometimes the equation of the population balance occurs in modified form (Bertrand et al. [89]) fpi ¼

i X

pij fsj

ð53Þ

j¼1

where pij — the transition function, which is the probability that the particle undergoes a change from state j to state i. Authors of paper (Zhukov et al. [50]) used the informational entropy to determine the transition function. In this solution less parameters applied due to lack of the selection function in explicit form. The results of the experimental identification of the entropy model confirm the accuracy of the theoretical analysis and the possibility of a satisfactory prediction of the particle size distribution of the single comminution product. On the basis of the comparative analysis of the results of identification of both comminution models, carried out for the same experimental data, it can be stated that the entropy model provides a more accurate prediction of the particle size distribution of the single comminution product.

b(x,y), bij e(x,y), eij e( y), ej E0 fp(x), fpi fs( y), fsj S( y), Sj x,y

breakage function density of used energy density of absorbed energy total absorbed energy particle size distribution of product particle size distribution of fed material selection function linear size of product and feed particles

Acknowledgements The author is grateful to Prof. Jakub Bodziony from Polish Academy of Sciences in Cracow for scientific support in this work. Appendix A. Variational problem of an informational entropy It is assumed that the breakage function b(x,y) realizes the extremum of the informational entropy functional h[b(x,y)], given by Eq. (36) h½b ðx; yÞ ¼

RR

Fðx; y; b; bx ; by Þdxdy Z ymax Z y ¼− bðx; yÞlnbðx; yÞdxdy D

9. Conclusions Analysis of the results of the experiment, the theoretical analysis of stochastic model and the formulated entropy model as well as the parametrical identification carried out for both comminution models, allows the following conclusions: – The stochastic model, based on the population mass balance of the particles for linear forms of selection and breakage functions, adequately predicts the particle size distribution of a single comminution product. – Introducing the energy balance, applying the maximum informational entropy method and taking into consideration the normalized condition, enables us to formulate and solve the variational problem and determine the breakage function by Lagrange's multipliers method. The sought form of the

0

ðA1Þ

0

and fulfils two additional conditions: – mass balance Z y Z K1 ½bðx; yÞ ¼ G1 ðx; yÞdx ¼ 0

y

bðx; yÞdx ¼ 1

ðA2Þ

0

– energy balance Z K2 ½bðx; yÞ ¼ ¼

ymax

Z

y

Z0 ymax Z0 y 0

0

G2 ðx; yÞdxdy bðx; yÞeðx; yÞfs ðyÞdxdy ¼ E0

ðA3Þ

H. Otwinowski / Powder Technology 167 (2006) 33–44

This is the isepiphanic case of variational problems (Tatarkiewicz [90]). In this case the Lagrange–Brunacci equation takes the following form BF B BF B BF BG1 BG2 − þ l2i − ¼ l1i ðyÞ Bb Bx Bbx By Bby Bb Bb |fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl} ¼0

ðA4Þ

¼0

On the left side of the equals sign in Eq. (A4) the second and the third summands are going to zero because the partial derivatives of the generating function F with respect to bx and by are equal to zero. After further transformations of Eq. (A4) one can obtain: −

B½bðx; yÞlnbðx; yÞ Bbðx; yÞ ¼ l1i ðyÞ Bbðx; yÞ Bbðx; yÞ Bbðx; yÞeðx; yÞfs ðyÞ þ l2i Bbðx; yÞ

ðA5Þ

−½1 þ lnbðx; yÞ ¼ l1i ð yÞ þ l2i eðx; yÞfs ð yÞ

ðA6Þ

lnbðx; yÞ ¼ −1−l1i ð yÞ −l2i eðx; yÞ fs ð yÞ |fflfflfflfflfflffl{zfflfflfflfflfflffl} |{z}

ðA7Þ

ln l1 ð yÞ

l2

Finally Eq. (39) for the fields of extremals is obtained bðx; yÞ ¼ l1 ð yÞel2 eðx; yÞ fs ð yÞ

ðA8Þ

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