Addressing an inverse problem of classifier size distributions

This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Powder Technology 176 (2007) 123 – 129 www.elsevier.com/locate/powtec Short communication py Addressing an inverse problem of classifier size distributions B. Venkoba Rao co Engineering and Industrial Services, Tata Consultancy Services Limited, 1 Mangaldas Road, Pune 411 001, India Received 2 November 2006; received in revised form 21 January 2007; accepted 23 February 2007 Available online 3 March 2007 al Abstract pe rs on One input and two output stream classifiers are commercially employed for the classification of particles. A mass balance equation for a classifier suggests that the feed size distribution can be evaluated from measured product size distributions if and only if the flow split of the feed particles to one of the product streams is also known. Moreover, the mass balance equation used to reconcile measured size distributions indicates that flow split of solid particles is in turn a function of all the three size distributions and is then redundantly expressed over the mass fraction of particles retained in various discrete size classes. Therefore for an operating classifier under steady state, the so far recognized approaches fail to address the profile of feed size distribution from the knowledge of measured fine and coarse product size distributions alone. In the forward approach of estimation of product size distributions, the feed distribution is integrated with efficiency curve of the classifier. Thus as an inverse problem, the feed distribution and efficiency curve need to be identified from the measured product size distributions. This paper attempts to address this inverse problem when flow split of feed particles to product streams is not known. However the method considers additional information regarding the functional forms of the classifier distributions due to inadequacy of product distributions alone to address the inverse problem. © 2007 Elsevier B.V. All rights reserved. Keywords: Classifier; Inverse problem; Particle size distributions r's 1. Introduction Au th o A variety of size classifiers such as screens, cyclones, mechanical screw and rake classifiers and hydraulic classifiers are industrially employed for classification of particles. The size distribution of classifier is a crucial information that depicts the performance of classifier and influences the downstream operations or affects the product quality wherein it is used. Plant audits and reconciliation of the measured size distributions help in the assessment of the plant performance as well as suggest possibilities for improvement of the performance of the circuit and hence are essential to be carried out in process industries on a regular basis [1–4]. Many a times the design of the plant does not permit the sampling of all the streams during plant audit and a compromise on the data collection points has to be made depending on the availability of sampling point for each stream. The missed out information for various streams of the circuit during a plant audit is rebuilt when permissible by way of reconciliation of the measured data subjected to mass balance constraints [5]. Classification of particles is generally studied with regard to the attributes like size and/or density, and classification results into two or more product streams. In case of multiple product streams, classification can be viewed as a network of one input and two output separations occurring in series [6,7]. A mass balance of one input and two output stream separator is written as FðdÞ ¼ Ssplit U ðdÞ þ ð1−Ssplit ÞOðdÞ where Ssplit represents solid flow split of feed particles to coarse stream, which is also called total-efficiency of separation and F(d ), U(d ) and O(d ) respectively represent cumulative percent finer size distributions of feed, coarse and fine streams. For discrete size distributions obtained from sieve analyses, Eq. (1) can be written as f ðdi Þ ¼ Ssplit uðdi Þ þ ð1−Ssplit Þoðdi Þ E-mail address: [email protected]. 0032-5910/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2007.02.028 ð1Þ ð2Þ where di denotes a representative size of the i-th size class of particles and f(di), u(di) and o(di) respectively represent the 124 B.V. Rao / Powder Technology 176 (2007) 123–129 mass fraction of particles retained in the i-th size class of feed, coarse and fine streams. Rewriting Eq. (2) in the following form expresses Ssplit redundantly over the size classes and this information is used in reconciliation of error prone measured data so as to make the distributions consistent with Eq. (2) [7]. f ðdi Þ−oðdi Þ uðdi Þ−oðdi Þ ð3Þ py Ssplit ¼ uðdi Þ f ðdi Þ th o r's pe rs Eqs. (1) and (2) suggest that feed size distribution can be obtained from the product size distributions only when the solid flow split of feed particles to one of the product streams, say to the coarse stream, Ssplit, is known. It appears from Eqs. (1) and (2) that there are infinitely many feed distributions that can produce, same coarse and fine distributions depending on the value of Ssplit. Therefore based on the chosen value of Ssplit to evaluate feed distribution, there exist a corresponding efficiency curve that varies between by-pass fraction of the classifier and 1, which when coupled with feed distribution gives the two product size distributions (refer Eqs. (7) and (8)). For example, Fig. 1 shows that the two product size distributions namely fine and coarse distributions can be obtained by respectively combining feed distributions (i), (ii) and (iii) in Fig. 1(a) with efficiency curves (i), (ii) and (iii) in Fig. 1(b). This non-unique solution makes this problem an ill posed inverse problem and the question is to identify the proper feed distribution and efficiency curve that gave the measured product size distributions for an operating classifier in the absence of measured Ssplit value. Since the measured data of product distributions are alone insufficient to address this problem, an additional hypothesis regarding the functional forms of the product distributions that are obtained by combining restricted functional forms of feed and actual efficiency curve are invoked. al ð4Þ on Ea ðdi Þ ¼ Ssplit co The actual separation efficiency also referred as grade efficiency of separation [8] can be calculated from the measured discrete mass fractions from the following expression. These efficiency factors are called partition coefficients for specified sizes. Au Plitt [9] proposed an actual efficiency curve of the following form that gives probability of feed of given size, d, reporting to the coarse stream. ð5Þ where the corrected efficiency curve, Ec(d), is given by

  d m Ec ðdÞ ¼ 1−exp −lnð2Þ d50c have been derived from the knowledge of feed and efficiency curve [10] by considering Z d ð1−Ea ðdÞÞf ðdÞdd 0 ð7Þ OðdÞ ¼ 100 dmax ð1−Ea ðdÞÞf ðdÞdd R 0 and U ðdÞ ¼ 100 2. Mathematical treatment Ea ðdÞ ¼ ð1−Rf ÞEc ðdÞ þ Rf Fig. 1. Characterization of inverse problem: Product size distributions in (a) can be obtained by combinations of feed distributions (i), (ii) and (iii) in (a) with corresponding efficiency curves (i), (ii) and (iii) in (b). ð6Þ Recently, truncated analytical expressions for cumulative percent passing distributions for the finer and coarser streams Z R d Ea ðdÞf ðdÞdd ð8Þ 0 dmax 0 Ea ðdÞf ðdÞdd Under restricted functional forms of feed distribution represented in terms of Gates–Gaudin–Schumann (GGS) function and classifier efficiency represented in terms of Plitt function [10], the cumulative form of feed, fine and coarse stream distributions of the classifier are respectively represented by 8
 d n > > < 100 dmax FðdÞ ¼ > > : 100 for 0VdVdmax for dNdmax ð9Þ B.V. Rao / Powder Technology 176 (2007) 123–129 for 0VdVdmax ð10Þ for d>dmax ð11Þ where the values of K and Kmax are given by
 d m K ¼ lnð2Þ d
50c  dmax m Kmax ¼ lnð2Þ d50c ð12Þ ð13Þ pe !
 n  ð1−Rf Þ d50c n  n ; K ¼ 1− g max m ðlnð2ÞÞðn=mÞ m dmax rs The flow split factor to the coarse stream is obtained by integrating all the feed particles with regard to their corresponding actual probabilities of report to coarse stream and is given by Ssplit co for 0VdVdmax al n 8 n  > > > m ;K −nð1−R mK Þg > f > > m > n < 100   U ðdÞ ¼ m −nð1−R Þg n ; K > mK > max f max > m > > > > : 100 py for dNdmax r's Further, it has been shown that the classifier distributions pivot at a size, d⁎, when expressed in density form, which corresponds to the maximum percentile separation of the cumulative passing distributions. Additionally, these distributions have been shown to be difference similar [11]. 3. Results and discussion size modulus, distribution modulus, sharpness index, cut size and by-pass fraction; and if evaluated from measured product distributions by functional fit can as well describe feed distribution in Eq. (9) and efficiency curve in Eq. (5). This method is explored in this paper to address the inverse problem. Eleven sets of measured product size distribution data from the works of Lynch et al. [12]; Austin et al. [13]; NapierMunn et al. [14]; Fuerstenau and Han [15] and Wills [7] have been considered and simultaneously fit to Eqs. (10) and (11) to obtain the five parameters. Table 1 provides the best-fit values along with sum of squared errors (SSE) for these data. Fig. 2 compares the measured feed size distribution with those estimated from optimized parameters for data sets # 1, # 4, # 5 and # 9 in Table 1 that belong to different types of classifiers. Fig. 3 compares the efficiency curve partition coefficients estimated from measured size distributions with those calculated from the parameters in Table 1 for the same data sets shown in Fig. 2. The match is quite satisfactory. Thus the approach helps in solving the inverse problem of classifier distributions to a reasonable accuracy. Present approach considers size distributions explicitly in continuous analytical form and thus has advantages over the discrete approach wherein the measurement errors have a bearing on the stability of the inverse solution [16,17]. Many a times the efficiency curve from reconciled data shows kinks [3] depicting the inadequacy of the reconciliation method to match up the efficiency curve to a monotonically non-decreasing form, although it satisfies Eq. (1). However, the estimated feed distribution and efficiency curve by the present approach are smooth as opposed to that derived from reconciled values and as well retain the mass balance continuously over the size range [0, dmax]. The flow split of solid particles to coarse stream can be conveniently obtained from Eq. (13). Fig. 4 compares the flow split obtained from this approach with that obtained from reconciliation of classifier measured size distributions for all data sets of Table 1. The main reasons for mismatch in feed distribution and efficiency curve in the present study arise because of two important reasons: one, the restricted GGS functional form of on n  8 > g ;K > > > <100  nm  OðdÞ ¼ g ; Kmax > m > > > : 100 125 th o The fine and coarse product size distributions represented in Eqs. (10) and (11) contain the five parameters namely, Au Table 1 Summary of distribution parameters and sum of squared errors (SSE) for various classifiers obtained by simultaneously fitting fine and coarse stream measured distribution data SI No. Classifier type Model parameters n dmax, μm d50c, μm m Rf, % 1 2 3 4 5 6 7 8 9 10 11 Hydrocyclone Air cyclone Dry screen Sieve bend DSF Dorr classifier Hydrocyclone Hydrocyclone Dry screen Dry screen Dry screen Hydrocyclone 0.229 0.773 0.541 0.567 0.746 0.409 0.662 0.632 0.537 0.558 0.520 899 105 40899 976 702 269 1053 17220 17026 17600 1151 298 41 17297 245 182 143 165 15764 15354 14026 245 3.018 2.451 3.884 1.664 1.699 1.516 1.172 13.323 12.377 11.188 2.134 14.944 60.274 0.468 24.593 30.473 15.856 56.693 0.560 0.927 1.390 56.592 SSE Reference 0.0004 0.0206 0.0229 0.0013 0.0096 0.0031 0.0089 0.0168 0.0075 0.0068 0.0219 Lynch et al. [12] Austin et al. [13] Austin et al. [13] Austin et al. [13] Austin et al. [13] Napier-Munn et al. [14] Fuerstenau and Han [5] Napier-Munn et al. [14] Napier-Munn et al. [14] Napier-Munn et al. [14] Wills [7] B.V. Rao / Powder Technology 176 (2007) 123–129 on al co py 126 rs Fig. 2. Comparison of measured feed distribution with that from model. because of a forced fit and two, the noise in the measured data affects the optimization of the parameters. Apart from these, the instances where the approach can fail is when the actual Au th o r's pe feed may not always conform to the measured feed distribution especially in the coarse size range and this misinterpretation is carried forward to the representation of product distributions Fig. 3. Comparison of calculated partition coefficients of efficiency curve from measured distributions with those from model. B.V. Rao / Powder Technology 176 (2007) 123–129 127 substituted in Eqs. (14) and (15) for a given set of evaluated model parameters suggest their importance towards the calculation of overall variances in VF(d) and VEa(d). d py

 AFðdÞ d n d ¼ 100 ln An
dmax n
dmax  AFðdÞ d n ¼ 100 Admax dmax d
 max m co − AEa ðdÞ ¼ 100 2 d50c ARf
m d − AEa ðdÞ d m ð1−Rf Þ ¼ −100 2 d50c mlnð2Þ Ad50c d50c d50c
m

 d − AEa ðdÞ d m d d 50c ¼ 100 2 ð1−Rf Þlnð2Þln Am d50c d50c ð Þ Fig. 4. Comparison of flow split of feed particles to coarse stream for data sets in Table 1, obtained from reconciliation of measured distributions as well as from model parameters. on Table 2 gives the estimated sensitivity coefficients for evaluated model parameters for data # 5 of Table 1 as a function of particle size. These results are typical and hold good for other sets of data as well. The sensitivity parameters indicate that for a given particle size the feed distribution is most sensitive to distribution modulus, n, while efficiency curve is most sensitive to by-pass fraction, Rf. The Eqs. (9) and (5) are least sensitive to variances in size modulus, dmax, and cut size, d50c. This fact is evident in Fig. 5, wherein the parameters of the inverse model are plotted against the corresponding parameters when all the three classifier-distributions are considered for fitting. The parameters for the later case are taken from an earlier study [10]. This shows that inverse model is most sensitive to estimation of parameters Rf and n in accordance with findings of Table 2. The above discussions demand a better analytical solution of product distributions under more general feed and efficiency curve representations. Nonetheless the present approach doesn’t limit the underlying principle of extracting the information of an unknown feed distribution and efficiency curve of an operating classifier to a fair accuracy from its measured product distributions. The method is as well applicable to identify feed distributions of an operating gravity separator pe rs efficiency curve does not conform to Plitt function, for example, the overall efficiency curve of a complex network of classifiers sometimes shows flat regions in-between distorting the sigmoid shape but with a monotonically non decreasing profile with increasing particle size. In order to verify which of the evaluated parameters affect the estimation more significantly, a sensitivity analysis is followed similar what has been presented for flotation recovery sensitivity [7]. The combined uncertainty in the evaluation of feed distribution as well as efficiency curve can be attributed to uncertainty in the estimation of parameters that represent these curves. Considering that these parameters are independent of one another, the variances of feed distribution and efficiency curve at a specified size due to variances in their parameter estimation are given by al ð16Þ and
VEa ðdÞ ¼
 AFðdÞ 2 Vn þ Vdmax Admax r's 2 
 AEa ðdÞ 2 AEa ðdÞ 2 VRf þ Vd50c ARf Ad50c
 AEa ðdÞ 2 þ Vm Am Au VFðdÞ ¼ AFðdÞ An ð14Þ th o
ð15Þ where VF(d), Vn, Vdmax, VEa(d), VRf, Vd50c and Vm respectively represent variances in F(d), n, dmax, Ea(d), Rf, d50c and m respectively. The squares of the partial derivatives are referred to as sensitivity coefficients and can be evaluated by partial differentiation of Eqs. (9) and (5) with respect to parameters contained in them. The simplified forms of these partial derivatives are given in Eq. (16) when the feed and efficiency curves are expressed in percent form. The derivatives when Table 2 Estimated sensitivity coefficients for model parameters as a function of particle size for data # 5 of Table 1 Particle size, d Sensitivity coefficients for estimated parameters n dmax d50c m Rf 86.3 76.0 57.7 43.5 32.8 26.7 22.1 1923.87 1790.30 1499.11 1218.32 969.93 812.52 685.66 4.95E-04 4.09E-04 2.71E-04 1.78E-04 1.17E-04 8.59E-05 6.48E-05 1.08E-02 7.54E-03 3.33E-03 1.37E-03 5.51E-04 2.80E-04 1.50E-04 69.37 66.50 50.70 32.47 18.67 11.90 7.67 6775.81 7307.77 8216.55 8855.32 9275.07 9483.27 9622.50 B.V. Rao / Powder Technology 176 (2007) 123–129 rs on al co py 128 pe Fig. 5. Comparison of estimated parameters by inverse model with those calculated when measured feed distribution was also considered along with product distributions. whose product distributions are expressed in terms of particle density instead of particle size. 4. Conclusions Au th o r's This paper addresses an inverse problem of identifying classifier feed size distribution and efficiency curve by utilizing closed-form solution to classifier distributions. The parameters also describe the efficiency curve smoothly without kinks as opposed to that derived from reconciled data and retain the mass balance over the entire size range. The flow split of feed particles to one of the product streams while they get classified can also be obtained from these parameters. Sensitivity analyses presented in this paper show that the inverse approach is most sensitive to distribution modulus and by-pass fraction parameters that respectively affect feed and efficiency curve representations. Nomenclature d Passing particle size, μm d50c Cut size of efficiency curve, μm dmax Feed maximum size also called size modulus of GGS function, μm d⁎ Pivot size of the classifier distributions Ea(d) Actual efficiency curve in terms of particle size Ec(d) Corrected efficiency curve in terms of particle size f(d ) Feed distribution in density form F (d ) K Kmax m n O(d ) Rf Ssplit U(d ) V Feed distribution expressed in cumulative percent passing form A function of particle size defined in Eq. (9) A constant defined in Eq. (9) Sharpness index Distribution modulus of GGS function Fine stream size distribution expressed in cumulative percent passing form By-pass fraction Solid flow split fraction to coarse stream, also called total-efficiency of separation Coarse stream size distribution expressed in cumulative percent passing form Variance of a variable or a parameter that are indicated in the subscript Greek symbols γ Gamma function Acknowledgements The author acknowledges Prof. G. H. Luttrell, Virginia Polytechnic Institute and State University, Blacksburg, VA for initiating the interest for this investigation. The author is grateful to Mr. Shivaram Kamat and Dr. Phanibhushan Sistu, EIS R&D, TCSL, Pune, India for their encouragement in this work. B.V. Rao / Powder Technology 176 (2007) 123–129 [10] B. Venkoba Rao, Analytical expressions for classifier product size distributions, Miner. Eng. 18 (2005) 557–560. [11] B. Venkoba Rao, The pivot phenomenon and difference–similarity of classifier particle distributions, Powder Technol. 168 (2006) 152–155. [12] A.J. Lynch, T.C. Rao, K.A. Prisbrey, The influence of hydrocyclone diameter on reduced efficiency curves, Int. J. Miner. Process. 1 (1974) 173–181. [13] L.G. Austin, R.R. Klimpel, P.T. Luckie, Process Engineering of Size Reduction: Ball Milling, SME, New York, 1984. [14] T.J. Napier-Munn, S. Morrell, R.D. Morrison, T. 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