A particle size distribution model for manufactured particulate solids of narrow and intermediate size ranges

Powder Technology 164 (2006) 117 – 123 www.elsevier.com/locate/powtec

A particle size distribution model for manufactured particulate solids of narrow and intermediate size ranges A.B. Nesbitt ⁎, W. Breytenbach Gravity Research Unit, Department of Chem. Eng., Cape Technikon, P.O. Box 652, Cape Town 8000, South Africa Received 4 October 2004; received in revised form 31 January 2006; accepted 22 March 2006 Available online 15 April 2006

Abstract Mass transport algorithms using particle size distribution equations are common where molecular mass flux across a hydrodynamic layer is present. In this paper, we suggest a unique model and prove its effectiveness in modelling particle size distributions that vary dramatically in character and propose that the model may be useful for distributions of narrow and intermediate size ranges, more specifically, for ion exchange resin and fluidised bed catalyst, both of which are manufactured particulates. The model has a single fitting parameter and can be used in the frequency form for finite-element transport phenomena calculations. © 2006 Elsevier B.V. All rights reserved. Keywords: PSD model; Ion exchange; Catalyst

1. Introduction Debate over different types of particle size distribution models and their effective portrayal of frequency and accumulative particle size distributions (PSD) is topical. Transport phenomena involving ion exchange between a particulate phase and an encompassing continuous phase can be found in typical physicochemical processes such as comminution, leaching, fluidised catalytic conversion, flotation and those using ion exchange resins or natural adsorbents. In all cases, transfer occurs across the particles' hydrodynamic boundary that will be closely associated with the particles' external surface area. If this area is to be incorporated into a discrete element transport model and a moderate range PSD is present, it can only be determined from the frequency distribution form of a PSD function. Manufactured particles tend to be of narrow to intermediate size range and tend to lack extended tails at the peripheries of the distribution. However, their involvement in mass transport

⁎ Corresponding author. E-mail address: [email protected] (A.B. Nesbitt). 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.03.015

systems dictates a comprehensive measurement of their PSD. Traditionally, PSDs are exhibited in the accumulative form, which is a far more exacting manner of presentation than that of a frequency histogram. If a frequency function is discerned from a histogram of the raw data, it will have an inherent element of averaging within the range of any particular class size interval. Experience shows that the frequency distribution form of presentation can be clumsy and almost always requires normalisation especially if it is based on uneven class intervals from which a particle frequency against particle size function is required. Transport phenomena, between a continuous fluid phase and a discontinuous solid phase, can be severely affected by shifts in the mode of the frequency, of the PSD, as rates of transport tend to be a strong function of particle Reynolds number. This is especially true where the Reynolds numbers are low, which is indicative of a surface-controlled mechanism of transfer, as is the case with most particulate systems under normal gravity. Often, use of an average particle size for the generalisation of transport phenomena calculations may not be sufficiently accurate if the PSD has a characteristic skewness that may alter, rendering any generalised model inaccurate.

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The classic manner of overcoming this difficulty is to use a calculation-intensive piecewise algorithm, such as a spline, piecewise polynomial or nodal network, so as to develop a continuous functional relationship describing the particle frequency distribution [6,4]. This approach, providing the PSD raw data acquisition is accurate and not withstanding the inaccuracies described above, will always be sounder than a parameter-fitted model of one or many parameters. The accuracy of the data will be paramount, as it is difficult to rule out the effects of an inaccurate measurement. Ultimately, the introduction of a piecewise algorithm to describe a frequency distribution requires intimate knowledge of the PSD and therein makes each distribution, or the fitting process, unique with very little possibility of generalisation. Furthermore, it is extremely difficult to combine a piecewise algorithm into a larger transport model for the purposes of analytical manipulation, or model fitting activities where the PSD has been poorly reported. Finally, the modelling of the transport phenomena of a particulate system, where the characteristics of the PSD undergo a change during the process, will be difficult, e.g., swelling of ion exchange resin beads in a packed or fluidised bed arrangement [11]. Recent work by Gbor et al. [2] proved that neglecting the effect of characteristic particle size distribution could, in a kinetic study, have the effect of incorrectly identifying the shift in control. A versatile, parameterised, PSD equation should be able to accommodate shifts in either mode or mean within the size range if fitted to PSD data in the accumulative form. In addition, it should be usable in the frequency distribution form for transport phenomena calculations. Particulate solids that are the result of a manufacturing process, e.g., ion exchange resin, which normally has a distribution within the 300- and 1200-μm boundaries, or fluidised bed catalyst with an equally narrow/intermediate range, traditionally have designated particle size ranges within which the mode or mean size can alter. This paper seeks to investigate the application of a suggested parameterised model to portray PSDs of these particulate solids and compare its performance to that of other popular PSD equations/models currently in use.

2. Theory If a particulate substance consists of particles of singular size and shape, the process of modelling transport phenomena becomes greatly simplified and the surface area normal to the direction of transport can be assumed to be a single specific value, dependent on particle population per unit volume. As this homogeneity is hardly ever present in reality, the simplest approach is to assume an average particle size. For batch or inline mixed-tank reactors, this approach may be sufficiently accurate, providing the characteristic PSD and shape do not alter significantly. However, if the system has very little longitudinal mix, which we define as being in the direction of continuous phase flow, making it more akin to a plug flow reactor, only a minor shift in mode of the frequency size distribution is likely to have an effect on transport rates. Such a set-up is common where the reactor constitutes a fluidised bed in which stratification of particle sizes may be present [7]. To accommodate these challenges, it was considered necessary that the PSD model/equation should meet with the following seven criteria: 2.1. Criteria 1 The model should be able to accommodate a bulk particle frequency Eq. (1) at any point within the fixed range of particle sizes, which indicates the peak frequency. d2 ðwÞ dðdpÞ2

¼0

ð1Þ

where ψ is the accumulative-particle size-passing distribution function and dp is a characteristic particle length. 2.2. Criteria 2 The model should be clearly defined in both the frequency, as well as the accumulative form, as model-fitting difficulties are less likely to be experienced in the accumulative form.

Frequency Distribution

2.4 K1= 0.5

2.0

K1= 1.0

K1= 1.5

1.6 1.2 0.8 0.4

K1 = 2.0

K 1 = 2.5

K 1 = 3.0

0.0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Normalised Screen Sizes Fig. 1. Frequency distributions for various values of K1.

0.8

0.9

1

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119

1.0

Cummulative Fraction

0.9 0.8

K1 = 1.5

0.7 0.6

K1 = 3.0

K1 = 1.0

0.5 0.4 K1 = 0.5

0.3

K1 = 2.5

0.2 K1 = 2.0

0.1 0.0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalised Screen Sizes Fig. 2. Accumulative curves for various values of K1.

However, its usefulness for transport phenomena will ultimately be as a frequency distribution.

form makes for easier manipulation, at the expense of linearisation.

2.3. Criteria 3

w ¼ f1 ðdpÞ−f2 ðdpÞ

The frequency form of the function should always meet with normalisation no matter what the value of the adjusted parameter (Eq. (2)). Z C¼ 0

þl

dðwÞ dðdpÞ ¼ wjdp¼l dðdpÞ

ð2Þ

ð4Þ

The combination of two functions proposed in Eq. (4) produces a synergy that returns the desired curve, providing the difference is initially zero and then eventually decays to a positive constant value. This is clearer in the differential or frequency form Eq. (5), where we see how the functions work in combination. dw df1 ðdpÞ df2 ðdpÞ ¼ − dðdpÞ dðdpÞ dðdpÞ

2.4. Criteria 4 For the purposes of insuring simplicity, there should only be a single characterising parameter (Eq. (3)). This will be especially useful if the model is to be part of a large transport model or algorithm, w ¼ f ðdp; K1 Þ

ð3Þ

where K1 is the single parameter, which when altered, should shift the mode of the PSD while the model adheres to the listed criteria. Traditionally, accumulative particle size distribution models have been combinations of functions, either in the form of a product or a function of a function. In this study, we break with tradition and propose the sum of two functions (Eq. (4)). This

dpmax > dp > dpmin

ð5Þ

where dpmax and dpmin are, respectively, the characteristic lengths of the largest and smallest particles in the distribution. Having decided on the difference of two functions as our basic model, further criteria can now be set regarding the function value returned at the smallest and largest particle size values. The first of these is that the differentials of both functions one f1′(dp) and two f2′(dp) should return to zero at the smallest particle size and should have equal values at the largest particle size. Therefore, the following two criteria should hold true: 2.5. Criteria 5  df1 ðdpÞ   dðdpÞ 

dp¼dpmin

 df2 ðdpÞ  ¼  dðdpÞ 

¼ 0:

ð6Þ

dp¼dpmin

Table 1 Single-parameter models including the proposed in this study

Table 2 Two-parameter models included in this study

Name

Model

Name

Model

Proposed

F(d) = 10 × ⌊(1/(K1 + 1)SK1 + 1) + (0.1−(1/(K1 + 1)) S1/(1/(K1 + 1))−0.1)⌋ F(d) = exp{−1/K21[ln(dp/dmax)]2} F(d) = 1 − (1 − dp/dmax)k1 F(d) = 1 − exp(− K,dp)

Fractal [3]

F(d)=exp{lnK2 + ((3K21 − 13K1 + 14)/ (K21 − 5K1 + 4) + 1)lndp} F(d) = ⌈1 + (K1/dp)K2⌉−m where m = 1 − (1/K2) F(d) = 1 − exp(− K1dpK2) F(d) = (K1/(1 − K2))dp1−K2

Jaky [1] Gaudin–Meloy [8] Exponential [8]

Van Genutchen [3] Rossin Rammler [9] Power Law [8]

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Table 3 Four-parameter models included in this study Name

Model

Weibull [3]

F(d) = K3 + (1 − K3){1 − exp(− K1KK2 4 )} where K4 = (dp − dmin)/(dmax − dmin) F(d) = K1 + K2exp{− exp[− K3(dp − K4)]}

Gompertz [5]

2.6. Criteria 6  df1 ðdpÞ   dðdpÞ 

dp¼dpmax

 df2 ðdpÞ  ¼  dðdpÞ 

ð7Þ dp¼dpmax

2.7. Criteria 7 Finally, adherence to Eq. (8) will ensure that the differential of our accumulation function is either zero or positive throughout the range. dpmax > dp > dpmin

A more functional form of the differential equation and the simplest that will meet with the listed criteria is Eq. (9) or in its accumulative/integral form is Eqs. (10a) and (10b). Z Z dw ¼ k1 ðS k2 −S k3 ÞdðSÞ ð9Þ   dw ¼ k1 S k2 −S k3 dðSÞ w ¼ k1

1:1k2 þ 0:1 0:9−0:1k2

k3 þ1

S S − k2 þ 1 k3 þ 1

1 dpmin − dpmax −dpmin dpmax −dpmin

2 6 w ¼ 104

ð10bÞ

where S is a value between 0 and 1 and is the normalised measure of the characteristic length of the particle. The differential of the second function f ′2(S) increases with increasing S at a slower rate than the differential of the first

ð12Þ

Eq. (13) gives the final equation in the frequency form and Eq. (14) is the accumulative. For values of k2 between 0 and 9, the model/equation gives a fairly versatile distribution presented in the frequency form in Fig. 1 and accumulative in Fig. 2. For the purposes of simplification, we now replace the k2 variable with K1, which we define as the single fitting variable for the model/equation.  # " 1:1K1 þ0:1 dw 0:9−0:1K 1 ¼ 10 S K1 −S ð13Þ dðdpÞ

ð10aÞ 

ð11Þ

To meet with criteria 5 and 6 normalisation is required and presupposes that the characteristic length of the smallest and largest particle present will be known or can be estimated. In this study, a simple linear normalisation was used (Eq. (12)) although it might be effective in other systems.

ð8Þ

2.8. Final function

k2 þ1

k3 ¼

S ¼ dp

df1 ðdpÞ df2 ðdpÞ z dðdpÞ dðdpÞ



f′1(S), but by using a suitable exponent it can be made to eventually increase more rapidly, effectively reducing the overall frequency function back to zero at S = 1. In this form of Eq. (9) it is possible to see that criteria 2 is easily met. However, to make the function meet with criteria 1, 3 and 4, the parameters k1, k2, and k3 had to be examined. The following interrelationships were discovered that effectively made the equation meet with these criteria i.e., k1, is made equal to 10 and the relationship between k2 and k3 is maintained as in Eq. (11).

1 K1 þ 1

K1 þ1 S þ 0:1−



1 S K1 þ 1



3 1

1 −0:1 K1 þ1

7 5

ð14Þ

3. Model testing The performance of the proposed model/equation was compared to the performance of a selection of published

Table 4 Statistical results for the fitting of the ion exchange resin PSD to each model Model (number of parameters) Proposed (1) Jaky (1) Gaudin Meloy (1) Exponential (1) Fractal (2) Van Genutchen (2) Rossin Rammler (2) Power Law (2) Weibull (4) Gompertz (4)

R2 correlation

SSE

AIC

Lowest

Highest Variance

Average

Lowest

Highest

Variance

Average

Lowest

Best

0.97962 0.96874 0.97192 0.93879 0.86192 0.97056 0.98631 0.89820 0.99575 0.99587

0.99775 0.99352 0.99978 0.99221 0.99891 0.99630 0.99770 0.99891 0.99933 1.00000

0.9925233 0.976775 0.9922033 0.9765333 0.9412817 0.9800683 0.9912283 0.9762983 0.998105 0.9989967

0.0007736 0.0042328 0.0001861 0.0019787 0.0006671 0.003066 0.0014777 0.0006671 0.0002741 6.09E−07

0.0197992 0.0362518 0.0345565 0.1104843 0.2651581 0.0204163 0.0095022 0.0647525 0.0025998 0.0024735

0.0069484 0.012813 0.0132427 0.0414173 0.0992713 0.0067085 0.003254 0.0251563 0.0009208 0.000961

0.0067997 0.0220691 0.0082289 0.0482097 0.0744074 0.0106567 0.0046618 0.0137989 0.0010797 0.0005268

− 17.6106 − 14.5863 − 14.8258 − 9.0144 − 2.6371 − 15.4571 − 19.2812 − 9.6859 − 19.7616 − 22.0106

− 33.8223 5.7298 − 25.3245 4.2404 − 40.9456 10.3567 − 29.1265 7.3166 − 32.5629 11.6528 − 24.9369 3.6132 − 28.5865 3.7404 − 33.8223 9.0357 − 31.0108 4.4330 − 63.5575 16.2297

0.0067868 0.0093955 0.0107736 0.0206448 0.059615 0.0094245 0.0043621 0.0389236 0.0013273 0.0015721

Variance Average −25.3489 −18.2230 −28.8387 −16.0181 −15.6137 −19.6885 −23.9522 −24.0818 −25.7476 −41.0059

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Table 5 Statistical results for the fitting of the catalyst PSD to each model Model (number of parameters) Proposed (1) Jaky (1) Gaudin Meloy (1) Exponential (1) Fractal (2) Van Genutchen (2) Rossin Rammler (2) Power Law (2) Weibull (4) Gompertz (4)

R2 correlation

SSE

AIC

Lowest

Highest

Variance

Average

Lowest

Highest

Variance

Average

Lowest

Best

Variance

Average

0.98103 0.99032 0.97635 0.99283 0.94381 0.99152 0.84109 0.84139 0.99232 0.99174

0.99463 0.99237 0.98683 0.99774 0.98426 0.99407 0.99501 0.99501 0.99602 0.99751

0.004539 0.000706 0.003951 0.001866 0.014568 0.001048 0.061255 0.078095 0.001554 0.002241

0.988573 0.991085 0.981568 0.994663 0.960865 0.992582 0.965847 0.942175 0.994353 0.994273

0.003147 0.007672 0.008545 0.016594 0.008676 0.003217 0.002583 0.002583 0.002064 0.001263

0.010383 0.011118 0.013527 0.066947 0.040070 0.004937 0.260044 0.260000 0.003897 0.004193

0.002395 0.001402 0.002215 0.018304 0.013097 0.000661 0.103968 0.132212 0.000772 0.001138

0.006451 0.009664 0.011332 0.035570 0.026945 0.003982 0.047924 0.089321 0.002886 0.002907

− 16.2702 − 15.9967 − 15.2124 − 8.81543 − 8.86847 − 17.2437 − 1.38761 − 1.38829 − 12.1906 − 13.8976

− 21.0453 − 17.4806 − 17.0499 − 14.3949 − 14.9886 − 18.9577 − 19.8358 − 19.8358 − 14.7332 − 18.6977

1.579339 0.601935 0.825845 1.982652 2.420417 0.669293 6.875468 8.813748 1.061364 1.827907

−18.42 −16.5943 −15.989 − 11.7614 −10.9796 −18.1498 −15.0787 −12.6709 −13.5092 −15.6752

equations, shown in Tables 1–3, by fitting each to a database of six PSDs of commercially available ion exchange resin and six solid catalyst PSDs, used in fluidised systems. The resin is Amberlite 252H, a general-purpose cation adsorption resin and a product of Rohm and Haas. Measurement of the resin PSDs was made by screen analysis with the volume in each screen interval being equated to the average of the interval. The six samples measured were randomly chosen for their extreme variance in PSDs. The catalyst (silver flake) is used in the adhesive and ink processing industries and is a product of Johnson Matthey Catalysts. The PSD data were sourced from the manufacturer and were measured by Lazar diffractometry as volume of particles versus size. The PSD of this particular catalyst is an important process criterion and, therefore, has to be accurately reported by the manufacturer. Six samples with widely diversified PSDs were specifically chosen for this exercise. A program was written in Turbo Pascal Version 7.0 that effectively fitted the model/equations to the raw data using the Hook Jeeves [10] search algorithm that iterated for the lowest sum of square errors (SSE) of each data record. In addition, the program calculated the R2 correlation coefficient (Eq. (15)) of the parity line between predicted/observed data. A further measurement was taken by using the Akaike's Information Criterion (AIC, Eq. (16)) which was developed by Professor Hirotugu Akaike in 1971 and proposed in 1974. The driving

force behind the AIC is to examine the complexity of the model together with goodness of its fit to the sample data and to produce a measure which balances between the two and discourages overfitting. 9 8 > > P P P = < N ð Yo Yp Þ−ð Yo Þð Yp Þ ffi ð15Þ R2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q > ; : t P Y 2 −ðP Yo Þ2 b tN P Y 2 −ðP Yp Þ2 > o p AIC ¼ N lnðSSEÞ þ 2P

ð16Þ

4. Results Tables 4 and 5 give a comprehensive breakdown of the results of the fitting process. The ion exchange resins' PSDs differed substantially in nature as a result of deliberate distribution slanting for the purposes of experimentation in the study of mass transport phenomena. The PSD of the catalyst, on the other hand, differs less but, nevertheless, represents distributions of other commercially available products. 5. Discussion Despite major differences in the nature of PSD of the ion exchange resin, the proposed model/equation effectively fitted the curves supplying suitably high R2 correlations with

Cummulative frac. passing

1 0.9 0.8 MODEL RESIN 2

0.7 0.6 0.5

RESIN 3

0.4 0.3

RESIN 5

0.2 0.1

RESIN 6

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalised particle size Fig. 3. Fitted proposed model/equation to selected resin data.

0.9

1

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A.B. Nesbitt, W. Breytenbach / Powder Technology 164 (2006) 117–123

Cummulative frac. passing

1 0.9 0.8

MODEL CATALYST 3

0.7 0.6 0.5

CATALYST 4

0.4 CATALYST 5

0.3 0.2

CATALYST 6

0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalised particle size Fig. 4. Fitted proposed model/equation to selected catalyst data.

acceptable variances. This is clearly seen in Fig. 3 where, for the sake of clarity, four of the resin distributions are shown. A similar trend was achieved for the catalyst PSD and is shown in Fig. 4. With respect to the resin distribution, it is clear that the performance of the model presented in this paper is superior to most of the listed models. Only the Gompertz and Weibull equations are statistically significantly better, but have four fitting parameters. Although the Gaudin–Meloy, which is also a single parameter model/equation, appears to achieve a similar R2 correlation, it has a higher variance. With respect to the commercial catalyst distributions, it would appear that the performance of the proposed model is slightly poorer than average. However, with this selection of data, the highest AIC value is attained, making it the best performing model in a combination of least sum of square errors and parameter values. Two difficulties can be observed with the proposed model/ equation. Firstly, the value of the second function becomes infinite at K1 = 9, which could halt the search algorithm prematurely should this value be approached, but can easily be accounted for. In reality, the frequency curve at K1 → 9 is so contorted to one side of the distribution, as to render the distribution impractical and therefore unlikely to exist in practice. Secondly it is essential to estimate the highest and lowest value of the distribution when applying the proposed model/ equation, as is the case with the Jaky, Gaudin–Meloy and Weibull model/equations. This is a common practice and does not detract from the proposed model. It is also notable that the technique used to acquire the relationship between k1 and k2, i.e., Eq. (11) was tailored to the system presented here, but this relationship could with some manipulation be altered to achieve a different type of function that may make the model/equation more useful in another application. The true versatility of the model/equation truly becomes apparent. 6. Conclusion The growth in popularity of mass and heat transfer studies involving particulate solids has necessitated the development of

a simple and versatile, equation-based empirical model that efficiently models a narrow/intermediate PSD as accurately as possible. An advantage of using a single equation is its potential for algebraic incorporation into a discrete transport model. Arguably, the researcher could incorporate the model/equation into his algorithm and, hence, improve on any particles size averaging approach. A further advantage of our simple two-term model/equation is in the ease with which it can be integrated/differentiated. Consider that the calculation of total flux through particle surfaces in a particulate mass could be calculated by integrating the product of the frequency PSD function and the flux equation Eq. (17), i.e., integration by parts. Our model/equation, on average, is more suited to this type of algebraic manipulation than others listed Z dpmax Q; J ¼ GðdpÞFðdpÞddp ð17Þ dpmin

where G(dp) F(dp)

is the PSD function is the flux function for each particle size

Nomenclature dp characteristic length of particle (L) dpmax characteristic length of largest particle in range (L) dpmin characteristic length of smallest particle in range (L) dpm characteristic length of the m particle size (L) dpn characteristic length of the n particle size (L) J mass transport flux (M L− 2 T− 1) K single fitting parameter for proposed model (dimensionless) k1,k2,k3 proposed model constants for manipulation (dimensionless) N number of data points (dimensionless) P number of parameters (dimensionless) Q heat transport flux (M L2 T− 3) S normalised characteristic length of particle 0 < S < 1 (dimensionless) SSE sum of squared error (dimensionless)

A.B. Nesbitt, W. Breytenbach / Powder Technology 164 (2006) 117–123

Subscripts 1,2,3,4 constant numbers o observed data points p predicted data points Greek Symbol Ψ accumulative function (dimensionless)

of

particle

size

range

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