Analysis of sieving data in reference to powder size distribution

PII: Acta mater. Vol. 46, No. 2, pp. 617±629, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00220-6 ANALYSIS OF SIEVING DATA IN REFERENCE TO POWDER SIZE DISTRIBUTION BING LI and E. J. LAVERNIA{ Department of Chemical & Biochemical Engineering and Materials Science, University of California, Irvine, CA 92697-2575, U.S.A. (Received 2 December 1996; accepted 30 June 1997) AbstractÐThe procedure conventionally employed to interpret experimentally determined sieving data was examined. It was demonstrated that the conventional procedure is inherently ¯awed in several aspects. Along with several implicit, hard-to-be-justi®ed assumptions associated with it, the conventional graphical representation procedure also failed to address the reliability of the results. To resolve these problems, a new procedure was formulated. Application of the formulated procedure to several sets of sample sieving data reveals that it is capable of extracting the total weight of powders from experimental data, of determining the nature of the size distribution, of establishing the characteristic parameters, and simultaneously, of determining the reliability of the results. # 1998 Acta Metallurgica Inc. 1. INTRODUCTION 2. OVERVIEW Experimental data on powder size distribution analysis may be tabulated or graphically presented. It is widely acknowledged that the most accurate representation of powder size distribution data is in tabular form, in which experimental data are listed as shown in Table 1. It is also realized, however, that graphical representation of experimental data has many advantages over tabulated results, as discussed in detail in [1] and [2]. Moreover, unless it is graphically presented, the tabulated experimental data provides limited insight into the nature of the size distribution of powders and limited information on size distribution, hence it hinders our ability to control the processing parameters. Therefore, interpretation of experimental data in powder size distribution analysis, such as sieving data, generally involves graphical representation. The reliability of graphical representation results depends on the suitability of the procedures used to obtain and to present the experimental data. A standard procedure has been established to obtain the experimental data in powder size distribution analysis (MPIF standard 05 and ASTM standard B214-92). The procedure to present and then interpret the experimental data, on the other hand, is far from complete. The purpose of this paper is to demonstrate that there are several serious shortcomings associated with the conventional procedure, and to formulate a new procedure for presenting the experimental data. It is helpful to describe, at ®rst, the graphical representations generally employed in practice, and the most widely used logarithmic±normal size distribution, before proceeding to address the problems associated with the conventional procedure in graphical representations. 2.1. Graphical representations Graphical representations of experimental data in powder size distribution analysis include [1]: (1) histogramsÐpresenting frequency of occurrence vs size range; (2) size frequency curveÐpresenting frequency of occurrence as a function of powder size, equivalent to a smoothed-out histogram; (3) cumulative plotÐpresenting percent of powders greater (or less) than a given powder size as a function of powder size; and (4) probability paper plotÐessentially the same as cumulative plot, except that the percentage is presented on a probability scale. 2.2. Logarithmic±normal distribution A logarithmic±normal distribution is closely related to the graphical representation of sieving data. It is commonly found that collections of atomized powders obey log±normal size distribution statistics. Accordingly, there are normally two main objectives of a graphical representation: to determine whether the sieving data obeys log±normal distribution or not, and, if it does, to determine the parameters characterizing a log±normal distribution. For powders obeying a log±normal distribution in terms of weight±size relationship, the probability {To whom all correspondence should be addressed. 617 618 LI and LAVERNIA: ANALYSIS OF SIEVING DATA Table 1. Cumulative weight undersize as a function of powder size [10] Opening (mm) Weight undersize (g) 212 180 150 125 106 90 75 63 53 45 38 129.8 128.7 125.7 117 103.8 87.3 66.5 48.5 31.6 18 7.2 density, P (D), corresponding to powder size D is given by [1±3]:   1 1 lnD ÿ Dm †2 p exp ÿ ; 1† P D† ˆ 2s2 D s 2p and Z 0 1 P D†dD ˆ 1; 2† where Dm is the mean mass powder diameter, s is the standard deviation. The probability density, P (D), corresponding to a powder size D is de®ned as P D† ˆ DWp =DD 3† where DWp is the weight percentage of powders with size falling in the range between D and D + DD. It is evident from equation (1) that powders obeying log±normal distributions may be characterized by two parameters, the standard deviation, s, and the mass mean droplet size, Dm. In practice, another three parameters are also frequently utilized to characterize the powder size distribution. The three parameters are the characteristic powder sizes, d16, d50, and d84, under which 16, 50, and 84 wt% of powders, respectively, are smaller than the stated sizes. These two characterizing methods are equivalent to each other, since 4† Dm ˆ d50 ; and s ˆ ln d50 =d16 † ˆ ln d84 =d50 †: 5† Graphically, d16, d50, and d84 may be determined from a cumulative plot or a probability paper plot, and s and Dm may be determined from a size±frequency plot. It is worth noting that the magnitudes of these characteristic parameters may be determined graphically either with or without rigorous curve ®tting involved. Without curve ®tting, however, the nature of powder size distribution is generally unknown. In this case, the characteristic parameters determined are of no physical signi®cance. In addition, even when the nature of size distribution is pre-known, curve ®tting is necessary to minimize the extensive experimental errors normally associated with sieving analysis. Therefore, graphical determination of these parameters should be preceded by curve ®tting of experimental results. 2.3. Curve ®tting in graphical representations All of the graphical representations involve the weight percentage, rather than weight, as evident from Section 2.1. Conversion of experimentally determined absolute weight into weight percentage necessitates the knowledge of the total weight of the powders under analysis. As will be shown in Sections 3 and 4, this parameter is generally unknown. Actually, it is this fact that complicates the procedure of graphical representation. In this section, the total weight of the powders is temporarily assumed to be a known parameter for the convenience of discussion. In this case, both the weight and weight percentage are readily used in graphical representations and curve ®tting. 2.3.1. Histogram and size frequency curves. In these two plots, curve ®tting may be established using equation (1). In sieving experiments, however, the direct experimental data are weight (or weight percentage) in a size range, or cumulative weight (or weight percentage) of powders under a given size, as shown in Table 1. In order to obtain the probability density, P(D), equation (3) has to be used. However, this may introduce extensive additional errors into the experimental data by either of the following two ways. First, if the size interval, DD, in equation (3), is selected as the same as that in sieving analysis (for example in Table 1, they are 32, 30, 25, ..., 8, and 7 mm in decreasing sequence), DD is generally too large to be used to accurately calculate P(D) through equation (3). If the size interval, DD, in equation (3) is selected to be a smaller value, on the other hand, such as in the range of 10 2 mm, subjective interpolation between the experimental data points would de®nitely be involved, introducing unexpected errors. Therefore, curve ®tting in either histogram or size frequency curve plots would not be considered in the present study. 2.3.2. Cumulative plot. Using equation (1), the cumulative percentage, Cp(D), under size D may be calculated as: ZD Cp D†% ˆ P D†dD 0   Z lnD 1 lnD ÿ lnDm †2 p exp ÿ d lnD ˆ 2 2 ÿ1  2 Z lnDÿlnD p m   2 1 ˆ p exp ÿt2 dt  ÿ1 Z lnDÿlnDm    1 ˆ p exp ÿt2 =2 dt: 6† 2 ÿ1 In sieving analysis, powders may be directly characterized by cumulative weight under size as a function of powder size. When the total weight of the powders is known, then the cumulative weight per- LI and LAVERNIA: ANALYSIS OF SIEVING DATA centage under size as a function of powder size, Cp(D), may be readily obtained. Accordingly, equation (6) may be employed to curve ®t the experimental data. Unfortunately, the form of equation (6) is so complex that no attempt was found in the literature to use it to curve ®t the experimental data. 2.3.3. Probability paper plot. By de®ning a function y = norm(x), for which y and x satisfy Zy 1 x% ˆ p exp ÿt2 =2†dt; 7† 2 ÿ1 equation (6) may be rearranged as lnD ÿ lnDm ˆ norm Cp †;  8† or log D ˆ log Dm ‡ 0:434
norm Cp †: 80 † In a probability paper plot, the abscissa is the powder size, generally in logarithmic scale, logD, while the ordinate is the cumulative weight percentage undersize in probability scale. It is worth noting that the probability scale is calculated using y = norm(x), or equation (7), i.e., x = 50 corresponds to y = 0; x = 60 corresponds to y = 0.25; x = 90, y = 1.28; x = 100, y = 1; x = 40, y = ÿ 0.25; x = 0, y = ÿ 1; and so on ... [4], as shown in Fig. 1. Therefore, equation (8), which indicates that logD is a linear function of norm(Cp), predicts a straight line for powders obeying log±normal distributions when the cumulative percentage undersize, Cp, is graphed vs the powder size, D, in a probability paper plot. This feature is well known and extensively utilized to determine whether a collection of powders obey a log±normal distribution or not, although its origin is not necessarily always well understood by the user. 3. PROBLEMS ASSOCIATED WITH GRAPHICAL REPRESENTATION PROCEDURE In this section, the problems associated with graphical representation procedures will be discussed. Essentially, all of these problems originate from the fact that the total weight of powders is generally unknown, which is closely related to the removal of coarse/®ne powders in practical atomization experiments. 3.1. Removal of coarse/®ne powders It is evident from Section 2.1. that all of the graphical representations involve weight percentage 619 (or frequency), rather than the absolute weight. However, experimental data generally involve absolute values. Therefore, the ®rst step of graphical representation is to convert the absolute weight, in the case of sieving experiments, into weight percentage. This necessitates the knowledge of the total weight for the powders under analysis. Unfortunately, this parameter is generally unknown in a lot of practical situations. This may appear unusual; however, any detailed examination of the powder size distribution analysis will indicate that this is generally beyond the capability of typically used experimental arrangements. Suce it to point out here that the powders subjected to experimental size distribution analysis, such as sieving, are di€erent from the powders formed because of dust separation by cyclones or/and removal of coarse particles. To make this point more clear, let us consider a dispersion of powders observing log±normal distribution, as shown in Fig. 2. Dust separation by cyclones and removal of coarse particles truncate the two tails from the bell shape distribution in Fig. 2 by introducing two size-limits, Dmin, the lower limit, and Dmax, the upper limit. While the total weight of the truncated distribution may be readily measured experimentally, the total weight of powders used for conversion from weight into weight percentage in a graphical representation of sieving data should be that of the entire powders, rather than the truncated one. Nevertheless, it is impossible to experimentally measure the total weight of the entire distribution of powders, since, by de®nition, the powder size ranges from zero to in®nity for a log± normal distribution. Removal of coarse/®ne powders, or the diculty in determining the total weight of powders experimentally, is not the problem associated with size distribution analysis. Actually, it is consistent with the basic concept to explore the real size distribution from limited experimental data [1]. The question is how to analyze the experimental data to minimize the e€ect of the intrinsic experimental dif®culties on the ®nal results. This is crucial to ensure that the ®nal results are reliable and of signi®cance. The conventional graphical representation procedure, however, hardly addressed this issue, as shown below. 3.2. Substitution by truncated distribution In a conventional procedure, the weight of a truncated distribution is generally employed as a substitution, for the purpose of conversion from weight into weight percentage, of the total weight Fig. 1. Probability scale y = norm(x). 620 LI and LAVERNIA: ANALYSIS OF SIEVING DATA Fig. 2. Schematic graph showing probability density, P(D), as a function of the powder size for a collection of log±normally distributed powders. of the powders. This substitution might be an acceptable approximation when the weight of the tails is insigni®cant relative to that of the entire distribution. As the tails become comparable to the central portion in terms of weight, the error resulting from this approximation may be large. The problem is complicated, however, by the fact that the relative contribution of the tails to the entire distribution, i.e., the criterion for the accuracy of the approximation, generally remains unknown until the total weight of the entire powders is determined. The conventional procedure is also ¯awed as a result of another factor. By assuming the weight of a truncated distribution as the total weight of the entire distribution, the experimentally obtained absolute weight may be readily converted into weight percentage. However, a natural result of this assumption is that the cumulative percentage undersize corresponding to Dmax is 100%. Since it is impossible to graph 100% in a probability paper plot, this data point is then ignored. This treatment is dicult to be justi®ed, but widely employed. 63 mm for Fig. 3(a)±(d), respectively. Moreover, from Table 1, the weight of these truncated distributions are 129.8, 125.7, 87.3, and 48.5 g for Fig. 3(a)±(d), respectively. The data were graphed on probability paper plot and curve ®tted, with the ®tting coecient, along with the characteristic parameters, summarized in Table 2. It is evident from Fig. 3 and Table 2 that introducing di€erent upper limits of powder size, Dmax, does not a€ect the nature of the size distribution; the data consistently followed log±normal distribution. The characteristic parameters, however, vary extensively with Dmax, rather than remain unchanged, as evident from Table 2. It is important to recall that the ultimate purpose of size distribution analysis is to determine the real size distribution, and the characteristic parameters for the real size distribution of the powders under analysis are unique. Therefore, at least some of the results in Table 2 provide misleading information as to the characteristics of the real size distribution. This conclusively reveals that data points obeying log±normal distribution alone cannot ensure the reliability of the results obtained. 3.3. Reliability 3.4. Diculty in determining reliability In the literature, an implicit concept that is widely used involves the assumption that, if the data points obey a log±normal distribution, then the results are reliable. To examine its validity, the experimental data in Table 1 were analyzed using the conventional procedure, with the results shown in Fig. 3(a)±(d). The upper limit of powder size, Dmax, was chosen to be 212, 150, 90, and 63 mm in Fig. 3(a)±(d), respectively. The magnitude of Dmin was temporarily set to be zero for all cases, as normally treated in conventional procedures. Accordingly, the size range of the truncated distributions are 0±212 mm, 0-150 mm, 0±90 mm, and 0± Since there are several dicult-to-be-justi®ed assumptions associated with the conventional procedure, no attempt will be made here to solve, or to prove it is impossible to solve, the reliability problem under the framework of conventional graphical representation procedure. Suce it to point out that the problem is complicated by the intrinsic dif®culty in using a probability paper plot. This may be brie¯y described as follows. Considering the bell shape for a log±normal distributed collection of powders in Fig. 2, as Dmax increases, the results are expected to gradually approach that of the real size distribution, becoming more and more reliable. This LI and LAVERNIA: ANALYSIS OF SIEVING DATA 621 Fig. 3. Probability paper plots for data in Table 1, using conventional procedure, with various Dmax introduced: (a) Dmax=212 mm; (b) Dmax=150 mm; (c) Dmax=90 mm; and (d) Dmax=63 mm. feature might enable one to judge the reliability of the results obtained. Unfortunately, this idealized situation is rarely realized in a probability paper plot. As Dmax deviates from the mean mass diameter, Dm (which means Dmax increases), the experimental error in the data points will be more and more exaggerated, such that a minor experimental error associated with the data points corresponding to large D values would completely change the ®nal results [5]. This could only be avoided by having the data points ``weighted'' (i.e., evaluating the importance of the data points) before graphical representation, as discussed in detail in Ref. [5]. Calculation and assignment of the ``weight'' for each data point, however, requires a knowledge of the weight percentage corresponding to each data point, which necessitates a knowledge of the total weight of the powders [5]. In the conventional procedure, the total weight of the powders is substi- tuted using the truncated distribution. The appropriateness of this substitution is, in turn, determined by the reliability of the results obtained. 3.5. Arti®cial distribution The e€ect stemming from removal of coarse/®ne powders on the ®nal results was addressed by Irani [1, 6] from another point of view. According to Irani [1, 6], because of the removal of coarse/®ne powders, the data points in a probability paper plot asymptotically approach a line parallel to the abscissa (the probability scaled axis), rather than fall onto a straight line as expected. To eliminate or minimize the e€ect of removal of coarse/®ne powders, Irani [1, 6] suggested that the total weight should be a value greater than the weight of the corresponding truncated distribution. To determine this value, a reiteration method is employed: assuming a total weight, converting weight into weight Table 2. Characteristic parameters and ®tting coecients obtained using the conventional procedure for data in Table 1 with di€erent Dmax(mm) introduced Parameters d16(mm) d50(Dm)(mm) d84(mm) s Dmax=212 Dmax=150 Dmax=90 Dmax=63 48.26 47.37 42.88 38.40 72.21 71.09 59.59 48.47 108.05 106.69 82.81 61.19 0.403 0.406 0.329 0.233 Fitting coecient 99.703% 99.713% 99.952% 99.996% 622 LI and LAVERNIA: ANALYSIS OF SIEVING DATA percentage based on the assumed total weight, then graphing the converted weight percentage on a probability paper plot vs powder size, and assuming a new total weight ..., and so on, until the data graphed on the probability paper plot satisfactorily fall onto a straight line. This method, however, has three drawbacks. Firstly, as discussed in Section 3.3, data points falling onto a straight line alone does not ensure that the results are reliable. Although the discussion in Section 3.3 is under the assumption that the total weight of powders may be substituted by that of the truncated distribution, it is generally true that data points obeying log±normal distribution alone does not assure the reliability of the results, as will be shown in Section 4.4. Secondly, the intrinsic dif®culty associated with probability paper plot cannot be solved using this method, and hence still a€ects this method. Finally, this method failed to formulate any equations in determining the total value of powders from curve ®tting the experimental data, rendering the entire procedure dubious. 4. PROPOSED APPROACH As evident from the above discussion, along with several implicit, hard-to-be-justi®ed, assumptions, the conventional graphical representation procedure failed to address the reliability of its results. In the following sections, a new procedure will be formulated. The proposed procedure is capable of extracting the total weight of powders from experimental data, of determining the nature of size distribution, of characterizing the characteristic parameters, and simultaneously, of determining the reliability of its results. 4.1. E€ect of removal of coarse powders In this section, with Dmin being temporarily set to be zero, only the e€ect of Dmax will be considered. The e€ect of introducing the lower limit of powder size into the distribution will be discussed in Section 44.2. 4.1.1. Mathematical formulation. equations (1)±(8) correlate powder size with probability (equation (1)), or cumulative weight percentage under size (equations (6) and (8)). Experimental data, however, are absolute weight and powder size (refer to Table 1). This necessitates development of equations directly correlating the powder size with the absolute weight. The development of these equations may be readily accomplished by incorporating the total weight of powders into the analysis in equations (1)±(8). Assuming the total weight to be Wt, the cumulative weight percentage undersize D may be calculated as 9† Cp D†% ˆ Wunder D†=Wt ; where Wunder(D) is the cumulative weight undersize D. When Dmin=0, Wunder(D) is equivalent to the experimentally obtained cumulative weight undersize, WEunder (D). Substituting equation (9) into equation (8) yields   E Wunder D† : log D ˆ log Dm ‡ 0:434s
norm 100 Wt 10† Similarly, substituting equation (9) into equation (6) gives: Z lnDÿlnDm s 1 E Wunder D† ˆ Wt p exp ÿt2 =2†dt: 11† 2p ÿ1 When the total weight is known, equations (10) and (11) are essentially equivalent to equations (6) and (8). In this case, development of equations (10) and (11) is of little signi®cance. When the total weight is unknown, however, these two sets of equations are totally di€erent. While equations (6) and (8) may not be utilized to curve ®t the experimental data (absolute value), equations (10) and (11) can. More importantly, equations (10) and (11) enables extraction of the unknown total weight of powders, Wt, by curve ®tting the sieving experimental data. 4.1.2. Selection of governing equation for curve ®tting. In the last section, two equations were developed to correlate the powder size with the cumulative weight undersize: equation (10) which curve ®ts D as a function of WEunder(D); and equation (11) which curve ®ts WEunder(D) a function of D. Mathematically, equation (10) is equivalent to equation (11). However, experimental data are unavoidably associated with some errors, making these two equations di€erent from each other in terms of curve ®tting. In the present study, equation (11) is uniquely selected to be the governing equation in curve ®tting because of the following two reasons. Firstly, in curve ®tting, if the governing equation is selected to be y = f(x), then it is generally assumed that the independent variable, x, is known to be without error [7]. All the errors are in the dependent variable y [7]. In a sieving experiment, the powder size D is generally predetermined, i.e., free of error. The experimental error normally arises from the measurement of the cumulative weight undersize WEunder(D). Accordingly, compared with equation (10), equation (11), which expresses WEunder(D) as a function of D, is more suitable to be employed as the governing equation. Secondly, in curve ®tting, it is generally assumed that the errors associated with the dependent variable, y, are random [7]. In a sieving experiment, the error arising from the measurement of the cumulative weight undersize may be reasonably taken to be random. Therefore, selecting equation (11) as the governing equation is consistent with the above assumption in curve ®tting. If equation (10) is used, on the other hand, WEunder(D) would be taken as LI and LAVERNIA: ANALYSIS OF SIEVING DATA without error. Instead, any errors arising from WE under(D) would be evaluated in terms of D. This makes the error no longer random, as elucidated as follows. It is evident from Fig. 1 that, as x% deviates gradually from 50%, y = norm(x) increases (positive) or decrease (negative) more and more rapidly. Suppose there is a deviation Dx in the independent variable x, the corresponding deviation in y would depend on the value of x. Let the deviation in y be Dyv50 for x% = 50%, it would become 15Dyv50 if x% = 99% (or 1%), and 28Dyv50 if x% = 99.5% (or 0.5%), and so on, progressively [5]. The situation discussed here applies to equation (10) by simply substituting (logD ÿ logDm)/0.434s as y, and 100WEunder(D)/Wt as x. If any errors originating from 100WEunder(D)/ Wt, or from WEunder(D), are evaluated in terms of (logD ÿ logDm)/0.434s, or D, the results would be dependent on the magnitude of 100WEunder(D)/Wt. The larger 100WE under(D)/Wt, the larger the error. Accordingly, if the errors in 100WEunder(D)/Wt or WEunder(D) are random, the evaluated errors in terms of (logD ÿ logDm)/0.434s, or D, would no longer be random. In this case, the experimental point should be ``weighted'' (i.e., evaluating the importance of the data points) before curve ®tting [5, 7]. Calculation and assignment of the ``weight'' for each data point, however, requires the knowledge of the weight percentage corresponding to each data point, which necessitates the knowledge of the total weight of the powders [5]. Unfortunately, the latter, i.e., the total weight of the powders, is a variable to be determined, making the assignment of the weight impossible. Finally, a remark should be made regarding the conventional probability paper plot. Since one axis of this plot is in probability scale, which is y = norm(x), the error would be analyzed in terms of D, or norm[100WEunder(D)/Wt], rather than 100WEunder(D)/Wt, or WEunder(D). Accordingly, 623 curve ®tting in probability paper plot always encounters the problem associated with equation (10). This is the reason behind the intrinsic diculty in the probability paper plot approach mentioned earlier. 4.1.3. Realization of curve ®tting. It is evident that equation (11) is, at least, as complex as equation (6). The practical usefulness of equation (11) relies on the availability of a quick and e€ective method to curve ®t the experimental data using this equation. This may be realized in a KaleidaGraph software (version 3.0 or above). In the KaleidaGraph, there are two normal distribution related functions available: y = norm(x) and y = inorm(x). Function y = norm(x) was discussed earlier. Function y = inorm(x) is related to equation (11) as follows: Z 100 x exp ÿt2 =2†dt; 12† inorm x† ˆ p 2 ÿ1 which transforms equation (11) into:    1 1 D E : D† ˆ Wunder Wt inorm ln 100  Dm Governing equation (13) may be de®ned under the General Curve Fitting menu in KaleidaGraph Window. To ensure the de®nition being complete, the partial derivative of equation (13) relative to Wt, s, and Dm, should be given. This may be readily accomplished if one notices that  2 @inorm x† 100 x : 14† ˆ p exp ÿ 2 @x 2 4.1.4. Applications. Six sets of sieving data (Tables 1 and 3) from di€erent sources were analyzed using the formulated procedure, with special attention to its capability of providing reliable results. Table 3. Experimental sieving data to be analyzed Cumulative weight undersize (g) Opening (mm) 600 425 300 250 180 150 149 125 106 105 90 75 74 63 53 45 44 38 Source A 1667.6 1656.3 1642.6 1630.7 1578.0 1509.9 Ð 1392.7 1172.2 Ð 775.84 608.76 Ð 374.38 277.54 142.95 Ð 67.67 [11] 13† B C D E Ð 129.52 Ð 128.25 127.11 126.63 Ð 126.13 125.56 Ð 124.66 121.82 Ð 111.12 86.27 68.961 Ð 44.876 [12] Ð Ð Ð Ð Ð Ð Ð 678.6 619.7 Ð 561.5 512.3 Ð 341.7 243.5 195.1 Ð 56.8 [13] Ð Ð Ð 100 Ð Ð 59.0 Ð Ð 41.1 Ð Ð 27.9 24.5 Ð Ð 12.2 Ð Run 73N [14] Ð Ð Ð 101 Ð Ð 95.6 Ð Ð 87.3 Ð Ð 75.8 70.9 Ð Ð 52.9 Ð Run 69N [14] 624 LI and LAVERNIA: ANALYSIS OF SIEVING DATA Fig. 4. Cumulative weight undersize vs powder size plot for the data in Table 1 using the newly formulated procedure. The Dmax is set to be equal to that employed in the experiment. Figure 4 shows the results for the data in Table 1 with Dmax=212 mm, which is the Dmax employed in the experiment. It is evident there that the experimental data obeys the log±normal distribution. The total weight extracted is 132.021.0 g, with s = 0.449 20.010 and Dm=74.0 20.6 mm (Table 4). The results, including Wt, s, d50(Dm), the standard errors associated with each of them, and the ®tting coecient, obtained for the data in Table 3 with Dmax equal to the value employed in each experiment, which are 600, 425, 125, 250, and 250 mm for the data in columns A, B, C, D, and E of Table 3, respectively, were also provided in Table 4. It is of interest to note that the total weight, Wt, extracted (Table 4) for the data in columns A and B, which are 1661.1 and 127.6 g, respectively, are smaller than the experimentally measured cumulative weight undersize Dmax, which are 1667.6 and 129.52 g for columns A and B in Table 3, respectively. This may be understood as follows. According to Fig. 2, as the powder size increases to be much larger than Dm, the cumulative weight undersize, WEunder(D), is expected to approach, but always remain smaller than, Wt. However, the experimental data is always associated with some errors, which makes WEunder(D) smaller or larger than the value it is supposed to be in the idealized situation. Consequently, when WEunder(D) is very close to Wt, any minor experimental error may raise WEunder(D) to exceed Wt. It is worth noting that this type of experimental error could never be tolerated by equation (10), as explained as follows. The ®rst step of curve ®tting using equation (10) is to calculate norm[100WEunder (D)/Wt] as a function of the independent variable WEunder(D). Function norm[100WEunder(D)/Wt], by de®nition, requires WEunder(D)/Wt to be smaller than 1, i.e., WEunder(D) < Wt. When WEunder(D) > Wt, over¯ow would occur, and the curve ®tting would be terminated. This further demonstrates that equation (10) is only of theoretical signi®cance. In order to investigate the capability of determining the reliability of the results, a method similar to that in Section 3.3 was employed, i.e., introducing di€erent Dmax. The results are summarized in Table 5. As evident from Table 5, the characteristic parameters obtained are, in general, a function of Dmax. This again raises the question: when would the results be reliable. It is evident from Table 5 (the ®tting coecient column) that introducing di€erent Dmax does not a€ect the nature of size distribution, i.e., in all cases, the data points obey log± normal distribution. This suggests that, under the newly formulated procedure, the reliability of the Table 4. Characteristic parameters and ®tting coecients obtained using equation (13) for data in Tables 1 and 3 with Dmax (mm) equal to that employed in experiment Parameters Data Table 1 A, Table 3 B, Table 3 C, Table 3 D, Table 3 E, Table 3 Dmax 212 600 425 125 250 250 Wt(g) d50(Dm)(mm) s 132.02 1.00 1661.1 2 23.6 127.6 2 0.8 696.52 43.0 1301.3 22358.7 103.0 2 1.2 74.0 20.6 86.5 21.6 43.7 20.4 61.3 22.9 4431.3 213022 42.3 20.7 0.449 20.010 0.435 20.024 0.351 20.017 0.411 20.057 2.014 20.703 0.866 20.044 Fitting coe€. (%) 99.971 99.754 99.778 99.363 99.907 99.932 LI and LAVERNIA: ANALYSIS OF SIEVING DATA 625 Table 5. Characteristic parameters and ®tting coecients obtained using equation (13) for data in Tables 1 and 3 with di€erent Dmax(mm) introduced. Dmin is assumed to be zero in all cases Parameters Data Table 1 Table 3 col. A Table 3 col. B Table 3 col. C Table 3 col. D 73N Table 3 col. E 69N Dmax 212 150 90 63 600 300 180 125 90 63 425 250 150 90 63* 53* 125 90 63* 250* 149* 105* 250 149 105 Wt(g) 132.0 134.8 118.7 74.3 1661.1 1660.3 1719.9 2955.8 1195.8 454.2 127.6 127.3 127.0 129.5 166.6 99.5 696.5 667.3 367.3 1301.3 147.1 61.0 103.0 102.4 97.3 Error (%) 0.8 1.6 8.9 11.3 1.4 2.2 5.5 41 26 15 0.6 0.7 1 3.2 40 1 6.2 20 30 180 73 38 1.2 3 5.9 d50(Dm)(mm) 74.0 75.1 69.6 56.0 86.5 86.5 88.7 129.8 75.6 49.9 43.7 43.6 43.6 44.0 51.5 39.3 61.3 59.7 46.2 4431.3 198.9 77.4 42.3 42.1 40.4 results may not be determined by the nature of size distribution either, similar to the situation encountered under the conventional framework as discussed in Section 3.3. However, there are two ways to evaluate the reliability of the results under the newly formulated procedure. The reliability of the results may be evaluated, to some extent, by the standard errors associated with the characteristic parameters determined. When the standard error is very large, such as 30% or above, it is highly unlikely that the results are reliable. When the standard error is very small, such as 1% or less, it is reasonable to take the results as reliable. With this criterion, several sets of results, which were marked by asterisks, in Table 5 may be readily determined to be unreliable. Nevertheless, it is impossible to provide a number to unambiguously judge the results: when the error is above it, the result is unreliable; otherwise, reliable. Accordingly, when the standard error is moderate, such as 5±15%, it is dicult to conclusively determine the reliability of the results using this method. A more e€ective but somewhat time-consuming way relative to the method mentioned above is associated with the intrinsic feature of the log±normal size distribution. For a collection of powders obeying log±normal distribution, as Dmax increases further and further, the cumulative weight would ®nally tend to ¯atten o€ (refer to Fig. 4). Once this region is reached, any further increase in Dmax would have only slight e€ect on the cumulative weight. Consequently, the results obtained are expected to be close to each other thereafter. Accordingly, if the results, which are Wt, s, and d50(Dm) in the present study, corresponding to the Error (%) 0.8 1.3 5.3 4.7 1.8 2.3 4.5 31 15 5.8 0.9 0.9 1.1 2 26 1 4.7 12 12 290 94 35 1.7 2.9 4.7 s 0.449 0.462 0.418 0.304 0.435 0.435 0.458 0.646 0.444 0.246 0.351 0.348 0.346 0.363 0.497 0.270 0.411 0.387 0.232 2.014 1.129 0.688 0.866 0.852 0.762 Error (%) Fitting coe€. (%) 2.2 2.9 8.4 9.4 5.5 6.7 10 21 20 20 4.8 4.9 5.8 10 40 1 14 29 59 35 36 36 5.1 9.5 16 99.971 99.971 99.937 99.981 99.754 99.702 99.621 99.681 99.718 99.826 99.778 99.793 99.776 99.73 99.76 100 99.363 98.89 98.075 99.907 99.781 99.698 99.932 99.903 99.897 introduced Dmax were plotted as a function of Dmax, it is anticipated that all of the curves representing each of the characteristic parameters would ¯atten o€ at certain Dmax=Dfmax. If Dfmax is smaller than the Dmax employed in the experiment, the ¯attened regions may be observed; and hence the results obtained may be reliable. Otherwise, the ¯attened regions would be absent, and the results would be thought unreliable. With this criterion, the reliability of the results may be evaluated. Figures 5(a)±(f) show the results obtained as a function of Dmax for the six sets of data under analysis. For the convenience of graphical representation, all of the results, for any given set of data, were normalized by their corresponding maximum magnitude. For example, the maximum extracted total weight for the data in column B of Table 3 is 166.6 g, corresponding to an introduced Dmax of 63 mm. Accordingly, in Fig. 5(c), all Wt values were normalized by 166.6 g. It is evident from Fig. 5 that, except those in Fig. 5(e), all of the curves exhibit a ¯attened out region as Dmax increases. Moreover, for each speci®c set of data, the Dmax at which the curve begins to ¯atten is almost the same for all of the three parameters, Wt, s, and d50(Dm). For example, in Fig. 5(c), all of the three parameters tend to be relatively insensitive to the change of Dmax after Dmax increases to 90 mm and beyond. The extended ¯attened regions in Fig. 5(a)± (c) suggest that the data corresponding to these ®gures, which are the data in Table 1, columns A and B of Table 3, respectively, are highly sucient to yield reliable results. The limited ¯attened regions in Fig. 5(d) and (f) implies that the results obtained with Dmax equal to that employed in ex- 626 LI and LAVERNIA: ANALYSIS OF SIEVING DATA Fig. 5. The characteristic parameters (normalized) obtained under di€erent introduced Dmax for data in Table 1, (a), columns A (b), B (c), C (d), D (e), and E (f) of Table 3. periment are almost reliable. In these cases, even though it is not mandatory, more data points, i.e., larger Dmax employed in the experiment, would be helpful to gain more con®dence on the results. The absence of a ¯attened region in Fig. 5(e) indicates that the data in column D of Table 3 are not sucient to give any reliable results. Finally, it is of interest to note that before the ¯attened region was reached, the results may monotonously increase (Fig. 5(a)) or decrease (Fig. 5(c)), or vibrate back and forth (Fig. 5(b)). It is worthwhile to point out that the intrinsic feature of a log±normal size distribution employed to determine the reliability of the above section remains to be the same in the conventional procedure. However, in the conventional procedure, the intrinsic diculty associated with the probability paper plot makes the utilization of this feature almost impossible, as discussed earlier. As a simple example, the data in column B of Table 3 were analyzed using the conventional procedure. The results were shown in Fig. 6. No ¯attened regions similar to that in Fig. 5(b) were observed in Fig. 6. As Dmax increases, the curve corresponding to Wt did ¯atten o€. The curves corresponding to s and d50(Dm), however, did not ¯atten o€ as that of LI and LAVERNIA: ANALYSIS OF SIEVING DATA 627 Fig. 6. The characteristic parameters (normalized) obtained, using the conventional procedures, for the data in column B of Table 3 as a function of Dmax. Wt did. Instead, s keeps increasing, while d50(Dm) gradually decreases. When the results obtained are determined to be unreliable, such as the case corresponding to the data in column D of Table 3, a larger upper limit of powder size, Dmax, should be used in the experiment. However, sometimes this may be very challenging to be achieved in sieving experiments. The fact that the largest opening in the U.S. standard sieving set is 600 mm limits the availability of sieves with openings larger than 600 mm. Moreover, selection of the upper limit of powder size, Dmax, in real experiments is generally based on another consideration. In many situations, the spherical powders formed are mixed with irregularly shaped particles such as splats and coalesced particles. It is unsuitable to consider these particles as spherical powders. While complete separation of spherical powders from splats and coalesced powders is almost impossible, it is noticed that the presence of these particles is sparse when the powder size is not too large, and may be neglected. When powder size increases, presence of these particles becomes more frequent. In some cases, before the powder size increases to 600 mm, presence of splats and coalesced powders becomes so frequent that it can no longer be neglected compared with the total quantity of powders. Accordingly, a value smaller than 600 mm, for example 300 mm, is set to be the upper limit of powder size, below which presence of splats or coalesced powders can be neglected. Following this criterion of selection of Dmax, it may be unacceptable in some practical situations to raise Dmax to a value larger than the previously selected one, even though the newly selected value is smaller than 600 mm. In these cases, alternative existing charac- terization techniques, such as light scattering, should be employed, or innovative techniques should be explored if meaningful results are anticipated. 4.2. E€ect of dust separation by cyclone As mentioned earlier, the size distribution of the powders formed initially is generally a€ected by dust separation, for example in cyclones, which removes some very ®ne powders. The e€ect of dust separation by cyclone on the distribution of the powders is somewhat di€erent from that of the removal of coarse powders. In the removal of coarse powders, the e€ect is discrete. For example, if the powders are topped using a 425 mm sieve (35 mesh), powders with size larger than 425 mm would be removed, while those with size smaller than 425 mm would be left una€ected. In dust separation by cyclone, the e€ect is continuous, as shown in Fig. 7. Powders with powder size smaller than 1 mm would be almost completely removed. As the powder size increase, the powders would be partially removed, and the percentage being removed would gradually decrease to zero. This continuous feature greatly complicates the problem. To make the problem tractable, a discrete removal, similar to removal of coarse powders, will be assumed. This may be an acceptable assumption if the collection eciency curve is very steep, such as curve a in Fig. 7. 4.2.1. Governing equations. When powders smaller than the lower limit of powder size, Dmin, are removed, the experimentally obtained cumulative weight undersize WEunder(D), is nominal (refer to Fig. 2). In this case, WEunder(D) is the cumulative weight of powders with size in the range from Dmin to D, rather than from 0 to D. Therefore, 628 LI and LAVERNIA: ANALYSIS OF SIEVING DATA Fig. 7. Collection eciency of a cyclone as a function of powder size for (a) XQ120 cyclone and (b) XQ465 cyclone under identical conditions: gas ¯ow rate (Q), gas density (lg), gas absolute viscosity (mg), and particle speci®c gravity (ld) [9]. E D† ‡ Wunder Dmin †; Wunder D† ˆ Wunder 15† where Wunder(Dmin) is the cumulative weight under size Dmin. The cumulative weight under size Dmin, Wunder(Dmin), may be further explicitly expressed, in terms of Dmin, as follows: Z lnDÿlnDm  1 exp ÿt2 =2†dt Wunder Dmin † ˆ Wt p 2 ÿ1    1 1 Dmin ˆ Wt inorm ln : 16† Dm 100  With equations (15) and (16), equations (10) and (13) would take the following form log D ˆ log Dm ‡ 0:434
norm   WE D† ‡ Wunder Dmin † 100 under Wt ˆ log Dm ‡ 0:434  norm     WE D† 1 Dmin ; 100 under ‡ inorm ln Wt Dm  17† and E Wunder D† ˆ    1 1 D Wt inorm ln 100  Dm ÿ Wunder Dmin † (    1 1 D Wt inorm ln ˆ 100  Dmin   ) 1 Dmin ; ÿ inorm ln Dm  18† respectively. In selection of the governing equation for curve ®tting from equations (17) and (18), arguments similar to those in Section 4.1.2 apply here. Moreover, equations (17) is much more complex than equation (18) in terms of computer manipulation. Accordingly, equation (18) is uniquely selected as the governing equation. Its usage in a computer is similar to that discussed in Section 4.1.3. 4.2.2. Evaluation of the e€ect of Dmin. The lower limit of powder size, Dmin, is determined by design of the cyclone [8]. It may range from 1 mm to 100 mm, depending on the details of the design [8]. To illustrate its possible e€ect, Dmin=5, 10, and 30 mm will be considered in the present study. The experimental data in Table 1 and in columns A, B, C, and E of Table 3 were analyzed using equation (18), with Dmin in it set to be 5, 10, and 30 mm, and Dmax equal to the ones employed in each experiment. The results, along with those in Table 4 which correspond to the case with Dmin=0 mm, were summarized in Table 6. Data in column D of Table 3 were excluded from further studies because of the incorrect selection of Dmax, as discussed earlier. It is evident from Table 6 that the e€ect of Dmin on the ®nal results varies with the magnitude of Dmin itself, and with the powder collections under study. For the data in Table 1 and in columns A, B, C of Table 3, Dmin has little e€ect on the ®nal results when it is less than 10 mm. As Dmin increases to 30 mm, distinct e€ects were observed for all of these data, with the most prominent e€ect on the data in column B of Table 3. For the data in column E of Table 3, Dmin has slight e€ect on the ®nal results when it is 5 mm. As it increases to 10 mm, the e€ect becomes much more pronounced. When it is set to be 30 mm, the results are completely di€erent from those corresponding to Dmin=0. Moreover, the standard errors associated with the determined parameters are so huge that the results appear to be of little signi®cance. LI and LAVERNIA: ANALYSIS OF SIEVING DATA 629 Table 6. Characteristic parameters and ®tting coecients obtained using equation (18) for data in Tables 1 and 3. Dmax is the same as that employed in experiment. Dmin is assumed to be 0, 5, 10 and 30 mm Parameters Data Table 1 Table 3 col. A Table 3 col. B Table 3 col. C Table 3 col. E Wt(g) Dmin 0, 5, 30 0, 5, 30 0, 5, 30 0, 5, 30 0 5 10 30 10 10 10 10 132.0 137.1 1661.1 1675.1 127.6 204.5 696.5 777.2 103.0 103.8 110.4 1.2  105 Error (%) 0.8 1 1.4 1.6 0.7 15 6.2 12 1.2 1.4 2.6 4000 d50(Dm)(mm) 74.0 72.6 86.5 86.0 43.7 34.7 61.3 59.2 42.3 42.0 39.5 0.04 Table 6 only demonstrates the possible e€ect of Dmin on the ®nal results obtained. In practice, to evaluate the e€ect of dust separation by cyclone, the true Dmin corresponding to the cyclone should be evaluated and used in equation (18). Moreover, similar to the removal of coarse powders, if the Dmin given by the cyclone has extensive e€ect on the ®nal results such that the results obtained by assuming Dmin=0 is no longer reliable, Dmin should be adjusted to a lower value by using a di€erent cyclone. This may be very dicult in many situations because of the cost of cyclone. An alternative but also costly method is to employ new techniques, such as in-situ characterization techniques. 5. SUMMARY The procedure conventionally employed to interpret the experimental sieving data was examined. It was demonstrated that the conventional procedure is ¯awed from several standpoints. Along with several implicit, hard-to-be-justi®ed, assumptions associated with it, the conventional graphical representation procedure also failed to address the reliability of its results. To resolve these problems, a new procedure was formulated. Application of the formulated procedure to several sets of sample sieving data reveals that it is capable of extracting the total weight of powders from experimental data, of determining the nature of size distribution, of characterizing the characteristic parameters, and simultaneously, of determining the reliability of its results. AcknowledgementsÐThis study was ®nancially supported by the Ford Research Center (Dr Dawn White) and the Error (%) s 0.8 0.8 1.8 1.9 0.9 9.2 4.8 4.7 1.7 1.8 2.9 4500 0.449 0.475 0.435 0.439 0.351 0.456 0.411 0.477 0.866 0.871 0.906 2.095 Error (%) 2.2 2.7 5.6 6.3 4.8 13 14 21 5.1 5.2 5.9 308 Fitting coe€. (%) 99.971 99.973 99.754 99.737 99.778 99.617 99.363 99.412 99.932 99.933 99.939 98.241 Air Force Oce of Scienti®c Research (Grant No. F49620-97-1-0301). The authors would like to thank Dr R. J. Perez and Ms M. L. Lau for making their original sieving data available to this study. In addition, Professor E. J. Lavernia would also like to acknowledge the Alexander von Humboldt Foundation in Germany for support of his sabbatical visit at the Max-Planck-Institut Fur Metallforschung, in Stuttgart, Germany. REFERENCES 1. Irani, R. R. and Callis, C. F., Particle size: measurement, interpretation, and application. John Wiley & Sons, New York, 1963. 2. Allen, T., Particle size measurement. Chapman and Hall, London, 1981. 3. Grant, P. S., Cantor, B. and Katgerman, L., Acta Metall. Mater., 1993, 41, 3109. 4. Je€rey, A., Handbook of mathematical formulas and integrals. Academic Press, San Diego, 1995. 5. Kottler, F., J. Franklin Inst., 1950, 250, 419. 6. Irani, R. R., J. phys. Chem., 1959, 63, 1603. 7. Danial, C. and Wood, F. S., Fitting equations to data. John Wiley & Sons, New York, 1980. 8. Storch, O., Industrial separators for gas cleaning. Elsevier, Amsterdam, 1979. 9. Fisher-Klosterman Inc., XQ series high performance cyclones. Bulletin 218-C, 1980. 10. German, R. M., Powder metallurgy science, Princeton, NJ, 1984. 11. Lau, M. L., Huang, B., Perez, R. J., Nutt, S. R. and Lavernia, E. J., in Processing and properties of nanocrystalline materials, ed. C. Suryanarayana, J. Singh and F. H. Froes. TMS, Cleveland, OH, 1995, p. 255. 12. Perez, R. J., Huang, B-L., Crawford, P. J., Sharif, A. A. and Lavernia, E. J., Nanostruct. Mater., 1996, 7, 47. 13. Yang, N., Guthrie, S. E., Ho, S. and Lavernia, E. J., J. Mater. Synth. Proc., 1996, 4, 15. 14. Grandzol, R. J. and Tallmadge, J. A., AIChE J., 1973, 19, 1149.