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Apparent density of compressible food powders under storage conditions
Journal of Food Engineering 276 (2020) 109897
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Apparent density of compressible food powders under storage conditions C. Lanzerstorfer a a
University of Applied Sciences Upper Austria, School of Engineering/Environmental Sciences, Stelzhamerstraße 23, A-4600, Wels, Austria
A R T I C L E I N F O
A B S T R A C T
Keywords: Food powders Silo storage Apparent density Powder compressibility Wall friction angle
The stress-dependence of the apparent density and the wall friction angle of various food powders like salt, sugar, flour, starch and protein powders were studied using a ring shear tester. The approximation of the stressdependence of the apparent density using a power function showed a very good correlation (r2 > 0.97). The two parameters of the approximation function are the apparent density at 1.0 kPa and the exponent, which characterizes the compressibility of the powder. The first correlates very well with the apparent density measured according to ISO 697 or EN ISO 60 (r2 ¼ 0.98), while the second shows some correlation (r2 ¼ 0.76) with the particle size. Combining the powder properties apparent density, the mass median diameter and the spread of the particle size distribution in a power function allows a reasonably good estimate of the exponent (r2 ¼ 0.93). The wall friction angle usually decreases at higher values of the wall normal stress. This dependence can often be approximated using a simple function. However, for powders with a low stress-dependence of the wall friction angle it is better to use a constant average value. Thus, the apparent density of compressible food powders under storage conditions can be described well.
1. Introduction Large quantities of powders are produced by the food industry either as final food products or as ingredients in processed food. In food pro duction the process unit operations storage, conveying, dosing and mixing are important steps (Cuq et al., 2011; Pordesimo et al., 2009). Concerning these process steps a simple but important property of powders is their apparent density or bulk density. It can be measured according to different standards, for example EN ISO 60 (1999), EN 1236 (1995) or ISO 697 (1981). However, the apparent density of powders is not constant. It increases with the consolidation stress. The stress-dependence of the apparent density can be considerable, espe cially for fine-grained powders. Various instruments can be used for the measurement of the apparent density as a function of the consolidation stress. In a uniaxial compression test the powder is compressed in a cylindrical die by a force applied to a flat punch. The force-displacement curve is obtained by measuring the displacement at different forces. From these data the apparent density can be obtained as a function of the consolidation stress (Thomson, 1997). For example, Bian et al. (2015) used a vented piston assembly that compressed the sample under increasing normal stress to measure the compressibility of wheat powder. However, in uniaxial compression tests steady state flow cannot be reached during consolidation because of wall friction resulting in a smaller Mohr stress
circle. As a result, the measured value of the apparent density ρb is too low (Schwedes, 2003). Contrary to uniaxial tests, in ring shear testers a steady state flow during consolidation can is reached, which ensures correct apparent density measurements under stress. For the approximation of the stress-dependence of the apparent density of granular material various types of equations have been applied. In the simplest type a linear dependence of the apparent density ρb on the vertical stress σ was assumed in the form of ρb ¼ ρ0þk⋅σ (Latincsics, 1985; Ooi et al., 1996), where ρ0 is the apparent density at a vertical stress of zero. Clower et al. (1973) adopted a parabolic rela tionship (ρb ¼ ρ0þh⋅σ0.5). More flexible approaches were used by Bag ster (1977): ρb ¼ ρ0⋅(1þσ)n, by (Tomas, 2001): ρb ¼ ρ0⋅(1 þ A/σ)n and by Ramirez et al. (2010): ρb ¼ ρ0þA⋅σn. Common to all these equations is that at a vertical stress of zero the apparent density is equal to ρ0. However, with the exception of the last study by Ramirez et al. (2010) the correlation between these functions and the measured data was not very good. Peleg et al. (1973) as well as Søgaard et al. (2014) used a different type of function: ρb ¼ A þ B⋅ln(σ). For this function they found a good correlation with the measured data. However, this function in not defined when the value of the stress is zero. The same applies to the slightly different notation of this function ρb ¼ A þ B⋅ln(σ/σ0) used by Ehlermann and Schubert (1987), where σ0 is the reference stress (typi cally 1 kPa).
E-mail address: [email protected]. https://doi.org/10.1016/j.jfoodeng.2019.109897 Received 17 July 2019; Received in revised form 4 December 2019; Accepted 26 December 2019 Available online 27 December 2019 0260-8774/© 2019 Elsevier Ltd. All rights reserved.
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Journal of Food Engineering 276 (2020) 109897
Under low compressive stress, as in powder storage, a good corre lation for the relationship between the apparent density and the consolidation stress was reported for the logarithmic function (Schubert, 1987; Thomson, 1997): � � � � ρ σ (1) log b ¼ c⋅log
Table 1 Food powder samples investigated. Sample no.
Flour and starch F1 Wheat flour, fine, type W 480 F2 Wheat flour, type W 480 F3 Wheat flour, coarse, type W 480 F4 Wheat flour, type 1050 Bio F5 Rey flour, type 1150 F6 Rice flour, full grain F7 Wheat semolina F8 Durum wheat semolina (69% carbohydrate, 11% protein) F9 Polenta (76% carbohydrate, 8% protein) F10 Maize starch F11 Potato starch Sugar and sweetener S1 Icing sugar S2 Crystal sugar, fine S3 Crystal sugar S4 Dextrose (91% carbohydrate) S5 Xylit (E967), fine S6 Xylit (E967) Inorganic (Salt and soda) I1 Baking-soda (Sodium bicarbonate) I2 Baking-powder (Disodium diphosphate and sodium bicarbonate; 22% starch) I3 Table salt (containing separating agent) I4 Sea-salt (without separating agent) Protein powder P1 Skimmed milk powder (37% protein, 1% fat, 51% carbohydrate) P2 Soy flour (41% protein, 21% fat) P3 Pea protein (80% protein) P4 Flax seed meal (33% protein, 10% fat, 35% fibre) Various food powders V1 Paprika powder (15% protein, 13% fat, 21% carbohydrate) V2 Cocoa powder (23% protein, 20% fat, 14% carbohydrate, 34% fibre) V3 Citric acid (E330)
σ0
ρb;0
Food powder
In this equation the apparent density ρb,0 is at a vertical stress of σ0. The equation was already used by Ooms (1981) in a different notation: � �c σ ρb ¼ ρb;0 ⋅ (2)
σ0
The constant c in equations (1) and (2) characterizes the mechanical compressibility of the powder. However, this function is also not defined at zero stress. Nevertheless, this equation was also used in a recent study (Lanzerstorfer, 2017) investigating the stress-dependence of the apparent density of various powders. A good correlation between the approximation function and the measured data was found. Additionally, it was demonstrated that equation (2) can be integrated into the deri vation of Janssen’s equation for the consolidation stress in a powder stored in a silo (Janssen, 1895). This resulted in a modified equation in which the stress-dependence of the apparent density is considered. �novas, In most particle flow considerations (Juliano and Barbosa-Ca 2010) as well as in the derivation of Janssen’s equation a constant wall friction angle is assumed. However, the few data available for the stress-dependence of the wall friction angle of food powders show a noticeable stress-dependence. Fitzpatrick (2007) found a slight depen dence of the wall friction angle of various types of milk powders with stainless steel 304. He found decreasing wall friction angles with increasing wall normal stress. Similar behaviour was reported for flour, whey and tea (Iqbal and Fitzpatrick, 2006). The aim of the present study was to determine the stress-dependence of the apparent density and the wall friction angle of various food powders. A shear tester was used for determination of the apparent density at various values of the consolidation stress in order to achieve reliable results. In contrast to other shear tests with food powders where a Jenike shear cell was used (Fitzpatrick et al., 2004; Iqbal and Fitzpa � ski et al., 2012), in this study a ring shear tester was trick, 2006; Opalin used. Afterwards simple equations were sought to approximate the stress-dependence of both parameters to be able to describe the apparent density of stored food powders in dependence of the distance to the surface.
a
Moisture content in % 10.7 11.0 10.4 9.7 9.5 13.2 11.4 12.7 8.1 11.7 19.0 0.3 0.1 0.2 8.7 0.16 0.04 0.02a 2.0a 0.03 0.05 3.0a 7.6 6.6 8.8 8.9a 1.5a 0.2
Dried in a desiccator under vacuum at room temperature for 72 h
mm was determined according to EN ISO 60 (1999). The bottom cover of a funnel is removed to allow 120 cm3 of powder stored in the funnel to flow by gravity into a coaxial 100 cm3 measuring cylinder. The excess material is removed by drawing a straight blade across the top of the measuring cylinder. For the coarser powders the apparent density was determined according to ISO 697 (1981), where a 500 cm3 measuring cylinder is used. Deviating from the standards, three instead of two measurements were carried out. The results show the arithmetic mean and the standard deviation. For determination of the apparent density under varying consolida tion stress σ1 a Schulze RST-XS ring shear tester with a 30 cm3 shear cell was used. Fig. 1a shows an illustration of the instrument. During the test the powder sample is loaded vertically with the normal force FN via the lid at a certain normal stress and then a shear deformation is applied to the sample by moving the shear cell at a constant angular velocity ω. The lid is kept in position by the two tie rods. The required horizontal force FH is measured for calculation of the shear stress in the sample. The volume of the sample is calculated from the measured height of the sample in the shear cell. Fig. 1b shows exemplarily the measured yield locus and the consolidation stress σ1. For powders with a d50 > 500 μm a larger shear tester was used (RST-01.pc with a 900 cm3 shear cell). The measurements were performed at four values of the normal stress (0.6 kPa, 2.0 kPa, 6.0 kPa and 20 kPa). The calibration of the shear tester was verified using the certified reference material BCR-116 (Limestone Powder from the Community Bureau of Reference). For four randomly selected powders (F4, S2, I4 and P2) the measurements were repeated twice. The maximum relative standard deviation for the density was
2. Materials and methods The various food powders investigated in this study are summarized in Table 1. The particle size distribution of the coarse powders was determined by sieve analysis (Fritsch ANALYSETTE 3 PRO laboratory sieve shaker). The used sieves were 2.0 mm, 1.7 mm, 1.25 mm, 1.0 mm, 800 μm, 630 μm, 500 μm, 400 μm and 315 μm. For powders with a maximum particle size less than 800 μm a laser diffraction instrument with dry sample dispersion was used (Sympatec HELOS/RODOS). The instrument was checked with a Sympatec SiC–P6000 06 standard. The mass median diameter d50 of the size distribution was calculated by interpolation using the two neighbouring points of the size distribution. The spread of the particle size distribution is the ratio of d90 to d10 (Rumpf, 1990), where d90 is the particle size with 90% of the mass of the powder consisting of particles smaller than this size. The d10 is defined in a similar way. The moisture content of most food powder samples was measured gravimetrically (Infrared moisture analyser Sartorius MA35M). The samples were dried at 105 � C until constant weight was reached. Food powders unstable at this condition were dried in a desiccator under vacuum at room temperature for 72 h. The apparent density of the food powder samples with a d50 < 1.0 2
C. Lanzerstorfer
Journal of Food Engineering 276 (2020) 109897
Fig. 1. Cut-away illustration of the Schulze RST-XS ring shear tester (a) and yield locus with consolidation stress σ1 (b).
4.4% and the average standard deviation was 1.7%. The wall friction angles were determined with the ring shear testers using wall friction shear cells, where the bottom ring was formed by a sample of stainless steel grade 1.4301. The values of the applied wall normal stress were 0.24 kPa, 0.6 kPa, 2.0 kPa, 6.0 kPa and 20 kPa. The maximum standard deviation for the wall friction angle of four randomly selected powders (F7, S5, I2 and P3) was 1.5� and the average standard deviation was 0.8� .
of the spread of the particle size distributions, are summarized in Table 2, columns 2 to 4. Fig. 2 shows the apparent density over the consolidation stress with both axes on a logarithmic scale. This stress-dependence of the apparent density was approximated by equations of the type of Eq. (2) with a value for σ0 of 1.0 kPa. As visible in Fig. 2 the approximation function fitted very well for all food powders. The values of the parameters of the approximation function, ρb,0 and c, as well as the respective correlation coefficients r2 are also shown in Table 2. All correlation coefficients were higher than 0.97 and the average correlation coefficient was 0.993 � 0.010. The estimate of the parameters ρb,0 and c by easy-to-measure powder characteristics like the powder properties apparent density measured according to EN ISO 60 or ISO 697, mass median diameter and spread of the particle size distribution was investigated.
3. Results and discussion 3.1. Dependence of the apparent density on the consolidation stress The apparent densities, ρb,N, measured according to EN ISO 60 or ISO 697, as well as the mass median diameters d50 and the calculated values
Table 2 Food powder samples: measured properties and coefficients of the approximation function. Sample no.
Apparent density ρb,N EN ISO 60 or ISO 697 in kg/m3
Mass median diameter d50 in μm
Spread d90/ d10
Approximation function
ρb,0 at a consolidation stress of 1.0 kPa in
c
r2
kg/m3
Flour and starch F1 579 � 2 F2 626 � 5 F3 704 � 6 F4 530 � 1 F5 496 � 2 F6 620 � 3 F7 755 � 1 F8 705 � 2 F9 756 � 2 F10 586 � 7 F11 696 � 1 Sugar and sweetener S1 535 � 2 S2 881 � 3 S3 824 � 2 S4 640 � 1 S5 760 � 5 S6 830 � 3 Inorganic (Salt and soda) I1 1140 � 4 I2 757 � 6 I3 1220 � 2 I4 1170 � 2 Protein powders P1 546 � 1 P2 338 � 7 P3 324 � 2 P4 390 � 2 Various food powders V1 461 � 2 V2 543 � 5 V3 886 � 1
64 88 130 55 37 180 660 300 840 13 38
9.2 10 11 11 39 8.4 1.5 7.4 2.4 2.8 3.3
636 670 734 566 534 591 759 688 726 575 646
0.0755 0.0654 0.0543 0.0949 0.0951 0.0593 0.0104 0.0476 0.0080 0.0695 0.0724
0.998 0.999 1.000 0.997 0.999 0.997 0.999 0.970 0.995 0.994 0.980
28 510 1040 230 92 640
40 3.2 2.2 5.8 46 4.0
575 865 818 644 762 837
0.1104 0.0080 0.0050 0.0235 0,0625 0.0236
1.000 0.997 0.979 0.990 0,999 0.991
97 20 480 570
5.0 17 2.2 3.9
1110 804 1230 1140
0.0354 0.0719 0.0032 0.0063
0.989 0.999 0.998 0.996
205 37 52 120
4.4 12.5 5.8 13.5
513 380 366 423
0,0339 0.1105 0.0650 0.0979
0,962 0.996 0.997 1.000
210 26 490
5.5 3.1 1.8
466 470 886
0.0749 0.0570 0.0145
0.999 0.999 0.974
3
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Journal of Food Engineering 276 (2020) 109897
Fig. 2. Stress-dependence of the apparent density of the different food powders; a: exponent c < 0.3; b: 0.3 < c < 0.6; c: 0.6 < c<0.8; d: c>0.8.
The apparent density ρb,0 at a σ0 of 1.0 kPa can be approximated by the apparent density of the material ρb,N measured according to EN ISO 60 or ISO 697. Fig. 3 shows this correlation which can be expressed by the following equation (Eq. (3)). For this approximation a correlation coefficient of 0.981 was obtained. (3)
ρb;0 ¼ 1:004⋅ρb;N
Exponent c in the approximation equations varied between 0.0032 and 0.11. For coarser powders lower values of the exponent were found (Fig. 4a). However, the correlation between the exponent and the mass median diameter of the powders is not very strong (r2 ¼ 0.76). Combining the powder properties apparent density measured according to EN ISO 60 or ISO 697, mass median diameter and spread of the particle size distribution in a power function according to Eq. (4), led to a considerably better fit. � �E3 d90 F ¼ ðρb;N ÞE1 ⋅ðd50 ÞE2 ⋅ (4) d10 For maximization of the correlation the values for the exponents were derived by minimization of the sum of the square of errors. The best correlation of c with F was found with the values of 1.0, 0.38 and 0.30 for the exponents E1, E2 and E3, respectively (Fig. 4b). The correlation for the approximation function (Eq. (5)) was 0.931. c ¼ 0:114⋅expð
0:000329 ⋅ FÞ
(5)
Fig. 3. Correlation of the apparent density at 1.0 kPa and the apparent density measured according to EN ISO 60 or ISO 697.
3.2. Dependence of the wall friction angle on the consolidation stress The wall friction measurements showed that for some food powders 4
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Journal of Food Engineering 276 (2020) 109897
Fig. 4. Correlation between the exponent c and the mass median diameter and correlation between the exponent c and F.
the wall friction angle strongly depends on the stress, while for others the stress-dependence is low and mainly limited to very low values of the wall normal stress (Fig. 5). This difference might be related to powder geometry. Generally, the wall friction angle ϕW decreased with increasing wall normal stress. Only in a few cases was a slight increase observed at
higher values of the stress. The stress-dependence of the wall friction angles was approximated by functions of the type � � σN k (6) tanðφW Þ ¼ W⋅
σ 0;N
Fig. 5. Wall friction angle as a function of the wall normal stress for various food powders: a) Flour and starch, b) Sugar and sweetener, c) Other food powders, d) Tangent of the wall friction angle and its approximation. 5
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Journal of Food Engineering 276 (2020) 109897
where σN is the wall normal stress, σ0,N is the reference wall normal stress (1.0 kPa) and the constant W and the exponent k are dimension less parameters experimentally obtained for each studied powder. Table 3 shows the results for W and k together with the correlation coefficients. The correlation was quite good (r2 > 0.9) for most food powders. For starch and icing sugar the correlation coefficients were between 0.8 and 0.9. Lower values for the correlation coefficient were found only for some coarse food powders with a mass median diameter larger than 480 μm (F7, S2, S3, S6, I3, I4, V3). The value of the exponent was in the range of 0.031–0.328. The higher the value of the exponent the stronger is the size dependence of the wall friction angle. No distinct correlation of the exponent with the particle size was found. However, the dependence seems to be less for inorganic and coarse powders. Also for parameter W of the approximation function for the wall friction angle no distinct relation between the parameter and the par ticle size of the powders was noticed. For several food powders (F3, F7, F9, F10, F11, S1, S2, S3, S4, S6, I2, I3, I4, P1, P3 and V3) the assumption of a constant wall friction angle would be quite feasible, because the stress-dependence is small for a wall stress higher than 2 kPa. For these powders the value of k was between zero and 0.12. Fortunately, all materials with a weak correlation of the proposed approximation function belong to this group of food powders. For some other food powders (F4, F5, F8, P2, P4, V1 and V2) the wall friction angle decreased by at least half when the wall normal stress was raised from 2 kPa to 20 kPa. For these materials the exponent in the approximation function, k, was higher than 0.29 and the correlation of the approximation function was very good (>0.96). For the powders F1, F2, F6, S5 and I1 the value of k was in a medium range (0.12 < k < 0.29) and the correlation of the approximation function was quite good (>0.91).
3.3. Calculation of the apparent density in stored compressible food powders The calculation of the apparent density of a compressible powder is based on the model introduced by Janssen (1895) extended by the stress-dependent apparent density and a stress-dependent wall friction angle based on the proposed approximation functions. First, it has to be distinguished between storage in a pile, where the stress results from the weight of the material above, and silo storage, where wall friction limits the stress in the bulk. In a pile of compressible powder the apparent density ρb as a function of the distance to the surface z results from equation (7): �
ρb ðzÞ ¼
Flour and starch F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 Sugar and sweetener S1 S2 S3 S4 S5 S6 Inorganic powders I1 I2 I3 I4 Protein powders P1 P2 P3 P4 Various food powders V1 V2 V3
W
k
r2
0.214 0.233 0.385 0.324 0.339 0.371 0.403 0.339 0.281 0.519 0.216
0.212 0.182 0.104 0.329 0.315 0.166 0.060 0.315 0.004 0.040 0.117
0.989 0.975 0.940 0.993 0.988 0.910 0.976 0.980 0.935 0.879 0.818
0.302 0.330 0.376 0.377 0.407 0.293
0.104 0.057 0.031 0.099 0.160 0.117
0.886 0.743 0.530 0.932 0.962 0.818
0.391 0.490 0.342 0.354
0.212 0.058 0.043 0.074
0.997 0.967 0.795 0.798
0.262 0.581 0.337 0.302
0.083 0.328 0.048 0.336
0.922 0.983 0.947 0.990
0.339 0.322 0.352
0.311 0.290 0.048
0.984 0.963 0.712
�1 1 c ⋅ðð1
cÞ⋅g⋅zÞ1
c
c
(7)
The exponent c is obtained from the stress-dependence of the apparent density. ρb,0 is the apparent density at a reference consolida tion stress σ0 of 1.0 kPa. In a silo with the cross section of the silo shaft A, the inner circum ference of the silo shaft U, a constant lateral stress ratio λ ¼ σh/σv and a constant wall friction angle the apparent density in the cylindrical shaft can be calculated using equation (8): 0 0 111 c c � �1 1 c ρb;0 g U C B ρb ðzÞ ¼ ⋅@ U ⋅@1 e λ⋅ A ⋅tanφw ⋅ð1 cÞ⋅z AA (8) λ⋅ A ⋅tanφw σc0 When a stress-dependence of the wall friction angle has also to be considered according to equation (6), the differential equation (9) cannot be solved analytically. However, the relation can be used in numerical simulations together with equations (2) and (6). Thereby, σ ¼ σv and σN ¼ σh. g⋅σρ0c ⋅σ cv
Table 3 Food powder samples: coefficients of the approximation function for the wall friction angle. Sample no.
ρb;0 σc0
0
1 λ1 k ⋅UA⋅σWk ⋅σ 1v
k
⋅ dσv ¼ dz
(9)
0;N
Fig. 6a shows exemplarily the calculated results for the apparent density of soy flour (P2) for different storage conditions. Its wall friction angle strongly depends on the stress. In Fig. 6b the results for maize starch (F10), a material with a nearly constant wall friction angle, are shown. For silo storage the following assumptions were made: an inner diameter of the silo of 3 m and a constant lateral stress ratio λ of 0.4. For the soy flour the density was calculated also for a constant average wall friction angle of 31� . The blue line shows the apparent density of the material in a pile as a function of the distance from the surface. The violet line shows the result for silo storage, assuming a constant wall friction angle. For soy flour a numerical calculation of the apparent density with the stress dependent wall friction angle was performed (dashed red line). The resulting values of the apparent density were slightly higher than those from the calcu lation with a constant average wall friction angle. 4. Conclusions The apparent density as well as the wall friction angle of food powders are stress-dependent, which can be expressed well using simple approximation functions. For the apparent density an approximation function already established in the literature fits very well. For approximation of the wall friction angle a new simple function was proposed. For materials with a high stress-dependence of the wall fric tion angle this function correlates well with the result of the measure ments. For materials with a low stress-dependence of the wall friction angle the fit was limited. However, for these materials a constant average value of the wall friction angle can be used. Depending on the storage circumstances, three different equations for calculation of the apparent density have to be used: 6
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Journal of Food Engineering 276 (2020) 109897
Fig. 6. Calculated results for the apparent density of soy flour (a) and maize starch (b).
- Storage of a compressible powder in a pile; the apparent density of the powder in dependence of the height can be calculated using the proper equation. - Storage of a compressible powder in a silo and approximately con stant wall friction angle: the apparent density can be calculated using the shown equation. - Storage of a compressible powder in a silo; stress-dependence of the wall friction angle has to be considered: the apparent density can be calculated numerically.
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Ooms, M., 1981. Determination of contents in storage bins and silos. Bulk. Solids Handl. 1 (4), 439–453. Opali� nski, I., Chutkowski, M., Stasiak, M., 2012. Characterizing moist food-powder flowability using a Jenike shear-tester. J. Food Eng. 108, 51–58. https://doi.org/ 10.1016/j.jfoodeng.2011.07.031. Peleg, M., Mannheim, C.H., Passy, N., 1973. Flow properties of some food powders. J. Food Sci. 38 (6), 959–964. Pordesimo, L.O., Onwulata, C.I., Carvalho, C.W.P., 2009. Food powder delivery through a feeder system: effect of physicochemical properties. Int. J. Food Prop. 12 (3), 556–570. https://doi.org/10.1080/10942910801947748. Ramirez, A., Moya, M., Ayuga, F., 2010. Determination of the mechanical properties of powdered agricultural products and sugar. Part. Part. Syst. Charact. 26 (4), 220–230. https://doi.org/10.1002/ppsc.200800016. Rumpf, H., 1990. Particle Technology. Chapman and Hall, London.
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Apparent density of compressible food powders under storage conditions C. Lanzerstorfer a a
University of Applied Sciences Upper Austria, School of Engineering/Environmental Sciences, Stelzhamerstraße 23, A-4600, Wels, Austria
A R T I C L E I N F O
A B S T R A C T
Keywords: Food powders Silo storage Apparent density Powder compressibility Wall friction angle
The stress-dependence of the apparent density and the wall friction angle of various food powders like salt, sugar, flour, starch and protein powders were studied using a ring shear tester. The approximation of the stressdependence of the apparent density using a power function showed a very good correlation (r2 > 0.97). The two parameters of the approximation function are the apparent density at 1.0 kPa and the exponent, which characterizes the compressibility of the powder. The first correlates very well with the apparent density measured according to ISO 697 or EN ISO 60 (r2 ¼ 0.98), while the second shows some correlation (r2 ¼ 0.76) with the particle size. Combining the powder properties apparent density, the mass median diameter and the spread of the particle size distribution in a power function allows a reasonably good estimate of the exponent (r2 ¼ 0.93). The wall friction angle usually decreases at higher values of the wall normal stress. This dependence can often be approximated using a simple function. However, for powders with a low stress-dependence of the wall friction angle it is better to use a constant average value. Thus, the apparent density of compressible food powders under storage conditions can be described well.
1. Introduction Large quantities of powders are produced by the food industry either as final food products or as ingredients in processed food. In food pro duction the process unit operations storage, conveying, dosing and mixing are important steps (Cuq et al., 2011; Pordesimo et al., 2009). Concerning these process steps a simple but important property of powders is their apparent density or bulk density. It can be measured according to different standards, for example EN ISO 60 (1999), EN 1236 (1995) or ISO 697 (1981). However, the apparent density of powders is not constant. It increases with the consolidation stress. The stress-dependence of the apparent density can be considerable, espe cially for fine-grained powders. Various instruments can be used for the measurement of the apparent density as a function of the consolidation stress. In a uniaxial compression test the powder is compressed in a cylindrical die by a force applied to a flat punch. The force-displacement curve is obtained by measuring the displacement at different forces. From these data the apparent density can be obtained as a function of the consolidation stress (Thomson, 1997). For example, Bian et al. (2015) used a vented piston assembly that compressed the sample under increasing normal stress to measure the compressibility of wheat powder. However, in uniaxial compression tests steady state flow cannot be reached during consolidation because of wall friction resulting in a smaller Mohr stress
circle. As a result, the measured value of the apparent density ρb is too low (Schwedes, 2003). Contrary to uniaxial tests, in ring shear testers a steady state flow during consolidation can is reached, which ensures correct apparent density measurements under stress. For the approximation of the stress-dependence of the apparent density of granular material various types of equations have been applied. In the simplest type a linear dependence of the apparent density ρb on the vertical stress σ was assumed in the form of ρb ¼ ρ0þk⋅σ (Latincsics, 1985; Ooi et al., 1996), where ρ0 is the apparent density at a vertical stress of zero. Clower et al. (1973) adopted a parabolic rela tionship (ρb ¼ ρ0þh⋅σ0.5). More flexible approaches were used by Bag ster (1977): ρb ¼ ρ0⋅(1þσ)n, by (Tomas, 2001): ρb ¼ ρ0⋅(1 þ A/σ)n and by Ramirez et al. (2010): ρb ¼ ρ0þA⋅σn. Common to all these equations is that at a vertical stress of zero the apparent density is equal to ρ0. However, with the exception of the last study by Ramirez et al. (2010) the correlation between these functions and the measured data was not very good. Peleg et al. (1973) as well as Søgaard et al. (2014) used a different type of function: ρb ¼ A þ B⋅ln(σ). For this function they found a good correlation with the measured data. However, this function in not defined when the value of the stress is zero. The same applies to the slightly different notation of this function ρb ¼ A þ B⋅ln(σ/σ0) used by Ehlermann and Schubert (1987), where σ0 is the reference stress (typi cally 1 kPa).
E-mail address: [email protected]. https://doi.org/10.1016/j.jfoodeng.2019.109897 Received 17 July 2019; Received in revised form 4 December 2019; Accepted 26 December 2019 Available online 27 December 2019 0260-8774/© 2019 Elsevier Ltd. All rights reserved.
C. Lanzerstorfer
Journal of Food Engineering 276 (2020) 109897
Under low compressive stress, as in powder storage, a good corre lation for the relationship between the apparent density and the consolidation stress was reported for the logarithmic function (Schubert, 1987; Thomson, 1997): � � � � ρ σ (1) log b ¼ c⋅log
Table 1 Food powder samples investigated. Sample no.
Flour and starch F1 Wheat flour, fine, type W 480 F2 Wheat flour, type W 480 F3 Wheat flour, coarse, type W 480 F4 Wheat flour, type 1050 Bio F5 Rey flour, type 1150 F6 Rice flour, full grain F7 Wheat semolina F8 Durum wheat semolina (69% carbohydrate, 11% protein) F9 Polenta (76% carbohydrate, 8% protein) F10 Maize starch F11 Potato starch Sugar and sweetener S1 Icing sugar S2 Crystal sugar, fine S3 Crystal sugar S4 Dextrose (91% carbohydrate) S5 Xylit (E967), fine S6 Xylit (E967) Inorganic (Salt and soda) I1 Baking-soda (Sodium bicarbonate) I2 Baking-powder (Disodium diphosphate and sodium bicarbonate; 22% starch) I3 Table salt (containing separating agent) I4 Sea-salt (without separating agent) Protein powder P1 Skimmed milk powder (37% protein, 1% fat, 51% carbohydrate) P2 Soy flour (41% protein, 21% fat) P3 Pea protein (80% protein) P4 Flax seed meal (33% protein, 10% fat, 35% fibre) Various food powders V1 Paprika powder (15% protein, 13% fat, 21% carbohydrate) V2 Cocoa powder (23% protein, 20% fat, 14% carbohydrate, 34% fibre) V3 Citric acid (E330)
σ0
ρb;0
Food powder
In this equation the apparent density ρb,0 is at a vertical stress of σ0. The equation was already used by Ooms (1981) in a different notation: � �c σ ρb ¼ ρb;0 ⋅ (2)
σ0
The constant c in equations (1) and (2) characterizes the mechanical compressibility of the powder. However, this function is also not defined at zero stress. Nevertheless, this equation was also used in a recent study (Lanzerstorfer, 2017) investigating the stress-dependence of the apparent density of various powders. A good correlation between the approximation function and the measured data was found. Additionally, it was demonstrated that equation (2) can be integrated into the deri vation of Janssen’s equation for the consolidation stress in a powder stored in a silo (Janssen, 1895). This resulted in a modified equation in which the stress-dependence of the apparent density is considered. �novas, In most particle flow considerations (Juliano and Barbosa-Ca 2010) as well as in the derivation of Janssen’s equation a constant wall friction angle is assumed. However, the few data available for the stress-dependence of the wall friction angle of food powders show a noticeable stress-dependence. Fitzpatrick (2007) found a slight depen dence of the wall friction angle of various types of milk powders with stainless steel 304. He found decreasing wall friction angles with increasing wall normal stress. Similar behaviour was reported for flour, whey and tea (Iqbal and Fitzpatrick, 2006). The aim of the present study was to determine the stress-dependence of the apparent density and the wall friction angle of various food powders. A shear tester was used for determination of the apparent density at various values of the consolidation stress in order to achieve reliable results. In contrast to other shear tests with food powders where a Jenike shear cell was used (Fitzpatrick et al., 2004; Iqbal and Fitzpa � ski et al., 2012), in this study a ring shear tester was trick, 2006; Opalin used. Afterwards simple equations were sought to approximate the stress-dependence of both parameters to be able to describe the apparent density of stored food powders in dependence of the distance to the surface.
a
Moisture content in % 10.7 11.0 10.4 9.7 9.5 13.2 11.4 12.7 8.1 11.7 19.0 0.3 0.1 0.2 8.7 0.16 0.04 0.02a 2.0a 0.03 0.05 3.0a 7.6 6.6 8.8 8.9a 1.5a 0.2
Dried in a desiccator under vacuum at room temperature for 72 h
mm was determined according to EN ISO 60 (1999). The bottom cover of a funnel is removed to allow 120 cm3 of powder stored in the funnel to flow by gravity into a coaxial 100 cm3 measuring cylinder. The excess material is removed by drawing a straight blade across the top of the measuring cylinder. For the coarser powders the apparent density was determined according to ISO 697 (1981), where a 500 cm3 measuring cylinder is used. Deviating from the standards, three instead of two measurements were carried out. The results show the arithmetic mean and the standard deviation. For determination of the apparent density under varying consolida tion stress σ1 a Schulze RST-XS ring shear tester with a 30 cm3 shear cell was used. Fig. 1a shows an illustration of the instrument. During the test the powder sample is loaded vertically with the normal force FN via the lid at a certain normal stress and then a shear deformation is applied to the sample by moving the shear cell at a constant angular velocity ω. The lid is kept in position by the two tie rods. The required horizontal force FH is measured for calculation of the shear stress in the sample. The volume of the sample is calculated from the measured height of the sample in the shear cell. Fig. 1b shows exemplarily the measured yield locus and the consolidation stress σ1. For powders with a d50 > 500 μm a larger shear tester was used (RST-01.pc with a 900 cm3 shear cell). The measurements were performed at four values of the normal stress (0.6 kPa, 2.0 kPa, 6.0 kPa and 20 kPa). The calibration of the shear tester was verified using the certified reference material BCR-116 (Limestone Powder from the Community Bureau of Reference). For four randomly selected powders (F4, S2, I4 and P2) the measurements were repeated twice. The maximum relative standard deviation for the density was
2. Materials and methods The various food powders investigated in this study are summarized in Table 1. The particle size distribution of the coarse powders was determined by sieve analysis (Fritsch ANALYSETTE 3 PRO laboratory sieve shaker). The used sieves were 2.0 mm, 1.7 mm, 1.25 mm, 1.0 mm, 800 μm, 630 μm, 500 μm, 400 μm and 315 μm. For powders with a maximum particle size less than 800 μm a laser diffraction instrument with dry sample dispersion was used (Sympatec HELOS/RODOS). The instrument was checked with a Sympatec SiC–P6000 06 standard. The mass median diameter d50 of the size distribution was calculated by interpolation using the two neighbouring points of the size distribution. The spread of the particle size distribution is the ratio of d90 to d10 (Rumpf, 1990), where d90 is the particle size with 90% of the mass of the powder consisting of particles smaller than this size. The d10 is defined in a similar way. The moisture content of most food powder samples was measured gravimetrically (Infrared moisture analyser Sartorius MA35M). The samples were dried at 105 � C until constant weight was reached. Food powders unstable at this condition were dried in a desiccator under vacuum at room temperature for 72 h. The apparent density of the food powder samples with a d50 < 1.0 2
C. Lanzerstorfer
Journal of Food Engineering 276 (2020) 109897
Fig. 1. Cut-away illustration of the Schulze RST-XS ring shear tester (a) and yield locus with consolidation stress σ1 (b).
4.4% and the average standard deviation was 1.7%. The wall friction angles were determined with the ring shear testers using wall friction shear cells, where the bottom ring was formed by a sample of stainless steel grade 1.4301. The values of the applied wall normal stress were 0.24 kPa, 0.6 kPa, 2.0 kPa, 6.0 kPa and 20 kPa. The maximum standard deviation for the wall friction angle of four randomly selected powders (F7, S5, I2 and P3) was 1.5� and the average standard deviation was 0.8� .
of the spread of the particle size distributions, are summarized in Table 2, columns 2 to 4. Fig. 2 shows the apparent density over the consolidation stress with both axes on a logarithmic scale. This stress-dependence of the apparent density was approximated by equations of the type of Eq. (2) with a value for σ0 of 1.0 kPa. As visible in Fig. 2 the approximation function fitted very well for all food powders. The values of the parameters of the approximation function, ρb,0 and c, as well as the respective correlation coefficients r2 are also shown in Table 2. All correlation coefficients were higher than 0.97 and the average correlation coefficient was 0.993 � 0.010. The estimate of the parameters ρb,0 and c by easy-to-measure powder characteristics like the powder properties apparent density measured according to EN ISO 60 or ISO 697, mass median diameter and spread of the particle size distribution was investigated.
3. Results and discussion 3.1. Dependence of the apparent density on the consolidation stress The apparent densities, ρb,N, measured according to EN ISO 60 or ISO 697, as well as the mass median diameters d50 and the calculated values
Table 2 Food powder samples: measured properties and coefficients of the approximation function. Sample no.
Apparent density ρb,N EN ISO 60 or ISO 697 in kg/m3
Mass median diameter d50 in μm
Spread d90/ d10
Approximation function
ρb,0 at a consolidation stress of 1.0 kPa in
c
r2
kg/m3
Flour and starch F1 579 � 2 F2 626 � 5 F3 704 � 6 F4 530 � 1 F5 496 � 2 F6 620 � 3 F7 755 � 1 F8 705 � 2 F9 756 � 2 F10 586 � 7 F11 696 � 1 Sugar and sweetener S1 535 � 2 S2 881 � 3 S3 824 � 2 S4 640 � 1 S5 760 � 5 S6 830 � 3 Inorganic (Salt and soda) I1 1140 � 4 I2 757 � 6 I3 1220 � 2 I4 1170 � 2 Protein powders P1 546 � 1 P2 338 � 7 P3 324 � 2 P4 390 � 2 Various food powders V1 461 � 2 V2 543 � 5 V3 886 � 1
64 88 130 55 37 180 660 300 840 13 38
9.2 10 11 11 39 8.4 1.5 7.4 2.4 2.8 3.3
636 670 734 566 534 591 759 688 726 575 646
0.0755 0.0654 0.0543 0.0949 0.0951 0.0593 0.0104 0.0476 0.0080 0.0695 0.0724
0.998 0.999 1.000 0.997 0.999 0.997 0.999 0.970 0.995 0.994 0.980
28 510 1040 230 92 640
40 3.2 2.2 5.8 46 4.0
575 865 818 644 762 837
0.1104 0.0080 0.0050 0.0235 0,0625 0.0236
1.000 0.997 0.979 0.990 0,999 0.991
97 20 480 570
5.0 17 2.2 3.9
1110 804 1230 1140
0.0354 0.0719 0.0032 0.0063
0.989 0.999 0.998 0.996
205 37 52 120
4.4 12.5 5.8 13.5
513 380 366 423
0,0339 0.1105 0.0650 0.0979
0,962 0.996 0.997 1.000
210 26 490
5.5 3.1 1.8
466 470 886
0.0749 0.0570 0.0145
0.999 0.999 0.974
3
C. Lanzerstorfer
Journal of Food Engineering 276 (2020) 109897
Fig. 2. Stress-dependence of the apparent density of the different food powders; a: exponent c < 0.3; b: 0.3 < c < 0.6; c: 0.6 < c<0.8; d: c>0.8.
The apparent density ρb,0 at a σ0 of 1.0 kPa can be approximated by the apparent density of the material ρb,N measured according to EN ISO 60 or ISO 697. Fig. 3 shows this correlation which can be expressed by the following equation (Eq. (3)). For this approximation a correlation coefficient of 0.981 was obtained. (3)
ρb;0 ¼ 1:004⋅ρb;N
Exponent c in the approximation equations varied between 0.0032 and 0.11. For coarser powders lower values of the exponent were found (Fig. 4a). However, the correlation between the exponent and the mass median diameter of the powders is not very strong (r2 ¼ 0.76). Combining the powder properties apparent density measured according to EN ISO 60 or ISO 697, mass median diameter and spread of the particle size distribution in a power function according to Eq. (4), led to a considerably better fit. � �E3 d90 F ¼ ðρb;N ÞE1 ⋅ðd50 ÞE2 ⋅ (4) d10 For maximization of the correlation the values for the exponents were derived by minimization of the sum of the square of errors. The best correlation of c with F was found with the values of 1.0, 0.38 and 0.30 for the exponents E1, E2 and E3, respectively (Fig. 4b). The correlation for the approximation function (Eq. (5)) was 0.931. c ¼ 0:114⋅expð
0:000329 ⋅ FÞ
(5)
Fig. 3. Correlation of the apparent density at 1.0 kPa and the apparent density measured according to EN ISO 60 or ISO 697.
3.2. Dependence of the wall friction angle on the consolidation stress The wall friction measurements showed that for some food powders 4
C. Lanzerstorfer
Journal of Food Engineering 276 (2020) 109897
Fig. 4. Correlation between the exponent c and the mass median diameter and correlation between the exponent c and F.
the wall friction angle strongly depends on the stress, while for others the stress-dependence is low and mainly limited to very low values of the wall normal stress (Fig. 5). This difference might be related to powder geometry. Generally, the wall friction angle ϕW decreased with increasing wall normal stress. Only in a few cases was a slight increase observed at
higher values of the stress. The stress-dependence of the wall friction angles was approximated by functions of the type � � σN k (6) tanðφW Þ ¼ W⋅
σ 0;N
Fig. 5. Wall friction angle as a function of the wall normal stress for various food powders: a) Flour and starch, b) Sugar and sweetener, c) Other food powders, d) Tangent of the wall friction angle and its approximation. 5
C. Lanzerstorfer
Journal of Food Engineering 276 (2020) 109897
where σN is the wall normal stress, σ0,N is the reference wall normal stress (1.0 kPa) and the constant W and the exponent k are dimension less parameters experimentally obtained for each studied powder. Table 3 shows the results for W and k together with the correlation coefficients. The correlation was quite good (r2 > 0.9) for most food powders. For starch and icing sugar the correlation coefficients were between 0.8 and 0.9. Lower values for the correlation coefficient were found only for some coarse food powders with a mass median diameter larger than 480 μm (F7, S2, S3, S6, I3, I4, V3). The value of the exponent was in the range of 0.031–0.328. The higher the value of the exponent the stronger is the size dependence of the wall friction angle. No distinct correlation of the exponent with the particle size was found. However, the dependence seems to be less for inorganic and coarse powders. Also for parameter W of the approximation function for the wall friction angle no distinct relation between the parameter and the par ticle size of the powders was noticed. For several food powders (F3, F7, F9, F10, F11, S1, S2, S3, S4, S6, I2, I3, I4, P1, P3 and V3) the assumption of a constant wall friction angle would be quite feasible, because the stress-dependence is small for a wall stress higher than 2 kPa. For these powders the value of k was between zero and 0.12. Fortunately, all materials with a weak correlation of the proposed approximation function belong to this group of food powders. For some other food powders (F4, F5, F8, P2, P4, V1 and V2) the wall friction angle decreased by at least half when the wall normal stress was raised from 2 kPa to 20 kPa. For these materials the exponent in the approximation function, k, was higher than 0.29 and the correlation of the approximation function was very good (>0.96). For the powders F1, F2, F6, S5 and I1 the value of k was in a medium range (0.12 < k < 0.29) and the correlation of the approximation function was quite good (>0.91).
3.3. Calculation of the apparent density in stored compressible food powders The calculation of the apparent density of a compressible powder is based on the model introduced by Janssen (1895) extended by the stress-dependent apparent density and a stress-dependent wall friction angle based on the proposed approximation functions. First, it has to be distinguished between storage in a pile, where the stress results from the weight of the material above, and silo storage, where wall friction limits the stress in the bulk. In a pile of compressible powder the apparent density ρb as a function of the distance to the surface z results from equation (7): �
ρb ðzÞ ¼
Flour and starch F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 Sugar and sweetener S1 S2 S3 S4 S5 S6 Inorganic powders I1 I2 I3 I4 Protein powders P1 P2 P3 P4 Various food powders V1 V2 V3
W
k
r2
0.214 0.233 0.385 0.324 0.339 0.371 0.403 0.339 0.281 0.519 0.216
0.212 0.182 0.104 0.329 0.315 0.166 0.060 0.315 0.004 0.040 0.117
0.989 0.975 0.940 0.993 0.988 0.910 0.976 0.980 0.935 0.879 0.818
0.302 0.330 0.376 0.377 0.407 0.293
0.104 0.057 0.031 0.099 0.160 0.117
0.886 0.743 0.530 0.932 0.962 0.818
0.391 0.490 0.342 0.354
0.212 0.058 0.043 0.074
0.997 0.967 0.795 0.798
0.262 0.581 0.337 0.302
0.083 0.328 0.048 0.336
0.922 0.983 0.947 0.990
0.339 0.322 0.352
0.311 0.290 0.048
0.984 0.963 0.712
�1 1 c ⋅ðð1
cÞ⋅g⋅zÞ1
c
c
(7)
The exponent c is obtained from the stress-dependence of the apparent density. ρb,0 is the apparent density at a reference consolida tion stress σ0 of 1.0 kPa. In a silo with the cross section of the silo shaft A, the inner circum ference of the silo shaft U, a constant lateral stress ratio λ ¼ σh/σv and a constant wall friction angle the apparent density in the cylindrical shaft can be calculated using equation (8): 0 0 111 c c � �1 1 c ρb;0 g U C B ρb ðzÞ ¼ ⋅@ U ⋅@1 e λ⋅ A ⋅tanφw ⋅ð1 cÞ⋅z AA (8) λ⋅ A ⋅tanφw σc0 When a stress-dependence of the wall friction angle has also to be considered according to equation (6), the differential equation (9) cannot be solved analytically. However, the relation can be used in numerical simulations together with equations (2) and (6). Thereby, σ ¼ σv and σN ¼ σh. g⋅σρ0c ⋅σ cv
Table 3 Food powder samples: coefficients of the approximation function for the wall friction angle. Sample no.
ρb;0 σc0
0
1 λ1 k ⋅UA⋅σWk ⋅σ 1v
k
⋅ dσv ¼ dz
(9)
0;N
Fig. 6a shows exemplarily the calculated results for the apparent density of soy flour (P2) for different storage conditions. Its wall friction angle strongly depends on the stress. In Fig. 6b the results for maize starch (F10), a material with a nearly constant wall friction angle, are shown. For silo storage the following assumptions were made: an inner diameter of the silo of 3 m and a constant lateral stress ratio λ of 0.4. For the soy flour the density was calculated also for a constant average wall friction angle of 31� . The blue line shows the apparent density of the material in a pile as a function of the distance from the surface. The violet line shows the result for silo storage, assuming a constant wall friction angle. For soy flour a numerical calculation of the apparent density with the stress dependent wall friction angle was performed (dashed red line). The resulting values of the apparent density were slightly higher than those from the calcu lation with a constant average wall friction angle. 4. Conclusions The apparent density as well as the wall friction angle of food powders are stress-dependent, which can be expressed well using simple approximation functions. For the apparent density an approximation function already established in the literature fits very well. For approximation of the wall friction angle a new simple function was proposed. For materials with a high stress-dependence of the wall fric tion angle this function correlates well with the result of the measure ments. For materials with a low stress-dependence of the wall friction angle the fit was limited. However, for these materials a constant average value of the wall friction angle can be used. Depending on the storage circumstances, three different equations for calculation of the apparent density have to be used: 6
C. Lanzerstorfer
Journal of Food Engineering 276 (2020) 109897
Fig. 6. Calculated results for the apparent density of soy flour (a) and maize starch (b).
- Storage of a compressible powder in a pile; the apparent density of the powder in dependence of the height can be calculated using the proper equation. - Storage of a compressible powder in a silo and approximately con stant wall friction angle: the apparent density can be calculated using the shown equation. - Storage of a compressible powder in a silo; stress-dependence of the wall friction angle has to be considered: the apparent density can be calculated numerically.
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Author contributions Single author work, All work has been done by the author except English proofreading, which is mentioned in the acknowledgement. Acknowledgement Proofreading by P. Orgill is gratefully acknowledged. References Bagster, D.F., 1977. The effect of compressibility on the assessment of the contents of a stockpile. Powder Technol. 16 (2), 193–196. Bian, Q., Sittipod, S., Garg, A., Ambrose, R.P.K., 2015. Bulk flow properties of hard and soft wheat flours. J. Cereal Sci. 63, 88–94. https://doi.org/10.1016/j. jcs.2015.03.010. Clower, R.E., Ross, I.J., White, G.M., 1973. Properties of compressible granular material as related to forces in bulk storage structures. Trans. ASAE 16 (3), 478–481.
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