1-s2.0-S0032591000003491-main

.Powder Technology 115 2001 243255www.elsevier.comrlocaterpowtec

On predicting roller milling performancePart II. The breakage function

G.M. Campbell), P.J. Bunn, C. Webb, S.C.W. HookSatake Centre for Grain Process Engineering, Department of Chemical Engineering, UMIST, P.O. Box 88, Manchester, M60 1QD, UK

Received 1 November 1999; received in revised form 1 June 2000; accepted 20 July 2000

Abstract

Paper I of this series showed that the relationship between the inlet and outlet particle-size distributions in a roller milling operationcan be found by integrating, over the range of input particles, the breakage of each individual particle as a function of its physicalcharacteristics. This is possible because particles break independently during roller milling. The pattern of breakage for individualparticles is called the breakage function. This paper considers the form of the breakage function and determines it experimentally forroller milling of wheat.

.The breakage function is shown to depend critically on the ratio of roll gap to input particle size the milling ratio . For a given ratio,the breakage function for wheat grains is linear with respect to output particle size, over a wide range. This is quite different from theparticle-size distribution produced by, e.g., hammer milling. It perhaps explains why roller milling is so suited to milling of wheat toproduce flour; the broad and even distribution of particle sizes produced allows effective separation of bran and efficient recovery ofwhite flour. Breakage functions depend on wheat variety and physical characteristics and on the design and operation of the roller mill.

Single-kernel testing is becoming widespread in wheat quality testing; distributions of individual kernel parameters, such as size, mass,hardness and moisture content, are measured. The breakage function approach potentially provides a link between single-kernel testingand milling performance. q 2001 Elsevier Science B.V. All rights reserved.

Keywords: Roller milling; Breakage equation; Breakage function; Wheat; Flour; Single-kernel characterisation

1. Introduction

In roller milling of, e.g., wheat grains, each grain passesw xthrough the mill independently of surrounding grains 1,2 .

The breakage patterns for each grain, therefore, dependonly on the interaction between the grains physico-chem-

.ical properties size, density, hardness, etc. and the rollermill design and operation roll diameter, fluting, roll gap,

.speed, differential, etc. , as illustrated in Fig. 1, and not oninteractions with surrounding grains. The particle-size dis-tribution produced by milling a mixture of grains can befound by integrating the breakage of each individual grain.Mathematically, this can be described by the breakage

.equation for roller milling, where r x is the probability2

) Tel.: q44-161-200-4406; fax: q44-161-200-4399. .E-mail address: [email protected] G.M. Campbell .

.density function mass-based of the output particle-sizedistribution:

Ds`r x s r x , D r D d D. 1 . . . .H2 1

Dsx

.Eq. 1 states that the value of the probability densityfunction for an output particle of size x can be determinedby integrating the mass of particles of size x produced by

breakage of an input particle of size D where D)x,.assuming only breakage and not aggregation occurs , mul-

tiplied by the mass of input particles of size D. Themass-based input particle-size distribution is described by

.the probability density function r D . The breakage func-1tion describes the mass fraction of particles of size xproduced by breakage of an input particle of size D, and is

. denoted r x, D strictly speaking, the mass fraction ofparticles in the range x, xqd x is described by r x,

. .D d x . Note that throughout this paper, x refers to thesize of an outlet particle, and D to the size of an inlet

0032-5910r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. .PII: S0032-5910 00 00349-1

( )G.M. Campbell et al.rPowder Technology 115 2001 243255244

Fig. 1. Raw material and process factors affecting wheat breakage during roller milling.

particle. The cumulative size distribution of the outlet .material is found by integrating Eq. 1 :

x

P x s r x d x 2 . . .H 20

2. Normalised breakage functions an invalid as-sumption

.To use Eq. 1 , the form of the breakage function isneeded. Various functions have been postulated and used.A common assumption is that the breakage function can benormalised, i.e. breakage patterns are independent of initial

w xparticle size, which acts only as a scaling factor 3 . Anw xexample is that used by Austin et al. 4,5 , Rogers and

w x w xShoji 6 and Klimpel and Austin 7 , shown here in itscumulative, continuous form:

xX XB x , D s r x , D d x . .H

0 3 .g bx xs f q 1yf . / /D D

.On a loglog graph, Eq. 3 is shown as the sum of two .straight lines corresponding to the two exponents. Eq. 3

is often used in a discretised form for convenient applica-tion to sieve analysis data. Another example of a nor-malised breakage function is that used by Broadbent and

w xCallcott 8 :

eyx rD

r x , D s 4 . .y1D 1ye .

. .Eqs. 3 and 4 imply that the proportion of milledmaterial smaller than x is a constant function of xrD. Forexample, if 50% of the output material is smaller than 500mm for a given input particle size D, then if D is doubled,50% of the output material will now be smaller than 1000mm, i.e. the average output particle size, x , will in-50crease.

The assumption of normalised breakage functions isw xrecognised as often invalid in ball milling 7,9,10 and

w xhammer milling 11 or in milling and breakage studiesw xgenerally 12,13 . They are used as a way of reducing the

number of parameters to be determined in milling studiesw x w x7 . Austin and Luckie 14 introduced a modification to

.Eq. 2 to account for nonnormalisable breakage distribu-tions, by making f a power function of xrD. Theyconcluded that Ait is often not possible to get accurate

results if the effect of nonnormalisable breakage distribu-.tions is neglectedB, and presented a method for determin-

ing the extent of nonnormalisation. They and other work-ers have also used the concept of selection functions togive some freedom to offset the invalidity of the assump-tion.

Surprisingly, several papers reporting studies of rollerw xmilling assume normalised breakage functions 46 . As

w xnoted in Paper I of this series 2 , it is not the case forroller milling that a normalised breakage function, such as

. .Eqs. 3 or 4 , can be applied; the assumption is ifanything considerably less justifiable than for ball or ham-mer milling. In roller milling, if the input particle size isincreased, the average output particle size is likely todecrease, as the particle must be broken more to pass

w xthrough the fixed roll gap 2 . A nonnormalised breakagefunction has not been suggested or measured previously

( )G.M. Campbell et al.rPowder Technology 115 2001 243255 245

for roller milling. Instead, normalised breakage functions .of the form of Eq. 3 have been combined with selection

functions and modifications, such as that of Austin andw xLuckie 14 .

w xPaper I 2 also questioned the applicability of or needfor selection functions in roller milling operations when allparticles are clearly broken, even if some remain in thesame size range as the feed material. If the breakagefunction has a simple mathematical form, there may be noneed to introduce the extra complexity of a selectionfunction. Also, if the inlet and outlet particle sizes arecharacterised using different techniques, then x and Dactually measure different things, invalidating the defini-tion of the selection function as the proportion of materialno longer remaining on the top sieve of the range. Forexample, in the work on wheat breakage that follows,

. wheat size D was measured as the thickness third.longest dimension using slotted sieves, as wheat breakage

depends on thickness. However, the size distribution ofmilled stocks was measured using square aperture sieves,so x is the second longest dimension. Ignoring selectionfunctions increases the generality of the breakage functionapproach; the breakage function can relate input and out-put size distributions even if they are measured differently.

3. Breakage matrices a step towards breakage func-tions

.It is possible to discretise Eq. 1 to give a matrixequation that describes the relationship between an inputsize vector, f , and the resulting output size vector, o:BP fso. 5 .

Vectors f and o describe the size distribution of thefeed and output, respectively, and could be determined bya sieve analysis, e.g., B is a matrix which, when multipliedby f using normal matrix multiplication, gives o. Theelements of the matrix can be determined by milling

w xmono-dispersed samples 1,2 or calculated by assuming a . . w xbreakage function, such as Eqs. 3 or 4 8,15,16 . These

latter workers introduced the breakage-matrix approach,applying it to describe cone milling of coal. Subsequent

workers have used the breakage-matrix approach, withw x w xvariations, to study milling of coal 4,6 , copper ore 7 ,

w x w xstone 4,13 and wheat 1 .The breakage-matrix approach is convenient for studies

where size distributions are measured discretely using,e.g., sieve analyses. Having determined the breakage ma-trix for a particular set of conditions, it can then be appliedto predict the output from any input size distribution underthose same conditions. However, the breakage matrix issomewhat inflexible and too unwieldy for use in determin-ing the effect of feed characteristics or mill operatingconditions on breakage. A breakage matrix determined forone set of conditions cannot be readily adapted for anotherset of conditions.

w xCampbell and Webb 2 , in Paper I of this series,showed that the columns of B are estimates of the break-

.age function, r x, D . The breakage function is a moreconvenient form for determining quantitatively and mecha-nistically how factors, such as particle size or roll gapaffect breakage.

In this paper, breakage matrices are constructed for twovarieties of wheat milled over a range of roll gaps. Mix-

.tures grists of the two wheats are milled, and the result-ing size distributions compared with breakage-matrix pre-dictions obtained by combining separate breakage matricesfor each wheat variety.

A mathematical form of the breakage function for rollermilling is then derived by careful analysis of the sizedistribution resulting from milling narrow size fractions ofeach variety over a range of roll gaps.

4. Materials and methods

Wheat was separated into size fractions and milled inthe Satake STR-100 experimental mill as described in

w xPaper I 2 . Two wheat varieties were investigated: Here-ward 11.4% protein, bulk density 78.1 kgrhl, moisture

. content 14.2% , a typical hard wheat; and Riband 8.8%.protein, bulk density 78.0 kgrhl, moisture content 14.2% ,

a typical soft wheat, both from the 1997 UK harvest,grown in the Humberside region.

Table 1Size fractions of Hereward and Riband wheatHereward Riband

Fraction Size range Percentage of Fraction Size range Percentage of . . . .reference mm sample % reference mm sample %

HA 3.003.50 6.00 RA )3.50 4.27HB 2.753.00 10.32 RB 3.003.50 8.37HC 2.502.75 22.15 RC 2.753.00 27.57HD 2.252.50 38.60 RD 2.502.75 46.82HE 2.002.25 22.93 RE 2.002.50 12.97

.Note that the size ranges used differ for the two varieties. Size refers to the thickness third largest dimension of wheat kernels.


Table 2Grists of Hereward and Riband milled experimentally to compare withbreakage matrix predictionsReference Percentage Percentage

. .of Hereward % of Riband %

HR30 30 70HR50 50 50HR70 70 30HR90 90 10

Note that the reference number indicates the percentage of Hereward.

Both wheats were separated into five narrow size frac- .tions by thickness using slotted sieves in a Satake SPU

125AU purifier. For the Hereward sample, fractions smallerthan 2 mm and larger than 3.5 mm were discarded ascontaining significant amounts of nonwheat or nonrepre-sentative material. Riband has bolder, plumper grains thanHereward and a larger average size, such that a significantproportion of the sample was larger than 3.5 mm indiameter. In this case, the top fraction was retained, whilematerial smaller than 2.00 was discarded as for the Here-ward. The smallest size fraction for Riband covered therange 2.002.50 mm, while for Hereward, this range wasseparated into two fractions. Table 1 shows the size frac-tions used and proportions of wheat obtained in each forboth varieties.

For the gristing trials, the two wheats were blended inthe proportions shown in Table 2.

One-kilogram batches of the native feed for each vari-ety and of each of the fractions and grists were conditionedto 16% moisture overnight. Samples were milled on theSatake STR-100 test roller mill, using a differential of 2.7

relative to a fast roll speed of 600 rpm. Fluted rolls 10.5.flutesrin., 100 mm in length were used in a sharp-to-sharp

disposition. Ten roll gap settings were studied, covering0.05 mm increments over the range 0.250.7 mm. Rollgaps were checked using a feeler gauge.

Samples of 100 g were milled, at a feed rate corre-sponding to 375500 kg h, and the entire milled samplewas collected for sieve analysis. Sieve analysis of theentire milled stock was performed on a Simon plansifterfor 3 min using a sieve stack comprising wire mesh sievesof size: 2057, 1676, 1204, 850, 600, 420 and 211 mm,along with a bottom collecting pan.

Breakage matrices were constructed from the nor-malised sieve analysis data for each roll gap. The breakagematrix for each roll gap was then used to predict the outputparticle-size distribution for the original whole sample.Predictions were compared with the actual milling outputfrom the original sample for each roll gap.

To predict the output particle-size distributions at aparticular roll gap from the gristing trials, breakage matri-ces were calculated for each grist by proportional additionof the corresponding breakage matrices for each variety.

5. Results from breakage-matrix prediction

As reported for a similar experiment in Paper I byw xCampbell and Webb 2 , breakage-matrix predictions were

again in excellent agreement with experimental results. Tocharacterise the particle-size distributions, the experimentaland predicted values of x were calculated, where x is50 50the particle size below which 50% by mass of the milledsample falls. Fig. 2 shows the comparison between pre-dicted and experimental x for the gristing trials, showing50excellent agreement and a good degree of precision. Thisconfirms that grains mill independently, and that theirbreakage patterns are not affected even when the surround-ing grains are of a different size and type of wheat. Thisalso demonstrates that having constructed breakage matri-ces for pure feed materials under particular operatingconditions, the matrices can then be combined to predictmilling of mixtures under those same conditions. This ispotentially valuable to flour millers, who continually blenddifferent grists for particular end uses.

Values of x were also calculated for each size frac-50tion milled. Fig. 3 shows how predicted and experimentalx varied with roll gap for each size fraction, for the two50wheat varieties. From Fig. 3, it is clear that as roll gap

increased, x increased particles were no longer broken50.to the same degree , and that as input particle size in-

creased, x decreased larger particles were broken to a50greater extent than smaller particles passing through the

.same roll gap . Also, x values were generally smaller,50 .and varied less with roll gap, for the Riband soft wheat

.samples compared with the Hereward hard wheat sam-ples. These results confirm what was noted earlier, that inroller milling, particles break differently depending on

Fig. 2. Comparison between predicted and experimental x for milling50of mixtures of Hereward and Riband.


.Fig. 3. Variation of x with roll gap and fraction size for a Hereward,50 .and b Riband.

input particle size, roll gap and feed material. The samebreakage function could not be applied to all systems.

6. Determining the form of the breakage function

It has been demonstrated that breakage matrices can beconstructed by milling mono-sized or narrow-sized frac-tions of input material, and then used to predict milling ofwhole samples or mixtures of feed materials. It has alsobeen shown that breakage patterns for roller milling de-pend on input particle size, roll gap and wheat variety, andby extrapolation, other material and process variables; anormalised breakage function as described in the literatureis not, therefore, appropriate. In this section, the mathemat-ical form of the breakage function for First Break millingof wheat is determined from the experimental data pre-sented above.

Considering Fig. 2, x increased with roll gap and50decreased with increasing feed particle size. Austin et al.w x4 reported that the breakage function depends on the ratio

of roll gap to feed particle size assuming the latter is.much smaller than the roll diameter . Fig. 4 shows, for

Hereward and Riband, x plotted against GrD,where G50is the roll gap and D the arithmetic mean size of an inletfraction. Fig. 4 also shows x , x and x , the output75 25 10particle sizes below which 75%, 25% and 10% of theparticles fall, by mass, respectively. Clearly, for bothHereward and Riband, all the data collapse onto the samelines, despite coming from a range of inlet size fractionsmilled over a range of roll gaps. This indicates that theparticle-size distribution resulting from milling a particularfeed size depends on the ratio of roll gap to input particle

Fig. 4. Particle size below which 75%, 50%, 25% and 10% of the milled .material falls vs. milling ratio for milling of fractions of a Hereward,

.and b Riband.


size, GrD, which we will term the milling ratio. As themilling ratio increases, particles are not forced to passthrough such a small roll gap, less breakage occurs, andthe output particles are larger.

The relationship between x , etc., and GrD could be50approximated by straight lines, as shown in Fig. 4. For agiven ratio of GrD, Riband tends to produce smalleroutput particles than Hereward, and the output particle-sizedistribution is less sensitive to the milling ratio; the slopesof the lines are less steep for Riband than for Hereward.This reflects the fact that Riband is a soft wheat andHereward a hard wheat; hard wheats tend to fracturecleanly during milling, yielding agglomerates of intactendosperm cells, while soft wheats tend to rupture morerandomly, disrupting the cellular endosperm structure. Thisknown difference in practical breakage characteristics evi-dently affects particle-size distribution and sensitivity tooperating conditions.

.The definition of the breakage function, r x, D , isthat it is the derivative of the cumulative probabilityfunction or the cumulative breakage function:

dr x , D s B x , D , 6 . . .d x

.where B x, D , the cumulative breakage function, is theproportion of particles smaller than size x produced by

breakage of a particle initially of size D. Values of B x,.D can be found from plots of the cumulative distributions,

.similarly to finding x ; in fact, B x, D is the inverse50concept to x , the proportion less than a given size, rather50than the size below which a given proportion falls.

. .Fig. 5 a shows how B x, D for various particle sizesx varies with GrD, for milling of Hereward samples. .Each value of B x, D was found by linear interpolationat 250 mm intervals of the cumulative distribution function

.constructed from sieve analysis data. The top curve, e.g.,shows the proportion of particles smaller than 2000 mm.As GrD increases, feed particles are no longer forcedthrough such a small gap, so less breakage occurs and theproportion smaller than 2000 mm decreases. The same is

.true for the curves for other particle sizes. Fig. 5 b showsthe same plot for Riband data. For both figures, all the datafall onto the same curves, even though they come from arange of inlet particle sizes and roll gaps.

Fig. 5 confirms the conclusion of Fig. 4, that theparticle-size distribution produced by roller milling ofwheat kernels of a given variety and moisture content

.depends on the ratio of roll gap to kernel size thickness . .By plotting the data in the form B x, D instead of x , it50

. is possible to determine r x, D by differentiating B x,. .D with respect to x. To do this, B x, D must be written

mathematically as a function of x.Inspecting Fig. 5 suggests that quadratic functions could

.adequately describe the variation in B x, D with GrDover the range covered. Quadratic functions cannot ulti-mately describe the physical process of particle breakage

realistically, as beyond the range shown in Fig. 5 quadraticfunctions would obtain values, which are physically mean-

ingless e.g. negative probabilities, or probabilities greater.than 100% . However, with that acknowledged limitation,

quadratic functions do adequately describe the data overthe range. Linear functions are not sufficient, and cubicfunctions were shown not to give a statistically improvedfit. The quadratic functions fitted for each curve are shownon the graphs, in the form:

2G GB x , D sa qa qa . 7 . .0 1 2 / /D D

The fitted quadratic functions describe the variation in .B x, D with D. What is needed in order to differentiate .B x, D is the variation with x. This can be found by

considering how the coefficients of the curves vary with x.Fig. 6 shows for Hereward and Riband how the three

.quadratic coefficients describing B x, D vary with x.Considering the Hereward data, the intercept, a , varies0from 0% up to 93%, suggesting that at the limit ofGrDs0, around 93% of the material would fall below2000 mm while, e.g., 79% would fall below 1000 mm.These figures are extrapolations beyond the range of thedata, but indicate the physical significance of the a0coefficient. The a coefficient indicates the slope of the1curves at the intercept, which initially decreases from

.around zero becomes more negative then increases backtowards zero. This reflects the fact that the particle-sizedistribution is more sensitive to roll gap in the middle thanat the two extremes. The second-order coefficient, a ,2initially increases for the Hereward data, and then de-creases to below zero. Negative values of this coefficientindicate that the curve has become concave with respect tothe abscissa. For Riband, all the curves considered re-mained convex with respect to the abscissa, and all thevalues of a stayed positive. A value of a close to zero2 2indicates more or less a straight line; this is the case, e.g.,

.for the Hereward B 1500, D curve, which is almostlinear with respect to GrD.

From Fig. 6, once again it appears that quadratic func-tions would adequately describe the variation of the coeffi-

.cients of Eq. 7 with x. Fitting quadratic functions asshown gives a set of three equations, each describing thevariation of the three coefficients:

a sa qb xqc x 20 0 0 0a sa qb xqc x 21 1 1 1a sa qb xqc x 2 . 8 .2 2 2 2

The overall equation then becomes:

G2 2B x , D sa qb xqc x q a qb xqc x . .0 0 0 1 1 1 /D

2G2q a qb xqc x . 9 . .2 2 2 /D


. .Fig. 5. Variation of proportion of material smaller than x with milling ratio for a Hereward, and b Riband.


.Fig. 6. Variation of quadratic coefficients with x for a Hereward, and .b Riband.

.Differentiating Eq. 9 with respect to x then gives:d

r x , D s B x , D . .d xG

sb q2c xq b q2c x .0 0 1 1 /D2G

q b q2c x 10 . .2 2 /DOn differentiating, the a coefficients are lost. The b and

c coefficients describe the particle-size distribution withinthe range 2502000 mm. The a coefficients allow inaddition the total proportion of the distribution below 250mm and above 2000 mm to be determined, but the shape

of the distribution beyond these limits cannot be deter-mined from these data.

. .Eqs. 9 and 10 , although they have arisen from curvefitting, are powerful equations, as they contain in their ninecoefficients information about the particle-size distributionproduced from milling any input particle of size D at any

. .roll gap G. Eq. 10 is in a suitable form for use in Eq. 1 ,so could be used to predict the particle-size distribution inthe range 2502000 mm resulting from any feed sizedistribution for any roll gap.

. .In matrix form, Eqs. 9 and 10 can be written:

a b c0 0 0 12

a b c xB x , D s 1 GrD GrD , . . . 1 1 12 0xa b c2 2 2

11 .

b 2c0 02 1b 2cr x , D s 1 GrD GrD . 12 . . . .1 1 x 0b 2c2 2

The values of the coefficients for Hereward and Ribandare given in Table 3. The a coefficients are smaller inmagnitude for Hereward than for Riband, while the b andc coefficients are larger.

.Eq. 10 indicates two important features of particlebreakage in roller milling, at least for First Break millingof wheat over a practical range of roll gaps. Firstly, theproportion of particles of size x varies as a quadraticfunction of GrD. This is helpful to millers, who use rollgap as the major control variable for controlling the flourmilling process. It is especially important for First Breakmilling, as the particle-size distribution produced here, andits subsequent separation, affect flows throughout the en-tire mill.

.The second point indicated by Eq. 10 is that the .probability density function r x, D is a linear function of

x, over the range 2502000 mm. Both this point and thelast arise as a mathematical consequence of the fact thatquadratic functions were used to fit the data; differentia-tion of a quadratic function gives a linear function. Never-theless, over the data range considered, the fits were verygood, and adequate to draw these conclusions.

Fig. 7 shows the particle-size distributions given by Eq. .10 for Hereward and Riband for milling ratios of 0.1, 0.2and 0.3, over the range 2502000 mm. As milling ratioincreases, particles are broken to a lesser extent, there aremore larger particles, and the slope of the line increases.But, in all cases, the particle-size distribution is linear overthis range. This is unusual, compared to hammer milling,e.g., which produces a narrower range of particles with adefinite peak at some midpoint. Fig. 8 illustrates this forhammer milling of the Hereward sample in a Perten fallingnumber mill using a range of screens, and in a Stenverthammer mill. Both hammer mills produced a bimodal


Table 3 . .Coefficients describing the breakage function, as defined in Eqs. 9 and 10 , for Hereward and Riband as a function of milling ratio

Hereward Ribandy5 y5a y1.709 b 0.1136 c y3.301=10 a 11.83 b 0.08236 c y2.143=100 0 0 0 0 0y4 y4a 8.301 b y0.6286 c 3.095=10 a y65.51 b y0.38710 c 1.890=101 1 1 1 1 1y4 y4a 60.294 b 0.8323 c y5.022=10 a 152.27 b 0.54941 c y2.915=102 2 2 2 2 2

distribution using a 2-mm screen, while in the fallingnumber mill, the distribution developed a single peak asscreen aperture size was reduced. But, in both cases, theparticle-size distribution of the milled stocks quite clearlyfalls into one or two narrow peaks, in contrast to the even,linear distributions of Fig. 7 given by roller milling.

A linear probability density function cannot ultimatelybe realistic, as the area under the pdf must equal 1.

.Fig. 7. Particle-size distributions given by Eq. 10 for various milling . .ratios for a Hereward, and b Riband.

Therefore, at some point beyond 2000 mm, the lines mustcurve down to zero. Nevertheless, the fact that the pdf islinear over that wide range, which covers typically 80% ormore of the total material, makes the breakage function inthis form very convenient to use. For flour millers, this isthe most important range of particle sizes from FirstBreak; particles smaller than about 212 mm are separatedas flour and not processed further, while bran particleslarger than 2000 mm are processed further to only alimited degree.

The linear breakage function is of the form ysAxqB,where A and B are given by:

2G GAs2c q2c q2c , 13 .0 1 2 / /D D

2G GBsb qb qb . 14 .0 1 2 / /D D

The b coefficients, thus, represent the variation in theintercept of the breakage function with GrD, while the ccoefficients describe the variation of the slope. Thesevariations indicate the sensitivity of the wheat to roll gap.From Fig. 7, the variation in the slopes of the lines fordifferent milling ratios is much less for Riband than forHereward, indicating, as noted earlier, that Riband is lesssensitive to roll gap than Hereward. The magnitude of theslopes is greater for Hereward than for Riband, at a given

Fig. 8. Particle-size distributions obtained from milling Hereward wheat .in a Perten falling number mill various screen apertures and in a

Stenvert mill with a screen aperture of 2 mm.


milling ratio, indicating that Hereward breaks to givelarger particles than Riband. This is in agreement with

w xwhat is expected for milling of hard wheats 1725 .The linear nature of the breakage function perhaps

explains why roller milling is so well suited to flourmilling. By producing an even range of particle sizes,subsequent separation is particularly effective, dividing thestocks from First Break milling into relatively even frac-tions. If the breakage function is close to a horizontal line,as for Riband at a milling ratio of 0.15, the same amountof material falls into each size interval if all intervals areof the same width. If the miller wanted to achieve a veryeven spread of stocks from First Break milling, he couldset the roll gap to 0.15 of the mean grain size to achievethis.

.Fig. 9. Cumulative size distributions predicted from Eq. 9 for various . .milling ratios, compared with experimental data for a Hereward, and b

Riband.

Fig. 10. Predicted cumulative size distributions for milling of Herewardand Riband native feed at roll gaps of 0.25 and 0.65 mm, compared withexperimental data.

More generally, using the information derived above,the miller could specify the size distribution required anddetermine the roll gap required to achieve this. He couldalso predict the effect of changing roll gap, or perhaps ofremoving a size fraction from the wheat before milling, orof milling a particular grist.

The above analysis gives an unambiguous basis onwhich to assess wheat breakage during roller milling.Clearly, Hereward and Riband wheats break differently,and this difference has been quantified. The next step is tobe able to relate the a, b and c coefficients to physico-chemical properties of wheat grains, such as moisture andprotein content, density and hardness.

Fig. 9 compares the cumulative size distribution given .by Eq. 9 with actual experimental data for several milling

ratios. The experimental data are averaged from severalreplicates at each milling ratio. The fitted curves clearlydescribe the experimental measurements very well.

In practical milling, a single roll gap is used, andreceives a range of input particle sizes, which break ac-cording to the milling ratio for each individual particle.

. .Eqs. 1 and 10 allow the outlet particle-size distributionto be predicted for a poly-disperse feed milled at a givenroll gap, even though the ratio of that roll gap to eachparticle size is different.

Fig. 10 shows the cumulative size distributions pre- . .dicted by integrating Eq. 1 using Eq. 10 as the breakage

function, for milling of native feed samples of Herewardand Riband wheat at roll gaps of 0.25 and 0.65 mm.Appendix A shows how the integration can be simplified.

.Eq. 9 provided the starting point for the integration atxs250 mm. The measured size distributions of Herewardand Riband given in Table 1 were used, assuming size tovary evenly within a size fraction. The agreement withexperimental results for a roll gap of 0.25 mm is very good


for both wheats, predicting correctly a narrower size distri-bution for the Hereward, with a similar x . At a roll gap50of 0.65 mm, the particle size of the milled stocks ispredicted to increase, and Hereward is predicted to give alarger x than Riband. The predictions are in the same50region as the experimental results, correctly predicting theeffect of roll gap; but, in this case, the discriminationbetween the two wheats is not so clear. This is undoubt-edly because of experimental error; the data are from asingle milling and sieve analysis for each wheat, and sufferfrom the inherent variability of this experimental proce-dure. Overall, the predicted outlet distributions agree fairlyclosely with the experimental results. This confirms thatthe quadratic functions fitted are adequate to describe FirstBreak milling of these wheats over this range of roll gaps.

Milling of Hereward and Riband has given breakagefunctions, which are linear with respect to particle sizeover a wide range, and which vary quadratically withmilling ratio. Paper III of this series confirms that thisfinding holds for a wider range of wheat varieties, andexamines further the usefulness of the approach for pre-dicting milling of mixtures of wheats.

7. Factors affecting the breakage function

The above derivation has allowed particle breakageduring First Break milling to be described in terms of thetwo most important parameters affecting breakage, inputparticle-size distribution and roll gap. Many other factorsaffect particle breakage during roller milling, as illustratedin Fig. 1. The coefficients given in Table 3 are useful forthese particular wheat varieties, at moisture contents of16%, using rolls of 250-mm diameter with 10.5 flutesrin.,run in a sharp-to-sharp disposition at a 2.7 differentialrelative to a fast roll speed of 600 rpm. Other wheats andmilling conditions would result in different breakage func-tions.

.The breakage function, r x, D , describes the particle-size distribution produced on breakage, given an initialparticle size D. Other factors affecting wheat kernel break-

. .age include hardness h , moisture content m , protein . .content p and density r . For given operating condi-

tions, the breakage function could be written more gener- .ally as r x given GrD, h, m, p, r . Knowing the

distribution of each of these factors, breakage could inprinciple be predicted for any wheat, without knowing thevariety, basing predictions entirely on measurement ofsingle-kernel characteristics.

Single-kernel testing is a growing trend in wheat qualityw xassessment 26,27 . Using an instrument, such as the Perten

.single-kernel characterisation system SKCS , each kernelin a sample of typically 300 grains is tested individually

for mass, moisture content, diameter and hardness as.indicated by the force deformation profile during crushing

w x2831 . The instrument measures the distributions of theseparameters. The difficulty facing users is then the interpre-tation of all this information, in terms of predicting millingperformance. The breakage function approach, which pre-dicts milling on the basis of individual kernel parameters,offers a sound foundation from which to interpret thedistributions produced by single-kernel testing.

8. Conclusions

Breakage functions for roller milling have been mea-sured experimentally for two wheat varieties, Herewardand Riband, using First Break fluted rolls. Having con-structed breakage matrices for each wheat variety, theproduct of milling mixtures of the two wheats could thenbe predicted by proportional addition of the individualmatrices. Breakage patterns depended critically on the ratioof roll gap to input particle size; large input particles werebroken to a greater extent, to produce smaller outputparticles. Breakage patterns for the two wheats were differ-

.ent, with Riband the soft wheat being less sensitive to .roll gap than Hereward the hard wheat . Results showed

clearly that a single normalised breakage function couldnot be applied to all feed materials, contrary to the ap-proach of previous workers studying other milling systems.This assumption was not, therefore, applicable to wheatbreakage by roller milling.

The particle-size distribution resulting from rollermilling of these wheats was found to be linear over a widerange. This perhaps indicates, in part, why roller milling iswell suited to flour milling; a wide and even range ofparticle sizes is produced, which can then be effectivelyseparated into size fractions for further processing.

The breakage function was found to change as aquadratic function of the milling ratio the ratio of roll gap

.to kernel size . Combined with the last point, this alloweda quantitative, unambiguous comparison to be made be-tween milling of the two wheats. It gave a basis forrelating wheat physical properties to breakage, to allowpredictions of milling performance irrespective of wheatvariety.

The finding that the breakage function was linear withrespect to particle size and a quadratic function of millingratio simplifies the study of wheat breakage considerably.Paper III of this series examines whether this finding holdsfor a wider range of wheat varieties, and applies thebreakage function approach to predict milling of mixtures.

The breakage function can be written more generally toinclude wheat kernel parameters, such as hardness, density,moisture content and protein content. Distributions of theseparameters are measured by such instruments as the SKCS.Linking the information produced from such a system toactual milling performance can in principle be achieved bythe breakage function approach presented here.


Acknowledgements

The authors gratefully acknowledge the Satake Corpora-tion of Japan for their generous support of these studies.

Appendix A. Integrating the breakage equation

.Eq. 10 , the breakage function, is in a form suitable for .use in Eq. 1 , the breakage equation for roller milling.

This requires integrating the equation numerically. How-ever, if the feed particles are all larger than the outletparticle sizes being considered, the integration can besimplified.

. .Substituting Eq. 10 into Eq. 1 gives:`

r x s r x , D r D d D . . .H2 1x

` Gs b q2c xq b q2c x .H 0 0 1 1 /Dx

2Gq b q2c x r D d D A1 . . .2 2 1 /D

.

Splitting the integral gives:`

w xr x s b q2c x r D d D . .H2 0 0 1x

` 1w xq b q2c x G r D d D .H1 1 1Dx

` 12w xq b q2c x G r D d D. A2 . .H2 2 12 /Dx

If the feed particles are all larger than the largest value` .of x being considered, as is the case here, then H r D d Dx

s1 and the other two integrals define the average value of . 2 .1rD and 1rD , respectively, giving:

1w x w xr x s b q2c x q b q2c x G .2 0 0 1 1 /D

12w xq b q2c x G . A3 .2 2 2 /D

In this form, the outlet particle-size distribution can becalculated without the need for numerical integration; in-

. 2 .stead, the mean values of 1rD and 1rD can becalculated for the feed particle-size distribution, and substi-

. .tuted directly into Eq. A3 . Eq. A3 is also linear withrespect to x, indicating that even for a poly-disperse inputof a particular wheat variety, the particle-size distributionproduced gives a straight line. This surprising result sim-plifies the prediction of First Break roller milling of wheatconsiderably.

.Eq. 10 only describes the outlet particle-size distribu-tion down to 250 mm. The integration requires a starting

.point at P 250 . Assuming, as seems likely, that the shapeof the size distribution below 250 mm varies as a quadratic

.function of 1rD , as it does above 250 mm, then thesame argument applies. The cumulative size distributionfor a poly-dispersed feed sample can, therefore, be calcu-

2 .lated directly by inserting 1rD and 1rD into Eq. 9 . . .To predict the output particle-size distribution for a

.mixture of wheats, Eq. A3 can be used for each variety,using the appropriate b and c coefficients and feed sizedistribution for that variety, and the resulting values of

.r x weighted according to the proportion of each variety2in the mixture. Again, this results in a straight line, irre-spective of the size distribution of either variety or of theirrelative proportions.

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