the optimisation of energy consumption and time in colour pigment grinding with pearl mills

Journal of Materials Processing Technology 171 (2006) 48–60

The optimisation of energy consumption and time incolour pigment grinding with pearl mills

Marjan Tusara,∗, Livija Tusarb,1, Jure Zupanc,2

a ZAG, National Building and Civil Engineering Institute, Dimiceva 12, SI-1109 Ljubljana, Sloveniab Ministry of Higher Education, Science and Technology, Trg OF 13, SI-1000 Ljubljana, Slovenia

c National Institute of Chemistry, Hajdrihova 19, SI-1000 Ljubljana, Slovenia

Received 15 April 2004; received in revised form 18 May 2005; accepted 15 June 2005

Abstract

Grinding with pearl mill capable of solid particles grinding in emulsions or greases from granulation of approximately 30–1�m wasstudied on the basis of statistically planned experiments. The fractional factorial design for five factors was implemented. The data wereused for modelling to develop back-propagation neural network and incomplete higher order polynoms. The obtained models were used fordetermination of the correlations among selected variables and for prediction of optimal values. Energy consumption and time were of ours nger millingt e predicted.T©

K

1

opptchamsa

p

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ption.ionslvedtion-ust betion.ld beplex

arelling

opedorksngsorsdels.ondeluatetheoly-

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pecial interest and were directly dependent on the granualisation of the particles, i.e. smaller particles demand more energy and loime. On the basis of the developed models and selected size of particles, the energy consumption and the time of milling could bhe problems inherent in the modelling with mentioned models were discussed in detail.2005 Elsevier B.V. All rights reserved.

eywords: Pigment grinding; Experimental design; Polynomials; Artificial neural network; Optimisation

. Introduction

Grinding of colour pigment with pearl mills is a processf milling pigment particles to a defined size. The millingrocess should fulfil the following demands to achievelanned machining operations: better accuracy of grinding

o achieve the selected size of pigment particles, more effi-ient grinding—in order to lower the consumption of energy,igher productivity by shortening the grinding times andutomation of the grinding processes. Some of these require-ents are contradictory and to fulfil all of them a compromise

hould be found through the development of adequate modelsnd the use of selected variables optimisation.

The optimisation of milling processes can be focused onartial technical problems, such as simulation of milling or on

∗ Corresponding author. Tel.: +386 1 2804 497; fax: +386 1 2804 264.E-mail addresses: [email protected] (M. Tusar), [email protected]

L. Tusar), [email protected] (J. Zupan).1 Tel.: +386 1 478 4681; Fax: +386 1 478 4719.2 Tel.: +386 1 4760 200; Fax: +386 1 4760 300.

general problems, such as influence on energy consumOptimisation could be done by using differential equatpresenting the dynamics of the milling system and soby numerical methods. In this case, some basic relaships between independent and dependent variables massumed in advance to find the proper analytical funcPolynomial models and response surface technology coualso very useful, such as linear models or some more commathematical models. Artificial neural networks (ANNs)often used in the case of optimisations of the complex miprocesses with many variables[1–3].

The present article is focused on comparison of develincomplete polynoms of higher orders and neural netw[4] and on determination of optimal conditions for milliwith pearl mills. The calibration procedure of used senis also described in detail as an example of indirect moThe preliminary research[5] was focused on the comparisof the linear polynomial models and neural network mobut it was found out that linear polynoms are not adeqto explain the complexity of the correlations. Therefore,procedure of the automatic determination of incomplete p

924-0136/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.jmatprotec.2005.06.063

M. Tusar et al. / Journal of Materials Processing Technology 171 (2006) 48–60 49

nomial models was developed later and used[6]. Empiricalmodelling was selected for studying the relationships amongall selected variables in the system at once.

2. Developing of automatic system, variablesselecting and experimental designing

In this work, we focused on the optimisation of energy con-sumption and grinding time in the colour pigment grindingwith pearl mills by using incomplete higher order polyno-mial and ANN models. Firstly, the variables effecting energyconsumption and processing time were defined and secondlywe focused our work on the determination of their optimalvalues.

The functioning of a pearl mill is shown inFig. 1togetherwith the positions where sensors were built in. The agitatordisc was rotating in the mill vessel that fitted to the shape ofthe agitator disc and only a small slot separated them. The slotwhere the grinding process took place contained mill pearls.The grinding mass was flowing out from the vessel at thetop through the separation slot and then horizontally through

the tube. The separation slot was thin enough to prevent thepearls to abandon the mill vessel. Redundant heat producedat the grinding process had to be led away. The mill vesselwas water-cooled.

The used pearl mill (CoBall-Mill MSZ-12) was capable ofsolid particle grinding in emulsions or greases to a granulationof 1�m. In our case, the starting pigment granulation wasaround 30�m. The mill pearls were from ZrO2 and theirsize ranged from 0.75 to 1 mm. It could be used in colour,paper, pharmaceutical, food, biochemical, etc. industries. Itsflow capability was 15–30 l/h. The theoretical mill volumewas 0.325 l. The critical point in the milling process wasthe pressure in the mill which could increase over 2.5 bar. Ifthe pressure in the mill increased over the critical value themill should be switched off. During the milling process themill was warming—therefore, it was necessary to monitor thetemperature in the mill vessel. There were the main reasonswhy the automatic system for measurement monitoring andacquisition of data was developed as it is shown inFig. 2.

The main elements of the developed automatic systemfor measurement monitoring and acquisition of data wereas follows: the personal computer, the multi-function

Fig. 1. Scheme of a pearl mill[3] and
built in positions of the sensors.

50 M. Tusar et al. / Journal of Materials Processing Technology 171 (2006) 48–60

Fig. 2. The automatic system for measurement and acquisition. T1 is the sensor for the measuring of the in-flow temperature of grinding mass. T2 is sensorfor the measuring of the out-flow temperature of grinding mass. T3 is sensor for the measuring of the input temperature of cooling water. T4 is sensor forthemeasuring of the output temperature of cooling water. T5 is sensor for the measuring of the temperature of grinding mass in the mill vessel. KPY55A is pressuretransducer. The positions where sensors were built in the pearl mill are marked inFig. 1. The other elements of the system are the turbine measurer, the scale,the multifunctional acquisition board, the multiplexer amplifier electronic circuit and the personnel computer.

acquisition board (i.e. 12 bit converter A/D with frequency8 MHz), the multiplexer amplifier electronic circuit, thesensors (i.e. resistance temperature sensors Pt 100), thescales for weighting the grinding mass (i.e. EB 15000G withintermediator, i.e. RS 232 C) and the computer programwritten in Basic. The sensors and the scales for weightingthe grinding mass were connected across multi-functionacquisition board (converter A/D)/multiplexer amplifierelectronic circuit with the personal computer.

The sensors in the pearl mill were measuring the in-flowand out-flow temperature of grinding mass, the input and out-put temperature of cooling water, the temperature of grindingmass in the mill vessel and the pressure in the mill vessel. Thesensors were built in pearl mill on places where variable mea-surements could be efficiently implemented by using specialparts made for this occasion.

2.1. Variables selection

Grinding with pearl mills is a well-known process so wecould assume which variables might be important for thementioned optimisation. The variables were classified intothree classes:

- independent variables;- dependent variables;- oring

stem

cedo input

data for modelling. The independent and dependent variablesrepresented the training values for ANN. The list of variablesis shown inTable 1.

Time dependent variables: the temperatures of differentparts of the milly4 andy7, the pressure in the vessely6, thethroughput of grinding massy5, and the time of grinding of1 kg of grinding massy1, were recorded by the automatic sys-tem for measurement monitoring and data acquisition. Thegranulation of the particlesy2 and the consumption of energyper 1 kg of the grinding massy3 were measured manually.

2.2. Calibration and testing of the elements of theautomatic measuring system

The elements were calibrated separately before they wereconnected to the automatic system and built in the mill.The calibration was done for temperature sensors, pressuretransducer and the multi-function acquisition board. Themeasurement errors were determined for the elements. Themeasurement error of the multi-function acquisition boardwas±0.0011 V.

2.2.1. Temperature sensor calibrationThe temperature range was from 0 to 100◦C, therefore,

the relationships between temperature and drop in voltage forfi erentt

i sorswd -

time dependent variables: sensors were used for monitand controlling of temperature and pressure in the syto prevent disorders in the milling process.

The values of independent variables directly influenn values of dependent variables and they were the

ve temperature sensors were measured for three diffemperatures: 1, 45 and 90◦C.

Firstly, the sensors were calibrated toT0 = 1◦C, the dropn voltageU0 was determined on 0.1 V. Secondly, the senere calibrated toT′ = 90◦C and the drop in voltageU′ wasetermined on 8.5 V. With sensors atT = 45◦C the linearisa


Table 1List of independent and dependent variables

Description of the independent/dependent variables Unit Minimal value Maximal value

Independent variablex1 Viscosity of the grinding mass s 20 60x2 Throughput of the cooling water l/h 200 800x3 Rotation frequency of the pump min−1 40 240x4 Rotation frequency of the mill min−1 1050 1750x5 Amount of the pearls in the mill % 30 70

Dependent variabley1 Time of grinding of one kilogram of grinding mass sy2 Granulation of the particles mmy3 Consumption of energy per 1 kg of the grinding mass Why4 Temperature of the out-flow grinding mass ◦Cy5 Throughput of the grinding mass kg/hy6 Pressure in the vessel bary7 Difference between the temperature of the in-flow and out-flow cooling water ◦C

tion was done. Because it was assumed that the relationshipwas linear, Eqs.(1)–(3)could be used:

k = U ′ − U0

T ′ − T0= 8.5 − 0.1

90− 1= 0.094382

(V/◦C

)(1)

n = U0 − kT0 = 0.1 − 0.094382× 1 = 5.618× 10−3 (V)

(2)

U(at 45◦C) = kT + n = 0.094382× 45+ 5.618× 10−3

= 4.2528 V (3)

Eq.(3)shows that the drop in voltageU atT = 45◦C should be4.253 V. All sensors were then calibrated at 45◦C on 4.253 V.One hundred measurements for every sensor at selected threetemperatures were done: 1, 45 and 90◦C. On the basis of themeasurements, the terms of linear polynomials were then cal-culated with the linear regression for every particular sensor.Eqs. from(4) to (8) show the calculated values of terms forparticular sensors.

The corresponding linear equations were as follows:

T1 = 10.629U1 − 0.192

temperature of the in-flow of grinding mass for general

)

T

t

T

i 6)

T

o

T5 = 10.656U5 − 0.017

temperature in the vessel, related to the pressurey6 (8)

The errors for the temperature sensors were as follows:

T1 =±0.267◦C;T2 =±0.545◦C;T3 =±0.177◦C;T4 =±0.48◦C;T5 =±0.267◦C;

2.2.2. Sensor calibration for pressure measurementsThe piezo resistor in pressure transducer KPY55A

(Siemens) was temperature dependent therefore the depen-dency of measured drop in voltageU2 for selected tem-peraturesTj and pressuresPi must be determined first byso called indirect model. The temperature characteristics ofthe pressure transducer prepared by producer is written inTable 2where the values of drop in voltageU0 at tempera-

Table 2Output of drop in voltageU0 (V) determined by pressure transducer KPY55Aat different pressuresPi (bar) and temperatureTj (◦C)

U0 (V)

P

932256.86.16.46.77.06.36.67

1 .96

T

monitoring and controlling (4

2 = 10.624U2 − 0.280

emperature of the out-flow of grinding massy4 (5)

3 = 10.640U3 − 0.293

n-flow temperature of the cooling water (

4 = 10.613U4 − 0.181

ut-flow temperature of the cooling watery7 = T3 − T4 (7)

i (bar) Tj (◦C)

1 5 10 15 20 25 30

0 1.78 1.79 1.82 1.84 1.87 1.9 1.1 2.09 2.09 2.12 2.15 2.17 2.2 2.2 2.43 2.44 2.46 2.48 2.5 2.53 2.3 2.75 2.76 2.78 2.79 2.81 2.83 24 3.08 3.09 3.10 3.11 3.12 3.14 35 3.39 3.39 3.41 3.42 3.42 3.44 36 3.72 3.72 3.73 3.73 3.74 3.75 37 4.04 4.04 4.05 4.05 4.05 4.06 48 4.36 4.35 4.35 4.35 4.35 4.35 49 4.68 4.66 4.66 4.66 4.66 4.66 40 5.00 4.88 4.98 4.97 4.97 4.96 4

ransducer producer supplied this data.


Table 3The sum of squares (SSr) and root mean squares (RMS) of differencesbetween measured and calculated values of drop in voltage with linear,quadratic, cubic polynomial and ANN models

Model RMS (V) SSr Number of parametersin the model

Fratioa

Linear 0.0241 0.0448 3 58.5790b

Quadratic 0.0129 0.0129 6 0.7482c

Cubic 0.0127 0.0123 10ANN 0.0253

a Fratio is defined with Eq.(9).b The Fratio for linear and quadratic model, correspondentFinv(0.01; 3;

71) = 4.07 < 58.5790.Conclusion: the quadratic model is more adequate.c The Fratio for quadratic and cubic model, correspondentFinv(0.01; 4;

67) = 3.61 > 0.7482.Conclusion: the quadratic model is more adequate.

turesTj and pressuresPi were measured at amplificationK0.The data inTable 2were used for the calculation of termsfor linear, quadratic, cubic polynoms and 2× 6× 1 error-backpropagation ANN.

Table 3shows the sum of squares and root mean squaresof differences (RMS) between measured and calculated dropin voltageU0. Fratio was used as selection criteria for themost adequate model[7,8]. Fratio is comparing the sums ofsquares of differences between measured and calculated val-ues for reduced model and full model. Linear model (no. ofparameters in the model isk = 3) and quadratic polynoms(no. of parameters in the model isp = 6) were compared, andsecondly quadratic (k = 6) and cubic polynoms (p = 10).

Fratio = (SSrreduced model− SSrfull model)/(p − k)

SSrfull model/(N − p)(9)

SSr means the sum of squares between the calculated andactual values of drop in voltageU0, k and p the numbersof parameters in the reduced and full model respectively,while N is the number of experiments,N = 70. Value ofFratiois compared toFinv, which is value from statistical tablesfor F distribution. Fratio for the comparison of linear andquadratic polynom is 58.5790 and it is much higher thenthe corresponding tabulatedFinv value.Fratio of the compar-ison between quadratic and cubic polynom is 0.7482 and iti ist omi thel odeli

U

E tageU lifier

Table 5The calculated values of drop in voltage atT = 22◦C for selected values ofreference pressures

P (bar) 0 1 2 3U1 (V) 1.8798 2.186 2.5076 2.8158

TheU1 was calculated with Eq.(10).

K0—the data are displayed inTable 2. But our system hadamplifier K1 and therefore we had to measure the drop involtageU2 in dependency of pressure and temperature oncemore. The measurements at the room temperature equal to22◦C are detailed inTable 4. The relationship between thedrop in voltageU2 and the pressurePi could be expressedby linear polynom in Eq.(11) determined with the linearregression on the basis of the data inTable 4.

U2 = 1.65392p + 3.37339

RMS = 0.0065 V(11)

The drop in voltageU1 at T = 22◦C for selected referencepressures were calculated by using Eq.(10). The calculatedU1 are written inTable 5. On the basis of the data inTable 5,the following Eq.(12)was calculated:

U1 = 0.31482p + 1.87265

RMS = 0.00001 V(12)

Eq.(13)was developed by linear transformation of Eqs. (11)and(12)as follows:

k = U1(1) − U1(0)

U2(1) − U2(0)= 0.31482

1.65392= 0.190348

n

U )

Ua raturem

dingm

oi redb

P

TT onstan

P .5U 7.52 3.53

s lower then the tabulatedFinv. Regarding the hypothesesting the first comparison in favour of quadratic polyns most likely. The ANN in this case is comparable withinear polynom which is not adequate. The quadratic ms as follows:

0 = 1.7657+ 0.3305p + 0.00393T − 0.0009p2

+ 0.00004T 2 − 0.00059pT (10)

q. (10) shows the relationship between the drop in vol0 in dependency of pressure and temperature for amp

able 4he measured drop in voltageU2 at different reference pressuresPref and c

ref (bar) 0 0.5 1 1.5 2 2

2 (V) 3.53 4.17 4.94 5.79 6.63

= U1(1) − kU2(1) = 2.18747− 0.190348× 5.0273

= 1.23122

0 = 0.190348U2 + 1.23122 (13

2: the drop of voltage measured by KPY55A;Tj: temper-ture measured by temperature sensor T5; the tempeeasurement error is±0.005 V.The measurement error of the throughput of the grin

ass is±2.02 kg/h.Eq.(13)was used for the calculation ofU0. Eq.(14)based

n Eq.(10)was used for the calculation of pressurePi (y6) bynputting the calculatedU0 values and temperature (measuy sensor T5).

i = −b − √b2 − 4ac

2a(14)

t temperatureT (22◦C) and amplifierK1

3 2.5 2 1.5 1 0.5 08.43 7.53 6.65 5.79 4.96 4.17


Table 6Values of independent and dependent variables (first and second row: signs and units) are presented

Expt. no. x1 (s) x2 (l/h) x3 (min−1) x4 (min−1) x5 (%) y1 (s/kg) y2 (�m) y3 (Wh/kg) y4 (◦C) y5 (kg/h) y6 (bar) y7 (◦C)

*1 20 232 234 1764 70 44.6 13.0 37.77 39.9 85.59 1.25 7.8*2 20 795 43 1746 70 220.8 9.0 208.21 33.0 17.10 0.08 3.9*3 60 800 230 1800 70 – – – – – >3.50 –*4 60 A202 40 1750 70 190.2 6.5 200.95 47.7 18.93 0.28 12.8*5 20 794 45 1075 30 254.2 16.0 83.08 20.8 12.35 0.13 1.6*6 20 188 232 1082 30 59.9 21.0 13.05 25.0 70.19 1.42 3.8*7 58 181 40 1060 30 218.2 17.0 93.05 27.8 16.26 0.52 5.3*8 58 804 235 1060 30 51.8 27.0 23.17 29.3 81.68 3.14 1.99 20 741 236 1046 70 53.1 12.0 36.29 39.2 72.26 1.42 3.010 20 194 42 1030 70 258.4 8.5 120.03 30.8 13.82 0.08 6.411 60 799 40 1020 70 217.4 10.0 173.74 30.9 16.69 0.56 3.412 60 212 232 1020 70 47.1 10.0 54.47 65.6 87.59 3.12 9.513 20 187 45 1813 30 258.6 14.0 127.74 32.1 13.75 0.11 6.114 20 735 232 1750 30 45.7 21.0 5.92 28.7 80.53 1.40 2.315 62 199 232 1791 30 40.2 33.0 35.88 44.4 95.56 3.14 12.016 62 776 42 1764 30 201.7 11.5 180.84 30.5 17.77 0.25 3.8*s1 43 474 125 1425 50 94.5 15.0 76.28 38.6 39.88 2.32 5.4*s2 43 464 125 1425 50 88.7 16.0 66.56 37.2 42.61 2.91 4.2*s3 42 520 133 1405 50 77.8 13.0 63.42 40.0 52.49 1.56 4.5s4 42 525 136 1382 50 68.9 12.0 58.90 40.6 54.07 1.60 5.2s5 42 552 139 1377 50 74.0 11.5 66.24 40.8 53.01 1.59 4.7t1 58 277 40 1774 30 214.8 12.0 143.62 35.7 19.23 0.33 7.1t2 58 807 236 1774 30 40.3 20.0 30.05 35.9 99.49 2.84 2.1t3 20 727 236 1030 70 48.6 13.0 38.75 42.8 75.37 1.87 2.9

Experiments from s1 to s5 are repetitions. Experiments marked with “*” were done in the first batch, the rest in second. Experiments from t1 to t3 are testexperiments. Experiment number 3 was terminated because the critical pressure has been exceeded.

a = 0.0009

b = 0.00059Tj − 0.3305

c = U0 − 1.7657− 0.00393Tj − 0.00004T 2j

2.3. Experimental design selection

The selection of the experimental design depends on twoconditions: the sample weight had to be 8 kg and the approxi-mate time of one experiment was 2 h. The time of experimentdepended on the size of particles (y2) that had to be less then10�m. Therefore, two-level fractional factorial design with16 experiments was selected and five repetitions of the cen-tral point for checking the repeatability were added[9]. Thisexperimental design enables determination of a polynomialwith at least 16 terms.

The measurements are written inTable 6. Experimentsmarked as “s” are repetitions. Experiments marked with “*”were done in the first batch, the rest in second. Experimentsmarked as t1, t2 and t3 are test experiments used for validationof developed models.

In the first batch of the experiments (marked with * inTable 6), three replications of the central point and eightexperiments were made, which enabled to calculate the linearmodels. In the second batch, two more replications were done(all together five marked with s1–s5 in Table 6) with the resto rmedl ilitys ment

the pressure in vessel was too high and the automatic systemstopped the milling process.

3. Modelling

In general, the methods for modelling could be dividedinto two groups: “black box models” (ANNs, i.e. error back-propagation[10–13], counter propagation[14] or Hopfieldneural network[15]) and defined models (polynomials andother mathematical equations).

The most adequate model of the milling process wassought, therefore, the polynomial and neural network modelswere developed and compared.

The simultaneous use of both types of models and thecomparison of the results of both models seemed to be themost efficient. For the optimisation the second order poly-nomial models and back-propagation ANN model were usedand compared. For the comparison of the calculated modelsPRMS expressed in percentages were used:

RMS =√∑m

i

∑nij (yij − yi)2

NPRMS = RMS× 100

ymax − ymin(15)

m is the number of different experiments in the experimentald enti les1 t

f the experiments. Three more experiments were perfoater for testing. Replications showed good reproducibo adequate factors were selected. At the third experi

esign,ni the number of replications of particular experimn experimental design,i the number of dependent variab–7, j the number of experiments 1–m, yij the dependen


Table 7Values ofPRMSm and PRMS calculated with Eq.(15), Fisher ratio,Flof ,regression factor,Freg, and correlation coefficient, CC, have been calculated

Dependent variable PRMSm PRMS Flof Freg CC

y1 4.33 12.44 15.38 19.91 0.8981y2 6.54 11.84 4.82 8.65 0.7583y3 2.82 9.90 21.44 21.32 0.9154y4 3.03 10.63 19.39 8.21 0.7438y5 7.14 4.42 0.39 133.31 0.9792y6 17.61 17.66 1.27 8.06 0.7581y7 3.95 9.49 8.76 19.68 0.8837

Freg have to be compared withFinv(α = 0.05, d.f.: 5, 14) = 2.958.Flof haveto be compared withFinv(α = 0.05, d.f.: 4, 10) = 5.964.

variable in the experimental design, ˜yi calculated dependentvariable,N the number of all experiments with replications(20 in our case) andymin and ymax are measured minimaland maximal dependent variables, i.e.y1min = 40.2 inTable 6,y1max= 258.6 inTable 6.

In the case ofPRMSm, yi is instead of ˜yi andyi is the aver-age of the replicated particular experiment in experimentaldesign.

3.1. Development of polynomial models

After performing the first batch of 11 experimentsthe preliminary linear polynomial models in the formyi = b0 + b1x1j + b2x2j + b3x3j + b4x4j + b5x5j were calculated.More reliable information for each of the seven dependentvariables was furthermore obtained on the basis of all 16experiments (j = 1–16).

The results of the statistical analysis for the above sevenlinear polynomial models are collected inTable 7. Fregvaluesin comparison withFinv(5, 14) values were not too high, thehighest being for they5 indicating that with the exception ofthe linear polynomial fory5 the models of higher order shouldbe considered. The comparison betweenFlof andFinv showeda more adequate, but similar picture: linear models were ade-quate fory2, y5 andy6, becauseFlof < Finv. Fory1, y3, y4 andy the F > F —the polynomials of higher orders shouldbP canb tef yno-m n oft bleso peri-m e only1 raticp iblet

vency delw osts sseda n

Table 8The list of the best-suited polynomial terms for they1 model determined onthe basis of testing experiments

No of iteration Terms Freg CC r2 PRMS

1 x3x3 169.1238 0.9465 0.9521 8.372 x1x4 236.9874 0.9739 0.9780 5.683 x1x3 226.3100 0.9794 0.9837 4.884 x1x5 290.2051 0.9870 0.9904 3.745 x2x4 275.1084 0.9886 0.9922 3.386 x1 262.2958 0.9897 0.9935 3.087 x4x5 389.2993 0.9939 0.9965 2.27

r2 is deterministic coefficient and CC is correlation coefficient. Bold valuesshow the incomplete polynom (incomplete polynom fory1 written inTable 9)wherePRMS are higher thanPRMSmand the first lower value fory1 in Table 7.

was:

y1 = b0 + b1x3 (16)

In the first step all 19 possible terms (x1, x2, x4, x5, x1x1,x2x2, x3x3, x4x4, x5x5, x1x2, x1x3, x1x4, x1x5, x2x3, x2x4, x2x5,x3x4, x3x5 andx4x5) were successively added to starting poly-nomial Eq.(16) for each of the 19 incomplete polynomialmodels with three terms.PRMS values were calculated andcompared withPRMSm calculated for measured dependentvariables. Among them the polynomial with the lowestPRMSvalues was selected for the second step (the termx3x3 in thecase ofy1):

y1 = b0 + b1x3 + b2x3x3 (17)

In the second step the remaining 18 terms (withoutx3x3) wereadded again and the same kind of evaluation was performed.

The procedure of adding terms was terminated afterPRMS(=3.74 fory1 in fourth iteration) turns out to be lower thenPRMSm= 4.33 inTable 7. In Table 8, the statistical parametersof models evaluated during the model building fory1 areshown.

The same procedure was applied to the remaining vari-ables and the following polynomials of the first order wereu sec-o

y

y

y

y

y

T arew ithA

7 lof inve more appropriate. Comparing the values ofPRMSm andRMS for each dependent variable, the same conclusione drawn. The linear model fory5 was statistically adequa

or all other dependent variables the higher order polial models had to be calculated. For the determinatio

he complete quadratic model with five independent variane had to determine 21 terms, therefore at least 22 exents equally spread in the space were necessary. Sinc6 different experiments were completed the full quadolynomial with five independent variables was not poss

o achieve.In order to obtain the best possible model within gi

onstrains the following procedure for each variabley1, y2,3, y4, y6 andy7 was started. The starting point for each moas the smallest possible linear equation involving the mignificant variable obtained from the linear models discubove. In the case ofy1 (grinding time) the starting equatio

sed for the determination of incomplete polynomials ofnd order:

2 = b0 + b1x1 + b2x2 + b3x3 + b4x4 + b5x5 (18)

3 = b0 + b1x1 + b2x2 + b3x3 + b4x4 + b5x5 (19)

4 = b0 + b1x1 + b3x5 (20)

6 = b0 + b1x1 + b2x2 + b3x3 + b4x4 + b5x5 (21)

7 = b0 + b1x1 + b2x2 + b3x3 + b4x4 + b5x5 (22)

he calculated incomplete polynomials of higher ordersritten inTable 9and they were used for the comparison wNN.


Tabl

e9

The

coef

ficie

nts

ofth

eob

tain

edpo

lyno

mia

lsof

high

eror

ders

y 1b 0

=38

1.40

4x 3

=−2

.914

x 3x 3

=0.

007

x 1x 4

=−0

.001

x 1x 3

=0.

004

x 1x 5

=−0

.008

y 2b 0

=−1

0.55

4x 1

=0.

369

x 2=

0.03

8x 3

=0.

114

x 4=

0.00

7x 5

=0.

113

x 3x 5

=−0

.001

x 1x 5

=−0

.005

y 3b 0

=−8

2.02

49x 1

=0.

7851

2x 2

=−0

.292

6x 3

=0.

3885

2x 4

=0.

1222

6x 5

=2.

0161

6x 2

x 2=

0.00

035

x 3x 4

=−0

.000

43x 2

x 3=

−0.0

0025

x 3x 5

=−0

.003

33x 1

x 3=

−0.0

0195

x 4x 5

=−0

.000

33y 4

b 0=

−55.

4146

x 1=

−0.0

2237

x 5=

0.31

438

x 1x 3

=0.

0013

3x 1

x 2=

−0.0

0027

x 4=

0.09

752

x 1x 5

=0.

0046

7x 3

x 5=

0.00

126

x 3x 4

=−0

.000

05x 4

x 5=

−0.0

0024

x 4x 4

=−0

.000

03y 5

b 0=

−19.

838

x 1=

0.20

468

x 2=

−0.0

0255

x 3=

0.35

177

x 4=

0.00

802

x 5=

0.05

511

y 6b 0

=−0

.764

70x 1

=0.

0023

4x 2

=−0

.000

04x 3

=0.

0257

5x 4

=−0

.000

18x 5

=−0

.000

4x 3

x 3=

−0.0

0008

x 1x 3

=0.

0001

8y 7

b 0=

12.1

4952

x 1=

−0.0

3381

x 2=

0.00

150

x 3=

−0.0

0567

x 4=

−0.0

1682

x 5=

0.06

066

x 1x 2

=−0

.000

1x 1

x 4=

0.00

009

x 4x 4

=0.

0000

1x 1

x 3=

0.00

021

Fig. 3. Graphical representation of RMS as dependence on number of iter-ations for two-level ANN with seven nodes on hidden layer is showed.

3.2. Development of ANN model

Additionally to the generation of the polynomial mod-els the modelling with error back propagation ANNs wasattempted. Due to the relatively small number of experimentsthe most suitable ANN was determined by systematic checkof the most promising layouts.

The one-layer and then five two-layer ANNs with three toseven nodes on the hidden layer were developed on the basisof usingyALL . The dependent variableyALL is expressed inEq. (23) and represents the cumulative of dependent vari-ables.

yALL =s∑

j=1

βjyj (23)

yALL is criteria for s dependent variables. ˜yj is calculatedvalue for j-dependent variable.βj is the share of ˜yj and itdeterminates its weight of importance in comparison withthe other dependent variables.

The following parameters of the ANNs were determinedto be most suitable: the learning rate 0.5 (from 0 to 2) andmomentum term 0.9 (from 0 to 0.99). The data fromTable 6were used for training. Each time after 100 iterations duringthe training procedure the root mean square of differencebetween experimental and calculated dependent variables( n-d NNo thes dt e andt itera-t wasi

ticalpla asmf wase t bet blesw um-

RMS) were calculated. InFig. 3, it can be seen the depeence of value RMS for the seven nodes hidden layer An the number of training iterations. At all other exampleslope was similar inTable 10. In Fig. 3, it could be observehat at 5000 iterations RMS was not decreasing any morhe training procedure was stopped, because additionalions were meaningless. So the final number of iterationsn all examples set to 5000.

In order to evaluate the ANNs with the same statisarameters Fisher ratioFlof , Factor of regressionFreg, corre-

ation coefficient CC, and deterministic coefficientr2 whichre valid for the polynomials the following consideration wade. The number of parameters in the ANN modelk, which

or the polynomials is determined with number of terms,qual to the number of connections. However, it mus

aken into account that in one ANN all dependent variaere incorporated. One-level ANN had a well defined n


Table 10Course of RMS at training of six ANNs

Number of iterations RMS at different number of nodes on hidden layer

Zero neuron Three neurons Four neurons Five neurons Six neurons Seven neurons

1000 0.337 0.327 0.309 0.295 0.336 0.2782000 0.332 0.322 0.298 0.279 0.294 0.2503000 0.328 0.319 0.294 0.264 0.267 0.2394000 0.328 0.319 0.293 0.262 0.264 0.2375000 0.326 0.319 0.293 0.262 0.263 0.236

Fig. 4. Architecture of a two-layer ANN used as model of grinding process.Five neurons on input layer and seven neurons on output layer represent fiveindependent variables,x1–x5, as seven dependent variablesy1–y7, respec-tively.

ber of parameters that contributed to the value of the outputneuron.

For a two-layer ANN model the value of parameterk wasnot defined so well because the connections between the hid-den and output nodes were common to all variables as shownin Fig. 4. For many-layer ANN the value of parameterk couldbe estimated using the presumption that all dependent vari-ables were influenced equally by all connections betweendifferent layers in the following way:

k = 1

noutput

output−1∑i=1

nini+1 (24)

Table 12Validation of ANN with five neurons on hidden layer by calculation of sta-tistical parameters as follows: Fisher ratioFlof , regression factorFreg anddeterministic coefficientr2

Dependent variable Flof Freg r2

y1 4.53 43.34 0.97y2 0.82 24.22 0.95y3 6.32 45.26 0.94y4 16.83 7.35 0.91y5 1.01 40.85 0.98y6 0.34 10.17 0.90y7 4.99 23.30 0.97

noutput is the number of output nodes,ni the number of nodeson layersi including the bias andni+1 is the number of nodeson layers (i + 1) without bias.

From Eq.(23) it can be seen that the value of parameterkfor one-layer ANN was equaln1 which was at the same timeequal to the number of parametersk in a linear model (withor without bias).

The most adequate ANN was determined on the basis ofthe values ofFlof . In Table 11, all values ofFlof for all sixANNs (after 5000 training iterations) are presented. Param-eter k was calculated using Eq.(24), while the other twoparameters had valuesN = 20 andp = 16.

Table 12demonstrates that almost all dependent variableswere simulated better with ANN with five nodes on a hiddenlayer. Even the ANN with one layer (results inTable 12) hadall valuesFlof lower than linear model (results inTable 10).The selected model was not sufficiently complex for y4 (tem-perature of out-flow grinding mass). The value ofFlof for y3(consumption of energy per kilogram of grinding mass) wasslightly higher than tabledFinv. FromFlof values calculated

Table 11Values ofFlof for seven dependent variables obtained from all six different ANNs and compared withFinv

Number of nodes onhidden layer + bias

Calculated parameterk Rounded value ofparameterk

Flof

y

0 6.00 6 .9643 + 1 6.67 7 .9994 + 1 8.43 8 .0415 + 1 10.29 10 .1636 + 1 12.14 12 .3887 + 1 14.00 14 .944

1 y2 y3 y4 y5 y6 y7 Finv

9.33 3.97 14.16 17.69 0.35 1.14 5.30 54.19 1.08 20.05 18.03 1.98 1.12 8.50 55.33 2.96 17.34 10.87 2.00 0.50 8.39 64.53 0.82 6.32 16.83 1.01 0.34 4.99 67.49 4.58 17.04 19.72 1.04 0.40 4.90 66.31 2.84 19.21 17.95 0.94 0.57 2.78 6


Table 13Correlation coefficients CC between the experimental dependent variablevalues and values predicted with best linear (CC1), higher order incompletepolynomials (CC2) and ANNs (CC3)

Dependent variable CC1 CC2 CC3

y1 0.86 0.99 0.94y2 0.67 0.90 0.90y3 0.89 0.99 0.88y4 0.65 0.96 0.84y5 0.98 0.98 0.97y6 0.67 0.88 0.81y7 0.84 0.96 0.94

for other dependent variables it could be assumed that theANN model was adequate. For the selected ANN with fivenodes on hidden layer some additional statistical parameterswere calculated and shown inTable 13, which proves theadequacy of the selected ANN model.

Two-layer architecture of the ANN with five neurons in thehidden layer was the most appropriate one because only in thiscase thePRMS for calculated values of dependent variableswere comparable with the measured ones.

4. Discussion

The results of the two-layer ANN were compared withthose obtained by the second orders polynomials. One-layerANN was tested as well and these results were comparablewith those obtained by the linear polynomials.

Tables 13 and 14show the results of the statistical anal-ysis of all developed models. On the basis of the devel-oped incomplete polynomials of higher orders, the correla-tions between the independent and dependent variables weredetermined.

Table 14The labelsPRMSm, PLP

RMS, PHPRMS andPANN

RMS stand for measurements, linear polynomials, incomplete higher order polynomials and ANNs, respectively

Dependent variable Dependent variables’ description Unit PRMSm PLPRMS PHP

RMS PANNRMS

y1 Grinding time of one kilogram of grinding mass s 4.3 12.4 3.7 5.7y2 Granulation of the particles mm 6.5 11.8 5.1 5.5y3 Consumption of energy per 1 kgof the grinding mass Wh 2.8 9.9 1.5 3.8y4 Temperature of the out-flow grinding mass ◦C 3.0 10.6 2.7 7.2y5 Throughput of the grinding mass kg/h 7.1 4.4 4.4 4.2y6 Pressure in the mill .7y7 Difference between the temperature of the in-flow

out-flow cooling wateryALL All dependent variables

Table 15Experiments used for test and dependent variables calculated with develope

Dependent variable FromTable 4 YHP

y1 t1 209.17 .80t2 36.86 30t3 52.93 60

y2 t1 17.82t2 20.41

00

y .620575

y 70

y

y

y

yys

t3 12.80

3 t1 143.49t2 31.85t3 31.60

4 t1 37.23
t2 34.39t3 39.67
5 t1 21.27t2 88.87t3 77.54

6 t1 0.33t2 2.93t3 1.39

7 t1 9.19t2 4.86t3 3.81

HP are calculated values of dependent variables with incomplete higher ord

m are measured values of dependent variables. LPI and UPI are lower andhowyANN values that are not within LPI and UPI limits.

bar 17.6 17.7 9.9 11and ◦C 4.0 9.5 4.2 6.2

6.8

d models

LPI UPI YANN ym

196.50 221.84 207.43 21423.92 49.80 31.31 40.40.85 65.02 42.97 48.14.96 20.69 14.63 12.0017.10 23.72 31.02 20.009.83 15.77 11.87 13.

136.17 150.81 146.57 14323.37 40.33 35.87 30.24.15 39.06 38.35 38.

34.27 40.20 38.15 35.
31.36 37.42 34.02 35.9036.80 42.53 38.16 42.80
15.77 26.77 15.00 19.2383.10 94.63 88.49 99.4971.94 83.13 74.63 75.37

−0.18 0.84 1.18 0.332.39 3.48 3.23 2.840.88 1.91 1.63 1.87

7.05 10.04 8.43 7.103.83 5.89 5.12 2.102.90 4.69 3.31 2.90

er polynomials.yANN are calculated values of dependent variables with ANN, andupper limits of prediction intervals calculated for dependent variables. Italic values


Fig. 5. Contours calculated from higher order incomplete polynomials (parameters inTable 9) showed the dependence of particular dependent variable: A (y1),B (y2), C (y3), D (y4), E (y5), F (y6) and G (y7) from x3 andx5. The values for other independent variables were constant:x1 = 20 s,x2 = 200 l/h,x4 = 1000 min−1.Ymax is the maximal value for particular dependent variable (fromy1 to y7). Ymin is the minimal value for particular dependent variable (fromy1 to y7).

In Table 15, the seven calculated dependent variables fortest experiments t1, t2 and t3 are presented and comparedwith the experimental ones. Lower and upper limits of theprediction intervals were used for checking the adequacyof calculated dependent variables which had to be insidethe limits. 90.5% of values predicted with polynomials andANNs were within the lower and upper limits of the calcu-lated prediction intervals. Dependent variablesyHP andyANNwere values calculated with incomplete higher polynomials(parameters inTable 9) and with ANN.

In order to show the applicability of the obtained modelsthe comparison of the calculated contours with the incom-plete second order polynomial and ANN models for all sevendependent variables are given inFigs. 5 and 6where thecontours are shown for two selected independent variablesx3 (pump rotation frequency) andx5 (amount of pearls in

Table 16Determination of two sets of optimal values for dependent variables withANN by using the simple genetic algorithm

Optimum 1 Optimum 2

Input data Output data Input data Output datax1 = 20 s y1 = 95 s/kg x1 = 40 s y1 = 209 s/kgx2 = 200 l/h y2 = 10.7�m x2 = 200 l/h y2 = 5.3�mx3 = 150 min−1 y3 = 47 Wh/kg x3 = 40 min−1 y3 = 200 Wh/kgx4 = 1000 min−1 y4 = 37◦C x4 = 1800 min−1 y4 = 52◦Cx5 = 70% y5 = 57 kg/h x5 = 70% y5 = 18 kg/h

y6 = 0.9 bar y6 = 0.3 bary7 = 5.2◦C y7 = 5.2◦C

For the determination of the optimum 1 the condition to determine the lowestenergy consumption was input in the genetic algorithm. For the determina-tion of the optimum 2 we input in the genetic algorithm one condition thaty2 has to be lower then 9�m.


Fig. 6. Contours calculated from ANN model showed the dependence of particular dependent variable: A (y1), B (y2), C (y3), D (y4), E (y5), F (y6), G (y7) andH (yALL ) from x3 andx5. The values for other independent variables were constant:x1 = 20 s,x2 = 200 l/h andx4 = 1000 min−1. Ymax is the maximal value forparticular dependent variable (fromy1 to y7 andyALL ). Ymin is the minimal value for particular dependent variable (fromy1 to y7 andyALL ).

the mill). The values for other three independent variableswere constant:x1 = 20 s, x2 = 200 l/h andx4 = 1000 min−1.Contours calculated with ANN and polynomial model weresimilar and had a similar position and value of maximal andminimal points. On the basis of contours obtained from theselected models and from statistical parameters, it can beconcluded that different models are comparable.

The optimal values of independent variables for optimalconsumption of energy were obtained from the ANN modelby using the simple genetic algorithm[16]. The criteria fordetermination of optimal values of variables were input inthe algorithm as intervals (lower and upper limits) or the

highest or lowest values. When the genetic algorithm reachesall criteria stops or more often when the selected numberof iterations is done. The model allowed determination ofoptimal values regarding the technological demands.

In our case the optimum depended on the energy consump-tion for achieving the required size of particles was sought.From the ANN model two sets of values for independent vari-ables were determined leading to optimal value of dependentvariables as it is shown inTable 16. For the determinationof the optimum 1 the genetic algorithm was used to find theindependent variable values giving the lowest energy con-sumption. Quite often the size of particlesy2 had to be less


then 9�m therefore for the determination of the second opti-mal point the genetic algorithm was used to find requiredvariable values. Actually, the algorithm was seeking the min-imal size value of the particles. The second optimal pointwas determined: the size of particles was 5.3�m but the con-sequences were the high consumption of energy and longermilling. The valueβ, taking into account Eq.(23), was equalto 1 for determination of both optimal points.

5. Conclusions

Several modelling procedures for prediction and optimi-sation of pigment grinding parameters with pearl mills waspresented and discussed. The optimal values found were con-firmed later on with the experiments and the models couldnow be used in the real production for the prediction ofrequired parameters as the response to the specific inputsvariables.

The reliability of predictions obtained from the calculatedincomplete higher order polynomials and those obtained fromANN models were comparable. The described models couldnow be used for optimisation purposes in daily routine workin the pilot laboratory. The determined correlations betweenselected variables enable prompt response to special techno-logical demands, which must be respected.

A

anyC ourr lE ond

R

orime

[2] F. Draganescu, M. Gheorghe, C.V. Doicin, Models of machine toolefficiency and specific consumed energy, J. Mater. Process. Technol.141 (2003) 9–15.

[3] V. Tandon, H. El- Mounayri, A novel artificial neural networks forcemodels for end milling, Int. J. Adv. Manuf. Technol. 18 (2001)693–700.

[4] W.C. Carpenter, J.-F.M. Barthelemy, A comparison of polynomialapproximations and artificial neural nets as response surfaces, Struct.Optimization 5 (1993) 166–174.

[5] I. Bajsic, M. Tusar, L. Tusar, A. Mismas, Optimizacija procesamletja barvnega pigmenta (Optimisation of colour pigment grindingwith pearl mills), Strojniski vestnik. (J. Mech. Eng.) 41 (1995) 81–98.

[6] M. Gasperlin, L. Tusar, M. Tusar, J. Smid-Korbar, J. Zupan, J.Kristl, Viscosity prediction of lipophilic semisolid emulsion sys-tems by neural network modelling, Int. J. Pharm. 196 (2000) 37–50.

[7] J. Banks, D. Goldsman, J.S. Carson II, in: H.M. Wadsworth Jr. (Ed.),Computer Simulation in Handbook of Statistical Methods for Engi-neers and Scientists, McGraw Hill Publishing Company, New York,1990.

[8] T.P. Ryan, in: H.M. Wadsworth Jr. (Ed.), Linear Regression in Hand-book of Statistical Methods for Engineers and Scientists, McGrawHill Publishing Company, New York, 1990.

[9] P.R. Nelson, in: H.M. Wadsworth Jr. (Ed.), Design and Analy-sis of Experiments in Handbook of Statistical Methods for Engi-neers and Scientists, McGraw Hill Publishing Company, New York,1990.

[10] D.E. Rumelhart, G.E. Hinton, R.J. Williams, Learning internal rep-resentations by error back-propagation, in: D.E. Rumelhart, J.L.MacClelland (Eds.), Distributed Parallel Processing: Explorations in

ge,

[ te on990)

[ tedDis-Press,

[ lgo-990)

[ duc-

[ EE

[ old,

cknowledgements

The authors would like to thanks to the chemical compOLOR from Medvode, Slovenia for giving support to

esearch and to Prof. Dr. Ivan Bajsic, Faculty of Mechanicangineering, University of Ljubljana, for fruitful cooperatiuring the experimental work.

eferences

[1] H.Z. Li, X.P. Li, X.Q. Chen, A novel chatter stability criterion fthe modelling and simulation of dynamic milling process in the tdomain, Int. J. Adv. Manuf. Technol. 22 (2003) 619–622.

the Microstructures of Cognition, vol. 1, MIT Press, CambridMA, USA, 1986, pp. 318–362.

11] H.L.J. van der Maas, P.F.M.J. Verschure, P.C.M. Molenaar, A nochaotic behavior in simple neural networks, Neural Netw. 3 (1119–122.

12] D.E. Rumelhart, J.L. McClelland, R.J. Williams, Parallel distribuprocessing, in: D.E. Rumelhart, J.L. McClelland (Eds.), Paraleltributed Processing, The Massachusetts Institute of TechnologyCambridge, 1987.

13] J. Leonard, M.A. Kramer, Improvement of backpropagation arithm for taining neural networks, Comput. Chem. Eng. 14 (3) (1337–341.

14] J. Zupan, J. Gasteiger, Neural Networks for Chemists: An Itrotion, VCH, Weinheim, 1993.

15] R.P. Lippman, An introduction to computing with neural nets, IEASSP Mag. (1987) 4–22.

16] L. Davis, Handbook of Genetic Algorithms, Van Nostrand ReinhNew York, 1991.

the optimisation of energy consumption and time in colour pigment grinding with pearl mills

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