International Journal of Mineral Processing 110–111 (2012) 53–61

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International Journal of Mineral Processing

j ourna l homepage: www.e lsev ie r .com/ locate / i jminpro

Prediction of terminal velocity of solid spheres falling through Newtonian andnon-Newtonian pseudoplastic power law fluid using artificial neural network

R. Rooki a, F. Doulati Ardejani a, A. Moradzadeh a, V.C. Kelessidis b,⁎, M. Nourozi c

a Faculty of Mining, Petroleum and Geophysics, Shahrood University of Technology, Shahrood, Iranb Mineral Recourses Engineering Department, Technical University of Crete, Chania, Greecec Faculty of Mechanic, Shahrood University of Technology, Shahrood, Iran

⁎ Corresponding author.E-mail address: kelesidi@mred.tuc.gr (V.C. Kelessidi

0301-7516/$ – see front matter © 2012 Elsevier B.V. Alldoi:10.1016/j.minpro.2012.03.012

a b s t r a c t

a r t i c l e i n f o

Article history:Received 26 November 2011Received in revised form 30 January 2012Accepted 19 March 2012Available online 3 April 2012

Keywords:Terminal velocityMineral processingNewtonian and power law fluidDrilling cuttings transportArtificial neural network

Prediction of the terminal velocity of solid spheres falling through Newtonian and non-Newtonian fluids isrequired in several applications like mineral processing, oil well drilling, geothermal drilling andtransportation of non-Newtonian slurries. An artificial neural network (ANN) is proposed which predictsdirectly the terminal velocity of solid spheres falling through Newtonian and non-Newtonian power lawliquids from the knowledge of the properties of the spherical particle (density and diameter) and of thesurrounding liquid (density and rheological parameters). With a combination of non-Newtonian data withNewtonian data taken from published data giving a database of 88 sets, an artificial neural network isdesigned. Analysis of the predictions shows that the artificial neural network could be used with goodengineering accuracy to directly predict the terminal velocity of solid spheres falling through Newtonian andnon-Newtonian power law liquids covering an extended range of power law values from 1.0 down to 0.06.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Knowledge of the terminal settling velocity of solids in liquids isrequired in many industrial applications. Typical examples includemineral processing, drilling for oil and gas, geothermal drilling,hydraulic transport systems, thickeners, solid–liquid mixing, andfluidization equipment. In many of these processes, it is the“hindered” falling velocity that is of interest, hindered by thepresence of walls or by other particles (Kelessidis and Mpandelis,2004). This velocity is proportional to the free (terminal) fallingvelocity of the solid particles so there has been a great interest inpredicting the free falling velocity of solid particles in liquids.

The type of movement of single solid sphere in Newtonian andnon-Newtonian liquids is well known; after a short acceleration time,it will fall at its terminal settling velocity V. For an unbounded liquid,V can be calculated from the knowledge of the liquid and solidphysical properties and from the drag coefficient, defined by

CD ¼ 43dg ρs−ρð Þ

ρV2 ð1Þ

Extensive work has been undertaken to relate the drag coefficientwith the Reynolds number of the particle. Previous work (Lali et al.,

s).

rights reserved.

1989; Kelessidis and Mpandelis, 2004) has shown that for a powerlaw fluid, with the rheological equation given by

τ ¼ K _γn ð2Þ

the Reynolds number can be properly defined as

Regen ¼ ρV2−ndpn

Kð3Þ

Various works have been done to create theoretical and semi-empirical relationships of the terminal settling velocity of solid spheresusing Re–CD relationship. There are over 50 correlations publishedrelating the Reynolds number to the drag coefficient for the case ofNewtonian and non-Newtonian fluids (Clift et al., 1978; Peden and Luo,1987; Koziol and Glowacki, 1988; Heider and Levenspiel, 1989;Reynolds and Jones, 1989; Kelessidis and Mpandelis, 2004; Chhabra,2006; Shah et al., 2007). In these correlations the terminal velocity isimplicitly derived, hence, resort must be made to trial and errorprocedure for deriving the terminal velocity. There are not as manyexplicit relationships to predict V, with few equations for Newtonianliquids (e.g. Turton and Clark, 1987; Hartman et al., 1989; Nguyen et al.,1997) and even fewer for non-Newtonian pseudoplastic power lawliquids which cover an extended range of Reynolds numbers (Chhabraand Peri, 1991; Kelessidis, 2004). Most of abovementioned correlationsare complex in form and they cover a special range of Reynolds number.

54 R. Rooki et al. / International Journal of Mineral Processing 110–111 (2012) 53–61

Therefore, a simple and reliable method which can be used withconfidence over the entire range of conditions is not yet available.

Artificial neural networks (ANNs) have gained an increasingpopularity in different fields of engineering in the past few decades,because of their capability of extracting complex and non-linearrelationships. Owing to their inherent nature to model and learn‘complexities’, ANNs have found wide applications in various areas,like, mineral processing (Van Der Walt et al., 1993; Moolman et al.,1995; Eren et al., 1997), of chemical engineering and related fields(Himmelblau, 2000; Sharma et al., 2004; Ibrehem and Hussain, 2009)and in oil industry (Ternyik et al., 1995; Ozbayoglu et al., 2002; Miri etal., 2007; Mohaghegh, 2000). Not much work has been done withANNs in the area of terminal velocity prediction except for the veryrecent work of Ghamari et al. (2010) who used ANN to relate seedsettling velocities with particular seed properties like size, seed typeand moisture content.

The aim of this work is to provide a different approach for theprediction of terminal unhindered velocity of solid spheres fallingthrough Newtonian and non-Newtonian pseudoplastic power lawliquids using artificial neural network.

2. Theory

2.1. Back propagation neural network design

Artificial neural networks (ANNs) are generally defined asinformation processing representation of the biological neuralnetworks. ANN has gained an increasing popularity in different fieldsof engineering in the past few decades, because of their ability ofresolving complex and non-linear relationships. The mechanism ofthe ANN is based on the following four major assumptions (Hagan etal, 1996), a) information processing occurs in many simple elementsthat are called neurons (processing elements), b) signals are passedbetween neurons over connection links, c) each connection link hasan associated weight, which, in a typical neural network, multipliesthe signal being transmitted and d) each neuron applies an activationfunction (usually nonlinear) to its net input in order to determine itsoutput signal.

Fig. 1 shows a typical neuron. Inputs (P) coming from anotherneuron are multiplied by their corresponding weights (w1, i), and

Fig. 1. A typical neuron (De

summed up (n). An activation function (f) is then applied to thesummation, and the output (a) of that neuron is now calculated andready to be transferred to another neuron (Demuth and Beale, 2002).

In this network, each element of the input vector P is connected toeach neuron input through the weight matrixW. The ith neuron has asummer that gathers its weighted inputs and bias to form its ownscalar output n (i). The various n (i) taken together form an S-elementnet input vector n. Finally, the neuron layer outputs form a columnvector derived from

nj ¼XRi¼1

piwij þ bj� �

; j ¼ 1;2;…; S ð4Þ

where

b ¼b1b2bS

24

35; P ¼

p1p2pR

24

35; W ¼

w1;1 w1;2 ::::w1;Rw2;1 w2;2 ::::w2;RwS;1 wS;2 ::::wS;R

24

35 ð5Þ

Then, final output of network is calculated by

aS ¼ f nSð Þ ð6Þ

Here, f is an activation function, typically a step function or a sigmoidfunction, which takes the argument n and produces the output a.Fig. 2 shows examples of various activation functions:

Back-propagation neural networks (BPNN) are recognized fortheir prediction capabilities and ability to generalize well on a widevariety of problems. These models are a supervised type of networks,in other words, trained with both inputs and target outputs. Duringtraining the network tries to match the outputs with the desiredtarget values. Learning starts with the assignment of randomweights.The output is then calculated and the error is estimated. This error isused to update the weights until the stopping criterion is reached. Itshould be noted that the stopping criteria is usually the average erroror epoch.

muth and Beale, 2002).

Fig. 2. Three examples of activation functions (Demuth and Beale, 2002).

55R. Rooki et al. / International Journal of Mineral Processing 110–111 (2012) 53–61

2.2. Network training: the over fitting problem

One of the most common problems in the training process is theover fitting phenomenon. This happens when the error on the trainingset is driven to a very small value, butwhennewdata is presented to thenetwork, the error is large. This problem occurs mostly in case of largenetworks with only few available data. Demuth and Beale (2002) haveshown that there are a number of ways to avoid over fitting problem.Early stopping and automated Bayesian regularization methods aremost common. However, with immediate fixing the error and thenumber of epochs to an adequate level (not too low/ not too high) anddividing the data into two sets; training and testing; one can avoid suchproblem by making several realizations and selecting the best of them.In this paper, we used the ANN Toolbox in MATLAB multi-purposecommercial software in order to implement the automated Bayesianregularization for training BPNN. In this technique, the available data isdivided into two subsets. The first subset is the training set, which isused for computing the gradient and updating the networkweights andbiases. The second subset is the test set. This method works bymodifying the performance function, which is normally chosen to bethe sum of squares of the network errors on the training set. The typicalperformance function that is used for training feed forward neuralnetworks is themean sumof squares of the network errors according to

mse ¼ 1N

XNi¼1

eiÞ2 ¼ 1N

XNi¼1

ti−aiÞ2�

ð7Þ

where, N represents the number of samples, ai is the predicted value, tidenotes the measured value and ei is the error. It is possible to improvethe generalization if we modify the performance function by adding aterm that consists of the mean of the sum of squares of the networkweights and biases which is given by

msereg ¼ γmseþ 1−γð Þmsw ð8Þ

where,msereg is themodified error, γ is the performance ratio, andmswcan be written as

msw ¼ 1N

XNi¼1

wi ð9Þ

Use of the performance function will cause the network to havesmaller weights and biases, and this will force the network responseto be smoother and less likely to over fit (Demuth and Beale, 2002).

3. Terminal velocity prediction using BPNN

The feed-forward neural networks with back propagation (BP)learning are very powerful in function optimization modeling

(Cybenko, 1989; Hornik et al., 1989). In this study, three-layer feed-forward neural networks with back propagation (BP) learning wereconstructed for calculation of terminal velocity.

In this study 88 data sets from literature, represented here inTable 1, were used. All experimental data refer to unhinderedvelocity. The properties of the spherical particles (density anddiameter) and of the surrounding liquid (density and rheologicalparameters) and acceleration of gravity data were selected as inputsof the network. The output of network was terminal velocity. In viewof the requirements of the neural computation algorithm, the data ofboth the inputs and output were normalized to an interval bytransformation process. In this study normalization of data (inputsand outputs) was done for the range of [−1, 1] using Eq. (10)

pn ¼ 2p−pmin

pmax−pmin−1 ð10Þ

where, pn is the normalized parameter, p denotes the actualparameter, pmin represents a minimum of the actual parameters andpmax stands for a maximum of the actual parameters. About 70% or 69out of 88 of the data sets were selected as train data and 25 data totest purposes, randomly.

Several architectures (varied numbers of neurons in hiddenlayer) with Automated Bayesian Regularization training algorithmand mean square error (MSE) performance function were tried topredict terminal velocity using BPNN. Two criteria were used inorder to evaluate the effectiveness of each network and its abilityto make accurate predictions, the root mean square error and thecoefficient of determination. The root mean square error (RMS),which measures the data dispersion around zero deviation, can becalculated as:

RMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1

yi−yið Þ2

N

vuuutð11Þ

where, yi is the measured value, yi denotes the predicted value, andN stands for the number of samples. RMS indicates the discrepancybetween the measured and predicted values. The lowest the RMS,the more accurate the prediction is. Furthermore, the coefficient ofdetermination, R2, given by

R2 ¼ 1−

PNi¼1

yi−yið Þ2

PNi¼1

y2i −PNi¼1

y2i

N

ð12Þ

Table 1Properties of fluid and solid spheres tested and experimental results of terminal fallingvelocities.

K(Pa*sn)

n(–)

dp(m)

ρs(kg/m3)

ρ(kg/m3)

V(m/s)

Re(–)

CD(–)

Kelessidis (2003)0.2648 0.7529 0.0015 2260 1000 0.0119 0.1125 174.5720.2648 0.7529 0.0021 2727 1000 0.0361 0.5678 35.5340.2648 0.7529 0.0023 2449 1000 0.0409 0.7234 26.0590.2648 0.7529 0.0030 2609 1000 0.0664 1.6292 14.4630.2648 0.7529 0.0035 2572 1000 0.0802 2.2735 11.0290.0353 0.8724 0.0015 2260 1000 0.0440 2.8774 12.7690.0165 0.9198 0.0015 2260 1000 0.0597 7.3105 6.9360.0353 0.8724 0.0021 2727 1000 0.1008 9.6228 4.5580.0353 0.8724 0.0023 2449 1000 0.1119 11.9690 3.4810.0165 0.9198 0.0021 2727 1000 0.1275 22.1153 2.8490.0353 0.8724 0.0030 2609 1000 0.1592 22.6541 2.5160.0165 0.9198 0.0023 2449 1000 0.1403 27.2608 2.2150.0353 0.8724 0.0035 2572 1000 0.1825 29.5950 2.1300.0165 0.9198 0.0030 2609 1000 0.1950 50.1299 1.6770.0165 0.9198 0.0035 2572 1000 0.2196 64.2225 1.471

Miura et al. (2001)0.5940 0.5610 0.0030 2500 1000 0.0314 0.4446 59.6980.5940 0.5610 0.0050 2500 1000 0.0881 2.6131 12.6390.5940 0.5610 0.0070 2500 1000 0.1594 7.4079 5.4050.1690 0.6250 0.0030 2500 999 0.1213 8.6137 4.0070.1770 0.6020 0.0050 2500 999 0.1972 24.0244 2.5270.1690 0.6250 0.0050 2500 999 0.2524 32.4644 1.5420.0675 0.6290 0.0030 2500 1000 0.2049 43.6445 1.4020.1770 0.6020 0.0070 2500 999 0.3031 53.6535 1.4970.1690 0.6250 0.0070 2500 999 0.3734 68.6443 0.9870.0299 0.7190 0.0030 2500 997 0.2673 94.4191 0.8280.0675 0.6290 0.0050 2500 1000 0.4051 153.2203 0.5980.0166 0.7510 0.0030 2500 998 0.3054 174.1574 0.6330.0299 0.7190 0.0050 2500 997 0.4391 257.4585 0.5110.0675 0.6290 0.0070 2500 1000 0.5235 269.0901 0.5010.0299 0.7190 0.0070 2500 997 0.5196 406.8381 0.5110.0166 0.7510 0.0050 2500 998 0.4437 407.5338 0.5000.0166 0.7510 0.0070 2500 998 0.6035 770.4722 0.378

Pinelli and Magelli (2001)0.0521 0.7300 0.0008 2470 1000 0.0306 1.2458 16.2220.0471 0.7300 0.0008 2470 1000 0.0392 1.8888 9.8850.0521 0.7300 0.0011 2900 1000 0.0718 4.7792 5.4470.0521 0.7300 0.0011 2900 1000 0.0818 5.6399 4.1970.0521 0.7300 0.0030 1470 1000 0.0734 9.9022 3.3660.0466 0.7300 0.0030 1470 1000 0.0887 14.0750 2.3050.0521 0.7300 0.0059 1170 1000 0.0829 19.2645 1.9220.0462 0.7300 0.0059 1170 1000 0.1013 28.0121 1.287

Ford and Oyeneyin (1994)9.1673 0.1714 0.0050 7949 1014.406 0.1200 0.9242 31.04719.7360 0.0623 0.0070 7744 1000 0.1900 1.4891 17.10519.7360 0.0623 0.0100 7796 1000 0.4100 6.7582 5.28819.7360 0.0623 0.0120 7730 1000 0.4200 7.1622 5.9884.9100 0.2075 0.0050 7949 1000 0.3200 8.7991 4.4389.1673 0.1714 0.0070 7744 1014.406 0.4400 10.5351 3.13711.4890 0.0614 0.0050 7949 1032.413 0.4000 10.9860 2.73816.1350 0.1580 0.0100 7796 1044.418 0.6100 12.5801 2.2724.0029 0.2867 0.0120 7730 1026.411 0.3800 13.7496 7.09911.2000 0.1113 0.0100 7796 1034.814 0.5800 19.7802 2.54016.1350 0.1580 0.0120 7730 1044.418 0.8000 21.3357 1.5704.0029 0.2867 0.0100 7796 1026.411 0.5800 26.9295 2.56411.2000 0.1113 0.0120 7730 1034.814 0.7100 29.5753 2.0159.1673 0.1714 0.0100 7796 1014.406 0.7500 29.6966 1.5554.9100 0.2075 0.0070 7744 1000.000 0.6600 34.5389 1.4186.5705 0.0796 0.0050 7949 1020.408 0.6100 39.4240 1.19311.4890 0.0614 0.0070 7744 1032.413 0.7900 41.9566 0.9549.1673 0.1714 0.0120 7730 1014.406 1.0000 51.8492 1.03911.4890 0.0614 0.0100 7796 1032.413 1.0800 78.6261 0.7354.9100 0.2075 0.0100 7796 1000.000 1.0600 86.9519 0.7916.5705 0.0796 0.0070 7744 1020.408 0.9100 87.2948 0.72911.4890 0.0614 0.0120 7730 1032.413 1.1800 94.4025 0.7316.5705 0.0796 0.0100 7796 1020.408 0.9800 103.5443 0.9044.9100 0.2075 0.0120 7730 1000.000 1.2100 114.4831 0.7216.5705 0.0796 0.0120 7730 1020.408 1.2400 165.0765 0.671

Kelessidis and Mpandelis (2004)0.0010 1 0.0032 2506 995.629 0.3692 1161.5720 0.4600.0010 1 0.0022 2668 997.066 0.2935 655.5112 0.5700.0010 1 0.0012 2314 1003.903 0.1763 215.9254 0.670

Table 1 (continued)

K(Pa*sn)

n(–)

dp(m)

ρs(kg/m3)

ρ(kg/m3)

V(m/s)

Re(–)

CD(–)

Kelessidis and Mpandelis (2004)0.0010 1 0.0026 11444 989.056 1.0660 2772.8972 0.3200.1350 1 0.0032 2506 1227.110 0.0420 1.2064 24.4200.1350 1 0.0022 2668 1238.396 0.0232 0.4767 62.8400.1350 1 0.0012 2314 1227.003 0.0072 0.0798 272.7000.1350 1 0.0026 11444 1226.688 0.1848 4.4163 8.3900.1350 1 0.0031 7859 1226.493 0.1656 4.6489 7.9700.1152 0.7449 0.0032 2506 999.591 0.1282 9.0398 3.7900.1152 0.7449 0.0022 2668 999.944 0.0835 4.0860 7.0100.1152 0.7449 0.0012 2314 999.985 0.0321 0.7828 20.3500.1152 0.7449 0.0026 11444 998.131 0.4657 39.7432 1.6600.0865 0.8610 0.0032 2506 999.592 0.1031 6.1128 5.8600.0865 0.8610 0.0022 2668 1000.211 0.0637 2.6282 12.0400.0865 0.8610 0.0012 2314 999.984 0.0225 0.4760 41.4200.0865 0.8610 0.0026 11444 999.089 0.3855 23.4289 2.4200.0865 0.8610 0.0031 7859 999.826 0.3399 23.3385 2.4000.0849 0.9099 0.0032 2506 999.835 0.0820 4.0910 9.2600.0849 0.9099 0.0022 2668 999.922 0.0493 1.7180 20.1100.0849 0.9099 0.0012 2314 1000.004 0.0164 0.2978 77.9600.0849 0.9099 0.0026 11444 999.267 0.3286 15.7112 3.3300.0849 0.9099 0.0031 7859 999.073 0.2828 15.4439 3.470

56 R. Rooki et al. / International Journal of Mineral Processing 110–111 (2012) 53–61

represents the percentage of the initial uncertainty explained by themodel. The best fitting between measured and predicted valueswould have a root mean square error of zero and a coefficient ofdetermination equal to one.

The best selected ANNmodel in this study, has one input layerwithsix inputs (ρ, K, n, g, dp, ρs) and one hidden layer with 12 neurons.Fletcher and Goss (1993) suggested that the appropriate number ofnodes in a hidden layer ranges from (2

ffiffiffik

p+m) to (2 k+1), where k is

the number of input nodes and m is the number of output nodes. Inthis study (k=6) and (m=1) and thus the appropriate number ofhidden layer neurons was chosen as 12. Fletcher and Goss (1993)further suggested that each neuron has a bias and is fully connected toall inputs and utilizes sigmoid hyperbolic tangent (tansig) activationfunction (Fig. 3). The output layer has one neuron (V) with linearactivation function without bias. Training function in this network isAutomated Bayesian Regularization algorithm (trainbr). Fig. 3.a showsthe back-propagation neural network architecture. In Fig. 3.b, Layer 1is hidden layer and Layer 2 is output layer. Fig. 3.c shows the detailedstructure of hidden layer.

4. Results and discussion

Using the approach described above, the predictions were made inMATLAB software. The matrix of inputs in training step is a k×Nmatrix, where k is the number of network inputs and N is the numberof samples used in training step; in this paper we used six inputvariables (ρ, K, n, g, dp, ρs), and 69 samples to train of the network,thus k=6 and N=69. The matrix of outputs in training step, is am×N matrix, where m is the number of outputs; in this paper m=1.The matrix of inputs for testing phase is k×N=6×19 and the outputmatrix is m×N=1×19. Comparison of the results of the proposedANN model with the two other models used which directly predictsettling velocities in power law fluids (Kelessidis, 2004; Chhabra andPeri, 1991) was performed using the coefficient of determination (R2)and RMS values. The latter parameters are affected by numbers ofdatasets (N) and number of parameters in the model.

In Fig. 4, the predicted velocities are compared with the measureddata for the training dataset of 69 data. The coefficient of determina-tion to the linear fit (y=ax) is 0.996 with an RMS value of 0.021 m/sgiving an almost perfect fit, something of course expected since it wasthis data set used for the training of the network. The very good fittingvalues indicate that the training was done very well.

Fig. 3. (a) Backpropagation neural network architecture, (b) general schematic diagram of network and its layers, (c) structure of hidden layer (Layer 1).

57R. Rooki et al. / International Journal of Mineral Processing 110–111 (2012) 53–61

The real test of course lays in the test dataset. The comparison ofthe predictions of the network with the measured values for the testdataset (population of 19) is shown in Fig. 5. The coefficient ofdetermination is 0.947 with an RMS of 0.072 m/s, indicating that thepredictions are not as good as with the training data set, but still withgood engineering accuracy. If we put all the data (88 data) togetherand compare predictions with measurements, we get the results in

Fig. 4. Comparison of the BPNN predicted and m

Fig. 6, which gives a coefficient of determination 0.986 and an RMSvalue of 0.038 m/s. Fig. 7 displays network predictions and measuredvelocity for all data using BPNN model.

It is very interesting to compare the predictions for the terminalvelocity using the ANN technique versus other approaches for velocitypredictions. A one-to-one comparison can bemade onlywith approacheswhich directly predict the terminal velocity without resorting to trial-

easured terminal velocity for training data.

Fig. 5. Comparison of the BPNN predicted and measured terminal velocity for test data.

Fig. 6. Comparison of the BPNN predicted and measured terminal velocity for all data.

58 R. Rooki et al. / International Journal of Mineral Processing 110–111 (2012) 53–61

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

sampels

V(m

/s)

Predicted

Measurement(all data)

Fig. 7. Comparison of the network predictions and measured velocity for all data usingBPNN model.

59R. Rooki et al. / International Journal of Mineral Processing 110–111 (2012) 53–61

and-error procedure. Asmentioned above, there are notmany equationswhich directly predict terminal settling velocity of solid falling in powerlaw liquids. One such equation has been proposed by Kelessidis (2004),which has been tested and derived from data (63 data) with power lawliquids and Newtonian fluids, with values of the power law exponenthigher than 0.50. If one restricts the comparison to such data points, onehas to remove the data of Ford and Oyeneyin (1994) which were usedin this work (Table A) and which span the range of n values between0.06 and 0.29. Another equation was suggested by Chhabra and Peri(1991). Such a comparison has then been made in Fig. 8. The analysisshows that for (n) values greater than 0.5, a coefficient of determinationfor the ANN set of 0.949 very close to the Kelessidis model (Kelessidis,2004) of 0.961 and much better than the only other direct velocity

Fig. 8. Comparison of ANN predictions with the predictions from the Kelessidis (2004) and Cwith 0.5bnb1.

determination of Chhabra and Peri (1991) of 0.778. The respective RMSvalues are, for the ANN model of 0.041 m/s, for the Kelessidis model of0.036 m/s and for Chhabra and Peri model of 0.580 m/s.

If the comparison between the ANN and the Kelessidis model isperformed for all experimental data with (n) values down to 0.06(1>n>0.06), then the results of Fig. 9 are derived which showcorrelation coefficients of 0.99 and 0.89 for the ANN and theKelessidis model respectively, while the respective RMS values are,for the ANN model 0.038 m/s and for the Kelessidis model 0.299 m/s.Of course the Kelessidis approach gives an equation which can beused in solving complex problems, but for n>0.5, while the ANNapproach results with a methodology and a software package whichmakes it a bit more difficult in using it for solving complex problemsbut can predict velocities down to very small power law indices.

This work has proven that it is very efficient to use neural networkto predict directly terminal settling velocity of solids. This is possiblebecause of the high capability of the ANN in deriving complex andnon-linear relationships and the value of the method is also oncovering an extended range of flow behavior index of 0.06 to 1 whichis not possible with other techniques. In order to apply this techniqueone should note that everyone can design a neural network inMATLAB multi-purpose commercial software using neural networkToolbox and using an experimental database of depended andindependent parameters. This network can then be applied for newdata with known depended parameters to predict the unknownindependent parameter (V).

5. Conclusion

A new method has been presented which allows prediction ofterminal settling velocity of solid spheres falling through Newtonianand non-Newtonian, power law liquids using ANN method. In thismethod all data from other investigators were divided into trainingdata (for training ANN) and test data (for validation ANN).

hhabra and Peri (1991) equations, for pseudoplastic power law fluids, restricted to data

Fig. 9. Comparison of ANN predictions with the predictions from the Kelessidis (2004) equation, for power law fluids, restricted to data with 1>n>0.06.

60 R. Rooki et al. / International Journal of Mineral Processing 110–111 (2012) 53–61

The predictions from the new model are compared with previouslyreported experimental data from other investigators which cover non-Newtonian and Newtonian liquids. The comparison is very acceptableand the coefficient of determination, R2, and RMS error in the terminalvelocity for all data points were 0.986 and 0.038 m/s respectively.Predictions with the ANN technique are similar to predictions of one ofthe two available equations for direct prediction of terminal velocity ofspheres falling in pseudoplastic power law liquids while it outperformsthe second available direct equation, while it covers an extended rangeof flowbehavior index. Therefore thismethod can be applied,with goodengineering accuracy, for this purpose. It is recommended that moreexperimental data should be used to train even better the ANN in orderto improve the validity of this method.

NomenclatureANN artificial neural network (–)dp particle diameter (m)R2 determination coefficient (–)CD drag coefficient (–)g acceleration of gravity (9.81 m/s)K consistency index of fluid (Pa*sn)k number of input nodes (–)m number of outputs (–)mse mean of sum squares error (–)msereg modified mse (–)msw mean of the sum of squares of the network weights (–)n flow behavior index (–) and number of inputs (–)N number of samples or data (–)Regn generalized Reynolds number, ρV

2−ndpn

K (–)RMS root mean squared error (–)V terminal velocity of solid spheres (m/s)

Greek lettersρ liquid density (kg/m3)

ρs solid density (kg/m3)γ performance ratio (–)_γ shear rate (s−1)τ shear stress (Pa)

References

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