Neural Network Computation for Cs Calculations

Neural network computation can be defined as a processing method which imitates the features of the human brain.11–13)The human brain contains approximately 10 billion nerve cells (neurons). A biological neuron which is a functioning unit of the nervous system consists of dendrite, soma, axon and synapse (Fig. 1(a)). A simplified artificial equivalent of a biological neuron is shown in Fig. 1(b). Neurons communicate via input signals. A neuron accepts inputs associated with different weights from multiple neurons. The summation of the inputs (intensity) is multiplied by their associated weight. When the intensity of the signal is high enough to pass over a certain critical value (threshold), then an output signal is transmitted through axon and synapse to the next neuron. The state that the intensity of signal exceeds the threshold is called “ignition” and it can be expressed in a sigmoid function shown in Eq. (1) in the neural network concept.

Y = f (x)=1/[1+exp(-Ƞ × x)]..................................................(1)

Here, x is an input value and y is the output. h is a coefficient which determines the shape of the sigmoid curve. Figure 2 represents a schematic diagram of the back propagation method in a three layers-type neural network computation which consists of an input layer, a middle (hidden) layer and an output layer. The units in the middle layer are connected with the input and output units. However, there are no connections within a layer. The final result “y” in the

Fig. 1. Schematic diagrams of a) a biological neuron and b) a simple artificial neuron.

 

output layer is expressed by using the above sigmoid function as follows:

………………………………(2)

……………………………….(3)

where xi is an input value of unit i in the input layer, Wki is a connection weight between unit i in the input layer and unit k in the middle layer, hi is a critical value for unit i. y is the final output, Vk is a connection weight between unit k in the middle layer and the final output, hk is a critical value for unit k. After values are applied to the units in the input layer, signals propagate through the middle layer to the output layer. Each link between neurons contains a unique weight value. A comparison is made between output values and the teaching values. The errors are calculated for each output unit and then propagated backwards through the network to correct the connection weights and the critical values in each unit. This “learning” process is repeated until the overall error value drops to acceptable levels.

Some studies exist on the estimation of some physical properties of molten slags by neural network approach in the literature. Tanaka et al. predicted viscosity and solidification temperature of mold fluxes in multi-component systems using neural network computing.14,15)Nakamoto et al. applied this approach to estimate the surface tension in ternary silicate melts.16)Both studies also discussed on the criteria for designing the number of units in the middle layer in order to obtain optimum results. In the present study, the computation was carried out by using SlagVis software. The SlagVis was designed by Research Center of Computational Mechanics Inc., Osaka University, and Sumitomo Metal Industries Ltd. to estimate first the physical properties of multi-component slags by neural network model.14–17)Since the calculation method is the same as that for those physical properties, the program was also found applicable for the Cs predictions. The following initial values were selected in the software for all calculations; number of middle unit 5, maximum iteration 1000000, learning rate 1.5 which is a constant used in artificial neural network learning algorithms to affect the speed of learning, and target relative error 0.02. These values, which were selected as the optimal parameters for the present study, were obtained by trial and error until an adequate match was achieved between experimental and calculated Cs values. In calculations, input values were mole fraction of the components, whereas the experimental Cs results in logarithmic scale were served as teaching values that provide feedback. Then, the sulfide capacity predictions of molten slags were calculated using this network.

 

 

4) 4.Conclusions

In the present work, we applied the neural network calculations to the sulfide capacity predictions in multi-component melts. The computation results were found in good agreement with the experimental values. It was also constructed iso-sulfide counters on ternary phase diagrams establishing a link among experimental Cs values of molten ternary and lower sub-systems. It can be concluded that neural network based computation is a very useful technique for predicting Cs values in molten melts, but attention needs to be paid to the quantity and accuracy of the experimental data. Acknowledgment Authors would like to thank the Osaka University Global COE Program, Center for Advanced Structural and Functional Materials Design for its kind support. The fruitful help of Daigen Fukayama in Research Center of Computational Mechanics Inc. in Japan is also greatly acknowledged.

3) the prediction. For this aim, experimentally obtained Cs values of molten state pure FeO, CaO–SiO2, FeO–SiO2, CaO–FeO, and CaO–FeO–SiO2melts were used. Figure 7 shows that an increase in the SiO2content results in a sharp decrease of the desulfurization power of slags and predicted iso-sulfide contour values within the large liquid region decreases from?1.8 to?4.5. Experimental sulfide capacity data of molten state CaO–CaCl2and CaO–CaCl2–CaF2 melts at 1273K were used as teaching value in order to evaluate iso-sulfide capacity counters on the ternary phase diagram. According to the calculations, the contours were found almost parallel to the CaCl2–CaF2axis, since CaO is the only Cs determinator in this melt. In the calculations, the number of parameters strongly depends on the component number of the slag. Temperature is also included to the input set when experimental Cs results obtained at different temperature are added to the calculations. For example, in order to calculate the Cs of a three-component slag system at a certain temperature, if the number of middle unit is selected as 5, the parameters needed are 15 of connection weight matrix of input to middle layer (Wki), 5 of threshold parameters of middle layer (critical value) (hi), 5 connection weight vector of middle to output layer (Vk) and 1 of threshold for output layer (hk), i.e. total 26 parameters according to Eqs. (2) and (3). The values of those parameters change sensitively with the number of input data, the number of iteration for learning and so on. As one example, the parameter values for the calculation of CaO–MgO–SiO2slag system at 1773K (Fig. 6) are tabulated in Table 2. The present work should be considered as a pioneering study that demonstrates a successful application of the neural network model for Cs prediction of some slags that may provide useful information for ferrous and non-ferrous metallurgy. In this study, for example, Cs values of high FeO regions in CaO–FeO–SiO2ternary system which were able to be estimated by the model, has a particular importance for some refining processes. This method was found to be an important tool for CaF2containing slags, since other empirical/theoretical models are not inadequate for their Cs estimations. It should be noted that in the present neural network calculations, not only composition change but also temperature was used as input values. Besides, a good regression between multicomponent melts and their subsystems was easily made. Moreover, we believe that a matrix which consists of components, temperature and physical properties such as impurity capacity, viscosity, surface tension, etc. can be calculated with the neural network model at once to estimate their inter-correlations, if enough number of required data is provided. ISIJ International, Vol. 50 (2010), No. 8

© 2010 ISIJ

Table 2.

List of parameter values for CaO–MgO–SiO2slag system at 1773K indicated in Fig. 6.

Fig. 7.

Neural network predicted iso-sulfide capacity contours in

liquid region of CaO–FeO–SiO2melts at 1773K.

Fig. 8.

Neural network predicted iso-sulfide capacity contours in

liquid region of CaO–CaCl2–CaF2melts at 1273K.

Fig. 6.

Neural network predicted iso-sulfide capacity contours in

liquid region of CaO–MgO–SiO2melts at 1773K.

the temperatures of 1723, 1773, 1823 and 1873K were

taken from the study of Nzotta et al.19)It is noted that for

Cs calculation of each composition, whereas optical basic-

ity model required mole fraction, temperature and theoreti-

cal or optimized optical basicity (L) values, these require-

ments were mole fraction, equilibrium constant, Keq, activ-

ity of metal oxides, aMeO, and temperature values for the

Reddy–Blander model. However, only one calculation was

carried out for neural network method to predict the whole

Cs values just inserting mole fraction and temperature as

input values and experimental Cs values as teaching values.

As a result of comparison of the these models with experi-

mental Cs values, it was shown in Fig. 3 that better regres-

sion can be obtained with neural network approach com-

pared to optical basicity and Reddy–Blander models.

Figures 4 and 5 represent the comparison between ex-

perimental and calculated values of multicomponent sili-

cate melts and halide containing multi-component oxide

melts, respectively. As seen in both figures, very good com-

pliance between neural network predicted values and the

experimental data points were found. Especially, in point of

halide containing melts view, only an optical basicity model

was tried for Cs prediction in the literature. However, the

results were not in good agreement with the experimental

data.31)This situation makes the neural network approach a

good engineering tool to estimate much reliable results.

Iso-sulfide capacities of the ternary slag systems for dif-

ferent compositions and temperatures were generated using

the neural network model. Some experimental Cs values in

Table 1, not only ternary but also of lower sub-systems

neighbor to liquidus region were taken into account for

each calculation. The experimental Cs values of the melt

compositions which are not in liquid region were neglected

to prevent erroneous Cs results. The iso-sulfide capacity

contours were inserted to phase diagrams which were gen-

erated by FactSage 6.0 using “Phase Diagram” module with

FToxide database.34)

Experimental Cs values of CaO–SiO2and CaO–MgO–

SiO2were used to generate iso-sulfide capacity contours in

liquid region of the ternary melt at 1773K. As seen in Fig.

6, logarithmic scaled capacity contours are in good agree-

ment with experimental data and vary between?3.7 and

?4.75. In order to perform more complicated regression

example, CaO–FeO–SiO2melt at 1773K was chosen for

ISIJ International, Vol. 50 (2010), No. 8

© 2010 ISIJ

Table 1.

Experimental data used for the Neural Network Cs

estimation.

Fig. 3.

Comparison between experimental and calculated Cs val-

ues obtained by different models for CaO–MgO–SiO2

slags at the temperatures of 1723–1873K.

Fig. 5.

Comparison between experimental and calculated Cs val-

ues obtained by neural network computation for halide

containing multicomponent melts at different tempera-

tures.

Fig. 4.

Comparison between experimental and calculated Cs val-

ues obtained by neural network computation for multi-

component silicate melts at different temperatures.

output layer is expressed by using the above sigmoid func-

tion as follows:

........................(2)

........................(3)

where xiis an input value of unit i in the input layer, Wkiis a

connection weight between unit i in the input layer and unit

k in the middle layer, hiis a critical value for unit i. y is the

final output, Vkis a connection weight between unit k in the

middle layer and the final output, hkis a critical value for

unit k. After values are applied to the units in the input

layer, signals propagate through the middle layer to the out-

put layer. Each link between neurons contains a unique

weight value. A comparison is made between output values

and the teaching values. The errors are calculated for each

output unit and then propagated backwards through the net-

work to correct the connection weights and the critical val-

ues in each unit. This “learning” process is repeated until

the overall error value drops to acceptable levels.

Some studies exist on the estimation of some physical

properties of molten slags by neural network approach in

the literature. Tanaka et al. predicted viscosity and solidifi-

cation temperature of mold fluxes in multi-component sys-

tems using neural network computing.14,15)Nakamoto et al.

applied this approach to estimate the surface tension in ter-

nary silicate melts.16)Both studies also discussed on the cri-

teria for designing the number of units in the middle layer

in order to obtain optimum results.

In the present study, the computation was carried out by

using SlagVis software. The SlagVis was designed by Re-

search Center of Computational Mechanics Inc., Osaka

University, and Sumitomo Metal Industries Ltd. to estimate

first the physical properties of multi-component slags by

neural network model.14–17)Since the calculation method is

the same as that for those physical properties, the program

was also found applicable for the Cs predictions. The fol-

lowing initial values were selected in the software for all

calculations; number of middle unit 5, maximum iteration

1000000, learning rate 1.5 which is a constant used in arti-

ficial neural network learning algorithms to affect the speed

of learning, and target relative error 0.02. These values,

which were selected as the optimal parameters for the pres-

ent study, were obtained by trial and error until an adequate

match was achieved between experimental and calculated

Cs values. In calculations, input values were mole fraction

of the components, whereas the experimental Cs results in

logarithmic scale were served as teaching values that pro-

vide feedback. Then, the sulfide capacity predictions of

molten slags were calculated using this network.

3.

Results and Discussions

Since the neural network model is based on an empirical

approach, the consistency of the experimentally determined

Cs values to be used as teaching values in the model is very

essential.

The main difficulties in the present study were the lack of

experimental data and/or inconsistent Cs values of some

similar slag compositions carried out by different authors.

For example, Cs values of MgO–SiO2slags at 1923K

found by Sharma and Richardson18)(7 data points) are not

only almost three times higher but also much more scat-

tered than the findings of Nzotta et al.19)(12 data points)

which were obtained more than 30 years later. In that case,

if both data are used in the same neural network calcula-

tions as teaching value, then the predicted results would in-

evitably be a failure. When the above concerns were taken

into account, where possible, comparatively reliable and/or

new data obtained with advanced measuring techniques

were selected from the literature in order to evaluate much

consistent results in the present calculations.

The experimental data collected from the literature was

used for the present neural network estimation of sulfide ca-

pacities of binary and multi-component melts at different

temperatures and listed in Table 1.

Initially, neural network estimation method was com-

pared with Sossinky and Sommerville’s optical basicity

equation and Reddy–Blander model which are often used

for predicting Cs values in silicate slag systems. For this

aim, experimental Cs values of CaO–MgO–SiO2slags at

y

f

a V

h

k

k

k

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(

)

⋅

∑

a

f

x W

h

k

i

ki

i

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?

(

)

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∑

ISIJ International, Vol. 50 (2010), No. 8

© 2010 ISIJ

Fig. 2.

Schematic diagram of the back-propagation approach in

3 layers-type neural network structure with 3 units in the

input layer and 3 units in the middle layer.

Fig. 1.

Schematic diagrams of a) a biological neuron and b) a

simple artificial neuron.

ISIJ International, Vol. 50 (2010), No. 8, pp. 1059–1063