dynamic structures in variance based data reconciliation adjustments for a chromite smelting furnace

Dynamic structures in variance based data reconciliation adjustmentsfor a chromite smelting furnace q

J.J. Eksteen a,*, S.J. Frank a, M.A. Reuter b

a Department of Chemical Engineering, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africab Department of Applied Earth Sciences, Delft University of Technology, Mijnbouwstraat 120, 2628 RX Delft, Netherlands

Received 19 April 2002; accepted 22 June 2002

Abstract

The characterisation of furnace product materials is discussed in relation to their impact on the material balance reconciliation. It

was found that the melts from open arc furnaces, as used in the chromite smelting industry, is not as homogenous as assumed. It was

found that, for silicon in the ferrochrome melt, the spatial composition variance could be related to the degrees sub cooling relative

to the alloy liquidus. The spatial variances in the furnaces were therefore incorporated in the variance-based reconciliation. The use

of different data reconciliation techniques as a tool towards furnace control in the pyrometallurgical industry is discussed. Both the

Lagrange multiplier, as well a direct method using the generalised reduced gradient method were evaluated. The adjustments to the

measurements, which may be viewed as a combination of structural and random error in the measurements as well as systematic

bias, were found to have dynamic structure. A comparison of time series models of the adjustments and their Fourier power spectra,

has shown that any given adjustment is auto-correlated with previous historic values, even though the reconciliations were per-

formed independently. Systematic biases were also apparent in the measured data, which were identified and subsequently con-

firmed.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Oxide ores; Pyrometallurgy; Reduction; Modelling; Process control; Mass balancing

1. Introduction

The modelling of industrial furnaces for the purpose

of metallurgical control is still in its infancy compared to

the model based control found in the process industries.Although many metallurgical models do exist, they sel-

dom lend themselves to proper decision support and

furthermore one seldom finds that the models have been

compared and validated against real-life industrial fur-

nace data. Current models tend to aid the metallurgist

more towards process analysis, design and development,

furnace design, and post-mortem diagnostics of ‘‘what

went wrong’’ in industrial scenarios, but they are nottargeted towards providing the plant metallurgist with

on-line guidance as to process and product control. A

need therefore exists to incorporate actual furnace data

in a metallurgical meaningful way into process models,

be it simple steady state mass balance models, or fully

dynamic models. Proper characterisation of all the ma-

terials fed to, and produced by, a furnace is required to

establish the statistical distributions in the materials.

Moreover the distributive properties of a material in- oroutput gives a good indication of the homogeneity of

that material stream. A first and necessary step towards

the development of steady-state predictive models is

to dynamically reconcile the data within their natural

measurement error. A degree of dynamic prediction

could be incorporated if consecutive measurements

show dynamic structure. It is the aim of this paper is to

indicate how this may be achieved for a chromitesmelting open arc furnace.

Data reconciliation, also referred to as measurement

error reconciliation, is the adjustment of a set of data in

order that the quantities derived from the data obey

natural laws, such as material and energy balances

(Abdul-el-zeet et al., 2002). The application of data

reconciliation techniques to process data from the py-

rometallurgical industry is quite recent. Bazin andTremblay (1999) applied data reconciliation techniques

qPresented at Pyromet �02, Cape Town, South Africa, March 2002.*Corresponding author. Tel.: +27-21-808-4485; fax: +27-21-808-

2059.

E-mail address: [email protected] (J.J. Eksteen).

0892-6875/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0892-6875 (02 )00131-0

Minerals Engineering 15 (2002) 931–943This article is also available online at:

www.elsevier.com/locate/mineng

mail to: [email protected]

to the induration of iron oxide pellets. It has also been

applied (Grund et al., 1999) to flash smelter data in

order to derive a smoothed mass and heat balance for

the processing of tin concentrates. In both cases the

generalised reduced gradient (GRG) method (which

implicitly uses Newton�s method) were used to findthe vector of reconciled values that minimised the sumsquare error of the adjustments. Measurements made on

the process are adjusted in proportion to the standard

error of the measurement. The adjustments are based on

the availability of high degrees of freedom associated

with redundant data. After adjustment one finds that the

material balances, and if considered, the energy balances

are exactly satisfied within the standard deviations of

the data (Bodington, 1995). The reconciled values maynow be used for more advanced furnace modelling

and may serve as a point of departure for the develop-

ment of model predictive control. Moreover, should

there by any dynamic structure in the error/adjustment

on a day-by-day, or tap-to-tap basis, it would imply that

an estimate of future adjustments may be made before

the future data is available to be reconciled. Dynamic

structure in the error would imply that the error doesnot follow a random or white noise sequence, that is

where any event in the sequence is completely indepen-

dent of any preceding event (Kanjilal, 1995; Bonekamp

et al., 1999; Reuter and Grund, 2001).

The application of data reconciliation techniques

using various error minimisation gross error detection

techniques have become established in the minerals

processing field and has been expanded to handle com-plex circuits (Cutting, 1976; Lynch, 1977; Hodouin and

Everell, 1980; Hodouin et al., 1981; Wills, 1986; Ragot

et al., 1999; Hodouin and Berton, 2000). The two

methods that appear to find favour among the re-

searchers in minerals processing are either the Lagrange

multiplier technique or a direct minimisation technique

of the variance weighted sum of square error which

work on the independent variables (Hodouin and Eve-rell, 1980). The direct minimisation techniques may

employ a number of non-linear programming algo-

rithms such as any of quadratic programming, sequen-

tial quadratic programming (SQP), the GRG method

which implicitly uses Newton�s method, or randomsearch methods, each algorithm having each own set of

advantages and disadvantages. An exhaustive evalua-

tion of the algorithms, based on efficiency, reliability,

global convergence, performance in solving degenerateproblems, performance in solving ill-conditioned prob-

lems, performance in solving indefinite problems, sen-

sitivity to variations in the problem and ease of use, has

shown that the GRG and SQP methods significantly

outperformed the other algorithms (Edgar and Him-

meblau, 1989).

However, all the methods require estimates of the

variances of the compositional and total flow variables.Establishing these variances for furnaces is in itself a

challenge, due to the hostile (hot, dangerous, even toxic)

conditions associated with the different material flows to

and from the furnace. It is therefore sensible to inves-

tigate the impact of sampling practice and material

characterisation which influence the reported variances

associated with the different components.

2. Sampling and materials characterisation

An open bath smelting furnace, such as a open arc

furnace, used for the manufacture of high carbon fer-

rochrome typically has multiple feed and product

streams. Table 1 lists some typical streams associated

with such a smelting operation.Furthermore a number of contaminant species such

as Mn, P, S, Zn, V in their elemental form as well as in

oxides of various oxidation states are present in most of

the phases.

2.1. Brief process description

Ferrochrome is produced by the carbothermic re-duction of chromite. The reduction process is highly

endothermic, which results in the need for the high

Table 1

Feed and product stream from a typical open arc chromite smelting furnace

Feed streams Product streams

� Lime/limestone (varying amounts of Ca in the forms of oxide, carbonate andhydroxide)

� High carbon ferrochrome alloy with Cr, Fe, C and Si asmain constituents

� Quarts (essentially pure SiO2) � Slag consisting mostly of the oxides of Si, Al, Ca, Mg, Feand Cr

� Chromite concentrate (chrome spinels with intergrown gangue) � Flue Gas, consisting mostly of H2, CO, CO2 and trace

amounts of purge gas (N2)

� Wet screenings associated with lumpy ore from submerged arc furnaces (SAF)(chrome spinels with intergrown gangue)

� Flue dust containing volatilised metal, gas-entrained feedfines and carbonaceous char

� Spills recycle (raw materials high in chromite)� Charge chrome fines from SAF (off-specification high carbon ferrochrome)

� Electrode (pure graphite)� Anthracite (which may include a portion bituminous coal and char)

932 J.J. Eksteen et al. / Minerals Engineering 15 (2002) 931–943

temperatures experienced in the furnace. The produc-

tion of ferrochrome and the mechanisms of reduction

has been described by a number of researchers (Curr

and Barcza, 1982; Howat, 1986; Curr, 1996; Smith et al.,

1996; Pei and Wijk, 1994; Demir and Eric, 1994; Maeda

et al., 1981; Aky€uuziu and Eric, 1992; Downing, 1975).The most important variables associated with the smelt-ing of chromite are the raw material particle size (more

so for submerged arc furnaces than for open arc fur-

naces), contact between the ore and the reducing agent,

the slag basicity, and the temperature of the melt.

Chromium ores consists of spinels of FeCr2O4, such as

FeO � Cr2O3, where the FeO can be replaced by MgO,

and the Cr2O3 by Al2O3 in the crystal lattice. The

MgO � Cr2O3 spinel melts at 2350 �C and therefore it isnecessary to flux the charge with silica to dissolve the

magnesia spinel and form a lower melting slag. The iron

oxides and chromium oxides dissolved in the slag are

reduced by carbon and carbon monoxide to produce

iron metal and chromium carbides (as in the following

reactions) and metallic chromium and iron. For disso-

lution of the chrome spinel in the slag phase at oxygen

partial pressure (PO2) values less than 10�8 atm, the

postulated reactions are:

Main slag forming reactions:

ðFe;MgÞOðCr;AlÞO3 þ COðgÞ) ðFeOÞ þ 2ðCrOÞ þ ðMgO �Al2O3Þ þ CO2ðgÞ

or

ðFe;MgÞOðCr;AlÞO3 þ CðsÞ) ðFeOÞ þ 2ðCrOÞ þ ðMgO �Al2O3Þ þ COðgÞ

and

ðFeOÞ þ ðCrOÞ þ ðCr2O3Þ þ ðAl2O3Þ þ ðMgOÞ þ ðSiO2Þþ ðCaOÞ ¼ ðSlagÞ

Furthermore,

2ðCr3þÞ þ ðO2�Þ () 2ðCr2þÞ þ ðOÞ

and

CðsÞ þ ðOÞ () COðgÞ

where the availability of oxide anions is determined by

the degree of silicate depolymerisation (only linear de-polymerisation shown).

ðSiO2Þ þ ð2O2�Þ () ðSiO4�4 Þ

ðSinþ1O2ðnþ2Þ�3nþ4 Þ þ ðO2�Þ () ðSiO4�4 Þ þ ðSinO2ðnþ1Þ�3nþ1 Þ

Main reduction reactions:

ðFeOÞ þ CðsÞ () Feþ COðgÞ7ðCrOÞ þ 10CðsÞ ) Cr7C3 þ 7COðgÞðSiO2Þ þ 2CðsÞ ) Siþ 2COðgÞ

The thermochemistry of phase equilibria of the slag

system pertaining to chromite smelting and the associ-

ated alloy system have been well studied (Toker et al.,

1991; De Villiers and Muan, 1992; Xiao and Holappa,

1993; Xiao, 1993; Xiao and Holappa, 1996; Wethmar

et al., 1975). Some SiO2 is reduced to SiO vapour which

reverts to SiO2 in the flue dust. Some metals with a highvapour pressure (such as Mn and Zn) leave the furnace

via the flue gas stream and revert to their oxides once the

gas has passed through a venturi scrubber and are

subsequently captured as part of the flue dust.

The feed streams are fed from hoppers on load cells,

via conveyor belts into the furnace. The alloy is tapped

into sand moulds and subsequently weighed, while the

slag is tapped into ladles, their inventory being gaugedby visual estimation. The flue gas is processed using a

venturi scrubber to remove most of the particulates. The

venturi scrubber effluent is thickened and filtered and

the dust is subsequently accumulated as a moist filter

cake.

2.2. Sampling and assaying

As the feeds are particulate, sample sizes can be de-

termined with Gy�s method (Gy, 1979), although goodsampling practise is not followed as a rule, introducing

uncertainties in the assays. Electrode consumption in

particular is difficult to gauge, and electrode replace-

ment frequencies are used to estimate the contribution

of electrode to the feed. Lime is often not pure CaO and

may contain various amounts of limestone (CaCO3) andslaked lime (Ca(OH)2). Intergrown gangue may signifi-

cantly alter the SiO2 component of the chromite con-

centrate and screenings. Despite all these inadequacies

the contribution of the various components of the feed

may be gauged with fair accuracy. Furthermore, the

oxidation state of the elements in the feed remains

constant and known.

However, the product streams leaving the furnaceshow a much larger variance and the reducible metal

oxides occur in a number of oxidation states. Current

practice at most open arc smelting operations in South

Africa reflects the inherent assumption that the melts in

open arc furnaces to be well mixed. The operating staff

felt that this assumption is well founded on the basis of:

• electro-hydrodynamic effects,• arc impact and momentum transfer to the melt,

• foaming or bubbling of the melt due to CO-gas re-

lease,

• buoyancy effects due to the significant thermal gradi-

ent.

Moreover, Gunnewiek and Tullis (1996) have claim

that, according to CFD models, melt flow in open arcfurnaces is fully three dimensionally developed and well

J.J. Eksteen et al. / Minerals Engineering 15 (2002) 931–943 933

mixed. To establish the validity of these assumptions, a

sampling campaign was initiated.

It was found that significant variance occurred in all

solute components in the melts where the solute com-

ponent made up less than 10% of the melt. Components

with larger concentration can be viewed as solvent

components (for example Cr and Fe in molten alloy).For a number of slag and alloy taps, 6 equaspaced

samples were taken per tap, once the melt flow through

the launder to the ladle was fully developed. These were

all handled identically, with regard to quenching and

assaying of the samples. All the sample sizes were similar

and of the order of about 50–60 cm3. The relative %

deviation in the assay of the silicon (relative to average

per tap) in the alloy and the Cr2O3 (relative to averageper tap) in the slag are shown in Figs. 1–5.

The variation in silicon, a contaminant in the alloy

that has a major effect on the alloy properties, therefore

appears to be significant, with some cases where the

relative deviation in % silicon varied between þ100%and )80% relative to the calculated arithmetic average

% silicon in the same tap. It immediately becomes ap-

parent that the assumption of obtaining an accuratealloy assay from a single sample is not correct. The

distributive nature can also be displayed as a histogram

of the frequencies of the % deviations relative to the

average % Si for all the 11 taps that were sampled as

described above, as shown in Fig. 2. The cumulative

distribution in Fig. 2 indicates that only about 35% of

the relative deviations occurred in the band )20% to

þ20% around the mean.It would appear reasonable to investigate the rela-

tionship of the relative standard deviation (as an indi-

cation of melt homogeneity) and the degrees of

superheat of sub cooling (relative to the alloy liquidus

temperature) associated with the alloy, as the greater thedegree of sub cooling, the more solid precipitation in the

melt one would expect. The precipitation of solid alloy/

carbide would lead to a significant increase in alloy

viscosity and an expected reduction in the degree of

mixing, all other factors remaining equal. To establish if

this was the case, 9 alloy taps were monitored using a

Mikron M90 H pyrometer, mounted on a tripod and

focussed on the molten alloy at the tap hole during thetapping. The pyrometer, with a range from 600 to 3000

�C and a precision of 0.4% was calibrated using dis-

posable dip thermocouples. The liquidus temperatures

for high carbon ferrochrome was obtained from litera-

ture [30]. The liquidus ranges investigated in the litera-

ture was for ferrochrome with assays between 0% and

8% C, 0% and 10% Si, 50% and 65% Cr, the remainder

being iron, and liquidus temperatures ranging from 1450to 1650 �C. The relative standard deviation (per alloytap) is defined as the absolute standard deviation in % Si

divided by the average % Si per tap, based on 6 equa-

spaced samples. Fig. 3 illustrates the effect of deviation

from the liquidus temperature on the % Si relative

standard deviation, each data point reflecting one tap.

The liquidus temperatures were estimated based on the

arithmetic average alloy assays for each tap.Fig. 3 shows that, for the alloy phase, the homoge-

neity increases (decreasing relative standard deviation)

as the sub cooling decreases and gradually switch to a

superheated alloy. The fact that such an observation can

be made despite the effect of other mixing driving forces

in the industrial furnace is significant and emphasises the

beneficial effect of operating close the liquidus.

The variation of Cr2O3 in the slag, as shown in Fig. 4,is less extreme than the case of Si in the alloy, but the

variation still reflects a high degree of melt inhomo-Fig. 1. % Deviation from the average in the % silicon in ferrochrome

alloy for 11 alloy taps with 6 samples per tap.

Fig. 2. Histogram of the frequency of % Si relative deviation in taps.


geneity. However, the effect of an unreliable Cr assay is

more severe as it impacts on the estimation of chrome

losses in the slag.

The effect of the different mixing driving forces (es-pecially CO bubble evolution and convection) in in-

creasing the slag mixedness is apparent––the maximum

deviations being between )25% and 20%, much lower

than was observed for silicon in the alloy. The reduction

in spread of deviations from the average can also be

observed from the deviation histogram in Fig. 5.

The slag samples were analysed using X-ray fluores-

cence spectrometry (XRF), while the alloy samples wereanalysed using a spark optical emission spectrometry

(spark OES). Carbon and sulphur in all feed and

product phases were analysed using a LECO CS200

analyser. It was found that the slag also contained car-

bon, even though no visible free carbon was present. It

was assumed that the carbon in the slag derives from

entrained alloy droplets which contain in the order of

8% carbon.

The dust was analysed based on grab samples from

the fresh filter cake as it was discharged from an auto-

matic plate-and-frame pressure filter. The filter cake was

dried at slightly above 100 �C to drive of the moisture.The heterogeneity of the dust is apparent in the SEM

photograph in Fig. 6. The presence of char particles isapparent (See composition of particle A, Table 2). The

other particles were also analysed using the SEM EDS,

with typical analyses as in Table 2. A concentration of

the oxides of volatile metals were noted, as was to be

expected. The dust was also analysed using XRF for the

metal oxides and the LECO analyser for carbon and

sulphur. The carbon content varied between 3% and

21% with an average carbon content of about 9%. Thedust is not analysed on a routine basis on the plant––a

Fig. 4. Variation in the % Cr2O3 in the chromite smelting slag for 9

slag taps with 6 samples per tap.

Fig. 5. Histogram of the frequency of % Cr2O3 relative deviation in

taps.

Fig. 6. SEM photograph of flue dust from a ferrochrome smelting

plant.

Fig. 3. Relationship between % Si assay relative standard deviation

per tap and the deviation from the liquidus.


significant variance may therefore be attributed to its

assay. The amounts of dust was estimated from the

known weight of filter cake released, the known mois-

ture level and the frequency of filter discharges.

The flue gas composition was fairly consistent al-

though the quantities varied significantly, depending on

the reductant to mineral feed ratio and the reactivity of

the reductant, which in turns is dependent on the ratioof volatiles to fixed carbon.

The compositions and variances were therefore es-

tablished for all streams entering and exiting the furnace

allowing a variance weighted data reconciliation to be

done.

3. Data reconcilation

As stated above, two general approaches are followed

to determine the require adjustments to process data,

being the direct minimisation of the sum square of thevariance-weighted adjustments and the application of

the Lagrange multiplier method which in this case is

applied to the minimisation of the weighted closure re-

siduals. Each method will be briefly discussed in turn.

Both are essentially weighted least squares techniques

which inherently assumes the errors (adjustments) to be

normally distributed and unbiased. Since it is not known

a priori that the errors are of the normal type and un-biased, and since the variances can, at best, only be es-

timated, the least squares approach is still expected to

give a reasonable solution (Wiegel, 1972). The bias and

normality of the adjustments will be evaluated post-

reconciliation.

3.1. Direct minimisation of the sum of squares of weighted

adjustments

In its most general form, the GRG approach to the

minimisation of a non-linear function JðxÞ subject tolinear (or linearised) constraints. In general, the mea-

surement vector (xm) may be written as:

xm ¼ xadjusted þ e ð1Þwhere xadjusted is the vector of the adjusted (true) valuesof the variables, e, the vector of unbiased random

measurement errors normally distributed with a zeromean and covariance matrix V.

The reconciliation problem can therefore be stated a

constrained least squares estimation problem where

weighted sum of adjustments is to be minimised to

constraints:

minxadjusted

ðxm � xadjustedÞTV �1ðxm � xadjustedÞ ð2Þ

subject to

f ðxadjustedÞ ¼ 0 ð3Þwhere V 2 Rnx�nx is the covariance matrix of the mea-sured variables xm. In our case the linear constraintsarise from the linear mass balance equations for the

different elements of the system, and the overall material

balance. When the data may be biased, the systematic

bias may be estimated as a parameter, where the ob-

jective function is reformulated as follows:

min JðxÞ ¼ minXk

i¼1

ð�xxi � ðxm � bbiÞÞ2

Við4Þ

subject to

Table 2

EDS analyses of the phases identified in Fig. 6

Particle Element Mass % Comment

A C 95.8 This particle is a piece of de-volatilised reductant. It is distinguishable due to its darker colour––due to the high

carbon content. As the flue dust is in-homogenous the quantity thereof will differ from sample to sample. Dust

with a high proportion of carbon is indicative of poor reductant utilization in the furnace. The texture of this

particle is more crystalline than the rest implying that this particle was entrained in the dust.

O 2.7

Si 0.3

S 0.7

K 0.5

B O 30.1 Particles of this nature make up the majority of the sample. This specific particle is a dust particle, however

partially reduced feed material has also been identified, but to a lesser degree than the dust particles––partially

reduced feed has a more crystalline texture than the dust. Typical dust particles have high proportions of Mg

and Si present. Analyses of other similar particles also yielded high proportions of Mn, Zn while Cr appeared

only in small amounts. This is primarily due to the high volatility of these elements when reduced to metal from

the slag. The high oxygen percentage is due to the fact that the elements are present in their oxidised state––the

elements are re-oxidised in the venturi scrubber

Na 2.0

Mg 22.4

Al 2.7

Si 21.5

S 3.3

K 2.0

Ca 2.1

Cr 5.4

Fe 3.7

Zn 4.8

C S 1.0 These bright particles are tiny specks of metal entrained in the dust. From the image it is clear that there size is

very small in comparison to the other particles. Particles with a high percent of Cr are also observed in the dust.Cr 1.0

Fe 98.0


f ð�xxiÞ ¼ 0�xxl;i 6�xxi 6�xxu;i 8ibbl;i 6 bbi 6 bbu;i 8i

ð5Þ

In the case where both total flows as well as composi-

tions are to be adjusted, Eq. (4) becomes:

min Jðx;QÞ ¼Xn

j¼1

ðQj � ðQm � bbjÞÞ2

Vj

þXn

j¼1

Xk

i¼1

ð�xxij � ðxm;j � bbijÞÞ2

Vijð6Þ

subject to the same constraints as (4) with the additional

total mass balance constraint:Xn

j¼1Qj ¼ 0 ð7Þ

The GRG algorithm, as well as the SQP algorithms for

minimisation is described by Edgar and Himmeblau

(1989) and is incorporated into most general optimi-

sation software, including the Premium Solver Plat-

forme for Microsoft Excel.

3.2. Minimisation of weighted closure residuals through

the application of the Lagrange multiplier method

This method determines the minimum of the variance

weighted closure residuals of the mass balance equations

through direct analytical differentiation of the Lagran-

gian, to be defined later. The basis for this method canbe found in many textbooks, and the nomenclature and

approach of Wills (1997) is used and expanded upon.

For the purposes of this paper, the furnace system is

defined in Fig. 7. Here the element mass balances can be

represented by the following equation:

Ffk � Cck �Mmk � Ttk � Llk ¼ rk ð8Þwhere k is a chemical element, rk is the difference fromthe balance for element k.

For the purpose of mass balance reconciliation, themass split ratios will be expressed as a fraction of the

total product or feed mass, and the elemental values will

be expressed as mass.

Eq. (8) may be further manipulated to give us the

result:

ðfk � lkÞ � Cðck � lkÞ �Mðmk � lkÞ � T ðtk � lkÞ ¼ rk

ð9ÞFollowing the concept of the least squares approach, wewant to minimize the sum of the difference from the

balance (rk) for each element. This enables us to obtainadjusted values for C, M, T, and L, and for the stream

compositions, thus satisfying the following equation:

f_

k � C_

c_k �M

_

m_

k � T_

t_

k � ð1� C_

�M_

� T_

Þ l_

k ¼ 0ð10Þ

On a mathematical basis the concept of the least squares

approach can be followed by the minimization of thefollowing equation, with taking into account the data

variance:

S ¼Xn

k¼1

ðrkÞ2

Vrkð11Þ

where

Vrk ¼orkofk

� �2Vfk þ

orkock

� �2Vck þ

orkotk

� �2Vtk þ

orkolk

� �2Vlk

ð12Þwhich equates to

Vrk ¼ Vfk þ Vck bCC2 þ Vmk bMM 2 þ Vtk bTT 2 þ Vlkð1� bCC � bMM � bTT Þ2ð13Þ

and

ðrkÞ2 ¼ ðfk � lkÞ2 � 2Cðck � lkÞðfk � lkÞ� 2Mðmk � lkÞðfk � lkÞ � 2T ðtk � lkÞðfk � lkÞþ C2ðck � lkÞ2 þ 2CMðmk � lkÞðck � lkÞþ 2CT ðtk � lkÞðck � lkÞ þM2ðmk � lkÞ2

þ 2MT ðtk � lkÞðmk � lkÞ þ T 2ðtk � lkÞ2 ð14Þ

By the minimization of Eq. (11), we are able to obtain

the adjusted values for C, M and T by finding the re-

spective partial derivatives for the variables and equal-

ling them to zero.

oSoC

¼ C_X ðck � lkÞ2

VrkþM

_ X ðmk � lkÞðck � lkÞVrk

þ T_X ðtk � lkÞðck � lkÞ

Vrk�X ðck � lkÞðfk � lkÞ

Vrk¼ 0

ð15Þ

oSoM

¼ C_X ðmk � lkÞðck � lkÞ

VrkþM

_ X ðmk � lkÞ2

Vrk

þ T_X ðtk � lkÞðmk � lkÞ

Vrk�X ðmk � lkÞðfk � lkÞ

Vrk¼ 0

ð16ÞFig. 7. Combined feed stream and product streams from the furnace

(daily basis).


oSoT

¼ C_X ðtk � lkÞðck � lkÞ

VrkþM

_ X ðtk � lkÞðmk � lkÞVrk

þ T_X ðtk � lkÞ2

Vrk�X ðtk � lkÞðfk � lkÞ

Vrk¼ 0 ð17Þ

It is possible to solve for the above variables by ex-

pressing the terms in the form of a matrix.

Solving by obtaining the matrix inverse and multiplying,

one may therefore obtain the ‘‘best fit’’ values for the

split fractions C, M, T and L.

To find the final reconciled values we have to

make incremental changes to the elemental composi-

tions of the streams. By making these changes the goal is

to further minimise the difference of the sum of the

balance for each element, for example for a componentin the feed (and similarly for the other phases):

f_

k ¼ fk � fka ð19ÞThe adjustments (subscript ka) can be calculated by

minimising the following Langrangian multiplier:

L ¼ Sa þ 2X

kkðrk � fka þ C_

c_

ka þM_

m_

ka þ T_

t_

ka

þ ð1� C_

�M_

� T_

ÞlkaÞ ð20Þ

where

Sa ¼Xn

k¼1

f 2kaVfk

�þ c2kaVck

þ m2kaVmk

þ t2kaVtk

þ l2kaVlk

�ð21Þ

Minimisation can be done by taking the partial deri-

vatives with respect to each variable, making it equal

zero and simplifying, for example with respect to thefeed components (and similarly for the other phases):

oLofka

¼ 2 fkaVfk

� 2kk ¼ 0) fka ¼ kkVfk ð22Þ

Through back substitution the above results into the

following equation:

rk ¼ fka � C_

c_ka �M

_

m_

ka � T_

t_

ka � ð1� C_

�M_

� T_

Þlkað23Þ

we obtain

rk ¼ kkhk ð24Þwhere

hk ¼ Vfk þ VckC_2

þ VmkM_ 2

þ VtkT_2

þ Vlkð1� C_

�M_

� T_

Þ2

ð25Þ

By manipulating Eq. (24) and substituting it into Eq.(22) (similarly for all the other phases), we obtain ex-

pressions for the amount by which each variable must be

adjusted, for example for the feed:

fka ¼rkVfkhk

ð26Þ

and similarly for all the product phases. Subtracting the

adjustments from the original values can thus represent

the reconciled values. This method therefore results in

explicit forms for the adjustments.

Despite the apparent differences between the above-mentioned approaches, it has been shown by Edgar and

Himmeblau (1989) that the Lagrange multiplier method

and the GRG method could be related to each other.

4. Reconcilation results

The data were reconciled for all elements flowing intoand out of the furnace within a period of one day. Data

reconciliation was performed on a daily basis versus a

tap-to-tap basis so as to eliminate the effect of variations

in the furnace freeze lining, smoothing of the error as-

sociated with electrode consumption estimation and

minimising the effect of residual inventories after tap-

ping which constitutes a feed stream from the perspec-

tive of the subsequent tap.Figs. 8–14 gives the output graphs for 68 consecutive

days of the measured amounts, and the reconciled

amounts using the GRG and Lagrangian methods. In

all the reconciled cases, mass balance closures were ob-

tained. Systematic biases did, however, occur (such as

the proportion silicon of the total feed fed to the fur-

nace). It is apparent that the two methods very often

gave significantly different reconciled values, but similartrends. The alloy, slag, flue dust and gas splits are based

on a unit total feed, where the total feed includes all the

chromite feedstocks, all fluxes and all reductants (in-

cluding electrode consumption). This approach is re-

quired for the Lagrange method as expressed above, but

X ðck � lkÞ2

Vrk

X ðmk � lkÞðck � lkÞVrk

X ðtk � lkÞðck � lkÞVrkX ðmk � lkÞðck � lkÞ

Vrk

X ðmk � lkÞ2

Vrk

X ðtk � lkÞðmk � lkÞVrkX ðtk � lkÞðck � lkÞ

Vrk

X ðtk � lkÞðmk � lkÞVrk

X ðtk � lkÞ2

Vrk

2666666664

3777777775

C_

M_

T_

2664

3775 ¼

X ðck � lkÞðfk � lkÞVrkX ðmk � lkÞðfk � lkÞVrkX ðtk � lkÞðfk � lkÞVrk

266666664

377777775

ð18Þ


not for the GRG direct method, where adjusted values

for all the feed streams may be calculated directly.

It is apparent from the time series in Figs. 8–11 that,

in general the reconciled data sequence is significantly

smoothed relative to the original measurements. Sig-nificant systematic biases were observed for nearly all

the cases where the Lagrange multiplier method was

used. These biases became particularly evident in the

calculated dust splits––due to it being calculated as

1� ðalloy splitþ slag splitþ gas splitÞ: a small changein any of the other spilt ratios resulted in a significant

change in the dust split ratio. The silicon fractions (split

fraction times assay) are discussed by way of example.

Both reconciliation methods adjusted the silicon in

the feed within the same band, but with a large bias

Fig. 8. Alloy split fractions.

Fig. 9. Slag split fractions.

Fig. 10. Gas split fractions.

Fig. 11. Flue dust split fractions.

Fig. 12. Si fraction in the feed.

Fig. 13. Si alloy fraction.


relative to the measurement. Smelter operating staff later

confirmed the probability of occurrence of a systematic

error in the feed assays pertaining to silicon (in the form

of SiO2). The reconciliation aids operating staff to gauge

if any measurement bias exist and to develop a reliable

feed recipe that could be accurately balanced with allproduct from the furnace. A reconciled estimate of the

dust production, and especially the carbon component

in the dust is useful to gauge the reductant utilisation

efficiency. Determination of the slag split ratio as frac-

tion of the feed allows determination of the slag pro-

duction and chrome losses with the slag.

The series of graphs presented in the figures above

emphasises that the Lagrange method led to larger ad-justments and a larger observed bias. The detection of

bias is always important as it indicates either that some

material is added or removed to the furnace, which is

not logged on the data acquisition system, or significant

error in the analysis instrumentation, or a very poor

sampling methodology. Once the mass balance is rec-

onciled an energy balance may be developed based on

the reconciled mass balance. Finally, data reconciliationis a tool to reconcile masses over a certain period. If this

period is longer than the inherent time constant of the

furnace melt, one would expect a loss in dynamic pre-

dictability. This may be the case in the reconciliation

done as above, as the period of reconciliation was 24 h,

much longer than the tap-to-tap time for open arc

furnaces. Consequently, the dynamics have to be char-

acterised to determine if some degree of dynamic pre-dictability is present (versus steady state predictions

made by the reconciled mass balances).

5. Dynamic patterns in the reconcilation adjustments

The errors/adjustments were analysed to determine if

they are truly random, normally distributed and inde-pendent. The adjustments calculated for any given day

were analysed to determined if they were dynamically

related, albeit not with a high accuracy, to the calculated

adjustments (measurements errors) of previous days. A

Fast Fourier Transform (FFT) power spectrum is useful

to determine if any such autocorrelation exists. That a

strong periodic component (and therefore a high degree

of autocorrelation) has been found, is demonstrated by

the amplitude occurring typically within the first fivefrequency components of the FFT power spectrum of

the adjustments time series. This is demonstrated for

silicon in the alloy and slag in Figs. 15 and 16 respec-

tively. The FFT for chrome adjustments in alloy and

slag, shown in Figs. 17 and 18 respectively, indicate that

very little periodicity occur for the chrome adjustments

and that the there is a very low signal to noise ratio. For

the silicon adjustments, the first 5 frequency componentsare particularly prominent. The FFT spectra shown are

all based on the adjustments time series as determined

by the Lagrange multiplier method.

To demonstrate this more clearly, fifth order linear

time series models of the adjustments were evaluated.

The objective was not prove that the adjustments defi-

nitely follows a fifth order and not higher order dy-

namics, but only to identify if the adjustment history isrelated to future adjustments. Should this be the case,

one may use time series models to estimate future ad-

Fig. 14. Si slag fraction.

Fig. 15. FFT spectrum of Si (alloy) errors.

Fig. 16. FFT of Si (slag) errors.


justments. A nth order time series model of the adjust-

ment, e, would have the following general structure:

eeðtÞ ¼ a0 þ a1eðt � 1Þ þ a2eðt � 2Þ þ a3eðt � 3Þþ a4eðt � 4Þ þ a5eðt � 5Þ þ � � � þ aneðt � nÞ ð27Þ

The R2, which determines which portion of the variancein the error can be explained by the model, and model

coefficients are listed in Table 3, which gives the best

least squares fit based on the simple time series model

where the first six terms of Eq. (27) were used.

The reasonably high autocorrelation that exists for

the adjustments is surprising given, that the daily data

reconciliation�s were done independently. Furthermore,it can be seen from the table that the error sequence, as

determined by the Lagrange multiplier method, always

led to similar or higher R2 values. This would be the casewhere a systematic bias contributes significantly to theadjustment, while the random component is relatively

small in comparison. Histograms confirmed the non-

zero mean (implying bias) and non-normality of the

adjustments for both algorithms. This is depicted in

Figs. 19 and 20 for the silicon error/adjustment in the

slag (S) and alloy phases (A) respectively based on the

Lagrange method.

The implications of these observations can be sum-marised as follows:

• While the adjustments to silicon (as silica) in the slag

was both positive and negative, the mean provided the

bias with which the silicon (silica) assay in the slag has

to be adjusted. The distribution of errors show a bell-

curve structure, skewed towards smaller adjustments,

Table 3

Coefficients of fifth order time series models (of the reconciliation adjustments)

Error minimisation method R2 a0 Coefficients

a1 a2 a3 a4 a5

GRG-based SSEa minimisation

Silicon adjustments (alloy phase): 0.46 0.0001 0.615 )0.129 0.146 )0.005 0.159

Chromium adjustments (alloy phase): 0.25 )0.0126 0.375 0.181 0.048 )0.236 0.009

Silicon adjustments (slag phase): 0.40 0.0012 0.382 0.132 0.017 0.386 )0.171Chromium adjustments (slag phase): 0.04 )0.0050 )0.051 )0.002 )0.060 0.173 0.028

Lagrange multiplier SSEa minimisation

Silicon adjustments (alloy phase): 0.56 )0.0005 0.578 0.027 0.224 )0.210 0.215

Chromium adjustments (alloy phase): 0.25 0.0048 0.271 0.315 )0.067 0.062 )0.231Silicon adjustments (slag phase): 0.57 0.0000 0.647 0.005 0.130 )0.176 0.235

Chromium adjustments (slag phase): 0.31 )0.1709 0.460 0.036 0.188 )0.345 0.085

a SSE: sum of square error.

Fig. 19. Error histogram for Si (S).

Fig. 18. FFT of Cr (slag) errors.

Fig. 17. FFT spectrum of Cr (alloy) errors.


but with a significant tail region towards larger posi-

tive adjustments.

• The adjustments of the silicon assay of the alloy

shows a strong non-normal trend (more of an expo-

nential distribution), with all adjustments (in the case

of the Lagrange multiplier method) being positive

and most of the adjustments being close to zero.

However, referring to the silicon fraction for the alloyin Fig. 13, it appears that the GRG method gives a

much more normal (equal positive and negative ad-

justments) distribution of adjustments.

• In general, the Lagrange method tend much more to

shift the reconciled values in a biased direction, that is

it would maintain a positive bias for one variable and

a negative for another, and not give a normal distri-

bution of errors. In doing so it does give a higher de-gree of dynamic predictability of the adjustments, but

do this at the expense of adjustment accuracy.

• The GRG method distributed the errors more ran-

domly around the measured values, but still made

measurement bias identification possible (as shown

for Fig. 12), where the inherent bias was significant.

The degree of predictability of the errors, was in gen-

eral much poorer. However, this in itself show thatthe allocation of adjustments were more random in

nature. It is still interesting that silicon maintained

a high level of dynamic predictability.

• As one are able to predict the adjustments, it becomes

possible to improve the recipe mixes of the feed to ob-

tain a specific product, lending itself therefore to a de-

gree of feed forward control.

• Should one develop semi-fundamental kinetic-ther-modynamic models for prediction and control, pref-

erably the reconciled values, and not the original

measurements should be used for parameter estima-

tion.

• The reconciliation models provide a useful tool to es-

timate the quantities and compositions of the dust

and flue gas, both of which are not frequently mea-

sured.

6. Conclusion

It has been shown that proper characterisation of the

state of homogeneity of the melt is important, not only

to establish the variance for data reconciliation but also

characterise the state of mixedness of the bath and to

which degree one sample is representative of the totalbath inventory. For silicon in ferrochrome alloy, it was

shown that the relative standard deviation decreases as

the degree of sub cooling is reduced. The assumption of

a homogenous melt was shown not to be necessarily

valid, and that significant variation in the melt assays

may occur. In general, one may expect a smaller devi-

ation for the slag phase due to higher mixing driving

forces in this melt phase.Characterising the amount of dust and the carbon

assay of the dust aids (in conjunction with the slag and

alloy), allows one to characterise the efficiency of re-

ductant utilisation.

Improved control is essentially about being able to

operate the furnaces at new required set points with an

improved stability (smaller variance). Knowledge of the

variances determines the benchmark for improved fur-nace control. Knowledge of how the variances maybe

reduced therefore becomes a primary lever towards

improved control. Material balance reconciliation con-

strains the set point determination to mass balance

constraints.

It is has been shown that data reconciliation of the

material flows for a open arc smelting furnace used for

chromite smelting led to material balance closure withinthe variances associated with the measurements.

Systematic biases did appear in the data and was

identified using traditional statistical methods. In gen-

eral, the bias appeared to be larger when the Lagrange

multiplier technique was used versus the more direct

method using the GRG method to minimise the sum of

square errors. Moreover, the contribution of the biases

to the total adjustments were in many cases significantenough (compared to the random component) to allow

a degree of dynamic prediction, as was demonstrated

using a simple fifth order time series model. This aspect

allows the a-priory estimation of dynamic adjustments

for mass balance based control models.

References

Abdul-el-zeet, Z.H., Roberts, P.D., Becerra, V.M., 2002. Enhancing

model predictive control using dynamic data reconciliation. AIChE

J. 48 (2), 324–333.

Aky€uuziu, T., Eric, R.H., 1992. Slag–metal equilibrium in the smelting

of high carbon ferrochromium. J. S. Afr. Inst. Min. Metall. 92 (4),

101–110.

Bazin, C., Tremblay, S.R., 1999, Reconciliation of flow, pressure and

temperature measurements for a gas solids reactor. In: Control and

Optimisation in Minerals, Metals and Materials Processing, 38th

Annual Conference of Metallurgists of CIM, Quebec.

Fig. 20. Error histogram for Si (A).


Bodington, C.E., 1995. Planning Scheduling and Control Integration

in the Process Industries. McGraw-Hill, London.

Bonekamp, I.H., Groeneveld, J.H., Reuter, M.A., 1999. Pyrometal-

lurgy: Quantification of the dynamics of the flash smelter. In:

Hancock, B.A., Pon, M.R.L. (Eds.), Proceedings of Copper 99––

Cobre 99, International Environment Conference, vol. VI. TMS,

Warrendale, USA, pp. 361–375.

Curr, T.R., 1996. A review of ferrochrome smelting technologies.

International Chrome Development Association: Spring Meeting,

Cape Town, South Africa.

Curr, T.R., Barcza, N.A., 1982. The production and treatment of

ferrochrome. South African Patent no. 827404, South Africa.

Cutting, G.W., 1976. Estimation of interlocking mass balances on

complex mineral beneficiation plants. Int. J. Miner. Process. 3,

207–218.

Demir, O.D., Eric, H.R., 1994. Reduction of chromite in liquid Fe–

Cr–Si–C alloys. Metall. Mater. Trans. B 25, 549–559.

De Villiers, J.P.R., Muan, A., 1992. Liquidus and solidus phase

relationships in the system CaO–CrO–Cr2O3–SiO2. J. Am. Ceram.

Soc. 75 (6), 1333–1341.

Downing, J.H., 1975. Smelting chrome ore. Geochim. Cosmol. Acta

39, 853–856.

Edgar, T.F., Himmeblau, D.M., 1989. Optimization of Chemical

Processes. McGraw-Hill, Singapore.

Grund, S.C., Reuter, M.A., Janssen, R.A.G., Nolte, A., 1999.

Optimization of a tin-lead-smelting furnace with the aid of

statistical modelling techniques. EPD Congress 1999, pp. 851–864.

Gunnewiek, L.H., Tullis, S., 1996. Prediction of heat and fluid flow in

the slag phase of an electric arc reduction furnace. In: Proceedings

of the International Symposium on Computational Fluid Dynam-

ics and Heat/Mass Transfer Modelling in the Metallurgical

Industry, Montreal, CIM, pp. 250–264.

Gy, P.M., 1979. Sampling of particulate materials, Theory and

practice. Elsevier, Amsterdam.

Hodouin, D., Berton, A., 2000. An algorithm for fault detection and

isolation using mass balance constraints. In: Proceedings of the

XXI IMPC Rome, vol. A3, pp. 59–65.

Hodouin, D., Everell, M.D., 1980. A hierarchical procedure for the

adjustment and material balancing of mineral processes data. Int.

J. Miner. Process. 7, 91–116.

Hodouin, D., Kasongo, T., Kouame, E., Everell, M.D., 1981.

BILMAT: An algorithm for material balancing mineral processing

circuits. CIM Bull. 74 (83), 123–131.

Howat, D.D., 1986. Chromium in South Africa. J. S. Afr. Inst. Min.

Metall. 86 (2), 37–50.

Kanjilal, P.P., 1995. Adaptive prediction and predictive control. IEE

Control Engineering Series 52. Peter Peregrinus, Exeter.

Lynch, A.J., 1977, Mineral crushing and grinding circuits: Their

simulation, optimisation, design and control. Developments in

Mineral Processing Volume 1, Elsevier Scientific Publishing Co.

Maeda, M., Sano, N., Mastushita, Y., 1981. Chromium recovery from

chromium containing slags. Conserv. Recycling 4 (3), 137–

144.

Pei, W., Wijk, O., 1994. Chromite ore smelting reduction by a carbon

saturated iron melt. Scand. J. Metall. 23, 216–223.

Ragot, J., Maquin, D., Adrot, O., 1999, LMI approach for data

reconciliation. In: Control and Optimisation in Minerals, Metals

and Materials Processing, 38th Annual Conference of Metallurgists

of CIM, Quebec.

Reuter, M.A., Grund, S.C., 2001. The use of data reconciliation as a

soft sensor in various hybrid modelling and control architectures

for pyrometallurgical furnaces. In: Taylor, P.R. (Ed.) Proceedings

Extraction & Processing Division EPD Congress, Warrendale, PA,

2001 TMS Annual Meeting, New Orleans, Louisana, USA, pp.

629–642.

Smith, H.L., Denton, G.M., Barcza, N.A., 1996. Ferrochromium

production. South African Patent no. 960877, South Africa.

Toker, N.Y., Darken, L.S., Muan, A., 1991. Phase relationships and

thermodynamics of the system Fe–Cr–O in the temperature range

of 1600 �C to 1825 �C under strongly reducing conditions. Metall.Mater. Trans. 22B, 689–703.

Wethmar, J.C.M., Howat, D.D., Jochens, P.R., 1975. Phase equilibria

in the Cr–Fe–Si–C system in the composition range representative

of high carbon ferrochromium alloys produced in South Africa.

Metal Sci. 9, 291–296.

Wiegel, R.L., 1972. Improving the plant metallurgical balance. Can.

Met. Q. 11 (2), 413–424.

Wills, B.A., 1986. Complex circuit mass balancing––A simple practi-

cal, sensitivity analysis method. Int. J. Miner. Process. 16, 245–262.

Wills, B.A., 1997. Mineral processing technology, sixth ed., Butter-

worth-Heinemann, Oxford, pp. 84–90.

Xiao, Y., 1993. Thermodynamic study of CrOx containing slags.

D.Tech dissertation, Helsinki University of Technology, Espoo,

Finland.

Xiao, Y., Holappa, L., 1993. Determination of activities in slags

containing chromium oxides. ISIJ Int. 33 (1), 66–74.

Xiao, Y., Holappa, L., 1996. Thermodynamic properties of chromium

bearing slags and minerals, A review. Laboratory of Metallurgy

Report TKK-V-B118, Helsinki University of Technology, Espoo,

Finland.


dynamic structures in variance based data reconciliation adjustments for a chromite smelting furnace

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