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Novel model for estimation of liquid levels in the blast furnace hearth
Chemical Engineering Science 59 (2004) 3423 – 3432
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Novel model for estimation of liquid levels in the blast furnace hearth Johnny Br'annbacka∗ , Henrik Sax,en Akademi University, Biskopsgatan 8, FIN-20500 Abo, Heat Engineering Laboratory, Abo Finland Received 27 January 2004; received in revised form 10 May 2004; accepted 11 May 2004
Abstract An advanced model has been developed to track iron and slag levels in the blast furnace hearth. The model is based on measurements of tapped quantities of iron and slag and standard blast furnace measurement variables. The hearth geometry is provided by a previously presented wear model of the hearth refractory, while the 5oating state of the hearth coke column—the dead man—is estimated from a simpli7ed force balance. The liquid level estimation problem is tackled by an extended Kalman 7lter, by which the variance of the measurements and parameters can be optimally considered. The results of the model clearly show the dramatic e9ect of the 5oating state of the dead man on the tap-cycle evolution of the liquid levels in the hearth, and therefore point out the importance of applying a proper estimate of the hearth geometry in the model. The 7ndings of the model have been analyzed with respect to the asymptotic limit of the descent below the taphole of the iron–slag interface and the required corrections of the material balances. The model has also been veri7ed by a comparison of its results with the tap-cycle trend of the gas pressure drop over the furnace. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Blast furnace hearth; Liquid levels; Tap cycle; Dead man 5oating; Kalman 7lter
1. Introduction The blast furnace is still the principal industrial unit for production of molten iron for primary steelmaking. Because of the emergence of new competing technologies, it is imperative to further optimize the furnace with respect to life length of the equipment, e>ciency of production, energy demand, product quality and emissions. A crucial region of the furnace is its lower part, the hearth, where the liquid iron and slag collect before being tapped out, since the campaign length of the furnace is usually limited by the hearth life length. The operation of the hearth is also important for product quality and for facilitating a high drainage rate. In particular, the state of the core of the coke bed, usually referred to as the dead man, is considered to be decisive for the operation of the hearth. Proper monitoring and control of the hearth state are, therefore, essential steps towards a successful operation of a blast furnace. In the absence of direct measurements of important internal hearth conditions, such as the extent of lining erosion, the iron and slag levels and 5ow conditions and the voidage, ∗
Corresponding author. Tel.: +358-2-215-31; fax: +358-2-215-4792. E-mail addresses: [email protected] (J. Br'annbacka), [email protected] (H. Sax,en). 0009-2509/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.05.007
shape and position of the hearth coke bed, the monitoring has usually to rely on indirect measurements. A qualitative picture of the evolution of the liquid levels can be obtained by measuring the electromotive force between electrodes attached to the furnace steel shell on the tuyere level and below the taphole level. The potential di9erence, which is believed to originate from an electrical circuit through the slag, iron, carbon hearth lining and steel shell (Dorofeev and Novokhatskii, 1984; Radilov, 1984) due to di9erences in the electrical conductivities of these materials, varies with changing liquid levels in the hearth. However, the signal is also in5uenced by other factors and usually exhibits strong drift, which makes it unsuitable as a measurement of the absolute liquid levels inside the hearth. The inner pro7le of the hearth changes dynamically due to erosion of, or growth of buildup material on, the sidewall and bottom. The lining erosion and sculling primarily depend on the 5ow rates of the liquids at the hearth wall and bottom. In model experiments and numerical simulations reported in the literature (Shibata et al., 1990; Preuer et al., 1992), the emergence of local high-velocity 5ow regions has been found to be due to the existence of coke-free zones, e.g. caused by a (partially) 5oating dead man. If a good estimate of the shape, position and state of the dead man is available in addition to an estimated inner pro7le of the hearth, it is
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80 Measured Slag Delay Modelled Slag Delay
Slag Delay (min)
70 60 50 40 30 20 10 200
400
600
800
1000 1200 1400 1600 1800 2000 2200 2400
Time (d) Fig. 1. Observed slag delay (dashed line) and slag delay estimated by the model with a 5oating dead man (solid line) for a 6.5-year period of a blast furnace (Br'annbacka and Sax,en, 2003).
“measurements”, which calls for application of correction terms in the balances. Such models, assuming a stationary dead man, have been presented in the literature (Nightingale et al., 2001; Br'annbacka and Sax,en, 2001). The present paper outlines a systematic method for estimation of the liquid levels in the hearth considering a time-varying inner hearth geometry and a (possibly) 5oating dead man. The model is based on an optimal estimation of the correction of the liquid levels to prevent drift, realized in the form of an extended Kalman 7lter. Applied on process data from an industrial blast furnace, the model illustrates the dramatic e9ect of the 5oating state of the dead man on the tap-cycle evolution of the liquid levels, as well as the in5uence of the hearth geometry. 2. Modeling the liquid levels 2.1. Tap cycle
possible to predict when and where regions with high risk of rapid erosion would occur. The voidage of the dead man is very di>cult to estimate during the operation of the furnace, but some approaches have been presented in the literature, e.g. core drilling of dead-man coke at the tuyere level (Steiler et al., 1991; Negro et al., 2001), residence time studies by tracer experiments (Negro et al., 2001), measurements of the static liquid pressure at the taphole (Desai, 1993; Danloy et al., 1999; Havelange et al., 2004), and interpretation of the time (called the “slag delay” in what follows) of iron-only 5ow in the beginning of each tap (Nightingale and Tanzil, 1997; Torrkulla and Sax,en, 2000). However, there are several drawbacks of these techniques: core drillings do not provide information about the voidage lower down in the hearth; sampling, tracer experiments and liquid pressure measurements are feasible only during short measuring campaigns; tracers, liquid pressure and direct slag delay interpretations are based on an assumption that the dead man be stationary (i.e. not 5oating). The over-all e9ect of a 5oating dead man has been investigated in a recent study (Br'annbacka and Sax,en, 2003), where the relation between liquid volumes and vertical levels was demonstrated to depend mainly on the 5oating state of the dead man. The variations in the slag delay observed in a one-taphole blast furnace were shown to strongly correlate with the 5oating state of the dead man; by estimating the 5oating state from a simpli7ed force balance and using an estimate of the hearth wear pro7le, the long-term evolution of the slag delay could be almost perfectly explained, as indicated in Fig. 1. If short-term information about production and tap rates is available, mass balance equations of iron and slag in the hearth can be applied to estimate the instantaneous liquid volumes, which, in turn, can be transformed into iron and slag levels if the inner hearth geometry and dead man properties are known. The level estimates are, however, corrupted by drifting disturbances caused by noise in the 5ow
Fig. 2 illustrates a typical tap cycle in a (one-taphole) blast furnace, which starts as the previous tap is 7nished, and the taphole is plugged with taphole mud. During the period when the mud is given time to solidify properly, neither iron nor slag are being drained so their volumes and levels in the hearth rise monotonously. When the taphole is opened, iron starts to 5ow out and the iron–slag interface (henceforth called the iron level) stays fairly horizontal under its descent due to the low iron viscosity and the large density di9erence between the two liquids. Slag will not be tapped until the iron level has descended (close) to the taphole, which gives rise to a slag delay. Once slag starts to drain, iron and slag 5ow out concurrently until the end of the tap, and iron will, thus, be “pumped” up from levels below the taphole. This is due to the large pressure gradient at the taphole that is caused by the 5ow of the highly viscous slag (Tanzil et al., 1984). The slag–gas interface (henceforth called the slag level) will eventually decline locally towards the taphole, causing outbursts of gas at the end of the tap even though the average slag level is still well above the taphole. This completes the tap cycle at a stage where the iron level is below and the slag level is above the taphole. 2.2. Model of liquid volumes in the hearth Assuming iron and slag to be immiscible liquids of constant densities, the mass balance equations can be expressed as volume balances of the liquids in the hearth. Therefore, the change in liquid volume from a time t0 to time t can be expressed by a time integral of the volume in- and out5ow rates (V˙ in and V˙ out , respectively). t Vliq (t) = Vliq (t0 ) + (V˙ liq; in − V˙ liq; out ) dt: (1) t0
The problem with this model is that practically all volume 5ow measurements contain noise. When integrating the
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
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noisy volume 5ow measurements, the resulting volume estimate contains a drifting noise which, if left uncorrected, causes it to take unrealistic values quite soon. Therefore this model is not useful in itself unless the volume estimates can be corrected to prevent drift.
Level
Fig. 2. Evolution of the iron and slag levels in the hearth during the tap cycle.
2.3. Measurements of in- and out;ows From the standard instrumentation of blast furnaces it is a well-established procedure to estimate the instantaneous production rates of iron (and slag) on the basis of top gas analysis and blast parameters, considering the oxide content of the burden. However, the technique is based on the assumption of quasi-stationary conditions and it is quite sensitive to measurement errors in the top gas analysis. As for the out5ow measurements (Br'annbacka and Sax,en, 2001), the blast furnace used to illustrate the model in Section 3 is equipped with facilities for iron level measurement by radar in the metallurgical vessel, called ladle, where the iron is tapped after the runner. Given the ladle geometry, the level measurement can be transformed into an instantaneous (5-min average) estimate of the iron out5ow rate. Errors arise because of changes in the geometry caused by ladle erosion and/or skulling. The slag 5ow rate, in turn, can be calculated either from mass and heat balances of the water used in the slag granulation unit or from the variations in hydraulic pressure in the bearings of the granulation drum. Both measurements are of quite poor accuracy, but they still provide valuable real-time information about of the out5ow term in the slag mass balance equation.
tss(k−1) tend(k−1)
tss(k)
tend(k)
tss(k+1) tend(k+1)
Time Fig. 3. Schematic illustration of the liquid levels (solid lines) as well as the level “measurement points” for iron (circles) and slag (squares). The taphole levels are indicated by horizontal dotted lines.
to the taphole level at the moment when the slag out5ow starts, tss . In Fig. 3 these measurement points have been indicated by small circles. As for the slag level, it can be estimated at the end of each tap, tend , from the relationship between the residual slag ratio and the 5ow-out coe>cient, FL . This dimensionless variable was proposed by Fukutake and Okabe (1976a,b). Later, Zulli (1991) studied the relationship between FL and the residual slag ratio in small-scale experiments, and extended the expression to include the effect of two liquid phases, i.e., to consider the fact that the lower surface of the upper liquid (slag) can be below the taphole. The 5ow-out coe>cient as de7ned by Zulli (1991) and applied on the blast furnace hearth is given by
1:4
2.4. Measurements of the iron and slag levels
1 sl w (1 − )2 FL = 180 3 ( dp )2 sl g
There are no direct measurements of iron and slag volumes in the hearth, but the liquid levels relative to the taphole level can be estimated at certain time instants of the tap cycle. On the basis of the over-all behavior of the tap cycle outlined in Section 2.1, the iron level is “measured” once every tap cycle (showing a su>ciently long slag delay), since the iron level can be taken to have descended
where is a shape factor, dp is the coke particle diameter, is the liquid viscosity, is the density, w is the mean super7cial velocity, dh is the hearth diameter, z is the vertical level and subscripts “sl” and “th” denote slag and taphole, respectively. From the experimentally determined relationship between FL and the residual slag ratio, the slag level at the end of the tap, denoted by the small squares in Fig. 3,
dh zsl − zth
;
(2)
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can be estimated. This is a prerequisite for “reconstructing” the liquid levels, illustrated schematically by the curves in the 7gure. In the 7gure, k denotes the tap number and subscripts “ss” and “end” refer to the moments of slag out5ow start and tap end respectively.
where d zi and dVj are the di9erentials of the liquid levels and volumes respectively, dVcf is the di9erential of the volume of the coke-free layer and A is the cross-sectional area of the hearth. The relationship dVcf =dVj is determined by a balance between the force pressing down the dead man and the buoyancy force acting on the submerged part of the dead man in the opposite direction (Br'annbacka and Sax,en, 2003). Instead of attempting to describe the downward-acting force in detail (Takahashi et al., 2002), it is here expressed by a parameterized pressure pro7le expression r 6 r0 ; pQ d ; n pd (r) = (4) r − r0 ; r ¿ r0 ; pQ d − a R where pQ d is an over-all term (independent of the radius r), a and n are model parameters and R is a scaling factor (e.g., the
i = j = sl 0.04
0.03
dzi / dVj (m/m3)
The relationship between liquid volumes and levels in the hearth depends strongly on the inner geometry of the hearth and on the properties of the dead man. In this work, an independent model based on thermocouple measurements in the hearth lining (Torrkulla and Sax,en, 2000) is used to estimate the wear/buildup pro7le of the hearth by solving an inverse heat-conduction problem. The model calculates the wear in two dimensions for di9erent sectors of the hearth, and the solution is interpolated to a three-dimensional internal pro7le. Since the changes in the hearth geometry are slow in comparison to the liquid level changes during the tap cycle, the short-term e9ect of a varying hearth geometry is minor. However, the long-term evolution is important mainly through its in5uence on the 5oating state of the dead man. The properties of the dead man that a9ect the volumes-tolevel functions are the dead-man voidage, , 5oating state and bottom shape. In lack of a reliable method for real-time estimation of the voidage, this variable was for the sake of simplicity assumed uniform and constant in the present study. If the dead man 5oats partially or completely, there is a coke-free region below it, the volume of which is a9ected by the total iron and slag volumes in the hearth, because the dead man moves with the liquid levels. The relationship between liquid volumes and levels is then given by d zi dVcf 1 − (1 − ) ; i; j = {ir; sl} and = dVj A dVj 0 if i = ir; j = sl; = (3) 1 otherwise;
0.05
i = { ir, sl }, j = ir
0.02
0.01
0 i = ir , j = sl −0.01
−0.02 60
V cf (m3)
2.5. Volumes–level coupling
0.06
40
20
0 50
60
70
80
90
100
110
120
130
140
pd (kPa) Fig. 4. Upper panel: volume derivatives of the liquid levels at di9erent dead man positions set by adjusting pQ d . Lower panel: volume of the coke-free layer.
radius of the unworn hearth). The dead man bottom shape and the free volume are thus determined by the shape of the hearth bottom and the downward-acting pressure pro7le, where pQ d is a9ected by the burden weight and the lifting force of the ascending gas. Br'annbacka and Sax,en (2003) have demonstrated that this model provides an adequate description of the long-term evolution of the slag delay (cf. Fig. 1). To illustrate the e9ect of the dead-man 5oating state on the volumes–level coupling functions, the derivatives d zi =dVj in Eq. (3) were calculated at di9erent dead man 5oating levels. The hearth geometry used for the illustration is estimated by the hearth erosion model for a speci7c day of the blast furnace presented in Section 3. The dead man bottom pro7le was determined by setting the parameters a = 112 kPa; n = 2; r0 = 2 m and R = 4 m in Eq. (4), using the over-all downward-acting pressure pQ d , which is directly proportional to the vertical position of the dead man, as a variable. The iron level was assumed to be at the taphole, and the slag layer was 1 m thick. The upper panel of Fig. 4 depicts the volume derivatives of the iron and slag levels at di9erent values of pQ d , while the volume of the coke-free layer, Vcf , is depicted in the lower panel. As can
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
be seen, the changes in the volumes–level coupling functions can be quite abrupt if the dead man starts to 5oat. As the dead man 5oating state changes from completely sitting to completely 5oating in the iron bath the volume derivatives of the level functions change between the asymptotic values given by dotted lines in the upper panel of the 7gure. The steepness of this transition depends on the shapes of the hearth and dead man bottoms; in the extreme case where the dead man and the hearth bottom have identical shapes, the transition from the completely sitting dead man state to the completely 5oating state takes place immediately when the buoyancy force exceeds the downward acting force. 2.6. State-space model of liquid volumes and levels in the hearth Since the iron and slag levels can be “measured” once every tap cycle, a discrete state-space model of the liquid volumes with a (variable) sampling time equaling the tap cycle length was formulated, using the moments for iron level measurements (i.e. the time instances when slag out5ow starts) as discretization points. The state-space model is given by Vir (tss (k + 1)) =Vir (tss (k)) +
tss (k+1)
tss (k)
(V˙ ir; in (t) − V˙ ir; out (t)) dt
+e1; ir (k + 1);
(5a)
Vsl (tss (k + 1)) =Vsl (tss (k)) +
tss (k+1)
tss (k)
(V˙ sl; in (t) − V˙ sl; out (t)) dt
+e1; sl (k + 1); zir (tss (k)) = fir (Vir; ss (k) ; Vsl; ss (k); : : : ; ) + e2; ir (k);
(5b) (5c)
zsl (tend (k − 1)) =fsl (Vir; ss (k) − SVir ; Vsl; ss (k) − SVsl ; : : : ; ) +e2; sl (k);
(5d)
where e1 and e2 represent disturbances in the system, and fir (·) and fsl (·), in turn, express the volumes–level coupling functions for the iron and slag levels. These functions include the e9ect of the irregular hearth geometry and the dead man bottom shape and 5oating state and are, therefore, rather complicated (Br'annbacka and Sax,en, 2003). However, the slag level zsl is not measured at the time of slag 5ow start, tss (k), but at the end of the previous tap, tend (k − 1), (cf. Fig. 2). This fact was considered by subtracting the
integrals SVir =
tss (k)
tend (k−1)
SVsl =
tss (k)
tend (k−1)
3427
(V˙ ir; in (t) − V˙ ir; out (t)) dt;
(6a)
(V˙ sl; in (t) − V˙ sl; out (t)) dt
(6b)
from the iron and slag volumes at the start of slag out5ow, Vir; ss (k) and Vsl; ss (k) respectively, before applying the volumes–level coupling function for the slag level. The above equations can be cast in standard discrete state-space form, where the liquid volumes in the hearth are “measured” by the liquid levels V(k + 1) = V(k) + u(k + 1) + e1 (k + 1); z(k) = f(V(k); : : : ; ) + e2 (k);
(7)
where the vectors V(k) = (Vir; ss (k); Vsl; ss (k))T ; z(k) = (zir; ss (k); zsl; end (k − 1))T ; u(k) contains the integrals of the iron and slag in- and out5ows between tss (k − 1) and tss (k) (cf. Eq. (5)a,b), and e1 (k) and e2 (k) contain the process and measurement noises, respectively. 2.7. State estimation with an extended Kalman ?lter If the function f in the state-space model can be linearized at a certain state, and the disturbances e1 and e2 are uncorrelated zero mean Gaussian noise, there is an optimal estimator, the Kalman 7lter (Haykin, 2001), of the state V based on the measurements of z at each time step. The Kalman 7lter is a robust estimator, so even if the above conditions are not fully satis7ed it will still produce fairly good (suboptimal) state estimates. The method requires that the variances of the white noise vectors e1 and e2 be given. The variances of the “process noise”, e1 = (e1; ir ; e1; sl )T , can fairly accurately be estimated by comparing the volume integrals of iron and slag (cf. Eq. (5)) with volumes calculated on the basis of hot metal weighing reports from the steel plant or the mass of charged burden materials in the blast furnace. As for e2 = (e2; ir ; e2; sl )T there is no direct measurement from which its variance could be calculated, so it has to be set on the basis of knowledge of the general behavior of the liquid levels in the hearth. The Kalman 7lter 7rst applies the model to estimate a new state based on the previous state estimate and the model inputs ˆ + 1 | k) = V(k|k) ˆ V(k + u(k + 1):
(8)
Next, the nonlinear function f is linearized at this state (cf. Eq. (3)), and the matrix F is formed as @f(V; : : : ; ) F(k + 1|k) = : (9) ˆ @V V=V(k+1|k) The Kalman gain matrix K is estimated by the recursive procedure P(k + 1|k) = P(k|k) + R2 ;
(10a)
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J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
K(k + 1) = P(k + 1|k)F(k + 1|k)T [F(k + 1|k)P(k + 1|k) ×F(k + 1|k)T + R1 ]−1 ;
(10b)
P(k + 1|k + 1) = [I − K(k + 1) ×F(k + 1|k)]P(k + 1|k);
(10c)
where P is the covariance matrix of the reconstruction error and R1 and R2 are the covariance matrixes of e1 and e2 , respectively. Finally, the optimal state estimate can be calculated by ˆ + 1 | k + 1) = V(k ˆ + 1|k) V(k +K(k + 1)[z(k + 1) ˆ + 1|k); : : : ; )]: −f(V(k
(11)
As the iron and slag volumes at time tss (k + 1) have been estimated, the corresponding levels are calculated through the volumes–level coupling functions (cf. Section 2.5 and Br'annbacka and Sax,en, 2003). The correction of the volumes can, e.g., be smeared over the short-term volume estimates between tss (k) and tss (k + 1) to produce smooth transitions of the estimated levels without discontinuities at the correction points. 2.8. Computational algorithm The computational procedure for estimating and correcting the liquid volumes and levels is summarized in the following algorithm: 1. Calculate the inner hearth geometry with the erosion model. 2. Estimate Vir; end (k) and Vsl; end (k) by integrating the measurements of the volume in- and out5ows between tss (k) and tend (k) (cf. Fig. 3). 3. Apply the volumes–level coupling function for the slag level to estimate zsl; end (k). 4. Apply the volume derivatives to the volumes–level coupling function for the slag level at this state to form the second row of the matrix F(k + 1|k). 5. Use the relationship between the 5ow-out coe>cient, Eq. (2), and the residual slag ratio to give a “measurement” of zsl; end (k). 6. Estimate Vir; ss (k + 1) and Vsl; ss (k + 1) by integrating the measurements of the volume in- and out5ows between tend (k) and tss (k + 1) (cf. Eqs. (6)). 7. Apply the volumes–level coupling function for the iron level to estimate zir; ss (k + 1). 8. Apply the volume derivatives to the volumes–level coupling function for the iron level at this state to form the 7rst row of the matrix F(k + 1|k). 9. “Measure” zir; ss (k +1) by assuming it to be at the taphole level, zth (k + 1). 10. Update the Kalman gain and use Eq. (11) to correct Vir; ss (k + 1) and Vsl; ss (k + 1) by considering the
di9erences between the estimated and the measured iron and slag levels. 11. Smear the volume corrections over the iron and slag volumes estimated on short-term basis between tss (k) and tss (k +1), and calculate the corrected short-term iron and slag levels. 12. Set k = k + 1. If new daily averages of the thermocouple measurements in the hearth lining are available go to 1, otherwise go to 2.
3. Results and discussion The model of the liquid levels has been evaluated on process measurements from a Finnish one-taphole blast furnace. In what follows, the results from a 3-month period will be used to illustrate the results. The data period is divided into two 40-day segments separated by a 2-week period in the middle (where information was lost because of a data backup problem). The 7rst segment of the period was characterized by a worn furnace hearth, and, hence, a quite well 5oating dead man. However, during certain parts of the period, particularly at its end, buildup material was formed on the hearth bottom, resulting in a reduced inner volume of the hearth and a lesser degree of dead man 5oating. 3.1. Noise covariances In the Kalman state estimator the only parameters not given by the model are the covariance matrixes R1 and R2 of the noise vectors e1 and e2 , and the optimal Kalman gain matrix K is estimated on the basis of this information. Studying the components of e1 , the largest noise source is clearly the out5ow measurements for both iron and slag. An examination of the tapped iron measurement was made by comparing it with the control weighing of each ladle at the steel plant (Voutilainen, 1999), and the estimated standard deviation for each ladle was 4:2 ton. In terms of iron volume of a tap, assuming each tap to 7ll three ladles, this yields a standard deviation of 1:8 m3 . The in5ow measurement is based on an oxygen balance over the furnace for the gas phase, and its most likely error is caused by drift in the top gas analysis, but these measurements are calibrated regularly. Considering possible minor stochastic disturbances in the in5ow measurement, the total standard deviation was estimated to 1; ir = 2:0 m3 . For the corresponding slag variable, e1; sl , no such calibration study has been conducted. Since the slag out5ow measurement must be considered slightly more uncertain than the iron out5ow measurement, and the tapped slag volume is about 60% of the iron volume, the standard deviation was assumed to be the same as for iron, i.e., 1; sl = 2:0 m3 . The interdependence between these two noise variables is quite weak, since only the production rate estimates are based on common factors. In lack of more accurate information
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
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9.5
8.5
70 50 7.5
7.5
9.5
9
30
9 70 8.5 50
Emf (%)
8
9.5
Level (m)
9 8.5 8
Level (m)
Level (m)
9
Emf (%)
Level (m)
9.5
8 30 7.5 07:12
8.5
09:36
12:00
14:24
16:48
19:12
21:36
00:00
Time (hour:minute) 8
7.5 06:00
07:12
08:24
09:36
10:48
12:00
Time (hour:minute) Fig. 5. Upper panel: iron and slag levels estimated assuming a standard deviation of 5 cm in the measurement noise. Lower panel: corresponding levels estimated assuming a standard deviation of 10 cm in the measurement noise. Dotted lines depict the taphole level.
the noise variables were therefore considered uncorrelated, yielding a diagonal R1 . The components of the measurement noise vector, e2 , cannot be measured but have still to be estimated a priori. Knowledge of the process would suggest standard deviations in the level measurements of about 5 cm. Using this in R2 , and assuming the disturbances to be uncorrelated, level estimates have been depicted for a short period of about three tap cycles in the upper panel of Fig. 5 (under the conditions described in the next subsection). The lower panel of the 7gure shows the corresponding liquid levels estimated with a standard deviation of 10 cm in the level measurement noise. The former noise level is clearly a more realistic estimate; a standard deviation of 10 cm was found to yield liquid level estimates that occasionally drifted strongly and exhibited unrealistic behavior, such as a slag level descending below the iron level or below the taphole, or an iron level well above the taphole during several consecutive tap cycles. Therefore, in what follows R1 = diag(4; 4) m6 and R2 = diag(25; 25) cm2 are used. 3.2. Dead man ;oating state The e9ect of dead man 5oating on the liquid levels is illustrated in Fig. 6, where the upper panel shows the evolutions of the slag and iron levels during 10 consecutive tap cycles for the case with a sitting dead man; this state was forced by assigning pQ d a high value (e.g., 300 kPa). The lower panel, in turn, illustrates the corresponding results for the case where the value of pQ d was calculated from the process data under the assumptions given in Br'annbacka and
Fig. 6. Estimated slag (upper solid curves) and iron (lower solid curves) levels in the BF hearth for a sitting (upper panel) and partially 5oating (lower panel) dead man. Scaled emf signals (dashed curves) have also been depicted, as well as the taphole level (dotted lines).
Sax,en (2003), yielding pQ d ≈ 102 kPa and a partially 5oating dead man. The amplitude of the liquid levels is clearly larger for the sitting dead man case. In terms of coincidence between the levels and the measured emf signal, the iron levels for both the sitting and partially 5oating dead man states show strong correlation. On the basis this observation and a comparison between the estimated iron levels and the emf for longer time periods, it seems di>cult to utilize the emf for detection of the 5oating state of the dead man. As for the veri7cation of the estimated amplitude of the liquid levels, no other direct measurement is available. However, an asymptotic maximum for the vertical distance that the iron level can descend below the taphole at the end of the tap is given by a simple pressure balance of the two-liquid 5ow system at hand (Tanzil et al., 1984). This maximum, expressed as Szir; end = zth − zir; end , depends on the height of the slag level at the end of the tap, Szsl; end = zsl; end − zth , which is a “measured” variable, according to sl Szir; end = Szsl; end (12) ir − sl and applies, strictly speaking, to the case where no iron 5ows out at the end of the tap. The upper panel of Fig. 7 shows the estimated Szir; end (solid line) and the asymptotic maximum value (dashed line) for the case with a sitting dead man, while the lower panel illustrates the corresponding curves for a partially 5oating dead man. In the former case, Szir; end is very close to its asymptotic maximum and clearly exceeds it during fairly long time periods. At least during the periods July 21–25 and August 6–25 the end iron levels assume completely unrealistic values. In the case where the dead man 5oating state was estimated from process data, Szir; end stays below the asymptotic limit throughout the data period, and it must therefore be considered a more plausible state of the furnace hearth. If the partially 5oating dead man model describes the actual volume–level relationships better than the sitting dead
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J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
9
0.6
Level (m)
∆zir,end (m)
3.
2.
1.
Estimated Asymptotic maximum
0.8
0.4 0.2
8.5 8
0 22:04 Estimated Asymptotic maximum
22:33
23:02
23:31
00:00
Time (hour:minute)
0.6
8
0.4
Level (m)
∆zir,end (m)
0.8
0.2 0 07/21
08/01
08/13
08/25
09/06
09/17
09/29
10/11
7.5 7
6 5.5
Time (month/day) Fig. 7. Estimated Szir; end (solid line) and its asymptotic maximum (dashed line) for a sitting dead man (upper panel) and a partially 5oating dead man (lower panel).
Table 1 Standard deviations of the volume corrections required by the state estimation model for iron and slag in the hearth Dead man state
Iron correction (m3 )
Slag correction (m3 )
Sitting Partially 5oating
3.23 2.80
4.33 3.92
man model, one would expect the corrections of the volume, ˆ + 1|k); : : : ; )] in Eq. (11), i.e., K(k + 1)[z(k + 1) − f(V(k to be smaller for this case. The standard deviations of the corrections of the iron and slag volumes for the sitting and partially 5oating dead man case, reported in Table 1, show that the necessary volume corrections to keep the liquid levels within reasonable bounds are smaller for the partially 5oating dead man case. None of the investigated indicators de7nitely prove that the dead man was 5oating during the studied period, but the studies of the asymptotic iron end level and the volume corrections given by the Kalman 7lter indicate that a partially 5oating state is more likely than a completely sitting state. Therefore the partially 5oating state or, more correctly, the 5oating state given by the estimated value of pQ d and the inner hearth pro7le, is considered a superior model for describing the volume–level coupling also on a short-term basis. 3.3. The coke-free layer An interesting “by product” of the state estimation model in combination with the model of the dead man 5oating state is the possibility to monitor the variations in size and shape of the coke-free layer during the tap cycles. Fig. 8 illustrates the liquid levels and lower surface of the dead man (dashed lines) for three di9erent time instants during one tap cycle. The elephant-foot erosion pro7le of the hearth lining is seen to gives rise to a circular coke-free region mainly at the
1. 2. 3.
6.5
4
3
2
1
0
1
2
3
4
Radius (m)
Fig. 8. Evolution of the iron and slag levels (upper panel) and the dead man bottom (lower panel) for a tap cycle during a period with a relatively eroded hearth.
1.
2.
3.
Fig. 9. Three-dimensional illustration of the coke-free regions in the cases 1–3 in Fig. 8. The black horizontal bar marks the location of the taphole.
hearth edges. Fig. 9 presents three-dimensional views of the coke-free regions for the three time instants of Fig. 8. The evolution of the minimum and maximum volume of the coke-free layer during each tap cycle has been illustrated in the upper panel of Fig. 10 for the 7rst 40-day data period of Fig. 7. Since the movement of the dead man a9ects the
Coke Free Volume (m3)
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432 40 Maximum Coke Free Volume Minimum Coke Free Volume b.
30 a. 20 10 0 13/07
18/07
23/07
28/07
02/08
07/08
12/08
17/08
Time (day/month)
1.2 10
1.1 1
5 0.9 0
09:36
12:00
14:24
Time (hour:minute)
16:48
0.8
Coke Free Volume (m3)
15
Gas Pressure Drop (bar)
Coke Free Volume (m3)
1.3
20 1.5 1.4
15
1.3 10 1.2 1.1
5 12:00
14:24
16:48
1 19:12
Gas Pressure Drop (bar)
Period b.
Period a.
Time (hour:minute)
Fig. 10. Upper panel: minimum and maximum volume of the coke-free layer. Lower panels: evolution of the volume of the coke-free layer during two short periods, a and b, and the gas pressure drop.
conditions in the upper part of the furnace (Nishio et al., 1977; Torrkulla et al., 2002), the changes in the volume of the coke-free layer may be correlated with short-term changes in other process variables. Torrkulla et al. (2002) presented the hypothesis that changes experienced in the gas pressure drop over the furnace during the tap cycle would re5ect the 5oating degree of the dead man. Focusing on two shorter periods, a and b indicated in the upper panel of Fig. 10, the estimated free volume (solid curves) and the gas pressure drop (dashed curves) for four consecutive tap cycles have been depicted in the lower panels of the 7gure. During period a, when the dead man barely 5oats, practically no correlation between the pressure drop and the free volume is evincible, while period b is characterized by a strong dependence between the pressure drop and the dead-man motion. It is not clear why the correlation arises, but possible reasons are that the upward movement of the dead man causes deformation of the raceways (e.g., elevation of the bird’s nest) and the coke slits in the cohesive zone, which a9ects the gas permeability of the bed.
3431
state of the hearth in the liquid level estimation problem. In particular, the 5oating state of the dead man has been demonstrated to be decisive for the results. The 7ndings of the model have been analyzed by comparing the estimated liquid levels with asymptotic limits given by the over-all pressure distribution in the system in question as well as by studying how the required 7ltering of the volume balance equations of the hearth region is a9ected by the hearth state. From this analysis it can be concluded that the dead man of the blast furnace studied in this work is likely to be in a partially 5oating state. This is also supported by the fact that the estimated evolution of the volume of the coke-free layer below the dead man shows correlation with the gas pressure drop over the furnace. The model is applicable on any blast furnace where an estimate of the hearth geometry and instantaneous measurements of the production and tap rates are available, and where there (generally) is a distinct period of iron-only 5ow in the beginning of the tap. In the forthcoming work, the level estimation model will be applied online to aid tapping control. The possibility to use the model to predict the end of the tap is especially interesting, since this would enable the operators to shut the taphole immediately when the gas bursts out, or even minutes before (Nightingale et al., 2001). This would aid tap planning, and considerably reduce splashing in the tap hall and erosion of the taphole. Furthermore, the model will be extended to include hysteresis in the dead man motion, which has been observed in recent cold model studies (Shinotake et al., 2003) as well as in operating blast furnaces (Havelange et al., 2004). The results of the level estimation model will be used to evaluate the quality of the emf signal, and the possibility to use it combined with the present model to more accurately estimate the iron level will be studied (Sax,en and Br'annbacka, 2004). The possibility to utilize knowledge of the dynamic behavior of the coke-free region below the dead man in CFD simulations will also be investigated in future research. This could shed welcome light on the mechanisms behind abrupt severe lining erosion that is occasionally observed in the operation of the blast furnace.
Notation 4. Conclusions
Romans
A model for estimating iron and slag levels in the blast furnace hearth has been developed on the basis of measurements of tapped quantities as well as on a description of the internal state of the hearth. The latter includes a hearth geometry estimate by a model of the hearth refractory wear and a force-balance model estimating the 5oating state of the hearth coke (dead man). The liquid level estimation problem is written in state space form and is solved by an extended Kalman 7lter. The model has been applied on measurements from an industrial blast furnace, and the results have illustrated the importance of considering the internal
A a d e f(·) FL g k n p
hearth cross-sectional area, m2 parameter in Eq. (4), kPa diameter, m disturbance to volume estimate or level measurement in Eq. (5), m3 or m volumes–level coupling function, m 5ow-out coe>cient, dimensionless gravitational acceleration, m=s2 tap number, dimensionless parameter in Eq. (4), dimensionless pressure, kPa
3432
r R t V w z
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
radius, m parameter in Eq. (4), m time, d; h; min volume, m3 super7cial velocity, m/s vertical level, m
Greek letters
parameter in Eq. (3), dimensionless dead man voidage, dimensionless viscosity, Pa s density, kg=m3 estimated standard deviation, m3 or m particle shape factor, dimensionless
Vectors and matrixes e F f(·) K P R u V z
disturbances to estimated iron and slag volumes or level measurements linearization matrix for the function f(·) levels measuring function Kalman gain covariance matrix of the reconstruction error covariance matrix of e measured changes in iron and slag volumes iron and slag volumes iron and slag levels
Subscripts 0 cf d end h i in ir j liq out p sl ss th
initial or reference state coke-free zone downward-acting tap end hearth iron or slag in5ow iron iron or slag liquid (iron and/or slag) out5ow coke particle slag slag5ow start taphole
References Br'annbacka, J., Sax,en, H., 2001. Modeling the liquid levels in the blast furnace hearth. ISIJ International 41, 1131–1138. Br'annbacka, J., Sax,en, H., 2003. Model Analysis of the operation of the blast furnace hearth with a sitting and 5oating dead man. ISIJ International 43, 1519–1527.
Danloy, G., Stolz, C., Crahay, J., Dubois, P., 1999. Measurement of iron and slag level in the blast furnace hearth. Proceedings of the 58th Ironmaking Conference, ISS, pp. 89–98. Desai, D., 1993. Analysis of blast furnace hearth drainage based on the measurement of liquid pressure inside of the hearth. Transactions of the Iron and Steel Society of AIME 14, 45–50. Dorofeev, V.N., Novokhatskii, A.M., 1984. Origin of di9erence in electric potentials on blast furnace shell. Steel in the USSR 14, 10–12. Fukutake, T., Okabe, K., 1976a. Experimental studies of slag 5ow in the blast furnace hearth during tapping operation. Transactions of the Iron and Steel Institute of Japan 16, 309–316. Fukutake, T., Okabe, K., 1976b. In5uences of slag tapping conditions on the amount of residual slag in the blast furnace hearth. Transactions of the Iron and Steel Institute of Japan 16, 317–323. Havelange, O., Danloy, G., Franssen, C., 2004. The dead man, 5oating or not? Revue de M,etallurgie-CIT 3, 195–201. Haykin, S. (Ed.), 2001. Kalman Filtering and Neural Networks. Wiley, New York. Negro, P., Petit, C., Urvoy, A., Sert, D., Pierret, H., 2001. Characterization of the permeability of the blast furnace lower part. Proceedings of the 60th Ironmaking Conference, ISS, 337–348. Nightingale, R.J, Tanzil, F.W.B.U., 1997. A novel approach in the estimation of blast furnace hearth voidage. Iron and Steelmaker 24, 35–37. Nightingale, R.J., Tanzil, F.W.B.U., Beck, A.J.G., Price, K., 2001. Blast furnace hearth condition monitoring and taphole management techniques. Revue de Metallurgie-CIT 98, 533–540. Nishio, H., Wenzel, W, Gudenau, H.W., 1977. Signi7cance of the “dead zone” in the blast furnace. Stahl Eisen 97, 867–875. Preuer, A., Winter, J., Hiebler, H., 1992. Computation of the erosion in the hearth of a blast furnace. Steel Research 63, 147–151. Radilov, S.V., 1984. Monitoring the melt level in a blast furnace hearth. Soviet Automatic Control 17, 76–78. Sax,en, H., Br'annbacka, J., 2004. Dynamic model of liquid levels in the blast furnace hearth. To be presented at Metal Separation Technologies III Symposium, June 20–24, 2004 , Copper Mountain, CO. Shibata, K., Kimura, Y., Shimizu, M., Inaba, S., 1990. Dynamics of dead-man coke and hot metal 5ow in a blast furnace hearth. ISIJ International 30, 208–215. Shinotake, A., Ichida, M., Ootsuka, H., Kurita, Y., 2003. Bottom shape of blast furnace deadman and its 5oating/sinking behavior by 3-dimensional model experiment. Tetso-to-Hagan,e 89, 77–84. Steiler, J., Nicolle, R., Negro, P., Helleisen, M., Jusseau, N., Metz, B., Thirion, C., 1991. Tuyere probing into the dead man of the blast furnace a way to assess hearth phenomena and coke behaviour. Proceedings of the 50th Ironmaking Conference, ISS, 715–723. Takahashi, H., Kawai, H., Suzuki, Y., 2002. Analysis of stress and buoyancy for solids 5ow in the lower part of a blast furnace. Chemical Engineering Science 57, 215–226. Tanzil, W.B.U., Zulli, P., Burgess, J.M., Pinczewski, W.V., 1984. Experimental model study of the physical mechanisms governing blast furnace hearth drainage. Transactions of the Iron and Steel Institute of Japan 24, 197–205. Torrkulla, J., Sax,en, H., 2000. Model of the state of the blast furnace hearth. ISIJ International 40, 438–447. Torrkulla, J., Br'annbacka, J., Sax,en, H., Waller, M., 2002. Indicators of the internal state of the blast furnace hearth. ISIJ International 42, 504–511. Voutilainen, J. 1999. Evaluation of measurements in an expert system for tapping control. M.Sc. Thesis, University of Oulu, Oulu, Finland (in Finnish). Zulli, P., 1991. Blast furnace hearth drainage with and without coke-free layer. Ph.D. Thesis, University of New South Wales, Sydney, Australia.
www.elsevier.com/locate/ces
Novel model for estimation of liquid levels in the blast furnace hearth Johnny Br'annbacka∗ , Henrik Sax,en Akademi University, Biskopsgatan 8, FIN-20500 Abo, Heat Engineering Laboratory, Abo Finland Received 27 January 2004; received in revised form 10 May 2004; accepted 11 May 2004
Abstract An advanced model has been developed to track iron and slag levels in the blast furnace hearth. The model is based on measurements of tapped quantities of iron and slag and standard blast furnace measurement variables. The hearth geometry is provided by a previously presented wear model of the hearth refractory, while the 5oating state of the hearth coke column—the dead man—is estimated from a simpli7ed force balance. The liquid level estimation problem is tackled by an extended Kalman 7lter, by which the variance of the measurements and parameters can be optimally considered. The results of the model clearly show the dramatic e9ect of the 5oating state of the dead man on the tap-cycle evolution of the liquid levels in the hearth, and therefore point out the importance of applying a proper estimate of the hearth geometry in the model. The 7ndings of the model have been analyzed with respect to the asymptotic limit of the descent below the taphole of the iron–slag interface and the required corrections of the material balances. The model has also been veri7ed by a comparison of its results with the tap-cycle trend of the gas pressure drop over the furnace. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Blast furnace hearth; Liquid levels; Tap cycle; Dead man 5oating; Kalman 7lter
1. Introduction The blast furnace is still the principal industrial unit for production of molten iron for primary steelmaking. Because of the emergence of new competing technologies, it is imperative to further optimize the furnace with respect to life length of the equipment, e>ciency of production, energy demand, product quality and emissions. A crucial region of the furnace is its lower part, the hearth, where the liquid iron and slag collect before being tapped out, since the campaign length of the furnace is usually limited by the hearth life length. The operation of the hearth is also important for product quality and for facilitating a high drainage rate. In particular, the state of the core of the coke bed, usually referred to as the dead man, is considered to be decisive for the operation of the hearth. Proper monitoring and control of the hearth state are, therefore, essential steps towards a successful operation of a blast furnace. In the absence of direct measurements of important internal hearth conditions, such as the extent of lining erosion, the iron and slag levels and 5ow conditions and the voidage, ∗
Corresponding author. Tel.: +358-2-215-31; fax: +358-2-215-4792. E-mail addresses: [email protected] (J. Br'annbacka), [email protected] (H. Sax,en). 0009-2509/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.05.007
shape and position of the hearth coke bed, the monitoring has usually to rely on indirect measurements. A qualitative picture of the evolution of the liquid levels can be obtained by measuring the electromotive force between electrodes attached to the furnace steel shell on the tuyere level and below the taphole level. The potential di9erence, which is believed to originate from an electrical circuit through the slag, iron, carbon hearth lining and steel shell (Dorofeev and Novokhatskii, 1984; Radilov, 1984) due to di9erences in the electrical conductivities of these materials, varies with changing liquid levels in the hearth. However, the signal is also in5uenced by other factors and usually exhibits strong drift, which makes it unsuitable as a measurement of the absolute liquid levels inside the hearth. The inner pro7le of the hearth changes dynamically due to erosion of, or growth of buildup material on, the sidewall and bottom. The lining erosion and sculling primarily depend on the 5ow rates of the liquids at the hearth wall and bottom. In model experiments and numerical simulations reported in the literature (Shibata et al., 1990; Preuer et al., 1992), the emergence of local high-velocity 5ow regions has been found to be due to the existence of coke-free zones, e.g. caused by a (partially) 5oating dead man. If a good estimate of the shape, position and state of the dead man is available in addition to an estimated inner pro7le of the hearth, it is
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J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
80 Measured Slag Delay Modelled Slag Delay
Slag Delay (min)
70 60 50 40 30 20 10 200
400
600
800
1000 1200 1400 1600 1800 2000 2200 2400
Time (d) Fig. 1. Observed slag delay (dashed line) and slag delay estimated by the model with a 5oating dead man (solid line) for a 6.5-year period of a blast furnace (Br'annbacka and Sax,en, 2003).
“measurements”, which calls for application of correction terms in the balances. Such models, assuming a stationary dead man, have been presented in the literature (Nightingale et al., 2001; Br'annbacka and Sax,en, 2001). The present paper outlines a systematic method for estimation of the liquid levels in the hearth considering a time-varying inner hearth geometry and a (possibly) 5oating dead man. The model is based on an optimal estimation of the correction of the liquid levels to prevent drift, realized in the form of an extended Kalman 7lter. Applied on process data from an industrial blast furnace, the model illustrates the dramatic e9ect of the 5oating state of the dead man on the tap-cycle evolution of the liquid levels, as well as the in5uence of the hearth geometry. 2. Modeling the liquid levels 2.1. Tap cycle
possible to predict when and where regions with high risk of rapid erosion would occur. The voidage of the dead man is very di>cult to estimate during the operation of the furnace, but some approaches have been presented in the literature, e.g. core drilling of dead-man coke at the tuyere level (Steiler et al., 1991; Negro et al., 2001), residence time studies by tracer experiments (Negro et al., 2001), measurements of the static liquid pressure at the taphole (Desai, 1993; Danloy et al., 1999; Havelange et al., 2004), and interpretation of the time (called the “slag delay” in what follows) of iron-only 5ow in the beginning of each tap (Nightingale and Tanzil, 1997; Torrkulla and Sax,en, 2000). However, there are several drawbacks of these techniques: core drillings do not provide information about the voidage lower down in the hearth; sampling, tracer experiments and liquid pressure measurements are feasible only during short measuring campaigns; tracers, liquid pressure and direct slag delay interpretations are based on an assumption that the dead man be stationary (i.e. not 5oating). The over-all e9ect of a 5oating dead man has been investigated in a recent study (Br'annbacka and Sax,en, 2003), where the relation between liquid volumes and vertical levels was demonstrated to depend mainly on the 5oating state of the dead man. The variations in the slag delay observed in a one-taphole blast furnace were shown to strongly correlate with the 5oating state of the dead man; by estimating the 5oating state from a simpli7ed force balance and using an estimate of the hearth wear pro7le, the long-term evolution of the slag delay could be almost perfectly explained, as indicated in Fig. 1. If short-term information about production and tap rates is available, mass balance equations of iron and slag in the hearth can be applied to estimate the instantaneous liquid volumes, which, in turn, can be transformed into iron and slag levels if the inner hearth geometry and dead man properties are known. The level estimates are, however, corrupted by drifting disturbances caused by noise in the 5ow
Fig. 2 illustrates a typical tap cycle in a (one-taphole) blast furnace, which starts as the previous tap is 7nished, and the taphole is plugged with taphole mud. During the period when the mud is given time to solidify properly, neither iron nor slag are being drained so their volumes and levels in the hearth rise monotonously. When the taphole is opened, iron starts to 5ow out and the iron–slag interface (henceforth called the iron level) stays fairly horizontal under its descent due to the low iron viscosity and the large density di9erence between the two liquids. Slag will not be tapped until the iron level has descended (close) to the taphole, which gives rise to a slag delay. Once slag starts to drain, iron and slag 5ow out concurrently until the end of the tap, and iron will, thus, be “pumped” up from levels below the taphole. This is due to the large pressure gradient at the taphole that is caused by the 5ow of the highly viscous slag (Tanzil et al., 1984). The slag–gas interface (henceforth called the slag level) will eventually decline locally towards the taphole, causing outbursts of gas at the end of the tap even though the average slag level is still well above the taphole. This completes the tap cycle at a stage where the iron level is below and the slag level is above the taphole. 2.2. Model of liquid volumes in the hearth Assuming iron and slag to be immiscible liquids of constant densities, the mass balance equations can be expressed as volume balances of the liquids in the hearth. Therefore, the change in liquid volume from a time t0 to time t can be expressed by a time integral of the volume in- and out5ow rates (V˙ in and V˙ out , respectively). t Vliq (t) = Vliq (t0 ) + (V˙ liq; in − V˙ liq; out ) dt: (1) t0
The problem with this model is that practically all volume 5ow measurements contain noise. When integrating the
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
3425
noisy volume 5ow measurements, the resulting volume estimate contains a drifting noise which, if left uncorrected, causes it to take unrealistic values quite soon. Therefore this model is not useful in itself unless the volume estimates can be corrected to prevent drift.
Level
Fig. 2. Evolution of the iron and slag levels in the hearth during the tap cycle.
2.3. Measurements of in- and out;ows From the standard instrumentation of blast furnaces it is a well-established procedure to estimate the instantaneous production rates of iron (and slag) on the basis of top gas analysis and blast parameters, considering the oxide content of the burden. However, the technique is based on the assumption of quasi-stationary conditions and it is quite sensitive to measurement errors in the top gas analysis. As for the out5ow measurements (Br'annbacka and Sax,en, 2001), the blast furnace used to illustrate the model in Section 3 is equipped with facilities for iron level measurement by radar in the metallurgical vessel, called ladle, where the iron is tapped after the runner. Given the ladle geometry, the level measurement can be transformed into an instantaneous (5-min average) estimate of the iron out5ow rate. Errors arise because of changes in the geometry caused by ladle erosion and/or skulling. The slag 5ow rate, in turn, can be calculated either from mass and heat balances of the water used in the slag granulation unit or from the variations in hydraulic pressure in the bearings of the granulation drum. Both measurements are of quite poor accuracy, but they still provide valuable real-time information about of the out5ow term in the slag mass balance equation.
tss(k−1) tend(k−1)
tss(k)
tend(k)
tss(k+1) tend(k+1)
Time Fig. 3. Schematic illustration of the liquid levels (solid lines) as well as the level “measurement points” for iron (circles) and slag (squares). The taphole levels are indicated by horizontal dotted lines.
to the taphole level at the moment when the slag out5ow starts, tss . In Fig. 3 these measurement points have been indicated by small circles. As for the slag level, it can be estimated at the end of each tap, tend , from the relationship between the residual slag ratio and the 5ow-out coe>cient, FL . This dimensionless variable was proposed by Fukutake and Okabe (1976a,b). Later, Zulli (1991) studied the relationship between FL and the residual slag ratio in small-scale experiments, and extended the expression to include the effect of two liquid phases, i.e., to consider the fact that the lower surface of the upper liquid (slag) can be below the taphole. The 5ow-out coe>cient as de7ned by Zulli (1991) and applied on the blast furnace hearth is given by
1:4
2.4. Measurements of the iron and slag levels
1 sl w (1 − )2 FL = 180 3 ( dp )2 sl g
There are no direct measurements of iron and slag volumes in the hearth, but the liquid levels relative to the taphole level can be estimated at certain time instants of the tap cycle. On the basis of the over-all behavior of the tap cycle outlined in Section 2.1, the iron level is “measured” once every tap cycle (showing a su>ciently long slag delay), since the iron level can be taken to have descended
where is a shape factor, dp is the coke particle diameter, is the liquid viscosity, is the density, w is the mean super7cial velocity, dh is the hearth diameter, z is the vertical level and subscripts “sl” and “th” denote slag and taphole, respectively. From the experimentally determined relationship between FL and the residual slag ratio, the slag level at the end of the tap, denoted by the small squares in Fig. 3,
dh zsl − zth
;
(2)
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J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
can be estimated. This is a prerequisite for “reconstructing” the liquid levels, illustrated schematically by the curves in the 7gure. In the 7gure, k denotes the tap number and subscripts “ss” and “end” refer to the moments of slag out5ow start and tap end respectively.
where d zi and dVj are the di9erentials of the liquid levels and volumes respectively, dVcf is the di9erential of the volume of the coke-free layer and A is the cross-sectional area of the hearth. The relationship dVcf =dVj is determined by a balance between the force pressing down the dead man and the buoyancy force acting on the submerged part of the dead man in the opposite direction (Br'annbacka and Sax,en, 2003). Instead of attempting to describe the downward-acting force in detail (Takahashi et al., 2002), it is here expressed by a parameterized pressure pro7le expression r 6 r0 ; pQ d ; n pd (r) = (4) r − r0 ; r ¿ r0 ; pQ d − a R where pQ d is an over-all term (independent of the radius r), a and n are model parameters and R is a scaling factor (e.g., the
i = j = sl 0.04
0.03
dzi / dVj (m/m3)
The relationship between liquid volumes and levels in the hearth depends strongly on the inner geometry of the hearth and on the properties of the dead man. In this work, an independent model based on thermocouple measurements in the hearth lining (Torrkulla and Sax,en, 2000) is used to estimate the wear/buildup pro7le of the hearth by solving an inverse heat-conduction problem. The model calculates the wear in two dimensions for di9erent sectors of the hearth, and the solution is interpolated to a three-dimensional internal pro7le. Since the changes in the hearth geometry are slow in comparison to the liquid level changes during the tap cycle, the short-term e9ect of a varying hearth geometry is minor. However, the long-term evolution is important mainly through its in5uence on the 5oating state of the dead man. The properties of the dead man that a9ect the volumes-tolevel functions are the dead-man voidage, , 5oating state and bottom shape. In lack of a reliable method for real-time estimation of the voidage, this variable was for the sake of simplicity assumed uniform and constant in the present study. If the dead man 5oats partially or completely, there is a coke-free region below it, the volume of which is a9ected by the total iron and slag volumes in the hearth, because the dead man moves with the liquid levels. The relationship between liquid volumes and levels is then given by d zi dVcf 1 − (1 − ) ; i; j = {ir; sl} and = dVj A dVj 0 if i = ir; j = sl; = (3) 1 otherwise;
0.05
i = { ir, sl }, j = ir
0.02
0.01
0 i = ir , j = sl −0.01
−0.02 60
V cf (m3)
2.5. Volumes–level coupling
0.06
40
20
0 50
60
70
80
90
100
110
120
130
140
pd (kPa) Fig. 4. Upper panel: volume derivatives of the liquid levels at di9erent dead man positions set by adjusting pQ d . Lower panel: volume of the coke-free layer.
radius of the unworn hearth). The dead man bottom shape and the free volume are thus determined by the shape of the hearth bottom and the downward-acting pressure pro7le, where pQ d is a9ected by the burden weight and the lifting force of the ascending gas. Br'annbacka and Sax,en (2003) have demonstrated that this model provides an adequate description of the long-term evolution of the slag delay (cf. Fig. 1). To illustrate the e9ect of the dead-man 5oating state on the volumes–level coupling functions, the derivatives d zi =dVj in Eq. (3) were calculated at di9erent dead man 5oating levels. The hearth geometry used for the illustration is estimated by the hearth erosion model for a speci7c day of the blast furnace presented in Section 3. The dead man bottom pro7le was determined by setting the parameters a = 112 kPa; n = 2; r0 = 2 m and R = 4 m in Eq. (4), using the over-all downward-acting pressure pQ d , which is directly proportional to the vertical position of the dead man, as a variable. The iron level was assumed to be at the taphole, and the slag layer was 1 m thick. The upper panel of Fig. 4 depicts the volume derivatives of the iron and slag levels at di9erent values of pQ d , while the volume of the coke-free layer, Vcf , is depicted in the lower panel. As can
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
be seen, the changes in the volumes–level coupling functions can be quite abrupt if the dead man starts to 5oat. As the dead man 5oating state changes from completely sitting to completely 5oating in the iron bath the volume derivatives of the level functions change between the asymptotic values given by dotted lines in the upper panel of the 7gure. The steepness of this transition depends on the shapes of the hearth and dead man bottoms; in the extreme case where the dead man and the hearth bottom have identical shapes, the transition from the completely sitting dead man state to the completely 5oating state takes place immediately when the buoyancy force exceeds the downward acting force. 2.6. State-space model of liquid volumes and levels in the hearth Since the iron and slag levels can be “measured” once every tap cycle, a discrete state-space model of the liquid volumes with a (variable) sampling time equaling the tap cycle length was formulated, using the moments for iron level measurements (i.e. the time instances when slag out5ow starts) as discretization points. The state-space model is given by Vir (tss (k + 1)) =Vir (tss (k)) +
tss (k+1)
tss (k)
(V˙ ir; in (t) − V˙ ir; out (t)) dt
+e1; ir (k + 1);
(5a)
Vsl (tss (k + 1)) =Vsl (tss (k)) +
tss (k+1)
tss (k)
(V˙ sl; in (t) − V˙ sl; out (t)) dt
+e1; sl (k + 1); zir (tss (k)) = fir (Vir; ss (k) ; Vsl; ss (k); : : : ; ) + e2; ir (k);
(5b) (5c)
zsl (tend (k − 1)) =fsl (Vir; ss (k) − SVir ; Vsl; ss (k) − SVsl ; : : : ; ) +e2; sl (k);
(5d)
where e1 and e2 represent disturbances in the system, and fir (·) and fsl (·), in turn, express the volumes–level coupling functions for the iron and slag levels. These functions include the e9ect of the irregular hearth geometry and the dead man bottom shape and 5oating state and are, therefore, rather complicated (Br'annbacka and Sax,en, 2003). However, the slag level zsl is not measured at the time of slag 5ow start, tss (k), but at the end of the previous tap, tend (k − 1), (cf. Fig. 2). This fact was considered by subtracting the
integrals SVir =
tss (k)
tend (k−1)
SVsl =
tss (k)
tend (k−1)
3427
(V˙ ir; in (t) − V˙ ir; out (t)) dt;
(6a)
(V˙ sl; in (t) − V˙ sl; out (t)) dt
(6b)
from the iron and slag volumes at the start of slag out5ow, Vir; ss (k) and Vsl; ss (k) respectively, before applying the volumes–level coupling function for the slag level. The above equations can be cast in standard discrete state-space form, where the liquid volumes in the hearth are “measured” by the liquid levels V(k + 1) = V(k) + u(k + 1) + e1 (k + 1); z(k) = f(V(k); : : : ; ) + e2 (k);
(7)
where the vectors V(k) = (Vir; ss (k); Vsl; ss (k))T ; z(k) = (zir; ss (k); zsl; end (k − 1))T ; u(k) contains the integrals of the iron and slag in- and out5ows between tss (k − 1) and tss (k) (cf. Eq. (5)a,b), and e1 (k) and e2 (k) contain the process and measurement noises, respectively. 2.7. State estimation with an extended Kalman ?lter If the function f in the state-space model can be linearized at a certain state, and the disturbances e1 and e2 are uncorrelated zero mean Gaussian noise, there is an optimal estimator, the Kalman 7lter (Haykin, 2001), of the state V based on the measurements of z at each time step. The Kalman 7lter is a robust estimator, so even if the above conditions are not fully satis7ed it will still produce fairly good (suboptimal) state estimates. The method requires that the variances of the white noise vectors e1 and e2 be given. The variances of the “process noise”, e1 = (e1; ir ; e1; sl )T , can fairly accurately be estimated by comparing the volume integrals of iron and slag (cf. Eq. (5)) with volumes calculated on the basis of hot metal weighing reports from the steel plant or the mass of charged burden materials in the blast furnace. As for e2 = (e2; ir ; e2; sl )T there is no direct measurement from which its variance could be calculated, so it has to be set on the basis of knowledge of the general behavior of the liquid levels in the hearth. The Kalman 7lter 7rst applies the model to estimate a new state based on the previous state estimate and the model inputs ˆ + 1 | k) = V(k|k) ˆ V(k + u(k + 1):
(8)
Next, the nonlinear function f is linearized at this state (cf. Eq. (3)), and the matrix F is formed as @f(V; : : : ; ) F(k + 1|k) = : (9) ˆ @V V=V(k+1|k) The Kalman gain matrix K is estimated by the recursive procedure P(k + 1|k) = P(k|k) + R2 ;
(10a)
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J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
K(k + 1) = P(k + 1|k)F(k + 1|k)T [F(k + 1|k)P(k + 1|k) ×F(k + 1|k)T + R1 ]−1 ;
(10b)
P(k + 1|k + 1) = [I − K(k + 1) ×F(k + 1|k)]P(k + 1|k);
(10c)
where P is the covariance matrix of the reconstruction error and R1 and R2 are the covariance matrixes of e1 and e2 , respectively. Finally, the optimal state estimate can be calculated by ˆ + 1 | k + 1) = V(k ˆ + 1|k) V(k +K(k + 1)[z(k + 1) ˆ + 1|k); : : : ; )]: −f(V(k
(11)
As the iron and slag volumes at time tss (k + 1) have been estimated, the corresponding levels are calculated through the volumes–level coupling functions (cf. Section 2.5 and Br'annbacka and Sax,en, 2003). The correction of the volumes can, e.g., be smeared over the short-term volume estimates between tss (k) and tss (k + 1) to produce smooth transitions of the estimated levels without discontinuities at the correction points. 2.8. Computational algorithm The computational procedure for estimating and correcting the liquid volumes and levels is summarized in the following algorithm: 1. Calculate the inner hearth geometry with the erosion model. 2. Estimate Vir; end (k) and Vsl; end (k) by integrating the measurements of the volume in- and out5ows between tss (k) and tend (k) (cf. Fig. 3). 3. Apply the volumes–level coupling function for the slag level to estimate zsl; end (k). 4. Apply the volume derivatives to the volumes–level coupling function for the slag level at this state to form the second row of the matrix F(k + 1|k). 5. Use the relationship between the 5ow-out coe>cient, Eq. (2), and the residual slag ratio to give a “measurement” of zsl; end (k). 6. Estimate Vir; ss (k + 1) and Vsl; ss (k + 1) by integrating the measurements of the volume in- and out5ows between tend (k) and tss (k + 1) (cf. Eqs. (6)). 7. Apply the volumes–level coupling function for the iron level to estimate zir; ss (k + 1). 8. Apply the volume derivatives to the volumes–level coupling function for the iron level at this state to form the 7rst row of the matrix F(k + 1|k). 9. “Measure” zir; ss (k +1) by assuming it to be at the taphole level, zth (k + 1). 10. Update the Kalman gain and use Eq. (11) to correct Vir; ss (k + 1) and Vsl; ss (k + 1) by considering the
di9erences between the estimated and the measured iron and slag levels. 11. Smear the volume corrections over the iron and slag volumes estimated on short-term basis between tss (k) and tss (k +1), and calculate the corrected short-term iron and slag levels. 12. Set k = k + 1. If new daily averages of the thermocouple measurements in the hearth lining are available go to 1, otherwise go to 2.
3. Results and discussion The model of the liquid levels has been evaluated on process measurements from a Finnish one-taphole blast furnace. In what follows, the results from a 3-month period will be used to illustrate the results. The data period is divided into two 40-day segments separated by a 2-week period in the middle (where information was lost because of a data backup problem). The 7rst segment of the period was characterized by a worn furnace hearth, and, hence, a quite well 5oating dead man. However, during certain parts of the period, particularly at its end, buildup material was formed on the hearth bottom, resulting in a reduced inner volume of the hearth and a lesser degree of dead man 5oating. 3.1. Noise covariances In the Kalman state estimator the only parameters not given by the model are the covariance matrixes R1 and R2 of the noise vectors e1 and e2 , and the optimal Kalman gain matrix K is estimated on the basis of this information. Studying the components of e1 , the largest noise source is clearly the out5ow measurements for both iron and slag. An examination of the tapped iron measurement was made by comparing it with the control weighing of each ladle at the steel plant (Voutilainen, 1999), and the estimated standard deviation for each ladle was 4:2 ton. In terms of iron volume of a tap, assuming each tap to 7ll three ladles, this yields a standard deviation of 1:8 m3 . The in5ow measurement is based on an oxygen balance over the furnace for the gas phase, and its most likely error is caused by drift in the top gas analysis, but these measurements are calibrated regularly. Considering possible minor stochastic disturbances in the in5ow measurement, the total standard deviation was estimated to 1; ir = 2:0 m3 . For the corresponding slag variable, e1; sl , no such calibration study has been conducted. Since the slag out5ow measurement must be considered slightly more uncertain than the iron out5ow measurement, and the tapped slag volume is about 60% of the iron volume, the standard deviation was assumed to be the same as for iron, i.e., 1; sl = 2:0 m3 . The interdependence between these two noise variables is quite weak, since only the production rate estimates are based on common factors. In lack of more accurate information
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
3429
9.5
8.5
70 50 7.5
7.5
9.5
9
30
9 70 8.5 50
Emf (%)
8
9.5
Level (m)
9 8.5 8
Level (m)
Level (m)
9
Emf (%)
Level (m)
9.5
8 30 7.5 07:12
8.5
09:36
12:00
14:24
16:48
19:12
21:36
00:00
Time (hour:minute) 8
7.5 06:00
07:12
08:24
09:36
10:48
12:00
Time (hour:minute) Fig. 5. Upper panel: iron and slag levels estimated assuming a standard deviation of 5 cm in the measurement noise. Lower panel: corresponding levels estimated assuming a standard deviation of 10 cm in the measurement noise. Dotted lines depict the taphole level.
the noise variables were therefore considered uncorrelated, yielding a diagonal R1 . The components of the measurement noise vector, e2 , cannot be measured but have still to be estimated a priori. Knowledge of the process would suggest standard deviations in the level measurements of about 5 cm. Using this in R2 , and assuming the disturbances to be uncorrelated, level estimates have been depicted for a short period of about three tap cycles in the upper panel of Fig. 5 (under the conditions described in the next subsection). The lower panel of the 7gure shows the corresponding liquid levels estimated with a standard deviation of 10 cm in the level measurement noise. The former noise level is clearly a more realistic estimate; a standard deviation of 10 cm was found to yield liquid level estimates that occasionally drifted strongly and exhibited unrealistic behavior, such as a slag level descending below the iron level or below the taphole, or an iron level well above the taphole during several consecutive tap cycles. Therefore, in what follows R1 = diag(4; 4) m6 and R2 = diag(25; 25) cm2 are used. 3.2. Dead man ;oating state The e9ect of dead man 5oating on the liquid levels is illustrated in Fig. 6, where the upper panel shows the evolutions of the slag and iron levels during 10 consecutive tap cycles for the case with a sitting dead man; this state was forced by assigning pQ d a high value (e.g., 300 kPa). The lower panel, in turn, illustrates the corresponding results for the case where the value of pQ d was calculated from the process data under the assumptions given in Br'annbacka and
Fig. 6. Estimated slag (upper solid curves) and iron (lower solid curves) levels in the BF hearth for a sitting (upper panel) and partially 5oating (lower panel) dead man. Scaled emf signals (dashed curves) have also been depicted, as well as the taphole level (dotted lines).
Sax,en (2003), yielding pQ d ≈ 102 kPa and a partially 5oating dead man. The amplitude of the liquid levels is clearly larger for the sitting dead man case. In terms of coincidence between the levels and the measured emf signal, the iron levels for both the sitting and partially 5oating dead man states show strong correlation. On the basis this observation and a comparison between the estimated iron levels and the emf for longer time periods, it seems di>cult to utilize the emf for detection of the 5oating state of the dead man. As for the veri7cation of the estimated amplitude of the liquid levels, no other direct measurement is available. However, an asymptotic maximum for the vertical distance that the iron level can descend below the taphole at the end of the tap is given by a simple pressure balance of the two-liquid 5ow system at hand (Tanzil et al., 1984). This maximum, expressed as Szir; end = zth − zir; end , depends on the height of the slag level at the end of the tap, Szsl; end = zsl; end − zth , which is a “measured” variable, according to sl Szir; end = Szsl; end (12) ir − sl and applies, strictly speaking, to the case where no iron 5ows out at the end of the tap. The upper panel of Fig. 7 shows the estimated Szir; end (solid line) and the asymptotic maximum value (dashed line) for the case with a sitting dead man, while the lower panel illustrates the corresponding curves for a partially 5oating dead man. In the former case, Szir; end is very close to its asymptotic maximum and clearly exceeds it during fairly long time periods. At least during the periods July 21–25 and August 6–25 the end iron levels assume completely unrealistic values. In the case where the dead man 5oating state was estimated from process data, Szir; end stays below the asymptotic limit throughout the data period, and it must therefore be considered a more plausible state of the furnace hearth. If the partially 5oating dead man model describes the actual volume–level relationships better than the sitting dead
3430
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
9
0.6
Level (m)
∆zir,end (m)
3.
2.
1.
Estimated Asymptotic maximum
0.8
0.4 0.2
8.5 8
0 22:04 Estimated Asymptotic maximum
22:33
23:02
23:31
00:00
Time (hour:minute)
0.6
8
0.4
Level (m)
∆zir,end (m)
0.8
0.2 0 07/21
08/01
08/13
08/25
09/06
09/17
09/29
10/11
7.5 7
6 5.5
Time (month/day) Fig. 7. Estimated Szir; end (solid line) and its asymptotic maximum (dashed line) for a sitting dead man (upper panel) and a partially 5oating dead man (lower panel).
Table 1 Standard deviations of the volume corrections required by the state estimation model for iron and slag in the hearth Dead man state
Iron correction (m3 )
Slag correction (m3 )
Sitting Partially 5oating
3.23 2.80
4.33 3.92
man model, one would expect the corrections of the volume, ˆ + 1|k); : : : ; )] in Eq. (11), i.e., K(k + 1)[z(k + 1) − f(V(k to be smaller for this case. The standard deviations of the corrections of the iron and slag volumes for the sitting and partially 5oating dead man case, reported in Table 1, show that the necessary volume corrections to keep the liquid levels within reasonable bounds are smaller for the partially 5oating dead man case. None of the investigated indicators de7nitely prove that the dead man was 5oating during the studied period, but the studies of the asymptotic iron end level and the volume corrections given by the Kalman 7lter indicate that a partially 5oating state is more likely than a completely sitting state. Therefore the partially 5oating state or, more correctly, the 5oating state given by the estimated value of pQ d and the inner hearth pro7le, is considered a superior model for describing the volume–level coupling also on a short-term basis. 3.3. The coke-free layer An interesting “by product” of the state estimation model in combination with the model of the dead man 5oating state is the possibility to monitor the variations in size and shape of the coke-free layer during the tap cycles. Fig. 8 illustrates the liquid levels and lower surface of the dead man (dashed lines) for three di9erent time instants during one tap cycle. The elephant-foot erosion pro7le of the hearth lining is seen to gives rise to a circular coke-free region mainly at the
1. 2. 3.
6.5
4
3
2
1
0
1
2
3
4
Radius (m)
Fig. 8. Evolution of the iron and slag levels (upper panel) and the dead man bottom (lower panel) for a tap cycle during a period with a relatively eroded hearth.
1.
2.
3.
Fig. 9. Three-dimensional illustration of the coke-free regions in the cases 1–3 in Fig. 8. The black horizontal bar marks the location of the taphole.
hearth edges. Fig. 9 presents three-dimensional views of the coke-free regions for the three time instants of Fig. 8. The evolution of the minimum and maximum volume of the coke-free layer during each tap cycle has been illustrated in the upper panel of Fig. 10 for the 7rst 40-day data period of Fig. 7. Since the movement of the dead man a9ects the
Coke Free Volume (m3)
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432 40 Maximum Coke Free Volume Minimum Coke Free Volume b.
30 a. 20 10 0 13/07
18/07
23/07
28/07
02/08
07/08
12/08
17/08
Time (day/month)
1.2 10
1.1 1
5 0.9 0
09:36
12:00
14:24
Time (hour:minute)
16:48
0.8
Coke Free Volume (m3)
15
Gas Pressure Drop (bar)
Coke Free Volume (m3)
1.3
20 1.5 1.4
15
1.3 10 1.2 1.1
5 12:00
14:24
16:48
1 19:12
Gas Pressure Drop (bar)
Period b.
Period a.
Time (hour:minute)
Fig. 10. Upper panel: minimum and maximum volume of the coke-free layer. Lower panels: evolution of the volume of the coke-free layer during two short periods, a and b, and the gas pressure drop.
conditions in the upper part of the furnace (Nishio et al., 1977; Torrkulla et al., 2002), the changes in the volume of the coke-free layer may be correlated with short-term changes in other process variables. Torrkulla et al. (2002) presented the hypothesis that changes experienced in the gas pressure drop over the furnace during the tap cycle would re5ect the 5oating degree of the dead man. Focusing on two shorter periods, a and b indicated in the upper panel of Fig. 10, the estimated free volume (solid curves) and the gas pressure drop (dashed curves) for four consecutive tap cycles have been depicted in the lower panels of the 7gure. During period a, when the dead man barely 5oats, practically no correlation between the pressure drop and the free volume is evincible, while period b is characterized by a strong dependence between the pressure drop and the dead-man motion. It is not clear why the correlation arises, but possible reasons are that the upward movement of the dead man causes deformation of the raceways (e.g., elevation of the bird’s nest) and the coke slits in the cohesive zone, which a9ects the gas permeability of the bed.
3431
state of the hearth in the liquid level estimation problem. In particular, the 5oating state of the dead man has been demonstrated to be decisive for the results. The 7ndings of the model have been analyzed by comparing the estimated liquid levels with asymptotic limits given by the over-all pressure distribution in the system in question as well as by studying how the required 7ltering of the volume balance equations of the hearth region is a9ected by the hearth state. From this analysis it can be concluded that the dead man of the blast furnace studied in this work is likely to be in a partially 5oating state. This is also supported by the fact that the estimated evolution of the volume of the coke-free layer below the dead man shows correlation with the gas pressure drop over the furnace. The model is applicable on any blast furnace where an estimate of the hearth geometry and instantaneous measurements of the production and tap rates are available, and where there (generally) is a distinct period of iron-only 5ow in the beginning of the tap. In the forthcoming work, the level estimation model will be applied online to aid tapping control. The possibility to use the model to predict the end of the tap is especially interesting, since this would enable the operators to shut the taphole immediately when the gas bursts out, or even minutes before (Nightingale et al., 2001). This would aid tap planning, and considerably reduce splashing in the tap hall and erosion of the taphole. Furthermore, the model will be extended to include hysteresis in the dead man motion, which has been observed in recent cold model studies (Shinotake et al., 2003) as well as in operating blast furnaces (Havelange et al., 2004). The results of the level estimation model will be used to evaluate the quality of the emf signal, and the possibility to use it combined with the present model to more accurately estimate the iron level will be studied (Sax,en and Br'annbacka, 2004). The possibility to utilize knowledge of the dynamic behavior of the coke-free region below the dead man in CFD simulations will also be investigated in future research. This could shed welcome light on the mechanisms behind abrupt severe lining erosion that is occasionally observed in the operation of the blast furnace.
Notation 4. Conclusions
Romans
A model for estimating iron and slag levels in the blast furnace hearth has been developed on the basis of measurements of tapped quantities as well as on a description of the internal state of the hearth. The latter includes a hearth geometry estimate by a model of the hearth refractory wear and a force-balance model estimating the 5oating state of the hearth coke (dead man). The liquid level estimation problem is written in state space form and is solved by an extended Kalman 7lter. The model has been applied on measurements from an industrial blast furnace, and the results have illustrated the importance of considering the internal
A a d e f(·) FL g k n p
hearth cross-sectional area, m2 parameter in Eq. (4), kPa diameter, m disturbance to volume estimate or level measurement in Eq. (5), m3 or m volumes–level coupling function, m 5ow-out coe>cient, dimensionless gravitational acceleration, m=s2 tap number, dimensionless parameter in Eq. (4), dimensionless pressure, kPa
3432
r R t V w z
J. Br'annbacka, H. Sax+en / Chemical Engineering Science 59 (2004) 3423 – 3432
radius, m parameter in Eq. (4), m time, d; h; min volume, m3 super7cial velocity, m/s vertical level, m
Greek letters
parameter in Eq. (3), dimensionless dead man voidage, dimensionless viscosity, Pa s density, kg=m3 estimated standard deviation, m3 or m particle shape factor, dimensionless
Vectors and matrixes e F f(·) K P R u V z
disturbances to estimated iron and slag volumes or level measurements linearization matrix for the function f(·) levels measuring function Kalman gain covariance matrix of the reconstruction error covariance matrix of e measured changes in iron and slag volumes iron and slag volumes iron and slag levels
Subscripts 0 cf d end h i in ir j liq out p sl ss th
initial or reference state coke-free zone downward-acting tap end hearth iron or slag in5ow iron iron or slag liquid (iron and/or slag) out5ow coke particle slag slag5ow start taphole
References Br'annbacka, J., Sax,en, H., 2001. Modeling the liquid levels in the blast furnace hearth. ISIJ International 41, 1131–1138. Br'annbacka, J., Sax,en, H., 2003. Model Analysis of the operation of the blast furnace hearth with a sitting and 5oating dead man. ISIJ International 43, 1519–1527.
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