Model for Fast Computation of Blast Furnace Hearth Erosion and

Ind. Eng. Chem. Res. 2008, 47, 7793–7801

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Model for Fast Computation of Blast Furnace Hearth Erosion and Buildup Profiles Johnny Bra¨nnbacka† and Henrik Saxeń* Heat Engineering Laboratory, Åbo Akademi UniVersity, Biskopsgatan 8, FIN-20500 Åbo, Finland

A model for estimation of the profiles of erosion and buildup material in the hearth of an ironmaking blast furnace has been developed. The model is based on thermocouple readings in the hearth wall and bottom and solves an inverse heat transfer problem for two-dimensional slices of the hearth geometry to estimate the inner profile. Special attention has been paid to the mathematical formulation of the problem at hand, yielding a general model optimized for fast computation. This includes a flexible formulation of the boundary conditions, a generic setup of the lining materials applied in the hearth refractory, and a sophisticated iterative procedure in the estimation of the location of the internal profile. These steps have led to a model that facilitates process analysis with estimation and reestimation of the furnace hearth conditions over whole campaigns using different parameter settings. The model has been applied to study the evolution of the hearth erosion and buildup formation processes in several industrial furnaces. 1. Introduction The hearth is a central part of the ironmaking blast furnace, because the operation of it significantly affects the performance of the whole furnace and also plays a key role in determining the hot metal quality. Furthermore, the erosion of the hearth lining is usually the main factor limiting the furnace campaign length. During the normal operation of the furnace, no measurements of the remaining lining thickness are available, but for managing the blast furnace this information would be very useful for strategic operation planning, for taking precautionary measures, and for scheduling stoppages, intermediate repairs, and full relining of the hearth. If the hearth relining is scheduled too late, there is a risk of an outbreak with potentially catastrophic consequences, while, on the other hand, a too-early relining of the hearth causes unnecessary loss of production and capital. To monitor the hearth lining conditions, modern blast furnaces have a large number of thermocouples embedded in the lining materials, and these temperature measurements are used to estimate the remaining thickness of the lining. However, the interpretation of the thermocouple readings in terms of remaining lining thickness is a nontrivial inverse heat transfer problem.1,2 Therefore, mathematical models tackling this problem2-8 have been developed, and are used on a daily basis at some steelworks. In its simplest form, a hearth erosion model utilizes only one pair of thermocouples at a time, and assumes the heat flux to be static and one-dimensional along the line of the thermocouples studied. All existing models are based on simplifying assumptions to make the problem manageable and possible to solve, and the accuracy of the estimated hearth profile depends both on the model formulation and on the quality and quantity of the temperature measurements available. Because of the geometric layout of the furnace hearth lining system, the heat transfer problem should be solved in at least two dimensions in order to obtain a reasonable estimate of the inner profile of the hearth. Several two-dimensional hearth * To whom correspondence should be addressed. Tel.: +358 2 215 4442. Fax: +358 2 215 4792. E-mail: [email protected]. † Present address: ArcelorMittal Global R & D East Chicago, 3001 E. Columbus Dr., East Chicago, IN 46312. E-mail: Johnny.Brannbacka@ mittalsteel.com.

erosion models have been presented in the literature. Quy and Cripps Clark3 were among the first to develop a two-dimensional model of the hearth and applied it to estimate the thickness of “good refractory” in different situations on the basis of thermocouple signals. Schulte et al.4 studied the progress of erosion in a blast furnace on the basis of information from sensors measuring the heat flux in the lining, and with these the investigators were able to reduce the inherent inaccuracies in solving inverse heat conduction problems by information from thermocouple pairs. Torrkulla and Saxeń5 developed a twodimensional model of the state of the blast furnace hearth by solving an inverse heat conduction problem with the finite element method (FEM), using a regularization of the solution to yield a smooth interpretation of the internal profile. The erosion and buildup formation were studied during several years of operation of two furnaces. More recently, Kumar6 also applied FEM to estimate refractory wear of the blast furnace hearth, including the coke bed and liquid iron in the computational domain, also considering the case with a coke-free layer (i.e., a partially floating dead man). By this procedure Kumar was able study the maximum hearth wear for different hearth designs. Kumar also compared the model’s findings with a set of thermocouple measurements from a blast furnace and drew conclusions about the likely internal state of the hearth. Gonzalez and Goldschmit7 used an inverse heat transfer formulation with a set of direction vectors ending in surface points, on the basis of which radial basis functions were applied to approximate the unknown hot face. The investigators used a regularized Gauss-Newton method for solving the problem, and illustrated the model’s performance on a set of temperature measurements with different degrees of noise and under different regularization schemes. Zhao et al.,8 using an approach similar to that of Kumar,7 formulated a two-dimensional finite-difference model of the furnace hearth, including the liquid metal and its possible phase change in the computational domain and evaluated different hearth designs, including all-carbon lining and ceramic cup configurations and their impact on the arising erosion profiles. In summary, a number of different models have been developed to throw light on the complex blast furnace hearth wear, either by off-line (“what-if”) analysis or by estimation on the basis of thermocouple measurements in the lining.

10.1021/ie800384q CCC: $40.75  2008 American Chemical Society Published on Web 09/10/2008

7794 Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008

0)

Figure 1. Schematic of the computational domain of the furnace hearth lining.

The erosion model described in this paper also applies a twodimensional static problem formulation, with a three-dimensional profile of the hearth being interpolated from several twodimensional profiles. However, the present model has a number of special features by which it differs from earlier ones. First, the model has been given a flexible structure, so it can fairly easily be set up for most blast furnaces (with a sufficient set of thermocouples in the hearth lining). This is facilitated by a special treatment of the heat conduction problem in the computational domain as well as an interpolation scheme for the reference or boundary temperatures. Second, the model formulation is optimized for fast computation by paying due attention to the numerical formulation of the estimation problem at hand. This includes a special treatment of the numerical discretization with its implication for computational speed and robustness of the solution procedure, a well-balanced parametrization of the problem, and a tailor-made formulation of the boundary to be estimated. A consequence of this is that the hearth wear during whole furnace campaigns, spanning several thousands of days, can be estimated in reasonable time. This makes it feasible to carry out detailed studies of the effect of different assumptions concerning, e.g., lining and buildup material properties, on the estimated hearth wear. The model has been evaluated on measurements from several blast furnaces. Its results have been indirectly verified by correlating them with independent hearth measurements, such as variables characterizing the tapping operation of the furnace,9,10 and in some cases it has been possible to compare the estimated hearth wear with the actual remaining lining at the end of a campaign. These results have demonstrated that the erosion model has been able to provide a reasonable picture of the progress of the wear of the furnace lining.

∂T ∂ ∂T 1 ∂ kr + k r ∂r ∂r ∂z ∂z

(

)

( )

(1)

where k is the thermal conductivity of the lining material. To fully mathematically describe the temperature field, the geometry and the boundary conditions must also be specified. The twodimensional geometry modeled is the cross section of the hearth at a specific angle relative to a reference point, e.g., a taphole. This problem is solved at K angles, where each cut is taken to represent a sector with the extent of 2π/K in the angular direction. The inner boundary of the geometry is determined by the central axis of the hearth and by the shape of the inner profile of the hearth, while the outer, upper, and lower boundaries are placed at positions where the boundary conditions (temperature or heat flux) can be measured or estimated. It should be noted that the assumption of steady state naturally introduces some errors in situations, where, e.g., the lining erodes rapidly and is in a transient thermal state. However, robustness and rapid execution were considered more important aspects for the model. 2.1.2. Thermal Conductivity and Boundary Conditions. The thermal conductivity, k, is not constant in the domain since different materials are used in different parts of the hearth lining. Therefore the position and thermal conductivity of each type of lining material must be specified. Furthermore, at the inner wall and bottom of the hearth a buildup material (“skull”) layer of varying thickness and composition is formed due to solidification of liquid iron as a result of heat being extracted through the hearth lining. The thermal conductivity of the skull should also be estimated in order to properly model the heat transfer since the layer is included in the calculation domain. Figure 2 illustrates an example of a possible calculation domain with the internal boundaries between different materials indicated by dashed lines; the skull is depicted by a and the different lining materials by b-d. The heat fluxes through the central axis and the upper boundary, 1 and 2, are assumed to be 0, and the temperature on the bottom boundary 3 is interpolated from thermocouples placed at this level (depicted by circles). The temperatures at the outer wall 4 are extrapolated from thermocouple pairs located near the outer surface of the wall. Alternatively, the outer boundary could be placed at the radius of the outer thermocouples in the thermocouple pairs, but including the full

2. Modeling 2.1. The Two-Dimensional Heat Transfer Problem. 2.1.1. Basic Equations. Figure 1 schematically illustrates the computational domain of the problem at hand, including the sidewall region below the taphole(s) as well as the hearth bottom refractory. The heat transfer in the lining is described by the equation for heat conduction in a solid material expressed in cylindrical coordinates. The heat transfer problem is locally approximated by a two-dimensional formulation, since the heat transfer in the angular (θ) direction can be assumed to be negligible compared to the heat transfer in the radial and axial directions. Due to the comparatively slow changes in the hearth state of the blast furnace, the steady state case is considered here. The temperature T in the hearth lining is described by a partial differential equation in the two spatial coordinates r and z:

Figure 2. Typical two-dimensional calculation domain with external (solid lines) and internal (dashed lines) boundaries and thermocouples (circles). The boundary conditions are referred to by 1-4, the skull by a, and lining materials by b-d.

Ind. Eng. Chem. Res., Vol. 47, No. 20, 2008 7795

Figure 5. Schematic of problem transformation with heat flow along the erosion line (dashed line), between the inner (thin line) and outer (thick line) boundaries of the skull layer.

Figure 3. Use of neighbor thermocouples in the calculation of integrated average temperatures in a sector.

Figure 6. Lining material setup window in the GUI.

Figure 4. Example of erosion lines (dashed lines) and points on these defining the inner profile (open circles), as well as computational mesh where solid circles denote the locations of the thermocouples used as reference temperatures in the lining.

wall thickness in the modeled geometry gives a better illustration of the actual remaining lining thickness when the results are depicted. Within each angle sector, there might be none, one, or several thermocouples representing the inner (“reference”) or outer (“boundary”) temperature positions needed for the calculation, depending on the locations of the thermocouples in the lining and the chosen number of sectors to calculate. The temperature reading needed for each position is calculated as an integrated average of all thermocouples at that position (e.g., wall, inner, vertical level 7.1 m) within the sector in question. To handle cases where there are no thermocouples for a given radial position within the sector, the nearest neighbor thermocouples outside the sector are used by interpolating one-dimensionally with respect to the angle, θ, the temperatures at the sector boundaries, and including these in the integrated average calculation for each radial position (cf. Figure 3). A trapezoidal numerical integration was applied. This formulation makes it very flexible to adapt the erosion model to different blast furnaces.

The temperature at the inner hearth boundary, Tb, is assumed to be given. In blast furnace hearth erosion models it is common to use Tb ) 1150 °C as the temperature of this boundary, which is the eutectic temperature of carbon-saturated iron, i.e., the lowest temperature at which iron can exist in liquid form. To solve the heat transfer problem, a mesh with triangular cells is created (cf. Figure 4), and a finite-element solver is used to calculate the temperature field in the discretized domain. In order to make it possible to allow for arbitrary shapes and penetration depths of the inner profile, the computational domain was treated as an entirety, but cells (triangles) in the subdomains were given different heat conductivities, in accordance with their location. In the present model formulation, the thermal conductivities were assumed time-invariant, even though it is known that they may change with time due to, e.g., iron penetration or formation of brittle layers. In principle, it would be possible to allow for time-variant thermal properties in the model, but this option was not implemented in the absence of a reliable means to quantify such changes. 2.2. The Inverse Problem. 2.2.1. Background. The calculation domain is chosen so that there are a number of thermocouples within the domain in addition to those used to specify the boundary conditions, so the calculated temperature field can be compared to measurements at specific points in the domain, called reference temperatures. The inverse heat transfer problem is then defined as finding the position and shape of the hearth inner profile that yields the closest match between calculated and measured reference temperatures. The general formulation of the problem is very complex and has large degrees of freedom. To optimize the model for fast computation and to yield a well-conditioned problem formulation, a number


Figure 7. Thermocouple location and database pointer setup window in the GUI.

Figure 8. Volume inside the skull line (lower curve) and inside the remaining lining (upper curve) during roughly 10 years of operation of BF A, with a midcampaign repair in 2002. The arrow indicates a period with a spike studied in more detail in Figure 9.

of measures have been taken, and these will be described in the subsections that follow. 2.2.2. Erosion Lines. In order to solve the inverse heat transfer problem properly, it is essential to select a suitable parametrization of the shape of the inner hearth profile. It is desirable to keep the number of parameters used for describing the inner profile to a minimum to facilitate the optimization, and also to make the solution unique. Therefore, a curve defining the hearth profile is parametrized by a number of points that

are allowed to move only along straight lines, here called erosion lines, analogous to the direction vectors used by Gonzalez and Goldschmit.7 A curve through these points (open circles in Figure 4) is expressed by a spline interpolation scheme to form a smooth inner profile. The number of erosion lines should not exceed the number of reference temperature measurements (solid circles in Figure 4) used to fit the temperature distribution to avoid overparametrization of the problem. In the furnaces modeled thus far, five to seven internal reference temperature


Figure 10. Observed hearth wear in BF A and the big scab found at the midcampaign repair in 2002. The wear (dashed line) and the buildup profiles (solid line) estimated by the model are also indicated.

Figure 9. Hearth inner profile during the three initial weeks of a “spike” in the hearth volume (indicated by an arrow in Figure 8). The colors express the physical extent of the inner profile, ranging from blue (skulled) to red (eroded).

measurements have been available, and therefore typically five erosion lines have been used per slice, as indicated in Figure 4. The two outermost erosion lines should be placed on the boundary of the computational domain (i.e., along the furnace central axis and horizontally at the highest vertical position of the wall) while the remaining erosion lines can be placed rather uniformly between these. Naturally, the locations of the erosion lines have some effect on the results, but the most important aspect (for the numerical conditioning of the problem) is to not use too many lines. 2.2.3. Problem Transformation. To solve the inverse heat transfer problem, an optimization routine is used to find the shape of the inner profile that minimizes the differences between the measured and calculated temperatures in the internal reference points. In a straightforward implementation, the shape of the calculation domain would be altered in every function execution, requiring a new computational mesh to be created, but this procedure would have two major drawbacks. First, the mesh generation accounts for a significant part of the total computation time, so this approach would severely impair computational speed. Second, and more important, the recreation of the mesh would induce numerical noise in the solution due to the changing positions of the mesh nodes. Even though the discrepancy between two solutions with different meshes is small, it can still disturb the numerical estimates of the gradient of the objective function, which, in turn, would be detrimental to the progress of the search. To overcome these problems, a transformation of the inner profile representation is applied. Since the inner lining is covered by a skull layer, the task of optimizing the inner hearth profile can be transformed into the problem of optimizing the skull

layer thickness, ∆ssk, along the erosion lines (see Figure 5). If the heat flux, q, through the skull layer is locally parallel to the erosion line over small distances, it can be approximated by q)

ksk (T - Tsl) ∆ssk b

(2)

where ksk is the skull layer thermal conductivity, Tb is the temperature at the inner boundary, and Tsl is the temperature at the interface between the skull layer and the lining. Changing the skull layer thickness thus directly affects the heat flux through the skull layer at this point (and/or the temperature Tsl). However, the same local effect can be accomplished by instead changing the skull layer thermal conductivity, ksk, using a fixed skull layer thickness, ∆ssk,0 (e.g., that of the previous solution). This approach has been applied to overcome the drawbacks of directly altering the geometry during the optimization of the inner hearth profile. The variables used in the optimization are, thus, the heat conductivities in the skull layer at the position of the erosion lines. Between the erosion lines, the thermal conductivity of the skull is interpolated from the neighbor values. After obtaining the optimal heat conductivities, ksk*, the results are transformed back to skull layer thickness by ∆ssk )

ksk,0 ∆s ksk* sk,0

(3)

where ksk,0 is the “original” thermal conductivity of the skull material. A slight drawback of this approach is that the transform does not yield the exact solution in terms of the optimized geometry if the heat flux is not parallel to the erosion lines or if there is a significant radial component to the change in thickness. This is still not considered a major problem, since the main purpose of the model is not to find the exact inner profile (which is impossible considering the limited information available) but only to find a reasonably good estimate of it. 2.2.4. Detection of Lining Erosion. The model captures the time evolution of the hearth erosion and buildup formation


Figure 11. Volume inside the skull line (lower curve) and inside the remaining lining (upper curve) during roughly 10 years of operation of BF B, with a midcampaign repair in 2004.

Figure 12. Observed hearth wear in BF B at the midcampaign repair in 2004. The wear (dashed line) and the buildup profiles (solid line) estimated by the model are also indicated.

processes by starting the calculation of the erosion profile from the state prevailing in the beginning of the campaign, and then repeating the calculation as new thermocouple information

enters. Due to the comparatively slow response of the furnace lining to changes, it is usually appropriate to use daily mean values of the lining temperatures as the source of information.


Figure 13. Volume inside the skull line (lower curve) and inside the remaining lining (upper curve) during almost 10 years of operation of BF C.

A common way in erosion models is to assume that the lining has eroded if the estimated inner profile crosses the current estimate of the remaining lining profile, i.e., that there is no skull. However, since the inner profile is here estimated by changing the skull layer thermal conductivity, another procedure must be applied. For this, the basic transformation procedure outlined in subsection 2.2.3 was modified as follows: An “erosion reserve”, ∆ser, was added to the skull layer before optimizing the inner geometry. Thus, before determining the optimal heat conductivities, the internal boundary between the skull layer and the lining was moved slightly “outward”, e.g., a few centimeters into the intact lining. To preserve the local one-dimensional heat flux along the erosion line, the relationship between the thermal conductivities and thicknesses of the composite layer and the skull and “erosion reserve” layers is given by ∆sco ∆ssk ∆ser ) + kco ksk ker

(4)

where subscript “co” refers to the composite layer and “er” to the erosion reserve zone of the lining material, and the thickness of the composite layer is ∆sco ) ∆ssk + ∆ser. The task is now to optimize the heat conductivity of the composite layer. When kco* has been found, the new thicknesses of the skull and erosion reserve layers are solved for by

{( ( { 0,

∆ssk )

ksk,0

∆sco,0 ∆ser,0 e kco* ker ∆sco,0 ∆ser,0 , if kco* ker if

ker

∆ser )

)

)

∆sco,0 , kco*

∆ser,0,

∆sco,0 ∆ser,0 > kco* ker

∆sco,0 ∆ser,0 e kco* ker ∆sco,0 ∆ser,0 > kco* ker

(5a)

if

if

(5b)

Thus, if ∆ser < ∆ser,0, the lining has eroded and there is no protective skull layer left.

2.2.5. Objective Function. The mathematical problem to determine the inner profile of the furnace hearth is realized by minimizing the deviations between the measured and simulated lining temperatures at certain reference points (solid circles in Figure 4) with respect to the skull heat conductivities, ksk. However, in solving inverse problems it is often necessary to apply regularization, which makes the solution well-defined and, furthermore, can help avoid completely irregular (and physically unlikely) solutions. A set of constraints on the profiles was therefore imposed in the points where the erosion lines end. Thus, the objective function to be minimized is expressed as

{

min E ) kco

M

∑ (T

N

m s 2 i -T i +γ

i)1

)

∑ (R - R˜ )

}

2

j

j)1

j

(6)

where superscript “m” refers to the measured and “s” to the simulated temperatures (at the M reference points), while the second term on the right-hand side is a regularization on the basis of the angles, Rj, between the straight lines connecting the N erosion line points. The second term is scaled by a weighing factor, γ, and expresses a deviation from a set of given “target” angles, R˜ j, as a quadratic penalty. A good philosophy of setting the target angles is to use the angles of the parametrization of the original uneroded inner profile. In order to yield solutions that are less sensitive to erroneous reference temperatures (that are occasionally encountered in the databases of any blast furnace), it is reasonable to assign a maximum rate of change of the skull layer thickness, (d∆ssk/dt)max, as well as max an upper limit for the skull layer thickness, ∆ssk . 2.2.6. Validity Checks. In order to avoid erroneous interpretation of the furnace hearth state caused by thermocouple malfunction, a set of validity checks was introduced. First, upper and lower limits of the thermocouple readings were imposed to eliminate totally irrelevant measurements. If a thermocouple temperature falls outside the feasible range, it is discarded. Second, for thermocouple pairs a condition was imposed that the “outer” thermocouple should always take on a lower temperature in order to avoid cases with a heat flow from the shell toward the center of the furnace. In these cases the signals from both thermocouples are discarded. Finally,


Figure 14. Estimated hearth wear in BF C in June 2001 along the second (middle) taphole line.

it is possible to permanently, or during certain time periods, eliminate information from a thermocouple that is considered inaccurate. Even though several thermocouple signals have been discarded, the interpolation routines applied to obtain the boundary and reference temperatures still make it possible to carry out a meaningful estimation of the state of the hearth lining. 2.3. Program Implementation and Visualization of Results. The model outlined in section 2.2 was implemented in Matlab, solving the heat conduction eq 1 by the finite element approach in FEMLAB. The optimization problem (eq 6) was solved by the Levenberg-Marquardt method. First, the optimal values of N unknowns (kco*) were solved for each of the K computational domains (cf. subsection 2.1.1 and Figure 2). Next, the composite layer thicknesses in the furnace center, ∆sco,c, i.e., in the only common point of the K solutions, were calculated and their arithmetic mean, ∆sco,c, was taken to be a fixed thickness in the center for all the domains. Finally, the remaining unknown N - 1 heat conductivities of the composite layers of each of the K domains were reestimated, and the solutions were converted back to layer thicknesses by eqs 5a and 5b, yielding the two-dimensional profiles. The user interacts with the program through the main three parts of the system, i.e., the setup part, calculation part, and visualization part, through a graphical user interface (GUI). Figure 6 illustrates the setup window for the lining material geometry and thermodynamic properties. The interface is flexible, allowing for different lining region configurations, where the thermal conductivities of the materials can be expressed in temperature-dependent form (k ) a + bT). With the thermocouple setup window, depicted in Figure 7, the locations of the lining thermocouples, and their pointers to the database, are specified. Furthermore, the user has to specify an estimate of the thermal conductivity of the skull layer, as well as maximum daily changes of the skull layer thickness and erosion line. In order to be able to create a view of the entire hearth erosion, the solutions to the two-dimensional problems are aggregated by interpolation to yield a three-dimensional picture of the inner profile of the hearth, which afterward can be illustrated and animated during periods of transition to throw light on the dynamics of the erosion and buildup formation phenomena. 3. Results The model has been applied to data from the campaigns of twin single-taphole furnaces, blast furnace A (BF A) and blast furnace B (BF B), as well as to data from a bigger three-taphole furnace, blast furnace C (BF C), and some of the results will next be illustrated. Even though the present formulation of the heat conduction equation neglects dynamic terms, the comparatively slow changes (with transition periods spanning several weeks) can still be tracked by this quasi-steady-state model applied to daily mean values of the lining temperatures. In the

hearth cross-section views to be presented, the taphole (referred to) is above the upper left part of the figures. The model was run with a erosion reserve of ∆ser ) 0.02 m and a maximum skull layer thickness change of (d∆ssk/dt)max ) 0.10 m/day, as max well as a maximum skull layer thickness of ∆ssk ) 1.0 m. The thermal conductivity of the skull layer was taken to be ksk,0 ) 2 W/(m · K). 3.1. Blast Furnace A. The campaign of BF A started in 1995 and was extended by a midcampaign hearth repair in 2002. Initially, two layers of mullite bricks were used in the hearth bottom while the rest of the hearth lining was made of high-conducting carbon bricks. At the repair 7 years later the two bottom layers were replaced by a single layer of carbon blocks, giving the hearth a larger sump depth. Figure 8 illustrates the evolution of the inner hearth volume (in the region considered by the model) from 1996 to 2006, where the lower line depicts the inner volume of the hearth, and the upper curve shows the inner volume enclosed by the lining material. Thus, the difference between the two curves is the skull volume. The former curve is characterized by spikes caused by thermal cycles in the hearth, while the latter shows a comparatively slow long-term wear that primarily occurs at the major spikes. The spikes are seen to be not entirely periodic, but to recur with periods of 1-4 months. To illustrate the typical hearth behavior at a spike, a three-dimensional picture of the hearth state is depicted in Figure 9 (with the taphole denoted by a black bar) with four snapshots of the inner profile taken at 1 week intervals starting from the beginning of the second spike after the midcampaign repair (indicated by an arrow in Figure 8). The first (top) panel shows the typical inner profile of a skulled hearth; the wall and bottom are covered with a thick skull layer except in the taphole region, where the liquid flow rate is higher. This can be interpreted as an indication that the iron flow would be concentrated in the taphole region, which leads to a short taphole length. Panel two illustrates the situation 1 week later. The wall skull has begun to melt, starting from the taphole and spreading “backward” around the wall. In panel three, 2 weeks after the start of the spike, the hearth wall skull has almost completely melted, but the bottom skull is still intact. In terms of the internal state, it seems that the flow in the hearth is now concentrated mainly at the wall, but the core of the dead man is still impermeable as the bottom skull has not started to melt. Three weeks after the start of the spike, the hearth volume has reached its maximum, as the bottom skull has also melted. However, by carefully studying the profile illustrated in panel four, it can be observed that new skull has already started forming on the wall. This may indicate that the dead man core has become more permeable, the flow is more uniformly distributed in the hearth, and the heat load on the wall decreases. Even though no major changes are made in the operation of the furnace, the hearth gradually returns to its skulled state, as can be seen in Figure 8. The reason for this almost cyclic behavior of the hearth has not been fully understood yet, but the distribution of the flows of liquids into the hearth seems to play an important role. A correlation


between the inner hearth volume and the sulfur accumulation/ depletion of the blast furnace has also been detected.11 As for the state at the midcampaign repair, a large scab rich in calcium sulfite was found in the hearth, and it is believed that the thermal cycles experienced during the operation would be connected to growth and partial remelting of the scab.12 Figure 10 illustrates the cross section of the hearth along the line of the taphole at the midcampaign repair with the scab indicated, as well as a dashed line illustrating the model’s estimate of the profile of the intact lining and the solid line illustrating the estimated buildup profile. The model is seen to have captured the erosion state of the hearth qualitatively, considering the fact that the maximum skull thickness of 1.0 m allowed in the estimation has been reached for the erosion lines further from the taphole. The lining wear is somewhat overestimated below the taphole and in the hearth center, but still quantitatively agrees with the measured profile of the intact lining. 3.2. Blast Furnace B. The fundamentally different behavior of the twin furnace BF B, with a campaign start in 1996 and a midcampaign hearth repair in 2004, is illustrated in Figure 11. This furnace hearth is seen to undergo a change that follows the response of a first-order system with a time constant of roughly 2 years, leading to an inner volume (in the domain considered by the model) stabilizing around 70 m3. The large changes in the skull layer thickness experienced in late 2000, lasting for about a year, were caused by a decrease in the CSR of the coke, which led to a partially clogged hearth. The extent of the skull growth estimated by the model has been verified by comparing the observed slag delay (i.e., the time of irononly flow from the taphole at the start of the cast) with the slag delay estimated by a force-balance model of the hearth coke, where the inner hearth volume is a central factor.9,13 It is also interesting to note that the removal of the two mullite brick layers in the midcampaign repair led to practically the same inner hearth volume before and after the repair. Figure 12 depicts the cross section of the hearth along the line of the taphole at the midcampaign repair in 2004 and the state estimated by the model, where the dashed line depicts estimated wear and the solid line the estimated buildup profile. Despite some discrepancies below the taphole and in the furnace center, the model has predicted the considerable progress of the erosion reasonably well. No scab was found in this furnace, so the gray region only depicts the remaining intact lining. 3.3. Blast Furnace C. Finally, the results of the model run on thermocouple data from the large three-taphole furnace BF C are briefly illustrated for a period spanning almost 10 years of operation. Figure 13, which illustrates the evolution of the hearth volumes, shows a rather slow and uniform erosion progress, and only occasionally formation of larger buildup (e.g., after February 2001) or rapid erosion (e.g., at the end of 2002 and in February 2004). The volume of the skull is seen to remain almost constant (at 10-15 m3). Overall, the evolution of the hearth volumes shows a resemblance to that of BF B (cf. Figure 11). As an example of the wear pattern in BF C, Figure 14 shows the estimated wear and buildup profiles in June 2001 along the line of the second (middle) taphole. This is seen to correspond to a state where the upper three sidewall blocks have been eroded, but only the center part of the uppermost hearth bottom layer.

4. Conclusions A novel model of the hearth erosion and buildup formation in the ironmaking blast furnace has been developed. The model is based on the solution of an inverse heat transfer problem, reconstructing the position of the hot interface using information from thermocouples embedded in the hearth lining, and it is optimized for fast computation, which makes it suitable for both offline analysis of data and online implementation at steelworks. Special attention has been devoted to the mathematical formulation of the problem, including a flexible formulation of the boundary conditions, hearth refractory materials, and thermocouple setup. By these measures, it is simple to adapt the model to estimate the hearth state of any blast furnace, and whole furnace campaigns, today typically spanning 15-20 years, can be analyzed with a computation time of a few days. The formulation of the boundary conditions guarantees robust estimates even for cases where the thermocouple information is erroneous and/or strongly limited. The model has been applied to study the evolution of the hearth erosion and buildup processes in three industrial furnaces, and the results have been found to agree reasonably well with observations in the operation and with the furnace hearth states found at midcampaign repairs. In a forthcoming work, the results of the model will be used for throwing light on the complex interaction between hearth liquid flow patterns, dead-man state,14 and erosion/buildup formation. Literature Cited ¨ zis¸ik, M. N. Heat Conduction; John Wiley & Sons: New York, (1) O 1993. (2) Kurpisz, K.; Novak, A. J. InVerse thermal problems; Computational Mechanics Publications: Southampton, U.K., 1995. (3) Quy, N.; Cripps Clark, C. Applications of Heat Transfer Studies to Analysis of Blast Furnace Hearth Erosion and Build-up. Proc. Aust. Chem. Eng. Conf. 1984, 2, 909. (4) Schulte, M.; Klima, R.; Ringel, D.; Voss, M. Wear monitoring in blast furnace hearths by means of heat flux sensors. Stahl Eisen 1997, 117, 57. (5) Torrkulla, J.; Saxeń, H. Model of the state of the blast furnace hearth. ISIJ Int. 2000, 40, 438. (6) Kumar, S. Heat Transfer Analysis and Estimation of Refractory Wear in an Iron Blast Furnace Hearth Using Finite Element Method. ISIJ Int. 2005, 45, 1122. (7) Gonzalez, M.; Goldschmit, M. B. Inverse heat transfer problem based on a radial basis functions geometry representation. Int. J. Numer. Methods Eng. 2006, 65, 1243. (8) Zhao, H.; Cheng, S.; Zhao, M. Analysis of All-Carbon Brick Bottom and Ceramic Cup Synthetic Hearth Bottom. J. Iron Steel Res. 2007, 14, 6. (9) Bra¨nnbacka, J.; Saxeń, H. Model analysis of the operation of the blast furnace hearth with a sitting and floating dead man. ISIJ Int. 2003, 43, 1519. (10) Bra¨nnbacka, J.; Saxeń, H.; Pomeroy, D. Detection and Quantification of the Dead Man Floating State In the Blast Furnace. Metall. Mater. Trans. B 2007, 38B, 443. (11) Helle, M.; Saxeń, H. Data-driven analysis of sulfur flows and behavior in the blast furnace Steel Res. Int. 2008, 79, 671. (12) Raipala, K. On Hearth Phenomena and Hot Metal Carbon Content in Blast Furnace. Doctoral Thesis, Helsinki University of Technology, Finland, 2003. (13) Bra¨nnbacka, J. Model Analysis of Dead-man Floating State and Liquid Levels in the Blast Furnace Hearth. Doctoral Thesis, Åbo Akademi University, Finland, 2004. (14) Helle, M.; Saxeń, H.; Kerkkonen, O. Assessment of the State of the Blast Furnace High Temperature Region by Tuyere Core Drilling. Submitted for publication.

ReceiVed for reView March 8, 2008 ReVised manuscript receiVed July 13, 2008 Accepted July 23, 2008 IE800384Q

Model for Fast Computation of Blast Furnace Hearth Erosion and

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