*2.3. Modeling of Dead Man State*

The dead-man state can be estimated by conducting a balance between the buoyancy of iron and slag in the hearth and the force pressing down on the dead man. The buoyancy force, which is a function of liquid level and dead man porosity, is relatively straightforwardly expressed. Nevertheless, the vertical force pressing down on the dead man is more complicated, since it is related to a set of furnace operating conditions, e.g., raceway length, gas drag and burden weight, as well as liquid holdup above the hearth liquids [34]. The BF lower part is schematically illustrated in Figure 4, where the dead man is divided into two particular regions based on the raceway length/gas drag intensity, i.e., a central region and a region under raceways [35].

**Figure 4.** Schematic sketch representing the lower part of the blast furnace. Reproduced with permission [35].

It should be stressed that both the buoyancy force and the downward-acting force are often expressed per unit area, i.e., in the form of pressure. The downward-acting stress at the tuyere level (cf. Figure 4), which was investigated by conducting both experimental runs and numerical calculations [34], is depicted in Figure 5. As can be seen in the figure, the stress is highly reduced in the region where the raceway is located. This can be explained by the drag of the upward-flowing gas from the raceway, which compensates for a portion of the burden weight above the tuyere level. In the region under the raceways, consequently, the dead man could float higher if the buoyancy force is sufficient.

**Figure 5.** Lateral distribution of the downward-acting stress at the tuyere level. Reproduced with permission [34].

As the radial distribution of the downward-acting force per area, *p*d, may vary with the operating conditions, Brännbacka et al. [25,26,36–38] suggested the simple but flexible parameterized expression

$$p\_{\rm d} = \begin{cases} \overline{p}\_{\rm d} & \text{if } r \le r\_0 \\ \overline{p}\_{\rm d} - a \left( \frac{r - r\_0}{R} \right)^{\rm n} & \text{if } r > r\_0 \end{cases} \tag{1}$$

where *p*d, *r* and *r*<sup>0</sup> are the overall downward pressure, radial position as well as radius of the central region where the downward-acting pressure is unaffected by the raceways, respectively. *R* is the hearth radius and *a* is a scaling factor, while *n* is a parameter that influences the arising shape of the dead man bottom under the raceways. The magnitude of the overall force can be obtained by calculating the burden weight as reduced by the lifting force of the gas drag and the friction of the furnace wall. The vertical position of the dead man bottom deduced from the force balance is

$$z\_{\rm dfm} = \begin{cases} z\_{\rm sl} - \frac{p\_{\rm d}}{\rho\_{\rm sl}\zeta(1-\varepsilon)} & \text{if } 0 \le p\_{\rm d} \le p\_{\rm b, sl}^{\max} \\ z\_{\rm ir} + \frac{\rho\_{\rm d}}{\rho\_{\rm ir}}(z\_{\rm sl} - z\_{\rm ir}) - \frac{p\_{\rm d}}{\rho\_{\rm r}\zeta(1-\varepsilon)} & \text{if } p\_{\rm b, sl}^{\max} < p\_{\rm d} \le p\_{\rm b, sl}^{\max} + p\_{\rm b, ir}^{\max} \\ z\_{\rm hb} & \text{if } p\_{\rm d} > p\_{\rm b, sl}^{\max} + p\_{\rm b, ir}^{\max} \end{cases} \tag{2}$$

with

$$p\_{\rm b,sl}^{\rm max} = \rho\_{\rm sl} g (1 - \varepsilon) (z\_{\rm sl} - z\_{\rm ir}); \ p\_{\rm b,ir}^{\rm max} = \rho\_{\rm ir} g (1 - \varepsilon) (z\_{\rm ir} - z\_{\rm hb}) \tag{3}$$

where ρir, ρsl, *g* and *z*hb are the densities of liquid iron and slag, gravitational acceleration and the vertical position of the hearth bottom, respectively, while ε, *z*ir and *z*sl are the dead-man porosity and the levels of liquid iron and slag, respectively.

If the parameters of Equation (1) are given, the dead man bottom profile can be calculated based on the quantities of hearth liquids and an average dead man porosity. Figure 6 shows the estimated evolution of the iron and slag levels and the bottom shape of the dead man during a tap cycle in a BF, where the inner hearth profile was estimated by solving an inverse heat transfer problem [25]. The corresponding three-dimensional coke-free zones beneath the dead man are depicted in Figure 7, where the black bar indicates the location of the taphole.

**Figure 6.** Evolution of the iron and slag levels (upper panel) and the dead-man bottom profile (lower panel) during a tap cycle in an eroded blast furnace. Reproduced with permission [25].

**Figure 7.** Three-dimensional illustration of the coke-free zones in cases 1–3 of Figure 6. The black horizontal bar marks the location of the taphole. Reproduced with permission [25].

### *2.4. E*ff*ect of Dead Man State on Hearth Performance*

#### 2.4.1. Liquid Flow Pattern and Flow-Induced Shear Stress

In the hearth, the floating state of the dead man plays a key role in determining the lining wear and the pattern of liquid flow [31]. Through dissections of quenched furnaces and based on observations at the campaign end when the hearth is relined, the wear profile of the hearth lining has been investigated. The profiles reported in such investigations usually indicate an elephant-foot-shaped profile with severe erosion of the lining in the lower periphery of the hearth. It has been recognized that an elephantfoot-shaped profile is caused by the intensive circumferential flow of hot metal that can occur when the permeability in the dead man's core deteriorates and/or the dead man floats partly, forming a coke-free zone ("gutter") at the hearth corner [39,40]. A bowl-shaped profile, where the lining in the middle of

the hearth bottom is excessively eroded, has also been reported. This pattern could be the result if the dead man floats completely, or if the porosity of the dead man is fairly uniform and it occupies the whole hearth. The latter can be expected for hearth designs where the sump depth is large.

The pattern of liquid flow in the BF hearth has been investigated by using both physical and numerical models. Physical studies utilizing scale models have usually considered only steady-state iron flow through a heterogeneous dead man with zones of different permeability. The modeling results can still give insightful information concerning the liquid flow close to the hearth bottom, where the lining erosion is mainly attributed to iron flow.

A number of computational fluid dynamics (CFD) models, focusing on the phenomena in the BF hearth and considering liquid flow and/or heat transfer, have also been built in the past. Usually, the dead man is taken as a fixed packed bed, and Darcy's/Ergun's equation can be applied. The influences of dead man properties, such as packing structure and floating state, on the liquid flow paths and distribution of temperature in the hearth lining have been thoroughly evaluated [41–50]. Figure 8 (based on unpublished results using the model outlined in [48]) illustrates the general streamlines of hot metal in one half of the hearth. These results indicate that a partly floating dead man leads to an intensive circumferential flow, thus exerting a strong heat load on the lining at the hearth corner. However, some simulation results have implied that the distribution of temperature at the hearth bottom is less sensitive to the dead man's properties, since the local heat transfer is controlled by the high thermal resistance of the hearth lining refractories (i.e., ceramic pad) [47].

**Figure 8.** General streamlines of iron flow in a hearth with a partially floating dead man.

The possible hearth lining wear mechanisms include chemical reactions between the lining materials and molten liquids, abrasion and friction caused by coke particles in the hearth, as well as thermo-mechanical stress and flow-induced shear stress. The last one that is caused by the near-wall flow field can lead to lining erosion alone, and a combination of it with other mechanisms could give rise to more wear, eventually resulting in severe hearth damage. Thus, it is imperative to understand the shear stress in terms of its generation and distribution so that appropriate precautions can be taken to reduce the shear stress in order to prolong the campaign life of the hearth. The flow-induced shear stress has been studied mainly using CFD models, since no direct measurements of the BF hearth state variables exist [48–53]. The contours of the shear stress on the hearth bottom under different dead man states, calculated in [48,49], are depicted in Figure 9. It can be seen that with a sitting dead man, the high shear stress zone emerges in the interior of the hearth bottom, particularly below the taphole entrance. Nevertheless, the zone moves to the peripheral region when the dead man partly floats, with a coke-free zone emerging at the hearth corner. In addition, a fully floating dead man state could mitigate the hearth bottom shear stress to some extent, since the shear stress distributes quite uniformly. This would imply that in furnaces where the dead man has started floating completely, the hearth bottom erosion would not progress much.

**Figure 9.** Influences of dead man floating state on the shear stress exerted on the hearth bottom. Reproduced with permission [48].

It was also reported that the high shear stress and heat load in the vicinity of the taphole can be effectively reduced with a longer taphole, since the overall liquid flow is forced to bend towards the center of the dead man, and the circumferential flow caused by a partly floating dead man, or a dead man with a blocked core, can be restrained [52]. This is actually the main reason why a long taphole is usually a prerequisite for protecting the hearth lining near the taphole from severe erosion. A long taphole is often associated with a high carbon content of the liquid iron, which supports the above hypothesis. Furthermore, it has been demonstrated [35] that to achieve a longer taphole, the injected mud must be in good contact with the dead man. Thus, if the dead man floats excessively at the wall, the taphole becomes short, and severe sidewall erosion may follow. In summary, the dead man state is strongly associated with hearth lining erosion, as discussed in the following subsection.
