2.1.2. Burden Profile Model

After the raw material moves from the chute tip, it is uniformly dumped to the burden surface of the furnace and forms a new one. For multi-ring charging programs, the material forms a new shape of the burden surface with several concentric piles. There is a certain width of the burden mass flow, after the material leaves the chute tip. Therefore, we describe the mass flow by two trajectories: The main trajectory of the material flow (mass trajectory) and the lower material flow, as shown in Figure 5.

**Figure 5.** Schematic diagram of a new surface formation on an original one, where δ is the repose angle of the burden. (**a**) the location of the inner foot B is the intersection of the lower trajectory with the previous burden profile; (**b**) the material keeps the repose angle slipping on the inner surface of the profile.

Along the radial direction of the furnace, the surface of a new burden profile can be described by three points as shown in Figure 5: Inner foot B of the pile, outer foot C of the pile, and apex A of the pile.

When the material settles down on the inner surface of a new pile, the angle of the new pile is less than the repose angle of the raw material, and the location of the inner foot B is the intersection of the lower trajectory with the previous burden profile as shown in Figure 5a. When the material slips on the inner surface of the pile, the inner surface reaches the repose angle of the material and will keep the repose angle, as shown in Figure 5b. Therefore, more material moves towards the center of the furnace and the inner foot B is decided by the volume conservation of the raw material. For the apex A calculation, it is given by the intersection of the main trajectory with the previous burden profile. The right-side material of the main trajectory falls to the outer side of the apex A and forms the outer surface of the pile. Because the particles have a velocity component in the radial direction of the furnace, they roll along the outer surface of the pile for a while. The rolling distance of the material is constant for different materials and is 0.7 m for coke and 0.5 m for ore [23].

The new burden profile is described by the A, B, and C points along the radial direction as shown in Figure 6. It is divided into several regions for integration to obtain the volume. Therefore, the new burden volume is calculated as follows and is equal to the volume of the material in each batch:

$$\mathcal{V} = 2\pi \int (f\_{\text{new}}(r) - f\_{\text{ori}}(r)) r dr = 2\pi \int\_{\eta\_i}^{\eta\_{i+1}} (f\_m(r) - f\_n(r)) r dr,\tag{18}$$

where *r*, *fnew*(*r*), *fori*(*r*), *fm*(*r*), and *fn*(*r*) are the distance from the center line (m), new burden profile function, original burden profile function, mth burden profile function, and nth burden profile function, respectively.

**Figure 6.** Burden volume partition integral.

#### 2.1.3. Burden Distribution Model

Tapping of hot metal and molten slag from the taphole in the hearth as well as combustion and gasification of coke give the space for the burden to descend in the furnace. The shape of the layer distribution changes during the descent. Therefore, it is necessary to modify the burden profile according to the burden descent. In practice, two to four mechanical stock rods are used to measure the descent of the burden at fixed locations.

According to the measurement, Nishida et al. [24] proposed a velocity distribution of the burden descent along the radial direction of the furnace (*V*(r)), which is measured directly by a profilometer and is a linear assumption based on the measured data. From the velocity distribution, the rate of the burden descent is slower in the center and faster near the wall of blast furnace:

$$V\_{\left(\mathbf{r}\right)} = a + br\tag{19}$$

$$\mathbf{a} = \frac{R\_1 V\_{cl} - \frac{2}{3} R\_0 V\_{rad}}{R\_1 - \frac{2}{3} R\_0} \tag{20}$$

$$\mathbf{b} = \frac{V\_{rad} - V\_{ch}}{R\_1 - \frac{2}{3}R\_0} \tag{21}$$

where *V*(r) is the velocity component of the burden descent at the radial position *r* (mm/s), *R*<sup>0</sup> is the throat radius (m), *R*<sup>1</sup> is the distance of the profilometer from the center line (m), *Vch* is the average velocity of the burden descent (mm/s), and *Vrod* is the descent velocity at the profilometer (mm/s).

The descent velocity of the burden is only vertical in the furnace throat (*Vr*), and divides into vertical (*Vh*) and horizontal (*Vr*) components in the shaft region as follows:

$$\boldsymbol{V}\_{\boldsymbol{h}} = \boldsymbol{V}\_{\left(\mathbf{r}\right)} \sin \boldsymbol{\beta}\_{\prime} \; \boldsymbol{V}\_{r = \boldsymbol{V}\_{\left(\mathbf{r}\right)}} \cos \boldsymbol{\beta}\_{\prime} \tag{22}$$

where β defines the shaft angle of the furnace (◦).

When different kinds of raw materials are charged into the throat of the furnace, the structure of alternating layers (ore layer and coke layer) forms. The burden still keeps its layer structure in the descent. Therefore, the layer structure of the burden can be used to study the distribution of the ratio of ore to coke and the permeability of the burden in the shaft.
