2.1.1. Burden Flow Trajectory Model

This part studies the movement of raw material particles and velocities of the material from the hopper to the trajectory of the burden flow. This part is divided into four sections and is shown in Figure 2. It includes the velocity of the burden at the hopper exit, velocity of the burden into the chute, velocity of the burden leaving the chute tip, and trajectory of the burden after leaving the chute tip.

**Figure 2.** Trajectory of the burden flow passing through the bell-less top.

The raw material flows out from the exit of the hopper in a funnel form, and its velocity (*V*0) can be described by the hydraulic formula [22]:

$$V\_0 = Q / \pi (2S / \mathbb{C} - d\_i / 2)^2,\tag{1}$$

where *S*, *C*, *di*, and *Q* express the projection area of the throttle valve (m2), circumference of the throttle (m), average particle size of the burden (m), and flow rate of the burden out the throttle valve (t/s), respectively.

According to the literature [23], the relationship between *Q* and *A* (throttle valve opening) is as follows:

Ore:

$$Q = 1 \times 10^{-5} A^3 - 7 \times 10^{-4} A^2 + 0.0366 A - 0.5;\tag{2}$$

Coke:

$$Q = 3 \times 10^{-5} A^3 - 2.7 \times 10^{-3} A^2 + 0.103 A - 1.29. \tag{3}$$

Before reaching the chute, raw material particles free fall with an initial velocity *V*<sup>0</sup> and collide with the wall of the downcomer (see Figure 2), causing a loss of energy (velocity), which can be calculated by the velocity attenuation factor *k*. Therefore, the velocity of the particles entering the chute is calculated by:

$$V\_1 = \sqrt{k \cos a (V\_0^2 + 2g(h + b/\sin a))},\tag{4}$$

where α, *h*, *b*, *g*, and *A* define the inclination angle of the chute in the vertical direction (◦) (see Figure 3), height of the downcomer (m), distance from the chute suspension point to the bottom of the chute (m), acceleration due to gravity (9.81m/s2), and throttle valve opening of the hopper's exit (◦), respectively.

The particles fall into the chute at the velocity (*V*1) and are mainly subjected to gravitational force (*F*1), supportive force (*F*2), frictional force (*F*3), and centrifugal force (*F*4) as shown in Figure 3. The applied forces on particles along the chute can be expressed by:

$$F\_1 = mg\tag{5}$$

$$F\_2 = mg\sin\alpha - \omega^2 lm\sin\alpha\cos\alpha\tag{6}$$

$$F\_3 = \mu F\_2 \tag{7}$$

$$F\_4 = \omega^2 m l \sin \alpha \tag{8}$$

$$
\sum F = F\_1 + F\_2 + F\_3 + F\_{4\nu} \tag{9}
$$

where <sup>ω</sup>, *<sup>m</sup>*, *<sup>l</sup>*, and <sup>μ</sup> express the rotation speed of the chute (r·S<sup>−</sup>1), mass of the burden (kg), length of the chute (m), and coefficient of dynamic friction (−), respectively.

**Figure 3.** Schematic diagrams of the applied forces on the particle flow along the chute.

According to Newton's second law, the velocity *V*<sup>2</sup> of particles leaving the chute end can be calculated and it is decomposed into the horizontal velocity *Vh*, vertical velocity *Vv*, and tangential velocity *Vt* as follows:

$$V\_2 = \sqrt{a^2 \sin a (\sin a + \mu \cos a) l^2 + 2g(\cos a - \mu \sin a)l + (V\_1 \cos a)^2},\tag{10}$$

$$V\_h = V\_2 \sin a$$

$$V\_{\mathbb{T}} = V\_{\mathbb{Z}} \cos a \tag{12}$$

$$V\_t = r\omega \tag{13}$$

After leaving the chute end, the burden moves with the velocity of *V*<sup>2</sup> in the throat until it falls onto the burden surface. In the movement, burden particles are subjected to gravitational force, buoyancy force, and the drag force of gas. The influence of the latter two forces on the movement of the burden is very small and is ignored [22]. Therefore, the movement of the burden is treated as a slant throw movement with gravity, as is shown in Figure 4.

The slant throw movement of particles can be decomposed into two directions: The radius of the throat (*Sr*) and the tangential direction of the radius (*St*). Therefore, the distance of particles from the center line of the furnace (*S*) to the falling point of particles with the burden profile can be calculated by:

$$S\_I = r + (V\_2 \sin \alpha)t\tag{14}$$

$$S\_t = art\tag{15}$$

$$\mathbf{S} = \sqrt{\mathbf{S}\_{r}^{2} + \mathbf{S}\_{l}^{2}} \tag{16}$$

where *r* and *t* are the radial distance from the chute tip to the center line of the blast furnace (m) and the movement time of particles between leaving the chute tip and reaching the burden profile (S).

When the burden moves below the zero value of the stock line, the vertical distance between material particles and the zero value of the stock line can be expressed by:

$$H = h\_2 - (h\_0 - h\_1),\tag{17}$$

where *h*0, *h*1, *h*2, and *H* define the distance from the chute suspension point to the zero stock line (m), distance from the chute suspension point to the end of the chute (m), vertical distance of the material

after leaving the chute tip (m), and distance between the material and zero value of the stock line (m), respectively.

**Figure 4.** The trajectory of the material in the cavity, where *h*<sup>0</sup> is the distance from the chute suspension point to the zero value stock of the line (m), *h*<sup>1</sup> is the distance from the chute suspension point to the end of the chute (m), *h*<sup>2</sup> is the vertical distance of the material after leaving the chute tip (m), and *H* is distance between the material and zero value of the stock line (m).
