#
Parametric Dimensional Analysis on a C-H_{2} Smelting Reduction Furnace with Double-Row Side Nozzles

^{*}

## Abstract

**:**

_{2}smelting reduction method. Although previous researchers have studied this through a large number of physical and numerical simulations, they have not yet refined general laws from the perspective of dimensional analysis. In this paper, a double-row side blow hydraulics simulation was carried out in the C-H

_{2}smelting reduction furnace, and an entire list of dimensionless groups of input and output parameters was proposed based on its hydraulics simulation data. The expressions between the dimensionless group of mixing time and dimensionless groups such as Capillary number (Ca) and Lagrange group (La

_{1}) were obtained by multiple linear regression based on the experimental research results and data analysis. By verifying the calculated and experimental values of the dimensionless group of mixing time, it can be seen that both have a good positive correlation. This study provides a better methodology for controlling key parameters and lays the foundation for the optimal design of the process parameters for the C-H

_{2}smelting reduction furnace.

## 1. Introduction

_{2}content in the ascending gas and the reaction rate constant of coal with CO

_{2,}and the way to minimize the direct reduction ratio was discussed with that diagram. You et al. [14] used Sn-bearing complex iron ore via reduction with mixed H

_{2}/CO gas to prepare Sn-enriched direct reduced iron (DRI). In the behavior of reduction, Park et al. [15,16] investigated the high-temperature behavior of a magnetite–coke composite pellet fluxed with dolomite by a customized thermogravimetric analyzer (TGA) at 1573 K (1300 °C) and the influence of different coal types on the reduction of the composite pellet. At the same time, previous research has proposed a new generation of C-H

_{2}smelting reduction furnace (cf. Figure 1).

_{2}smelting reduction furnace through physical simulation and mathematical simulation, the experimental data and results have not been quantitatively analyzed. This paper is aimed at providing a quantitative method for examining how each parameter affects the mixing time in the C-H

_{2}smelting reduction furnace, and proposes a dimensionless input and output parameter based on the Buckingham theory, a complete list of derived dimensionless groups. This is helpful for establishing a single design standard for C-H

_{2}smelting reduction furnaces. This study provides a means to understand critical parameters better and lays the foundation for the optimal design of the side blowing parameters of the C-H

_{2}smelting reduction furnace. The conclusions obtained are also widely applicable to the engineering design and design analysis of the smelting reduction furnace.

## 2. Experimental Setup and Methods

_{2}smelting reduction furnace was carried out. Based on the similarity principle, the smelting reduction iron-making process under the high temperature conditions in the prototype was studied by hydraulic simulation at room temperature in this experiment. The schematic diagram of the C-H

_{2}smelting reduction model apparatus is shown in Figure 2. The model was simulated by the scale ratio of 1:1 to the prototype, in which the molten iron of the prototype was 200 kg. The experiment was carried out in a cylindrical transparent plexiglass furnace with a diameter of about 0.4 m and a height of 1.68 m. In the experiment, the molten iron was simulated by water, high vacuum oil was used to simulate the slag, and oxygen-enriched air was blown on the top and bottom and side nozzle to simulate the flux injection and the bottom blowing hydrogen. The volume ratio of water to high vacuum oil in the model was 1:2 [25,26], in which the water phase was 0.246 m and the height of the oil phase was 0.492 m. The physical parameters of the experiment are shown in Table 1. In the prototype, the combined top, bottom, and side blowing of the C-H

_{2}smelting reduction process occurs. This paper is the first phase of the project, aiming to carry out the physical simulation and dimensional analysis of the single-side blowing of double-row side nozzles. The side blowing nozzle was divided into the upper side nozzle and the lower side nozzle, and the diameter of the side nozzle was 0.004 m. The upper side nozzle was 0.574 m from the bottom and was located at 1/3 above the slag phase. The lower side nozzle was 0.492 m from the bottom and was located in the middle of the slag layer. The prototype and dimensions of the water model are shown in Figure 3a,b, respectively.

_{2}smelting reduction bath, the mixing time was regarded as an important index. The mixing time [27] was defined as the period required for an instantaneous tracer concentration to settle within ±5% deviation around the final tracer concentration in the C-H

_{2}smelting reduction reactor bath. This definition is referred to as the 95% mixing time. In the C-H

_{2}smelting reduction bath, the mixing time was measured by the conductivity of three electrodes, which was 0.05 m away from the bottom of the bath. Figure 4b is the position and angle of the sensor and tracer. In the prototype, the feed port was used for feed preheating ore and flux. In this experiment, in order to simulate the effect of different raw material positions on the mixing time of the molten pool, a saturated Sodium chloride (NaCl) aqueous solution (75 mL) was fed from the intermediate to the C-H

_{2}smelting reduction furnace. The conductivity of water was measured by three DSS-IIA conductivity meters and recorded automatically by a computer software recorder. For each physical simulation test site in each mode of operation, the measurements were taken at least three times and the arithmetic mean of the average residence time was obtained. Through the orthogonal test and analysis, the relationship between the average residence time and various experimental parameters can be obtained. These results would eventually be organized into a functional relationship between the dimensionless groups.

## 3. Dimensional Analysis

_{1}~π

_{8}is a dimensionless group, and $\phi $ is a functional symbol. After substituting in each variable, it gives

_{s}and V

_{s}, we get

_{oil}, m

^{2}·s

^{−1}. The tracer feeding position and the insertion position of the side nozzle are closely related to the total height of the water phase and oil phase, the D

_{S}in the tracer feeding position number and the insertion depth number of side nozzle are replaced by the total height of the water phase and oil phase H

_{oil}

_{+w}, which can be converted into the following equation:

_{s}is flow velocity of the side nozzle, m·s

^{−1}. D

_{f}is the diameter of the furnace, m, and Q

_{s}is the volumetric flow rate, m

^{3}·s

^{−1}.

_{oil}is the kinematic viscosity of the high vacuum oil, m

^{2}·s

^{−1}, σ is the surface tension of the high vacuum oil, kg·s

^{−2}, g is the acceleration of gravity, m·s

^{−2}, ε is the stirring energy of the side nozzle, kg·m

^{2}·s

^{−3}, δ is the dimensionless groups of momentum, q

_{s}is the flow rate of the single side nozzle, m

^{3}·s

^{−1}, h

_{s}is the insertion depth of the side nozzle, m, H

_{oil}

_{+w}is the total height of the high vacuum oil and water, m, D

_{s}is the diameter of the single side nozzle, m, and V

_{s}is the flow velocity of the side nozzle, m·s

^{−1}.

_{jm}is named in order to simplify the expression of $\frac{{\mathrm{j}}_{\mathrm{M}}\times {\mathrm{N}}_{\mathrm{sh}}}{{\mathrm{N}}_{\mathrm{sc}}^{\frac{5}{3}}}$, that is ${\mathrm{N}}_{\mathrm{jm}}=\frac{{\mathrm{j}}_{\mathrm{M}}\times {\mathrm{N}}_{\mathrm{sh}}}{{\mathrm{N}}_{\mathrm{sc}}^{\frac{5}{3}}}$. According to Table 3, Table 4 and Table 5 and the actual working conditions of this study, the equation of dimensionless groups can be expressed as follows:

## 4. Results and Discussions

#### 4.1. The Effects of Ca and La_{1}

_{1}can be obtained, respectively:

_{us}is the injection velocity of the upper side nozzle, m·s

^{−1}, V

_{ls}is the injection velocity of the lower side nozzle, m·s

^{−1}.

_{1}, τ

_{2}and τ

_{3}of the upper and lower side nozzle, respectively. It can also be seen that it has a higher consistency with the τ (cf. Figure 5).

#### 4.2. The Effects of Ca, La_{1} and N_{jm}

_{1}, the N

_{jm}dimensionless group is added here for the purpose of investigating the relationship between the dimensionless group of the mixing time and the dimensionless groups such as density, dynamic viscosity, stirring energy, flow rate, and so on based on the variable of velocity. The following table contains the multiple linear regression data based on the three dimensionless numbers of upper side nozzle and lower side nozzle, respectively:

_{1}and N

_{jm}for the upper and lower side nozzles can be obtained, respectively, as shown below:

_{1}, τ

_{2}, τ

_{3}is also shown a high consistency with τ (cf. Figure 7), respectively.

#### 4.3. The Effects of Ca, La_{1}, N_{jm}, and δ

_{1}, τ

_{2}, τ

_{3}(cf. Figure 10) and τ (cf. Figure 9).

#### 4.4. The Effects of Ca, La_{1}, N_{jm}, δ, H_{i} and h_{i}

## 5. Conclusions

_{1}were obtained by multiple linear regression. It can be seen from the expressions that the indexes of the dimensionless groups have a higher identity when they have more than three dimensionless groups. By verifying the calculated and experimental values of the dimensionless group of mixing time, it can be seen that both have a good positive correlation. At the same time, it can also be seen from the comparison between the calculated values of τ

_{1}, τ

_{2}, τ

_{3}and the experimental values that they are in good agreement with the corresponding τ, which indicates that the fitting expressions have higher reliability. Because density, surface tension, and other parameters of the medium have not been changed in this study, Equations (15) and (16) are more suitable for the study of the side nozzle velocity and related angle. Equations (21) and (22) will be of great significance when the density, viscosity, surface tension, and furnace diameter of the medium are changed in further work. This conclusion will better provide help for the control of key parameters, help to establish the design standard of C-H

_{2}smelting reduction furnaces, and lay a foundation for the optimization of side nozzle parameters of C-H

_{2}smelting reduction furnaces.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Prototype (

**a**) and dimensions (

**b**) of water model, the relative angle between the upper side nozzle and the lower side nozzle (

**c**), the horizontal angle of the upper side nozzle and the lower side nozzle (

**d**).

**Figure 4.**Different tracer feeding positions (

**a**), and the position and angle of the sensor and tracer (

**b**).

**Figure 5.**Comparison of experimental lgτV

_{s}/D

_{s}with calculated ones for upper side nozzle (

**a**) and lower side nozzle (

**b**), respectively, using proposed Equations (15) and (16).

**Figure 6.**Comparison of experimental lgτV

_{s}/D

_{s}with calculated ones for upper side nozzle (

**a**–

**c**) and lower side nozzle (

**d**–

**f**), respectively, using ${\tau}_{1},{\tau}_{2},{\tau}_{3}$.

**Figure 7.**Comparison of experimental lgτV

_{s}/D

_{s}with calculated ones for upper side nozzle (

**a**) and lower side nozzle (

**b**), respectively, using proposed Equations (17) and (18).

**Figure 8.**Comparison of experimental lgτV

_{s}/D

_{s}with calculated ones for upper side nozzle (

**a**–

**c**) and lower side nozzle (

**d**–

**f**) respectively using τ

_{1}, τ

_{2}, τ

_{3}.

**Figure 9.**Comparison of experimental lgτV

_{s}/D

_{s}with calculated ones for upper side nozzle (

**a**) and lower side nozzle (

**b**), respectively, using proposed Equations (19) and (20).

**Figure 10.**Comparison of experimental lgτV

_{s}/D

_{s}with calculated ones for upper side nozzle (

**a**–

**c**) and lower side nozzle (

**d**–

**f**), respectively, using τ

_{1}, τ

_{2}, τ

_{3}.

**Figure 11.**Comparison of experimental lgτV

_{s}/D

_{s}with calculated ones for upper side nozzle (

**a**) and lower side nozzle (

**b**), respectively, using proposed Equations (21) and (22).

**Figure 12.**Comparison of experimental lgτV

_{s}/D

_{s}with calculated ones for upper side nozzle (

**a**–

**c**) and lower side nozzle (

**d**–

**f**), respectively, using τ

_{1}, τ

_{2}, τ

_{3}.

Density (kg/m^{3}) | Kinematic Viscosity (m^{2}/s) | |
---|---|---|

Iron | 7020 | 9 × 10^{−5} |

Slag | 3000 | 1.33 × 10^{−4} |

Air | 1.205 | 1.506 × 10^{−5} |

High vacuum oil (25 °C) | 860 | 8.75 × 10^{−5} |

Water | 1000 | 1 × 10^{−6} |

Category | Variable | Symbol | Unit | M | L | t |
---|---|---|---|---|---|---|

Physical Parameters | Density | ρ | kg·m^{−3} | 1 | −3 | 0 |

Kinematic viscosity | υ | m^{2}·s^{−1} | 0 | 2 | −1 | |

Surface tension | σ | kg·s^{−2} | 1 | 0 | −2 | |

Mixing time | τ | s | 0 | 0 | 1 | |

Acceleration of gravity | g | m·s^{−2} | 0 | 1 | −2 | |

Stirring energy | ε | kg·m^{2}·s^{−3} | 1 | 2 | −3 | |

Model Size | Height of furnace | H_{f} | m | 0 | 1 | 0 |

Diameter of furnace | D_{f} | m | 0 | 1 | 0 | |

Height of high vacuum oil | H_{oil} | m | 0 | 1 | 0 | |

Height of water | H_{w} | m | 0 | 1 | 0 | |

Height of upper side nozzle | H_{us} | m | 0 | 1 | 0 | |

Height of lower side nozzle | H_{ls} | m | 0 | 1 | 0 | |

Diameter of side nozzle | D_{s} | m | 0 | 1 | 0 | |

Tracer Position | Height of tracer | H_{i} | m | 0 | 1 | 0 |

Distance from tracer to the center of the circle | R_{i} | m | 0 | 1 | 0 | |

Measuring Point | Height of measuring point | H_{m} | m | 0 | 1 | 0 |

Distance from measuring point to the center of the circle | R_{m} | m | 0 | 1 | 0 | |

Working Condition | Insertion depth of side nozzle | h_{s} | m | 0 | 1 | 0 |

Flow velocity of side nozzle | V_{s} | m·s^{−1} | 0 | 1 | −1 | |

Flow rate of single side nozzle | q_{s} | m^{3}·s^{−1} | 0 | 3 | −1 |

Dimensionless Groups | Expressions |
---|---|

${\pi}_{1}$ | $\frac{{\upsilon}_{oil}}{{D}_{S}{V}_{S}}$ |

${\pi}_{2}$ | $\frac{\sigma}{\rho {D}_{s}{V}_{s}^{2}}$ |

${\pi}_{3}$ | $\frac{\tau {V}_{s}}{{D}_{s}}$ |

${\pi}_{4}$ | $\frac{g{D}_{s}}{{V}_{s}^{2}}$ |

${\pi}_{5}$ | $\frac{\epsilon}{\rho {D}_{s}^{2}{V}_{s}^{3}}$ |

${\pi}_{6}$ | $\delta $ |

${\pi}_{7}$ | $\frac{{q}_{s}}{{D}_{s}^{2}{V}_{s}}$ |

${\pi}_{8}$ | $\frac{{H}_{i}}{{H}_{oil+w}}$ |

${\pi}_{9}$ | $\frac{{h}_{s}}{{H}_{oil+w}}$ |

**Table 4.**The dimensionless groups associated with Equation (11) based on the physical chemistry handbook [31].

Symbol | Name | Expression |
---|---|---|

Re | Reynolds number | $\frac{{D}_{s}{V}_{s}}{{\upsilon}_{oil}}$ |

Ca | Capillary number | $\frac{\rho {\mathsf{\upsilon}}_{oil}{V}_{s}}{\sigma}$ |

La_{1} | Lagrange group | $\frac{\epsilon {\upsilon}_{oil}}{\rho {D}_{s}^{3}{V}_{s}^{4}}$ |

H_{o1} | Homochronous number | $\frac{\tau {V}_{s}}{{D}_{s}}$ |

K_{F} | Capillarity-buoyancy number | $\frac{g{\rho}^{4}{\mathsf{\upsilon}}_{oil}^{4}}{\rho {\sigma}^{3}}$ |

Z | Ohnesorge number | $\frac{\rho {\upsilon}_{oil}}{{\left(\rho \sigma {D}_{s}\right)}^{\frac{1}{2}}}$ |

$\mathsf{\delta}$ | Diameter group | ${\left(\frac{\pi}{4}\right)}^{\frac{1}{2}}\times {(2\times {({V}_{s}\times \mathrm{cos}\alpha )}^{2})}^{\frac{1}{4}}\times (\frac{{D}_{f}}{{\left({Q}_{s}\right)}^{\frac{1}{2}}})$ |

j_{M} | J-factor | ${(\frac{{q}_{s}{\upsilon}_{oil}^{8}}{{D}_{s}^{10}{V}_{s}^{9}})}^{\frac{1}{3}}$ |

N_{sh} | Sherwood number | $\frac{{V}_{s}{D}_{s}^{2}}{{q}_{s}}$ |

N_{sc} | Schmidt number | $\frac{{\upsilon}_{oil}{D}_{s}}{{q}_{s}}$ |

Dimensionless Groups | Expression |
---|---|

${\pi}_{1}=\frac{{\upsilon}_{oil}}{{D}_{S}{V}_{S}}$ | $\frac{1}{\mathrm{Re}}$ |

${\pi}_{2}=\frac{\sigma}{\rho {D}_{s}{V}_{s}^{2}}$ | $\frac{1}{\mathrm{Ca}\times \mathrm{Re}}$ |

${\pi}_{3}=\frac{\tau {V}_{s}}{{D}_{s}}$ | ${\mathrm{H}}_{o1}$ |

${\pi}_{4}=\frac{g{D}_{s}}{{V}_{s}^{2}}$ | $\frac{{\mathbf{K}}_{\mathbf{F}}}{{\mathbf{Z}}^{\mathbf{2}}}\times \frac{\mathbf{1}}{\mathbf{Ca}}$ |

${\pi}_{5}=\frac{\epsilon}{\rho {D}_{s}^{2}{V}_{s}^{3}}$ | ${\mathrm{La}}_{1}\times \mathrm{Re}$ |

${\pi}_{6}=\delta $ | $\mathsf{\delta}$ |

${\pi}_{7}=\frac{{q}_{s}}{{D}_{s}^{2}{V}_{s}}$ | $\frac{{\mathrm{j}}_{\mathrm{M}}\times {\mathrm{N}}_{\mathrm{sh}}}{{\mathrm{N}}_{\mathrm{sc}}^{\frac{5}{3}}}\times \mathrm{Re}$ |

**Table 6.**Value and standard error of lg Ca and lg La

_{1}by multiple linear regression for the upper side nozzle and lower side nozzle.

Y Axis | X Axis | Value | Standard Error | |
---|---|---|---|---|

Upper side nozzle | lgτ_{exp}V_{us}/D_{s} | “lgCa” | −0.9564 | 0.0684 |

“lgLa_{1}” | −0.6496 | 0.0006 | ||

Lower side nozzle | lgτ_{exp}V_{ls}/D_{s} | “lgCa” | −0.7403 | 0.0647 |

“lgLa_{1}” | −0.6685 | 0.0058 |

**Table 7.**Value and standard error of lg Ca, lg La

_{1}, and lgN

_{jm}by multiple linear regression for upper side nozzle and low side nozzle.

Y Axis | X Axis | Value | Standard Error | |
---|---|---|---|---|

Upper side nozzle | lgτ_{exp}V_{us}/D_{s} | “lgCa” | −1.8971 | 1.3603 |

“lgLa_{1}” | −2.6093 | 2.8302 | ||

“lgN_{jm}” | 5.0122 | 7.2387 | ||

Lower side nozzle | lgτ_{exp}V_{ls}/D_{s} | “lgCa” | −1.5112 | 1.2876 |

“lgLa_{1}” | −2.2745 | 2.679 | ||

“lgN_{jm}” | −4.1078 | 6.8521 |

**Table 8.**Value and standard error of lg Ca, lg La

_{1}, lg N

_{jm}and lg δ by multiple linear regression for the upper side nozzle and lower side nozzle.

Y Axis | X Axis | Value | Standard Error | |
---|---|---|---|---|

Upper side nozzle | lgτ_{exp}V_{us}/D_{s} | “lgCa” | −1.8802 | 1.3685 |

“lgLa_{1}” | −2.7131 | 2.8806 | ||

“lgN_{jm}” | 5.3305 | 7.4090 | ||

“lgδ” | 0.1140 | 0.5086 | ||

Lower side nozzle | lgτ_{exp}V_{ls}/D_{s} | “lgCa” | −1.8503 | 1.3312 |

“lgLa_{1}” | −2.6127 | 2.7000 | ||

“lgN_{jm}” | 4.7726 | 6.8838 | ||

“lgδ” | −0.4049 | 0.4034 |

**Table 9.**Value and standard error of lg Ca, lg La

_{1}, lg N

_{jm}, lgδ, lgH

_{i}/(H

_{oil+w}), and lg(h

_{s}/H

_{oil+w}) by multiple linear regression for upper side nozzle and lower side nozzle.

Y Axis | X Axis | Value | Standard Error | |
---|---|---|---|---|

Upper side nozzle | lgτ_{exp}V_{us}/D_{s} | “lgCa” | −1.8704 | 1.2735 |

“lgLa_{1}” | −2.7167 | 2.6804 | ||

“lgN_{jm}” | 5.3508 | 6.8942 | ||

“lgδ” | 0.1180 | 0.4733 | ||

“lgH_{i}/(H_{oil+w})” | −0.3671 | 0.0872 | ||

“lg(h_{us}/H_{oil+w})” | 0.0103 | 0.0168 | ||

Lower side nozzle | lgτ_{exp}V_{ls}/D_{s} | “lgCa” | −1.8389 | 1.2257 |

“lgLa_{1}” | −2.6102 | 2.4860 | ||

“lgN_{jm}” | 4.7765 | 6.3382 | ||

“lgδ” | −0.4019 | 0.3715 | ||

“lgH_{i}/(H_{oil+w})” | −0.3632 | 0.0808 | ||

“lg(h_{ls}/H_{oil+w})” | 0.0108 | 0.0157 |

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## Share and Cite

**MDPI and ACS Style**

Xie, J.; Wang, B.; Zhang, J.
Parametric Dimensional Analysis on a C-H_{2} Smelting Reduction Furnace with Double-Row Side Nozzles. *Processes* **2020**, *8*, 129.
https://doi.org/10.3390/pr8020129

**AMA Style**

Xie J, Wang B, Zhang J.
Parametric Dimensional Analysis on a C-H_{2} Smelting Reduction Furnace with Double-Row Side Nozzles. *Processes*. 2020; 8(2):129.
https://doi.org/10.3390/pr8020129

**Chicago/Turabian Style**

Xie, Jinyin, Bo Wang, and Jieyu Zhang.
2020. "Parametric Dimensional Analysis on a C-H_{2} Smelting Reduction Furnace with Double-Row Side Nozzles" *Processes* 8, no. 2: 129.
https://doi.org/10.3390/pr8020129