**1. Introduction**

The steel industry is one of the most important raw material industries. The iron-making process in the early steel process is mainly done through a blast furnace [1]. With changes in the times, a series of drawbacks of the blast furnace have gradually emerged. One is the heavy dependence on coke resources [2], and the second is the increasing pressure on environmental protection. With the rapid depletion of natural resources and awareness regarding environmental protection, the smelting reduction process came into being. This significantly reduces energy consumption and investment by recycling the waste heat and eliminating sintering and coking plants [3]. Among the variants of this process, representative ones are COREX [4], jointly developed by VAI and Korf, FINEX [5], jointly developed by POSCO Steel and VAI Engineering, HISMELT [6], jointly developed by CRA Australia and Kloekner, and DIOS [7], jointly developed by the Japan Iron and Steel Federation to organize eight domestic steel companies.

In the field of smelting reduction, many researchers have done considerable research on physical simulation and mathematical simulation. Zou et al. [8,9] proposed a two-step three segment iron bath smelting reduction process with a thick slag layer, which was developed for the gradient separation of the oxidation zone and the reduction zone by a thick slag layer with physical simulation. Srishilan et al. [10] proposed a predictive thermochemical model of the COREX process which enables the rapid computation of process parameters. The model helps in predicting the variations in process

parameters with respect to the degree of metallization and post-combustion ratio for the given raw material conditions. A series of recent representative research results have also been obtained in smelting reduction and hydrocarbon reduction. In the field of plasma reduction, for example, Behera et al. [11] succeeded in the smelting reduction of iron ore (hematite) in thermal hydrogen plasma. Mandal et al. [12] described a new concept for maintaining an inert atmosphere with a high temperature of about 1973 K (1700 ◦C) inside the furnace during smelting reduction. In terms of direct reduction, Shim et al. [13] reported that the direct reduction rate in a melter–gasifier was roughly drawn as the product of the CO2 content in the ascending gas and the reaction rate constant of coal with CO2, 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 H2/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-H2 smelting reduction furnace (cf. Figure 1).

**Figure 1.** A new generation of C-H2 iron bath smelting reduction process configuration.

The theory of similarity is the theory of experimentation. For the formulas of complex phenomena that cannot be solved by mathematical analysis, the similarity theory provides an experimental solution. The similar first theorem specifies the physical quantities that need to be measured during the experiment, as well as the conditions that the model experiment should follow. The π theorem is a universal dimensional analysis method which was proposed by E. Buckinghan in 1915, so it is also known as the Buckingham Theorem. The Buckingham Theorem is described as F(x1, x2, ... , xn) = 0, if there are n variables that are mutual functions for a physical phenomenon. If these variables contain m basic quantities, they can be arranged into the functional relation ϕ(π 1, π 2, ... , π n ≤ m) of (n ≤ m) dimensionless numbers, and then n physical quantities can be combined into (n ≤ m) dimensionless π numbers. In this paper, the dimensional analysis is based on the Buckingham theorem, which specifies how to organize the experimental results. The experimental results need to be organized into a functional relationship between dimensionless groups. Finally, the dimensionless groups and the coefficients of similarity are determined according to the experimental results. The similarity theory is used to solve the problem in various fields. Zhen et al. [17] focuses on providing a quantitative methodology on how each parameter affects the structural response of the subsurface tension leg

platform (STLP), which will facilitate establishing the unique design dimensionless groups as regards to STLP. Vatankhah et al. [18] predicted the rate of discharge through different side holes in irrigation and hydraulic engineering. Sharp-crested side triangular orifices were studied experimentally and analytically, and several models were derived for the discharge coefficient based on Buckingham's theorem of dimensional analysis. Meng et al. [19] used the π theorem, an improved dimensional analysis method, and the dimensionless quantity, which can effectively reflect the relationship between the non-sinusoidal vibration parameters of a continuous casting mold, was given. The dimensionless function correlation formula, which can objectively describe the actual phenomenon, was obtained.

Side blowing is one of the key processes of smelting reduction. Fuel and enriched oxygen are injected into the furnace by a side nozzle to provide the heat required for the reduction reaction and the melt is stirred at the same time. By designing and optimizing the parameters such as the arrangement, the angle, and the flow rate of the side nozzle, this provides better reaction conditions for melting in the furnace and prolongs the service life of the side nozzle, and finally achieves the combination with the top and bottom blowing. For side blowing, many scholars had conducted relevant research in this field. For example, Feng [20] outlined the production overview of the side-blown melting reduction furnace, introduced the construct design of main components, and listed the practice and summarized the characteristics of the side-blown reduction furnace. Due to a large number of side nozzle parameters and furnace parameters, the experimental results were not regular on the surface, so it was necessary to sort out the data through dimensional analysis and provide an analytical formula, so as to obtain the quantitative law. Wang et al. [21] investigated the penetration behavior of immersion side-blowing gas flow in a slag lade water model by photography and the dimensional analysis method, in order to provide a theoretical base for improving the reaction speed between the gas–liquid interface and the oxygen gun jet distribution. The maximum penetration depth calculated by the empirical formula is in agreement with the measured data from the experiments.

Although many studies [22–24] have been carried out on the C-H2 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-H2 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-H2 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-H2 smelting reduction furnace. The conclusions obtained are also widely applicable to the engineering design and design analysis of the smelting reduction furnace.
