Capillary Infiltration of Slag in Hydrogen-Direct Reduced Iron and Influence on Melting

Abstract

The infiltration of slag into the porous structure of Hydrogen Direct Reduced Iron (H-DRI) was studied with computational fluid dynamics to determine the rate of slag infiltration and its effect on heating and melting of H-DRI. It was found that liquid slag rising in an iron walled capillary quickly exchanges its heat and solidifies within 5 capillary radii from the inlet. The resulting effect is that slag infiltration into a porous structure can only occur when the local temperature of the iron matrix is higher than the slag melting temperature of 1523 K.

The convective heating of the porous structure by infiltrating hot slag and the resulting effective thermal diffusivity of the infiltrated structure was used to simulate the heating and melting of 60 and 65% porosity H-DRI for three different infiltration assumptions: no-infiltration, time-dependent infiltration along the slag solidification temperature isotherm, or instant-infiltration. It was found that time-dependent-infiltration and instant slag infiltration results in longer melting time for 60% porosity H-DRI regardless of heat transfer coefficient, and for 65% porosity H-DRI when the heat transfer coefficient is 500 Wm⁻²K⁻¹. If a significantly higher heat transfer coefficient of 2500 Wm⁻²K⁻¹ from slag to H-DRI is applied, a slight decrease in melting time was observed for slag infiltrated 65% porosity H-DRI. These results indicate that slag infiltration into H-DRI is not beneficial for melting.

1. Introduction

To reduce carbon dioxide emissions in connection with steel production, several steel companies have started transitions from the current Blast Furnace-based iron production to a process based on direct reduction with hydrogen and Electric Arc Furnaces (EAFs). This transition will reduce Sweden’s total carbon dioxide emissions by 7% and Finland’s by 10% when it is completed. Furthermore, the closure of blast furnaces and coke plants will help to reduce emissions of particles, sulphur oxides and nitrogen oxides, which will improve the local environment around the steel plants.¹⁾

Current industrial direct reduction plants use natural gas to produce an iron sponge product containing 1–4.5% carbon, referred to as Direct Reduced Iron (DRI). This product and its melting process have been thoroughly studied in previous research and are currently used to produce approximately 127 million tonnes of steel per year (2022).^1,2,3,4,5) When DRI is melted in EAF a continuous feeding via conveyor belts is preferred to avoid formation unmelted agglomerates or piles of DRI. These unmelted piles are colloquially called “Ferrobergs” and occur due a combination of low heat transfer to the DRI and low internal heat conductivity of DRI. The loose agglomerate of DRI in a Ferroberg melts slowly and result in drastically reduced productivity and increased risk of damage to electrodes and furnace lining.

The properties of hydrogen direct reduced iron (H-DRI) differ from commercial DRI, with a significant difference being the absence of Carbon. Previous research has shown that carbon-free DRI does not generate gas during melting and the EAF process must therefore be supplemented with the addition of biochar or other foaming reagent to enable a foaming slag and reduction of the iron oxides in the slag.⁶⁾ Despite the lack of gas generation, lab-scale tests have shown that heating of H-DRI can be faster than heating of carbon-containing DRI and that the melting time therefore can be shorter.^2,7) The melting of H-DRI has been studied in both experimental and numerical works to determine what factors control the melting time.⁸⁾

During experimental melting of H-DRI in molten slag it has been observed that slag will infiltrate the porous structure before melting, and it has been hypothesized that this may contribute to the increased melting rate. Huss et al. found that the slag infiltrated the entire porous volume of a compacted H-DRI agglomerate after only 5 s in slag. Additionally, for partially infiltrated H-DRI, a precipitated autogenous slag was found at a greater depth than the slag infiltration.⁹⁾ O’Malley studied compacted DRI spheres and found that a solidified shell formed on the surface upon submersion in the slag, and once the solidified shell was melted, the porous structure was quickly filled with infiltrating slag. This type of infiltration behavior was observed for both reactive carbon-containing DRI and unreactive H-DRI.³⁾ Ammasi et al. also found indications of slag infiltration into the porous structure of H-DRI during melting and examined its chemical reactions with the iron and gangue inside H-DRI.¹⁰⁾ The infiltration of molten slag into the porous structure is believed to cause significantly increased effective heat transfer to H-DRI from the slag and within H-DRI due to the extra convective contribution.⁷⁾ The melting time of the H-DRI in the mentioned experimental works vary from a few seconds up to 40 s depending on DRI type, melting medium, and laboratory setup.

Numerical simulations of H-DRI melting have been performed by Gonzalez et al., Pineda-Martínez et al., Govro et al., Calderon Hurtado et al. and Svantesson et al.^{8,11,12,13,14)} These CFD model have all been developed to describe the melting of H-DRI in slag or steel in order to evaluate how the properties of the H-DRI and slag, as well as how different process conditions such as flow and temperature affect their melting rate. The numerical studies have found melting times ranging from 4.56 s to more than 40 s depending on H-DRI size and bath conditions. The effect of infiltration in the porous structure has been included by alteration of the initial temperature, but to the authors knowledge, no efforts have been made to simulate the flow of slag into the porous structure to study how it may affect the heating rate and ultimately melting time of the H-DRI.

The slag infiltration into H-DRI is believed to be governed by the capillary force from surface tension in the capillary sized pores of the structure. As the H-DRI is small and filled with air, the hydrostatic pressure from the surrounding slag is almost equal from all sides of the sphere. For slag infiltration to occur the gas in the porous structure must first either be removed or compressed due to the increasing pressure. The gas may be evacuated via some of the pores during submersion, or even partially dissolved in the slag. The gravitational effect on slag rise in the capillaries will likely be small as the H-DRI is not fixed in the same direction during melting and may “roll” around in the slag due to both outside flow and internal lopsided buoyancy. However, the infiltration phenomenon has not only been observed in industrial scale, but also in laboratory scale, where the submersion depth is very small and the hydrostatic pressure is limited.^9,15) Additionally, the driving force from chemical reactions between the infiltrating slag and oxides in the H-DRI is expected to aid the infiltration.⁹⁾ The contribution of capillary flow on the infiltration behavior and resulting melting rate is unknown and one of the main subjects of this study.

The rate of infiltration and the effect on the heating of the H-DRI will be dependent on the characteristics of the porous structure, such as the size of pores, their interconnectivity, and total porosity. The shape and size of the pores inside the H-DRI has been found to differ depending on the production route, and as noted by Fogelström et al. the heating rate of the H-DRI during reduction significantly impacts the pore structure from a fully porous structure to one with both porous and dense characteristics.¹⁶⁾ The size of the porous structure within H-DRI has been found to be in the range of 10–20 μm.¹⁷⁾ The tortuosity of H-DRI has not been experimentally determined, but a general estimation based on the empirical formula for porous materials according to Eq. (1) can be used, where ϵ is the porosity fraction, τ is the tortosity, and m is the empirical exponent (usually 0.5–1.5 for porous materials), and l (m) is the straight distance (H-DRI radius). In combination with Eq. (2) the effective length L_eff (m) of the porous network to the center of the H-DRI can be estimated.¹⁸⁾

  

$$\tau \approx {ϵ}^{-m}$$(1)

  

$$L_{eff}=l\cdot \tau$$(2)

The imbibition of liquid into pores occur for a wide range of pore-sizes, and as the length scale changes the relative magnitude of the factors deciding the rate of capillary action change. At smaller length scales the surface roughness on walls, chemical reactions at walls, effect of solid particles in the liquid, all have an increased importance compared to the mm length scale, where capillary pressure and gravity dominate.¹⁹⁾

This study uses numerical models, validated with experimental data, to explore the magnitude of slag infiltration and its possible influence on heating and melting rate of H-DRI under steelmaking conditions present in the EAF. By assessing the possible impact of slag infiltration on H-DRI melting rates it is possible to determine if it is a significant mechanism for melting and if it can be exploited to increase the melting rate of H-DRI. To facilitate this, the rate of infiltration of slag into straight capillaries is first investigated using analytical and numerical models, followed by an investigation of the rate of heat transfer from slag in a pore to the surrounding iron matrix. Based on this it is possible to determine the propagation of the heating front caused by the combined conduction through iron matrix and slag and the convection from the inflowing hot slag.

1.1. Capillary Flow Theory

Capillary flow has been studied extensively and analytical equations have been formulated to calculate both capillary pressure (Young-Laplace Equation), the equilibrium rise height (Jurin’s Law Eq. (3)),²⁰⁾ and rate of capillary flow (Lucas-Washburn Equation ).^21,22) These formulations are functions of surface tension σ (N m⁻¹), contact angle θ (deg), gravity g (m s⁻²), density ρ (kg m⁻³), and radius of the capillary r (m).

  

$$\begin{array}{c}{L}_{jur}=\frac{2\sigma \text{cos}\left(\theta \right)}{\rho gr}\end{array}$$(3)

However, all of these formulations assume a straight cylindrical capillary under constant conditions. Additionally the LW equation does not account for the effect of inertia at the early stages of flow and is therefore only valid for fully developed flows.²³⁾ For a more accurate analytical representation of the time-dependent flow within a capillary, the momentum equation for a fluid element in the capillary can be used to derive the Bosanquet equation which takes into account both surface tension, viscous forces, and gravitational forces as function of the current height. The analytical solution to this equation is commonly called “the classical model” and has been reviewed thoroughly since its inception by Das. S et al., Wang et al., Zhmud et al., and many more.^22,24,25) Recent improvements to the Bosanquet equation have been proposed by Wang et al. where the inertia of the fluid in the reservoir outside of the capillary and a correction to the height term to avoid the singularity at h = 0 in the classical model. The resulting analytical formulation in Eq. (4) found good agreement to experimental results performed in the same study.²⁵⁾

  

$$\begin{array}{c}\begin{array}{c}\frac{2\sigma cos\theta }{\rho r}=\left(h+h_0\right)\frac{d^{2}L}{d{t}^2}+\left(1+\frac{\xi }{2}\right){\left(\frac{dL}{dt}\right)}^2\\ +\frac{8\mu }{\rho r^2}\left(L+L_0\right)\left(\frac{dL}{dt}\right)+gL\end{array}\end{array}$$(4)

Where L (m) is the height of the fluid pillar, μ (Pa s) is the dynamic viscosity of the fluid, ξ is 0.5 for rising meniscus in most reservoir-capillary systems, and −3 during falling meniscus to account for the resistance to outflow via entrance and exit effects. This type of analytical equation is solved numerically using the ODE for of the equation together with a numerical scheme like the Runge-Kutta-Fehlberg (RK45) model.

The solution of the analytical model by Wang et al. shows an oscillatory behavior for most fluids unless the viscosity is high, as was first showed by Quéré and then followed by many others.^{22,26,27,28,29)} This fact can be used to categorize capillary rise into oscillating and non-oscillating flows based on the material properties of the fluid via a dampening value. The dampening value (Ω) can be used to calculate the oscillatory nature of a fluid in a domain (Eq. (5)) for a 3D system as derived by Gründig.³⁰⁾ Where a value below 2 implies oscillation around the capillary height, and above 2 is dampened sufficiently to approach the capillary height asymptotically as visualized in Fig. 1.

  

$$\begin{array}{c}\mathrm{\Omega }=\sqrt{\frac{128\sigma cos\theta {\mu }^2}{{\rho }^3{g}^2{r}^5}}\end{array}$$(5)

Fig. 1. Rise height scaled against final height as function of time for varying Ω to visualize onset of oscillation. (Online version in color.)

As the fluids studied in this work are metallurgical slags with high viscosities with high surface tension and in pore structures on the μm scale, the dampening value is very high compared to most experimental work on capillary rise, which has been conducted with water on millimeter scale capillaries. A review of the dampening values for fluids in μm pores shows that they are all well in the overdampened region with Ω > 2, as shown in Table 1. For fluids in the overdampened region the flow never changes direction, which simplifies the inlet boundary conditions and the treatment of the wall effects near the meniscus. The analytical solution of Eq. (4) does not account for the intricacies of near wall effects such as surface roughness or slip-length, nor is it applicable for other geometries than straight capillaries. It is therefore beneficial to employ numerical models to resolve the flow in more detail.³⁰⁾

Table 1. Dampening value for fluids for decreasing capillary size.

Pore size	1 mm	0.1 mm	10 μm
Water	0.12	38.5	12000
Slag	15.8	5000	1.5·106

1.2. Numerical Modeling of Capillary Flows

While calculating the capillary infiltration behavior in simple geometries is trivial with analytical equations, the introduction of complex geometries as present in the porous internal geometry of a H-DRI require more powerful calculation tools. The CFD modeling software Ansys Fluent can be used to calculate the capillary flow in simple and complex geometries using the explicit formulation for the Volume of Fluid (VOF) model.³²⁾ In addition to the material flow the energy transport of a hot slag in the H-DRI porous network can be simulated to provide a description of how the infiltrating slag will affect the heat transfer inside the pore-network and how it affects the melting rate of H-DRI.

Simulations of capillary flows using numerical models has previously been performed by Gründig, and Naghashnejad.^31,33,34) A thorough discussion of previous work in both analytical and numerical solutions of the capillary rise problem is provided by Gründig.²⁹⁾ For a numerical model of capillary rise, the application of the Navier-slip condition on walls at the meniscus instead of the standard no-slip condition is important to avoid the contact line paradox at the wetting of the walls as confirmed by several authors.^35,36,37) The contact line paradox arises when the propagation of the liquid front by wetting is hindered by the no-slip condition as it states that the velocity at the wall is zero, effectively hindering all movement. The Navier-slip condition sets the velocity at the wall to that of the flow at a set distance from the wall called the free-slip length (λ). The slip velocity at the wall (u_slip) is defined in Eq. (6) as the normal gradient of the velocity tangential to the wall times the free slip length.

  

$$u_{sli{p}_x}=\frac{du}{dy}\lambda$$(6)

The influence of the free-slip length on the accuracy to the analytical models was studied by Gründig et al. who tested and found a values close to 1/5 of the capillary radius to be more accurate to the classical model than a slip length of 1/50 of the capillary radius.³¹⁾

In the present work the Navier-slip condition at the meniscus was implemented as a moving wall boundary condition with the components of velocity calculated and adjusted by a User Defined Function (UDF) in Ansys Fluent. The meniscus region was identified by the Volume of Fluid (VOF) gradient and the boundary faces in the identified region has their velocity set to the corresponding slip velocity, while the remaining boundary faces are set to 0 velocity to maintain the no-slip condition for the bulk flow. The inspiration and structure of the UDF was adapted from the thesis of Eriksson and Vanky and further built upon for this specific case.³⁸⁾ The similarity between numerical and analytical models for capillary flow can be high but will change depending on the length scale.

2. Method

2.1. Material Parameters

Capillary flow is controlled by the wetting characteristics between the rising liquid and the wall material of the capillary. The wetting of water to a glass capillary is well documented at room temperature and is used as a reference in the current study. However, the wetting angle between metallurgical slags and reduced iron ore has not been studied experimentally and therefore has to be estimated. Since previous experimental work has shown infiltration, it is reasonable to assume that wetting does occur and that the wetting angle therefore is less than 90°. The wetting angle is determined by the balance of forces at the three-phase boundary between gas, liquid, and solid and is dependent on both temperature, surface roughness, chemical reactions, and the surface tension of the liquid. Due to the uncertainty of this property, the resulting rise rates from the simulations in this study are not to be seen as quantitative values but rather be used as a tool for understanding the mechanisms at play during slag infiltration. As the model does not employ any melting model, the latent heat of the materials is instead included in the simulations by modification of the specific heat over the range of the melting interval. The occurrence of oxidizing reactions between the infiltrating slag and the grains of iron ore in the porous structure may have significant effects on the wetting angle but was not accounted for in this study.

2.2. Capillary Rise Modeling

The modeling of capillary rise dynamics was performed by CFD simulations in ANSYS fluent 19.1. The simulation domains were constructed using SpaceClaim as 2D axisymmetric capillaries with gravity acting along the axis. Outside the capillary a solid region was added with a symmetry condition to replicate the heat transfer through the iron matrix surrounding the pores. The relative size of the capillary and solid region was adapted to resemble a porosity of 60%, in line with earlier findings.¹⁶⁾ The domain was meshed in the ANSYS meshing module to form a regular rectangular mesh. The mesh refinement was defined as the number of cells over the radius of the capillary. The size of the geometries is scaled to fit the desired capillary fluid, with a base size of 1 mm radius for water capillaries and 10 μm radius for slag capillaries. A sample image of a small part of the 10 μm radius geometry with an ongoing capillary rise of slag is depicted in Fig. 2 for a mesh with 20 cells over the radius of the capillary with an added external wall region.

Fig. 2. Section of geometry and mesh from 10 μm capillary during filling with slag indicating inlet (A), outlet (B), capillary wall (C), and outer wall (D), gravity is acting in the -x direction with 9.82 m s−2. (Online version in color.)

The capillary rise model used the explicit Volume of Fluid (VOF) model to resolve the interface between rising fluid and air in the capillaries. The geo-reconstruct scheme was used to reconstruct the meniscus based on the volume fraction of phases. The pressure velocity coupling used the PISO scheme without skewness correction. The spatial discretization uses the least squares cell based approach for the gradient, PRESTO! for pressure, and 2nd order upwind for the momentum. The transient formulation was a first order implicit scheme.

The inlet (A) was defined as a pressure inlet with a small gauge pressure of 3000 Pa to represent a submersion depth of 10 cm in slag and facilitate the initial entry of fluid into the domain. For cases considering energy the temperature was set to 1873 K at the inlet. The outlet (B) was defined as a pressure outlet with 0 gauge pressure, a volume fraction of slag as 0, and a temperature of 300 K. The walls of the capillary (C) were defined as moving walls modified by a UDF which sets the wall velocity in every cell on the wall depending on the Navier-slip condition. The outer wall of the solid (D) was set as a symmetry condition with no heat flux. The UDF controlling the wall velocity uses a slip length of 1/5th of the capillary radius, the complete UDF is included in the appendix of the manuscript.

In addition to the varying size and length of the porosities the expansions and necks in the structure may influence the flowrate or stop it completely in certain regions of the H-DRI. The model for flow through necks in pores uses a of 5 μm capillary radius with a V-shaped taper of constant angle over 20 times the capillary radius down to a minimum neck of 0.5 times the pore thickness. After the neck the capillary opens up again using the same V-shaped taper.

To validate the model, the capillary rise of water was compared to the analytical calculations. The model was then used to study the capillary rise rate for slag using the material properties and contact angles for gas, liquid, and solid as defined in the material properties, Table 2. The actual flowrate of slag into the porous capillary structure is difficult to accurately predict due to the possible variations in slag properties, interactions between slag and iron matrix, as well as pore size and structure. Therefore, the general “best case” flow situation of a straight capillary with constant wetting angle and slag material parameters is studied to show the higher end of possible flowrates.

Table 2. Material properties of Water, Slag, Iron, and Air. *Values vary by temperature within the specified range. **Assumption based on best available literature.

Phase	Water	Slag	Iron	Air
Density [kg m−3]	1000	3000	7450–7850*	1.225
Viscosity [kg m−1 s−1]	0.0001	0.143)	–	1.78*10−5
Surface Tension [N m−1]	0.072	0.8	–	–
Wetting Angle [deg]	30	70**	–	–
Thermal conductivity [W m−1 K−1]		0.93)	30–80*	0.0242
Specific Heat [J kg−1 K−1]		12003)	450–820*	1006.43
Latent Heat [J kg−1 mol−1]		600000	288000
Solidus Temperature [K]		15233)	1801
Liquidus Temperature [K]		15833)	1810

For the lower limit of slag infiltration, very small capillaries with intricate porous structure and thin necks in the structure will create the slowest possible infiltration, if additionally chemical reactions change the wetting angle between the slag and the iron matrix, even slower infiltration rates are possible.

The influence of inlet pressure on the capillary rise rate was studied to determine the magnitude of the hydrostatic pressure on capillary infiltration. Additionally, the heat transfer in the system was studied to determine the rate of heat transfer between the iron matrix and the infiltrating slag to evaluate the possible solidification of slag in the porous structure.

2.3. Validation

The validation of the capillary rise model was done via comparison to the analytical formulations of capillary rise as described previously in Eq. (3) for the equilibrium height in a capillary, and Eq. (4) for the time-dependent rise rate in a capillary. The equilibrium rise height of water in a 1 mm parallel plate capillary for increasing mesh refinements is presented in Table 3 according to the procedure established by Celik and Richardson.^39,40) The grid convergence index (GCI) shows that refinement to mesh 3 provides a convergence index of 1% indicating sufficient mesh independence, further refinement to mesh 4 only provides slight improvements in accuracy. The analytical equilibrium rise height of water in such a system is 12.72 mm according to Jurin’s equation (Eq. (3)).

Table 3. Mesh sensitivity analysis for water rise in 1mm distance parallel plate capillary.

Mesh	Avg cell size [mm]	Cells over width	Refinement	Rise height, Φ [mm]	GCIfine
1	0.15	7	–	12
2	0.10	10	1.43	12.2
3	0.075	14	1.33	12.5	0.01
4	0.050	20	1.5	12.6	0.0049

For the rise of slag in a cylindrical 10 μm radius capillary the analytical equilibrium rise height is 1.8 m. Since this is many times the radius of the H-DRI it is of greater interest to ensure appropriate rise rate in the early stages of capillary rise. By comparison to the analytically calculated value of the modified classical model in Eq. (4) we can ensure appropriate resolution of the effect of inertia as well as the effect of mesh refinement. The meniscus heights as functions of time for mesh resolution of 5, 10, and 20 cells over the radius are compared to the analytical value in Fig. 3. Improved accuracy to the analytical value can be seen when moving from 5 to 10 cells over the radius, but further refinement does not significantly improve accuracy.

Fig. 3. Meniscus height for slag in a cylindrical 5μm radius capillary as function of time for 5, 10, 20 cells over radius. (Online version in color.)

2.4. Heat Transfer Modeling of Spherical H-DRI

Heat transfer modeling of H-DRI was performed on a spherical H-DRI to evaluate how the heat from the infiltrating slag and the changing average material properties of the H-DRI during heating and infiltration will affect the heating and melting of the H-DRI. The boundary of the H-DRI was set as a convection boundary condition with a free stream temperature of 1873 K and a heat transfer coefficient of 500 or 2500 W m⁻² K⁻¹. These values were based on the natural convection of a 1873 Kslag around a 300 K sphere and can be considered as initial values and fully developed flow, similar h-values were also found in previous research.⁸⁾ A wall thickness of slag is applied on the outside of the simulation domain to imitate the frozen layer of slag which forms when the cold H-DRI is submerged in hot slag. The thickness of the slag layer is set to 1 mm when the wall temperature is below 1523 K and 0 mm when the temperature is higher, as it is impossible for a frozen slag to be present on the surface while infiltration is occurring, see schematic model in Fig. 4. In reality, the thickness of the frozen slag layer will grow and shrink dynamically when the H-DRI is submerged in slag, depending on the heat flux, H-DRI temperature, and slag properties.

Fig. 4. Schematic image of heat transfer model with indicated frozen slag layer, H-DRI zone, and the applied surface temperature (Tsurf). (Online version in color.)

The model considers the H-DRI as a single phase where the effective material properties of the porous iron matrix filled with air or slag are used. This simplification does not account for the convective heat transfer of the infiltrating slag, which is instead added by an energy source term for the domain but restricted to distribute the energy only where slag is present. The magnitude of the source term is calculated from the volume change of the infiltrated zone during each timestep, assuming that the released heat from slag cooling from 1873 to 1523 K is all absorbed by the H-DRI during infiltration. The calculation of infiltrated volume, released energy, and distribution of it are all managed by a set of UDFs listed in appendix.

The material properties of the heat transfer model are calculated based on the values in Table 2 to describe how the mixture of properties in matrix, pore, and infiltrated pores combine to average material properties. The density and the specific heat of the combined material properties are calculated according to volume fraction with weighting according to the assumed porosity. The effective thermal conductivity is estimated based on Landauer’s relation (Eq. (7)) which considers the microstructure for an assumed isotropic porous material. The equation uses the conductivity of the material in the porosity (k_p), thermal conductivity of the matrix (k_m), and porous fraction (f), to calculate the effective thermal conductivity (k_eff) for the porous material.^41,42)

  

$$\begin{array}{c}{k}_{eff}=k_m\frac{2k_p+k_m-2f\left(k_m-k_p\right)}{2k_p+k_m+f\left(k_m-k_p\right)}\end{array}$$(7)

The heat transfer model was used to compare the heating rate of H-DRI for three different assumptions of infiltration. It is assumed that the H-DRI will either experience no-infiltration, a time-dependent-infiltration along the iso-temperature line of 1523 K (slag melting temperature), or a near instant slag infiltration only hindered by capillary flowrate.

For the assumption of no-infiltration the material properties are set to the combination of iron and air in the porosities. For the second case of time-dependent-infiltration along the 1523 K iso-line, the H-DRI is considered to be comprised of iron matrix and air up until the melting temperature of slag at 1523 K, and then gradually transition to a structure filled with slag and iron at 1583 K and above. The latent heat of slag is not added to the calculation as it is assumed to come in as liquid, then solidify, and then remelt, resulting in a net zero effect of the latent heat. For the third infiltration case where slag is assumed to fully infiltrate the H-DRI before any heating is done, the initial temperature of the H-DRI is set to 1250 K for a 60% porosity and 1350 K for a 65% porosity, according to the equilibrium temperature of a 1873 K slag solidifying and cooling in a 300 K iron matrix. The latent energy for melting of the slag is then accounted for by increasing the specific heat capacity of the effective material properties between 1523 and 1583 K.

The latent heat of iron during melting is added to the model by increasing the specific heat capacity (c_p) of iron over the melting interval of 1800 – 1810 K. When the temperature is above the melting temperature of iron at 1811 K, the material in the model should be considered completely molten. This is implemented in the model by increasing the conductivity to 100 W m⁻¹ K⁻¹, and decreasing the density and specific heat to 1 kg m⁻³ and 1 J kg⁻¹ K⁻¹ respectively. This removes most resistance to heat transfer by imposing a very high thermal diffusivity and effectively allows the surface of the solid H-DRI to experience the boundary condition without moving the boundary of the simulation domain.

3. Results

3.1. Capillary Rise Modeling

3.1.1. Flowrates

The infiltration of slag into a straight 10 um radius cylindrical capillary was described accurately by the model as indicated by the validation presented in Fig. 3. For such a system the rise height over time and thus the flowrate can be estimated by solving the analytical Eq. (4) for any set of capillary size, slag material properties, and interaction coefficients. In Fig. 5 the rise height of slag over time for circular capillaries ranging from 2 μm to 20 μm radius is presented to visualize the impact of pore size.

Fig. 5. Analytical rise height of slag as function of time in capillaries with radius from 2 to 20 μm. (Online version in color.)

For the slag to infiltrate to the center of a H-DRI the effective pore network length as defined in Eq. (2) needs to be covered. For a 65% porosity H-DRI with a radius of 5 mm the effective pore network length is estimated to 7.5 mm assuming a medium exponent coefficient of 1. According to Fig. 5, the slag may then fill the H-DRI in 2 s for large pore sizes while requiring an estimated 16 s for the smallest studied pores. Additionally, the spherical geometry of the H-DRI requires the coalescence of pores to maintain constant porosity as we approach the center, which reduces the volumetric flowrate of slag required to fill the pores, this significantly speeds up the possible fill-rate of the H-DRI.

In addition to the size of the pores, the shape and structure of the pores may also affect the infiltration rate. For a pore with a V-neck taper to 0.5 times the radius of the pore, the rise-rate deviates from the analytical values, as shown in Fig. 6. The slag rise rate initially follows the analytical rise rate for the full radius capillary until the necking starts. In the necking area the flow is concentrated and the effective rise rate is significantly faster than the analytical values as the flow is forced through a smaller orifice by its previous inertia. When the meniscus passes the throat of the neck the rise rate slows down and eventually stabilizes in the second straight region of the pore. The evolution of the rise height then approaches the analytical rise height of a capillary with a constant width of the neck radius, indicating that the smallest neck in a pore will control the total rise rate due to constriction of flow. In contrast to pipe flow following Bernoulli’s principle, the Poiseuille flow in capillaries is dominated by viscous drag and the volumetric flow rate, Q is proportional to r⁴ as described by Poiseuilles’s law for flow in Eq. (8) where L is the length of the capillary.²¹⁾

  

$$\begin{array}{c}Q=\frac{\pi r^4\mathrm{\Delta }P}{8\mu L}\end{array}$$(8)

Fig. 6. Rise height of slag in a 5 μm radius cylindrical capillary with a 2,5 μm neck as function of time compared to the analytical rise height of the full radius and the neck radius capillary. (Online version in color.)

The external hydrostatic pressure applied to the pores from submerging H-DRI in slag is approximately 3 kPa at 0.1 m depth and 30 kPa at 1 m depth. The effect of applied external pressure on the flowrate of slag into a 5 μm radius cylindrical capillary is presented in Fig. 7, where the hydrostatic pressure is shown to provide an increase in slag infiltration rate of approximately 20%. This is proportional to the increase in total pressure from hydrostatic pressure and capillary pressure (Eq. (3)) which for this capillary configuration is 30 kPa + 109 kPa = 139 kPa. This aligns well with Poiseuille’s law in Eq. (8) where the total flow is proportional to the total pressure difference over the capillary. In reality the air which fills the H-DRI will also present a counter pressure to the infiltration unless it is evacuated from the structure. However, this was not accounted for in the simulations.

Fig. 7. The rise height of slag in a 5 μm radius cylindrical capillary as function of time when subjected to external hydrostatic pressure. (Online version in color.)

3.1.2. Heat Exchange in Capillary Flow

Simulations with thermal energy show that the infiltrating slag quickly exchanges energy with the iron matrix and equilibrates in temperature, indicating a very rapid solidification of the infiltrating slag. In Fig. 8 the meniscus height from the inlet of the capillary is plotted over the infiltration time and compared to the location of the isotherm of 1523 K representing the slag solidification temperature. The difference between the capillary infiltration length and the temperature front indicates that the meniscus temperature is lower than the slag solidification temperature, and would therefore be solidified if solidification was enabled in the model. However, the design of the model does not allow implementation of the solidification module in an accurate way as the wall movement used for the Navier-slip UDF implementation will create erroneous movement in the solidified frozen layer, and the contact angle between wall and liquid slag will not be present if the slag has solidified on the wall, as it is defined on the wall boundary. Since the solidification module is not used, the slag movement will not be inhibited by the partial blocking of the capillary from solidification of the walls of the capillary. Neither will it be completely blocked in the event that full solidification occurs at any point in the capillary. The resulting behavior is that the model allows flow of the slag even below its solidification temperature. Nevertheless, the 1523 K isotherm has been used as a measure of slag infiltration speed.

Fig. 8. Infiltration length of slag in 5um radius cylindrical capillary compared to the distance from the inlet to the iso-temperature of 1523K in the center of the capillary and the center of the wall. (Online version in color.)

The temperature in the fluid and in the solid, in Fig. 8, appear equal but in reality the iso-temperature line of 1523 K in slag is slightly higher compared to the solid, as clarified in Fig. 9. This small difference will cause a convective heat transport by the slag into the H-DRI, which contributes to the local heating of the matrix, albeit very little, due to the small difference in temperature. The distance from the capillary entrance to the iso-temperature line of 1523 K in the slag also represents the possible infiltration distance before solidification starts, which according to Fig. 9 is approximately 25 μm, only 5 times the capillary radius. As the model suggests that solidification starts almost instantaneously, small variations in slag properties and slag to wall contact angles are assumed to have minimal effect on the infiltration behavior.

Fig. 9. Distance from inlet to iso-temperature line of 1523 K in fluid slag and solid. (Online version in color.)

3.2. Heat Transfer Modeling of Spherical H-DRI

3.2.1. Calculated Thermal Diffusivity

The effective thermal diffusivity of the combined material properties of iron matrix and pore provide indications of the heat transfer in the different simulation cases. For the base slag with properties according to Table 2 and a porosity of 60 or 65% the effective thermal diffusivity varies over the entire temperature range according to Fig. 10. The porosities are there combined with the three possible infiltration cases outlined previously: no-infiltration (iron matrix and air), time-dependent-infiltration of slag above 1523 K (iron matrix and air up to 1523 K, then iron matrix and slag), and instant-infiltration (iron matrix and slag).

Fig. 10. Thermal diffusivity as function of temperature for the combined material properties of iron, slag, and air, associated with no-infiltration, time-dependent-infiltration, or instant-infiltration cases into 60 or 65% porosity H-DRI. Note: time-dependent-infiltration does not start until 1523 K. (Online version in color.)

For a 60% porosity H-DRI the thermal diffusivity is equal or higher in the air-filled iron matrix than one filled with slag over the entire temperature range. If a higher porosity of 65% is assumed the thermal diffusivity of the slag infiltrated structure is higher than the air-filled porosity over the entire temperature interval, but the time-dependent-infiltration case will only experience the higher thermal diffusivity above 1523 K when the slag is assumed to enter the structure.

3.2.2. Heating and Melting of H-DRI

When the effective material properties are used for the heating of a 10 mm H-DRI it is possible to track the temperature front in the material and associate it to infiltration and melting. In Figs. 11, 12, 13, 14 the H-DRI radius is the distance from the core of the H-DRI to the temperature front of the iron melting temperature of 1810 K, and the slag infiltration depth is the distance from the surface to the iso-temperature of slag solidus at 1523 K. In these cases the heat transfer coefficient is 500 or 2500 W m⁻² K⁻¹ from a free stream temperature of 1873 K and with a slag thermal conductivity of 0.9 W m⁻¹ K⁻¹, applied on 60 or 65% porosity H-DRI. In Figs. 11, 12, 13, 14 the temperature at the center and the unmelted radius of the H-DRI is shown for three different infiltration assumptions: no-infiltration, time-dependent-infiltration along the melting temperature of slag, and instant-infiltration.

Fig. 11. H-DRI radius and center temperature as functions of time of a 60% porosity H-DRI heated with h=500 for: no-infiltration, time-dependent-infiltration, and instant-infiltration. (Online version in color.)

Fig. 12. H-DRI radius and center temperature as functions of time of a 65% porosity H-DRI heated with h=500 for: no-infiltration, time-dependent-infiltration, and instant-infiltration. (Online version in color.)

Fig. 13. H-DRI radius and center temperature as functions of time of a 60% porosity H-DRI heated with h=2500 for: no-infiltration, time-dependent-infiltration, and instant-infiltration. (Online version in color.)

Fig. 14. H-DRI radius and center temperature as functions of time of a 65% porosity H-DRI heated with h=2500 for: no-infiltration, time-dependent-infiltration, and instant-infiltration. (Online version in color.)

If the infiltration is assumed to be instantaneous, all the heat from the infiltrating slag will be distributed in the H-DRI and can be represented as a higher initial temperature of the H-DRI. The 60% porosity H-DRI has an initial temperature of 1250 K and the 65% porosity H-DRI starts at 1350 K, in accordance with the heating from the infiltrating slag.

For the heating of the 60% porosity H-DRI with 500 W m⁻² K⁻¹ in Fig. 11 the instant-infiltration case has a higher initial temperature of 1250 K but the time-dependent-infiltration and no-infiltration cases show significantly higher rates of heating and quickly surpass the instant-infiltration temperature. The time-dependent-infiltration case results in a faster center temperature increase than the no-infiltration sample above the infiltration temperature of 1523 K. The melting of the H-DRI starts earliest in the time-dependent-infiltration case, and the instantly infiltrated case is the last to start melting. However, despite the addition of heat from the infiltration, the case without infiltration reaches full melting first after 57.5 s, before the time-dependent-infiltration case which melts after 73.3 s, and lastly the instantly infiltrated case after 93 s.

For the 65% porosity H-DRI with 500 W m⁻² K⁻¹ shown in Fig. 12 the heating effect of the infiltrating slag is more prominent than for the 60% porosity H-DRI, but in principle the results are comparable. The case with no-infiltration melts first at 68.3 s followed by the time-dependent-infiltration case at 72 s, and lastly the instantly infiltrated case at 88 s.

For the 60% porosity H-DRI with 2500 W m⁻² K⁻¹ in Fig. 13 the H-DRI heats up and melts significantly faster while maintaining the same relative difference between the infiltration assumption cases. All the cases start melting at approximately the same time but the case with no-infiltration finishes melting first after 21.0 s, followed by the time-dependent-infiltration at 23.3 s and lastly the instant-infiltration case melts at 27.1 s. For the 65% porosity H-DRI with 2500 W m⁻² K⁻¹ in Fig. 14 the melting behavior changes slightly. The instantly infiltrated case is the first to start melting but is then surpassed by both other cases. The case with time-dependent-infiltration finishes melting first at 22.3 s, compared to 23.2 s for the case with no-infiltration and 24.5 s for the instantly infiltrated case. Comparison of Figs. 11, 12, 13, 14 show that the total melting time for an instantly infiltrated H-DRI is always longer than for time-dependent-infiltration or no-infiltration H-DRI regardless of porosity or heat transfer coefficient. Comparison of final melting times presented in Table 4 with previous melting of H-DRI in experimental and numerical works indicate that the melting times from the current heat transfer model fall within the normal range of 10–40 s for the 2500 W m⁻² K⁻¹ heat transfer cases, but the 500 W m⁻² K⁻¹ heat transfer coefficient cases the presented melting times are longer than what has been previously reported.

Table 4. Melting times for varying porosity, heat transfer coefficient, and infiltration assumption.

Porosity [%]	Heat Transfer Coefficient [Wm−2K−1]	No-infiltration	Time-dependent-infiltration	Instant-infiltration
60	500	57.5	73.3	93.0
65	500	68.3	72.0	87.9
60	2500	21.0	23.3	27.1
65	2500	23.2	22.3	24.5

3.2.3. Slag Infiltration Rate

The infiltrating slag front is tracked by the 1523 K iso-temperature and the results for varying h values of the inflowing heat are shown in Fig. 15 for 60% porosity and in Fig. 16 for 65% porosity. The figures show that the infiltration is very slow at first and then almost unhindered once the bulk of the H-DRI reaches the slag infiltration temperature. Comparison of Figs. 15 and 16 also shows that the total melting time is very similar for 60 and 65% porosity H-DRI despite the significant difference in slag infiltration effect and thermal diffusivity of the infiltrated areas. Increasing the heat transfer coefficient to the H-DRI by a factor 5 from 500 to 2500 W m⁻² K⁻¹ cuts the infiltration time in half and reduces the total melting time for the H-DRI from 72–73.3 s to approximately 23 s.

Fig. 15. Infiltration depth of slag along the 1523 K iso-temperature and H-DRI size for varying h-values in 60% porosity. (Online version in color.)

Fig. 16. Infiltration depth of slag along the 1523 K iso-temperature and H-DRI size for varying h-values in 65% porosity. (Online version in color.)

4. Discussion

The slag infiltration into porous H-DRI prior to melting has been studied using numerical simulations to determine the possible infiltration depth, the rate of slag infiltration, the energy exchange between slag and iron matrix within the capillary, and the effective thermal properties of the H-DRI during infiltration and how these affect the melting rate of H-DRI.

4.1. Capillary Rise of Slag in Pores

The infiltration rate of slag in porous structures depends on the combination of slag properties, slag-H-DRI interaction coefficients, and the shape and size of the pore structure. For the capillary rise modeling it was assumed that straight uniform pores provide the fastest capillary rise and can therefore be considered the upper bounds of slag infiltration rate. For a cylindrical pore the rise height as a function of time was calculated using the analytical equation in Fig. 5. For the slag to infiltrate to the center of an average H-DRI of radius 5 mm, it has to cover a distance of 7.5 mm due to the assumed tortuosity of the pore structure. For the studied slag with the material properties according to Table 2, a capillary of 20 μm reaches 7.5 mm after 2 s, and a 10 μm pore after 4 s. The spherical geometry of the H-DRI also results in a decreasing pore volume needing to be filled when approaching the center of the H-DRI. As the pore structure is not likely to have perfectly uniform diameter the effect of necks in the pore on the rise rate was studied. A neck in the pore will significantly slow down the rise rate of slag, to what appears to be the rise rate of the smallest diameter, according to Fig. 6.

The effect of external pressure on the rise rate of slag was studied and visualized in Fig. 7 where it was observed that an external pressure will increase the rise rate relative to the increase in total pressure. An added external hydrostatic pressure of 30 kPa resulted in an increased rise rate of 20% in a 5 μm radius capillary. The observed effect of necks and hydrostatic pressure on the total flow can both be related to theory in Poiseuille’s law of flow in Eq. (8). The hydrostatic and capillary pressure will in part be counteracted by the internal pressure from the gas trapped inside the pore structure. For simplicity it is assumed that the trapped gas will escape though the top part of the H-DRI as soon as slag infiltration starts.

In addition to the structure and pressure, the composition of slag and its chemical reactions with the iron matrix may affect the contact angle and in turn the infiltration rate. The possible solidification of slag on the walls of the capillary will likely also significantly affect the contact angle and infiltration rate, but neither of these behaviors were studied in this work.

4.2. Solidification of Slag in Pores

The slag infiltration simulations with heat showed that a slag at 1873 K infiltrating a 300 K iron matrix driven by a small hydrostatic pressure and capillary forces was cooled to its solidification temperature within 5 capillary radii from the inlet due to the rapid heat transfer from the slag to the iron, as illustrated in Figs. 8 and 9. The solidifying slag on the wall of the pore reduce the open radius of the pore which impedes bulk flow and further promotes solidification. The solidified slag will hinder any further infiltration until the temperature of the surrounding iron matrix reaches the melting temperature of slag.

A similar behavior has been observed in experimental environments, where slag or steel will quickly solidify on the surface of H-DRI during submersion due to the rapid heat transfer to the H-DRI and insufficient conductive and convective heat transfer in the slag.^13,14) Since the heat transfer from a free flowing slag to H-DRI is sufficiently fast to enable the formation of a solidified layer on the outside of a H-DRI while in a molten bath, it is reasonable that slag flowing within small porosities in H-DRI will quickly solidify if the temperature of the surrounding iron matrix is low. However, since the conductivity of iron is high, the heat will spread through the iron matrix using the connected pathways, and also through the slag. Eventually the temperature of the iron matrix and the slag in the surface region of the H-DRI will be above the melting point of the slag, allowing the infiltration to proceed. At this point the molten slag in the pore will contribute to the remaining heating of the iron matrix by convective flow of hot slag and bridging of insulated pores in the matrix structure.

With this reasoning the infiltrating slag is expected to show a time-dependent-infiltration where it must be preceded by conductive heating of the iron matrix to a temperature above the slag melting temperature. This will significantly hinder the infiltration rate and render the heating almost solely dependent on the conductive heat transfer through iron matrix and slag filled porosities. The time-dependent-infiltration behavior may explain why experimental work has found varying degrees of slag infiltration despite very long submersion times.⁷⁾ I.e. if the iron matrix never reaches the slag melting temperature, infiltration would not be possible. Despite the low effective thermal conductivity of H-DRI the local thermal conductivity of iron is significantly higher than any metallurgical slag, meaning that without flow, the heat will dissipate and spread through the iron matrix faster than in slag filled pores. However, if the slag is moving, the convective contribution will increase the heat transfer within the slag phase, allowing it to maintain a higher temperature than the iron matrix as was seen in Fig. 9.

4.3. Effect of Slag Infiltration on Heat Transfer

The heat transfer modeling in a spherical H-DRI assumes a constant heat transfer coefficient and free stream temperature outside the simulation domain. It also assumes that the frozen layer on the outside of the H-DRI has a constant thickness of 1 mm as long as the surface of the H-DRI is below the slag melting temperature. The addition of the frozen slag layer will significantly reduce the heat transfer to the H-DRI, leading to increased melting times. Additionally, in the molten zone of the domain the material properties are set to produce a very high thermal diffusivity to make the unmelted part of the domain experience similar heat transfer coefficient regardless of size. In reality the heat transfer coefficient, frozen layer thickness, and slag temperature will all change dynamically during the melting process, both due to changing process conditions, but also due to the changing size of the H-DRI.

The thermal diffusivity of iron filled with air, or iron filled with slag change significantly with temperature as in Fig. 10. In a 60% porosity H-DRI the thermal diffusivity increases in both the time-dependent-infiltration and no-infiltration structures when the temperature reaches 1523 K, with a larger increase in the no-infiltration structure because the thermal conductivity of iron is increasing. However, for a 65% porosity H-DRI the time-dependent-infiltration results in a higher thermal diffusivity as compared to the no-infiltration structure. This shows that slag infiltration may be beneficial to the heat transfer in porous structures if the porosity is high enough. A higher porosity of the structure increases the beneficial effect of infiltrating slag as it helps bridge the gaps in the iron matrix, which would otherwise be isolated by the low conductivity of stagnant air. However, it is important to note that a porosity of 65% results in a significantly lower thermal diffusivity than 60% porosity, despite the more pronounced beneficial effect of slag infiltration. Comparison shows 170% higher thermal diffusivity in the no-infiltration structure and 20% higher thermal diffusivity in the time-dependent-infiltration structure for 60% porosity H-DRI compared to 65% porosity H-DRI above 1523 K.

The differences in material properties were also apparent in the heat transfer simulations. In Fig. 11 the infiltrating slag provided a noticeable increase in the center temperature for both instant infiltration and time-dependent-infiltration, but the total melting time was still shorter for the H-DRI without infiltration. For 65% porosity H-DRI in Fig. 12 the slag infiltration provided an even greater increase to the center temperature, but still did not result in a reduction in melting time. The resulting heating rate, infiltration rate, and melting time are all dependent on the material properties of the slag, and if a higher conductivity is assumed, the heat transfer to the H-DRI increases, along with the thermal diffusivity in the slag infiltrated structure. Which would both increase the heating rate and shorten the time for infiltration and melting.

The convective heating from the inflowing slag together with the higher thermal diffusivity in the slag infiltrated structure allows a fast heating of the H-DRI. However, if the infiltrating slag deposits its heat in the surface of the H-DRI, the inner parts of the iron matrix and the cooled slag are left to be heated by conduction. In the model the convective heating of the inflowing slag is added uniformly over the slag infiltrated volume for the cases with time-dependent-infiltration. In reality, the convective heat of the infiltrating slag will contribute to the heating of the iron matrix mainly to the surface layer of the H-DRI. The radial distribution of this heat contribution will depend on the rate of heat transfer from inflowing slag to iron matrix, which in turn is dependent on the slag infiltration rate. With the increased mass of the infiltrated H-DRI as compared to an empty H-DRI the heat requirement to be supplied via conduction increases further. As is seen in Fig. 14 when the surface heat transfer coefficient is increased to 2500 W m⁻² K⁻¹ , the case with time-dependent-infiltration heats up and melts faster than the case with no-infiltration for 65% porosity.

If it is assumed that the H-DRI is instantly infiltrated by slag it will assume a higher initial temperature. But despite the higher initial temperature the melting time is longer than for both time-dependent-infiltration and no-infiltration. The infiltration of the slag gives a quick initial heating of the H-DRI, but also increases the requirement of conductive heating through the structure to heat up the combined mass of slag and iron matrix to the melting temperature. The higher initial temperature also reduces the thermal gradients within the structure, which reduces the heat transported by conductivity. This results in a significantly longer melting time for the H-DRI as the conduction is a limiting factor.

The net heat provided by the inflowing slag over the course of the melting of the H-DRI will be the difference in sensible heat from the superheat of the slag to the melting temperature of the H-DRI, all other heat contributed by the slag is simply an exchange which later needs to be reintroduced to the slag by conduction to cause melting. The effective contribution of a full slag infiltration is approximately 72 J for a 60% porosity 10 mm H-DRI, and 78 J for a 65% porosity H-DRI, equating to approximately 30 K of heating for a H-DRI. As this is a relatively small fraction of the total 1500 K heating required for melting, it will not compensate for the increased need for conduction through the shell. In fact, the increased heating of the surface of the H-DRI from the infiltration will decrease the conductive heating from bath to H-DRI due to the lower temperature difference, further diminishing the possible positive effect of slag infiltration. If the heat supplied by conduction through the shell is high, as in Figs. 13 and 14, the instant slag infiltration is not very detrimental to the final melting time, but still results in the longest melting time of the three infiltration assumptions. As was visualized in Fig. 10, the thermal diffusivity in the slag filled iron matrix for a 65% porosity H-DRI is higher than the not infiltrated structure. This shows that if the heat transfer coefficient to the H-DRI is large enough, and porosity is high, the slag infiltration can result in shorter melting times.

In previous experimental melting of H-DRI, autogenous slag was observed within the porous structure ahead of the slag infiltration front.⁹⁾ This indicates that the temperature of the iron matrix was sufficiently high to allow the formation of autogenous slag before the slag reached that point, supporting the conclusion that slag infiltration is temperature dependent. With the assumption of time-dependent-infiltration only when the local temperature of the H-DRI exceeds the slag melting temperature of 1523 K, the melting time and infiltration rate will be very dependent on the heat transfer coefficient from bath to H-DRI. As shown in Figs. 15 and 16 it takes 12 s for the slag to fully infiltrate the H-DRI when the heat transfer coefficient is 2500 W m⁻² K⁻¹, and approximately 23 s when the heat transfer coefficient is 500 W m⁻² K⁻¹. Within the studied range the porosity does not have a significant effect on the infiltration rate. For all observed cases the infiltration will initially be very slow, with only limited surface infiltration, and then once the bulk temperature of the H-DRI is sufficiently high it will quickly infiltrate the entire structure in 2–3 s. At this point the infiltration flow rate may be a limiting factor for the slag infiltration rate according to the previously presented capillary infiltration rates. These heat transfer coefficients were chosen to represent a range of possible heat transfer coefficients for convection in metallurgical slags, but significantly higher heat transfer coefficients are possible if the material properties of the slag are different, or forced convection is applied. Even for natural convection, small changes in viscosity, thermal conductivity, and thermal expansion coefficient can double or triple the heat transfer coefficient from slag to H-DRI which facilitates significantly faster heating, infiltration, and melting. A higher heat transfer coefficient to the H-DRI will progressively shift the rate limiting step from the heat transfer coefficient to the internal conductivity. This can be seen when considering the Biot number for the system, which starts as low as 0.1 for a 500 W m⁻² K⁻¹ heat transfer coefficient if the conduction of the slag infiltrated iron matrix is high. This increases to 0.5 for a 2500 W m⁻² K⁻¹ heat transfer coefficient, which is still within the range where the H-DRI can be considered a lumped system for heat transfer. Further increase in heat transfer coefficient or significantly lower internal conductivity would change the rate limiting step to a combined external and internal heat transfer problem.

5. Conclusions

The infiltration of slag into the porous structure of H-DRI and effect on melting time was evaluated using numerical models of capillary flow and heat transfer. The capillary flow models show that slag rising in an iron walled capillary quickly exchanges heat with the wall and assumes equilibrium temperature after as little as 5 capillary radii, resulting in solidification. This indicates that slag infiltration may only occur in areas where the iron matrix temperature is above the slag melting temperature. When the temperature of the iron matrix is above the slag melting temperature, the rate of slag infiltration into H-DRI is limited by the capillary rise, based on the pore size and wetting properties of the slag. It is also limited by the shape of the pores, where necking will reduce the maximum flowrate to that of a capillary equal to the smallest neck.

The contribution to heating from the infiltrating slag is the convective flow of heat with the inflowing slag and the increased effective thermal conductivity in the H-DRI structure due to the bridging of insulating air-filled pores in the iron matrix. The total contribution of convective heat transport from the infiltrating slag will be equal to the difference in sensible heat of the slag from the superheat temperature down to the melting temperature of the H-DRI, which represents approximately 30 K increase for a 10 mm H-DRI filled with slag. The convection may transfer more energy than this to the iron matrix, but any additional heat transferred from slag to iron must then be reintroduced via conduction to heat up the slag to its melting temperature. The increased effective thermal conductivity of the infiltrated structure is counteracted by the increased effective density and specific heat capacity, resulting in relatively small changes in thermal diffusivity. For a 60% porosity H-DRI this results in a lower thermal diffusivity for the slag infiltrated structure as compared to a not infiltrated structure, while for a 65% porosity H-DRI the thermal diffusivity increases slightly.

It was shown that a 65% porosity H-DRI may see reductions in melting time from slag infiltration if the heat transfer coefficient from bath to H-DRI is large, while a 60% porosity pellet will always melt slower if it experiences slag infiltration. However, the lowest melting time within the tested range is achieved for a 60% porosity pellet without infiltration. Additionally, melting high porosity H-DRI results in a lower total bulk melting rate of iron due to the lower weight of the more porous H-DRI given the same size, providing another slight benefit of using more dense H-DRI.

Future Work

For further evaluation of infiltration into porous structures a more detailed description of actual pore structures would be required. With a 3D scan of the porous structure the flow behavior into a representative structure could be modeled to visualize how necks, splitting pores, and connecting pore networks interact with the infiltrating slag. Additionally, the effect of chemical reactions and solidification on the contact angle between slag and iron is currently not well studied and deserve further attention.

For further refinement of the model the effect of the viscosity of the slag on its ability to flow into porosities, how the degree of foaming of a slag affects its infiltration, and how both affect how the slag solidifies on the surface of the H-DRI and inside the porosities should be studied.

Statement for Conflict of Interest

The authors declare that there is no conflict of interest in this work.

Acknowledgements

This work was performed in collaboration between SSAB, HYBRIT, and KTH Royal Institute of Technology. Part of the simulations were performed on resources provided by the Hillert Modelling Laboratory at KTH funded by the “Hugo Carlssons Stiftelse för vetenskaplig forskning”.

Nomenclature

τ: Tortuosity

ϵ: Porosity fraction

m: Empirical exponent

L_eff: Effective pore length

r: Pore radius

Δp_cap: Capillary pressure

γ: Surface Tension

θ: Contact Angle

Δp_g: Hydrostatic pressure from capillary fluid

g: Gravity

L_jur: Jurin Height

Δp_poiseuille: Poiseuille Pressure

μ: Dynamic viscosity

u: Flow velocity

t: Time

ρ: Density

ξ: Meniscus flow direction correction term

Ω: Dampening value

λ: Free slip length

k: Thermal Conductivity

f: Porosity fraction

Q: Volumetric flowrate

References

• 1)  Annual Report 2022, SSAB, https://mb.cision.com/Main/980/3740173/1939986.pdf, (accessed 2025-05-22).
• 2)   J.  Elliot,  J.  Nauman and  K.  Sadrnezhaad: Proceedings of the 3rd International Iron Steel Congress, ASM 1978, Chicago, (1978), 397. https://docs.sadrn.com/papers/001.pdf, (accessed 2025-05-22).
• 3)  R. O’Malley: Ph.D. Thesis, Massachusetts Institute of Technology, Philadelphia, (1983). http://hdl.handle.net/1721.1/44645, (accessed 2025-05-22).
• 4)   K.  Sadrnezhaad: Ph.D. Thesis, Massachusetts Institute of Technology, Massachusetts, (1979). http://hdl.handle.net/1721.1/44235, (accessed 2025-05-22).
• 5)  Midrex Technologies: Midrex STATS Book 2022, (2023). https://www.midrex.com/wp-content/uploads/MidrexSTATSBook2022.pdf, (accessed 2025-05-22).
• 6)   A.  Pfeiffer,  G.  Wimmer and  J.  Schenk: Materials, 15 (2022), 5691. https://doi.org/10.3390/ma15165691
• 7)   J.  Huss,  A.  Vickerfält and  N.  Kojola: Steel Res. Int., 95 (2024), 1. https://doi.org/10.1002/srin.202300325
• 8)   J.  Svantesson,  M.  Ersson and  N.  Kojola: Metall. Mater. Trans. B, (2025), 2846. https://doi.org/10.1007/s11663-025-03504-z
• 9)   J.  Huss: Ph.D. Thesis, KTH Royal Institute of Technology, Stockholm, (2023). ISBN: 978-91-8040-685-7
• 10)   A.  Ammasi,  P. M.  Rahul Karthik and  D.  Vishal: J. Sustain. Metall., 10 (2024), 542. https://doi.org/10.1007/s40831-024-00802-9
• 11)   O.  Gonzalez,  Y.  Guzman,  M.  Ramirez-Argaez and  A.  Conejo: Arch. Metall. Mater., 53 (2008), 359. https://api.semanticscholar.org/CorpusID:136638798
• 12)   E.  Pineda-Martinez,  C.  Hernandez-Bocanegra,  A.  Conejo and  M.  Ramirez-Argaez: ISIJ Int., 55 (2015), 1906. https://doi.org/10.2355/isijinternational.ISIJINT-2015-190
• 13)   J.  Govro,  A.  Meena and  S.  Chakraborty: Iron Steel Technol., 20 (2023), 58.
• 14)   F.  Calderon Hurtado,  J.  Govro,  A.  Emdadi and  R.  O’Malley: Metals, 14 (2024), 821. https://doi.org/10.3390/met14070821
• 15)   A.  Pfeiffer,  D.  Ernst,  H.  Zheng,  G.  Wimmer and  J.  Schenk: Metals, 13 (2023), 978. https://doi.org/10.3390/met13050978
• 16)   J.  Fogelström and  J.  Martinsson: ISIJ Int., 65 (2024), 193. https://doi.org/10.2355/isijinternational.ISIJINT-2024-288
• 17)   J.  Huss,  A.  Vickerfält and  N.  Kojola: Steel Res. Int., 94 (2023), 10. https://doi.org/10.1002/srin.202300064
• 18)   B.  Ghanbarian,  A. G.  Hunt,  E.  Ewing and  M.  Sahimi: Soil Sci. Soc. Am. J., 77 (2013), 1461. https://doi.org/10.2136/sssaj2012.0435
• 19)   G.  Bamorovat Abadi and  M.  Bahrami: Sci. Rep., 12 (2022), 14867. https://doi.org/10.1038/s41598-022-19111-w
• 20)   G.  Batchelor: An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, (1967), 67. ISBN 8185618240
• 21)   J.  Bikerman: Surface Chemistry: Theory and Applications, Academic Press Inc., New York, (2013), 23. ISBN ‎0120978563
• 22)   B. V.  Zhmud,  F.  Tiberg and  K.  Hallstensson: J. Colloid Interface Sci., 228 (2000), 263. https://doi.org/10.1006/jcis.2000.6951
• 23)   M. A.  Oliva and  D. D.  Joye: J. Colloid Interface Sci., 51 (1975), 509. https://doi.org/10.1016/0021-9797(75)90147-2
• 24)  S. Das and S. Mitra: Phys. Rev. E, 87 (2013), 063005. https://doi.org/10.1103/PhysRevE.87.063005
• 25)   Q.  Wang,  L.  Li,  J.  Gu and  N.  Weng: Chem. Eng. Sci., 209 (2019), 115212. https://doi.org/10.1016/j.ces.2019.115220
• 26)   D.  Quéré,  E.  Raphael and  J.  Ollitrault: Langmuir, 15 (1999), 3679. https://doi.org/10.1021/la9801615
• 27)   N.  Fries and  M.  Dreyer: J. Colloid Interface Sci., 320 (2008), 259. https://doi.org/10.1016/j.jcis.2008.01.009
• 28)   N.  Fries and  M.  Dreyer: J. Colloid Interface Sci., 338 (2009), 514. https://doi.org/10.1016/j.jcis.2009.06.036
• 29)   D.  Gründig: Int. J. Multiphase Flow, 128 (2020), 103230. https://doi.org/10.1016/j.ijmultiphaseflow.2020.103210
• 30)   D.  Gründig: Ph.D. Thesis, Technische Universität Darmstadt, Darmstadt, (2019). https://doi.org/10.25534/tuprints-00011442, (accessed 2025-05-22).
• 31)   D.  Gründig,  M.  Smuda,  T.  Antritter,  M.  Fricke,  D.  Rettenmaier,  F.  Kummer,  P.  Stephan,  H.  Marschall and  D.  Bothe: Appl. Math. Model., 86 (2020), 142. https://doi.org/10.1016/j.apm.2020.04.020
• 32)  ANSYS, Fluent 19.1 User Guide, Canonsburg, (2019).
• 33)   M.  Naghashnejad and  H.  Shabgard: 5th-6th Thermal and Fluids Engineering Conference, Virtual Conference, (2021), 269. https://doi.org/10.1615/TFEC2021.cmd.036699
• 34)   M.  Naghashnejad,  H.  Shabgard and  T.  Bergman: Int. J. Heat Mass Transf., 186 (2022), 122438. https://doi.org/10.1016/j.ijheatmasstransfer.2021.12249
• 35)   C.  Huh and  L. E.  Scriven: J. Colloid Interface Sci., 35 (1971), 85. https://doi.org/10.1016/0021-9797(71)90188-3
• 36)   J.  Sprittles and  Y.  Shikhmurzaev: Int. J. Numer. Meth. Fluids, 68 (2012), 1257. https://doi.org/10.1002/fld.2603
• 37)   J.  Lowndes: J. Fluid Mech., 101 (1980), 631. https://doi.org/10.1017/S0022112080001838
• 38)   E.  Love and  V.  Patricia: Msc. Thesis, Chalmers University of Technology, Gothenburg, (2020). https://hdl.handle.net/20.500.12380/301605, (accessed 2025-05-22).
• 39)   I.  Celik,  U.  Ghia,  P.  Roache,  C.  Freitas,  H.  Coleman and  P.  Raad: J. Fluids Eng., 130 (2008), 078001. https://doi.org/10.1115/1.2960953
• 40)   L.  Richardson: Philos. Trans. R. Soc. Lond. A, 210 (1910), 307. https://doi.org/10.1098/rsta.1911.0009
• 41)   R.  Landauer: J. Appl. Phys., 23 (1952), 779. https://doi.org/10.1063/1.1702301
• 42)   D.  Smith,  A.  Alzina,  J.  Bourret,  B.  Nait-Ali,  F.  Pennec and  N.  Tessier-Doyen: J. Mater. Res., 28 (2013), 2260. https://doi.org/10.1557/jmr.2013.179
• 43)   E.  Turkdogan and  R.  Fruehan: Fundamentals of Iron and Steelmaking - The Making, Shaping and Treating of Steel, Vol. 2, AISE Steel Foundation, Pittsburgh, (1998), 100. ISBN: 0930767039

Appendix

1. User Defined Functions

1.1. Navier Slip Condition on Capillary Walls

#include “udf.h”

enum {

    VOF_G_X,

    VOF_G_Y,

    VOF_G_Z,

    CONTACT_N_REQ_UDM,

};

/* Define slip length in meters */

#define C_VOF_G_X(C, T) C_UDMI(C, T, VOF_G_X) /*VOF gradient in x*/

#define C_VOF_G_Y(C, T) C_UDMI(C, T, VOF_G_Y) /*VOF gradient in y*/

#define C_VOF_G_Z(C, T) C_UDMI(C, T, VOF_G_Z) /*VOF gradient in y*/

#define VOF_GRAD_THRESHOLD 1e-5 /* Threshold for detecting meniscus region */

#define SLIP_LENGTH 0.0000001 /* [m] */

#define MU_WALL 0.1 /* [kg/m-s] */

#define theta 70

#define sigma 0.8

DEFINE_ADJUST(vofgrad, domain) {

    Thread *ct, *pct;

    cell_t c;

    int phase_domain_index = 1;

    Domain *pDomain = DOMAIN_SUB_DOMAIN(domain, phase_domain_index);

    if (first_iteration) {

        /* Check UDMs have been set up */

        if (N_UDM < CONTACT_N_REQ_UDM) {

            Message0(“\nWARNING: Requires at least x UDMs to be set up for dynamic contact angle model.\n”, CONTACT_N_REQ_UDM);

            return;

        }

        /* Calculate VOF gradient */

        Alloc_Storage_Vars(pDomain, SV_VOF_RG, SV_VOF_G, SV_NULL); /* Primary storage of variables being calculated */

        Scalar_Reconstruction(pDomain, SV_VOF, -1, SV_VOF_RG, NULL);

        Scalar_Derivatives(pDomain, SV_VOF, -1, SV_VOF_G, SV_VOF_RG, Vof_Deriv_Accumulate);

        /* Store VOF gradient in user memory */

        thread_loop_c(ct, domain) {

            if (FLUID_THREAD_P(ct)) {

                pct = THREAD_SUB_THREAD(ct, phase_domain_index); /* Use this instead of pt[] */

                begin_c_loop(c, ct) {

                ND_V(C_VOF_G_X(c, ct), C_VOF_G_Y(c, ct), C_VOF_G_Z(c, ct), =, C_VOF_G(c, pct)); /* Set 3 components to a vector */

                }

                end_c_loop(c, ct)

            }

        }

        /* Free memory used for VOF gradient */

        Free_Storage_Vars(pDomain, SV_VOF_RG, SV_VOF_G, SV_NULL);

    }

}

DEFINE_PROFILE(wallvelx, t, i)

{

    face_t f;

    cell_t c0;

    Thread *t0;

    real vof_grad[ND_ND], vof_grad_mag, velocity;

    begin_f_loop(f, t)

    {

        /* Get the adjacent cell and its thread */

        c0 = F_C0(f, t);

        t0 = F_C0_THREAD(f, t);

        /* Fetch VOF gradient from UDMs to find meniscus region*/

        vof_grad[0] = C_VOF_G_X(c0, t0);

        vof_grad[1] = C_VOF_G_Y(c0, t0); /* For 2D; extend for 3D if needed */

        vof_grad_mag = sqrt(vof_grad[0] * vof_grad[0] + vof_grad[1] * vof_grad[1]);

        /* Fetch tangential velocity gradient components */

        if (vof_grad_mag > VOF_GRAD_THRESHOLD)

        {

            /* Navier slip shear stress */

            real weight = vof_grad_mag / (vof_grad_mag + VOF_GRAD_THRESHOLD);

            velocity = weight * C_DUDY(c0, t0) * SLIP_LENGTH;

        }

        else

        {

            /* No-slip shear stress (proportional to velocity gradients) */

            velocity = 0;

        }

        F_PROFILE(f, t, i) = velocity;

    }

    end_f_loop(f, t)

}

DEFINE_PROFILE(wallvely, t, i)

{

    face_t f;

    cell_t c0;

    Thread *t0;

    real vof_grad[ND_ND], vof_grad_mag, velocity;

    begin_f_loop(f, t)

    {

        /* Get the adjacent cell and its thread */

        c0 = F_C0(f, t);

        t0 = F_C0_THREAD(f, t);

        /* Fetch VOF gradient from UDMs to find meniscus region*/

        vof_grad[0] = C_VOF_G_X(c0, t0);

        vof_grad[1] = C_VOF_G_Y(c0, t0); /* For 2D; extend for 3D if needed */

        vof_grad_mag = sqrt(vof_grad[0] * vof_grad[0] + vof_grad[1] * vof_grad[1]);

        /* Fetch tangential velocity gradient components */

        if (vof_grad_mag > VOF_GRAD_THRESHOLD)

        {

            /* Navier slip shear stress */

            real weight = vof_grad_mag / (vof_grad_mag + VOF_GRAD_THRESHOLD);

            velocity = weight * C_DVDX(c0, t0) * SLIP_LENGTH;

        }

        else

        {

            /* No-slip shear stress (proportional to velocity gradients) */

            velocity = 0;

        }

        F_PROFILE(f, t, i) = velocity;

    }

    end_f_loop(f, t)

}

1.2. Convective Heating from Slag Infiltration

#include “udf.h”

#include “mpi.h”

real previous_hot_volume = 0.0, delta_hot_volume = 0.0, current_hot_volume = 0.0;

real global_volume_primary = 0.0, global_volume_secondary = 0.0, computed_source_value = 0.0;

#define REFERENCE_VOLUME 5.095455e-07

#define TIME_STEP 0.005

#define HEAT_CAPACITY 1200

#define TEMPERATURE_DIFFERENCE (1875.0 - 1523.0)

#define DENSITY 3000

DEFINE_EXECUTE_AT_END(calculate_hot_volume) {

    Domain *domain = Get_Domain(1);

    Thread *t = Lookup_Thread(domain, 346);

    if (!t) { Message(“Error: Could not find thread for zone ID 346\n”); return; }

    cell_t c; real local_hot_volume = 0.0;

    begin_c_loop(c, t) if (C_T(c, t) > 1523.0) local_hot_volume += C_VOLUME(c, t); end_c_loop(c, t)

    real global_hot_volume = PRF_GRSUM1(local_hot_volume);

    if (global_hot_volume > REFERENCE_VOLUME) global_hot_volume = REFERENCE_VOLUME;

    delta_hot_volume = global_hot_volume - previous_hot_volume;

    previous_hot_volume = global_hot_volume;

    if (MPT_I_AM_NODE_ZERO_P) {

        Message(“Total hot volume: %g m^\3n”, global_hot_volume);

        Message(“Hot volume increase: %g m^\3n”, delta_hot_volume);

    }

}

DEFINE_ADJUST(compute_heat_source, domain) {

    Thread *t = Lookup_Thread(domain, 346);

    if (!t) { Message(“Error: Could not find thread for zone ID 346\n”); return; }

    cell_t c; real local_primary = 0.0, local_secondary = 0.0;

    real global_delta_hot_volume = PRF_GRSUM1(delta_hot_volume);

    begin_c_loop(c, t) {

        if (C_T(c, t) >= 1800.0 && C_T(c, t) <= 1810.0) local_primary += C_VOLUME(c, t);

        else if (C_T(c, t) >= 1523.0) local_secondary += C_VOLUME(c, t);

    } end_c_loop(c, t)

    global_volume_primary = PRF_GRSUM1(local_primary);

    global_volume_secondary = PRF_GRSUM1(local_secondary);

    real heat_source = global_delta_hot_volume > 0 ? global_delta_hot_volume * 0.6 * HEAT_CAPACITY * TEMPERATURE_DIFFERENCE * DENSITY / TIME_STEP : 0.0;

    heat_source = (global_volume_primary > 0) ? heat_source / global_volume_primary : (global_volume_secondary > 0 ? heat_source / global_volume_secondary : 0.0);

    heat_source /= 30;

    if (MPT_I_AM_NODE_ZERO_P) {

        RP_Set_Real(“user-defined-real-0”, heat_source);

        Message(“Heat source set to: %g W\n”, heat_source);

    }

    int udm_id = 0;

    thread_loop_c(t, domain) if (FLUID_THREAD_P(t)) begin_c_loop(c, t) C_UDMI(c, t, udm_id) = heat_source; end_c_loop(c, t)

    if (MPT_I_AM_NODE_ZERO_P) {

        Message(“Total energy before distribution: %g W\n”, RP_Get_Real(“user-defined-real-0”));

        Message(“Primary volume: %g m^3\n”, global_volume_primary);

        Message(“Secondary volume: %g m^3\n”, global_volume_secondary);

        Message(“Final applied heat source: %g W/m\³n”, heat_source);

    }

}

DEFINE_SOURCE(energy_source, c, t, dS, eqn) {

    int udm_id = 0;

    real source = C_UDMI(c, t, udm_id);

    return (global_volume_primary > 0 && C_T(c, t) >= 1800.0 && C_T(c, t) <= 1810.0) ||

        (global_volume_primary == 0 && global_volume_secondary > 0 && C_T(c, t) >= 1523.0) ? source : 0.0;

}