COLD-FINGER MEASUREMENT OF HEAT TRANSFER THROUGH SOLIDIFIED MOLD FLUX LAYERS
Karina Lara Santos Assis1, P. Chris Pistorius1
1Center for Iron and Steelmaking Research, Department of Materials Science & Engineering, Carnegie Mellon University; 5000 Forbes Avenue; Pittsburgh, PA, 15213, U.S.A.
Keywords: Mold flux, Continuous Casting, Heat Transfer, Cold Finger
Abstract
The thermal resistance between the cast strand and the copper mold in continuous casting is dominated by the conduction resistance through the partially solidified mold flux layer and the contact resistance between the solidified mold flux and the mold. In the cold-finger approach, a freeze layer of mold flux is grown on a water-cooled probe immersed in molten flux. In principle, the thickness of the solid layer and the steady-state heat flux can be used to estimate conductivity and contact resistance. Lower-basicity fluxes generally give somewhat lower heat fluxes under these conditions and result in formation of glassy films. Glassy films are generally significantly thinner than crystalline films, because of the higher thermal conductivity of crystalline films. A potential approach to estimate thermal conductivity and contact resistance from transient changes in solid film thickness and heat flux is outlined.
Introduction
Of the multiple roles that mold flux performs in the continuous caster mold, control of heat transfer is possibly the least understood. In part, this is due to the significant experimental difficulty of measuring the heat flux through a controlled thickness (of the order of millimeters thick) of mold flux, under conditions relevant to caster molds. Heat fluxes in casters range from around 1 MW/m2 for slab casters, to 4 MW/m2 for thin-slab casters, and the typical temperature difference across the thickness of the mold flux is around 1000°C. Current mold flux compositions are also unstable at temperature, tending to lose fluorides by evaporation. The mold flux consists of multiple phases: liquid next to the strand surface, often glassy solid next to the mold, fully or partially crystalline in between, and with gas bubbles captured within the flux layer and at the flux-mold interface.
Given this difficult and complex combination, it is not surprising that there is disagreement regarding the main factors which control heat transfer: whether radiative heat transfer through the flux is significant compared with conduction, and whether conduction or the interfacial resistance (between mold flux and mold) is the main resistance to heat transfer.
As examples of the apparently contradictory results reported in the literature, Yamauchi et al. [1]
stated that the interfacial resistance (affected by roughness of the mold flux film at its interface with the mold) is the main factor controlling heat transfer, whereas Tsutsumi et al. [2] found the conduction resistance to be larger, at least for films in the practically important thickness range
Advances in Molten Slags, Fluxes, and Salts: Proceedings of The 10th International Conference on Molten Slags, Fluxes and Salts (MOLTEN16) Edited by: Ramana G. Reddy, Pinakin Chaubal, P. Chris Pistorius, and Uday Pal TMS (The Minerals, Metals & Materials Society), 2016
of thicker than 0.5 mm, and Kromhout et al. [3] concluded that surface roughness and interfacial resistance are of lesser importance. Of course, some of this lack of agreement may reflect differences in the mold fluxes which were studied, but there does appear to be a gap in the fundamental knowledge of what controls heat transfer, and how this can be measured.
Cold-finger measurements [4] have been used to study solidification of mold flux under more realistic casting conditions, and correlate its thermal properties with morphology and composition. In this method, a water-cooled copper probe is immersed in molten mold flux with a controlled flux temperature. The temperature differential of the cooling water is used to determine the heat transfer rate at the cold face of the copper probe; under steady-state conditions, the heat transfer rate is the same at the hot and cold faces of the copper probe, and at the hot face of the solidified mold flux layer.
For heat transfer across mold flux, the boundary conditions in cold-finger measurements differ in important ways from those in continuous casters, as illustrated schematically in Figure 1. The main difference is that the total thickness of the mold flux layer is around 1 mm in continuous casters, whereas in the cold-finger measurements the solid film thickness is unconstrained; the solid film grows until steady-state heat transfer is attained.
Figure 1. Schematic of the main differences in boundary conditions for heat transfer across the mold flux in the continuous caster (upper drawing) and in cold-finger measurements (lower drawing).
At steady state, the heat transfer rate across the solidified film is equal to the rate of convective heat transfer between the liquid flux and solid layer. Analysis of a range of mold flux compositions (spanning the range of basicities of industrially used fluxes) indicated that the temperature at the solid-liquid interface is close to the solidus temperature [5], similar to what
was concluded by Fallah-Mehrjardi et al. [6]. This means that the steady-state heat transfer rate is given by Equation (1):
q = hflux(Tbulk-Tsolidus) (1) where q is the heat flux (W/m2), hflux (W/m²K) is the convective heat transfer coefficient between the solid and liquid mold flux, Tbulk (K) is the (controlled) temperature of the liquid mold flux, and Tsolidus (K) is the solidus temperature of the mold flux.
As shown by Equation (1), the steady-state heat flux contains no direct information on the thermal conductivity of the solid mold flux layer, nor the contact resistance between the solid mold flux layer and the copper probe. These values, which are the ones of importance to heat transfer in the continuous caster mold, can be extracted from cold-finger measurements, but this requires additional analysis of the results. The directly measured heat transfer rate in cold-finger measurements is not useful in itself for comparing heat transfer between different mold fluxes. In this paper, results of a simple one-dimensional heat transfer model is used to illustrate the principles of the methods which can be used to obtain the thermal resistance of solid mold flux layers from cold-finger measurements.
Heat transfer model
The expected response of the cold-finger measurement to changes in the heat transfer conditions was predicted with a simple transient, finite difference, one-dimensional purely heat transfer model; the main features of the model are shown in Figure 2. This was a pure explicit finite- difference heat-transfer model. The temperature at the liquid-solid interface was taken to be the melting point (solidus temperature) of the mold flux; the position of the solid-liquid interface was tracked explicitly.
Figure 2. Main elements of 1D transient model of cold-finger measurement.
The default conditions assumed in the calculations are given in Table I. The initial conditions were taken to be those when the cold finger is first immersed into the molten mold flux, i.e. cold copper (at the temperature of the cooling water) with zero thickness of solidified flux.
Results and discussion
Typical results are shown in Figures 3 and 4. The change with time of the calculated heat flux at the cold face of the copper has the same shape as that measured experimentally [4,7,8], namely an increase to a peak value, with a subsequent decrease to a steady-state value. The initial increase in heat flux results from the thermal inertia of the copper layer between the cooling water and the mold flux; the heat flux at the interface between the copper probe and the mold flux is highest at time zero and decreases over time to approach steady state. Figure 3 illustrates that the steady-state heat flux is proportional to the convective heat transfer coefficient between the molten flux and solid layer (see also Equation [1]); the interfacial resistance (between the probe and the solid flux layer) and the thermal conductivity of the solid layer have no effect on the steady-state heat transfer rate but do affect the peak rate somewhat (Figure 4).
Because the thermal resistances (conduction and interfacial resistances) of the solidified mold flux do not affect the steady-state heat flux, additional measurements are required to obtain these resistances. Based on the relationship in Equation (2), the steady-state thickness of the solid flux film is a useful measurement:
Lsolid = ksolid[(Tsolidus-Twater)/q - LCu/kCu - 1/hwater – Rint] (2) where ksolid is the thermal conductivity of the solid flux layer, Tsolidus the solidus temperature of the mold flux, Twater the cooling water temperature, q the steady-state heat flux, LCu the thickness of the copper probe and kCu is thermal conductivity, hwater the convective heat transfer coefficient between the cooling water and the copper probe, and Rint=1/hint the contact resistance between the solid flux film and the copper probe.
Table I. Default conditions for 1D cold-finger model calculations Copper
kCu (W/mK) c (J/kgK) (kg/m3) LCu (m) Ti (°C)
400 385 8960 0.005 25
Mold flux
ksolid (W/mK) Hf (kJ/kg) (kg/m3) Tf (°C) Tbulk (°C) hflux (W/m2K)
1 559 2500 1200 1500 1500
Interface Cooling water hint (W/m2K) Tw (°C) hwater (W/m2K)
1000 25 20000
Figure 3. Calculated effect of the heat transfer coefficient between solid and liquid flux on the measured heat flux at the cold face of the copper probe.
Figure 4. Predicted effects of differences in the thermal conductivity of the solidified flux (left) and interfacial resistance (right) on the measured heat flux.
Equation (2) shows that, for a given set of measurements of steady-state heat flux and solid film thickness, different pairs of values of interfacial resistance and thermal conductivity of solid can fit the results; it is not possible to find interfacial resistance and thermal conductivity from a single steady-state measurement.
However, one possible approach is to perform measurements at different bulk temperatures. The solid layer is thinner for higher bulk liquid flux temperatures; the change in solid-layer thickness with temperature depends on the balance between conduction and interfacial resistances; see Figure 5.
This predicted effect of bulk temperature on solid-layer thickness was tested experimentally using two mold fluxes, with higher and lower basicity; compositions are given in Table II.
Details of the measurement approach are given elsewhere [5,7]. The solid layers formed from the higher-basicity mold flux were largely crystalline (with cuspidine the major phase) whereas the
lower-basicity mold flux formed glassy solid layers. Measured film thicknesses are given in Figure 6, with fitted thermal properties listed in Table III.
Table II. Compositions (mass percentages) of mold fluxes used to illustrate effect of bulk temperature on cold-finger measurements
CaO CaF₂ SiO₂ Li₂O Na₂O MgO Al₂O₃ Fe₂O₃ MnO
Higher B 33.1 24.5 30 5.4 0.9 0.5 3.4 2.1 0
Lower B 15 20.5 38.7 0 17.5 1.3 3.3 0 3.1
Table III. Fitted values for Figure 6.
hflux
(W/m²K) ksolid
(W/mK)
Rint
(m²K/W)
Higher B 1500 2.8 6×10-4
Lower B 1400 1 3×10-4
Figure 5. Illustration of the principle behind using different bulk temperatures to evaluate the film conductivity (ksolid) and interfacial resistance (Rint): the change in steady-state solid film thickness with bulk temperature depends on the balance of ksolid
and Rint (calculated ksolid=2.8 W/mK and Rint=0.6×10-3 m²K/W, or ksolid=3.5 W/mK and Rint=1×10-3 m²K/W; hflux=1500 W/m²K;
Tsolidus=1000°C; other values as in Table I).
Figure 6. Measured solid layer thickness (data points) with fitted relationships (lines; see Table III), for higher- and lower-basicity mold fluxes tested at 1250°C, 1325°C and 1400°C.
In choosing the thermal properties to fit the experimental results (Figure 6 and Table III), the assumption is that the effective thermal conductivity and interfacial resistance do not depend on the bulk temperature of the liquid mold flux. It seems reasonable to assume that the effective thermal conductivity is the same, since the temperature range across the solid layer is similar in all cases (this range is from the solidus temperature to the much lower temperature adjacent to the copper probe). However, roughness measurements of the surface of the solid film which had been adjacent to the probe showed that the interface was rougher for crystalline (higher-basicity) solid films formed at higher temperatures [5]. The rougher interface is expected to result in a higher interfacial resistance, which is why the measured thickness for the highest bulk temperature in Figure 6 falls significantly below the fitted line, for the crystalline (higher- basicity) mold flux.
The dependence of interfacial roughness on the bulk liquid temperature likely results from the rate at which the film grows. For a higher liquid temperature, the film grows more slowly, and the interface is rougher. This effect of the bulk temperature invalidates the assumptions which underpin this approach, and another method is needed to estimate the solid-film thermal conductivity and interfacial resistance. A promising approach is to use the transient heat flux and change in solid-film thickness as the system approaches steady state. Equation (2) shows that different combinations of solid-film thermal conductivity (ksolid) and interfacial resistance (Rint) could fit the measured steady-state heat flux and solid film thickness, for a single bulk liquid temperature. However, for such a set of steady-state measurements, Equation (2) defines a fixed relationship between ksolid and Rint; for different pairs of values of ksolid and Rint which fit this relationship, the transient changes in measured heat flux (especially the peak heat flux) and solid layer thickness can be quite different. An example of this effect is shown in Figure 7.
Figure 7. Principle of using measured transient heat transfer at one liquid-flux temperature to fit the solid-layer thermal conductivity and interfacial resistance: combinations of ksolid and Rint which yield the same steady-state thickness (for the given heat flux) give very different peak heat flux values, and different film growth rates. Values plotted for 1325°C bulk temperature, ksolid=3 W/mK or 2 W/mK, and Rint=6.9×10-4 m2K/W or 0.62×10-4 m2K/W.
The results of Wen et al. [8] illustrate just such a relationship. In that work, mold fluxes with the same overall composition were prepared using different raw materials; the choice of raw material affected the crystallization behavior. As expected from the relationships presented here, the measured steady-state heat flux was the same for mold fluxes prepared with different raw materials [8] – this is expected, because the steady-state heat flux is controlled by liquid-flux heat transfer. However, the peak (transient) heat flux was lower for more-crystalline films (of the same composition) [8]; again as expected, since more-crystalline films would tend to have a higher thermal conductivity and a higher interfacial resistance – resulting (as shown in Figure 6) in a lower peak heat flux.
Conclusions
The cold-finger method is useful in yielding heat fluxes and solid-film microstructures which are similar to those encountered in industrial continuous casters, as shown in previous work [7]. The heat flux measured with the cold finger does not give direct information on the thermal resistances of the solidified mold flux. However, measuring the steady-state heat-flux and the steady-state thickness of the solid film constrains the values of the solid-film thermal conductivity and the interfacial resistance to one functional relationship. Using this relationship together with the transient changes in measured heat flux should allow reliable determination of the solid-film properties.
Acknowledgements
Support of this work by the industrial members of the Center for Iron and Steelmaking Research is gratefully acknowledged.
References
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