Experimental considerations

Metal droplets in slag

The behaviour of metal droplets in slag is important for many metallurgical unit processes [356]. In this case, the dynamic viscosity of the continuous phase is considerably higher than that of the dispersed metal phase. Metal droplets with diameters in the range of 1 to 5 mm have Reynolds numbers in the slag between 100 and 1000 [356]. Under such circumstances the wake outside the metal droplet is accompanied by a circulatory pattern inside the metal droplet [356].

A case involving iron and copper droplets of 2–6 mm diameter in CaO–SiO2–Al2O3

slag is discussed based on an example in [357].²²In the first case, it was assumed that the mass transfer resistance lies solely in the dispersed metal phase (see Fig. 19a). The analytical solutions of Kronig and Brink [328] , Calderbank and Korchinski [335], and Newman [323] are in reasonable agreement with the measured mass transfer coefficients of Mn in Fe and Cu droplets. However, the measured mass transfer rate of S in Fe droplets is considerably larger than that predicted by the analytical correlations, being in the order of Sh = 100. On the other hand, Si in Fe droplets exhibit slower mass transfer than those obtained from the analytical solutions and are close to Sh = 1. Oeters [357]

suggests that sulphur is likely to cause interfacial instability, which permits interfacial convection and thereby explains the observed high mass transfer rate. Moreover, it is suggested the slow mass transfer rate of Si in Fe droplets might be caused by the suppressing effect of Si on CO formation [357].

In the second case, it was assumed that the mass transfer limitation is solely in the surrounding slag phase,i.e.in the continuous phase (see Fig. 19b). The Higbie [226]

correlation (Eq. 158) appears to overestimate the Sherwood number at low ReSc values, but converges towards the experimental values as the ReSc increases. The predictions obtained with the Calderbank correlation (Eq. 157) are in very good agreement with the experimental values; as stated earlier, this correlation assumes a laminar flow according to the Hadamard stream function. The Ranz-Marschall [252, 253] correlation (Eq. 151) underestimates the Sherwood number throughout the studied range, which suggests the assumption of rigid sphere behaviour holds poorly for external mass transfer in the

22The metal droplets were saturated with carbon and in some cases the metal droplets contained 0.5 wt-% S and the slag contained MnO [357]. The experiments were conducted at two temperatures: 1773 K (1500^◦C) and 1873 K (1600^◦C) [357].

studied droplet size range. However, all three correlations fail to describe the high mass transfer rate of S in Fe droplets.

1 E - 0 6 1 E - 0 5 1 E - 0 4 1 E - 0 3 1 E - 0 2

Mass transfer coefficient according to Kronig and Brink [m/s]

M e a s u r e d m a s s t r a n s f e r c o e f f i c i e n t [ m / s ]

S h = 1 0 0

S h = 1↓

(a) Measured and calculated [323, 328, 335] internal mass transfer coefficients.

4 . 0 4 . 5 5 . 0 5 . 5 6 . 0 6 . 5 7 . 0

(b) Measured and calculated [226, 252, 253, 352] external Sherwood numbers.

Fig 19. Mass transfer rates of Mn, Si, and S in Fe droplets and Mn in Cu droplets.

Adapted from [1].

Rhamdhaniet al.[358] studied kinetics of reactions between Fe–Al alloy droplets and CaO–SiO2–Al2O3slag. In their study, dispersed phase mass transfer was suggested as the rate controlling step. Accounting for the changes in the interfacial area, the

measured mass transfer rate was found to be close to the asymptotic value of the Newman solution for mass transfer within a stagnant sphere.

In conclusion, it may be stated that that small metal droplets behave similarly to stagnant fluid spheres, while large metal droplets may exhibit a circulatory flow [1, 356].

It should be noted that the critical droplet diameter depends on the physical properties of the metal and slag phases as well as the amount of surface active elements. Nevertheless, the circulation is not as vigorous as in the case of aqueous and organic systems [356].

This deviation may be attributable to the high surface tension of liquid steel [356].

Metal droplets in gas flow

Wuet al.studied the decarburisation kinetics of levitated Fe–Cr–C droplets with top-blown O2–Ar [359] and CO2–Ar [355] gas mixtures. In both cases, the experimentally determined mass transfer rates (Sh < 2) at low Re values were lower than those permitted by pure diffusion alone or those predicted by the Steinberger-Treybal [354] correlation.

However, in the case of CO2–Ar gas with moderate Re values [355], the Ranz-Marschall [252, 253] and the Steinberger-Treybal [354] correlations were found to under predict the mass transfer in the gas phase (see Fig. 20).

1 1 0 1 0 0

05

1 0 1 5 2 0 2 5

M e a s u r e d ( W u e t a l .) S t e i n b e r g e r - T r e y b a l c o r r e l a t i o n R a n z - M a r s c h a l l c o r r e l a t i o n

Sherwood number

R e y n o l d s n u m b e r

Fig 20. Measured and calculated [252, 253, 354] Sherwood numbers for CO2in CO₂–Ar gas surrounding Fe–Cr–C droplets. Adapted from [355].

Bakeret al.studied levitated Fe–C droplets (0.7 g) with top-blown O2–He [360]

and CO2–He [361] gas mixtures. They found that the predictions obtained with the Steinberger and Treybal [354] correlation were in very good agreement with the experimental values from experiments with CO2–He [360] gas (see Table 18).

However, in the case of O2–He gas, the predicted mass transfer rates at high O2partial pressures were considerably lower than the experimentally measured values [360].

It was postulated that the discrepancy is caused by a CO combustion zone, which shrouds the specimen and increases the temperature of the gas [360]. An excellent agreement with the experimental values at high O2concentration was obtained when the gas film temperature was taken as equal to the temperature of the metal droplet [360].

Experiments similar to Bakeret al. [360, 361] were conducted later by Distinet al.

[362] with slightly larger metal droplets (1 g and 2 g). In the case of O2injection, the mass transfer coefficients obtained with the Steinberger-Treybal were much higher than the measured values [362]. The deviation decreased with higher gas flow rates [362].

Table 18. Decarburisation of levitated Fe–C droplets [360]. Adapted from [116].

Parameter Unit CO2tests O2tests

x_CO2 – 0.0081 0.884 0.097 – – –

x_O2 – – – – 0.0078 0.926 0.942

x_CO – 0.038 0.116 0.903 0.0043 0.074 0.058

x_He – 0.988 – – 0.988 – –

p_G atm 1 1 1 1 1 0.1

Gr – 5.12 236 155 4.43 183 1.83

Decarburisation rate

Observed _m^mol2s×10² 2.88 96 20.7 5.65 520 400

Predicted – a.) _m^mol2s×10² 2.52 98 18.4 5.72 354 276

Predicted – b.) _m^mol2s×10² – – – – 495 386

Gas properties defined at: a.)T_G= 0.5·(Tin+T^∗); b.)T_G=T^∗.

El-Kaddah and Robertson [363] studied the kinetic of decarburisation of levitated Fe–C droplets in CO–CO2gas mixtures. With respect to mass transfer, the metal droplets behaved similarly to stagnant spheres [363]. A qualitative confirmation of the internal concentration gradients was obtained from microscope analysis of rapidly quenched metal droplets [363]. The effective diffusivity of carbon was found to be three times the molecular value [363], which corresponds to Sh≈20. This value is in good agreement with the asymptotic value of the Kronig-Brink [328] solution (Sh≈17.9) for rigid spheres with internal circulation.

Widlundet al.[364] studied the decarburisation of relatively large (dmd≈8 mm) Fe–C–Si droplets in O2–He mixtures (10–20 % O2). Their experimental results suggest that the mass transfer within metal droplets becomes a rate-limiting mechanism, when the carbon contents of the metal droplets decreases to less than 0.5 wt-% [364]. Based on their data it is possible to deduce that the Sherwood number of the internal mass transfer should be Sh > 100, which indicates rapid circulatory mixing within the droplets.

Slag droplets in a metal bath

Mietzet al.[365] studied emulsification and mass transfer in ladle metallurgy and found that the rate constant was increased by higher rates of gas injection. The increase was attributed to the higher emulsified volume of the top phase as well as to smaller droplet size, which enables faster mass transfer.

Nakasugaet al. [366, 367] found that the reduction of Cr2O3 from slag was controlled by mass transport in the slag phase. The overall rate was found to be higher with higher amounts of CaF2and lower with a high sulphur content in the metal bath [366]. In the case of solid Cr2O3, the overall rate was found to be controlled by a chemical reaction rate at the interface [366]. In contrast to this result, Sevinc and Elliott [368] found when using rotational cylinders that the rate-limiting step for the reduction of solid Cr2O3by liquid Fe–Cr–C alloys was the mass transfer of oxygen in the metal phase. Fruehan [223] studied the rate of reaction of Cr2O3particles with Fe–Cr–C alloys in Ar–O2gas mixtures, and postulated that the rate of reaction is controlled by the diffusive mass transfer of dissolved carbon.

Gas bubbles in liquid metal

In the case of reactions between rising gas bubbles and liquid metal, the mass transfer resistance of the gas phase can usually be ignored [369]. The exact treatment of the continuous phase mass transfer coefficient is possible for small bubbles which are roughly spherical and rise in straight paths [352, 369]. The shapes and rising paths of larger bubbles are irregular and vary over time; reasonable approximates of the mass transfer rate, however, can be obtained even if this behaviour is ignored [352, 369].

At high Reynolds numbers (Re > 5000), the gas bubbles rising in the metal bath are spherical caps and their rising velocity is virtually independent of the liquid metal properties [116]. The findings by Lochiel and Calderbank [353] were later confirmed

by Guthrie and Bradshaw [370], who found that the experimentally determined mass transfer coefficients for oxygen bubbles in liquid silver were in good agreement with predictions in which mass transfer was assumed to take place only through the front of the spherical cap.

The findings of Calderbank and Moo-Young [337] suggest that the diameter and free rising velocity of the bubbles have no effect on the continuous phase mass transfer coefficient in gas–liquid dispersions. This is due to the mutually compensating effect of bubble size and slip velocity [337]. However, a distinction can be made between small gas bubbles (db< 2.5 mm), which behave similarly to rigid spheres, and large gas bubbles (db> 2.5 mm), which exhibit unhindered surface renewal [337].