6 Mathematical modelling of the AOD process
6.2 A model for reactions during top-blowing (Article I)
6.2.5 Splashing of metal droplets
If the momentum flux of the top-blown gas jet is sufficiently high, the liquid surface becomes unstable and the splashing of metal droplets occurs [411, 492]. Owing to similar physical properties of carbon and stainless steels, it is reasonable to expect that the mechanism of droplet generation due to top-blowing, excluding the foaming of slag, is essentially similar to the BOF process, for which quantitative descriptions are available in the literature [411, 420, 425, 437–441]. Standish and He [493] identified two regions of droplet generation:
1. the dropping region, in which single droplets are gradually formed and ejected. This is the mechanism of droplet generation, when the gas flow rate is relatively low.
2. the swarming region, in which not only single droplets but large tears of liquid phase are ejected from the bath. This mechanism takes place when the gas flow rate is increased past a certain limit.
The secondary break-up of the metal droplets can occur due to various reasons, e.g.due to the aerodynamic forces of the gas jet [494, 495], impact on the slag layer [496], bursting resulting from spontaneous CO nucleation within the droplet [410], or interfacial instability caused by chemical reactions involving surface-active elements [497]. On the other hand, the very smallest droplets are likely to be carried away with the flue gas [73, 498]. However, in the absence of suitable quantitative descriptions for the break-up mechanisms and due to uncertainties related to the trajectories of the droplets, the effect of the various break-up mechanisms on the droplet size distribution was not accounted for.
The generation of metal droplets is also caused by side-blowing through a mechanism referred to as bubble bursting [282, 320, 499–503]. This phenomenon occurs when a rising gas bubble reaches the surface of the steel and bursts creating very small metal droplets from the thin film of steel around the bubble [282, 320, 501–504]. A related mechanism is the entrainment of large droplets due to jet formation, which is caused by the collapsing of the cavity after the rupture of the iron film [320, 502]. The experimental results of Han and Holappa [503] suggest that the entrainment of metal droplets in slag increases as the bubble size increases, but decreases as the metal–slag interfacial tension and slag viscosity increase. In this work, it was assumed that the effect of these phenomena on the reaction rates is essentially captured by following the splashing generated by the top-blowing.
Trajectory
In this work, the lifespan of the metal droplets was assumed to consist of three successive steps. At first, the metal droplets are generated in the vicinity of the cavity, from which they are ejected onto a gas–metal–slag emulsion. This also includes metal droplets, which have been ejected into the atmosphere and land on the emulsion. Thereafter, the metal droplets pass though the emulsion layer, reacting simultaneously with gas and slag species. Finally, the metal droplets return to the metal bath, where they mix with the metal bath immediately. The metal droplets were assumed to be spherical in geometry;
this assumption should hold well for small metal droplets [505], which make up most of the surface area. The total mass and surface area of the metal droplets residing in the emulsion were defined according to Eqs. 241 and 242, respectively.
mmd=
∑
i
mmd,i, (241)
Amd=
∑
i
6mmd,i
dmd,iρL, (242)
wheremmd,ianddmd,iare the mass and diameter of size classi, respectively. The mass of the metal droplets in the size classiresiding in the emulsion can be solved from
mmd,i=fmd,im˙mdmin(tmd,i,t), (243) where fmd,i, ˙mmd, andtmd,iare the mass fraction of size classiat the place of origin, the metal droplet generation rate, and the residence time of size classi, respectively, and tis the time.33Because the terminal velocity of the droplets increases with increasing droplet size [507, 508], the residence time of large droplets is likely to be shorter than that of smaller droplets. In this work, the residence time of each size class was determined in relation to an ideal trajectory through the emulsion layer [509]. The terminal velocity of the metal droplets was calculated according to Subagyo and Brooks [507].
33The relation shown in Eq. 243 is essentially a reformulation ofLittle’s law[506], which has been proven to hold for any stable and non-preemptive system.
Droplet generation rate
The concept of the blowing number has been employed for describing the onset of splashing. Blocket al.[411] defined the blowing number as follows:
˜NB= V˙G,lance
dlance(hcav+hlance), (244)
where ˙VG,lanceis the volumetric gas flow rate through the top lance anddlance is the nozzle diameter of the top lance. It should be noted that in Eq. 244, ˙VG,lanceis in units of Nm3/h, whiledlance,hcav, andhlanceare in cm, and thus ˜NB is expressed in Nm3/(h·cm2). Blocket al. [411] suggested that splashing starts, when a critical value of the blowing number is exceeded. The critical blowing number was found to be dependent on the surface tension of the liquid phase so that in the case of molten steel, a typical value of the surface tension (1.5 N/m) yields a critical blowing number of ˜NB,crit= 2.0 Nm3/(h·cm2). This value is in close agreement with the experimental values reported later by Koria [431] and Kochet al.[445]. According to Blocket al.
[411], the droplet generation rate (in kg/h) can be expressed as a function of the blowing number as follows [411]:
˙
mmd=KV˙G,lance ˜NB−˜NB,crit2
, (245)
whereKis constant, which was reported to be 180 kg·h2·cm4·(Nm3)−3for liquid iron [411]. On the basis for the definitions of blowing number given in Eqs. 247 and 244 it can be deduced that –ceteris paribus– the intensity of splashing increases as the dynamic pressure of the gas phase increases, surface tension decreases, density of the metal phase decreases, gas flow rate increases, or lance height decreases.
Koria and Lange [431, 448] derived an expression for the generation rate of metal droplets based on hot modelling experiments. The droplet generation rate in the immediate vicinity of the cavity was expressed as follows [431, 448]:
˙
mmd=0.8863mbath
m˙tcosθ nlanceρLgh3lance
0.4
, (246)
wherenlanceis the number of exit ports in the top lance, ˙mt is the total momentum flow rate of the gas jet, andθis the inclination angle of the gas jet relative to the lance axis. It should be noted that a considerably lower prefactor of 0.0288 was calculated on the basis of slag samples collected from an industrial scale BOF vessel [431]. The large discrepancy between the calculated values was attributed to the sampling position [431].
Subagyoet al.[509] defined the blowing number as a dimensionless quantity, which relates the intensity of the jet momentum to the properties of the liquid steel and is defined by [509]:
NB= ρGu2G
2√σLgρL= η2pd
√σLgρL where η=uG
uj , (247)
whereuGis the critical gas velocity,ηis constant, andpdis the dynamic pressure of the gas jet. NBis essentially a reformulated Weber number (We), which is normalised for the context of metal droplet generation. The blowing number is defined so that NB
= 1 represents the criterion for the onset of droplet formation [509]. The results of other studies suggest thatηis not independent of the lance height [510–514] or the gas jet angle [515]. Therefore, although splashing increases with the increasing blowing number in the splashing mode, an increase in the blowing number will lead to a decrease in the splashing rate if the cavity mode changes from a splashing mode to a penetrating mode [511–513, 516, 517]. Making use of the blowing number concept, Subagyoet al.
[509] proposed a correlation, which applies to the splashing cavity mode and is based on hot- and cold modelling data:
˙ mmd
V˙G,lance = (NB)3.2
h2.6×106+2.0×10−4(NB)12i0.2 (R2=0.97), (248)
where ˙VG,lanceis the volumetric gas flow rate through the top lance (in Nm3/s). One of the main differences between Eqs. 244 and 247 is that the former suggests that the blowing number is dependent on the geometry of the cavity. As noted by Sarkar et al.[441] and Routet al. [516], Eq. 248 yields droplet generation rates which are considerably below the values estimated from plant data. Routet al.[516] have argued that the discrepancy is caused partly by the fact that Eq. 248 has been derived for conditions corresponding to room temperature.
In this work, the droplet generation rate was calculated according to a modified expression proposed by Routet al.[516]. More specifically, Routet al.[516] modified Eq. 248 so that the blowing number NBand the volumetric gas flow rate ˙VG,lanceare temperature corrected for the conditions at the point of impact:
˙ mmd
V˙G,lance0 = (N0B)3.2
h2.6×106+2.0×10−4(N0B)12i0.2, (249)
where N0Bis the modified blowing number and ˙VG,lance0 is the modified volumetric gas flow rate, which is calculated by converting ˙VG,lanceto the pressure (pG) and temperature (TG) of the gas at the impact point. The modified blowing number N0Bcan be obtained from Eq. 247 by employing the dynamic pressure at the point of impact. In this work, the dynamic pressure at the point of impact was calculated according to a correlation proposed by Deo and Boom [126]. Furthermore, the variation ofη as a function of the gas jet angle was approximated by the method employed by Alamet al.[515].
Routet al.[516] also suggested that due to the low lance height, the experiments conducted by Subagyoet al.[509] did not actually correspond to the splashing mode, but rather the penetrating mode of jet interaction, which – as noted earlier – is characterised by a lower droplet generation rate than the splashing mode. For this reason, the parameter Jeffwas introduced similarly to Sarkaret al.[441] in order to calculate the effective droplet generation rate:
˙
mmd,eff=Jeff·m˙md. (250)
It needs to be kept in mind that the value ofJeffdepends on the system observed, and is thus essentially a fitting parameter, which is evaluated based on plant data. In their BOF model, Sarkaret al.[441] calculated the droplet generation rate based on the expression shown in Eq. 248 and found that a reasonable agreement with the measured metal content of the emulsion in BOF processing was obtained with a value ofJeff= 15.
Employing the same data, Routet al.[516] found the modified expression (Eq. 249) yields comparable results without the parameterJeff.
It should be noted that surface active elements can have a significant effect on the surface tension of the metal phase, thereby affecting the splashing behaviour. For example, sulphur may increase splashing in the case of Fe–C melts [518]. In this work, however, the effect of surface active elements on the splashing behaviour was not accounted for.
Size distribution
An important aspect of the splashing phenomenon is the size distribution of the metal droplets generated. A substantial contribution to understanding the interaction of molten metal and gas jets was published by Koria and Lange [280, 281, 431, 462, 494, 495, 519–
523]. As a part of their work, they found that the size distribution of the metal droplets at their place of birth could be described well using a slightly modified version of the
Rosin-Rammler-Sperling(RRS) distribution [280]:34
RF= (0.001)
d
dlimitmd,i
n
, (251)
whereRFis the cumulative weight-fraction,nis a distribution parameter, anddlimit
is the limiting diameter, which corresponds toRF= 0.001. More specifically,ndescribes the homogeneity of the droplet distribution, whiledlimitis a measure of the fineness of the distribution [280]. In this work, Eq. 251 was employed for the size distribution of metal droplets at their place of origin. For the distribution exponent, a value ofn= 1.26 was taken from Koria and Lange [280], because it represents an arithmetic mean for relatively large data. Modifying the expression presented by Deoet al.[437] to a more general form, the mass fraction of size classiat place of birth was calculated as follows:
fmd,i=−ln(0.001)n RFdmd,in−1
dlimitn . (252)
Following the assumption of non-coalescing jets, the limiting diameter was obtained from [281]:
dlimit=5.513×10−6×
10 dt2
h2lance
pamb
1.27 p0
pamb−1
cosθ1.206
, (253) wherep0andpambare the lance supply pressure (in Pa) and ambient pressure (in Pa), respectively.