As the melt is tapped from the furnace into the ladle the refining process is performed continuously, and continues for a set amount of time after the ladle is full. The purge gas is introduced through a porous plug located at the bottom of the ladle. This creates a bubble column rising through the ladle, driving a flow while mixing the melt. On its ascent the O2 in the bubbles will react with Si to form SiO2 and SiOg. SiOg will be transported into the bubble saturating the gas phase, while SiO2 will nucleate at the bubble interface. Tang  writes that SiO2 forms a droplet on the surface, while Schei et al.  writes that it can form a film around the bubble. SiO2 then reacts with the impurities in the melt, primarily Al and Ca, to form a predominantly SiO2-CaO-Al2O3 slag. When the slag is formed there will be a driving force to approach equilibrium between the present phases driving a mass transport to the slag so their respective equilibrium concentration can be reached. Since the temperature of the system is high it is often assumed that the reactions can be considered instantaneous . As the diffusion of species in the metal is faster than in the slag, the rate determining step for the transport of Al and Ca is considered to be the transport in the slag [3, 4, 6]. To sum up the overall reactions can be expressed as:
- 1. Si F O2 = SiO2
- 2. Si F 2 O2 = SiOg
- 3. 2Ca F SiO2 = 2CaO F Si
Fig. 1 The whole system including the falling jet. Bubbles of entrapped air are displayed in green
- 4. 4A + 3SiO2 = 3Si + 2A/2O3
- 5. 3CaO + 2A = 3Ca + A/2O3
with Fig. 1 illustrating the whole process.
To express the system it has been proposed to use a standard  batch reactor model . By applying two-film theory to the interface the molar transfer for species i, Ni, from the metal to slag can be expressed:
where kip [m] is the mass transfer coefficient for species i in phase p, pp [mg] is the density of phase p, and Mi [kkgj denotes the molar mass of species i. xi;P denotes here the mass fraction of species i in phase p, and the superscript1 expresses that it is at the interface. Due to the complexity of determining the mass transfer coefficient, ki, ki m and ki;S may be combined in an overall mass transfer coefficient ki;t .
where Ki denotes the equilibrium constant Ki = between the two phases for
species i. Here yi is the raultion activity coefficient for the oxide of species i and fi is the henrian activity coefficient of species i in the melt. By assuming that the slag and metal is completely mixed the mass balance can be expressed as :
Here mp [kg] is the mass of phase p, and As [m2] is the reaction area. In Eq. 3 x^ is the hypothetical concentration in the metal at equilibrium with the actual concentration in the slag xi;S given by x^Kf = yixi;S. Assuming that xi;St=0 = 0 and lim xim = x1 Eq. 3 can be written as:
Equation 4 poses a problem due to most of the parameters on the RHS are functions with respect to time . There is also an additional problem in the fact that ladle refining of silicon is done while tapping and therefore cannot be considered a true batch process for its full duration. Kero et al.  writes that allowing the RHS in Eq. 4 to be constant with respect to time will still give an equation which can be useful in comparing the refining kinetics in this system. This is due to the fact that mm, ms, As and pm all change in the same way with respect to time for every species in the melt allowing it to serve as a crude approximation, according to Kero et al.
Cussler  has gathered multiple expressions for k and from this it can be seen that some common dependencies are k = f (p, U, L, D, r, p). Here U [m] is the
terminal rise velocity, L [m] is some characteristic length, D m is the diffusion
coefficient, r U is the surface energy density, and p [ms] is the dynamic viscosity. This states that k is a property depending not only on the thermophysical properties of the phases involved, but also on the flow and geometries. If these properties can be considered constant or their behavior similar then ki;t might be considered constant, or similar in behavior as well.
An interesting point when looking at Eq. 4 is that the exponential transient behavior of the RHS, after integration, is shared with the much more complex CFD models from Ashrafian et al.  and Olsen et al.  on the same system. In addition it shows a decent fit with the industrial sample set from Kero et al. , as seen in Fig. 2, for many elements, like Ca.
The experiments performed in this work differ from the industrial case in that a clean metal is used instead of a slag. This leads to the transport of Al and Ca from the slag, with a known slag concentration, to the clean metal. However, the mass transfer behavior should not be affected by this as the steps of transport are the same in both cases. Clean metal is used due to it being considerably easier to create a synthetic slag with an accurate composition, than doping pure silicon with specific
Fig. 2 Normalized Ca, Al, Mg and B concentrations as functions of time. From Kero et al. . Printed with permission under the Creative Commons Attribution License as stated in Kero et al.  amounts of Ca and Al. By applying the same theory as used earlier on this system, with a RHS which is not a function of time, gives: