The fourth case to be considered is that where the resist ance is found in the continuous phase with an internally
circulating dispersed phase. The mass transfer coefficient is again described through the Sherwood number and a number of correlations have appeared (157,158). Heertjes (157) has reviewed most of the proposed models and notes that the main
Most of the above models for either continuous or dis continuous phase control, requires the knowledge of specific data in order to enable mass transfer coefficients or dim- ensionless groups to be calculated. In the system under study most of the data cannot be as yet definitely estab
lished although attempts have been made on the preceding pages. The drop velocity, variations of contact area, changing contact times, and the fact that the so-called continuous phase is in reality a foam which contains a high proportion of a dynamic gas phase, allow only approximations to be made as shown in the previous calculations. The results of the experiments carried out here indicate that these chemical engineering ■ type models based upon the existence of a well defined inter
face, are difficult to apply to the complex interactions observed between metal alloys and slags.
Another situation which may arise for transport control is that known as mixed control. The situation arises when resistances to transport in both .the continuous (slag] and discontinuous (metal] phase are significant. The rates of transport to the interface are equivalent. It is usually assumed that equilibrium is established, so that in terms of slag metal phases:
■jjp = km .ACCb ’m - C 1 ’™) = kS L Q .A. CC1,slg-Cb,slgl S.5.4.1 In the present metal/slag reactions the concentrations at the interface for both metal and slag are not known and there fore the equation cannot be solved.
5.5.5 Chemical Reaction.
The majority of processes by which steel is currently produced have received considerable attention in terms of
thermodynamic analysis. The criteria are based upon the
concept that every process corresponds to an equilibrium state in which, phenomenologically, the process does not proceed any
further. The criteria for phase equilibria in systems of
steelmaking interest are well documented and it is possible to predict with confidence the theoretical limits of refining
or recovery attainable in any given situation. However, frcm
the practical point of view, it is the rate at which reactions
take place that is of greater importance. The rate at which
a chemical reaction occurs may be a function of several factors including concentration of reactants and products, pressure
and temperature. The observed behaviour of many reactions
has been explained by two main theories known as the simple
collision and the absolute reaction rate theory. In the
Simple Collision Theory it is assumed that molecules behave
as non-attracting hard spheres. By combining the forward
and reverse velocities of reaction an equilibrium constant is determined which is expressed by an Arrhenius type
equation:
kR = A^ exp (-E/RT) 5.5.5.1
Two factors are involved - a frequency factor, A^, and an exponential factor which includes the activation energy, E. The latter factor is assumed to represent an energy barrier which has to be overcome if the reactions are to proceed.
The frequency factor represents the number of collisions between potentially reactive molecules, and therefore the exponential term becomes the fraction of collisions
possessing the ability to undergo reaction.
The Absolute Reaction Theory considers the formation of an intermediate compound known as activated complex. At equilibrium, the flows through the configuration of the activated complex in the forward and backward direction are equal. It is generally assumed that the activated complex is in equilibrium with the species of origin although react ants and products may not be in equilibrium with one another. The mathematical expression, in the Arrhenius form, yields a relation of:
b E°
kp = aT exp (- — ) 5.5.5.2
R T
and compared with for simple collision theory indicates
that E = E° + bRT and only when h RT >> E° may any difference be detected.
According to Darken (160) the above theories, applied to classical molecular reactions, differ from liquid metal systems because the molecular state as such does not exist and the
electron bond is very much, higher in liquid metals. It has been generally accepted that most chemical
reactions taking place at steelmaking temperatures occur at such high rates that local equilibrium generally exists and it is improbable for chemical kinetics to be the rate controlling
step. Figure 73 shows a diagram representing chemical
reaction control. The basis for this concept has been
( I
established from calculated actiyation energies for typical chemical reactions which, rarely exceed 60 k cal, and are generally in the region of 20 to 30 kcal (1611. Only if the activation energy is much in excess of 250 kcal is chemistry likely to he rate controlling (161).
The influence of temperature on the rate of reaction is expressed by the Arrhenius equation:
d In i .
E
^ •„ „ ,
tfT r t2” ' ' S.S.S.3
This equation is applicable to either homogeneous or hetero
geneous type reactions. On integrating equation (5.5.5.3)
the rate constant is expressed as E
k = const, exp (- 5.5.5.4
By plotting In k vs 1/T a linear relation should be obtained (with, a gradient equal to - E/R] provided the factors which control the rate of reaction remain the same throughout the
temperature range under study. The order of a reaction is
the measure by which the rate of reaction is proportional to the number of atoms or molecular groups taking part in the
reaction. Considering the decarburization data as plotted
on a 1C vs time basis, a zero-order reaction would be expressed as
-k ' 5.5.5.5
and the carbon drop should be linear with time, i.e. (Co -C^)= which was not found in the present study.
For a first-order reaction dC It kc 5 . S • S • 6 by integrating: Cf t=t dC ~C k dt Co t—o and k t 5 . 5 . .5. 7
From the plotted data of |C vs time, the values for k were found from the slopes of the curve which were obtained. A constant time of 360 seconds evaluated by the following
technique, was taken for each calculation from graphs as typified in figure 74.
The fall in carbon content was read between the time
interval considered. Because’the graphs do not join the
% C
axis, the iterations were initiated after either one minute (e.g. at 1400° C) or three minutes, depending on the parti
cular situation of each graph. Values of k were calculated
along the curve from equation (5.5.3.7)
The average value of all the k calculations was then
taken to represent the overall rate of reaction. This pro
cedure was repeated for all the decarburization graphs. The values calculated for all the decaiburization experiments are given in table V.
Due to the limitations under the present experimental conditions, the temperatures at which reactions were made are
of a restricted range. It was nevertheless attempted to
determine an approximation for the activation energy values
from the calculated values. The logarithm of k was plotted
against the reciprocal of temperature in figure 75.
Fayalite decarburization was performed at 1240°C, 1300 and 1^00°C, and the slope of the plot gave an activation
energy of 29.436 Kcal. This figure is in close agreement
with activation energies found from literature C100> 101,
102) for graphite-iron oxide reactions which are in the range
of 32 to 40 Kcal. The slight discrepancy between the figures
may stem from the fret that the presence of silica may have
an effect on the bonding of FeO to form fayalite. Therefore,
a decrease in the activation energy from the pure iron oxide
case may be expected. Three other slags containing lime
additions of about 26% to 32% were similarly analysed. The slopes of the straight lines allowed values of activation
energies to be found and are given with the figure. The
lowest value found was 13.8 Kcal. The results, although
limited, indicate that as a second fSi02) or third (TaO) component is added to FeO, the characteristics for reaction
become more towards transport control. The experiments
carried out by Tarby and Philbrook (102) on slag-metal-
graphite reactions contained alumina from 8.9 to 50.6%. For alumina contents of 8.9% in the slag, the apparent
activation'energy for slag-metal reaction was found by these authors to be 12 Kcal/mole.
It may therefore he concluded that the values obtained in the present experiments are in agreement w i t h those
observed with other workers, who in turn, concluded that chemical reaction control was not applicable. For such systems to have chemical reaction as the rate controlling mechanism, values of activation energies in excess of 250 Kcal are necessary (161] fsee section 2.7.2).
From the preceding calculations it is concluded that chemical reaction within the present experiments is not rate controlling. Chemical reaction will not be further discussed. Therefore, mass transfer control of both continuous and dis persed phase will he discussed in more detail in the following section.