Scrap Dissolution in Molten Iron Containing Carbon for the Case of Coupled Heat and Mass Transfer Control

Abstract

The scrap dissolution in an actual process like the BOF is affected both by mass transfer and heat transfer. In this paper, the mass transfer of carbon in liquid melt is considered along with heat transfer. The approaches used in this paper to model the scrap dissolution phenomenon include the application of Green’s function, quasi-static, integral profile, and the finite difference approach for different Biot numbers. Mass transfer coefficients are calculated using the Chilton–Colburn’s analogy for the case of forced convection. Since the quasi-static approach requires the least computational time, it is used for a detailed parametric study, including the effect of other parameters like different scrap ratios and heating rates of liquid melt. The region of control of heat transfer vs mass transfer is also identified. The dissolution of mixed scrap (light and heavy scrap) is investigated for different scrap ratios and the autogenous heating rates of liquid melt, with the help of mathematical models. The heat transfer coefficient is estimated as a function of mixing energy and the mass transfer coefficient by invoking the Chilton–Colburn analogy. The permissible limits of light scrap, which can be charged into the BOF, are also suggested from the results of this model. The Artificial Neural Network (ANN) model is trained on the dataset (patterns) generated by the coupled heat and mass transfer model. The accuracy of the results obtained using different ANN topologies is discussed followed by a recommendation for selecting the best approach.

Previous Work on Scrap Dissolution Through Simultaneous Heat and Mass Transfer

Several investigations have been done where the problem of scrap dissolution is solved by considering simultaneous heat and mass transfer[1–19] (of carbon). The special problems in this category include the case of scrap melting in the BOF steelmaking process. Asai and Muchi[1] presented a detailed model for dissolution of scrap in the BOF. Temperature and carbon concentration of the melt were considered as important variables and their influence on the progress of melting of the scrap was investigated for different operating conditions. The temperature and carbon concentration profiles were assumed to exist only inside the liquid close to the interface. As a simplification, the composition of the solidified shell formed on the scrap was assumed to be the same as that of the scrap. The temperature inside the scrap was considered to be uniform and the velocity of the moving interface was considered to be constant. It may be noted that such approximations are, however, valid only for the case of thin-sized scraps. Hartog et al.[2] developed an interesting numerical procedure based on the solution of simultaneous, unsteady-state heat transfer and mass transfer of carbon equations solved at the bath–scrap interface. They considered a moving boundary layer approach to estimate the heat transfer coefficient and used the Chilton–Colburn analogy to estimate the mass transfer coefficient. The value of the heat transfer coefficient was estimated as 47,500 W/m²/K. The temperature of the scrap–melt interface and the carbon composition at the interface was deduced from the Fe-C phase diagram. The rates of energy generation due to chemical reactions were calculated from off-gas analysis and bath sampling. The heat losses from the BOF were calculated by a posteriori calculations in real heats. Solutions were thus obtained for different sizes of scrap and carbon contents. The results showed that the evolution of temperature of the bath and complete dissolution of the scrap depended upon the scrap size distribution. For example, in the presence of a higher proportion of light scrap, the temperature of the bath was predicted to be almost near the liquidus temperature, while the dissolution of the colder heavy scrap had to wait until the light scrap dissolved. The diffusion of carbon inside the solid scrap was ignored.

In another parallel approach, Gaye et al.[3] expressed heat transfer coefficient as a function of input energy to the system and thus incorporated the effect of blowing conditions in the BOF on heat and mass transfer coefficients. They also studied (similar to Hartog et al.[2]) the dissolution behavior of scrap of different thicknesses and scrap mixes. The permissible limits of scrap size to achieve complete melting by the time of in-blow sublance measurement were recommended. As expected, the bottom stirring improved the melting rate significantly. An interesting finding was that the preheating of the heavy scrap did not improve its dissolution rate.

Brabie and Kawakami[4] reviewed the melting phenomena of scrap in a steel bath. They suggested that when the temperature of the melt is lower than the melting point of the scrap, and the carbon content is higher in the melt than the scrap, the melting is exclusively controlled by the mass transfer. According to them, the heat transfer resistance between the molten steel and the solidified crust is still an unsolved problem.

Mori and Nomura[5] developed a mathematical model for scrap melting in the steelmaking process based on the concept of simultaneous heat transfer and mass transfer of carbon. They solved an unsteady-state heat transfer equation in the solid scrap and a heat and mass transfer equation in the boundary layer between the solid and liquid. The overall problem was solved by the finite difference method.

Kawakami et al.[6] investigated the dissolution of a cylindrical steel rod in high carbon melt by performing laboratory experiments. The estimated heat transfer coefficients were found to be in the range of 27,700 to 77,200 W/m²/K and mass transfer in the range of 0.83 to 2.08e−4 m/s. The dimensionless correlations for heat and mass transfer were developed:

$$ \begin{gathered} Nu = 0.017 Re^{0.8} Pr^{0.33} \hfill \\ Sh = 0.017Re^{0.8} Sc^{0.33} . \hfill \\ \end{gathered} $$

Isobe et al.[7] conducted experiments on melting of a rotating steel rod in a 300-kg furnace and also on scrap melting in a 5-ton experimental (pilot scale) converter. Correlations between heat and mass transfer coefficients coincided closely with the Chilton–Colburn analogy, and the nondimensional correlation for the mass transfer given was

$$ Sh = 0.163 Re^{0.78} Sc^{0.356} . $$

The melting time was found to depend on mixing energy, but became independent of mixing energy beyond a certain point. In the pilot plant 5-ton converter tests, the estimated heat transfer coefficients were found to be in the range of 23,000 to 46,000 W/m²/K and the mass transfer coefficients in the range of 0.8 to 1.95e−4 m/s where the mixing energies were in the range of 3500 to 12,000 W/ton.

Kosaka and Minova[8] studied the dissolution solid steel cylinders in high carbon melts in the temperature range of 1573 K to 1773 K (1300 °C to 1500 °C) and proposed the nondimensional correlations for the static as well as the rotating cylinder. Kim and Pehlke[9] investigated transient heat transfer during the initial stages of steel scrap melting under nonisothermal conditions. The solid cylindrical specimens were preheated to various temperatures below the liquid bath temperature prior to immersion. The effects of heat transfer were then evaluated. The formation of a frozen layer of carbon-rich iron on the specimen, which subsequently remelted, was also considered. The results of the model were in good agreement with the experimental results. Yorucu and Rolls[10] employed the finite difference procedure and considered the diffusion of carbon inside the scrap. The diffusion coefficient was found to depend on temperature. They assumed that carbon composition at the scrap surface was the same as that of the bath and that the temperature was equal to the melting point of the bath. Zhang and Oeters[11,12] considered the dissolution of sponge iron and scrap in high carbon melts. Three simultaneous equations involving heat transfer, carbon mass transfer, and the carbon–temperature relationship (as given by the liquidus line in Fe-C diagram) were solved for the three unknowns, for the different stages of solidification, to calculate carbon and temperature at the interface and the velocity of the moving interface. Interface carbon content in liquid and solid phases was related to the Fe-C phase diagram during frozen shell development. The thermal boundary layer was assumed to form as soon as the scrap came into contact with the liquid, whereas the development of the concentration boundary layer took some time due to the slow process of mass transfer. It was shown that heat transfer was almost 40 times faster than mass transfer during the initial period of shell formation. Since the initial solidification velocity was high due to a steep gradient of temperature in scrap, the initial frozen layer had the same composition as that of the melt. As the velocity of the shell formation became smaller, the concentration boundary layer developed at the interface. In this manner, the time beyond which the contribution of thermal gradient inside the scrap could be neglected was calculated. Wright[13] studied the dissolution rates of commercial black iron rods in iron-carbon melt under isothermal conditions. The effects of melt carbon content, temperature, natural convection, and gas stirred forced convection conditions were investigated and correlations were derived. Under the condition of bottom gas stirring, the mass transfer coefficient was found to depend upon the gas flow rate. Gupta and coworkers[14,15] considered a uniform temperature and carbon composition inside the scrap, but a steep temperature and composition gradient at the interface for a semi-infinite body (similar to Asai and Muchi[1]). The composition of the solidified layer was assumed to change as a function of time, consistent with the mass transfer process and the changes in bulk liquid metal composition. Though the interfacial velocity was calculated at each time step, it was found to be nearly constant during the major period of scrap dissolution, except toward the end of dissolution. The effects of various parameters on the peak value of the ratio of solid (scrap + solidified shell) to liquid for a given scrap to hot metal ratio and for different preheat temperatures and heat generation factors were studied. The amount of buildup of solidified shell was predicted to be up to 33 pct of the initial scrap mass, which is rather high. Specht and Jeschar[16] assumed bath temperature and composition to be constant. The diffusion of the carbon inside the solid scrap was considered by using Fick’s law. Solidification of metal on scrap (chilling effect) was not considered. The temperature profile at the interface was calculated by considering the solid as a semi-infinite body. The heat of fusion included an additional term to account for the energy required to raise the temperature of melting mass, starting from the interface temperature and rising to the bulk temperature. The heat and mass transfer coefficients were estimated in terms of Sherwood and Nusselt numbers for spherical, cylindrical, and laminar slab-shaped solids under forced convection conditions. The time for complete melting was proportional to the initial size of the scrap. Guthrie performed separate investigations for scrap melting in pig iron under mass transfer control and in steel ladle under heat transfer control.[17,18]

Coupling of Heat and Mass Transfer Equation

In the present work, the heat and mass transfer equations are been coupled together.[19] The following assumptions are made to formulate the simultaneous heat and mass transfer model:

• 1. Thermal conductivity and density of scrap are considered to be uniform for solid scrap as well as for the liquid melt. The physical properties of steel melt and steel scrap used in this work are given in Table I.

  Table I Physical Properties of Steel Scrap and Liquid Melt

• 2. The actual values of heat and mass transfer coefficients are calculated on the basis of a moving boundary layer concept (for details, see Appendix A).

• 3. The latent heat of solidification does not include the additional term to account for the energy required to raise the temperature of melting mass from the interface temperature to the bulk temperature. This effect is taken into account while defining the actual heat transfer coefficient for the moving boundary (Appendix A).

• 4. During the initial stages, when solidification of liquid metal on solid scrap takes place due to heat extraction at the interface, the rate of heat transfer is faster as compared to the mass transfer and hence the temperature boundary layer forms immediately. The concentration boundary layer forms later as the process of mass transfer is much slower. Therefore, due to dominance of heat transfer and the very slow rate of mass transfer, the solidified shell is assumed to be of the same composition as that of the bulk liquid melt. During the melting period of the solidified shell and melting of the parent scrap, the composition of liquid melt at the interface is assumed to be given by (as explained in Appendix A)

  $$ C_{\text{i}} = C_{\text{sc}} + (C_{\text{b}} - C_{\text{sc}} )\exp \left( {\frac{\nu }{{\beta_{\text{o}} }}} \right) $$

  (1)

  where $C_{\text{sc}}$ is the carbon content of the already solidified shell (for the case of melting of the solidified shell). It is also the composition of the parent scrap for the case of melting of the parent scrap. The parent scrap gets carburized at the interface and the composition of the parent scrap at the interface is given by

  $$ C_{\text{s}} = \gamma C_{\text{i}} $$

  (2)

  where $\gamma$ is the distribution ratio of carbon between solid and liquid and is obtained from the Fe-C phase diagram as

  $$ \begin{array}{ll} {\gamma = \frac{{1533 - T_{\text{m}} }}{{1563 - T_{\text{m}} }}\;\frac{93}{225.65}} & {0.18 < C_{\text{i}} \le 4.5} \\ {\gamma = 0.2} & {C_{\text{i}} \le 0.18} \\ \end{array} $$

  (3)

  where T _m is the liquidus temperature, which depends upon carbon content of the melt at interface C _i:

  $$ \begin{array}{ll} {T_{\text{m}} = 1563 - 93\;C_{\text{m}} } & {0.18 < C_{\text{m}} \le 4.5} \\ {T_{\text{m}} = \,1535 - 84\;C_{\text{m}} } & {C_{\text{m}} \le 0.18} \\ \end{array} . $$

  (4)

• 5. The velocity of the moving interface is assumed to be constant within a small time step.

• 6. The correlation between heat and mass transfer coefficients is given by the Chilton–Colburn analogy for the case of forced convection as follows:

  $$ \beta_{\text{o}} = \left( {\frac{{D_{\text{c}} }}{\alpha }} \right)^{0.6667} \left( {\frac{{h_{\text{o}} }}{{\rho_{\text{m}} C_{\text{pm}} }}} \right) $$

  (5)

  where D _C is the diffusion coefficient of carbon in liquid melt and it is given as a function of carbon composition and temperature of liquid melt at the interface as follows[20]:

  $$ D_{\text{C}} = 1.0e{-}4\;\exp \left\{ { - \left( {\frac{12100}{{T_{\text{m}} + 273}} + 2.568} \right) + \left( {\frac{1320}{{T_{\text{m}} + 273}} - 0.554} \right)\;\left( {C_{\text{m}} } \right)} \right\}. $$

  (6)

  The value of D _C calculated by the above equation lies in the range of 1.40e−8 to 2.40e−8 m²/s.

• 7. The standard temperature and carbon composition trajectories are assumed in the absence of any scrap. The overall heat and carbon mass balance is performed at each time step to estimate the bulk liquid temperature and bulk carbon composition. The equation for the heat balance is

$$ C_{\text{pm}} W_{\text{m}} \frac{{dT_{\text{b}} }}{dt} = - h\;A\;(T_{\text{b}} - T_{\text{m}} ) + C_{\text{pm}} \;W_{\text{m}} \;\left. {\frac{{dT_{\text{b}} }}{dt}} \right|_{\text{woscrap}} . $$

(7)

The rate of carbon removal is calculated from

$$ \frac{{dC_{\text{b}} }}{dt} = \left. {\frac{{dC_{\text{b}} }}{dt}} \right|_{\text{woscrap}} \;\;\;\;\left( {{\text{during}}\;{\text{solidification}}} \right) $$

(8)

$$ W_{\text{m}} \frac{{dC_{\text{b}} }}{dt} = - \beta A\;(C_{\text{b}} - C_{\text{i}} ) + W_{\text{m}} \;\left. {\frac{{dC_{\text{b}} }}{dt}} \right|_{\text{woscrap}} \;\;\;\;\left( {{\text{during}}\;{\text{the}}\;{\text{melting}}\;{\text{of}}\;{\text{solidified}}\;{\text{shell}}\;{\text{and}}\;{\text{melting}}\;{\text{of}}\;{\text{the}}\;{\text{parent}}\;{\text{scrap}}} \right) $$

(9)

where $\left. {\frac{{dT_{\text{b}} }}{dt}} \right|_{\text{woscrap}}$ and $\left. {\frac{{dC_{\text{b}} }}{dt}} \right|_{\text{woscrap}}$ are the rates of change in the bulk liquid melt temperature (heating rate of the liquid melt as expressed in later sections) and carbon composition due to the standard trajectories (in the absence of any scrap charge).

Different Approaches Used to Solve Scrap Dissolution

The different approaches adopted to solve the coupled heat and mass transfer equation in this paper include (a) Green’s function approach; (b) quasi-static approach; (c) integral profile method (IPM); and (d) finite difference approach. The details of all these approaches are given in my recent work.[19]

Results and Discussion

The simulations carried out to study the dissolution of pure steel scrap in high carbon melt under the influence of time-dependent standard temperature and carbon composition trajectory are presented here. For the sake of simplicity, the temperature trajectory of bulk liquid melt is assumed to be

$$ T_{\text{b}} = 1310 + \frac{45}{60}\;t. $$

(10)

Equation [10] implies a rise of bulk metal temperature at the rate of 40 to 50 °C/min. A standard carbon trajectory (change in carbon content of metal with time) is also assumed:

$$ \begin{array}{ll} {C_{\text{b}} = 4.5 - \frac{0.2667}{6}\left( \frac{t}{60} \right)^{2} } & {{\text{if}}\;0 \le t \le 180} \\ {C_{\text{b}} = 4.1 - 0.2667\left( {\frac{t}{60} - 3} \right)} & {{\text{if}}\;180 < t \le 1000} \\ {C_{\text{b}} = 0.442\;\exp \left( { - 0.6\left( {\frac{t}{60} - 16.7} \right)} \right)} & {{\text{if}}\;t \ge 1000} \\ \end{array} $$

(11)

where t is the time in seconds.

Comparison of Results of Different Approaches

The simulation runs are done for three different Biot numbers of 50, 100, and 500. The scrap ratio is assumed to be 0.20. The time step values used for simulation are summarized in Table II. The appropriate value of the time step was chosen by performing calculations for different values of time steps and then selecting the value in the range where the results do not change considerably, giving a stable solution.

Table II Time Step Values Selected for Simulations in Fig. 1

Location of the moving boundary for different approaches

The results of different approaches are compared by plotting the boundary location for different times and Biot numbers in Figure 1. The calculated temperature trajectory in the bulk liquid melt is plotted on the same graph in Figure 1. It can be seen from Figure 1 that the results of the Green’s function approach are close to that obtained by the quasi-static approach for all Biot numbers. The FDM Explicit (fixed grid) method results in 5 to 10 pct lower total time of dissolution as compared to the Green’s function approach. The total dissolution time calculated by the IPM approach is almost the same as that obtained by the Green’s function approach for the Biot number 50, but it deviates by approximately 4 pct for the Biot number 100, and by approximately 15 pct for the Biot number 500, as compared to the Green’s function approach. The path of dissolution obtained with the IPM approach deviates from other approaches throughout the process for Biot numbers 50 and 100, whereas it follows nearly the same path as that in other approaches up to 75 pct of dissolution time for Biot number 500.

Fig. 1

Boundary location vs time for different Biot numbers and different approaches

The Green’s function approach has the highest computational time and the quasi-static approach has the minimum computational time. Due to minimum computational time, finally, the quasi-static approach is used for other comparisons. As far as the accuracy of different approaches is concerned, it was already established that the Green’s function approach is more accurate for a pure heat transfer case based upon overall heat balance.[19] The overall heat balance-based error analysis is not possible for a coupled heat and mass transfer case because interface temperature is also changing due to carbon mass transfer. However, presuming that Green’s function should be the most accurate approach, another approach the results of which are the closest to Green’s function is the quasi-static approach. Moreover, the quasi-static approach has the least computational time as compared to other approaches. Therefore, the quasi-static approach is finally recommended for calculations because of the least computational time along with reasonable accuracy.

Computed C-T trajectory in different approaches

Bulk carbon vs bulk temperature trajectory is superimposed on the Fe-C phase diagram for all the approaches (Figure 2). Quasi-static and Green’s function approaches follow nearly the same C-T trajectory, whereas the trajectory in the IPM approach deviates slightly from the Green’s function trajectory. The C-T trajectory for the FDM Explicit (fixed grid) approach is close to the Green’s function approach for the Biot number 100 and 500, whereas it deviates from Green’s function for the Biot number 50. The C-T trajectory is close to the liquidus line when the Biot number is 50, but moves away from the liquidus line as the Biot number is increased from 50 to 100 or from 100 to 500.

Fig. 2

Bulk temperature and carbon trajectory plotted on Fe-C phase diagram for different Biot numbers and different approaches

Computed temperature profile inside scrap from different approaches

Temperature profiles are computed at an interval of (1/11^th) of the total dissolution time for all the approaches (Figures 3 through 5). Faster penetration of heat is observed for all Biot numbers in the case of the quasi-static approach. Temperature profiles generated by the IPM method are identical to that obtained by the Green’s function approach when the Biot number is 500. However, a faster penetration of heat is observed as compared to other approaches for the Biot numbers 50 and 100.

Fig. 3

Temperature profile inside the scrap as predicted by different approaches (Biot no = 50)

Fig. 4

Temperature profile inside the scrap as predicted by different approaches (Biot no = 100)

Fig. 5

Temperature profile inside the scrap as predicted by different approaches (Biot no = 500)

Significance of heat transfer vs mass transfer controlled regions

The calculated carbon and temperature trajectories for the different Biot numbers given in Figures 6 through 8 show the results estimated by the quasi-static approach. The significant points to be noted are as follows:

Fig. 6

The bulk and interface carbon and temperature trajectories calculated by different approaches (Biot no = 50)

Fig. 7

The bulk and interface carbon and temperature trajectories calculated by different approaches (Biot no = 100)

Fig. 8

The bulk and interface carbon and temperature trajectories calculated by different approaches (Biot no = 500)

• (a) All the scrap dissolves before the bulk liquid melt attains the liquidus temperature of the scrap [1809 K (1536 °C)] for the Biot numbers 50 and 100.

• (b) The difference between the temperature of bulk metal and that at the interface is observed to be small for the Biot numbers 50 and 100 throughout the melting process. However, for Biot number of 500, the difference between the temperature of the bulk metal and interface is small as long as the bulk liquid temperature is below the liquidus temperature of the scrap; hereafter, the difference widens.

• (c) The following different zones of control of the process (with respect to heat and mass transfer) can be defined on the basis of the calculated results:

  • If $(T_{\text{b}} > T_{\text{m}} )$ and $(C_{\text{b}} > C_{\text{sc}} )$, the process is controlled by heat and mass transfer for small differences of T _b and T _m. As the differences between T _b and T _m increase, heat transfer control begins to dominate.

  • If $(T_{\text{b}} > T_{\text{m}} )$ and $(C_{\text{b}} < C_{\text{sc}} )$, the process is controlled by heat transfer only.

  • If $(T_{\text{b}} < T_{\text{m}} )$ and $(C_{\text{b}} > C_{\text{sc}} )$, the process is controlled by mass transfer only.

  • If $(T_{\text{b}} < T_{\text{m}} )$ and $(C_{\text{b}} < C_{\text{sc}} )$, the dissolution of scrap does not take place.

Thus, the rate of scrap dissolution is slow and is controlled by mass transfer when the temperature of the melt is lower than the liquidus temperature of the scrap. When the temperature of the melt becomes higher than the liquidus temperature of the scrap, the rate of scrap dissolution increases rapidly and it is then controlled by heat transfer.

Effect of Different Scrap Ratios

The location of the solid–liquid boundary calculated from the quasi-static approach is plotted in Figures 9 through 11 for different melting times and Biot numbers along with the calculated temperature trajectory in the bulk liquid melt, for different scrap ratios. The following observations can be made:

Fig. 9

Boundary location and velocity of moving boundary vs time for different scrap ratios (Biot no = 50)

Fig. 10

Boundary location and velocity of moving boundary vs time for different scrap ratios (Biot no = 100)

Fig. 11

Boundary location and velocity of moving boundary vs time for different scrap ratios (Biot no = 500)

• (a) The total dissolution time increases (Figures 9 through 11) with scrap ratio and the effect is greater for a lower Biot number.

• (b) The bulk liquid metal temperature trajectory follows the lower temperature path for high scrap ratios.

• (c) Bulk carbon vs bulk temperature trajectory is superimposed on the Fe-C phase diagram for different scrap ratios and Biot numbers in Figure 12. The C-T trajectory approaches the liquidus line of the Fe-C phase diagram when the scrap ratio increases.

  Fig. 12

  

  Bulk temperature and carbon trajectory plotted on Fe-C phase diagram for different Biot numbers and scrap ratios

• (d) For a lower Biot number (say Bi = 50) and higher scrap ratio (say scrap ratio = 0.3), the C-T trajectory nearly touches the liquidus line of Fe-C phase diagram during most of the dissolution process.

• (e) The physical properties of steel, as listed in Table I, are used for estimating the dimensional values. The heating rate of liquid melt is kept as 45 deg/min for all these calculations. Based upon the results shown in the Figures 9 through 11, the limiting size of scrap which can be dissolved completely within 20 minutes and the values of the corresponding heat transfer coefficient are plotted in Figure 13.

  Fig. 13

  

  Limiting size of scrap vs heat transfer coefficient for different scrap ratios

Effect of Different Heating Rates of Liquid Melt

The location of the solid–liquid boundary calculated from the quasi-static approach is plotted in Figures 14 through 16 for different times and Biot numbers along with the calculated temperature trajectory in the bulk liquid melt, for different assumed heating rates of liquid melt. The total dissolution time is seen to decrease with an increase in heating rate of liquid melt (Figures 14 through 16) and the effect is greater for a lower Biot number. The bulk liquid metal temperature trajectory follows a lower path for the lower heating rate of liquid melt. The bulk carbon vs bulk temperature trajectory is also superimposed on the Fe-C phase diagram for different heating rates of liquid melt and Biot numbers (Figure 17). The C-T trajectory approaches the liquidus line of the Fe-C phase diagram when the heating rate of liquid melt decreases. For a lower Biot number (say Bi = 50) and lower heating rate of liquid melt (say heating rate = 30 deg/min), the C-T trajectory nearly touches the liquidus line of Fe-C phase diagram during most of the dissolution process. Based upon the results of Figures 14 through 16, the limiting size of scrap which can be dissolved completely in 20 minutes and the corresponding heat transfer coefficient values are plotted in Figure 18. The physical properties of steel as listed in Table I are used for estimating the dimensional values. The scrap ratio is kept as 0.2 for all these calculations.

Fig. 14

Boundary location and velocity of the moving boundary vs time for different heating rates of liquid metal (Biot no = 50)

Fig. 15

Boundary location and velocity of the moving boundary vs time for different heating rates of liquid metal (Biot no = 100)

Fig. 16

Boundary location and velocity of the moving boundary vs time for different heating rates of liquid metal (Biot no = 100)

Fig. 17

Bulk temperature and carbon trajectory plotted on the Fe-C phase diagram for different Biot numbers and rates of heating of liquid metal

Fig. 18

Limiting size of scrap vs heat transfer coefficient for different rates of heating of liquid metal

Relative Importance of Mass Transfer (Dissolution) and Heat Transfer (Melting)

The effect of the mass transfer vs heat transfer controlled process is shown clearly in Figures 9 through 11 and Figures 14 through 16 where the velocity of the moving boundary is also plotted. It can be seen that the fraction of control by mass transfer (with lower velocity of melting) is greater for a lower Biot number. Since L is proportional to the Biot number for fixed k and h, it can be concluded that the scrap with a smaller size will dissolve under a mass transfer controlled process (before the melt temperature reaches the melting temperature of the parent scrap), whereas the scrap of a bigger size will initially dissolve by the mass transfer controlled process followed later by the heat transfer controlled process (after the melt temperature becomes more than the melting temperature of the parent scrap). The melting rate of large-sized scrap increases in the last stages of a heat transfer controlled process, as can be seen from the Figures 9 through 11 and Figures 14 through 16.

Figures 6 through 8 show that the difference between the temperature of bulk liquid melt and scrap interface remains small for most of the dissolution period for the case of Biot numbers 50 and 100, whereas the difference widens during the last stage of melting for the case of Biot number 500. It also supports the fact that the scrap with a smaller size (smaller Biot number) dissolves under a mass transfer controlled process, whereas the scrap of a bigger size (higher Biot number) initially dissolves by the mass transfer controlled process followed by the heat transfer controlled process in the later stages.

Melting of Mixed Scrap in High Carbon Melt

The term “mixed scrap” means that both light scrap (<2 cm thick) and heavy scrap (>20 cm thick) are mixed in various proportions. Besides the thickness of the scrap, the heat transfer coefficient varies with time in an actual BOF steelmaking process mainly due to the change in fluid flow conditions. It is practically not possible to simulate the exact conditions of scrap dissolution in the BOF process in a mathematical model. However, only as an approximation, the heat transfer coefficient can be calculated as a function of mixing energy. The mixing energy can be calculated for a standard blowing pattern and decarburization profile of a typical BOF steelmaking process (Figure 19). The detailed calculation procedure is given in Appendix B. The estimated heat transfer coefficient, mass transfer coefficient, and mixing energy are plotted in Figure 20. The estimated heat transfer coefficient lies in the range of 30,000 to 55,000 W/m²/K, the mass transfer coefficient in the range of 0.958 to 3.65e−4 m/s, and the mixing energy in the range of 2000 to 28,000 W/ton. The average values are 52,000 W/m²/K, 2.59e−4 m/s, and 20,000 W/ton, respectively. These values are in close agreement with the values reported in References 2, 3, 6, and 7. The practical aspects of using different scrap mixes with different total scrap ratios for different hot metal compositions and temperatures are discussed in detail in my recent publication.[21]

Fig. 19

Standard blowing pattern and decarburization profile

Fig. 20

Calculated heat transfer coefficient, mass transfer coefficient, and mixing energy for a standard blowing pattern and decarburization profile

Effect of ratio of light to heavy scrap on melting rate

The simultaneous dissolution of heavy-sized scrap (thickness 30 cm) and light-sized scrap (thickness 2 cm) is investigated in this section for different total scrap ratios as well as the ratio of light to heavy scrap for different heating rates of the liquid melt. The investigation is done only for the first 7 minutes (400 seconds) of the process. The standard heating rates of liquid melt are considered in the range of 30 to 60 °C/min and scrap ratio between 0.1 and 0.3. The calculated sizes of light scrap, bulk liquid metal temperature trajectories are plotted for different times, different total scrap ratios and different ratios of light to heavy scrap, and heating rates of liquid melt (Figure 21). Bulk carbon vs bulk temperature trajectories are also superimposed on the Fe-C phase diagram for the same conditions. As the amount of light scrap increases, the solidification of bulk metal on the light scrap increases and the instantaneous temperature of bulk metal goes down. This results in an increase in total dissolution time of the light scrap. Based upon the through analysis of the results plotted in Figure 21, a limiting ratio of light to heavy scrap can be estimated for the given total scrap ratio and the heating rate of liquid melt in such a way that all the light scrap dissolves completely in the first 7 minutes and the temperature trajectory of bulk liquid melt does not touch the liquidus line of Fe-C phase diagram during the first 7 minutes. The permissible percentage of light scrap estimated in such a manner for different total scrap ratios and heating rates of liquid melt is plotted in Figure 22.

Fig. 21

Calculated results of simultaneous dissolution of heavy and light scraps for different scrap ratios and heating rates

Fig. 22

Permissible light scrap vs total scrap ratio for different heating rates of liquid metal

Application of Artificial Neural Network model

The total computation time required for solving the simultaneous heat and mass transfer equations is large (>30 minutes) in all the approaches. The Artificial Neural Network (ANN) model can be trained on the data generated by the fundamental models for different conditions. The details of the ANN model are given in Appendix C.

In the present work, approximately 50,000 patterns have been generated from a fundamental model of coupled heat and mass transfer approach for the case of mixed scrap; of these, 5000 patterns are selected for training and 2500 patterns are selected for testing purposes (through random selection). The sum of square of the error is minimized (using MATLAB software) during the training phase. The parameters selected for developing the neural network model are given in Table III.

Table III Parameters Selected for the ANN Model of Scrap Dissolution Process

The range of input and output parameters and their mean and standard deviations are given in Table IV.

Table IV Range, Mean, and Standard Deviation of the Data Selected for ANN

The results of different ANN topologies (with the backpropagation method in terms of R ² and standard errors) are plotted in Figure 23.

Fig. 23

Performance with different ANN topologies

It is clear from Figure 23 that network 6-30-3 gives the best result.

The actual vs predicted and residual plots for the best performing 6-30-3 ANN network are shown in Figure 24. Accuracy of prediction decreases for smaller size of scrap (Figure 24).

Fig. 24

Performance of 6-30-3 ANN network

Recommendations for Using Different Approaches

The process of scrap dissolution has been studied in detail for the case of plate geometry using various approaches (Green’s Function, quasi-static approach, IPM (with parabolic profile), and FDM Explicit (fixed grid)) for the case of coupled heat transfer and mass transfer (of carbon) in a high carbon melt. The results of all approaches are compared for different Biot numbers. The results of the quasi-static approach are found to be nearly the same as that of the Green’s function approach. There is a small deviation with the IPM and FDM approach calculated results. It is recommended to use the quasi-static approach, which requires the least computation time. It can be employed to study the effect of other parameters (like scrap ratios, heating rates of liquid melt). The IPM method can also be used to investigate. The dissolution of scrap mix is also investigated with the help of mathematical models for different total scrap ratios and heating rates of liquid melt. The ANN model has been developed with the help of data generated by the fundamental coupled heat and mass transfer controlled model. It is recommended to use a 6-30-3 ANN topology which gives the best results.

Relevance of this Work to the Industrial Steelmaking Process

As we can understand, the industrial steelmaking process is very complex and it is also difficult to measure the scrap weights dissolved during the process. An attempt has been made in the current study to quantitatively introduce the effect of various parameters in a simplified manner for getting the insight of the scrap dissolution process under industrial conditions based on coupled heat and mass transfer control. As far as the validation of our model calculated results is concerned, considering the average heat transfer coefficient of the BOF process as 47,500 (the literature reported as well as verified by calculations, Figure 20), and also considering the physical properties as given in Table I, and estimating the total dissolution time in dimensional form from Figures 9 through 11 and Figures 14 through 16, the total dissolution time for the scrap is listed Table V.

Table V Total Time for Scrap Dissolution for Different Thickness, Scrap Ratio, and Heating Rates

As far as a comparison with the industrial work is concerned, one test could be that all the scrap should get dissolved during the blowing time (16 minutes). It has been my observation that even heavy scrap (40 cm thick) can be dissolved in 16 minutes and light scrap (<2 cm thick) gets dissolved within 5 minutes of the process which was verified by converter turndowns. There have been few experimental data for industrial operations available in the literature. Oya[22] reported that with a scrap ratio of 0.25, a square billet of 50 cm thickness could be dissolved completely in 20 minutes. Hill[23] reported that the scrap even as thick as 70 to 90 cm gets dissolved in 23 minutes. Bondarenko et al.[24] reported that scrap of size between 17 and 23 cm gets dissolved completely within 17 minutes. The calculated data in Table IV match well with these observations. Therefore, our model can be used as a guideline for deciding the scrap ratios and scrap size mix for effective process control.

Appendices

Appendix A: Estimation of Heat and Mass Transfer Coefficients as a Function of Velocity of the Moving Interface

Stagnant concentration and thermal boundary layers in liquid melt are assumed to exist adjacent to the interface. The heat transfer in the boundary layer takes place only through conduction, whereas the mass transfer occurs through diffusion.

Governing Mathematical Equations for Mass Transfer

It is assumed that the mass transfer in the boundary layer occurs only due to the diffusion process. The mass transfer equation in 1-D in the liquid melt adjacent to the scrap interface is given by

$$ D\frac{{d^{2} C}}{{dX^{2} }} + v\frac{dC}{dX} = 0 $$

(A1)

with the following boundary conditions:

$$ C = C_{\text{i}} \;\;\;\;{\text{at}}\;x = 0 $$

$$ C = C_{\text{b}} \;\;\;\;{\text{at}}\;x = \delta_{\text{C}} $$

where the thickness of the concentration boundary layer $\delta_{\text{c}}$ is estimated from

$$ D\frac{{C_{\text{b}} - C_{\text{i}} }}{{\delta_{\text{C}} }} = \beta_{\text{o}} (C_{\text{b}} - C_{\text{i}} ) $$

which means that the thickness of the boundary layer is given by

$$ \delta_{\text{C}} = \frac{D}{{\beta_{\text{o}} }} $$

(A2)

where D is the diffusion coefficient for mass transfer in the liquid melt and $\beta_{0}$ is the mass transfer coefficient.

The solution of Eq. [A1] is

$$ \frac{{C - C_{\text{i}} }}{{C_{\text{b}} - C_{\text{i}} }} = \frac{\exp ( - vX/D) - 1.0}{{\exp ( - v\delta_{\text{C}} /D) - 1.0}}. $$

(A3)

The mass transfer of carbon from the melt to the interface per unit time unit area is calculated from

${\text{Mass}}\;{\text{flux}}\;{\text{of}}\;{\text{carbon}}\;\left( {{\text{unit}}\;{\text{time}}\;{\text{unit}}\;{\text{area}}} \right) = \left. {D\frac{dC}{dX}} \right|_{X = 0} = D(C_{\text{b}} - C_{\text{i}} )\left( \frac{v}{D} \right)\left. {\frac{\exp ( - vX/D)}{{1 - \exp ( - v\delta_{\text{C}} /D)}}} \right|_{X = 0} = (C_{\text{b}} - C_{\text{i}} )\frac{v}{1 - \exp ( - v\delta /D)}.$

Substituting the value of $\delta_{\text{C}}$,

the mass flux of carbon can be calculated from

$$ J_{\text{C}} = (C_{\text{b}} - C_{\text{i}} )\frac{v}{{1 - \exp ( - v/\beta_{\text{o}} )}}. $$

(A4)

Equation [A4] can be rewritten as

$$ J_{\text{C}} = \beta (C_{\text{b}} - C_{\text{i}} ) $$

(A5)

where β is the mass transfer coefficient for the case of the moving boundary.

From Eqs. [A4] and [A5], the mass transfer coefficient for the moving boundary is estimated from

$$ \beta = \frac{v}{{1 - \exp ( - v/\beta_{0} )}}. $$

(A6)

Governing Equation for Heat Transfer

It is assumed that the heat transfer in the boundary layer occurs due to conduction. The heat transfer equation in 1-D in the liquid melt adjacent to the scrap interface is given by

$$ \alpha \frac{{d^{2} C}}{{dX^{2} }} + v\frac{dC}{dX} = 0 $$

(A7)

where

$$ \alpha = \frac{k}{{\rho C_{\text{p}} }} $$

with following conditions:

$$ T = T_{\text{m}} \;\;\;\;{\text{at}}\;x = 0 $$

$$ T = T_{\text{b}} \;\;\;\;{\text{at}}\;x = \delta_{T} $$

where $\delta_{T}$ is the thickness of the thermal boundary layer which is estimated from the following equation:

$$ k\frac{{T_{\text{b}} - T_{\text{m}} }}{{\delta_{T} }} = h_{\text{o}} (T_{\text{b}} - T_{\text{m}} ). $$

It implies that the thickness of the boundary layer is given by

$$ \delta_{T} = \frac{k}{{h_{\text{o}} }} $$

(A8)

where k is the thermal conductivity in the liquid melt and $h_{\text{o}}$ is the heat transfer coefficient.

The solution of Eq. [A7] is

$$ \frac{{T - T_{\text{m}} }}{{T_{\text{b}} - T_{\text{m}} }} = \frac{\exp ( - vX/\alpha ) - 1.0}{{\exp ( - v\delta_{T} /\alpha ) - 1.0}}. $$

(A9)

The heat transfer flux at the interface (J _H) is given by

$$ J_{\text{H}} = \left. {k\frac{dT}{dX}} \right|_{X = 0} = (T_{\text{b}} - T_{\text{m}} )(kv/\alpha )\left. {\frac{\exp ( - vX/\alpha )}{{1 - \exp ( - v\delta_{T} /\alpha )}}} \right|_{X = 0} = (T_{\text{b}} - T_{\text{m}} )(kv/\alpha )\frac{1.0}{{1 - \exp ( - v\delta_{T} /\alpha )}}. $$

Putting the value of $\delta_{T}$ and $\alpha$,

$$ J_{\text{H}} = (T_{\text{b}} - T_{\text{m}} )(\lambda v/\alpha )\frac{1.0}{{1 - \exp ( - \rho Cpv/h_{\text{o}} )}}. $$

(A10)

Equation [A10] may be rewritten as

$$ J_{\text{H}} = h(T_{\text{b}} - T_{\text{m}} ) $$

(A11)

where h is the heat transfer coefficient for the moving boundary.

From Eqs. [A8], [A10], and [A11], the heat transfer coefficient for the moving boundary can be estimated from

$$ h = \frac{{\rho C_{\text{p}} v}}{{1 - \exp ( - \rho C_{\text{p}} v/h_{0} )}}. $$

(A12)

Heat and Mass Balance at the Interface

Heat balance at the interface is given as

$$ \rho \nu \Updelta H_{0} = \left[ {\left. {k\frac{\partial T}{\partial x}} \right|_{\text{interface}} - h(T_{\text{b}} - T_{\text{m}} )} \right]. $$

(A13)

On substituting the value of h from Eq. [A12], the above equation can be written as

$$ \rho \nu \Updelta H_{0} = \left[ {\left. {k\frac{\partial T}{\partial x}} \right|_{\text{interface}} - \frac{{\rho C_{\text{p}} \nu }}{{\left( {1 - \exp \left( { - \frac{{\rho C_{\text{p}} v}}{{h_{\text{o}} }}} \right)} \right)}}(T_{\text{b}} - T_{\text{m}} )} \right]. $$

(A14)

Mass balance of carbon at the interface is given by

$$ v(C_{\text{i}} - C_{\text{sc}} ) = \beta (C_{\text{i}} - C_{\text{b}} ). $$

(A15)

On substituting the value of β from Eq. [A6], the above equation becomes

$$ \nu = \beta_{\text{o}} \ln \left( {\frac{{C_{\text{i}} - C_{\text{sc}} }}{{C_{\text{b}} - C_{\text{sc}} }}} \right). $$

(A16)

Equations [A14] and [A16] are coupled together as

$$ \nu = \beta_{\text{o}} \ln \left( {\frac{{C_{\text{i}} - C_{\text{sc}} }}{{C_{\text{b}} - C_{\text{sc}} }}} \right) = \frac{1}{{\rho \Updelta H_{\text{o}} }}\left[ {\left. {k\frac{\partial T}{\partial x}} \right|_{\text{interface}} - \frac{{\rho C_{\text{p}} \nu }}{{\left( {1 - \exp \left( { - \frac{{\rho C_{\text{p}} v}}{{h_{\text{o}} }}} \right)} \right)}}\left( {T_{\text{b}} - T_{\text{m}} } \right)} \right]. $$

(A17)

Appendix B

Estimation of Heat Transfer Coefficient as a Function of Total Mixing Energy to the BOF Steelmaking System[25,26]

The following equations are used to estimate heat transfer coefficient as a function of total mixing energy to the BOF steelmaking system which is considered under combined influence of top lance, bottom stirring flow rate, and decarburization reaction.

$$ \begin{gathered} E_{t}^{0} = 6.32 \times 10^{ - 7} \cos \phi \frac{{Q_{t}^{3} M}}{{Wn^{2} d_{\text{t}}^{3} X}} \hfill \\ Q_{\text{decarb}} = \frac{d[C]}{dt}W\frac{{10^{6} \times 22.4 \times 60}}{12} \hfill \\ E_{\text{decarb}}^{0} = 6.18 \times \frac{{Q_{\text{decarb}} T_{\text{l}} }}{W}\left( {\ln \left[ {1 + \frac{{\rho g{\text{H}}h\_{\text{frac}}}}{{p_{\text{atm}} }}} \right] + \left[ {1 - \frac{{T_{0} }}{{T_{\text{l}} }}} \right]} \right) \hfill \\ E_{\text{b}}^{0} = 6.18 \times \frac{{Q_{\text{bottom}} T_{\text{l}} }}{W}\left( {\ln \left[ {1 + \frac{\rho gH}{{p_{\text{atm}} }}} \right] + \left[ {1 - \frac{{T_{\text{o}} }}{{T_{\text{l}} }}} \right]} \right) \hfill \\ E_{\text{total}}^{0} = E_{\text{top}}^{0} \times 0.10 + E_{\text{b}}^{0} + E_{\text{decarb}}^{0} \hfill \\ h = 5000\left( {7.0E_{\text{total}}^{0} } \right)^{0.20} \hfill \\ \end{gathered} $$

where $\phi$ is the angle of the lance tip from vertical, $Q_{\text{t}}$ is the oxygen flow rate, W is the weight of steel, n is the number of openings of the lance, X is the lance height above the metal bath during blowing, H is the bath depth, h_frac is the average depth fraction at which CO bubble formation takes place, $d_{\text{t}}$ is the throat diameter, $T_{\text{o}}$ is the temperature of the bottom stirring gas at input, $T_{\text{l}}$ is the average temperature of the liquid steel, $p_{\text{atm}}$ is the atmospheric pressure, and h is the heat transfer coefficient.

Process Conditions Used in Simulation

Lance angle = 14 deg Number of openings in lance = 6 Throat diameter = 2.46 cm Bath depth = 1.30 m

bottom stirring flow rate = 2 Nm3/min (first 30 pct and last 30 pct of the blow) blowing pattern is defined as follows: oxygen flow rate = 500 NM3/min if 0 < t < 180 seconds; Lance height = 2.20 m if 180 < t < 300 seconds; Lance height = 2.00 m if 300 < t < 400 seconds; Lance height = 1.80 m if t > 400 seconds; Lance height = 1.60

Appendix C: Artificial Neural Network

The concept of ANN originally comes from the mechanisms for information processing in the human brain system. ANN models have been applied to a wide range of complex metallurgical processes[27–30] and proved to be successful due to their ability to develop nonlinear relationships. ANNs are the mathematical patterns constructed by several neurons arranged in different layers interconnected through the complex networks. The layers are defined as input layer, output layer, and at least one hidden layer. A multilayer feed forward backpropagation ANN network has been used in the present work. The typical ANN topology is presented in Figure A1.

Fig. A1

Architecture of feed forward backpropagation ANN

The output of a neuron (k) in the network (y _k ) is the summation of all signals from a previous layer multiplied by weights (w _k,j ) and a bias (b _k ) which is activated by a transfer function (tanh sigmoid) in the following way:

$$ y_{k} = f\left( {\sum\limits_{j = 1}^{N} {\left( {w_{k,j} x_{j} } \right) + b_{k} } } \right)\;\;\;\;{\text{where}}\;f(z) = \frac{2}{1 - \exp ( - 2z)} - 1. $$

(A18)

The sum of the square of the errors (between the training output data and output data obtained using ANN) is minimized for getting the correct values of weights.