Numerical simulation on the melting kinetics of steel scrap in iron-carbon bath

Steel scrap melting kinetics is a key factor that affects the scrap ratio and temperature trajectory of Basic Oxygen Furnace steelmaking process and the productivity and energy utilization of Electric Arc Furnace steelmaking process. In this paper, the interface between steel scrap and melt is analyzed and a novel theoretical model of steel scrap melting in iron-carbon bath is established. Moreover, the effects of bath temperature, bath carbon content, preheating temperature and characteristic length of steel scrap on the melting rate of steel scrap, mass transfer coefficient of carbon, heat transfer coefficient and interface carbon concentration were investigated. The results show that the ultimate steady melting rate increases from (cid:0) 2.04 × 10 (cid:0) 4 to (cid:0) 3.32 × 10 (cid:0) 3 m/s with the increases of bath temperature from 1793 K to 1873 K, the ultimate steady heat and mass transfer coefficient first increase and subsequently decrease, are in the range of 1.04 × 10 5 to 2.86 × 10 5 W/m 2 /K and 0 to 2.89 × 10 (cid:0) 4 m/s, respectively. As the bath carbon content increase from 1.0 wt% to 4.0 wt%, the ultimate steady melting rate increases from (cid:0) 8.85 × 10 (cid:0) 5 to (cid:0) 3.23 × 10 (cid:0) 4 m/s, the ultimate steady heat transfer coefficient first increases and subsequently decreases, is in the range of 7.59 × 10 4 to 1.01 × 10 5 W/m 2 /K. Furthermore, the ultimate steady mass transfer coefficient and the ultimate steady interface carbon content gradually decreases from 1.40 × 10 (cid:0) 4 to 5.21 × 10 (cid:0) 5 m/s and from 0.73 to 0.69 wt%, respectively. With the increase of preheating temperature and characteristic length of steel scrap, the ultimate steady melting rate, heat transfer coefficient between steel scrap and melt, mass transfer coefficient and interface carbon concentration remain almost constant, about (cid:0) 8.85 × 10 (cid:0) 5 m/s, 8.40 × 10 4 W/m 2 /K, 1.40 × 10 (cid:0) 4 m/s and 0.70 wt%, respectively.


Introduction
It is reported that it can reduce the consumption of 0.4 t coke or 1 t raw coal, 1.7 t fresh water and 1.7 t iron ore per ton steel scrap used in the ferrous metallurgy process.Meanwhile, 70% wastewater emission, 1.6 t CO 2 , 0.35 t waste slag and 2.6 t tailings can also be decreased [1][2][3].Therefore, improving the utilization ratio of steel scrap has favorable economic and environmental benefits, and is of great significance to the sustainable development of iron and steel industry.
The scrap ratio in Basic Oxygen Furnace (BOF) is influenced by steel scrap melting rate, mainly due to the low bath temperature and the high viscosity of the metal in the early melting time, resulting in the low melting rate of steel scrap [4][5][6].Moreover, steel scrap is one of the major raw feedstocks for Electric Arc Furnace (EAF) steelmaking process [7][8][9].Heating and melting the charge consumes approximately 60% of the overall energy, and charge melting time contributes more than 50% of overall smelting time [10].Therefore, steel scarp melting kinetics is an essential factor in governing the productivity and energy consumption of EAF process.
Predicting the steel scrap melting rate, heat transfer coefficient and mass transfer coefficient of carbon can supply a theoretical foundation for BOF and EAF process modeling [7,11,12].Several research on the steel scrap melting behavior in iron-carbon melt have been conducted.Based on thermal model experiments, it was discovered that a solidified layer formed surrounding the parent steel scrap when it was immersed in melt.Subsequently, with the increases of immersion time, solidified layer and parent steel scrap began to melt [10,[13][14][15].The melting rate of steel scrap increases gradually with the increase in bath temperature, bath carbon content, and the decrease in characteristic length of steel scrap.The increase in preheating temperature of steel scrap can improve melting rate in the early melting time, but has less effect on the late melting time.Shukla et al. [16] investigated the mass transfer of carbon and heat transfer during steel scrap melting process through the concept of moving boundary layer, and calculated the variation of heat and mass transfer coefficient with time, are in the range of 3.0 × 10 4 to 5.5 × 10 4 W/m 2 /K and 0.958 × 10 − 4 to 3.65 × 10 − 4 m/s, respectively.However, this model ignored the existence of solid-liquid coexistence areas.Kruskopf et al. [6,11] established a steel scrap melting model that considered the existence of solid-liquid coexistence zone, which was equated as the thermal and carbon concentration boundary layer.The variation of steel scrap melting rate over time and the exchange of heat and mass between steel scrap and melt were predicted and the results show that there are sharp temperature and carbon concentration gradients adjacent to the melting interface, however, the carbon concentration inside the solid steel scrap is essentially the same as the initial carbon concentration of steel scrap.
The melting process of steel scrap was numerically simulated by developing a theoretical model in this study and the solid-liquid coexistence region, thermal boundary layer and carbon concentration boundary layer was analyzed.The melting rate of steel scrap, heat transfer coefficient, mass transfer coefficient of carbon and interface carbon concentration with time were calculated.Moreover, the effects of various parameters, such as bath temperature, bath carbon content, characteristic length and preheating temperature of steel scrap on the steel scrap melting behavior were investigated.The purpose of this work is to investigate the melting mechanism of steel scrap in iron-carbon bath, which can provide a theoretical foundation for the establishment of BOF or EAF steelmaking process model.

Theoretical analysis
Steel scrap melting kinetics is fundamentally a moving boundary problem [6,16,17].In addition, due to the uncertainty of accurately defining the boundary conditions, mathematical modeling of metallurgical procedure is extremely challenging.Therefore, accurate boundary condition definition is critical in mathematical modeling for solving differential equations [18,19].
When steel scrap comes into contact with liquid iron-carbon melt, a thermal boundary layer formed.However, the formation of the concentration boundary layer is delayed by the sluggish mass transfer mechanism.Temperature and carbon content profiles derived from the concept of moving boundary layer are shown in Fig. 1 and Fig. 2 [15].Meanwhile, the steel scrap melting process includes solid-liquid coexistence region, thermal boundary layer and concentration boundary layer, as shown in Fig. 1.
Previous research has indicated that the steel scrap melting process can be classified into three stages [1,20], i.e., (a) the formation of a solidified layer on the steel scrap surface by liquid melts; (b) the melting of solidified layer; (c) the melting of parent steel scrap.During the process of solidification and melting, the following heat balance equation exists: Heat flux provided by bath (H 1 ) = Heat flux expended for melting steel scrap (H 2 ) + Heat flux dissipated in steel scrap (H 3 ).
When steel scrap is immersed into liquid melt, the thermal gradient is great enough that H 3 exceeds H 1 .According to the heat balance, H 2 is negative and this results in the formation of solidified layer.The temperature gradient decreases, with the increase of immersion time, and the solidified layer reaches its maximum thickness when H 1 is balanced with H 3 .Namely, liquid melt solidifies into a solidified layer owing to chilling effect, which is the first stage of steel scrap melting [20].Subsequently, H 1 starts to be larger than H 3 , and solidified layer remelts, which is the second stage of steel scrap melting.The liquid melt then carburizes to steel scrap surface accompanied by heat transfer.Once the melting point of carburized layer does not exceed liquidus temperature, it melts into liquid melt [21].The melt starts to re-carburize new solid steel scrap surface until parent steel scrap is completely melted, which is the third stage of steel scrap melting.

Theoretical model of steel scrap melting
For the purpose of simplicity, the following hypothesis has been made.
(1) The melt and steel scrap consist of iron and carbon.
(2) The physical properties of liquid melt and solid steel scrap are constants and these parameters are presented in Table 1.
(3) The carbon mass transfer within parent steel scrap is not considered owing to its slow rate.(4) The influence of melt flow on the melting process of steel scrap is not considered.

The mathematical equations controlling carbon mass transfer
A one-dimensional mass transfer equation for the liquid melt next to steel scrap interface is gained from Fick's second law of diffusion [16]: These boundary conditions are present: Meanwhile, the δ C is gained by the concept of non-moving boundary: The mass flux of carbon from melt to interface per unit area unit time can be found by Fick's first law of diffusion: Therefore, the mass transfer coefficient of moving boundary is derived:

The mathematical equations controlling heat transfer
A one-dimensional heat transfer equation for the liquid melt next to steel scrap interface is gained from Fourier Equation state [16]: These boundary conditions are present: Meanwhile, the δ T is gained by the concept of non-moving boundary.
According to Fourier Equation state, the heat flux from melt to interface can be calculated: Therefore, the heat transfer coefficient of moving boundary is derived: In addition, the Fourier equation can be resolved by the method of separated variables: It is assumed that the solution takes the form of [16]: Taking: The following boundary conditions are present: It can be obtained that: Therefore, the relationship between temperature at a point in radial direction of steel scrap with time and the temperature gradient versus time at the interface can be represented as follows, respectively.

The heat and mass balance equations of interface
The heat balance equation on the interface is presented as [15]: The physical meaning of Equation ( 15) is: The mass balance equation on the interface is presented as [16]: As the mass transfer of carbon within parent steel scrap is not considered in this work, Equation ( 16) can be written as follows: Therefore, the C i can be derived: Meanwhile, the C s is provided by Equation (19).
Where γ is the carbon distribution ratio between liquid melt and solid steel scrap, which can be acquired from Fe-C phase diagram (Fig. 2).The correlation between h 0 and β 0 is provided by Chilton-Colburn analogy, and D C is the function of C i and T m [6,16,23].
T m is depended on the C i , as shown in Equation ( 22) [16].
The above Eqs.( 4), ( 8) and ( 14) ~ 22) form the theoretical model of steel scrap melting.The steel scrap shape can be a plate, a cylinder or a sphere in this model.It was programmed to calculate the melting rate, heat transfer coefficient, mass transfer coefficient of carbon, and interface carbon concentration at various times when the bath temperature, bath carbon content, characteristic length and preheating temperature of steel scrap are 1773-1873 K, 1.0-4.0wt%, 0.01-0.05m, 300-1100 K, respectively.Moreover, the influence of bath temperature, bath carbon content, preheating temperature and characteristic length of steel scrap on the melting behavior in iron-carbon bath were also studied.

Model validation
To test the accuracy of this model, the experimental conditions in previous studies were substituted into program to compare the steel scrap melting rate in the late melting time [5,10,17,20,24].Fig. 3 verifies the influence of bath temperature, bath carbon content, preheating temperature and characteristic length of steel scrap on the melting rate, respectively.As shown in Fig. 3, with the increase of bath temperature and bath carbon content, the melting rate of steel scrap gradually increases.As the characteristic length of steel scrap increase, the melting rate gradually decreases.The preheating temperature of steel scrap has less influence on the melting rate of steel scrap in the later stages of melting.Meanwhile, it can be found that the melting rate calculated by this model is in accordance with the results of literature data, which indicates that this model can predict the influence of above process factors on the steel scrap melting rate.melting of solidified layer or parent steel scrap and v > 0 means the solidification of liquid melt), then it turns into a negative number and trends towards becoming a steady constant.This result can be explained by the fact that when the steel scrap is initially immersed in the liquid melt, the melt solidifies on the steel scrap surface to form a solidified layer.Subsequently, H 1 begins to be greater than H 3 , the solidified layer and steel scrap start to melt.With the prolonging of immersion time, H 2 gradually increases and H 3 gradually decreases, therefore, v gradually increases.During the late time of steel scrap melting, the temperature gradient is almost unchanged.Therefore, the disparity between H 1 and H 3 remains almost unchanged and v becomes nearly constant [14,25].

The influence of bath temperature on steel scrap melting behavior
With the increases of T b , the solidification time of solidified layer and the time to attain final steady melting rate decreases.Simultaneously, the final steady melting rate increases from − 2.04 × 10 − 4 to − 3.32 × 10 − 3 m/s, which is basically consistent with previous thermal simulation experiments [4,25].This can be explained by the fact that as the T b increases, H 1 gradually increases.It is noteworthy that when the T b is low (1793 K), the melt cannot supply sufficient heat for melting, and the melting process is delayed [26,27].
Fig. 4(b) reveals the effect of T b on the h.As it is verified from Fig. 4(b) that h first decreases and subsequently increases over time and ultimately converges to a steady constant.As the immersion time increases, the temperature gradient between liquid melt and solidified layer gradually decreases, which leads to decrease in h.As the solidified layer and parent steel scrap begin to melt, H 3 decreases and H 2 increases, hence, h gradually increases.When the steel scrap is "heat through", the disparity between H 1 and H 3 remains almost unchanged and h becomes constant.
With the increases of T b , H 1 gradually increases, which resulting in the solidified layer formation time and h decreases.Moreover, the final steady h first increases and subsequently decreases, is in the range of 1.04 × 10 5 to 2.86 × 10 5 W/m 2 /K.Fig. 4(c) and (d) display the effect of T b on the β and C i , respectively.It can be found in Fig. 4(c) and (d) that there is no carbon mass transfer process in the formation and remelting process of solidified layer, and C i is the same as that of melt.As the steel scrap starts to melt, some of H 1 is utilized to heat steel scrap, resulting in β gradually increases and approaching the steady constant, moreover, C i gradually decreases and then approaches a stable constant.As the T m increases, carbon mass transfer is quickened, which leads to the increase of β and the decrease of C i .When the steel scrap is "heat through", the entire system becomes a steady temperature field, meanwhile, β and C i remain steady.
With the increases of T b , the final steady β first increases and subsequently decreases, is in the range of 0-2.89 × 10 − 4 m/s, moreover, the time to reach the lowest C i and the ultimate steady C i gradually decreases.It is worth noting that when T b is 1873 K, β is 0, and C i is equal to the same as the content of parent steel scrap.The cause of this is as the T b increases, H 1 increases gradually, T m increases, and carbon mass transfer process speeds up, resulting in the increases of β.As the T b increase further, the difference between T b and T m becomes greater and the steel scrap melting process is primarily driven by heat transfer, so that β decreases and approaches zero [6,28].

The influence of bath carbon content on the steel scrap melting behavior
Fig. 5(a) represents the effect of C b on the v.It can be found in Fig. 5(a) that as the C b increases, the formation time of solidified layer decrease, nevertheless, the ultimate steady v increases, which is well correlated with the previous study of thermal simulation experiment [4,29].It is noteworthy that when C b is more than 2.0 wt%, the melt does not form a solidified layer on steel scrap surface, namely, there are no the first and second stage in which the steel scrap is melted.This is due to as C b increases, interface concentration gradient increases, simultaneously, the mass flux of carbon from melt to interface increases per unit area per unit time, namely, the carbon content transmitted from melt to steel scrap surface increases.In consequence, the melting point of local steel scrap is reduced to a greater extent, the formation time of solidified layer is reduced, H 3 is decreased, H 2 is increased and the ultimate steady melting rate is increased from − 8.85 × 10 − 5 to − 3.23 × 10 − 4 m/s.[17].As a result, the formation time of solidified layer is reduced, and the h is decreased gradually, as shown in Fig. 5(b).Moreover, the ultimate steady h first increases and then decreases, is in the range of 7.59 × 10 4 to 1.01 × 10 5 W/m 2 /K.Fig. 5(c) and (d) indicate the effect of C b on the β and C i , respectively.As it is verified from Fig. 5(c) and (d) that with the increases of C b , the ultimate steady β gradually decreases from 1.40 × 10 − 4 to 5.21 × 10 − 5 m/s, the ultimate steady C i decreases slightly from 0.73 to 0.69 wt%.This is due to an increase in the interface concentration gradient and mass flux of carbon from melt to interface as C b increase.

The influence of characteristic length of steel scrap on the steel scrap melting behavior
Fig. 6(a) indicates the effect of R on the v.As it is verified from Fig. 6(a), the formation time of solidified layer increase and the time to reach maximum v prolong, with the increase of R.This can be attributed to as R increases, the specific surface area decreases, H 3 increases.Moreover, in the latter time of steel scrap melting, the variation in temperature gradient is minimal and the variation between H 1 and H 3 stays nearly unchanged.Thus, the v becomes nearly constant, about − 8.85 × 10 − 5 m/s.It is instructive to note that when R is 0.01 m, the melt does not develop a solidified layer on the steel scrap surface.It can be explained by the large specific surface area of steel scrap, which is easy to "heat through", H 3 decreases and H 2 increases.
Fig. 6(b) reveals the effect of R on the h.As shown in Fig. 6(b), with the increases of R, the h first gradually decreases and then increases, tends to the same constant finally, meanwhile, the time to reach ultimate steady h is prolonged.This is due to with the increases of R, the specific surface area decreases, and the steel scrap is not easy to heat through.However, the ultimate steady h is basically unchanged, about 8.40 × 10 4 W/m 2 /K.The reason for this is the variation in the temperature gradient becomes almost minimal and the variation between H 1 and H 3 remains unchanged.Hence, h becomes almost constant in the late time of steel scrap

melting.
Fig. 6(c) and (d) indicate the effect of R on the β and C i , respectively.As it is verified from Fig. 6(c) and (d) that as the R increase, the time to reach maximum β and minimum C i increases, however, the ultimate steady β and C i remain basically unchanged at 1.40 × 10 − 4 m/s and 0.70 wt%, respectively.This is due to the fact that as R increases, the specific surface area decreases, the increase rate of T m decreases, and carbon mass transfer process slow down.When the steel scrap is "heat through", the entire system becomes a steady temperature field, β and C i remain stable constant at this moment.
4.4.The influence of preheating temperature of steel scrap on the steel scrap melting behavior Fig. 7(a) displays the effect of T y on the v.As it is verified from Fig. 7(a), with the increase of T y , the formation time of solidified layer decrease, the v at early melting time gradually increases, moreover, the ultimate steady melting rate remain stable constants, about − 8.85 × 10 − 5 m/s, which is in satisfactory consistent with the previous study of thermal simulation experiment [10,20].This can be explained by the fact that as T y increases, the heat of parent steel scrap gradually increases, the temperature gradient of interface decreases, which resulting in H 3 decreases and H 2 increases gradually.During the late time of steel scrap melting, the temperature gradient is almost unchanged.Therefore, the variation between H 1 and H 3 remains almost unchanged and v becomes nearly constant.Fig. 7(b) represents the effect of T y on the h.It can be found in Fig. 7(b) that as the T y increase, the formation time of solidified layer decreases, and the h decreases gradually.However, the ultimate stable h remains basically unchanged at 8.40 × 10 4 W/m 2 /K.The cause of this is as T y increases, the heat of parent steel scrap gradually increases, the temperature gradient of interface decreases, thus, the h of early melting time of steel scrap gradually decreases.During the late time of steel scrap melting, the temperature gradient is almost unchanged, the disparity between H 1 and H 3 remains almost unchanged.Therefore, the h becomes nearly constant.Fig. 7(c) and (d) indicate the effect of T y on the β and C i , respectively.As it is verified from Fig. 7(c) and (d), with the increase of T y , β in early melting time gradually increases, and interfacial carbon content gradually decreases.However, ultimate steady β and interface carbon concentration are basically unchanged at 1.40 × 10 − 4 m/s and 0.70 wt%, respectively.This is due to the fact that as the T y increase, the carbon mass transfer process is accelerated, β gradually increases and carbon content at the interface gradually decreases.When the steel scrap is "heat through", H 3 = 0, H 1 is equal to H 2 , β and C i remain unchanged at this moment.
In addition, it can be seen from the above research that the preheating temperature of steel scrap has less influence on the steel scrap melting behavior compared with bath temperature, bath carbon content and characteristic length of steel scrap.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Latent heat of melting (J/kg)

Fig. 3 .
Fig. 3. Verification of (a) bath temperature, (b) bath carbon content, (c) characteristic length of steel scrap, (d) preheating temperature of steel scrap on the melting rate of steel scrap.

Fig. 4 (Fig. 4 .
Fig. 4(a) indicates the effect of T b on the v.As it is verified from Fig. 4(a) that v is a steady positive number at first (v < 0 means the

Fig. 5 .
Fig. 5.The (a) melting rate of steel scrap, (b) heat transfer coefficient, (c) mass transfer coefficient of carbon, (d) interface carbon concentration under various bath carbon content.

Fig. 6 .
Fig. 6.The (a) melting rate of steel scrap, (b) heat transfer coefficient, (c) mass transfer coefficient of carbon, (d) interface carbon concentration under various characteristic length of steel scrap.

Fig. 5 (
Fig.5(b) reveals the effect of C b on the h.As the C b increases, the melting point of solidified layer decreases, making it easy to melt[17].As a result, the formation time of solidified layer is reduced, and the h is decreased gradually, as shown in Fig.5(b).Moreover, the ultimate steady h first increases and then decreases, is in the range of 7.59 × 10 4 to 1.01 × 10 5 W/m 2 /K.Fig.5(c) and (d) indicate the effect of C b on the β and C i , respectively.As it is verified from Fig.5(c) and (d) that with the increases of C b , the ultimate steady β gradually decreases from 1.40 × 10 − 4 to 5.21 × 10 − 5 m/s, the ultimate steady C i decreases slightly from 0.73 to 0.69 wt%.This is due to an increase in the interface concentration gradient and mass flux of carbon from melt to interface as C b increase.

Fig. 7 .
Fig. 7.The (a) melting rate of steel scrap, (b) heat transfer coefficient, (c) mass transfer coefficient of carbon, (d) interface carbon concentration under various preheating temperatures of steel scrap.

( 1 )
With increasing the bath temperature, the solidified layer formation time, the time to attain ultimate steady melting rate, the time to attain ultimate steady interface carbon concentration, and the ultimate steady interface carbon concentration decreases.Meanwhile, the ultimate steady melting rate increases from − 2.04 × 10 − 4 to − 3.32 × 10 − 3 m/s, the ultimate steady heat transfer coefficient and mass transfer coefficient first increase and subsequently decrease, are in the range of 1.04 × 10 5 to 2.86 × 10 5 W/m 2 /K and 0 to 2.89 × 10 − 4 m/s, respectively.(2) With the increase of bath carbon content, the solidified layer formation time decrease, the ultimate steady melting rate increases from − 8.85 × 10 − 5 to − 3.23 × 10 − 4 m/s, moreover, the ultimate steady heat transfer coefficient increases initially and then decreases, is in the range of 7.59 × 10 4 to 1.01 × 10 5 W/m 2 /K.Furthermore, the ultimate steady mass transfer coefficient and interface carbon concentration gradually decreases from 1.40 × 10 − 4 to 5.21 × 10 − 5 m/s and from 0.73 to 0.69 wt%, respectively.(3) As the characteristic length of steel scrap increases, the solidified layer formation time increases.In the melting process of steel scrap, the time to reach ultimate steady melting rate, mass transfer coefficient and interface carbon concentration are prolonged, but the ultimate steady melting rate, heat transfer coefficient, mass transfer coefficient of carbon and interface carbon concentration remain almost constant, at − 8.85 × 10 − 5 m/s, 8.40 × 10 4 W/m 2 /K, 1.40 × 10 − 4 m/s and 0.70 wt%, respectively.(4) As the preheating temperature of steel scrap increase, the solidified layer formation time decrease.The melting rate of steel scrap and mass transfer coefficient of carbon in the early time of melting gradually increase, interface carbon concentration is gradually reduced.The ultimate steady melting rate, heat transfer coefficient, mass transfer coefficient of carbon and interface carbon concentration remain almost constant, and they were found to be − 8.85 × 10 − 5 m/s, 8.40 × 10 4 W/m 2 /K, 1.40 × 10 − 4 m/s and 0.70 wt%, respectively.(5) Compared with the preheating temperature of steel scrap, bath temperature, bath carbon content and characteristic length of steel scrap have greater influence on the steel scrap melting behavior.
Carbon distribution ratio between liquid melt and solid steel scrap h 0 Heat transfer coefficient for the concept of non-moving boundary (W/m 2 /K) hHeat transfer coefficient of moving boundary (W/m 2 /K)