_{1}

^{*}

A theoretical investigation of heat flow, solidification and solid shell resistance “**I**_{c}” has been undertaken by using a mathematical model and previous plant trials. The ultimate purpose is to develop operating conditions and therefore to improve the surface quality for continuously cast steel slabs. A new simple criterion called mold thermomechanical rigidity “MTMR” has been proposed to evaluate and to improve these purposes. The parameters of MTMR and its non-dimensional number which use to control the surface defects are present in this investigation. Previous plant trails of slab surface defects formation have been investigated thermo-mechanically with this criterion. The predications show that this criterion is very sensitive of operating parameters and is a significant qualitative tool to evaluate the surface quality. From examination of the behavior of MTMR, the susceptibility and mechanism of surface defects formations with MTMR have been primarily discussed.

Nowadays the continuous casting technology is still undergoing important developments due to the facts that the requirements on product quality, increasing the productivity, saving the energy and recently on the needs of clean environment are continuously being increased. These developments incorporate not only equipment revamps or updates in the installation setups and in their process controls but also in the strategies developments of zero defects by helping the computational models. Computational models of heat transfer, solid shell resistance, interdendritic strain, and micro/macrosegregation are powerful and reliable tools to help of avoiding the defects formation in different cooling zones.

The first requirement to develop a successful zero defects strategy to avoid defects formation is to gain insight into thermal fields during solidification especially in the mold zone and to have an in-depth knowledge of mechanisms of many complex phenomena associated with mold heat transfer and therefore the mechanisms behind the surface defects formation and breakouts.

In the present work, it has been undertaken to develop a fast, simple and flexible criterion to predict and evaluate the surface defects formations in continuously cast steel slabs. Subsequently, this study has been conducted as theoretical investigation with previous metallurgical studies on the plant trials of early solidification stages in continuously cast steel slabs to clarify the mechanism of “MTMR” with different effects of operating conditions based on the nature of surface defects formation.

The present work computes 1-D transient heat flow, so-

lidification and solid shell resistance through the solidifying shell and mushy zone. This because the Pèclet number [_{mold}rC_{p}/l) in the directions of z and y of this process are high and are equal to (Pe_{z} = 2.406 × 10^{3}) and (Pe_{y} = 5.697 × 10^{3}), respectively [

Superheat from liquid steel was evaluated by using the same approach developed by Huang et al. [_{sh}” tabulated in

In order to calculate the different phases of low and peritectic carbon steels, the solidification behaviors of different steels can be classified into three modes as shown in Figures 2(a)-(c). The same approach developed by Rogberg [

ing solid phase transformation or during peritectic reactions [9,10]. Also, Miyazawa and Schwerdtferger [

the effect of different solid dendritic phases on the solid shell resistance as illustrated in Figures 3(c) and (d). Tables 1-3 summarize the model governing equations and its supplementary relations. Only few explanations are provided here, and the reader is referred to the originnal references for the details of the model and assumptions made in the derivations of its governing equations.

However, In order to examine a mold quality criterion of continuously cast steel slab, a new criterion called mold thermo-mechanical rigidity “MTMR” has been proposed to reveal slab macro/micro-surface defects level such as macrosegregation level, interdendritic cracks, oscillation marks, bulging and related defects formed in the mold zone [

The first part of Equation (1) is pure thermal part which consists of the ratio between steel solidus temperature “T_{s}” and the surface temperature “T_{sur}”. T_{s} covers the effects of composition and macro segregation whereas T_{sur} includes the effects of cooling conditions and steel type on MTMR and therefore on the thermo-mechanical resistance of different stresses. The

metallurgical studies and measurements by Brimacombe et al. [_{s} and therefore to the level of macrosegregation which it agrees well with Brimacombe et al. metallurgical examinations [

The second part is thermo-mechanical part and consists of several parameters in the nominator and dominator of Equation (1). In the nominator of Equation (1), it includes I_{mold} which contains the indirect effects of cool-

ing conditions [

In dominator of the second part of Equation (1), the mold powder viscosity which normally adds to the free surface of the liquid steel affects MTMR. It melts and then flows into the mold wall with liquid steel to act as lubricant which results in decreasing in the thermal conductivity of steel compound and reduces the heat flow from steel compound into mold wall. This causes to delay the dissipation of superheat stored in the liquid steel which affects the surface temperature and the solid shell growth. Based on the melting temperature and viscosity of mold powder, therefore, the square of m_{lub} is inversely proportional to the degree of resistance of thermo-mechanical stresses or to MTMR. The support for this mechanism was found in the scanning cracked samples and their micrographs by Brimacombe et al. [_{s}” which it’s increasing decreases MTMR. This agrees well with reports by Larsen and Moss [_{mold}. Previous work by Dippenaar et al. [

Finally, the material part in Equation (1) reveals mold dwell time “t” and casting speed “v” in nominator and dominator of Equation (1), respectively, for non-dimensional mathematical requirements. Also, different materials formula of exponent creep material m appeared in both the nominator and dominator of Equation (1) represented the effect of different types of steels on MTMR. Wherever, Morozenskii et al. [

Heat transfer at ingot surface was assumed to follow a generalised Newtonian law and the initial and boundary conditions were described in detail in Refs. [15,25,26] and the equations used in these computations are summarized in

The model requires simultaneous solution of governing equations in Tables 1-3. The simulation starts by stetting the initial steel to the pouring temperature. Then, each time step begins by estimating the new casting temperature distribution Different dendritic phase fractions and thermo-physical properties of steel were then calculated. The initial Newtonian heat transfer coefficient is then estimated based on the previous experience. In order to evaluate thermal fields, the heat flow equations in

Also, work adopts temperature-dependent steel properties chosen to be as realistic as possible. The actual liquidus and solidus temperatures of multi-components alloy are summarized in Ref. [

The simulated slab casters are based on two actual Industrial casting machines and the reader is referred to the original references for the details of cooling and operating conditions [_{s}” was taken constant and is equal to 20˚C for different heats. Two mold powders were examined in these trials and their viscosity were summarized in

The model developed in the previous sections in this paper was applied to simulate the continuous casting of steel in two different molds where previous experimental measurements had been made as described in the a series of previous papers [16,25].

Mold thickness is average, 240 mm.

Figures 4(a)-(d) show the comparisons between the predicted heat fluxes “Q_{f}” surface temperatures “Tf”, coherent solid shell resistances “I_{c} and mold thermo-mechanical rigidity criterion “MTMR”, respectively, for case 1.

_{f}” with distance from meniscus for different heats. At pouring temperature, it is considered that heat transfer controlled by turbulent convection streams in liquid region where liquid metal flows into the mold through a submerged entry nozzle and directed by the nozzle characteristics. Therefore, it can be seen that the initial value of Q_{f} of the initial sub-region of BAGF is always within 1200 kW/m^{2} for different heats 1 and 2. These values decrease slightly and reach a minimum value of 800 kW/m^{2 }at distance 175 mm beneath the meniscus where the direction of the steel jet controls and concentrates the heat flux of turbulent fluid flow on the solidified shell. [1,2] This affects delivery of superheat to the solid/liquid interface of the growing shell. This agrees with the predications and measurements of Panaras et al. [_{f}_{ }increases slowly based on the solidification behavior, cooling and fluid flow conditions until the coherent temperature. At the initial coherent sub-region, it is obvious that Q_{f} rises rapidly from initial contact values of 800 until peak values 2000 and 1620 kW/m^{2} at 200 mm for heats 1 and 2, respectively. This is due to a changing in the heat transfer mode from convection/conduction into a conduction mode [^{2} for different heats as shown in _{f} until minimum value at mold exit 100 kW/m^{2}. In general, the predications point out that there is no observed difference between the heat fluxes of heats 1 and 2 expect in initial coherent region where the difference is observed between peak values and also at the mold exit.

The surface temperature profiles shown in

Simulation of Ic is also often used as criterion for measuring the resistance against different thermo-metallurgical and mechanical stresses subjected to the slab in different cooling zones [11,29]. Recently, this criterion affects significantly the mold heat transfer especially in AAGF and therefore the air gap width profile. This quantity was studied here under various mold cooling conditions and solidification phenomena for its importance. Therefore, to examine the influence of various operating conditions of different heats on the ability of coherent shell to resist different thermo-mechanical stresses and therefore, to control the mold thermo-mechanical rigidity, the model was used to simulate Ic and its results are shown in

It has been suggested that the mold thermo-mechanical rigidity criterion “MTMR” for different mold cooling conditions can be related to surface quality of continuously cast steel slabs [_{c} growth shown in _{c} on MTMR is clearly observed at mold exit where the value of MTMR of heat 1 is higher than its value of heats 2 and the effect of mechanical stress due to ferro-static head begins to consider [

In the case of peritectic carbon steels, the comparisons between the behaviors of the heat fluxes “Q_{f}” surface temperatures “Tf”, coherent solid shell resistances “I_{c}” and mold thermo-mechanical rigidity “MTMR” for case 2 are shown in Figures 5(a)-(d), respectively.

_{f} in the mold zone for different heats 3 and 4. Overall, it can be seen from this figure that Q_{f} follows the same trend in various mold cooling regions. Consequently, the quantitative differences appear due to the changes in type of steel alloy and low casting speeds [42,43]. In initial BAGF, the predications demonstrate that the initial value Q_{f} decreases slightly from initial value 1200 into 800 kW/m^{2} within 200 mm for different heats due to the effect of turbulent flow and superheat. Consequently, the situation changes completely at the coherent region where Q_{f} raises rapidly from initial value of 800 until peak values of 1400 and 1600 kW/m^{2} at 250 mm for heats 3 and 4, respectively. This is followed by a steep drop in Q_{f} into 720 kW/m^{2} in AAGF. As air gap continues to grow, Q_{f} decreases gradually by different rates into minimum values at mold exit associated with slight fluctuations. Not surprisingly, Q_{f} reveals no difference between the heat fluxes of heats 3 and 4 expect in initial coherent region where the difference only is observed between the peak values. This agrees with measurements by Singh and Blazek [

The profiles of Tf shown in

The calculations of profile of Ic for heats 3 and 4 are shown in