Numerical Modeling and Optimization of Steelmaking Tundish Design by using Response Surface Methodology Maqusud Alam1, Md Irfanul Haque Siddiqui2*, Man Hoe Kim3, Vineet Chak4 1
Department of Mechanical Engineering, Aligarh Muslim University, Aligarh, India-202002 Research Institute of Engineering Design Technology, Kyungpook National University, Daegu, South Korea-701702 3 School of Mechanical Engineering, Kyungpook National University, Daegu, South Korea-701702 4 National Institute of Foundry & Forge Technology, Ranchi, India-834003 Email:
[email protected] 2
Abstract In the present work, a three-dimensional numerical investigation has been carried out on two strand slab caster tundish. The flow control devices (FCDs) were considered for optimization of characteristic parameters of steelmaking tundish by using response surface methodology (RSM). The FCD parameters are shroud depth, APB wall inclination angle and dam height in tundish. The twenty numerical CFD simulations were performed according to the experimental design. Three responses, in terms of volume fraction of dead region, mixed volume of melt and plug flow were monitored. A second-order quadratic equation was developed for each response based on interactions of each parameter. It was found that the model was significant and confirmatory simulations were carried out to validate the predictive model. It was seen that APB wall inclination angle has most significant impact on the melt flow characteristic. The shroud depth and dam height were seen the most significant parameters affecting the melt flow inside the tundish. Keywords: steelmaking, CFD, RSM, tundish 1. Introduction Steel cleanliness and strict composition control, together with surface and internal quality of strands, are now becoming the primary concern of steelmakers. Tundish is the last metallurgical vessel through which molten metal flows before solidifying in the continuous casting mold[1]. During the transfer of metal through the tundish, molten steel interacts with refractories, slag, and atmosphere. Thus, proper design and operation of a tundish are important for delivering steel of correct composition, quality, and temperature. The numerical simulation is an inexpensive method to study process under very high temperature conditions[2], [3]. On another hand, the physical modeling investigation using a water model is an appreciable and efficient method[4]. The use of control devices is an appreciable solution to improve the fluid flow behavior inside tundish. In the present work, a three-dimensional numerical investigation has been carried out on a strand slab caster tundish. The flow control devices (FCDs) were considered for optimization of characteristic parameters of steelmaking tundish by using response surface methodology (RSM). RSM is a collection of mathematical and statistical techniques useful for the modelling and analysis of problems in which a response of interest is influenced by several variables and the objective is to optimize this response. The RSM based design of experiments is implemented due to its usefulness for accurately predicting optimised fluid flow behaviour for all possible combinations of FCDs and it is highly suggested to steelmakers. The effect of 3 FCD’s parameters are shroud depth, APB wall inclination angle and dam height in tundish has been studied by analyzing the various fluid flow characteristic of the tundish. The twenty
numerical CFD simulations were performed according to the experimental design. Three responses, in terms of volume fraction of dead region, mixed volume of melt and plug flow were monitored. 2. Physical Description Figure 1 shows schematic diagram of experimental setup and dimensions of full-scale tundish as well as advanced Pouring box (APB) respectively. A reduced scale model (scale factor of λ= 1/3) of two strand slab caster tundish made of the transparent thermoplastic acrylic sheet has been used for water modeling. The dimensional analysis requires similarity of different dimensionless numbers such as Re, Fr, Ri and We etc for reduced scale laboratory setup. The flow behavior inside tundish is largely affected by these dimensionless numbers. It is assumed that flow phenomena in tundish are largely dominated by the inertial and gravitational forces (i.e., Froude number) rather than the viscous forces (Reynolds number)[1]. In this case, geometric similarity has been maintained by keeping the dimensions of the model and industrial tundish in the same ratio and dynamic similarity is achieved by considering the inertial, viscous and gravitational forces. Pure water was used as a fluid medium (at room temperature) as it has equivalent kinematic viscosity to molten steel. The water model has a close resemblance of flow dynamic properties to the industrial size tundish and hence, sufficient accuracy of the iso-thermal study of steel flow in tundish is expected[5]. .
(a)
(b) Figure 1. Diagram and Experimental Setup
3. Mathematical Model Following equations of mass, momentum and energy have been solved for numerical model. Continuity: u i 0 x i
(1)
Momentum: u u j P u i u j eff i x xi xi xi j xi
Where, eff 0 t 0 C
k2
(2)
Energy equation: T keff ui h xi xi xi Concentration,
C ui C t xi xi
(3)
eff C x i c
(4)
Theoretical mean residence time,
Volume of the tundish Volumetric flow rate
(5)
Actual Residence Time, C avi ti (6) tr C avi In equation (6), the integration is carried over a time span 2τ with an equal interval of time step; Fraction of dead volume of the tundish = 1 - t
r
Break through time, tp = First appearance of tracer at the exits V
Fraction of plug volume =
t
p p V
Fraction of Mixed Volume = V
m
V
V p Vd 1 V V
(7) (8) (9) (10)
Turbulence Models: The standard k-ε[6] turbulence model in which the solution of two separate transport equations, allows the turbulent velocity and length scales to be independently determined. Governing equations for turbulent kinetic energy (K) and that for rate of dissipation of turbulent kinetic energy(𝛆) are as follows: ∂ μt (ρK) + div(ρKU) = div [( ) grad K] + 2μt Sij . Sij − ρε ∂t σk ∂ μt ε ε2 (ρε) + div(ρεU) = div [( ) gradε] + C1ε 2μt Sij . Sij − C2ε ρ ∂t σε K K
(11) (12)
The turbulent (or eddy) viscosity, µt, is computed by combining k and ε as K2 μt = ρCμ ε Where, Cµ is a dimensionless constant. According to the recommendations of Launder et al. for turbulent flows, following values have been adapted for constants [7]. C1ε=1.44, C2ε= 1.92, Cµ=0.09, σk=1 and σε=1.30 Reynolds stresses are computed by the following Boussinesq relationship[8]. 𝜕𝑈𝑖 𝜕𝑈𝑗 2 2 −𝜌𝑢̅𝑖′ 𝑢̅𝑗′ = 𝜇𝑡 ( + ) − 𝜌𝑘𝛿𝑖𝑗 = 2𝜇𝑡 𝑆𝑖𝑗 − 𝜌𝑘𝛿𝑖𝑗 𝜕𝑥𝑗 𝜕𝑥𝑖 3 3
(13)
Numerical Details: A three-dimensional numerical investigation has been carried out on two strand slab caster tundish which is used for numerical simulations. A control volume based technique has been used to convert the governing equations to algebraic equations. Secondorder upwind discretization scheme was used to discretize the transport equations. The SIMPLE algorithm was used for pressure-velocity coupling and body force (due to gravity) has been considered. The species equation was solved in the complete flow domain Computation was carried out for half of the tundish because of prevalence of symmetry at the center plane. Fluid flow in tundish was considered predominantly turbulent, hence, the standard k-ε turbulence model was considered. The species equation was solved in the complete flow domain. The CFD software Ansys Fluent was used for solving the set of equations and generating the results. The side walls and bottom were set to a no slip condition with zero velocity and the turbulent quantity has been set from the logarithmic law of the wall of k-ε models. Turbulence intensity at the inlet was specified at 2 % and at the outlets, atmospheric pressure was assumed to be present. At inlet nozzle, the mean vertical velocity was assumed to be uniform through its cross section and other two perpendicular velocities are assumed to be zero. All velocities were set to zero at any wall. The upper top surface was assumed as a free surface. The top surface was considered as a shear-stress free plane. In RSM, the quantitative form of relationship can be given between the desired response and independent input variables (Montgomery [12]) by following equation: yi= f(z1, z2, z3……zn) ± ε (14) where, f is response function while Є is a fitting error. In present work, a quadratic model of yi for 3 input parameters has been considered. 4. Results and Discussions: The fluid flow behavior was calculated outlet from CFD simulation. The results were investigated for analysis of variance (ANOVA) for fitted RSM quadratic model and ANOVA for each term on the performance characteristics. By applying multiple regression analysis on the CFD simulation data, the response variable and test variable were related by second-order polynomial equations. Table 1: The Characteristic operating condition of one third scale tundish at steady state operating condition. Characteristic Parameters Units Full scale tundish Inlet velocity m/s 1.4 Bath height (Hb) mm 830 Length (Ls) mm 2940 Theoretical residence time sec 934 Shroud diameter mm 80 3 Molten steel density kg/m 7020 The Statistical testing of regression equation of each parameter was checked by F-value. The studied models are significant. The suitability of the model for the present was confirmed of the basis of p-Value of the model which was smaller than 0.0001.The determination coefficient (R2= 0.9560 to 0.9782 for each response) was close to 1, manifesting that the model could explain more than 95% of the response value changes. Meanwhile, a low value of the coefficient of the variation (C.V.) in each case has showed a high degree of precision and a good deal of reliability of the CFD results. The numerical investigations have been carried out
on various design parameters of two strand slab caster tundish. The table 1 shows the characteristic operating parameters of tundish. The 20 simulations have been carried out by changing the different FCD’s parameter which we get from response surface methodology. The table 2 shows the variable parameters range of shroud depth, APB angle, dam height. Table 2: Variable parameters range. Variable Parameters symbol -1 0 +1 Shroud depth A (=Ls/Hb) 20% 35% 50% APB wall angle B (θ) 80o 105o 130o Dam height C (=Dh/Hb) 0.2 0.35 0.5
Figure 2. Schematics of locations of FCDs in a two strand slab caster tundish.
Figure. 3 Response surface showing dependency of dead volume upon APB angle and shroud depth. The minimization of intermixed amount from each outlet is one of the main aims during sequential casting. The optimum conditions of various parameters to minimize the average of intermixed grade of steel from three outlets were obtained from Design-Expert 7.0.0 software. To verify the suitability of predicted optimized model, three CFD simulations were carried on
the basis of new design. Table 3 shows the optimized design of tundish FCDs in order to minimize the dead volume fraction of steel melt. Table 3: Optimized design of FCDs S.No. Shroud Depth 1 20.0 2 20.0 3 20.0 4 20.2 5 20.0
APB Angle 112.8 113.2 112.5 112.9 112.1
Dam Height 0.500 0.500 0.498 0.500 0.492
Vm
Vd
Vp
0.635 0.634 0.633 0.632 0.628
0.034 0.035 0.035 0.037 0.040
0.36 0.36 0.36 0.36 0.36
Figure. 4 Response surface showing dependency of dead volume upon APB angle and dam height. Analysis of the tundish dead volume at fixed dam height: Figure 3 represents the effect of shroud depth and APB wall inclination angle on dead volume when dam height is 50% of total height of tundish. It is seen here that the increase in shroud depth there is no significant effect on dead volume. In contrast to this, an increase in APB wall inclination angle causes decrease in dead volume but after certain value its value increases, its value is minimum at 112o. Analysis of the dead volume at fixed shroud depth: Figure 4 represents the effect of APB angle and dam height on dead volume when shroud depth is 50% of bath height. It is seen that the increase in APB angle and dam height, dead volume decreases sharply and then after certain value it increases and if dam height increases dead volume increases very slowly till 0.275 times of bath height. Moreover, it has been observed that there is insignificant variation of dead volume caused by shroud length and dam height at any certain APB angle. The value of dead volume was found to be minimum at 20% total height of tundish and is maximum at 50% of total bath height.
5. Conclusion The flow control devices (FCDs) of tundish were considered for optimization by using response surface methodology. The studied parameters are shroud depth, APB wall inclination angle,
dam height and dam position in tundish. A three level and four factorial statistical analyses were implemented using Central Composite Design (CCD). The 20 numerical CFD simulations were performed according to the experimental design. Four responses, in terms of intermixed amount from each out three and average of all were monitored. A second-order quadratic equation was developed for each response based on interactions of each parameter. It was seen that APB wall inclination angle has most significant impact on dead volume fraction. The optimum value of the parameter which resulted into minimum amount of dead volume were shroud depth of 20%, APB angle of 112o, and dam height of 0.5. 6. Reference [1] D. Mazumdar, “Tundish Metallurgy: Towards Increased Productivity and Clean Steel,” Transactions of the Indian Institute of Metals, vol. 66, no. 5–6, pp. 597–610, Jul. 2013. [2] M. I. H. Siddiqui and P. K. Jha, “Numerical simulation of flow-induced wall shear stresses in three different shapes of multi-strand steelmaking tundishes,” Steel Research International, vol. 85, no. 9999, pp. 1–12, 2014. [3] K. Chattopadhyay, M. Isac, and R. I. L. Guthrie, “Physical and Mathematical Modelling of Steelmaking Tundish Operations: A Review of the Last Decade (1999-2009),” ISIJ International, vol. 50, no. 3, pp. 331–348, 2010. [4] M. I. H. Siddiqui and P. K. Jha, “Assessment of turbulence models for prediction of intermixed amount with free surface variation using CLSVOF method,” ISIJ International, vol. 54, no. 11, pp. 2578–2587, 2014. [5] D. Mazumdar and J. W. Evans, Modeling of Steelmaking Processes. CRC Press, 2009. [6] B. E. Launder and D. B. Spaulding, Mathematical Models of Turbulence. Academic Press, 1972. [7] B. E. Launder, A. Morse, W. Rodi, and D. B. Spalding, “The prediction of shear flows - A comparison of performance of six turbulence models,” in NASA Conference on Free Shear Flows, Langley, 1972. [8] J. O. Hinze, Turbulence. New York: McGraw-Hill Publishing Co., 1975.
