Pressure Distribution and Flow Rate Behavior in Continuous-Casting Slide-Gate Systems: PFSG

Abstract

Quantifying the pressure distribution and flow rate in the metal delivery system from the tundish to the mold is important to understand nozzle clogging due to air aspiration, argon bubble size, and other important phenomena in continuous casting of steel. A one-dimensional pressure energy model, PFSG, is presented here to calculate pressure distribution and flow rate in slide-gate nozzle systems for argon-molten steel flow systems including argon gas expansion by solving a system of Bernoulli equations. PFSG is a user-friendly MATLAB-based software package with an intuitive Graphical User Interface capable of inverse solutions that enables users to conduct parametric studies in seconds. The model predictions is verified with computational fluid dynamics simulations and validated with plant measurements and water model measurements, with errors of less than 6 pct. The verified and validated PFSG model is then applied to investigate the effect of different casting conditions on the flow rate and pressure distribution. For a given slide gate opening, the flow rate increases with increasing distance from the tundish level to the mold level, increasing SEN diameter, and decreasing argon gas injection rate. The slide-gate opening fraction that produces the minimum pressure is almost independent of casting conditions, except for SEN bore diameter. This critical opening fraction, which is most detrimental for air aspiration, increases from 50 to 70 pct with increasing SEN diameter by 13 pct for the system studied. The minimum pressure worsens (decreases) with increasing tundish height and/or nozzle bore diameter.

Appendix: Wall Friction Calculation for Tapered Pipe

Wall friction equation for a pipe with constant diameter (i.e. not tapered), discussed in fluid mechanics reference book,^[28] is defined as:

$$h= \frac{1}{2}f\frac{L}{D}\frac{{V}^{2}}{g}$$

(A1)

where h is pressure head loss due to wall friction [m], f friction factor, L pipe length [m], D pipe diameter [m], V fluid velocity [m/s], and g Earth’s gravitation acceleration [m/s²].

Now, consider a tapered pipe as shown in Figure A1.

Fig. A1

Schematic of tapered length of nozzle (pipe)

For a very small portion of pipe with the length of dx, it is true to say:

$$\text{d}h= \frac{1}{2}f\frac{\text{d}x}{D}\frac{{V}^{2}}{g}$$

(A2)

Consider the continuity equation:

$$V= \frac{{V}_{1}{D}_{1}^{2}}{{D}^{2}}$$

(A3)

where V and D are velocity and diameter of the section [m/s] and [m], respectively.

Also from Figure A1:

$$x= \frac{{D}_{1}-D}{2\tan\theta }$$

(A4)

where $\theta$ is the taper angle [rad].

Therefore:

$$\text{d}x= \frac{-\text{d}D}{2\tan\theta }$$

(A5)

Substituting Eqs. [A3] and [A5] into Eq. [A2] gives:

$$dh= \frac{1}{2}f\frac{{V}_{1}^{2}{D}_{1}^{4}}{{D}^{5}g}\frac{-dD}{2\tan\theta }$$

(A6)

Integrating Eq. [A6] over the whole domain (from ${D}_{1}$ to ${D}_{2}$) and simplifying gives:

$$h= \frac{1}{16}\frac{f}{g}\frac{{V}_{1}^{2}}{{D}_{2}^{4}}\frac{\left({D}_{1}^{4}-{D}_{2}^{4}\right)}{\tan\theta }$$

(A7)

Insert into Figure A1:

$$\tan\theta = \frac{{D}_{1}-{D}_{2}}{2L}$$

(A8)

Finally, substituting Eq. [A8] into Eq. [A7] gives the pressure loss in a tapered pipe due to wall friction:

$$h= \frac{1}{8}f\frac{L}{g}\frac{{V}_{1}^{2}}{{D}_{2}^{4}}\left({D}_{1}+ {D}_{2}\right)\left({D}_{1}^{2}+ {D}_{2}^{2}\right)$$

(A9)