Pressure Distribution and Flow Rate Behavior in Continuous-Casting Stopper-Rod Systems: PFSR

Abstract

A new simple model of pressure and flow in the liquid-metal delivery system of continuous casting operations with stopper-rod flow control, PFSR, is introduced. This one-dimensional model calculates the gauge pressure distribution and flow rate in the complete tundish, stopper-rod, and nozzle system by solving a set of pressure-energy balance Bernoulli-type equations. It includes the effects of argon gas injection and its expansion according to the local pressure. PFSR is a MATLAB-based software package with a user-friendly graphical user interface (GUI). It employs an inverse model to solve the system of governing equations for any unknown chosen by the user. This enables fast and efficient parametric studies to investigate the effects of casting conditions and nozzle geometry under realistic conditions. The model is verified with three-dimensional computational fluid dynamics (CFD) simulations and validated with both water model and plant measurements. To overcome unrealistically low minimum pressure predictions in steel casters, two other physical phenomena should be considered: cavitation and non-primed annular/slug (waterfall-type) flow with large gas pockets. Preliminary results that include these two new phenomena into the PFSR model show that cavitation and air pockets (non-primed flow) can explain steel plant measurements and likely occur for most casting conditions in real casters with stopper-rod control systems.

Appendix: gap area calculation

The minimum cross-sectional area of the gap formed between the stopper-rod and tundish floor region, ${A}_{gap}$, [m²] controls the flow rate and pressure in the system. It is calculated based on 6 input parameters shown in Figure A1.

Fig. A1

Stopper-rod schematic showing parameters associated with general nozzle tip defined with 3 radii

These include stopper-rod nose (tip) radius, ${r}_{tip}$, stopper-rod bend 1 radius, ${r}_{b1}$, stopper-rod bend 2 radius, ${r}_{b2}$, stopper-rod height from tip to the straight section, ${H}_{stopper}$, stopper-rod diameter, ${D}_{stopper}$, all in [mm] are input by the user to PFSR to define the stopper shape. To fully define the stopper geometry, 8 more parameters are required: the coordinates of the two tangent points between part-circles ${p}_{1}$ and ${p}_{2}$ , and their centers (${x}_{c1}$,${y}_{c1}$) and (${x}_{c2}$,${y}_{c2}$). This geometry is defined relative to the nose (tip) radius center at the origin, (0,0). The following 8 independent equations are solved simultaneously for these 8 unknowns:

$${y}_{c2}={H}_{stopper}-{r}_{tip}$$

(A1)

$${\left(\frac{{D}_{stopper}}{2}- {x}_{c2}\right)}^{2}+ {\left(\left({H}_{stopper}-{r}_{tip}\right)- {y}_{c2}\right)}^{2}={r}_{b2}^{2}$$

(A2)

$${\left({x}_{p2}- {x}_{c2}\right)}^{2}+ {\left({y}_{p2}- {y}_{c2}\right)}^{2}={r}_{b2}^{2}$$

(A3)

$${\left({x}_{p2}- {x}_{c1}\right)}^{2}+ {\left({y}_{p2}- {y}_{c1}\right)}^{2}={r}_{b1}^{2}$$

(A4)

$${\left({x}_{p1}- {x}_{c1}\right)}^{2}+ {\left({y}_{p1}- {y}_{c1}\right)}^{2}={r}_{b1}^{2}$$

(A5)

$${\left({x}_{p1}\right)}^{2}+ {\left( {y}_{p1}\right)}^{2}={r}_{tip}^{2}$$

(A6)

$$\frac{{y}_{c2}-{y}_{p2}}{{x}_{c2}-{x}_{p2}}=\frac{{y}_{c2}-{y}_{c1}}{{x}_{c2}-{x}_{c1}}$$

(A7)

$$\frac{{y}_{c1}-{y}_{p1}}{{x}_{c1}-{x}_{p1}}=\frac{{y}_{c1}}{{x}_{c1}},$$

(A8)

where ${x}_{ci}$ and ${y}_{ci}$ (i = 1,2) are the $x$-coordinate and $x$ y-coordinate of the bend $i$.

The next step is to define the tundish floor geometry. The user must input: tundish floor radius, ${r}_{t}$, and diameter of the UTN entry, ${D}_{UTNa}$, both in [mm]. Assuming a single radius of curvature of this floor, this requires the coordinates of the circular segment (${x}_{t}$,${y}_{t}$) with given radius ${r}_{t}$ and the coordinates of the tangent point (${x}_{m}$,${y}_{m}$) where the stopper touches the tundish floor when the system is fully closed (i.e., no flow), as shown in Figure A2(a). These 4 parameters are found by solving the following 4 equations simultaneously:

$${x}_{t}={r}_{t}+\frac{{D}_{UTNa}}{2}$$

(A9)

$${\left({x}_{c1}- {x}_{m}\right)}^{2}+ {\left({y}_{c1}- {y}_{m}\right)}^{2}={r}_{b1}^{2}$$

(A10)

$${\left({x}_{t}- {x}_{m}\right)}^{2}+ {\left({y}_{t}- {y}_{m}\right)}^{2}={r}_{b1}^{2}$$

(A11)

$$\frac{{y}_{c1}-{y}_{t}}{{x}_{c1}-{x}_{t}}=\frac{{y}_{m}-{y}_{c1}}{{x}_{m}-{x}_{c1}}$$

(A12)

Fig. A2

Geometrical relationship between stopper-rod and tundish floor circular segment at (a) fully closed (b) partially open position

From these parameters, the gap opening distance, $S$, for any stopper-rod opening, ${h}_{sro}$, is calculated according to the geometry shown in Figure A2(b) as follows:

$$S=\sqrt{{\left({x}_{0}\right)}^{2}+ {\left({y}_{0}+ {h}_{sro}\right)}^{2}}-\left({r}_{t}+{r}_{b1}\right)$$

(A13)

and the angle $\theta$ , shown in Figure A2(b) is found from

$$\theta =acos\left(\frac{{x}_{0}}{S + {r}_{t} + {r}_{b1}}\right),$$

(A14)

where the horizontal and vertical distances between the center of the circular segment of the tundish floor and the center of the first bend, ${x}_{0}$ and ${y}_{0}$, are defined as follows:

$${x}_{0}={x}_{c1}-{x}_{t}$$

(A15)

$${y}_{0}={y}_{c1}-{y}_{t}$$

(A16)

Fig. A3

Stopper geometry showing minimum gap

Finally, the area of the gap, which is the lateral surface area of the conical frustum shown in Figure A3, is calculated as follows:^[40]

$${A}_{gap}=\pi S\left({R}_{1,x}+{R}_{2,x}\right),$$

(A17)

where

$${R}_{1,x}={x}_{c1}+{r}_{b1}cos\theta $$

(A18)

$${R}_{2,x}={x}_{t}-{r}_{t}cos\theta $$

(A19)